Further Reading

The basic stability and controllability concepts from Section 15.1 appear in many control textbooks, especially ones that specialize in nonlinear control; see [523,846] for an introduction to nonlinear control. More advanced concepts appear in [156]. For illustrations of many convergence properties in vector fields, see [44]. For linear system theory, see [192]. Brockett's condition and its generalization appeared in [143,996]. For more on stabilization and feedback control of nonholonomic systems, see [156,846,964]. For Lyapunov-based design for feedback control, see [278].

For further reading on the Hamilton-Jacobi-Bellman equation, see [85,95,492,789,912]. For numerical approaches to its solution (aside from value iteration), see [2,253,707]. Linear-quadratic problems are covered in [28,570]. Pontryagin's original works provide an unusually clear explanation of the minimum principle [801]. For other sources, see [95,410,789]. A generalization that incorporates state-space constraints appears in [927].

Works on which Section 15.3 is based are [64,127,211,294,814,903,904,923]. Optimal curves have been partially characterized in other cases; see [227,903]. One complication is that optimal curves often involve infinite switching [370,1000]. There is also interest in nonoptimal curves that nevertheless have good properties, especially for use as a local planning method for car-like robots [31,358,520,794,848]. For feedback control of car-like robots, see [112,663].

For further reading on nonholonomic system theory beyond Section 15.4, there are many excellent sources: [83,112,113,156,478,725,741,846]. A generalization of the Chow-Rashevskii theorem to hybrid systems is presented in [724]. Controllability of a car pulling trailers is studied in [594]. Controllability of a planar hovercraft with thrusters is considered in [669]. The term holonomic is formed from two Greek words meaning ``integrable'' and ``law'' [135].

Section 15.5 is based mainly on the steering methods in [574] (Section 15.5.1) and [142,727] (Section 15.5.2). The method of Section 15.5.1 is extended to time-varying systems in [299]. A multi-rate version is developed in [713]. In [480], it was improved by using a Lyndon basis, as opposed to the P. Hall basis. Another steering method that involves series appears in [154,155]. For more on chained-form systems, see [858,902]. For a variant that uses polynomials and the Goursat normal form, instead of sinusoids, see [846]. For other steering methods, see the references suggested in Section 15.5.3.

Steven M LaValle 2012-04-20