FA-87956 / Inverse kinematics solvers / Open access

CCD reports a sweep for the convergence check that stops it · case 01

Solves that start inside tolerance report one sweep instead of zero, and every count is one high after convergence.

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

ROOT CAUSE

The sweep counter is incremented before the tolerance test.

VERIFIED REPAIR

Increment the counter only after the tolerance test has decided to sweep.

Unsuccessful approach: Assigning the loop index to the counter lags by one after each real sweep.

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)
        used+=1
        ex,ey=pts[-1]
        if math.hypot(t[0]-ex,t[1]-ey)<=tol: break
        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]]], [['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]], ['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]]], [['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['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]], ['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]], ['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]], ['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]]]]
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], 1, 0.0]`	`[[0.0, 90.0], 0, 0.0]`	Failed
converges early	`[[0.0, 90.0], 2, 0.0]`	`[[0.0, 90.0], 1, 0.0]`	Failed
unreachable stretch	`[[8.551, -17.103], 3, 2.0222]`	`[[8.551, -17.103], 3, 2.0222]`	Passed

SHA-256 / f0c3ca00c4f078b7411a80a1187b0f51db5b875a285e2b72886963534e8e32d9

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=it
        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]]], [['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]], ['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]]], [['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['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]], ['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]], ['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]], ['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]]]]
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], 0, 0.3586]`	`[[5.996, 17.348, 108.435], 1, 0.3586]`	Failed
three links several sweeps	`[[63.011, 58.047, 42.064], 3, 0.0426]`	`[[63.011, 58.047, 42.064], 4, 0.0426]`	Failed
target behind base	`[[-121.779, -101.253], 2, 0.2445]`	`[[-121.779, -101.253], 3, 0.2445]`	Failed
tight limits	`[[36.006, 45.0, 37.294], 2, 0.0448]`	`[[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], 0, 0.0]`	`[[0.0, 90.0], 1, 0.0]`	Failed
unreachable stretch	`[[8.551, -17.103], 2, 2.0222]`	`[[8.551, -17.103], 3, 2.0222]`	Failed

SHA-256 / 338f944c8463403a723545ff791a8856f0df90200a1dade9609b58808ff09159

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]]], [['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]], ['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]]], [['target behind base', [[[1, 1], [0, 0], [-1.5, -0.2], 3, 0.001, 175]], [[-121.779, -101.253], 3, 0.2445]], ['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]], ['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]], ['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]], ['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]]]]
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 / 10f1dc6cc59eaa6006994c18f5281cba9dfb5bd74f80d059b6f8cb33f904592d

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 2026-09-29T14:51:03.671321+00:00.

Case digest / cbcaca92c9861684aeaea49c5411dbab3ddef8bc7b3bdf733c22fd7ca29e81ed