# Nonholonomic Motion Planning

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This chapter provides a survey of some of the techniques which have been used in planning paths for nonholonomic systems. Conventional (holonomic) path planners implicitly assume that arbitrary motion in the configuration space is allowed as long as obstacles are avoided. If a system contains nonholonomic constraints, many of these path planners cannot be directly applied since they generate paths which violate the constraints. For this reason, it is important to understand how to efficiently compute paths for nonholonomic systems.

## Chapter Summary

1. Optimal controls (minimizing integral least squares cost) for steering a system with growth vector <amsmath>(2,3)</amsmath> of the form
<amsmath>

\aligned \dot q_1 &= u_1 \\ \dot q_2 &= u_2 \\ \dot q_3 &= q_1 u_2 - q_2 u_1 \endaligned

</amsmath>

are sinusoidal. Further more when <amsmath>q_1(0) = q_2 (0) = q_1 (1) = q_2 (1) = 0</amsmath>, the optimal inputs are sinusoids at frequency <amsmath>2 \pi</amsmath>.

2. For a system of the form
<amsmath>

\aligned \dot q &= u \\ \dot Y &= q u^T - u q^T \endaligned

</amsmath>

the optimal steering inputs, minimizing the integral least squares cost, are sinusoidal. Further, when <amsmath>q(0) = q (1) = 0</amsmath> the optimal inputs are sinusoids at integrally related frequencies.

3. Using integrally related sinusoids as (sub-optimal) inputs, one can steer chained form systems. A one-chain system is one of the form
<amsmath>
 \aligned
\dot q_1 & = u_1 \\
\dot q_2 &= u_2 \\
\dot q_3 & = q_2 u_1 \\
\dot q_4 & = q_3 u_1 \\
&\makebox[1.35em][c]{\vdots} \\
\dot q_n & = q_{n-1} u_1
\endaligned

</amsmath>

Generalizations to multi-chain systems also exist. Involutivity conditions for converting given control systems into the chained form may be given.

4. While it is difficult to give closed form expressions for the optimal controls associated with solving the least squares steering problem for a nonholonomic control system, one can derive formulas for the time derivatives of the optimal inputs. Further, numerical techniques, such as the Ritz approximation algorithm, may be used to derive approximate algorithms for generating the optimal controls.
5. Piecewise constant inputs can be used to steer a nonholonomic control system in the Philip Hall basis coordinates when the controllability Lie algebra is nilpotent.
6. Dynamic finger repositioning on the surface of an object can be carried out using sinusoids. In the special case of a spherical finger rolling on the surface of a flat object, the geometry of the Gauss-Bonnet theorem may be used to position the finger on the object surface and adjust the angle of contact.