FA-88246 / Inverse kinematics solvers / Open access

Trajectory jump test is applied to the first solution rather than the chosen one · case 01

Waypoints are rejected as jumps even though the other branch is close.

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

ROOT CAUSE

The jump check measures the first computed branch instead of the best one, or treats the limit as exclusive.

VERIFIED REPAIR

Reject only when the chosen solution changes by more than maxjump.

Unsuccessful approach: Rejecting a change equal to maxjump contradicts the inclusive limit.

Case contract

Input [l1,l2,q0,targets,maxjump]: planar 2R arm with continuous joints following a list of targets from joint state q0 (degrees). For each target compute the elbow +/- solutions; unreachable targets append "hold". Choose the solution with the smallest max wrapped joint change from the previous state; if that change exceeds maxjump append "jump" and keep the state. Otherwise unwrap the solution to lie within 180 of the previous angles, append it (3 decimals) and count a flip whenever the elbow sign differs from the last accepted one. Return [outputs, flips].

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,q0,targets,maxjump=x
    prev=list(q0)
    out=[]
    flips=0
    branch=None
    for px,py in targets:
        c=(px*px+py*py-l1*l1-l2*l2)/(2*l1*l2)
        if abs(c)>1.0:
            out.append('hold')
            continue
        sols=[]
        for sgn in (1,-1):
            s=sgn*math.sqrt(1-c*c)
            t2=math.degrees(math.atan2(s,c))
            t1=math.degrees(math.atan2(py,px)-math.atan2(l2*s,l1+l2*c))
            sols.append([sgn,t1,t2])
        def dist(sol):
            return max(abs((sol[1]-prev[0]+180.0)%360.0-180.0),abs((sol[2]-prev[1]+180.0)%360.0-180.0))
        best=min(sols,key=dist)
        if dist(sols[0])>maxjump:
            out.append('jump')
            continue
        if branch is not None and best[0]!=branch: flips+=1
        branch=best[0]
        t1=prev[0]+((best[1]-prev[0]+180.0)%360.0-180.0)
        t2=prev[1]+((best[2]-prev[1]+180.0)%360.0-180.0)
        prev=[t1,t2]
        out.append([round(t1,3),round(t2,3)])
    return [out,flips]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['smooth arc', [[1, 1, [0, 90], [[1, 1], [0.9, 1.1], [0.8, 1.2]], 30]], [[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]], ['seam crossing', [[1, 1, [170, 20], [[-1.5, 0.3], [-1.5, -0.3], [-1.4, -0.6]], 40]], [['jump', 'jump', 'jump'], 0]], ['multi-turn start', [[1, 1, [360, 90], [[1, 1], [1.1, 0.9]], 30]], [[[360.0, 90.0], [354.576, 89.427]], 0]], ['dropout in middle', [[1, 1, [0, 90], [[1, 1], [3, 0], [0.9, 1.1]], 30]], [[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]], ['dropout then branch change', [[1, 1, [0, 60], [[1.5, 0.8], [3, 3], [1.6, -0.6]], 200]], [[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]], ['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]]], [['dropout in middle', [[1, 1, [0, 90], [[1, 1], [3, 0], [0.9, 1.1]], 30]], [[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]], ['dropout then branch change', [[1, 1, [0, 60], [[1.5, 0.8], [3, 3], [1.6, -0.6]], 200]], [[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]], ['large jump rejected', [[1, 1, [0, 90], [[1, 1], [-1, -1], [1, 0.9]], 20]], [[[0.0, 90.0], 'jump', [-5.738, 95.451]], 0]], ['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['flip through straight', [[1, 1, [0, 30], [[1.9, 0.3], [2, 0], [1.9, -0.3], [1.8, -0.6]], 60]], [[[-6.922, 31.788], [0.0, 0.0], [-24.867, 31.788], [-36.87, 36.87]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['flip through straight', [[1, 1, [0, 30], [[1.9, 0.3], [2, 0], [1.9, -0.3], [1.8, -0.6]], 60]], [[[-6.922, 31.788], [0.0, 0.0], [-24.867, 31.788], [-36.87, 36.87]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['big jump budget', [[1, 1.5, [0, 0], [[-1, 1.5], [1.5, -1], [0.5, 2]], 360]], [[[67.38, 90.0], [-90.0, 90.0], [32.65, 70.529]], 0]], ['negative seam elbow', [[1, 1, [-170, -20], [[-1.5, -0.3], [-1.5, 0.3], [-1.3, 0.8]], 45]], [['jump', 'jump', 'jump'], 0]], ['all unreachable', [[1, 1, [0, 0], [[3, 3], [0, 5]], 30]], [['hold', 'hold'], 0]], ['dropout near base', [[1, 1, [146, 142], [[-0.1, -1.0], [3, 3], [-0.3, 0.2]], 120]], [[[204.455, 119.67], 'hold', [225.924, 200.772]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['all unreachable', [[1, 1, [0, 0], [[3, 3], [0, 5]], 30]], [['hold', 'hold'], 0]], ['jump then recover', [[1, 1, [0, 120], [[0.2, 1.0], [1.8, 0.2], [0.3, 1.0]], 60]], [[[19.347, 118.685], 'jump', [14.768, 117.065]], 0]], ['second target flips', [[1.5, 1, [0, 45], [[2, 1.2], [2.2, -0.4]], 150]], [[[13.925, 43.114], [-31.604, 54.315]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]], ['down to up flip', [[1, 1, [69, -23], [[0.3, -0.8], [1.9, -0.9], [0.1, 0.9]], 164]], [[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['second target flips', [[1.5, 1, [0, 45], [[2, 1.2], [2.2, -0.4]], 150]], [[[13.925, 43.114], [-31.604, 54.315]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]], ['down to up flip', [[1, 1, [69, -23], [[0.3, -0.8], [1.9, -0.9], [0.1, 0.9]], 164]], [[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]], ['wide sweep flips', [[1, 1, [-5, -54], [[0.1, -1.7], [0.7, 1.5], [-1.2, 0.5]], 154]], [[[-55.005, -63.256], [30.841, 68.284], [107.922, 98.917]], 1]], ['rejected then flip', [[1, 1, [171, 101], [[0.8, -1.3], [0.8, 0.7], [0.2, -1.1]], 56]], [['jump', 'jump', [224.293, 112.024]], 0]], ['rejection mid sweep', [[1, 1, [37, 154], [[0.4, 1.3], [-0.5, -0.6], [-0.8, 1.4]], 82]], [[[25.747, 94.301], 'jump', [83.474, 72.542]], 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
smooth arc	`[[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]`	`[[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]`	Passed
seam crossing	`[['jump', 'jump', 'jump'], 0]`	`[['jump', 'jump', 'jump'], 0]`	Passed
multi-turn start	`[[[360.0, 90.0], [354.576, 89.427]], 0]`	`[[[360.0, 90.0], [354.576, 89.427]], 0]`	Passed
dropout in middle	`[[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]`	`[[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]`	Passed
dropout then branch change	`[[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]`	`[[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]`	Passed
jump exactly at limit	`[[[0.0, 0.0], [0.0, 90.0]], 0]`	`[[[0.0, 0.0], [0.0, 90.0]], 0]`	Passed
start near down branch	`[['jump', 'jump'], 0]`	`[[[78.451, -77.291], [72.456, -75.522]], 0]`	Failed

SHA-256 / 354169ece4c20ef711b4c6e70ba3821e3360b5a8cefa92effc185fe1a2ccd0e5

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,q0,targets,maxjump=x
    prev=list(q0)
    out=[]
    flips=0
    branch=None
    for px,py in targets:
        c=(px*px+py*py-l1*l1-l2*l2)/(2*l1*l2)
        if abs(c)>1.0:
            out.append('hold')
            continue
        sols=[]
        for sgn in (1,-1):
            s=sgn*math.sqrt(1-c*c)
            t2=math.degrees(math.atan2(s,c))
            t1=math.degrees(math.atan2(py,px)-math.atan2(l2*s,l1+l2*c))
            sols.append([sgn,t1,t2])
        def dist(sol):
            return max(abs((sol[1]-prev[0]+180.0)%360.0-180.0),abs((sol[2]-prev[1]+180.0)%360.0-180.0))
        best=min(sols,key=dist)
        if dist(best)>=maxjump:
            out.append('jump')
            continue
        if branch is not None and best[0]!=branch: flips+=1
        branch=best[0]
        t1=prev[0]+((best[1]-prev[0]+180.0)%360.0-180.0)
        t2=prev[1]+((best[2]-prev[1]+180.0)%360.0-180.0)
        prev=[t1,t2]
        out.append([round(t1,3),round(t2,3)])
    return [out,flips]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['smooth arc', [[1, 1, [0, 90], [[1, 1], [0.9, 1.1], [0.8, 1.2]], 30]], [[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]], ['seam crossing', [[1, 1, [170, 20], [[-1.5, 0.3], [-1.5, -0.3], [-1.4, -0.6]], 40]], [['jump', 'jump', 'jump'], 0]], ['multi-turn start', [[1, 1, [360, 90], [[1, 1], [1.1, 0.9]], 30]], [[[360.0, 90.0], [354.576, 89.427]], 0]], ['dropout in middle', [[1, 1, [0, 90], [[1, 1], [3, 0], [0.9, 1.1]], 30]], [[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]], ['dropout then branch change', [[1, 1, [0, 60], [[1.5, 0.8], [3, 3], [1.6, -0.6]], 200]], [[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]], ['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]]], [['dropout in middle', [[1, 1, [0, 90], [[1, 1], [3, 0], [0.9, 1.1]], 30]], [[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]], ['dropout then branch change', [[1, 1, [0, 60], [[1.5, 0.8], [3, 3], [1.6, -0.6]], 200]], [[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]], ['large jump rejected', [[1, 1, [0, 90], [[1, 1], [-1, -1], [1, 0.9]], 20]], [[[0.0, 90.0], 'jump', [-5.738, 95.451]], 0]], ['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['flip through straight', [[1, 1, [0, 30], [[1.9, 0.3], [2, 0], [1.9, -0.3], [1.8, -0.6]], 60]], [[[-6.922, 31.788], [0.0, 0.0], [-24.867, 31.788], [-36.87, 36.87]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['flip through straight', [[1, 1, [0, 30], [[1.9, 0.3], [2, 0], [1.9, -0.3], [1.8, -0.6]], 60]], [[[-6.922, 31.788], [0.0, 0.0], [-24.867, 31.788], [-36.87, 36.87]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['big jump budget', [[1, 1.5, [0, 0], [[-1, 1.5], [1.5, -1], [0.5, 2]], 360]], [[[67.38, 90.0], [-90.0, 90.0], [32.65, 70.529]], 0]], ['negative seam elbow', [[1, 1, [-170, -20], [[-1.5, -0.3], [-1.5, 0.3], [-1.3, 0.8]], 45]], [['jump', 'jump', 'jump'], 0]], ['all unreachable', [[1, 1, [0, 0], [[3, 3], [0, 5]], 30]], [['hold', 'hold'], 0]], ['dropout near base', [[1, 1, [146, 142], [[-0.1, -1.0], [3, 3], [-0.3, 0.2]], 120]], [[[204.455, 119.67], 'hold', [225.924, 200.772]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['all unreachable', [[1, 1, [0, 0], [[3, 3], [0, 5]], 30]], [['hold', 'hold'], 0]], ['jump then recover', [[1, 1, [0, 120], [[0.2, 1.0], [1.8, 0.2], [0.3, 1.0]], 60]], [[[19.347, 118.685], 'jump', [14.768, 117.065]], 0]], ['second target flips', [[1.5, 1, [0, 45], [[2, 1.2], [2.2, -0.4]], 150]], [[[13.925, 43.114], [-31.604, 54.315]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]], ['down to up flip', [[1, 1, [69, -23], [[0.3, -0.8], [1.9, -0.9], [0.1, 0.9]], 164]], [[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['second target flips', [[1.5, 1, [0, 45], [[2, 1.2], [2.2, -0.4]], 150]], [[[13.925, 43.114], [-31.604, 54.315]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]], ['down to up flip', [[1, 1, [69, -23], [[0.3, -0.8], [1.9, -0.9], [0.1, 0.9]], 164]], [[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]], ['wide sweep flips', [[1, 1, [-5, -54], [[0.1, -1.7], [0.7, 1.5], [-1.2, 0.5]], 154]], [[[-55.005, -63.256], [30.841, 68.284], [107.922, 98.917]], 1]], ['rejected then flip', [[1, 1, [171, 101], [[0.8, -1.3], [0.8, 0.7], [0.2, -1.1]], 56]], [['jump', 'jump', [224.293, 112.024]], 0]], ['rejection mid sweep', [[1, 1, [37, 154], [[0.4, 1.3], [-0.5, -0.6], [-0.8, 1.4]], 82]], [[[25.747, 94.301], 'jump', [83.474, 72.542]], 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
smooth arc	`[[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]`	`[[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]`	Passed
seam crossing	`[['jump', 'jump', 'jump'], 0]`	`[['jump', 'jump', 'jump'], 0]`	Passed
multi-turn start	`[[[360.0, 90.0], [354.576, 89.427]], 0]`	`[[[360.0, 90.0], [354.576, 89.427]], 0]`	Passed
dropout in middle	`[[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]`	`[[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]`	Passed
dropout then branch change	`[[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]`	`[[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]`	Passed
jump exactly at limit	`[[[0.0, 0.0], 'jump'], 0]`	`[[[0.0, 0.0], [0.0, 90.0]], 0]`	Failed
start near down branch	`[[[78.451, -77.291], [72.456, -75.522]], 0]`	`[[[78.451, -77.291], [72.456, -75.522]], 0]`	Passed

SHA-256 / b7448e91cda935d023f8a5d88c3911059cc3c0ca7f65bbb458a575f0177cade0

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,q0,targets,maxjump=x
    prev=list(q0)
    out=[]
    flips=0
    branch=None
    for px,py in targets:
        c=(px*px+py*py-l1*l1-l2*l2)/(2*l1*l2)
        if abs(c)>1.0:
            out.append('hold')
            continue
        sols=[]
        for sgn in (1,-1):
            s=sgn*math.sqrt(1-c*c)
            t2=math.degrees(math.atan2(s,c))
            t1=math.degrees(math.atan2(py,px)-math.atan2(l2*s,l1+l2*c))
            sols.append([sgn,t1,t2])
        def dist(sol):
            return max(abs((sol[1]-prev[0]+180.0)%360.0-180.0),abs((sol[2]-prev[1]+180.0)%360.0-180.0))
        best=min(sols,key=dist)
        if dist(best)>maxjump:
            out.append('jump')
            continue
        if branch is not None and best[0]!=branch: flips+=1
        branch=best[0]
        t1=prev[0]+((best[1]-prev[0]+180.0)%360.0-180.0)
        t2=prev[1]+((best[2]-prev[1]+180.0)%360.0-180.0)
        prev=[t1,t2]
        out.append([round(t1,3),round(t2,3)])
    return [out,flips]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['smooth arc', [[1, 1, [0, 90], [[1, 1], [0.9, 1.1], [0.8, 1.2]], 30]], [[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]], ['seam crossing', [[1, 1, [170, 20], [[-1.5, 0.3], [-1.5, -0.3], [-1.4, -0.6]], 40]], [['jump', 'jump', 'jump'], 0]], ['multi-turn start', [[1, 1, [360, 90], [[1, 1], [1.1, 0.9]], 30]], [[[360.0, 90.0], [354.576, 89.427]], 0]], ['dropout in middle', [[1, 1, [0, 90], [[1, 1], [3, 0], [0.9, 1.1]], 30]], [[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]], ['dropout then branch change', [[1, 1, [0, 60], [[1.5, 0.8], [3, 3], [1.6, -0.6]], 200]], [[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]], ['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]]], [['dropout in middle', [[1, 1, [0, 90], [[1, 1], [3, 0], [0.9, 1.1]], 30]], [[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]], ['dropout then branch change', [[1, 1, [0, 60], [[1.5, 0.8], [3, 3], [1.6, -0.6]], 200]], [[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]], ['large jump rejected', [[1, 1, [0, 90], [[1, 1], [-1, -1], [1, 0.9]], 20]], [[[0.0, 90.0], 'jump', [-5.738, 95.451]], 0]], ['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['flip through straight', [[1, 1, [0, 30], [[1.9, 0.3], [2, 0], [1.9, -0.3], [1.8, -0.6]], 60]], [[[-6.922, 31.788], [0.0, 0.0], [-24.867, 31.788], [-36.87, 36.87]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['flip through straight', [[1, 1, [0, 30], [[1.9, 0.3], [2, 0], [1.9, -0.3], [1.8, -0.6]], 60]], [[[-6.922, 31.788], [0.0, 0.0], [-24.867, 31.788], [-36.87, 36.87]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['big jump budget', [[1, 1.5, [0, 0], [[-1, 1.5], [1.5, -1], [0.5, 2]], 360]], [[[67.38, 90.0], [-90.0, 90.0], [32.65, 70.529]], 0]], ['negative seam elbow', [[1, 1, [-170, -20], [[-1.5, -0.3], [-1.5, 0.3], [-1.3, 0.8]], 45]], [['jump', 'jump', 'jump'], 0]], ['all unreachable', [[1, 1, [0, 0], [[3, 3], [0, 5]], 30]], [['hold', 'hold'], 0]], ['dropout near base', [[1, 1, [146, 142], [[-0.1, -1.0], [3, 3], [-0.3, 0.2]], 120]], [[[204.455, 119.67], 'hold', [225.924, 200.772]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['start near down branch', [[1, 1, [60, -60], [[1.2, 1.0], [1.3, 0.9]], 50]], [[[78.451, -77.291], [72.456, -75.522]], 0]], ['all unreachable', [[1, 1, [0, 0], [[3, 3], [0, 5]], 30]], [['hold', 'hold'], 0]], ['jump then recover', [[1, 1, [0, 120], [[0.2, 1.0], [1.8, 0.2], [0.3, 1.0]], 60]], [[[19.347, 118.685], 'jump', [14.768, 117.065]], 0]], ['second target flips', [[1.5, 1, [0, 45], [[2, 1.2], [2.2, -0.4]], 150]], [[[13.925, 43.114], [-31.604, 54.315]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]], ['down to up flip', [[1, 1, [69, -23], [[0.3, -0.8], [1.9, -0.9], [0.1, 0.9]], 164]], [[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]]], [['jump exactly at limit', [[2, 1, [0, 0], [[3, 0], [2, 1]], 90]], [[[0.0, 0.0], [0.0, 90.0]], 0]], ['second target flips', [[1.5, 1, [0, 45], [[2, 1.2], [2.2, -0.4]], 150]], [[[13.925, 43.114], [-31.604, 54.315]], 0]], ['dropout across branch change', [[1, 1, [-93, 83], [[1.9, -0.3], [3, 3], [-1.5, -1.3]], 120]], [[[-24.867, 31.788], 'hold', [-132.051, -14.07]], 1]], ['down to up flip', [[1, 1, [69, -23], [[0.3, -0.8], [1.9, -0.9], [0.1, 0.9]], 164]], [[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]], ['wide sweep flips', [[1, 1, [-5, -54], [[0.1, -1.7], [0.7, 1.5], [-1.2, 0.5]], 154]], [[[-55.005, -63.256], [30.841, 68.284], [107.922, 98.917]], 1]], ['rejected then flip', [[1, 1, [171, 101], [[0.8, -1.3], [0.8, 0.7], [0.2, -1.1]], 56]], [['jump', 'jump', [224.293, 112.024]], 0]], ['rejection mid sweep', [[1, 1, [37, 154], [[0.4, 1.3], [-0.5, -0.6], [-0.8, 1.4]], 82]], [[[25.747, 94.301], 'jump', [83.474, 72.542]], 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
smooth arc	`[[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]`	`[[[0.0, 90.0], [5.997, 89.427], [12.456, 87.708]], 0]`	Passed
seam crossing	`[['jump', 'jump', 'jump'], 0]`	`[['jump', 'jump', 'jump'], 0]`	Passed
multi-turn start	`[[[360.0, 90.0], [354.576, 89.427]], 0]`	`[[[360.0, 90.0], [354.576, 89.427]], 0]`	Passed
dropout in middle	`[[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]`	`[[[0.0, 90.0], 'hold', [5.997, 89.427]], 0]`	Passed
dropout then branch change	`[[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]`	`[[[-3.716, 63.577], 'hold', [-51.862, 62.613]], 0]`	Passed
jump exactly at limit	`[[[0.0, 0.0], [0.0, 90.0]], 0]`	`[[[0.0, 0.0], [0.0, 90.0]], 0]`	Passed
start near down branch	`[[[78.451, -77.291], [72.456, -75.522]], 0]`	`[[[78.451, -77.291], [72.456, -75.522]], 0]`	Passed

SHA-256 / 67ac318ed956d3f9403512bc6555804ed0d28156fcdee7dd4855183fd7e496ea

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 2026-09-29T14:51:06.370330+00:00.

Case digest / 815ae8290c60cfd3287dcfce19e83aa47ebcaee761b8fa0c607ca444ddd3edbf