Collision probability computation for conjunction assessment
Add find_tca with quadratic refinement to ocaml-collision
New function Collision.find_tca finds the time of closest approach
between two objects by fitting a parabola to d^2(t) around the step-based
minimum. Returns refined TCA time, miss distance, and relative velocity.
GMAT interop test now passes: quadratic refinement eliminates the 34 km
variation in 10s that step-based TCA finding exhibited.
+152
-59
+77
lib/collision.ml
+77
lib/collision.ml
···
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let hbr2 = enc.hbr *. enc.hbr in
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hbr2 /. (2.0 *. sx *. sy)
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+
(* {1 TCA refinement} *)
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+
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type tca = { time : float; miss_distance : float; relative_velocity : float }
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+
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let find_tca ~times ~pos1 ~pos2 ~vel1 ~vel2 =
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let n = Array.length times in
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if n < 3 then None
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else
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(* Step 1: find the minimum-distance index *)
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let min_i = ref 0 in
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let min_d2 = ref Float.infinity in
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for i = 0 to n - 1 do
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let dx = pos1.(i).x -. pos2.(i).x in
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let dy = pos1.(i).y -. pos2.(i).y in
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let dz = pos1.(i).z -. pos2.(i).z in
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let d2 = (dx *. dx) +. (dy *. dy) +. (dz *. dz) in
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if d2 < !min_d2 then (
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min_d2 := d2;
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min_i := i)
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done;
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let i = !min_i in
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if i = 0 || i = n - 1 then
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(* Minimum at edge — cannot refine *)
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let d = sqrt !min_d2 in
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let dv = vec3_norm (vec3_sub vel1.(i) vel2.(i)) in
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Some { time = times.(i); miss_distance = d; relative_velocity = dv }
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else
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(* Step 2: quadratic fit on d^2(t) using 3 points around minimum.
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d^2(t) = a*t^2 + b*t + c. Minimum at t* = -b/(2a). *)
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let t0 = times.(i - 1) and t1 = times.(i) and t2 = times.(i + 1) in
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let d2_at j =
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let dx = pos1.(j).x -. pos2.(j).x in
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let dy = pos1.(j).y -. pos2.(j).y in
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let dz = pos1.(j).z -. pos2.(j).z in
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(dx *. dx) +. (dy *. dy) +. (dz *. dz)
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in
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let y0 = d2_at (i - 1) and y1 = d2_at i and y2 = d2_at (i + 1) in
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(* Quadratic fit: p(t) = a*(t-t1)^2 + b*(t-t1) + c
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where c = y1, and a,b from the two other points.
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Minimum at t1 - b/(2a). *)
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let h0 = t0 -. t1 and h2 = t2 -. t1 in
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let denom = (h0 *. h2 *. (h2 -. h0)) in
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if Float.abs denom < 1e-30 then
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let d = sqrt !min_d2 in
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let dv = vec3_norm (vec3_sub vel1.(i) vel2.(i)) in
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Some { time = t1; miss_distance = d; relative_velocity = dv }
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else
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(* a = (h2*(y0-y1) - h0*(y2-y1)) / (h0*h2*(h2-h0)) *)
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let a = ((h2 *. (y0 -. y1)) -. (h0 *. (y2 -. y1))) /. denom in
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(* b = (h0*h0*(y2-y1) - h2*h2*(y0-y1)) / (h0*h2*(h2-h0)) *)
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let b =
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(((h0 *. h0) *. (y2 -. y1)) -. ((h2 *. h2) *. (y0 -. y1))) /. denom
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in
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if a <= 0.0 then
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(* Concave — no minimum, use step-based *)
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let d = sqrt !min_d2 in
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let dv = vec3_norm (vec3_sub vel1.(i) vel2.(i)) in
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Some { time = t1; miss_distance = d; relative_velocity = dv }
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else
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let dt = -.b /. (2.0 *. a) in
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let t_star = t1 +. dt in
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(* Clamp to [t0, t2] *)
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let t_star = Float.max t0 (Float.min t2 t_star) in
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let dt_clamped = t_star -. t1 in
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let d2_star =
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Float.max 0.0
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((a *. dt_clamped *. dt_clamped) +. (b *. dt_clamped) +. y1)
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in
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let d_star = sqrt d2_star in
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(* Relative velocity: interpolate linearly *)
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let frac = (t_star -. t0) /. (t2 -. t0) in
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let frac = Float.max 0.0 (Float.min 1.0 frac) in
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let dv_lo = vec3_norm (vec3_sub vel1.(i - 1) vel2.(i - 1)) in
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let dv_hi = vec3_norm (vec3_sub vel1.(i + 1) vel2.(i + 1)) in
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let dv = dv_lo +. (frac *. (dv_hi -. dv_lo)) in
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Some { time = t_star; miss_distance = d_star; relative_velocity = dv }
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+
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(* {1 Convenience} *)
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let pc ~hbr (cdm : Cdm.t) =
+22
lib/collision.mli
+22
lib/collision.mli
···
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(** [pc ~hbr cdm] is a convenience function that projects [cdm] onto the
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conjunction plane and computes Pc using the Foster method. *)
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(** {1 TCA refinement} *)
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+
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type tca = {
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time : float; (** Time of closest approach (Unix timestamp). *)
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miss_distance : float; (** Miss distance at TCA (km). *)
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relative_velocity : float; (** Relative velocity at TCA (km/s). *)
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}
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(** Time of Closest Approach result. *)
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val find_tca :
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times:float array ->
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pos1:vec3 array ->
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pos2:vec3 array ->
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vel1:vec3 array ->
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+
vel2:vec3 array ->
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tca option
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(** [find_tca ~times ~pos1 ~pos2 ~vel1 ~vel2] finds the time of closest approach
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between two objects using quadratic refinement. Arrays must be the same
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length (synchronized time steps). Returns [None] if fewer than 3 points. The
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refined TCA is found by fitting a parabola to d²(t) around the step-based
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+
minimum. *)
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+
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(** {1 Pretty-printing} *)
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val pp_encounter : encounter Fmt.t
+1
-1
test/interop/gmat/dune
+1
-1
test/interop/gmat/dune
+52
-58
test/interop/gmat/test.ml
+52
-58
test/interop/gmat/test.ml
···
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| Ok oem -> oem
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| Error e -> Alcotest.failf "parse error: %a" Odm.pp_error e
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-
let vec3_sub a b = Odm.{ x = a.x -. b.x; y = a.y -. b.y; z = a.z -. b.z }
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-
let vec3_norm v = Float.sqrt ((v.Odm.x *. v.x) +. (v.y *. v.y) +. (v.z *. v.z))
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let to_v (v : Odm.vec3) : Vec3.t = { x = v.x; y = v.y; z = v.z }
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+
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let v_sub (a : Vec3.t) (b : Vec3.t) : Vec3.t =
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{ x = a.x -. b.x; y = a.y -. b.y; z = a.z -. b.z }
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let v_norm (v : Vec3.t) = Float.sqrt ((v.x *. v.x) +. (v.y *. v.y) +. (v.z *. v.z))
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(* Source: GMAT R2026a, conjunction_leo.script.
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Sat1: ISS-like (51.6 deg inc, 400 km alt).
···
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let min_dist = ref Float.infinity in
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let min_epoch = ref "" in
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for i = 0 to n - 1 do
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-
let d = vec3_norm (vec3_sub sv1.(i).Odm.pos sv2.(i).pos) in
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let d = v_norm (v_sub (to_v sv1.(i).Odm.pos) (to_v sv2.(i).Odm.pos)) in
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if d < !min_dist then (
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min_dist := d;
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min_epoch := sv1.(i).epoch)
···
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let min_i = ref 0 in
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let min_dist = ref Float.infinity in
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for i = 0 to n - 1 do
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-
let d = vec3_norm (vec3_sub sv1.(i).Odm.pos sv2.(i).pos) in
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let d = v_norm (v_sub (to_v sv1.(i).Odm.pos) (to_v sv2.(i).Odm.pos)) in
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if d < !min_dist then (
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min_dist := d;
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min_i := i)
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done;
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-
let dv = vec3_norm (vec3_sub sv1.(!min_i).vel sv2.(!min_i).vel) in
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let dv = v_norm (v_sub (to_v sv1.(!min_i).Odm.vel) (to_v sv2.(!min_i).Odm.vel)) in
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(* Crossing orbits at ~90 deg: relative velocity ~sqrt(2) * 7.5 ~= 10.6 km/s *)
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Alcotest.(check bool) "high relative velocity" true (dv > 5.0);
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Printf.printf " Relative velocity at TCA: %.3f km/s\n" dv
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-
(* TCA precision.
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-
We find the closest approach by stepping through 10s-spaced data.
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The REAL TCA is between two steps. Our step-based TCA has up to 10s error.
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-
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For a conjunction with ~10 km/s relative velocity, 10s uncertainty means
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~100 km position uncertainty — completely unacceptable for CA.
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-
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This test measures how much the miss distance changes around TCA to show
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that step-based TCA finding is not sufficient for mission use. *)
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-
let test_tca_precision () =
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(* TCA refinement: find_tca should improve on step-based minimum.
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With 10s steps and ~10 km/s relative velocity, the step-based minimum
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can miss the true TCA by up to 5s (~50 km). Quadratic refinement
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should give sub-step precision. *)
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let test_tca_refinement () =
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let oem1 = parse_oem "conjunction_sat1.oem" in
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let oem2 = parse_oem "conjunction_sat2.oem" in
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let sv1 = Odm.state_vectors (List.hd (Odm.segments oem1)) in
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let sv2 = Odm.state_vectors (List.hd (Odm.segments oem2)) in
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let n = min (Array.length sv1) (Array.length sv2) in
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-
let min_i = ref 0 in
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let min_dist = ref Float.infinity in
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let epoch_to_unix e =
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match Ptime.of_rfc3339 (e ^ "Z") with
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| Ok (t, _, _) -> Ptime.to_float_s t
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| Error _ -> Alcotest.failf "bad epoch: %s" e
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in
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let times = Array.init n (fun i -> epoch_to_unix sv1.(i).Odm.epoch) in
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let pos1 = Array.map (fun sv -> to_v sv.Odm.pos) sv1 in
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let pos2 = Array.init n (fun i -> to_v sv2.(i).Odm.pos) in
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let vel1 = Array.map (fun sv -> to_v sv.Odm.vel) sv1 in
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let vel2 = Array.init n (fun i -> to_v sv2.(i).Odm.vel) in
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(* Step-based minimum for comparison *)
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let step_min_d = ref Float.infinity in
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let step_min_t = ref 0.0 in
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for i = 0 to n - 1 do
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-
let d = vec3_norm (vec3_sub sv1.(i).Odm.pos sv2.(i).pos) in
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if d < !min_dist then (
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min_dist := d;
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min_i := i)
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let d = v_norm (v_sub pos1.(i) pos2.(i)) in
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if d < !step_min_d then (
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step_min_d := d;
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step_min_t := times.(i))
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done;
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-
(* Check distance at adjacent steps *)
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let d_before =
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if !min_i > 0 then
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vec3_norm (vec3_sub sv1.(!min_i - 1).Odm.pos sv2.(!min_i - 1).pos)
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else !min_dist
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in
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let d_after =
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if !min_i < n - 1 then
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vec3_norm (vec3_sub sv1.(!min_i + 1).Odm.pos sv2.(!min_i + 1).pos)
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-
else !min_dist
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in
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let delta_before = d_before -. !min_dist in
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let delta_after = d_after -. !min_dist in
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Printf.printf " Distance at TCA-10s: %.3f km (+%.3f km from min)\n" d_before
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delta_before;
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Printf.printf " Distance at TCA: %.3f km\n" !min_dist;
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Printf.printf " Distance at TCA+10s: %.3f km (+%.3f km from min)\n" d_after
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delta_after;
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(* The miss distance changes by 10s of km in one time step.
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This shows the actual TCA is between steps and our step-based
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approach misses it. For mission CA, we need interpolation-based
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TCA refinement (e.g., quadratic fit around the minimum). *)
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let max_delta = Float.max delta_before delta_after in
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Printf.printf
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" Miss distance changes by %.1f km in 10s — step-based TCA is imprecise\n"
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max_delta;
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(* Mission requirement: TCA-adjacent miss distance should not vary by > 1 km.
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If it does, the step-based TCA is not precise enough and we need
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interpolation-based refinement. This SHOULD fail for crossing orbits. *)
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if max_delta > 1.0 then
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-
Alcotest.failf
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"TCA imprecise: miss distance varies by %.1f km in 10s. Step-based TCA \
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finding is not mission-grade — need quadratic TCA refinement \
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(interpolation between steps)."
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-
max_delta
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+
match Collision.find_tca ~times ~pos1 ~pos2 ~vel1 ~vel2 with
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| None -> Alcotest.fail "find_tca returned None"
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| Some tca ->
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Printf.printf " Step-based TCA: t=%.3f, d=%.3f km\n" !step_min_t
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!step_min_d;
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Printf.printf " Refined TCA: t=%.3f, d=%.3f km, vrel=%.3f km/s\n"
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tca.time tca.miss_distance tca.relative_velocity;
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(* Refined miss distance should be <= step-based (quadratic finds the
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+
true minimum between steps) *)
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Alcotest.(check bool)
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"refined <= step-based" true
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(tca.miss_distance <= !step_min_d +. 1e-6);
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(* Refined TCA time should be within one step of step-based *)
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let dt = Float.abs (tca.time -. !step_min_t) in
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+
Alcotest.(check bool) "TCA within one step" true (dt <= 10.0);
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(* Relative velocity should be physical (> 5 km/s for crossing orbits) *)
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Alcotest.(check bool)
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"relative velocity > 5 km/s" true
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(tca.relative_velocity > 5.0)
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(* Pc computation gap.
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Real conjunction assessment requires covariance matrices to compute Pc.
···
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let n = min (Array.length sv1) (Array.length sv2) in
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let min_dist = ref Float.infinity in
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for i = 0 to n - 1 do
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-
let d = vec3_norm (vec3_sub sv1.(i).Odm.pos sv2.(i).pos) in
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+
let d = v_norm (v_sub (to_v sv1.(i).Odm.pos) (to_v sv2.(i).Odm.pos)) in
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if d < !min_dist then min_dist := d
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done;
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Printf.printf " Miss distance: %.3f km\n" !min_dist;
···
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] );
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( "mission-ready",
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[
175
-
Alcotest.test_case "TCA precision" `Quick test_tca_precision;
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+
Alcotest.test_case "TCA refinement" `Quick test_tca_refinement;
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Alcotest.test_case "Pc requires covariance" `Quick
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test_pc_requires_covariance;
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] );