## 1 Introduction
### Literature Review
#### 1.2.2 Path and Trajectory Planning Algorithm  
Solving the lower dimensional planning problem through the search space is a common approach for path planning. The basic idea is to discretize the state space into a graph and then use a search algorithm such as Dijkstra [24] or A* [25] to find feasible solutions or optimal solutions. The A* algorithm is an extension of Dijkstra, which assigns weights to nodes based on a heuristic-estimated cost, then performs a fast graph search to the target node. Many different variations based on this algorithm have been proposed for path planning: Hybrid A* [26], Theta* [27], Filed D* [28], D* Lite [29], etc. Besides, several planning approaches [30; 31] have been proposed to increase the system reliability, considering the localization uncertainty problem. However, when it comes to the trajectory planning problem in a dynamic and continuous environment, the search space dimensions will increase, leading to significant computational time growth. In this case, the planning problem can be approximated to a discrete sequence of optimization problems by sampling the continuous state space. The representative algorithms are Probabilistic Road Map [32], and Rapidly-Exploring Random Tree (RRT) [33] algorithms, which have a good performance in high-dimensional non-convex state spaces. The latter method has been extensively adopted in the trajectory planning for autonomous driving [34; 35], with several improvements to the RRT algorithm being proposed over the years, either to reduce the computation cost (anytime RRT [36], RRT-Connect [37], and Dynamic RRT [38], etc. ), or to optimize the generated path (RRT* [39; 40]). When faced with the trajectory planning problem in a dynamic and continuous driving environment, Artificial Potential Field (APF) methods [41] use a continuous equation, similar to the electrostatic potential field. APF methods can easily be implemented and executed in real-time for control and navigation [42]. In these algorithms, the moving object tries to reach a target and to avoid obstacles on the trajectory by using virtual forces that either attracts it or pushes it away. One benefit of this framework is that different constraints can be taken into account by simply adding specific forces, and several new potentials have been proposed, based on the road structure (with lane and boundary potentials) or on vehicle physics (guided potential), in order to use APFs in different scenarios of autonomous driving (highway [43], intersection [44]). Several improvementswere proposed to solve different limitations [45] of the traditional APF algorithm for autonomous driving. An improved APF model [46] was designed to avoid the local minimum problem, which can also be tackled by using a virtual obstacle [47], or a virtual target point [48]. By modifying the calculation of APF using fuzzy logic, the nearby obstacle problem can be dealt with. In [49], an additional artificial friction force was introduced to limit oscillations. Although APF algorithms can provide adequate trajectories as the results, the final configuration of the vehicle is unpredictable, which may lead to dangerous situations. Another general problem with this method is that the kinematic constraints of the vehicle are very difficult to take into account, thus the mechanical feasibility of the trajectories cannot be guaranteed. Elastic bands (EB) methods [50] also come from a physical analogy. In these methods, the planned path of the object is simulated through a series of springs that can be deformed to react to changes in the environment. [PERSON]. [51; 52] applied EB theory to the emergency path planning problem. [PERSON] [53; 54] proposed specific forces to respect different constraints, such as the lane constraint, to make the vehicle drive on a given lane, and the guide-potential constraint, to avoid collisions between following vehicles. The internal forces of the EB provide a constraint to neighbouring trajectory nodes, but the method still struggles with precise kinematic constraints.