PRONTO.jl

A Julia implementation of the PRojection-Operator-Based Newton’s Method for Trajectory Optimization (PRONTO). PRONTO is a numerical method for trajectory optimization which leverages variational calculus to solve the optimal control problem directly in infinite-dimensional function space. Consider the optimal control problem:

$$\min h(ξ) = p(x(T)) + \int_0^T l(x(t),u(t)) dt$$ where $h$ is $\mathcal{C}^2$ convex function, and $ξ(t) = (x(t),u(t))$ is a trajectory restricted by the dynamics of the system, which lies on the trajectory manifold:

$$\mathcal{T} = \{ ξ ∈(\mathcal{X}\times\mathcal{U})\ |\ \dot{x} = f(x,u),\ x(0)=x_0 \}$$

To explore beyond this trajectory manifold, we consider an arbitrary curve $η ∈(\mathcal{X}\times\mathcal{U})$ where $η(t) = (α(t),μ(t))$. The projection operator $\mathcal{P}:(\mathcal{X}\times\mathcal{U})\rightarrow \mathcal{T}$, which maps curves onto the trajectory manifold of the system, is used to convert the above constrained optimization problem to the unconstrained problem:

$$g(η) = h(\mathcal{P}(η))$$

Given the current trajectory iterate $ξ_k$, we wish to compute the descent direction $ζ_k$, which lies in the tangent space of $\mathcal{T}$. This is done by solving the optimization problem:

$$ζ_k = \arg\min Dg(ξ_k)\cdotζ + \frac{1}{2}D^2g(ξ_k)\cdot(ζ,ζ)$$

where $Dg$ and $D^2g$ are the first and second Fréchet derivatives of $g$.

This optimization problem is first converted into a linear-quadratic optimal control problem, and then solved in one of two ways. First, we try to solve a Newton descent direction, which does not have a guaranteed solution, but provides quadratic convergence. If this fails, we switch to a quasi-Newton descent direction, which has a guaranteed solution and provides super-linear convergence.

Because $ζ_k$ is computed by local second order approximation, there is no guarantee that $g(ξ_k + ζ_k) < g(ξ_k)$. Consequently we find a step size $γ_k ≤ 1$ via backtracking line search which enforces the Armijo rule.

After calculating a descent direction $ζ_k$ and a step size $γ_k$ the projection operator is used to ensure the next solution iterate lies on the trajectory manifold.

$$ ξ_{k+1} = \mathcal{P}(ξ_k + γ_k ζ_k)$$

For the mathematical details of the PRONTO algorithm, please refer to: (Hauser 2002), (Hauser 2003), and (Shao et al. 2022)

Usage

Please see the examples folder for usage examples. This API is still very likely to change – especially regarding the symbolic generation of jacobians/hessians and passing them to the solver. Note that upcoming changes in Julia 1.9 should substantially improve the compile time of large generated functions (#45276, #45404).

To see the generated definition of any model function, run, eg. methods(PRONTO.f) and open the temporary file where the definition is stored.

Upcoming Changes

• Fine grained control of verbosity/runtime-feedback.
• Support for SVector parameters (eg. regulator Q/R matrix diagonals).
• Support for constraints (Dearing et al. 2022), (Hauser & Saccon 2006)
• Easy access to differential equation solver options (algorithm, tolerance, etc.).
• More options for guess trajectories (eg. smooth atan)