Hovercraft Path Planning

In this case study, we seek to determine an optimal control policy for the trajectory of a hovercraft that travels to a set of dynamic waypoints while trying to minimize the thrust input.

Formulation

The corresponding dynamic optimization problem is expressed:

\[\begin{aligned} &&\underset{x(t), v(t), u(t)}{\text{min}} &&& \int_{t \in T} |u(t)|_2^2 dt \\ &&\text{s.t.} &&& v(0) = v0\\ &&&&& \frac{dx}{dt} = v(t), && t \in T\\ &&&&& \frac{dv}{dt} = u(t), && t \in T\\ &&&&& x(t_i) = xw_i, && i \in I \end{aligned}\]

where $x(t)$ is the Cartesian position, $v(t)$ is the velocity, $u(t)$ is the thrust input, $xw_i, \ i \in I,$ are the waypoints, and $T$ is the time horizon.

Model Definition

Let's implement this in InfiniteOpt and first import the packages we need:

using InfiniteOpt, Ipopt

Next we'll specify our waypoint data:

xw = [1 4 6 1; 1 3 0 1] # positions
tw = [0, 25, 50, 60];    # times

We initialize the infinite model and opt to use the Ipopt solver:

m = InfiniteModel(optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0));

Let's specify our infinite parameter which is time $t \in [0, 60]$:

@infinite_parameter(m, t in [0, 60], num_supports = 61)

t

Now let's specify the decision variables:

@variables(m, begin
    # state variables
    x[1:2], Infinite(t)
    v[1:2], Infinite(t)
    # control variables
    u[1:2], Infinite(t), (start = 0)
end)

(GeneralVariableRef[x[1](t), x[2](t)], GeneralVariableRef[v[1](t), v[2](t)], GeneralVariableRef[u[1](t), u[2](t)])

Specify the objective:

@objective(m, Min, ∫(u[1]^2 + u[2]^2, t))

∫{t ∈ [0, 60]}[u[1](t)² + u[2](t)²]

Set the initial conditions with respect to the velocity:

@constraint(m, [i = 1:2], v[i](0) == 0)

2-element Vector{InfOptConstraintRef}:
 v[1](0) = 0
 v[2](0) = 0

Define the point physics ODEs which serve as our system model:

@constraint(m, [i = 1:2], ∂(x[i], t) == v[i])
@constraint(m, [i = 1:2], ∂(v[i], t) == u[i])

2-element Vector{InfOptConstraintRef}:
 ∂/∂t[v[1](t)] - u[1](t) = 0, ∀ t ∈ [0, 60]
 ∂/∂t[v[2](t)] - u[2](t) = 0, ∀ t ∈ [0, 60]

Ensure we hit all the waypoints:

@constraint(m, [i = 1:2, j = eachindex(tw)], x[i](tw[j]) == xw[i, j])

2×4 Matrix{InfOptConstraintRef}:
 x[1](0) = 1  x[1](25) = 4  x[1](50) = 6  x[1](60) = 1
 x[2](0) = 1  x[2](25) = 3  x[2](50) = 0  x[2](60) = 1

Problem Solution

Optimize the model:

optimize!(m)

Extract the results:

x_opt = value.(x);

Plot the results:

using Plots
scatter(xw[1,:], xw[2,:], label = "Waypoints", background_color = :transparent)
plot!(x_opt[1], x_opt[2], label = "Trajectory")
xlabel!("x_1")
ylabel!("x_2")

That's it, now we have our optimal trajectory!

Maintenance Tests

These are here to ensure this example stays up to date.

using Test
@test termination_status(m) == MOI.LOCALLY_SOLVED
@test has_values(m)
@test x_opt isa Vector{<:Vector{<:Real}}

Test Passed

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