nonlinear control of an electric vehicle using chebyshev ... · pdf fileintroduction in recent...

1

Project Title

Nonlinear Control of an Electric Vehicle Using Chebyshev Neural Network

Number of Students 2 Supervisor Assoc Prof Dr Waree Kongprawechnon

Instructor Muhammad Usman Jamil

Objective

Electrical Vehicles are highly nonlinear system The objective of this work is to compare and

examine the performance of Chebyshev neural network (CNN) based backstepping controller with

a CNN based optimal adaptive controller which forces the plant output to track a specified

reference trajectory in the presence of time varying mass and varying armature winding resistance

that is lim119905rarrinfin

(119910 minus 119910119889) = 0

Proposed Methodology

Two nonlinear controllers are proposed for an electric vehicle Chebyshev neural network based

backstepping controller and Chebyshev neural network based optimal adaptive controller The

electric vehicle (EV) will be driven by switched reluctance (SR) motor Both the controllers will

use Chebyshev neural network (CNN) to estimate the unknown nonlinearities The unknown

nonlinearities arise as it is not possible to precisely model the dynamics of an EV Mass of

passengers resistance in the armature winding of the SR motor aerodynamic drag coefficient and

rolling resistance coefficient are assumed to be varying with time The learning algorithms will be

derived from Lyapunov stability analysis so that system-tracking stability and error convergence

can be assured in the closed-loop system

The block diagram of the proposed controllers are shown in Figure 1 and Figure 2 respectively

Figure 1 Block diagram of CNN based backstepping controller

2

Figure 2 Block diagram of CNN based optimal adaptive controller

Requirements

Basic skill in Machine Drives

Strong background in Control Systems

Strong skills in MATLAB (Simulink)

Some knowledge about Artificial Neural Networks (ANN)

References

1 Zhang Ruiwei Qian Xisen ldquoAn adaptive sliding mode current control for switched

reluctance motorrdquo IEEE Conference and Expo Transportation Electrification Asia-Pacific

(ITEC Asia-Pacific) Aug 2014

2 Vikas Sharma and Shubhi Purwar ldquoNonlinear Controllers for a Light-Weighted All-

Electric Vehicle Using Chebyshev Neural Networkrdquo International Journal of Vehicular

Technology Hindawi 2014

Research ArticleNonlinear Controllers for a Light-Weighted All-Electric VehicleUsing Chebyshev Neural Network

Vikas Sharma and Shubhi Purwar

Department of Electrical Engineering Motilal Nehru National Institute of Technology Allahabad 211004 India

Correspondence should be addressed to Vikas Sharma vikassharmaecbgmailcom

Received 24 October 2013 Revised 19 March 2014 Accepted 20 March 2014 Published 22 April 2014

Academic Editor Tang-Hsien Chang

Copyright copy 2014 V Sharma and S Purwar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Two nonlinear controllers are proposed for a light-weighted all-electric vehicle Chebyshev neural network based backsteppingcontroller and Chebyshev neural network based optimal adaptive controller The electric vehicle (EV) is driven by DCmotor Boththe controllers use Chebyshev neural network (CNN) to estimate the unknown nonlinearitiesThe unknown nonlinearities arise asit is not possible to precisely model the dynamics of an EVMass of passengers resistance in the armature winding of the DCmotoraerodynamic drag coefficient and rolling resistance coefficient are assumed to be varying with time The learning algorithms arederived from Lyapunov stability analysis so that system-tracking stability and error convergence can be assured in the closed-loopsystemThe control algorithms for the EV system are developed and a driving cycle test is performed to test the control performanceThe effectiveness of the proposed controllers is shown through simulation results

1 Introduction

In recent years due to stricter emission standards and globalfuel supply issues researchers in the automobile industryare paying more attention to zero-polluting electric vehicles(EVs) for energy conservation and environmental protectionEV has emerged as a promising alternative to improve fueleconomywhile meeting the tightened emission standards [1]EV is used in many applications particularly for the purposeof patrolling and other short-range transportations A lot ofwork has been reported in the literature for reducing thecost extending the driving range and sophisticated energymanagement strategies to improve the performance andusage of energy [2ndash6] Direct current (DC) power is suppliedfrom the battery and therefore EVs driven by DCmotor are afavorable selection The control of DC motor is simple and itcan provide comparatively larger startup torque In additionto the primary function of propulsion the DCmotor can alsobe used effectively as the braking device because of its fasttorque response characteristics and capability of regeneration[7 8]

The dynamics of EV are inherently nonlinear and it is wellknown that nonlinear control techniques are superior to the

conventional linear controlmethods [9 10]The performanceof nonlinear control techniques specifically the differential-geometric approach to the control of EV is presented in [11]Due to the complex operating conditions of EV intelligentor fuzzy control is suggested in [12 13] In [12] the fuzzylogic controller (FLC) is modeled to be capable of increasingthe initial torque required for the electric vehicle drive withvariable speed characteristics and high efficiency In [13]fuzzy logic based controller to control the wheel slip forelectric vehicle antilock braking systems (ABSs) is developedA methodology to generate stochastic drive cycles for thedesign and control optimization of EVs is detailed in [14]An algorithm for determining online energy based drivingguidance for an EV using particle swarm optimization isdeveloped in [15]

Neural networks have been used for approximationof nonlinear systems for classification of signals and forassociative memory Chebyshev neural network (CNN) hasbeen shown to be able to approximate any continuousfunctions over a compact set to arbitrary accuracy [16ndash18]CNN is a functional link neural network (FLN) based onChebyshev polynomials The efficacy of CNN in the areas ofonline system identification [19 20] and tracking controller

Hindawi Publishing CorporationInternational Journal of Vehicular TechnologyVolume 2014 Article ID 867209 14 pageshttpdxdoiorg1011552014867209

2 International Journal of Vehicular Technology

for nonlinear systems has been established [21ndash23] Theproposed controller does not necessitate exact knowledge ofthe unknown nonlinearities The CNN is used for estimatingthe unknown nonlinearities of the system The adaptationlaws for the CNN weights are such that they guarantee thestability of the system The tracking error mainly dependson the CNN feedback functions to be used for the weightadaptation law and other design parameters

It is not possible to precisely model the dynamics ofan EV as some parameters may vary with timeconditionsFor example the resistance in the armature winding ofthe DC motor changes as the temperature varies and theaerodynamic drag coefficient 119862119889 and the rolling resistancecoefficient 120583119903119903 are varying because of wind and road con-ditions respectively In this paper the resistance in thearmature winding (119877119886) of the DC motor the aerodynamicdrag coefficient 119862119889 the rolling resistance coefficient 120583119903119903 andthemass of the passengers (Δ119872) are considered to be varyingwith time resulting in unknown nonlinearities The aim ofthis paper is to design CNN based backstepping controllerand CNN based optimal adaptive controller for EV in thepresence of unknownnonlinearities and test the performanceof the overall system on NEDC drive cycle test

The paper is organized as follows In Section 2 thedescription of the complete EV system and the structure ofCNNwill be presentedThe problem statement is introducedin Section 3 The design of conventional backstepping con-troller and CNN based backstepping controller is describedin Section 4 In Section 5 we give an optimal control designfor EV systems using the H-J-B equation followed by a CNNbased optimal adaptive controller Section 6 validates theperformance of the proposed controllers through simulationsand drive cycle test followed by conclusion

2 EV System Description and CNN Structure

An EV system dynamics mainly comprises two parts thevehicle dynamics and dynamics of the motor system asshown in Figure 1 Motor system is connected to EV systemthrough transmission unit which includes the gearing sys-tem In the actual EVs the driver provides the commandsignal through the acceleratorbrake pedal in the form ofaccelerationdeceleration to the controller of the propulsionsystem The DC motor is used in the proposed EV systemfor propulsion and DC motor system is connected to EVsystem through transmission unit which includes the gearingsystem Accordingly the speed of DC motor is controlled soas to control the actual EV system

21 Vehicle Dynamics The major factors that affect thevehicle dynamics are road condition aerodynamic drag hillclimbing acceleration and so forth After these factors aretaken into account vehicle dynamics can bewritten as follows[1]

119865 = 120583119903119903119898119892⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1

+1

2120588119860119862119889V

2

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2

+ 119898119892 sin120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

3

+ 119898119889V119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟

4

(1)

Transmissionunit

Motor dynamics

Vehicledynamics

+minus

Figure 1 EV system

where 120583119903119903 is the rolling resistance coefficient119898 is the mass ofthe EV 119892 is the gravity acceleration 120588 is the air density 119860 isthe frontal area of the vehicle 119862119889 is the drag coefficient V isthe driving velocity of the vehicle and 120601 is the hill climbingangle In this paper the aerodynamic drag coefficient 119862119889 andthe rolling resistance coefficient 120583119903119903 are assumed to be varyingwith time In [1] 119898 is a constant which is a very stringentassumption In the proposed work 119898 includes the mass ofvehicle 119872 and the mass of passengers Δ119872 that is 119898 =

119872 + Δ119872 Thus119898 is varying with time and not a constantIn the vehicle dynamics (1) the first term assimilates to

the rolling resistance force the second term assimilates tothe aerodynamic drag force the third term assimilates to thehill climbing force and the fourth term assimilates to theacceleration forceThis resultant traction force119865will producea counterproductive torque to the driving motor which isrepresented by the following relationship

119879119871 = 119865119903

119866 (2)

where119879119871 is the torque produced by the drivingmotor 119903 is thetyre radius of the EV and 119866 is the gearing ratio

22 Motor Dynamics The EV is driven by a DC motor andthe dynamics of which are given by [11]

119869119889120596

119889119905= 119871119886119891119894

2minus 119861120596 minus 119879119871

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894120596

(3)

where 119869 is the inertia of the motor including the gearingsystem and the tyres 120596 is the motor angular speed 119894 is thearmature current (also field current) 119871119886 119877119886 119871field and 119877119891

are the armature inductance armature resistance field wind-ing inductance and field winding resistance respectively119861 is the viscous coefficient 119879119871 corresponds to the externaltorque 119881 is the control input voltage and 119871119886119891 is the mutualinductance between the armature winding and the fieldwinding generally nonlinear because of saturation In thispaper the resistance in the armature winding 119877119886 of the DCmotor is considered to be varying as the armature windingresistance of theDCmotor changes as the temperature varies

International Journal of Vehicular Technology 3

Table 1 Parameters of the EV system [11]

Motor Vehicle119871119886+ 119871field (mH) 6008 119872 (kg) 800

119877119886 + 119877119891 (Ω) 012 119860 (m2) 18119861 (NMs) 00002 120588 (kgm3) 125119869 (kgm2) 005 119862119889 03119871119886119891 (mH) 1766 120601 (∘) 0119881 (V) 0sim48 120583rr 0015119894 (A) 78A (250max) 119903 (m) 025120596 (rmin) 2800 (V = 25 kmh) 119866 11Δ119872 = 150 (kg) for 119905 le 195 and 119905 ge 780 (kg) Δ119872 = 220 for 119905 gt 195 and 119905 le 585 and Δ119872 = 300 (kg) for 119905 gt 585 and 119905 lt 780

0 20 40 60 80 100

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

minus10

Figure 2 Open-loop response of the EV system

23 Complete Dynamics and Open-Loop Response Withvehicle dynamics (1) and motor dynamics (3) the completedynamics of the EV system becomes [11]

119869 + 1198981199032

1198662119889120596

119889119905= 119871119886119891119894

2minus 119861120596

minus119903

119866

120583119903119903119898119892

+1

2120588119860119862119889V

2+ 119898119892 sin120601

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894 120596

(4)

By using (4) the open-loop response of a light-weightedall-electric vehicle is studied The parameters used on alight-weighted all-electric vehicle are specified in Table 1Thesimulation result of open-loop response is shown in Figure 2The plot shows the full power speed characteristics As givenin Table 1 the desired nominal speed is V = 25 kmhr whereasin open-loop conditions the speed is beyond 40 kmhr whichis not acceptable Therefore it is required to design propercontroller

The relation between the driving velocity of the vehicle Vand the motor angular speed 120596 is given as

V =119903

119866120596 (5)

where 119903 is the tyre radius of the EV and119866 is the gearing ratio

Functionalexpansion

sum

x1

x2

W

y

Figure 3 Structure of CNN

24 Structure of Neural Network In this paper a single layerCNN is considered for the NN structure CNN consistsof a functional expansion (FE) block and a single-layerperceptron network The purpose of the FE block is toincrease the dimension of the input pattern so as to improverepresentation of the input pattern in a higher dimensionalspace Chebyshev expansions are frequently used for approx-imations to functions as they are much more efficient thanother power series expansion of the same degree Amongorthogonal polynomials the Chebyshev polynomials whichare derived from the solution of the Chebyshev differentialequation occupy an important place since in the case of abroad class of functions expansions in Chebyshev polyno-mials converge more rapidly than expansions in other set ofpolynomials Hence we consider the Chebyshev polynomialsas basis functions for the neural network

The Chebyshev polynomials can be generated by thefollowing recursive formula [17]

119879119894+1 (119909) = 2119909119879119894 (119909) minus 119879119894minus1 (119909) 1198790 (119909) = 1 (6)

where 119879119894(119909) is a Chebyshev polynomial 119894 is the order ofChebyshev polynomials chosen and here 119909 is scalar quantity1198791(119909) can be chosen as 119909 2119909 2119909 minus 1 or 2119909 + 1 In this paper1198791(119909) is chosen as 119909 For example an enhanced pattern usingthe Chebyshev polynomials for 119909 isin R2 is obtained as

120601 (119909) = [1 1198791 (1199091) 1198792 (1199091) sdot sdot sdot 1198791 (1199092) 1198792 (1199092) sdot sdot sdot ]119879

(7)

where 119879119894(119909119895) is a Chebyshev polynomial 119894 is the order of theselected Chebyshev polynomial and 119895 = 1 2 120601(119909) denotesthe Chebyshev polynomial basis function

Referring to Figure 3 the architecture of the CNN con-sists of two parts [17] namely numerical transformation partand learning part The numerical transformation is the FE


of the input pattern consisting of a finite set of Chebyshevpolynomials Consequently the Chebyshev polynomial basiscan be considered as a new input vector The learning partinvolves functional-link neural network based on Chebyshevpolynomials The CNN is a single-layered neural networkand in general its learning is fast [16 17]

On the basis of approximation property of CNN [19]a general nonlinear function 119910(119909) can be approximated byCNN as

119910 (119909) =W119879120601 + 120576 (8)

where 120576 is the CNN functional reconstruction error vectorand 120576 le 120576119873 which is bounded W is the optimal weightmatrix and 120601 denotes the Chebyshev polynomial basisfunction The output of the CNN is given by

=W119879

120601 (9)

where W is the estimate of the optimal weight matrixW

3 Problem Statement

The complete dynamics in (4) can be described as

X = 119891 (X) + 119892 (X) 119906

119910 = ℎ (X) (10)

where

X = [11990911199092] = [

120596

119894]

119891 (X) =[[[[

[

1

119869 + 119898 (11990321198662 )

11987111988611989111990922minus 1198611199091 minus

119903

119866

times(120583119903119903119898119892 +

1

2120588119860119862119889

1199032

119866211990912

+119898119892 sin120601)

minus119877119886 + 119877119891

119871119886 + 119871field1199092 minus

119871119886119891

119871119886 + 119871field11990911199092

]]]]

]

(11)

119892 (X) = [[

0

1

119871119886 + 119871field

]

]

ℎ (X) = 1199091 (12)

Assuming the hill climbing angle 120601 to be zero the nonlinearfunctions 119891(X) and 119892(X) become

119891 (X) = [[

1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

minus11987071199092 minus 119870811990911199092

]

]

119892 (X) = [ 01198709] ℎ (X) = 1199091

(13)

where 1198701 = 11986621199032 1198702 = (119866

21199032)119869 1198703 = 119871119886119891 1198704 = 119861 1198705 =

(12)120588119860119862119889(11990331198663) 1198706 = (119903119866)120583119903119903119892 1198707 = (119877119886 + 119877119891)(119871119886 +

119871field) 1198708 = 119871119886119891(119871119886 + 119871field) and1198709 = 1(119871119886 + 119871field)As a result the EV system becomes

1 =1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

2 = minus11987071199092 minus 119870811990911199092 + 1198709119906

119910 = 1199091

(14)

The aim of this paper is to compare and test the perfor-mance of CNN based backstepping controller with a CNNbased optimal adaptive controller which forces the plantoutput 119910 to track a specified reference trajectory 119910119889 in thepresence of time varying mass ldquo119898rdquo and varying armaturewinding resistance (119877119886) aerodynamic drag coefficient 119862119889and the rolling resistance coefficient 120583119903119903 that is

lim119905rarrinfin

(119910 minus 119910119889) = 0 (15)

4 Nonlinear Backstepping Controller Design

The nonlinear backstepping controller is designed in thefollowing ways

41 Conventional Backstepping Controller In this subsectionthe steps involved in the development of backstepping con-troller for system (14) are discussed

By selecting the following state transformation

1199111 = 1199091

1199112 = 1

(16)

the system (14) becomes

1 = 1199112

2 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922minus11987011198704

119898 + 1198702

1199112

minus211987011198705

119898 + 1198702

11991111199112 +2119870111987031198709

119898 + 1198702

1199092119906

119910 = 1199111

(17)

which is in the strict feedback form [24 25] In this caseall the nonlinear functions are considered known The errordynamics is defined as

1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889 (18)

1198902 = 1199112 minus 1199112119889 (19)

The time derivative of 1198901 can be found as

1198901 = 1 minus 119910119889 (20)

From (17) and (19) we have (20) as

1198901 = 1198902 + 1199112119889 minus 119910119889 (21)


By selecting 1199112119889 = 119910119889 minus 11989611198901 (21) yields

1198901 = 1198902 minus 11989611198901 (22)

where 1198961 is the positive constantDifferentiating (19)

1198902 = 2 minus 2119889 (23)

Substituting 2 from (17) (23) becomes

1198902 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922

minus11987011198704

119898 + 1198702

1199112 minus211987011198705

119898 + 1198702

11991111199112

+2119870111987031198709

119898 + 1198702

1199092119906 minus 2119889

(24)

To stabilize (24) the control effort 119906 is chosen as

119906 =

2119889 minus 1198901 +2119870111987031198707

119898 + 1198702

11990922+2119870111987031198708

119898 + 1198702

119911111990922

+11987011198704

119898 + 1198702

1199112 +211987011198705

119898 + 1198702

11991111199112 minus 11989621198902

times1

(2119870111987031198709) (119898 + 1198702) 1199092

(25)

where 1198962is the positive constantSubstituting (25) in (24) gives

1198902 = minus1198901 minus 11989621198902 (26)

To prove the convergence of the EV system a Lyapunovfunction is chosen as

1198711 =1

21198902

1+1

21198902

2 (27)

The time derivative of (27) is

1 = 1198901 1198901 + 1198902 1198902 (28)

Using (22) and (26) we get

1 = minus11989611198902

1minus 11989621198902

2 (29)

Thus it can be easily seen that the system is globallyasymptotically stable

42 CNN Based Backstepping Controller The EV system in(17) is rewritten in the following form for the implementationof the CNN based backstepping controller

1 = 1199112

2 = 119891 (119909 119911) + 119892 (119909 119911) 119906

119910 = 1199111

(30)

Here the nonlinear functions 119891(119909 119911) and 119892(119909 119911) areunknown as119898 119877119886 119862119889 and 120583119903119903 are varying with time

The error dynamics is defined as

1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889

1198902 = 1199112 minus 1199112119889

(31)

The CNN based backstepping controller is designed in thefollowing two steps

Step 1The time derivative of 1198901 is given by

1198901 = 1 minus 119910119889 (32)

Using (30) and (31) (32) becomes

1198901 = 1198902 + 1199112119889 minus 119910119889 (33)

Stabilize (33) by choosing

1199112119889 = 119910119889 minus 11989611198901 (34)

where 1198961is the positive constantSubstituting (34) in (33) 1198901 becomes

1198901 = 1198902 minus 11989611198901 (35)

Step 2The time derivative of 1198902 can be obtained as

1198902 = 2 minus 2119889 (36)

By using (30) (36) results in

1198902 = 119891 (119909 119911) + 119892 (119909 119911) 119906 minus 2119889 (37)

The unknown nonlinear functions 119891(119909 119911) and 119892(119909 119911)will be estimated by CNN 1 and CNN 2 respectively Thenonlinear functions119891(119909 119911) and 119892(119909 119911) can be represented bya CNN as

119891 =W11987911206011+ 1205761

119892 =W11987921206012+ 1205762

(38)

where 1205761 and 1205762 are the bounded CNN approximation errorsW1 and W2 are the optimal weight matrices and 120601

1and 120601

2

are the basis functionsThe estimate 119891 of 119891 and 119892 of 119892 can be written as

119891 = W11987911206011

119892 = W11987921206012

(39)

where W1 is the estimate of theW1 and W2 is the estimate oftheW2

Adding and subtracting119892(119909 119911)119906 and rearranging (37) wehave

1198902 = 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 2119889 + 119892 (119909 119911) 119906

(40)


+ +

+

+minusminus

minus



119906 =1

119892 (119909 119911)minus119891 (119909 119911) + 2119889 minus 11989621198902 minus 1198901 (41)

where 1198962 is the positive constant The block diagram of theoverall system is presented in Figure 4

Substituting (41) in (40) 1198902 becomes

1198902 = 119891 (119909 119911) minus 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 11989621198902 minus 1198901

(42)

Define the estimation error as119891 = 119891 minus 119891

119892 = 119892 minus 119892

(43)

Using (38) and (39) in (43) gives

119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)

where W1 =W1 minus W1 and W2 =W2 minus W2 are weight errorsNow by applying (44) in (42) 1198902 becomes

1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)

Two standard assumptions which are commonly used inthe neural networks literature are given below [26]

Assumption 1 The optimal weightsW1 andW2 are boundedby known positive values so that

1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)

We only need to know that ideal weights exist to prove theconvergence analysis The exact value of the ideal weightsneed not be knownThe symbol ∙ 119865 denotes the Frobeniusnorm that is given amatrixA the Frobenius norm is definedby

A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)

Assumption 2 Thedesired trajectory119910119889 and its derivatives upto second order are bounded

Based on the above Assumptions 1 and 2 the stabilityanalysis is given in Section 43

43 Stability Analysis

Theorem 3 Consider the EV system (30) and control input(41) satisfying Assumptions 1 and 2 If the weights of the CNN1 and CNN 2 are updated according to adaptation law given in(48) and (49) respectively

W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)

where 1205781 and 1205782 are the learning rate and 120588 is dampingcoefficient then the weight errors W1 = W1 minus W1 W2 =W2 minus W2 and the errors 1198901 and 1198902 are uniformly ultimatelybounded (UUB)

Proof Consider the Lyapunov function

1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)

Now substitute 1198901 and 1198902 from (35) and (45) respectively andperform a simple manipulation (ie using 119909119879119910 = tr119909119879119910 =tr119910119909119879 for placing weight matrices inside a trace operator)Then we have

2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)

With the adaptation law given in (48) and (49) (52) becomes

2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)


Apply the following inequality [27]

tr [W119879 (W minus W)] = ⟨WW⟩119865minus10038171003817100381710038171003817W100381710038171003817100381710038172

119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)

And assume that the upper bounds are as follows100381710038171003817100381712057611003817100381710038171003817 le 1205761119872

100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)

Now we can express (53) as

2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)

As the first term is always negative now defining nonnegativeterm and completing the square terms in (56) yield

2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)

which is negative as long as either (58) (59) or (60) holds

[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)

Thus 2 is negative outside a compact set According toa standard Lyapunov theorem extension [28] this demon-strates uniform ultimate boundedness of weight errors W1W2 and errors 1198901 1198902

5 Nonlinear Optimal ControllerDesign Using CNN

The following approach is used to design the nonlinearoptimal controller for the EV system (17)

The tracking errors are defined as

119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)

And the filtered tracking error is defined as

119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)

where Λ is the positive constantDifferentiating (62) and rearranging it we have

119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)

Substituting 2(119905) from EV system (17) (63) becomes

119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112

and 1198651119909 = ((2119870111987031198709)(119898 + 1198702))1199092System (64) may be written as

119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)

Now we define an auxiliary control input 119906(119905) which is tobe optimized in the next subsection as

119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)

with 119906(119905) as the control input The closed-loop systembecomes

119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)

51 Optimal Controller Using Hamilton-Jacobi-Bellman (H-J-B) Optimization The augmented system [27] is achievedusing (62) and (68)

[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)

or with shorter notation

z (119905) = Az (119905) + B119906 (119905) (70)

with z(119905) being defined as z(119905) = [119890(119905) 119903(119905)]119879 A = [minusΛ 119868

0 Λ] and B = [0 119868]119879 A quadratic performance measure

119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)

Theobjective is to find the auxiliary control input119906(119905) thatminimizes the quadratic performance measure 119869(119906) subject


to the constraints imposed by (69) which will be denoted by119906lowast(119905) in the presence of known nonlinearitiesA necessary and sufficient condition for 119906lowast(119905) is that there

exist a function 119881 = 119881(z 119905) which is accredited as the valuefunction and satisfies the H-J-B equation [29]

120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)

where the Hamiltonian of optimization is described as

119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)

and 119881(z 119905) satisfies the partial differential equation

minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)

The minimum is achieved for 119906(119905) = 119906lowast(119905) and the Hamil-tonian is then given by

119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)

Lemma 4 (see [27]) The function 119881 composed of z and 119870satisfies the H-J-B equation

119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)

whereΛ and119870 in (62) and (77) respectively can be found fromthe Riccati differential equation

PA + A119879P119879 minus PB119877minus1B119879P + P +Q = 0 (78)

The optimal control 119906lowast(119905) that minimizes (71) subject to (70) is

119906lowast(119905) = minus119877

minus1B119879Pz = minus119877minus1119903 (119905) (79)

LetQ 119877 be chosen such that

Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879

12lt 0 [27] Then the Λ and 119870 required in

Lemma 4 can be calculated as given below

119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)

with (82) solved for ΛUsing (67) and (79) the input to the EV system is given

as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)

where ℎ(119909) is given by (66) and is assumed to be knownThefollowing subsection details the stability analysis

52 Stability Analysis Suppose that119870 andΛ exist that satisfyLemma 4 and in addition there exist constants 1198961015840

1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840

2lt infin and the spectrum of119875 is bounded

in the sense that 11989610158401119868 lt P lt 119896

1015840

2119868 on (1199050infin) Then using

the feedback control (79) into (70) results in the controllednonlinear system becomes

z (119905) = A minus B119877minus1B119879P z (119905) (84)

The function 119881(z 119905) is chosen as a suitable Lyapunovfunction it shows that 119889119881119889119905 lt 0 for all z = 0 The timederivative of 119881(z 119905) is given by

119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)

From the solution of the H-J-B equation (75) and using (85)it results that

119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)

Substituting optimal control law (79) into (72) the timederivative of 119881(z 119905) becomes

119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)

lt 0 just forall119905 gt 0 z = 0

(87)

The time derivative of the Lyapunov function is negativedefinite implying global exponential stability

53 CNN Based Optimal Adaptive Controller In Section 51the nonlinear function (66) is assumed to be known Thisassumption is relaxed and ℎ(119909) is treated as an unknownnonlinear function The function ℎ(119909) is estimated usingCNN neural network The nonlinear function ℎ(119909) can berepresented by a CNN as

ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)



where 120593(119909) is a basis function for the CNN The blockdiagram in Figure 5 demonstrates the CNN neural controllerbased on H-J-B optimization

The estimate ℎ(119909) of ℎ(119909) can be written as

ℎ (119909) = W119879120593 (119909) (89)

Using (67) (79) and (89) the input is given by

119906 (119905) = (inV (1198651199091)) 119906lowast(119905) minus W119879120593 (119909) minus ] (119905) (90)

where ](119905) is a robustifying term which is given by

] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)

with 119896119911 le 119887119889 and 119903(119905) being defined as the filtered trackingerror in (62) Using (88) and (90) (65) becomes

119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)

where W =Wminus W is the weight-estimation error Using (92)in (70) yields

z (119905) = Az (119905) + B [119906lowast (119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905)] (93)

with A B and z being given in (69) and (70)Using the optimal control law (79) into (93) we have

z (119905) = (A minus B119877minus1B119879P) z (119905) + B W119879120593 (119909) + 120576 (119909) minus ] (119905) (94)

Theorem 5 Suppose the optimal control law 119906lowast(119905) given by(79)minimizes the quadratic performance measure 119869(119906) givenin (71) If the weights of the CNN are updated according toadaptive learning law given by

W = 120593 (119909) z119879PBΓ minus 119896 z W (95)

with Γ gt 0 and 119896 gt 0 then the errors 119890(119905) 119903(119905) and W(119905) areuniformly ultimately bounded

Proof Consider the following Lyapunov function

1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)

where 119870 is positive given by (81) The time derivative 3 of(96) becomes

3 = z119879P z + 12z119879Pz + tr (W119879Γminus1 W) (97)


3 = z119879PAz minus z119879PB119877minus1B119879Pz + 12z119879Pz

+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)

+ tr (W119879Γminus1 W)

(98)

Using z119879PAz = (12)z119879A119879P + PAz and from the Riccatiequation (78) we obtain

1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)

Then applying (99) in (98) and performing a simple manipu-lation for placing weight matrices inside a trace operator wehave

3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)


Now substitute the robustifying term (91) the adaptivelearning law (95) and the following inequality


119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)

The time derivative 3 becomes

3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)

Completing the square terms yields

3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2

minus 120576119872 minus1

41198961198822

119872]

(103)

which is guaranteed to be negative as long as either (104) or(105) holds

(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)

Thus 3 is negative outside a compact set According to astandard Lyapunov theory extension [28] this demonstratesuniform ultimate boundedness of 119890(119905) 119903(119905) and W(119905)

6 Simulation Results

Thedrive cycle tests that are currently used for light-weightedEVs are new European driving cycle (NEDC) Federal TestProcedure (FTP-75) and JC08The NEDC is used in Europeand the low powered EV version of this cycle is used in IndiaThe FTP 75 cycle is used in USA and the JC08 in JapanIn order to show the validity of the proposed controllersthe NEDC is used for testing the performance The NEDCis a driving cycle consisting of four repeated ECE-15 drivingcycles and an extra-urban driving cycle (EUDC) [1] Themaximum speed of NEDC is 120 kmh but it is scaled to50 kmh when applied in this paper [11]

The simulation is implemented inMATLAB 780 (2009a)with m-file programming The controller design parametersfor conventional backstepping are chosen as 1198961= 15 and1198962 = 15 The controller design parameters for CNN basedbackstepping are chosen as 1198961 = 08 and 1198962 = 035 For updateof parameters in (48) and (49) 1205781 1205782 and 120588 are chosen as 100001 and 015 respectively For approximating 119891(119909 119911) theinput to the CNN 1 is 1199111 1199112 and for approximating 119892(119909 119911)the input to the CNN 2 is 1199111 1199112 The order of Chebyshevpolynomial is chosen as 1 for both 1199111 and 1199112 The parameters

of CNNare initialized to zeroThus1206011and120601

2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1

andW2 have dimension of (5 times 1) For optimal controller thedesign parameters are chosen as Λ = 125 119870 = 4 For CNNbased optimal adaptive controller parameter 119896119911 is chosen as00001 For update of parameters in (95) Γ and 119896 are chosen as001 For approximating ℎ(119909) the input to the CNN is 1199111 1199112 119890119910119889 119910119889 and 119910119889 The order of Chebyshev polynomial is chosenas 1 for all inputs to the CNN The parameters of CNN areinitialized to zero Thus 120593 andW have dimension of (13 times 1)The initial conditions for [1199111(0) 1199112(0)]

119879= [001 01]119879Performance of designed controllers for mass varia-

tion as given in (106) is considered Passengers mass isincreaseddecreased at different point of time in the drivingcycleThe variation in armature winding resistance of theDC motor due to temperature changes the variation inthe aerodynamic drag coefficient and the variation in therolling resistance coefficient are considered as given in (107)(108) and (109) respectivelyThe variation inmass armaturewinding resistance aerodynamic drag coefficient and rollingresistance coefficient are assumed to be known in conven-tional backstepping and optimal controller The variationin mass armature winding resistance aerodynamic dragcoefficient and rolling resistance coefficient are consideredunknown in CNN based backstepping controller and CNNbased optimal adaptive controller

Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)

The drive cycle test performances and tracking errorsfor conventional backstepping controller and CNN basedbackstepping controller are shown in Figures 6 and 7respectively It is clear that the conventional backsteppingcontroller has better tracking performance than the CNNbased backstepping controller in the range of speed belowdesigned nominal speed (V = 25 kmhr) The CNN basedbackstepping controller performs much better in high speedrangeThe amp-hour consumption for CNN based backstep-ping controller is 44834 km1075AH

The drive cycle test performances and tracking errorsfor optimal controller and CNN based optimal adaptivecontroller are shown in Figures 8 and 9 respectively Theamp-hour consumption for CNN based optimal adaptivecontroller is 44795 km1070AH It can be seen that the opti-mal controller has better tracking performance in the speed


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

NEDC standardBacksteeping controller

minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)

Figure 6 Performance and tracking error of conventional backstepping controller

0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

NEDC standardCNN based backsteeping controller

minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)

Figure 7 Performance and tracking error of CNN based backstepping controller

0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

NEDC standardOptimal controller

minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)

Figure 8 Performance and tracking error of optimal controller


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

NEDC standardCNN based optimal adaptive controller

minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)

Figure 9 Performance and tracking error of CNN based optimal adaptive controller

0 200 400 600 800 1000 1200Time (s)

Nonlinear optimalNEDC standard

60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)

Nonlinear robustNEDC standard

60

40

20

0Spee

d (k

mh

r)

(b)

Figure 10 Results of NEDC test of [11]

Table 2 Comparative results of driving cycle test

Controller Amp-hour consumptionNonlinear optimal [11] 448 km1197AHNonlinear robust [11] 44825 km1078AHCNN based backstepping 44834 km1075AHCNN based optimal adaptive 44795 km1070AH

range below designed nominal speed than conventionalbackstepping controller and CNN based backstepping con-troller However for the high speed range the performanceof CNN based backstepping controller is comparable to theoptimal controller CNN based optimal adaptive controllergives the best tracking performance on the entire speedrange of the drive cycle test as compared to CNN basedbackstepping controller designed in current work and theresults presented in [11] which are reproduced in Figure 10The comparative result of the driving cycle test is presentedin Table 2 and to provide detailed quantitative analysis ofthe designed controllers root mean squared (RMS) trackingerror of controllers is presented in Table 3 The proposedCNN based optimal adaptive learning shows both robustnessand adaptation to changing system dynamics and unknownnonlinearities

Table 3 RMS tracking error of controllers designed

Controller RMS value of tracking errorsConventional backstepping 00301CNN based backstepping 00279Optimal 00165CNN based optimal adaptive 00040

The traction force is required to propel the EV inforward direction It is produced by the DC motor torqueand transferred through transmission unit which includesthe gearing system and finally drive the vehicle While thevehicle is in motion there are forces that try to stop itsmovementThese forces usually include rolling resistance andaerodynamic drag force The simulation result of tractionforce of the EV is shown in Figure 11

In present work all the parameters are specified on alight-weighted all-electric vehicle and are given in Table 1The mass of the vehicle is considered as 800 kg This includesthe body mass of the vehicle accessories weight of theDC motor and batteries For modeling light-weighted EVswe have considered four batteries each of 12 volts and thisrestricts the control effort in the range of 0 sim 48V (seeTable 1) The control effort is shown in Figure 12


600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)

Traction forceAerodynamic drag forceRolling resistance force

Acceleration force

Figure 11 Traction force

7 Conclusion

Two nonlinear controllers for a light-weighted all-electricvehicle CNN based backstepping controller and CNN basedoptimal adaptive controller are presented in this paper Theunknown nonlinearities in EV system arise due to varyingmass of passengers varying resistance in the armature wind-ing of the DC motor and variation in aerodynamic dragcoefficient and the rolling resistance coefficient are estimatedby CNN The CNN weights are updated online accordingto the adaptive-learning algorithm which is obtained fromLyapunov stability analysis so that system-tracking stabilityand error convergence can be assured in the closed-loopsystem The salient feature of the proposed design method-ologies demonstrates that the control objective is obtainedwith unknown nonlinear dynamics of the EV system TheNEDC is used for testing the performance of the proposedcontrollers It is shown that the tracking performance of thecontrollers designed in this paper is satisfactory in both thecasesThe test results for CNN based backstepping controllerand the CNN based optimal adaptive controller have bettertracking performance than that reported by Huang et al [11]and amp-hour consumption is also less than the nonlinearcontrollers described in [11] From simulation results and thecomparative and quantitative results presented in Tables 2and 3 respectively it is clear that the CNN based optimaladaptive controller gives better performance as compared toother controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)

Figure 12 Control effort 119906(119905)

Acknowledgment

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments and constructivesuggestionswhich have helped in improving the quality of thepaper

References

[1] J Larminie and J Lowry Electric Vehicle Technology ExplainedJohn Wiley amp Sons West Sussex UK 2003

[2] Y Cheng J Van Mierlo P Van Den Bossche and P LataireldquoEnergy sources control and management in hybrid electricvehiclesrdquo inProceedings of the 12th International Power Electron-ics and Motion Control Conference (EPE-PEMC rsquo06) pp 524ndash530 Portoroz Slovenia September 2006

[3] D F Opila X Wang R McGee R B Gillespie J A Cook andJ W Grizzle ldquoAn energy management controller to optimallytrade off fuel economy and drivability for hybrid vehiclesrdquo IEEETransactions on Control Systems Technology vol 20 no 6 pp1490ndash1505 2012

[4] Y L Murphey J Park Z ChenM L KuangM AMasrur andA M Phillips ldquoIntelligent hybrid vehicle power control-part Imachine learning of optimal vehicle powerrdquo IEEE Transactionon Vehicular Technology vol 61 pp 3519ndash3530 2012

[5] Y L Murphey J Park L Kiliaris et al ldquoIntelligent hybridvehicle power control-part II online intelligent energymanage-mentrdquo IEEE Transaction on Vehicular Technology vol 62 pp69ndash79 2013

[6] H Zhang LM Tolbert andBOzpineci ldquoImpact of SiC deviceson hybrid electric and plug-in hybrid electric vehiclesrdquo IEEETransactions on Industry Applications vol 47 no 2 pp 912ndash9212011

[7] J Dixon I Nakashima E F Arcos and M Ortuzar ldquoElectricvehicle using a combination of ultracapacitors and ZEBRAbatteryrdquo IEEE Transactions on Industrial Electronics vol 57 no3 pp 943ndash949 2010

[8] M Ye Z Bai and B Cao ldquoRobust control for regenerativebraking of battery electric vehiclerdquo IET Control Theory andApplications vol 2 no 12 pp 1105ndash1114 2008

[9] M Pahlevaninezhad P Das J Drobnik GMoschopoulos P KJain and A Bakhshai ldquoA nonlinear optimal control approachbased on the control-lyapunov function for an ACDC con-verter used in electric vehiclesrdquo IEEE Transaction on IndustrialInformatics vol 8 pp 596ndash614 2012


[10] R Wang and J Wang ldquoPassive actuator fault-tolerant controlfor a class of overactuated nonlinear systems and applicationsto electric vehiclesrdquo IEEE Transaction on Vehicular Technologyvol 62 pp 972ndash985 2013

[11] Q Huang Z Huang and H Zhou ldquoNonlinear optimal androbust speed control for a light-weighted all-electric vehiclerdquoIET Control Theory and Applications vol 3 no 4 pp 437ndash4442009

[12] S Poorani K Udaya Kumar and S Renganarayanan ldquoIntelli-gent controller design for electric vehiclerdquo in Proceedings of the57th IEEE Semiannual Vehicular Technology Conference (VTCrsquo03) vol 4 pp 2447ndash2450 Jeju Korea April 2003

[13] P Khatun C M Bingham N Schofield and P H MellorldquoApplication of fuzzy control algorithms for electric vehicleantilock brakingtraction control systemsrdquo IEEE Transactionson Vehicular Technology vol 52 no 5 pp 1356ndash1364 2003

[14] V Schwarzer and R Ghorbani ldquoDrive cycle generation fordesign optimization of electric vehiclesrdquo IEEE Transaction onVehicular Technology vol 62 pp 89ndash97 2013

[15] S Kachroudi M Grossard and N Abroug ldquoPredictive drivingguidance of full electric vehicles using particle swarm optimiza-tionrdquo IEEE Transaction on Vehicular Technology vol 61 pp1309ndash3919 2012

[16] A Namatame and N Ueda ldquoPattern classification with Cheby-shev neural networksrdquo International Journal of Neural Networksvol 3 pp 23ndash31 1992

[17] T-T Lee and J-T Jeng ldquoThe Chebyshev-polynomials-basedunified model neural networks for function approximationrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 28 no 6 pp 925ndash935 1998

[18] J C Patra and A C Kot ldquoNonlinear dynamic system iden-tification using Chebyshev functional link artificial neuralnetworksrdquo IEEE Transactions on Systems Man and CyberneticsB Cybernetics vol 32 no 4 pp 505ndash511 2002

[19] S Purwar I N Kar and A N Jha ldquoOn-line system identifi-cation of complex systems using Chebyshev neural networksrdquoApplied Soft Computing Journal vol 7 no 1 pp 364ndash372 2007

[20] J C Patra ldquoChebyshev neural network-based model for dual-junction solar cellsrdquo IEEE Transactions on Energy Conversionvol 26 no 1 pp 132ndash139 2011

[21] S Purwar I N Kar and A N Jha ldquoAdaptive output feedbacktracking control of robot manipulators using position measure-ments onlyrdquo Expert Systems with Applications vol 34 no 4 pp2789ndash2798 2008

[22] A-M Zou K D Kumar and Z-G Hou ldquoQuaternion-basedadaptive output feedback attitude control of spacecraft usingchebyshev neural networksrdquo IEEE Transactions on NeuralNetworks vol 21 no 9 pp 1457ndash1471 2010

[23] A-M Zou K D Kumar Z-G Hou and X Liu ldquoFinite-timeattitude tracking control for spacecraft using terminal slidingmode and chebyshev neural networkrdquo IEEE Transactions onSystems Man and Cybernetics B Cybernetics vol 41 no 4 pp950ndash963 2011

[24] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[25] H K Khalil Nonlinear Systems Prentice Hall Upper SaddleRiver NJ USA 3rd edition 2002

[26] C Kwan and F L Lewis ldquoRobust backstepping control ofnonlinear systems using neural networksrdquo IEEETransactions on

Systems Man and Cybernetics A Systems and Humans vol 30no 6 pp 753ndash766 2000

[27] Y H Kim and F L Lewis ldquoOptimal design of CMAC neural-network controller for robot manipulatorsrdquo IEEE Transactionson Systems Man and Cybernetics C Applications and Reviewsvol 30 no 1 pp 22ndash31 2000

[28] K S Narendra and A M Annaswamy ldquoA new adaptivelaw for robust adaptation without persistent excitationrdquo IEEETransactions on Automatic Control vol 32 no 2 pp 134ndash1451987

[29] F L Lewis and V L Syrmos Optimal Control John Wiley ampSons New York NY USA 2nd edition 1995

Senior Project

Main Topics

Biomedical Engineering Biomechanics Biomechanics of Level Ground Walking

Sub Topics offered are as follow

(1) Design of Experimental System to Capture

Gait Data

(2) Simulation of Human Walking using

MATLAB based on captured data

Abstract

Gait data is needed in gait analysis that can be use to certain area in both clinical and

engineering For this research the gait data is needed to develop a model of stance phase in level

ground human walking The scope of work of this research will include but not limited to the

experimental system design that will capture gait data of the subject ie knee angle hip angle ankle

angle acceleration and ground reaction force and simulation of level ground walking in MATLAB (1)Subject will walk on a treadmill with force sensors attached indirectly to both of the feet by shoes

Reflective markers will also attached to the hip knee and ankle joint of the subject and a high speed

camera will capture the image of the subject during walking on a treadmill Angles and acceleration

data will be obtained by employing image processing technique while ground reaction force data will

be obtained directly by force sensors (2)After the data are obtained a proper model will be

constructed based on mathematical equations that derived from dynamical system model of stance

phase The simulation will run on MATLAB Simulink with Simscape toolbox

Required Competencies

For student who want to take sub topic (1)

Microcontroller Electronic Circuit Design

Image processing


MATLAB MATLAB Simulink Dynamic System

Modelling

Picture shown is just an illustration as the actual experimental system and simulation may not be the exact same

Courtesy of intechopencom

WK2

WK2 - ref

WK3

2


Requirements

Basic skill in Machine Drives

Strong background in Control Systems

Strong skills in MATLAB (Simulink)

Some knowledge about Artificial Neural Networks (ANN)

References

1 Zhang Ruiwei Qian Xisen ldquoAn adaptive sliding mode current control for switched

reluctance motorrdquo IEEE Conference and Expo Transportation Electrification Asia-Pacific

(ITEC Asia-Pacific) Aug 2014

2 Vikas Sharma and Shubhi Purwar ldquoNonlinear Controllers for a Light-Weighted All-

Electric Vehicle Using Chebyshev Neural Networkrdquo International Journal of Vehicular

Technology Hindawi 2014









1 Introduction













119865 = 120583119903119903119898119892⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1

+1

2120588119860119862119889V

2

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2

+ 119898119892 sin120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

3

+ 119898119889V119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟

4

(1)

Transmissionunit

Motor dynamics

Vehicledynamics

+minus

Figure 1 EV system




119879119871 = 119865119903

119866 (2)



119869119889120596

119889119905= 119871119886119891119894

2minus 119861120596 minus 119879119871

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894120596

(3)







0 20 40 60 80 100

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

minus10



119869 + 1198981199032

1198662119889120596

119889119905= 119871119886119891119894

2minus 119861120596

minus119903

119866

120583119903119903119898119892

+1

2120588119860119862119889V

2+ 119898119892 sin120601

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894 120596

(4)



V =119903

119866120596 (5)


Functionalexpansion

sum

x1

x2

W

y




119879119894+1 (119909) = 2119909119879119894 (119909) minus 119879119894minus1 (119909) 1198790 (119909) = 1 (6)



(7)






119910 (119909) =W119879120601 + 120576 (8)


=W119879

120601 (9)


3 Problem Statement


X = 119891 (X) + 119892 (X) 119906

119910 = ℎ (X) (10)

where

X = [11990911199092] = [

120596

119894]

119891 (X) =[[[[

[

1

119869 + 119898 (11990321198662 )

11987111988611989111990922minus 1198611199091 minus

119903

119866

times(120583119903119903119898119892 +

1

2120588119860119862119889

1199032

119866211990912

+119898119892 sin120601)

minus119877119886 + 119877119891

119871119886 + 119871field1199092 minus

119871119886119891

119871119886 + 119871field11990911199092

]]]]

]

(11)

119892 (X) = [[

0

1

119871119886 + 119871field

]

]

ℎ (X) = 1199091 (12)


119891 (X) = [[

1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

minus11987071199092 minus 119870811990911199092

]

]

119892 (X) = [ 01198709] ℎ (X) = 1199091

(13)

where 1198701 = 11986621199032 1198702 = (119866

21199032)119869 1198703 = 119871119886119891 1198704 = 119861 1198705 =

(12)120588119860119862119889(11990331198663) 1198706 = (119903119866)120583119903119903119892 1198707 = (119877119886 + 119877119891)(119871119886 +


1 =1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

2 = minus11987071199092 minus 119870811990911199092 + 1198709119906

119910 = 1199091

(14)


lim119905rarrinfin

(119910 minus 119910119889) = 0 (15)





1199111 = 1199091

1199112 = 1

(16)


1 = 1199112

2 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922minus11987011198704

119898 + 1198702

1199112

minus211987011198705

119898 + 1198702

11991111199112 +2119870111987031198709

119898 + 1198702

1199092119906

119910 = 1199111

(17)


1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889 (18)

1198902 = 1199112 minus 1199112119889 (19)


1198901 = 1 minus 119910119889 (20)


1198901 = 1198902 + 1199112119889 minus 119910119889 (21)



1198901 = 1198902 minus 11989611198901 (22)


1198902 = 2 minus 2119889 (23)


1198902 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922

minus11987011198704

119898 + 1198702

1199112 minus211987011198705

119898 + 1198702

11991111199112

+2119870111987031198709

119898 + 1198702

1199092119906 minus 2119889

(24)


119906 =

2119889 minus 1198901 +2119870111987031198707

119898 + 1198702

11990922+2119870111987031198708

119898 + 1198702

119911111990922

+11987011198704

119898 + 1198702

1199112 +211987011198705

119898 + 1198702

11991111199112 minus 11989621198902

times1

(2119870111987031198709) (119898 + 1198702) 1199092

(25)


1198902 = minus1198901 minus 11989621198902 (26)


1198711 =1

21198902

1+1

21198902

2 (27)


1 = 1198901 1198901 + 1198902 1198902 (28)


1 = minus11989611198902

1minus 11989621198902

2 (29)



1 = 1199112

2 = 119891 (119909 119911) + 119892 (119909 119911) 119906

119910 = 1199111

(30)



1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889

1198902 = 1199112 minus 1199112119889

(31)



1198901 = 1 minus 119910119889 (32)


1198901 = 1198902 + 1199112119889 minus 119910119889 (33)


1199112119889 = 119910119889 minus 11989611198901 (34)


1198901 = 1198902 minus 11989611198901 (35)


1198902 = 2 minus 2119889 (36)


1198902 = 119891 (119909 119911) + 119892 (119909 119911) 119906 minus 2119889 (37)


119891 =W11987911206011+ 1205761

119892 =W11987921206012+ 1205762

(38)


1and 120601

2


119891 = W11987911206011

119892 = W11987921206012

(39)



1198902 = 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 2119889 + 119892 (119909 119911) 119906

(40)


+ +

+

+minusminus

minus



119906 =1





(42)


119892 = 119892 minus 119892

(43)


119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)


1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)



1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)


A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)





W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)



1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3









1 Introduction













119865 = 120583119903119903119898119892⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1

+1

2120588119860119862119889V

2

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2

+ 119898119892 sin120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

3

+ 119898119889V119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟

4

(1)

Transmissionunit

Motor dynamics

Vehicledynamics

+minus

Figure 1 EV system




119879119871 = 119865119903

119866 (2)



119869119889120596

119889119905= 119871119886119891119894

2minus 119861120596 minus 119879119871

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894120596

(3)







0 20 40 60 80 100

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

minus10



119869 + 1198981199032

1198662119889120596

119889119905= 119871119886119891119894

2minus 119861120596

minus119903

119866

120583119903119903119898119892

+1

2120588119860119862119889V

2+ 119898119892 sin120601

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894 120596

(4)



V =119903

119866120596 (5)


Functionalexpansion

sum

x1

x2

W

y




119879119894+1 (119909) = 2119909119879119894 (119909) minus 119879119894minus1 (119909) 1198790 (119909) = 1 (6)



(7)






119910 (119909) =W119879120601 + 120576 (8)


=W119879

120601 (9)


3 Problem Statement


X = 119891 (X) + 119892 (X) 119906

119910 = ℎ (X) (10)

where

X = [11990911199092] = [

120596

119894]

119891 (X) =[[[[

[

1

119869 + 119898 (11990321198662 )

11987111988611989111990922minus 1198611199091 minus

119903

119866

times(120583119903119903119898119892 +

1

2120588119860119862119889

1199032

119866211990912

+119898119892 sin120601)

minus119877119886 + 119877119891

119871119886 + 119871field1199092 minus

119871119886119891

119871119886 + 119871field11990911199092

]]]]

]

(11)

119892 (X) = [[

0

1

119871119886 + 119871field

]

]

ℎ (X) = 1199091 (12)


119891 (X) = [[

1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

minus11987071199092 minus 119870811990911199092

]

]

119892 (X) = [ 01198709] ℎ (X) = 1199091

(13)

where 1198701 = 11986621199032 1198702 = (119866

21199032)119869 1198703 = 119871119886119891 1198704 = 119861 1198705 =

(12)120588119860119862119889(11990331198663) 1198706 = (119903119866)120583119903119903119892 1198707 = (119877119886 + 119877119891)(119871119886 +


1 =1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

2 = minus11987071199092 minus 119870811990911199092 + 1198709119906

119910 = 1199091

(14)


lim119905rarrinfin

(119910 minus 119910119889) = 0 (15)





1199111 = 1199091

1199112 = 1

(16)


1 = 1199112

2 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922minus11987011198704

119898 + 1198702

1199112

minus211987011198705

119898 + 1198702

11991111199112 +2119870111987031198709

119898 + 1198702

1199092119906

119910 = 1199111

(17)


1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889 (18)

1198902 = 1199112 minus 1199112119889 (19)


1198901 = 1 minus 119910119889 (20)


1198901 = 1198902 + 1199112119889 minus 119910119889 (21)



1198901 = 1198902 minus 11989611198901 (22)


1198902 = 2 minus 2119889 (23)


1198902 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922

minus11987011198704

119898 + 1198702

1199112 minus211987011198705

119898 + 1198702

11991111199112

+2119870111987031198709

119898 + 1198702

1199092119906 minus 2119889

(24)


119906 =

2119889 minus 1198901 +2119870111987031198707

119898 + 1198702

11990922+2119870111987031198708

119898 + 1198702

119911111990922

+11987011198704

119898 + 1198702

1199112 +211987011198705

119898 + 1198702

11991111199112 minus 11989621198902

times1

(2119870111987031198709) (119898 + 1198702) 1199092

(25)


1198902 = minus1198901 minus 11989621198902 (26)


1198711 =1

21198902

1+1

21198902

2 (27)


1 = 1198901 1198901 + 1198902 1198902 (28)


1 = minus11989611198902

1minus 11989621198902

2 (29)



1 = 1199112

2 = 119891 (119909 119911) + 119892 (119909 119911) 119906

119910 = 1199111

(30)



1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889

1198902 = 1199112 minus 1199112119889

(31)



1198901 = 1 minus 119910119889 (32)


1198901 = 1198902 + 1199112119889 minus 119910119889 (33)


1199112119889 = 119910119889 minus 11989611198901 (34)


1198901 = 1198902 minus 11989611198901 (35)


1198902 = 2 minus 2119889 (36)


1198902 = 119891 (119909 119911) + 119892 (119909 119911) 119906 minus 2119889 (37)


119891 =W11987911206011+ 1205761

119892 =W11987921206012+ 1205762

(38)


1and 120601

2


119891 = W11987911206011

119892 = W11987921206012

(39)



1198902 = 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 2119889 + 119892 (119909 119911) 119906

(40)


+ +

+

+minusminus

minus



119906 =1





(42)


119892 = 119892 minus 119892

(43)


119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)


1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)



1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)


A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)





W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)



1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3








119865 = 120583119903119903119898119892⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1

+1

2120588119860119862119889V

2

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2

+ 119898119892 sin120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

3

+ 119898119889V119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟

4

(1)

Transmissionunit

Motor dynamics

Vehicledynamics

+minus

Figure 1 EV system




119879119871 = 119865119903

119866 (2)



119869119889120596

119889119905= 119871119886119891119894

2minus 119861120596 minus 119879119871

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894120596

(3)







0 20 40 60 80 100

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

minus10



119869 + 1198981199032

1198662119889120596

119889119905= 119871119886119891119894

2minus 119861120596

minus119903

119866

120583119903119903119898119892

+1

2120588119860119862119889V

2+ 119898119892 sin120601

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894 120596

(4)



V =119903

119866120596 (5)


Functionalexpansion

sum

x1

x2

W

y




119879119894+1 (119909) = 2119909119879119894 (119909) minus 119879119894minus1 (119909) 1198790 (119909) = 1 (6)



(7)






119910 (119909) =W119879120601 + 120576 (8)


=W119879

120601 (9)


3 Problem Statement


X = 119891 (X) + 119892 (X) 119906

119910 = ℎ (X) (10)

where

X = [11990911199092] = [

120596

119894]

119891 (X) =[[[[

[

1

119869 + 119898 (11990321198662 )

11987111988611989111990922minus 1198611199091 minus

119903

119866

times(120583119903119903119898119892 +

1

2120588119860119862119889

1199032

119866211990912

+119898119892 sin120601)

minus119877119886 + 119877119891

119871119886 + 119871field1199092 minus

119871119886119891

119871119886 + 119871field11990911199092

]]]]

]

(11)

119892 (X) = [[

0

1

119871119886 + 119871field

]

]

ℎ (X) = 1199091 (12)


119891 (X) = [[

1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

minus11987071199092 minus 119870811990911199092

]

]

119892 (X) = [ 01198709] ℎ (X) = 1199091

(13)

where 1198701 = 11986621199032 1198702 = (119866

21199032)119869 1198703 = 119871119886119891 1198704 = 119861 1198705 =

(12)120588119860119862119889(11990331198663) 1198706 = (119903119866)120583119903119903119892 1198707 = (119877119886 + 119877119891)(119871119886 +


1 =1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

2 = minus11987071199092 minus 119870811990911199092 + 1198709119906

119910 = 1199091

(14)


lim119905rarrinfin

(119910 minus 119910119889) = 0 (15)





1199111 = 1199091

1199112 = 1

(16)


1 = 1199112

2 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922minus11987011198704

119898 + 1198702

1199112

minus211987011198705

119898 + 1198702

11991111199112 +2119870111987031198709

119898 + 1198702

1199092119906

119910 = 1199111

(17)


1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889 (18)

1198902 = 1199112 minus 1199112119889 (19)


1198901 = 1 minus 119910119889 (20)


1198901 = 1198902 + 1199112119889 minus 119910119889 (21)



1198901 = 1198902 minus 11989611198901 (22)


1198902 = 2 minus 2119889 (23)


1198902 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922

minus11987011198704

119898 + 1198702

1199112 minus211987011198705

119898 + 1198702

11991111199112

+2119870111987031198709

119898 + 1198702

1199092119906 minus 2119889

(24)


119906 =

2119889 minus 1198901 +2119870111987031198707

119898 + 1198702

11990922+2119870111987031198708

119898 + 1198702

119911111990922

+11987011198704

119898 + 1198702

1199112 +211987011198705

119898 + 1198702

11991111199112 minus 11989621198902

times1

(2119870111987031198709) (119898 + 1198702) 1199092

(25)


1198902 = minus1198901 minus 11989621198902 (26)


1198711 =1

21198902

1+1

21198902

2 (27)


1 = 1198901 1198901 + 1198902 1198902 (28)


1 = minus11989611198902

1minus 11989621198902

2 (29)



1 = 1199112

2 = 119891 (119909 119911) + 119892 (119909 119911) 119906

119910 = 1199111

(30)



1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889

1198902 = 1199112 minus 1199112119889

(31)



1198901 = 1 minus 119910119889 (32)


1198901 = 1198902 + 1199112119889 minus 119910119889 (33)


1199112119889 = 119910119889 minus 11989611198901 (34)


1198901 = 1198902 minus 11989611198901 (35)


1198902 = 2 minus 2119889 (36)


1198902 = 119891 (119909 119911) + 119892 (119909 119911) 119906 minus 2119889 (37)


119891 =W11987911206011+ 1205761

119892 =W11987921206012+ 1205762

(38)


1and 120601

2


119891 = W11987911206011

119892 = W11987921206012

(39)



1198902 = 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 2119889 + 119892 (119909 119911) 119906

(40)


+ +

+

+minusminus

minus



119906 =1





(42)


119892 = 119892 minus 119892

(43)


119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)


1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)



1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)


A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)





W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)



1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3





0 20 40 60 80 100

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)

minus10



119869 + 1198981199032

1198662119889120596

119889119905= 119871119886119891119894

2minus 119861120596

minus119903

119866

120583119903119903119898119892

+1

2120588119860119862119889V

2+ 119898119892 sin120601

(119871119886 + 119871field)119889119894

119889119905= 119881 minus (119877119886 + 119877119891) 119894 minus 119871119886119891119894 120596

(4)



V =119903

119866120596 (5)


Functionalexpansion

sum

x1

x2

W

y




119879119894+1 (119909) = 2119909119879119894 (119909) minus 119879119894minus1 (119909) 1198790 (119909) = 1 (6)



(7)






119910 (119909) =W119879120601 + 120576 (8)


=W119879

120601 (9)


3 Problem Statement


X = 119891 (X) + 119892 (X) 119906

119910 = ℎ (X) (10)

where

X = [11990911199092] = [

120596

119894]

119891 (X) =[[[[

[

1

119869 + 119898 (11990321198662 )

11987111988611989111990922minus 1198611199091 minus

119903

119866

times(120583119903119903119898119892 +

1

2120588119860119862119889

1199032

119866211990912

+119898119892 sin120601)

minus119877119886 + 119877119891

119871119886 + 119871field1199092 minus

119871119886119891

119871119886 + 119871field11990911199092

]]]]

]

(11)

119892 (X) = [[

0

1

119871119886 + 119871field

]

]

ℎ (X) = 1199091 (12)


119891 (X) = [[

1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

minus11987071199092 minus 119870811990911199092

]

]

119892 (X) = [ 01198709] ℎ (X) = 1199091

(13)

where 1198701 = 11986621199032 1198702 = (119866

21199032)119869 1198703 = 119871119886119891 1198704 = 119861 1198705 =

(12)120588119860119862119889(11990331198663) 1198706 = (119903119866)120583119903119903119892 1198707 = (119877119886 + 119877119891)(119871119886 +


1 =1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

2 = minus11987071199092 minus 119870811990911199092 + 1198709119906

119910 = 1199091

(14)


lim119905rarrinfin

(119910 minus 119910119889) = 0 (15)





1199111 = 1199091

1199112 = 1

(16)


1 = 1199112

2 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922minus11987011198704

119898 + 1198702

1199112

minus211987011198705

119898 + 1198702

11991111199112 +2119870111987031198709

119898 + 1198702

1199092119906

119910 = 1199111

(17)


1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889 (18)

1198902 = 1199112 minus 1199112119889 (19)


1198901 = 1 minus 119910119889 (20)


1198901 = 1198902 + 1199112119889 minus 119910119889 (21)



1198901 = 1198902 minus 11989611198901 (22)


1198902 = 2 minus 2119889 (23)


1198902 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922

minus11987011198704

119898 + 1198702

1199112 minus211987011198705

119898 + 1198702

11991111199112

+2119870111987031198709

119898 + 1198702

1199092119906 minus 2119889

(24)


119906 =

2119889 minus 1198901 +2119870111987031198707

119898 + 1198702

11990922+2119870111987031198708

119898 + 1198702

119911111990922

+11987011198704

119898 + 1198702

1199112 +211987011198705

119898 + 1198702

11991111199112 minus 11989621198902

times1

(2119870111987031198709) (119898 + 1198702) 1199092

(25)


1198902 = minus1198901 minus 11989621198902 (26)


1198711 =1

21198902

1+1

21198902

2 (27)


1 = 1198901 1198901 + 1198902 1198902 (28)


1 = minus11989611198902

1minus 11989621198902

2 (29)



1 = 1199112

2 = 119891 (119909 119911) + 119892 (119909 119911) 119906

119910 = 1199111

(30)



1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889

1198902 = 1199112 minus 1199112119889

(31)



1198901 = 1 minus 119910119889 (32)


1198901 = 1198902 + 1199112119889 minus 119910119889 (33)


1199112119889 = 119910119889 minus 11989611198901 (34)


1198901 = 1198902 minus 11989611198901 (35)


1198902 = 2 minus 2119889 (36)


1198902 = 119891 (119909 119911) + 119892 (119909 119911) 119906 minus 2119889 (37)


119891 =W11987911206011+ 1205761

119892 =W11987921206012+ 1205762

(38)


1and 120601

2


119891 = W11987911206011

119892 = W11987921206012

(39)



1198902 = 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 2119889 + 119892 (119909 119911) 119906

(40)


+ +

+

+minusminus

minus



119906 =1





(42)


119892 = 119892 minus 119892

(43)


119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)


1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)



1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)


A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)





W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)



1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3




119910 (119909) =W119879120601 + 120576 (8)


=W119879

120601 (9)


3 Problem Statement


X = 119891 (X) + 119892 (X) 119906

119910 = ℎ (X) (10)

where

X = [11990911199092] = [

120596

119894]

119891 (X) =[[[[

[

1

119869 + 119898 (11990321198662 )

11987111988611989111990922minus 1198611199091 minus

119903

119866

times(120583119903119903119898119892 +

1

2120588119860119862119889

1199032

119866211990912

+119898119892 sin120601)

minus119877119886 + 119877119891

119871119886 + 119871field1199092 minus

119871119886119891

119871119886 + 119871field11990911199092

]]]]

]

(11)

119892 (X) = [[

0

1

119871119886 + 119871field

]

]

ℎ (X) = 1199091 (12)


119891 (X) = [[

1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

minus11987071199092 minus 119870811990911199092

]

]

119892 (X) = [ 01198709] ℎ (X) = 1199091

(13)

where 1198701 = 11986621199032 1198702 = (119866

21199032)119869 1198703 = 119871119886119891 1198704 = 119861 1198705 =

(12)120588119860119862119889(11990331198663) 1198706 = (119903119866)120583119903119903119892 1198707 = (119877119886 + 119877119891)(119871119886 +


1 =1198701

119898 + 1198702

119870311990922minus 11987041199091 minus 11987051199091

2minus 1198706119898

2 = minus11987071199092 minus 119870811990911199092 + 1198709119906

119910 = 1199091

(14)


lim119905rarrinfin

(119910 minus 119910119889) = 0 (15)





1199111 = 1199091

1199112 = 1

(16)


1 = 1199112

2 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922minus11987011198704

119898 + 1198702

1199112

minus211987011198705

119898 + 1198702

11991111199112 +2119870111987031198709

119898 + 1198702

1199092119906

119910 = 1199111

(17)


1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889 (18)

1198902 = 1199112 minus 1199112119889 (19)


1198901 = 1 minus 119910119889 (20)


1198901 = 1198902 + 1199112119889 minus 119910119889 (21)



1198901 = 1198902 minus 11989611198901 (22)


1198902 = 2 minus 2119889 (23)


1198902 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922

minus11987011198704

119898 + 1198702

1199112 minus211987011198705

119898 + 1198702

11991111199112

+2119870111987031198709

119898 + 1198702

1199092119906 minus 2119889

(24)


119906 =

2119889 minus 1198901 +2119870111987031198707

119898 + 1198702

11990922+2119870111987031198708

119898 + 1198702

119911111990922

+11987011198704

119898 + 1198702

1199112 +211987011198705

119898 + 1198702

11991111199112 minus 11989621198902

times1

(2119870111987031198709) (119898 + 1198702) 1199092

(25)


1198902 = minus1198901 minus 11989621198902 (26)


1198711 =1

21198902

1+1

21198902

2 (27)


1 = 1198901 1198901 + 1198902 1198902 (28)


1 = minus11989611198902

1minus 11989621198902

2 (29)



1 = 1199112

2 = 119891 (119909 119911) + 119892 (119909 119911) 119906

119910 = 1199111

(30)



1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889

1198902 = 1199112 minus 1199112119889

(31)



1198901 = 1 minus 119910119889 (32)


1198901 = 1198902 + 1199112119889 minus 119910119889 (33)


1199112119889 = 119910119889 minus 11989611198901 (34)


1198901 = 1198902 minus 11989611198901 (35)


1198902 = 2 minus 2119889 (36)


1198902 = 119891 (119909 119911) + 119892 (119909 119911) 119906 minus 2119889 (37)


119891 =W11987911206011+ 1205761

119892 =W11987921206012+ 1205762

(38)


1and 120601

2


119891 = W11987911206011

119892 = W11987921206012

(39)



1198902 = 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 2119889 + 119892 (119909 119911) 119906

(40)


+ +

+

+minusminus

minus



119906 =1





(42)


119892 = 119892 minus 119892

(43)


119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)


1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)



1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)


A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)





W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)



1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3



1198901 = 1198902 minus 11989611198901 (22)


1198902 = 2 minus 2119889 (23)


1198902 = minus2119870111987031198707

119898 + 1198702

11990922minus2119870111987031198708

119898 + 1198702

119911111990922

minus11987011198704

119898 + 1198702

1199112 minus211987011198705

119898 + 1198702

11991111199112

+2119870111987031198709

119898 + 1198702

1199092119906 minus 2119889

(24)


119906 =

2119889 minus 1198901 +2119870111987031198707

119898 + 1198702

11990922+2119870111987031198708

119898 + 1198702

119911111990922

+11987011198704

119898 + 1198702

1199112 +211987011198705

119898 + 1198702

11991111199112 minus 11989621198902

times1

(2119870111987031198709) (119898 + 1198702) 1199092

(25)


1198902 = minus1198901 minus 11989621198902 (26)


1198711 =1

21198902

1+1

21198902

2 (27)


1 = 1198901 1198901 + 1198902 1198902 (28)


1 = minus11989611198902

1minus 11989621198902

2 (29)



1 = 1199112

2 = 119891 (119909 119911) + 119892 (119909 119911) 119906

119910 = 1199111

(30)



1198901 = 1199111 minus 1199111119889 = 1199111 minus 119910119889

1198902 = 1199112 minus 1199112119889

(31)



1198901 = 1 minus 119910119889 (32)


1198901 = 1198902 + 1199112119889 minus 119910119889 (33)


1199112119889 = 119910119889 minus 11989611198901 (34)


1198901 = 1198902 minus 11989611198901 (35)


1198902 = 2 minus 2119889 (36)


1198902 = 119891 (119909 119911) + 119892 (119909 119911) 119906 minus 2119889 (37)


119891 =W11987911206011+ 1205761

119892 =W11987921206012+ 1205762

(38)


1and 120601

2


119891 = W11987911206011

119892 = W11987921206012

(39)



1198902 = 119891 (119909 119911) + 119892 (119909 119911) minus 119892 (119909 119911) 119906 minus 2119889 + 119892 (119909 119911) 119906

(40)


+ +

+

+minusminus

minus



119906 =1





(42)


119892 = 119892 minus 119892

(43)


119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)


1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)



1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)


A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)





W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)



1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3


+ +

+

+minusminus

minus



119906 =1





(42)


119892 = 119892 minus 119892

(43)


119891 = W11987911206011+ 1205761

119892 = W11987921206012+ 1205762

(44)


1198902 = W11987911206011+ 1205761 + (W

119879

21206012+ 1205762) 119906 minus 11989621198902 minus 1198901 (45)



1003817100381710038171003817W11003817100381710038171003817119865 leW1119872

1003817100381710038171003817W21003817100381710038171003817119865 le 1198822119872 (46)


A2119865= tr (A119879A) = sum

119894119895

1198862

119894119895 (47)





W1 = 120578112060111198902 minus 1205881205781100381710038171003817100381711989021003817100381710038171003817 W1 (48)

W2 = 120578212060121199061198902 minus 1205881205782100381710038171003817100381711989021003817100381710038171003817 |119906| W2 (49)



1198712 =1

21198902

1+1

21198902

2+1

2tr W119879

1120578minus1

1W1 +

1

2tr W119879

2120578minus1

2W2

(50)


2 = 1198901 1198901 + 1198902 1198902 + tr W119879

1120578minus1

1

W1

+ tr W1198792120578minus1

2

W2 (51)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(12060111198902 + 120578

minus1

1

W1)

+ tr W1198792(12060121198902 + 120578

minus1

2

W2)

(52)


2 = minus11989611198902

1+ 1198902 (1205761 + 1205762119906 minus 11989621198902)

+ tr W1198791(120588100381710038171003817100381711989021003817100381710038171003817 W1)

+ tr W1198792(120588100381710038171003817100381711989021003817100381710038171003817 |119906| W2)

(53)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865W119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(54)


100381710038171003817100381712057621003817100381710038171003817 le 1205762119872 |119906| le 119906119878 (55)


2 le minus1198961100381710038171003817100381711989011003817100381710038171003817

2+100381710038171003817100381711989021003817100381710038171003817 (1205761119872 + 1205762119872119906119878 minus 1198962

100381710038171003817100381711989021003817100381710038171003817)

+ 120588100381710038171003817100381711989021003817100381710038171003817 (

10038171003817100381710038171003817W1100381710038171003817100381710038171198651198821119872 minus

10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865)

+ 120588100381710038171003817100381711989021003817100381710038171003817 119906119878 (

10038171003817100381710038171003817W2100381710038171003817100381710038171198651198822119872 minus

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865)

(56)


2 le minus100381710038171003817100381711989021003817100381710038171003817 [1198962

100381710038171003817100381711989021003817100381710038171003817 minus 1205761119872 minus 1205762119872119906119878

+ 120588(10038171003817100381710038171003817W110038171003817100381710038171003817

2

119865minus1198821119872

2)

2

minus 1205881198822

1119872

4+ 120588119906119878(

10038171003817100381710038171003817W210038171003817100381710038171003817

2

119865minus1198822119872

2)

2

minus120588119906119878

1198822

2119872

4]

(57)


[1205761119872 + 1205762119872119906119878 + 120588 (1198822

11198724 + 119906119878119882

2

21198724)]

1198962

lt100381710038171003817100381711989021003817100381710038171003817

(58)

1198821119872

2+ radic(

1198822

1119872

4) +

1205761119872

120588le10038171003817100381710038171003817W110038171003817100381710038171003817119865 (59)

Or

1198822119872

2+ radic119906119878 (

1198822

2119872

4+1205762119872

120588) le

10038171003817100381710038171003817W210038171003817100381710038171003817119865 (60)





119890 (119905) = 1199111 (119905) minus 119910119889 (119905)

119890 (119905) = 1 (119905) minus 119910119889 (119905) = 1199112 (119905) minus 119910119889 (119905)

(61)


119903 (119905) = 119890 (119905) + Λ119890 (119905) (62)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 2 (119905) minus Λ2119890 (119905) (63)


119903 (119905) = Λ119903 (119905) minus 119910119889 (119905) + 1198831119909 + 1198651119909119906 minus Λ2119890 (119905) (64)

where 1198831119909 = minus(2119870111987031198707(119898 + 1198702))11990922- (2119870111987031198708(119898 +

1198702))119911111990922- ((1198701 1198704)(119898 + 1198702))1199112- ((211987011198705)(119898 + 1198702))11991111199112


119903 (119905) = Λ119903 (119905) + 1198651119909119906 (119905) + ℎ (119909) (65)

where

ℎ (119909) = 1198831119909 minus 119910119889 (119905) minus Λ2119890 (119905) (66)


119906 (119905) = ℎ (119909) + 1198651119909119906 (119905) (67)


119903 (119905) = Λ119903 (119905) + 119906 (119905) (68)


[119890

119903] = [

minusΛ 119868

0 Λ] [119890

119903] + [

0

119868] 119906 (119905) (69)


z (119905) = Az (119905) + B119906 (119905) (70)



119869(119906) is as follows

119869 (119906) = int

infin

1199050

119871 (z 119906) 119889119905 (71)

with the Lagrangian

119871 (z 119906) = 12z119879 (119905)Qz (119905) + 1

2119906119879(119905) 119877119906 (119905)

=1

2[119890 119903] [

11987611 11987612

119876119879

1211987622

][119890

119903] +

1

2119906119879119877119906

(72)





120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3




120597119881 (z 119905)120597119905

+min119906

[119867(z 119906 120597119881 (z 119905)120597119905

119905)] = 0 (73)


119867(z 119906 120597119881 (z 119905)120597119905

119905) = 119871 (z 119906) + 120597119881 (z 119905)120597119905

z (74)


minus120597119881 (z 119905)120597119905

= 119871 (z 119906lowast) + 120597119881 (z 119905)120597119905

z (75)


119867lowast= min119906

[119871 (z 119906) + 120597119881 (z 119905)120597119905

z]

= 119867(z 119906lowast 120597119881 (z 119905)120597119905

119905)

= minus120597119881 (z 119905)120597119905

(76)


119881 =1

2z119879Pz = 1

2z119879 [119870 0

0 119868] z (77)




119906lowast(119905) = minus119877



Q = [11987611 11987612119876119879

1211987622] gt 0 119877

minus1= 11987622 (80)

with 11987612 + 119876119879



119870 = minus1

2(11987612 + 119876

119879

12) gt 0 (81)

Λ119879119870 + 119870Λ = 11987611 (82)


as

119906lowast(119905) = (inV (1198651199091)) 119906

lowast(119905) minus ℎ (119909) (83)



1and 1198961015840

2

such that 0 lt 11989610158401lt 1198961015840



1015840





119889119881 (z 119905)119889119905

=120597119881 (z 119905)120597119905

+120597119881 (z 119905)120597119905

z (85)


119889119881 (z 119905)119889119905

= minus119871 (z 119906lowast) (86)


119889119881 (z 119905)119889119905

= minus1

2z119879Qz + (B119879Pz)

119879

119877minus1(B119879Pz)


(87)



ℎ (119909) =W119879120593 (119909) + 120576 (119909) 120576 (119909) le 120576119872 (88)





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3





ℎ (119909) = W119879120593 (119909) (89)




] (119905) =minus119896119911119903 (119905)

119903 (119905)(91)


119903 (119905) = Λ119903 (119905) + 119906lowast(119905) + W119879120593 (119909) + 120576 (119909) minus ] (119905) (92)






W = 120593 (119909) z119879PBΓ minus 119896 z W (95)



1198713 =1

2z119879 [119870 0

0 1] z + 1

2tr (W119879Γminus1W) (96)





+ z119879PB W119879120593 (119909) + 120576 (119909) minus ] (119905)


(98)


1

2A119879P + 1

2PA + 1

2P = minus1

2Q + 1

2PB119877minus1B119879P (99)


3 = minus1

2z119879Qz minus 1

2z119879PB119877minus1B119879Pz

+ z119879PB 120576 (119909) minus ] (119905)

+ tr W119879 (Γminus1 W + 120593z119879PB)

(100)




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3




119865

le10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865

(101)


3 le minus1

2z2 120582min (Q) + 120582min (119877

minus1)

+ z 120576119872 + 119896 z (10038171003817100381710038171003817W10038171003817100381710038171003817119865119882119872 minus

10038171003817100381710038171003817W100381710038171003817100381710038172

119865)

(102)


3 le minus1

2z [ z 120582min (Q) + 120582min (119877

minus1)

+ 119896(10038171003817100381710038171003817W10038171003817100381710038171003817119865 minus

1

2119882119872)

2


41198961198822

119872]

(103)


(120576119872 + (14) 1198961198822

119872)

120582min (Q) + 120582min (119877minus1)

le z (104)

radic120576119872

119896+1

41198822

119872+1

2119882119872 le

10038171003817100381710038171003817W10038171003817100381710038171003817119865 (105)






2have dimension

(5 times 1) where 1206011= 1206012= [1 1199111 2119911

2

1minus 1 1199112 2119911

2

2minus 1]119879 W1




Choosing

119898 =

950 0 lt 119905 le 195

1 020 195 lt 119905 le 585

1 080 585 lt 119905 lt 780

950 780 le 119905 le 1180

(106)

119877119886 = 00867 300 lt 119905 le 500

00947 500 lt 119905 le 780(107)

119862119889 = 027 200 lt 119905 le 300

033 600 lt 119905 le 780(108)

120583119903119903 = 00165 200 lt 119905 le 300

00135 600 lt 119905 le 780(109)




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus2

minus15

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)



0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3


0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

Spee

d (k

mh

r)


minus10

(a)

0 200 400 600 800 1000 1200Time (s)

Trac

king

erro

r (km

hr)

minus1

minus05

0

05

1

(b)


0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(a)

0 200 400 600 800 1000 1200Time (s)


60

40

20

0Spee

d (k

mh

r)

(b)










600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3


600

500

400

300

200

100

0

Trac

tion

forc

eF(N

)

0 600 1200

Time (s)


Acceleration force


7 Conclusion




0 200 400 600 800 1000 1200

0

10

20

30

40

50

Time (s)

minus10

Con

trol e

ffortu

(V)


Acknowledgment


References
































Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3























Senior Project

Main Topics




Gait Data



Abstract















Image processing



Modelling



WK2

WK2 - ref

WK3

nonlinear control of an electric vehicle using chebyshev ... · pdf fileintroduction in recent...

Documents