Open-Source, Object-Oriented, Multi-Phase Pseudospectral Optimization Using Pyomo

Trajectory Planning with Deep Reinforcement Learning in High-Level Action Spaces

“…In most cases the optimal control problem cannot be solved analytically. Rather, so-called direct transcription methods [1,42] or indirect methods [43] can be applied to solve the problem numerically, in which case the dynamics (1) can be approximated with a numerical discretization of the form…”

“…Trajectory generation is often formulated as an optimal control problem and then converted into a parameter optimization problem through direct transcription methods [1,42]. This produces a problem formulation that is compatible with modern nonlinear programming techniques [54].…”

“…This produces a problem formulation that is compatible with modern nonlinear programming techniques [54]. For many problems this method works well [1,2], but long solution times and convergence stability can limit real-time application.…”

“…The following shows that the best approximation error grows at a bounded linear rate along the horizon in the case where 𝑧 = 𝑦 𝑑𝑒𝑠 . (1) which is optimal with respect to (2). Define x(𝑡) = 𝑧 * (𝑡), where 𝑧 * (𝑡) is a piecewise polynomial as described in (9) over 𝑡 ∈ [𝑡 0 , 𝑡 𝐻 ] and best approximates 𝑓 (𝑥 * (𝑡), 𝑢 * (𝑡)) over 𝑡 ∈ [𝑡 0 , 𝑡 𝐻 ] in some sense.…”

“…Many advances have been made in trajectory planning and optimization. Examples include nonlinear programming methods [1,2], sampling based methods [3][4][5],…”