FA-88146 / Inverse kinematics solvers / Open access

Hexapod plane distance ignores the coxa link · case 01

Feet land a coxa length short of the commanded position.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The horizontal distance to the femur pivot is taken as hypot(x,y) without subtracting the coxa length.

THE FAILURE

The horizontal distance to the femur pivot is taken as hypot(x,y) without subtracting the coxa length.

Unsuccessful approach: Subtracting the coxa from x alone only works for feet straight ahead.

Case contract

Input [coxa,femur,tibia,x,y,z,fmin,fmax]: foot target in the leg root frame (z up). Coxa yaw is atan2(y,x); the femur/tibia pair works in the vertical plane at horizontal distance hypot(x,y)-coxa. With D the distance from the femur pivot to the foot, D>femur+tibia or D<|femur-tibia| or D==0 is "unreachable". Femur pitch is atan2(z,h) plus the triangle angle at the femur pivot (knee up); tibia angle is the interior knee angle minus 180 (0 = straight). Femur pitch outside [fmin,fmax] returns "limit". Degrees rounded to 4.

Why this case matters

Inverse kinematics code turns task-space goals into joint commands; a wrong branch, sign, limit or update order sends a real arm to the wrong pose.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    cx,fe,ti,px,py,pz,fmin,fmax=x
    ca=math.atan2(py,px)
    h=math.hypot(px,py)
    D=math.hypot(h,pz)
    if D>fe+ti or D<abs(fe-ti) or D==0: return 'unreachable'
    a1=math.atan2(pz,h)
    a2=math.acos(max(-1.0,min(1.0,(fe*fe+D*D-ti*ti)/(2*fe*D))))
    fa=math.degrees(a1+a2)
    ta=math.degrees(math.acos(max(-1.0,min(1.0,(fe*fe+ti*ti-D*D)/(2*fe*ti)))))-180.0
    if fa<fmin or fa>fmax: return 'limit'
    return [round(math.degrees(ca),4),round(fa,4),round(ta,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['nominal stance', [[3, 6, 8, 10, 2, -6, -90, 90]], [11.3099, 18.0529, -97.294]], ['foot straight out', [[3, 6, 8, 12, 0, -3, -90, 90]], [0.0, 38.5667, -95.9792]], ['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['foot out of reach', [[3, 6, 8, 18, 0, -2, -90, 90]], 'unreachable'], ['foot inside coxa reach', [[3, 6, 8, 16, 0, 0, -90, 90]], [0.0, 25.3317, -44.0486]], ['high foot', [[3, 6, 8, 9, 3, 4, -90, 90]], 'limit']], [['foot straight out', [[3, 6, 8, 12, 0, -3, -90, 90]], [0.0, 38.5667, -95.9792]], ['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot inside coxa reach', [[3, 6, 8, 16, 0, 0, -90, 90]], [0.0, 25.3317, -44.0486]], ['high foot', [[3, 6, 8, 9, 3, 4, -90, 90]], 'limit'], ['femur limit exceeded', [[3, 6, 8, 5, 0, 6, -90, 30]], 'limit'], ['deep foot', [[3, 6, 8, 5, 5, -9, -90, 90]], [45.0, -11.6013, -91.4483]], ['tight femur limits', [[3, 6, 8, 10, 2, -6, -10, 10]], 'limit']], [['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['deep foot', [[3, 6, 8, 5, 5, -9, -90, 90]], [45.0, -11.6013, -91.4483]], ['tight femur limits', [[3, 6, 8, 10, 2, -6, -10, 10]], 'limit'], ['long tibia near', [[2, 4, 9, 6, 1, -8, -90, 90]], [9.4623, 14.4416, -103.1099]], ['diagonal foot', [[2.5, 5, 6, -6, 6, -3, -90, 90]], [135.0, 33.0338, -105.6409]], ['hole too close', [[3, 6, 12, 4, 0, -1, -90, 90]], 'unreachable']], [['nominal stance', [[3, 6, 8, 10, 2, -6, -90, 90]], [11.3099, 18.0529, -97.294]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['deep foot', [[3, 6, 8, 5, 5, -9, -90, 90]], [45.0, -11.6013, -91.4483]], ['diagonal foot', [[2.5, 5, 6, -6, 6, -3, -90, 90]], [135.0, 33.0338, -105.6409]], ['hole too close', [[3, 6, 12, 4, 0, -1, -90, 90]], 'unreachable'], ['foot far below', [[3, 6, 8, 15, 0, -10, -90, 90]], 'unreachable'], ['femur at lower bound', [[3, 6, 8, 11, 0, -8, -60, 90]], [0.0, -2.44, -73.0422]]], [['nominal stance', [[3, 6, 8, 10, 2, -6, -90, 90]], [11.3099, 18.0529, -97.294]], ['foot straight out', [[3, 6, 8, 12, 0, -3, -90, 90]], [0.0, 38.5667, -95.9792]], ['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['foot inside coxa reach', [[3, 6, 8, 16, 0, 0, -90, 90]], [0.0, 25.3317, -44.0486]], ['long tibia near', [[2, 4, 9, 6, 1, -8, -90, 90]], [9.4623, 14.4416, -103.1099]], ['femur at lower bound', [[3, 6, 8, 11, 0, -8, -60, 90]], [0.0, -2.44, -73.0422]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
nominal stance	`[11.3099, 7.4551, -65.3757]`	`[11.3099, 18.0529, -97.294]`	Failed
foot straight out	`[0.0, 18.5972, -56.4899]`	`[0.0, 38.5667, -95.9792]`	Failed
foot to the side	`[83.6598, 17.7923, -69.949]`	`[83.6598, 29.4436, -96.745]`	Failed
foot behind	`[-159.444, 11.2882, -69.949]`	`[-159.444, 20.4684, -95.0617]`	Failed
foot out of reach	`unreachable`	`unreachable`	Passed
foot inside coxa reach	`unreachable`	`[0.0, 25.3317, -44.0486]`	Failed
high foot	`[18.4349, 73.7133, -86.4167]`	`limit`	Failed

SHA-256 / 13232cd02d3f3b180ba0f2fcbdc85f5ec256390f7e2451cbd8b1b4cdbbbb2c7f

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    cx,fe,ti,px,py,pz,fmin,fmax=x
    ca=math.atan2(py,px)
    h=px-cx
    D=math.hypot(h,pz)
    if D>fe+ti or D<abs(fe-ti) or D==0: return 'unreachable'
    a1=math.atan2(pz,h)
    a2=math.acos(max(-1.0,min(1.0,(fe*fe+D*D-ti*ti)/(2*fe*D))))
    fa=math.degrees(a1+a2)
    ta=math.degrees(math.acos(max(-1.0,min(1.0,(fe*fe+ti*ti-D*D)/(2*fe*ti)))))-180.0
    if fa<fmin or fa>fmax: return 'limit'
    return [round(math.degrees(ca),4),round(fa,4),round(ta,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['nominal stance', [[3, 6, 8, 10, 2, -6, -90, 90]], [11.3099, 18.0529, -97.294]], ['foot straight out', [[3, 6, 8, 12, 0, -3, -90, 90]], [0.0, 38.5667, -95.9792]], ['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['foot out of reach', [[3, 6, 8, 18, 0, -2, -90, 90]], 'unreachable'], ['foot inside coxa reach', [[3, 6, 8, 16, 0, 0, -90, 90]], [0.0, 25.3317, -44.0486]], ['high foot', [[3, 6, 8, 9, 3, 4, -90, 90]], 'limit']], [['foot straight out', [[3, 6, 8, 12, 0, -3, -90, 90]], [0.0, 38.5667, -95.9792]], ['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot inside coxa reach', [[3, 6, 8, 16, 0, 0, -90, 90]], [0.0, 25.3317, -44.0486]], ['high foot', [[3, 6, 8, 9, 3, 4, -90, 90]], 'limit'], ['femur limit exceeded', [[3, 6, 8, 5, 0, 6, -90, 30]], 'limit'], ['deep foot', [[3, 6, 8, 5, 5, -9, -90, 90]], [45.0, -11.6013, -91.4483]], ['tight femur limits', [[3, 6, 8, 10, 2, -6, -10, 10]], 'limit']], [['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['deep foot', [[3, 6, 8, 5, 5, -9, -90, 90]], [45.0, -11.6013, -91.4483]], ['tight femur limits', [[3, 6, 8, 10, 2, -6, -10, 10]], 'limit'], ['long tibia near', [[2, 4, 9, 6, 1, -8, -90, 90]], [9.4623, 14.4416, -103.1099]], ['diagonal foot', [[2.5, 5, 6, -6, 6, -3, -90, 90]], [135.0, 33.0338, -105.6409]], ['hole too close', [[3, 6, 12, 4, 0, -1, -90, 90]], 'unreachable']], [['nominal stance', [[3, 6, 8, 10, 2, -6, -90, 90]], [11.3099, 18.0529, -97.294]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['deep foot', [[3, 6, 8, 5, 5, -9, -90, 90]], [45.0, -11.6013, -91.4483]], ['diagonal foot', [[2.5, 5, 6, -6, 6, -3, -90, 90]], [135.0, 33.0338, -105.6409]], ['hole too close', [[3, 6, 12, 4, 0, -1, -90, 90]], 'unreachable'], ['foot far below', [[3, 6, 8, 15, 0, -10, -90, 90]], 'unreachable'], ['femur at lower bound', [[3, 6, 8, 11, 0, -8, -60, 90]], [0.0, -2.44, -73.0422]]], [['nominal stance', [[3, 6, 8, 10, 2, -6, -90, 90]], [11.3099, 18.0529, -97.294]], ['foot straight out', [[3, 6, 8, 12, 0, -3, -90, 90]], [0.0, 38.5667, -95.9792]], ['foot to the side', [[2, 5, 7, 1, 9, -4, -90, 90]], [83.6598, 29.4436, -96.745]], ['foot behind', [[2, 5, 7, -8, -3, -5, -90, 90]], [-159.444, 20.4684, -95.0617]], ['foot inside coxa reach', [[3, 6, 8, 16, 0, 0, -90, 90]], [0.0, 25.3317, -44.0486]], ['long tibia near', [[2, 4, 9, 6, 1, -8, -90, 90]], [9.4623, 14.4416, -103.1099]], ['femur at lower bound', [[3, 6, 8, 11, 0, -8, -60, 90]], [0.0, -2.44, -73.0422]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
nominal stance	`[11.3099, 18.3872, -98.9893]`	`[11.3099, 18.0529, -97.294]`	Failed
foot straight out	`[0.0, 38.5667, -95.9792]`	`[0.0, 38.5667, -95.9792]`	Passed
foot to the side	`[83.6598, -4.2615, -144.5168]`	`[83.6598, 29.4436, -96.745]`	Failed
foot behind	`limit`	`[-159.444, 20.4684, -95.0617]`	Failed
foot out of reach	`unreachable`	`unreachable`	Passed
foot inside coxa reach	`[0.0, 25.3317, -44.0486]`	`[0.0, 25.3317, -44.0486]`	Passed
high foot	`limit`	`limit`	Passed

SHA-256 / e83734c64841ac1869b5bf02a6485e63bfa313cc8ed17fed889bb174a187cdc3

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

Deterministic planar or low-dimensional teaching model with a stipulated convention; not a general robotics library. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:51:05.549989+00:00.

Case digest / 4f7a8d1f1e8caa96dc096600e2ea0642cfe917e448db01d6a41ab39936ec7c68