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
Unknowns (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"#f#684"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xf83be1e2, 0xd28dc8dc, 0x29fb5b80, 0xb6cd3fac, 0xb88f70e0), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xe9f74804, 0x76f7e014, 0x2e776db9, 0xc35ec679, 0xbb2e68ce), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, ModelingToolkit.var"#_jac#689"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x43cd8278, 0xfe3e9f5d, 0xcf882a1c, 0x61939334, 0x509e9882), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x39f0e9a4, 0x27f62ecb, 0xb7c70372, 0xa08459e1, 0xca74ee4e), Nothing}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.var"#835#generated_observed#693"{Bool, ODESystem, Dict{Any, Any}, Vector{SymbolicUtils.BasicSymbolic{Real}}}, Nothing, ODESystem, Nothing, Nothing}) (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}:
 -809.6645094237803
 -616.723163326895
    0.17644338969842763
   19.594026245707806