FA-93371 / Solar tracker geometry / Open access
Ground-coverage-ratio backtracking: coverage ratio scaling · case 01
Rows backtrack at noon or never backtrack at all.
ROOT CAUSE
The cosine is multiplied by the coverage ratio instead of divided by it.
VERIFIED REPAIR
Divide by gcr: temp = |cos(ideal)|/gcr.
Unsuccessful approach: Dividing by the gap fraction 1-gcr models the wrong geometry.
Case contract
Horizontal single-axis rows with ground coverage ratio gcr in (0, 1]. temp = |cos(ideal)|/gcr. When temp < 1 the rows would shade each other and the angle is corrected by -sign(ideal)*acos(temp) degrees (backtracking, flag True). gcr <= 0 disables backtracking. Return [angle rounded to 3, backtracking flag].
Why this case matters
Single-axis and dual-axis solar trackers turn a sun direction into actuator commands; a sign, frame or limit mistake points a whole plant away from the sun or into a mechanical stop.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(ideal, gcr):
if gcr <= 0:
return [round(ideal, 3), False]
temp = abs(math.cos(math.radians(ideal))) * gcr
if temp < 1:
corr = -math.copysign(1, ideal) * math.degrees(math.acos(temp))
return [round(ideal + corr, 3), True]
return [round(ideal, 3), False]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-10, 0.4], [-10, False]],
['regression: coverage ratio scaling (partial repair)', [-44.5, 0.25], [-44.5, False]],
['control 1', [30, 0.25], [30, False]], ['control 2', [-29.5, 0.5], [-29.5, False]],
['control 3', [-30, 0.4], [-30, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-44.5, 0.6], [-44.5, False]],
['regression: coverage ratio scaling (partial repair)', [-79.5, 0.4], [-16.603, True]],
['control 1', [-80, 0.35], [-19.745, True]], ['control 2', [-10, 0.35], [-10, False]],
['control 3', [75, 0.6], [10.554, True]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-70, 0.4], [-38.765, True]],
['regression: coverage ratio scaling (partial repair)', [-79.5, 0.3], [-26.905, True]],
['control 1', [0, 0.25], [0, False]], ['control 2', [10.5, 0], [10.5, False]],
['control 3', [70, 0.35], [57.742, True]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-45, 0.25], [-45, False]],
['regression: coverage ratio scaling (partial repair)', [60, 0.25], [60, False]],
['control 1', [-45, 0.5], [-45, False]], ['control 2', [45, 0], [45, False]],
['control 3', [-9.5, 0.35], [-9.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [60.5, 0.25], [60.5, False]],
['regression: coverage ratio scaling (partial repair)', [85.5, 0.4], [6.812, True]],
['control 1', [10, 0.35], [10, False]], ['control 2', [-60, 0.3], [-60, False]],
['control 3', [70.5, 0.25], [70.5, False]]]]
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 |
|---|---|---|---|
| boundary: noon never backtracks | [-66.422, True] | [0, False] | Failed |
| normal: steep morning angle backtracks | [9.058, True] | [-25.319, True] | Failed |
| regression: coverage ratio scaling | [56.801, True] | [-10, False] | Failed |
| regression: coverage ratio scaling (partial repair) | [35.229, True] | [-44.5, False] | Failed |
| control 1 | [-47.496, True] | [30, False] | Failed |
| control 2 | [34.703, True] | [-29.5, False] | Failed |
| control 3 | [39.732, True] | [-30, False] | Failed |
SHA-256 / 467c41947a01144edc0d979e33cd678ea2dedd6126f137e4154ececb996abe23
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(ideal, gcr):
if gcr <= 0:
return [round(ideal, 3), False]
temp = abs(math.cos(math.radians(ideal))) / (1 - gcr)
if temp < 1:
corr = -math.copysign(1, ideal) * math.degrees(math.acos(temp))
return [round(ideal + corr, 3), True]
return [round(ideal, 3), False]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-10, 0.4], [-10, False]],
['regression: coverage ratio scaling (partial repair)', [-44.5, 0.25], [-44.5, False]],
['control 1', [30, 0.25], [30, False]], ['control 2', [-29.5, 0.5], [-29.5, False]],
['control 3', [-30, 0.4], [-30, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-44.5, 0.6], [-44.5, False]],
['regression: coverage ratio scaling (partial repair)', [-79.5, 0.4], [-16.603, True]],
['control 1', [-80, 0.35], [-19.745, True]], ['control 2', [-10, 0.35], [-10, False]],
['control 3', [75, 0.6], [10.554, True]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-70, 0.4], [-38.765, True]],
['regression: coverage ratio scaling (partial repair)', [-79.5, 0.3], [-26.905, True]],
['control 1', [0, 0.25], [0, False]], ['control 2', [10.5, 0], [10.5, False]],
['control 3', [70, 0.35], [57.742, True]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-45, 0.25], [-45, False]],
['regression: coverage ratio scaling (partial repair)', [60, 0.25], [60, False]],
['control 1', [-45, 0.5], [-45, False]], ['control 2', [45, 0], [45, False]],
['control 3', [-9.5, 0.35], [-9.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [60.5, 0.25], [60.5, False]],
['regression: coverage ratio scaling (partial repair)', [85.5, 0.4], [6.812, True]],
['control 1', [10, 0.35], [10, False]], ['control 2', [-60, 0.3], [-60, False]],
['control 3', [70.5, 0.25], [70.5, False]]]]
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 |
|---|---|---|---|
| boundary: noon never backtracks | [0, False] | [0, False] | Passed |
| normal: steep morning angle backtracks | [-10.554, True] | [-25.319, True] | Failed |
| regression: coverage ratio scaling | [-10, False] | [-10, False] | Passed |
| regression: coverage ratio scaling (partial repair) | [-26.49, True] | [-44.5, False] | Failed |
| control 1 | [30, False] | [30, False] | Passed |
| control 2 | [-29.5, False] | [-29.5, False] | Passed |
| control 3 | [-30, False] | [-30, False] | Passed |
SHA-256 / e2d74ac93a1166ccb596b6ef31a922a93d9d09019783d79f33d15053f97740b9
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(ideal, gcr):
if gcr <= 0:
return [round(ideal, 3), False]
temp = abs(math.cos(math.radians(ideal))) / gcr
if temp < 1:
corr = -math.copysign(1, ideal) * math.degrees(math.acos(temp))
return [round(ideal + corr, 3), True]
return [round(ideal, 3), False]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-10, 0.4], [-10, False]],
['regression: coverage ratio scaling (partial repair)', [-44.5, 0.25], [-44.5, False]],
['control 1', [30, 0.25], [30, False]], ['control 2', [-29.5, 0.5], [-29.5, False]],
['control 3', [-30, 0.4], [-30, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-44.5, 0.6], [-44.5, False]],
['regression: coverage ratio scaling (partial repair)', [-79.5, 0.4], [-16.603, True]],
['control 1', [-80, 0.35], [-19.745, True]], ['control 2', [-10, 0.35], [-10, False]],
['control 3', [75, 0.6], [10.554, True]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-70, 0.4], [-38.765, True]],
['regression: coverage ratio scaling (partial repair)', [-79.5, 0.3], [-26.905, True]],
['control 1', [0, 0.25], [0, False]], ['control 2', [10.5, 0], [10.5, False]],
['control 3', [70, 0.35], [57.742, True]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [-45, 0.25], [-45, False]],
['regression: coverage ratio scaling (partial repair)', [60, 0.25], [60, False]],
['control 1', [-45, 0.5], [-45, False]], ['control 2', [45, 0], [45, False]],
['control 3', [-9.5, 0.35], [-9.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['regression: coverage ratio scaling', [60.5, 0.25], [60.5, False]],
['regression: coverage ratio scaling (partial repair)', [85.5, 0.4], [6.812, True]],
['control 1', [10, 0.35], [10, False]], ['control 2', [-60, 0.3], [-60, False]],
['control 3', [70.5, 0.25], [70.5, False]]]]
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 |
|---|---|---|---|
| boundary: noon never backtracks | [0, False] | [0, False] | Passed |
| normal: steep morning angle backtracks | [-25.319, True] | [-25.319, True] | Passed |
| regression: coverage ratio scaling | [-10, False] | [-10, False] | Passed |
| regression: coverage ratio scaling (partial repair) | [-44.5, False] | [-44.5, False] | Passed |
| control 1 | [30, False] | [30, False] | Passed |
| control 2 | [-29.5, False] | [-29.5, False] | Passed |
| control 3 | [-30, False] | [-30, False] | Passed |
SHA-256 / 3f260ed14014fd55eb45bf364a8efb1081f7da6c2d8b966ae8bf060d432dfb90
Verification & scope
Deterministic stipulated toy contract for teaching; no claim of conformance with any standard, vendor protocol or production controller. 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 2026-09-29T14:51:54.472127+00:00.
Case digest / 62ae164c4815f0d1e28c5425d17521469801d4213c46f3a9a185f74995a18c11