Self-Motion Graph in Path Planning Problem with End-effector Path Specified

Zhenwang YaoSchool of Engineering Science, Simon Fraser University

Abstract

Redundant robots have good dexterity in avoiding obstacles when performing tasks. We propose an enhanced probabilistic method to solve the problem of path planning along a specified end-effector path. Introducing a data structure named Self-Motion Graph, we allow robot self-motion along the end-effect path. Computer simulations show that this enhancement improves performance in some simulated cases.

Problem Definition

As shown in the figure above, given an end-effector path in workspace, xg(t); t[0; T], and a start robot configuration q0, determine an entire joint space path q(t), such that F(q(t)) = xg(t); t[0; T], where F(q) is the robot forward kinematics function.

Related Work

The approaches for this problem:1. On-line, Local

• Jacobian-based methods [1]Drawback: Local minimum issues

2. Off-line, Global• Extension from Jacobian-based methods [2]

Drawback: High computation• Probabilistic methods [3]

Of our main interest

Probabilistic Methods

The basic idea is to establishe connections among semi-randomly generated configurations for the sequential sampling poses:• Configuration Generation: (semi-random) techniques for closed-chain robots

• Active-passive link decomposition [4]• Random loop generator [5]

• Planning Strategies• Greedy

Drawback: Depth-first search characteristic• RRT-Like

Drawback: Slow convergence• Combinations

RRT-Connect-Like, RRT-Greedy-Like, etc.

Copyright © Zhenwang Yao, 2004

RoboticsLabortrary

Planners Case (a) Case (b) Case (c)

Greedy 43.5 19.4 29.9

RRT-C 43.6 3.8 18.1

RRT-G-C 11.6 5.7 7.6

SMG-Greedy 8.7 2.5 6.2

SMG-RRT-C 9.4 3.7 6.8

SMG-RRT-C2 11.1 3.0 13.0

(a)

(b)

(c)

Comparison in Planning Time (s)

Planners Case (a) Case (b) Case (c)

Greedy 24457 99366 69552

RRT-C 6038 80573 35818

RRT-G-C 8659 21380 14242

SMG-Greedy 3528 16007 11061

SMG-RRT-C 5462 15330 10812

SMG-RRT-C2 3852 11210 10818

Comparison in Collision Checking (# of calls)

Current

Planners

(a)

Success/Failure

(b)

Success/Failure

Greedy 0/20 8/12

RRT-C 0/20 15/5

RRT-G-C 0/20 19/1

(a) Start with a “bad” conf. (b) Start with a “good” conf.

Inefficiency in Existing Methods and Enhancement Proposed

Data Structure and Implementation

Experimental Results

Note:1. The running time is measured on a Pentium-4 2.0G PC.2. The experiments were done with our in-house developed software library MPK, the Motion Planning Kernel [6].3. The first three planners are the planners in [3], and the last three planners are enhanced by Self-Motion Graph.

Inefficiency:As shown on the left, with current planners and the same parameters, the two scenes attempt to find a path starting with different configurations for the first pose. The experimental result demonstrates that a “bad” configuration may result in failure to find a path.

Explanation:In the current planners, no self-motion is allowed, therefore the number of configurations in the final found joint path is limited to the number of poses in the end-effector path. At the same time two consecutive configurations should be close enough, such that the end-effector does not move off the specified path. These two constraints restrict the robot to move out of a bad configuration by a limited number of small movements, which may be difficult in some cases.

Proposed enhancement:To allow the self-motion.

What is self-motion:Self-motion is a movement that does not affect the end-effector pose (position and orientation). Mathematically, self-motion is in the null-space of the Jacobian matrices. Self-motion can be used to satisfy additional constraints including obstacle avoidance, joint limit avoidance, and singularity avoidance. In our enhancement, we explore self-motion in a probabilistic fashion.

Self-Motion Graph:In the tree-type data structure used by the current planners, each level of the tree corresponds to a pose along the end-effector path. Self-motion is incorporated into the planners by further expanding a node in the tree into an equivalent group of nodes corresponding to the same pose, which is represented by a connected graph, called the Self-Motion Graph (SMG).

Self-Motion Graph Exploration:To reduce the computation to build and search the explored tree, we only create SMGs when required. Only for the poses where the robot has difficulty extending further the SMGs are propagated; for the poses where the planners can extend to the next pose within a fixed number of retries, simple nodes are retained.

Applications

The problem, the path planning problem with end-effector path specified, has wide applications, such as cutting, painting, inspection, etc. SMGs help in applications where time requirement along the end-effector path is loose, like inspection robots. On the other hand, some applications may not benefit from this enhancement. For example, in spray painting applications, a constant end-effector velocity is required along the end-effector path, and self-motion generates an inconstant end-effector velocity (thereby uneven paint deposition may result).

Conclusion

By introducing the Self-Motion Graph, an enhancement is proposed based on current probabilistic planners for path planning problem with end-effector path specified. To explore robot self-motion, we adopt configuration generation techniques for closed-chain robots. Computer simulations prove that this enhancement indeed improves the performance, in terms of finding a collision-free path.

Future Work

The future work will focus on path smoothing. The path found by probabilistic methods normally has some jerky movements, and path smoothing is to reduce the jerky movements and optimize the path for later execution stage.

Reference[1] A. Maciejewski and C. Klein. Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments. International Journal of Robotics Research, 4:109–117, 1985.

[2] S. Seereeram and J.T. Wen. A global approach to path planning for redundant manipulators. IEEE Transactions on Robotics and Automation, 11:152–160, 1995.

[3] G. Oriolo, M. Ottavi, and M. Vendittelli. Probabilistic motion planning for redundant robots along given end-effector paths. In IEEE/RSJ International Conference on Intelligent Robots and Systems, pages1657–1662, 2002.

[4] L. Han and N. Amato. A kinematics-based probabilistic roadmap method for closed kinematic chains. In B. Donald, K. Lynch, and D. Rus, editors, Workshop on Algorithmic Foundations of Robotics, pages 233–246, March 2000.

[5] J. Cortes, T. Simeon, and J.P. Laumond. A random loop generator for planning the motions of closed kinematic chains with PRM methods. In IEEE International Conference on Robotics and Automation, pages 2141–2146, 2002.

[6] I. Gipson, K. Gupta, and M. Greenspan. MPK: An open extensible motion planning kernel. Journal of Robotic Systems, 18(8):433–443, Aug. 2001.

[7]. Z. Yao. Self-motion graph in path planning problems with end-effector path specified. In ENSC-894 Course Transactions, Simon Fraser University, Summer, 2005.

Incorporation into current planners:• SMG-Greedy, the Greedy planner proposed in [3] enhanced

by SMG, has the following procedure:1. Explore the path as the original Greedy planner.2. Create a new SMG for the pose where the planner fails

to extend further.3. Repeatedly explore the SMG until a feasible

configuration is found to achieve the next pose, or the size of the SMG reaches the limit.

The data structure used to store the generated configurations now is different. As shown on the left, (a) is the data structure in the original Greedy planner, and (b) is the data structure in SMG-Greedy.

• More sophisticated planners are proposed, including:1. SMG-RRT-C;2. SMG-RRT-C2;

for more details, please refer to [7].

(a) Before

(b) After

Fig 1. The problem.