15.5 Steering Methods for Nonholonomic Systems

This section briefly surveys some methods that solve the BVP for nonholonomic systems. This can be considered as a motion planning problem under differential constraints but in the absence of obstacles. For linear systems, optimal control techniques can be used, as covered in Section 15.2.2. For mechanical systems that are fully actuated, standard control techniques such as the acceleration-based control model in (8.47) can be applied. If a mechanical system is underactuated, then it is likely to be nonholonomic. As observed in Section 15.4, it is possible to generate motions that appear at first to be prohibited. Suppose that by the Chow-Rashevskii theorem, it is shown that a driftless system is STLC. This indicates that it should be possible to design an LPM that successfully connects any pair of initial and goal states. The next challenge is to find an action trajectory that actually causes to reach upon integration in (14.1). Many methods in Chapter 14 could actually be used, but it is assumed that these would be too slow. The methods in this section exploit the structure of the system (e.g, its Lie algebra) and the fact that there are no obstacles to more efficiently solve the planning problem.

- 15.5.1 Using the P. Hall Basis
- Formal calculations
- The exponential map
- The Chen-Fliess series
- Returning to the system vector fields
- The Chen-Fliess-Sussmann equation
- Using the original action variables

- 15.5.2 Using Sinusoidal Action Trajectories
- 15.5.2.1 Steering the nonholonomic integrator
- 15.5.2.2 First-order controllable systems
- 15.5.2.3 Chained-form systems

- 15.5.3 Other Steering Methods
- Differentially flat systems
- Decoupling vector fields
- Averaging methods
- Variational techniques
- Pontryagin's minimum principle
- Dynamic programming

- Further Reading
- Exercises