Kinematic bicycle model

Overview

A kinematic model describes how a vehicle moves from its positions, angles and speeds, without modelling the forces that cause the motion. The bicycle simplification treats each axle as a single wheel at the axle's centre 2606.31889 §2. The invariant every variant shares is the no-slip constraint: each block's velocity lies along its own heading [arxiv:2606.31889 §2]. The authors judge this accurate enough for vehicles that run at low accelerations, and their target vehicles are slow field machines [arxiv:2606.31889 §1].

Two papers with cards use this model. 2606.31889 §2 writes the full state model and integrates it inside a controller. 2412.03174 §3.1 uses the same turning-radius relation, k = L / tan φ, to expand nodes in a search, so it needs only a closed-form pose update, not a time integration.

Variants

Articulated tractor-trailer model

A tractor with front steering angle φ and a central articulation joint with angle γ, pulling a trailer hitched at distance d_1 behind the rear axle. The state is the trailer's xy position, the rear and trailer yaws, γ and φ. The inputs are the front speed and the two joint rates 2606.31889 §2. The model's denominator is nonzero only while |γ| < π/2 2606.31889 §2. Card: arXiv 2606.31889, tractor-trailer path tracking.

Tractor-only model for front-block control

The same embedded tractor, controlled through its front block's xy position with no trailer in the model. The authors' controller knows nothing about the trailer. In headland turns this gives a front-block error below 3.8 m, and the trailer turns on a tighter circle than the reference 2606.31889 §5.1. The paper uses it as the baseline for the trailer-controlled version 2606.31889 §5.

Arc motion primitives for hybrid A* search

A search node advances by one circular arc at a fixed steering angle, not by integrating the state over time. The steering set is a fraction r_s of the maximum steering angle, and the arc length is a fraction r_a of a scale s_max. The paper's values are r_s = 0.3, r_a = 0.5 and s_max = 1.0 [arxiv:2412.03174 §3.1.1]. The turning radius is the wheelbase divided by the tangent of the steering angle 2412.03174 §3.1. The paper uses these arcs in a hybrid A* search that builds a drivable car-like path when the local trajectory is infeasible 2412.03174 §3. Its step cost penalises forward and backward arc length, steering in proportion to arc length, and steering changes. Its priority adds a weighted straight-line distance to the goal [arxiv:2412.03174 §3.1.1]. The wheelbase and the steering limit are not stated in the paper. Card: arXiv 2412.03174, resilient TEB planner. See a-star-search and motion-primitives for the general methods.

When to use which

Implementation sketch

The shared core is the state derivative of the articulated tractor-trailer. Angles are in radians, and the state order follows the paper [arxiv:2606.31889 §2]. The tractor-only variant is the same model with the trailer states removed 2606.31889 §5.1. Geometry values come from the paper's scenario 2606.31889 §5.

# State derivative of the articulated tractor-trailer  [arxiv:2606.31889 eq. 13]
# State:   x_t, y_t, theta_r, theta_t, gamma, phi    [arxiv:2606.31889 §2]
# Input:   v_f, omega_1 = gamma_dot, omega_2 = phi_dot  [arxiv:2606.31889 §2]
# Geometry: L_r, L_f, d_1, d_2 (fixed lengths)     [arxiv:2606.31889 §2]

def model_rhs(state, control, geometry):
    # Unpack the state, the input and the geometry  [arxiv:2606.31889 §2]
    x_t, y_t, theta_r, theta_t, gamma, phi = state
    v_f, omega_1, omega_2 = control
    L_r, L_f, d_1, d_2 = geometry
    # Rear yaw rate from the constraints, needs |gamma| < pi/2  [arxiv:2606.31889 eq. 7]
    theta_r_dot = (
        (v_f * sin(gamma + phi) - L_f * cos(gamma) * omega_1)
        / (L_r + L_f * cos(gamma))
    )
    # Speed of the rear axle centre  [arxiv:2606.31889 eq. 14]
    v_r = (
        v_f * cos(gamma + phi)
        + L_f * (theta_r_dot + omega_1) * sin(gamma)
    )
    # Trailer yaw rate  [arxiv:2606.31889 eq. 9]
    theta_t_dot = (
        v_r / d_2 * sin(theta_r - theta_t)
        - d_1 / d_2 * theta_r_dot * cos(theta_r - theta_t)
    )
    # Trailer axle xy velocity  [arxiv:2606.31889 eq. 13]
    x_t_dot = (
        v_r * cos(theta_r)
        + d_1 * theta_r_dot * sin(theta_r)
        + d_2 * theta_t_dot * sin(theta_t)
    )
    y_t_dot = (
        v_r * sin(theta_r)
        - d_1 * theta_r_dot * cos(theta_r)
        - d_2 * theta_t_dot * cos(theta_t)
    )
    # The joint angles integrate their rates directly  [arxiv:2606.31889 eq. 13]
    return [x_t_dot, y_t_dot, theta_r_dot, theta_t_dot, omega_1, omega_2]

For tracking, the paper discretises this model with three collocation points per sampling period and solves a finite-horizon problem at each step. Only the first input is applied, then the horizon shifts 2606.31889 §3 2606.31889 §4. The cost is quadratic in the trailer xy error and the inputs, with a terminal term 2606.31889 §4. The paper's receding-horizon parameters are a 6 s horizon and a 0.1 s sampling period 2606.31889 §5. See model-predictive-control and direct-collocation for the general methods.

The search variant needs no integration. One expansion from a pose is a closed-form arc, which the paper writes as a discrete transition 2412.03174 §3.1.1:

# One arc motion primitive of a car-like robot  [arxiv:2412.03174 §3.1.1]
# Pose: x, y, theta (radians); steer (radians); arc length in metres
# Wheelbase is not stated in the paper, so it is an input here
def arc_step(pose, steer, arc, wheelbase):
    x, y, theta = pose
    # Straight motion leaves the heading unchanged  [arxiv:2412.03174 §3.1.1]
    if steer == 0:
        return (x + arc * cos(theta), y + arc * sin(theta), theta)
    # Turning radius from the wheelbase and the steering angle  [arxiv:2412.03174 §3.1]
    k = wheelbase / tan(steer)
    # Heading advances by the arc divided by the radius  [arxiv:2412.03174 §3.1.1]
    theta_new = theta + arc / k
    # Position on the circle, measured from the start pose  [arxiv:2412.03174 §3.1.1]
    x_new = x + k * (sin(theta_new) - sin(theta))
    y_new = y - k * (cos(theta_new) - cos(theta))
    return (x_new, y_new, theta_new)

Contradictions

None between papers on the model itself; the two sources use it for different jobs, a controller and a search. Within 2606.31889, Definition 1 sets the articulation joint limit at ±π/4, while the MPC bound is ±60° 2606.31889 §5. The authors' code differs from the paper in state order, cost terms and the joint limit. See the card's Paper vs code section for the details. Within 2412.03174, the stated forward arc range [s_max, 2 s_max] does not contain r_a · s_max, which is 0.5 with the paper's own values [arxiv:2412.03174 §3.1.1]. Either r_a, the range or s_max is wrong, and the paper does not say which.

Papers

• arXiv 2412.03174, resilient TEB planner: uses car-like arc primitives in a hybrid A* search to recover a drivable path for a TEB local planner, with author-reported gains in cluttered settings.
• arXiv 2606.31889, tractor-trailer path tracking: derives the articulated tractor-trailer kinematic model and tracks the trailer's xy path with a nonlinear MPC, in simulation.