Stanford AA203 Optimal and Learning-Based Control | Spring 2026 | Lecture 6: Optimal Control
Stanford Online · 1:20:28 · 2 days ago
Direct methods have largely superseded indirect methods in robotics because they simplify the design process by transforming complex control problems into standard numerical optimization tasks. While indirect methods prioritize deriving analytical optimality conditions, direct methods prioritize discretization and iterative solving, with advanced approaches like sequential convex programming providing reliable paths to optimal solutions.
- Time rescaling — To solve problems with unknown end times, analysts define time between 0 and 1 and introduce a dummy variable with static dynamics to lock the final time .
- Direct method workflow — By discretizing continuous time into intervals, the problem becomes a nonlinear optimization task, which is then handled by standard numerical solvers .
- Shooting method — Treating control inputs as the only variables allows the system to compute states recursively, resulting in smaller optimization problems that are often easier to manage .
- Collocation method — Including both state and control values as variables makes the system dynamics function as constraints, which is necessary when handling complex boundary limits .
- Sequential Convex Programming (SCP) — This strategy transforms a difficult nonlinear problem into a sequence of easier convex problems by linearizing dynamics around a current guess .
- Optimization stability — To prevent the solver from failing, engineers often add slack variables to relax constraints or use trust regions to restrict how far the solver updates the trajectory in one step .
How do slack variables influence the final cost calculation in optimization routines?