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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, :x, :u, traj)
BilinearIntegrator{DirectTrajOpt.Integrators.var"#7#8"{Main.var"#2#3"}}(DirectTrajOpt.Integrators.var"#7#8"{Main.var"#2#3"}(Main.var"#2#3"()), :x, :u, 2, 6, 98)

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)
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.

Number of nonzeros in equality constraint Jacobian...:      776
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:     1254

Total number of variables............................:      196
                     variables with only lower bounds:        0
                variables with lower and upper bounds:      100
                     variables with only upper bounds:        0
Total number of equality constraints.................:       98
Total number of inequality constraints...............:        0
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        0

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.3846842e-01 2.86e+00 2.86e-08   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.8765294e-01 1.43e-01 9.27e-01  -1.3 2.58e+00    -  9.92e-01 1.00e+00h  1
   2  7.3077355e-02 9.80e-02 7.14e-01  -1.0 6.06e+00    -  4.55e-01 3.17e-01f  1
   3  1.0631861e-01 6.09e-02 5.23e-01  -2.1 4.08e+00    -  4.00e-01 3.78e-01h  1
   4  1.7495983e-01 3.29e-02 1.83e-01  -1.7 2.18e+00    -  8.65e-01 4.59e-01h  1
   5  2.9968831e-01 1.98e-02 3.96e-01  -1.8 1.02e+00    -  8.77e-01 4.08e-01h  1
   6  5.3626179e-01 1.20e-02 2.70e+00  -1.8 1.49e+00    -  9.82e-01 4.59e-01h  1
   7  6.9621250e-01 9.79e-03 1.40e+01  -1.4 2.02e+00    -  1.00e+00 1.94e-01h  1
   8  7.9905556e-01 8.62e-03 6.17e+01  -1.0 2.53e+00    -  1.00e+00 1.23e-01h  1
   9  8.9377647e-01 7.85e-03 8.69e+02  -0.7 3.96e+00    -  1.00e+00 9.08e-02h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  9.1655557e-01 7.64e-03 8.59e+03  -0.1 2.09e+00    -  1.00e+00 2.67e-02h  1
  11  9.2012149e-01 7.61e-03 1.78e+06   0.5 6.02e-01    -  1.00e+00 4.87e-03h  1
  12  9.2019915e-01 7.61e-03 1.73e+10   2.3 2.10e+00    -  1.00e+00 1.02e-04h  1
  13  9.2019834e-01 7.58e-03 1.72e+08   2.7 1.94e-05  10.0 9.90e-01 1.00e+00f  1
  14  9.2017595e-01 7.46e-03 3.42e+05   2.2 1.03e-04   9.5 1.00e+00 1.00e+00f  1
  15  9.3891413e-01 7.46e-03 1.66e+08  -4.0 1.10e+04    -  8.56e-05 8.70e-05h  1
  16r 9.3891413e-01 7.46e-03 9.80e+02   2.7 0.00e+00    -  0.00e+00 4.57e-07R  4
  17r 9.1160917e-01 4.93e-03 9.47e+00   0.4 5.41e-01    -  9.90e-01 9.90e-01f  1

Number of Iterations....: 18

Number of objective function evaluations             = 23
Number of objective gradient evaluations             = 19
Number of equality constraint evaluations            = 23
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 19
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 18
Total seconds in IPOPT                               = 5.532

EXIT: Invalid number in NLP function or derivative detected.

Step 5: Access the Solution

Let's look at the results.

plot(prob.trajectory)
Example block output

The optimized trajectory is stored in prob.trajectory:

println("Final state: ", prob.trajectory.x[:, end])
println("Control norm: ", norm(prob.trajectory.u))
Final state: [1.0, 0.0]
Control norm: 6.83981471214229

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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