Mechanical Prosthetic Arm Adaptive I-PD Control Model ... · Figure 1. (a) Block diagram representations of IPD system Figure 1.(b)Block Diagram of Model reference adaptive I-PD control

J.Mech.Cont.& Math. Sci., Vol.-13, No.-2, May-June (2018) Pages 43-55

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Mechanical Prosthetic Arm Adaptive I-PD Control Model Using

MIT Rule Towards Global Stability

1Sudipta Paul,

2Swati Barui,

3Pritam Chakraborty,

4Dipak Ranjan Jana,

*5Biswarup Neogi and

6Alexey Nazarov

1Electronics and Communication Engineering, Techno India - Batanagar,

Batanagar, W.B. -700141, India

2Electronics and Communication Engineering, Narula Institute of Technology,

Agarpara, W.B.- 700019, India.

3,5Electronics and Communication Engineering, JIS College of Engineering,

Nadia, W.B. -741235, India.

4Mechanical Engineering, JIS College of Engineering,

Nadia, W.B. -741235, India

6Software Engineering, National Research University Higher School of

Economics (HSE), Moscow, Russia

[email protected],

[email protected],

[email protected],

[email protected],

[email protected] and

[email protected]

*Corresponding author: Biswarup Neogi5

E-mail:[email protected]

Abstract

The development of prosthetic arm in accordance with the stable control

mechanism is the blooming field in the engineering study. The analysis of Model

Reference Adaptive Control (MRAC) for Prosthetic arm utilizing Gradient

method MIT rule has been presented using controlling system parameters of the

D.C motor. Adaptive tuning and performance analysis has been done for

controlling hand prosthesis system using Adaptive I-PD controller constraints

rationalized time to time in response with variations in D.C motor parameters to

track the desired reference model and application of Gradient Method MIT-Rule.

Further on, Lyapunov rule has been implemented towards closed loop asymptotic

tracking to ensure global stability on nonconformity of plant parameters because

adaptive controller design based on MIT rule doesn’t guarantee convergence or

stability. Computer-aided control system design (CACSD) and analysis has been

done using MATLAB-Simulink towards adaptive controller design and estimation

of adaptation gain.

ISSN (Online) : 2454 -7190 ISSN (Print) 0973-8975

mailto:[email protected]


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Keywords : Mechanical Prosthetic Arm, Model Reference Adaptive Control

(MRAC), Adaptive I-PD control, Gradient method MIT rule, Lyapunov rule.

I. Introduction

Disturbance rejection stands the most important objective of any feedback control

system. A constraint of milder effect on overall plant process due to set-point

changes and also need for sluggish modifications in the plant variable with no or

less overshoot. A novel effort on Model Reference Adaptive Control of a direct

drive D.C motor was presented by Hans Butler et al. [I,VII], promises approximate

time-optimal performance of the motor through the application of step input and

improves the performance over conventional PID control when load inertia

changes. Lillie Dewan et.al presents a technique of design MRAC based PI

controller (MRAC+PI) for speed control of D.C. motor using Lab VIEW software

tool improves the performance over MRAC [XVI]. Ramadan A. Elmoudi et al

[XIII] showed that the D.C motor speed response can be controlled and persistent at

specified performance specifications using MRAC with adaptation law based on

PID control for D.C motor speed control system. In the field Biomechanics

presented by Edmund Y S Chao et.al discussed the method of controlling the hand

prosthesis using neuro-signal that excites the neuromotor to provide mechanical

force for gripping or holding the object to prosthetic arm [VI]. Moreover recently

ANN, EMG signals are used towards controlling hand prosthesis and through some

electronic control algorithm transformed signals are provided to D.C motor and the

corresponding joint angles are generated. But for the part of controlling D.C motor

to achieve the desired performance, there remains lack of systematic methods.

This present study deals with the design of Model Reference Adaptive I-PD

Controller (MRAI-PDC) for D.C motor used in prosthetic hand to track the desired

reference model and the performance comparison is evaluated by means of

employing different input signals. Since several motors are required for position

control and gripping of hand and a linear second order time-invariant differential

equation is selected as the reference model for each degree of freedom such as

flexion, rotation, abduction and gripping of the prosthetic arm [VI]. Motor produces

high torque that is required for linear and smooth movement of the prosthetic arm. In

this paper, a reference model is selected that is a second-order differential equation of

D.C motor for position control of prosthetic arm with quick movement and little or

no vibrations. Different model reference of D.C motor that gives the desired

performance should be selected for each degree of freedom[XV]. Moreover, the

study of MRAC with I-PD control based adaptation law can give better performance

in the presence of disturbances and uncertainties than MRAC or MRAC with PID

control [III, V, XII, XVII].

II. Mathematical Background of Mechanical Prosthetic Arm Control

Modelling Using Gradient Method MIT Rule

In this section, a model reference adaptive control is designed for artificial arm

utilizing MIT rule-based adaptation law. Each link of the prosthetic arm is gear


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driven by a D.C servomotor is considered as the plant used in the simulation. The

main goal is to design a controller that makes gear angle θ to follow desired

trajectory i.e. gear angleθ_m. A Perfect tracking of model reference is achieved by

developing parameter adaptation laws for control algorithm utilizing MIT rule.

Here, we are considering a second order differential equation transfer function in s-

domain as a reference model in the design of Mechanical Prosthetic Arm and can be

represented as [XVIII, XIV,IX,X].

(1)

Figure 1. (a) Block diagram representations of IPD system

Figure 1.(b)Block Diagram of Model reference adaptive I-PD control

system

I-PD controllers are the forms of PID controllers where only integral components

directly propagate changes in to compared to other two Proportional and

derivative components. The adaptive I-PD controller for prosthetic arm has three


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controller parameters which can beupdated to track the desired

trajectory The control law of MRAC structure taken as in the following form,

(2)

Where adaptive feed forward-feedback controller parameter can be written in vector

form as Adaptive gain will be adjusted to track the reference

model. Adaptation or tracking error which is defined as the difference

between desired output and plant output is used to adjust the controller parameters in

the adjustment mechanism shown in Figure. 1(a)-(b).

The input output relationship for the prosthetic arm can be simplified by considering

load torque to zero and a very small circuit inductance given by[1]-[2],

(3)

By taking the parameters as the eqn. (3) can be written

as,

(4)

Substituting eq. (4) into eq. (2) yields

(5)

The equation (5) represents overall transfer function of plant with I-PD

controller[XI]. The values of plant parameter a, b & c are unknown and varying, so

adaptation mechanism for each controller parameter have to be found that

are uniquely based on measurable quantity. Update law for each of the adaptive I-PD

controller parameters are derived mathematically utilizing MIT rule and

experimentally evaluated utilizing MATLAB simulation.

Since, our main goal is to make the plant come closer to the model. The

mathematical equations which are derived needs to be approximated for applying the

exact MIT rule. When the parameters of the plant are very close enough to the

desired model values, the plant characteristics can be replaced by model

characteristics as,


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(6)

Generalized MIT Rule conveys that,

(7)

Here; is the adaptation gain.

Hence, Cost function can be defined as, , which has to be

minimized. Thus, the sensitivity derivatives for each of the controller

parameters can be written as,

(8)

Utilizing, eqn. [7- 10], update law for each of the controller parameter

as follows,

Similarly,

And

Where adaptive gain for each controller parameters are

, , (11)

III. Result and Analysis: I-PD Control Modelling Using Gradient

Method MIT Rule

The simulation of closed loop model reference adaptive control scheme was done

using MATLAB software (Simulink). Second order modelling of arm controller can


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be calculated by considering each parameter values in eqn. (4). For design and

simulation physical parameters values of motor measured as follows,

Motor RPM , =Armature current= , Armature

resistance=10 , Equivalent moment of inertia , Viscous

friction coefficient= Motor torque = , Motor

torque constant = , Backe.m.f constant = =

Angular velocity = Angular acceleration = .

Substituting the above values in eqn.4, the calculated transfer function is given by,

The desired transfer function of prosthetic arm controller chosen in such a

manner that it could respond quickly to the reference input as follows,

The reference model has settling time of 0.5714 sec, damping factor of 0.8488

and percentage overshoot of 0.6470%.Figure 2-(a) shows the generalised block

diagram of D.C geared motor used for prosthetic arm and Figure 2-(b) with adaptive

I-PD controller and adjustment mechanism corresponds to sets of input signal. The

desired gear angle and corresponding output for variations in input signal and its

amplitude, Figure 3(a)-(b), shows different values of adaptive gains for step input of

unit amplitude and tracking error corresponds to adaptive gain respectively. Table I,

values of adaptive gain for each controller parameters i.e. , and corresponds

to step input for desired gear angle.

Table I. Values of adaptive gain for controller parameters

Controller adaptive

gain parameter ( )

Desired gear angle gain*

Set 1 Set 2

gammai ( -25 -50

gammap ( 1 2

gammad ( 0.5 1

*output corresponds to Step input


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Figure. 2 (a)Block

representation of the D.C motor

system

Figure.2 (b)Simulink Block Diagram of

MIT rule Based MRAC Prosthetic arm.

Figure 3.(b) Adaptation error between

real gear angle and desired gear angle for

different values of adaptive gains.

Figure 3.(a) The desired gear angle and

corresponding output set 1 & 2

represent real gear angle for different

values of adaptive gains for step input.


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4(a) 4(b)

4(c)

Table II. Time characteristic values for gear angle of D.C motor using step input

Specifications Reference Model Set of Gammas

1 2

Settling Time (sec) 0.569 0.961 0.958

% Overshoot 0.6 0.3 0.7

Peak Time (sec) 0.727 1.397 1.189

IV. Global Stability Analysis Utilizing Lyapunov Criterion

This section is same as above for designing model reference adaptive I-PD control

design but here Lya-punov stability theory has been used for originating the

adaptation law. The designing of adaptive controller based on MIT rule does not

guarantee convergence or stability. Lya-punov based design can be effective to make

the adaptive system stable. By choosing a proper Lya-punov function candidate a

Figure 4. (a), (b) and (c) Represents the different Adaptive I-PD controller

parameters for different set of adaptive gains


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stable adaptation mechanism is designed and I-PD control gains are adjusted in such

a manner that it drives the tracking error to zero [VIII,II].

The closed loop differential equation of the D.C motor with I-PD controller is given

as follows,

And the model reference is given by,

Describe,

Subtracting equation (34) from (33) get.

Yields,

Which is a positive definite function, and then its derivative is

For the positive definite function , should be negative definite for the closed

loop system to be stable. The derivative of Lyapunov function will be negative if the

terms in the brackets are zero,

This gives,

(16)


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Where,

The above equation represents adaptation law of adaptive I-PD controller parameter

which can be implemented on MATLAB Simulink.

Equation (36), , and taking;

Substituting these values in equation (15) and by transformation yields,

That gives the desired model reference for stable adaptive I-PD controller design.

As discussed above there are more options to select the values of reference model

parameters in the stable range which might have possibility to choose the nearby

desired reference model.

The negativeness of the derivative indicates the stable adaptive controller design.

Figure 5. Simulink Block Diagram of Lyapunov Based Model Reference Adaptive

Control of Prosthetic arm


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Figure 6. Desire gear angle step input and corresponding output real gear angle for

adaptive gains α_p=0.1,α_i= -10,α_d=0.01

V. Conclusions

The design of Model Reference Adaptive I-PD controller (MRA I-PDC) for a

prosthetic arm using adaptation law based on gradient method -MIT rule and system

stability analysis utilizing Lyapunov stability criterion we discussed in this article.

Incorporation of I-PD controller leading towards reduced system percentage (%)

overshoot, settling time of around 0.3%, 0.961 sec and 0.7%, 0.958 sec corresponds

to system gain in comparison with reference plant model. Both the MIT rule and

Lyapunov based model reference adaptive I-PD controller analysis is done for

utilizing Matlab® and Simulink®. Design utilizing Lyapunov stability gives the

stable adaptive controller and more options to select the values of reference model

parameters closer to desired reference model. On the future aspects, this research

extended towards the aim of performance study of multi motor prosthetic arm with

high degree of freedom. In additions, other advanced tuning methods will be

introduced to carry the performance on view of high efficacy.

Acknowledgement

Authors would like to dedicate this research in the memory of Late. N.G Nath and his

present research team for their kind concern towards the development of this research

work. We would also acknowledge AICTE (Gov. of India) project grant no.

20/AICTE/RFID/RPS (POLICY-IV) 24/2012-13 for funding this research work.


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Documents

Mechanical Prosthetic Arm Adaptive I-PD Control Model ... · Figure 1. (a) Block diagram representations of IPD system Figure 1.(b)Block Diagram of Model reference adaptive I-PD control