FA-87891 / Inverse kinematics solvers / Open access

Joint wrap saturates at +/-180 instead of wrapping · case 01

Multi-turn heading requests yield wrist angles pinned at 180 or -180.

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

ROOT CAUSE

Each joint is clamped to [-180,180] rather than reduced modulo 360.

THE FAILURE

Each joint is clamped to [-180,180] rather than reduced modulo 360.

Unsuccessful approach: Wrapping only values whose magnitude exceeds 360 leaves 180..360 unreduced.

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=math.radians(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=max(-180.0,min(180.0,d))
        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'], ['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]]], [['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]], ['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]]], [['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]], ['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]], ['near base with long tool', [[2, 2, 2, 0.5, 0.5, 10, 'up']], [105.7453, 136.645, 127.6097]]], [['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'], ['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 300', [[2.5, 1.5, 0.8, 1, -3, 300, 'up']], [-111.095, 112.0608, -60.9658]]], [['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]], ['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	`[73.7398, -90.0, 106.2602]`	`[73.7398, -90.0, 106.2602]`	Passed
tool pointing back	`[30.0, 120.0, 30.0]`	`[30.0, 120.0, 30.0]`	Passed
tool pointing down	`[-48.0197, 128.6822, -170.6625]`	`[-48.0197, 128.6822, -170.6625]`	Passed
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
multi-turn heading	`[32.3158, 101.7795, 180.0]`	`[32.3158, 101.7795, -89.0953]`	Failed

SHA-256 / eae4f65433e2580161cda5c3abb4ef72bfc70a969322f6f4d63e3fb2d8ee28fe

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=math.radians(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 if abs(d)>360.0 else d
        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'], ['multi-turn heading', [[2, 1.5, 0.5, 1, 2.5, 405, 'up']], [32.3158, 101.7795, -89.0953]]], [['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]], ['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]]], [['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]], ['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]], ['near base with long tool', [[2, 2, 2, 0.5, 0.5, 10, 'up']], [105.7453, 136.645, 127.6097]]], [['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'], ['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 300', [[2.5, 1.5, 0.8, 1, -3, 300, 'up']], [-111.095, 112.0608, -60.9658]]], [['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]], ['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	`[73.7398, -90.0, 106.2602]`	`[73.7398, -90.0, 106.2602]`	Passed
tool pointing back	`[30.0, 120.0, 30.0]`	`[30.0, 120.0, 30.0]`	Passed
tool pointing down	`[-48.0197, 128.6822, -170.6625]`	`[-48.0197, 128.6822, -170.6625]`	Passed
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
multi-turn heading	`[32.3158, 101.7795, 270.9047]`	`[32.3158, 101.7795, -89.0953]`	Failed

SHA-256 / 10edfc00fcb1fe9dd9b910739d5209ed546d96d7aa265856ca2f4cd251d7b925

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 2026-09-29T14:51:03.135199+00:00.

Case digest / d92fc550d88a9a3de8efc94537b80a5d8098306dbcef912202da4fbfe97ec580