## 1 Introduction  
The growing of transportation demand has brought great challenges to road safety and efficiency. Automatic driving is regarded as the most promising solution due to its potential to replace human driver by an intelligent system [1, 2]. As a key part of the automatic driving system, motion planner has been designed by several types of methods including artificial potential field (APF) [3, 4], sampling-based method [5, 6] and optimal control method [7, 8]. The APF generates the control force by the gradient of the potential field composed of the target's attraction and the repulsion energy of obstacles. The advantage of APF is the low cost of computation, but it can hardly consider the future driving performance and the constraints of vehicle dynamics. The fundamental of the sampling-based method is to convert the continuous planning problem into a discrete space and then search for a sequence of points meeting the task requirements. The rapidly-exploring random tree is one of the commonly used ones because of its better real time performance [9]. However, the constraints of vehicle dynamics also cannot be dealt with effectively because of its inherent random mechanism. Moreover, the generated control command by the motion planner easily exceeds the vehicle's executable ability without consideration of vehicle dynamics. Accordingly, the actual state trajectory of vehicle will deviate from the desired one, which causes the degradation of driving performances and even traffic accidents [10]. To solve this problem, optimal control methods have been widely used to design the motion planner in recent years. Among them, model predictive control (MPC) has received more attention, because it can optimize multiple objectives in finite time horizon, meanwhile handle nonlinear dynamics and inequality constraints [11]. Essentially, MPC is a model based approach, whose performance is greatly affected by the accuracy and complexity of the model. A high-order nonlinear vehicle model is often used to study the handle limits of vehicles [12]. To analyze the influence of the arrangement of the propulsion system, the mass and the suspension on the lap time of racing cars, [PERSON] et al. used a 14-DOF vehicle model running on a parallel computation platform with 8 CPUs on each node [13]. For the application to motion planning, [PERSON] et al. compared three types of vehicle dynamics models, i.e., bicycle model, 8-DOF model and 14-DOF model, with the commercial model \"CarSim\" and concluded that the high-fidelity model is beneficial to the performance of motion planner, but its requirement of computation resource goes beyond the ability of on-board processors significantly [14]. To reduce the consumption of computation resources, some researchers devoted to the development of more efficient numerical methods for motion planning. [PERSON] et al. discretized the bicycle model with variable time-steps [15] and [PERSON] et al. accelerated the numerical optimization by selection a better initial value, which is roughly optimized with less steps [16]. Another way is to select a simple and low order vehicle model according to the working condition. When designing the automatic parking system, [PERSON] et al. optimized the parking trajectory using the kinematic model, regardless of the tire's slip and cornering characteristic [17]. This model was also used in the design of motion planner for obstacle avoidance, roundabout entry, intersection crossing and lane change [18]. However, the kinematic model only can describe the vehicle behavior finely under the condition that both longitudinal and lateral accelerations are small enough. For the application to motion planning, [PERSON] et al. studied the limitation of the kinematic model and gave out its application range quantitatively [19]. Another vehicle model widely used in motion planning is the vehicle dynamics model. [PERSON] et al. combined a modified nonlinear Fiala tire model with the bicycle model to design the lane keep controller at high speed [20].