5.3 Design and Implementation of ETAC over Controller- to-Robot Channel with Input Delay
5.3.3 Experimental Results
Proposition 5. Under the action of the designed controller (5.26) and adaptation law (5.29) with the event-triggered scheme (5.13), the robot dynamics (5.12) is stable in the sense of Lyapunov and all states of closed-loop system are bounded. Furthermore, the tracking errors converge to the origin as t → ∞. Moreover, Zeno behavior is excluded under the proposed ET scheme.
Proof. Based on the previous analysis, the triggering function (5.35) was finally defined in a way that guarantees the time derivative of Lyapunov function V2 = 12 Pn
i=1ZiTZi+
1
2Θ˜TΓ−11 Θ +˜ 12P˜TΓ−12 P˜ to be negative semi-definite. Thus, the stability in the sense of Lyapunov is ensured under proposed event-triggered scheme and all closed-loop signals are bounded. On the other hand, the boundedness of the auxiliary compensating system (5.14) is also required to complete the proof. Let us takeVη = 12η1Tη1+12η2Tη2as Lyapunov function candidate whose derivative is given as
V˙η =−η1Tκ1η1+ηT1Sη2+η2TGu(t)ˆ −η2TGu(tˆ −τ)
≤ −κ1kη1k2 +kη1kkSkkη2k+kηT2Gˆkku(t)k+kη2TGˆkku(t−τ)k
≤ −(κ1 −1
2)kη1k2+ ¯O (5.36)
where ¯O = ku(t)k2 2+ku(t−τ)k2 2+kη22k2+kGηˆ 2k2. Since the control input (5.26) is a function of all bounded signals, the term ¯O is bounded. Moreover, by choosing κ1 > 12, boundedness of auxiliary variables is ensured.
The convergence of the tracking errors can be directly proved similar to the analysis in Proposition 3 by invoking Barbalat lemma [97]. Furthermore, the proof of Zeno exclusion presented in Proposition 1 is still valid and can be directly applied in this scenario.
5.3 Design and Implementation of ETAC over Controller-to-Robot Channel with Input Delay
Figure 5.3: Experiment setup
A lemniscate path is chosen as a reference trajectory and it is defined as xref = arefsin(wreft)
1 + sin2(wreft) , yref = arefsin(wreft) cos(wreft)
1 + sin2(wreft) , (5.37) where aref = 1.2 and wref = 0.1. The initial position of the robot is mainly chosen as (0,0,10). Other parameters are chosen asC1 = diag(0.8, 0.9, 1.2), C2 = diag(4, 4), κ1 = 6, Γ1 = I4, Γ2 = I3 and τ = 0.1 [s]. Three different values are considered for the ad- justable parameter of event-triggered condition (ζ = 0.4, 0.6 and 0.8).
The obtained experimental results under proposed controller (5.26), adaptive law (5.29), and triggering function (5.35) on robot dynamics (5.12) are illustrated in Fig.
5.4–5.9. The tracking performance in the X-Y plane for different values of ζ is shown in Fig. 5.4. It can be observed that the designed controller is able to compensate for input delays while achieving accurate tracking. Furthermore, the proposed event-triggered and time-triggered control signals are illustrated in Fig. 5.5 and Fig. 5.6. To evaluate the tracking performance under the proposed scheme, the integral square error (ISE) and the maximum of the absolute tracking errors are provided. Moreover, the L2 norm of the
x [m]
-0.6 -0.4 -0.2 0 0.2 0.4
y[m]
-0.4 -0.2 0 0.2 0.4 0.6
Simulation
Experiment (ζ= 0.4) Experiment (ζ = 0.6) Experiment (ζ= 0.8)
Figure 5.4: Trajectory tracking in X-Y plane for different values ofζ.
control signal is also provided for each value ofζ. The experiments are repeated 15 times (5 for each value of ζ) and the average values of the obtained results are summarized in Table 5.1.
In order to further show the effectiveness of the proposed control scheme in handling the effects of parametric uncertainties and input delays, several simulations are also con- ducted under different scenarios where these effects are mutually deactivated. In the first scenario, the effect of uncertainties in system parameters is evaluated assuming ideal communications (τ = 0). The parametric uncertainties are defined as Θ(t) = Θ(0) + ∆Θ wherein ∆Θ = 12Θ(0) sin(t). The system performance under this scenario using the kinematic controller and dynamic controller with/without adaptation is shown in Figs.
5.7a–5.7c. The tracking performance under different initial conditions is also shown in Fig. 5.7d and the obtained results are summarized in Table 5.2. On the other hand, the induced delay is considered in the second scenario assuming known parameters (∆Θ = 0).
The effect of input delay on system performance and stability is illustrated in Fig. 5.8, in which the trajectories in the X-Y plane with and without the incorporation of the auxil- iary compensation variables are depicted in Fig. 5.8a and Fig. 5.8b, respectively. It can be observed from the obtained results that the auxiliary system is capable to compensate for input delays and achieve accurate tracking. However, as the input delay increases, the system performance degrades and it may become unstable for larger input delays. From the zoomed figure, it can also be observed that the overshoot is increasing for the case of
5.3 Design and Implementation of ETAC over Controller-to-Robot Channel with Input Delay
0 10 20 30 40 50 60
vin[m/s]
0 0.1
0.2 ET TT
Time [s]
0 10 20 30 40 50 60
vin[m/s]
0 0.1
0.2 ET TT
0 10 20 30 40 50 60
vin[m/s]
0 0.1
0.2 ET TT
ζ= 0.4
ζ= 0.6
ζ= 0.8
Figure 5.5: Event-triggered and time-triggered linear velocities for different values ofζ.
τ >0.3. Moreover, the performance is further degraded and a large tracking error is ob- tained for the case ofτ >1, which is undesirable for practical applications. On the other hand, the actual trajectory converges to the desired one comparatively faster in case of τ <0.3. Besides, Fig. 5.9 depicts the triggering time instants for each value of ζ where the actual control is updated/transmitted through the network. It can be observed that, as the value of ζ increases and approaches 1, the number of triggering/transmissions decreases. Hence, the communication burden is significantly reduced. However, these savings in resources are achieved at the cost of tracking accuracy and control efforts. Yet, a trade-off between the resource utilization in tandem with an acceptable performance could be achieved by adjusting ζ. The channel usage is calculated as
Channel Usage = Number of Updates× dt
T ×100 % (5.38)
where dt and T are sampling time and the total time of experiment run, respectively.
As compared to time-triggered (TT) implementation, it is worth mentioning that 3000 control updates/transmissions are needed for 60 seconds of the experiment run under 0.02 sampling interval for the TT scenario. It can be observed that this number is further decreased under the proposed ET strategy yielding a significant saving in the usage of network resources.
0 10 20 30 40 50 60 win[deg/s]
-50 0 50
ET TT
0 10 20 30 40 50 60
win[deg/s]
-50 0 50
ET TT
Time [s]
0 10 20 30 40 50 60
win[deg/s]
-50 0 50
ET TT
ζ= 0.6 ζ= 0.4
ζ= 0.8
Figure 5.6: Event-triggered and time-triggered angular velocities for different values ofζ.
Table 5.1: Comparison of results for different values of the adjustable parameterζ.
Resource utilizations Performance metrics Tracking error
#Control update %Channel usage %Saving ISE Max Absolute Error L2norm (u) [0-T/2] [T/2-T]
ζ= 0.4 1774 59.13 40.87 0.083 0.14 0.036 18.00
ζ= 0.6 726 24.21 75.79 0.095 0.16 0.040 18.38
ζ= 0.8 271 9.04 90.96 0.102 0.17 0.047 19.68
Table 5.2: Comparison of results for different initial conditions.
Resource utilizations Performance metrics
Initial Tracking error
condition
#Control update %Channel usage %Saving ISE Max Absolute Error L2 norm (u) [0-T/2] [T/2-T]
(0.1,0) 231 6.60 93.40 0.09 0.13 0.032 19.96
(-0.4,-0.1) 243 6.95 93.05 0.23 0.41 0.031 20.80
(-0.1,-0.2) 216 6.17 93.83 0.37 0.29 0.032 20.82
(-0.4,0.2) 219 6.26 93.74 0.45 0.41 0.030 20.32
5.3 Design and Implementation of ETAC over Controller-to-Robot Channel with Input Delay
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y[m]
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Reference Actual
(a) Kinematic controller
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y[m]
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Reference Actual
(b) Dynamic controller without adaptation
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y[m]
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Reference Actual
(c) Dynamic controller with adaptation
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y[m]
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X: 0.1 Y: 0
X: -0.1 Y: -0.2 X: -0.4
Y: -0.1 X: -0.4 Y: 0.2
(d)Initial condition trials Figure 5.7: The effect of system uncertainties and different initial conditions.
x [m]
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y[m]
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Reference τ= 0.3 τ= 0.5 τ= 1
(a) With auxiliary compensation
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y[m]
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Reference Actual
(b) Without auxiliary compensation Figure 5.8: The effect of input delay.
Time [s]
0 10 20 30 40 50 60
(4) (3) (2) (1)
(1): ζ= 0.8 (2): ζ= 0.6 (3): ζ= 0.4 Time-triggered
Figure 5.9: Illustration of triggering events for different values ofζ in comparison with time-triggered implementation (TT).
5.4 Design and Implementation of Predictor-based Event-triggered Controller over Robot-to-Controller Channel with State/Input Delays