Abstract
We present a new algorithm for computing globally optimal topology and trajectory jointly for 2D planar linkages. Planar linkage structures can generate complex end-effector trajectories using only a single rotational actuator, which is very useful in building low-cost robots. We address the problem of searching for the optimal topology and geometry of these structures and present new optimization methods that consider topology changes that are non-smooth and non-differentiable. We formulate this problem as a mixed-integer convex programming (MICP) problem for which a global optimum can be found using the branch-and-bound (BB) algorithm. As a result, within a finite amount of time, our method can find planar linkage structures with end-effector trajectories that closely match the user-specified target trajectories. We tested our method to search for planar linkages with 5–7 rigid bodies. Compared with sampling-based methods or simulated annealing, our method improves the quality of the solution by at most \(7{\times }\) and the optimized planar linkage structure has been tested on a 4-legged walking robot.
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
References
Bächer, M., Coros, S., Thomaszewski, B.: LinkEdit: interactive linkage editing using symbolic kinematics. ACM Trans. Graph. (TOG) 34(4), 1–8 (2015). Article no: 99
Bohigas, O., Manubens, M., Ros, L.: Singularities of non-redundant manipulators: a short account and a method for their computation in the planar case. Mech. Mach. Theory 68, 1–17 (2013)
Bommes, D., Zimmer, H., Kobbelt, L.: Mixed-integer quadrangulation. ACM Trans. Graph. (TOG) 28(3), 1–10 (2009). Article no. 77
Byrd, R.H., Nocedal, J., Waltz, R.A.: Knitro: an integrated package for nonlinear optimization. In: Di Pillo, G., Roma, M. (eds.) Large-Scale Nonlinear Optimization. NOIA, vol. 83, pp. 35–59. Springer, Boston (2006). https://doi.org/10.1007/0-387-30065-1_4
Conforti, M., Di Summa, M., Eisenbrand, F., Wolsey, L.A.: Network formulations of mixed-integer programs. Math. Oper. Res. 34(1), 194–209 (2009)
Dai, H., Izatt, G., Tedrake, R.: Global inverse kinematics via mixed-integer convex optimization. In: International Symposium on Robotics Research, Puerto Varas, Chile, pp. 1–16 (2017)
Ding, H., Reißig, G., Groß, D., Stursberg, O.: Mixed-integer programming for optimal path planning of robotic manipulators. In: 2011 IEEE International Conference on Automation Science and Engineering, pp. 133–138. IEEE (2011)
Gurobi Optimization, LLC: Gurobi optimizer reference manual (2018)
Ha, S., Coros, S., Alspach, A., Bern, J.M., Kim, J., Yamane, K.: Computational design of robotic devices from high-level motion specifications. IEEE Trans. Robot. 99, 1–12 (2018)
Ha, S., Coros, S., Alspach, A., Kim, J., Yamane, K.: Joint optimization of robot design and motion parameters using the implicit function theorem. In: Robotics: Science and Systems (2017)
Hernández, A., Gómez, C., Crespo, J., Barber, R.: A home made robotic platform based on Theo Jansen mechanism for teaching robotics. In: INTED 2016 Proceedings: 10th International Technology, Education and Development Conference, IATED, 7–9 March 2016, pp. 6689–6698 (2016)
Kanno, Y.: Topology optimization of tensegrity structures under compliance constraint: a mixed integer linear programming approach. Optim. Eng. 14(1), 61–96 (2013). https://doi.org/10.1007/s11081-011-9172-0
Kecskemethy, A., Krupp, T., Hiller, M.: Symbolic processing of multiloop mechanism dynamics using closed-form kinematics solutions. Multibody Sys. Dyn. 1(1), 23–45 (1997). https://doi.org/10.1023/A:1009743909765
Lawler, E.L., Wood, D.E.: Branch-and-bound methods: a survey. Oper. Res. 14(4), 699–719 (1966)
Liberti, L.: Reformulation and convex relaxation techniques for global optimization. Ph.D. thesis, Springer (2004)
Liu, J., Ma, Y.: A survey of manufacturing oriented topology optimization methods. Adv. Eng. Softw. 100, 161–175 (2016)
Lobato, E., Echavarren, F., Rouco, L., Navarrete, M., Casanova, R., Lopez, G.: A mixed-integer LP based network topology optimization algorithm for overload alleviation. In: 2003 IEEE Bologna Power Tech Conference Proceedings, vol. 2, pp. 5–pp. IEEE (2003)
Nansai, S., Elara, M.R., Iwase, M.: Dynamic analysis and modeling of Jansen mechanism. Procedia Eng. 64, 1562–1571 (2013)
Saar, K.A., Giardina, F., Iida, F.: Model-free design optimization of a hopping robot and its comparison with a human designer. IEEE Robot. Autom. Lett. 3(2), 1245–1251 (2018)
Song, P., et al.: Computational design of wind-up toys. ACM Trans. Graph. (TOG) 36(6), 1–13 (2017). Article no. 238
Spielberg, A., Araki, B., Sung, C., Tedrake, R., Rus, D.: Functional co-optimization of articulated robots. In: 2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 5035–5042. IEEE (2017)
Thomaszewski, B., Coros, S., Gauge, D., Megaro, V., Grinspun, E., Gross, M.: Computational design of linkage-based characters. ACM Trans. Graph. 33(4), 64:1–64:9 (2014)
Trespalacios, F., Grossmann, I.E.: Improved Big-M reformulation for generalized disjunctive programs. Comput. Chem. Eng. 76, 98–103 (2015)
Vielma, J.P.: Mixed integer linear programming formulation techniques. SIAM Rev. 57(1), 3–57 (2015)
Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128(1–2), 49–72 (2011). https://doi.org/10.1007/s10107-009-0295-4
Zhang, H., Kumar, A.S., Fuh, J.Y.H., Wang, M.Y.: Design and development of a topology-optimized three-dimensional printed soft gripper. Soft Robot. 5(5), 650–661 (2018)
Zhang, H., Wang, M.Y., Chen, F., Wang, Y., Kumar, A.S., Fuh, J.Y.: Design and development of a soft gripper with topology optimization. In: 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 6239–6244. IEEE (2017)
Zhu, B., Skouras, M., Chen, D., Matusik, W.: Two-scale topology optimization with microstructures. ACM Trans. Graph. (TOG) 36(5), 164 (2017)
Zhu, L., Xu, W., Snyder, J., Liu, Y., Wang, G., Guo, B.: Motion-guided mechanical toy modeling. ACM Trans. Graph. 31(6), 127:1–127:10 (2012)
Acknowledgement
This research is supported in part by ARO grant W911NF-18-1-0313, and Intel.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
1 Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Pan, Z., Liu, M., Gao, X., Manocha, D. (2022). Globally Optimal Joint Search of Topology and Trajectory for Planar Linkages. In: Asfour, T., Yoshida, E., Park, J., Christensen, H., Khatib, O. (eds) Robotics Research. ISRR 2019. Springer Proceedings in Advanced Robotics, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-030-95459-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-95459-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-95458-1
Online ISBN: 978-3-030-95459-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)