Trajectory Optimization

Writing ‘how to move’ as an optimization problem: satisfy the dynamics and constraints while minimizing a cost.

Trajectory optimization casts ‘how to move’ as a mathematical optimization problem: the decision variables are the state and control input over a stretch of time, the objective is to minimize a cost (energy, time, deviation from a target), and the constraints include the dynamics equations, torque and joint limits, obstacle avoidance, and start/end conditions. It is essentially open-loop optimal control solved for a single initial state, which is much cheaper than solving for a feedback law over the entire state space. Numerically, direct methods are most common: direct shooting optimizes only the control, with the state obtained by simulating forward; direct collocation treats both state and control as variables, enforcing the dynamics ẋ = f(x, u) only at collocation points; multiple shooting sits between the two. Gradient-based methods generally find only a local optimum, so the initial guess matters a great deal. Re-solving every control cycle and executing only the first step is model predictive control (MPC).

ExampleSrinivasan and Ruina's 2006 study in Nature used trajectory optimization to find the most energy-efficient way for a simplified biped model to move, finding that ‘walking’ was cheaper at low speed and ‘running’ cheaper at high speed.

Also called

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Related

Optimal Control · Direct Collocation · Multiple Shooting · Model Predictive Control · Iterative Linear Quadratic Regulator · Contact-Implicit Trajectory Optimization

Sources

Wikipedia: Trajectory optimization
Russ Tedrake, Underactuated Robotics: Trajectory Optimization
Lynch & Park, Modern Robotics (2017 preprint), 10.7 Nonlinear Optimization