A General 3D Road Model for Motorcycle Racing

TL;DR

The paper addresses the gap in control-oriented trajectory planning for motorcycles on truly nonplanar tracks by introducing a general nonplanar road model that integrates motorcycle camber dynamics. It formulates a differential-algebraic motorcycle model that couples road geometry with vehicle forces via a comprehensive momentum-balance framework and Pacejka tire modeling, enabling direct-collocation optimization of minimum-time racelines. Key contributions include the camber-axis motorcycle formulation, a tractable DAE representation, and a demonstrated 3D raceline computation on a curved track, revealing intuitive 3D racing behavior. The work advances practical trajectory planning for high-performance motorcycles on complex terrains, with potential implications for safety-critical maneuvers and race optimization.

Abstract

We present a novel control-oriented motorcycle model and use it for computing racing lines on a nonplanar racetrack. The proposed model combines recent advances in nonplanar road models with the dynamics of motorcycles. Our approach considers the additional camber degree of freedom of the motorcycle body with a simplified model of the rider and front steering fork bodies. We demonstrate the effectiveness of our model by computing minimum-time racing trajectories on a nonplanar racetrack.

Figures (5)

• Figure 1: Schematic of motorcycle geometry as viewed from behind. The motorcycle body is assumed to camber with angle $c$ about a point located a distance $r$ above the road surface. The center of mass (COM) is located at height $h-r$ and lateral offset $d$ in the frame of the motorcycle due to motion of the rider's body. $\boldsymbol{e}^b_{1} = \boldsymbol{e}^m_{1}$ and point into the page. We define $c>0$ and $d>0$ when they shift to the left from the driver's perspective. ($c>0$ and $d<0$ as shown)
• Figure 2: Schematic of motorcycle geometry as viewed from side with no camber. The front steering assembly has rake angle $\epsilon$ and offset $\delta$. The center of mass is a distance $h$ above the road surface, with the reference location directly below it. The front and rear wheels make contact with the surface at distances $l_f$ and $l_r$ along the motorcycle relative to the reference location. $\boldsymbol{e}^b_{1}$ and $\boldsymbol{e}^m_{1}$ are always equal while $\boldsymbol{e}^b_{3}$ and $\boldsymbol{e}^m_{3}$ are only equal at zero camber (see Figure \ref{['fig:motorcycle_dims']}).
• Figure 3: Tire diagram, with the tire cross-sectioned through its plane of symmetry. Tire camber and steering angle $c^t$ and $\gamma^t$ are positive as shown, and differ from the steering angle of the motorcycle steering assembly and camber angle of the motorcycle body. Tire forces $F_{x,y,z}^t$ are discussed in Section \ref{['sec:vehicle_tire_forces']}.
• Figure 4: Schematic of parametric road surface model. The surface is defined by a function $\boldsymbol{x}^p(s,y)$ and a vehicle reference location is located at a fixed normal offset $n$ from the surface. The reference location is equipped with the basis $\left\{ \boldsymbol{e}^b_{1},\boldsymbol{e}^b_{2},\boldsymbol{e}^b_{3} \right\}$, which remains tangent to the surface at all times ($\boldsymbol{e}^b_{3}=\boldsymbol{e}^p_n$). With these assumptions, position and orientation of the reference location and basis are fully determined by $\left( s,y,\theta^s \right)$.
• Figure 5: Nonplanar racetrack with snapshots of the motorcycle raceline shown every tenth of a second.