{"abstract":"A target exactly at l1+l2 is reported unreachable although the straight arm touches it.","category":"Inverse kinematics solvers","checks":7,"contract":"Input [l1,l2,px,py,elbow] with positive link lengths. Return [q1,q2] in degrees (4 decimals) for a planar two-link arm whose end effector reaches (px,py); elbow \"up\" gives q2>=0, \"down\" gives q2<=0; q1 is wrapped to [-180,180). Points outside the annulus |l1-l2|<=r<=l1+l2 (1e-9 slack on r squared) return \"unreachable\".","contract_signature":"x","evaluation_group":"w2-inverse_kinematics_solvers-two-link-analytic","failed_approach":"Comparing r squared against the unsquared reach mixes units and admits targets beyond the arm.","family":"w2-inverse_kinematics_solvers-two-link-analytic-outer-reach-boundary","id":"FA-87841","implementations":{"attempt":{"sha256":"121dd2be5a3bcd0c4e38964d3f95c7151c95dd0a836bcbe9968575954f00805f","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x):\n    l1,l2,px,py,elbow=x\n    r2=px*px+py*py\n    reach=l1+l2\n    if r2>reach+1e-9: return 'unreachable'\n    if r2<(l1-l2)**2-1e-9: return 'unreachable'\n    c=(r2-l1*l1-l2*l2)/(2*l1*l2)\n    c=max(-1.0,min(1.0,c))\n    s=math.sqrt(1-c*c)\n    if elbow=='down': s=-s\n    q2=math.atan2(s,c)\n    q1=math.atan2(py,px)-math.atan2(l2*s,l1+l2*c)\n    d1=math.degrees(q1)\n    d1=(d1+180.0)%360.0-180.0\n    return [round(d1,4),round(math.degrees(q2),4)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['general elbow up', [[2, 1, 2, 1, 'up']], [0.0, 90.0]], ['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable']], [['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable'], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]]], [['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]], ['wrap past minus 180', [[1, 2, -2.2, -0.3, 'up']], [123.5259, 91.0027]], ['upper left down', [[3, 2, -1, 4, 'down']], [131.2512, -70.5288]], ['near full reach', [[5, 3, 7.9, 0.5, 'down']], [10.0523, -17.1893]]]]\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":"c8bc4c1d2b62b916a3a8653a499dc14b1d3207fb22708fcdfb8448f2356b3d42","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x):\n    l1,l2,px,py,elbow=x\n    r2=px*px+py*py\n    reach=l1+l2\n    if r2>=reach*reach: return 'unreachable'\n    if r2<(l1-l2)**2-1e-9: return 'unreachable'\n    c=(r2-l1*l1-l2*l2)/(2*l1*l2)\n    c=max(-1.0,min(1.0,c))\n    s=math.sqrt(1-c*c)\n    if elbow=='down': s=-s\n    q2=math.atan2(s,c)\n    q1=math.atan2(py,px)-math.atan2(l2*s,l1+l2*c)\n    d1=math.degrees(q1)\n    d1=(d1+180.0)%360.0-180.0\n    return [round(d1,4),round(math.degrees(q2),4)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['general elbow up', [[2, 1, 2, 1, 'up']], [0.0, 90.0]], ['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['outside workspace', [[2, 1, 3, 1, 'up']], 'unreachable'], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable']], [['general elbow down', [[2, 1, 2, 1, 'down']], [53.1301, -90.0]], ['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['between reach and reach squared', [[1, 1, 1.9, 1.2, 'up']], 'unreachable'], ['inside inner hole', [[3, 1, 1, 0, 'up']], 'unreachable'], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['inner boundary folded', [[3, 1, 2, 0, 'down']], [0.0, -180.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['second quadrant target', [[2, 1.5, -2, 1, 'down']], [-165.5619, -102.0247]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]]], [['fully extended diagonal', [[3, 4, 4.2, 5.6, 'up']], [53.1301, 0.0]], ['negative y axis', [[1.5, 1, 0, -2, 'up']], [-118.955, 75.5225]], ['behind base elbow down', [[2, 1, -2.5, -0.5, 'down']], [-147.3681, -67.9757]], ['equal links near base', [[1, 1, 0.5, 0.2, 'up']], [-52.5785, 148.7597]], ['long forearm hole', [[1, 3, 0.5, 1, 'down']], 'unreachable'], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]]], [['fully extended on x axis', [[2, 1, 3, 0, 'up']], [0.0, 0.0]], ['third quadrant target', [[2, 2, -2, -1, 'up']], [150.5529, 112.0243]], ['long forearm reachable', [[1, 3, 2.5, 0.5, 'up']], [-95.7981, 125.6853]], ['positive y axis', [[2, 2, 0, 3, 'down']], [131.4096, -82.8192]], ['wrap past minus 180', [[1, 2, -2.2, -0.3, 'up']], [123.5259, 91.0027]], ['upper left down', [[3, 2, -1, 4, 'down']], [131.2512, -70.5288]], ['near full reach', [[5, 3, 7.9, 0.5, 'down']], [10.0523, -17.1893]]]]\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-two-link-analytic-outer-reach-boundary","generated_at":"2026-09-29T14:51:02.619645+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":"The outer workspace comparison uses a strict >= against reach squared without the tolerance, so boundary targets fail.","sha256":"86ecd08f694595ec7f818a9dc93e21b62f0a10f5bdf969cb80c377bb5d4ff701","title":"Two-link reach test rejects the fully extended pose · 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":39.883,"exit_code":1,"observations":[{"actual":"unreachable","check":"general elbow up","expected":[0.0,90.0],"passed":false},{"actual":"unreachable","check":"general elbow down","expected":[53.1301,-90.0],"passed":false},{"actual":"unreachable","check":"fully extended on x axis","expected":[0.0,0.0],"passed":false},{"actual":"unreachable","check":"fully extended diagonal","expected":[53.1301,0.0],"passed":false},{"actual":"unreachable","check":"outside workspace","expected":"unreachable","passed":true},{"actual":"unreachable","check":"between reach and reach squared","expected":"unreachable","passed":true},{"actual":"unreachable","check":"inside inner hole","expected":"unreachable","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"general elbow up\", \"actual\": \"unreachable\", \"expected\": [0.0, 90.0], \"passed\": false}, {\"check\": \"general elbow down\", \"actual\": \"unreachable\", \"expected\": [53.1301, -90.0], \"passed\": false}, {\"check\": \"fully extended on x axis\", \"actual\": \"unreachable\", \"expected\": [0.0, 0.0], \"passed\": false}, {\"check\": \"fully extended diagonal\", \"actual\": \"unreachable\", \"expected\": [53.1301, 0.0], \"passed\": false}, {\"check\": \"outside workspace\", \"actual\": \"unreachable\", \"expected\": \"unreachable\", \"passed\": true}, {\"check\": \"between reach and reach squared\", \"actual\": \"unreachable\", \"expected\": \"unreachable\", \"passed\": true}, {\"check\": \"inside inner hole\", \"actual\": \"unreachable\", \"expected\": \"unreachable\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.149,"exit_code":1,"observations":[{"actual":[0.0,90.0],"check":"general elbow up","expected":[0.0,90.0],"passed":true},{"actual":[53.1301,-90.0],"check":"general elbow down","expected":[53.1301,-90.0],"passed":true},{"actual":"unreachable","check":"fully extended on x axis","expected":[0.0,0.0],"passed":false},{"actual":"unreachable","check":"fully extended diagonal","expected":[53.1301,0.0],"passed":false},{"actual":"unreachable","check":"outside workspace","expected":"unreachable","passed":true},{"actual":"unreachable","check":"between reach and reach squared","expected":"unreachable","passed":true},{"actual":"unreachable","check":"inside inner hole","expected":"unreachable","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"general elbow up\", \"actual\": [0.0, 90.0], \"expected\": [0.0, 90.0], \"passed\": true}, {\"check\": \"general elbow down\", \"actual\": [53.1301, -90.0], \"expected\": [53.1301, -90.0], \"passed\": true}, {\"check\": \"fully extended on x axis\", \"actual\": \"unreachable\", \"expected\": [0.0, 0.0], \"passed\": false}, {\"check\": \"fully extended diagonal\", \"actual\": \"unreachable\", \"expected\": [53.1301, 0.0], \"passed\": false}, {\"check\": \"outside workspace\", \"actual\": \"unreachable\", \"expected\": \"unreachable\", \"passed\": true}, {\"check\": \"between reach and reach squared\", \"actual\": \"unreachable\", \"expected\": \"unreachable\", \"passed\": true}, {\"check\": \"inside inner hole\", \"actual\": \"unreachable\", \"expected\": \"unreachable\", \"passed\": true}], \"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."}}