= 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.<Robotic Systems by Kris Hauser>[Chapter 8, introductory definition, Stating the motion planning problem and Point-to-point path planning]{ref}
Let $Q_{\rm free}$ be the configurations allowed by joint limits and <collision detection>. A geometric solution is a continuous curve
$$
\gamma:[0,1]\longrightarrow Q_{\rm free},\qquad
\gamma(0)=q_{\rm start},\quad\gamma(1)=q_{\rm goal}.
$$
Each 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 length
$$
L=2\sqrt{0.5^2+0.2^2}\approx1.0770\,\mathrm m.
$$
The perpendicular distance from the obstacle center to either segment is
$$
d_{\min}=\frac{0.5(0.20)}{\sqrt{0.5^2+0.2^2}}
\approx0.1857\,\mathrm m,
$$
so 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.<Robotic Systems by Kris Hauser>[Chapter 8, Completeness and optimality; Explicit vs implicit plans]{ref}
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