Parallel Path Planning with Multiple Evasion Strategies

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1 Parallel Path Planning with Multiple Evasion Strategies Stefano Caselli, Monica Reggiani, Roberto Sbravati Dipartimento di Ingegneria dell Informazione Università di Parma Parma - Italy {caselli,reggiani,sbravati}@ce.unipr.it Abstract Probabilistic path planning driven by a potential field is a well established technique and has been successfully exploited to solve complex problems arising in a variety of domains. However, planners implementing this approach are rather inefficient in dealing with certain types of local minima occurring in the potential field, especially those characterized by deep or large attraction basins. In this paper, we present a potential field planner combining smart escape motions from local minima with parallel computation to improve overall performance. The results obtained show significant improvement in planning time, along with remarkable reduction in standard deviation. A performance comparison on a benchmark problem of the potential field planner and an existing, state-of-the-art planner is also included. Our investigation confirms the effectiveness of potential field as heuristic to solve difficult path planning problems. 1 Introduction One of the major approaches toward solving difficult, high dimensional path planning problems involves the definition of a suitable artificial potential field whose local negated gradient helps in guiding the robot configuration toward the goal. Since potential fields for practical problems unavoidably include local minima, the approach must be supplemented with a strategy for dealing with these local minima. Well-established potential field-based planners, such as the Random Path Planner (RPP) algorithm by Barraquand and Latombe [4] which has pioneered the approach, deal with local minima with brownian motion phases. While the effectiveness of potential field planners has been shown by their early application on highly relevant industrial case studies [10, 12, 21, 22], their performance is still inadequate to provide solution of many useful problems within acceptable times. Associated shortcomings include excessive variability in solution times, difficulty in dealing with deep local minima of the potential field, and need to compute the heuristic for each planning environment. Yet, given the computational difficulty of path planning, potential field planners, such as RPP, remain among the algorithms of choice for high dimensional problems. An alternative approach to probabilistic path planning is represented by roadmap methods, initiated by the Probabilistic RoadMap (PRM) algorithm [19, 20, 25]. Roadmap methods randomly sample collision-free configurations and connect them into a graph during a preprocessing phase; this graph is exploited as a roadmap to efficiently answer subsequent path queries [1, 14, 19, 25, 24]. In the last few years, probabilistic roadmap methods have become very popular. However, they suffer a major difficulty in dealing with narrow passages offering minimal visibility between connected components of C-free [16, 15]. It should be noted that both potential field and roadmap methods are prone to inefficient situations owing to the random decisions they make. This issue has been addressed in a few recent extensions of the basic roadmap approach [1, 24, 14], where random sampling of configurations is biased to make it more effective. One of the goals of this paper is to develop and evaluate a similar extension for a potential field planner. Several researchers have proposed to exploit parallel computing techniques to solve motion planning problems [13], and, specifically, in order to overcome shortcomings and extend the range of applicability of potential field planners. First, among others, Gini et al. [9] have developed an OR-parallel formulation of RPP based on random competition parallel search [11]. Qin and Henrich [26] have pursued an AND-parallel approach which generates random subgoals and then tries to connect them in parallel with the initial and final configurations. We have experimentally investigated a parallel RPP implementation, based on message passing, designed according to a random competition scheme, and found it very suitable for PC clusters, thanks to its minimal communication requirements [5]. These parallel planning approaches, along with those relying on parallel probabilistic roadmaps, e.g. [2], have further extended the range of planning problems successfully dealt with [23]. In this paper we describe a parallel potential field planner which aims to overcome some of the limitations of the potential field approach and thus expand its range of applicability. First, we enhance the efficiency of potential field planning by supplementing the random escape motion from local minima with more informed search strategies, while preserving the probabilistic completeness property of the algorithm. Second, we integrate the search strategies of the potential field planner into a parallel computation scheme which proves highly effective for many degree of freedom (d.o.f.) problems. These enhancements are being integrated in a new path planning tool, called parallel Potential Field Planner (ppfp), under development at the University of Parma, which also includes exploitation of past experience [6] and pluggable distance computation functions [27]. The paper is organized as follows. Section 2 reviews the operation of potential field planners, along with the extensions we propose to more efficiently deal with local minima. Section 3 describes a parallel computation framework which allows integration of multiple heuristics for

2 path planning. Section 4 reports experimental results for a set of 11 d.o.f. problems assessing the effectiveness of potential field planning when the proposed local minima heuristics are supported by parallel computation. This section also includes a performance comparison of ppfp with a state-of-the-art probabilistic roadmap planner on a benchmark problem proposed in the literature. A final section summarizes the contributions of the paper. 2 Potential Field Planners Algorithm 1 is a simplified version of a probabilistic complete algorithm exploiting a potential field U as heuristic for the solution of a path planning problem. Algorithm 1 Path planning using a potential field. 1: Execute a gradient motion 2: while current local minimum global minimum do 3: repeat 4: Execute an escape motion 5: until escaped OR max number of tries reached 6: if escaped then 7: Execute a gradient motion 8: else 9: backtracking 10: end if 11: if reached max number of gradient-escape motions then 12: return failure 13: end if 14: end while 15: return success The potential function U is obtained from a potential attracting the robot toward the goal (placed in the global minimum of U) and a repulsive potential around obstacles. Starting from the initial configuration q init of the robot, the algorithm executes a gradient motion, following the negated gradient of the potential U until it finds a local minimum q loc. If q loc is the global minimum q goal (i.e. U = 0), the algorithm terminates with success since a path has been found; otherwise the algorithm tries to escape from q loc executing an escape motion. Once the region of attraction of the local minimum has been successfully escaped, the escape motion is followed by a gradient motion that will find another (possibly new) local minimum. The algorithm therefore alternates gradient motions with escape motions until the goal configuration is reached. To guarantee the probabilistic completeness of the planning algorithm, a random motion is usually executed as escape motion. The idea of using a random motion as a way to escape from local minima has been first proposed in the RPP algorithm that pioneered this stream of research [4]. However, this random escape phase is quite inefficient, owing to its expensive exploration of the configuration space. The chaotic brownian search, indeed, often moves back to previously explored areas. Profiling data from RPP execution on several problems, including the problems described later in this paper, show that the time spent in the aggregated random motion phases is more than three times the time required by the gradient motion phases. Moreover, the average time spent executing random motion with respect to the total computation time increases from 43% for 7 d.o.f. problems to 75% for 11 d.o.f. problems, suggesting that the inefficiency of random escape may aggravate with the complexity of the problem. Barraquand, Langlois, and Latombe [3] have shown that the number of steps t(q l ) required to escape from the attraction domain of a local minimum by a random motion has a quadratic dependence on the average radius of the attraction basin R(q l ), i.e. t(q l ) = E [( R i (q l )] ) 2, i where i is the step size along the i-th axis. When the local minimum has a large attraction basin, i.e. high E [( R i (q l )] ) i, and at the same time a large passage to exit, this motion becomes inefficient. The random motion, indeed, must always execute a number of steps at least equal to the square of the attraction radius, in order to depart enough from the minimum and thus have sufficient chances to escape its attraction basin. In summary, improvements in the escape phase can significantly improve overall planning efficiency, especially in many d.o.f. problems. 2.1 Escape motion for easy local minima We contend that in typical planning problems, most local minima possess broad passages and are more efficiently managed using an optimistic approach. This is the rationale at the base of the approach presented in [7] and summarized in the following. Instead of executing expensive random motions guaranteeing probabilistic completeness, unguaranteed but fast motions are exploited to escape from easy local minima. Two such fast motions are briefly described in the following. A more detailed analysis, along with performance data obtained by sequential computation, is presented in [7]. Algorithm StraightLine (SL) simply selects a random direction from q loc in the n-dimensional C-space, and follows it until it reaches an obstacle or a configuration such that U(q) < U(q loc ). In case a joint limit is reached, a new admissible random direction is chosen leading to a zig-zag path. With either terminating conditions, SL is followed by a gradient motion which eventually leads to a local minimum. If the original local minimum or a minimum with higher potential is reached, the move is discarded and SL tries a new random direction. Up to a maximum number of random directions from q loc are searched until the local minimum is successfully escaped or the algorithm fails. StraightLineSelect (SLS) is a modified straight line escape motion. SLS tries to further optimize the basic idea by early pruning non-promising candidate directions. During each SLS move, the potential U(q) along the straight line or zig-zag path is tracked by a simple state machine. If along the chosen direction the potential increases monotonically, with high chance the ensuing down motion would bring to the same local minimum. These directions are thus discarded without actually performing the down motion. More promising are those directions where U(q) exhibits a non-monotone behavior a clue for a valley in the potential driving to a new local minimum. Only these promising random directions are thus actually followed by the down motion search in SLS. The average cost of an escape motion using the SL or SLS algorithms depends on the average number of attempts required to escape from the local minimum using the cho-

3 sen algorithm, E[X S], and the average cost of every attempt, R(q l )C cc + E[C pd ]q, where C cc is the cost of a collision detection operation, E[C pd ] the average cost of the down motion, and q the probability to execute the down motion. In SL, q = 1 because the down motion is always executed, whereas in SLS q depends on the probability that the SLS motion crosses a potential valley. The main computational advantage of SL and SLS with respect to brownian search stems from the linear rather than quadratic number of collision check operations they perform in escaping the local minimum. Additionally, SLS only performs the down path computation for a fraction q of the explored directions. However, for complex local minima these factors could be offset by an increase in the number of attempts E[X S]. The SL and SLS heuristics turn out to be effective in many easy local minima, with high probability to escape from the local minimum with a straight line motion and low E[X S], that are dealt with quite inefficiently by brownian search. These easy local minima are likely to occur rather frequently in many planning problems. For difficult local minima, however, the probability to escape with straight line motion decreases, possibly reaching zero, and SL and SLS could become more expensive than a random motion. Hence, we have modified SL and SLS, limiting the number of attempts they perform to a value empirically chosen based on statistical data. The planner, once reached this limit, autonomously classifies the problem as too difficult for the optimistic escape motion. It then reverts to random motion, thereby guaranteeing also the probabilistic completeness of the planning algorithm. 2.2 Comparing escape techniques We have evaluated the StraightLine and StraightLine- Select, along with the random motion in RPP (labeled as R in the following tables), on the four problems shown in Figures 1, all referring to a 7 d.o.f. robot consisting of 2 links with a free base. These planning problems have been designed to cover the workspace classification proposed by Hwang and Ahuja [17]. Problem 1 (Figure 1(a)) belongs to the set of easy problems, and fits the first class in Hwang and Ahuja s classification. The space among obstacles is large compared to robot dimensions and no narrow passage is present. Problems 2 and 3 (Figures 1(b) and 1(c)) belong to the second class, i.e. intermediate difficulty problems. The planner must find a path for the robot when its movements are restricted in a narrow space among close obstacles. Problem 4 (Figure 1(d)) belongs to the final class, comprising the most difficult problems. The robot is required to enter a narrow slot, where the goal is located. If the entering orientation is incorrect, the robot might be forced to exit the slot in order to re-orient itself, a situation likely involving a very deep local minimum. Table 1 reports average and standard deviation of the planning times achieved by the modified heuristics for the four problems, where each problem has been solved 100 times. Performance results in this section have been obtained with a 450 MHz Pentium-II PC with 256 MB main memory, under the Linux operating system. Table 1 shows the improvement attained by the SL and SLS heuristics over the standard random motion. The improvement is about a factor across the span of problems, and tends to be larger for the most difficult ones, problem R SL SLS 1 avg std avg std avg std avg std Table 1: Sequential planning times for 7 d.o.f. problems. (Times in seconds; 100 independent solutions per problem). where RPP random search indeed requires an increased fraction of time. 3 Parallel motion planning Given the complexity of motion planning, the goal of a parallel approach is both to decrease the time needed for the solution of a given problem and to improve the quality of the solution. The fundamental phase of the algorithms exploiting a potential field heuristic is the construction and search in the graph of the local minima. We have chosen not to pursue an AND-parallel approach because of the difficulty to attain a satisfactory subdvision of configuration space exploration among processes without incurring in high communication overhead. Instead, we have chosen to investigate an OR-parallel random competition strategy [11], with concurrent processes exploiting different heuristics in their search for a solution [8, 6]. Briefly, random competition exploits a set of independent parallel processes executing different algorithms or the same algorithm with different initial search seeds. The execution of the planning algorithm is coordinated by a master process. Each processing node executes one worker process. Thus, parallel path planning consists of the following phases: 1. startup phase: The master process transmits the entire task to all the worker processes and sets up the algorithms or the random seeds of the algorithm in order to differentiate the search paths. 2. working phase: Each worker attempts to solve the entire problem according to the master s instructions. A worker process interrupts its search in case of success or when it receives a message from the master. 3. termination: When a worker identifies a solution of the problem or fails (i.e. it does not find a solution within the predefined time), it communicates to the master the result of its own execution. After an arbitration among received messages, the master interrupts all workers and returns the final result. This approach always improves the solution time of the problem when random algorithms are used, or when different algorithms are available with problem-dependent relative performance. In these situations, execution times are variable and it is impossible to establish in advance which

4 (a) (b) (c) (d) Figure 1: A set of 7 d.o.f. problems: (a) easy; (b) and (c) intermediate difficulty; (d) difficult. algorithm or which initialization seed will yield the best performance. Moreover, since every process can solve the problem, even if one or more processes fail the remaining ones can still find the solution. Random competition is also flexible and simple to implement, since the addition of one processor does not imply changes to the code. As in random competition communication among processors is limited to the startup and termination phases, the approach introduces a minimal communication overhead and hence is well suited even for PC clusters. Algorithm 2 Parallel path planning in a potential field with Random competition. 1: if master then 2: for all workers do 3: Send initialization seed or choose the algorithm for the worker 4: end for 5: repeat 6: Receive a message from a worker 7: until winner s message OR all workers failed 8: if winner then 9: for all workers do 10: Stop the worker 11: end for 12: Output solution 13: else 14: Output failure message 15: end if 16: else 17: Execute a gradient motion 18: while current local minimum global minimum do 19: Check for a winner message from master 20: repeat 21: Execute an escape motion 22: until escaped OR max number of tries reached 23: Check for a winner message from master 24: if escaped then 25: Execute a gradient motion 26: else 27: backtracking 28: end if 29: if reached max number of gradient-escape motions then 30: Send a failure message to master 31: end if 32: end while 33: Send success message with solution to master 34: end if The random competition paradygm can be adapted in two different ways in order to introduce concurrency in algorithm 1. In the global random competition approach, all worker processes compete in the exploration of the whole search space without any cooperation until one of them finds a solution (algorithm 2). In the initialization phase the master starts a different algorithm or sends a different seed to each worker process (line 3). In the working phase each worker executes the path planning algorithm (lines 17-33). Finally, in the termination phase a winner worker sends the solution to the master (line 33), and the master terminates the other workers. In the local random competition approach, all worker processes compete in every escape phase from a local minimum. A competition round ends when one of the workers finds a new local minimum with a lower potential value. A new competition is then started from the new local minimum. While appealing in principle, local random competition requires expensive startup and termination phases for each local minimum, thus increasing the cost for collaboration among workers, which reduces the benefit of a parallel implementation. Furthermore, this approach tends to develop paths connecting many close local minima, thereby decreasing the effectiveness of the new SL and SLS heuristics. We have tested a POSIX thread-based implementation of local random competition on an SMP shared memory computer and found it rather inefficient indeed. The experimental results of the global approach are presented in the next section. 4 Experimental results This section reports the results obtained combining the smart techniques to escape from local minima described in Section 2 with the parallel computation approach described in Section 3. The data in Tables 2 and 3 refer to experiments involving an 11 d.o.f. robot in an environment similar to Problem 2 (Figure 1(b)) solved using three workers. Analogous results have been obtained for robots with different number of d.o.f. s and using more workers. The results relying on parallel computation reported in this section have been obtained on a low-cost parallel architecture consisting of a NOW platform of 450 MHz Pentium-II PCs, with 256 MB main memory per node. Four sets of experiments have been carried out on ten

5 Table 2: Average execution time for a set of ten 11 d.o.f. problems solved by random competition among three workers. (Times in seconds; 20 independent solutions per problem). goal 3R 3SL 3SLS R+SL+SLS Execution time (sec.) R+R+R SL+SL+SL SLS+SLS+SLS R+SL+SLS Figure 2: Average execution time for the ten problems with an 11 d.o.f. robot (20 independent solutions per problem). Table 3: Standard deviation from the mean for a set of ten 11 d.o.f. problems solved by random competition among three workers. (Times in seconds; 20 independent solutions per problem.) goal 3R 3SL 3SLS R+SL+SLS SLS SL R problems, all sharing the same environment of Figure 1(b) and the 11 d.o.f. robot but differing in the goal position (generated randomly). The first three sets of experiments solve the problems with a random competition among three workers, all using the same escape technique (random motion, StraightLine, and StraightLineSelect, respectively) but differing in the initialization seed. In the fourth experiment, instead, each worker uses a different escape technique. The ten problem routine is identical in the four experiments and each experiment comprises 20 independent executions with a fresh set of initial random seeds. Tables 2 and 3 report the average execution time and the standard deviation from the mean resulting from the experiments. These experimental results show the remarkable improvement in execution time and standard deviation obtained by the proposed escape techniques. The average planning time reduces, across the set of 11 d.o.f. problems investigated, of a factor between 10 (for SL) and 13 (for SLS) compared to the planning time of three workers exploiting the standard random motion heuristic. The additional gains of reduced variance in planning time and improved path quality are also essential for practical path planning applications. Random competition, indeed, increases the likelihood that at least one of the workers can Figure 3: Percentage of problem instances solved by Random motion, SL, and SLS, on the set of ten 11 d.o.f. problems (20 solutions per problem). reach the goal going through a reduced number of difficult local minima. This situation vastly reduces execution time, as random motions now become the exception rather than the rule. While both SL and SLS show large improvement over random motion (see Figure 2, showing graphically the same data of Table 2), the marginal advantage of SLS over SL is inconclusive until supported by additional evidence. Figure 3 refers to the final set of experiments previously discussed, i.e. to R+SL+SLS concurrent searches. The winner process has been traced during the 20 solutions of the ten problems. The percentage of success of the different techniques in finding the path for each problem is shown in Figure 3. Usually, the worker process exploiting the random motion during its escape phase is slower than the other workers. Only for a few problems, such as the eighth, random motion is the winner for a significant number of solutions. This analysis confirms the effectiveness of the SL and SLS heuristics.

6 Table 4: Average execution time and standard deviation for 30 independent sequential solutions of the alpha1.5 benchmark. (Times in seconds; 30 independent solutions). solver avg std PRM PFP Figure 4: The alpha puzzle. 4.1 Comparing performance against alternative approaches A hovering question is whether a potential field planner such as ppfp is indeed a competitive planner, considering the availability of advanced path planners based on alternative approaches. This section reports a performance comparison of PFP, the sequential build of ppfp, with a state-of-the-art probabilistic roadmap planner on a benchmark problem proposed in the literature. The comparison is admittedly very limited (based on a single problem, a single alternative package, and only for sequential solution), owing to the many idiosyncrasies and input format incompatibilities of the different planners. The chosen reference planner is the wellknown Probabilistic RoadMap planner (PRM) for rigid objects in 3D [18]. The PRM package has been made available by prof. L. Kavraki of Rice University at: We have compared the performance of PFP and PRM in solving the alpha puzzle problem shown in Figure 4. This problem has been proposed by prof. Amato of (Texas A&M University) as a standard benchmark [28] for comparing path planning performance and is available at benchmarks/. One of the alpha rings must be moved with respect to the other until they are separated. The original problem, even though only 6 d.o.f., turns out to be very difficult, owing to the very small clearance between the rings. Scaled problem versions are obtained by proportionally increasing the gap between the two wings of a ring. Performance results in Table 4 refer to the scaled 1.5 version. PRM has been executed with its default parameter settings. Apparently, PRM solution time exhibits very high variance on this problem. Indeed, in a few instances PRM has been able to solve the problem in a short time with the first set of random nodes an no enhanced nodes, whereas in other cases many additional random and enhanced nodes are required, leading to very high computation times. We should also emphasize that PRM is designed to efficiently solve multiple query problems, rather than a single problem in a given workspace. We reiterate that results in Table 4 should be looked at with some caution, as they refer to a single problem; however, they suggest that advanced planners based on potential field should at least be regarded as a credible alternative to the prevailing roadmap planners for single shot problems. 5 Conclusions and future work The paper has presented a potential field planner combining parallel computation and smart escape techniques from local minima in order to attain good performance in solving complex path planning problems. The results reported in the paper confirm the effectiveness of potential field as heuristic for the solution of path planning problems. Moreover, they show that significant improvement in performance can be obtained without compromising completeness. In the heuristics proposed in this paper, random motion is only resorted to when the minimum is classified as a difficult one, thereby guaranteeing the probabilistic completeness of the proposed algorithm. As far as the future developments are concerned, we plan to extend the parallel implementation of our path planner to allow different levels of collaboration among the processes. We conjecture that the optimal competition level among workers should be somewhere between the two alternatives investigated, i.e. global path search and competition at each local minimum. Acknowledgments We thank prof. Lydia Kavraki (Rice University) for making available to us her roadmap planner, and prof. Nancy Amato (Texas A&M University) for making available to the research community the alpha puzzle benchmark. Work on this paper has been supported in part by MURST, Italy (Project ISIDE, Sviluppo di sistemi di elaborazione reattivi ed affidabili per applicazioni industriali) and by a research project supported by ENEA on parallel visualization techniques for robot systems. References [1] N. Amato and Y. Wu. A randomized roadmap method for path and manipulation planning. In IEEE International Conference on Robotics and Automation, pages , Minneapolis, MN, [2] N. M. Amato and L. K. Dale. Probabilistic roadmap methods are embarrassingly parallel. In IEEE International

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