FA-88196 / Inverse kinematics solvers / Open access
Redundant arm forward kinematics uses relative angles as absolute · case 01
The task error is computed from the wrong tip position.
ROOT CAUSE
Link directions use each relative joint angle instead of the cumulative angle.
VERIFIED REPAIR
Use cumulative angles q0, q0+q1, q0+q1+q2 for link directions.
Unsuccessful approach: Accumulating only the preceding joints shifts every link by one joint.
Case contract
Input [lengths(3), q(3 deg), target, qc(3 deg), k]: planar 3R arm, position task only. One step dq = J+ e + (I - J+ J) z, J+ = J^T (J J^T)^-1, e = target - fk(q), z = k*(qc-q) in radians. Return the updated q in degrees rounded to 4, or "singular" when |det(J J^T)|<1e-12.
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):
ls,q,t,qc,k=x
qr=[math.radians(v) for v in q]
cum=[sum(qr[:i+1]) for i in range(3)]
px=sum(ls[i]*math.cos(qr[i]) for i in range(3))
py=sum(ls[i]*math.sin(qr[i]) for i in range(3))
J=[[-sum(ls[j]*math.sin(cum[j]) for j in range(i,3)) for i in range(3)],
[sum(ls[j]*math.cos(cum[j]) for j in range(i,3)) for i in range(3)]]
a=sum(v*v for v in J[0])
b=sum(J[0][i]*J[1][i] for i in range(3))
c=sum(v*v for v in J[1])
det=a*c-b*b
if abs(det)<1e-12: return 'singular'
inv=[[c/det,-b/det],[-b/det,a/det]]
P=[[J[0][i]*inv[0][0]+J[1][i]*inv[1][0],J[0][i]*inv[0][1]+J[1][i]*inv[1][1]] for i in range(3)]
e=[t[0]-px,t[1]-py]
z=[k*math.radians(qc[i]-q[i]) for i in range(3)]
N=[[(1.0 if i==j else 0.0)-sum(P[i][r]*J[r][j] for r in range(2)) for j in range(3)] for i in range(3)]
dq=[P[i][0]*e[0]+P[i][1]*e[1]+sum(N[i][j]*z[j] for j in range(3)) for i in range(3)]
return [round(q[i]+math.degrees(dq[i]),4) for i in range(3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]]]
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 |
|---|---|---|---|
| task only | [9.3316, 57.6188, 75.1212] | [-9.6761, 48.859, 73.9791] | Failed |
| centering only | [42.3877, 49.4099, 43.3623] | [23.38, 40.6501, 42.2202] | Failed |
| task and centering | [-14.5181, 96.1595, -12.4732] | [-62.5399, 148.4425, -29.151] | Failed |
| straight arm singular | singular | singular | Passed |
| unequal links | [75.7308, -160.8733, 142.1595] | [87.5837, -214.0176, 178.4237] | Failed |
| negative config | [-27.7128, -64.324, -87.8622] | [-7.4068, -55.9156, -92.0392] | Failed |
| strong centering | [124.5737, 96.2513, 3.9645] | [43.7121, 92.5758, 40.7197] | Failed |
SHA-256 / 416889c93a2aa216c8437dd14b3ddfeede32835a5816c7346a93b2c5444f94c6
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
ls,q,t,qc,k=x
qr=[math.radians(v) for v in q]
cum=[sum(qr[:i]) for i in range(3)]
px=sum(ls[i]*math.cos(cum[i]) for i in range(3))
py=sum(ls[i]*math.sin(cum[i]) for i in range(3))
J=[[-sum(ls[j]*math.sin(cum[j]) for j in range(i,3)) for i in range(3)],
[sum(ls[j]*math.cos(cum[j]) for j in range(i,3)) for i in range(3)]]
a=sum(v*v for v in J[0])
b=sum(J[0][i]*J[1][i] for i in range(3))
c=sum(v*v for v in J[1])
det=a*c-b*b
if abs(det)<1e-12: return 'singular'
inv=[[c/det,-b/det],[-b/det,a/det]]
P=[[J[0][i]*inv[0][0]+J[1][i]*inv[1][0],J[0][i]*inv[0][1]+J[1][i]*inv[1][1]] for i in range(3)]
e=[t[0]-px,t[1]-py]
z=[k*math.radians(qc[i]-q[i]) for i in range(3)]
N=[[(1.0 if i==j else 0.0)-sum(P[i][r]*J[r][j] for r in range(2)) for j in range(3)] for i in range(3)]
dq=[P[i][0]*e[0]+P[i][1]*e[1]+sum(N[i][j]*z[j] for j in range(3)) for i in range(3)]
return [round(q[i]+math.degrees(dq[i]),4) for i in range(3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]]]
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 |
|---|---|---|---|
| task only | [-15.6662, 74.9817, 107.4888] | [-9.6761, 48.859, 73.9791] | Failed |
| centering only | [-2.9637, 84.7595, 105.555] | [23.38, 40.6501, 42.2202] | Failed |
| task and centering | [-11.4087, 55.0819, 34.7015] | [-62.5399, 148.4425, -29.151] | Failed |
| straight arm singular | singular | singular | Passed |
| unequal links | [-28.4949, 214.2316, -183.7851] | [87.5837, -214.0176, 178.4237] | Failed |
| negative config | [45.7555, -144.3816, -130.2409] | [-7.4068, -55.9156, -92.0392] | Failed |
| strong centering | [52.5051, 127.8538, 71.6012] | [43.7121, 92.5758, 40.7197] | Failed |
SHA-256 / 4886735bacd1c490587836351955be3797ece36cfca1c7cd32e8e66243ab511a
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
ls,q,t,qc,k=x
qr=[math.radians(v) for v in q]
cum=[sum(qr[:i+1]) for i in range(3)]
px=sum(ls[i]*math.cos(cum[i]) for i in range(3))
py=sum(ls[i]*math.sin(cum[i]) for i in range(3))
J=[[-sum(ls[j]*math.sin(cum[j]) for j in range(i,3)) for i in range(3)],
[sum(ls[j]*math.cos(cum[j]) for j in range(i,3)) for i in range(3)]]
a=sum(v*v for v in J[0])
b=sum(J[0][i]*J[1][i] for i in range(3))
c=sum(v*v for v in J[1])
det=a*c-b*b
if abs(det)<1e-12: return 'singular'
inv=[[c/det,-b/det],[-b/det,a/det]]
P=[[J[0][i]*inv[0][0]+J[1][i]*inv[1][0],J[0][i]*inv[0][1]+J[1][i]*inv[1][1]] for i in range(3)]
e=[t[0]-px,t[1]-py]
z=[k*math.radians(qc[i]-q[i]) for i in range(3)]
N=[[(1.0 if i==j else 0.0)-sum(P[i][r]*J[r][j] for r in range(2)) for j in range(3)] for i in range(3)]
dq=[P[i][0]*e[0]+P[i][1]*e[1]+sum(N[i][j]*z[j] for j in range(3)) for i in range(3)]
return [round(q[i]+math.degrees(dq[i]),4) for i in range(3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]]]
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 |
|---|---|---|---|
| task only | [-9.6761, 48.859, 73.9791] | [-9.6761, 48.859, 73.9791] | Passed |
| centering only | [23.38, 40.6501, 42.2202] | [23.38, 40.6501, 42.2202] | Passed |
| task and centering | [-62.5399, 148.4425, -29.151] | [-62.5399, 148.4425, -29.151] | Passed |
| straight arm singular | singular | singular | Passed |
| unequal links | [87.5837, -214.0176, 178.4237] | [87.5837, -214.0176, 178.4237] | Passed |
| negative config | [-7.4068, -55.9156, -92.0392] | [-7.4068, -55.9156, -92.0392] | Passed |
| strong centering | [43.7121, 92.5758, 40.7197] | [43.7121, 92.5758, 40.7197] | Passed |
SHA-256 / 2d5cec270683ca06a09674580ce1d3b81b6f8e23e3f10e4553c893fdaa4d2cdc
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 2026-09-29T14:51:05.865778+00:00.
Case digest / 5e52f029f47c1077bc5d19bc83a8b5cf7b942b105d6cbf63d5f638da0eebe06b