A Quasi-Steady-State Black Box Simulation Approach for the Generation of g-g-g-v Diagrams

TL;DR

The paper tackles generating g-g-g-v diagrams under non-planar driving by introducing a quasi-steady-state, black-box simulation approach. It leverages external inertial forces and a Milliken Moment Method-inspired framework to hold constant speed while steering ramps reveal tire-load transfer, enabling open-loop-stable regions to be extracted without differentiating or simplifying the vehicle model. Key contributions include an open-source, parallelized toolchain, a virtual inertial force concept to decouple longitudinal acceleration from load transfer, and a robust procedure to filter out open-loop unstable behavior while using high-fidelity or proprietary models. The method is validated on both a simple force-constrained model and a high-fidelity two-track model with variable aerodynamics, demonstrating accurate g-g-g-v envelopes and practical applicability for trajectory planning on non-planar roads.

Abstract

The classical g-g diagram, representing the achievable acceleration space for a vehicle, is commonly used as a constraint in trajectory planning and control due to its computational simplicity. To address non-planar road geometries, this concept can be extended to incorporate g-g constraints as a function of vehicle speed and vertical acceleration, commonly referred to as g-g-g-v diagrams. However, the estimation of g-g-g-v diagrams is an open problem. Existing simulation-based approaches struggle to isolate non-transient, open-loop stable states across all combinations of speed and acceleration, while optimization-based methods often require simplified vehicle equations and have potential convergence issues. In this paper, we present a novel, open-source, quasi-steady-state black box simulation approach that applies a virtual inertial force in the longitudinal direction. The method emulates the load conditions associated with a specified longitudinal acceleration while maintaining constant vehicle speed, enabling open-loop steering ramps in a purely QSS manner. Appropriate regulation of the ramp steer rate inherently mitigates transient vehicle dynamics when determining the maximum feasible lateral acceleration. Moreover, treating the vehicle model as a black box eliminates model mismatch issues, allowing the use of high-fidelity or proprietary vehicle dynamics models typically unsuited for optimization approaches. An open-source version of the proposed method is available at: https://github.com/TUM-AVS/GGGVDiagrams

Figures (8)

• Figure 1: Representation of the four-dimensional g-g-g-v diagram visualized at a vertical acceleration $a_z$ of $8$ and $15$$\frac{\mathrm{m}}{\mathrm{s}^2}$. This g-g-g-v diagram was generated with the proposed method, using the validated vehicle model of Section \ref{['sec:two_track_model_aero']}.
• Figure 2: Virtual longitudinal and vertical external forces $(F_{x,\mathrm{ext}},F_{z,\mathrm{ext}})$ used in our simulation-based generation of the g-g-g-v diagram. We perform ramp steer maneuvers, using a longitudinal controller to counterbalance $F_{x,\mathrm{ext}}$ and keep a constant vehicle speed $v$, thus ensuring QSS conditions. In Section \ref{['sec:two_track_model_aero']}, we apply our algorithm to a high-fidelity model of the autonomous race car depicted in the image.
• Figure 3: Block diagram of the QSS black box simulation approach.
• Figure 4: Structure of the "Open Car Dynamics" vehicle model according to sagmeister2024. Each vehicle is composed of a vehicle dynamics, a steering actuator, and a drivetrain model.
• Figure 5: Top: Step steer test followed by ramp steer. Middle: Corresponding wheel torque evolution. Bottom: Lateral acceleration for understeer and oversteer cases for a rear-wheel-driven race car at a velocity of $30\,\mathrm{m/s}$ and target $a_x$ of $0\,\mathrm{m/s^2}$. The detected open-loop stable maximum $a_{y}$ is marked with a black cross as described in Section \ref{['subsec:detection']}.