FA-88011 / Inverse kinematics solvers / Open access

Branch cost ignores per-joint weights · case 01

Moving a heavy base joint is charged the same as moving a light wrist joint.

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

ROOT CAUSE

The weight vector is unpacked but never multiplied into the cost.

THE FAILURE

The weight vector is unpacked but never multiplied into the cost.

Unsuccessful approach: Using w_i * d_i makes the metric linear and changes which branch wins.

Case contract

Input [candidates, current, limits, weights] with joint angles in degrees. A limits entry is null for a continuous joint (distance is the shortest wrap-around difference) or [lo,hi] for a limited joint (inclusive bounds, distance is the plain difference because motion cannot pass the stop). Cost is sum w*d^2. Return [index into candidates, cost rounded 4] for the cheapest feasible candidate, earliest wins ties within 1e-9, or "none".

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):
    cands,cur,lims,w=x
    best=None
    bi=None
    for k,c in enumerate(cands):
        ok=all(l is None or l[0]<=q<=l[1] for q,l in zip(c,lims))
        if not ok: continue
        cost=0.0
        for q,p,wi,l in zip(c,cur,w,lims):
            if l is None:
                d=abs(q-p)%360.0
                d=min(d,360.0-d)
            else:
                d=abs(q-p)
            cost+=d*d
        if best is None or cost<best-1e-9:
            best=cost
            bi=k
    if bi is None: return 'none'
    return [bi,round(best,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['limit boundary candidate', [[[[170, 10], [0, 10]], [160, 10], [[-170, 170], None], [1, 1]]], [0, 100.0]], ['all infeasible', [[[[200, 0], [-200, 0]], [0, 0], [[-170, 170], None], [1, 1]]], 'none'], ['continuous wrap shorter', [[[[0, 170], [0, -170]], [0, 175], [[-90, 90], None], [1, 1]]], [0, 25.0]], ['multi-turn limited joint', [[[[-200, 0], [150, 0]], [170, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['wide limited joint far', [[[[250, 0], [-100, 0]], [-120, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['exact tie keeps first', [[[[10, 0], [-10, 0]], [0, 0], [[-90, 90], None], [1, 1]]], [0, 100.0]]], [['continuous wrap shorter', [[[[0, 170], [0, -170]], [0, 175], [[-90, 90], None], [1, 1]]], [0, 25.0]], ['wide limited joint far', [[[[250, 0], [-100, 0]], [-120, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['exact tie keeps first', [[[[10, 0], [-10, 0]], [0, 0], [[-90, 90], None], [1, 1]]], [0, 100.0]], ['near tie within tolerance', [[[[10, 0], [0, 10.0000000001]], [0, 0], [None, None], [1, 1]]], [0, 100.0]], ['slightly better later', [[[[10, 0], [9.999, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 1]]], [1, 99.98]], ['three joints mixed', [[[[10, 350, -20], [15, -5, -25], [100, 0, 0]], [12, 0, -22], [[-120, 120], None, [-45, 45]], [2, 1, 1]]], [1, 52.0]]], [['multi-turn limited joint', [[[[-200, 0], [150, 0]], [170, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['near tie within tolerance', [[[[10, 0], [0, 10.0000000001]], [0, 0], [None, None], [1, 1]]], [0, 100.0]], ['slightly better later', [[[[10, 0], [9.999, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 1]]], [1, 99.98]], ['infeasible first candidate', [[[[95, 0], [30, 0], [20, 5]], [0, 0], [[-90, 90], None], [1, 1]]], [2, 425.0]], ['lower bound inclusive', [[[[-90, 0], [60, 60]], [-80, 0], [[-90, 90], [-90, 90]], [1, 1]]], [0, 100.0]], ['continuous across 180', [[[[0, -179], [0, 100]], [0, 179], [None, None], [1, 1]]], [0, 4.0]], ['heavy weight flips choice', [[[[5, 40], [30, 0]], [0, 0], [None, None], [10, 1]]], [0, 1850.0]]], [['limit boundary candidate', [[[[170, 10], [0, 10]], [160, 10], [[-170, 170], None], [1, 1]]], [0, 100.0]], ['wide limited joint far', [[[[250, 0], [-100, 0]], [-120, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['continuous across 180', [[[[0, -179], [0, 100]], [0, 179], [None, None], [1, 1]]], [0, 4.0]], ['three joints mixed', [[[[10, 350, -20], [15, -5, -25], [100, 0, 0]], [12, 0, -22], [[-120, 120], None, [-45, 45]], [2, 1, 1]]], [1, 52.0]], ['heavy weight flips choice', [[[[5, 40], [30, 0]], [0, 0], [None, None], [10, 1]]], [0, 1850.0]], ['continuous large difference', [[[[0, 300], [0, 200]], [0, 0], [[-10, 10], None], [1, 1]]], [0, 3600.0]]], [['limit boundary candidate', [[[[170, 10], [0, 10]], [160, 10], [[-170, 170], None], [1, 1]]], [0, 100.0]], ['all infeasible', [[[[200, 0], [-200, 0]], [0, 0], [[-170, 170], None], [1, 1]]], 'none'], ['continuous wrap shorter', [[[[0, 170], [0, -170]], [0, 175], [[-90, 90], None], [1, 1]]], [0, 25.0]], ['multi-turn limited joint', [[[[-200, 0], [150, 0]], [170, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['three joints mixed', [[[[10, 350, -20], [15, -5, -25], [100, 0, 0]], [12, 0, -22], [[-120, 120], None, [-45, 45]], [2, 1, 1]]], [1, 52.0]], ['continuous large difference', [[[[0, 300], [0, 200]], [0, 0], [[-10, 10], None], [1, 1]]], [0, 3600.0]]]]
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
limit boundary candidate	`[0, 100.0]`	`[0, 100.0]`	Passed
all infeasible	`none`	`none`	Passed
continuous wrap shorter	`[0, 25.0]`	`[0, 25.0]`	Passed
multi-turn limited joint	`[1, 400.0]`	`[1, 400.0]`	Passed
wide limited joint far	`[1, 400.0]`	`[1, 400.0]`	Passed
weighted wrist	`[1, 1600.0]`	`[0, 350.0]`	Failed
exact tie keeps first	`[0, 100.0]`	`[0, 100.0]`	Passed

SHA-256 / 74e98efadbe836752b88b114b94dcc7cadc68adaf67a0bb35221fa0d8f7cb88a

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    cands,cur,lims,w=x
    best=None
    bi=None
    for k,c in enumerate(cands):
        ok=all(l is None or l[0]<=q<=l[1] for q,l in zip(c,lims))
        if not ok: continue
        cost=0.0
        for q,p,wi,l in zip(c,cur,w,lims):
            if l is None:
                d=abs(q-p)%360.0
                d=min(d,360.0-d)
            else:
                d=abs(q-p)
            cost+=wi*d
        if best is None or cost<best-1e-9:
            best=cost
            bi=k
    if bi is None: return 'none'
    return [bi,round(best,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['limit boundary candidate', [[[[170, 10], [0, 10]], [160, 10], [[-170, 170], None], [1, 1]]], [0, 100.0]], ['all infeasible', [[[[200, 0], [-200, 0]], [0, 0], [[-170, 170], None], [1, 1]]], 'none'], ['continuous wrap shorter', [[[[0, 170], [0, -170]], [0, 175], [[-90, 90], None], [1, 1]]], [0, 25.0]], ['multi-turn limited joint', [[[[-200, 0], [150, 0]], [170, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['wide limited joint far', [[[[250, 0], [-100, 0]], [-120, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['exact tie keeps first', [[[[10, 0], [-10, 0]], [0, 0], [[-90, 90], None], [1, 1]]], [0, 100.0]]], [['continuous wrap shorter', [[[[0, 170], [0, -170]], [0, 175], [[-90, 90], None], [1, 1]]], [0, 25.0]], ['wide limited joint far', [[[[250, 0], [-100, 0]], [-120, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['exact tie keeps first', [[[[10, 0], [-10, 0]], [0, 0], [[-90, 90], None], [1, 1]]], [0, 100.0]], ['near tie within tolerance', [[[[10, 0], [0, 10.0000000001]], [0, 0], [None, None], [1, 1]]], [0, 100.0]], ['slightly better later', [[[[10, 0], [9.999, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 1]]], [1, 99.98]], ['three joints mixed', [[[[10, 350, -20], [15, -5, -25], [100, 0, 0]], [12, 0, -22], [[-120, 120], None, [-45, 45]], [2, 1, 1]]], [1, 52.0]]], [['multi-turn limited joint', [[[[-200, 0], [150, 0]], [170, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['near tie within tolerance', [[[[10, 0], [0, 10.0000000001]], [0, 0], [None, None], [1, 1]]], [0, 100.0]], ['slightly better later', [[[[10, 0], [9.999, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 1]]], [1, 99.98]], ['infeasible first candidate', [[[[95, 0], [30, 0], [20, 5]], [0, 0], [[-90, 90], None], [1, 1]]], [2, 425.0]], ['lower bound inclusive', [[[[-90, 0], [60, 60]], [-80, 0], [[-90, 90], [-90, 90]], [1, 1]]], [0, 100.0]], ['continuous across 180', [[[[0, -179], [0, 100]], [0, 179], [None, None], [1, 1]]], [0, 4.0]], ['heavy weight flips choice', [[[[5, 40], [30, 0]], [0, 0], [None, None], [10, 1]]], [0, 1850.0]]], [['limit boundary candidate', [[[[170, 10], [0, 10]], [160, 10], [[-170, 170], None], [1, 1]]], [0, 100.0]], ['wide limited joint far', [[[[250, 0], [-100, 0]], [-120, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['continuous across 180', [[[[0, -179], [0, 100]], [0, 179], [None, None], [1, 1]]], [0, 4.0]], ['three joints mixed', [[[[10, 350, -20], [15, -5, -25], [100, 0, 0]], [12, 0, -22], [[-120, 120], None, [-45, 45]], [2, 1, 1]]], [1, 52.0]], ['heavy weight flips choice', [[[[5, 40], [30, 0]], [0, 0], [None, None], [10, 1]]], [0, 1850.0]], ['continuous large difference', [[[[0, 300], [0, 200]], [0, 0], [[-10, 10], None], [1, 1]]], [0, 3600.0]]], [['limit boundary candidate', [[[[170, 10], [0, 10]], [160, 10], [[-170, 170], None], [1, 1]]], [0, 100.0]], ['all infeasible', [[[[200, 0], [-200, 0]], [0, 0], [[-170, 170], None], [1, 1]]], 'none'], ['continuous wrap shorter', [[[[0, 170], [0, -170]], [0, 175], [[-90, 90], None], [1, 1]]], [0, 25.0]], ['multi-turn limited joint', [[[[-200, 0], [150, 0]], [170, 0], [[-270, 270], None], [1, 1]]], [1, 400.0]], ['weighted wrist', [[[[10, 50], [40, 0]], [0, 0], [[-90, 90], [-90, 90]], [1, 0.1]]], [0, 350.0]], ['three joints mixed', [[[[10, 350, -20], [15, -5, -25], [100, 0, 0]], [12, 0, -22], [[-120, 120], None, [-45, 45]], [2, 1, 1]]], [1, 52.0]], ['continuous large difference', [[[[0, 300], [0, 200]], [0, 0], [[-10, 10], None], [1, 1]]], [0, 3600.0]]]]
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
limit boundary candidate	`[0, 10.0]`	`[0, 100.0]`	Failed
all infeasible	`none`	`none`	Passed
continuous wrap shorter	`[0, 5.0]`	`[0, 25.0]`	Failed
multi-turn limited joint	`[1, 20.0]`	`[1, 400.0]`	Failed
wide limited joint far	`[1, 20.0]`	`[1, 400.0]`	Failed
weighted wrist	`[0, 15.0]`	`[0, 350.0]`	Failed
exact tie keeps first	`[0, 10.0]`	`[0, 100.0]`	Failed

SHA-256 / da935c705437f16c8c0b89229c9c30430e8677fca92192723b176ba203f80e2e

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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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 2026-09-29T14:51:04.265289+00:00.

Case digest / b3591000b53d58a3f112775e74fdb06116781d8c6088854dbe14f5ce7b85b8b3