FA-88326 / Inverse kinematics solvers / Open access

Adaptive gain uses J^T J^T e instead of J J^T e · case 01

The step length is tuned to a vector that is not the predicted task-space motion.

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

ROOT CAUSE

The second product applies the transpose again when forming the predicted end-effector motion.

VERIFIED REPAIR

Form h = J g with g = J^T e.

Unsuccessful approach: Transposing only the first component still mixes joint and task indices.

Case contract

Input [l1,l2,q1,q2,tx,ty]: one Jacobian-transpose step for a planar 2R arm with the error-optimal gain alpha = <e, J J^T e> / <J J^T e, J J^T e>, dq = alpha J^T e. When <J J^T e, J J^T e> < 1e-15 return ["stationary", q1, q2]. Otherwise return ["step", alpha (6 decimals), q1 and q2 updated in degrees (4 decimals)].

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=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)
    g1=j11*ex+j21*ey
    g2=j12*ex+j22*ey
    h1=j11*g1+j21*g2
    h2=j12*g1+j22*g2
    den=h1*h1+h2*h2
    if den<1e-15: return ['stationary',q1,q2]
    alpha=(ex*h1+ey*h2)/den
    return ['step',round(alpha,6),round(q1+math.degrees(alpha*g1),4),round(q2+math.degrees(alpha*g2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]], [['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]]]
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
moderate error	`['step', 0.20936, 37.6248, 64.3906]`	`['step', 0.263207, 39.5859, 65.5199]`	Failed
at target	`['stationary', 0, 90]`	`['stationary', 0, 90]`	Passed
large error	`['step', -1.414214, -113.6085, -109.6986]`	`['step', 0.287793, 23.1194, 76.4812]`	Failed
unequal links	`['step', 0.292337, 105.8443, -21.4608]`	`['step', 0.11371, 53.3906, -32.7888]`	Failed
negative angles	`['step', 0.186104, -42.7751, -65.3212]`	`['step', 0.183895, -42.6234, -65.2581]`	Failed
behind base	`['step', 0.098389, 135.4013, 40.5193]`	`['step', 0.217338, 154.0209, 53.2367]`	Failed
small error	`['step', -0.003474, 9.9652, 9.9826]`	`['step', 0.201469, 12.0193, 11.0115]`	Failed

SHA-256 / 7b840f18fd46d053dd067dc18639374c45fb8ec0ea52abf960891b35f74b8c20

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=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)
    g1=j11*ex+j21*ey
    g2=j12*ex+j22*ey
    h1=j11*g1+j12*g2
    h2=j12*g1+j22*g2
    den=h1*h1+h2*h2
    if den<1e-15: return ['stationary',q1,q2]
    alpha=(ex*h1+ey*h2)/den
    return ['step',round(alpha,6),round(q1+math.degrees(alpha*g1),4),round(q2+math.degrees(alpha*g2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]], [['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]]]
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
moderate error	`['step', 0.195518, 37.1207, 64.1003]`	`['step', 0.263207, 39.5859, 65.5199]`	Failed
at target	`['stationary', 0, 90]`	`['stationary', 0, 90]`	Passed
large error	`['step', 1.305929, 104.9097, 187.8536]`	`['step', 0.287793, 23.1194, 76.4812]`	Failed
unequal links	`['step', 0.665001, 215.2762, 2.1725]`	`['step', 0.11371, 53.3906, -32.7888]`	Failed
negative angles	`['step', 0.19514, -43.3953, -65.5796]`	`['step', 0.183895, -42.6234, -65.2581]`	Failed
behind base	`['step', 0.174458, 147.3087, 48.6522]`	`['step', 0.217338, 154.0209, 53.2367]`	Failed
small error	`['step', 0.323953, 13.2469, 11.6264]`	`['step', 0.201469, 12.0193, 11.0115]`	Failed

SHA-256 / 092a9c48a2ebab5861d35732a0aa24dd7327c88f0c5a0e5f8a9608c5aaced58b

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=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)
    g1=j11*ex+j21*ey
    g2=j12*ex+j22*ey
    h1=j11*g1+j12*g2
    h2=j21*g1+j22*g2
    den=h1*h1+h2*h2
    if den<1e-15: return ['stationary',q1,q2]
    alpha=(ex*h1+ey*h2)/den
    return ['step',round(alpha,6),round(q1+math.degrees(alpha*g1),4),round(q2+math.degrees(alpha*g2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]], [['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['large error', [[1, 1, 0, 45, -1.5, 0.2]], ['step', 0.287793, 23.1194, 76.4812]], ['unequal links', [[2, 1, 20, -40, 1.5, 2.0]], ['step', 0.11371, 53.3906, -32.7888]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]]], [['moderate error', [[1, 1, 30, 60, 0.5, 1.6]], ['step', 0.263207, 39.5859, 65.5199]], ['at target', [[1, 1, 0, 90, 1, 1]], ['stationary', 0, 90]], ['negative angles', [[1.5, 1, -30, -60, 0.8, -2]], ['step', 0.183895, -42.6234, -65.2581]], ['behind base', [[1, 1, 120, 30, -1, -1]], ['step', 0.217338, 154.0209, 53.2367]], ['small error', [[1, 1, 10, 10, 1.9, 0.6]], ['step', 0.201469, 12.0193, 11.0115]], ['long forearm', [[0.5, 2, 70, -20, -0.5, 2.2]], ['step', 0.098879, 93.8254, -1.5747]], ['straight arm radial error', [[1, 1, 0, 0, 1.5, 0]], ['stationary', 0, 0]]]]
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
moderate error	`['step', 0.263207, 39.5859, 65.5199]`	`['step', 0.263207, 39.5859, 65.5199]`	Passed
at target	`['stationary', 0, 90]`	`['stationary', 0, 90]`	Passed
large error	`['step', 0.287793, 23.1194, 76.4812]`	`['step', 0.287793, 23.1194, 76.4812]`	Passed
unequal links	`['step', 0.11371, 53.3906, -32.7888]`	`['step', 0.11371, 53.3906, -32.7888]`	Passed
negative angles	`['step', 0.183895, -42.6234, -65.2581]`	`['step', 0.183895, -42.6234, -65.2581]`	Passed
behind base	`['step', 0.217338, 154.0209, 53.2367]`	`['step', 0.217338, 154.0209, 53.2367]`	Passed
small error	`['step', 0.201469, 12.0193, 11.0115]`	`['step', 0.201469, 12.0193, 11.0115]`	Passed

SHA-256 / 1b923daa195fa5466b9f7ae2e04ebbccafbf5b2dfab356a024694f41bb50c6ef

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 2026-09-29T14:51:07.106336+00:00.

Case digest / 0496a2a2b9560b63fbddad5ce0680b5ecfde95d9209a65f2daacf3264bf17b25