Optimal orbit

opt_orbit.jl

### A Pluto.jl notebook ###
# v0.16.1

using Markdown
using InteractiveUtils

# ╔═╡ b2663d22-8605-11eb-17fa-e90e625ac922
using ComponentArrays

# ╔═╡ 932b2cb6-8609-11eb-27d7-a393d81f6d8a
using ControlSystems: care

# ╔═╡ 94e89f78-8609-11eb-0688-a745741d702f
using LinearAlgebra: I

# ╔═╡ eab4fe16-8605-11eb-3b11-cf58c2dffa78
using DifferentialEquations

# ╔═╡ eab57056-8605-11eb-17de-a3ee0f3df254
using GalacticOptim

# ╔═╡ 2e95444e-8609-11eb-1bef-13cce8dce38f
using ModelingToolkit

# ╔═╡ 33322012-8609-11eb-2abe-eb001e8ca823
using ModelingToolkit.Symbolics: value

# ╔═╡ 42ccef75-b452-4791-b088-d24901f0b2fe
using Optim

# ╔═╡ eab62458-8605-11eb-3397-4536351c04d1
using UnPack

# ╔═╡ 1644185a-860b-11eb-369a-67edd730560b
using Latexify

# ╔═╡ eabe038a-8605-11eb-2543-fd874aaa7da8
using Plots; gr();

# ╔═╡ eabe62d0-8605-11eb-0bd4-19fdfcb5d688
using Plots: scatter!

# ╔═╡ 3953dbb0-270d-43cc-aa15-86d0cadd2154
using Setfield: @set, @set!

# ╔═╡ 459a7146-879e-11eb-2f8f-0bfc1f9619ad
using PlutoUI

# ╔═╡ 0d63f2f8-8617-11eb-042c-c5505db57b49
md"""
Jonnie Diegelman

EAS 6415 | Guidance Navigation and Control

March 15, 2021
"""

# ╔═╡ a21a5dfc-8616-11eb-207a-0be8279d50e5
md"# Project 4"

# ╔═╡ 26dd9d90-879f-11eb-0baf-27a2e6de9445
md"All handwritten derivations are at the bottom of this notebook in the Extra Stuff section."

# ╔═╡ 465328aa-8607-11eb-0a33-ad2e91536374
md"## Problem 1 | Forced Duffing Oscillator"

# ╔═╡ 6288f1c2-860d-11eb-2051-ab1863135f10
md"### Part 1: Open-loop optimal control"

# ╔═╡ cf0ed7e4-8791-11eb-3900-a981dd1104d6
md"We'll start by defining the equations of motion for a forced Duffing oscillator"

# ╔═╡ eeeeb6fc-8605-11eb-10c7-9d0b84b80b7f
function duffing!(ẋ, x, p, t; u=0)
    @unpack α, β = p
    ẋ[1] = x[2]
    ẋ[2] = -α*x[1] - β*x[1]^3 + u
end

# ╔═╡ e02e5962-8791-11eb-2a4d-ef8e44062de1
md"Now we can use the Pontryagin minimum principle to derive the costate equations for an optimally controlled Duffing oscillator"

# ╔═╡ 322743da-8606-11eb-13e1-5773eb2c6dfe
function duffing_opt!(D, vars, params, t)
    @unpack x, λ = vars
    @unpack α, β = params

    u = -λ[2]
    duffing!(D.x, x, params, t; u)

    D.λ[1] = (α + 3β*x[1]^2)*λ[2]
    D.λ[2] = -λ[1]
end

# ╔═╡ 0565356a-8792-11eb-0026-b33c2218b8be
md"We'll define a `solve_with` function to solve the optimal equations of motion for a given `λ₀`."

# ╔═╡ 3234beb6-8606-11eb-0f7e-83af77b041cf
function solve_with(λ₀, p; kwargs...)
    ic = ComponentArray(x=p.x0, λ=λ₀)
    prob = ODEProblem(duffing_opt!, ic, (0.0, 2π), p)
    return solve(prob; dt=0.1, kwargs...)
end

# ╔═╡ 6899353a-8792-11eb-16e8-4b0281b6cca2
md"The cost function for the optimization problem is
``\frac{1}{2} \gamma (x(t_f)-x_f)^T (x(t_f)-x_f) + \frac{1}{2} u^T u`` where ``\gamma`` is set to 1, but can be adjusted to favor better accuracy at the expense of more control effort."

# ╔═╡ 323ade9a-8606-11eb-225c-29e144b240ac
function cost(λ, p)
    Δt = 0.01
    sol = solve_with(λ, p; saveat=Δt)
    Δxf = sol[end].x - p.xf
    u = map(state -> -state.λ[2], sol)
    return p.γ*0.5Δxf'*Δxf + Δt*0.5u'*u
end

# ╔═╡ 72fc69a6-8793-11eb-1302-6fab7a8b976d
md"The final conditions costate variables are the constraints for the optimization problem."

# ╔═╡ 323ec24e-8606-11eb-0829-f9e02af63a9a
function constraint(λ, p)
    sol = solve_with(λ, p)
    return sol[end].λ - sol[end].x + p.xf
end

# ╔═╡ 3645bb4a-8606-11eb-3437-8b0a09ec65ed
p = (
	# Parameters
    α = 1,
    β = 0.5,
    γ = 1,
	# Initial and desired final conditions of the state
    x0 = [0.0, 0.0],
    xf = [5.0, 2.0],
);

# ╔═╡ 3f81ee04-8606-11eb-38e9-a9ed9887f583
λ = [1.0, 1.0];

# ╔═╡ fcd23eec-8793-11eb-0671-1f45f7b14200
md"The first method we will try is direct optimization using the cost function and the constraint. We will use Newton's method with a trust and forward-mode automatic differentiation for efficient and exact (to numerical precision) jacobian and hessian calls."

# ╔═╡ 3e696e16-8606-11eb-3f8e-d917185fea3e
λ⁺, opt_time = let
	opt_fun = OptimizationFunction(cost, GalacticOptim.AutoForwardDiff();
		cons = constraint,
	)
	prob = OptimizationProblem(opt_fun, λ, p)
	λ⁺ = solve(prob, NewtonTrustRegion())
	
	time = @elapsed solve(prob, NewtonTrustRegion())
	λ⁺, time
end;

# ╔═╡ 1b0ef3d8-8795-11eb-1ff0-8b94a636c5c3
md"The previous method was a bit wasteful since the costate equations are already taking care of the optimality conditions for us. We really just need to satisfy the constraints. This time, we'll solve the problem using Newton's method directly."

# ╔═╡ e2063c3e-8606-11eb-0caf-d93244fc6522
λ⁺nl = solve(NonlinearProblem(constraint, λ, p));

# ╔═╡ a08c069a-879a-11eb-2c7b-dba00bfd0553
newton_time = @elapsed solve(NonlinearProblem(constraint, λ, p));

# ╔═╡ f70c24a0-8795-11eb-2ac5-e5ffc09d8da3
md"""
Using Newton's method directly to find the costate initial conditions ended up being  three orders of magnitude faster than the optimization method in this case.

|Method  |Time (s) |
|:--|:--|
|**Constrained Optimization** | $opt_time |
|**Newton's Method** | $newton_time |

As we can see in the plot below, the results are almost identical.
"""

# ╔═╡ 41da74e0-8609-11eb-1363-01a6d57da93c
md"#### Alternative method: Automation with symbolic modeling"

# ╔═╡ c3df0828-8609-11eb-3d70-2b1f83de8baa
md"Deriving the optimality conditions by hand is time consuming and error-prone. Let's instead make a function that automates the process of finding the necessary conditions for optimality for any given system."

# ╔═╡ 2b6f5432-860a-11eb-3ce0-45be1ea1c8e5
@parameters t

# ╔═╡ 2d5f7bf2-860a-11eb-10e7-eb2fedc5a703
D = Differential(t);

# ╔═╡ c35f19a2-8798-11eb-32d9-977f2a75b70d
∂_∂(x) = y -> expand_derivatives(Differential(x)(y))

# ╔═╡ 87f730a6-8609-11eb-2c3e-ab3bf3cf0550
function Pontryagin(; name, sys_fun, L, x, u, ps...)
	x = collect(x)
	u = collect(u)
	ps = collect(ps)
    @named sys = sys_fun(; x, u, ps...)

    # Get right hand side of equations
    f = [eq.rhs for eq in sys.eqs]

    # Lagrange multipliers / costate variables
    @variables λ[1:length(x)](t)
	λ = collect(λ)

    # Hamiltonian
    H = λ'*f + L(x, u, t)

    # Costate equations
    λ_eqs = [D(λᵢ) ~ -∂_∂(xᵢ)(H) for (xᵢ, λᵢ) in zip(x, λ)]
    
    # Control optimality
    opt_eqs = [0.0 ~ ∂_∂(uᵢ)(H) for uᵢ in u]

    defaults = Dict(λ .=> 0.0)

    ODESystem([opt_eqs..., λ_eqs..., sys.eqs...], t, [u..., λ..., x...], sys.ps; name, defaults)
end

# ╔═╡ 11612608-860a-11eb-34ef-1becabce3712
md"We'll make our forced duffing oscillator again, this time with symbolic modeling"

# ╔═╡ 09f2c2b4-860a-11eb-322e-596ed42559e0
function Duffing(; name, x, u, α, β)
    u = only(u)

    eqs = [
        D(x[1]) ~ x[2]
        D(x[2]) ~ -α*x[1] - β*x[1]^3 + u
    ]

    ODESystem(eqs, t, [x..., u], [α, β]; name)
end

# ╔═╡ 6f8daa3a-860a-11eb-22e2-751d7e15bb4e
md"Our Lagrangian for minimum control effort"

# ╔═╡ 60e490de-860a-11eb-1bd1-0103169054bf
L(x, u, t) = 0.5sum(u.^2)

# ╔═╡ 5edad0c8-860a-11eb-3877-8975ec0576c2
md"The final state cost derivative could be handled symbolically as well too, but I'm lazy."

# ╔═╡ 9e13b098-860a-11eb-2c22-ef7070c11b03
∂Φ_∂xf(x⁺f, xf, γ) = γ * (x⁺f .- xf)

# ╔═╡ 347069b0-860a-11eb-2d31-8bc44d7bc8f9
@variables x[1:2](t) u(t)

# ╔═╡ b7242fe0-860a-11eb-3221-5931477ac2c6
@parameters α, β

# ╔═╡ bf7d2c0a-860a-11eb-057d-157aa490c4d7
@named opt_duff = Pontryagin(; sys_fun=Duffing, x, u=[u], L, α, β);

# ╔═╡ cd6021a6-860a-11eb-20b7-bf7e9c0307f1
md"`structural_simplify` reduces our differential algebraic equations to plain differential equations (if possible) and reduces alias variables like control variables."

# ╔═╡ c2476068-860a-11eb-1f7a-b7e9b59a4b8e
simplified_duff = structural_simplify(opt_duff)

# ╔═╡ 497ebb2a-860c-11eb-0dac-c97a5b06feab
md"Since we already solved this problem as both a constrained optimization problem and a root finding problem, we'll do it a third way this time by solving it directly as a boundary value problem."

# ╔═╡ 01e9aca2-8611-11eb-2052-314cb6d609db
params = [
	α => 1.0
	β => 0.5
];

# ╔═╡ 689af84e-860b-11eb-2e2e-b557abae2c83
bvp_sol, bvp_time = let
	# Initial conditions
	ic = [
		opt_duff.λ[1] => 1.0
		opt_duff.λ[2] => 1.0
		x[1] => 0.0
		x[2] => 0.0
	]
	
	# Desired final state
	xf = [5.0, 2.0]
	
	# Final state vs control cost weight (higher favors final state accuracy)
	γ = 1.0
	
	# The boundary conditions can't be handled symbolically (yet)
	function bc!(residual, vars, params, t)
		vars0, varsf = @views vars[begin], vars[end]
		λ⁺f, x⁺f, x⁺0 = @views varsf[1:2], varsf[3:4], vars0[3:4]
		x0 = last.(@view ic[3:4])

		# Initial condition of state variables
		residual[1:2] .= x⁺0 .- x0

		# Final condition of costate variables
		residual[3:4] .= λ⁺f .- ∂Φ_∂xf(x⁺f, xf, params[3])
	end
	
	# Build two-point boundary problem from a regular ODE problem
	prob = ODEProblem(simplified_duff, ic, (0.0, 2π), params)
	bvp = TwoPointBVProblem(prob.f, bc!, prob.u0, (0.0, 2π), [prob.p..., γ])
	
	# Solve the BVP
	sol = solve(bvp, Shooting(Tsit5()), dt=0.1)
	time = @elapsed solve(bvp, Shooting(Tsit5()), dt=0.1)
	
	sol, time
end;

# ╔═╡ 021214fe-860d-11eb-0d34-3f2b09cf90b8
md"""
This method is the fastest of all. Updating our table:

|Method  |Time (s) |
|:--|:--|
|**Constrained Optimization** | $opt_time |
|**Newton's Method** | $newton_time |
|**Shooting Method** | $bvp_time |


We can plot it over our last plot to see that the three methods produce the same answer
"""

# ╔═╡ 9daeae98-860d-11eb-04fa-0f00f858ba96
md"### Part 2: LQR"

# ╔═╡ a9611c9e-860d-11eb-1a30-57f452ff3711
md"Since the automated approach worked so well last time, let's do something similar to build a linear quadratic regulator problem symbolically from any arbitrary system."

# ╔═╡ e62115b2-860d-11eb-0219-e159be379433
# Equilibrium to linearize about
xeq = [0.0, 0.0]

# ╔═╡ e3dffc78-860d-11eb-36fd-63a99005485a
function LQR(; name, sys_fun, x, u, p, Q, R,
			   operating_point=zeros(size(Q,1)), linear=true, ps...)
	x = collect(x)
    # Create system to be controlled
    @named sys = sys_fun(; x, u, ps...)
    
    # State and control variables
    @variables δx[1:length(x)](t)
	δx = collect(δx)

    # Linearize by calculating state and control jacobians
	# Note: We have to convert to floating point because the
	#   `care` function doesn't work symbolically.
    J = substitute.(calculate_jacobian(sys), Ref([(x.=>operating_point)..., p...]))
    A = J[:, eachindex(x)] .|> value .|> float
    B = J[:, length(x).+eachindex(u)] .|> value .|> float

    # Solve continuous algebraic Ricatti equation and calculate Kalman gain
    P = care(A, B, Q, R)
    K = R\B'*P

    # Alias equations
    eqs = [
        u .~ -K*δx
        δx .~ x - xeq
    ]

    diff_eqs = if linear
        D.(δx) .~ A*δx + B*u
    else
        diff_eqs = [
			substitute(eq.lhs, x.=>δx) ~ substitute(eq.rhs, [x.=>δx; p])
			for eq in sys.eqs
		]
    end

    return ODESystem([eqs..., diff_eqs...], t, [δx..., x..., u...], []; name)
end

# ╔═╡ 4beb0de4-860e-11eb-2cb6-6130ea972f3e
Q = 6I(2)

# ╔═╡ 561104fe-860e-11eb-1e71-b1c4ff5c439a
R = 1

# ╔═╡ 64fccbba-860e-11eb-1ee7-17509c864c0d
@named lqr_duff = LQR(;
	sys_fun = Duffing,
	x,
	u = [u],
	α,
	β,
	operating_point = xeq,
	p = params,
	Q,
	R,
);

# ╔═╡ 3d4fcdcc-8610-11eb-3c22-dbfbd6e83d29
simplified_lqr = structural_simplify(lqr_duff)

# ╔═╡ ebde1b5e-8611-11eb-0bb3-5dfb08904d7e
lqr_ic = collect(simplified_lqr.states .=> 0.5ones(2));

# ╔═╡ 580f083a-8610-11eb-3467-19bc305be25b
lqr_sol = let
	prob = ODEProblem(simplified_lqr, lqr_ic, (0.0, 2π))
	solve(prob)
end;

# ╔═╡ af92f2ce-8610-11eb-3c8b-abe9fd946cb9
plt_lqr = plot(lqr_sol;
    vars = [x..., u],
    labels = "linear " .* string.([x... u]),
	lw = 1.5,
    title = "Duffing oscillator with LQR control",
    legend = :bottomright,
)

# ╔═╡ b4ccb52c-8610-11eb-1a17-372edb74b41f
md"Let's now use the same method to control our nonlinear system"

# ╔═╡ e533bdc8-8610-11eb-3e38-b5a4f234957a
@named lqr_duff2 = LQR(;
	sys_fun = Duffing,
	x,
	u = [u],
	α,
	β,
	operating_point = xeq,
	p = params,
	Q,
	R,
	linear = false,
);

# ╔═╡ e78d0174-8610-11eb-0465-33ca3c964058
simplified_lqr2 = structural_simplify(lqr_duff2)

# ╔═╡ c8fbc514-8611-11eb-0505-9b6c572761f0
lqr_sol2 = let
	prob = ODEProblem(simplified_lqr2, lqr_ic, (0.0, 2π), params)
	solve(prob)
end;

# ╔═╡ 104c2292-8612-11eb-10d6-fbe4bc54a2ed
md"We see that the controller for the linear system works pretty well for the nonlinear as well"

# ╔═╡ a7eff61a-8611-11eb-33b6-43681a57bed0
plot!(plt_lqr, lqr_sol2;
    vars = [x..., u],
    color = [1 2 3],
    linestyle = :dash,
	lw = 1.5,
    labels = "nonlinear " .* string.([x... u]),
)

# ╔═╡ 6732c5b4-8607-11eb-0bea-3fd32f4f9b97
md"## Problem 2 | Optimal Time Orbit Transfer"

# ╔═╡ 488a2528-860d-11eb-13d6-c5611a8c376d
md"We'll pick a default maximum time span, in case the terminal conditions aren't reached"

# ╔═╡ 8a098bf2-8607-11eb-0430-230257194511
tspan = (0.0, 1000.0);

# ╔═╡ 2e42e77c-8612-11eb-10f3-77688f74dbce
md"Let's switch back to the old method of modeling for now and write our equations of motion and their optimality conditions by hand."

# ╔═╡ 8c03bde2-8607-11eb-3740-19e4a3e3cdfe
function two_body_system!(D, x, p, t; ϕ=0.0)
    @unpack μ, aₜ, ṁ = p
    @unpack r, vᵣ, vₛ, θ = x

    D.r = vᵣ
    D.vᵣ = vₛ^2/r - μ/r^2 + aₜ*sin(ϕ)/(1+ṁ*t)
    D.vₛ = -vᵣ*vₛ/r + aₜ*cos(ϕ)/(1+ṁ*t)
    D.θ = vₛ/r
    return D
end

# ╔═╡ 8c0420a2-8607-11eb-3887-eb8293663b10
function two_body_system_opt!(D, vars, params, t)
    @unpack x, λ = vars
    @unpack r, vᵣ, vₛ, θ = x
    @unpack μ, aₜ, ṁ = params

    ϕ = atan(λ.vᵣ, λ.vₛ)
    two_body_system!(D.x, x, params, t; ϕ)

    D.λ.r = λ.vᵣ*(vₛ^2/r^2 - 2*μ/r^3) - λ.vₛ*vᵣ*vₛ/r^2 + λ.θ*vₛ/r^2
    D.λ.vᵣ = -λ.r + λ.vₛ*vₛ/r
    D.λ.vₛ = -λ.vᵣ*2*vₛ/r + λ.vₛ*vᵣ/r - λ.θ/r
    D.λ.θ = 0
end

# ╔═╡ 6cc9dd9a-8612-11eb-000a-8f06d32b5925
md"We will need a terminal condition to stop the simulation. We can use the radial position for this."

# ╔═╡ 903d2d82-8607-11eb-19b7-658f01c97580
terminal_condition(vars, t, integrator) = vars.x.r - integrator.p.xf.r

# ╔═╡ 99c6e45e-8607-11eb-1cac-9dccb2bc2ed4
cb = ContinuousCallback(terminal_condition, terminate!);

# ╔═╡ 99c8b1a8-8607-11eb-1190-01b653284c5e
function solve_with2(λ, p; tspan=tspan, callback=cb, kwargs...)
    ic = ComponentArray(; x=p.x0, λ)
    prob = ODEProblem(two_body_system_opt!, ic, tspan, p)
    return solve(prob; callback, kwargs...)
end

# ╔═╡ 8a47b6c4-8612-11eb-2cc4-01582b1cf43a
md"We will need the hamiltonian this time for the final conditions."

# ╔═╡ 99c77798-8607-11eb-0f25-334e076024a5
function H(vars, p, t)
    ẋ = two_body_system!(copy(vars.x), vars.x, p, t)
    return 1 + vars.λ' * ẋ
end

# ╔═╡ b78c7e7c-8607-11eb-2996-a334ca23a0a0
function final_condition(λ, p; kwargs...)
    out = zero(p.x0)
    sol = solve_with2(λ, p; kwargs...)
    out[1:3] .= -sol[end].x[1:3] + p.xf
    out.θ = H(sol[end], p, sol.t[end])
    return out
end

# ╔═╡ d7cdc074-8607-11eb-35f2-81bf9b96713d
p2 = let
	μ = 1
	r0 = 1
	rf = 1.523
	p = (;
		μ,
		aₜ = 0.115, 
		ṁ = -0.250154,
		x0 = ComponentArray(
			r = r0,
			vᵣ = 0,
			vₛ = sqrt(μ/r0),
			θ = 0,
		),
		xf = ComponentArray(
			r = rf,
			vᵣ = 0.0,
			vₛ = sqrt(μ/rf),
		),
	)
end;

# ╔═╡ 017d2f72-8608-11eb-28f0-bf589670eca7
λ0 = -one.(p2.x0);

# ╔═╡ c75efa22-8612-11eb-2dbb-c5cbd415d489
md"We will use the rootfinding method this time, since that was the fastest method last time."

# ╔═╡ 0474c990-8608-11eb-2f39-d9a9a253f382
λ⁺2 = NonlinearProblem(final_condition, λ0, p2) |> solve;

# ╔═╡ bca70996-8615-11eb-01e2-d97291bf1821
nlsol = solve_with2(λ⁺2.u, p2);

# ╔═╡ 5cf35f3e-8608-11eb-38ed-f1509c6991bb
plt_orb_states, x_lims = let
	p = p2
	t = range(0, nlsol.t[end], length=200)
	ϕ = map(sol->atan(sol.λ.vᵣ, sol.λ.vₛ), nlsol(t))
	θ = map(sol->sol.x.θ, nlsol(t))
	r = map(sol->sol.x.r, nlsol(t))
	var_strings = permutedims(string.([keys(p.x0)...]))
	xlims = (t[1],t[end]+0.1)
	
	plt = plot(nlsol; vars=1:4, labels=var_strings, title="Orbit Transfer State Variables", lw=1.5, legend=:bottomleft)
	plot!(t, ϕ, label="ϕ", lw=1.5)
	scatter!(fill(t[end], 3)', p.xf';
		color = [1 2 3],
		xlims = xlims,
		markersize = 5,
		markerstrokewidth = 0,
		labels = var_strings[:,1:3] .* " target",
	)
	plt, xlims
end; plt_orb_states

# ╔═╡ d0796336-8608-11eb-3830-69d130422b78
let
	p = p2
	t = range(0, nlsol.t[end], length=200)
	θ = map(sol->sol.x.θ, nlsol(t))
	r = map(sol->sol.x.r, nlsol(t))
	θs = range(0, 2π; length=200)
	
	plot(θ, r;
		proj=:polar,
		color=3,
		framestyle=:zerolines,
		label="trajectory",
		title="Optimal Orbit Transfer",
		lw=1.5
	)
	
	scatter!([p.x0.θ], [p.x0.r];
		color=1,
		markersize=10,
		markerstrokewidth=0,
		label="Earth",
	)
	scatter!([nlsol[end].x.θ], [p.xf.r];
		color=2,
		markersize=6,
		label="Mars",
		legend=:bottomleft,
		markerstrokewidth=0,
	)
	
	orbit_args = (linestyle=:dash, label=nothing)
	plot!(θs, fill(p.x0.r, length(θs)); color=1, orbit_args...)
	plot!(θs, fill(p.xf.r, length(θs)); color=2, orbit_args...)
end

# ╔═╡ de66b890-8612-11eb-3e8f-bfcc58a9487a
md"#### Back to symbolics"

# ╔═╡ 0a9cd610-8613-11eb-3870-df95f9f4e8ac
md"Using the same framework from part 1, we can let our `Pontryagin` function automatically symbolically derive our optimality conditions for us. But first we need to redefine our two body system in symbolic form."

# ╔═╡ 3ac31c28-8613-11eb-30e3-3f8c97f23ce9
function TwoBodySystem(; name, x, u, aₜ, ṁ, μ)
	x = collect(x)
	u = collect(u)
	r, vᵣ, vₛ, θ = x
    ϕ = only(u)
    
    eqs = [
        D(r)  ~  vᵣ
        D(vᵣ) ~  vₛ^2/r - μ/r^2 + aₜ*sin(ϕ)/(1+ṁ*t)
        D(vₛ) ~ -vᵣ*vₛ/r + aₜ*cos(ϕ)/(1+ṁ*t)
        D(θ)  ~  vₛ/r
    ]

    return ODESystem(eqs; name)
end

# ╔═╡ 3f9a304c-8613-11eb-2e81-6366cd964685
md"Our Lagrangian this time will just be 1 because this is a optimal time problem."

# ╔═╡ 60bf8f60-8613-11eb-1645-d363a3f41f3d
L2(x, u, t) = 1

# ╔═╡ 76c78fba-8613-11eb-366d-89f27751a8e6
@parameters μ aₜ ṁ

# ╔═╡ 79172776-8613-11eb-30cd-25fe09b55e94
@variables r(t) vᵣ(t) vₛ(t) θ(t) ϕ(t)

# ╔═╡ 7c9fe4fa-8613-11eb-0505-8ff68d1ac588
x2 = [r, vᵣ, vₛ, θ];

# ╔═╡ 893c11c2-8613-11eb-093a-e95929c3ff5e
@named opt_two_body = Pontryagin(; sys_fun=TwoBodySystem, x=[r, vᵣ, vₛ, θ], u=[ϕ], L=L2, μ, aₜ, ṁ)

# ╔═╡ cc70c65a-8613-11eb-17c9-8319e6491e43
md"Looking at the generated equations, we can see that the symbolic tools weren't smart enough to simplify our control variable ``\phi(t)`` into a tangent law. We could just let things be and send it through an implicit DAE solver, but since we know the answer already, we might as well change that equation ourselves."

# ╔═╡ 4a5302a6-8614-11eb-1dca-15cea504e8d1
simplified_two_body = let sys = opt_two_body
	@nonamespace λ = sys.λ
	eqs = collect(sys.eqs)
	eqs[1] = ϕ ~ atan(λ[2], λ[3])
	@set! sys.eqs = eqs
	structural_simplify(sys)
end

# ╔═╡ 73e6fe38-8614-11eb-152d-0ff202e7c557
md"We see now that it is able to get rid of the algebraic equation and give us a simple system of differential equations, which can be solved by much more efficient explicit solvers. Let's plug in our optimal costate initial condition from last time and see how well our automated method worked for this problem."

# ╔═╡ baea530c-8614-11eb-3f4c-0b0a9cfe5f52
two_body_sol = let sys = simplified_two_body
	@nonamespace λ = collect(sys.λ)
	
	p = [
		μ => 1.0
		aₜ => 0.115
		ṁ => -0.250154
	]

	r₀ = 1
	ic = [
		r => r₀
		vᵣ => 0
		vₛ => sqrt(last(p2[1])/r₀)
		θ => 0
		λ .=> λ⁺2
	]

	r₁ = 1.523
	fc = ComponentArray(
		r = r₁,
		vᵣ = 0,
		vₛ = sqrt(last(p2[1])/r₁),
	)

	prob = ODEProblem(sys, ic, tspan, p)

	terminal_condition(vars, t, integrator) = vars[5] - fc.r
	cb = ContinuousCallback(terminal_condition, terminate!)

	solve(prob; callback=cb)
end;

# ╔═╡ 8980db28-8615-11eb-291c-b3bbdcc2f831
plot!(plt_orb_states, two_body_sol;
	vars = [x2..., ϕ],
	color = (1:5)',
	lw = 2,
	linestyle = :dash,
	labels = false,
	widen = true,
	xlims = x_lims,
)

# ╔═╡ 23067f8a-8616-11eb-0fb9-0d3b73db6068
md"We can see that we get the same state trajectory as we did before, but without having to go through the effort of deriving our optimality conditions by hand."

# ╔═╡ 5c9ce030-8607-11eb-1f66-033a19d0698a
md"## Extra Stuff"

# ╔═╡ 4d377e4e-879e-11eb-1ac0-b11ec2877488
md"#### Worked problems"

# ╔═╡ 5b1d9f2a-879e-11eb-16db-c5de544fdd2f
PlutoUI.LocalResource("Project4Notes-1.png")

# ╔═╡ 1ba2ae20-879f-11eb-20c5-33c2f83ef2d3
PlutoUI.LocalResource("Project4Notes-2.png")

# ╔═╡ 802768c0-8616-11eb-0399-9342f0dd67af
md"#### All of the usings"

# ╔═╡ 72b560b6-8616-11eb-3a40-7d0b4fdece74
md"#### Plot functions for part 1"

# ╔═╡ 32287af2-8606-11eb-3e21-2117ef1aa0da
plot_sol!(p, sol; kwargs...) = (p; plot_sol!(sol; kwargs...))

# ╔═╡ 323462d6-8606-11eb-1083-3d9b5d2a3ae1
function plot_sol!(sol; label_postfix="", kwargs...)
    xticks = 0:1//2:2

    plot!(sol;
        vars = 1:2,
        color = [1 2],
        legend = :topleft,
        labels = ["x₁" "x₂"] .* label_postfix,
        xticks = (xticks*π, string.(xticks) .* " π"),
        kwargs...,
    )

    plot!(sol;
        vars = ((t,λ₂)->(t,-λ₂), 0, 4),
        color = 3,
        label = "u " * label_postfix,
        xlims = (0.0, 2π+0.03),
        kwargs...,
    )
end

# ╔═╡ 32279948-8606-11eb-25e2-e3472a9b7646
function plot_sol(sol, xf; kwargs...)
    plot()
    scatter!(fill(sol.t[end], 2)', xf';
        color = [1 2],
        labels = ["x₁ target" "x₂ target"],
        xlims = (0.0, 2π+0.03),
        markersize = 6,
		markerstrokewidth = 0,
    )

    plot_sol!(sol; kwargs...)
end

# ╔═╡ 0413a2d0-8607-11eb-066f-c7025bb553c4
begin
	plt1 = plot_sol(solve_with(λ⁺, p), p.xf; lw=1, label_postfix=" const opt")
	plot_sol!(solve_with(λ⁺nl, p); label_postfix=" nlsolve", lw=2, linestyle=:dash)
end

# ╔═╡ 8b74bebc-860c-11eb-14f2-a78ec57974f8
let plt = deepcopy(plt1)
	xf = [5.0, 2.0]
	plot!(plt)
	plot!(bvp_sol;
		vars = simplified_duff.states[3:4],
		label = ["x₁" "x₂"] .* " bvp",
		color = [1 2],
		linestyle = :dot,
		lw = 3,
		# title = "Optimally Forced Duffing Oscillator",
	)
	plot!(bvp_sol;
		vars = ((t, λ₂) -> (t, -λ₂), 0, 2),
		label = "u bvp",
		color = 3,
		linestyle = :dot,
		lw = 3,
	)
	xlims!(0.0, 2π+0.05)
	# scatter!(fill(2π, 1)', xf';
	# 	color = [1 2],
	# 	labels = ["x₁ target" "x₂ target"],
	# 	xlims = (0.0, 2π+0.03),
	# )
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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[compat]
ComponentArrays = "~0.11.5"
ControlSystems = "~0.10.1"
DifferentialEquations = "~6.19.0"
GalacticOptim = "~2.0.3"
Latexify = "~0.15.6"
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"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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