Trajectory planning

The term trajectory planning has been used for decades in robotics to refer mainly to the problem of determining both a path and velocity function for a robot arm (e.g., PUMA 560). This corresponds to finding a path in the phase space $X$ in which $x \in X$ is defined as $x = (q,{\dot q})$. Most often the problem is solved using the refinement approach mentioned in Section 1.4 by first computing a path through ${\cal C}_{free}$. For each configuration $q$ along the path, a velocity ${\dot q}$ must be computed that satisfies the differential constraints. An inverse control problem may also exist, which involves computing for each $t$, the action $u(t)$ that results in the desired ${\dot q}(t)$. The refinement approach is often referred to as time scaling of a path through ${\cal C}$ [456]. In recent times, trajectory planning seems synonymous with kinodynamic planning, assuming that the constraints are second-order ($x$ includes only configuration and velocity variables). One distinction is that trajectory planning still perhaps bears the historical connotations of an approach that first plans a path through ${\cal C}_{free}$.