Planar Entry Dynamics

Also known as canonical entry dynamics!

Overview

The Planar Entry model assumes a spacecraft moving in an exponential atmosphere about a spherical planet. Acceleration due to gravity is ignored. The equations of motion are shown below.

\[\begin{aligned} \dot{\gamma} &= \frac{1}{v} \left( L_m - (1 - \frac{v^2}{v_c^2}) g \cos{\gamma} \right) \\ \dot{v} &= -D_m - g \sin{\gamma} \\ \dot{r} &= v \sin{\gamma} \\ \dot{\theta} &= \frac{v}{r} \cos{\gamma} \\ \end{aligned}\]

Examples

julia> model = PlanarEntrySystem()Model PlanarEntry with 4 equations
States (4):
  γ(t): flight path angle in degrees
  v(t): airspeed in meters per second
  r(t): polar distance relative to planet center in meters
  θ(t): polar angle relative to planet horizontal in degrees
Parameters (7):
  R: spherical planet radius
  P: atmospheric density at sea level in kilograms per meter cubed
  H: scale factor for exponential atmosphere in meters
  m: entry vehicle mass in kilograms
⋮

Let's compute the Jacobian for these dynamics.

julia> J = calculate_jacobian(PlanarEntrySystem())4×4 Matrix{Num}:
 (sin(γ(t))*((R / r(t))^2)*(1 - ((v(t) / sqrt(μ / r(t)))^2))*μ) / ((R^2)*v(t))  …  0
                                         (-((R / r(t))^2)*cos(γ(t))*μ) / (R^2)     0
                                                                v(t)*cos(γ(t))     0
                                                      (-sin(γ(t))*v(t)) / r(t)     0

Finally, let's construct a Julia function which implements these dynamics!

julia> f = PlanarEntryFunction()(::ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var"#k#545"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x510be63b, 0x5375dd43, 0x02383486, 0x537f71a3, 0x878b2e96), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xba97c8c8, 0xbca3bd0a, 0xd551efee, 0x49ce5ec8, 0x7921545c), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, ModelingToolkit.var"#___jac#551"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xb736dc56, 0x5f8c4599, 0xd86410d6, 0x8139d429, 0xb1d95b54), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xf9d68983, 0x99e2bc90, 0x11795c0a, 0x1d37149f, 0xedb90c33), Nothing}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Vector{Symbol}, Symbol, Vector{Symbol}, ModelingToolkit.var"#630#generated_observed#555"{Bool, ODESystem, Dict{Any, Any}, Vector{SymbolicUtils.BasicSymbolic{Real}}}, Nothing, ODESystem}) (generic function with 1 method)
julia> let u = abs.(randn(4)), p = abs.(randn(7)), t = 0
           f(u, p, t)
       end4-element Vector{Float64}:
  4.521159689425291
 -0.37666511412162096
  0.9749285590613717
  0.2785620346557619