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Fig. 11 Error index comparison of the feedforward lineariza tion and
the observer based control scheme
Fig. 9 Comparison of the tracking errors in the Cartesian space
Fig. 10 Trajectory tracking behavior of the feedforward linear zation
and the observer based control scheme
Fig. 8 Experimental trajectory tracking
2
0
2
0
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Simulation and experimental results on a laboratory prototype demonstrate
quite satisfactory and coincident performances.
robust position control and reference trajectory tracking tasks in nonholonomic
wheeled robots. The fundamental ideas of the GPI philosophy can be used to
advantage in robustness enhancing tasks, as demonstrated in this work and,
specially, for those systems undergoing unknown perturbation inputs. Although
the proposed GPI controller was designed to reject constant disturb ance
inputs, the compensation law can be adequately manage unmodeled
disturbances affecting the nonholonomic car.
This approach can be improved by means of a trajectory planning, which,
dynamically, assigns an interpolating desired trajectory smoothly connecting
the actual state with the real desired trajec tory in a certain fixed interval of
time, thus overcoming the prob lem of having “nonsufficiently close ”Initial
conditions. Once this requirement is achieved, the controller is capable of
regulating the system, even in the case of rather complex desired output refer
ence trajectories.
According to the exact linearization methodology, here explored, the control
system performance is sensitive to initial conditions.
Most common mobile robot models, treated in the literature, turn out to be
differentially flat
[19,21].
This property is the key for the exact feedforward
linearization feedback controller design task.
In this work, an exact feedforward linearization approach, in combination
with a GPI control, was found to be quite suitable for
^ x_iðtÞ ¼ qiðtÞ þ ki1ðxiðtÞ x ^ iðtÞÞ
q_iðtÞ ¼ ki0ðxiðtÞ x ^ iðtÞÞ
þ ðx2ðsÞ x 2ðsÞÞ2 ds. The second, J2 (u), is defined as J2ðuÞ ¼ Ðt ðu1ðsÞ
þ u2ðsÞ Þds. Figure
11
shows that, even though both control schemes work
well, the feedforward linearization method achieves a better error
performance with respect to the observer based controller and,
moreover, the feedforward linearization con troller
uses less energy than the observer based control to achieve better results .
MAY 2015, Vol. 137 / 051001-7
Also, two performance indexes were proposed to compare both control
schemes, the first one, J1 (e), consisting in the integral of the square error
norm, that is, J1ðeÞ: ¼ Ðt ðx1ðsÞ x 1ðsÞÞ2
and the observer estimates were computed as follows:
Journal of Dynamic Systems, Measurement, and Control
for i ¼ 1, 2. The observer injection gains were k11 ¼ k21 ¼ 7.1, k10¼ k20 ¼
9. The controller gains were chosen as kd1 ¼ kd2 ¼ 6.6, kp1 ¼ kp2 ¼ 13.05,
and ki1 ¼ ki2 ¼ 6.75, respectively. Figure
10
compares the results of both
control schemes. Both control laws achieve good tracking performance for
similar experimental tests.
7 Conclusion
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The exact feedforward linearization approach exhibits a smooth
behavior with low complexity in the implementation, reducing the
computational costs.
The discussed feedback scheme can be extended to include
some other physical frameworks of position sensing (GPS, 3D
vision schemes, etc.) and other classes of trajectories which may
be required to overcome singularity points.
This article was supported by SIP-IPN under research Grant
No. 20140373 (Alberto Luviano-Ju arez).
051001-8 / Vol. 137, MAY 2015
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References
Acknowledgment
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