Notation · Principles of Robot Motion via Rust

The book follows the notation of Choset, Lynch, Hutchinson, Kantor, Burgard, Kavraki and Thrun so that readers can move between this text and the literature without translating. Where the field has moved on — asymptotically optimal planners, trajectory optimization — the notation is extended rather than replaced. Each chapter adds only the symbols it introduces, in its own notation table.

Workspace and configuration space

Notation used in this chapter
Symbol	Meaning	Note
$\W \subset \R^2 \text{ or } \R^3$	The workspace: the ambient space the robot and obstacles live in
$\WO_i,; \Wfree$	The i-th workspace obstacle; the free workspace
$\Q,; q$	The configuration space and one configuration; dim Q is the number of degrees of freedom
$\Rq \subset \W$	The set of workspace points occupied by the robot at configuration q
$\QO_i = \{ q : \Rq \cap \WO_i \neq \emptyset \}$	The C-obstacle: every configuration in which the robot touches obstacle i
$\Qfree = \Q \setminus \bigcup_i \QO_i$	Free configuration space
$c : [0,1] \to \Qfree$	A path — a continuous curve in free space; a trajectory when the parameter is time	Path planning finds c; trajectory planning finds c(t) with velocities and accelerations
$\qstart,; \qgoal$	The query endpoints
$d(x, y),; d_i(q),; D(x)$	A metric; the distance from q to C-obstacle i; the distance from a workspace point to the nearest obstacle
$\rho(x, s),; \rho_R(x, s)$	Raw range along the ray at angle s from x; the same saturated at the sensor range R

Topology and kinematics

Notation used in this chapter
Symbol	Meaning
$S^1,; T^n,; \SOtwo,; \SEtwo,; \SOthree,; \SEthree$	The circle, the n-torus, and the rotation and rigid-motion groups of the plane and of space
$\varphi : \Q \to \R^n,; D\varphi,; J(q)$	The forward kinematic map, its differential, the Jacobian
$\Tq$	The tangent space at q: the velocities available there
$f(q),; g_i(q)$	Drift vector field and control vector fields of a control system
$\lie{f}{g},; \Dist,; \overline{\Dist}$	The Lie bracket; a distribution; its involutive closure
$\omega(q)\,\dot q = 0$	A Pfaffian (velocity) constraint

Potentials, graphs, and samplers

Notation used in this chapter
Symbol	Meaning
$U(q) = U_{att}(q) + U_{rep}(q),; \nabla U$	An artificial potential and its gradient
$G = (V, E),; g(v),; h(v),; f(v) = g(v) + h(v)$	A graph; cost-to-come; heuristic; the A* priority
$n,; k,; r_n$	Sample count, neighbor count, and the connection radius of a sampling-based planner
$(\epsilon, \alpha, \beta)$	The expansiveness parameters of free space, which govern how fast a roadmap captures its connectivity

Dynamics and trajectories

Notation used in this chapter
Symbol	Meaning
$M(q)\ddot q + C(q, \dot q)\dot q + g(q) = \tau$	The standard form of a robot's equations of motion: inertia, Coriolis/centrifugal, gravity, joint torques
$s,; \dot s$	Path parameter and its rate — the phase plane of time-optimal trajectory planning
$\tau_{\min},; \tau_{\max}$	Actuator torque limits

Where estimation appears

Chapters 15–16 and the capstone use the sister book\'s vocabulary unchanged: the state x_t, control u_t, measurement z_t, and belief bel(x_t). For Rusty the planner\'s q and the filter\'s x_t are the same numbers; the text says so once, where they meet.

The color code

Figures, color-coded equation terms, widget controls, and code comments all use the same seven planning roles. Where a belief is drawn, the sister book\'s five estimation colors are imported unchanged.

Start
Goal
Obstacle
C-obstacle
Roadmap / tree
Path
Robot

Prior
Prediction
Measurement
Posterior
Truth