FA-88091 / Inverse kinematics solvers / Open access

Delta towers are placed at 0, 120 and 240 degrees · case 01

Carriage heights belong to a rotated tower layout, so moves in y tilt the effector.

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

ROOT CAUSE

The tower table starts at the x axis instead of at 90 degrees.

VERIFIED REPAIR

Place towers at 90, 210 and 330 degrees in that order.

Unsuccessful approach: Listing the 330 degree tower second keeps positions right but swaps two carriage outputs.

Case contract

Input [R,e,L,x,y,z,hmax]: linear delta printer, towers at 90, 210 and 330 degrees (in that output order) on radius R, effector joints offset e toward the centre so the effective radius is R-e, diagonal rods of length L. Carriage height is z+sqrt(L^2-dx^2-dy^2); a negative radicand returns "unreachable", a carriage above hmax returns "out-of-travel". Return heights rounded to 4.

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):
    R,e,L,px,py,pz,hmax=x
    r=R-e
    S=math.sqrt(3.0)/2.0
    hs=[]
    for tx,ty in [(r,0.0),(-r/2,r*S),(-r/2,-r*S)]:
        q=L*L-(px-tx)**2-(py-ty)**2
        if q<0: return 'unreachable'
        h=pz+math.sqrt(q)
        if h>hmax: return 'out-of-travel'
        hs.append(round(h,4))
    return hs
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['centre', [[10, 2, 12, 0, 0, 5, 40]], [13.9443, 13.9443, 13.9443]], ['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['beyond rod reach', [[10, 2, 10, 0, -3, 0, 40]], 'unreachable'], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]]], [['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['high z low travel', [[10, 2, 12, 0, 0, 25, 30]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]]], [['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['high z low travel', [[10, 2, 12, 0, 0, 25, 30]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['large offset', [[12, 5, 11, -1, 2, 4, 40]], [13.7468, 12.07, 10.3934]], ['asymmetric point', [[10, 2, 12, 3, 4, 1, 40]], 'unreachable'], ['negative z', [[10, 2, 12, -2, 1, -3, 40]], [6.5394, 6.7321, 3.2679]]], [['centre', [[10, 2, 12, 0, 0, 5, 40]], [13.9443, 13.9443, 13.9443]], ['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['asymmetric point', [[10, 2, 12, 3, 4, 1, 40]], 'unreachable'], ['negative z', [[10, 2, 12, -2, 1, -3, 40]], [6.5394, 6.7321, 3.2679]]], [['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['beyond rod reach', [[10, 2, 10, 0, -3, 0, 40]], 'unreachable'], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['large offset', [[12, 5, 11, -1, 2, 4, 40]], [13.7468, 12.07, 10.3934]]]]
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
centre	`[13.9443, 13.9443, 13.9443]`	`[13.9443, 13.9443, 13.9443]`	Passed
off centre	`[14.9499, 8.071, 14.6213]`	`[9.3589, 12.9553, 15.8955]`	Failed
rod horizontal at tower one	`[5.6569, 2.0706, 7.7274]`	`[0.0, 6.9282, 6.9282]`	Failed
beyond rod reach	`unreachable`	`unreachable`	Passed
carriage exactly at top	`[7.0, 7.0, 7.0]`	`[7.0, 7.0, 7.0]`	Passed
carriage over top	`out-of-travel`	`out-of-travel`	Passed
toward tower two	`[10.2111, 14.35, 16.8267]`	`[12.3808, 17.2952, 11.9245]`	Failed

SHA-256 / 12ff18f225d08242c09ac6e0d06e4d5ba293ccb5f3cade864dc12bf8884d279f

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    R,e,L,px,py,pz,hmax=x
    r=R-e
    S=math.sqrt(3.0)/2.0
    hs=[]
    for tx,ty in [(0.0,r),(r*S,-r/2),(-r*S,-r/2)]:
        q=L*L-(px-tx)**2-(py-ty)**2
        if q<0: return 'unreachable'
        h=pz+math.sqrt(q)
        if h>hmax: return 'out-of-travel'
        hs.append(round(h,4))
    return hs
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['centre', [[10, 2, 12, 0, 0, 5, 40]], [13.9443, 13.9443, 13.9443]], ['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['beyond rod reach', [[10, 2, 10, 0, -3, 0, 40]], 'unreachable'], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]]], [['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['high z low travel', [[10, 2, 12, 0, 0, 25, 30]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]]], [['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['high z low travel', [[10, 2, 12, 0, 0, 25, 30]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['large offset', [[12, 5, 11, -1, 2, 4, 40]], [13.7468, 12.07, 10.3934]], ['asymmetric point', [[10, 2, 12, 3, 4, 1, 40]], 'unreachable'], ['negative z', [[10, 2, 12, -2, 1, -3, 40]], [6.5394, 6.7321, 3.2679]]], [['centre', [[10, 2, 12, 0, 0, 5, 40]], [13.9443, 13.9443, 13.9443]], ['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['asymmetric point', [[10, 2, 12, 3, 4, 1, 40]], 'unreachable'], ['negative z', [[10, 2, 12, -2, 1, -3, 40]], [6.5394, 6.7321, 3.2679]]], [['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['beyond rod reach', [[10, 2, 10, 0, -3, 0, 40]], 'unreachable'], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['large offset', [[12, 5, 11, -1, 2, 4, 40]], [13.7468, 12.07, 10.3934]]]]
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
centre	`[13.9443, 13.9443, 13.9443]`	`[13.9443, 13.9443, 13.9443]`	Passed
off centre	`[9.3589, 15.8955, 12.9553]`	`[9.3589, 12.9553, 15.8955]`	Failed
rod horizontal at tower one	`[0.0, 6.9282, 6.9282]`	`[0.0, 6.9282, 6.9282]`	Passed
beyond rod reach	`unreachable`	`unreachable`	Passed
carriage exactly at top	`[7.0, 7.0, 7.0]`	`[7.0, 7.0, 7.0]`	Passed
carriage over top	`out-of-travel`	`out-of-travel`	Passed
toward tower two	`[12.3808, 11.9245, 17.2952]`	`[12.3808, 17.2952, 11.9245]`	Failed

SHA-256 / 262912104f0fc8492bb91343c8a9ba20a572e8bce1a677841424f116a4d41995

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    R,e,L,px,py,pz,hmax=x
    r=R-e
    S=math.sqrt(3.0)/2.0
    hs=[]
    for tx,ty in [(0.0,r),(-r*S,-r/2),(r*S,-r/2)]:
        q=L*L-(px-tx)**2-(py-ty)**2
        if q<0: return 'unreachable'
        h=pz+math.sqrt(q)
        if h>hmax: return 'out-of-travel'
        hs.append(round(h,4))
    return hs
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['centre', [[10, 2, 12, 0, 0, 5, 40]], [13.9443, 13.9443, 13.9443]], ['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['beyond rod reach', [[10, 2, 10, 0, -3, 0, 40]], 'unreachable'], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]]], [['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['high z low travel', [[10, 2, 12, 0, 0, 25, 30]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]]], [['toward tower two', [[10, 1, 15, -4, -2, 3, 40]], [12.3808, 17.2952, 11.9245]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['high z low travel', [[10, 2, 12, 0, 0, 25, 30]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['large offset', [[12, 5, 11, -1, 2, 4, 40]], [13.7468, 12.07, 10.3934]], ['asymmetric point', [[10, 2, 12, 3, 4, 1, 40]], 'unreachable'], ['negative z', [[10, 2, 12, -2, 1, -3, 40]], [6.5394, 6.7321, 3.2679]]], [['centre', [[10, 2, 12, 0, 0, 5, 40]], [13.9443, 13.9443, 13.9443]], ['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['toward tower three', [[10, 1, 15, 4, -2, 3, 40]], [12.3808, 11.9245, 17.2952]], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['asymmetric point', [[10, 2, 12, 3, 4, 1, 40]], 'unreachable'], ['negative z', [[10, 2, 12, -2, 1, -3, 40]], [6.5394, 6.7321, 3.2679]]], [['off centre', [[10, 2, 12, 2, -3, 5, 40]], [9.3589, 12.9553, 15.8955]], ['rod horizontal at tower one', [[10, 2, 10, 0, -2, 0, 40]], [0.0, 6.9282, 6.9282]], ['beyond rod reach', [[10, 2, 10, 0, -3, 0, 40]], 'unreachable'], ['carriage exactly at top', [[12, 0, 13, 0, 0, 2, 7]], [7.0, 7.0, 7.0]], ['carriage over top', [[12, 0, 13, 0, 0, 2.5, 7]], 'out-of-travel'], ['no effector offset', [[8, 0, 12, 1, 1, 2, 40]], [11.6954, 9.4929, 11.1573]], ['large offset', [[12, 5, 11, -1, 2, 4, 40]], [13.7468, 12.07, 10.3934]]]]
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
centre	`[13.9443, 13.9443, 13.9443]`	`[13.9443, 13.9443, 13.9443]`	Passed
off centre	`[9.3589, 12.9553, 15.8955]`	`[9.3589, 12.9553, 15.8955]`	Passed
rod horizontal at tower one	`[0.0, 6.9282, 6.9282]`	`[0.0, 6.9282, 6.9282]`	Passed
beyond rod reach	`unreachable`	`unreachable`	Passed
carriage exactly at top	`[7.0, 7.0, 7.0]`	`[7.0, 7.0, 7.0]`	Passed
carriage over top	`out-of-travel`	`out-of-travel`	Passed
toward tower two	`[12.3808, 17.2952, 11.9245]`	`[12.3808, 17.2952, 11.9245]`	Passed

SHA-256 / c6bf90e742116c0d9b0cc691005a49700433274da03a32efd488a7d003b1e77b

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 2026-09-29T14:51:04.939915+00:00.

Case digest / 2b95b0d1db3d70f831a7b273f0740a259c9896a0d4ff352cd8567df6d2bbf35e