FAILURE MAP
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FA-87906 / Inverse kinematics solvers / Open access

Forearm Jacobian column uses the shoulder angle · case 01

Steps move the wrist in a direction that does not reduce the task error whenever q2 is non-zero.

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

ROOT CAUSE

The x-derivative with respect to q2 is written -l2 sin(q1) instead of -l2 sin(q1+q2).

VERIFIED REPAIR

Differentiate the forearm term at the absolute angle q1+q2.

Unsuccessful approach: Flipping the sign of the entry keeps the right angle but points the column the wrong way.

Case contract

Input [l1,l2,q1,q2,tx,ty,lam,maxstep], angles in degrees, maxstep in radians. Perform one damped least squares step dq = J^T (J J^T + lam^2 I)^-1 e with e = target - fk(q), scale dq so its Euclidean norm is at most maxstep, and return the updated [q1,q2] in degrees rounded to 4 places; a near-singular damped matrix (|det|<1e-12) returns "singular".

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,q1,q2,tx,ty,lam,maxstep=x
    a=math.radians(q1)
    b=math.radians(q1+q2)
    ex=tx-(l1*math.cos(a)+l2*math.cos(b))
    ey=ty-(l1*math.sin(a)+l2*math.sin(b))
    j11=-l1*math.sin(a)-l2*math.sin(b)
    j12=-l2*math.sin(a)
    j21=l1*math.cos(a)+l2*math.cos(b)
    j22=l2*math.cos(b)
    m11=j11*j11+j12*j12+lam*lam
    m12=j11*j21+j12*j22
    m22=j21*j21+j22*j22+lam*lam
    det=m11*m22-m12*m12
    if abs(det)<1e-12: return 'singular'
    w1=(m22*ex-m12*ey)/det
    w2=(-m12*ex+m11*ey)/det
    d1=j11*w1+j21*w2
    d2=j12*w1+j22*w2
    n=math.hypot(d1,d2)
    if n>maxstep and n>0: d1,d2=d1*maxstep/n,d2*maxstep/n
    return [round(q1+math.degrees(d1),4),round(q2+math.degrees(d2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['straight arm damped', [[1, 1, 0, 0, 1.5, 0.5, 0.5, 1.0]], [10.9135, 5.4567]], ['undamped singular', [[1, 1, 0, 0, 1.5, 0.5, 0.0, 1.0]], 'singular'], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['already on target', [[1, 1, 0, 90, 1, 1, 0.1, 1.0]], [0.0, 90.0]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]]], [['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['already on target', [[1, 1, 0, 90, 1, 1, 0.1, 1.0]], [0.0, 90.0]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['long upper arm', [[3, 1, 60, -30, 1, 3, 0.3, 0.3]], [66.6639, -45.8444]], ['target behind', [[1, 1, 45, 45, -1.2, -0.4, 0.2, 2.0]], [2.6992, 151.4982]]], [['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['long upper arm', [[3, 1, 60, -30, 1, 3, 0.3, 0.3]], [66.6639, -45.8444]], ['target behind', [[1, 1, 45, 45, -1.2, -0.4, 0.2, 2.0]], [2.6992, 151.4982]], ['lightly damped folded', [[1, 1, 30, 170, 0.5, 0.2, 0.01, 0.5]], [1.4116, 171.8446]], ['mid budget', [[2, 2, -120, 60, -1, -2, 0.1, 0.4]], [-133.1333, 78.7821]], ['upper reach', [[1, 2, 90, -45, -1, 2.5, 0.05, 1.5]], [151.7219, -104.8057]]], [['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]], ['lightly damped folded', [[1, 1, 30, 170, 0.5, 0.2, 0.01, 0.5]], [1.4116, 171.8446]], ['mid budget', [[2, 2, -120, 60, -1, -2, 0.1, 0.4]], [-133.1333, 78.7821]], ['upper reach', [[1, 2, 90, -45, -1, 2.5, 0.05, 1.5]], [151.7219, -104.8057]], ['unit damping', [[1, 1, 15, 15, 0.2, 1.9, 1.0, 1.0]], [44.712, 37.4827]]], [['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['straight arm damped', [[1, 1, 0, 0, 1.5, 0.5, 0.5, 1.0]], [10.9135, 5.4567]], ['undamped singular', [[1, 1, 0, 0, 1.5, 0.5, 0.0, 1.0]], 'singular'], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['unit damping', [[1, 1, 15, 15, 0.2, 1.9, 1.0, 1.0]], [44.712, 37.4827]]]]
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 fixtureActualExpectedOutcome
small correction[21.2264, 80.426][22.9688, 62.7571]Failed
large correction clamped[11.4592, 90.0][-1.7958, 101.3176]Failed
straight arm damped[10.9135, 5.4567][10.9135, 5.4567]Passed
undamped singularsingularsingularPassed
heavy damping[32.4429, 62.9881][31.2311, 68.0501]Failed
already on target[0.0, 90.0][0.0, 90.0]Passed
negative angles[-25.8287, -94.8973][-22.919, -92.471]Failed

SHA-256 / f54b8c1ec91140ee1a7c0a872cc0cadff2534cd58dd1a864b33340087f1e641b

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,q1,q2,tx,ty,lam,maxstep=x
    a=math.radians(q1)
    b=math.radians(q1+q2)
    ex=tx-(l1*math.cos(a)+l2*math.cos(b))
    ey=ty-(l1*math.sin(a)+l2*math.sin(b))
    j11=-l1*math.sin(a)-l2*math.sin(b)
    j12=l2*math.sin(b)
    j21=l1*math.cos(a)+l2*math.cos(b)
    j22=l2*math.cos(b)
    m11=j11*j11+j12*j12+lam*lam
    m12=j11*j21+j12*j22
    m22=j21*j21+j22*j22+lam*lam
    det=m11*m22-m12*m12
    if abs(det)<1e-12: return 'singular'
    w1=(m22*ex-m12*ey)/det
    w2=(-m12*ex+m11*ey)/det
    d1=j11*w1+j21*w2
    d2=j12*w1+j22*w2
    n=math.hypot(d1,d2)
    if n>maxstep and n>0: d1,d2=d1*maxstep/n,d2*maxstep/n
    return [round(q1+math.degrees(d1),4),round(q2+math.degrees(d2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['straight arm damped', [[1, 1, 0, 0, 1.5, 0.5, 0.5, 1.0]], [10.9135, 5.4567]], ['undamped singular', [[1, 1, 0, 0, 1.5, 0.5, 0.0, 1.0]], 'singular'], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['already on target', [[1, 1, 0, 90, 1, 1, 0.1, 1.0]], [0.0, 90.0]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]]], [['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['already on target', [[1, 1, 0, 90, 1, 1, 0.1, 1.0]], [0.0, 90.0]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['long upper arm', [[3, 1, 60, -30, 1, 3, 0.3, 0.3]], [66.6639, -45.8444]], ['target behind', [[1, 1, 45, 45, -1.2, -0.4, 0.2, 2.0]], [2.6992, 151.4982]]], [['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['long upper arm', [[3, 1, 60, -30, 1, 3, 0.3, 0.3]], [66.6639, -45.8444]], ['target behind', [[1, 1, 45, 45, -1.2, -0.4, 0.2, 2.0]], [2.6992, 151.4982]], ['lightly damped folded', [[1, 1, 30, 170, 0.5, 0.2, 0.01, 0.5]], [1.4116, 171.8446]], ['mid budget', [[2, 2, -120, 60, -1, -2, 0.1, 0.4]], [-133.1333, 78.7821]], ['upper reach', [[1, 2, 90, -45, -1, 2.5, 0.05, 1.5]], [151.7219, -104.8057]]], [['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]], ['lightly damped folded', [[1, 1, 30, 170, 0.5, 0.2, 0.01, 0.5]], [1.4116, 171.8446]], ['mid budget', [[2, 2, -120, 60, -1, -2, 0.1, 0.4]], [-133.1333, 78.7821]], ['upper reach', [[1, 2, 90, -45, -1, 2.5, 0.05, 1.5]], [151.7219, -104.8057]], ['unit damping', [[1, 1, 15, 15, 0.2, 1.9, 1.0, 1.0]], [44.712, 37.4827]]], [['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['straight arm damped', [[1, 1, 0, 0, 1.5, 0.5, 0.5, 1.0]], [10.9135, 5.4567]], ['undamped singular', [[1, 1, 0, 0, 1.5, 0.5, 0.0, 1.0]], 'singular'], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['unit damping', [[1, 1, 15, 15, 0.2, 1.9, 1.0, 1.0]], [44.712, 37.4827]]]]
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 fixtureActualExpectedOutcome
small correction[28.8272, 35.885][22.9688, 62.7571]Failed
large correction clamped[-1.7958, 78.6824][-1.7958, 101.3176]Failed
straight arm damped[10.9135, 5.4567][10.9135, 5.4567]Passed
undamped singularsingularsingularPassed
heavy damping[31.7349, 51.7959][31.2311, 68.0501]Failed
already on target[0.0, 90.0][0.0, 90.0]Passed
negative angles[-24.3183, -56.4607][-22.919, -92.471]Failed

SHA-256 / ef749489f7c34d95a3ea84c3b5a87a709c713bd8ad089985726db96f498fb8ce

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,q1,q2,tx,ty,lam,maxstep=x
    a=math.radians(q1)
    b=math.radians(q1+q2)
    ex=tx-(l1*math.cos(a)+l2*math.cos(b))
    ey=ty-(l1*math.sin(a)+l2*math.sin(b))
    j11=-l1*math.sin(a)-l2*math.sin(b)
    j12=-l2*math.sin(b)
    j21=l1*math.cos(a)+l2*math.cos(b)
    j22=l2*math.cos(b)
    m11=j11*j11+j12*j12+lam*lam
    m12=j11*j21+j12*j22
    m22=j21*j21+j22*j22+lam*lam
    det=m11*m22-m12*m12
    if abs(det)<1e-12: return 'singular'
    w1=(m22*ex-m12*ey)/det
    w2=(-m12*ex+m11*ey)/det
    d1=j11*w1+j21*w2
    d2=j12*w1+j22*w2
    n=math.hypot(d1,d2)
    if n>maxstep and n>0: d1,d2=d1*maxstep/n,d2*maxstep/n
    return [round(q1+math.degrees(d1),4),round(q2+math.degrees(d2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['straight arm damped', [[1, 1, 0, 0, 1.5, 0.5, 0.5, 1.0]], [10.9135, 5.4567]], ['undamped singular', [[1, 1, 0, 0, 1.5, 0.5, 0.0, 1.0]], 'singular'], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['already on target', [[1, 1, 0, 90, 1, 1, 0.1, 1.0]], [0.0, 90.0]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]]], [['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['already on target', [[1, 1, 0, 90, 1, 1, 0.1, 1.0]], [0.0, 90.0]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['long upper arm', [[3, 1, 60, -30, 1, 3, 0.3, 0.3]], [66.6639, -45.8444]], ['target behind', [[1, 1, 45, 45, -1.2, -0.4, 0.2, 2.0]], [2.6992, 151.4982]]], [['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['long upper arm', [[3, 1, 60, -30, 1, 3, 0.3, 0.3]], [66.6639, -45.8444]], ['target behind', [[1, 1, 45, 45, -1.2, -0.4, 0.2, 2.0]], [2.6992, 151.4982]], ['lightly damped folded', [[1, 1, 30, 170, 0.5, 0.2, 0.01, 0.5]], [1.4116, 171.8446]], ['mid budget', [[2, 2, -120, 60, -1, -2, 0.1, 0.4]], [-133.1333, 78.7821]], ['upper reach', [[1, 2, 90, -45, -1, 2.5, 0.05, 1.5]], [151.7219, -104.8057]]], [['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['negative angles', [[1.5, 1, -40, -70, 1, -1.5, 0.2, 0.5]], [-22.919, -92.471]], ['lightly damped folded', [[1, 1, 30, 170, 0.5, 0.2, 0.01, 0.5]], [1.4116, 171.8446]], ['mid budget', [[2, 2, -120, 60, -1, -2, 0.1, 0.4]], [-133.1333, 78.7821]], ['upper reach', [[1, 2, 90, -45, -1, 2.5, 0.05, 1.5]], [151.7219, -104.8057]], ['unit damping', [[1, 1, 15, 15, 0.2, 1.9, 1.0, 1.0]], [44.712, 37.4827]]], [['small correction', [[1, 1, 30, 45, 1.0, 1.4, 0.1, 1.0]], [22.9688, 62.7571]], ['large correction clamped', [[1, 1, 0, 90, -1.5, 0.5, 0.1, 0.2]], [-1.7958, 101.3176]], ['straight arm damped', [[1, 1, 0, 0, 1.5, 0.5, 0.5, 1.0]], [10.9135, 5.4567]], ['undamped singular', [[1, 1, 0, 0, 1.5, 0.5, 0.0, 1.0]], 'singular'], ['heavy damping', [[2, 1, 20, 60, 1, 2, 2.0, 1.0]], [31.2311, 68.0501]], ['tiny step budget', [[1, 1, 10, 20, 0, 1.8, 0.05, 0.01]], [9.7971, 20.5358]], ['unit damping', [[1, 1, 15, 15, 0.2, 1.9, 1.0, 1.0]], [44.712, 37.4827]]]]
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 fixtureActualExpectedOutcome
small correction[22.9688, 62.7571][22.9688, 62.7571]Passed
large correction clamped[-1.7958, 101.3176][-1.7958, 101.3176]Passed
straight arm damped[10.9135, 5.4567][10.9135, 5.4567]Passed
undamped singularsingularsingularPassed
heavy damping[31.2311, 68.0501][31.2311, 68.0501]Passed
already on target[0.0, 90.0][0.0, 90.0]Passed
negative angles[-22.919, -92.471][-22.919, -92.471]Passed

SHA-256 / 31c5de0ead136e5b901b02d97a61fadeb30bcb42ddccdada602ab9246317e3c5

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 2026-09-29T14:51:03.227889+00:00.

Case digest / 87ffcbb4d637146afd9e0fe6bd92d10746f73d98bbc3162ff181265c02c406f0