The General Trajectory Optimization Problem

Topics: Convex Optimization

Trajectory optimization essentially consists of solving the following optimal control problem via optimization methods:

$$\begin{aligned} \underset{u, p,t_f}{\text{min}} \quad & J(x,u,p,t_f) \\ \text{s.t.} \quad & \dot{x}(t) = f(x,u,p,t) \\ & (x(t), u(t), p, t_f) \in \mathcal{C}(t), \quad \forall t \in [0,t_f] \\ & (x(0), p) \in \mathcal{X}_0 \\ & (x(t_f),p) \in \mathcal{X}_f \end{aligned}$$

where

• $J$ is the cost/objective function
• $f$ is the dynamics function
• $\mathcal{C}$ is the state and control constraint set
• $\mathcal{X}_0$ is the initial condition set
• $\mathcal{X}_f$ is the terminal condition set
• $p$ is a static decision variable

This problem formulation is in general form and is infinite dimensional. To solve the problem, especially in real-time, different methods are adopted, which target the following concepts:

• Formulation: specify how the functions $J$ and $f$ and the sets $\mathcal{C}$, $\mathcal{X}_0$, and $\mathcal{X}_f$ are expressed
• Discretization: approximate the infinite-dimensional problem into a finite and discrete problem
• Numerical Optimization: denotes the iterative computational algorithms to obtain an optimal solution of the discretized problem

References

1. Convex Optimization for Trajectory Generation - A Tutorial on Generating Dynamically Feasible Trajectories Reliably and Efficiently