Constraints

Constraints restrict the feasible region beyond dynamics. DirectTrajOpt supports bounds, boundary conditions, and nonlinear path constraints.

using DirectTrajOpt
using NamedTrajectories
using LinearAlgebra

N = 50
traj = NamedTrajectory(
    (x = randn(2, N), u = randn(1, N), Δt = fill(0.1, N));
    timestep=:Δt, controls=:u
)

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

Bounds (Cheapest)

Box constraints on variables:

traj_bounds = NamedTrajectory(
    (x = randn(2, N), u = randn(2, N), Δt = fill(0.1, N));
    timestep=:Δt, controls=:u,
    bounds=(
        x = 5.0,                          # -5 ≤ x ≤ 5
        u = (-1.0, 2.0),                  # -1 ≤ u ≤ 2
        Δt = (0.01, 0.5)                  # 0.01 ≤ Δt ≤ 0.5
    )
)

N = 50, (x = 1:2, u = 3:4, → Δt = 5:5)

Per-component bounds:

traj_component_bounds = NamedTrajectory(
    (x = randn(2, N), u = randn(2, N), Δt = fill(0.1, N));
    timestep=:Δt, controls=:u,
    bounds=(u = ([-1.0, -2.0], [1.0, 3.0]),)  # Different bounds per component
)

N = 50, (x = 1:2, u = 3:4, → Δt = 5:5)

Nonlinear Constraints

Inequality: c(x, u) ≥ 0 (preferred - easier to satisfy)

constraint_ineq = NonlinearKnotPointConstraint(
    u -> [1.0 - norm(u)],  # ||u|| ≤ 1
    :u, traj; times=1:N, equality=false
)

NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#31#32"{Main.var"#2#3"}}(DirectTrajOpt.Constraints.var"#31#32"{Main.var"#2#3"}(Main.var"#2#3"()), [:u], false, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 1, 50)

Equality: c(x, u) = 0 (more restrictive)

constraint_eq = NonlinearKnotPointConstraint(
    x -> [x[1] - 0.5],  # x₁ = 0.5
    :x, traj; times=[25], equality=true
)

NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#31#32"{Main.var"#5#6"}}(DirectTrajOpt.Constraints.var"#31#32"{Main.var"#5#6"}(Main.var"#5#6"()), [:x], true, [25], [nothing], 1, 2, 1)

Multiple variables:

constraint_multi = NonlinearKnotPointConstraint(
    (x, u) -> [x[1]^2 + x[2]^2 - u[1]],
    [:x, :u], traj; equality=false
)

NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#39#40"{Main.var"#8#9", Vector{UnitRange{Int64}}}}(DirectTrajOpt.Constraints.var"#39#40"{Main.var"#8#9", Vector{UnitRange{Int64}}}(Main.var"#8#9"(), UnitRange{Int64}[1:2, 3:3]), [:x, :u], false, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 3, 50)

Common Patterns

Obstacle avoidance:

obs_center, obs_radius = [0.5, 0.5], 0.2
constraint_obstacle = NonlinearKnotPointConstraint(
    x -> [norm(x - obs_center)^2 - obs_radius^2],
    :x, traj; times=1:N, equality=false
)

NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#31#32"{Main.var"#11#12"}}(DirectTrajOpt.Constraints.var"#31#32"{Main.var"#11#12"}(Main.var"#11#12"()), [:x], false, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 2, 50)

Multiple obstacles:

constraints_obstacles = [
    NonlinearKnotPointConstraint(
        x -> [norm(x - center)^2 - radius^2],
        :x, traj; equality=false
    )
    for (center, radius) in [([0.3, 0.3], 0.15), ([0.7, 0.7], 0.15)]
]

2-element Vector{NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#31#32"{Main.var"#17#18"{Float64, Vector{Float64}}}}}:
 NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#31#32"{Main.var"#17#18"{Float64, Vector{Float64}}}}(DirectTrajOpt.Constraints.var"#31#32"{Main.var"#17#18"{Float64, Vector{Float64}}}(Main.var"#17#18"{Float64, Vector{Float64}}(0.15, [0.3, 0.3])), [:x], false, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 2, 50)
 NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#31#32"{Main.var"#17#18"{Float64, Vector{Float64}}}}(DirectTrajOpt.Constraints.var"#31#32"{Main.var"#17#18"{Float64, Vector{Float64}}}(Main.var"#17#18"{Float64, Vector{Float64}}(0.15, [0.7, 0.7])), [:x], false, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 2, 50)

State-dependent control limits:

constraint_state_dep = NonlinearKnotPointConstraint(
    (x, u) -> [1.0 - u[1] / (1.0 + abs(x[1]))],
    [:x, :u], traj; equality=false
)

NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#39#40"{Main.var"#20#21", Vector{UnitRange{Int64}}}}(DirectTrajOpt.Constraints.var"#39#40"{Main.var"#20#21", Vector{UnitRange{Int64}}}(Main.var"#20#21"(), UnitRange{Int64}[1:2, 3:3]), [:x, :u], false, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 3, 50)

Energy constraints:

E_max = 2.0
constraint_energy = NonlinearKnotPointConstraint(
    (x, u) -> [E_max - (0.5 * norm(x)^2 + 0.5 * norm(u)^2)],
    [:x, :u], traj; equality=false
)

NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#39#40"{Main.var"#23#24", Vector{UnitRange{Int64}}}}(DirectTrajOpt.Constraints.var"#39#40"{Main.var"#23#24", Vector{UnitRange{Int64}}}(Main.var"#23#24"(), UnitRange{Int64}[1:2, 3:3]), [:x, :u], false, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10  …  41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 3, 50)

Time Selection

All times, specific times, or ranges:

constraint_all = NonlinearKnotPointConstraint(
    u -> [1.0 - norm(u)], :u, traj; times=1:N, equality=false
)

constraint_specific = NonlinearKnotPointConstraint(
    x -> [x[1]^2 + x[2]^2 - 1.0], :x, traj;
    times=[1, 10, 20, 30, 40, 50], equality=false
)

constraint_range = NonlinearKnotPointConstraint(
    u -> [0.5 - norm(u)], :u, traj; times=10:40, equality=false
)

NonlinearKnotPointConstraint{DirectTrajOpt.Constraints.var"#31#32"{Main.var"#32#33"}}(DirectTrajOpt.Constraints.var"#31#32"{Main.var"#32#33"}(Main.var"#32#33"()), [:u], false, [10, 11, 12, 13, 14, 15, 16, 17, 18, 19  …  31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing  …  nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing], 1, 1, 31)

Creating a Problem

G_drift = [-0.1 1.0; -1.0 -0.1]
G_drives = [[0.0 1.0; 1.0 0.0]]
G = u -> G_drift + sum(u .* G_drives)
integrator = BilinearIntegrator(G, traj, :x, :u)
obj = QuadraticRegularizer(:u, traj, 1.0)

constraints = [constraint_obstacle, constraint_ineq]
prob = DirectTrajOptProblem(traj, obj, integrator; constraints=constraints)

Summary

Performance Tips

Troubleshooting

If optimizer struggles:

• Infeasible start: Initial guess violates constraints → improve initial guess
• Over-constrained: Too many/conflicting constraints → relax or remove some
• Poorly scaled: Values span many orders of magnitude → rescale to O(1)
• Tight constraints: Little feasible space → relax bounds or use soft constraints

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