Motion planning computes a feasible collision-free path or trajectory from a start state to a goal under geometry, dynamics, and constraints. Robots, autonomous vehicles, drones, manipulators, CNC tools, warehouses, animation, and assistive systems need plans that are not only geometrically valid but executable and safe. Configuration space represents robot degrees of freedom; obstacles map to forbidden configurations. Path planning finds geometry, trajectory planning adds time, velocity, acceleration, dynamics, and control. Completeness, optimality, real-time replanning, and uncertainty depend on the algorithm and assumptions. An engineering definition states variables, units, assumptions, domains, initial and boundary conditions, sampling or update rate, uncertainty, stability or error objective, and implementation constraints. Mathematical guarantees apply to the stated model; they do not automatically cover unmodeled dynamics, finite precision, sensor faults, saturation, delay, concurrency, or hostile inputs.

Architecture, representation, and operating mechanism. Graph/grid search uses A or D over discretized states; sampling planners include RRT/RRT*, PRM, and kinodynamic variants; potential fields create local guidance; optimization methods such as TrajOpt/CHOMP refine trajectories; MPC repeatedly solves a constrained horizon; task-and-motion planning combines symbolic actions with geometry. The planner builds or searches candidate states, checks collisions and constraints, scores distance/time/energy/risk, connects feasible motions, and returns a path. Smoothing and time parameterization make it executable; a local planner or controller tracks it while monitoring obstacles and replanning. Success rate, planning time and tail, path length, clearance, smoothness, dynamic feasibility, control effort, collision probability, optimality gap, completeness conditions, memory, replanning rate, tracking error, and fallback frequency matter. Sensors, actuators, sampling clocks, quantizers, communication, memory, processors, power, thermal behavior, software scheduling, safety interlocks, and operators affect the delivered result. End-to-end design allocates error and latency budgets to named components instead of assuming ideal data and unlimited compute. Results report accuracy or error, stability and robustness margins where applicable, convergence, latency, throughput, memory, numerical conditioning, precision, energy, coverage, false alarms, and behavior at operating limits. Reference models, analytic cases, independent implementations, and confidence bounds make numerical or test evidence interpretable.

Implementation, hardware, and failure modes. Occupancy grids, signed-distance fields, meshes, kinematics, collision libraries, heuristics, sampling distributions, nearest-neighbor indexes, steering functions, constraint projection, trajectory gradients, warm starts, and learned proposal models shape results. CPU graph search and collision checks, GPU parallel sampling/distance fields, FPGA or accelerator collision kernels, map memory, sensor-to-plan latency, and real-time OS scheduling determine deadline behavior. Learned/diffusion planners add tensor inference and verification. Discretization misses narrow passages, RRT samples inefficiently, potential fields trap locally, optimization starts in bad basins, dynamics or actuator limits are omitted, maps are stale, predicted agents behave differently, and learned plans violate rare constraints. Engineering must include data movement, finite precision, resource contention, numerical or physical limits, error propagation, and deterministic behavior when assumptions are violated. Requirements, mathematical model, discretization, algorithm, numerical format, implementation, calibration, verification, deployment, monitoring, update, and incident response form one lifecycle. Versions of coefficients, transforms, test corpora, compiler settings, hardware kernels, tolerances, and assumptions remain linked to measurements.

Evaluation, verification, and deployment. Use diverse start/goal pairs, narrow and cluttered spaces, moving obstacles, uncertainty, map/calibration perturbation, timing overload, blocked goals, actuator limits, independent collision checking, simulation and hardware trials, and safe-stop tests. Perception, localization, maps, prediction, mission/task planning, global and local planning, control, vehicle dynamics, safety envelope, and operator intent form the autonomy stack. Interface latency and frame consistency matter as much as planner runtime. Plans in public or human-shared spaces require risk policy, right-of-way, accessibility, privacy-aware maps, explainable stops, event logging, remote support boundaries, and rigorous safety assurance. Verification uses analytic identities, invariants, dimensional checks, deterministic unit cases, randomized and property tests, Monte Carlo uncertainty, worst-case boundaries, high-precision references, formal reasoning where tractable, extracted or hardware models, fault injection, and closed-loop or production replay. Independent evidence is essential when one model is used to validate itself. Requirements, mathematical model, discretization, algorithm, numerical format, implementation, calibration, verification, deployment, monitoring, update, and incident response form one lifecycle. Versions of coefficients, transforms, test corpora, compiler settings, hardware kernels, tolerances, and assumptions remain linked to measurements. Results report accuracy or error, stability and robustness margins where applicable, convergence, latency, throughput, memory, numerical conditioning, precision, energy, coverage, false alarms, and behavior at operating limits. Reference models, analytic cases, independent implementations, and confidence bounds make numerical or test evidence interpretable.

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Selection and practical application. Use A for structured low-dimensional maps, RRT or PRM for high-dimensional geometry, kinodynamic search for dynamics, trajectory optimization for smooth constrained refinement, and MPC for receding-horizon execution; hybrid planners are common. Robot arms, mobile robots, autonomous cars, drones, spacecraft, surgery, warehouse routing, game characters, and manufacturing motion use planning. Sensors, actuators, sampling clocks, quantizers, communication, memory, processors, power, thermal behavior, software scheduling, safety interlocks, and operators affect the delivered result. End-to-end design allocates error and latency budgets to named components instead of assuming ideal data and unlimited compute. An engineering definition states variables, units, assumptions, domains, initial and boundary conditions, sampling or update rate, uncertainty, stability or error objective, and implementation constraints. Mathematical guarantees apply to the stated model; they do not automatically cover unmodeled dynamics, finite precision, sensor faults, saturation, delay, concurrency, or hostile inputs. CFS connects this topic to semiconductor architecture, implementation, verification, manufacturing, packaging, test, and deployed AI-system tradeoffs across the platform.