Single-track models of an A-double heavy vehicle combination
This report presents single-track models representing an A-double vehicle combination. The equations of motion are expressed relative the center of mass for the first vehicle unit and are derived using Lagrangian formalism. The symbolic manipulations of the equations have been done using the software Mathematica.
The model derived in this report is intended to be valid for studies of vehicle cornering in high speed with moderate lateral and longitudinal acceleration levels. In order to keep the model complexity as low as possible, the derived equations of motion are simplified. The steering angle and the articulation angles of the towed vehicle units are assumed to be small. All products of the steering angle, the yaw rate of the first unit and the articulation angles and their time derivatives are set to zero. Further, the representation of longitudinal vehicle velocity leads to different level of model complexity. Firstly, the longitudinal velocity is assumed to be constant. Together with the use of a linear lateral tyre model this results in a fully linear vehicle model, consisting of eight states and one input. Secondly, the longitudinal velocity is allowed to be varying and is represented as a model state variable. Also here a linear lateral tyre model is used. The resulting vehicle model becomes non-linear, consisting of nine states and six inputs. Further simplifications of the longitudinal dynamics is carried out by assuming e.g. slowly varying longitudinal velocity and longitudinal forces only on the towing unit. This results in a non-linear model consisting of nine states and two inputs. This model is to be compared with the fully linear model.
Realistic linear tyre parameters are found by tuning the derived linear single-track model with results from a high fidelity vehicle model developed at Volvo Group Trucks Technology (VGTT). The tuning was made using the software Matlab and a built-in Genetic Algorithm(GA), where the summarized squared difference in yaw rate gain for vehicle units was minimized.
The performance of the derived vehicle model, including the difference in treatment of the longitudinal velocity, is evaluated in a comparison with the high fidelity model developed at VGTT. The open-loop analysis carried out in the comparison are step steer response analysis, single-sine steer response analysis, braking in a turn analysis (ISO14794).
Long vehicle combinations
Single track model