Next: 15.4.2.4 The Frobenius Theorem Up: 15.4.2 Determining Whether a Previous: 15.4.2.2 Distributions## 15.4.2.3 Lie bracketsThe 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! Algebraically, the motions of (15.71) appear to be checking for commutativity. Recall from Section 4.2.1 that a group is called 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
Finally, the Taylor series expansion after all four primitives have been applied is
Taking the limit yields which is called the Lie bracket of and and is denoted by . Similar to (15.74), the th component can be expressed asThe Lie bracket is an important operation in many subjects, and is related to the Poisson and Jacobi brackets that arise in physics and mathematics. Example 15..9 (Lie Bracket for the Differential Drive) The Lie bracket should indicate that sideways motions are possible for the differential drive. Consider taking the Lie bracket of the two vector fields used in (15.54). Let and. Rename and to and to allow subscripts to denote the components of a vector field.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. Example 15..10 (Lie Bracket of Linear Vector Fields) Suppose that each vector field is a linear function of . The Jacobians and are constant.As a simple example, recall the nonholonomic integrator (13.43). In the linear-algebra form, the system is Let and . The Jacobian matrices are
Using (15.80), This result can be verified using (15.81). Next: 15.4.2.4 The Frobenius Theorem Up: 15.4.2 Determining Whether a Previous: 15.4.2.2 Distributions |