CiteSeerX — Balancing Exploration and Exploitation in Motion Planning

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Balancing Exploration and Exploitation in Motion Planning

Markus Rickert^†, Oliver Brock^‡, Alois Knoll^†

†Robotics and Embedded Systems Lab, Department of Computer Science, Technische Universit¨at M¨unchen

‡Robotics and Biology Laboratory, Department of Computer Science, University of Massachusetts Amherst

Abstract— Computationally efficient motion planning must avoid exhaustive exploration of high-dimensional configuration spaces by leveraging the structure present in real-world planning problems. We argue that this can be accomplished most effectively by carefully balancing exploration and exploitation. Exploration seeks to understand configuration space, irrespective of the planning problem, and exploitation acts to solve the problem, given the available information obtained by exploration. We present an exploring/exploiting tree (EET) planner that balances its exploration and exploitation behavior.

The planner acquires workspace information and subsequently uses this information for exploitation in configuration space. If exploitation fails in difficult regions the planner gradually shifts to its behavior towards exploration. We present experimental results to demonstrate that adaptive balancing of exploration and exploitation leads to significant performance improvements, compared to other state-of-the-art sampling-based planners.

I. INTRODUCTION

Sampling-based motion planners routinely solve complex, high-dimensional planning problems. This may seem surpris- ing, considering that the general motion planning problem [1]

is PSPACE-hard [2]. However, the configuration spaces of many practical problems contain considerable structure that may help in solving a planning problem. In addition, not all parts of configuration space have to be explored to solve a particular motion planning problem. Today’s sampling- based planners achieve computational efficiency by leveraging these properties of practical motion planning problems.

We cast motion planning as a state space search problem, similar to work in reinforcement learning [3], [4], to gain insights on how to improve the computational efficiency of motion planners. In this formulation, there are two competing goals of planning: exploration and exploitation.

Exploration seeks to understand the connectivity of the configuration space, irrespective of a particular motion planning problem. Exploration thus does not assess if a region of configuration space is relevant for a particular task. It explores to improve the planner’s understanding of configuration space. A typical example from motion planning is the initial roadmap building phase of the basic PRM planner with uniform random sampling [5].

Exploitation strives to determine a valid path for a specific task, leveraging available information. Exploration thus assumes that the information available suffices to solve the problem and just begins to act. For example, the artificial potential field approach performs pure exploitation [6].

Guided exploration, a technique performed by most sampling-based motion planners, improves on pure exploration by leveraging available information to guide exploita-

Fig. 1. EET: Exploring/Exploiting Trees

tion based on characteristics of the underlying state space.

However, this guidance is directed at achieving a complete understanding of the space and not at accomplishing a particular task.

Motion planning will be most efficient if exploration and exploitation are adequately balanced. Exploration is required to understand the connectivity of relevant configuration space regions. Exploitation should be used whenever greedy ac- tions can solve sub-problems quickly. A planner can achieve over computational efficiency by employing exploration and exploitation when appropriate, given the local and global properties of a particular configuration space.

We present a motion planner based on Explor- ing/Exploiting Trees (EETs). This planner leverages its understanding of the planning problem to balance exploration and exploitation. To our knowledge, it is the first motion planner capable of doing so. The planner begins by acquiring information about the workspace. It leverages this information for exploitation in configuration space. If exploitation fails in difficult regions, the planner gradually shifts to exploration. Our experimental results show that active balancing of exploration and exploitation results significant performance improvements, up to several orders of magnitude, when compared to state-of-the-art planners.

II. RELATEDWORK

We classify related work based on how understanding of configuration space is obtained. We consider the following categories: exploration, guided exploration, exploitation, and combinations thereof.

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The earliest approaches to sampling-based motion planning perform pure exploration. The original PRM planner with uniform random sampling [5] and the original formulation of RRTs [7] fall into this category. The exploration of these methods does not depend on the task or on information obtained during the exploration. Note, however, that the refinement step of PRM planners constitutes guided exploration.

A large number of sampling-based multi-query motion planners perform guided exploration. They assess properties of regions of configuration space to guide exploration. These properties can depend on obstacles [8], [9], visibility [10], or narrow passages [11]. Other planners use workspace information to adapt exploration [12], [13], [14], [15]. In another planner, global information about the entire configuration space is used to guide exploration [16].

Artificial potential field approaches employ pure exploitation [6]. The complete elimination of exploration makes these methods computationally efficient but also susceptible to local minima. This is also true for potential field approaches that are applied to entire paths [17].

Many sampling-based motion planners combine exploration and exploitation. However, all of these planners perform (possibly guided) exploration and exploitation as distinct steps of the planning process, rather than deliberately balancing the two within a unified framework.

Fuzzy PRM [18] and Lazy PRM planners [19] initially perform pure exploitation in a randomly generated configuration space graph. Only after this exploitation has failed, they perform guided exploration to augment the graph in difficult regions.

A series of RRT-based planners alternate between exploration and exploitation. The RRT connect [20] performs undirected exploration using two trees. It augments every exploration step with an exploitation attempt, i.e. the planner attempts to connect the two trees using a straight-line path.

Improvements of this planner replace the unguided exploration with guided exploration [21], [22]. Another variant of this type of planner uses obstacle information to guide exploration [23].

Other planners combine exploration and exploitation in the workspace and in configuration space. These planners perform efficient exploitation in the low-dimensional workspace.

The obtained information is subsequently used to perform exploitation [24], [25] or guided exploration [26] in configuration space.

III. EXPLORING/EXPLOITINGTREEPLANNER

We develop a tree-based motion planner, called exploring/exploiting (EET) planner, that exploits whenever possible and gradually transitions to exploration when necessary. The planner performs a tree expansion in configuration space, similar to RRT methods [20]. However, in contrast to the purely exploratory behavior of RRT methods, our objective is to carefully balance exploratory and exploitative behavior so as to leverage the structure inherent in the planning problem for rending motion planning as efficient as possible.

To be able to perform meaningful exploitation, the planner has to possess information about the planning problem.

The EET planner leverages three sources of information to perform exploitation and to balance between exploitation and exploration.

To guide exploitation, the planner acquires acquires global connectivity information for relevant portions of the workspace. This is achieved by a sphere-based wavefront expansion in workspace [24]. This results in a tree of workspace spheres. The branches of the tree capture the connectivity of the workspace. The size of the spheres captures the extent of the workspace. (See Figure 3(a) for an example.) These spheres and their connectivity define an approximate navigation function over parts of the workspace [24]. The gradient of this navigation function defines an attractive force for a point on the robot. This force

“pulls” the robot towards the goal in the workspace. The workspace force is projected into a direction in the robot’s configuration space using the Jacobian matrix. Exploitation uses this direction to move in configuration space.

Exploitation is likely to be successful, if it is based on accurate information. We therefore attempt to augment the information represented in the workspace spheres with more accurate information about the best configuration space directions for exploitation. This second source of information is derived from repulsive forces exerted by obstacles onto the robot [6]. We combine the repulsive forces with the attractive force derived from the workspace navigation function. The combined force can also be projected into a direction in configuration space using the Jacobian matrix.

To balance exploitation and exploration, the planner leverages a second source of information, namely the information obtained during the tree expansion steps. When exploitation fails, tree expansion attempts will become increasingly exploratory. Successful tree expansions will lead to increasingly exploitative behavior.

We now describe the the EET planner in detail (numbers in parenthesis refer to lines in Algorithm 1). The planner builds a configuration space tree, much like RRT-based planners [20]. However, every vertex q in this tree T is associated with a workspace frame F , consisting of the position and orientation of a control point on the robot. This point can be the end-effector in the case of articulated robots, or an arbitrary point on the robot in the case of rigid body robots. Our EET planner requires this information to leverage workspace-based information for exploitation.

The sphere-based workspace expansion determines a tree S of workspace spheres that capture workspace connectivity (1). The starting configuration qstartis added to the configuration space tree T . We process the tree of workspace spheres S in depth-first fashion (3). (Note that it is possible that multiple paths int the tree lead to the goal configuration. We can perform backtracking at this step, if a particular path fails.)

We attempt to expand the tree, while balancing exploitation and exploration. Similarly to RRT methods, we create a configuration towards which we want the tree to expand. In

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Algorithm 1: EET(qstart, qgoal) S ← EXPLORE(qstart, qgoal)

1

ADD VERTEX(T, qstart)

2

forall s ∈ S do

3

σ ← s.radius/3

4

repeat

5

pnew← GAUSS(s.center, σ)

6

Rnew← RAND()

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qnear← SELECT(T, pnew, Rnew)

8

if σ < s.radius ∗ β/3 then

9

Rdelta← GAUSS(0, π/3)

10

Rnew← RnearRdelta 11

if qnew← CONNECT(T, qnear, pnew, Rnew) then

12

ADD VERTEX(T, qnew)

13

ADD EDGE(T, qnear, qnew)

14

σ ← (1 − α) · σ

15

else

16

σ ← (1 + α) · σ

17

if σ > s.radius ∗ γ/3 then

18

s ← i − 1

19

until kpnew− s.centerk < s.radius

20

contrast to RRTs, however, we determine this configuration based on the workspace information contained in S. We either want to “pull” the entire robot (in the case of rigid bodies) or the robot’s end-effector (in the case of articulated robots) into the direction indicated by the workspace connectivity information stored in S.

First, we determine the variance σ for a Gaussian around the center of sphere s so that the majority of samples will fall inside the sphere (4). This parameter σ is critical:

it balances exploration and exploitation, as we will see below. We can now sample a workspace point inside the sphere, distributed around the sphere’s center (6). We also sample a random orientation (7). Together, these parameters define a workspace frame F . We obtain from the tree T the configuration q_near for which the associated frame most closely matches F (8).

If the current variance σ is relatively small, we are performing mostly exploitation (9). The parameter β determines where this threshold lies. In this case, we replace the randomly generated orientation Rrandwith an orientation that is derived by slightly modifying the original orientation Rnear

(11) with a normally distributed rotation matrix R_delta (10).

This ensures that during exploitation the orientation of the frame is only modified slightly.

Next, we attempt to connect this new frame (pnew, R_new) to the tree T (12). The connect step is described in Algorithm 2;

it is basically identical to the connect step for RRT-based planners [20]. If this connect step is successful, we add a vertex (13) and an edge (14) to the tree T . A successful addition results in the reduction of σ (15); we shift slightly towards exploitation. If the connection attempt fails, we increase σ and shift the balance towards exploration (17).

If this balance would lead our Gaussian sampling of pnew to extend beyond the boundary of the current sphere s (18), we backtrack to the previous sphere (19).

The series of these expansion steps ends when the boundary of the sphere s has been reached (20). The workspace information has now been fully leveraged and the next sphere is selected from S (3). When the final sphere is reached, the planning problem has been solved.

Algorithm 2: CONNECT(T, qnear, p_new, R_new) while qnew← EXTEND(T, qnear, pnew, Rnew) do

1

qnear← qnew 2

return qnew 3

To complete the description of the EET planner, we now discuss the extend step (Algorithm 3), used by the connect algorithm (Algorithm 2). The extend step moves the frame F associated with configuration q_near towards the frame (pnew, Rnew) and determines the corresponding new configuration qnew. This is accomplished by first determining the vector δx, pointing from the existing frame towards the new frame 1). This displacement is translated into a displacement in configuration space using the pseudo-inverse of the Jacobian (2). Based on the resulting δq, a new configuration qnew is determined (3). If it is collision free, we return qnew, otherwise we report failure.

Algorithm 3: EXTEND(T, qnear, pnew, Rnew) δx ← |(pnear, Rnear)(pnew, Rnew)|

1

δq ← J^∗(qnear)δx

2

qnew← qnear+ δq

3

if FREE(qnew) then

4

return

5

else

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return qnew 7

IV. EXPERIMENTALRESULTS

Our goal is to demonstrate the importance and the positive effect of carefully balancing exploration and exploitation in motion planning. We validate the proposed EET planner by comparing its performance with that of a PRM planner with uniform sampling (PRM in Table 3) [5], PRM with Gaussian sampling (Gaussian PRM) [9], PRM with bridge test (Bridge Test PRM) [11], RRTConnect with one tree (RRTCon- nect), RRTConnect with two trees (RRTConnect2) [20], and adaptive-domain RRT (AD-RRT) [22].

Our experiments are performed in four different scenarios.

The first scenario (see Figure 3(a)) consists of a 30m×30m×

2m large maze with walls that are 2m apart from each other.

The robot is a free-flying rigid box (6 DOF) of dimensions 3m × 0.5m × 0.5m moving from the lower left side of the maze to a position on the top right. Due to the length of the robot and the distance to the walls (see Figure 2(a))), traveling through the corners of the maze is difficult and requires exploration. The straight corridors can be solved by exploitation, provided the robot is aligned with the walls.

The maze also contains large regions irrelevant to the task;

this will distract some of the planners.

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(a) (b) (c) (d) Fig. 2. Robots used in our experiments

In the second scenario, a much larger box (1.5m × 1.5m × 1.5m) moves through same maze (see Figure 3(b)). The clearance to the walls is very small (see Figure 2(b))) so that motion through the straight corridors becomes much more difficult.

The third scenario is based on a stationary 6 DOF St¨aubli TX-60L manipulator (see Figure 3(c)). The task consists of finding a way out of the first hole in the wall (see Figure 2(c)) and then into a different hole. Both holes represent difficult narrow passages in the configuration space.

In the last experimental scenario, the holonomic, 10 DOF mobile manipulator UMan (UMass Mobile Manipulator) is placed in a 5m × 5m × 2.5m large room (see Figure 3(d)).

The robot has to perform a similar task as the stationary manipulator. However, the holes are further apart so that the robot has to move its mobile base to reach from one hole to the other (see Figure 2(d)).

The tables in Figure 3 summarize our results. The reported numbers are averaged over 20 trials; numbers in parenthesis indicate the standard deviation. If a planner was not able to solve a problem with in 20min, the experiment was aborted, we report a total time of n/a. We report the number of vertices, edges and collision detections up to that point.

Scenario 1: All planners except the single tree RRTCon- nect are capable of solving this problem within the time limit of 20min. The single query methods have the advantage of focusing on the correct area of the maze, but still spend considerable time exploring straight corridors, instead of performing exploitation. The multi-query approaches waste computational resources exploring irrelevant configuration space regions. Due to the use of workspace connectivity information, the EET planner is able to perform exploitation in the straight corridors. In the tighter turns, it slightly shifts towards exploration. The resulting EET roadmap has far fewer vertices than that of any other method. The computed path is smooth and does not exhibit the zigzaggy behavior commonly observed in sampling-based motion planning.

Scenario 2: The EET planner is the only planner capable of solving this task in less than 20min. The reduced clearance around the robot makes it difficult for other tree-based methods to move through the straight, narrow corridors. The PRMs have difficulties placing valid samples in the corridors.

Again, the EET planner benefits from its ability to balance

exploitation and exploration. In the straight corridors, it performs exploitation using the workspace information. In the more difficult turns, the planner increases exploratory behavior, as indicated by the increased number of vertices and edges in the resulting roadmap.

For this most difficult scenario, we tested an additional source of information for exploitation. As described in Section III, we combine the attractive force provided by the workspace spheres with repulsive forces from obstacles.

This additional information should further guide exploration towards the correct direction in configuration space. Indeed, as indicated by the results reported for EETRepulsive, this additional information reduces the number of vertices and edges required to solve the planning problem, relative to EET. This demonstrates that the EET planner is able to balance exploration and exploitation adequately. When more accurate information is available for exploitation, the planner shifts further towards exploitative behavior. However, in this case the overall planning time only improves marginally, as the computation of repulsive forces requires additional distance computations.

Scenario 3: The PRM with Gaussian sampling is the only PRM variant able to solve this scenario; however, the planner takes a long time, generating a very large roadmap.

The roadmap mainly covers the open configuration space regions around the robot’s base. The resulting path was of poor quality. In contrast, the RRT variants with two trees are able to solve this scenario much more efficiently. The adaptive dynamic-domain RRT requires less vertices and edges than RRTConnect, but planning times are very similar.

The EET planner again benefits from the balancing of exploitation and exploration. Initially, the robot is in a narrow passage and exploration is required. However, the exploitative behavior can direct this exploration so that the robot is able to withdraw from narrow passage with relatively little effort. The influence of exploitation directs exploration to quickly find the path through the narrow passage. The resulting path is very smooth, indicating that little unnecessary exploration was performed.

Scenario 4: In this scenario, the PRM with bridge test succeeds in 60% of the trials, the PRM with Gaus- sian sampling in 25% of the trials. Both use considerable computational resources; it is the only PRM variant able to

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solve the problem. The RRTConnect variants with two trees solve this task faster than the the third scenario. This is due to the fact that robot has more degrees of freedom and a suitable direction for expansion is easier to find. The single tree variant is unable to solve the narrow passage, leading to the goal configuration. Again, the EET planner’s ability to balance exploration and exploitation results in the best planning performance.

Summary: The behavior of PRM planners is governed by exploration. As a result, they waste computational resources in an attempt to understand the entire configuration space.

RRTConnect planners with two trees benefit from the fact that their exploration is seeded with the initial and final configuration of the robot, both of which are inside the two only narrow passages. Once the RRTConnect planners have found a way out of the narrow passage, the connect step (exploitation) allows them to solve scenarios 3 and 4 efficiently. However, as scenario 3 illustrates, the pure exploratory behavior of RRT variants causes difficulties in finding paths into narrow passages. The failure of RRTConnect with a single tree to solve scenarios 3 and 4 supports this observation.

The EET planner outperforms all other planners in all of the experiments. It is able to overcome the difficulties of PRM-based and RRT-based planners by carefully balancing exploration and exploitation. Using workspace information, it performs exploitation without trapped in irrelevant regions of configuration space. The ability to balance exploration and exploitation during tree expansion based on the difficulty of local configuration space leads to highly effective motion planning. The paths generated by the EET planner are of much higher quality that those generated by other sampling- based planners and do not require smoothing prior to execu- tion. This indicates that exploitation was able to direct the planning process effectively.

V. CONCLUSION

We propose a new categorization of sampling-based planning methods. This categorization relies on the concepts of exploration and exploitation, commonly used to characterize reinforcement learning techniques [3], [4]. Our discussion reveals that existing sampling-based motion planners do not deliberately balance between exploration and exploitation.

We argue that balancing exploration and exploitation is the most effective way of minimizing the amount of configuration space exploration required to solve a particular motion planning problem. If exploitation is successful, it sig- nificantly advances the planner towards a solution with neg- ligible computational cost. However, overuse of exploitation leads to a waste of computational resources, as the planner begins to explore irrelevant regions of configuration space.

An efficient planner should thus perform sufficient global exploration to allow exploitation whenever possible, and it should gradually switch to exploration when exploitation alone cannot solve a locally difficult part of the configuration space.

We present exploring/exploiting tree (EET) planners, a motion planning method that deliberately balances exploration and exploitation based on information about the workspace and configuration space. Workspace information is acquired prior to the planning process and used to avoid getting trapped globally. Information about specific regions of the configuration space is acquired during configuration space sampling. This information is used to adjust the balance between exploration and exploitation in accor- dance with the difficulty of the local planning problem.

Our experimental results demonstrate that the balancing of exploration and exploitation performed by the EET planner leads to greatly improved planning performance in a variety of difficult, real-world motion planning problems, compared to state-of-the-art sampling-based planners.

ACKNOWLEDGMENTS

We gratefully acknowledge support by the National Sci- ence Foundation (NSF) under grants CNS-0454074, IIS- 0545934, CNS-0552319, and CNS-0647132.

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(a) (b) (c) (d)

Planner Vertices Edges Collision Detection Queries Workspace Time (s) Total Time (s)

PRM 10,172 (2,199) 10,092 (2,182) 2,513,741 (544,651) n/a 112.52 (26.3)

Gaussian PRM 8,127 (2,526) 7,815 (2,428) 3,230,438 (1,006,646) n/a 114.49 (39.42) Bridge PRM 10,939 (4,671) 7,749 (3,325) 9,281,244 (4,011,784) n/a 225.23 (105.77) RRTConnect

RRTConnect2 2,665 (760) 2,663 (760) 91,986 (22,510) n/a 57.93 (26.38)

ADD-RRT 2,847 (463) 2,845 (463) 52,978 (8,150) n/a 76.49 (23.22)

EET 437 (5.21) 436 (55.21) 12,788 (729) 0.54 (0.05) 1.09 (0.19)

(a) Free-flying box (6 DOF, 3m × 0.5m × 0.5m) in a maze

Planner Vertices Edges Collision Detection Queries Workspace Time (s) Total Time (s) PRM

Gaussian PRM 63,879 (559) 56,793 (543) 37,395,409 (240,963) n/a n/a

Bridge PRM 38,438 (132) 23,235 (105) 59,036,085 (142,949) n/a n/a

RRTConnect 3487 3486 1092407 n/a n/a

RRTConnect2 4928 4926 649104 n/a n/a

ADD-RRT 1,651 1,649 23,759 n/a n/a

EET 1,805 (290) 1,803 (290) 19,240 (4,658) 0.009 (0.017) 11.98 (5.89)

EETRepulsive 1,381 (302) 1,379 (302) 17,439 (4,672) 0.006 (0.008) 9.52 (6.35)

(b) Free-flying box (6 DOF, 1.5m × 1.5m × 1.5m) in a maze

PRM 13,740 13,657 1,618,396 n/a n/a

Gaussian PRM 9,305 (588) 8,691 (640) 1,652,267 (115,149) n/a 957 (83)

Bridge PRM 9,688 (38) 8,314 (2.12) 2,416,521 (4,992) n/a n/a

RRTConnect 18,017 (285) 18,016 (285) 1,026,929 (11,307) n/a n/a

RRTConnect2 781 (328) 779 (328) 37,437 (20,835) n/a 11.22 (6.36)

ADDRRT 584 (203) 582 (203) 30,288 (14,131) n/a 10.07 (4.22)

EET 23.75 (18.07) 22.75 (18.07) 2,015 (1,388) 0.06 (0.01) 0.71 (0.41)

(c) Fixed holonomic manipulator (6 DOF) reaching through two holes in a wall

PRM 21,878 6,147 7,97,470 n/a n/a

Gaussian PRM 3,520 (2,667) 3,462 (2,714) 990,864 (774,581) n/a n/a

Bridge PRM 3,848 (2,135) 3,458 (1,930) 1,252,420 (717,738) n/a 735 (416)

RRTConnect 10,018 10,017 953,757 n/a n/a

RRTConnect2 143 (64) 141 (64) 7,238 (4,446) n/a 4.59 (2.83)

ADD-RRT 130 (41) 128 (41.36) 6,332 (4,003) n/a 4.64 (2.27)

EET 28.9 (17.83) 27.9 (17.83) 1,862 (1264) 0.18 (0.03) 1.45 (0.82)

(d) UMan (10 DOF) reaching through two holes in a wall

Fig. 3. Comparison of planner performance in four experimental scenarios

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