Chapter 4 Control of a Hydraulic Actuated Stewart Platform
4.2 Controller Design
4.4.2 Controller Performance
The second part is to show the performance of the observer-based backstepping control which is as shown in Figure 4.4. The actuator forces obtained from pressure sensors are used as input to the observer and are also used in the actuator controller simultaneously. In
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(b) Difference between estimation and Newton’s method
the experiments, feedback signals need to be filtered since the parameters of the hydraulic dynamics may enlarge the noises greatly and cause the control signal obstructed by the noises.
The motion in first experiment is to track a ramp signal with slope π for π seconds and then to regulate at 10 m along z-axis. Figure 4.9 shows the results of regulation in z-axis. In Figure 4.9(a), the estimated trajectory tracks and converges to the desired trajectory. The tracking error is shown in Figure 4.9(b) where the maximum error is about 0.007 m. To obtain good position tracking performance and to avoid larger vibration at the same time, we increase the position gains Kd, Kp and reduce the force gain Kτ. The comparison of the desired force and the actual force of one actuator is shown in Figure 4.9(c).
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Figure 4.9 Experimental results of regulation
In the second case, the platform first moves along z-axis for 5 cm from the initial position and is regulated at 0.52 m. The following is a sinusoidal motion with 5 cm amplitude and 2 rad/s frequency. The experimental results are shown in Figure 4.10.
Because the feedback signals are generated from the incorporated nonlinear observer, apparently some time lag between the estimated trajectories and the desired ones have been revealed. In this experiment, we also decrease the force feedback gain to avoid large vibration of the platform and obtain a smooth tracking trajectory. Up to now, this experiment successfully validates the feasibility of the proposed control law built upon the observer based real-time forward kinematics solver.
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-600 -400 -200 0 200 400 600 800 1000 1200
Time (sec)
Force (N)
(c) Desired and measured forces
Desired Measured
Figure 4.10 Position tracking of the sinusoidal wave
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0.48 0.5 0.52 0.54 0.56
Time (sec)
Position (m)
(a) Position tracking
Estimated Desired
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-0.015 -0.01 -0.005 0 0.005 0.01 0.015
(b) Position error
Time(sec)
Error (m)
Chapter 5
Washout Filter Design
Washout filter plays an important role in the integrated driving simulator system. It is the connection of the Virtual Environment (computer graphics, vehicle dynamics and physics simulation) and the motion platform. The motion of the vehicle in the virtual world does not directly input the motion platform. There should be a converter that transforms the motion of vehicle to simulator for there is limited space of the motion platform. The washout filter extracts necessary information to the motion platform. The schematic diagram can be seen in Figure 5.1. The Operator interacts with the virtual environment, and is affected by both virtual environment and motion platform in the visual and vestibular aspects respectively. Thus, the design of washout filter will decide how the operator’s perception from the aspect of vestibular system is.
Figure 5.1 Schematic diagram of the driving simulator
5.1 Classical Washout Filter
The motion cueing algorithm so called as washout filter is a transformation that transforms the motion from actual vehicles to simulators. The purpose of the washout filter is to provide the motion cues representing real human perception, besides the motion must stay in the bounds of the simulator workspace. One of its functions is to “wash out” the position of the simulator back to its neutral position. In order to make a simulation more realistic, linear accelerations and angular rates are exerted on the pilot by moving the platform on which the mock-up vehicle is located. This has to be accomplished without driving the simulator out of its workspace.
Many researches of washout filter have been represented in the last three decades.
Classical washout filter was the first scheme that has been proposed. The scheme is
composed of linear low-pass and high pass filters and is simplicity and easy to adjust. It is characterized by an empirically determined combination of linear high- and low-pass filters, whose break frequencies and damping ratios can be adjusted off-line through trial and error.
The scheme of the classical washout algorithm is shown in Figure 5.2.
Figure 5.2 Classical washout algorithm
The notations in Figure 5.2 are as follows:
fAA: body-axis components of the vehicle specific force at the cockpit reference point,
AA AA A
f =a −g ;
aAA: body-axis components of the vehicle acceleration at the cockpit reference point;
SI, I
a S : inertial components of the simulator reference point acceleration and position;
a : intermediate acceleration; c 1, 2, L
f f f : intermediate acceleration;
, A, I
g g g :gravitational acceleration expressed as a scalar and as component in the body and
inertial frames. gI =
[
0 0 g]
T;The inputs to the algorithm are the vehicle specific forces, respectively, along the three
axes in the body coordinate frame, defined as fAA =aAA−gI, and the vehicle angular rates about three axes, ωAA. The vehicle’s specific forces fAA are filtered by a high-pass filter to yield the simulator’s translational accelerations which are to be double integrated to obtain displacement of the platform, S. However, the longitudinal and lateral specific forces are also passed through low-pass filters, scaled, and rate-limited to produce pitch and roll tilt angles, respectively. The purpose of this “tilt-coordination” mechanism is to orient the gravity vector in the simulator with regard to the operator in the same way that the low-frequency force is oriented with regard to the vehicle, thus allowing sustained vehicle accelerations to be simulated. On the other hand, the vehicle’s angular motion is high-pass filtered to yield the high-frequency component of the simulator’s angular motion, while the tilt mechanism described above supplies the low-frequency component from the specific force.
One of the drawbacks of this scheme is its fixed parameters. The parameters must be designed for worst-case maneuvers and often yields the minimum motion under gentle maneuvering. For different operators, the perception errors between individuals cannot be made up for the fixed parameters. To solve this problem, various algorithms have been proposed, such as the adaptive washout filter [34], the optimal design [35], or washout filters that are composed of an empirically determined combination of filters. Unlike the classical framework, some of the parameters in these filters can be systematically varied to
achieve better performance and workspace usage.
5.2 Optimal Washout Filter
The optimal washout filter design applies a scheme different from the classical one. In Figure 5.3, two paths are separated to compare the perceptions due to actual vehicle and simulator, respectively. The upper path symbolizes that the actual vehicle input u is A transformed by the human vestibular model to the signal y , which represents the A sensation of the operator driving the actual vehicle. In the lower path, the vehicle input u A is transformed by the washout filter to the reference input of the simulator u , then to the S output of the simulator, u , and finally through the vestibular model to the sensation of the P simulator's operator, namely, y . To determine the structure and the parameters of the S washout filter means to find a mapping from the actual vehicle input u to the simulator A input u . Here, we discuss the ideal case where the control error could be neglected to S results in the output of the simulator, u , which intends to approximate the command P u . S To compare the signals from two paths will lead to the sensation error e which is meant to be minimized in our optimal design.
e
u
Au
S+
−
u
PyA
y
SFigure 5.3 Optimal washout filter design problem
5.2.1 Model for the Vestibular System
The washout filter model can be separated to 4 sub models: surge/pitch, sway/roll, heave, and yaw. The first two models are one acceleration paired with one angular velocity and the derivations are similar. The third and forth models consider only independent acceleration and angular velocity respectively. Here, we use a linear model for the
vestibular system (surge/pitch) as shown in Figure 5.4, where ˆf and x θˆɺ are the sensed specific force and the angular velocity, respectively, a and x θɺ are the actual vehicle inputs, i.e., the longitudinal (surge) acceleration and the pitch angular velocity, and f is x the specific force stimulus. Figure 5.5 shows the influence of pitch rotation on surge's specific force. For the centroid of the motion platform, the specific force is
cos sin
x x x
f =a θ+g θ ≅a +gθ (5.1)
where θ is assumed to be small so that sinθcan be approximated by θ and cosθ by 1,
respectively. In Figure 5.4, the sensed specific force ˆf is related to the specific force x
transform of (5.1), and we obtain
( ) ( ) 1 ( )
x x
f s a s g s
sθ
= + ɺ (5.3)
which is substituted into (5.2) to result in
2
Figure 5.5 Influence of rotation on specific force in longitudinal mode
which then can be realized in state space form as
ˆ
ˆ 2
expressed in state space form as
ˆ
where xsc is the state vector of the semicircular canals model, and
[ ]
The representations in (5.6) and (5.9) can be integrated to form a single representation as
ˆ
5.2.2 Integrated System
We assume the human vestibular model can be applied to both the vehicle operator and simulator operator as shown in Figure 5.3. Then, the vestibular state error is defined as
e S A
x =x −x where x and S x are the vestibular states for simulator and vehicle, A respectively. Then, the pilot's sensation error e can be calculated in the form
e ve e ve S ve A
Owing to the limited workspace of the simulator, the motion of the platform should be constrained. For this purpose, we want to incorporate the platform states into the cost function. Here, the velocity, displacement, and rotation angle of the platform are considered to be accessable. The additional terms are included in the state equation:
c c c c S
The vehicle input u consists of filtered noise, and can be represented as A
n n n n
where xn is the filtered white noise state, ω is the white noise, and
each degree-of-freedom. By combining (5.11), (5.12), and (5.13), we can obtain the desired
system equation combined states, and the combined system matrices A, B, C, D and H are then given as
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5.2.3 Derivation of Optimal Design
The cost function for the optimal design is defined as
{
0}
From the cost function, it can be seen that the design considers reducing the sensation error and the limited workspace of the platform.
The system equation and cost function can be transformed to comply with the optimal control formulation as:
The cost function is minimized when
1
' 2 T w w
u = −R B P x− (5.17) where Pw is the solution of the algebraic Riccati equation
1
0 .
Substitution of (5.20) into (5.21) then leads to
1 2 3 following equation can readily be derived in the Laplace domain:
( ) ( ) ( )
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
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(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.
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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.
Bibliography
[1] D. Stewart, "A Platform with Six Degrees of Freedom," Proceeding of the Institute of Mechanical Engineers, vol. 180, pp. 371-386, 1965.
[2] L. R. Harris, M. Jenkin, and D. C. Zikovitz, “Visual and non-visual cues in the perception of linear self motion,” Experimental Brain Research, 2000.
[3] V. E. Gough and S. G. Whitehall, "Universal type test machine," Proceeding of 9th International Technical Congress FISITA, pp. 117–137, 1962.
[4] G. Lebret, K. Liu, and F. L. Lewis, "Dynamic analysis and control of a Stewart platform manipulator," Journal of Robotic Systems, vol. 10, pp. 629–655, 1993.
[5] Z. Geng, L. S. Haynes, J. D. Lee, and R. L. Carroll, "On the dynamics model and kinematics analysis of a class of Stewart platform," Robotics and Autonomous Systems, vol. 9, pp. 237–254, 1992.
[6] W. Q. D. Do and D. C. H. Yang, "Inverse dynamic analysis and simulation of a platform type of robot," Journal of Robotic Systems, vol. 5, pp. 209-227, 1998.
[7] B. Dasgupta and T. S. Mruthyunjaya, "Closed-form dynamic equations of the General Stewart platform through the Newton–Euler approach," Mechanism and Machine Theory, vol. 33, pp. 993–1012, 1998.
[8] B. Dasgupta and T. S. Mruthyunjaya, "A Newton–Euler formulation for the inverse
[8] B. Dasgupta and T. S. Mruthyunjaya, "A Newton–Euler formulation for the inverse