FA-88181 / Inverse kinematics solvers / Open access
Secondary motion is projected onto the task space · case 01
The joint-centering objective disturbs the end effector instead of moving in the self-motion manifold.
ROOT CAUSE
The projector is built as J+ J instead of I - J+ J.
VERIFIED REPAIR
Use N = I - J+ J.
Unsuccessful approach: Adding J+ J to the identity doubles task-space leakage rather than removing it.
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(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=[[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]]], [['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['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]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['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 | [16.3387, 18.3661, 28.8469] | [23.38, 40.6501, 42.2202] | Failed |
| task and centering | [-55.5967, 145.4425, -42.7815] | [-62.5399, 148.4425, -29.151] | Failed |
| straight arm singular | singular | singular | Passed |
| unequal links | [79.4024, -201.5325, 172.2439] | [87.5837, -214.0176, 178.4237] | Failed |
| negative config | [7.7384, 0.2006, -61.6816] | [-7.4068, -55.9156, -92.0392] | Failed |
| strong centering | [-49.6212, -80.7575, -25.9469] | [43.7121, 92.5758, 40.7197] | Failed |
SHA-256 / 24a640fa9f8059836722557ff2568acb0b33ee0fac7591140db2d8c490ee30fe
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+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]]], [['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['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]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['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 | [6.3387, 3.3661, 8.8469] | [23.38, 40.6501, 42.2202] | Failed |
| task and centering | [-58.5967, 142.4425, -27.7815] | [-62.5399, 148.4425, -29.151] | Failed |
| straight arm singular | singular | singular | Passed |
| unequal links | [73.4024, -192.5325, 160.2439] | [87.5837, -214.0176, 178.4237] | Failed |
| negative config | [37.7384, 30.2006, -11.6816] | [-7.4068, -55.9156, -92.0392] | Failed |
| strong centering | [-169.6212, -200.7575, -145.9469] | [43.7121, 92.5758, 40.7197] | Failed |
SHA-256 / 520a8bcaea4012c2ca01a49970c0fa662e845ce9c709e1985c4a7f49a2466583
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]]], [['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['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]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['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 / e88d9b16de1ec44666b8df86c1843a3f1ee82f6ab5aaa6e40105ef9a09a38dbb
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 2026-09-29T14:51:05.730462+00:00.
Case digest / 7cb2e20cc5de3d4e68bb938b4eb2560c1d672d5c5f9b33b224b48875f6aa44e1