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Trajectory vs Path in Robotics: Key Differences and Pure Pursuit Path Planning

Trajectory vs Path in Robotics: Key Differences and Pure Pursuit Path Planning

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Nov 1, 2026

In robotics and motion control, the terms Path and Trajectory are closely related but not identical. A Path captures where the robot should go (spatial geometry), while a Trajectory captures how the robot moves along that geometry over time (time law), often including references for position and derivatives such as velocity and acceleration. This “time law” distinction is repeatedly emphasized in robotics terminology discussions and textbooks/surveys. 2

A helpful decomposition used in robotics is: Path first defines a curve in configuration space parameterized by a path parameter ss, and then a Trajectory is obtained by mapping ss to time via a scaling function s(t)s(t).

Footnotes

  1. Rigorous terminology clarification - Path Planning, Path Following, Trajectory Planning, Trajectory Following and Motion Planning - Explains path vs trajectory and that trajectory adds time information to a geometric path. ↩

  2. robot path planning - an overview - States that a trajectory is a geometric path with a specified time law; includes comparison distinctions. ↩

  3. 9.1 and 9.2. Point-to-Point Trajectories (Part 1 of 2) – Modern Robotics - Defines a path as a curve parameterized by ss and a trajectory by time scaling s(t)s(t). ↩

Pure Pursuit (path following) overview

Core differences: Path vs Trajectory

Path and Trajectory differ along several dimensions:

  1. Representation

    • Path: a geometric curve in joint space or Cartesian space (configuration space geometry), typically parameterized by s∈[0,1]s \in [0,1].
    • Trajectory: a sequence/reference of robot states parameterized by time tt (e.g., position x(t)x(t), and often velocity/acceleration via time derivatives). 2

  2. Time information

    • Path: no explicit time law is specified (purely geometric).
    • Trajectory: time-law explicitly assigned (how fast/when the robot progresses along the path). 2

  3. Motion-control implications

    • When you have only a Path, a controller still must decide speeds/temporal behavior to traverse it safely/feasibly.
    • With a Trajectory, the controller can track a time-indexed reference and/or respect time-dependent constraints such as limits on velocity/acceleration and smoothness requirements (because derivatives w.r.t. time are defined). 2

  4. Common robotics workflow framing

    • A common pipeline is: plan a geometric Path, then convert it into a Trajectory by adding a time scaling s(t)s(t) (and possibly dynamics/comfort constraints).

Key terminology you will see in this area:

  • Configuration space
  • [Path parameter ss]{def="A dimensionless parameter that indexes progress along a geometric path"}
  • Time scaling
  • Time law

Footnotes

  1. 9.1 and 9.2. Point-to-Point Trajectories (Part 1 of 2) – Modern Robotics - Defines a path as a curve parameterized by ss and a trajectory by time scaling s(t)s(t). ↩ ↩2

  2. Rigorous terminology clarification - Path Planning, Path Following, Trajectory Planning, Trajectory Following and Motion Planning - Explains path vs trajectory and that trajectory adds time information to a geometric path. ↩ ↩2 ↩3

  3. robot path planning - an overview - States that a trajectory is a geometric path with a specified time law; includes comparison distinctions. ↩ ↩2 ↩3 ↩4

Memory anchor

"If you can describe the plan with a curve but you never assigned when to be at each point, it’s a Path. If you also specify s(t)s(t) (or equivalent) and thus get x(t)x(t), it’s a Trajectory." 2

Footnotes

  1. robot path planning - an overview - States that a trajectory is a geometric path with a specified time law; includes comparison distinctions. ↩

  2. 9.1 and 9.2. Point-to-Point Trajectories (Part 1 of 2) – Modern Robotics - Defines a path as a curve parameterized by ss and a trajectory by time scaling s(t)s(t). ↩

Path planning technique (one technique): Pure Pursuit

What “pure pursuit” does

pure pursuit is a Path tracking/control technique commonly used for ground vehicles. It repeatedly:

  1. chooses a look-ahead point a fixed distance in front of the vehicle along the path, and
  2. computes a steering command (e.g., curvature) that drives the vehicle toward that point.

The “chasing a moving point” intuition is standard: as the vehicle advances, the selected look-ahead point changes, producing continual corrections that keep the vehicle near the path. 2

A mathematically common representation uses the vehicle geometry to relate steering angle to the angle from the vehicle to the look-ahead point; in one survey presentation, the steering command is derived using the bicycle kinematic geometry with a look-ahead distance parameter.

Footnotes

  1. Implementation of the Pure Pursuit Path Tracking Algorithm (PDF) - Describes pure pursuit as chasing a point ahead on the path using look-ahead distance. ↩

  2. Pure Pursuit Controller - MATLAB & Simulink - Defines pure pursuit as computing curvature to reach a look-ahead point; explains the role of LookAheadDistance. ↩

  3. A Comprehensive Survey of PID and Pure Pursuit Control Algorithms for Autonomous Vehicle Navigation - Provides a geometric/bicycle-model-based derivation involving steering angle δ\delta, wheelbase LL, look-ahead distance ldl_d, and angle α\alpha. ↩

Pure Pursuit Path Tracking (algorithmic loop)

  1. 1
    Step 1

    You need the desired Path (a set of waypoints/segments), the vehicle pose (position and heading), and a look-ahead distance ldl_d.

  2. 2
    Step 2

    Compute the point on the path that lies approximately ldl_d ahead of the vehicle along the path (often via intersection of a circle of radius ldl_d with path segments).

  3. 3
    Step 3

    Determine the angle (or relative heading) between the vehicle’s current direction and the line to the look-ahead point; denote this geometry-dependent angle (commonly α\alpha in bicycle-model derivations).

  4. 4
    Step 4

    "Use vehicle geometry (e.g., bicycle kinematics) to compute steering or curvature that will move the vehicle toward the look-ahead point. One commonly cited form is based on δ=tan⁡−1(2Lsin⁡αld)\delta = \tan^{-1}\left(\frac{2L\sin\alpha}{l_d}\right), where LL is wheelbase."

    Footnotes

    1. A Comprehensive Survey of PID and Pure Pursuit Control Algorithms for Autonomous Vehicle Navigation - Provides a geometric/bicycle-model-based derivation involving steering angle δ\delta, wheelbase LL, look-ahead distance ldl_d, and angle α\alpha. ↩

  5. 5
    Step 5

    "Command steering (and often assume constant linear velocity for the basic controller) until the next control update."

    Footnotes

    1. Pure Pursuit Controller - MATLAB & Simulink - Defines pure pursuit as computing curvature to reach a look-ahead point; explains the role of LookAheadDistance. ↩

  6. 6
    Step 6

    "As the vehicle moves, repeat: update pose, recompute the look-ahead point, and update steering. The goal is to “track” the path by continually steering to the moving target on the path." 2

    Footnotes

    1. Implementation of the Pure Pursuit Path Tracking Algorithm (PDF) - Describes pure pursuit as chasing a point ahead on the path using look-ahead distance. ↩

    2. Pure Pursuit Controller - MATLAB & Simulink - Defines pure pursuit as computing curvature to reach a look-ahead point; explains the role of LookAheadDistance. ↩

Why pure pursuit works (conceptual justification)

Pure pursuit is essentially a geometric guidance strategy: it does not explicitly optimize a global cost every step. Instead, it enforces a local geometric constraint—your instantaneous motion is chosen so the vehicle heads toward a point ahead on the desired path.

This approach is often described as a curvature command to reach the look-ahead point, with the look-ahead distance ldl_d acting as the key tuning parameter: changing ldl_d changes the responsiveness/smoothness tradeoff.

Footnotes

  1. Pure Pursuit Controller - MATLAB & Simulink - Defines pure pursuit as computing curvature to reach a look-ahead point; explains the role of LookAheadDistance. ↩

Tuning parameter

"The look-ahead distance ldl_d strongly affects behavior: larger ldl_d tends to smooth motion but can reduce responsiveness; smaller ldl_d can increase responsiveness but may make tracking less smooth. Pure pursuit explicitly uses ldl_d to choose the target point ahead on the path." 2

Footnotes

  1. Pure Pursuit Controller - MATLAB & Simulink - Defines pure pursuit as computing curvature to reach a look-ahead point; explains the role of LookAheadDistance. ↩

  2. Implementation of the Pure Pursuit Path Tracking Algorithm (PDF) - Describes pure pursuit as chasing a point ahead on the path using look-ahead distance. ↩

Path vs Trajectory: quick comparison table

AspectPathTrajectory
MeaningSpatial curve (geometry)Spatial curve + time law
ParameterizationTypically indexed by ss (path parameter)Indexed by tt (time) with s(t)s(t)
Output formWaypoints/curve in state/config spaceReference x(t)x(t) (and possibly x˙(t),x¨(t)\dot{x}(t), \ddot{x}(t))
Example pipelinePlan geometry ignoring speedAdd time scaling + respect time-based constraints
Standard distinction“Without time information”“With time information”

Path vs Trajectory: what each includes

Qualitative breakdown of what is specified

Common confusion checks

Knowledge Check

Question 1 of 4
Q1Single choice

Which statement best characterizes the difference between a Path and a Trajectory?

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