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Modeling and simulation of minimum time-handling inverse dynamics of a vehicle

Advances in Mechanical Engineering 2018, Vol. 10(7) 1–11 Ó The Author(s) 2018 DOI: 10.1177/1687814018786089 journals.sagepub.com/home/ade

Li-Xia Zhang , Fu-Quan Pan, Hui Zhang and Ting Feng

Abstract The performance of a vehicle in minimum time handling is highly important for the safety of the vehicle. In this study, a vehicle motion state equation with 3 degrees of freedom was established on the basis of the lateral, yaw, and longitudinal motions of the vehicle. Equations on the linear tire and motion trajectory were established with consideration of longitudinal load transfer to establish the vehicle-handling dynamics model. Steering-wheel angle, driving force equation set, and yaw angle equation had been introduced to convert the vehicle-handling dynamics model into the vehicle-handling inverse dynamics model. By introducing performance index, control set, and several constraint conditions, an optimal control model of the vehicle minimum time handling was established, which was solved by improved direct multipleshooting nonlinear programming method. A comparison of the simulation results of ADAMS/Car and MATLAB showed that both of the optimal routes input were in tangent with the road boundary. We can observe through the longitudinal velocity that the MATLAB simulation results are more similar to a straight line than that of the ADAMS/Car simulation results, which meet the psychological expectation of a driver. Thus, the inverse dynamics model on minimum time handling of the vehicle is reasonable and feasible. Keywords Vehicle, minimum time handling, inverse dynamics, sequential quadratic programming, optimal control

Date received: 2 October 2017; accepted: 30 May 2018 Handling Editor: Mario Terzo

Introduction Vehicle-handling stability under high speed determines the safety and driving performance of a vehicle. The safety of vehicle is a social focus issue. Thus, vehiclehandling stability under high speed is a significant research topic. Currently, two methods are used in the research on vehicle-handling stability: one is called open loop and the other is called closed loop. In the open-loop method, feedback from the driver is not considered, whereas in the closed-loop method, the driver model parameter is difficult to identify. In order to solve the bottleneck problem in the modeling of the driver, corresponding research has been conducted on vehicle-handling inverse dynamics since the mid1980s.

In vehicle-handling inverse dynamics, which belongs to the inverse problem in dynamics, the vehicle under high-speed driving is taken as a research object. The aim is to obtain the permissible driver-handling input based on the already known vehicle-handling dynamics model and vehicle motion response. The method of vehicle-handling inverse dynamics, which calculates the handling input for the driver under the designated School of Automobile and Transportation, Qingdao University of Technology, Qingdao, China Corresponding author: Li-Xia Zhang, School of Automobile and Transportation, Qingdao University of Technology, Qingdao 266520, China. Email: [email protected]

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2 handling performance, is different from vehiclehandling dynamics. The designated handling performance refers to passing through the given route at the fastest possible speed and not deviating from the given route boundary, thereby taking the shortest time; this issue reflects the minimum time-handling problem. Fujioka and Kimura1 used the simple vehicle transient dynamics model to express vehicles with different driving means and different steering configurations. They used the conjugate gradient descent to solve the problem of minimum time handling whose start point, end point, and trace were not constrained. However, this problem was not universal. Using the optimal control theory and Pontryagin’s minimum principle, Hendrikx et al.2 investigated the problem of minimum time and optimal handling based on vehicle dynamic model with 2 degrees of freedom (DOFs). An adjoint equation should be established by using this method and then transformed to a nonlinear two-point boundary value problem. The major limitation was that although robustness could be achieved for the imprecise start point by using gradient descent to solve the optimal control problem, the convergence rate was slow. Casanova et al.3–5 studied the problem of minimum time handling. Although the general solution was not provided, sequential quadratic programming algorithm was tried to solve the constrained minimum value of a multivariable function. With regard to the effect of yawing moment of inertia on vehicle perform