1 Generalized Sampling based Motion Planners S. Chakravorty, S. Kumar Abstract—In this paper, generalized versions of the probabilis- tic sampling based planners, Probabilistic Road Maps (PRM) and Rapidly exploring Random Tree (RRT), are presented. The generalized planners, Generalized Probabilistic Road Map (GPRM) and the Generalized Rapidly Exploring Random Tree (GRRT), result in hybrid hierarchical feedback planners that are robust to the uncertainties in the robot motion model as well as uncertainties in the robot map/ workspace. The proposed planners are analyzed and shown to be probabilistically complete. The algorithms are tested on fully actuated, as well as under- actuated robots, on several maps of varying degrees of difficulty, and the results show that the generalized methods have a significant advantage over the traditional methods when planning under uncertainty. I. I NTRODUCTION In this paper, generalized versions of the traditional probabilistic sampling based planners, the probabilistic roadmap (PRM) and the rapidly exploring random tree (RRT), are presented. The traditional techniques are generalized to take into account uncertainties in the robot motion model as well as uncertainties in the obstacle locations in the map. These techniques result in hybrid hierarchical feedback planners in the state space of the robot. The algorithms are analyzed to show that they are probabilistically complete, i.e., they generate hybrid planners with a guaranteed minimum probability of success if such a planner exists. Experiments are performed on an idealized planar holonomic point robot, as well as a nonholonomic unicycle robot, and results show that the performance of the generalized planners, in terms of their probability of success, is significantly improved when compared to the traditional techniques. Motion Planning of robots while avoiding obstacles in the workspace has been an active area of research for the past several decades. Classical motion planning can be roughly divided into three different deterministic approaches [1]: cell decomposition, roadmaps and potential field methods. The cell decomposition and roadmap techniques are deterministic methods in the sense that the environment of the robot is sampled/ discretized in a deterministic fashion. However, the problems are PSPACE-Hard [2] and in order to circumvent this computational complexity, randomized sampling based methods known as probabilistic roadmaps (PRM) were introduced [3], [4]. PRM techniques usually do not take the dynamics of robotic platform into account and this can lead to serious performance issues. To address these issues, the rapidly exploring random tree (RRT) was introduced as a Suman Chakravorty is an Assistant Professor, Department of Aerospace En- gineering, Texas A&M University, College Station, schakrav@aero.tamu.edu S. Kumar is a Graduate Research Assistant, Department of Aerospace Engineering, Texas A&M University, College Station randomized sampling based planner that takes into account the dynamics of the mobile robot [2], [5] while building a tree of dynamically feasible trajectories in the free space of the robot. The randomized PRM and RRT techniques have resulted in the solution of motion planning problems in very high dimensional state spaces which were hitherto unsolvable using deterministic motion planning techniques. However, both PRM and RRT are open loop planners designed for perfectly known robot models/ workspaces and our primary motivation in this paper is to generalize these two techniques in order to generate feedback motion planners robust to uncertainties in the robot motion model as well as the map. Just as PRM/ RRT helped solve motion planning problems in high dimensions, we expect that the generalized techniques, the generalized probabilistic roadmap (GPRM) and the generalized rapidly exploring random tree (GRRT), will help us solve feedback motion planning problems in high dimensional state spaces under uncertainty (in fact, using existing techniques, these problems can only be solved in low dimensional / discrete state and control spaces). The GPRM and GRRT techniques are closely related to Markov Decision Processes (MDP), Sequential Composition (SC) and other generalized versions of the PRM. In the following, we examine the relationship between our techniques and the seemingly disparate planning techniques: MDPs, SC and other generalized PRMs. The robot motion planning problem can be formulated as a Markov decision problem (MDP) if the uncertainties in the robot and the environment are modeled probabilistically. However, MDPs are virtually intractable for anything but small to moderate state spaces as they are subject to the famous “curse of dimensionality”. In particular, it is nearly impossible to solve these problems in continuous state and control spaces even without constraints. In the presence of constraints, there are no well established techniques to accomplish the planning. One approach to resolving the issue of dimensionality is through the use of hierarchical methods, an approach that is seen in most biological systems. A variety of methods to solve large MDPs in a hierarchical, model-free fashion have been developed, and the field of research is known as hierarchical Reinforcement Learning (Hierarchical RL) [6], [7]. These methods, instead of taking actions invoke policies/ options at each state which continue until termination. Moreover, if it is assumed that these temporally abstract policies can terminate only at one of a few “distinguished” states, then the original large MDP can be transformed into a significantly smaller semi Markov Decision Problem (SMDP) that needs to be solved only at the distinguished states and thus, drastically reduces the computational burden of the Dynamic Programming