On the Configuration Space of Planar Closed Kinematic Chains

Year 2020, Volume: 13 Issue: 1, 74 - 86, 30.01.2020

Gerhard Zangerl

https://doi.org/10.36890/iejg.611106

Cited By: 1

Abstract

A planar kinematic chain consists of $n$ links connected by joints. In this work we investigate the space of configurations, described in terms of joint angles, that guarantee that the kinematic chain is closed. We give explicit formulas expressing the joint angles that guarantee closedness by a new set of parameters. Moreover, it turns out that this parameters are contained in a domain that possesses a simple structure. We expect that the new insight can be applied for several issues such as motion planning for closed kinematic chains or singularity analysis of their configuration spaces. In order to demonstrate practicality of the new method we present numerical examples.

Keywords

configuration space, closed kinematic chain, path planning

References

• [1] Suthakorn, J., Chirikjian, G.: A new inverse kinematics algorithm for binary manipulators with many actuators. Adv. Robotics 15 (2), 225-244 (2000). https://doi.org/10.1163/15685530152116245
• [2] Hinokuma, T., Shiga, H.: Topology of the Configuration Space of Polygons as a Codimension One of a Torus. Publ. RIMS, Kyoto Univ. 34 (4), 313-324 (1998). https://doi.org/10.2977/prims/1195144628
• [3] Hausmann, J. C., Knutson, A.: The cohomology ring of polygon spaces. Ann. Inst. Fourier 48 (1), 281-321 (1998). https://doi.org/10.5802/aif.1619
• [4] Cortes J., Simeon, T.:Sampling-based motion planning under kinematic loop closure constraints. In: Algorithmic Foundations of Robotics VI. Springer (2004). https://doi.org/10.1007/10991541_7
• [5] Celaya E., Creemers, T., Ros, L.: Exact interval propagation for the efficient solution of position analysis problems on planar linkages. Mech. Mach. Theory 54, 116-131 (2012). https://doi.org/10.1016/j.mechmachtheory.2012.03.005
• [6] La Valle, S. M., Yakey, J. Kavraki, L.: A probabilistic roadmap approach for systems with closed kinemtic chains. In Proc. IEEE Int. Conf. Robot. Autom. (ICRA) 3, 1671-1676 (1999). https://10.1109/ROBOT.1999.770349
• [7] La Valle, S. M.:Planing Algorithms. Cambridge University Press (2006).
• [8] Han, L., Rudolph, L., Blumenthal, J., Valodzin, I.:Algorithmic Foundation of Robotics VII. Springer, 47, 235-250 (2008).
• [9] Han, L., Rudolph, L.: Inverse Kinematics for a Serial Chain with Joints Under Distance Constraints. Robotics: Science and systems, 177-184 (2006).
• [10] Han, L., Rudolph, L., Blumenthal, J., Valodzin, I.: Convexly stratified deformation spaces and efficient path planning for planar closed chains with revolute joints. Int. J. R. Res. 27, 1189-1212 (2008). https://doi.org/10.1177/0278364908097211
• [11] Milgram, R. J., Trinkle, J. C.: The geometry of configuration spaces for closed chains in two and three dimensions. Homol. Homotopy Appl. 6 (1), 237-267 (2004).
• [12] Milgram, R. J., Trinkle, J. C.: Complete path planning for closed kinematic chains with spherical joints. Int. J. R. Res. 21 (9), 773-789 (2002). https://doi.org/10.1177/0278364902021009119
• [13] Jaillet, L., Porta, J. M.: Path planning under kinematic constraints by rapidly exploring manifolds. IEEE Trans. Robot. 29 (1), 105-117 (2012). https://doi.org/10.1109/TRO.2012.2222272
• [14] Kapovich, M., Millson, J.: On the moduli spaces of polygons in the euclidean plane. J. Differential Geom. 42 (1), 133-164 (1995).
• [15] Liu, G. F., Trinkle, J. C., Milgram, R. J.: Toward complete path planning for planar 3r-manipulators among point obstacles. Algorithmic Foundations of Robotics VI (2004).
• [16] Luenberger D., Yinyu, Y.:Linear and nonlinear programming. Springer, (2004).
• [17] Yakey, J. H., LaValle, S. M., 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
• [18] Yajia, Z., Hauser, K., Jingru L.: Robotics and Automation. (ICRA), 2013 IEEE International Conference on (2013).