Logarithmic control, trajectory tracking, and formation for nonholonomic vehicles on Lie group SE(2)

ABSTRACT

This paper studies the problem of set-point stabilisation, trajectory tracking, and formation tracking for unicycle-type vehicles by taking advantage of the exponential coordinates and the other mathematical tools provided by the Lie group setting of the special Euclidean group SE(2). Motivated by recent developments in geometric control approaches for holonomic systems, we first study the stabilisation problem for nonholonomic constrained kinematic systems by improving the logarithmic stabilising control laws in such a way that they can satisfy nonholonomic constraints too. We then extend the control design to the problem of trajectory tracking by proposing auxiliary systems and investigating the conditions for which the adjoint map in the Lie algebra preserves the velocity constraint. This leads to a global control law which is valid for general time-varying as well as non-smooth trajectories. The tracking control law is also applied to the problem of leader-follower formation tracking by constructing nonholonomic virtual leaders in the desired formation. Finally, the kinematic control law is translated to a differential drive robot using the backstepping technique. To demonstrate the effectiveness of the controllers, the numerical results are presented and the performances are compared with the other three controllers in the literature.

KEYWORDS:

• Nonholonomic planar vehicle
• set-point stabilisation
• trajectory tracking
• formation tracking
• exponential coordinates

Acknowledgments

The authors would like to thank the reviewers for the valuable comments, specifically the suggestions on the comparative study.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation (NNSF) of China [grant number 61374033].