Ackermann Steering Model — Robotics/Planar Kinematics

Ackermann Steering Model

Robotics / Planar Kinematics

The kinematic bicycle model: a steering angle and a speed become a path curvature and a yaw rate. κ = tan(δ) / L and ω = v · κ, with δ first clamped to the vehicle's mechanical steering limit.

A real Ackermann linkage steers its two road wheels through different angles; this model collapses them onto one virtual wheel at the centre of the front axle, which is the form every path tracker and every vehicle-dynamics text is written against. It is the source of the yaw rate that Unicycle Odometry integrates into a pose.

Ports

• v – the longitudinal speed of the rear axle (m/s), any size [m,n]. Negative is reverse, and is meaningful: the curvature is unchanged and the yaw rate reverses with it.
• delta – the virtual front-wheel steering angle (rad), positive to the left (counter-clockwise, per the family's convention). It must be the same size as v; the two are paired entry by entry as one vehicle's command, and neither is broadcast over the other.
• omega – the yaw rate v·tan(δ)/L (rad/s), the same size as the inputs.
• kappa – the path curvature tan(δ)/L (1/m), the same size as the inputs. It is independent of speed, which is what makes it the quantity a pursuit controller compares against: the geometry of the path the vehicle is currently on, whatever speed it is travelling at.

Parameters

• Wheelbase (m) – L, the distance from rear axle to front axle, a positive scalar. Default 2.7, an ordinary passenger car.
• Max Steer Angle (rad) – the mechanical stop, a scalar strictly inside (0, π/2). delta is clamped to ±this before the tangent is taken. Default 0.6 (about 34°), which is a typical car's lock. This is not a safety net bolted on: see the note below on why the clamp is part of the model.
• 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 wheelbase reaches the core as the folded constant 1/L and the steering limit as a literal; neither is retunable on the generated core, so re-export to change a vehicle.

The three HDL targets are simulation-only: they carry the arithmetic in real and quantize only at the port boundary. A tangent has no Q16.16 form to call, so offering them as synthesizable would be a claim the generated core could not keep.

No target carries a domain guard, and none needs one – the clamp happens before the tangent in all eleven implementations, so the argument is always inside the range the configuration pinned.

Simulink bridge

No equivalent (Support::None). Bicycle-model blocks ship in the Automated Driving Toolbox and the Robotics System Toolbox, and neither is installed on this machine (measured – isfolder([matlabroot '/toolbox/driving']) and the robotics probe both come back false). To rebuild it there: a Saturation on the angle, a Trigonometric Function set to tan, a Gain of 1/L, and a Product against the speed – in that order, since saturating anywhere else is a different curve.

Notes

• Algebraic and stateless: both outputs depend only on the current inputs, so the block cannot break an algebraic loop. It carries no tyre model, no slip and no lateral dynamics – it is the kinematic bicycle, and it stops being accurate where the tyres start sliding.
• The clamp is part of the model, not a guard. tan is unbounded as |δ| → π/2, and a single sample near the asymptote does not produce a large error but a different kind of object: a curvature of 10¹⁵, which saturates the fixed-point targets and moves a downstream pose by kilometres in one step. The mechanical stop is where a real linkage ends, so clamping to it is what the vehicle does.
• The clamp is on the ANGLE, never on the curvature. tan is nonlinear, so clamping the output afterwards would leave a different and steeper curve between the two limits – the same endpoints, a different vehicle.
• Nonlinear, and deliberately carries no state space – nonlinear in delta and bilinear in (v, delta). A fabricated linear form would let model reduction merge matrices that do not describe this block.
• Straight ahead is exact. At δ = 0 the curvature is exactly zero in every target, so a vehicle commanded straight does not drift.
• Pairs with Unicycle Odometry: feed v and this block's omega into it to dead-reckon the pose, and keep Angle Wrap out of the path unless you want the heading wrapped – that block is the only one that wraps.