FA-93386 / Solar tracker geometry / Open access
Ground-coverage-ratio backtracking: dense row coverage · case 01
Rows with a coverage ratio of 1 track steeply and shade each other.
ROOT CAUSE
A coverage ratio of 1 is rejected as if backtracking were disabled.
VERIFIED REPAIR
Only gcr <= 0 disables backtracking; gcr = 1 is valid.
Unsuccessful approach: Using a 0.99 cut-off still treats continuous rows as unconstrained.
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 or gcr >= 1:
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]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [45, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [85.5, 1.0], [0.0, True]],
['control 1', [30, 0.25], [30, False]], ['control 2', [-29.5, 0.5], [-29.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-79.5, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [85.5, 1.0], [0.0, True]],
['control 1', [-80, 0.35], [-19.745, True]], ['control 2', [-10, 0.35], [-10, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-80, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [60.5, 1.0], [0.0, True]],
['control 1', [0, 0.25], [0, False]], ['control 2', [10.5, 0], [10.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-60, 1.0], [-0.0, True]],
['regression: dense row coverage (partial repair)', [30, 1.0], [0.0, True]],
['control 1', [-59.5, 1.0], [0.0, True]], ['control 2', [-45, 0.5], [-45, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-69.5, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [-60, 1.0], [-0.0, True]],
['control 1', [10, 0.35], [10, False]], ['control 2', [-60, 0.3], [-60, 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 |
| boundary: continuous rows | [60, False] | [0.0, True] | Failed |
| regression: dense row coverage | [45, False] | [0.0, True] | Failed |
| regression: dense row coverage (partial repair) | [85.5, False] | [0.0, True] | Failed |
| control 1 | [30, False] | [30, False] | Passed |
| control 2 | [-29.5, False] | [-29.5, False] | Passed |
SHA-256 / 56d6f4641eed0d12a2e40de83c79eb2a9954aa1928f9a9e48f2f534fa51ffbf4
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 or gcr > 0.99:
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]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [45, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [85.5, 1.0], [0.0, True]],
['control 1', [30, 0.25], [30, False]], ['control 2', [-29.5, 0.5], [-29.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-79.5, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [85.5, 1.0], [0.0, True]],
['control 1', [-80, 0.35], [-19.745, True]], ['control 2', [-10, 0.35], [-10, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-80, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [60.5, 1.0], [0.0, True]],
['control 1', [0, 0.25], [0, False]], ['control 2', [10.5, 0], [10.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-60, 1.0], [-0.0, True]],
['regression: dense row coverage (partial repair)', [30, 1.0], [0.0, True]],
['control 1', [-59.5, 1.0], [0.0, True]], ['control 2', [-45, 0.5], [-45, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-69.5, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [-60, 1.0], [-0.0, True]],
['control 1', [10, 0.35], [10, False]], ['control 2', [-60, 0.3], [-60, 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 |
| boundary: continuous rows | [60, False] | [0.0, True] | Failed |
| regression: dense row coverage | [45, False] | [0.0, True] | Failed |
| regression: dense row coverage (partial repair) | [85.5, False] | [0.0, True] | Failed |
| control 1 | [30, False] | [30, False] | Passed |
| control 2 | [-29.5, False] | [-29.5, False] | Passed |
SHA-256 / eff72ae44eecd010352030a778c7bef1fd821402a179884e3c8eef2370d7117f
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]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [45, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [85.5, 1.0], [0.0, True]],
['control 1', [30, 0.25], [30, False]], ['control 2', [-29.5, 0.5], [-29.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-79.5, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [85.5, 1.0], [0.0, True]],
['control 1', [-80, 0.35], [-19.745, True]], ['control 2', [-10, 0.35], [-10, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-80, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [60.5, 1.0], [0.0, True]],
['control 1', [0, 0.25], [0, False]], ['control 2', [10.5, 0], [10.5, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-60, 1.0], [-0.0, True]],
['regression: dense row coverage (partial repair)', [30, 1.0], [0.0, True]],
['control 1', [-59.5, 1.0], [0.0, True]], ['control 2', [-45, 0.5], [-45, False]]],
[['boundary: noon never backtracks', [0, 0.4], [0, False]],
['normal: steep morning angle backtracks', [-75, 0.4], [-25.319, True]],
['boundary: continuous rows', [60, 1.0], [0.0, True]],
['regression: dense row coverage', [-69.5, 1.0], [0.0, True]],
['regression: dense row coverage (partial repair)', [-60, 1.0], [-0.0, True]],
['control 1', [10, 0.35], [10, False]], ['control 2', [-60, 0.3], [-60, 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 |
| boundary: continuous rows | [0.0, True] | [0.0, True] | Passed |
| regression: dense row coverage | [0.0, True] | [0.0, True] | Passed |
| regression: dense row coverage (partial repair) | [0.0, True] | [0.0, True] | Passed |
| control 1 | [30, False] | [30, False] | Passed |
| control 2 | [-29.5, False] | [-29.5, False] | Passed |
SHA-256 / d3d89e19715ea967b1ed8f39da71d529e77f1bf14d64f0bd59a2f628bdf3d629
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 2026-09-29T14:51:54.648212+00:00.
Case digest / 1b991ff462ffd36333c6cae5a03962a16fc8e077a4daab2a27a25d2512bb20f8