FA-87951 / Inverse kinematics solvers / Open access

CCD clamps the per-step rotation instead of the joint angle · case 01

Joints drift beyond their limits over repeated sweeps.

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

ROOT CAUSE

The limit is applied to the incremental angle rather than to the resulting joint value.

VERIFIED REPAIR

Clamp qs[j]+ang into [-lim, lim].

Unsuccessful approach: Clamping the old joint value before adding the increment still allows the sum to exceed the limit.

Case contract

Input [lengths, angles, target, iters, tol, lim]: planar serial chain with relative joint angles in degrees and a symmetric joint limit +/-lim degrees. Each iteration first stops if the tip is within tol of target, else counts one sweep that visits joints from the tip joint down to the base, re-evaluating forward kinematics before each joint, rotating by the signed angle between tip and target vectors and clamping the joint. Return [angles (3 decimals), sweeps performed, final error (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):
    ls,qs,t,iters,tol,lim=x
    qs=[math.radians(q) for q in qs]
    lim=math.radians(lim)
    def fk(qs):
        pts=[(0.0,0.0)]
        a=0.0
        for l,q in zip(ls,qs):
            a+=q
            px,py=pts[-1]
            pts.append((px+l*math.cos(a),py+l*math.sin(a)))
        return pts
    used=0
    for it in range(iters):
        pts=fk(qs)
        ex,ey=pts[-1]
        if math.hypot(t[0]-ex,t[1]-ey)<=tol: break
        used+=1
        for j in range(len(ls)-1,-1,-1):
            pts=fk(qs)
            jx,jy=pts[j]
            ex,ey=pts[-1]
            ve=(ex-jx,ey-jy)
            vt=(t[0]-jx,t[1]-jy)
            ang=math.atan2(ve[0]*vt[1]-ve[1]*vt[0],ve[0]*vt[0]+ve[1]*vt[1])
            qs[j]=qs[j]+max(-lim,min(lim,ang))
    pts=fk(qs)
    err=math.hypot(t[0]-pts[-1][0],t[1]-pts[-1][1])
    return [[round(math.degrees(q),3) for q in qs],used,round(err,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['converges early', [[[2, 1], [0, 0], [2, 1], 10, 0.01, 170]], [[0.0, 90.0], 1, 0.0]], ['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]]], [['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['converges early', [[[2, 1], [0, 0], [2, 1], 10, 0.01, 170]], [[0.0, 90.0], 1, 0.0]], ['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]], ['four links curl', [[[1, 0.8, 0.6, 0.4], [20, 20, 20, 20], [0.3, 1.2], 3, 0.001, 120]], [[36.469, 53.792, 70.335, 118.779], 3, 0.001]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['zero iterations', [[[1, 1], [0, 0], [0, 2], 0, 0.001, 170]], [[0.0, 0.0], 0, 2.8284]]], [['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]], ['four links curl', [[[1, 0.8, 0.6, 0.4], [20, 20, 20, 20], [0.3, 1.2], 3, 0.001, 120]], [[36.469, 53.792, 70.335, 118.779], 3, 0.001]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['zero iterations', [[[1, 1], [0, 0], [0, 2], 0, 0.001, 170]], [[0.0, 0.0], 0, 2.8284]], ['target crossing pi', [[[1, 1, 1], [170, 0, 0], [-2, -0.5], 2, 0.001, 180]], [[168.982, -6.514, 102.831], 2, 0.0176]], ['loose tolerance', [[[1, 1, 1], [5, 5, 5], [2.9, 0.4], 3, 0.2, 170]], [[5.0, 5.0, 5.0], 0, 0.1285]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]]], [['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['target crossing pi', [[[1, 1, 1], [170, 0, 0], [-2, -0.5], 2, 0.001, 180]], [[168.982, -6.514, 102.831], 2, 0.0176]], ['loose tolerance', [[[1, 1, 1], [5, 5, 5], [2.9, 0.4], 3, 0.2, 170]], [[5.0, 5.0, 5.0], 0, 0.1285]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]], ['tight elbow limit', [[[1, 1, 1], [0, 0, 0], [-1, 1], 3, 0.001, 60]], [[60.0, 60.0, 60.0], 3, 0.7321]]], [['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]], ['tight elbow limit', [[[1, 1, 1], [0, 0, 0], [-1, 1], 3, 0.001, 60]], [[60.0, 60.0, 60.0], 3, 0.7321]]]]
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
three links one sweep	`[[5.996, 17.348, 108.435], 1, 0.3586]`	`[[5.996, 17.348, 108.435], 1, 0.3586]`	Passed
three links several sweeps	`[[63.011, 58.047, 42.064], 4, 0.0426]`	`[[63.011, 58.047, 42.064], 4, 0.0426]`	Passed
target behind base	`[[-121.779, -101.253], 3, 0.2445]`	`[[-121.779, -101.253], 3, 0.2445]`	Passed
tight limits	`[[34.995, 46.61, 36.873], 3, 0.0594]`	`[[36.006, 45.0, 37.294], 3, 0.0448]`	Failed
already converged	`[[0.0, 90.0], 0, 0.0]`	`[[0.0, 90.0], 0, 0.0]`	Passed
converges early	`[[0.0, 90.0], 1, 0.0]`	`[[0.0, 90.0], 1, 0.0]`	Passed
unreachable stretch	`[[8.551, -17.103], 3, 2.0222]`	`[[8.551, -17.103], 3, 2.0222]`	Passed

SHA-256 / 868a3d317a077ca67482e2c2a2c4f269aadd3efc186604aba380e016ca9da2b0

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,qs,t,iters,tol,lim=x
    qs=[math.radians(q) for q in qs]
    lim=math.radians(lim)
    def fk(qs):
        pts=[(0.0,0.0)]
        a=0.0
        for l,q in zip(ls,qs):
            a+=q
            px,py=pts[-1]
            pts.append((px+l*math.cos(a),py+l*math.sin(a)))
        return pts
    used=0
    for it in range(iters):
        pts=fk(qs)
        ex,ey=pts[-1]
        if math.hypot(t[0]-ex,t[1]-ey)<=tol: break
        used+=1
        for j in range(len(ls)-1,-1,-1):
            pts=fk(qs)
            jx,jy=pts[j]
            ex,ey=pts[-1]
            ve=(ex-jx,ey-jy)
            vt=(t[0]-jx,t[1]-jy)
            ang=math.atan2(ve[0]*vt[1]-ve[1]*vt[0],ve[0]*vt[0]+ve[1]*vt[1])
            qs[j]=max(-lim,min(lim,qs[j]))+ang
    pts=fk(qs)
    err=math.hypot(t[0]-pts[-1][0],t[1]-pts[-1][1])
    return [[round(math.degrees(q),3) for q in qs],used,round(err,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['converges early', [[[2, 1], [0, 0], [2, 1], 10, 0.01, 170]], [[0.0, 90.0], 1, 0.0]], ['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]]], [['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['converges early', [[[2, 1], [0, 0], [2, 1], 10, 0.01, 170]], [[0.0, 90.0], 1, 0.0]], ['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]], ['four links curl', [[[1, 0.8, 0.6, 0.4], [20, 20, 20, 20], [0.3, 1.2], 3, 0.001, 120]], [[36.469, 53.792, 70.335, 118.779], 3, 0.001]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['zero iterations', [[[1, 1], [0, 0], [0, 2], 0, 0.001, 170]], [[0.0, 0.0], 0, 2.8284]]], [['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]], ['four links curl', [[[1, 0.8, 0.6, 0.4], [20, 20, 20, 20], [0.3, 1.2], 3, 0.001, 120]], [[36.469, 53.792, 70.335, 118.779], 3, 0.001]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['zero iterations', [[[1, 1], [0, 0], [0, 2], 0, 0.001, 170]], [[0.0, 0.0], 0, 2.8284]], ['target crossing pi', [[[1, 1, 1], [170, 0, 0], [-2, -0.5], 2, 0.001, 180]], [[168.982, -6.514, 102.831], 2, 0.0176]], ['loose tolerance', [[[1, 1, 1], [5, 5, 5], [2.9, 0.4], 3, 0.2, 170]], [[5.0, 5.0, 5.0], 0, 0.1285]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]]], [['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['target crossing pi', [[[1, 1, 1], [170, 0, 0], [-2, -0.5], 2, 0.001, 180]], [[168.982, -6.514, 102.831], 2, 0.0176]], ['loose tolerance', [[[1, 1, 1], [5, 5, 5], [2.9, 0.4], 3, 0.2, 170]], [[5.0, 5.0, 5.0], 0, 0.1285]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]], ['tight elbow limit', [[[1, 1, 1], [0, 0, 0], [-1, 1], 3, 0.001, 60]], [[60.0, 60.0, 60.0], 3, 0.7321]]], [['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]], ['tight elbow limit', [[[1, 1, 1], [0, 0, 0], [-1, 1], 3, 0.001, 60]], [[60.0, 60.0, 60.0], 3, 0.7321]]]]
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
three links one sweep	`[[5.996, 17.348, 108.435], 1, 0.3586]`	`[[5.996, 17.348, 108.435], 1, 0.3586]`	Passed
three links several sweeps	`[[63.011, 58.047, 42.064], 4, 0.0426]`	`[[63.011, 58.047, 42.064], 4, 0.0426]`	Passed
target behind base	`[[-121.838, -101.135], 3, 0.2429]`	`[[-121.779, -101.253], 3, 0.2445]`	Failed
tight limits	`[[56.443, 44.547, -22.35], 3, 0.3013]`	`[[36.006, 45.0, 37.294], 3, 0.0448]`	Failed
already converged	`[[0.0, 90.0], 0, 0.0]`	`[[0.0, 90.0], 0, 0.0]`	Passed
converges early	`[[0.0, 90.0], 1, 0.0]`	`[[0.0, 90.0], 1, 0.0]`	Passed
unreachable stretch	`[[8.551, -17.103], 3, 2.0222]`	`[[8.551, -17.103], 3, 2.0222]`	Passed

SHA-256 / de8d9b25f0b66f9ed7521392ea1baf715c5ac38673c9563c580a7ce4fe53e1ca

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,qs,t,iters,tol,lim=x
    qs=[math.radians(q) for q in qs]
    lim=math.radians(lim)
    def fk(qs):
        pts=[(0.0,0.0)]
        a=0.0
        for l,q in zip(ls,qs):
            a+=q
            px,py=pts[-1]
            pts.append((px+l*math.cos(a),py+l*math.sin(a)))
        return pts
    used=0
    for it in range(iters):
        pts=fk(qs)
        ex,ey=pts[-1]
        if math.hypot(t[0]-ex,t[1]-ey)<=tol: break
        used+=1
        for j in range(len(ls)-1,-1,-1):
            pts=fk(qs)
            jx,jy=pts[j]
            ex,ey=pts[-1]
            ve=(ex-jx,ey-jy)
            vt=(t[0]-jx,t[1]-jy)
            ang=math.atan2(ve[0]*vt[1]-ve[1]*vt[0],ve[0]*vt[0]+ve[1]*vt[1])
            qs[j]=max(-lim,min(lim,qs[j]+ang))
    pts=fk(qs)
    err=math.hypot(t[0]-pts[-1][0],t[1]-pts[-1][1])
    return [[round(math.degrees(q),3) for q in qs],used,round(err,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['converges early', [[[2, 1], [0, 0], [2, 1], 10, 0.01, 170]], [[0.0, 90.0], 1, 0.0]], ['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]]], [['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['converges early', [[[2, 1], [0, 0], [2, 1], 10, 0.01, 170]], [[0.0, 90.0], 1, 0.0]], ['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]], ['four links curl', [[[1, 0.8, 0.6, 0.4], [20, 20, 20, 20], [0.3, 1.2], 3, 0.001, 120]], [[36.469, 53.792, 70.335, 118.779], 3, 0.001]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['zero iterations', [[[1, 1], [0, 0], [0, 2], 0, 0.001, 170]], [[0.0, 0.0], 0, 2.8284]]], [['unreachable stretch', [[[1, 1], [30, 30], [4, 0], 3, 0.001, 170]], [[8.551, -17.103], 3, 2.0222]], ['four links curl', [[[1, 0.8, 0.6, 0.4], [20, 20, 20, 20], [0.3, 1.2], 3, 0.001, 120]], [[36.469, 53.792, 70.335, 118.779], 3, 0.001]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['zero iterations', [[[1, 1], [0, 0], [0, 2], 0, 0.001, 170]], [[0.0, 0.0], 0, 2.8284]], ['target crossing pi', [[[1, 1, 1], [170, 0, 0], [-2, -0.5], 2, 0.001, 180]], [[168.982, -6.514, 102.831], 2, 0.0176]], ['loose tolerance', [[[1, 1, 1], [5, 5, 5], [2.9, 0.4], 3, 0.2, 170]], [[5.0, 5.0, 5.0], 0, 0.1285]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]]], [['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['limit blocks tip joint', [[[1, 1], [0, 0], [0, -1.8], 2, 0.001, 30]], [[-30.0, -30.0], 2, 1.4333]], ['target crossing pi', [[[1, 1, 1], [170, 0, 0], [-2, -0.5], 2, 0.001, 180]], [[168.982, -6.514, 102.831], 2, 0.0176]], ['loose tolerance', [[[1, 1, 1], [5, 5, 5], [2.9, 0.4], 3, 0.2, 170]], [[5.0, 5.0, 5.0], 0, 0.1285]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]], ['tight elbow limit', [[[1, 1, 1], [0, 0, 0], [-1, 1], 3, 0.001, 60]], [[60.0, 60.0, 60.0], 3, 0.7321]]], [['three links one sweep', [[[1, 1, 1], [0, 0, 0], [1.5, 1.5], 1, 0.001, 170]], [[5.996, 17.348, 108.435], 1, 0.3586]], ['three links several sweeps', [[[1, 1, 1], [10, 10, 10], [-1, 2], 4, 0.001, 170]], [[63.011, 58.047, 42.064], 4, 0.0426]], ['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['tight limits', [[[1, 1, 1], [0, 0, 0], [0.5, 2.5], 3, 0.001, 45]], [[36.006, 45.0, 37.294], 3, 0.0448]], ['already converged', [[[1, 1], [0, 90], [1, 1], 5, 0.001, 170]], [[0.0, 90.0], 0, 0.0]], ['negative limits start', [[[1.5, 1], [-60, -60], [1, 1.5], 2, 0.001, 90]], [[90.0, -90.0], 1, 0.0]], ['tight elbow limit', [[[1, 1, 1], [0, 0, 0], [-1, 1], 3, 0.001, 60]], [[60.0, 60.0, 60.0], 3, 0.7321]]]]
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
three links one sweep	`[[5.996, 17.348, 108.435], 1, 0.3586]`	`[[5.996, 17.348, 108.435], 1, 0.3586]`	Passed
three links several sweeps	`[[63.011, 58.047, 42.064], 4, 0.0426]`	`[[63.011, 58.047, 42.064], 4, 0.0426]`	Passed
target behind base	`[[-121.779, -101.253], 3, 0.2445]`	`[[-121.779, -101.253], 3, 0.2445]`	Passed
tight limits	`[[36.006, 45.0, 37.294], 3, 0.0448]`	`[[36.006, 45.0, 37.294], 3, 0.0448]`	Passed
already converged	`[[0.0, 90.0], 0, 0.0]`	`[[0.0, 90.0], 0, 0.0]`	Passed
converges early	`[[0.0, 90.0], 1, 0.0]`	`[[0.0, 90.0], 1, 0.0]`	Passed
unreachable stretch	`[[8.551, -17.103], 3, 2.0222]`	`[[8.551, -17.103], 3, 2.0222]`	Passed

SHA-256 / e6b523ea0f79c218c47435f8c2e4e1bdabc4d707d440ebcecd7ff4cfac5d78d3

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 2026-09-29T14:51:03.665508+00:00.

Case digest / 2f5b14d6a3970fc5e41c1243b0860572e6a1a7fc38bde3ba484f80c6878f7aa4