FA-87881 / Inverse kinematics solvers / Open access

Tool heading in degrees is used directly as radians · case 01

Headings other than zero produce wild wrist offsets and wrist joint angles.

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

ROOT CAUSE

The phi input is never converted, so 90 degrees is treated as 90 radians.

THE FAILURE

The phi input is never converted, so 90 degrees is treated as 90 radians.

Unsuccessful approach: Scaling by pi/360 converts half the angle.

Case contract

Input [l1,l2,l3,px,py,phi,elbow]: planar three-link arm, phi is the required tool heading in degrees. Subtract l3 along phi to get the wrist centre, solve the two-link sub-chain (elbow "up" gives q2>=0), set q3=phi-q1-q2 and wrap each joint to [-180,180) in degrees rounded to 4 places. A wrist centre outside |l1-l2|..l1+l2 returns "unreachable".

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):
    l1,l2,l3,px,py,phi,elbow=x
    a=phi
    wx=px-l3*math.cos(a)
    wy=py-l3*math.sin(a)
    r2=wx*wx+wy*wy
    if r2>(l1+l2)**2+1e-9 or r2<(l1-l2)**2-1e-9: return 'unreachable'
    c=(r2-l1*l1-l2*l2)/(2*l1*l2)
    c=max(-1.0,min(1.0,c))
    s=math.sqrt(1.0-c*c)*(1 if elbow=='up' else -1)
    q1=math.atan2(wy,wx)-math.atan2(l2*s,l1+l2*c)
    q2=math.atan2(s,c)
    q3=a-q1-q2
    out=[]
    for q in (q1,q2,q3):
        d=math.degrees(q)
        d=(d+180.0)%360.0-180.0
        out.append(round(d,4))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['tool pointing along x', [[2, 1.5, 0.5, 3, 1, 0, 'up']], [-11.5172, 80.4059, -68.8887]], ['tool pointing up', [[2, 1.5, 0.5, 2, 2, 90, 'down']], [73.7398, -90.0, 106.2602]], ['tool pointing back', [[2, 2, 1, -1, 2, 180, 'up']], [30.0, 120.0, 30.0]], ['tool pointing down', [[2, 1, 0.5, 1.5, -1, -90, 'up']], [-48.0197, 128.6822, -170.6625]], ['wrist just reachable', [[1, 1, 1, 2.5, 0, 0, 'up']], [-41.4096, 82.8192, -41.4096]], ['tip in reach but wrist not', [[1, 1, 1, 2.5, 0, 180, 'up']], 'unreachable'], ['diagonal heading', [[2, 1, 1, 2, 2, 45, 'down']], [74.8541, -114.4698, 84.6157]]], [['tool pointing back', [[2, 2, 1, -1, 2, 180, 'up']], [30.0, 120.0, 30.0]], ['wrist just reachable', [[1, 1, 1, 2.5, 0, 0, 'up']], [-41.4096, 82.8192, -41.4096]], ['tip in reach but wrist not', [[1, 1, 1, 2.5, 0, 180, 'up']], 'unreachable'], ['diagonal heading', [[2, 1, 1, 2, 2, 45, 'down']], [74.8541, -114.4698, 84.6157]], ['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]], ['negative heading', [[3, 2, 1, 3, -2, -30, 'down']], [5.9393, -121.0875, 85.1482]], ['third quadrant wrist', [[2, 2, 0.5, -2, -2, 225, 'up']], [170.5989, 108.8023, -54.4011]]], [['tool pointing down', [[2, 1, 0.5, 1.5, -1, -90, 'up']], [-48.0197, 128.6822, -170.6625]], ['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]], ['negative heading', [[3, 2, 1, 3, -2, -30, 'down']], [5.9393, -121.0875, 85.1482]], ['third quadrant wrist', [[2, 2, 0.5, -2, -2, 225, 'up']], [170.5989, 108.8023, -54.4011]], ['zero length tool', [[2, 1, 0, 2, 1, 60, 'up']], [0.0, 90.0, -30.0]], ['heading 150', [[2, 1.5, 1, -1, 2.5, 150, 'down']], [137.8289, -111.8395, 124.0106]], ['wrist in inner hole', [[3, 1, 1, 2.2, 0, 0, 'up']], 'unreachable']], [['diagonal heading', [[2, 1, 1, 2, 2, 45, 'down']], [74.8541, -114.4698, 84.6157]], ['zero length tool', [[2, 1, 0, 2, 1, 60, 'up']], [0.0, 90.0, -30.0]], ['heading 150', [[2, 1.5, 1, -1, 2.5, 150, 'down']], [137.8289, -111.8395, 124.0106]], ['wrist in inner hole', [[3, 1, 1, 2.2, 0, 0, 'up']], 'unreachable'], ['wrist behind tip beyond reach', [[1, 1, 1, 1.5, 0, 180, 'up']], 'unreachable'], ['wrist stretched', [[2, 1, 1, 4, 0, 0, 'down']], [0.0, 0.0, 0.0]], ['heading -135', [[1.5, 1.5, 0.5, -1.2, -1.7, -135, 'down']], [-64.1702, -115.9708, 45.141]]], [['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]], ['wrist behind tip beyond reach', [[1, 1, 1, 1.5, 0, 180, 'up']], 'unreachable'], ['wrist stretched', [[2, 1, 1, 4, 0, 0, 'down']], [0.0, 0.0, 0.0]], ['heading -135', [[1.5, 1.5, 0.5, -1.2, -1.7, -135, 'down']], [-64.1702, -115.9708, 45.141]], ['near base with long tool', [[2, 2, 2, 0.5, 0.5, 10, 'up']], [105.7453, 136.645, 127.6097]], ['heading 300', [[2.5, 1.5, 0.8, 1, -3, 300, 'up']], [-111.095, 112.0608, -60.9658]], ['heading -300', [[2.5, 1.5, 0.8, 1, 3, -300, 'down']], [111.095, -112.0608, 60.9658]]]]
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
tool pointing along x	`[-11.5172, 80.4059, -68.8887]`	`[-11.5172, 80.4059, -68.8887]`	Passed
tool pointing up	`[67.8456, -79.3568, 128.1314]`	`[73.7398, -90.0, 106.2602]`	Failed
tool pointing back	`[53.1852, 89.9449, 90.1102]`	`[30.0, 120.0, 30.0]`	Failed
tool pointing down	`[-47.6865, 115.4975, 175.5688]`	`[-48.0197, 128.6822, -170.6625]`	Failed
wrist just reachable	`[-41.4096, 82.8192, -41.4096]`	`[-41.4096, 82.8192, -41.4096]`	Passed
tip in reach but wrist not	`unreachable`	`unreachable`	Passed
diagonal heading	`[67.6355, -112.1001, 102.7746]`	`[74.8541, -114.4698, 84.6157]`	Failed

SHA-256 / 4e933e76447dca54694a18dcd929bb3b83c5aa542df23b8168471e89739e889d

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    l1,l2,l3,px,py,phi,elbow=x
    a=phi*math.pi/360
    wx=px-l3*math.cos(a)
    wy=py-l3*math.sin(a)
    r2=wx*wx+wy*wy
    if r2>(l1+l2)**2+1e-9 or r2<(l1-l2)**2-1e-9: return 'unreachable'
    c=(r2-l1*l1-l2*l2)/(2*l1*l2)
    c=max(-1.0,min(1.0,c))
    s=math.sqrt(1.0-c*c)*(1 if elbow=='up' else -1)
    q1=math.atan2(wy,wx)-math.atan2(l2*s,l1+l2*c)
    q2=math.atan2(s,c)
    q3=a-q1-q2
    out=[]
    for q in (q1,q2,q3):
        d=math.degrees(q)
        d=(d+180.0)%360.0-180.0
        out.append(round(d,4))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['tool pointing along x', [[2, 1.5, 0.5, 3, 1, 0, 'up']], [-11.5172, 80.4059, -68.8887]], ['tool pointing up', [[2, 1.5, 0.5, 2, 2, 90, 'down']], [73.7398, -90.0, 106.2602]], ['tool pointing back', [[2, 2, 1, -1, 2, 180, 'up']], [30.0, 120.0, 30.0]], ['tool pointing down', [[2, 1, 0.5, 1.5, -1, -90, 'up']], [-48.0197, 128.6822, -170.6625]], ['wrist just reachable', [[1, 1, 1, 2.5, 0, 0, 'up']], [-41.4096, 82.8192, -41.4096]], ['tip in reach but wrist not', [[1, 1, 1, 2.5, 0, 180, 'up']], 'unreachable'], ['diagonal heading', [[2, 1, 1, 2, 2, 45, 'down']], [74.8541, -114.4698, 84.6157]]], [['tool pointing back', [[2, 2, 1, -1, 2, 180, 'up']], [30.0, 120.0, 30.0]], ['wrist just reachable', [[1, 1, 1, 2.5, 0, 0, 'up']], [-41.4096, 82.8192, -41.4096]], ['tip in reach but wrist not', [[1, 1, 1, 2.5, 0, 180, 'up']], 'unreachable'], ['diagonal heading', [[2, 1, 1, 2, 2, 45, 'down']], [74.8541, -114.4698, 84.6157]], ['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]], ['negative heading', [[3, 2, 1, 3, -2, -30, 'down']], [5.9393, -121.0875, 85.1482]], ['third quadrant wrist', [[2, 2, 0.5, -2, -2, 225, 'up']], [170.5989, 108.8023, -54.4011]]], [['tool pointing down', [[2, 1, 0.5, 1.5, -1, -90, 'up']], [-48.0197, 128.6822, -170.6625]], ['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]], ['negative heading', [[3, 2, 1, 3, -2, -30, 'down']], [5.9393, -121.0875, 85.1482]], ['third quadrant wrist', [[2, 2, 0.5, -2, -2, 225, 'up']], [170.5989, 108.8023, -54.4011]], ['zero length tool', [[2, 1, 0, 2, 1, 60, 'up']], [0.0, 90.0, -30.0]], ['heading 150', [[2, 1.5, 1, -1, 2.5, 150, 'down']], [137.8289, -111.8395, 124.0106]], ['wrist in inner hole', [[3, 1, 1, 2.2, 0, 0, 'up']], 'unreachable']], [['diagonal heading', [[2, 1, 1, 2, 2, 45, 'down']], [74.8541, -114.4698, 84.6157]], ['zero length tool', [[2, 1, 0, 2, 1, 60, 'up']], [0.0, 90.0, -30.0]], ['heading 150', [[2, 1.5, 1, -1, 2.5, 150, 'down']], [137.8289, -111.8395, 124.0106]], ['wrist in inner hole', [[3, 1, 1, 2.2, 0, 0, 'up']], 'unreachable'], ['wrist behind tip beyond reach', [[1, 1, 1, 1.5, 0, 180, 'up']], 'unreachable'], ['wrist stretched', [[2, 1, 1, 4, 0, 0, 'down']], [0.0, 0.0, 0.0]], ['heading -135', [[1.5, 1.5, 0.5, -1.2, -1.7, -135, 'down']], [-64.1702, -115.9708, 45.141]]], [['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]], ['wrist behind tip beyond reach', [[1, 1, 1, 1.5, 0, 180, 'up']], 'unreachable'], ['wrist stretched', [[2, 1, 1, 4, 0, 0, 'down']], [0.0, 0.0, 0.0]], ['heading -135', [[1.5, 1.5, 0.5, -1.2, -1.7, -135, 'down']], [-64.1702, -115.9708, 45.141]], ['near base with long tool', [[2, 2, 2, 0.5, 0.5, 10, 'up']], [105.7453, 136.645, 127.6097]], ['heading 300', [[2.5, 1.5, 0.8, 1, -3, 300, 'up']], [-111.095, 112.0608, -60.9658]], ['heading -300', [[2.5, 1.5, 0.8, 1, 3, -300, 'down']], [111.095, -112.0608, 60.9658]]]]
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
tool pointing along x	`[-11.5172, 80.4059, -68.8887]`	`[-11.5172, 80.4059, -68.8887]`	Passed
tool pointing up	`[84.6459, -97.9362, 58.2903]`	`[73.7398, -90.0, 106.2602]`	Failed
tool pointing back	`[65.7048, 138.5904, -114.2952]`	`[30.0, 120.0, 30.0]`	Failed
tool pointing down	`[-55.4058, 144.7798, -134.374]`	`[-48.0197, 128.6822, -170.6625]`	Failed
wrist just reachable	`[-41.4096, 82.8192, -41.4096]`	`[-41.4096, 82.8192, -41.4096]`	Passed
tip in reach but wrist not	`unreachable`	`unreachable`	Passed
diagonal heading	`[85.7008, -107.8522, 44.6514]`	`[74.8541, -114.4698, 84.6157]`	Failed

SHA-256 / 1a5b4e96962734e6a33728b3a979171d2db850c5ebc1c3901da400af4d3a29d4

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 2026-09-29T14:51:03.015081+00:00.

Case digest / 80aeb0261f8e4cef3af841def44a1ef4c27ac6a726ce28c790062107c7282ee4