# Phases of a Trajectory#

Dymos uses the concept of phases to support intermediate boundary constraints and path constraints on variables in the system. Each phase represents the trajectory of a dynamical system, and may be subject to different equations of motion, force models, and constraints. Multiple phases may be assembled to form one or more trajectories by enforcing compatibility constraints between them.

For implicit and explicit phases, the equations-of-motion or process equations are defined via an ordinary differential equation.

An ODE is of the form

(86)#\begin{align} \frac{\partial \textbf x}{\partial t} = \textbf f(t, \textbf x, \textbf u) \end{align}

where $$\textbf x$$ is the vector of state variables (the variable being integrated), $$t$$ is time (or time-like), $$\textbf u$$ is the vector of parameters (an input to the ODE), and $$\textbf f$$ is the ODE function.

Dymos can treat the parameters $$\textbf u$$ as either static parameters or dynamic controls. In addition, Dymos automatically calculates the first and second time-derivatives of the controls. These derivatives can then be utilized as via constraints or as additional parameters to the ODE. Subsequently, the optimal control problem as solved by Dymos can be expressed as:

(87)#\begin{align} \textrm{Minimize}:& \quad J = \textbf f_{obj}(t, \textbf x, \textbf u, \dot{\textbf u}, \ddot{\textbf u}) \\ \textrm{subject to:}& \\ &\textrm{system dynamics} \quad &\frac{\partial \textbf x}{\partial t} &= \textbf f_{ode}(t, \textbf x, \textbf u, \dot{\textbf u}, \ddot{\textbf u}) \\ &\textrm{initial time bounds} \quad &t_{0,lb} &\,\le\, t_0 \,\le\, t_{0,ub} \\ &\textrm{elapsed time bounds} \quad &t_{p,lb} &\,\le\, t_p \,\le\, t_{p,ub} \\ &\textrm{state bounds} \quad &\textbf x_{lb} &\,\le\, \textbf x \,\le\, \textbf x_{ub} \\ &\textrm{control bounds} \quad &\textbf u_{lb} &\,\le\, \textbf u \,\le\, \textbf u_{ub} \\ &\textrm{nonlinear boundary constraints} \quad &\textbf g_{b,lb} &\,\le\, \textbf g_{b}(t, \textbf x, \textbf u, \dot{\textbf u}, \ddot{\textbf u}) \,\le\, \textbf g_{b,ub} \\ &\textrm{nonlinear path constraints} \quad &\textbf g_{p,lb} &\,\le\, \textbf g_{p}(t, \textbf x, \textbf u, \dot{\textbf u}, \ddot{\textbf u}) \,\le\, \textbf g_{p,ub} \\ \end{align}

The ability to utilize control derivatives in the equations of motion provides some unique capabilities, namely the ability to easily solve problems using differential inclusion, which will be demonstrated in the examples.

The solution techniques used by the Phase classes in Dymos generally fall into two categories: implicit and explicit phases. They differ in underlying details but both allow for the same general form of the optimal control problem.

Segments

Variables

Constraints

Objective

Timeseries Outputs