Examples.md

@def title = "Examples" @def hascode = true @def hasmath = true @def tags = ["examples", "tutorials"]

Examples

This page contains complete, runnable JuDO examples. Each example uses:

• DynModel(Interesso.Optimizer) to create a model backed by the Interesso solver
• JuDO macros (@phase, @variable, @constraint, @objective) for problem formulation
• JuDO.warmstart! to provide an initial guess
• dyn_value(model, x) to extract solution trajectories as callable functions

Shared setup

Both examples below use a simple linear warm-start interpolant. Define it once per Julia session before running either example:

using JuDO, Interesso, DynOptInterface
using Plots

struct LinearInterpolant <: DynOptInterface.AbstractDynamicSolution
    y_a::Float64
    y_b::Float64
    t_0::Float64
    t_f::Float64
end
(li::LinearInterpolant)(t::Real) = li.y_a + (t - li.t_0) * (li.y_b - li.y_a) / (li.t_f - li.t_0)

---

Cart-Pole Swing-Up

The cart-pole swing-up is a classic control benchmark. A pole is attached via a pivot to a cart that moves along a frictionless track. Starting from the hanging position ($\theta = 0$), the goal is to swing the pole to the upright position ($\theta = \pi$) by applying a horizontal force to the cart, while minimising total control effort.

Problem formulation

Parameters

Symbol	Value	Description
$m_1$	1.0 kg	Cart mass
$m_2$	0.3 kg	Pole mass
$l$	0.5 m	Pole half-length
$g$	9.81 m/s²	Gravitational acceleration
$u_{\max}$	20 N	Maximum control force
$r_{\max}$	2 m	Maximum cart displacement

Optimal control problem

$$ \min_{u(\cdot)} \quad \int_0^{2} u(t)^2 , \mathrm{d}t $$

subject to the Lagrangian equations of motion:

$$ \dot r = \nu, \qquad \dot \nu = \frac{l , m_2 \sin\theta \cdot \omega^2 + u + m_2 g \cos\theta \sin\theta}{m_1 + m_2 \sin^2\theta} $$

$$ \dot \theta = \omega, \qquad \dot \omega = \frac{-l , m_2 \cos\theta \sin\theta \cdot \omega^2 - u \cos\theta - (m_1 + m_2) g \sin\theta}{l (m_1 + m_2 \sin^2\theta)} $$

with boundary conditions $r(0)=\nu(0)=\theta(0)=\omega(0)=0$ and $r(2)=1,;\nu(2)=0,;\theta(2)=\pi,;\omega(2)=0$.

JuDO code

using JuDO, Interesso, DynOptInterface
using Plots

const g     = 9.81
const l     = 0.5
const m_1   = 1.0
const m_2   = 0.3
const t_0   = 0.0
const t_f   = 2.0
const u_max = 20.0
const r_max = 2.0

dop = DynModel(Interesso.Optimizer)

# Independent variable (time)
@phase(dop, t)
@constraint(dop, initial(t) == 0)
@constraint(dop, final(t)   == 2)

# States
@variable(dop, 0 <= r <= r_max, DefinedOn(t))   # cart position (m)
@variable(dop, ν,               DefinedOn(t))   # cart velocity (m/s)
@variable(dop, θ,               DefinedOn(t))   # pole angle (rad)
@variable(dop, ω,               DefinedOn(t))   # pole angular velocity (rad/s)

# Control
@variable(dop, -u_max <= u <= u_max, DefinedOn(t))  # horizontal force (N)

# Boundary conditions
@constraint(dop, initial(r) == 0);  @constraint(dop, final(r) == 1)
@constraint(dop, initial(ν) == 0);  @constraint(dop, final(ν) == 0)
@constraint(dop, initial(θ) == 0);  @constraint(dop, final(θ) == pi)
@constraint(dop, initial(ω) == 0);  @constraint(dop, final(ω) == 0)

# Equations of motion
@constraint(dop, derivative(r) == ν)
@constraint(dop, derivative(ν) == (l*m_2*sin(θ)*ω^2 + u + m_2*g*cos(θ)*sin(θ)) /
                                  (m_1 + m_2*sin(θ)^2))
@constraint(dop, derivative(θ) == ω)
@constraint(dop, derivative(ω) == (-l*m_2*cos(θ)*sin(θ)*ω^2 - u*cos(θ) - (m_1 + m_2)*g*sin(θ)) /
                                  (l*(m_1 + m_2*sin(θ)^2)))

# Minimise control effort
@objective(dop, Min, integral(u^2))

# Warm-start with linear guesses
JuDO.warmstart!(dop, LinearInterpolant(0.0, 1.0, t_0, t_f), r)
JuDO.warmstart!(dop, LinearInterpolant(0.0, pi,  t_0, t_f), θ)

JuDO.optimize!(dop)

# Extract solution trajectories (each is a callable t -> value)
r_sol = dyn_value(dop, r)
θ_sol = dyn_value(dop, θ)
ν_sol = dyn_value(dop, ν)
ω_sol = dyn_value(dop, ω)
u_sol = dyn_value(dop, u)

ts = collect(range(t_0, t_f; length=200))

# Plot pole angle trajectory
p1 = plot(ts, θ_sol.(ts);
    xlabel="Time (s)", ylabel="Pole angle θ (rad)",
    legend=false, lw=1.5, lc=:black)

# Plot cart position trajectory
p2 = plot(ts, r_sol.(ts);
    xlabel="Time (s)", ylabel="Cart position r (m)",
    legend=false, lw=1.5, lc=:black)

# Plot control force
p3 = plot(ts, u_sol.(ts);
    xlabel="Time (s)", ylabel="Control force u (N)",
    legend=false, lw=1.5, lc=:black)

display(plot(p1, p2, p3; layout=(3,1), size=(600, 700)))

---

Space Shuttle Reentry {#space_shuttle_reentry}

This benchmark problem, due to Betts (2010), optimises the reentry trajectory of a space shuttle to maximise the crossrange (final latitude $\theta(t_f)$), subject to full six-state atmospheric flight dynamics and terminal boundary conditions. The reference optimal crossrange is approximately 34.14°.

Problem formulation

Physical constants

Symbol	Value	Description
$m$	203000/32.174 slug	Shuttle mass
$S$	2690 ft²	Reference area
$R_e$	20902900 ft	Earth radius
$\mu$	0.14076539 × 10¹⁷ ft³/s²	Gravitational parameter
$\rho_0$	0.002378 slug/ft³	Sea-level atmospheric density
$h_r$	23800 ft	Density scale height

Aerodynamic model

$$ C_L(\alpha) = a_0 + a_1 \alpha_{\deg}, \qquad C_D(\alpha) = b_0 + b_1 \alpha_{\deg} + b_2 \alpha_{\deg}^2 $$

where $\alpha_{\deg} = (180/\pi),\alpha$ and $(a_0, a_1) = (-0.20704,;0.029244)$, $(b_0, b_1, b_2) = (0.07854,;-0.61592 \times 10^{-2},;0.621408 \times 10^{-3})$.

Optimal control problem

$$ \max_{\alpha(\cdot),,\beta(\cdot)} \quad \theta(t_f) $$

subject to the six-state 3-DOF equations of motion over a spherical Earth:

$$ \dot h = v \sin\gamma, \qquad \dot\theta = \frac{v \cos\gamma \cos\psi}{r}, \qquad \dot\Phi = \frac{v \cos\gamma \sin\psi}{r \cos\theta} $$

$$ \dot v = -\frac{D}{m} - g\sin\gamma, \qquad \dot\gamma = \frac{L\cos\beta}{mv} + \cos\gamma!\left(\frac{v}{r} - \frac{g}{v}\right), \qquad \dot\psi = \frac{L\sin\beta}{mv\cos\gamma} + \frac{v\cos\gamma\sin\psi\tan\theta}{r} $$

where $r = R_e + h$, $g = \mu/r^2$, $\rho = \rho_0 e^{-h/h_r}$, $L = \tfrac{1}{2}S\rho v^2 C_L$, $D = \tfrac{1}{2}S\rho v^2 C_D$.

Boundary conditions

Variable	Initial	Final
$h$	260 000 ft	80 000 ft
$v$	25 600 ft/s	2 500 ft/s
$\theta$	0	free (maximised)
$\Phi$	0	free
$\gamma$	−1°	−5°
$\psi$	90°	free
$t$	0	$\leq$ 2500 s

JuDO code

using JuDO, Interesso, DynOptInterface
using Plots

dop = DynModel(Interesso.Optimizer)

# Physical constants
const m             = 203000 / 32.174
const ρ_0, h_r, R_e = 0.002378, 23800.0, 20902900.0
const μ             = 0.14076539e17
const a_0, a_1      = -0.20704, 0.029244
const b_0, b_1, b_2 = 0.07854, -0.61592e-2, 0.621408e-3
const S             = 2690

# Warm-start interpolant
struct LinearInterpolant <: DynOptInterface.AbstractDynamicSolution
    y_a::Float64;  y_b::Float64
end
(li::LinearInterpolant)(t::Real) = li.y_a + t * (li.y_b - li.y_a) / 2500

# Independent variable (time)
@phase(dop, t)
@constraint(dop, initial(t) == 0)
@constraint(dop,   final(t) ≤ 2500)

# States
@variable(dop,              0 ≤ h,              DefinedOn(t))  # altitude (ft)
@variable(dop, deg2rad(-89) ≤ θ ≤ deg2rad(89),  DefinedOn(t))  # latitude (rad)
@variable(dop,               Φ,                 DefinedOn(t))  # longitude (rad)
@variable(dop,           1e-4 ≤ v,              DefinedOn(t))  # velocity (ft/s)
@variable(dop, deg2rad(-89) ≤ γ ≤ deg2rad(89),  DefinedOn(t))  # flight path angle (rad)
@variable(dop,               ψ,                 DefinedOn(t))  # azimuth (rad)

# Controls
@variable(dop, deg2rad(-90) ≤ α ≤ deg2rad(90), DefinedOn(t))  # angle of attack (rad)
@variable(dop, deg2rad(-90) ≤ β ≤ deg2rad(1),  DefinedOn(t))  # bank angle (rad)

# Boundary conditions
@constraint(dop, initial(h) == 2.6e5);  @constraint(dop, final(h) == 0.8e5)
@constraint(dop, initial(θ) == 0)
@constraint(dop, initial(Φ) == 0)
@constraint(dop, initial(v) == 25600);  @constraint(dop, final(v) == 2500)
@constraint(dop, initial(γ) == deg2rad(-1));  @constraint(dop, final(γ) == deg2rad(-5))
@constraint(dop, initial(ψ) == deg2rad(90))

# Intermediate expressions (aerodynamics)
@expression(dop, r,     R_e + h)
@expression(dop, g,     μ / r^2)
@expression(dop, ρ,     ρ_0 * exp(-h / h_r))
@expression(dop, α_deg, (180 / pi) * α)
@expression(dop, L,     0.5 * S * ρ * v^2 * (a_0 + a_1 * α_deg))
@expression(dop, D,     0.5 * S * ρ * v^2 * (b_0 + b_1 * α_deg + b_2 * α_deg^2))

# Equations of motion
@constraint(dop, derivative(h) == v * sin(γ))
@constraint(dop, derivative(θ) == v * cos(γ) * cos(ψ) / r)
@constraint(dop, derivative(Φ) == v * cos(γ) * sin(ψ) / (r * cos(θ)))
@constraint(dop, derivative(v) == -D / m - g * sin(γ))
@constraint(dop, derivative(γ) == L * cos(β) / (m * v) + cos(γ) * (v / r - g / v))
@constraint(dop, derivative(ψ) == L * sin(β) / (m * v * cos(γ)) +
                                  v * cos(γ) * sin(ψ) * sin(θ) / (r * cos(θ)))

# Warm-start with linear guesses
JuDO.warmstart!(dop, LinearInterpolant(2.6e5,       0.8e5),        h)
JuDO.warmstart!(dop, LinearInterpolant(0.0,         deg2rad(45)),  θ)
JuDO.warmstart!(dop, LinearInterpolant(0.0,         deg2rad(50)),  Φ)
JuDO.warmstart!(dop, LinearInterpolant(25600.0,     2500.0),       v)
JuDO.warmstart!(dop, LinearInterpolant(deg2rad(-1), deg2rad(-5)),  γ)
JuDO.warmstart!(dop, LinearInterpolant(deg2rad(90), deg2rad(-20)), ψ)

# Maximise crossrange (final latitude)
@objective(dop, Max, final(θ))

JuDO.optimize!(dop)

# Extract solution trajectories
h_sol = dyn_value(dop, h)
θ_sol = dyn_value(dop, θ)
Φ_sol = dyn_value(dop, Φ)
v_sol = dyn_value(dop, v)
γ_sol = dyn_value(dop, γ)
ψ_sol = dyn_value(dop, ψ)
α_sol = dyn_value(dop, α)
β_sol = dyn_value(dop, β)

tf = phase_final(t)
ts = collect(range(0, tf; length=500))

# 3-D trajectory plot
traj = plot(
    rad2deg.(Φ_sol.(ts)),
    rad2deg.(θ_sol.(ts)),
    h_sol.(ts) ./ 1000;
    linewidth=1, legend=nothing, lc=:black,
    xlabel="Longitude (deg)", ylabel="Latitude (deg)", zlabel="Altitude (km)")
display(traj)

# Individual state and control trajectories
configs = [
    (h_sol, "Altitude (km)",           y -> y/1000),
    (θ_sol, "Latitude (deg)",          rad2deg),
    (Φ_sol, "Longitude (deg)",         rad2deg),
    (v_sol, "Velocity (km/s)",         y -> y/1000),
    (γ_sol, "Flight path angle (deg)", rad2deg),
    (ψ_sol, "Azimuth (deg)",           rad2deg),
    (α_sol, "Angle of attack (deg)",   rad2deg),
    (β_sol, "Bank angle (deg)",        rad2deg),
]
for (sol, ylabel, scale) in configs
    p = plot(ts, t -> scale(sol(t));
             legend=false, ylabel=ylabel, xlabel="Time (s)", lw=1, lc=:black)
    display(p)
end

Reference: Betts, J.T. (2010). Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed. SIAM.