path-tracking control of 4ws4wd electric vehicles and control...path-tracking control of 4ws4wd...

Path-tracking Control of 4WS4WD ElectricVehicles

Ramprasad Potluri

Department of Electrical Engineering, Indian Institute of Technology Kanpur,Kanpur 208016, Uttar Pradesh, INDIA

August 04 & 05, 2015

TEQIP School on Systems & ControlIndian Institute of Technology Kanpur

04 – 09 August, 2015.

Overview of the presentation

The presentation is in 2 parts:

PART 1: Activites in the Networked Control Systems Laboratory, Department of Electri-cal Engineering, IIT Kanpur.

SUBPART I: Overview of the Networked Control Systems Laboratory at IIT Kanpur.

SUBPART II: Path-Tracking Control of an Autonomous 4WS4WD Electric Vehicle.

SUBPART III: Path-Tracking Control of a Moon Rover for ISRO.

SUBPART IV: Networking Issues in the Path-Tracking Control Problems.

PART II: Application of input-to-state stability theory in an electric vehicle.

TEQIP School on Systems & Control, IITK, August 04 — 09, 2015 2 of 91 Potluri

SUBPART I: Overview of the NetworkedControl Systems Laboratory at IIT Kanpur.


Recent activities (2006 onwards)

Use NCS to build civilian applications for +ve impact on society.

I started the NCS Lab

Control NetworksCAN, LonWorksM.Tech. Theses

DSP, µCEmbedded systems

DAVR, DMBB

AlgorithmsNew Control Lab

EE380 experiments

4WS4WD EVMoon Rover for ISRO

PhDs theses

Positive impact on society

DAVR: Digital Automatic Voltage Regulator DMBB: Dual Motor Ball BeamCAN: Controller Area Network NCS: Networked Control Systems

MWDMWS EV: Multi Wheel Drive Multi Wheel Steer Electric Vehicle


What is NCS? Answer as of 2007

SerialBus

Other devices

Controller

Actuator-1

Sensor-1

Plant

Feedback control system 2

···

Feedback control system 1

Feedback control system k

Actuator-m

Sensor-n

Traditional distributed control system (DCS):

Controller

Sensor Actuator

Plant

Controller

Sensor Actuator

Plant

Feedback controlSystem # 1

Feedback controlSystem # n

Serial Bus

Other DevicesSupervisoryController

• DCS is a type of NCS.

• NCS can be more versatile than DCS: DCS networks sys-tems; NCS may network devices.

• E.g., one sensor can be used by multiple controllers in NCS.


Facilities in the NCS Lab

PhD studentsManavaalan Gunasekaran:Developed and demonstratedpath-tracking control algo-rithms for ISRO’s MoonRover.

Arun Kant Singh: Developingrobust path-tracking controlstrategy for 4WS4WD electricvehicle.

Arunava Karmakar: Workingon consensus and coopera-tion problems.

Equipment

1. Yokogawa DSO with CAN analyzer soft-ware

2. ezDSP kits (F2808 and F2812)3. 8051 microcontrollers, dsPIC microcon-

trollers4. USB-CAN converters5. Lonworks’ miniEVK6. EZDSK91C111 Ethernet network daugh-

ter boards for ezDSP kits7. PMDC motor control setups


Activities completed in NCS Lab (1/5)

1. Digital automatic voltage regulator (DAVR) for BHEL Bhopal.

IITK'spart

Mypart

Figu

reis

from

Bas

ler

Elec

tric

’sIN

STRU

CTI

ON

MA

NU

AL

FOR

DIG

ITA

LEX

CIT

ATIO

NC

ON

TRO

LSY

STEM

DEC

S-30

0



2. Human machine interface (HMI) for DAVR: Karthik Thota.

3. Graphical user interface (GUI) for DAVR: Prashant Srivastava.

Figure is from Basler Electric’sINSTRUCTION MANUAL FOR DIGITAL EXCITATION CONTROL SYSTEM DECS-300



4. DC motor NCS: Awadhesh Chaudhury.

DCDCconverter

Motor

SensorNode

Encoder

CAN Bus

Regulated DC

Speed

PWMSignal

QEPA & QEPB

Signals

MotorNode

ControllerNode

Each

node

impl

emen

ted

ona

Spec

trum

Dig

ital’s

ezD

SPki

tfo

rTI

’sTM

S320

F280

8.



5. Dual-motor ball-beam (DMBB) (Manavaalan Gunasekaran) — helps study controlallocation, multi-actuator coordination.

Brushed dc motor

Planetary gear

Steel rod

Plexi glass rodBallSpur gear

Potentiometer Nichrome wire

Arm

steel rod nichrome wire

Vin

Vout

ball



6. A restructured Control Systems Lab set up in 2009 is an offshoot of activities in theNCS Lab. Currently running experiments for a compulsory UG course (EE380).Manavaalan Gunasekaran & Yash Pant & Ramabhatla Sirisha.


Current activities in NCS Lab (1/2)

1. EE380 Control Systems experiments aimed at 4WS4WD EV.

What is this EV? It has 4 wheels@ 1 driving motor & 1 steeringmotor per wheel.

Advantages:

• 0 turning radius⇒ maneuver-able.

• No transmission ⇒ ≈ 20%more efficient than with con-ventional drive train.

• No transmission and axles ⇒CG lower⇒ more stable.

• EV ⇒ does not pollute sur-roundings.

Platform

motorSteeringmotor

Tractionmotor

Wheel

Serial bus

Serial link

ControllerSupervisory


Current activities in NCS Lab (2/2)

2. Role in ISRO-IITK project on Moon Rover: Coordinating 6 driving motors & 4 steeringmotors of rover for path tracking.

• All motors, gears, con-trollers, calculated and se-lected in NCS Lab.

• Kinematics of rover workedout in NCS Lab.

• CANopen-based network ofcontrollers built.

• Kinematics-based controldesigned.

• Designed experiments forEE380 to test torque ob-servers.

Rover built by and picture shot by Dr. Ashish Dutta of ME, IITK.


Some results thus far

1. Ramprasad Potluri and Arun Kant Singh. Path-Tracking Control of an Autonomous4WS4WD Electric Vehicle Using its Natural Feedback Loops. IEEE Transactions onControl Systems Technology.

2. Manavaalan Gunasekaran graduated with PhD degree. PhD thesis titled

Path tracking control of a moon rover: modeling, design, and implementation.

3. Manavaalan Gunasekaran and Ramprasad Potluri. Low-cost undergraduate controlsystems experiments using microcontroller-based control of a dc motor. IEEE Trans-actions on Education, 55.4 (2012): 508-516.


Conclusion

In short, here is a recap of recent activities in NCS Lab:

I started the NCS Lab

Control NetworksCAN, LonWorksM.Tech. Theses

DSP, µCEmbedded systems

DAVR, DMBB

AlgorithmsNew Control Lab

EE380 experiments

4WS4WD EVMoon Rover for ISRO

PhDs theses

Positive impact on society

Back to opening slide


SUBPART II: Path-Tracking Control of anAutonomous 4WS4WD Electric Vehicle


What is a 4WS4WD EV?

←− Image borrowed fromhttp://www.savage.gr/

↑ J. Ploeg, H.E. Schouten, and H. Nijmeijer. Positioncontrol of a wheeled mobile robot including tire behav-ior. IEEE Trans. Intell. Transp. Sys-s, 10(3):523 – 533,Sept. 2009.

• A driving motor for each wheel; mostly a hub motor.• A steering motor for each wheel.• Total 8 motors for driving and steering.


Positives of a 4WS4WD EV

• In comparison to internal combustion engine vehicles —

Has all the advantages of conventional electrical vehicles:

– Not a distributed source of pollution.

– Low-noise.

– Quick to respond.

• In comparison to conventional electric vehicles —

– Transmission removed; so 20% more efficient.Willie D. Jones. Putting electricity where the rubber meets the road. IEEE Spectrum, pages 12 – 13, July 2007.

– Maneuverable; so can save real estate.


Envisaged applications of 4WS4WD EV

Agricultural robotics Transportation Other robots

C. Liu, M. Wang, and J. Zhou. Coordinatingcontrol for an agricultural vehicle with indi-vidual wheel speeds and steering angles. IEEEControl Systems Magazine, 28(5):21 24, Oc-tober 2008.

Agricultural Robotics Portalhttp://www.unibots.com/AgriculturalRobotics Portal.htm

• Weedy robot, Faculty of Engineering &Computer Science, Univ. of Applied Sci-ences Osnabrueck, Germany.

• Supportive Autonomous Vehicle forAGriculturE (SAVAGE), MSc thesis at Pi-raeus Institute of Technology, Greece incollaboration with Univ. of Thessaly.

• Autonomous Platform & Information sys-tem. MSc thesis, Danish Technical Univ.

• Hortibot, Danish Institute of AgriculturalSciences, Denmark.

• Weeding robot, Wageningen Univ. PhDthesis of Dr. Tijmen Bakker.

• AgRover, Iowa State Univ., USA.• Skinny Boy, USP san Carlos, with EM-

BRAPA (stands for “Brazilian AgriculturalResearch Corporation”), Brazil.

Smartwheel (Australia)http://www.smartwheel.com.au/

Nissan Pivo

Siemens VDO e-Corner

Michelin Active Wheel

J. Ploeg, H.E. Schouten, and H. Ni-jmeijer. Position control of a wheeledmobile robot including tire behavior.IEEE Trans. Intell. Transp. Sys-s,10(3):523 – 533, Sept. 2009.

A. Percy, I. Spark, Y. Ibrahim, L.Hardy. A numerical control algo-rithm for navigation of an operator-driven snake-like robot with 4WD-4WS segments. Robotica, 29:471 –482, 2011.


Path-tracking problem: 2 formulationsS.T. Peng et al

OR

Desired

path

Tangent todesired path

Ocγ

~vx

y

yc

β

φ

x

y

x0y0

ψ

ψt

Want(

φ(t)yc(t)

)−→ 0.


Path-tracking problem: 2 formulationsS.T. Peng et al J. Ploeg et al

OR

Desired

path

Tangent todesired path

Ocγ

~vx

y

yc

β

φ

x

y

x0y0

ψ

ψt

x

y

CM

γ

β

ψ

V

xl

yl1

2

3

4

Want(

φ(t)yc(t)

)−→ 0. Want

x(t)y(t)ψ(t)

−→

xref(t)yref(t)ψref(t)

.


Peer-reviewed solutions

S.T. Peng et al

Prof. Shou-Tao Peng, Department of Mechanical En-gineering, Southern Taiwan Univ. of Technology.• 2 conference papers.• S.T. Peng. On one approach to constraining the com-

bined wheel slip in the autonomous control of a 4WS4WDvehicle. IEEE Trans. Control Sys. Tech., 15(1):168 – 175, May2007.• S.T. Peng, C.C. Chang, and J.J. Sheu. On robust

bounded control of the combined wheel slip with integralcompensation for an autonomous 4WS4WD vehicle. VehicleSystem Dynamics, 45(5):477 – 503, May 2007.

Features of the solution

• Solution uses singular perturbation theory,linearization, very complicated math manipula-tions.• Math maze: physics unclear in the course of

design.∴ Practical tuning of controller can be difficult.

J. Ploeg, H. Nijmeijer et al

• Dr. J. Ploeg, Netherlands Organization forApplied Scientific Research TNO.• Prof. H. Nijmeijer, Mech. Eng., Eindhoven

Univ. of Technology, Netherlands. Fellow, IEEE .• J. Ploeg, J.P.M. Vissers, and H. Nijmeijer. Control de-

sign for an overactuated wheeled mobile robot. In 4th IFACSymp. on Mechatronic Systems, pages 127 – 132, Germany,2006.• J. Ploeg, H.E. Schouten, and H. Nijmeijer. Position con-

trol of a wheeled mobile robot including tire behavior. IEEETrans. Intell. Transp. Sys-s, 10(3):523 – 533, Sept. 2009.

Features of the solution

• Unaware of Peng et al’s work.• Uses feedback linearization, Kalman filter.• Feedback linearization does not cancel out

undesired nonlinear dynamics completely unlessperfect model. ∴ Solution not robust.• Physics more (not entirely) visible in the de-

sign.


Potluri-Singh (PS) solution

PSsolution

Modifyvery

slightly

Drawblock

diagram

Crucial idea

S.T. Peng’s math model

Ploeg et al’s formulation of path-tracking problem

Simple controllers,physics clear,implementation easier,only observer is DOB

Disturbance observer (DOB) is easy to tune.


PS: Vehicle dynamics from Peng et al

xl

yl fx1

fy1

fy2

fy3

fy4

fx2

fx3

fx4

1

2

3

4

m(vxl − γvyl

)= fx1 + fx2 + fx3 + fx4

m(vyl + γvxl

)= fy1 + fy2 + fy3 + fy4

Jzγ = ld (− fx1 + fx2− fx3 + fx4)

+ l f(

fy1 + fy2)− lr

(fy3 + fy4

),

PS: Driving motor dynamics from Peng et al

Peng et al call these the wheel dynamics. These are simply torque balance equations.

Jmjωj = −rej(

fxj cos δj + fyj sin δj)+ Tj,

j = 1, . . . , 4.

Denote

J−1m ,

1Jm1

0 0 00 1

Jm20 0

0 0 1Jm3

00 0 0 1

Jm4


PS: Other equations from Peng et al

vxl = ‖V‖ cos β, vyl = ‖V‖ sin β.

V1 =

[vx1vy1

]=

[vxl − ldγvyl + l f γ

]

V2 =

[vx2vy2

]=

[vxl + ldγvyl + l f γ

]

V3 =

[vx3vy3

]=

[vxl − ldγvyl − lrγ

]

V4 =

[vx4vy4

]=

[vxl + ldγvyl − lrγ

]

x

y

CM

γ

β

ψ

V

xl

yl1

2

3

4

Sj =

[SLjSSj

]=

rejωj cos αj− ‖Vj‖max

(rejωj cos αj, ‖Vj‖

)rejωj sin αj

max(rejωj cos αj, ‖Vj‖

)

Here,

αj = δj− β j,β j = ∠

(vxj + ivyj

),

i =√−1.

[fxjfyj

]= fzj

[cos β j −ksj sin β jsin β j ksj cos β j

]µRes

(‖Sj‖, χ

)

‖Sj‖Sj.

xl

yl

βj

αj

δj

Vj

SLj

SSj

Sj

Uwe Kiencke and Lars Nielsen. AutomotiveControl Systems For Engine, Driveline, andVehicle. 2nd ed., Springer 2005.

How to understand these equations?


PS: Peng et al do not try to make sense

• Peng et al don’t tryto make sense of theseequations.• Treat equations as

x = f (x, u)y = g(x, u)

and• Linearize, and• Apply variant of LQRthat handles input satu-ration.

• Ploeg et al do better.• Visualize 4WS4WD EV ascollection of 4 unicycles.• Each unicycle tracks itsown reference path.• Write equations for eachunicycle,

• BUT, treat equations as

x = f (x, u),y = g(x, u),

and• Apply feedback lineariza-tion — a math technique.• Then, PID control.

Other works: Mainly Mas-ter’s theses. Also, treat equa-tions as

x = f (x, u),y = g(x, u).

There is a master’s thesisa

that works similar to our ap-proach, though there are im-portant differences.

aRoel Leenen. Motion control designfor a 4ws and 4wd overactuated vehicle.Masters thesis, Eindhoven University ofTechnology, Department of MechanicalEngineering, 2004.

Instead, we take the following approach . . .


PS: Combine driving-steering dynamics

X︷︸︸︷

ω1ω2ω3ω4

δ1

δ2

δ3

δ4

= −

D(X)︷︸︸︷

re1 cos δ1Jm1

re1 sin δ1Jm1

0 0 0 0 0 00 0 re2 cos δ2

Jm2

re2 sin δ2Jm2

0 0 0 00 0 0 0 re3 cos δ3

Jm3

re3 sin δ3Jm3

0 00 0 0 0 0 0 re4 cos δ4

Jm4

re4 sin δ4Jm4

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

Ffric︷︸︸︷

fx1fy1fx2fy2fx3fy3fx4fy4

+

+

[J−1m 00 I4

]

︸︷︷︸J −1

[T1 T2 T3 T4 ωs1 ωs2 ωs3 ωs4

]ᵀ︸︷︷︸

U

The slight modifications are:• The introduction of the equations of steering motors (δj = ωsj).• Jmj, j = 1, . . . , 4 represent moment of inertia of driving motors, instead of wheels.


PS: Crucial idea: Draw block diagram of math model

Loading on the 8 motors

J −1U + ∫X X

Sj =

rejωj cos αj − ‖Vj‖max


)rejωj sin αj


)

[fxjfyj

]= fzj


]µRes(‖Sj‖, χ)

‖Sj‖Sj

S1S2S3S4

−D(X)

+

m(vxl − γvyl

)= fx1 + fx2 + fx3 + fx4

m(vyl + γvxl

)= fy1 + fy2 + fy3 + fy4

Jzγ = ld (− fx1 + fx2− fx3 + fx4)

+ l f(

fy1 + fy2)− lr

(fy3 + fy4

)


V1 =

[vx1vy1

]=


]

V2 =

[vx2vy2

]=


]

V3 =

[vx3vy3

]=


]

V4 =

[vx4vy4

]=


]

V =

V1V2V3V4

vxlvylγ

Block diagram helps see the physics! 2 loops: 2 physical insights.


PS: 1st physical insight: 1st loop

Insight: The big bad equations areonly loading the motors.

J −1U + ∫X X

Loading on the8 motors

∂(X)

+




J −1U + ∫X X


∂(X)

+

∴ Can use DOB to help motors overcome load.

J −1U + ∫X X


∂(X)

+r +

ΞJ X+−

KJ ∂

−

Good disturbance rejection obtained as K → I.




J −1U + ∫X X


∂(X)

+

∴ Can use DOB to help motors overcome load.

J −1U + ∫X X


∂(X)

+r +

ΞJ X+−

KJ ∂

−

Good disturbance rejection obtained as K → I.

On the 4WS4WD EV, DOB-based control scheme for driving and steering motors is:

DOB-based rejection of ∂(X) = −AFfric

r +4WS4WD EV

U X

ΞJ X+−

KJ ∂

−CONXref


PS: Practical DOB-based disturbance rejection (DOB-DR)

r +4WS4WD EV

U X

ΞJ X+−

KJ ∂

−CONXref

Here, only driving motorsneed DOB-DR. Loads on steer-ing motors are assumed tobe small enough to not needDOB-DR.

A practical speed control system with DOB-DR for the j-th PMDC or BLDC motor:

Motor

ωjref +Kωj(s)

idj + +Kij(s)

+ 1RΣj

ijKtj

Tj +

TLj = fxjrej cos δj + fyjrej sin δj

− 1Jmjs + Bj

ωj

Kbj

−−

Jmjs + Bj

Ktj(τjs + 1)

1τjs + 1

y1 + y2−iLj

+−

T. Umeno & Y. Hori. Robust speed control of dc servomotorsusing modern two degrees-of-freedom controller design. IEEETrans. Ind. Electronics, 38(5):363 – 368, Oct. 1991.

M. Gunasekaran & R. Potluri. Low-cost undergraduatecontrol systems experiments using microcontroller-basedcontrol of a dc motor. IEEE Trans. Edu., Nov. 2012.


For BLDC motors & SM too exist load torque observer

Synchronous motors

1. S. Komada, M. Ishida, K. Ohnishi, andT. Hori. Disturbance observer-based motion con-trol of direct drive motors. IEEE Transactions onEnergy Conversion, 6(3):553–559, 1991.

2. Faa-Jeng Lin. Real-time IP position controllerdesign with torque feedforward control for PM syn-chronous motor. IEEE Transactions on IndustrialElectronics, 44(3):398 – 407, June 1997.

3. Kooksun Lee, Ick Choy, Juhoon Back, andJuyeop Choi. Disturbance observer based sensor-less speed controller for pmsm with improved ro-bustness against load torque variation. In PowerElectronics and ECCE Asia (ICPE & ECCE), 2011IEEE 8th International Conference on, pages 2537–2543. IEEE, 2011.

BLDC motors

1. Jiancheng Fang, Xinxiu Zhou, and Gang Liu.Precise accelerated torque control for small induc-tance brushless dc motor. Power Electronics, IEEETransactions on, 28(3):1400–1412, 2013.

2. Y. Hori, Y. Chun, and H. Sawada. Experi-mental evaluation of disturbance observer-based vi-bration suppression and disturbance rejection con-trol in torsional system. Proc. of PEMC’96, 1:120–124, 1996.


PS: Philosophy for PTC based on load-torque compensation

• Irrespective of payload, only tyres provide resources for traction.

xl

yl fx1

fy1

fy2

fy3

fy4

fx2

fx3

fx4

1

2

3

4

• Forces felt by tyres eventually supported by driving & steering motors.

Jmjωj + rej(

fxj cos δj + fyj sin δj)= Tj, j = 1, . . . , 4.

∴ Focus on motors.


PS: How to generate Xref?

r +4WS4WD EV

U X

v = J X+−

KJ ∂

−CONXref

What should Xref be for the vehicle CG to travel with desired V =

[vxlvyl

]and γ?

x

y

CM

γ

β

ψ

V

xl

yl1

2

3

4


PS: How to generate Xref?

r +4WS4WD EV

U X

v = J X+−

KJ ∂

−CONXref

What should Xref be for the vehicle CG to travel with desired V =

[vxlvyl

]and γ?

x

y

CM

γ

β

ψ

V

xl

yl1

2

3

4

Why think of V and γ?∵ [ vxl vyl γ ] transforms to [ x y ψ ],

and∵ Want [ x y ψ ] to track

[ xref yref ψref ].

cos ψ − sin ψ 0sin ψ cos ψ 0

0 0 1

vxlvylγ

∫

xyψ

xyψ


PS: What the desired [vxl vyl γ] means for the wheels . . 1

Rigid body with yaw rate γ & velocity of CG V. O is instantaneous center of motion(ICM).

γ

V3

xl

yl

lr

ld ld

lfβV

V1

V2

V4

r4

r

r1

r2

r3

O




γ

V3

xl

yl

lr

ld ld

lfβV

V1

V2

V4

r4

r

r1

r2

r3

O

• Distance from ICM is r =‖V‖

γ;

• Velocities of 4 corners of body are

V1 =

[vx1vy1

]=


]

V2 =

[vx2vy2

]=


]

V3 =

[vx3vy3

]=


]

V4 =

[vx4vy4

]=


]

vxl = ‖V‖ cos β, vyl = ‖V‖ sin β.




γ

V3

xl

yl

lr

ld ld

lfβV

V1

V2

V4

r4

r

r1

r2

r3

O

• That is, for a rigid body,

Want[

Vrefγref

]⇔Want

V1refV2refV3refV4ref

• For 4WS4WD EV approximated asrigid body, fixing Vref, γref fixesdesired velocities of wheel-groundcontact points as V1ref, V2ref, V3ref,V4ref.

• Assuming zero slips, V1ref, V2ref,V3ref, V4ref give Xref.


PS: Generate Xref for [vxlref vylref γref] assuming zero slipsVref

Assume wheel slips absent:• αj = 0 (zero slip)⇒ δjref = ∠(vxjref + ivyjref).• SLj = 0 (free rolling)

⇒ ωjref =√

vx1ref2 + vy1ref

2/

rej.xl

yl

βj

αj

δj

Vj

SLj

SSj

Sj

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

Xref

V1 =

[vx1

vy1

]=

[vxl − ldγ

vyl + l f γ

]

V2 =

[vx2

vy2

]=

[vxl + ldγ

vyl + l f γ

]

V3 =

[vx3

vy3

]=

[vxl − ldγ

vyl − lrγ

]

V4 =

[vx4

vy4

]=

[vxl + ldγ

vyl − lrγ

]

vx1ref

vy1ref

vx2ref

vy2ref

vx3ref

vy3ref

vx4ref

vy4ref

vxlref

vylref

γref

This scheme coordinates the 8 motors for the desired [vxlref vylref γref].The assumption of absence of wheel slips borrowed from J. Ploeg, J.P.M. Vissers, and H. Nijmeijer. Control design for an

overactuated wheeled mobile robot. In 4th IFAC Symp. on Mechatronic Systems, pp. 127 – 132, Germany, Sept. 2006.


PS: How will the vehicle body behave?

Coordinatorof 8 motors

vxlref

vylref

γref

Regulatorof XusingDOB

Xref Vehiclebody

X

vxl

vyl

γ

If (a) slips = 0, and (b) X = Xref, then

vxlvylγ

=

vxlrefvylrefγref

If (a) slips ≈ 0 (maybe / 2%), and (b) X ≈ Xref, then

vxlvylγ

≈

vxlrefvylrefγref


PS: Overall path-tracking control scheme

The plant to control is


vxlref

vylref

γref


Xref Vehiclebody

XT

vxl

vyl

γ

∫

xyψ

xyψ


PS: Overall path-tracking control scheme

The plant to control is


vxlref

vylref

γref


Xref Vehiclebody

XT

vxl

vyl

γ

∫

xyψ

xyψ

The path-tracking control scheme is

Path-tracking

con-troller

xref(t)yref(t)ψref(t)


vxlref

vylref

γref


Xref Vehiclebody

XT

vxl

vyl

γ

∫

xyψ

xyψ

Transformation from vehicle-fixed coordinatesystem to ground-fixed coordinate system:

T =

cos ψ − sin ψ 0sin ψ cos ψ 0

0 0 1


Finally: Simulation diagram

xref +

yref +

ψref +

K T1s+1T2s+1

K T1s+1T2s+1

K T1s+1T2s+1

ex

ey

eψ

T −1

Coo

rdin

ator

of8

mot

ors

vxlref

vylref

γref

Controller1

Controller2

Controller3

Controller4

Wheel 1dynamics

Wheel 2dynamics

Wheel 3dynamics

Wheel 4dynamics

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

ω1

δ1

ω2

δ2

ω3

δ3

ω4

δ4

TL1

TL2

TL3

TL4

Ave

rage

r

T

vxl

vyl

γ

∫x

∫y

∫ψ

x

y

ψ

−

−

−

Vehicle dynamicsof Loop 2

• Ramprasad Potluri Arun Kant Singh, Path-Tracking Control of an Autonomous 4WS4WD Electric Vehicle Using its NaturalFeedback Loops. IEEE Transactions on Control Systems Technology. 2015.

• Ramprasad Potluri Arun Kant Singh, Path-Tracking Control of an Autonomous 4WS4WD Electric Vehicle Using its NaturalFeedback Loops, IEEE Multi-Conference on Systems and Control (MSC 2013), Hyderabad, India. 28 - 30 August 2013.

• Ramprasad Potluri and Arun Kant Singh. Path-tracking control of an autonomous 4WS4WD electric vehicle using drivingmotors’ dynamics, 7th IEEE International Conference on Industrial and Information Systems (ICIIS). Aug. 2012, IIT Madras,Chennai, India.


PS: 2nd physical insight: 2nd loop


J −1U + ∫X X

Sj =



)rejωj sin αj


)

[fxjfyj

]= fzj



‖Sj‖Sj

S1S2S3S4

−D(X)

+

m(vxl − γvyl

)= fx1 + fx2 + fx3 + fx4

m(vyl + γvxl

)= fy1 + fy2 + fy3 + fy4

Jzγ = ld (− fx1 + fx2− fx3 + fx4)

+ l f(

fy1 + fy2)− lr

(fy3 + fy4

)


V1 =

[vx1vy1

]=


]

V2 =

[vx2vy2

]=


]

V3 =

[vx3vy3

]=


]

V4 =

[vx4vy4

]=


]

V =

V1V2V3V4

vxlvylγ

Loop 2 can be seen as a feedback control system!


Results from Loop 2

• Loop 2 is a nonlinear feedback control system⇒ We examined its stability usinginput-to-state stability theory.• The following constraint helps select maximum value of Vjref(t) such that ‖Sj‖2 is

restricted to specified values:

‖Sj‖2 ≤ −√

2c

ln

(1− 1

θgµsatRes

supt0≤τ≤t

‖Vjref(τ)‖2

),

µRes(‖Sj‖ 2,χ

)

‖Sj‖20 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2 χ1 = Asphalt dry

χ2 = Concrete dry µsat

Res (χ1)

µsatRes (χ2)

Upp

erbo

und

on‖Sj‖ 2

supt0≤τ≤t

‖Vjref(τ)‖2,[m/s2

]0 1 2 3 4 5 6 7 8

0.05

0.10

0.15


Simulation results: Values of parameters used

Parameter Value

krφ 20972 N-mrad

k f φ 50539 N-mrad

ld 0.75 mlr = l f 1.42 mh 0.42 mm 1000 kgKtj = Ktj 2 N-m/AJmj = Jmj 0.36 kg-m2

τj 0.001 s

Parameter Value

Bj = Bj 0.57 N-mrad/s

Jz 1950 kg-m2

rej 0.32 m

Cx(s), Cy(s), Cψ(s) 100s+110s+1

Kωj(s) 30 + 60s

g 9.8 m/s2

Sampling period 0.0001 s


Simulation results: Tracking a figure 8 . . . . . . . . . . . . . . . . . 1/5

−20 0 20 40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

x [m]

y[m

]

desired pathactual path

Start PointEnd Point



0 5 10 15 20 25 30 35 40

−1

0

1

2

ex[m

]

0 5 10 15 20 25 30 35 40

−1

0

1

ey[m

]

0 5 10 15 20 25 30 35 40

−0.06

−0.04

−0.02

0

0.02

0.04

time [sec]

eψ[rad]



0 5 10 15 20 25 30 35 40

20

40

[rad/sec]

ω1ω1ref

0 5 10 15 20 25 30 35 40

20

40

[rad/sec]

ω2ω2ref

0 5 10 15 20 25 30 35 40

20

40

[rad/sec]

ω3ω3ref

0 5 10 15 20 25 30 35 40

20

40

time(t) [sec]

[rad/sec]

ω4ω4ref



0 5 10 15 20 25 30 35 40

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

time [sec]

δj[rad],

j=

1...4

δ1δ2δ3δ4



0 5 10 15 20 25 30 35 40

0.05

0.1

0.15

0.2

‖S1‖ 2

0 5 10 15 20 25 30 35 40

0.05

0.1

0.15

0.2

‖S2‖ 2

0 5 10 15 20 25 30 35 40

0.05

0.1

0.15

0.2

‖S3‖ 2

0 5 10 15 20 25 30 35 40

0.05

0.1

0.15

0.2

time [sec]

‖S4‖ 2


Simulation results: Tracking a circle

• Question needed to be answered through this simulation is whether the maximumvalue of ‖Sj‖2 at each value of centripetal acceleration is as predicted by the solid ordashed curves.• Results of this simulation are plotted as a dotted curve.• Curve is closer to the dashed curve, and suggests that the solid curve is conserva-

tive near its knee.

Upp

erbo

und

on‖Sj‖ 2

supt0≤τ≤t


]0 1 2 3 4 5 6 7 8

0.05

0.10

0.15


Summary: Output so far from this effort

1. PTC of 4WS4WD EV:

1.1. New path-trackingcontrol scheme devel-oped that uses simplecontrollers.

1.2. Sufficient constraintdeveloped on driving& steering rates ofwheels to bound slips— used input-to-statestability theory fromnonlinear control.

1.3. Simulations resultsencouraging.

2. Contribution made to theliterature on math modelof 4WS4WD EVs:

A.K. Singh and R. Potluri.Comments on “Model-Independent Adaptive Fault-Tolerant Output Tracking Con-trol of 4WS4WD Road Vehi-cles. IEEE Transactions onIntelligent Transportation Sys-tems.

3. Hardware experience:

3.1. Prototype 4WS4WDEV scale testbed built.

3.2. Efforts underway toimplement the devel-oped PTC on this pro-totype.

Video of our 4WS4WDEV scale testbed


Future directions

1. The PTC needs to be carefully tested on our scale 4WS4WD EV test bed.

2. Predictions need to be made for how a full-size EV will perform.

3. The dotted curve in the figure says that the PTC is stable only for ‖Vjref‖2 . 6 m/s2

or ‖Sj‖2 . 3.5%: presence of slips causing the vehicle to deviate from its nominalmodel assumed in the design of the PID controllers of the PTC. This deviation isenough to destabilize the PTC. What is the worst case deviation that can be toleratedby the best PTC? We will attempt to answer this question using robust control theory.

Upp

erbo

und

on‖Sj‖ 2

supt0≤τ≤t


]0 1 2 3 4 5 6 7 8

0.05

0.10

0.15



SUBPART III: Path-Tracking Control of aMoon Rover for ISRO


Automatic Control of ATRs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/2

Desired Path Tracking Actuators

Sensors

EstimatorPosition

Rover

Path Controller

Block diagram of motion control of rover.


Automatic Control of ATRs . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/2

Path TrackingController

PathPlanning

DesiredPathPosition

Desired

EstimatorPosition

Terrain Sensor

Terrain

Rover-TerrainInteraction

PositionRoverActuators

Sensors

Block diagram of rover with autonomous mobility.


Prototype moon rover at IIT Kanpur


Problem Statement

• The methodologies for modeling and controlling stationary manipulators and WMRson 2D hard and smooth surfaces are available in the literature.

• On the other hand, the methodologies for modeling and controlling ATRs are underresearch and development.

• Planetary rovers operate on unknown terrain.

• Speeds are limited by insufficient knowledge of the terrain, and possibly by the exist-ing technologies.

• Kinematics model-based control is sufficient to control the motion of these rovers.


Contributions

• A general procedure for modeling the kinematics of ATRs undergoing 3D motionwith wheel slips.

– This procedure is applied to derive the kinematics model of the moon rover.

• A path tracking control which includes the wheel slips.

– A kinematics-based motion estimator that includes the turn slip and translationalslip.

• A dynamics model of the moon rover.

– Wheel speed slip estimator using motor current and speed sensors.


A side view of the Moon rover

ρ ψ1

θ1θ3θ5

β1

• Six independently driven wheels con-nected to the body of rover using rocker-bogie mechanism.

• Front two wheels are Steerable.

• Pitch averaging mechanism to transferreaction load on wheels of one side towheels on other side.

• Rocker joint angle ρ = ρ1 = −ρ2, whereρ1, ρ2 are right and left rocker angles..

• β1, β2 are right and left bogie angles..

• ψi, θi are steering and wheel-rolling an-gles of ith wheel.


Coordinate frames for rover body and right side

k2

y

k1

zx

M

R

xy

z

ρ

k3

z

yS5

x

ρ

k9

z

y S3 A1

β1B1

k5

zx

A5

y

yA3

zx

k8

x

S1

k8

z

ψ1 x

k6

y

z

y

xk73

k4

k71

z

y

x


Control task

Path TrackingController Rover

Estimator

J†θ

[xdyd

] [XY

][ΨdΘd

][XdYd

]

XY

ΦZ

qΘImD

T ′(.)

• The control problem is that R has to track the desired path (Xd(t), Yd(t)) defined inW.

• Ji relates the driving motors angular velocity to the velocity of R with respect to R.

• The control problem has been redefined to track (xd(t), yd(t)) defined in R coordi-nate frame.


Path Tracking Controller

• Eight wheel controllers – two steering angle controller and six angular velocity con-troller.

• PID controller is used to control the steering motor to achieve the desired steeringangle ψid.

• PI controller is used to control the driving motor to achieve the desired angular ve-locity of the wheel θid.

• The desired steering angle for the different wheels are determined using steeringwheel-kinematics.

• PTC provides the desired task to the rover wheel controllers.


Position Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/3

of Jθ and Jx,y

Computation

∫T (.)

Pseudo-inverse

arctan(

yx

) ∫Slip Estimator

d(.)dt

Jx,y

[qD

]

[xy

] [xy

] [XY

]

J†θJθ

φz ΦZJx,y

KinematicsForwardΘ

˙E

Recall:

Path TrackingController

PathPlanning

DesiredPathPosition

Desired

EstimatorPosition

Terrain Sensor

Terrain

Rover-TerrainInteraction

PositionRoverActuators

Sensors



of Jθ and Jx,y

Computation

∫T (.)

Pseudo-inverse

arctan(

yx

) ∫Slip Estimator

d(.)dt

Jx,y

[qD

]

[xy

] [xy

] [XY

]

J†θJθ

φz ΦZJx,y

KinematicsForwardΘ

˙E

Recall:

Path TrackingController Rover

Estimator

J†θ

[xdyd

] [XY

][ΨdΘd

][XdYd

]

XY

ΦZ

qΘImD

T ′(.)



of Jθ and Jx,y

Computation

∫T (.)

Pseudo-inverse

arctan(

yx

) ∫Slip Estimator

d(.)dt

Jx,y

[qD

]

[xy

] [xy

] [XY

]

J†θJθ

φz ΦZJx,y

KinematicsForwardΘ

˙E

Wheel Speed Slip Estimator:

1mis

r

ωmi

Imi Tmi Fdias+a

αi migsin(.)

vi

θi1Rg

a(Jmis+Bmi)s+a

KtiRgr

si


Control structure for the motion control of the Moon rover.

RTW Controller

Speed

Controller

Position Steering

Motors

Computation

of v, φz

Steering

Kinematics

v

φz 4

ψid

4

ψi

Kinematics Jx,y

ForwadW TR

Computation

of RTW

Computation

of φz

x

y

x

y

φz

X(nTo)

Y (nTo)

ΦZ(nTo)

To

∫

Computation

of Jθ

1To

θiXd(nTo)Yd(nTo) yd(nTo)

xd(nTo)

6

RTW

Slips

θid

6 Wheel Dynamics

Driving Motors &

6ωmi

Jθ

xd(nTo)

yd(nTo)

Outer Loop

Controller

Imi

6

Slip Estimators


Practical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/6

Specifications of the motors used in the rover.

Specification SM DM UnitsMaxon order number 339268 272766 -Nominal torque 22.8 33.9 mNmNominal speed 2850 6670 rpmNumber of pole pairs 4 1 -Gear ratio 318:1 318:1 -Sensor resolution @ motor shaft 24 6 pulses/turn



CAN-based networked control system

CAN Bus Terminator

EPOS-24/1 SM1SM2

EPOS-24/5DM2

Laptop

DM4 EPOS-24/5

CAN Bus Terminator

DM6 EPOS-24/5

EPOS-24/1

EPOS-24/5

EPOS-24/5

EPOS-24/5

DM1

DM3

DM5

NI-8473sUSB-CAN

CAN Bus

USB



0 2 4 6 8 10 12 14 16 18 200

500

1000

1500

2000

2500

t [s]

DM

speed[rpm]

NmdNmNmf

0 2 4 6 8 10 12 14 16 18 200

200

400

600

t [s]

DM

curren

t[m

A]

ImImf

Speed control data for a driving motor, where Nmd, Nm and Nm f are the desired, sensed and filtered motor speeds, Im and Im f

are the sensed and filtered motor currents.


Observations

• The low resolution of the speed sensor⇒ noise in the sensed ωmi and Imi.

• The sensed current has additional noise due to the switching circuits.

• This noisy data is used in PTC and slip estimator for the given setup and results inunstable operation.

• Digital filter with cut-off frequency of π rad/s is implemented in laptop for both speedand current data.

• Even after addition of filter the current data is not improved, ⇒ poor estimate si ofspeed slip.

• si is assumed as zero.

• Each sampling interval of the PTC which is 0.1 s length, comprises sending 8 CANmessages containing commands, 14 CAN messages to request each of 14 senseddata, receiving 14 CAN messages, and filtering the current and speed data.


PTC–2D

0 0.5 1 1.5 2 2.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

X [m]

Y[m

]

Rd

R1R2

• The average maximum path tracking error for multiple tests is found to be 0.12 m.

• This error takes place after the rover moves for a distance more than 3 m.


Future Directions

• In 3D – wheel struck with some moving object, need of torque sensing to be in safe.

• Improve the path-tracking using a current based speed-slip estimation.

• NCS – effects of the time delay.

• Control based on dynamic model for high speed rovers

videos of 2D motion of moon rover



SUBPART IV: Networking Issues in thePath-Tracking Control Problems


Revisit the simulation diagram of the PTC of 4WS4WD EV

xref +

yref +

ψref +

Cx(s)

Cy(s)

Cψ(s)

ex

ey

eψ

T −1

Coo

rdin

ator

of8

mot

ors

vxlref

vylref

γref

MCS 1

MCS 2

MCS 3

MCS 4

Wheel 1dynam-

ics

Wheel 2dynam-

ics

Wheel 3dynam-

ics

Wheel 4dynam-

ics

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

ω1

δ1

ω2

δ2

ω3

δ3

ω4

δ4

TL1

TL2

TL3

TL4

Ave

rage

r

T

vxl

vyl

γ

∫x

∫y

∫ψ

x

y

ψ

−

−

−

Vehicle dynam-ics of Loop 2

MCS: Motor Control System

Cx(s) = Cy(s) = Cψ(s) = KT1s + 1T2s + 1

, T1 = 100, T2 = 10, K = 1.

Here is how we plan to practically implement this control system . . .


Planned practical implementation of the PTC . . . . . . . . . .1/2

xref +

yref +

ψref +

Cx(s)

Cy(s)

Cψ(s)

ex

ey

eψ

T −1

Coo

rdin

ator

of8

mot

ors

vxlref

vylref

γref

MCS 1

MCS 2

MCS 3

MCS 4

Wheel 1dynam-

ics

Wheel 2dynam-

ics

Wheel 3dynam-

ics

Wheel 4dynam-

ics

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

ω1

δ1

ω2

δ2

ω3

δ3

ω4

δ4

Ave

rage

r

T

vxl

vyl

γ

∫x

∫y

∫ψ

x

y

ψ

−

−

−

Averager: Inverts theoperation performed

by Coordinatorof 8 motors, and

averages the results

means this link absent.

Shaded part is implemented in supervisor.


Planned practical implementation of the PTC . . . . . . . . . .2/2

xref +

yref +

ψref +

Cx(s)

Cy(s)

Cψ(s)

ex

ey

eψ

T −1

Coo

rdin

ator

of8

mot

ors

vxlref

vylref

γref

MCS 1

MCS 2

MCS 3

MCS 4

Wheel 1dynam-

ics

Wheel 2dynam-

ics

Wheel 3dynam-

ics

Wheel 4dynam-

ics

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

ω1

δ1

ω2

δ2

ω3

δ3

ω4

δ4

Ave

rage

r

T

vxl

vyl

γ

∫x

∫y

∫ψ

x

y

ψ

−

−

−




means this link absent.

Shaded part is implemented in supervisor.

Controller Area Network (CAN) bus


An approximate continuous-time analysis . . . . . . . . . . . . . 1/6

xref +

yref +

ψref +

Cx(s)

Cy(s)

Cψ(s)

ex

ey

eψ

T −1

Coo

rdin

ator

of8

mot

ors

vxlref

vylref

γref

MCS 1

MCS 2

MCS 3

MCS 4

Wheel 1dynam-

ics

Wheel 2dynam-

ics

Wheel 3dynam-

ics

Wheel 4dynam-

ics

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

ω1

δ1

ω2

δ2

ω3

δ3

ω4

δ4

Ave

rage

r

T

vxl

vyl

γ

∫x

∫y

∫ψ

x

y

ψ

−

−

−




CAN introduces delays. The up-per bound on the delays of jth pri-ority message of length l bits in aCAN network with baud rate of Rkbps with a message cycle lengthci of the ith priority message (the

period after which the message isrepeated) is

dj =(j + 2)l

R−∑j−1i=0(l/ci)

.

[Klehmet et al. Delay bounds for CAN

communication in automotive applica-

tions. In 14th GI/ITG conference mea-

surement, modelling and evaluation of

computer and communication systems.

Dortmund, Germany. 2008.]



xref +

yref +

ψref +

Cx(s)

Cy(s)

Cψ(s)

ex

ey

eψ

T −1

Coo

rdin

ator

of8

mot

ors

vxlref

vylref

γref

MCS 1

MCS 2

MCS 3

MCS 4

Wheel 1dynam-

ics

Wheel 2dynam-

ics

Wheel 3dynam-

ics

Wheel 4dynam-

ics

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

ω1

δ1

ω2

δ2

ω3

δ3

ω4

δ4

Ave

rage

r

T

vxl

vyl

γ

∫x

∫y

∫ψ

x

y

ψ

−

−

−




The delays dj in CAN-based con-trol system are claimed to be uni-formly distributed in the interval[0, 1.7Ts], where Ts is the sam-

pling period of the control system.[Klehmet et al. Combined AFS and DYC

control of four-wheel independent-drive

electric vehicles over CAN network with

time-varying delays. IEEE Trans. Vehic-

ular Technology, vol. 63, No. 2, Feb.

2014]

For starters, analyze this CAN-based system as a continuous-time system with delays!



xref +

yref +

ψref +

Cx(s)

Cy(s)

Cψ(s)

ex

ey

eψ

T −1

Coo

rdin

ator

of8

mot

ors

vxlref

vylref

γref

MCS 1

MCS 2

MCS 3

MCS 4

Wheel 1dynam-

ics

Wheel 2dynam-

ics

Wheel 3dynam-

ics

Wheel 4dynam-

ics

ω1ref

δ1ref

ω2ref

δ2ref

ω3ref

δ3ref

ω4ref

δ4ref

ω1

δ1

ω2

δ2

ω3

δ3

ω4

δ4

Ave

rage

r

T

vxl

vyl

γ

∫x

∫y

∫ψ

x

y

ψ

−

−

−




• Set xref = yref = ψref = 0 to analyze closed-loopstability.

• Use the fact that Cx(s) = Cy(s) = Cψ(s).• Assume that MCS 1 — MCS 4 do a good job:

ωj = ωjref and δj = δjref.

• Assume that [ω1 δ1 ω2 δ2 ω3 δ3 ω4 δ4] →[vxl vyl γ] transformation is exactlythe inverse of [vxlref vylref γref] →[ω1ref δ1ref ω2ref δ2ref ω3ref δ3ref ω4ref δ4ref] trans-formation.

Then, this block diagram can be redrawn as . . .



1e−tδ1refs e−tδ1sδ 1

ref

δ 1


ref

δ 2


ref

δ 3


ref

δ 4

1e−tω1refs e−tω1sω

1ref

ω1


2ref

ω2


3ref

ω3


4ref

ω4

Generator of ωjref, δjref Generator of vxl, vyl, γ

−K T1s+1T2s+1

1s

−K T1s+1T2s+1

1s

−K T1s+1T2s+1

1s

vxlref

vylref

γref

vxl

vyl

γ

tωjref, tδjref, tωj, tδj, where j = 1, . . . , 4, are delays introduced by CAN. Further conversion . . .



e−(tδ1ref+tδ1)s

−K T1s+1T2s+1

1s

vx1ref vx1e−(tω1ref+tω1)s

−K T1s+1T2s+1

1s

vy1ref vy1e−(tδ2ref+tδ2)s

−K T1s+1T2s+1

1s


−K T1s+1T2s+1

1s


−K T1s+1T2s+1

1s


−K T1s+1T2s+1

1s


−K T1s+1T2s+1

1s


−K T1s+1T2s+1

1s

vy4ref vy4 • The [vxjref vyjref] → [ωjref δjref] transfor-mation is approximately reversed by the[ωj δj]→ [vxj vyj] transformation.

• So, replaced the [ωjref δjref] vectorat the input of the delay block with[vxjref vyjref], and the [ωj δj] at the out-put of the delay block with [vxj vyj].

• The loops are identical except for the de-lays.



e−θs

−Ks

T1s+1T2s+1

• To select Ts, let us analyze the stabilityof any one loop assuming the maximumdelay of θ = 2× 1.7Ts. (Or, should wesay θ = (2× 1.7Ts)× 8?)

• Plot the Bode plot of the loop transferfunction K

sT1s+1T2s+1e−θs for Ts = 2 ms.

• For this Ts, gain crossover frequency isωg = 10 rad/s, and PM = 90◦.

• So, sampling frequency of closed-loopsystem is chosen as at least 10×ωg . . .

• . . . and as at most such that the super-visor can send out whatever messages it

needs to send out within Ts.

• So, we have Ts min ≤ Ts ≤ Ts max.

• For example, in our system Ts min = 2ms, and Ts max = 0.068 s.

• However, in simulation, PTC is stableonly until Ts max = 0.01 s.

• Ts max = 0.0628 s in simulation at K =0.5, that is, . . .

• . . . for increasing θ, closed-loop stabil-ity achieved in simulation by decreas-ing K, although path-tracking error in-creases. ⇒Our simplistic analysis is giv-ing expected results.

• Of course, these results need to be veri-fied on the actual CAN network.


Conclusions: Research opportunities in CAN and 4WS4WD

• CAN allows to have a fixed physical topology: the bus topology. So, wiring is mini-mal.

• This bus topology allows a choice of logical topologies: master-slave (supervisorycontrol, as done by us), multi-master, peer-to-peer, etc.

• For example, can treat the PTC problem in the framework of consensus and cooper-ation among the 4 unicycles.

• Finally, actual problems to be solved in the continuous-time analysis of CAN-inducedtime delays are

e−θ1s 0 · · · 00 e−θ2s · · · 0... ... . . . 00 0 · · · e−θ8s

8× 8 matrix.But, what is it?

Will the stability analysis of this MIMOloop give us the maximum permissiblesampling interval that we saw in simula-tion?



PART II: Application of input-to-statestability theory in an electric vehicle.


Recall this figure . . .


J −1U + ∫X X

Sj =



)rejωj sin αj


)

[fxjfyj

]= fzj



‖Sj‖Sj

S1S2S3S4

−D(X)

+

m(vxl − γvyl

)= fx1 + fx2 + fx3 + fx4

m(vyl + γvxl

)= fy1 + fy2 + fy3 + fy4

Jzγ = ld (− fx1 + fx2− fx3 + fx4)

+ l f(

fy1 + fy2)− lr

(fy3 + fy4

)


V1 =

[vx1vy1

]=


]

V2 =

[vx2vy2

]=


]

V3 =

[vx3vy3

]=


]

V4 =

[vx4vy4

]=


]

V =

V1V2V3V4

vxlvylγ

Loop 2 can be seen as a feedback control system!


2nd loop excerpted

(5), (6)with j =1, . . . , 4

Vref(3), (4)

with j =1, . . . , 4

Sj(1)

TLj = rej (fxj cos δj + fyj sin δj)j = 1, . . . , 4.

TLj , with

j = 1, . . . , 4

Ffric

(5)

vxlvylγ

V


Decompose into wheel subsystems

Vjref

(5)&(6)

vxjref

vyjref (3), (4)Sj

mj vxj = fxjmj vyj = fyj

mj , fzjg

rej [cos δj sin δj ]TLj

[fxjfyj

] [vxjvyj

]

[ExjEyj

]= −µRes(·)g

Exj√Exj

2+Eyj2

Eyj√Exj

2+Eyj2

+

[vxjrefvyjref

].

Examine this equation’s input-to-state stability.


Results from Loop 2

• Loop 2 is a nonlinear feedback control system⇒ We examined its stability usinginput-to-state stability theory.• The following constraint helps select maximum value of Vjref(t) such that ‖Sj‖2 is

restricted to specified values:

‖Sj‖2 ≤ −√

2c

ln

(1− 1

θgµsatRes

supt0≤τ≤t

‖Vjref(τ)‖2

),

µRes(‖Sj‖ 2,χ

)

‖Sj‖20 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2 χ1 = Asphalt dry

χ2 = Concrete dry µsat

Res (χ1)

µsatRes (χ2)

Upp

erbo

und

on‖Sj‖ 2

supt0≤τ≤t


]0 1 2 3 4 5 6 7 8

0.05

0.10

0.15



path-tracking control of 4ws4wd electric vehicles and control...path-tracking control of 4ws4wd...

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