{"abstract":"Joint velocities are computed for the wrong lever arms and the task error is not reduced.","category":"Inverse kinematics solvers","checks":7,"contract":"Input [lengths(3), q(3 deg), target, qc(3 deg), k]: planar 3R arm, position task only. One step dq = J+ e + (I - J+ J) z, J+ = J^T (J J^T)^-1, e = target - fk(q), z = k*(qc-q) in radians. Return the updated q in degrees rounded to 4, or \"singular\" when |det(J J^T)|<1e-12.","contract_signature":"x","evaluation_group":"w2-inverse_kinematics_solvers-null-space-redundancy","failed_approach":"Correcting only the x row leaves the y row summing the wrong links.","family":"w2-inverse_kinematics_solvers-null-space-redundancy-jacobian-column-span","id":"FA-88176","implementations":{"attempt":{"sha256":"dc1c519f66074225d10b1c96e72805a92d2b4f691727b345c7f9e548e8ab68ab","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x):\n    ls,q,t,qc,k=x\n    qr=[math.radians(v) for v in q]\n    cum=[sum(qr[:i+1]) for i in range(3)]\n    px=sum(ls[i]*math.cos(cum[i]) for i in range(3))\n    py=sum(ls[i]*math.sin(cum[i]) for i in range(3))\n    J=[[-sum(ls[j]*math.sin(cum[j]) for j in range(i,3)) for i in range(3)],\n       [sum(ls[j]*math.cos(cum[j]) for j in range(i+1)) for i in range(3)]]\n    a=sum(v*v for v in J[0])\n    b=sum(J[0][i]*J[1][i] for i in range(3))\n    c=sum(v*v for v in J[1])\n    det=a*c-b*b\n    if abs(det)<1e-12: return 'singular'\n    inv=[[c/det,-b/det],[-b/det,a/det]]\n    P=[[J[0][i]*inv[0][0]+J[1][i]*inv[1][0],J[0][i]*inv[0][1]+J[1][i]*inv[1][1]] for i in range(3)]\n    e=[t[0]-px,t[1]-py]\n    z=[k*math.radians(qc[i]-q[i]) for i in range(3)]\n    N=[[(1.0 if i==j else 0.0)-sum(P[i][r]*J[r][j] for r in range(2)) for j in range(3)] for i in range(3)]\n    dq=[P[i][0]*e[0]+P[i][1]*e[1]+sum(N[i][j]*z[j] for j in range(3)) for i in range(3)]\n    return [round(q[i]+math.degrees(dq[i]),4) for i in range(3)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"8c118f7fa339ab26d49db0e9f0c5ab71372f0cb4e7265218813f203bdacf283d","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x):\n    ls,q,t,qc,k=x\n    qr=[math.radians(v) for v in q]\n    cum=[sum(qr[:i+1]) for i in range(3)]\n    px=sum(ls[i]*math.cos(cum[i]) for i in range(3))\n    py=sum(ls[i]*math.sin(cum[i]) for i in range(3))\n    J=[[-sum(ls[j]*math.sin(cum[j]) for j in range(i+1)) for i in range(3)],\n       [sum(ls[j]*math.cos(cum[j]) for j in range(i+1)) for i in range(3)]]\n    a=sum(v*v for v in J[0])\n    b=sum(J[0][i]*J[1][i] for i in range(3))\n    c=sum(v*v for v in J[1])\n    det=a*c-b*b\n    if abs(det)<1e-12: return 'singular'\n    inv=[[c/det,-b/det],[-b/det,a/det]]\n    P=[[J[0][i]*inv[0][0]+J[1][i]*inv[1][0],J[0][i]*inv[0][1]+J[1][i]*inv[1][1]] for i in range(3)]\n    e=[t[0]-px,t[1]-py]\n    z=[k*math.radians(qc[i]-q[i]) for i in range(3)]\n    N=[[(1.0 if i==j else 0.0)-sum(P[i][r]*J[r][j] for r in range(2)) for j in range(3)] for i in range(3)]\n    dq=[P[i][0]*e[0]+P[i][1]*e[1]+sum(N[i][j]*z[j] for j in range(3)) for i in range(3)]\n    return [round(q[i]+math.degrees(dq[i]),4) for i in range(3)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]], [['task only', [[[1, 1, 1], [20, 30, 40], [1.5, 1.5], [20, 30, 40], 0.0]], [-9.6761, 48.859, 73.9791]], ['centering only', [[[1, 1, 1], [20, 30, 40], [1.0911, 2.3209], [0, 0, 0], 0.5]], [23.38, 40.6501, 42.2202]], ['task and centering', [[[1, 1, 1], [10, 40, -20], [2, 1], [0, 30, 30], 0.3]], [-62.5399, 148.4425, -29.151]], ['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]]], [['straight arm singular', [[[1, 1, 1], [0, 0, 0], [2, 1], [0, 0, 0], 0.5]], 'singular'], ['unequal links', [[[2, 1, 0.5], [30, -45, 60], [1.5, 1], [0, 0, 0], 0.2]], [87.5837, -214.0176, 178.4237]], ['negative config', [[[1, 1.5, 0.7], [-30, -20, -60], [1, -2], [0, 10, -10], 1.0]], [-7.4068, -55.9156, -92.0392]], ['strong centering', [[[1, 1, 1], [60, 60, 60], [-1, 1.5], [0, 0, 0], 2.0]], [43.7121, 92.5758, 40.7197]], ['behind base', [[[1, 1, 1], [120, 30, 20], [-2, 0.5], [90, 0, 0], 0.4]], [75.3463, 86.5441, 80.8124]], ['small error', [[[1.2, 0.8, 0.5], [45, -30, 15], [1.6, 1.1], [45, -30, 15], 0.0]], [102.7802, -138.4201, 15.5818]], ['asymmetric centre', [[[1, 1, 1], [5, 80, -40], [1.2, 1.9], [10, 20, 30], 0.8]], [-22.1921, 110.3115, 3.6813]]]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"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.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-inverse_kinematics_solvers-null-space-redundancy-jacobian-column-span","generated_at":"2026-09-29T14:51:05.865778+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"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.","root_cause":"Column i sums links 0..i instead of the links from joint i to the tip.","sha256":"7ce6e13ad0dae9d001a6d056d95fd5dea75f5e4429a22a50319d1a08f9d95fd5","title":"Redundant arm Jacobian sums links proximal to each joint · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":43.62,"exit_code":1,"observations":[{"actual":[41.6611,22.0906,13.031],"check":"task only","expected":[-9.6761,48.859,73.9791],"passed":false},{"actual":[30.4129,36.1096,35.4131],"check":"centering only","expected":[23.38,40.6501,42.2202],"passed":false},{"actual":[33.7449,44.0835,-42.1457],"check":"task and centering","expected":[-62.5399,148.4425,-29.151],"passed":false},{"actual":"singular","check":"straight arm singular","expected":"singular","passed":true},{"actual":[122.6636,-75.6553,32.7289],"check":"unequal links","expected":[87.5837,-214.0176,178.4237],"passed":false},{"actual":[-45.4932,-30.4063,-28.5412],"check":"negative config","expected":[-7.4068,-55.9156,-92.0392],"passed":false},{"actual":[70.1623,39.6753,78.3767],"check":"strong centering","expected":[43.7121,92.5758,40.7197],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"task only\", \"actual\": [41.6611, 22.0906, 13.031], \"expected\": [-9.6761, 48.859, 73.9791], \"passed\": false}, {\"check\": \"centering only\", \"actual\": [30.4129, 36.1096, 35.4131], \"expected\": [23.38, 40.6501, 42.2202], \"passed\": false}, {\"check\": \"task and centering\", \"actual\": [33.7449, 44.0835, -42.1457], \"expected\": [-62.5399, 148.4425, -29.151], \"passed\": false}, {\"check\": \"straight arm singular\", \"actual\": \"singular\", \"expected\": \"singular\", \"passed\": true}, {\"check\": \"unequal links\", \"actual\": [122.6636, -75.6553, 32.7289], \"expected\": [87.5837, -214.0176, 178.4237], \"passed\": false}, {\"check\": \"negative config\", \"actual\": [-45.4932, -30.4063, -28.5412], \"expected\": [-7.4068, -55.9156, -92.0392], \"passed\": false}, {\"check\": \"strong centering\", \"actual\": [70.1623, 39.6753, 78.3767], \"expected\": [43.7121, 92.5758, 40.7197], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":41.205,"exit_code":1,"observations":[{"actual":[-8.9095,5.1645,59.9864],"check":"task only","expected":[-9.6761,48.859,73.9791],"passed":false},{"actual":[9.777,27.3913,56.3852],"check":"centering only","expected":[23.38,40.6501,42.2202],"passed":false},{"actual":[-98.4425,46.0089,8.8025],"check":"task and centering","expected":[-62.5399,148.4425,-29.151],"passed":false},{"actual":"singular","check":"straight arm singular","expected":"singular","passed":true},{"actual":[208.632,-147.3671,47.3363],"check":"unequal links","expected":[87.5837,-214.0176,178.4237],"passed":false},{"actual":[21.629,0.221,-100.3191],"check":"negative config","expected":[-7.4068,-55.9156,-92.0392],"passed":false},{"actual":[24.4696,82.2348,55.5304],"check":"strong centering","expected":[43.7121,92.5758,40.7197],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"task only\", \"actual\": [-8.9095, 5.1645, 59.9864], \"expected\": [-9.6761, 48.859, 73.9791], \"passed\": false}, {\"check\": \"centering only\", \"actual\": [9.777, 27.3913, 56.3852], \"expected\": [23.38, 40.6501, 42.2202], \"passed\": false}, {\"check\": \"task and centering\", \"actual\": [-98.4425, 46.0089, 8.8025], \"expected\": [-62.5399, 148.4425, -29.151], \"passed\": false}, {\"check\": \"straight arm singular\", \"actual\": \"singular\", \"expected\": \"singular\", \"passed\": true}, {\"check\": \"unequal links\", \"actual\": [208.632, -147.3671, 47.3363], \"expected\": [87.5837, -214.0176, 178.4237], \"passed\": false}, {\"check\": \"negative config\", \"actual\": [21.629, 0.221, -100.3191], \"expected\": [-7.4068, -55.9156, -92.0392], \"passed\": false}, {\"check\": \"strong centering\", \"actual\": [24.4696, 82.2348, 55.5304], \"expected\": [43.7121, 92.5758, 40.7197], \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}