FA-87866 / Inverse kinematics solvers / Open access

Target bearing reflects the goal across the diagonal · case 01

Targets off the 45 degree diagonal are reached at their mirror image.

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

ROOT CAUSE

The target bearing is evaluated as atan2(px,py) with the arguments swapped.

VERIFIED REPAIR

Use atan2(py,px) for the bearing of the target.

Unsuccessful approach: Replacing atan2 by atan(py/px) loses the quadrant for targets with negative x.

Case contract

Input [l1,l2,px,py,elbow] with positive link lengths. Return [q1,q2] in degrees (4 decimals) for a planar two-link arm whose end effector reaches (px,py); elbow "up" gives q2>=0, "down" gives q2<=0; q1 is wrapped to [-180,180). Points outside the annulus |l1-l2|<=r<=l1+l2 (1e-9 slack on r squared) return "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,px,py,elbow=x
    r2=px*px+py*py
    reach=l1+l2
    if r2>reach*reach+1e-9: return 'unreachable'
    if 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-c*c)
    if elbow=='down': s=-s
    q2=math.atan2(s,c)
    q1=math.atan2(px,py)-math.atan2(l2*s,l1+l2*c)
    d1=math.degrees(q1)
    d1=(d1+180.0)%360.0-180.0
    return [round(d1,4),round(math.degrees(q2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['general elbow up', [[2, 1, 2, 1, 'up']], [0.0, 90.0]], ['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]]], [['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable'], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]]], [['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]]], [['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]], ['wrap past minus 180', [[1, 2, -2.2, -0.3, 'up']], [123.5259, 91.0027]], ['upper left down', [[3, 2, -1, 4, 'down']], [131.2512, -70.5288]], ['near full reach', [[5, 3, 7.9, 0.5, 'down']], [10.0523, -17.1893]]]]
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
general elbow up	`[36.8699, 90.0]`	`[0.0, 90.0]`	Failed
general elbow down	`[90.0, -90.0]`	`[53.1301, -90.0]`	Failed
fully extended on x axis	`[90.0, 0.0]`	`[0.0, 0.0]`	Failed
fully extended diagonal	`[36.8699, 0.0]`	`[53.1301, 0.0]`	Failed
outside workspace	`unreachable`	`unreachable`	Passed
between reach and reach squared	`unreachable`	`unreachable`	Passed
third quadrant target	`[-172.5772, 112.0243]`	`[150.5529, 112.0243]`	Failed

SHA-256 / 8451b191113d74b1504119d0e178047466b5a93379546fd8ec56ab8c918e95e0

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,px,py,elbow=x
    r2=px*px+py*py
    reach=l1+l2
    if r2>reach*reach+1e-9: return 'unreachable'
    if 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-c*c)
    if elbow=='down': s=-s
    q2=math.atan2(s,c)
    q1=(math.atan(py/px) if px else 0.0)-math.atan2(l2*s,l1+l2*c)
    d1=math.degrees(q1)
    d1=(d1+180.0)%360.0-180.0
    return [round(d1,4),round(math.degrees(q2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['general elbow up', [[2, 1, 2, 1, 'up']], [0.0, 90.0]], ['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]]], [['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable'], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]]], [['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]]], [['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]], ['wrap past minus 180', [[1, 2, -2.2, -0.3, 'up']], [123.5259, 91.0027]], ['upper left down', [[3, 2, -1, 4, 'down']], [131.2512, -70.5288]], ['near full reach', [[5, 3, 7.9, 0.5, 'down']], [10.0523, -17.1893]]]]
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
general elbow up	`[0.0, 90.0]`	`[0.0, 90.0]`	Passed
general elbow down	`[53.1301, -90.0]`	`[53.1301, -90.0]`	Passed
fully extended on x axis	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
fully extended diagonal	`[53.1301, 0.0]`	`[53.1301, 0.0]`	Passed
outside workspace	`unreachable`	`unreachable`	Passed
between reach and reach squared	`unreachable`	`unreachable`	Passed
third quadrant target	`[-29.4471, 112.0243]`	`[150.5529, 112.0243]`	Failed

SHA-256 / ac35d448d3b8fc1af04d4b51e16a4e07ca22238623eef2e334eb80b2e7cb4af2

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    l1,l2,px,py,elbow=x
    r2=px*px+py*py
    reach=l1+l2
    if r2>reach*reach+1e-9: return 'unreachable'
    if 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-c*c)
    if elbow=='down': s=-s
    q2=math.atan2(s,c)
    q1=math.atan2(py,px)-math.atan2(l2*s,l1+l2*c)
    d1=math.degrees(q1)
    d1=(d1+180.0)%360.0-180.0
    return [round(d1,4),round(math.degrees(q2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['general elbow up', [[2, 1, 2, 1, 'up']], [0.0, 90.0]], ['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]]], [['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable'], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]]], [['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]]], [['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]], ['wrap past minus 180', [[1, 2, -2.2, -0.3, 'up']], [123.5259, 91.0027]], ['upper left down', [[3, 2, -1, 4, 'down']], [131.2512, -70.5288]], ['near full reach', [[5, 3, 7.9, 0.5, 'down']], [10.0523, -17.1893]]]]
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
general elbow up	`[0.0, 90.0]`	`[0.0, 90.0]`	Passed
general elbow down	`[53.1301, -90.0]`	`[53.1301, -90.0]`	Passed
fully extended on x axis	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
fully extended diagonal	`[53.1301, 0.0]`	`[53.1301, 0.0]`	Passed
outside workspace	`unreachable`	`unreachable`	Passed
between reach and reach squared	`unreachable`	`unreachable`	Passed
third quadrant target	`[150.5529, 112.0243]`	`[150.5529, 112.0243]`	Passed

SHA-256 / b600a6294716243912a2ecf6d412df62f85e613aa1a1ac51e8c5c55c949cb150

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 2026-09-29T14:51:02.982648+00:00.

Case digest / d0099da88e9027602e13c5dacc0268c0967a8e1f8e01e4e499fafa1bc235194e