Motion Planning

April 9, 2026

4 min read

Motion planning is the study of how an agent — a robot arm, a mobile base, a drone, a self-driving car — decides how to move from where it is to where it needs to be, without colliding with anything, without violating its physical limits, and ideally without wasting energy or time. It sits at the hinge between geometry, optimisation, control, and, increasingly, machine learning.

This series is written as a standalone textbook on the subject. It starts from the geometric picture that underlies every planner ever built, develops classical discrete and sampling-based algorithms, introduces the optimisation-based and learning-based methods that dominate modern practice, and closes with current research frontiers. Each chapter builds on the preceding ones but is self-contained enough to be read in isolation once the foundations are in place.

How to read this book

Readers with no background in motion planning should work through the chapters in order. The first four establish the vocabulary — configuration spaces, collision checking, discrete and reactive planning — that the rest of the book depends on. Readers with a specific goal in mind (a 7-DoF arm, an autonomous vehicle, a learning-based pipeline) can follow one of the shortcut paths below.

Chapters

Part I — Foundations

The geometric and algorithmic language of motion planning — the vocabulary the rest of the book depends on.

What Motion Planning Is — problem statement, planner families, complexity.
Configuration Spaces and Robot Geometry — configuration spaces, manifolds, Minkowski sums, $SO(3)$ .
Collision Checking and Environment Models — occupancy, SDFs, GJK, bounding volume hierarchies, narrow passages.
Discrete Planning — Dijkstra, A*, weighted A*, Theta*, JPS, state lattices.

Part II — Classical Planning

The sampling-based and reactive algorithms that made motion planning practical in the real world.

Reactive and Local Planning — artificial potential fields, navigation functions, DWA, velocity obstacles, ORCA.
Probabilistic Roadmaps — PRM, probabilistic completeness, Gaussian and bridge samplers.
Rapidly-Exploring Random Trees — RRT, Voronoi bias, RRT-Connect, shortcutting.
Optimal and Informed Sampling-Based Planning — RRT*, Informed RRT*, FMT*, BIT*, AIT*/ABIT*.

Part III — Optimisation and Dynamics

From finding a path to finding a good one, and from purely geometric planning to planning with physics.

Trajectory Optimisation — CHOMP, STOMP, TrajOpt, direct collocation, graphs of convex sets.
Kinodynamic Planning — Dubins and Reeds–Shepp curves, LQR-RRT*, KPIECE, SST/SST*.
Planning Under Uncertainty — robust planning, chance constraints, belief-space planning, MPPI.

Part IV — Extensions

Planning in settings richer than a single robot reaching a single goal.

Multi-Robot Motion Planning — MAPF, conflict-based search, continuous multi-robot planning, MMD.
Task and Motion Planning — PDDL, PDDLStream, logic-geometric programming, LLM-guided TAMP.
Software, Benchmarks, and Reproducibility — OMPL, MoveIt, Drake, benchmark hygiene.

Part V — Modern and Learning-Based Planning

Where the field is moving: neural samplers, diffusion policies, and the open research frontier.

Learning-Based Motion Planning — learned samplers, MPNet, MPiNets, distribution shift, DAgger.
Neural Motion Planning — Diffuser, MPD, Neural MP, PRESTO, $SE(3)$ -DiffusionFields.
Frontier Seminar — open problems, recent papers, active research directions.

Reading paths for specific goals

If you already know what you’re trying to build, you don’t need to read the whole book. Each path below is a curated minimum sequence, written in the order you should read it.

Path A · planning for a 7-DoF manipulator arm

The pragmatic pipeline: enough geometry to understand the configuration space of the arm, fast sampling-based planning, and trajectory smoothing for final execution.

Path B · theoretical foundations

For readers who care why algorithms work: completeness, optimality, convergence rates, and the structure of the planning problem itself.

Path C · learning-based motion planning

The route to diffusion policies, neural samplers, and the learned planners that are pushing the state of the art.

Path D · autonomous driving and ground vehicles

Collision checking in cluttered outdoor environments, search on roadmaps, kinodynamic planning for non-holonomic vehicles, and multi-agent interaction.

Path E · humanoid and whole-body planning

Many degrees of freedom, dynamics, contact, and the long horizons that humanoid manipulation demands.

Notation

The symbols below are used consistently throughout the book. Later chapters introduce additional notation as needed.

$q \in \mathcal{C}$ — a robot configuration in the configuration space.
$\mathcal{C}_{\text{free}}$ and $\mathcal{C}_{\text{obs}}$ — the free and obstacle regions of $\mathcal{C}$ .
$x \in \mathcal{X}$ — a full state (configuration together with its time derivatives).
$u \in \mathcal{U}$ — a control input.
$\tau$ — a path (geometric) or trajectory (time-parameterised).
$\pi$ — a policy, i.e. a map from state to action.
$q_s, q_g, \mathcal{G}$ — the start configuration, goal configuration, and goal set.
$J(\tau)$ — the cost of a path or trajectory.
$d(q_1, q_2)$ — a metric on $\mathcal{C}$ .
$f(x, u)$ — the system dynamics, $\dot{x} = f(x, u)$ .

Cite this note

@online{zandonati2026motion,
  author    = {Ben Zandonati},
  title     = {Motion Planning},
  year      = {2026},
  month     = {apr},
  url       = {https://benzandonati.co.uk/notes/mp_notes/mp-index/},
  urldate   = {2026-05-10},
  note      = {Online; accessed 2026-05-10},
  keywords  = {motion-planning, index, series},
}

Ben Zandonati (2026). Motion Planning. Retrieved from https://benzandonati.co.uk/notes/mp_notes/mp-index/

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