Quickstart Guide

Welcome to DirectTrajOpt.jl! This guide will get you up and running in minutes.

What is DirectTrajOpt?

DirectTrajOpt.jl solves trajectory optimization problems - finding optimal control sequences that drive a dynamical system from an initial state to a goal state while minimizing a cost function.

Installation

First, install the package:

using Pkg
Pkg.add("DirectTrajOpt")

You'll also need NamedTrajectories.jl for defining trajectories:

using DirectTrajOpt
using NamedTrajectories
using LinearAlgebra
using CairoMakie

A Minimal Example

Let's solve a simple problem: drive a 2D system from [0, 0] to [1, 0] with minimal control effort.

Step 1: Define the Trajectory

A trajectory contains your states, controls, and time information:

N = 50  # number of time steps
traj = NamedTrajectory(
    (
        x = randn(2, N),    # 2D state
        u = randn(1, N),    # 1D control
        Δt = fill(0.1, N)   # time step
    );
    timestep=:Δt,
    controls=:u,
    initial=(x = [0.0, 0.0],),
    final=(x = [1.0, 0.0],),
    bounds=(Δt = (0.05, 0.2), u = 1.0)
)

N = 50, (x = 1:2, u = 3:3, → Δt = 4:4)

Step 2: Define the Dynamics

Specify how your system evolves. For bilinear dynamics ẋ = (G₀ + u₁G₁) x:

G_drift = [-0.1 1.0; -1.0 -0.1]   # drift term
G_drives = [[0.0 1.0; 1.0 0.0]]   # control term
G = u -> G_drift + sum(u .* G_drives)

integrator = BilinearIntegrator(G, traj, :x, :u)

Step 3: Define the Objective

What do we want to minimize? Let's penalize control effort:

obj = QuadraticRegularizer(:u, traj, 1.0)

QuadraticRegularizer(:u, [1.0], [0.0 0.0 … 0.0 0.0], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50])

Step 4: Create and Solve

Combine everything into a problem and solve:

prob = DirectTrajOptProblem(traj, obj, integrator)
solve!(prob; max_iter=100, verbose=false)

Step 5: Access the Solution

Let's look at the results.

plot(prob.trajectory)

The optimized trajectory is stored in prob.trajectory:

println("Final state: ", prob.trajectory.x[:, end])
println("Control norm: ", norm(prob.trajectory.u))

What You Can Do

• Multiple objectives: Combine regularization, minimum time, terminal costs
• Flexible dynamics: Linear, bilinear, time-dependent systems
• Add constraints: Bounds, path constraints, custom nonlinear constraints
• Smooth controls: Penalize derivatives for smooth, implementable controls
• Free time: Optimize trajectory duration

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