Swing-up optimization

The optimizer works in continuous angles, with phase state \([x,v,\theta_1,\omega_1,\ldots]\) and ideal cart force as the control. It discretizes a finite time horizon into plan_nodes intervals and applies Hermite–Simpson direct collocation. It enforces initial conditions, sampled dynamics, sampled cart bounds \(|x|\leq x_{\max}-\text{cart_margin}\), force bounds, and a terminal upright pose with small terminal motion. The cost is the configured control and trajectory objective in the solver; plan.J is useful for comparing valid candidates of the same problem, not as a hardware energy rating.

Each integer seed builds a different angle-path initial guess. The terminal winding number is not fixed by that seed: an optimized link can end at any integer multiple of \(2\pi\) consistent with upright. CasADi/IPOPT solves a local nonlinear program for each seed. plan.attempts includes solver convergence and separate numerical validity, with cost, terminal error, peak cart travel, and peak force when available. The cheapest valid attempt is selected. planning_failed means none passed; it does not prove physical infeasibility.

The optimization assumes ideal horizontal force and samples constraints at its collocation nodes. It does not embed the command-to-force transfer function, delay, command clipping, or a guarantee between sampled nodes. The JSON runner tracks the plan with TVLQR and then simulates the configured actuator. Compare plan.csv, trajectory.csv, summary.plan, and summary.catch. A valid plan can miss the catch or cross the rail in that second stage.

Solve time can rise sharply with the number of links, nodes, seeds, and initial conditions. Change one parameter at a time and record the attempt diagnostics. See model and control workflow.