Convex MPC

Simplifying a legged robot to a single rigid body so MPC becomes a quadratic program that solves quickly to the global optimum.

Convex MPC generally refers to the quadruped control method published by MIT Biomimetic Robotics Lab's Di Carlo, Wensing, Kim, et al. at IROS 2018. Model predictive control needs to solve, every cycle, an optimization subject to dynamics constraints; a full-body nonlinear model is too slow. They instead treat the robot as a single rigid body pushed by ground reaction forces (ignoring leg mass) and apply three approximations: small roll and pitch angles, the state staying close to the commanded trajectory (linearizing around commanded yaw and planned foothold positions), and dropping nonlinear terms involving angular velocity. With each foot's contact timing fixed in advance by the gait scheduler, the only decision variables left are the ground reaction forces, and with friction-cone constraints added, the problem becomes a convex quadratic program: it's guaranteed to find the global optimum, and it solves fast. It outputs the reaction forces over a future window, which joint torque control or whole-body control then executes. The approximation breaks down for large rolling motions, where nonlinear MPC is used instead.

ExampleOn Cheetah 3, a force plan with a prediction horizon up to 0.5 seconds solves in under 1 millisecond, running at 20–30 Hz; the same set of gains produced standing, trotting, flying trot, bounding, pronking, pacing, a three-leg gait, and 3D galloping, reaching forward speeds up to 3 m/s. After porting to Mini Cheetah in 2019, paired with a 500 Hz whole-body impulse controller (WBIC), it reached 3.7 m/s.

Also called

Single Rigid-Body MPC

Related

Model Predictive Control · Single Rigid Body Dynamics Model · Nonlinear Model Predictive Control · Contact Force Optimization (Force Distribution) · Gait Planning · Whole-Body Control

Sources

Di Carlo et al., Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive Control (IROS 2018)
Kim et al., Highly Dynamic Quadruped Locomotion via Whole-Body Impulse Control and Model Predictive Control (arXiv:1909.06586)