Design, Modelling, and Control of a Portable Leg Rehabilitation

System

Khaled M Goher1 and Sulaiman O Fadlallah2

1Lincoln University, New Zealand, e-mail: khaled.goher@lincoln.ac.nz

2Auckland University of Technology, New Zealand, e-mail: sulaiman.fadlallah@aut.ac.nz

Abstract: In this work, a novel design of a portable leg rehabilitation system (PLRS) is presented. The main purpose of this paper is to

provide a portable system, which allows patients with lower limb disabilities to perform leg and foot rehabilitation exercises anywhere

without any embarrassment compared to other devices that lack the portability feature. The model of the system is identified by inverse

kinematics and dynamics analysis. In kinematics analysis, the pattern of motion of both leg and foot holders for different modes of

operation has been investigated. The system is modeled by applying Lagrangian dynamics approach. The mathematical model derived

considers calf and foot masses and moment of inertias as important parameters. Therefore, a gait analysis study is conducted to calculate

the required parameters to simulate the model. PD controller and PID controller are applied to the model and compared. The PID

controller optimized by Hybrid Spiral-Dynamics Bacteria-Chemotaxis (HSDBC) algorithm provides the best response with a reasonable

settling time and minimum overshot. The robustness of the HSDBC-PID controller is tested by applying disturbance force with various

amplitudes. A setup is built for the system experimental validation where the system mathematical model is compare with the estimated

model using System Identification Toolbox. A significant difference is observed between both models when applying the obtained

HSDBC-PID controller for the mathematical model. The results of this experiment are used to update the controller parameters of the

HSDBC-optimized PID.

Keywords: Modelling, leg rehabilitation, simulation, control, PD, PID, HSDBC.

1. INTRODUCTION

Lower-Limb disabilities have attracted several research studies, especially in the rehabilitation field. Over the past years,

innovative ideas of robotics rehabilitation systems have been developed in order to serve different and better leg

rehabilitation purposes. Díaz et al. [1] presented different types of robotic systems for lower-limb rehabilitation including

treadmill gait trainers; foot-plate-based gait trainers; over-ground gait trainers; stationary gait trainers; and ankle

rehabilitation systems.

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Monaco et al. [2] proposed a stationary gait trainer NEUROBike which is designed to provide neuro-rehabilitative

treatments to bedridden patients. Hwang and Jeon [3] developed a wheelchair integrated lower limb exercise/rehabilitation

system comprising a wheelchair, a body lifter, and a lower limb exoskeleton.

Agrawal and Fatta [4]; Banala et al [5]; and Agrawal et al. [6] investigated gravity-balancing leg orthotic device which

used later by Banala et al. [7] to design ALEX; gravity balancing exoskeleton. As a further improvement for the active leg

exoskeleton, also known as ALEX; Banala et al. [8] put in their concerns that the gait of every adult is unique. Therefore,

adapting the ALEX to the patients was their target. Duschau-Wicke et al. [9] presented a patient-cooperative strategy which

allows patients to influence the timing of their leg movement along a meaningful path called Path Control method.

Another contribution in the field of rehabilitation was proposed by Zhang and Li [10]; a lower limb 4-degrees of

freedom rehabilitation mechanism is proposed using MATLAB®/Simulink software. Homma et al. [11] proposed a new

concept for a multiple degrees of freedom leg rehabilitation system. The system was based on manipulating the patient's leg

by wires where two degrees of freedom rehabilitation exercise was performed. Homma et al. [12] developed an improved

system for the wire-driven system with four degrees of freedom. The system targets knee flexion/extension, hip

flexion/extension, hip abduction/adduction, and hip internal/external rotation. Yakimovich et al. [13] developed an

electromechanical stance-control knee-ankle-foot orthosis (SCKAFO). The system allowed free knee motion during swing

and other non-weight-bearing activities and inhibits knee flexion while allowing knee extension during weight bearing. On

the other hand; Park et al. [14] presented a novel active soft orthotic device powered by pneumatic artificial muscles for use

in treating gait pathologies associated with neuromuscular disorders.

Other researchers focused their studies on providing a proper control for leg movement. McNeal et al. [15] proposed an

open-loop control of the freely-swinging paralyzed leg. An experimental model has been used to study the issues that are

related to the use of electrical stimulation for helping patients with disabilities walk normally. Spek et al. [16] designed a

neuro fuzzy control strategy to control the cyclical leg movements of paraplegic subjects. The proposed control is a

combination of fuzzy logic controller and a neural network, which makes the controller adaptive and self-tuning. Tsukahara

et al. [17] proposed an estimation algorithm that infers the intention related to the forward leg-swing. This was for the

purpose of supporting the gait for complete spinal cord injury patients wearing an exoskeleton system called a Hybrid

Assistive Limb (HAL). Aguirre-Ollinger et al. [18] proposed an inertia compensation control for a one degree of freedom

exoskeleton to assist lower limb disabilities. The controller is based on adding a feedback loop consisting of low-pass filtered

angular acceleration multiplied by a negative gain.

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1.1 Paper Contribution

Despite the previously mentioned research contributions focused on rehabilitation using lower limb rehabilitation devices;

these systems suffer from the following: the lack of portability function, the lack of comfort while being used and the

hardships associated with their costs. In this paper, a novel portable leg rehabilitation mechanism is proposed to help the user

to perform the extension/flexion exercises for both the knee and ankle simultaneously or consecutively as per prescribed on

the treatment plan. The portable device will also allow the user to practise physical and rehabilitation exercises anywhere

without the hassles associated with effort and cost of regular visits to hospitals and rehabilitation clinics. In this paper, a

portable leg rehabilitation system is introduced and a mathematical model of the system is developed using Lagrangian

approach. The system model is developed considering joints friction and the user’s actual calf and foot masses.

1.2 Paper organisation

The rest of the paper is organized as follows: Section 2 describes in details the proposed leg rehabilitation system including the

modes of operation and SolidWorks design. Section 3 introduces detailed derivation of the system dynamic equations based on

the Lagrangian dynamic formulation technique. An open-loop system response is investigated in section 4 in order to study the

behaviour of the developed model. Different control strategies are implemented and compared for the purpose of obtaining a

suitable response for the system. The process of visualizing the system is described in section 5 including software, hardware,

and circuit design. This followed by estimating the prototype transfer functions using System Identification and the design of

the control algorithm. Section 6 concludes the paper by highlighting the findings of the research.

2. SYSTEM DESCRIPTION

The design of the portable leg rehabilitation system (PLRS) using SolidWorks, shown in Figure 1, consists of seat and

thigh bases, thigh cover, and calf and foot carriers. The System has two degrees of freedom (2DOF) which represents the

angular motion of the knee and the ankle joints. By applying a lifting torque on the calf and foot carriers, different

rehabilitation exercises can be carried out. The system consists of two links connected with two revolute joint in order to lift

the patient's leg to a desired position. The system has the possibility to lift the patient's leg up by applying a lifting torque, T 1,

which will change θ1 and also has a feature to move the foot to a desired angle by applying a torque, T2, which will change

the angle θ2. Figure 2 represents the schematics diagram of the PLRS and the system parameters are described in Table 1.

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Being a two-arm manipulator is the key feature of the mechanism to fit with the project main purpose of simplicity,

portability and cost. Although the design seems simple and relying on a traditional concept but fit for purpose. The two-arm

structure, which simulates the leg and foot architecture and is used for developed mechanism.

The system provides different modes of operation, shown in Figure 3, which can be described as follows:

Mode (a): The knee joint motor will be activated to provide the required torque to lift the leg to the desired position

to perform calf exercise. The speed of lifting the knee is controlled to ensure safe exercising for the user.

Mode (b): The motor of the ankle joint will be activated and the user can practice certain exercise for the foot by

raising it up and down.

Mode (c): This is a simultaneous exercise for both the leg and the foot using both motors at the joints to provide

motion for knee and ankle exercises.

Given that the rationale behind this new design as compared to previous designs is the lack portability, comfort, and costs,

exact metrics have been targeted. In terms of portability, the PLRS’s weight must not exceed 10 kg. Moving to the

comfortability feature, the design must provide the necessary flexibility to fit persons with various body sizes. As for cost, the

proposed system’s cost should be below 400 US dollars.

The SolidWorks design for the system is transferred to ADAMS software and simulated to obtain a displacement-torque

relation for both calf and foot. Understanding the relationship between the torque and the displacement is important in terms

of motor selection for the purpose of building the prototype. The torque required for knee and ankle joints were simulated as

shown in Figure 4. For the calf motion, as the leg carrier reaches an angular displacement of 90 degrees, the maximum torque

calculated was approximately 20 N-m. On the other hand, the ankle joint torque was found to be 0.425 N-m for a 30 degrees’

orientation.

3. SYSTEM MODELING

3.1 Lagrangian Modeling Approach

Applying Lagrange-Euler equations leads to a set of coupled second-order ordinary differential equations and provides a

formulation of the dynamic equations of motion.

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Using Lagrange formulation for the leg rehabilitation system dynamics, equations of motion for the system can be

expressed as follows:

ddt ( ∂L

∂ θ̇1 )−( ∂L∂θ1 )=T 1−T f 1

(1)

ddt ( ∂L

∂ θ̇2 )−( ∂L∂θ2 )=T2−T f 2

(2)

where L is the Lagrangian function and represents the difference between the system kinetic and potential energy, T 1 and T 2

are the motor torque applied at each joint, and T f 1 and

T f 2 are the frictional moments at the two joints. The frictional

moments are based on Coulomb's friction model and assuming same damping characteristics for both joints, can be represented

as

T f 1=cv θ̇1+cc sin { θ̇1¿ (3)

T f 2=cv θ̇2+cc sin { θ̇2¿ (4)

where cv and cc represents the viscous and Coulomb friction coefficients the two joints respectively. Using Equation (1) and

Equation (2) as well as the frictional moments at the two joints, the systems equation of motion can be described as follows:

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m1( L12 θ1

¿⋅¿)+m2( L

12θ1

¿⋅¿)+ I1

θ1

¿⋅¿−12 m

2(L

1L

2θ

1¿

θ

2¿

sin(θ

2−θ

1))

¿

−12

m2

L1

L2((θ

2¿

−θ1¿

)(θ2¿

)sin (θ2−θ

1)+ θ

2

¿⋅¿cos( θ2

−θ1

)) ¿

+12

m1(gL

1sin θ

1)+m

2( gL

1sin θ

1)=T

1−T

f 1

¿

¿

(5)

12

m2( L1 L2 θ1

¿

θ2

¿

sin(θ2−θ1 ))+12

m2 L1 L2( θ1

¿⋅¿cos (θ2−θ1))

¿−12

m2 L1 L2((θ2

¿

−θ1

¿

)(θ1

¿

)sin(θ2−θ1)+ θ1

¿⋅¿cos (θ2−θ1 )) ¿

+12

m2 (gL2sin θ2 )+14

m2( L22 θ2

¿⋅¿ )+ I2 θ2

¿⋅¿=T2

−Tf 2

¿

¿

(6)

where θ1

¿⋅¿¿andθ2

¿⋅¿¿ are the angular accelerations for link 1 and link 2, g is the gravitational acceleration, and I 1 and I 2 are the

mass moments of inertia for both links.

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3.2 Mass and Inertia Calculation

Anthropometric data are the measurements of human body parts such as total height, body mass, and segments length. Since

the mathematical model derived in the previous part considers calf and foot masses and moments of inertia as input

parameters, calculating those parameters is essential in order to simulate the model. The anthropometric measurements are

listed in Table 2 have been adapted from Vaughan et al. [19].

The masses of each segment for the lower limb can be calculated based on the following formulas Vaughan et al. [19]:

mfoot=( 0.0083 )TM +(254 . 5 )FL∗MH∗MW −0 . 065 (7)

mcalf =(0 .0226 )TM+(31. 33)CL∗CC 2−0 .016 (8)

To calculate the moment of inertia for human body segments, a linear regression based formulas are applied to obtain the

inertia for the both the calf and the foot about the three orthogonal axes. The formulas can be expressed as the following:

For the Calf Vaughan et al. [19]

I calfX=(0 .00347)TM∗(CL2+0 . 076∗CC2 )+0. 00511 (9)

I calfY =(0 . 00387)TM∗(CL2+0 .076∗CC2 )+0. 00138 (10)

I calfZ=(0 . 00041)TM∗CC 2+0 . 00012 (11)

As for the Foot [19]

I footX=(0 .00023 )TM∗( 4∗MH2+3∗FL2 )+0 . 00022 (12)

I footY=(0 . 00021)TM∗(4∗MW 2+3∗FL2 )+0 . 00067 (13)

I footZ=(0 . 00141)TM∗( MH 2+MW 2 )+0.00067 (14)

The resultant moment of inertia for each segment in the three directions can be calculated according to the following formula:

I Total=√Ix2+I

y2+ Iz2

(15)

Table 3 lists the anthropometric data of the of a 23 old age volunteer which are needed for masses and moments of inertia. The

calculated masses and moments of inertia for both calf and foot used on the derived mathematical model are illustrated in

Table 4.

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4. CONTROL SYSTEM DESIGN

The system is tested for open loop response at initial stage. Different control strategies are implemented and compared for the

purpose of obtaining a suitable response for the system.

4.1 Open loop system response

An open-loop system is investigated to study the behaviour of the developed model. The obtained model is simulated using

MATLAB environment. Table 5 shows the numerical parameters used in the simulation. m1 and m2 are the equivalent calf and

foot masses, respectively, including the mass of each associated link. The system considered to orient the calf by 90 degrees

and the foot by 30 degrees. The desired values for the system are 1 = 90o and 2 = 30o. A step input signal is applied to the

system with two different gains in order to identify the system's behavior. The open-loop response of the system is illustrated

in Figure 5.

Based on the results of the simulation, Figure 5(a), the system stabilise subject to the application of a unit step input. The

system seems to be stable but the desired angular position has not been reached with such an input. However, if the input is a

step with a gain of 100, as shown in Figure 5(b), the system behaves linearly with both outputs reaching infinity.

4.2 Control design

A control approach, shown in Figure 6, is designed to control the angular orientation of the calf and foot. The system was

simulated using Matlab Simulink environment and different controllers were tested in order to obtain the desired response

with minimum error and overshoot while reaching steady state response.

4.2.1 PD algorithm

Table 6 represents the values for the PD controller parameters for both calf and foot which are obtained by trial and error.

The system response for both calf and knee is shown in Figure 7. As noticed in the figure, the system does not reach the

desired orientation for both calf and foot. The maximum desired angles obtained from both systems are 73.5349 and 21.6504

degrees. Another drawback associated with applying PD controller is that both systems have a settling time around 1.585

second for calf response and 0.6866 seconds for foot system response. Applying such a controller on the rehabilitation system

might cause a serious damage to the patient due to the fast response. As noticed from system response, it is experiencing an

overshoot and this physically indicates that foot and calf will go beyond the nature limit of their motion boundary. The

system performance characteristics parameters for PD controller are tabulated in Appendix A in Table A1.

4.2.2 PID algorithm

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As observed previously, applying PD controller caused the system to response fast with an offset of 18% for the calf and 28%

for the foot. Proportional integral derivative (PID) controller is designed and implemented on the system with the gain values

available in Table 6. Based on the output response, shown in Figure 8, the system approaches the desired position for both

calf and foot. As for the calf, the response settles in 44.72 seconds. In addition to that, the system reaches the desired position

with an error calculated to be 0.0167 %. On the other hand, for foot response, the system settles at 26.01 seconds. No

overshoot is observed for this response. Detailed information regarding system performance characteristics with PID

implementation is included in Table A2 in Appendix A.

Considering the previous mentioned comparison between the implemented two control approaches, PID controller was able

to result in more suitable system performance considering the physical and health conditions of the user which primarily

requires smooth and comfortable operation of the used rehabilitation devices. Although the response is smooth and

experiences no overshoot, the response takes more than 40 seconds to settle for calf movement and around 25 second in the

case of foot orientation. Further improvement on the system performance will be targeted using PID-optimised controller.

4.3 HSDBC-optimized PID

4.3.1 Overview of HSDBC

Tuning PID controllers remains a field of interest in various disciplines of control engineering. Over the past decades, several

optimization techniques and algorithms have been developed for optimizing PID parameters including the spiral dynamics

algorithm (SDA) proposed by Tamura and Yasuda [20], bacterial forging algorithm (BFA) developed by Korani et al. [21],

and a recently developed algorithm by Nasir et al. [22] which represents a fusion between the previously mentioned

algorithms known as the Hybrid Spiral-Dynamics Bacteria-Chemotaxis (HSDBC) algorithm. This hybrid algorithm has been

implemented in multiple applications by Nasir et al. [23] and Nasir et al. [24]. Using HSDBC optimization tool, the gain

parameters of the PID controller used on the PLRS has been optimized in order to obtain a better performance than manual

tuning.

Table 7 lists the parameters of the algorithm. In HSDBC, shown in Figure 9, the bacterial chemotaxis strategy is applied

in step 2 for exploration and exploitation balance and enhancement of the search space. As for bacteria movement, the

movement starts from low nutrient locations to high nutrient ones, which are located at the middle of the spiral. In HSDBC

algorithm, the bacteria are located at the low nutrient locations and this results in large step movement which results in quick

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convergence. However, the bacteria move with smaller step size in case of approaching locations with rich nutrients and this

results in oscillations around the optimal point.

4.3.2 Optimization constraintsand objective functions

This process of optimization was constrained within certain boundary limits considered as the stability region of the system

identified earlier. Based on the minimum mean squared error (MSE), the performance index has been chosen. The MSE can

be calculated using Equation 16 and Equation 17, for each control loop of the rehabilitation system:

Objective Function 1=min[ 1N ∑

i=1

N

(θ1 d−θ1 m)2 ]

(16)

Objective Function 2=min[ 1N ∑

i=1

N

(θ2 d−θ2m)2 ]

(17)

The total objective function of the leg rehabilitation system is calculated based on the total MSE which is shown in Equation

18.

J=∑i=1

J

(Objective Function i ) (18)

The calculated optimized controller gain parameters are tabulated in Table 8 associated with the cost function.

4.3.3 Simulation results

Figure 10 presents the system response comparison between manual tuning and HSDBC optimization for both calf and foot.

The performance is significantly improved by applying HSDBC. The settling time for both systems has been reduced to

approximately 17 seconds for calf response, and 11 seconds for foot response. Additional information regarding system

performance for HSDBC-PID controller is tabulated in Table A3 in Appendix A.

Figure 11 shows the characteristics Parameters performance for each controller. As can be seen, HSDBC optimization

tool enhanced the performance of the system for both calf and foot. As for calf movement, the system reached its desired

position with no noticed error. The settling time for the system is located within a safe region since the system deals with

patients suffering from lower limb disabilities. On the other hand, for foot movement, the settling time has been improved as

well by applying HSDBC. A minor drawback is that the system experiences a very small overshoot which can be simply

neglected. Details are available in Table A4 in Appendix A.

4.3.4 Control effort

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In order to minimize energy consumption, the system's control effort is investigated. Figure 12 presents the control effort for

the calf and foot movement for the three implemented control techniques. For PD controller, the system experienced a big

oscillation, up to 160 N.m, till it reached the stabilized value of torque. The same controller failed to achieve the desired

position in system performance analysis.

Using PID controller, the system reached a control effort up to 35 N.m similar to the case of applying HSDBC-PID. HSDBC-

PID controller takes less time to settle compared to the PID controller. On the other hand, for foot movement, the PD

controller experienced oscillations. However, HSDBC performs better than the other controllers in terms of less oscillations

and the time taken by the control signal.

4.3.5 Testing HSDBC-PID controller against external disturbance

Disturbance force was applied on the system to examine its stability and robustness against variable disturbance amplitudes

as shown in Figure 13. The disturbance levels used were 0 N, 0.5 N, 1 N, and 2 N. The disturbance was applied on both calf

and foot. The disturbance forces applied on the system effects the stability of the system, as shown in Figure 14, while

seeking its reference orientation for both calf and foot. However, the controllers were able to cope with it and to maintain the

stability of the system and return to the desired position.

5. SYSTEM IDENTIFICATION (SI)

5.1 Introduction about SI

System Identification is a methodology for building mathematical models of dynamic system using measurements of the

system's input and output signals. Matlab has provided a Graphical User Interface (GUI) toolbox for which the user can insert

the measured data and estimate the transfer function which represents the estimated model. Usually it is done by adjusting

parameters within a given model until the estimated output becomes closer to the measured output. Figure 15 shows the GUI

for the system identification toolbox.

5.2 Prototype design

The realised PLRS system is shown in Figure 16. The system has two degrees of freedoms represented by the angular

rotation of both the calf and foot around an axis perpendicular to the sagittal plane and this makes it two-link serial arm.

Figure 17 shows the schematic diagram for the circuit which is used for operating the prototype. In order to control the speed

and orientation, an analog signal is produced by putting a potentiometer between UART TXD and GND and connecting the

center terminal of the potentiometer to the SCL/An analog input line of the motor. The range of the analog voltage for speed

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control is from 0-5V DC. The motor speed will be zero at the center voltage of 2.5V. Two values of voltages are considered,

3.1V (lifting) and 1.9V (descending) for the process of rotating the motor in both directions during rehabilitation exercises.

Both calf and foot orientations represent the outputs of the system. In order to measure the desired angles, an ADXL 335

accelerometer sensor is used. The accelerometers' output is processed Using data acquisition board (Humosoft MF624)

connected to Matlab Simulink. The output signals of the sensor are used to measure the system outputs. A combination of

software and hardware filtering were considered to compensate for noise and vibrations that may disturb the sensor signals.

A software filtering with a combination of low pass and Kalman filtering is used firstly. However, for assured noise

filtering and better reliable readings, a hardware low pass is designed with the circuit design shown in Figure 18.

5.3 Prototype transfer function

The mathematical modelling of the proposed system and results are obtained from the model as described in Section 3.

However, these results need to be compared with the actual prototype constructed for verification. In this part, the prototypes'

transfer functions will be estimated using the system identification technique. Figure 19 shows the schematics for which the

portable system. The system is divided into three main parts:

Analog Input: the variation in voltage from the potentiometer.

Analog Output: the angles from both calf and foot measured by the accelerometer.

Transfer Function: describes the input/output mathematical relationship.

In Figure 20, two different modes of operation the prototype are presented for the calf and foot angular rotation. Based on the

potentiometer orientation, the system responses to the input and provides the orientation with different speeds as mentioned

earlier. In order to consider the expected weight of the human lower limb as calculated in Table 4, external weights

equivalent to the calculated masses of both calf and foot were attached to the developed prototype.

For the purpose of rehabilitation, the process is divided into three main stages numbered respectively: extension, stability,

and flexion regions. The measured output of the calf and foot orientation for an actual extension-flexion practice is

represented in Figure 21. As can be noticed, the maximum desired position which can be reached with the actual prototype is

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measured to be around 86 degrees. The system takes around 12 seconds to reach the maximum desired position. In regard to

foot movement, it can be observed that the system takes less than 5 seconds to reach the desired position to reach an angle of

32 degrees.

The input-output relation, shown in Figure 22, demonstrates shows how the system performance changes with variation in

the voltage. The system remains constant when the input voltage is 2.5 volts. By varying the value to approximately 3.1 volts,

the system starts moving till it reaches the desired orientation. When the moment of the system reaches its desired output, the

voltage returns back to the initial input voltage (2.5 volts).

5.3.1 Calf transfer function estimation

As per the calf, the measured data obtained is considered as inputs to System Identification (SI) Toolbox. The desired number

of poles and zeros are also fed into the SI toolbox in order to estimate the appropriate transfer function to for the system. In

order to obtain the best fit from the measured date, multiple combinations are considered for the number of poles and zeros.

Figure 23 shows a comparison between the measured output and two estimated models with different combinations of poles

and zeros. As can be noticed, the estimation with the combination of two poles and one zero gives better fitting (90.28%)

compared to the four poles-two zeros model (89.89%). Considering the previous analysis, the transfer function for the calf is

identified as follows:

Calf Transfer Function= 1 . 449 s+0 .6824s2+0. 1549+0 . 02833 (19)

The HSDBC-PID controller previously obtained for the mathematical model is implemented on the system using the

numerical parameters in Table 8. Figure 24 summarizes the comparison between the mathematical model and the actual

prototype by applying HSDBC-PID controller for both systems. As can be noticed, the controller performs well with the

mathematical model. However, it causes the estimated model of the actual prototype to have with under-damped oscillations.

A modified HSDBC-PID controller is designed to enhance the performance of the actual system.

The HSDBC-PID controller for the actual prototype model has been updated. Table 9 lists the updated controller gain

parameters. Based on the modified PID controller parameters, the system was simulated and compared with the obtained

mathematical model PID controller. The modified controller enhanced the system performance as shown in Figure 25. The

maximum desired position reaches approximately 89.9 degrees and the response settled after 12.3 seconds. No overshoot has

been noticed for this run.

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5.3.2 Foot transfer function estimation

As per the foot motion analysis, the measured input-output data are impeded into MATLAB's SI Toolbox. Two estimated

models have been considered in order to get the best fit. Figure 26 summarizes the comparison between the measured output

and the two estimated models. Based on the results obtained and to the fact that the estimated model with the combination of

four poles and two zeros is fitted better than the other proposed estimation, 92.75% fitting percentage compared to 88.87%,

the estimated transfer function for the foot system is represented as the following

Foot Transfer Function=65. 63 s2+174 s+0.9918s4+5 . 659 s3+13 .38 s2+0 .6049 s+0 .2041 (20)

Before proceeding to design a PID controller for the estimated transfer function shown in Equation 20, it is important to

investigate whether the obtained HSDBC-PID controller will be working properly with model of the actual system. Figure 27

demonstrates a comparison between the mathematical model and the actual prototype transfer functions by applying

HSDBC-PID controller where the parameters are listed in Table 8. As can be noticed in the figure, both responses reached

the desired orientation after almost 10 seconds. An overshoot is experienced in the actual prototype response which can harm

the user of the rehabilitation device. In order to compensate for this overshoot, the PID controller parameters and

implemented on the system where the updated optimized controller gains are tabulated in Table 10. Figure 28 shows a

comparison between old and new configurations for HSDBC-PID controller. The system performance has improved by

minimizing the settling time and cancelling the overshoot. Based on the extracted information regarding the improved

response, mentioned in Appendix A, the system reaches approximately an angle of 30.017 degrees, which is 0.0566% greater

than the desired orientation. Although the plot shows a smooth response, however there is a small overshoot detected around

0.539%.

6. DESIGN FEASIBILITY AND POTENTIAL IMPROVEMENTS

Since the main objective of this work is to design a portable leg rehabilitation system that allows patients with various body-

sized with lower limb disabilities to perform leg and foot rehabilitation exercises anywhere, it is necessary to analyse if this

system has achieved the targeted requirements (i.e. portability, comfortability).

In terms of the portability feature, the developed prototype’s weight is 8 kg. Although the achieved weight is less than the

targeted aim (10 kg), further modifications can be considered including the use of high strength lightweight composite

materials for the PLRS’s chassis instead of conventional aluminum.

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In terms of comfortability, the system’s seat base has an adjustable width to fit persons with various body sizes. However,

and given the fact that users may have different sizes of calf and foot, both calf and foot carriers’ dimensions are fixed. This

can be solved by designing adjustable carriers and include feedback linear actuators to control the process of modifying both

calf and foot carriers’ sizes.

7. CONCLUSION

The proposed PLRS provides the portability feature which are not available in current lower limb rehabilitation devices.

PLRS has two degrees of freedom that provides knee and ankle rehabilitation exercises with different modes of operation.

The proposed system has been described and designed using SolidWorks. The system design has been transferred to ADAMS

software to obtain a displacement-torque relation for the calf and foot. The maximum torque calculated was approximately 20

N-m for knee joint and 0.425 N-m for the ankle joint. Based on Lagrangian-Euler formulation, the system has been derived.

The required parameters, such as the masses and moments of inertia for both calf and foot, have been calculated and used to

simulate the derived mathematical model. Furthermore, the mathematical model has been investigated along with a design

and implementation of a proper control strategy. Considering the obtained system performance was not satisfactory, another

approach has been designed to obtain a better system performance. HSDBC optimization algorithm has been applied on the

proposed PID controller in order to enhance the performance of the system and to obtain a response with minimum overshoot

and a reasonable settling time. Implementation of the HSDBC has resulted in reducing the settling time significantly

compared to the manual-tuned PID controller. The robustness of the HSDBC-PID controller implemented in the system has

been tested by applying disturbance force with various amplitudes. An experimental setup has been built including software,

hardware, and circuit design for the purpose of estimating its transfer function and comparing it to the derived mathematical

model. The prototype transfer functions have been estimated using System Identification Toolbox. A significant variation has

been observed between the mathematical model and the estimated model when applying the HSDBC-PID controller for the

mathematical model. The reason behind this variation is that the mathematical model transfer function was considering the

torque as input to the system. On the other hand, the real model considers voltage as a system input. Based on that, the PID

controller parameters have been modified. Based on the comparison of implementation of various approaches, the updated

HSDBC-PID controller has significantly improved the system response compared to the one used for the mathematical

model. Future work will consider further various optimization techniques for optimizing the PID controller. The system

response of the actual prototype will be examined against real disturbance forces generated from muscle retractions. The

device developed in this paper is for lab experiments. Further studies around the clinical trials of the device will be

14

conducted. Number of patients will be used to test and validate the developed technology. The realized prototype has been

developed as a test rig and to prove the concept. The initial setup has been used for lab-based testing and validation. Should

the device prove usability its design and appearance will be improved and refined. The prototype itself has been built using

Aluminum to assure lightweight. The part attached to the seat has an adjustable width to fit persons with various body size

15

Appendix A: System step response performance characteristics parameters using various controllers

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17

Table caption list

Table 1 Parameters and descriptionTable 2 Anthropometric parameters (Vaughan et al, 1999)Table 3 Volunteer detailsTable 4 Volunteer calculated parametersTable 5 Simulation parametersTable 6 PD and PID parameter configurationTable 7 HSDBC optimization algorithm parameters [22]Table 8 Optimized controller gain parameters and HSDBC cost function Table 9 Calf optimized controller gain parameters Table 10 Foot optimized controller gain parameters

18

Figure caption list

Fig. 1 System Overall Design using Solidworks Fig. 2 System Schematics DiagramFig. 3 Modes of OperationFig. 4 Knee and Ankle Joint Displacement-Torque Relation in ADAMSFig. 5a System open loop response: Angular positions with a unit step inputFig. 5b System open loop response: Angular positions for a step input with a gain of 100Fig. 6 PLRS schematic description of the control strategyFig. 7 Calf and foot system response using PD controllerFig. 8 Calf and foot system response using PID controllerFig. 9 HSDBC optimization algorithm Fig. 10 Calf and foot system response comparison (Manual Tuning vs. HSDBC)Fig. 11 Characteristics parameters of calf and foot movementFig. 12 Calf and foot control effort comparisonFig. 13 Disturbance signals applied on the systemFig. 14 Calf and foot system response with external disturbanceFig. 15 System identification toolbox GUIFig. 16 PLRS prototypeFig. 17 Designed circuit for motor orientation controlFig. 18 Hardware low-pass filterFig. 19 Prototype system identificationFig. 20 Prototype modes of operationFig. 21 Calf and foot measured outputsFig. 22 System behaviour while input variationFig. 23 Calf measured and simulated model output comparisonFig. 24 Comparison between mathematical model and actual prototype by applying HSDBC-PID controller for calfFig. 25 Angular calf displacement (Math. Model HSDBC-PID controller vs. Updated HSDBC-PID controller)Fig. 26 Foot measured and simulated model output comparisonFig. 27 Comparison between mathematical model and actual prototype by applying HSDBC-PID controller for footFig. 28 Angular foot displacement (Math. Model HSDBC-PID controller vs. Updated HSDBC-PID controller

19

Fig. 1 System Overall Design using Solidworks

20

Fig. 2 System Schematics Diagram

21

Fig. 3 Modes of Operation

22

Fig. 4 Knee and Ankle Joint Displacement-Torque Relation in ADAMS

23

Fig. 5a System open loop response: Angular positions with a unit step input

24

Fig. 5b System open loop response: Angular positions for a step input with a gain of 100

25

Fig. 6 PLRS schematic description of the control strategy

26

Fig. 7 Calf and foot system response using PD controller

27

Fig. 8 Calf and foot system response using PID controller

28

Fig. 9 HSDBC optimization algorithm

29

Fig. 10 Calf and foot system response comparison (Manual Tuning vs. HSDBC)

30

Fig. 11 Characteristics parameters of calf and foot movement

31

Fig. 12 Calf and foot control effort comparison

32

Fig. 13 Disturbance signals applied on the system

33

Fig. 14 Calf and foot system response with external disturbance

34

Fig. 15 System identification toolbox GUI

35

Fig. 16 PLRS prototype

36

Fig. 17 Designed circuit for motor orientation control

37

Fig. 18 Hardware low-pass filter

38

Fig. 19 Prototype system identification

39

Fig. 20 Prototype modes of operation

40

Fig. 21 Calf and foot measured outputs

41

Fig. 22 System behaviour while input variation

42

Fig. 23 Calf measured and simulated model output comparison

43

Fig. 24 Comparison between mathematical model and actual prototype by applying HSDBC-PID controller for calf

44

Fig. 25 Angular calf displacement (Math. Model HSDBC-PID controller vs. Updated HSDBC-PID controller)

45

Fig. 26 Foot measured and simulated model output comparison

46

Fig. 27 Comparison between mathematical model and actual prototype by applying HSDBC-PID controller for foot

47

Fig. 28 Angular foot displacement (Math. Model HSDBC-PID controller vs. Updated HSDBC-PID controller

48

Table 1 Parameters and description

Terminology Description UnitsL1 Length of link 1 (calf carrier) m

L2 Length of link 2 (foot carrier) m

m1 Mass of link 1 kg

m2 Mass of link 2 kg

θ1 Angular position of link 1 rad

θ̇1 Angular velocity of link 1 rad/s

θ2 Angular position of link 2 rad

θ̇2 Angular velocity of link 2 rad/s

49

Table 2 Anthropometric parameters (Vaughan et al, 1999)

Variable DescriptionTM Total body massCL Calf lengthCC Calf circumferenceFL Foot lengthMH Malleolus heightMW Malleolus width/breadth

50

Table 3 Volunteer detailsGender MaleAge 23 yearsTotal body mass (TM) 89 kgCalf length (CL) 0.43 mCalf circumference (CC) 0.47 mFoot length (FL) 0.27 mMalleolus height (MH) 0.08 mMalleolus width/breadth (MW) 0.11 m

51

Table 4 Volunteer calculated parameters

Parameter Value Unit

mcalf 4.96 kg

mfoot 1.31 kg

I Calf Total 0.0970742 kg.m2

I footTotal 0.0083514 kg.m2

52

Table 5 Simulation parameters

Variable Value Unitg 9.81 m/s2

I1 0.1 kg.m2

I2 0.01 kg.m2

L1 0.3 mL2 0.15 mm1 7.5 kgm2 4.5 kgcv 0.6 -cc 0.61 -

53

Table 6 PD and PID parameter configuration

Calf Foot

PD ControllerKp 100 5Kd 50 1

PID ControllerKp 1 0.2Ki 2 0.3Kd 3 0.8

54

Table 7 HSDBC optimization algorithm parameters [22]

Parameter Descriptionθi , j Bacteria angular displacement on xi - xj

plane around the originR Spiral radiusm Number of search points

kmax Maximum iteration numberNs Maximum number of swim

xi(k) Bacteria positionRn n x n matrix

55

Table 8 Optimized controller gain parameters and HSDBC cost function

HSDBC optimized parameters

Minimum cost function

Calf

(θ1 )

Kp1 0.3800196

2.1918e-9

Ki1 5.1325Kd1 1.3055

Foot

(θ2 )

Kp2 0.2960407Ki2 0.7046196Kd2 0.2436158

56

Table 9 Calf optimized controller gain parameters

HSDBC(Math. model)

HSDBC(Actual prototype model)

Calf

(θ1 )

Kp1 0.3800196 0.401Ki1 5.1325 0.01Kd1 1.3055 1.398

Table 10 Foot optimized controller gain parametera

57

HSDBC(Math. model)

HSDBC(Actual prototype model)

Foot

(θ2 )

Kp1 0.2960407 3.012Ki1 0.7046196 0.01Kd1 0.2436158 5.103

58

Table A1: Mathematical model system performance characteristics parameters (PD)

Variable (deg)

Characteristic Parameters

Values

θ1

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

0.88431.585066.193373.5349

2.2204e-130

73.534995.4088

θ2

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

0.44180.686619.344821.65040.7676

021.65041.7168

59

Table A2: Mathematical model system performance characteristics parameters (PID)

Variable (deg)

Characteristic Parameters

Values

θ1

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

25.400544.721680.994589.9849

00

89.9849100

θ2

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

14.620226.012427.002130.0000

00

30.0000100

60

Table A3: Mathematical model system performance characteristics parameters (HSDBC-PID)

Variable (deg)

Characteristic Parameters

Values

θ1

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

9.543117.031681.005690.0000

00

90.0000100

θ2

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

6.331611.286927.006730.0000

22.2045e-150

30.000098.7652

61

Table A4: Actual prototype system performance characteristics parameters (HSDBC-PID)

Variable (deg)

Characteristic Parameters

Values

θ1

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

7.815912.308080.926289.9094

00

89.9094100.000

θ2

Rise TimeSettling TimeSettling, MinSettling, MaxOvershootUndershootPeakPeak Time

3.59046.049726.876630.0177

0.53935580

30.017712.5039

62