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Medial Axis Planning

Motion planning is a difficult and widely studied problem in robotics. Current research aims not ony to find feasible paths, but to ensure paths have certain properties, e.g., shortest or safest paths. This is difficult for current state-of-the-art sampling-based techniques as they typically focus on simply finding any path.

The medial axis is the set of all points equidistant between two or more obstacle boundaries. For an n-dimensional space, the medial axis is an n-1-dimensional manifold which represents the connectivity of the space. Thus, the medial axis is a useful tool for computing high clearance motions and paths.

We use the observation that any point, free or not, may be retracted to the medial axis by monitoring when the witness point (for collision) changes during the retraction. We use this function to apply medial axis retraction two two large classes of sampling based motion planning algorithms: PRMs and RRTs. We develop two sampling-based algorithms: MAPRM and MARRT and show their effectiveness in solving narrow passage problems and generating high clearance solutions.


MAPRM

MARRT

Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages in the free space. We propose a general framework for sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. This framework provides a template encompassing both exact and approximate retraction approaches.

The fundamental primitive operations are the computation of penetration depth and clearance in configuration space. As these quantities can only be computed efficiently in low-dimensional C-space, the application to high-dimensional problems employs approximate clearance and penetration depth calculations. Thus, our framework supports methods that exactly or approximately retract a given configuration onto the medial axis. Exact methods provide fast and accurate retraction in low (2 or 3) dimensional space, while approximate methods extend the method to high dimensional problems, such as many DOF articulated robots.

Algorithm Clearance Computation Penetration Computation
MAPRMexactexact
MAPRM~exactapproximate
MAPRM~~approximateapproximate

Theoretical and experimental results show improved performance on problems requiring traversal of narrow passages. We also study the trade-offs between accuracy and efficiency for different levels of approximation, and investigate how the level of approximation affects the quality of the resulting roadmap.

2D Experimental Results

We compare uniform sampling to MAPRM in two 2D environments, one in which MAPRM shows significant improvement and one in which it does not. For each experiment, we sampled 1000 valid nodes. In the first example, MAPRM produces many nodes in the corridor because the obstacles are "thick". In the second example, MAPRM is less successful because the corridor is sourronded by "thin" obstacles.

PRM

MAPRM

PRM

MAPRM

2D Experimental Results

We compare uniform sampling to MAPRM in 3D environments, both for rigid bodies and articulated linkages.

S-tunnel. Uniform random sampling was unable to solve the query with 11 hours of computation time. In contrast, all MAPRM variants were able to produce a valid solution path. MAPRM~ takes slightly longer than MAPRM because the penetration approximation calculation takes longer than the exact calculation. MAPRM~~ is the slowest variant because it approximates both clearance and penetration calculation.
Hook. MAPRM cannot be used in this environment because it contains non-convex objects. MAPRM~~ was able to solve the query in just over one minute, an order of magnitude faster than uniform random sampling, requiring only 1354.3 nodes, an order of magnitude smaller than uniform random sampling. For environments like this, it is critical that nodes are generated in the narrow corridor. MAPRM~~ performs better than MAPRM~ because MAPRM~~ considers robot rotation, in addition to translation, in computating clearance and penetration depth.
Walls. For the stick robot, both MAPRM~ and MAPRM~~ out-performed uniform random sampling. Although MAPRM~ finds a solution faster than MAPRM~~, the difference is not as pronounced as in the S-tunnel environment. Since the robot needs to maintain certain orientations to pass through the holes, we conclude that constraints on rotation decrease the performance of MAPRM~.

For the articulated robot, only MAPRM~~ can be applied because of the robots high degrees of freedom. MAPRM~~ again beat uniform random samply by solving the query with a roadmap half the size. MAPRM~~ requires more time to generate a node (33.74ms on average) than uniform random sampling (0.92ms on average). The time MAPRM~~ lost during node generation was more than made up for during node connection.

Here we apply our medial axis planning framework to Rapidly Exploring Random Trees (RRTs). RRTs search the planning space by biasing exploration toward unexplored regions. We introduce a novel RRT variant, Medial Axis RRT (MARRT), which biases tree exploration to the medial axis of the free space by pushing all configurations from expansion steps towards the medial axis. We prove that this biasing increases the tree's clearance from obstacles. Improving obstacle clearance is useful where path safety is important, e.g., path planning for robots performing tasks in close proximity to the elderly. We also experimentally analyze MARRT, emphasizing its ability to effectively map difficult passages while increasing obstacle clearance, and compare it to contemporary RRT techniques. MARRT differs from RRT in that instead of growing from the tree directly toward a random configuration, it constrains the growth near the medial axis. It does so by taking a small step towards the random configuration and then pushes the resulting intermediate configuration to the medial axis. It repeats this expansion process until it reaches the maximum expansion length or fails to make progress. Below is an example of this process:

Below are examples of different tree growth in the Tunnel environment. MARRT's growth is nicely constrained to the medial axis.

RRT
RRTObst
RRT*
OBRRT
MARRT

We also see that MARRT increases both roadmap and path clearances.

Average Roadmap Clearance
Average Path Clearance

MARRT: Medial Axis Biased Rapidly-Exploring Random Trees, Jory Denny, Evan Greco, Shawna L. Thomas, Nancy M. Amato, In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), pp. 90 - 97, Hong Kong, China, Jun 2014.
Proceedings(ps, pdf, abstract)

A General Framework for Sampling on the Medial Axis of the Free Space, Jyh-Ming Lien, Shawna L. Thomas, Nancy M. Amato, In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), pp. 4439-4444, Taipei, Taiwan, Sep 2003.
Proceedings(ps, pdf, abstract)

A Probabilistic Method for Rigid Body Motion Planning Using Sampling from the Medial Axis of the Free Space, Steven A. Wilmarth, Ph.D. Thesis, Department of Mathematics, Texas A&M University, Dec 1999.
Ph.D. Thesis(ps, pdf, abstract)

Motion Planning for a Rigid Body Using Random Networks on the Medial Axis of the Free Space, Steven A. Wilmarth, Nancy M. Amato, Peter F. Stiller, In Proc. ACM Symp. Comput. Geom., pp. 173-180, Miami Beach, FL, Jun 1999. Also, Technical Report, TR98-028, Department of Computer Science and Engineering, Texas A&M University, Dec 1998.
Proceedings(ps, pdf, abstract) Technical Report(ps, pdf, abstract)

MAPRM: A Probabilistic Roadmap Planner with Sampling on the Medial Axis of the Free Space, Steven A. Wilmarth, Nancy M. Amato, Peter F. Stiller, In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), pp. 1024-1031, Detroit, MI, May 1999. Also, Technical Report, TR98-0022, Department of Computer Science and Engineering, Texas A&M University, Nov 1998.
Proceedings(ps, pdf, abstract) Technical Report(ps, pdf, abstract)

Supported by NSF

Project Alumni:Evan Greco,Jyh-Ming Lien,Peter Stiller,Steve Wilmarth

Related Projects
UMAPRM