13.1.1.2 Parametric constraints

The parametric way of expressing velocity constraints gives a different interpretation to $U(q)$. Rather than directly corresponding to a velocity, each $u \in U(q)$ is interpreted as an abstract action vector. The set of allowable velocities is then obtained through a function that maps an action vector into $T_q({\cal C})$. This yields the configuration transition equation (or system)

$${\dot q}= f(q,u) ,$$ (13.2)

in which $f$ is a continuous-time version of the state transition function that was developed in Section 2.1. Note that (13.2) actually represents $n$ scalar equations, in which $n$ is the dimension of ${\cal C}$. The system will nevertheless be referred to as a single equation in the vector sense. Usually, $U(q)$ is fixed for all $q \in {\cal C}$. This will be assumed unless otherwise stated. In this case, the fixed action set is denoted as $U$.

There are two interesting ways to interpret (13.2):

1. Subspace of the tangent space: If $q$ is fixed, then $f$ maps from $U$ into $T_q({\cal C})$. This parameterizes the set of allowable velocities at $q$ because a velocity vector, $f(q,u)$, is obtained for every $u \in U(q)$.
2. Vector field: If $u$ is fixed, then $f$ can be considered as a function that maps each $q \in {\cal C}$ into $T_q({\cal C})$. This means that $f$ defines a vector field over ${\cal C}$ for every fixed $u \in U$.

Example 13..1 (Two Interpetations of ${\dot q}=f(q,u)$)   Suppose that ${\cal C}= {\mathbb{R}}^2$, which yields a two-dimensional velocity vector space at every $q = (x,y) \in {\mathbb{R}}^2$. Let $U = {\mathbb{R}}$, and ${\dot q}=f(q,u)$ be defined as ${\dot x}= u$ and ${\dot y}= x$.

To obtain the first interpretation of ${\dot q}=f(q,u)$, hold $q = (x,y)$ fixed; for each $u \in U$, a velocity vector $({\dot x},{\dot y}) = (u,x)$ is obtained. The set of all allowable velocity vectors at $q = (x,y)$ is

$$\{ ({\dot x},{\dot y}) \in {\mathbb{R}}^2 \;\vert\; {\dot y}= x\} .$$ (13.3)

Suppose that $q = (1,2)$. In this case, any vector of the form $(u,1)$ for any $u \in {\mathbb{R}}$ is allowable.

To obtain the second interpretation, hold $u$ fixed. For example, let $u=1$. The vector field $({\dot x},{\dot y}) = (1,x)$ over ${\mathbb{R}}^2$ is obtained. $\blacksquare$

It is important to specify $U$ when defining the configuration transition equation. We previously allowed, but discouraged, the action set to depend on $q$. Any differential constraints expressed as ${\dot q}=f(q,u)$ for any $U$ can be alternatively expressed as ${\dot q}= u$ by defining

$$U(q)= \{ {\dot q}\in {\mathbb{R}}^n \;\vert\; \exists u \in U {\dot q}= f(q,u)\}$$ (13.4)

for each $q \in {\cal C}$. In this definition, $U(q)$ is not necessarily a subset of $U$. It is usually more convenient, however, to use the form ${\dot q}=f(q,u)$ and keep the same $U$ for all $q$. The common interpretation of $U$ is that it is a set of fixed actions that can be applied from any point in the C-space.

In the context of ordinary motion planning, a configuration transition equation did not need to be specifically mentioned. This issue was discussed in Section 8.4. Provided that $U$ contains an open subset that contains the origin, motion in any direction is allowed. The configuration transition equation for basic motion planning was simply ${\dot q}= u$. Since this does not impose constraints on the direction, it was not explicitly mentioned. For the coming models in this chapter, constraints will be imposed on the velocities that restrict the possible directions. This requires planning algorithms that handle differential models and is the subject of Chapter 14.