### 14.6.3.1 Expressing systems in terms of , , and

Suppose that a system is given in the form

 (14.31)

in which there are action variables . It may be helpful to glance ahead to Example 14.6, which will illustrate the coming concepts for the simple case of double integrators . The acceleration in is determined from the state and action . Assume , in which is an -dimensional subset of . If is nonsingular at , then an -dimensional set of possible accelerations arises from choices of . This means it is fully actuated. If there were fewer than action variables, then there would generally not be enough freedom to follow a specified path. Therefore, must be -dimensional. Which choices of , however, constrain the motion to follow the given path ? To determine this, the , , and variables need to be related to the path domain and its first and second time derivatives and , respectively. This leads to a subset of that corresponds to actions that follow the path.

Suppose that , , , and a path are given. The configuration is

 (14.32)

Assume that all first and second derivatives of exist. The velocity can be determined by the chain rule as

 (14.33)

in which the derivative is evaluated at . The acceleration is obtained by taking another derivative, which yields

 (14.34)

by application of the product rule. The full state can be recovered from using (14.32) and (14.33).

The next step is to obtain an equation that looks similar to (14.31), but is expressed in terms of , , and . A function can be obtained from by substituting for and the right side of (14.33) for :

 (14.35)

This yields

 (14.36)

For a given state (which can be obtained from and ), the set of accelerations that can be obtained by a choice of in (14.36) is the same as that for the original system in (14.31). The only difference is that is now constrained to a 2D subset of , which are the states that can be reached by selecting values for and .

Applying (14.34) to the left side of (14.36) constrains the accelerations to cause motions that follow . This yields

 (14.37)

which can also be expressed as

 (14.38)

by moving the first term of (14.34) to the right. Note that equations are actually represented in (14.38). For each in which , a constraint of the form

 (14.39)

is obtained by solving for .

Steven M LaValle 2012-04-20