FA-93631 / Solar tracker geometry / Open access
Cross-axis slope aware backtracking: correction angle units · case 01
Backtracking barely changes the angle and rows keep shading each other.
ROOT CAUSE
The acos correction is applied in radians to an angle in degrees.
VERIFIED REPAIR
Convert the acos correction to degrees.
Unsuccessful approach: Using asin yields the complementary correction angle.
Case contract
Rows on terrain with cross-axis slope cross_tilt degrees. dist = 1/(gcr*cos(cross_tilt)); temp = dist*cos(ideal - cross_tilt). When temp < 1 the angle becomes ideal - sign(ideal)*acos(max(temp, -1)) degrees, else ideal. The result is then clamped to +-max_angle. Return the angle rounded to 3.
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, cross_tilt, max_angle):
dist = 1 / (gcr * math.cos(math.radians(cross_tilt)))
temp = dist * math.cos(math.radians(ideal - cross_tilt))
if temp < 1:
theta = ideal - math.copysign(math.acos(max(temp, -1.0)), ideal)
else:
theta = ideal
theta = max(-max_angle, min(max_angle, theta))
return round(theta, 3)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-80, 0.5, 0, 75], -10.322],
['regression: correction angle units (partial repair)', [80, 0.3, -10, 55], -10.0],
['control 1', [-45, 0.5, -5, 50], -45], ['control 2', [-60, 0.3, 0, 60], -60]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [60, 0.5, -5, 55], 28.045],
['regression: correction angle units (partial repair)', [-80, 0.3, 0, 75], -25.368],
['control 1', [80, 0.4, 0, 60], 15.729], ['control 2', [-80, 0.4, 15, 75], 23.037]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-70, 0.3, 15, 60], 2.496],
['regression: correction angle units (partial repair)', [80, 0.4, 5, 55], 30.505],
['control 1', [0, 0.4, -5, 55], 0], ['control 2', [70, 0.4, -5, 75], 20.505]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-80, 0.4, -10, 75], -50.255],
['regression: correction angle units (partial repair)', [80, 0.3, -5, 55], 6.955],
['control 1', [-60, 0.4, -5, 50], -50], ['control 2', [0, 0.4, -5, 60], 0]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-70, 0.3, 5, 75], -40.0],
['regression: correction angle units (partial repair)', [-80, 0.4, 0, 75], -15.729],
['control 1', [-80, 0.5, 0, 75], -10.322], ['control 2', [-45, 0.4, 15, 50], -45]]]
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 |
|---|---|---|---|
| normal: flat terrain matches plain backtracking | -60 | -38.765 | Failed |
| boundary: slope-facing noon | 0 | 0 | Passed |
| boundary: clamp after backtracking | 50 | -10.0 | Failed |
| regression: correction angle units | -75 | -10.322 | Failed |
| regression: correction angle units (partial repair) | 55 | -10.0 | Failed |
| control 1 | -45 | -45 | Passed |
| control 2 | -60 | -60 | Passed |
SHA-256 / 99d14b7de9dfb56e1647cb8d1d2be0347107b0bf101c6f6ff1b99bfd49c6661c
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, cross_tilt, max_angle):
dist = 1 / (gcr * math.cos(math.radians(cross_tilt)))
temp = dist * math.cos(math.radians(ideal - cross_tilt))
if temp < 1:
theta = ideal - math.copysign(math.degrees(math.asin(max(temp, -1.0))), ideal)
else:
theta = ideal
theta = max(-max_angle, min(max_angle, theta))
return round(theta, 3)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-80, 0.5, 0, 75], -10.322],
['regression: correction angle units (partial repair)', [80, 0.3, -10, 55], -10.0],
['control 1', [-45, 0.5, -5, 50], -45], ['control 2', [-60, 0.3, 0, 60], -60]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [60, 0.5, -5, 55], 28.045],
['regression: correction angle units (partial repair)', [-80, 0.3, 0, 75], -25.368],
['control 1', [80, 0.4, 0, 60], 15.729], ['control 2', [-80, 0.4, 15, 75], 23.037]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-70, 0.3, 15, 60], 2.496],
['regression: correction angle units (partial repair)', [80, 0.4, 5, 55], 30.505],
['control 1', [0, 0.4, -5, 55], 0], ['control 2', [70, 0.4, -5, 75], 20.505]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-80, 0.4, -10, 75], -50.255],
['regression: correction angle units (partial repair)', [80, 0.3, -5, 55], 6.955],
['control 1', [-60, 0.4, -5, 50], -50], ['control 2', [0, 0.4, -5, 60], 0]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-70, 0.3, 5, 75], -40.0],
['regression: correction angle units (partial repair)', [-80, 0.4, 0, 75], -15.729],
['control 1', [-80, 0.5, 0, 75], -10.322], ['control 2', [-45, 0.4, 15, 50], -45]]]
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 |
|---|---|---|---|
| normal: flat terrain matches plain backtracking | -11.235 | -38.765 | Failed |
| boundary: slope-facing noon | 0 | 0 | Passed |
| boundary: clamp after backtracking | 50 | -10.0 | Failed |
| regression: correction angle units | -59.678 | -10.322 | Failed |
| regression: correction angle units (partial repair) | 55 | -10.0 | Failed |
| control 1 | -45 | -45 | Passed |
| control 2 | -60 | -60 | Passed |
SHA-256 / 53c5ec795504c1672c8192c87e8fd9905aea6cb077f0a71c6590fb288aa0ec17
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, cross_tilt, max_angle):
dist = 1 / (gcr * math.cos(math.radians(cross_tilt)))
temp = dist * math.cos(math.radians(ideal - cross_tilt))
if temp < 1:
theta = ideal - math.copysign(math.degrees(math.acos(max(temp, -1.0))), ideal)
else:
theta = ideal
theta = max(-max_angle, min(max_angle, theta))
return round(theta, 3)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-80, 0.5, 0, 75], -10.322],
['regression: correction angle units (partial repair)', [80, 0.3, -10, 55], -10.0],
['control 1', [-45, 0.5, -5, 50], -45], ['control 2', [-60, 0.3, 0, 60], -60]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [60, 0.5, -5, 55], 28.045],
['regression: correction angle units (partial repair)', [-80, 0.3, 0, 75], -25.368],
['control 1', [80, 0.4, 0, 60], 15.729], ['control 2', [-80, 0.4, 15, 75], 23.037]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-70, 0.3, 15, 60], 2.496],
['regression: correction angle units (partial repair)', [80, 0.4, 5, 55], 30.505],
['control 1', [0, 0.4, -5, 55], 0], ['control 2', [70, 0.4, -5, 75], 20.505]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-80, 0.4, -10, 75], -50.255],
['regression: correction angle units (partial repair)', [80, 0.3, -5, 55], 6.955],
['control 1', [-60, 0.4, -5, 50], -50], ['control 2', [0, 0.4, -5, 60], 0]],
[['normal: flat terrain matches plain backtracking', [-70, 0.4, 0, 60], -38.765],
['boundary: slope-facing noon', [0, 0.4, 10, 60], 0],
['boundary: clamp after backtracking', [80, 0.3, -10, 50], -10.0],
['regression: correction angle units', [-70, 0.3, 5, 75], -40.0],
['regression: correction angle units (partial repair)', [-80, 0.4, 0, 75], -15.729],
['control 1', [-80, 0.5, 0, 75], -10.322], ['control 2', [-45, 0.4, 15, 50], -45]]]
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 |
|---|---|---|---|
| normal: flat terrain matches plain backtracking | -38.765 | -38.765 | Passed |
| boundary: slope-facing noon | 0 | 0 | Passed |
| boundary: clamp after backtracking | -10.0 | -10.0 | Passed |
| regression: correction angle units | -10.322 | -10.322 | Passed |
| regression: correction angle units (partial repair) | -10.0 | -10.0 | Passed |
| control 1 | -45 | -45 | Passed |
| control 2 | -60 | -60 | Passed |
SHA-256 / 186c68c2219b639e295795f75a5461e8a8a9e6574fe14ae92965b4976b84e6f9
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 2026-09-29T14:51:56.873039+00:00.
Case digest / 8028cac23b3acce7135b5385762f19f63ed574b36dbb9b7343bcb1c0b8d85171