Probabilistic model based path planning

Highlights

• A transition probabilistic matrix is estimated based on a modified RRT algorithm.

• By introducing the safe coefficient, the path can keep a safe distance from obstacles.

• The well-known push-based dynamic programming is used to obtain optimal paths.

Abstract

Path planning shows great potential for exploring indoor and outdoor environments. In this paper, a probabilistic method is proposed to design path planners based on transition probabilistic matrices and signed distance functions. The transition probabilistic matrix is constructed by collecting path sequence data generated by performing a modified RRT with many times. Moreover, the signed distance function is introduced to simulate the safety coefficient which can guarantee a suitable distance between robots and obstacles. By combining the transition probability and the safety coefficient, our path planning task is modeled as a maximal probability sequence decision problem which in essence is equivalent to a minimal cost path problem, and then the dynamic programming solver is achieved by using the push-based efficient implementation of Bellman–Ford’s algorithm Kleinberg and Tardos (2006). Several path evaluation criteria are also used to evaluate path planning results, and plenty of experimental results illustrate the effectiveness of the proposed method.

Introduction

Path planning is one of fundamental and important components for robots and Unmanned Aerial Vehicles (UAVs). In the sensing–planning–actuation pipeline, sensing module captures the global or partial environment information by sensors, and planning module accepts the information and outputs a decision which is then pushed to the actuation module, then control commands are generated and guide motions of agents. Therefore path planning module acts as a central role in the pipeline, and it is thus extensively used by many real world applications, including autonomous driving [1], smart service robots [2], [3], UAVs [4], and micro-flow based bio-molecular computation (MF-BMC) [5].

Due to extensive real-world applications, currently many path planning algorithms have been proposed to tackle with various task requirements [6], [7], [8], [9]. However, There is no universal solution that is capable of dealing with all path planning tasks in practice. In general, path planning algorithms can be coarsely divided into two categories, i.e., complete methods and probabilistic complete methods. Complete methods can return a result which is either a collision-free path joining given start and target positions or a flag that the path does not exist. A classical complete method is the graph-based path planning algorithms [10]. However, complete methods need explicit representation of the configuration space which is time-consuming especially in higher-dimensions. Probabilistic complete methods, on the other hand, can always return a valid solution if there exists solutions, but the solution is possibly not optimal. For example, Probabilistic Road Map (PRM) [8] and rapidly-exploring RandomTree (RRT) [9] are two classical probabilistic complete path planning methods, but both always show different results, i.e., path planning results are not deterministic.

Path planning algorithms designed on grid world state space are (resolution) complete, but local minima (e.g., artificial potential field [11]) or resolution complete results will be achieved. In contrast, sampling based stochastic planners, such as Expansive Space Tree (EST [12]), PRM [8] and RRT [9], do not need to discrete the state space, and then they are fairly effective even in higher dimensions since obstacle space needs not be constructed anymore. Unfortunately, randomness nature results in random paths which restricts their vast applications. Therefore the goal of this paper is to take advantage of randomness of RRT algorithm [9]. In concrete, we perform $RRT$ procedure many times and then extract deterministic path information according to these experimental results. Especially we construct a transition probabilistic matrix which help us determine a maximal probabilistic path from many stochastic paths.

Path evaluation criteria are crucial for path designing algorithms. Minimal time cost [13] and shortest path length [14] are two common requirements for many path planning tasks. A large number of repeated experimental results illustrate that RRT algorithm tends to generate moderate length paths, and the histogram (see Fig. 1) verifies the statement. The path length data is collected by running RRT algorithm with 2000 times on the map (see Fig. 4(a)), and the start and goal positions are (40,440) and (440,440), respectively. But minimum distances between robots and obstacles are very small in most cases, and the minimum distance is a major indicator for path safety. In intelligence service robots and UAVs, an enough safety distance is desired [3]. Thus RRT algorithm needs to be improved to maintain a user preferred safety distance and also a proper path length.

Efficient and flexible representation of obstacle space is always required by various path planning algorithms. Trajectory optimization-based planning methods can run very fast even in very complex environments, but distance information and distance gradients to obstacles need to be computed for all points in a map [13]. As an analytic function, the signed distance field can conveniently represent obstacles in Euclidean space [15], therefore distance information and distance gradients can be efficiently computed with an analytic manner. Recently incremental signed distance field functions [7], [16] updating strategies have been proposed to represent obstacle space for real-time dynamic environments. In this paper, we compute signed distance functions using RBFs to get the distance information. We then transform the distance information into a safety coefficient map which is then combined with our transition probabilistic matrix, and a probabilistic model is built to guide the path designing.

Noting that our probabilistic model in essence is a directed diagram with cycles, and the size of maps may be very large. Therefore the dynamic programming solver based on the push-based efficient implementation of Bellman–Ford’s algorithm [17] is adopted.

The rest of this paper is organized as follows. In the next Section some related work are introduced. The probabilistic model of path planning is present in Section 3. In the next section the dynamic programming solution for our path planning model is also proposed. In Section 5, several path evaluation criteria are presented. At last, plenty of experimental results are shown and conclusions are made.