hybrid genetic algorithm pid control for a five-fingered smart

Hybrid Genetic Algorithm PID Control for

a Five-Fingered Smart Prosthetic Hand

CHENG-HUNG CHEN

Measurement and Control

Engineering Research Center

Department of Electrical Engineering

and Computer Science

School of Engineering

Idaho State University

Pocatello, ID 83209, USA

[email protected]

D. SUBBARAM NAIDU

Measurement and Control

Engineering Research Center

Department of Electrical Engineering

and Computer Science

School of Engineering

Idaho State University

Pocatello, ID 83209, USA

[email protected]

Abstract: A hybrid of soft control technique of adaptive neuro-fuzzy inference system (ANFIS) and genetic algo-

rithm (GA) and hard control technique of proportional-integral-derivative (PID) for a five-fingered, smart prosthetic

hand is presented. The ANFIS is used for inverse kinematics and GA is used for tuning the PID parameters with

the objective of minimizing the error squared between desired and actual angles of the links of the fingers of the

prosthetic hand. Simulation results for all the five fingers with GA-tuned PID controller exhibit superior perfor-

mance compared to the PID control without GA.

Key–Words: Prosthetic Hand, PID Control, Genetic Algorithm, Adaptive Neuro-Fuzzy Inference System, Hybrid

Control

1 Introduction

Hard control (HC) methodologies are used at lowerlevels for accuracy, precision, stability and robust-

ness. HC comprises proportional-derivative (PD) con-

trol [1], proportional-integral-derivative (PID) control

[2, 3], optimal control [3–6], adaptive control [7–9]

etc. with specific applications to prosthetic devices.

However, our previous works [1–3, 10] for a smart

prosthetic hand showed that PID controller resulted in

overshooting and oscillation because the system dy-

namics are sensitive to the rigidity of the target ob-ject and the used gain parameters of PD or PID con-

troller [11].

Soft computing (SC) or computational intelli-

gence (CI) is an emerging field based on synergy and

seamless integration of neural networks (NN), fuzzy

logic (FL) and optimization methods, such as genetic

algorithms (GA), particle swarm (PS) [1,12,13], tabu

search (TS) [13] and so on. The methodology basedon SC can be used at upper levels of the overall mis-

sion whereas the HC can be used at lower levels for

accuracy, precision, stability and robustness. Hence,

we propose the GA-based PID controller to solve

problems that cannot be solved satisfactorily by us-

ing either HC or SC methodology alone with specific

applications to prosthetics.

In this paper, we first consider briefly trajec-

tory planning and kinematics problems. Then, adap-

tive neuro-fuzzy inference system (ANFIS) is used to

solve inverse kinematics problem for three-link fin-

gers (index, middle, ring, and little). Next, the dy-

namics of the hand is derived and feedback lineariza-

tion technique is used to obtain linear tracking error

dynamics. Then we propose the GA-based PID con-

trol, which uses GA to tune all PID parameters by

minimizing the tracking errors, for the five-fingeredprosthetic hand. The resulting overall hybrid system

incorporating both soft and hard control techniques is

simulated with practical data for the hand and found

to be superior to that using PID alone. We finally pro-

vide conclusions and future work.

2 Modeling

2.1 Trajectory Planning and Kinematics

The trajectory planning using cubic polynomial was

discussed in our previous work [1, 2, 5, 8, 9, 14] for

a two-fingered (thumb and index finger) smart pros-

thetic hand. The inverse and differential kinematics

of two-link thumb and three-link fingers were dis-

cussed in our previous publications [1,2,5,8,9,14] for

a two-fingered (thumb and index finger) smart pros-

thetic hand.

Recent Researches in Applications of Electrical and Computer Engineering

ISBN: 978-1-61804-074-9 57

For five fingers shown in Figure 1, XG, Y G, and

Figure 1: The Relationship between Global Coordi-

nate and Local Coordinates

ZG are the three axes of the global coordinate. The

local coordinate xt-yt-zt of thumb can be reached by

rotating through angles α and β to XG and Y G of the

global coordinate, subsequently. The local coordinate

xi-yi-zi of index finger can be obtained by rotating

through angle α to XG and then translating the vector

di of the global coordinate; similarly, the local coor-

dinate xj-yj-zj of middle finger (j = m), ring finger(j = r), and little finger (j = l) can be obtained by ro-

tating through angle α to XG and then translating the

vector dj (j = m, r and l) of the global coordinate.

2.2 Adaptive Neuro-Fuzzy Inference System

(ANFIS)

The inverse kinematics problems can be solved by us-

ing adaptive neuro-fuzzy inference system (ANFIS)

method [15] where the input of fuzzy-neuro system is

the Cartesian space and the output is the joint space.

ANFIS tunes the membership function and identifies

the coefficients by the backpropagation gradient de-scent and least-squares methods, respectively. Fig-

ure 2 (a) shows a two input first-order Sugeno fuzzy

model with two rules and Figure 2 (b) depicts the

equivalent ANFIS structure for all the computation

below. Sugeno-type fuzzy system has the following

Rule Base [15].

If x is A1 and y is B1, then f1 = p1x + q1y + r1.

If x is A2 and y is B2, then f2 = p2x + q2y + r2.

Here, x and y are inputs to constitute the premise pa-rameters A1, A2, B1, and B2 (Layer 1 in Figure 2

(b)). pi, qi, and ri (i = 1,2) are the consequent param-

eters. We evaluate the rules by choosing product∏

for T-Norm (Layers 2 and 3) which results in

wi = µAi(x) µBi

(y), i = 1, 2. (1)

X Y

X Y

x y

w1

w2

f p x q y r= + +1 1 1 1

f p x q y r= + +2 2 2 2

(a) A Two Input First-OrderSugeno Fuzzy Model with Two Rules

f =

w f +1 1 2 2w f

w +1 2w

= +w f1 1 2 2w f

(b) Equivalent ANFIS Structures

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5

S

x y

w1

wf11

w2

Desired output fd

f

x

y wf

22

x y

A1

A2

B1

B2

w2

BackpropagationAlgorithm

+

+

S-

+

w1

Figure 2: ANFIS Architecture: (a) A Two Input

First-Order Sugeno Fuzzy Model with Two Rules (b)

Equivalent ANFIS Structure [15]

Here, µAi(x) and µBi

(y) are designed fuzzy mem-

bership functions. Now after leaving the arguments

(Layer 4), we get the output f(x, y) by Rule Conse-

quences.

f(x, y) =w1(x, y)f1(x, y) + w2(x, y)f2(x, y)

w1(x, y) + w2(x, y). (2)

f (Layer 5) can be written as

f =w1f1 + w2f2

w1 + w2

= w1f1 + w2f2, (3)

where

w1 =w1

w1 + w2

, w2 =w2

w1 + w2

. (4)

2.3 Dynamics of the Prosthetic Hand

The dynamic equations of hand motion are derived viaLagrangian approach using kinetic energy and poten-

tial energy as [7,14,16] and can be written as below.

M(q)q + N(q, q) = τ , (5)

where M(q) is the inertia matrix and N(q, q) =C(q, q)+G(q) represents nonlinear terms, including


ISBN: 978-1-61804-074-9 58

Coriolis/centripetal vector C(q, q) and gravity vector

G(q). The dynamic relations for the two-link thumb

and the remaining three-link fingers are quite lengthy

and omitted here due to lack of space [14].

3 Control Techniques

3.1 Feedback Linearization

The nonlinear dynamics represented by (5) is to be

converted into a linear state-variable system using

feedback linearization technique [7]. Let us suppose

the prosthetic hand is required to track the desiredtrajectory qd(t) described under path generation or

tracking. Then, the tracking error e(t) is defined as

e(t) = qd(t) − q(t). (6)

Here, qd(t) is the desired angle vector of joints and

can be obtained by trajectory planning [1, 2, 5, 8, 14];q(t) is the actual angle vector of joints. Differentiat-

ing (6) twice, to get

e(t) = qd(t) − q(t), e(t) = qd(t) − q(t). (7)

Substituting (5) into (7) yields

e(t) = qd(t) + M−1(q(t)) [N(q(t), q(t)) − τ (t)] (8)

from which the control function can be defined as

u(t) = qd(t) + M−1(q(t)) [N(q(t), q(t)) − τ (t)] . (9)

This is often called the feedback linearization control

law, which can also be inverted to express it as

τ (t) = M(q(t)) [qd(t)− u(t)) + N(q(t), q(t)] . (10)

Using the relations (7) and (9), and defining state vec-tor x(t) = [e′(t) e′(t)]′, the tracking error dynamics

in the form of a linear system can be written as

x(t) =

[

0 I

0 0

]

x(t) +

[

0

I

]

u(t). (11)

3.2 GA-Based PID Hybrid Control

Figure 3 shows the block diagram of a hybrid GA-

based PID controller for the presented five-fingered

prosthetic hand with control signal as

u(t) = −KPe(t) −KI

∫

e(t)dt − KDe(t) (12)

with the proportional KP, integral KI, and derivative

KD diagonal gain matrices. We then rewrite (10) as

τ (t) = M(q(t))[qd(t) + KPe(t) + KI

∫

e(t)dt

+KDe(t)) + N(q(t), q(t)]. (13)

Figure 3: Block Diagram of the Hybrid GA-Based

PID Controller for 14-DOF Five-Fingered Prosthetic

Hand

Then we use GA to tune all gain coefficients KP,

KD and KD of PID controller. Figure 4 shows the

flowchart of GA and the procedure is briefly stated

below.

1. Define the GA parameters: include initial pop-ulation, population at the end of the first gener-

ation, number of chromosomes kept for mating,

mutation rate, and tolerance ε so on.

2. Create a homogeneous population: generate Nelements (chromosomes) and N is the initial

population.

3. Evaluate cost (fitness) function of each chromo-

some: calculate the fitness value of the ith mem-

ber in the population.

4. Select mate based on the performance of eachgene: create a new population from the current

population based on the ranking of the current

fitness value, e.g. determine which parents par-

ticipate in producing offspring for the next gen-

eration.

5. Reproduce the generation by crossover: use the

single or multiple crossover points to generate

new chromosomes that retain the good feature

and discard the bad feature.


ISBN: 978-1-61804-074-9 59

Figure 4: The Flowchart of Genetic Algorithm (GA)

6. Mutate: utilize the mutation rate which can ran-

domly mutate the gene to avoid falling into the

local minima area.

7. Repeat steps 3 to 6 until it reaches the maximum

number of iterations or stopping condition de-

fined by ε is satisfied.

4 Simulation Results and Discussion

We present simulations with a PID controller and GA-tuned PID controller for the 14 DOFs five-fingered

smart prosthetic hand grasping a rectangular object as

shown in Figure 5. All parameters of the smart pros-

thetic hand selected for the simulations are the same

as our previous works [3, 9]. All initial actual angles

are zero and the diagonal coefficients, KP, KI and

KD, for the PID controller alone are arbitrarily cho-

sen as 100. From the derived dynamic and control

models, after the parameters (KP, KI and KD) aredetermined, the torque matrix τ can be computed, and

then the squared-tracking errors eji (t) of the joint i of

the finger j are obtained. Therefore, the total error

E(t), which is a time-dependent function, can be de-

scribed as

E(t) =

∫ tf

t0

(eji (t))

2dt, (14)

Figure 5: A Five-Finger Prosthetic Hand Grasping a

Rectangular Object

where t0 and tf are initial and terminal time, respec-

tively. The tuned diagonal parameters (KP, KI and

KD) and the total error E(t) of PID controller by

GA are listed in Table 1. To study whether the tuned

Table 1: Parameter Selection of GA-Tuned PID Con-

troller and Computed Total Errors

Input Output

Fingers KP KI KD E(t)Case I [976,956] [779,279] [170,236] 0.3107

Case II [988,999] [ 78,848] [ 80,109] 0.1557

Case III [199,198] [127,157] [104,102] 0.8100

Index [794,398,960] [960,918,914] [15,59,242] 0.0465

Middle [794,398,960] [960,918,914] [15,59,242] 0.1003

Ring [794,398,960] [960,918,914] [15,59,242] 0.0465

Little [794,398,960] [960,918,914] [15,59,242] 0.0607

parameter range influences total tracking errors, we

design three different cases with altering lower and

upper bounds of tuned parameter ranges for two-link

thumb. Cases I, II, and III for the thumb represent

that the PID parameters KP, KI and KD are con-

stricted in three different bounded ranges [100,1000],

[50,1000], and [100,200], respectively. Figure 6 and

Figure 7 show that tracking errors and desired/actual

angles of joints 1 and 2 of PID and GA-based PIDcontrollers for Thumb. These simulations show that

the large ranges [100,1000] (Case I) and [50,1000]

(Case II) provide better results than the PID controller

parameters arbitrarily chosen as 100. However, the

small range [100,200] (Case III) gives worse result

than the PID controller alone. These results suggest

that the bigger parameter range, the smaller the to-

tal error. Cases I and II explain that GA finds some

parameter values ∈ [100,1000] and [50,100] escaping


ISBN: 978-1-61804-074-9 60

0 5 10 15 20−20

0

20

40

60

80

100

Time (seconds)

Tra

ckin

g E

rrors

of

Join

t 1 (

deg

rees

)

PID

GA+PID (Case I)

GA+PID (Case II)

GA+PID (Case III)

0 5 10 15 20−60

−50

−40

−30

−20

−10

0

10

20

Time (seconds)

Tra

ckin

g E

rrors

of

Join

t 2 (

deg

rees

)

PID

GA+PID (Case I)

GA+PID (Case II)

GA+PID (Case III)

Figure 6: Tracking Errors of Joint 1 (left) and Joint

2 (right) of PID and GA-Based PID Controllers for

Thumb

0 5 10 15 200

20

40

60

80

100

120

Time (seconds)

Tra

ckin

g A

ngle

s of

Join

t 1 (

deg

rees

)

PID

GA+PID (Case I)

GA+PID (Case II)

GA+PID (Case III)

0 5 10 15 20−80

−70

−60

−50

−40

−30

−20

−10

0

Time (seconds)

Tra

ckin

g A

ngle

s of

Join

t 2 (

deg

rees

)

PID

GA+PID (Case I)

GA+PID (Case II)

GA+PID (Case III)

Figure 7: Tracking Angles of Joint 1 (left) and Joint

2 (right) of PID and GA-Based PID Controllers for

Thumb

the local minimum area. Case III covers the value 100

in lower bound, but both total error and convergent

speed are even worse than PID alone, suggesting that

GA performs better for a large range, but is poor for

searching on the boundary. To further consider the

convergent speed, Case I gives smaller total error, but

does not improve its convergent speed when compar-

ing to PID control alone. Yet, Case II gives good totalerror and convergent speed. Case III gives poor total

error and convergent speed. Taken together, these re-

sults imply that the global minimum could be located

in the ranges [50,100] and [200,1000] and the parame-

ter ranges play an important role in GA tuning. Based

on these findings, we use the range [50,1000] for the

remaining three-link fingers. Figures 8 to 11 show the

simulations of PID and GA-based PID controllers for

the remaining three-link fingers.

0 5 10 15 20−10

−5

0

5

10

15

20

25

30

35

Time (seconds)

Tra

ckin

g E

rrors

(deg

rees

)

Joint 1 of Index Finger (PID)



Joint 1 of Index Finger (GA+PID)



0 5 10 15 200

10

20

30

40

50

60

70

80

90

Time (seconds)

Join

t A

ngle

s (d

egre

es)

Desired Joint 1

Desired Joint 2

Desired Joint 3

Actual Joint 1 (PID)



Actual Joint 1 (GA+PID)



Figure 8: Tracking Errors (left) and Joint Angles

(right) of PID and GA-Based PID Controllers for In-

dex Finger

0 5 10 15 20−20

−10

0

10

20

30

40

50

Time (seconds)

Tra

ckin

g E

rrors

(deg

rees

)

Joint 1 of Middle Finger (PID)



Joint 1 of Middle Finger (GA+PID)



0 5 10 15 200

10

20

30

40

50

60

70

80

90

100

Time (seconds)

Join

t A

ngle

s (d

egre

es)

Desired Joint 1

Desired Joint 2

Desired Joint 3








(right) of PID and GA-Based PID Controllers for

Middle Finger

5 Conclusions and Future Work

A hybrid control technique combining soft controlwith adaptive neuro-fuzzy inference system (ANFIS)

and genetic algorithm (GA) and hard control with

proportional-integral-derivative (PID) was presented

for a five-fingered smart prosthetic hand. The AN-

FIS is used for inverse kinematics and GA is used

for tuning the PID parameters with the objective of

minimizing the error squared between desired and ac-

tual angles of the links of the fingers. Simulation re-

sults for all the five fingers with GA-tuned PID con-troller showed superior performance compared to the

PID control alone. A real-time implementation of

this technique on a prototype of a prosthetic hand is

planned for future work.

Acknowledgements: The research was sponsored by

the U.S. Department of the Army, under the award

number W81XWH-10-1-0128 awarded and adminis-

tered by the U.S. Army Medical Research Acquisi-

tion Activity, 820 Chandler Street, Fort Detrick, MD

21702-5014. The information does not necessarily


ISBN: 978-1-61804-074-9 61

0 5 10 15 20−10

−5

0

5

10

15

20

25

30

35

Time (seconds)

Tra

ckin

g E

rrors

(deg

rees

)

Joint 1 of Ring Finger (PID)



Joint 1 of Ring Finger (GA+PID)



0 5 10 15 200

10

20

30

40

50

60

70

80

90

Time (seconds)

Join

t A

ngle

s (d

egre

es)

Desired Joint 1

Desired Joint 2

Desired Joint 3








(right) of PID and GA-Based PID Controllers for Ring

Finger

0 5 10 15 20−15

−10

−5

0

5

10

15

20

25

30

35

Time (seconds)

Tra

ckin

g E

rrors

(deg

rees

)

Joint 1 of Little Finger (PID)



Joint 1 of Little Finger (GA+PID)



0 5 10 15 200

10

20

30

40

50

60

70

80

90

Time (seconds)

Join

t A

ngle

s (d

egre

es)

Desired Joint 1

Desired Joint 2

Desired Joint 3








(right) of PID and GA-Based PID Controllers for Lit-

tle Finger

reflect the position or the policy of the Government,

and no official endorsement should be inferred. For

purposes of this article, information includes news

releases, articles, manuscripts, brochures, advertise-

ments, still and motion pictures, speeches, trade asso-

ciation proceedings, etc.

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