Abstract
Path planning has long been one of the major research areas in robotics, with PRM and RRT being two of the most effective path planners. Though they are generally very efficient, these two sample-based planners can become computationally expensive in the important special case of narrow passage problems. This paper develops a path planning paradigm which uses ellipsoids and superquadrics to respectively encapsulate the rigid parts of the robot and obstacles. The main benefit in doing this is that configuration-space obstacles can be parameterized in closed form, thereby allowing prior knowledge to be used to avoid sampling infeasible configurations, in order to solve the narrow passage problem efficiently. Benchmark results for single-body robots show that, remarkably, the proposed method outperforms the sample-based planners in terms of the computational time in searching for a path through narrow corridors. Feasible extensions that integrate with sample-based planners to further solve the high dimensional multi-body problems are discussed, which will require substantial additional theoretical development in the future.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Here the word “arena” denotes the bounded area in which the robot and obstacles are contained.
- 2.
Explicitly, \(\varvec{\omega }= \log ^\vee (R)\), \(R \in \text {SO}(n)\) and \(\mathbf{t} \in \mathbb {R}^n\) (\(\mathbf{t}\) is the actual translation as seen in the world reference frame). And the pair \((R,\mathbf{t})\) forms the “Pose Change Group”, i.e. \(\text {PCG}(n) \doteq \text {SO}(n) \times \mathbb {R}^n\), with the group operation being a direct product, which is different than \(\text {SE}(n) \doteq \text {SO}(n) \rtimes \mathbb {R}^n\) (for more details, see [33]).
References
Kavraki, L.E., Svestka, P., Latombe, J.C., Overmars, M.H.: Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. Robot. Autom. 12(4), 566–580 (1996)
LaValle, S.M.: Rapidly-exploring random trees: a new tool for path planning (1998)
Kuffner, J.J., LaValle, S.M.: RRT-connect: an efficient approach to single-query path planning. In: Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 2000, vol. 2, pp. 995–1001. IEEE (2000)
Bohlin, R., Kavraki, L.E.: Path planning using lazy PRM. In: Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 2000, vol. 1, pp. 521–528. IEEE (2000)
Varadhan, G., Manocha, D.: Star-shaped roadmaps - a deterministic sampling approach for complete motion planning. In: Robotics: Science and Systems, vol. 173. Citeseer (2005)
Rodriguez, S., Tang, X., Lien, J.M., Amato, N.M.: An obstacle-based rapidly-exploring random tree. In: Proceedings 2006 IEEE International Conference on Robotics and Automation, ICRA 2006, pp. 895–900. IEEE (2006)
Hsu, D., Jiang, T., Reif, J., Sun, Z.: The bridge test for sampling narrow passages with probabilistic roadmap planners. In: Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 2003, vol. 3, pp. 4420–4426. IEEE (2003)
Shi, K., Denny, J., Amato, N.M.: Spark PRM: using RRTs within PRMs to efficiently explore narrow passages. In: 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 4659–4666. IEEE (2014)
Yan, Y., Ma, Q., Chirikjian, G.S.: Path planning based on closed-form characterization of collision-free configuration-spaces for ellipsoidal bodies obstacles and environments. In: Proceedings of the 1st International Workshop on Robot Learning and Planning, pp. 13–19 (2016)
Barr, A.H.: Superquadrics and angle-preserving transformations. IEEE Comput. Graphics Appl. 1(1), 11–23 (1981)
Jaklic, A., Leonardis, A., Solina, F.: Segmentation and Recovery of Superquadrics, vol. 20. Springer, Dordrecht (2013)
Agba, E.I., Wong, T.L., Huang, M.Z., Clark, A.M.: Objects interaction using superquadrics for telemanipulation system simulation. J. Robot. Syst. 10(1), 1–22 (1993)
Guibas, L., Ramshaw, L., Stolfi, J.: A kinetic framework for computational geometry. In: 24th Annual Symposium on Foundations of Computer Science, pp. 100–111. IEEE (1983)
Latombe, J.C.: Robot Motion Planning, vol. 124. Springer, Boston (2012)
Kavraki, L.E.: Computation of configuration-space obstacles using the fast Fourier transform. IEEE Trans. Robot. Autom. 11(3), 408–413 (1995)
Agarwal, P.K., Flato, E., Halperin, D.: Polygon decomposition for efficient construction of Minkowski sums. Comput. Geom. 21(1), 39–61 (2002)
Nelaturi, S., Shapiro, V.: Configuration products and quotients in geometric modeling. Comput. Aided Des. 43(7), 781–794 (2011)
Lien, J.M.: Hybrid motion planning using Minkowski sums. In: Proceedings of Robotics: Science and Systems IV (2008)
Nelaturi, S., Shapiro, V.: Solving inverse configuration space problems by adaptive sampling. Comput. Aided Des. 45(2), 373–382 (2013)
Fogel, E., Halperin, D.: Exact and efficient construction of Minkowski sums of convex polyhedra with applications. Comput. Aided Des. 39(11), 929–940 (2007)
Hachenberger, P.: Exact Minkowksi sums of polyhedra and exact and efficient decomposition of polyhedra into convex pieces. Algorithmica 55(2), 329–345 (2009)
Behar, E., Lien, J.M.: Dynamic Minkowski sum of convex shapes. In: 2011 IEEE International Conference on Robotics and Automation (ICRA), pp. 3463–3468. IEEE (2011)
Kurzhanskiy, A.A., Varaiya, P.: Ellipsoidal Toolbox (ET). In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 1498–1503 (2006). https://doi.org/10.1109/CDC.2006.377036
Hartquist, E.E., Menon, J., Suresh, K., Voelcker, H.B., Zagajac, J.: A computing strategy for applications involving offsets, sweeps, and Minkowski operations. Comput. Aided Des. 31(3), 175–183 (1999)
Yan, Y., Chirikjian, G.S.: Closed-form characterization of the Minkowski sum and difference of two ellipsoids. Geom. Dedicata 177(1), 103–128 (2015)
Chirikjian, G.S., Yan, Y.: The kinematics of containment. In: Lenarčič, J., Khatib, O. (eds.) Advances in Robot Kinematics, pp. 355–364. Springer, Cham (2014)
Ma, Q., Chirikjian, G.S.: A closed-form lower bound on the allowable motion for an ellipsoidal body and environment. In: Proceedings of 2015 IDETC/CIE, pp. V05CT08A055–V05CT08A055. American Society of Mechanical Engineers (2015)
Ruan, S., Ding, J., Ma, Q., Chirikjian, G.S.: The kinematics of containment for N-dimensional ellipsoids. J. Mech. Rob. 11(4) (2019)
Sucan, I.A., Moll, M., Kavraki, L.E.: The open motion planning library. IEEE Robot. Autom. Mag. 19(4), 72–82 (2012)
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2008)
Yan, Y.: Geometric motion planning methods for robotics and biological crystallography. Ph.D. thesis (2014)
Choset, H.M., Hutchinson, S., Lynch, K.M., Kantor, G., Burgard, W., Kavraki, L.E., Thrun, S.: Principles of Robot Motion: Theory, Algorithms, and Implementation. MIT Press, Cambridge (2005)
Chirikjian, G.S., Mahony, R., Ruan, S., Trumpf, J.: Pose changes from a different point of view. J. Mech. Robot. 10(2), 021008 (2018)
Pan, J., Chitta, S., Manocha, D.: FCL: a general purpose library for collision and proximity queries. In: 2012 IEEE International Conference on Robotics and Automation (ICRA), pp. 3859–3866. IEEE (2012)
Acknowledgements
The authors would like to thank Dr. Fan Yang, Mr. Thomas W. Mitchel and Mr. Zeyi Wang for useful discussions. This work was performed under National Science Foundation grant IIS-1619050 and Office of Naval Research Award N00014-17-1-2142. The ideas expressed in this paper are solely those of the authors.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Ruan, S., Ma, Q., Poblete, K.L., Yan, Y., Chirikjian, G.S. (2020). Path Planning for Ellipsoidal Robots and General Obstacles via Closed-Form Characterization of Minkowski Operations. In: Morales, M., Tapia, L., Sánchez-Ante, G., Hutchinson, S. (eds) Algorithmic Foundations of Robotics XIII. WAFR 2018. Springer Proceedings in Advanced Robotics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-44051-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-44051-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-44050-3
Online ISBN: 978-3-030-44051-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)