FA-93561 / Solar tracker geometry / Open access
Fixed-tilt row pitch for a design sun: shadow length function · case 01
Row spacing is too tight and rows shade each other in winter.
ROOT CAUSE
The shadow uses the sine of the elevation instead of the tangent.
VERIFIED REPAIR
Shadow length = height / tan(elevation).
Unsuccessful approach: Multiplying by cos(elevation) gives a length that shrinks as the sun gets lower.
Case contract
Rows of slant length length (mm) tilted by tilt degrees face row_az. The shadow reaches forward height*max(cos(sun_az - row_az), 0)/tan(sun_el). Minimum unshaded pitch = length*cos(tilt) + shadow. Return [pitch rounded 3, length/pitch rounded 4]; None when sun_el <= 0.
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(length, tilt, sun_el, sun_az, row_az):
if sun_el <= 0:
return None
t = math.radians(tilt)
rel = math.cos(math.radians(sun_az - row_az))
depth = length * math.cos(t)
height = length * math.sin(t)
shadow = height * max(rel, 0.0) / math.sin(math.radians(sun_el))
pitch = depth + shadow
return [round(pitch, 3), round(length / pitch, 4)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 25, 15, 250, 180], [3325.232, 0.6917]],
['regression: shadow length function (partial repair)', [2000, 25, 20, 225, 200],
[3917.305, 0.5106]],
['control 1', [1700, 30, 30, 90, 200], [1472.243, 1.1547]],
['control 2', [1700, 30, 0, 90, 180], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 10, 10, 100, 170], [3039.753, 0.7566]],
['regression: shadow length function (partial repair)', [2000, 30, 30, 135, 180],
[2956.796, 0.6764]],
['control 1', [1700, 30, 0, 225, 170], None],
['control 2', [1700, 30, 20, 315, 0], [3123.589, 0.5442]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 35, 20, 200, 200], [5508.593, 0.4175]],
['regression: shadow length function (partial repair)', [1000, 30, 20, 60, 0], [1552.895, 0.644]],
['control 1', [2000, 0, -2, 315, 0], None], ['control 2', [2000, 30, 0, 30, 0], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2000, 10, 15, 200, 180], [3187.577, 0.6274]],
['regression: shadow length function (partial repair)', [1000, 25, 10, 135, 200],
[1919.234, 0.521]],
['control 1', [2300, 35, 30, 60, 0], [3026.533, 0.7599]],
['control 2', [1000, 30, 20, 250, 180], [1335.872, 0.7486]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [1700, 10, 20, 100, 180], [1815.012, 0.9366]],
['regression: shadow length function (partial repair)', [2000, 20, 10, 100, 170],
[3206.213, 0.6238]],
['control 1', [2000, 35, -2, 225, 170], None],
['control 2', [1700, 20, 20, 90, 170], [1874.877, 0.9067]]]]
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: winter noon design sun | [5595.754, 0.3574] | [5464.102, 0.366] | Failed |
| boundary: flat rows | [2000.0, 1.0] | [2000.0, 1.0] | Passed |
| boundary: sun behind the rows | [1732.051, 1.1547] | [1732.051, 1.1547] | Passed |
| regression: shadow length function | [3369.0, 0.6827] | [3325.232, 0.6917] | Failed |
| regression: shadow length function (partial repair) | [4052.38, 0.4935] | [3917.305, 0.5106] | Failed |
| control 1 | [1472.243, 1.1547] | [1472.243, 1.1547] | Passed |
| control 2 | None | None | Passed |
SHA-256 / 4519eaef77ba10bb435eb274f661edce558ef79f80f34ea46f4746fc1d98ebaa
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(length, tilt, sun_el, sun_az, row_az):
if sun_el <= 0:
return None
t = math.radians(tilt)
rel = math.cos(math.radians(sun_az - row_az))
depth = length * math.cos(t)
height = length * math.sin(t)
shadow = height * max(rel, 0.0) * math.cos(math.radians(sun_el))
pitch = depth + shadow
return [round(pitch, 3), round(length / pitch, 4)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 25, 15, 250, 180], [3325.232, 0.6917]],
['regression: shadow length function (partial repair)', [2000, 25, 20, 225, 200],
[3917.305, 0.5106]],
['control 1', [1700, 30, 30, 90, 200], [1472.243, 1.1547]],
['control 2', [1700, 30, 0, 90, 180], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 10, 10, 100, 170], [3039.753, 0.7566]],
['regression: shadow length function (partial repair)', [2000, 30, 30, 135, 180],
[2956.796, 0.6764]],
['control 1', [1700, 30, 0, 225, 170], None],
['control 2', [1700, 30, 20, 315, 0], [3123.589, 0.5442]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 35, 20, 200, 200], [5508.593, 0.4175]],
['regression: shadow length function (partial repair)', [1000, 30, 20, 60, 0], [1552.895, 0.644]],
['control 1', [2000, 0, -2, 315, 0], None], ['control 2', [2000, 30, 0, 30, 0], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2000, 10, 15, 200, 180], [3187.577, 0.6274]],
['regression: shadow length function (partial repair)', [1000, 25, 10, 135, 200],
[1919.234, 0.521]],
['control 1', [2300, 35, 30, 60, 0], [3026.533, 0.7599]],
['control 2', [1000, 30, 20, 250, 180], [1335.872, 0.7486]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [1700, 10, 20, 100, 180], [1815.012, 0.9366]],
['regression: shadow length function (partial repair)', [2000, 20, 10, 100, 170],
[3206.213, 0.6238]],
['control 1', [2000, 35, -2, 225, 170], None],
['control 2', [1700, 20, 20, 90, 170], [1874.877, 0.9067]]]]
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: winter noon design sun | [2697.977, 0.7413] | [5464.102, 0.366] | Failed |
| boundary: flat rows | [2000.0, 1.0] | [2000.0, 1.0] | Passed |
| boundary: sun behind the rows | [1732.051, 1.1547] | [1732.051, 1.1547] | Passed |
| regression: shadow length function | [2405.631, 0.9561] | [3325.232, 0.6917] | Failed |
| regression: shadow length function (partial repair) | [2532.462, 0.7897] | [3917.305, 0.5106] | Failed |
| control 1 | [1472.243, 1.1547] | [1472.243, 1.1547] | Passed |
| control 2 | None | None | Passed |
SHA-256 / cd2e4497862181951e602a6524a5a6ca09f575ec13e91d39d32bbe80937fb8e9
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(length, tilt, sun_el, sun_az, row_az):
if sun_el <= 0:
return None
t = math.radians(tilt)
rel = math.cos(math.radians(sun_az - row_az))
depth = length * math.cos(t)
height = length * math.sin(t)
shadow = height * max(rel, 0.0) / math.tan(math.radians(sun_el))
pitch = depth + shadow
return [round(pitch, 3), round(length / pitch, 4)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 25, 15, 250, 180], [3325.232, 0.6917]],
['regression: shadow length function (partial repair)', [2000, 25, 20, 225, 200],
[3917.305, 0.5106]],
['control 1', [1700, 30, 30, 90, 200], [1472.243, 1.1547]],
['control 2', [1700, 30, 0, 90, 180], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 10, 10, 100, 170], [3039.753, 0.7566]],
['regression: shadow length function (partial repair)', [2000, 30, 30, 135, 180],
[2956.796, 0.6764]],
['control 1', [1700, 30, 0, 225, 170], None],
['control 2', [1700, 30, 20, 315, 0], [3123.589, 0.5442]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2300, 35, 20, 200, 200], [5508.593, 0.4175]],
['regression: shadow length function (partial repair)', [1000, 30, 20, 60, 0], [1552.895, 0.644]],
['control 1', [2000, 0, -2, 315, 0], None], ['control 2', [2000, 30, 0, 30, 0], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [2000, 10, 15, 200, 180], [3187.577, 0.6274]],
['regression: shadow length function (partial repair)', [1000, 25, 10, 135, 200],
[1919.234, 0.521]],
['control 1', [2300, 35, 30, 60, 0], [3026.533, 0.7599]],
['control 2', [1000, 30, 20, 250, 180], [1335.872, 0.7486]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: shadow length function', [1700, 10, 20, 100, 180], [1815.012, 0.9366]],
['regression: shadow length function (partial repair)', [2000, 20, 10, 100, 170],
[3206.213, 0.6238]],
['control 1', [2000, 35, -2, 225, 170], None],
['control 2', [1700, 20, 20, 90, 170], [1874.877, 0.9067]]]]
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: winter noon design sun | [5464.102, 0.366] | [5464.102, 0.366] | Passed |
| boundary: flat rows | [2000.0, 1.0] | [2000.0, 1.0] | Passed |
| boundary: sun behind the rows | [1732.051, 1.1547] | [1732.051, 1.1547] | Passed |
| regression: shadow length function | [3325.232, 0.6917] | [3325.232, 0.6917] | Passed |
| regression: shadow length function (partial repair) | [3917.305, 0.5106] | [3917.305, 0.5106] | Passed |
| control 1 | [1472.243, 1.1547] | [1472.243, 1.1547] | Passed |
| control 2 | None | None | Passed |
SHA-256 / 8bd3ec1f09f679e07aa28756fe2b0b1baf8c50b772d4248347276a9b25dfefb9
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 2026-09-29T14:51:56.237520+00:00.
Case digest / 05cc66a89e5c65f12ba4f67bf9d121bf03d1520b1d695b437a61a248670a1c24