trajectoryoptimization

Trajectory optimization is the process of computing a time-parametrized trajectory and control inputs for a dynamical system that minimize a cost function while satisfying dynamics and constraints. It is used to plan feasible, efficient motion for robotics, aerospace, autonomous vehicles, and computer animation. Trajectory optimization differs from static path planning by enforcing dynamical feasibility and temporal aspects.

A typical formulation: x ∈ R^n state, u ∈ R^m control, dynamics xdot = f(x,u,t). Objective J = ∫_0^T L(x,u,t) dt + Φ(x(T)). Subject to initial condition x(0)=x0, possible terminal constraint x(T) ∈ X_T; path constraints h(x,u,t) ≤ 0; control limits u ∈ U, state bounds; the horizon T may be fixed or free with time optimization.

Solution approaches: indirect methods exploit necessary conditions from optimal control (Pontryagin's principle) and solve a boundary-value problem; direct methods discretize the decision variables and solve an NLP. Direct collocation and direct transcription are common discretization techniques; multiple shooting is another. Widely used software includes CasADi, IPOPT, and ACADO Toolkit.

Challenges and variants: nonconvex problems may have multiple local minima; high dimensionality; real-time or receding-horizon applications require fast solvers; collision avoidance can be handled via penalties or hard constraints; extensions address robustness, stochastic dynamics, and learning-based integration.