Tuning and transfer functional modelling of a prosthetic arm

Abstract

This work presents performance comparison of different Tuning schemes applying the optimization technique named as Fitness Based Adaptive Differential Evolution (FBADE) for a prosthetic arm. From the dexterous point of view, the mathematical modelling is the fundamental requirement to improve the performance level. Here, we define the optimum transfer function applying fuzzy logic to establish the link of the robustness and the controllability metrics between amputated body parts and prosthetic arm. A proportional integral derivative (PID) controller is chosen for the purpose of continuously modulated control. Also, the role of most influential tuning parameters have been illustrated to measure the performance level of different tuning schemes. In the end, Lyapunov method is applied to analyze the stability of the designed model.

Keywords

Prosthetic arm transfer function FBADE optimization technique comparative response analysis Lyapunov method

1. Introduction

The development of prosthetic [9, 21] arm modelling is continuously growing over the decades, and many important and fruitful research work already been made till date. One of the recent trends of research is to control a biomechanical arm connected with the nervous system by linking electrodes to human nerves. The dynamic mathematical modelling of prosthetic arm controlled is demonstrated in [4] and it is acquired in this study to extend further. The four sets of transfer functions are developed for the prosthetic arm model and they are tuned through Ziegler-Nichols (ZN) rule [30], Genetic Algorithm (GA), Particle Swarm Optimization (PSO) rule [10, 30] and Bacterial Foraging Optimization (BFO) methods [11, 23]. The schemes in this work are initially passed through Fitness Based Adaptive Differential Evolution (FBADE) technique, and then comparative step response analysis and control parameter analysis have been performed applying Fuzzy Logic to obtain the most appropriate transfer function. Besides, to achieve more suitable result, the stability analysis is also performed applying Lyapunov method.

Figure 1.

Block diagram of the proposed Prosthetic arm gripper.

Figure 2.

Mechanical prosthetic arm used for control modelling determination.

2. Control modelling of prosthetic arm

The mechanism of the experimental process of Prosthetic arm gripper is presented in Fig. 1. The force experienced by the gripper is fed to the error detector to compare with the reference input signal at the input end. The error produces at each and every instant of time is processed with Fuzzy controller. Thereafter, the corrected output of the controller is applied on the dynamometer of the gripper. The physical movement of the arm gripper is operated as per the controller corrective action. During the functioning of this mechanism friction, the shaft movement may cause some disturbance to the system. The mechanical arm used for the present study is shown in Fig. 2. The displacement of fingers of the prosthetic arm is measured with change in load using dynamometer. Figure 3 illustrates how more force can be applied to the fingers to move away from each other in step by step manner [4]. The four sets of transfer functions from the dynamic modelling of the prosthetic arm are as follows:

$\displaystyle G(s)=\frac{1}{0.21s^{2}+0.4038s+0.0411},$ (1) $\displaystyle G(s)=\frac{1}{0.21s^{2}+0.1193s+0.0245},$ (2) $\displaystyle G(s)=\frac{1}{0.21s^{2}+0.0434s+0.0299},$ (3) $\displaystyle G(s)=\frac{1}{0.21s^{2}+0.2369s+0.0054}.$ (4)

Figure 3.

Experimental set up with mechanical arm using dynamometer.

3. Comparative analysis

The soft computing techniques are adopted for setting the optimized PID controllers constants. The satisfactory decision making is achieved through the PID tuning approaches with simple control configuration. In this work, 4 sets of transfer functions with unity feedback are considered and then ZN rule [6], GA [9], BFO methods are applied to tune the transfer functions. The ZN tuning rule has a large impact in making PID feedback controls [1] acceptable to control engineers [10, 21, 30]. In genetic algorithm, a population of candidate solutions to an optimization problem is evolved toward better result. When the optimized PID parameters [14] are highly correlated, GA cannot provide proper efficiency, whereas the PSO rule optimizes a problem by iteration and trying to improve a candidate solution with regard to a given measure of quality. Also, the tuning with BFO has been widely accepted for distributed optimization and control [5]. In the present study, another optimization technique that is FBADE technique is applied for better efficacy and comparative result analysis. Measured tuning procedures are transcendent for the selection of proportional, integral and derivative constants. This will be furnished further to obtain efficient mathematical model for prosthetic arm.

Table 1
PID parameters after tuning by five tuning methods [2, 3]

Set no	Values of PID parameters applying Z-N rule	Values of PID parameters applying PSO rule	Values of PID parameters applying GA	Values of PID parameters applying BFO	Values of PID parameters applying FBADE technique
1	$K_{p}=0.0471$ , $T_{d}=1.046$ , $T_{i}=4.161$	$K_{p}=1.8255$ , $T_{d}=2.816$ , $T_{i}=0.2219$	$K_{p}=1.2365$ , $T_{d}=2.529$ , $T_{i}=5.859$	$K_{p}=9.82741$ , $T_{d}=2.826$ , $T_{i}=0.2196$	$K_{p}=1.235$ , $T_{d}=2.213$ , $T_{i}=0.2091$
2	$K_{p}=0.0083$ , $T_{d}=1.8383$ , $T_{i}=7.3535$	$K_{p}=1.9045$ , $T_{d}=0.8013$ , $T_{i}=1.0019$	$K_{p}=2.1023$ , $T_{d}=0.7929$ , $T_{i}=6.9427$	$K_{p}=1.983$ , $T_{d}=1.7938$ , $T_{i}=6.9938$	$K_{p}=1.9045$ , $T_{d}=0.6921$ , $T_{i}=1.00061$
3	$K_{p}=0.0028$ , $T_{d}=2.165$ , $T_{i}=8.66$	$K_{p}=1.255$ , $T_{d}=1.8728$ , $T_{i}=1.215$	$K_{p}=1.8937$ , $T_{d}=1.7856$ , $T_{i}=1.0105$	$K_{p}=4.199$ , $T_{d}=2.8665$ , $T_{i}=0.189$	$K_{p}=0.623$ , $T_{d}=1.6228$ , $T_{i}=1.0213$
4	$K_{p}=0.0036$ , $T_{d}=3.37$ , $T_{i}=13.48$	$K_{p}=4.478$ , $T_{d}=1.821$ , $T_{i}=3.545$	$K_{p}=3.146$ , $T_{d}=2.278$ , $T_{i}=4.335$	$K_{p}=5.0257$ , $T_{d}=1.147$ , $T_{i}=2.726$	$K_{p}=0.7$ , $T_{d}=0.968$ , $T_{i}=3.303$

The different PID tuning methods such as ZN rule, PSO method, GA, BFO and FBADE technique tuning parameter values are observed for the previously mentioned transfer functions in Table 1.

4. Comparative step response analysis of each closed loop transfer function with different tuning methods

For the selected transfer function, the corresponding step responses are illustrated in Figs 4–7 applying the conventional and advanced tuning methods respectively to take on characteristics performance analysis. All the graphical plots are given to show the output and scale adjustments are chosen according to the requirement. For each graph within the specified scale, the system is showing its stability. The parametric analysis is adopted as it is difficult to select the best suited transfer function applying graphical analysis for the proposed model.

Figure 4.

Step responses of set 1 transfer function with different tuning.

Figure 5.

Step responses of set 2 transfer function with different tuning.

Figure 6.

Step responses of set 3 transfer function with different tuning.

Figure 7.

Step responses of set 4 transfer function with different tuning.

5. Control parameters model of each closed loop transfer function with different tuning methods

The different control parameters such as rise time, settling time, overshoot and peak time (see Figs 8–11) are observed to determine the most suitable model through the application of PID tuning via various types advanced methods of tuning.

Figure 8.

The comparison of Rise time for 4 sets, where Fig. 8a stands for set 1, Fig. 8b stands for set 2, Fig. 8c stands for set 3, and Fig. 8d stands for set 4, of transfer functions applying different methods such as ZN, PSO, GA, BFO, FBADE.

Figure 9.

The comparison of Settling time for 4 sets, where Fig. 9a stands for set 1, Fig. 9b stands for set 2, Fig. 9c stands for set 3, and Fig. 9d stands for set 4, of transfer functions applying different methods such as ZN, PSO, GA, BFO, FBADE.

Figure 10.

The comparison of overshoot for 4 sets, where Fig. 10a stands for set 1, Fig. 10b stands for set 2, Fig. 10c stands for set 3, and Fig. 10d stands for set 4, of transfer functions applying different methods such as ZN, PSO, GA, BFO, FBADE.

Figure 11.

The comparison of Peak time for 4 sets, where Fig. 11a stands for set 1, Fig. 11b stands for set 2, Fig. 11c stands for set 3, and Fig. 11d stands for set 4, of transfer functions applying different methods such as ZN, PSO, GA, BFO, FBADE.

By reviewing the control parameters, more than one preferable case can be observed, but it is difficult to select the most optimum case. Therefore, fuzzy logic concept is incorporated to select the most suitable mathematical model.

6. The selection of optimum transfer function applying fuzzy logic

In the recent age of technology, the applications of fuzzy logic have increased in huge number. This work is concentrated on the development of software package for the automatic tuning of myoelectric prosthesis [7, 12, 13]. To improve the response of the flexible arm depending on the vibration feedback, PID is introduced for controlling of a single-link flexible manipulator [26, 27, 28]. Point to be noted, Fuzzy logic is a multi-valued logical system. Fuzzy logic (FL) relates classes of objects with uncap boundaries in which membership is a matter of degree [15, 16, 17]. Here a flexible-link robot arm control using fuzzy logic and a singular perturbation approach are considered. A brief on singular perturbation approach is incorporated to derive the slow and fast sub-systems, and thus reduce the effect of spillover [18, 19, 20].

To find the best stable method, the five tuning methods are compared applying the suitable rule as given below:

RT	ST	OV	PT
L	L	M	L

where, L stands for Low, M stands for Medium, H stands for High, RT stands for Rise Time, ST stands for Settling Time, OV stands for Overshoot, PT stands for Peak Time.

The different control parameter status for various tuning methods are illustrated in the following Tables 3–7.

Table 2
Control parameter analysis for different tuning method

Parameter	Tuning method	Model transfer function (set no.)	Fig. nos.
Rise time	FBADE	Set 1	8(a)
Settling time	GA	Set 4	9(d)
Overshoot	BFO and FBADE	Sets 1 and 2	10(a) and 10(b)
Peak time	PSO	Set 4	11(d)

Table 3

Control parameter status for ZN method

ZN method	RT	ST	OV	PT
Set 1	H	H	H	H
Set 2	L	L	M	L
Set 3	L	L	L	M
Set 4	L	L	L	H

Table 4

Control parameter status for PSO method

PSO method	RT	ST	OV	PT
Set 1	H	H	L	H
Set 2	L	L	H	L
Set 3	L	L	M	L
Set 4	L	L	L	L

Figure 12.

Graphical presentation of fuzzy membership function of tuning parameters.

Table 5

Control parameter status for GA method

GA method	RT	ST	OV	PT
Set 1	H	H	L	H
Set 2	L	L	H	L
Set 3	L	L	H	L
Set 4	L	L	M	L

Table 6

Control parameter status for BFO method

BFO method	RT	ST	OV	PT
Set 1	H	H	H	H
Set 2	L	L	L	L
Set 3	L	L	L	L
Set 4	L	L	L	L

Table 7

Control parameter status for FBADE method

FBADE method	RT	ST	OV	PT
Set 1	M	L	L	H
Set 2	M	M	H	L
Set 3	L	L	L	L
Set 4	M	H	H	L

ZN method of set 2, PSO Method of set 3, and GA Method of set 4 are concluded as the best models from the analysis. Hence, they are again considered for further selection of most optimum one. In Figs 13–15, the step responses of the selected three models are demonstrated. According to step response and control parametric analysis of the selected mathematical models; for set 2 model using ZN, the settling time is too high among others, so it will take more time to reach to steady state value that would affect the speed of the response and also it is showing undershoot characteristic. Between remaining two transfer functions, set 3 transfer function using PSO method, the overshoot value is much higher compared to another. Increasing overshoot will degrade the stability. Reduced overshoot shows how quickly the response is changing over time. Finally, set 4 transfer function using GA method is opted for its most appropriate characteristics as it is nearly satisfied with the rule mentioned in Table 2 compared with other two methods ZN and PSO.

Figure 13.

Step response of closed loop transfer function for set 2 using ZN method.

In Section 6, fuzzy logic is applied to achieve the optimum transfer function among all other methods. As per fuzzy rule base selection, the control parameters such as rise time, settling time, peak time should be low and overshoot should be medium which have taken into account to select the most suitable method. All the specified tuning parameters of different methods are verified with the suitable fuzzy rule based methods and the results are shown in Tables 3 to 7 respectively. The ZN method of set 2, PSO method of set 3, and GA method of set 4 are perfectly matched with the expertise knowledge based system for achieving the outcome. As the remaining sets are entirely matched, thus they have not incorporated for further study.

$\displaystyle G(s)=\frac{2.275s^{2}+1.525s+1}{0.255s^{3}+0.53s^{2}+0.36s}$ (5)

Table 8

The tuning parameters of PSO method

SET3

0.2295

1.4397

4.9305

0.6883

Figure 14.

Step response of closed loop transfer function for set 3 using PSO method.

Figure 15.

Step response of closed loop transfer function for set 4 using GA method.

7. Stability analysis applying Lyapunov Stability method

The Lyapunov Stability method for open loop transfer function have been implemented for paraplegic and fatigued condition of human arm with control representations [8]. The learning and exploiting the power of Control Lyapunov Function (CLF) scheme have been incorporated to ensure global asymptotic stability of nonlinear dynamic system [24]. Here, the block diagram representation of the optimum model is illustrated in Fig. 16. for further study through Lyapunov Stability method. Now, the information about Lyapunov basic is given for the controlling of a robot [22]. The qualitative study between optional and prescribed prosthetic model is analyzed using Lyapunov logic [25, 29].

Figure 16.

Control block of the optimum model.

The analysis of the Lyapunov stability of the closed loop linear system can be followed as: From Fig. 16, we get:

$\displaystyle e(s)G_{c}(s)G_{p}(s)=Y(s)$ $\displaystyle or,e(s)\frac{2.275s^{2}+1.525s+1}{0.255s^{3}+0.53s^{2}+0.36s}=Y(s)$ (6) $\displaystyle or,(s^{3}+11s^{2}+7.39s+3.92)e(s)=0.$

Applying inverse Laplace transform, we get:

$\displaystyle\dddot{e}+11\ddot{e}+7.39\dot{e}+3.92e=0.$ (7)

Let, $x_{1}=e$ , $x_{2}=\dot{e}$ , $x_{3}=\ddot{e}$

Thus, $\dot{x_{1}}=x_{2}$ , $\ddot{x_{2}}=x_{3}$

$\displaystyle\dot{x_{3}}=-3.92x_{1}-7.39x_{2}-11x_{3}.$ (8)

Equation (8) can be re-expressed as:

$\displaystyle\dot{x_{3}}=-ax_{1}-bx_{2}-cx_{3},$ (9)

where

$\displaystyle a=3.32,b=7.39,c=11$ (9a)

Now, scalar positive definite function can be expressed as:

$\displaystyle V(x)=\frac{1}{2}S_{1}x_{1}^{2}+\frac{1}{2}S_{2}x_{2}^{2}+\frac{1% }{2}S_{3}x_{3}^{2}+S_{4}x_{1}x_{2}+S_{5}x_{2}x_{3}+S_{6}x_{3}x_{1},$ (10)

where $S_{1},S_{2},S_{3},S_{4},S_{5}>0$ .

Now, taking the derivative with respect to time t, we get:

$\displaystyle\dot{V(x)}=S_{1}x_{1}\dot{x_{1}}+S_{2}x_{2}\dot{x_{2}}+S_{3}x_{3}% \dot{x_{3}}+S_{4}(\dot{x_{1}}x_{2}+\dot{x_{2}}x_{1})\!+S_{5}(\dot{x_{2}}x_{3}+% \dot{x_{3}}x_{2})\!+S_{6}(\dot{x_{3}}x_{1}+\dot{x_{1}}x_{3})=(S_{1}-aS_{5}-bS_% {6})x_{1}x_{2}+(S_{2}-bS_{3}-cS_{5}+S_{6})x_{2}x_{3}+(S_{4}-cS_{6}-aS_{3})x_{3% }x_{1}{}+(S_{5}+cS_{3})x_{3}^{2}+(S_{4}+bS_{5})x_{2}^{2}-ax_{1}S_{6}$ (11)

For the positive definite function $V$ , we need another positive definite function $U$ such that, $\dot{V(x)}=-U(x)$ .

Now we take the coefficients in such a manner that,

$\displaystyle\dot{V(x)}=-U(x).$ (12)

Thus taking,

$\displaystyle(S_{1}-aS_{5}-bS_{6})=0,$ (13) $\displaystyle((S_{2}-bS_{3}-cS_{5}+S_{6}))=0,$ (14) $\displaystyle((S_{4}-cS_{6}-aS_{3}))=0,$ (15) $\displaystyle((S_{5}+cS_{3}))=0,$ (16) $\displaystyle S_{6}=0$ (17)

Here we show, $(S_{4}+bS_{5})<0$ , such that

$\displaystyle\dot{V(x)}=-(S_{4}+bS_{5})x_{2}^{2}.$ (18)

Substituting Eq. (17) into Eqs (13)–(15), we get:

$\displaystyle S_{1}=aS_{5}$ (19) $\displaystyle S_{2}=bS_{3}+cS_{5}$ (20) $\displaystyle S_{4}=aS_{3}$ (21) $\displaystyle S_{5}=cS_{3}$ (22)

Now, substituting Eq. (22) into Eqs (19) and (20), we get:

$\displaystyle S_{1}=acS_{3}$ (23) $\displaystyle S_{2}=(b+c^{2})S_{3}$ (24)

Now substituting Eqs (17), (21)–(24) into Eq. (10) we get:

$\displaystyle V(x)=\frac{1}{2}acS_{3}x_{1}^{2}+\frac{1}{2}(b+c^{2})S_{3}x_{2}^% {2}+\frac{1}{2}S_{3}x_{3}^{2}+aS_{3}x_{1}x_{2}+cS_{3}x_{2}x_{3}$ (25)

Let $S_{3}=1$ , substituting value of a, b, c from Eq. (9a) into Eq. (25) we obtain:

$\displaystyle V(x)=21.56x_{1}^{2}+64.195x_{2}^{2}+0.5x_{3}^{2}+3.92x_{1}x_{2}+% 11x_{2}x_{3}$ (26)

From Eq. (26), we clearly see that $V(x)$ is a positive definite function and $(S_{4}-bS_{5})=(aS_{3}bcS_{3})=(a-bc)S_{3}$ .

As $S_{3}=1$ , therefore

$\displaystyle(S_{4}-bS_{5})=(a-bc)=-77.37<0$ (27)

Hence, from Eqs (18) and (27), we get:

$\displaystyle\dot{V(x)}=-77.37x_{2}^{2}<0,$ (28)

which is negative semi-definite and $\dot{V(0)}=0$ .

From Eq. (26), we obtain: $V(x)->\infty$ as $x->\infty$ and $\dot{V(x)}\leqslant 0.$

By showing that $\dot{V(x)}$ being negative semi-definite, it is sufficient for asymptotic stability condition to show that $x_{1}$ axis is not a trajectory of the system.

$x_{1}=m$ (constant) $\forall x_{3}=0,\dot{x_{1}}=x_{2}=0$ and $\dot{x_{2}}=x_{3}=0$ .

The equilibrium state is at the origin of the system. So that the system is asymptotically stable.

8. Conclusion

This work is a basic approach for generating efficient prosthetic arm model for adopting different advanced tuning procedures. As the comparative step response analysis is not adequate to decide the optimum case. therefore control parameter selection by fuzzy logic approach is implemented to formulate the best result of the present study. Also the stability analysis by Lyapunov Stability method is adopted to emphasize on the selected model. The further extension of this work is definitely going to be a benchmark in the field of Limb prosthesis.

References

Altınten

Ketevanlioğlu

Erdoğan

Hapoğlu

and Alpbaz

, Self-tuning PID control of jacketed batch polystyrene reactor using genetic algorithm, Chemical Engineering Journal 138(1) (2008), 490–497.

Biswas

Das

Dasgupta

and Abraham

, Stability analysis of the reproduction operator in bacterial foraging optimization, in: Proceedings of the 5th International Conference on Soft Computing as Transdisciplinary Science and Technology, 2008, pp. 564–571.

Sharma

and Noel

M.M.

, Design of a low-cost five-finger anthropomorphic robotic arm with nine degrees of freedom, Robotics and Computer-Integrated Manufacturing 28(4) (2012), 551–558.

Neogi

Ghosal

Ghosh

Bose

T.K.

and Das

, Dynamic Modeling and Optimizations of Mechanical Prosthetic Arm by Simulation Technique, in: 1𝑠𝑡 International Conference on Recent Advances in Information Technology, 2012, DOI: 10.1109/RAIT.2012.6194542.

Hang

C.C.

Åström

K.J.

and Ho

W.K.

, Refinements of the Ziegler-Nichols tuning formula, IEEE Control Theory and Applications 138(2) (1991), 111–118.

Liu

C.H.

and Hsu

Y.Y.

, Design of a self-tuning PI controller for a STATCOM using particle swarm optimization, IEEE Transactions on Industrial Electronics 57(2) (2010), 702–715.

Bonivento

Davalli

Fantuzzi

Sacchetti

and Terenzi

, Automatic tuning of myoelectric prostheses, Journal of Rehabilitation Research and Development 35(3) (1998), 294–304.

Semasinghe

C.L.

Ranaweera

R.K.P.S.

Prasanna

J.L.B.

Kandamby

H.M.

Madusanka

D.G.K.

and Gopura

R.A.R.C.

, HyPro: A Multi-DoF Hybrid-Powered Transradial Robotic Prosthesis, Journal of Robotics, 2018, https://doi.org/10.1155/2018/8491073.

Datta

and Brain

N.D.

, Clinical applications of myoelectrically controlled prostheses, Critical Rev in Phys and Rehab Medicine, 1992, 215-239.

10.

Kim

D.H.

, GA-PSO based vector control of indirect three phase induction motor, Applied Soft Computing 7(2) (2007), 601–611.

11.

Kim

D.H.

, Hybrid GA-BF based intelligent PID controller tuning for AVR system, Applied Soft Computing 11(1) (2011), 11–22.

12.

Rivera

D.E.

Morari

and Skogestad

, Internal model control: PID controller design, Industrial and Engineering Chemistry Process Design and Development 25(1) (1986), 252–265.

13.

Kwek

and Choi

, Is a prosthetic arm customized PRADA? A critical perspective on the social aspects of prosthetic arms, Disability and Society 31(8) (2016).

14.

Hotson

McMullen

D.P.

Fifer

M.S.

Johannes

M.S.

Katyal

K.D.

Para

M.P.

Armiger

Anderson

W.S.

Thakor

N.V.

and Wester

B.A.

, Individual finger control of a modular prosthetic limb using high-density electrocorticography in a human subject, Journal of Neural Engineering 13(2) (2016).

15.

Patra

G.R.

Singha

Choudhury

S.S.

and Das

, A Fitness-Based Adaptive Differential Evolution Approach to Data Clustering, in: Proceedings of the International Conference on Frontiers of Intelligent Computing: Theory and Applications (FICTA), 2013, pp. 469–475.

16.

Nightingale

J.M.

, Microprocessor control of an artificial arm, Journal of Microcomputer Applications 8(2) (1985), 67–173.

17.

Åström

K.J.

Hägglund

Hang

C.C.

and Ho

W.K.

, Automatic tuning and adaptation for PID controllers-a survey, Control Engineering Practice 1(4) (1993), 699–714.

18.

Kuo

K.Y.

and Lin

, Fuzzy logic control for flexible link robot arm by singular perturbation approach, Applied Soft Computing 2(1) (2002), 24–38.

19.

Das

Paul

Ghosh

PalBhowmik

Neogi

and Ganguly

, An Approach Towards the Representation of Sign Language by Electromyo-graphy Signals with Fuzzy Implementation, International Journal of Sensors, Wireless Communications and Control 7(1) (2017), 26–32.

20.

Cominos

and Munro

, PID controllers: recent tuning methods and design to specification, IEEE Proceedings-Control Theory and Applications 149(1) (2002), 46–53.

21.

Scott

R.N.

and Parker

P.A.

, Myoelectric prostheses: state of the art, Journal of Medical Engineering and Technology 12(4) (1988), 143–151.

22.

Murray

R.M.

and Sastry

S.S.

, A Mathematical Introduction to Robotic Manipulation, CRC Press, 1993.

23.

Das

Paul

Ojha

S.K.

Neogi

Nazarov

Ghosh

and Ghosh

, On Design and Implementation of an Artificial Lower Limb, International Journal of Sensors, Wireless Communications and Control 8(2) (2018), 100–108.

24.

Mohammad Khansari-Zadeh

and Billard

, Learning Control Lyapunov Function to Ensure Stability of Dynamical System-based Robot Reaching Motions, Robotics and Autonomous Systems 62(6) (2014), 752–765.

25.

Wurdeman

S.R.

Myers

S.A.

Jacobsen

A.L.

and Stergiou

, Prosthesis preference is related to stride-to-stride fluctuations at the prosthetic ankle, J. Rehabil Res. Dev. 50(5) (2013), 671–686.

26.

Mansour

Konno

and Uchiyama

, Neural Network Based Tuning Algorithm for MPID Control, Tohoku University Japan.

27.

Schweitzer

Thali

M.J.

and Egger

, Case-study of a user-driven prosthetic arm design: bionic hand versus customized body-powered technology in a highly demanding work environment, Journal of Neuro Engineering and Rehabilitation 15(1) (2018).

28.

Chang

W.D.

, A multi-crossover genetic approach to multivariable PID controllers tuning, Expert Systems with Applications 33(3) (2007), 620–626.

29.

Mitsukura

Yamamoto

and Kaneda

, A design of self-tuning PID controllers using a genetic algorithm, in: IEEE Proceedings of the American Control Conference, 1999, pp. 1361–1365.

30.

Gaing

Z.L.

, A particle swarm optimization approach for optimum design of PID controller in AVR system, IEEE Transactions on Energy Conversion 19(2) (2004), 384–391.

Tuning and transfer functional modelling of a prosthetic arm

Abstract

Keywords

1. Introduction

Table 1 PID parameters after tuning by five tuning methods [2, 3]

Table 2 Control parameter analysis for different tuning method

References

Table 1
PID parameters after tuning by five tuning methods [2, 3]

Table 2
Control parameter analysis for different tuning method