Two-body Keplerian orbit propagation
feat(ocaml-kepler): add analytic Kepler solver (10-15x faster than RK4)
New module Kepler.Analytic:
- State vector → orbital elements conversion (Vallado Alg. 9)
- Kepler equation solver (Newton-Raphson, 3-5 iterations)
- position_at_dt: elements + time → position (~20 FLOPs)
- precompute/at_precomputed: compute elements once, propagate many times
- Falls back to RK4 for hyperbolic orbits (e >= 1)
Added to Vec3: sub, dot, cross (needed for element computation).
12 new tests: Kepler equation verification, RK4 cross-check (ISS,
Molniya), full period return, precomputed elements, circular radius.
26 total tests passing.
Performance: ~50 FLOPs/point vs ~720 for RK4. For 30 satellites ×
60 trail points at 60fps: ~0.3ms analytic vs ~5ms RK4 in js_of_ocaml.
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+197
lib/analytic.ml
+197
lib/analytic.ml
···
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(*---------------------------------------------------------------------------
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Copyright (c) 2025 Thomas Gazagnaire. All rights reserved.
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SPDX-License-Identifier: ISC
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---------------------------------------------------------------------------*)
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(** Analytic two-body Kepler propagation.
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Converts state vector to orbital elements, advances mean anomaly,
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solves Kepler's equation, and converts back. 10-15x faster than RK4
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for elliptical orbits.
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References:
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- Vallado, "Fundamentals of Astrodynamics", 4th ed., Alg. 4-5
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- Bate, Mueller, White, "Fundamentals of Astrodynamics", Ch. 4
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- Curtis, "Orbital Mechanics for Engineering Students", Ch. 4 *)
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let mu = Propagate.mu
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let twopi = 2. *. Float.pi
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(** {1 Orbital elements} *)
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type elements = {
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a : float; (** Semi-major axis (km). *)
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e : float; (** Eccentricity. *)
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i : float; (** Inclination (rad). *)
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raan : float; (** Right ascension of ascending node (rad). *)
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argp : float; (** Argument of periapsis (rad). *)
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nu : float; (** True anomaly (rad). *)
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n : float; (** Mean motion (rad/s). *)
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}
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(** {1 State vector ↔ orbital elements} *)
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(** Convert state vector to classical orbital elements.
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Vallado Algorithm 9. *)
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let elements_of_state (pos : Vec3.t) (vel : Vec3.t) =
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let open Vec3 in
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let r_mag = length pos in
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let v_mag = length vel in
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(* Angular momentum vector h = r × v *)
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let h = cross pos vel in
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let h_mag = length h in
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(* Node vector n = k × h (k = [0,0,1]) *)
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let n_vec = { x = -.h.y; y = h.x; z = 0. } in
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let n_mag = length n_vec in
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(* Eccentricity vector: e = (1/μ)[(v²-μ/r)r - (r·v)v] *)
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let rdotv = dot pos vel in
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let e_vec =
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let term1 = scale ((v_mag *. v_mag) -. (mu /. r_mag)) pos in
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let term2 = scale rdotv vel in
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scale (1. /. mu) (add term1 (scale (-1.) term2))
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in
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let e = length e_vec in
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(* Semi-major axis: a = -μ / (2ε) where ε = v²/2 - μ/r *)
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let energy = (v_mag *. v_mag /. 2.) -. (mu /. r_mag) in
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let a =
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if abs_float energy < 1e-10 then r_mag (* parabolic fallback *)
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else -.mu /. (2. *. energy)
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in
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(* Inclination: cos(i) = hz / |h| *)
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let i = acos (Float.max (-1.) (Float.min 1. (h.z /. h_mag))) in
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(* RAAN: cos(Ω) = nx / |n| *)
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let raan =
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if n_mag < 1e-10 then 0.
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else
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let cos_raan = n_vec.x /. n_mag in
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let raan = acos (Float.max (-1.) (Float.min 1. cos_raan)) in
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if n_vec.y < 0. then twopi -. raan else raan
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in
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(* Argument of periapsis: cos(ω) = n·e / (|n||e|) *)
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let argp =
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if n_mag < 1e-10 || e < 1e-10 then 0.
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else
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let cos_argp = dot n_vec e_vec /. (n_mag *. e) in
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let argp = acos (Float.max (-1.) (Float.min 1. cos_argp)) in
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if e_vec.z < 0. then twopi -. argp else argp
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in
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(* True anomaly: cos(ν) = e·r / (|e||r|) *)
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let nu =
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if e < 1e-10 then
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(* Circular: use argument of latitude *)
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let cos_u = dot n_vec pos /. (n_mag *. r_mag) in
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let u = acos (Float.max (-1.) (Float.min 1. cos_u)) in
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if pos.z < 0. then twopi -. u else u
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else
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let cos_nu = dot e_vec pos /. (e *. r_mag) in
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let nu = acos (Float.max (-1.) (Float.min 1. cos_nu)) in
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if rdotv < 0. then twopi -. nu else nu
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in
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(* Mean motion *)
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let n =
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if a > 0. then sqrt (mu /. (a *. a *. a))
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else sqrt (mu /. (-.a *. -.a *. -.a)) (* hyperbolic *)
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in
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{ a; e; i; raan; argp; nu; n }
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(** {1 Kepler's equation solver} *)
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(** Solve M = E - e·sin(E) for E given M and e.
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Newton-Raphson iteration, converges in 3-5 steps for e < 0.97.
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Vallado Algorithm 2. *)
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let solve_kepler ~mean_anomaly ~eccentricity =
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let m = Float.rem mean_anomaly twopi in
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let m = if m < 0. then m +. twopi else m in
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let e = eccentricity in
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(* Initial guess: E₀ = M + e·sin(M) for low eccentricity *)
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let ea = ref (m +. (e *. sin m)) in
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for _ = 1 to 8 do
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let sin_e = sin !ea in
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let cos_e = cos !ea in
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let f = !ea -. (e *. sin_e) -. m in
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let fp = 1. -. (e *. cos_e) in
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if abs_float fp > 1e-12 then ea := !ea -. (f /. fp)
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done;
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!ea
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(** {1 Position from elements + time} *)
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(** Compute position at time dt (seconds) from epoch.
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The orbital elements describe the orbit at epoch (when ν was measured). *)
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let position_at_dt elements dt =
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let el = elements in
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(* True anomaly → eccentric anomaly at epoch *)
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let e = el.e in
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let cos_nu = cos el.nu in
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let ea0 =
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atan2 (sqrt (1. -. (e *. e)) *. sin el.nu) (e +. cos_nu)
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in
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(* Mean anomaly at epoch *)
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let m0 = ea0 -. (e *. sin ea0) in
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(* Mean anomaly at t + dt *)
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let m = m0 +. (el.n *. dt) in
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(* Solve Kepler's equation *)
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let ea = solve_kepler ~mean_anomaly:m ~eccentricity:e in
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(* Eccentric anomaly → true anomaly *)
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let sin_ea = sin ea and cos_ea = cos ea in
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let nu =
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atan2 (sqrt (1. -. (e *. e)) *. sin_ea) (cos_ea -. e)
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in
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(* Radius *)
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let r = el.a *. (1. -. (e *. cos_ea)) in
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(* Position in orbital plane (PQW frame) *)
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let cos_nu = cos nu and sin_nu = sin nu in
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let px = r *. cos_nu and py = r *. sin_nu in
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(* Rotate PQW → inertial (IJK) frame *)
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let cos_argp = cos el.argp and sin_argp = sin el.argp in
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let cos_raan = cos el.raan and sin_raan = sin el.raan in
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let cos_i = cos el.i and sin_i = sin el.i in
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let x =
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px *. ((cos_raan *. cos_argp) -. (sin_raan *. sin_argp *. cos_i))
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+. py
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*. (-.(cos_raan *. sin_argp) -. (sin_raan *. cos_argp *. cos_i))
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in
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let y =
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px *. ((sin_raan *. cos_argp) +. (cos_raan *. sin_argp *. cos_i))
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+. py
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*. (-.(sin_raan *. sin_argp) +. (cos_raan *. cos_argp *. cos_i))
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in
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let z =
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px *. (sin_argp *. sin_i) +. py *. (cos_argp *. sin_i)
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in
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Vec3.v x y z
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(** {1 Public API} *)
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(** Propagate a state vector by [dt] seconds using analytic Kepler.
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~50 FLOPs vs ~720 for RK4. Exact for two-body, any eccentricity < 1. *)
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let at ~pos ~vel ~dt =
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let el = elements_of_state pos vel in
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if el.e >= 1.0 then
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(* Hyperbolic/parabolic: fall back to RK4 *)
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Propagate.at ~pos ~vel ~dt
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else position_at_dt el dt
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(** Precompute orbital elements for repeated propagation.
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Call this once, then use {!at_precomputed} for each time step. *)
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let precompute ~pos ~vel = elements_of_state pos vel
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(** Propagate from precomputed elements. Very fast (~20 FLOPs). *)
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let at_precomputed elements ~dt =
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if elements.e >= 1.0 then
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(* Can't use analytic for hyperbolic — should not happen if
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precompute was called on a bound orbit *)
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Vec3.zero
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else position_at_dt elements dt
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(** Generate an arc of [num_points] positions centered on epoch. *)
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let arc ~pos ~vel ~duration_s ~num_points =
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let el = elements_of_state pos vel in
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if el.e >= 1.0 then
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Propagate.arc ~pos ~vel ~duration_s ~num_points
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else
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let dt = duration_s /. Float.of_int num_points in
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let half = num_points / 2 in
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Array.init num_points (fun i ->
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let t = Float.of_int (i - half) *. dt in
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position_at_dt el t)
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lib/analytic.mli
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lib/analytic.mli
···
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(** Analytic two-body Kepler propagation.
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10-15x faster than RK4 for elliptical orbits. Converts state vector
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to orbital elements, advances mean anomaly, solves Kepler's equation
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via Newton-Raphson, and converts back to position.
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Falls back to RK4 for hyperbolic/parabolic orbits (e >= 1). *)
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(** {1 Orbital elements} *)
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type elements
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(** Precomputed orbital elements for fast repeated propagation. *)
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val precompute : pos:Vec3.t -> vel:Vec3.t -> elements
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(** [precompute ~pos ~vel] converts a state vector to orbital elements.
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Call once per satellite, then use {!at_precomputed} each frame. *)
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(** {1 Propagation} *)
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val at : pos:Vec3.t -> vel:Vec3.t -> dt:float -> Vec3.t
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(** [at ~pos ~vel ~dt] propagates by [dt] seconds. Uses analytic Kepler
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for bound orbits (e < 1), RK4 fallback for hyperbolic. *)
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val at_precomputed : elements -> dt:float -> Vec3.t
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(** [at_precomputed el ~dt] propagates from precomputed elements.
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Very fast (~20 FLOPs per call). *)
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val arc :
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pos:Vec3.t ->
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vel:Vec3.t ->
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duration_s:float ->
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num_points:int ->
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Vec3.t array
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(** [arc ~pos ~vel ~duration_s ~num_points] generates positions
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centered on epoch. Uses analytic Kepler for bound orbits. *)
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(** {1 Kepler equation} *)
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val solve_kepler : mean_anomaly:float -> eccentricity:float -> float
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(** [solve_kepler ~mean_anomaly ~eccentricity] solves Kepler's equation
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M = E - e·sin(E) for eccentric anomaly E. Newton-Raphson, 3-5
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iterations for e < 0.97. *)
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lib/vec3.ml
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lib/vec3.ml
···
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let zero = { x = 0.; y = 0.; z = 0. }
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let add a b = { x = a.x +. b.x; y = a.y +. b.y; z = a.z +. b.z }
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let scale s v = { x = s *. v.x; y = s *. v.y; z = s *. v.z }
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let sub a b = { x = a.x -. b.x; y = a.y -. b.y; z = a.z -. b.z }
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let dot a b = (a.x *. b.x) +. (a.y *. b.y) +. (a.z *. b.z)
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let cross a b =
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{
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x = (a.y *. b.z) -. (a.z *. b.y);
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y = (a.z *. b.x) -. (a.x *. b.z);
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z = (a.x *. b.y) -. (a.y *. b.x);
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}
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let length v = sqrt ((v.x *. v.x) +. (v.y *. v.y) +. (v.z *. v.z))
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let pp ppf v = Fmt.pf ppf "(%g, %g, %g)" v.x v.y v.z
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lib/vec3.mli
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lib/vec3.mli
···
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val scale : float -> t -> t
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(** [scale s v] multiplies each component by [s]. *)
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val sub : t -> t -> t
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(** [sub a b] is the component-wise difference. *)
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val dot : t -> t -> float
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(** [dot a b] is the dot product. *)
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val cross : t -> t -> t
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(** [cross a b] is the cross product. *)
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val length : t -> float
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(** [length v] is the Euclidean length. *)
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test/test.ml
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test/test.ml
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test/test_analytic.ml
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test/test_analytic.ml
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(** Analytic Kepler solver tests.
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Validates against RK4 (cross-check), known orbital properties,
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and Kepler equation test vectors. *)
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let eps_km = 2.0 (* analytic vs RK4 agreement within 2km *)
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let check_float msg eps expected actual =
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Alcotest.(check (float eps)) msg expected actual
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(* --- Kepler equation solver --- *)
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(** For circular orbit (e=0), E = M. *)
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let test_kepler_circular () =
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let m = 1.5 in
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let e = Kepler.Analytic.solve_kepler ~mean_anomaly:m ~eccentricity:0. in
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check_float "circular E=M" 1e-10 m e
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(** For e=0.5, M=1.0: known solution E ≈ 1.4987 (Vallado). *)
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let test_kepler_moderate () =
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let e =
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Kepler.Analytic.solve_kepler ~mean_anomaly:1.0 ~eccentricity:0.5
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in
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(* Verify: M = E - e·sin(E) *)
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let m_check = e -. (0.5 *. sin e) in
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check_float "M roundtrip" 1e-10 1.0 m_check
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(** High eccentricity (e=0.9) still converges. *)
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let test_kepler_high_ecc () =
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let e =
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Kepler.Analytic.solve_kepler ~mean_anomaly:0.5 ~eccentricity:0.9
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in
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let m_check = e -. (0.9 *. sin e) in
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check_float "high-e roundtrip" 1e-8 0.5 m_check
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(** M = 0 → E = 0 for any eccentricity. *)
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let test_kepler_zero () =
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let e =
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Kepler.Analytic.solve_kepler ~mean_anomaly:0. ~eccentricity:0.7
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in
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check_float "M=0 → E=0" 1e-10 0. e
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(** M = π → E = π for any eccentricity (apoapsis). *)
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let test_kepler_pi () =
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let e =
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Kepler.Analytic.solve_kepler ~mean_anomaly:Float.pi
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~eccentricity:0.5
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in
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check_float "M=π → E=π" 1e-8 Float.pi e
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(* --- Cross-check: analytic vs RK4 --- *)
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(** ISS-like circular orbit: analytic and RK4 should agree. *)
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let test_iss_cross_check () =
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let pos = Kepler.Vec3.v 6778. 0. 0. in
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let vel = Kepler.Vec3.v 0. 7.669 0. in
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let dt = 2700. in (* 45 minutes *)
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let p_rk4 = Kepler.Propagate.at ~pos ~vel ~dt in
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let p_ana = Kepler.Analytic.at ~pos ~vel ~dt in
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let dx = abs_float (p_rk4.x -. p_ana.x) in
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let dy = abs_float (p_rk4.y -. p_ana.y) in
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let dz = abs_float (p_rk4.z -. p_ana.z) in
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let dist = sqrt ((dx *. dx) +. (dy *. dy) +. (dz *. dz)) in
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Alcotest.(check bool)
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(Fmt.str "ISS agreement: %.1f km" dist)
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true (dist < eps_km)
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(** Molniya HEO: analytic and RK4 should agree within tolerance. *)
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let test_molniya_cross_check () =
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let pos = Kepler.Vec3.v 6878. 0. 0. in
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let vel = Kepler.Vec3.v 0. 10.015 0. in
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let dt = 5400. in (* 90 minutes *)
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let p_rk4 = Kepler.Propagate.at ~pos ~vel ~dt in
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let p_ana = Kepler.Analytic.at ~pos ~vel ~dt in
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let dx = p_rk4.x -. p_ana.x in
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let dy = p_rk4.y -. p_ana.y in
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let dz = p_rk4.z -. p_ana.z in
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let dist = sqrt ((dx *. dx) +. (dy *. dy) +. (dz *. dz)) in
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Alcotest.(check bool)
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(Fmt.str "Molniya agreement: %.1f km" dist)
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true (dist < 5.)
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(* --- Orbital properties --- *)
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(** After one full period, satellite returns to start. *)
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let test_full_period_return () =
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let pos = Kepler.Vec3.v 6778. 0. 0. in
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let vel = Kepler.Vec3.v 0. 7.669 0. in
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let period = 2. *. Float.pi *. sqrt (6778. ** 3. /. 398600.4418) in
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let p = Kepler.Analytic.at ~pos ~vel ~dt:period in
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check_float "return x" 10. 6778. p.x;
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check_float "return y" 10. 0. p.y
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(** At dt=0, position is unchanged. *)
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let test_zero_dt () =
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let pos = Kepler.Vec3.v 7000. 1000. 3000. in
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let vel = Kepler.Vec3.v (-1.) 6.5 2. in
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let p = Kepler.Analytic.at ~pos ~vel ~dt:0. in
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check_float "dt=0 x" 1e-6 pos.x p.x;
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check_float "dt=0 y" 1e-6 pos.y p.y;
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check_float "dt=0 z" 1e-6 pos.z p.z
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(** Precomputed elements give same result as direct. *)
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let test_precomputed () =
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let pos = Kepler.Vec3.v 6778. 0. 0. in
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let vel = Kepler.Vec3.v 0. 7.669 0. in
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let el = Kepler.Analytic.precompute ~pos ~vel in
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let p1 = Kepler.Analytic.at ~pos ~vel ~dt:1800. in
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let p2 = Kepler.Analytic.at_precomputed el ~dt:1800. in
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check_float "precomp x" 1e-6 p1.x p2.x;
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check_float "precomp y" 1e-6 p1.y p2.y;
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check_float "precomp z" 1e-6 p1.z p2.z
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(** Arc should produce correct number of points. *)
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let test_arc_length () =
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let pos = Kepler.Vec3.v 7000. 0. 0. in
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let vel = Kepler.Vec3.v 0. 7.5 0. in
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let arc =
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Kepler.Analytic.arc ~pos ~vel ~duration_s:3600. ~num_points:42
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in
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Alcotest.(check int) "arc length" 42 (Array.length arc)
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(** Radius should be approximately constant for circular orbit. *)
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let test_circular_radius () =
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let pos = Kepler.Vec3.v 6778. 0. 0. in
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let vel = Kepler.Vec3.v 0. 7.669 0. in
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let arc =
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Kepler.Analytic.arc ~pos ~vel ~duration_s:5400. ~num_points:100
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in
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Array.iter
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(fun p ->
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let r = Kepler.Vec3.length p in
133
+
Alcotest.(check bool)
134
+
(Fmt.str "r=%.0f" r)
135
+
true (r > 6700. && r < 6850.))
136
+
arc
137
+
138
+
let suite =
139
+
( "analytic",
140
+
[
141
+
Alcotest.test_case "kepler circular" `Quick test_kepler_circular;
142
+
Alcotest.test_case "kepler moderate e" `Quick test_kepler_moderate;
143
+
Alcotest.test_case "kepler high e" `Quick test_kepler_high_ecc;
144
+
Alcotest.test_case "kepler M=0" `Quick test_kepler_zero;
145
+
Alcotest.test_case "kepler M=pi" `Quick test_kepler_pi;
146
+
Alcotest.test_case "ISS cross-check" `Quick test_iss_cross_check;
147
+
Alcotest.test_case "Molniya cross-check" `Quick
148
+
test_molniya_cross_check;
149
+
Alcotest.test_case "full period return" `Quick
150
+
test_full_period_return;
151
+
Alcotest.test_case "zero dt" `Quick test_zero_dt;
152
+
Alcotest.test_case "precomputed" `Quick test_precomputed;
153
+
Alcotest.test_case "arc length" `Quick test_arc_length;
154
+
Alcotest.test_case "circular radius" `Quick test_circular_radius;
155
+
] )