Projection-based first-order constrained optimization solver for robotics (2306.17611v1)
Abstract: Robot programming tools ranging from inverse kinematics (IK) to model predictive control (MPC) are most often described as constrained optimization problems. Even though there are currently many commercially-available second-order solvers, robotics literature recently focused on efficient implementations and improvements over these solvers for real-time robotic applications. However, most often, these implementations stay problem-specific and are not easy to access or implement, or do not exploit the geometric aspect of the robotics problems. In this work, we propose to solve these problems using a fast, easy-to-implement first-order method that fully exploits the geometric constraints via Euclidean projections, called Augmented Lagrangian Spectral Projected Gradient Descent (ALSPG). We show that 1. using projections instead of full constraints and gradients improves the performance of the solver and 2. ALSPG stays competitive to the standard second-order methods such as iLQR in the unconstrained case. We showcase these results with IK and motion planning problems on simulated examples and with an MPC problem on a 7-axis manipulator experiment.
- P. E. Gill, W. Murray, and M. A. Saunders, “Snopt: An sqp algorithm for large-scale constrained optimization.” SIAM J. Optimization, vol. 12, no. 4, pp. 979–1006, 2002.
- D. Kraft, “A software package for sequential quadratic programming,” Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt, 1988.
- A. Wachter, “An interior point algorithm for large-scale nonlinear optimization with applications in process engineering,” Ph.D. dissertation, Carnegie Mellon University, 2002.
- Y. Aoyama, G. Boutselis, A. Patel, and E. A. Theodorou, “Constrained differential dynamic programming revisited,” in Proc. IEEE Intl Conf. on Robotics and Automation (ICRA), 2021, pp. 9738–9744.
- V. Sindhwani, R. Roelofs, and M. Kalakrishnan, “Sequential operator splitting for constrained nonlinear optimal control,” in American Control Conference (ACC), 2017, pp. 4864–4871.
- J. Schulman, J. Ho, A. X. Lee, I. Awwal, H. Bradlow, and P. Abbeel, “Finding locally optimal, collision-free trajectories with sequential convex optimization.” in Proc. Robotics: Science and Systems (RSS), vol. 9, no. 1, 2013, pp. 1–10.
- N. Ratliff, M. Zucker, J. A. Bagnell, and S. Srinivasa, “Chomp: Gradient optimization techniques for efficient motion planning,” in Proc. IEEE Intl Conf. on Robotics and Automation (ICRA), 2009, pp. 489–494.
- T. A. Howell, B. E. Jackson, and Z. Manchester, “Altro: A fast solver for constrained trajectory optimization,” in Proc. IEEE/RSJ Intl Conf. on Intelligent Robots and Systems (IROS), 2019, pp. 7674–7679.
- E. G. Birgin, J. Martinez, and M. Raydan, “Spectral projected gradient methods,” Encyclopedia of Optimization, vol. 2, 2009.
- E. G. Birgin, J. M. Martínez, and M. Raydan, “Spectral projected gradient methods: review and perspectives,” Journal of Statistical Software, vol. 60, pp. 1–21, 2014.
- R. Andreani, E. G. Birgin, J. M. Martínez, and M. L. Schuverdt, “On augmented lagrangian methods with general lower-level constraints,” SIAM Journal on Optimization, vol. 18, no. 4, pp. 1286–1309, 2008.
- X. Jia, C. Kanzow, P. Mehlitz, and G. Wachsmuth, “An augmented lagrangian method for optimization problems with structured geometric constraints,” Mathematical Programming, pp. 1–51, 2022.
- M. Schmidt, E. Berg, M. Friedlander, and K. Murphy, “Optimizing costly functions with simple constraints: A limited-memory projected quasi-newton algorithm,” in Artificial intelligence and statistics. PMLR, 2009, pp. 456–463.
- H. H. Bauschke, “Projection algorithms and monotone operators,” Ph.D. dissertation, Theses (Dept. of Mathematics and Statistics)/Simon Fraser University, 1996.
- I. Usmanova, M. Kamgarpour, A. Krause, and K. Levy, “Fast projection onto convex smooth constraints,” in International Conference on Machine Learning. PMLR, 2021, pp. 10 476–10 486.
- H. H. Bauschke and V. R. Koch, “Projection methods: Swiss army knives for solving feasibility and best approximation problems with halfspaces,” Contemporary Mathematics, vol. 636, pp. 1–40, 2015.
- J. P. Boyle and R. L. Dykstra, “A method for finding projections onto the intersection of convex sets in hilbert spaces,” in Advances in order restricted statistical inference. Springer, 1986, pp. 28–47.
- G. Torrisi, S. Grammatico, R. S. Smith, and M. Morari, “A projected gradient and constraint linearization method for nonlinear model predictive control,” SIAM Journal on Control and Optimization, vol. 56, no. 3, pp. 1968–1999, 2018.
- M. Giftthaler and J. Buchli, “A projection approach to equality constrained iterative linear quadratic optimal control,” in 2017 IEEE-RAS 17th International Conference on Humanoid Robotics (Humanoids). IEEE, 2017, pp. 61–66.