FA-88241 / Inverse kinematics solvers / Open access

Trajectory branch distance ignores joint wrap-around · case 01

Near the +/-180 seam the solver picks the far branch or reports a spurious jump.

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

ROOT CAUSE

Joint changes are measured as plain differences for continuous joints.

THE FAILURE

Joint changes are measured as plain differences for continuous joints.

Unsuccessful approach: Wrapping only the shoulder difference still misjudges elbow changes across the seam.

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]),abs(sol[2]-prev[1]))
        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]], ['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]], ['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]]], [['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]], ['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]], ['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]], ['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]], ['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]], ['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]]], [['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]], ['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]], ['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]]], [['multi-turn start', [[1, 1, [360, 90], [[1, 1], [1.1, 0.9]], 30]], [[[360.0, 90.0], [354.576, 89.427]], 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	`[['jump', 'jump'], 0]`	`[[[360.0, 90.0], [354.576, 89.427]], 0]`	Failed
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
large jump rejected	`[[[0.0, 90.0], 'jump', [-5.738, 95.451]], 0]`	`[[[0.0, 90.0], 'jump', [-5.738, 95.451]], 0]`	Passed
down to up flip	`[[[-4.734, -129.42], 'hold', [146.738, -126.157]], 0]`	`[[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]`	Failed

SHA-256 / db7543b2f5be7fe2daea714fa9c228fc8715f175cd6d85495ca989be76310e26

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]))
        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]], ['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]], ['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]]], [['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]], ['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]], ['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]], ['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]], ['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]], ['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]]], [['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]], ['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]], ['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]]], [['multi-turn start', [[1, 1, [360, 90], [[1, 1], [1.1, 0.9]], 30]], [[[360.0, 90.0], [354.576, 89.427]], 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
large jump rejected	`[[[0.0, 90.0], 'jump', [-5.738, 95.451]], 0]`	`[[[0.0, 90.0], 'jump', [-5.738, 95.451]], 0]`	Passed
down to up flip	`[[[-4.734, -129.42], 'hold', [146.738, -126.157]], 0]`	`[[[-4.734, -129.42], 'hold', [20.581, -233.843]], 1]`	Failed

SHA-256 / 69c5ff4d22ffa2d599e72384af8fc5c505cdffc163f040f4a06f5effb5d47b12

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:51:06.467877+00:00.

Case digest / e4ef41fab1c93c2c713259e302181eaefc1982bc0b75c7eab5a34752e315d77b