Chapter 1 Introduction Introduction
4.4 Numerical Simulation and Experimental Validation
4.4.2 Validation of Adaptive SMC Using SMP II
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Table 4.5. List of Specifications of Mechanical and Control Parameters.
Mechanical parameters
Inertia mC = 500kg, Ia, It = 344kg•m2, g = 9.8m/s2, PG = [0.0789 ‒0.0173 ‒0.2112]T m Radius rw = 0.075m, rs = 0.15m, RC = 1.2m
Limit qhome = 1.055m, qreverse limit/qforward limit = 1.075/0.467m Control parameters (Orientation)
k1 = 2, k2 = 1, k3 = 1.5, k4 = 3, k5 =0.01,η1 = 207, η2 = 209, c = 1.5 Control parameters (Position)
k1 = 3, k3 = 3, k4 = 0.01, k5 =0.01,η1 =97, η2 =109, c = 2
A urethane ball as the small sphere has been used to minimize slipping motion and assure high rigidity. Each urethane ball with a hardness of 90~95 can support heavy loads of at least 1000kg and endure friction force during repetitive operations. The motors to rotate each small sphere are installed with a 20:1 gear ratio to increase torque; MSMF152LG5 and ABR115-020 are chosen for the motor and gearbox, respectively. The same brake-typed motor without a gearbox is used for translational motion. The motor is connected to the ball screw actuator, PSA210H-600, to move the linear stage.
Furthermore, linear guides, SBL-55L in Fig. 4.15, support both sides of the linear stages, helping the linear movement. Three proximity sensors, EE-SX674 are installed on each ball screw actuator for reference as homing and safety to limit the range of motion such as qforward limit and qreverse limit in Table 4.5. The motion capability for the SMP is calculated from the kinematics with the mechanical limits in Table 4.6. All eight motors are connected to each servo driver with encoder feedback; MDDLT55SF is used for the servo driver. A real-time embedded programmable controller, NI-cRIO, is used as the main controller for all motor drivers simultaneously. The sampling period is 0.02s for the controller. The average calculation burden for each DOF move is lower than 1ms so that the platform can be controlled without delay due to the calculation burden. The eight modules are used to interface stepper servo drivers and control angular position, velocity, acceleration, and deceleration of each motor. It also has an incremental encoder for position feedback and a full set of motion, including inputs from the home, forward limit, and reverse limit switches. The mechanical parameters for experiments are detailed in Table 4.5.
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Table 4.6. Motion capability of full-scale SMP II.
Rotational motion ϕ θ ψ
Max. orientation (deg) ±180 ±90 ±180
Max. angular velocity (deg/s) ±173.1 ±173.1 ±115.7
Translational motion x y z
Max. position (m) ±0.3 ±0.3 0.42
Max. linear velocity (m/s) ±0.8 ± 0.8 ±0.88
4.4.2.2 Orientation control including unlimited rotation
The orientation of the cockpit sphere is controlled by both classical proportional-integral- differential (PID) and the SMC. An input for tracking control of the cockpit sphere is sinusoidal functions, [90sin(0.08πt)+50 0 0]T deg with zero initial condition. The orientation of the cockpit sphere is estimated from sensor fusion with both optical sensors and IMUs. Fig. 4.16(a) and (b) show the comparison of the trajectory tracking performance from the PID controller and the SMC. The maximum error (eMax) and RMSE is 4.07 deg and 1.36 deg for the SMC, respectively, but 13.15 deg and 8.24 deg for the PID controller, as shown in Table 4.7. The RMSE and eMax for orientation control are evaluated in a steady-state response. The RMSE for orientation control is evaluated as follows:
2 2 2
1
( ) ( ) ( )
RMSE N
k
e k e k e k
N
.As a result, the SMP with SMC realizes more natural motion minimizing tracking error against disturbances and uncertainties than the PID controller.
In addition, unlimited rotation for ϕ and ψ with the angular velocity of 10 deg/s is controlled by the SMC from the experiments in Fig. 4.17. The eMax and RMSE are 1.82 deg and 1.25 deg, respectively, for ϕ tracking control in Table 4.7. The unlimited rotation of ψ is also ensured with eMax and RMSE of 4.43 deg and 2.84 deg. As a result, the percentage of RMSE during 360 deg rotation is smaller than 1%, and it can be demonstrated the feasibility of control for unlimited rotation.
100 (a)
(b)
Figure 4.16. Comparison of tracking performance for orientation control. (a) Tracking results (PID).
(b) Tracking results (SMC).
Table 4.7. Orientation Control Performance for PID and SMC.
sinusoidal input PID SMC
8.24/13.15 1.36/4.07 unlimited rotation (SMC) ϕ tracking ψ tracking 1.25/1.82 2.84/4.43 RMSE/eMax (deg)
(a) (b)
Figure 4.17. Comparison of tracking performance for orientation control. (a) Unlimited rotation for ϕ (SMC). (b) Unlimited rotation for ψ (SMC).
0 25 50 75 100 125
-50 0 50 100 150 200
0 25 50 75 100 125
-10 0 10 20
0 25 50 75 100 125
-2 0 2 4
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As the cockpit sphere linearly moves, undesired rotation is incurred due to kinematic coupling so that rotational motion should be controlled simultaneously. The desired command inputs for position and orientation are set as [xd yd zd]T = [0.15sin(0.04πt) 0.15cos(0.04πt) 0]T m and [ϕd θd ψd]T = [0 0 0]T, respectively. The cockpit sphere position can be computed from the motor encoder in (2) and (3). The RMSE and eMax for the control are similarly evaluated in a steady-state response, and the RMSE for position control is computed in the same way:
2 2 2
1
( ) ( ) ( )
RMSE N x y z
k
e k e k e k
N
.Fig. 4.18 shows the comparison of position control along x- and y-axes between PID and SMC. The eMax and RMSE are 0.011m and 0.0055m for the SMC, respectively, smaller than 0.022m and 0.014m for the PID controller, as shown in Fig. 4.18 and Table 4.8. The experimental results verify better tracking performance against uncertainties and disturbances compared with the PID controller.
(a) (b)
Figure 4.18. Comparison of tracking performance for xy position control. (a) Tracking result. (b) Tracking error.
Table 4.8. Position Control Performance for PID and SMC.
controller PID SMC
xy tracking 0.014/0.022 0.0055/0.011 xz tracking 0.022/0.027 0.0037/0.0040 RMSE/eMax (m)
0 20 40 60 80
-0.05 0 0.05 0.1 0.15
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The undesired coupled rotation due to position control should be compensated for orientation control. The SMC is implemented to both position and orientation control, while the ADC in (4.21) is utilized to compensate for the coupled rotation. Fig. 4.19 shows the orientation induced from the position control, comparing three cases: i) PID without ADC, ii) SMC without ADC, and iii) SMC with ADC. The RMSE in orientation tracking is about 0.49 deg for the SMC with the ADC, but 1.78 deg and 1.06 deg for the PID and the SMC without the ADC, respectively. The detailed results are summarized in Table 4.9. The results indicate that the ADC improves the robustness of SMC against coupled rotation.
Thus, the cockpit sphere can be controlled independently between translational and rotational motion by applying the SMC with the ADC.
(a) (b)
Figure 4.19. Comparison of results of compensated coupled rotation. (a) |eϕ|. (b) |eθ|.
Table 4.9. Orientation Control Performance during Position Control.
PID w/o ADC SMC w/o ADC SMC w/ ADC
1.78/2.02 1.06/1.5 0.49/0.67
RMSE/eMax (deg)
Fig. 4.20 shows the comparison of the position control along x- and z-axes between the PID and the SMC. The desired command inputs for position and orientation are given as [xd yd zd]T = [0.15sin(0.04πt) 0 0.1cos(0.04πt)]T deg and [ϕd θd ψd]T = [0 0 0]T, respectively. Similar to the orientation, the SMC has better performance with 0.004m and 0.0037m along all axes than the PID with 0.027m and 0.022m of eMax and RMSE shown in Figs. 4.20(a) and (b) and Table 4.8, respectively.
Moreover, Figs. 4.20(c) and (d) show C1 and C2 in (2.67) and (2.68) for the SMC compared to the PID.
0 20 40 60 80
0
1
2
3
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The smaller output indicates less slip occurrence and power loss by four contacts between the SPWs and the cockpit.
(a) (b)
(c) (d)
Figure 4.20. Comparison of tracking performance for xz position control. (a) Tracking result. (b) Tracking error. (c) PID result. (d) SMC result.
0 20 40 60 80
0 2 4 6 8 10-3
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