The key to establishing whether a system is nonholonomic is to construct motions that combine the effects of two action variables, which may produce motions in a direction that seems impossible from the system distribution. To motivate the coming ideas, consider the differential-drive model from (15.54). Apply the following piecewise-constant action trajectory over the interval :
The result of all four motion primitives in succession from is shown in Figure 15.16. It is fun to try this at home with an axle and two wheels (Tinkertoys work well, for example). The result is that the differential drive moves sideways!15.9 From the transition equation (15.54) such motions appear impossible. This is a beautiful property of nonlinear systems. The state may wiggle its way in directions that do not seem possible. A more familiar example is parallel parking a car. It is known that a car cannot directly move sideways; however, some wiggling motions can be performed to move it sideways into a tight parking space. The actions we perform while parking resemble the primitives in (15.71).
Algebraically, the motions of (15.71) appear to be checking for commutativity. Recall from Section 4.2.1 that a group is called commutative (or Abelian) if for any . A commutator is a group element of the form . If the group is commutative, then (the identity element) for any . If a nonidentity element of is produced by the commutator, then the group is not commutative. Similarly, if the trajectory arising from (15.71) does not form a loop (by returning to the starting point), then the motion primitives do not commute. Therefore, a sequence of motion primitives in (15.71) will be referred to as the commutator motion. It will turn out that if the commutator motion cannot produce any velocities not allowed by the system distribution, then the system is completely integrable. This means that if we are trapped on a surface, then it is impossible to leave the surface by using commutator motions.
Now generalize the differential drive to any driftless control-affine system that has two action variables:
Suppose that the commutator motion (15.71) is applied to (15.72) as shown in Figure 15.17. Determining the resulting motion requires some general computations, as opposed to the simple geometric arguments that could be made for the differential drive. If would be convenient to have an expression for the velocity obtained in the limit as approaches zero. This can be obtained by using Taylor series arguments. These are simplified by the fact that the action history is piecewise constant.
The coming derivation will require an expression for under the application of a constant action. For each action, a vector field of the form is obtained. Upon differentiation, this yields
Now the state trajectory under the application of (15.71) will be determined using the Taylor series, which was given in (14.17). The state trajectory that results from the first motion primitive can be expressed as
The idea of substituting previous Taylor series expansions as they are needed can be repeated for the remaining two motion primitives. The Taylor series expansion for the result after the third primitive is
By applying (15.81), the Lie bracket is
As a simple example, recall the nonholonomic integrator (13.43). In the linear-algebra form, the system is
Steven M LaValle 2020-08-14