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 action trajectory is a sequence of four motion primitives: 1) translate forward, 2) rotate forward, 3) translate backward, and 4) rotate backward.

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:

Using the notation of (15.53), the vector fields would be and ; however, and are chosen here to allow subscripts to denote the components of the vector field in the coming explanation.

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

This follows from the chain rule because is a function of , which itself is a function of . The derivative is actually an Jacobian matrix, which is multiplied by the vector . To further clarify (15.73), each component can be expressed as

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

which makes use of (15.73) in the second line. The Taylor series expansion for the second primitive is

An expression for can be obtained by using the Taylor series expansion in (15.75) to express . The first terms after substitution and simplification are

To simplify the expression, the evaluation at has been dropped from every occurrence of and and their derivatives.

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

(15.78) |

Finally, the Taylor series expansion after all four primitives have been applied is

(15.79) |

Taking the limit yields

which is called the

The Lie bracket is an important operation in many subjects, and is related to the Poisson and Jacobi brackets that arise in physics and mathematics.

By applying (15.81), the Lie bracket is

The resulting vector field is , which indicates the sideways motion, as desired. When evaluated at , the vector is obtained. This means that performing short commutator motions wiggles the differential drive sideways in the direction, which we already knew from Figure 15.16.

As a simple example, recall the nonholonomic integrator (13.43). In the linear-algebra form, the system is

Let and . The Jacobian matrices are

(15.84) |

Using (15.80),

This result can be verified using (15.81).

Steven M LaValle 2020-08-14