Pure Pursuit Controller

The controller continuously calculates a circular arc connecting the vehicle's rear axle to a look-ahead point on the desired path. The steering angle is derived purely from the geometry of this arc and the vehicle's wheelbase. The look-ahead distance acts as a tuning parameter: shorter distances create aggressive, oscillatory tracking, while longer distances produce smoother but lazier cornering. This geometric solution avoids solving complex differential equations, making it computationally trivial for embedded systems.

Pure Pursuit is specifically designed for Ackermann-steered vehicles, which use a four-bar linkage to ensure the inner front wheel turns at a sharper angle than the outer wheel during a turn. This geometry prevents tire scrubbing. The algorithm assumes a bicycle model simplification, collapsing the two front and two rear wheels into single virtual wheels at the axle midpoints. This abstraction is valid for low-speed, non-holonomic motion where lateral slip is negligible.

The look-ahead distance (L) is the critical hyperparameter governing tracking behavior. Common tuning strategies include:

• Constant distance: Simple but fails at varying speeds.
• Speed-proportional: L = k * v, where k is a gain. This naturally dampens oscillations at high speed.
• Adaptive curvature: L decreases in tight turns and increases on straights, often using cross-track error or path curvature as inputs. Incorrect tuning leads to cutting corners (L too short) or overshooting (L too long).

Pure Pursuit is a path tracking algorithm, not a trajectory tracker. It ignores temporal constraints like velocity profiles and arrival times. The controller only cares about spatial convergence to the geometric path. This makes it suitable for applications where precise timing is secondary to spatial accuracy, such as agricultural row following or warehouse AGV navigation. For dynamic obstacle avoidance requiring time-parameterized trajectories, Model Predictive Control (MPC) is a more appropriate choice.

The algorithm exhibits well-known failure modes:

• Singularity at zero cross-track error: When the vehicle is perfectly on the path, the look-ahead point can become undefined, requiring a fallback strategy.
• Reverse driving instability: The geometric relationship inverts, causing the controller to steer away from the path. A modified formulation or separate controller is needed for reverse maneuvers.
• High-speed instability: The bicycle model assumption breaks down as lateral tire slip becomes significant, requiring a dynamic model like the Stanley controller or MPC.

A geometric path tracking controller developed at Stanford University that calculates steering based on cross-track error and heading error relative to the reference path. Unlike Pure Pursuit, which uses a look-ahead point, the Stanley controller uses the closest point on the path to the front axle. The steering law combines a nonlinear term proportional to cross-track error divided by velocity and a linear term for heading alignment. It is particularly effective at high speeds and is the primary alternative to Pure Pursuit for Ackermann-steered vehicles.

An advanced control strategy that solves a finite-horizon optimization problem at each timestep to compute steering and velocity commands. Unlike Pure Pursuit's geometric simplicity, MPC explicitly models vehicle dynamics, actuator limits, and future path preview. It predicts the vehicle's trajectory over a receding horizon and minimizes a cost function balancing path tracking error, control effort, and constraint violations. MPC handles nonlinear tire dynamics and input saturation that geometric controllers cannot, but requires significantly more onboard computation.

The kinematic steering mechanism used by most passenger vehicles and many mobile robots, where the inner front wheel turns at a sharper angle than the outer front wheel to avoid tire scrub during cornering. This geometry introduces a nonholonomic constraint: the vehicle cannot move sideways instantaneously. Pure Pursuit implicitly respects this constraint by computing a constant curvature arc to the look-ahead point, making it naturally suited for Ackermann platforms. The turning radius is bounded by the vehicle's minimum turning radius, a hard constraint that Pure Pursuit must respect when selecting look-ahead distances.

Velocity-level constraints that cannot be integrated into equivalent position constraints. For wheeled vehicles, the classic nonholonomic constraint is the rolling-without-slipping condition: the lateral velocity at each wheel contact point must be zero. This restricts the vehicle's instantaneous motion to a one-dimensional manifold of velocities aligned with its heading. Pure Pursuit generates feasible steering commands because it outputs circular arcs that satisfy this constraint by construction. In contrast, holonomic platforms like omnidirectional robots can move in any direction and do not require specialized path tracking geometry.

The lateral distance between the vehicle's reference point and the closest point on the desired path. It is the primary error metric minimized by path tracking controllers. Pure Pursuit does not directly minimize cross-track error; instead, it converges to the path by continuously steering toward a look-ahead point ahead on the reference. The relationship between look-ahead distance and cross-track error correction is nonlinear: shorter look-ahead produces aggressive correction but risks oscillation, while longer look-ahead yields smoother convergence but slower error reduction. Adaptive look-ahead schemes vary this distance based on velocity and cross-track error magnitude.

A simplified vehicle model that approximates a four-wheeled Ackermann vehicle as a two-wheeled bicycle with a single front steerable wheel and a single rear fixed wheel. The state is defined by position (x, y), heading (θ), and steering angle (δ). This model captures the essential nonholonomic constraints while remaining analytically tractable. Pure Pursuit is derived directly from the bicycle model's geometry: the circular arc connecting the rear axle to the look-ahead point defines the required steering angle. The model assumes no slip, rigid body motion, and instantaneous steering actuation, which limits accuracy at high lateral accelerations.