 Robot motion planning

Robot motion planning computes a robot's proposed motion from its modeled state, environment and task constraints. A geometric planning problem asks for a path through configuration space; a timed motion can additionally constrain velocity, acceleration, forces and actuator effort. Hauser's primary text explicitly distinguishes geometric path planning from the broader problem involving dynamics and execution.^[ref]

Let $Q_{\mathrm{free}}$ be the configurations allowed by joint limits and collision detection. A geometric solution is a continuous curveEach point represents a complete robot configuration. A configuration-space obstacle is a forbidden set of configurations, not necessarily the same shape as the physical obstacle in the workspace.

An original planar example uses a point robot starting at $(0,0)$ and aiming for $(1,0)$, in metres. A circular obstacle centered at $(0.5,0)$ with radius $0.10\mathrm{m}$ blocks the straight path. A two-segment detour through $(0.5,0.20)$ has lengthThe perpendicular distance from the obstacle center to either segment isso the detour clears the circular obstacle. Checking only the start and goal would instead miss the collision on the straight segment.

Now give the robot a circular footprint of radius $0.05\mathrm{m}$, with translation as its only degree of freedom. The forbidden center positions form a disk of radius $0.15\mathrm{m}$: the obstacle is expanded by the footprint through Minkowski addition. The same detour still passes, but its center-path margin above the expanded obstacle is only about $0.0357\mathrm{m}$. The geometric reduction depends on the stated footprint and permitted motion.

A path's corner does not supply a dynamically smooth trajectory. Traversing this sharp detour at nonzero constant speed would instantaneously change velocity at its midpoint. A physical controller must stop, smooth the corner or choose a different feasible trajectory, while continuing to respect the obstacle and actuator constraints. A larger path length can be an acceptable cost if it permits the required motion.

A proposed validation compares the continuous planned geometry, sampled collision tests and measured execution on a controlled obstacle course. Report the robot geometry, constraints, computational budget and observed clearance. Failure to find a path in a limited run does not alone prove none exists; success on a numerical path does not alone establish accurate execution. Hauser treats completeness and the distinction between paths and feedback policies separately.^[ref]