Pure Pursuit Steering — Robotics/Trajectory Generation

Pure Pursuit Steering

Robotics / Trajectory Generation

The classic path-tracking law: aim the vehicle at a point on the path a fixed lookahead distance ahead of it, and steer along the circular arc that reaches it.

1. Project the pose onto every segment of the waypoint path and keep the nearest.
2. Walk L_d forward along the path from there – that is the goal point.
3. κ = 2·y_local / L_d², where y_local is the goal's lateral offset in the vehicle frame.
4. δ = atan(L·κ) for a vehicle of wheelbase L.

The lookahead is the whole tuning: short tracks corners tightly and oscillates, long is smooth and cuts them.

Ports

• pose – the vehicle pose (x, y, θ) as a [3,1] column, per the family's SE(2) convention. θ is in radians, measured counter-clockwise from +x, and is used raw – wrapping is Angle Wrap's job and sin/cos do not care.
• kappa – the commanded path curvature (1/m), a scalar. Positive is a left turn, matching the counter-clockwise convention.
• delta – the commanded steering angle (rad), a scalar, for the configured wheelbase.

Parameters

• Waypoints – the path as an [N,2] matrix of (x, y) rows, N ≥ 2. Consecutive duplicate points are rejected rather than skipped, since a zero-length segment has no direction. The path is open: it is not closed back to the first point.
• Lookahead Distance (m) – L_d, positive.
• Wheelbase (m) – L, positive; used only for the delta output.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The whole path is baked in as constants and the segment search is fully unrolled, so no target performs a data-dependent array index. The waypoints are therefore not retunable on the generated core – re-export to change a route.

No square root is taken at run time in any target. The textbook implementation intersects a circle of radius L_d with the path, which needs one root per segment; this block works in arc length instead, with every segment length and reciprocal computed once at export. The nearest-segment search compares squared distances for the same reason.

The three HDL targets are nonetheless simulation-only: the pose must be rotated into the vehicle frame (a sine and a cosine of the heading every sample) and delta is an arctangent. The kappa path on its own would have been synthesizable.

Simulink bridge

No equivalent (Support::None). Simulink's Pure Pursuit block ships in the Robotics System Toolbox, which is not installed on this machine (measured). Note its interface differs in any case: it takes the waypoints on a port and returns a linear and an angular velocity, where this block takes the path as configuration and returns the geometry – curvature and steering angle – leaving the speed policy to the model.

Notes

• Algebraic, stateless and memoryless. The goal point is recomputed from scratch every sample, so the block cannot get stuck on a stale segment – and equally, it does not remember progress: a path that crosses itself is tracked by whichever branch is nearer.
• The end of the path is a clamp, and it is a convention. When the lookahead runs past the final waypoint the goal stays at that waypoint, so the vehicle steers toward the end and the curvature falls away as it arrives. Because L_d in the denominator stays at its configured value while the goal is nearer than that, the command there is conservative – it under-steers slightly rather than over-steering into the last point.
• There is no "no intersection" case to fall back from, unlike a circle-intersection implementation: the nearest point on the path always exists, so a vehicle far off the route still receives a sensible command that pulls it back toward the path.
• It is the exact inverse of Ackermann Steering Model: this block produces delta from a curvature, that one consumes it. Chained, they return the curvature unchanged – provided both carry the same wheelbase.
• Nonlinear, and deliberately carries no state space.