Chapter 5 Washout Filter Design
5.3 Fuzzy Compensation Design
e u
AuS +
−
uP
yA
yS
+
+
uF
Figure 5.6 Optimal washout filter design with fuzzy control compensation
The scheme of the fuzzy control based optimal washout filter proposed in this paper is shown in Figure 5.6. A feedback loop was added in the original optimal scheme, Figure 5.3.
Considering the sensation errors e of the operator and the time derivative of the input u , A the fuzzy logic produces an additional compensating signal u to eliminate the errors. The F human body can not precisely tell the quantity of the motion, but only can discriminate large, medium, or small in different directions. In the fuzzy logic, the motion sensed by the human body in different direction is divided to 5 conditions: NL (negative large), NS (negative small), ZO (zero), PS (positive small), and PL (positive large). The classification of the derivation of input u is the same. According to the direction and magnitude of the A sensation errors e and the time derivative of the input u , we can determine the variation A
of simulator input u by the fuzzy logic. The table of the fuzzy control rules is shown in F Table 5.1.
Table 5.1 Fuzzy control rules
u F uɺ A
e NL NS ZO PS PL
NL ZO PS PS PL PL
NS NS ZO PS PS PL
ZO NS NS ZO PS PS
PS NL NS NS ZO PS
PL NL NL NS NS ZO
For example, when uɺ is PL (large in positive) and A e is NL (large in negative), u is F determined as PL. Because e= yS −yA and uS ∝yS , the variation u should be in F opposite direction to restrain the error. Nevertheless, when the time derivative of u is A positive-large, the error is negative-large, namely uɺA ∝ −e, so the variation of u should S be the same direction to restrain the error. Figure 5.7 shows the Membership functions in the fuzzy logic.
(a)
(b)
(c)
Figure 5.7 Membership functions (a) Sensation error (b) Time derivative of u (c) Output A uF
5.4 Simulation Results
The simulations present linear acceleration in surge and angular velocity in pitch, which are coupled as in Figure 5.5. The parameters of human perception model are shown by Table 2.1. The numerical values of other parameters are shown as follows:
1 0.01 rad/s, 2 0.025 rad/s
The elements W sij( ), ,i j=1, 2 turn out to be transfer function of nine dimensions, which
corresponds to the dimension of the matrix A. This transfer function maps the actual motion to simulator motion which are 2×2 signals. The selection of the parameters Qw, Rw, Rc
depends on the emphasis on which part, and determines the coefficients of the washout filter.
We compare the sensation errors of the classical washout filter, the optimal design, and the optimal one with fuzzy compensation. The case is with the actual motion that accelerated forward at 2 m/s2 for 10 sec and then held the constant speed for 25 sec. The results of the simulation are plotted in Figures 5.8~5.11. Figures 5.8(a) and (b) show the comparison of the actual and simulator inputs for linear acceleration and angular velocity, respectively. The washout filter transforms the actual input to simulator input. In Figure 5.8(a), the transformed acceleration decreases rapidly and vibrates substantially in the change of actual acceleration (dash). The transformation of washout filter conserves the high frequency motion and filters others to keep the platform moving in the workspace.
Note that the signal with fuzzy feedback (dotted) has peaks that are closer to 2 m/s2 than optimal-only (solid) washout filter. In Figure 5.8(b), although the actual angular velocity (dash) holds on zero, the simulator input (solid) produces motions to present the low frequency linear acceleration motion. This phenomenon can be revealed by comparing Figures 5.8(a) and (b). In the simulations, the fuzzy feedback was only used on the
acceleration signal.
(a)
(b)
Figure 5.8 Actual and Simulator inputs (a) Acceleration (b) Angular velocity
0 5 10 15 20 25 30 35
(a)
Figure 5.10 Sensation errors of classical, optimal, and fuzzy-based optimal washout filters
Figure 5.9 shows the sensation of actual vehicle operator y (dash) comparing to the A sensation of simulator operator y referred to Figure 5.3. The sensed accelerations of S optimal (solid) and fuzzy-based optimal (dotted) washout filters are both shown in Figure 5.9(a), and a room-in figure is shown in 5.9(b). In Figure 5.9(b), obviously, the trajectory with fuzzy compensation is more close to actual sensation than optimal algorithm. Figure 5.10 shows the sensation error e of the classical (dash), optimal (solid), and fuzzy-based optimal (dotted) washout filters. The RMS of sensation error for the acceleration of optimal design is 0.0237 which is smaller than that of classical design 0.1107. The fuzzy-based optimal washout filter can further decrease the RMS to 0.0046, which is 19% and 4% of optimal and classical washout filters respectively. The results reveal the fuzzy control can compensate the optimal design and perform more realistic human perception signals.
0 5 10 15 20 25 30 35
Chapter 6 Conclusion
The vehicle simulator is an integrated system that is composed of a motion platform system, computer graphs, 3-dimensional physics, and the motion cue algorithm. The motion platform mentioned here is the Stewart platform which includes the hydraulic actuators and the parallel structure. In this thesis, the issues discussed are the system of Stewart platform, including dynamics and controller, and the motion cue algorithm. The main results and the future works can be summarized as below:
The first contribution made in this thesis is to propose an output feedback posture control via observer based-forward kinematics for the Stewart platform. The nonlinear observer provides a practical method to solve the forward kinematics problem in real time, i.e., alleviating computational burden of the numerical iteration and the complexity of the elimination algorithm. After the posture state of the moving platform is estimated from the observer, it further generates some observed states which are fed back to the sliding mode controller to achieve successful posture control. The Lyapunov stability analysis ensures convergence of the trajectory tracking error of the whole system. Not only the simulation results reveal that the actual motion trajectories will converge to the desired ones, but also
the experiments validate the feasibility of this control scheme promisingly in practical applications.
Another contribution is to develop a backstepping controller incorporating a nonlinear observer based real-time forward kinematics solver to tackle the overall hydraulic manipulator which consists of : 1) the mechanical system, and 2) the hydraulic actuators.
The nonlinear dynamics of the hydraulic actuator makes the design of the control method different from the ones of the standard robot manipulators. Thus, to consider the hydraulic dynamics in the control design of the robot manipulator becomes inevitable. So far, most researches on control of Stewart platform fail to consider the integration of the mechanical part and the hydraulic dynamics. One critical issue to combine these two sub-systems is that the state in the platform dynamics (task-space) is different from that in the actuator dynamics (joint-space) and the forward kinematics problem is difficult to solve in real time.
An observer based solution is proposed in this work to make the posture feedback feasible.
Specifically, the estimated posture state is fed back to the backstepping controller to achieve the control of the overall system. We also use the Lyapunov stability analysis to ensure convergence of the trajectory tracking error of the entire system. Simulations and experiments reveal that the actual motion trajectories will converge to the desired ones. The feasibility of this control scheme has been validated in practical applications.
Compared with conventional control on Stewart platform, the hereby proposed new schemes control the platform posture directly without requiring conversion of the posture command to the leg length commands in the first place. As a result, tracking the desired posture becomes real control objective.
The motion cue algorithm, namely washout filter, is the connection between the motion platform and the virtual environment. The optimal design with fuzzy compensation proposed here has been applied to minimize the sensation error and to conserve the workspace. By building the human vestibular system, we can obtain the sensed acceleration and angular velocities which are defined as the human perception. For our designing purpose, the cost function contains the pilot’s sensation error, the motion of platform, and the input. The fuzzy controller helps to compensate the drawback of fixed parameters of the optimal algorithm. With sensation error and the time derivative of input feedback, the fuzzy logic can provide an adjustment term to further eliminate the perception errors. In the simulations, the sensation errors of classical, optimal, and fuzzy-based optimal washout filters are compared to show the performance of this algorithm.
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