An Iterative Algorithm to Symbolically Derive Generalized n-Trailer Vehicle Kinematics

TL;DR

The work tackles the challenge of deriving control-oriented kinematic models for generalized $n$-trailer articulated vehicles by developing an iterative, symbolic kernel-computation approach that leverages holonomic hitch constraints and nonholonomic Pfaffian constraints. It extends Ackermann steering to multi-axle trailer configurations, yielding a generalized steering law and limiting independent kinematic controls per unit through algebraic dependencies. The authors validate the resulting first-order models via partial real-driver data and comparison to a high-fidelity dynamic model, analyzing rearward yaw rate amplification and offtracking across configurations and road geometries. While the models are robust under low-slip conditions and useful for trajectory planning and supervisory control, they underperform under high-slip and when hitch friction is non-negligible, highlighting the need to couple with richer tire and joint models for stability-focused control. Overall, the symbolic, modular framework offers a tractable, controllability-friendly tool for planning and control of complex multi-trailer vehicles in 2-D motion.

Abstract

Articulated multi-axle vehicles are interesting from a control-theoretic perspective due to their peculiar kinematic offtracking characteristics, instability modes, and singularities. Holonomic and nonholonomic constraints affecting the kinematic behavior is investigated in order to develop control-oriented kinematic models representative of these peculiarities. Then, the structure of these constraints is exploited to develop an iterative algorithm to symbolically derive yaw-plane kinematic models of generalized $n$-trailer articulated vehicles with an arbitrary number of multi-axle vehicle units. A formal proof is provided for the maximum number of kinematic controls admissible to a large-scale generalized articulated vehicle system, which leads to a generalized Ackermann steering law for $n$-trailer systems. Moreover, kinematic data collected from a test vehicle is used to validate the kinematic models and, to understand the rearward yaw rate amplification behavior of the vehicle pulling multiple simulated trailers.

Figures (19)

• Figure 1: Challenges involved in control of articulated vehicle systems
• Figure 2: Articulated vehicle system and holonomic constraints: (Left): The articulated vehicle system is schematically shown, and its various frames of reference are defined here. The inertial frame $\{s\}$ is placed arbitrarily, and body frames $\{i\}$ are shown for each vehicle unit. (Right): The illustration shows the holonomic constraint between two adjacent vehicle units due to hitch connectivity.
• Figure 3: Illustration: Nonholonomic constraint. Notice that due to the assumption of rolling without slipping, the velocity vector for each wheel must be oriented along the longitudinal axis of the wheel frame of reference.
• Figure 4: Ackermann steering for a car with two axles and front wheel steering
• Figure 5: Illustration: Generalized Ackermann steering law for a vehicle unit: Under no-slip condition, the $\hat{\mathrm{y}}$-axes for all wheel frames of reference in a multi-axle vehicle unit $\mathcal{V}_i$ shall pass through the instantaneous center of rotation $O_i$. For a laterally symmetrical chassis unit as shown here, the bicycle approximation provides an accurate representation of the steering geometry.

Theorems & Definitions (4)

• Proposition 1
• proof
• Corollary 1
• proof