FA-88266 / Inverse kinematics solvers / Open access

Articulated arm ignores the shoulder height above the base · case 01

Every solution is computed for a target shifted up by d1.

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

ROOT CAUSE

The vertical coordinate uses pz instead of pz-d1.

THE FAILURE

The vertical coordinate uses pz instead of pz-d1.

Unsuccessful approach: Adding d1 shifts the target the other way.

Case contract

Input [d1,a2,a3,px,py,pz,reach,elbow]: base yaw t1, shoulder t2, elbow t3 with shoulder height d1. reach "front" uses t1=atan2(py,px) and r=hypot(px,py); "back" turns t1 by 180 degrees and uses -r. With s=pz-d1, cos t3=(r^2+s^2-a2^2-a3^2)/(2 a2 a3); beyond +/-(1+1e-12) is "unreachable", else clamp. elbow "up" takes sin t3<=0. t2=atan2(s,r)-atan2(a3 sin t3, a2+a3 cos t3). Angles in degrees wrapped to [-180,180), rounded 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):
    d1,a2,a3,px,py,pz,reach,elbow=x
    t1=math.atan2(py,px)
    r=math.hypot(px,py)
    if reach=='back':
        t1=t1+math.pi
        r=-r
    s=pz
    c3=(r*r+s*s-a2*a2-a3*a3)/(2*a2*a3)
    if c3>1.0+1e-12 or c3<-1.0-1e-12: return 'unreachable'
    c3=max(-1.0,min(1.0,c3))
    s3=math.sqrt(1.0-c3*c3)
    if elbow=='up': s3=-s3
    t3=math.atan2(s3,c3)
    t2=math.atan2(s,r)-math.atan2(a3*s3,a2+a3*c3)
    res=[]
    for t in (t1,t2,t3):
        d=math.degrees(t)
        d=d-360.0*math.floor((d+180.0)/360.0)
        res.append(round(d,4))
    return res
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['front nominal', [[1, 2, 2, 2, 1, 2, 'front', 'up']], [26.5651, 76.3336, -104.4775]], ['back nominal', [[1, 2, 2, 2, 1, 2, 'back', 'up']], [-153.4349, -151.8561, -104.4775]], ['front elbow down', [[1, 2, 2, 1, 2, 3, 'front', 'down']], [63.4349, 0.4007, 82.8192]], ['back on x axis', [[1, 2, 2, 1, 0, 2, 'back', 'down']], [-180.0, 65.7048, 138.5904]], ['stretched diagonal', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'front', 'up']], [45.0, 0.0, -0.0]], ['stretched diagonal back', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'back', 'down']], [-135.0, -180.0, 0.0]], ['out of reach', [[1, 1, 1, 2, 2, 1, 'front', 'up']], 'unreachable']], [['back nominal', [[1, 2, 2, 2, 1, 2, 'back', 'up']], [-153.4349, -151.8561, -104.4775]], ['stretched diagonal', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'front', 'up']], [45.0, 0.0, -0.0]], ['stretched diagonal back', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'back', 'down']], [-135.0, -180.0, 0.0]], ['out of reach', [[1, 1, 1, 2, 2, 1, 'front', 'up']], 'unreachable'], ['below base', [[2, 2, 1.5, 1.5, -1, 0.5, 'front', 'up']], [-33.6901, -0.3729, -97.1808]], ['high tall base', [[3, 1.5, 1.5, 1, 1, 5, 'front', 'down']], [45.0, 19.4712, 70.5288]], ['back third quadrant', [[1, 2, 2, -1, -2, 0, 'back', 'up']], [63.4349, -103.6664, -104.4775]]], [['front elbow down', [[1, 2, 2, 1, 2, 3, 'front', 'down']], [63.4349, 0.4007, 82.8192]], ['below base', [[2, 2, 1.5, 1.5, -1, 0.5, 'front', 'up']], [-33.6901, -0.3729, -97.1808]], ['high tall base', [[3, 1.5, 1.5, 1, 1, 5, 'front', 'down']], [45.0, 19.4712, 70.5288]], ['back third quadrant', [[1, 2, 2, -1, -2, 0, 'back', 'up']], [63.4349, -103.6664, -104.4775]], ['front negative y', [[0.5, 2, 1, 1, -1.5, 1, 'front', 'down']], [-56.3099, -14.2024, 112.0243]], ['base column below', [[4, 2, 2, 0.5, 0.5, 1, 'front', 'up']], [45.0, -37.1408, -79.1931]], ['back reach below shoulder', [[1.4, 2, 2, -0.7, -1.7, -0.1, 'back', 'down']], [67.6199, 165.5945, 107.2325]]], [['back on x axis', [[1, 2, 2, 1, 0, 2, 'back', 'down']], [-180.0, 65.7048, 138.5904]], ['front negative y', [[0.5, 2, 1, 1, -1.5, 1, 'front', 'down']], [-56.3099, -14.2024, 112.0243]], ['base column below', [[4, 2, 2, 0.5, 0.5, 1, 'front', 'up']], [45.0, -37.1408, -79.1931]], ['back reach below shoulder', [[1.4, 2, 2, -0.7, -1.7, -0.1, 'back', 'down']], [67.6199, 165.5945, 107.2325]], ['back reach low front quadrant', [[1.4, 2, 2, 1.5, -0.6, 1.0, 'back', 'down']], [158.1986, 128.4939, 130.8249]], ['stretched with rounding residue', [[0, 1, 1, 1.977371993328519, 0, 0.3, 'front', 'up']], [0.0, 8.6269, -0.0]], ['reach past base height only', [[3, 1, 1, 1, 0, 3.5, 'front', 'up']], [0.0, 82.5772, -112.0243]]], [['front nominal', [[1, 2, 2, 2, 1, 2, 'front', 'up']], [26.5651, 76.3336, -104.4775]], ['back nominal', [[1, 2, 2, 2, 1, 2, 'back', 'up']], [-153.4349, -151.8561, -104.4775]], ['front elbow down', [[1, 2, 2, 1, 2, 3, 'front', 'down']], [63.4349, 0.4007, 82.8192]], ['below base', [[2, 2, 1.5, 1.5, -1, 0.5, 'front', 'up']], [-33.6901, -0.3729, -97.1808]], ['back reach low front quadrant', [[1.4, 2, 2, 1.5, -0.6, 1.0, 'back', 'down']], [158.1986, 128.4939, 130.8249]], ['stretched with rounding residue', [[0, 1, 1, 1.977371993328519, 0, 0.3, 'front', 'up']], [0.0, 8.6269, -0.0]], ['reach past base height only', [[3, 1, 1, 1, 0, 3.5, 'front', 'up']], [0.0, 82.5772, -112.0243]]]]
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
front nominal	`[26.5651, 83.2199, -82.8192]`	`[26.5651, 76.3336, -104.4775]`	Failed
back nominal	`[-153.4349, 179.5993, -82.8192]`	`[-153.4349, -151.8561, -104.4775]`	Failed
front elbow down	`[63.4349, 32.596, 41.4096]`	`[63.4349, 0.4007, 82.8192]`	Failed
back on x axis	`[-180.0, 60.5529, 112.0243]`	`[-180.0, 65.7048, 138.5904]`	Failed
stretched diagonal	`[45.0, 0.0, -0.0]`	`[45.0, 0.0, -0.0]`	Passed
stretched diagonal back	`[-135.0, -180.0, 0.0]`	`[-135.0, -180.0, 0.0]`	Passed
out of reach	`unreachable`	`unreachable`	Passed

SHA-256 / f082a53a7a994c6396ea4a9b4296d5a5bf44f0ee082827ec632d5e538c74c91f

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    d1,a2,a3,px,py,pz,reach,elbow=x
    t1=math.atan2(py,px)
    r=math.hypot(px,py)
    if reach=='back':
        t1=t1+math.pi
        r=-r
    s=pz+d1
    c3=(r*r+s*s-a2*a2-a3*a3)/(2*a2*a3)
    if c3>1.0+1e-12 or c3<-1.0-1e-12: return 'unreachable'
    c3=max(-1.0,min(1.0,c3))
    s3=math.sqrt(1.0-c3*c3)
    if elbow=='up': s3=-s3
    t3=math.atan2(s3,c3)
    t2=math.atan2(s,r)-math.atan2(a3*s3,a2+a3*c3)
    res=[]
    for t in (t1,t2,t3):
        d=math.degrees(t)
        d=d-360.0*math.floor((d+180.0)/360.0)
        res.append(round(d,4))
    return res
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['front nominal', [[1, 2, 2, 2, 1, 2, 'front', 'up']], [26.5651, 76.3336, -104.4775]], ['back nominal', [[1, 2, 2, 2, 1, 2, 'back', 'up']], [-153.4349, -151.8561, -104.4775]], ['front elbow down', [[1, 2, 2, 1, 2, 3, 'front', 'down']], [63.4349, 0.4007, 82.8192]], ['back on x axis', [[1, 2, 2, 1, 0, 2, 'back', 'down']], [-180.0, 65.7048, 138.5904]], ['stretched diagonal', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'front', 'up']], [45.0, 0.0, -0.0]], ['stretched diagonal back', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'back', 'down']], [-135.0, -180.0, 0.0]], ['out of reach', [[1, 1, 1, 2, 2, 1, 'front', 'up']], 'unreachable']], [['back nominal', [[1, 2, 2, 2, 1, 2, 'back', 'up']], [-153.4349, -151.8561, -104.4775]], ['stretched diagonal', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'front', 'up']], [45.0, 0.0, -0.0]], ['stretched diagonal back', [[0, 1, 1, 1.4142135623730951, 1.4142135623730951, 0, 'back', 'down']], [-135.0, -180.0, 0.0]], ['out of reach', [[1, 1, 1, 2, 2, 1, 'front', 'up']], 'unreachable'], ['below base', [[2, 2, 1.5, 1.5, -1, 0.5, 'front', 'up']], [-33.6901, -0.3729, -97.1808]], ['high tall base', [[3, 1.5, 1.5, 1, 1, 5, 'front', 'down']], [45.0, 19.4712, 70.5288]], ['back third quadrant', [[1, 2, 2, -1, -2, 0, 'back', 'up']], [63.4349, -103.6664, -104.4775]]], [['front elbow down', [[1, 2, 2, 1, 2, 3, 'front', 'down']], [63.4349, 0.4007, 82.8192]], ['below base', [[2, 2, 1.5, 1.5, -1, 0.5, 'front', 'up']], [-33.6901, -0.3729, -97.1808]], ['high tall base', [[3, 1.5, 1.5, 1, 1, 5, 'front', 'down']], [45.0, 19.4712, 70.5288]], ['back third quadrant', [[1, 2, 2, -1, -2, 0, 'back', 'up']], [63.4349, -103.6664, -104.4775]], ['front negative y', [[0.5, 2, 1, 1, -1.5, 1, 'front', 'down']], [-56.3099, -14.2024, 112.0243]], ['base column below', [[4, 2, 2, 0.5, 0.5, 1, 'front', 'up']], [45.0, -37.1408, -79.1931]], ['back reach below shoulder', [[1.4, 2, 2, -0.7, -1.7, -0.1, 'back', 'down']], [67.6199, 165.5945, 107.2325]]], [['back on x axis', [[1, 2, 2, 1, 0, 2, 'back', 'down']], [-180.0, 65.7048, 138.5904]], ['front negative y', [[0.5, 2, 1, 1, -1.5, 1, 'front', 'down']], [-56.3099, -14.2024, 112.0243]], ['base column below', [[4, 2, 2, 0.5, 0.5, 1, 'front', 'up']], [45.0, -37.1408, -79.1931]], ['back reach below shoulder', [[1.4, 2, 2, -0.7, -1.7, -0.1, 'back', 'down']], [67.6199, 165.5945, 107.2325]], ['back reach low front quadrant', [[1.4, 2, 2, 1.5, -0.6, 1.0, 'back', 'down']], [158.1986, 128.4939, 130.8249]], ['stretched with rounding residue', [[0, 1, 1, 1.977371993328519, 0, 0.3, 'front', 'up']], [0.0, 8.6269, -0.0]], ['reach past base height only', [[3, 1, 1, 1, 0, 3.5, 'front', 'up']], [0.0, 82.5772, -112.0243]]], [['front nominal', [[1, 2, 2, 2, 1, 2, 'front', 'up']], [26.5651, 76.3336, -104.4775]], ['back nominal', [[1, 2, 2, 2, 1, 2, 'back', 'up']], [-153.4349, -151.8561, -104.4775]], ['front elbow down', [[1, 2, 2, 1, 2, 3, 'front', 'down']], [63.4349, 0.4007, 82.8192]], ['below base', [[2, 2, 1.5, 1.5, -1, 0.5, 'front', 'up']], [-33.6901, -0.3729, -97.1808]], ['back reach low front quadrant', [[1.4, 2, 2, 1.5, -0.6, 1.0, 'back', 'down']], [158.1986, 128.4939, 130.8249]], ['stretched with rounding residue', [[0, 1, 1, 1.977371993328519, 0, 0.3, 'front', 'up']], [0.0, 8.6269, -0.0]], ['reach past base height only', [[3, 1, 1, 1, 0, 3.5, 'front', 'up']], [0.0, 82.5772, -112.0243]]]]
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
front nominal	`[26.5651, 74.0056, -41.4096]`	`[26.5651, 76.3336, -104.4775]`	Failed
back nominal	`[-153.4349, 147.404, -41.4096]`	`[-153.4349, -151.8561, -104.4775]`	Failed
front elbow down	`unreachable`	`[63.4349, 0.4007, 82.8192]`	Failed
back on x axis	`[-180.0, 70.6737, 75.5225]`	`[-180.0, 65.7048, 138.5904]`	Failed
stretched diagonal	`[45.0, 0.0, -0.0]`	`[45.0, 0.0, -0.0]`	Passed
stretched diagonal back	`[-135.0, -180.0, 0.0]`	`[-135.0, -180.0, 0.0]`	Passed
out of reach	`unreachable`	`unreachable`	Passed

SHA-256 / 80cc79cc4d2aa4544158d5d8794c79d5b246b19ba2f903f72cfda59a751c3cf9

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 2026-09-29T14:51:06.511515+00:00.

Case digest / 4fda2ba142ba6e2e83c11365209c94b6d94eb5fa7e9412641169bf13e3579080