hybrid genetic algorithm pid control for a five-fingered smart
TRANSCRIPT
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
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
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.
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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
Recent Researches in Applications of Electrical and Computer Engineering
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.
Recent Researches in Applications of Electrical and Computer Engineering
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
Recent Researches in Applications of Electrical and Computer Engineering
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 2 of Index Finger (PID)
Joint 3 of Index Finger (PID)
Joint 1 of Index Finger (GA+PID)
Joint 2 of Index Finger (GA+PID)
Joint 3 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 2 (PID)
Actual Joint 3 (PID)
Actual Joint 1 (GA+PID)
Actual Joint 2 (GA+PID)
Actual Joint 3 (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 2 of Middle Finger (PID)
Joint 3 of Middle Finger (PID)
Joint 1 of Middle Finger (GA+PID)
Joint 2 of Middle Finger (GA+PID)
Joint 3 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
Actual Joint 1 (PID)
Actual Joint 2 (PID)
Actual Joint 3 (PID)
Actual Joint 1 (GA+PID)
Actual Joint 2 (GA+PID)
Actual Joint 3 (GA+PID)
Figure 9: Tracking Errors (left) and Joint Angles
(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
Recent Researches in Applications of Electrical and Computer Engineering
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 2 of Ring Finger (PID)
Joint 3 of Ring Finger (PID)
Joint 1 of Ring Finger (GA+PID)
Joint 2 of Ring Finger (GA+PID)
Joint 3 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
Actual Joint 1 (PID)
Actual Joint 2 (PID)
Actual Joint 3 (PID)
Actual Joint 1 (GA+PID)
Actual Joint 2 (GA+PID)
Actual Joint 3 (GA+PID)
Figure 10: Tracking Errors (left) and Joint Angles
(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 2 of Little Finger (PID)
Joint 3 of Little Finger (PID)
Joint 1 of Little Finger (GA+PID)
Joint 2 of Little Finger (GA+PID)
Joint 3 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
Actual Joint 1 (PID)
Actual Joint 2 (PID)
Actual Joint 3 (PID)
Actual Joint 1 (GA+PID)
Actual Joint 2 (GA+PID)
Actual Joint 3 (GA+PID)
Figure 11: Tracking Errors (left) and Joint Angles
(right) of PID and GA-Based PID Controllers for Lit-
tle Finger
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