*1.1. Related Work*

The most prevalent path planners in robotics may be classified into sampling-, search-, optimization-, and learning-based methods.

Sampling-based planners generate candidate path/trajectory primitives and then connect selected ones to form a complete solution. Such primitives are typically formed using polynomials [12–14], state lattices [15–17], and closed-loop tracking [18].

Search-based methods divide a continuous solution space into nodes in a graph and then search for a link between the nodes in the graph. Typical searchers include dynamic programming (DP) [19,20], A\* [16,21], and rapidly exploring random tree (RRT) [18,22].

An optimization-based planner formulates the concerned planning task as an optimal control problem (OCP) before solving the OCP numerically. Herein, solving an OCP numerically refers to discretizing it into a mathematical programming problem and solving it using a gradient-based optimizer. Typical mathematical programming problems include quadratic programming (QP) [20,23,24], quadratically constrained QP (QCQP) [25], nonlinear programming (NLP) [26–28], and unconstrained optimization problems [12,29]. Most gradient-based optimizers suitable for path planning only exhibit local optimization capabilities because global convergence requires a much longer runtime than one can afford. Given this property, the finally derived optimal path is close to the initial guess [30,31]. Therefore, identifying a good initial guess with global optimality contributes considerably toward finding a high-quality optimum. A sampling-/search-based planner is commonly used to provide a good initial guess [32,33].

A learning-based method generates vehicle motions based on trained data. Artificial neural networks [34], spline-constrained policy networks [35], and double deep Q-networks [36] are used in offline training. Reinforcement learning-based methods [37,38] are applied to online training.

The aforementioned four types of planners have their respective strengths and limitations. Sampling- and search-based methods do not handle the configuration space in its continuous form. Thus, they find suboptimal rather than optimal solutions. They even fail to find any feasible solution if the complexity of the planning scheme is beyond the sampling/search resolution level [23]. Increasing the resolution level is inapplicable when using a sampling-/search-based planner due to the curse of dimensionality. Despite their drawbacks, sampling-/search-based methods can efficiently describe the non-differentiable occlusion-related cost [39,40]. An optimization-based planner finds optimal paths in the continuous solution space; however, it has two typical limitations: (1) occlusion-related constraints/costs cannot be handled due to non-differentiability, and (2) the runtime is considerably longer than that of a sampling-/search-based method. Learning-based methods work fast online after a long offline training process, but they lack interpretability, and thus are rarely tested on real-world vehicles [41,42]. According to the above analysis, concluding that one type of planner fully outperforms another is difficult; instead, maximizing the advantages of each planner type while combining multiple types has been a common practice in this community.

Combining a sampling-/search-based planner with an optimization-based one renders a hierarchical planning framework; however, the optimization layer is time-consuming if the planning scheme is described in the Cartesian frame. At this point, the majority of path planners for on-road cruising scenarios describe the movements of an autonomous vehicle in the Frenet frame to enhance real-time performance. The Frenet frame, also known as the curvilinear frame, has been widely used to standardize the irregular trend of a road [43]. When using the Frenet frame, the ego vehicle is regarded as driving in a straight tunnel with left and right bounds. The Frenet frame helps convert an NLP problem into an easier form, making an optimization-based planner faster. However, the paths optimized in the Frenet frame may violate vehicle kinematic constraints because the conversions between

the Cartesian and Frenet frames are neither unique nor uniform [44]. The situation worsens when the road is curvy, as it might be in our warehouse scenario. Therefore, a perfect solution that simultaneously considers kinematic feasibility, optimality, runtime efficiency, and occlusion-avoidance performance is not yet available.
