FA-87841 / Inverse kinematics solvers / Open access
Two-link reach test rejects the fully extended pose · case 01
A target exactly at l1+l2 is reported unreachable although the straight arm touches it.
ROOT CAUSE
The outer workspace comparison uses a strict >= against reach squared without the tolerance, so boundary targets fail.
VERIFIED REPAIR
Reject only when r squared exceeds reach squared plus the 1e-9 slack.
Unsuccessful approach: Comparing r squared against the unsquared reach mixes units and admits targets beyond the arm.
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: 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]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['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]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['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 | unreachable | [0.0, 0.0] | Failed |
| fully extended diagonal | unreachable | [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 / c8bc4c1d2b62b916a3a8653a499dc14b1d3207fb22708fcdfb8448f2356b3d42
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+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]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['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]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['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 | unreachable | [0.0, 90.0] | Failed |
| general elbow down | unreachable | [53.1301, -90.0] | Failed |
| fully extended on x axis | unreachable | [0.0, 0.0] | Failed |
| fully extended diagonal | unreachable | [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 / 121dd2be5a3bcd0c4e38964d3f95c7151c95dd0a836bcbe9968575954f00805f
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]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['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]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['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 / b32d9a0bd751cf84aba5571fcdf741647e06c5818efeb516485779e16b98236d
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 2026-09-29T14:51:02.619645+00:00.
Case digest / 38f0e43aa4d287bddff4208cdf03bc524867b7e3a8e6e6dd32e94ec6331c9317