Designing hedge algebraic controller and optimizing by genetic algorithm for serial robots adhering trajectories

gn relationship as shown in Table 2.
Table 2. The sign relationship of hedges and generating elements
V L N P
V + + − +
L − − + −
Interpolation based on the Semantic Distance Weighting method (ISDMd) is selected
as in [28]. This is a linear interpolation method that allows interpolation in n-dimensional
space. It ensures the systematic monotony in case of interpolation/extrapolation and needs a
small number of calculations that meets the requirements of real-time control of the system.
3. OPTIMIZATION OF PARAMETERS BY GENETIC SOLUTION
The genetic algorithm is a method of optimal random search that follows the evolution
and selection of biological populations in nature [29]. Operations in algorithm-based evo-
lution include crossover, mutation, and selection. Each individual is expressed simply as
a chromosome consisting of multiple gene segments. Each gene segment is encoded for an
optimal parameter. Then each individual is a solution of the problem with an optimal set
of parameters. After an evolutionary process (repetition) is large enough, individuals will
gradually adapt to the adaptive conditions evaluated by the fitness function.
The fuzziness parameters need to be optimized for the controller.
For fuzzy controllers, the variable domain of the input and output is usually symmetric.
When expressing the value on this domain in language, the rank of Zero (ZE) has semantics
that refers to the real value of 0. This is the element corresponding to the neutral element
W of hedge algebra, v(W ) = θ. When mapping all linguistic values to the semantic domain
in the range [0, 1], semantic value (ZE) = 0.5. Therefore, according to (19), we choose fixed
variables of values fm (N) = v(ZE) = 0.5.
The set of hedges in hedge algebras is built with only 2 hedges, V (Very) and L (Little),
so we only need to optimize µ (L) =α (because of µ (V ) =β= 1−α). So we only need to
optimize 3 parameters αe, αce and αu corresponding to 3 variables e, ce, and u.
The objective function is chosen according to the absolute value integration standard of
the tracking error (IAE)
fitness =
n∑
k=1
|e (k)| → min (27)
where, e(k) is the sample of deviation data at the simulation cycle k, n is the total number
of data samples of a simulation run.
In Matlab, GA is an available tool that allows us to easily use it. In this study, we use
the GA() function in Matlab with gene encoding using real numbers of type double. The
values set for GA include:
278 NGUYEN TIEN DUY, VU DUC VUONG
• Population size: Population Size = 100;
• The number of Generation: Generation = 4*Population Size;
• Maximum limit for searching time: Time Limit = 86400, (24 hours);
• The target function is used as in formula (27).
Results are the optimal parameters set for the controller as shown in Table 3.
Table 3. The optimal parameters of HAC based on GA
AX e : αe = µ(Le) AX ce : αce = µ(Lce) AY : αu = µ(Lu)
0.520936 0.508062 0.499074
Figure 5. Surface of input/output relationship S3 of optimal HAC
With optimal fuzziness parameters found in Table 3, the SQMs function (19) - (22) is
used to calculate the semantic value of the elements in LRBS in Table 1, we obtain QRBS
and the surface S3 of HAC correspondingly. Figure 5 is an input-output relationship of the
rule based system corresponding to the optimal QRBS.
4. SIMULATION RESULTS AND DISCUSSION
The diagram simulating the control system of 2-DOF serial robot with HAC on the
Matlab-Simulink environment as shown in Figure 6. In the simulation diagram, we use 2
sets of HAC with common parameter sets to control for 2 motors with the same parameters.
The control program code is written as Level-2 M-file S-Function. Blocks DC1 and DC2
are mathematical representations for brushless DC motors. These selected motor are the
same. The “Robot” block calculates the inverse dynamics problem. “Trajectory” is a block
that generates reference data for moving robots. The “forward kinematic” block performs
forward kinematic computation to convert data from joint space to workspace.
DESIGNING HEDGE ALGEBRAIC CONTROLLER AND OPTIMIZING BY... 279
Some simulations with a 2-DOF serial robot arm which moves in the horizontal plane
driven by 2 actuators are implemented. The numerical simulations are carried out in Simu-
link. These trjectories include circle, spriral and square shapes. The results are shown in
Figure 7 to Figure 12.
Figure 6. Diagram of 2-DOF serial robot arm simulation to HAC
Circle trajectory
In this simulation, the center of the end effector will be moved along a circular trajectory,
The trajectory has a center at (xC; yC) = (0.45;−0.45) [m] and radius R = 0.2 [m]. This
trajectory is chosen to be in the task space of the robot. The starting point to move is
(0.6022; -0.5127). Simulation results are shown in Figure 7 and Figure 8.
Figure 7. The desired path vs. the actual path
of the end effector with the circle path
Figure 8. Tracking errors of x, y coordinates with
the circle path
Spiral trajectory
In this simulation, the center of the end effector will be moved along a spiral trajectory,
The trajectory has a center at (xC; yC) = (0.45;−0.45) [m] and the boundary radius R = 0.2
[m]. This trajectory is chosen to be in the task space of the robot. The starting point to
280 NGUYEN TIEN DUY, VU DUC VUONG
move (0.6022; -0.5127). Simulatinon results are shown in Figure 9 and Figure 10.
Figure 9. The desired path vs. the actual
path of the end effector with the spiral path
Figure 10. Tracking errors of x, y coordinates
with the spiral path
Square trajectory
In this simulation, the center of the end effector will be moved along a square trajectory
with the following coordinates
A(0.65,−0.45); B(0.45,−0.25); C(0.25,−0.45); D(0.45,−0.65) (28)
is at the starting point to move (0.6022; -0.5127). Simulation results are shown in Figure 11
and Figure 12.
Figure 11. The desired path vs. the actual
path of the end effector with the square path
Figure 12. Tracking errors of x, y coordinates
with the square path
Simulation with noise
DESIGNING HEDGE ALGEBRAIC CONTROLLER AND OPTIMIZING BY... 281
In this simulation, the center of the end effector will be moved along a spiral trajectory
with the effection of noise. The result is shown in Figure 13.
Figure 13. The desired path vs. the actual path of the end effector moves along the spiral path with
noise
It can be seen in Figure 7, the end effector of the robot moves from the initial position at
the coordinate of (0.6022; -0.5127). The control trajectory of HAC is closer to the reference
trajectory than the control trajectory by PID-Controller. Figure 8 shows the tracking error
in the x and y directions by time is rapidly decreasing to the value which is close to 0
in the range of 0.5 [s] with one small overshoot/undershoot. Still at the original position
at (0.6022; -0.5127), perform simulations with more complex trajectories such as Spiral
Trajectory (Figure 9, Figure 10) and Square Trajectory (Figure 11, Figure 12). The control
trajectory of HAC always gives deviations and response time is better than the control
trajectory of the PID-controller. Simulated with the case of white noise impacting the
angles q1 and q2 at the output of the robot with the spiral reference trajectory, the control
trajectory of the HAC has a small fluctuation but does not lose control and still close to the
trajectory reference (see Figure 13). The results show the efficiency of the hedge algebras
controller optimized by GA with tracking control 2-DOF serial robot arm.
5. CONCLUSION
In this study, we designed HAC for the 2-DOF serial robot arm. The controller has a
fairly simple structure with a rule control system consisting of only 25 rules. The structure
of hedge algebras for the input-output variables has only two hedges, including a negative
hedge - Little and a positive hedge - Very. The controller has only 3 parameters that measure
the fuzziness of the hedge. The application of available GA in Matlab to optimize these
parameters is very effective. With the optimal set of parameters, through the simulation, it
was found that HAC worked very well.
This result shows that the potential of HAC application in robot control is very wide and
282 NGUYEN TIEN DUY, VU DUC VUONG
promising. In the near future, we will expand the research on HAC applications for more
complex robots such as robots with 3 degrees of freedom or higher, parallel robots, etc.
Reference trajectories will be set up with many complex forms, with major turning points.
Parameter optimization will also be performed in parallel and separately for each controller.
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Received on August 23, 2020
Revised on April 25, 2020