Chapter 1 Introduction Introduction
5.5 Numerical Simulation and Experimental Validation
5.5.1 Numerical Simulation
The numerical simulations validate that the proposed PPC can achieve desired tracking performance showing better robustness than existing methods. Moreover, the application to the motion platform demonstrates the feasibility of the actual application. The control tracking performance is evaluated from the orientation control for the platform. The orientation of the cockpit sphere in Fig. 5.1(a) is steered into the desired trajectory, and it is set to be as xd(t) = [xd,1T xd,2T]T, where xd,1(t) = [30e‒
t/20sin(0.1πt) 60sin(0.1π(t+10)) 30sin(0.2πt)]T deg and xd,2xd,1.Then, the system dynamics (5.2) can be represented as
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3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3
0 0 0
( ) ( ) ( ) ( )
0 0
t I t t h t
I I
x x u d (5.61)
where x = [x1 x2 x3 x4 x5 x6]T = [ ] Twith an initial condition as x0 = [06×1]. The control input is represented as u(t) = [J]‒1U(t), where J = diag(Ia,It,It). For the simulation with motion platform, it should be considered that the disturbance d is composed of viscous friction, gravitational torque, and constant disturbance. The viscous friction df is incurred in rolling contact between the cockpit and small spheres in Fig. 1. It is also unavoidable that uncertain gravitational torque dG occurs from the eccentricity caused by equipment and human inside the cockpit. The constant disturbance dc
can also be incurred by uncertain dynamics. Thus, the disturbance d is represented as d = df + dG + dc, where df Cf[ ] T and dG = [0 ‒glcosθ 0]T. In the simulation, the input delay h is chosen equal to 1s twice the larger value presented in [81]‒[84]. The mechanical and control parameters are listed in Table 5.2.
Table 5.2. List of Specifications of Mechanical and Control Parameters for Numerical Simulation.
Mechanical parameters
Moment of inertia of cockpit: Ia = 344 kg•m2, It = 344 kg•m2 Viscous frictional coefficient: Cf = 0.2 Nm•s
Mass eccentricity: l = 0.2 m
Constant disturbance: dc = [2 2 2]T Nm Control parameters
DOB L = 5[03×3 I3×3] LQ controller Q = I6×6, R = I6×6
Predictor p0 = 5, r = 1, q0 = 10, m = 2
5.5.1.1 Comparison of State Prediction
The robustness against disturbance is compared between three prediction methods, such as i) predictive control (PC) [81] with xpˆ 2(5.22), ii) disturbance observer-based predictive control (DOB- PC) [82] with xpˆ 3(5.23), and iii) proposed SP with χ (5.34). The performance of the controllers is first compared without the application of previewed trajectory. For the simulation, the same LQ optimal control is implemented to three controllers to stabilize the state x, and control inputs are computed as
123 (5.57) with Q and R. The tracking error is defined as
ˆ 2 ˆ 2 ,
xp p d
e x x expˆ 3xpˆ 3xd,and eχ = χ ‒ xd, and control inputs are obtained as (5.62).
ˆ 2
ˆ 3
1
1
1
( ) ( ) ˆ( ) ( )
( ) ( ) ˆ( ) ( )
( ) ( ) ˆ( ) ( )
p
p
T
PC x d
T
DOB PC x d
T
SP d
t t t t
t t t h t
t t t h t
Bu BR B Pe Bd u
Bu BR B Pe Bd u
Bu BR B Pe Bd u
(5.62)
where ud( )t xd( )t Axd( ).t For the DOB-PC and the SP, the DOB (5.24) and predictor (5.33) are implemented with the same parameter. Fig. 5.5(a) shows the estimation result of disturbance ˆ( )dt using DOB. The results are turned into the predictor to calculate ˆd(t h ) with (5.25)‒(5.30) and (5.33) as follows:
0 0
1 1
ˆ ( ) ( ) 1
1 ˆ
( ) ( ) ( )
0
T
i i i
t h t h
c c
t t d t
c c
d Φ
Φ Φ (5.63)
where Φ = [Φ1 Φ2]T. c0 = 10 and c1 = 25 calculated from p0 and r. The predicted disturbance ˆd(t h ) is calculated by (5.63) and used to predict the state and control input for the exact cancellation of disturbance. Fig. 5.5(b) shows the prediction result of d(t+h), ˆd(t h ). It is noticed that the disturbance is well predicted with small error, and expected that the accurate prediction improves the desired tracking control performance, minimizing the prediction errors εp. Moreover, εv and εT can decrease according to the minimization of εp in (5.36) and (5.47), respectively. Thus, the desired tracking error ev in (5.46) will tend to zero.
Fig. 5.6 and Table 5.3 show that the SP (χ) results in RMSE of 0.038 deg for ϕ, but 2.14 deg, the PC (xˆ 2p ) and 0.11 deg, the DOB-PC (xˆ 3p). In the prediction xpˆ 3 and χ, the future disturbance
ˆ (t h )
d improves disturbance attenuation compared withxˆ 2p.In addition, the proposed SP χ enhances robustness against disturbance compared with the single prediction xpˆ 3. The RMSE is evaluated at steady-state after 10s. In addition, Table 5.3 shows the tracking results of θ and ψ for the controllers, verifying that the SP better attenuates the disturbance d despite time delay than the existing methods.
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(a) (b)
Figure 5.5. Simulation results. (a) Disturbance estimation. (b) Prediction result of disturbance.
Figure 5.6. Comparison of performance for state prediction.
Table 5.3. Comparison of RMSE for Tracking Control (degree/degree/s).
controller ev,1/ev,4 ev,2/ev,5 ev,3/ev,6
PC 2.14/0.19 3.62/1.14 4.57/2.55
DOB-PC 0.11/0.033 0.58/0.24 1.38/0.88
SP 0.038/0.015 0.27/0.14 0.82/0.53
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First, considering that previewed reference trajectory is not used, tracking results of the proposed prediction are analyzed in Fig. 5.7. Note that eχ(t) converges to zero due to the control input (5.62), as shown in Fig. 5.7. Then, according to the convergence, it is guaranteed that ev(t) will tend to zero according to Proposition 5.3 for the time-varying reference trajectory. However, the convergence of x(t) to vd(t) (= xd(t‒h)) can cause a mismatch between motion and visual cues in motion simulation.
Thus, it will lead to the use of the preview control for time-varying trajectory.
(a)
(b)
Figure 5.7. Tracking results without previewed trajectory. (a) Orientation tracking. (b) Angular velocity tracking.
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5.5.1.2 Performance Verification of Prediction-Based Preview Control
In this section, the effectiveness of PPC is verified. In preview control (5.58), the information for future reference should be first obtained from the current xd(t) and the predictor. Then, aˆ (d t h )can be computed as follows:
2
0 0
1 1 ,
2 2
ˆ ( ) ( ) 1 /2
1 0
( ) 0 1 ( ) ( )
0 0
T d
i i d i
t h t h h
c c
t c t c a t
c c
a Θ
Θ Θ (5.64)
where Θ = [Θ1 Θ2 Θ3]T. c0 = 30, c1 = 300, and c2 = 1000 calculated from q0 and m. The previewed trajectory is computed by (5.48) and (5.64). Note that the conventional preview controller [28] utilizes ad(t) instead ofaˆ (d s h )to compute χd(t).
( )
( ) d ( ) d ( )
t
h t s
d d d d
t h
t e t e ds t
A
Aχ x B a (5.65)
The obtained future reference trajectory is computed from only current information, and it can incur a large prediction error εd(t). Note that in (5.60), the error εd disturbs that x correctly tracks xd. Fig. 5.8 shows better prediction results for the predictor-based previewed trajectory (5.48) than the conventional (5.65). The RMSE of εd is 0.041deg for the proposed but 0.53deg, the conventional method. Thus, the proposed prediction for future trajectory can then lead the state x to converge to the desired xd.
In Fig. 5.9, it is verified that ed(t) firstly converges to zero by the PPC, and then e(t) tends to zero, verifying Theorem 5.1. In addition, the SP (5.34) improves disturbance attenuation by the DOB and the predictor. In order to guarantee the boundedness of ed and e with small error, the control input is computed from (5.58), composed of state feedback, feedforward control, and DOB for disturbance cancellation. Fig. 5.9 also verifies that the PPC can make e(t) converge zero for time-varying reference trajectory xd(t) in spite of disturbance d(t) and time delay h.
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Figure 5.8. Simulation results for comparison of previewed trajectories.
(a)
(b)
Figure 5.9. Simulation results for prediction-based preview control. (a) Orientation tracking. (b) Angular velocity tracking.
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5.5.2 Experimental Verification
5.5.2.1 Experimental Setup
In order to verify the feasibility and effectiveness of the PPC from the perspective of practical implementation, experiments are performed using a full-scale spherical motion platform. The platform in Fig. 5.10 consists of the cockpit sphere and base. The base system includes four linear stages with SPWs composed of an active wheel, small sphere, and roller in Fig. 5.10(c), where these are made from urethane material to minimize slipping between rolling contact. The rotation of the cockpit sphere is controlled by four SPWs. The torque generated by a motor is consecutively transferred from the active wheel to the small sphere and cockpit.
VN-110 as an IMU is selected to measure the orientation (ϕ, θ, ψ) and body angular velocity (ωx, ωy, ωz) of the cockpit sphere. The IMU is installed inside the cockpit sphere, and the data is transmitted by a wireless communication module. NI-cRIO 9039 is used to drive the control algorithms at 50 Hz, where the computation time of the algorithm for each step is smaller than 0.001s.
5.5.2.2 Experimental Results
The sinusoidal reference trajectory is set to be as xd,1 = [ϕd θd ψd]T = [20sin(0.04πt) 20cos(0.04πt) 10sin(0.02πt)]T deg and xd,2xd,1( )t . The state is defined as x[ ] T with an initial condition as x0 = [06×1]. In the experiment, the input delay h is set equal to 1s. Similar to numerical simulation, LQ optimal control is implemented to compare the performance of PC and PPC.
The control input for PC is computed by (5.62) and for PPC (5.58). In addition, notice that the tracking error in control input for PPC is defined as ed = χ ‒ χd, using future trajectory, but
ˆ 2 xp
e for PC, similar to numerical simulation. The detailed control parameters are listed in Table 5.4. In order to compare performance, the same parameters are utilized for DOB and LQ controller.
129 (a)
(b) (c)
Figure 5.10. Spherical motion platform based on SPWs. (a) Overall structure. (b) Driving mechanism. (c). SPW.
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Table 5.4. List of Specifications of Control Parameters.
DOB L = 5[03×3 I3×3] LQ controller Q = I6×6, R = 5[I6×6]
Predictor p0 = 1, r = 1, q0 = 10, m = 1
Table 5.5. Comparison of Tracking Performance.
Orientation (deg) e1 e2 e3
controller PC 3.85/8.42 3.83/8.44 1.33/3.20
PPC 0.55/1.39 0.33/0.70 0.70/1.60
Angular velocity (deg/s) e4 e5 e6
controller PC 0.77/1.75 0.77/1.68 0.26/0.84
PPC 0.31/0.66 0.29/0.55 0.11/0.29
RMSE/eMAX
Fig. 5.11 shows the comparison of tracking performance in orientation and angular velocity for the spherical motion platform applying the controllers. The RMSE and maximum error (eMAX) are evaluated at steady-state after 10s to compare the performance. The RMSE and eMAX of orientation for the PPC are smaller than 0.70 and 1.60 deg, respectively. However, for the PC, the RMSE and eMAX are larger than 3.80 and 8.40 deg, respectively, as shown in Table 5.5. In addition, for θ (x2), PPC shows better robust tracking at the initial state than PC despite the difference of 20 deg. Similarly, for ϕ and ψ, PPC better tracks the reference trajectories compared with PC. It demonstrates that the PPC better attenuates the disturbance by predicted future disturbance ˆd(t h ) (5.33) and the SP (5.34) compared with PC.
Based on Theorem 5.1, it is also noticed that the time-delay effect is minimized by the previewed trajectories (5.48). Moreover, the angular velocity more correctly tracks the desired velocity trajectory for the PPC with smaller RMSE and eMAX than the PPC. As a result, the experimental results validate that the PPC can be applied to multi-DOF systems, including motion simulators, to minimize the tracking error caused by delay and disturbance and simultaneously maximize reality. It is noteworthy that the experimental validation dealing with the time delay is firstly introduced for the multi-DOF motion simulator, even if the delay is still a common and severe problem in practical applications.
131 (a)
(b)
Figure 5.11. Experimental results for the sinusoidal response. (a) Tracking results for orientation. (b) Tracking results for angular velocity.
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