Unconstrained mechanical systems

A lattice can even be obtained for the general case of a fully actuated mechanical system, which for example includes most robot arms. Recall from (13.4) that any system in the form ${\dot q}=f(q,u)$ can alternatively be expressed as ${\dot q}= u$, if $U(q)$ is defined as the image of $f$ for a fixed $q$. The main purpose of using $f$ is to make it easy to specify a fixed action space $U$ that maps differently into the tangent space for each $q \in {\cal C}$.

A similar observation can be made regarding equations of the form ${\ddot q}= h(q,{\dot q},u)$, in which $u \in U$ and $U$ is an open subset of ${\mathbb{R}}^n$. Recall that this form was obtained for general unconstrained mechanical systems in Sections 13.3 and 13.4. For example, (13.148) expresses the dynamics of open-chain robot arms. Such equations can be expressed as ${\ddot q}= u'$ by directly specifying the set of allowable accelerations. Each $u$ will map to a new action $u'$ in an action space given by

$$U'(q,{\dot q})= \{ {\ddot q}\in {\mathbb{R}}^n \;\vert\; \exists u \in U {\ddot q}= h(q,{\dot q},u)\}$$ (14.24)

for each $q \in {\cal C}$ and ${\dot q}\in {\mathbb{R}}^n$.

Each $u' \in U'(q,{\dot q})$ directly expresses an acceleration vector in ${\mathbb{R}}^n$. Therefore, using $u' \in U(q,{\dot q})$, the original equation expressed using $h$ can be now written as ${\ddot q}= u'$. In its new form, this appears just like the multiple, independent double integrators. The main differences are

1. The set $U'(q,{\dot q})$ may describe a complicated region in ${\mathbb{R}}^n$, whereas $U$ in the case of the true double integrators was a cube centered at the origin.
2. The set $U'(q,{\dot q})$ varies with respect to $q$ and ${\dot q}$. Special concern must be given for this variation over the time sampling interval $\Delta t$. In the case of the true double integrators, $U$ was fixed.

Figure: (a) The set, $U'(q,{\dot q})$, of new actions and grid-based sampling. (b) Reducing the set by some safety margin to account for its variation over time.

The first difference is handled by performing grid sampling over ${\mathbb{R}}^n$ and making an edge in the reachability graph for every grid point that falls into $U'(q,{\dot q})$; see Figure 14.15a. The grid resolution can be improved along with $\Delta t$ to obtain resolution completeness. To address the second problem, think of $U'(q(t),{\dot q}(t))$ as a shape in ${\mathbb{R}}^n$ that moves over time. Choosing $u'$ close to the boundary of $U'(q(t),{\dot q}(t))$ is dangerous because as $t$ increases, $u'$ may fall outside of the new action set. It is often possible to obtain bounds on how quickly the boundary of $U'(q,{\dot q})$ can vary over time (this can be determined, for example, by differentiating $h$ with respect to $q$ and ${\dot q}$). Based on the bound, a thin layer near the boundary of $U'(q,{\dot q})$ can be removed from consideration to ensure that all attempted actions remain in $U'(q(t),{\dot q}(t))$ during the whole interval $\Delta t$. See Figure 14.15b.

These ideas were applied to extend the approximation algorithm framework to the case of open-chain robot arms, for which $h$ is given by (13.148). Suppose that $U$ is an axis-aligned rectangle, which is often the case for manipulators because the bounds for each $u_i$ correspond to torque limits for each motor. If $q$ and ${\dot q}$ are fixed, then (13.140) applies a linear transformation to obtain ${\ddot q}$ from $u$. The rectangle is generally sheared into a parallelepiped (a $n$-dimensional extension of a parallelogram). Recall such transformations from Section 3.5 or linear algebra.