On the Configurations of Closed Kinematic Chains in Three-dimensional Space

Year 2022, , 96 - 115, 30.04.2022

Gerhard Zangerl , Alexander Steinicke

https://doi.org/10.36890/iejg.972576

Abstract

A kinematic chain in three-dimensional Euclidean space consists of $n$ links that are connected by spherical joints. Such a chain is said to be within a closed configuration when its link lengths form a closed polygonal chain in three dimensions. We investigate the space of configurations, described in terms of joint angles of its spherical joints, that satisfy the the loop closure constraint, meaning that the kinematic chain is closed. In special cases, we can find a new set of parameters that describe the diagonal lengths (the distance of the joints from the origin) of the configuration space by a simple domain, namely a cube of dimension $n-3$. We expect that the new findings can be applied to various problems such as motion planning for closed kinematic chains or singularity analysis of their configuration spaces. To demonstrate the practical feasibility of the new method, we present numerical examples.

Keywords

Configuration space, closed kinematic chain, closed continues random walk, path planning

References

• [1] Baker, M.: Matt Baker’s Math Blog. https://mattbaker.blog/2018/06/25/the-balanced-centrifuge-problem/. Accessed: 2021-07-15.
• [2] Brunnthaler, K., Schröcker, H., Husty, M.: A new method for the synthesis of bennett mechanisms. Proceedings of CK2005, Cassino, (2005).
• [3] Chansu, S., Taewoong, T., Beobkyoon, K., Hakjong, N., Munsang K., Park, F.: Tangent space rrt: A randomized planning algorithm on constraint manifolds. In Robotics and Automation (ICRA), 2011 IEEE International Conference on, pages 4968–4973,(2011).
• [4] Chirikjian, G.: A new inverse kinematics algorithm for binary manipulators with many actuators. Advanced Robotics, 15:20, (2000). https://doi.org/10.1163/15685530152116245
• [5] Cortes, J., Simeon, T.: Sampling-based motion planning under kinematic loop closure constraints. In Proc. of Workshop on Algorithmic Foundations of Robotics, (2004).
• [6] Creemers, T., Celaya, E., Ros, L.: Exact interval propagation for the efficient solution of planar linkages. In Proc. of the 12th World Conference in Mechanism and Machine Science, (2007).
• [7] David, B., Nir, S.: Generic singular configurations of linkages. Topol. Appl. 159(3),877–890 (2012). https://doi.org/10.1016/j.topol.2011.12.003
• [8] Hausmann, C., Knutson, A.: The cohomology ring of polygon spaces. Ann. Inst: Fourier (Grenoble), 48,281–321 (1998).
• [9] Hegedüs, G., Schicho, J., Schröcker, H.: Four-pose synthesis of angle-symmetric 6r linkages. J. Mech. Robot. 7(4), (2015). https://doi.org/DOI: 10.1115/1.4029186
• [10] Hinokuma, T., Shiga, H.: Topology of the configuration Space of polygons as a codimension one submanifold of a torus. Publ. RIMS, Kyoto Univ.. 34, 313–324 (1998). https://doi.org/10.2977/PRIMS/1195144628
• [11] Husty, M., Pfurner, M., Schröcker, H.: A new and efficient algorithm for the inverse kinematics of a general serial 6r manipulator. Mech. Mach Theory. 42(1),66–81 (2007). https://doi.org/10.1016/j.mechmachtheory.2006.02.001
• [12] Husty, M., Pfurner, M., Schröcker, H., Brunnthaler, K.: Algebraic methods in mechanism analysis and synthesis. Robotica. 25(6),661–675 (2007).
• [13] Iosif, B., Husty, M., Calin, V., Bogdan, G., Tucan, P., Doina, P.: Joint-space characterization of a medical parallel robot based on a dual quaternion representation of SE(3). Mathematics, 8(7), 1086 (2020). https://doi.org/10.3390/math8071086
• [14] Jaillet, L., Porta, J.: Path planning under kinematic constraints by rapidly exploring manifolds. IEEE Trans. Robot.. 29(1),105–117 (2013). https://doi.org/10.1109/TRO.2012.2222272
• [15] Kamiyama, Y.: The homology of the configuration space of a singular arachnoid mechanism. JP J. Geom. Topol. 7,385–395 (2007).
• [16] Kapovich, M., Millson, J.: On the moduli spaces of polygons in the euclidean plane. J. of Differential Geometry. 42,133–164 (1995). https://doi.org/10.4310/jdg/1214457034
• [17] La Valle, S., M.: Planing algorithms. Cambridge University Press, (2006).
• [18] Li, H., Lee, R.: Inverse kinematics for a serial chain with joints under distance constraints. In: Robotics: Science and systems, pages 177–184, (2006).
• [19] Li, H., Lee, R.; A unified geometric approach for inverse kinematics of a spatial chain with spherical joints. In: Proceedings 2007 IEEE International Conference on Robotics and Automation, pages 4420–4427, (2007).
• [20] Li, H., Lee, R., Blumenthal, J., Ihar, V.: Convexly stratified deformation spaces and efficient path planning for planar closed chains with revolute joints. Int. J. Rob. Res. 27(11-12), 1189–1212 (2008). https://doi.org/10.1177/0278364908097211
• [21] Li, H., Lee, R., Blumenthal, J., Ihar, V.: Stratified deformation space and path planning for a planar closed chain with revolute joints. In: Algorithmic Foundation of Robotics VII, pages 235–250, (2008).
• [22] Liu, G., Trinkle, J., Yang, Y., Luo, S.: Motion planning of planar closed chains based on structural sets. IEEE Access. 8,117203–117217 (2020). https://doi.org/10.1109/ACCESS.2020.3004229
• [23] Luenberger, D., Yinyu, Y.: Linear and nonlinear programming, volume 228. Springer, (2015).
• [24] Milgram, J., Trinkle, J.: The geometry of configuration spaces for closed chains in two and three dimensions. Homol. Homotopy Appl. 6(1),237–267 (2004).
• [25] Ruggiu, M., Müller, A.: Investigation of cyclicity of kinematic resolution methods for serial and parallel planar manipulators. Robotics. 10(1),9 (2021). https://doi.org/10.3390/robotics10010009
• [26] Trinkle, J., C., Liu, G., F., Milgram, R., J.: Toward complete path planning for planar 3r-manipulators among point obstacles. Algorithmic Foundations of Robotics VI, pages 329–344, (2005).
• [27] Trinkle, J., C., Milgram, J., R.: Complete path planning for closed kinematic chains with spherical joints. Int. J. Rob. Res. 21(9),773–789 (2002). https://doi.org/10.1177/0278364902021009119
• [28] Tondu, B.: Closed-form redundancy solving of serial chain robots with a weak generalized inverse approach. Rob. Auton. Sys. 74, 360–370 (2015). https://doi.org/10.1016/j.robot.2015.08.003
• [29] Yakey, J., La Valle, S., Kavraki, L.: A probabilistic roadmap approach for systems with closed kinemtic chains. In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), (1999).
• [30] Yajia, Z., Hauser, K., Jingru, L.: Unbiased, scalable sampling of closed kinematic chains. In: Robotics and Automation (ICRA), 2013 IEEE International Conference on, pages 2459–2464. IEEE, (2013).
• [31] Yakey, J., H., La Valle, S., Kavraki, L., E.: Randomized path planning for linkages with closed kinematic chains. IEEE Trans. Robot. Autom. 17(6),951–958 (2001). https://doi.org/10.1109/70.976030
• [32] Zangerl, G.: On the configuration space of planar closed kinematic chains. Int. Electron. J. Geom. 13(1),74–86 (2019). https://doi.org/10.1177/0278364902021009119