FA-87861 / Inverse kinematics solvers / Open access
Shoulder angle adds the forearm offset instead of subtracting it · case 01
The shoulder points past the target by twice the forearm offset angle.
ROOT CAUSE
q1 is computed as atan2(py,px)+atan2(l2 s,l1+l2 c).
VERIFIED REPAIR
Subtract the offset angle atan2(l2 sin q2, l1+l2 cos q2) from the target bearing.
Unsuccessful approach: Swapping sine and cosine inside the offset atan2 measures the offset from the wrong axis.
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(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'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable']], [['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]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['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]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['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 | [53.1301, 90.0] | [0.0, 90.0] | Failed |
| general elbow down | [0.0, -90.0] | [53.1301, -90.0] | Failed |
| 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 |
| inside inner hole | unreachable | unreachable | Passed |
SHA-256 / 0b657171c43cf1231506a2bca2027af9bd7a8e10891b4b78e8f99b4e4a786cc5
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.atan2(py,px)-math.atan2(l2*c,l1+l2*s)
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'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable']], [['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]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['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]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['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 | [26.5651, 90.0] | [0.0, 90.0] | Failed |
| general elbow down | [26.5651, -90.0] | [53.1301, -90.0] | Failed |
| fully extended on x axis | [-26.5651, 0.0] | [0.0, 0.0] | Failed |
| fully extended diagonal | [0.0, 0.0] | [53.1301, 0.0] | Failed |
| outside workspace | unreachable | unreachable | Passed |
| between reach and reach squared | unreachable | unreachable | Passed |
| inside inner hole | unreachable | unreachable | Passed |
SHA-256 / b86c7dc970b2113318dbc3278901b9ef38f6ad8047f26e542d7427f0e5ec26ae
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'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable']], [['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]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['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]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['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 |
| inside inner hole | unreachable | unreachable | Passed |
SHA-256 / a6c248822f8f3d89fa456d043f7e4b21b0215f39dee065750858981efd903b8b
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 2026-09-29T14:51:02.844649+00:00.
Case digest / c41e25dd7a866187f4c535f0b4d774755de0e2d7c2b527820faaecdf2d9ef748