FAILURE MAP
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FA-88306 / Inverse kinematics solvers / Open access

Encoder counts use banker's rounding · case 01

Exact half counts round to even, so +0.5 counts sometimes round down.

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

ROOT CAUSE

round() is used where the contract rounds half up.

THE FAILURE

round() is used where the contract rounds half up.

Unsuccessful approach: int(x + 0.5) truncates toward zero and is wrong for negative counts.

Case contract

Input [geo,off,dirs,ratio,lines,lims], all per-joint lists: IK joint angles (degrees) become encoder counts per joint as counts = floor(dirs*(geo-off)*ratio/360*lines*4 + 0.5) (quadrature gives 4 counts per line, half rounds up). lims holds inclusive [lo,hi] count limits; the first violating joint returns ["limit", i, counts].

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):
    geo,off,dirs,ratio,lines,lims=x
    out=[]
    for i in range(len(geo)):
        m=dirs[i]*(geo[i]-off[i])*ratio[i]
        cnt=round(m/360.0*lines[i]*4)
        lo,hi=lims[i]
        if cnt<lo or cnt>hi: return ['limit',i,cnt]
        out.append(cnt)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['two joints nominal', [[[10, 20], [5, -5], [1, -1], [50, 30], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [278, -833]], ['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['negative half rounds up', [[[-45], [0], [1], [1], [11], [[-10, 10]]]], [-5]], ['reversed joint with offset', [[[30], [90], [-1], [2], [180], [[-10000, 10000]]]], [240]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['within limit exactly', [[[0, 9], [0, 0], [1, 1], [1, 1], [10, 10], [[-1, 1], [-1, 1]]]], [0, 1]]], [['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['negative half rounds up', [[[-45], [0], [1], [1], [11], [[-10, 10]]]], [-5]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['within limit exactly', [[[0, 9], [0, 0], [1, 1], [1, 1], [10, 10], [[-1, 1], [-1, 1]]]], [0, 1]], ['three joint arm', [[[45, -30, 12], [0, 10, -3], [1, -1, 1], [100, 80, 50], [500, 500, 256], [[-1000000, 1000000], [-1000000, 1000000], [-1000000, 1000000]]]], [25000, 17778, 2133]], ['gear down joint', [[[180, 45], [0, 0], [1, 1], [0.5, 4], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [100, 200]], ['offset makes negative', [[[10], [40], [1], [3], [360], [[-5000, 5000]]]], [-360]]], [['two joints nominal', [[[10, 20], [5, -5], [1, -1], [50, 30], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [278, -833]], ['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['three joint arm', [[[45, -30, 12], [0, 10, -3], [1, -1, 1], [100, 80, 50], [500, 500, 256], [[-1000000, 1000000], [-1000000, 1000000], [-1000000, 1000000]]]], [25000, 17778, 2133]], ['gear down joint', [[[180, 45], [0, 0], [1, 1], [0.5, 4], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [100, 200]], ['offset makes negative', [[[10], [40], [1], [3], [360], [[-5000, 5000]]]], [-360]], ['motor degrees small counts large', [[[5], [0], [1], [100], [1000], [[-5000, 5000]]]], ['limit', 0, 5556]], ['geo within range counts out', [[[20], [0], [1], [50], [100], [[-100, 1000]]]], ['limit', 0, 1111]]], [['two joints nominal', [[[10, 20], [5, -5], [1, -1], [50, 30], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [278, -833]], ['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['negative half rounds up', [[[-45], [0], [1], [1], [11], [[-10, 10]]]], [-5]], ['reversed joint with offset', [[[30], [90], [-1], [2], [180], [[-10000, 10000]]]], [240]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['geo within range counts out', [[[20], [0], [1], [50], [100], [[-100, 1000]]]], ['limit', 0, 1111]]], [['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['reversed joint with offset', [[[30], [90], [-1], [2], [180], [[-10000, 10000]]]], [240]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['within limit exactly', [[[0, 9], [0, 0], [1, 1], [1, 1], [10, 10], [[-1, 1], [-1, 1]]]], [0, 1]], ['three joint arm', [[[45, -30, 12], [0, 10, -3], [1, -1, 1], [100, 80, 50], [500, 500, 256], [[-1000000, 1000000], [-1000000, 1000000], [-1000000, 1000000]]]], [25000, 17778, 2133]], ['gear down joint', [[[180, 45], [0, 0], [1, 1], [0.5, 4], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [100, 200]]]]
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
two joints nominal[278, -833][278, -833]Passed
half count rounding up[2][3]Failed
negative fractional count[-3][-3]Passed
negative half rounds up[-6][-5]Failed
reversed joint with offset[240][240]Passed
limit exceeded['limit', 0, 100000]['limit', 0, 100000]Passed
within limit exactly[0, 1][0, 1]Passed

SHA-256 / 807ac94d4564894b233ecd6d551088a2053657cbccc6a74b4e63aadf9830d63e

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    geo,off,dirs,ratio,lines,lims=x
    out=[]
    for i in range(len(geo)):
        m=dirs[i]*(geo[i]-off[i])*ratio[i]
        cnt=int(m/360.0*lines[i]*4+0.5)
        lo,hi=lims[i]
        if cnt<lo or cnt>hi: return ['limit',i,cnt]
        out.append(cnt)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['two joints nominal', [[[10, 20], [5, -5], [1, -1], [50, 30], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [278, -833]], ['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['negative half rounds up', [[[-45], [0], [1], [1], [11], [[-10, 10]]]], [-5]], ['reversed joint with offset', [[[30], [90], [-1], [2], [180], [[-10000, 10000]]]], [240]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['within limit exactly', [[[0, 9], [0, 0], [1, 1], [1, 1], [10, 10], [[-1, 1], [-1, 1]]]], [0, 1]]], [['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['negative half rounds up', [[[-45], [0], [1], [1], [11], [[-10, 10]]]], [-5]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['within limit exactly', [[[0, 9], [0, 0], [1, 1], [1, 1], [10, 10], [[-1, 1], [-1, 1]]]], [0, 1]], ['three joint arm', [[[45, -30, 12], [0, 10, -3], [1, -1, 1], [100, 80, 50], [500, 500, 256], [[-1000000, 1000000], [-1000000, 1000000], [-1000000, 1000000]]]], [25000, 17778, 2133]], ['gear down joint', [[[180, 45], [0, 0], [1, 1], [0.5, 4], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [100, 200]], ['offset makes negative', [[[10], [40], [1], [3], [360], [[-5000, 5000]]]], [-360]]], [['two joints nominal', [[[10, 20], [5, -5], [1, -1], [50, 30], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [278, -833]], ['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['three joint arm', [[[45, -30, 12], [0, 10, -3], [1, -1, 1], [100, 80, 50], [500, 500, 256], [[-1000000, 1000000], [-1000000, 1000000], [-1000000, 1000000]]]], [25000, 17778, 2133]], ['gear down joint', [[[180, 45], [0, 0], [1, 1], [0.5, 4], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [100, 200]], ['offset makes negative', [[[10], [40], [1], [3], [360], [[-5000, 5000]]]], [-360]], ['motor degrees small counts large', [[[5], [0], [1], [100], [1000], [[-5000, 5000]]]], ['limit', 0, 5556]], ['geo within range counts out', [[[20], [0], [1], [50], [100], [[-100, 1000]]]], ['limit', 0, 1111]]], [['two joints nominal', [[[10, 20], [5, -5], [1, -1], [50, 30], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [278, -833]], ['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['negative half rounds up', [[[-45], [0], [1], [1], [11], [[-10, 10]]]], [-5]], ['reversed joint with offset', [[[30], [90], [-1], [2], [180], [[-10000, 10000]]]], [240]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['geo within range counts out', [[[20], [0], [1], [50], [100], [[-100, 1000]]]], ['limit', 0, 1111]]], [['half count rounding up', [[[45], [0], [1], [1], [5], [[-10, 10]]]], [3]], ['negative fractional count', [[[-45], [0], [1], [1], [5.4], [[-10, 10]]]], [-3]], ['reversed joint with offset', [[[30], [90], [-1], [2], [180], [[-10000, 10000]]]], [240]], ['limit exceeded', [[[90, 0], [0, 0], [1, 1], [100, 1], [1000, 1000], [[-1000, 1000], [-10, 10]]]], ['limit', 0, 100000]], ['within limit exactly', [[[0, 9], [0, 0], [1, 1], [1, 1], [10, 10], [[-1, 1], [-1, 1]]]], [0, 1]], ['three joint arm', [[[45, -30, 12], [0, 10, -3], [1, -1, 1], [100, 80, 50], [500, 500, 256], [[-1000000, 1000000], [-1000000, 1000000], [-1000000, 1000000]]]], [25000, 17778, 2133]], ['gear down joint', [[[180, 45], [0, 0], [1, 1], [0.5, 4], [100, 100], [[-100000, 100000], [-100000, 100000]]]], [100, 200]]]]
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
two joints nominal[278, -832][278, -833]Failed
half count rounding up[3][3]Passed
negative fractional count[-2][-3]Failed
negative half rounds up[-5][-5]Passed
reversed joint with offset[240][240]Passed
limit exceeded['limit', 0, 100000]['limit', 0, 100000]Passed
within limit exactly[0, 1][0, 1]Passed

SHA-256 / 759aee915730a2bf0db05252dd86b062313ebe579bc0d0128781744152412cec

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 2026-09-29T14:51:06.903563+00:00.

Case digest / 452ec8fe11097d4f71d41311aad8974cf2b8cdb5bfeed0f32f5e181d0f47f746