FA-93356 / Solar tracker geometry / Open access
Horizontal single-axis true tracking: rotation angle function · case 01
The commanded rotation is the complement of the correct angle.
ROOT CAUSE
atan2 receives the vertical component first.
VERIFIED REPAIR
Use atan2(perpendicular, up).
Unsuccessful approach: asin of the unnormalised perpendicular component ignores the along-axis sun component.
Case contract
Sun azimuth clockwise from north and elevation in degrees. The horizontal axis points to axis_az; rotation is positive toward azimuth axis_az+90. Sun vector east=cos(el)sin(az), north=cos(el)cos(az), up=sin(el). Rotation = atan2(component along axis_az+90, up). Clamp to +-max_angle (status 'limited', else 'tracking'); sun_el <= 0 returns [0.0, 'night']. Round to 3 decimals.
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(sun_az, sun_el, axis_az, max_angle):
if sun_el <= 0:
return [0.0, 'night']
az = math.radians(sun_az)
el = math.radians(sun_el)
e = math.cos(el) * math.sin(az)
n = math.cos(el) * math.cos(az)
u = math.sin(el)
pa = math.radians(axis_az + 90)
perp = e * math.sin(pa) + n * math.cos(pa)
rot = math.degrees(math.atan2(u, perp))
status = 'tracking'
if abs(rot) > max_angle:
rot = math.copysign(max_angle, rot)
status = 'limited'
return [round(rot, 3), status]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [30, 10, 180, 55], [-55.0, 'limited']],
['regression: rotation angle function (partial repair)', [315, 75, 20, 45], [-13.65, 'tracking']],
['control 1', [150, 10, 190, 45], [-45.0, 'limited']],
['control 2', [180, 89, 0, 55], [0.0, 'tracking']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [100, 3, 20, 60], [60.0, 'limited']],
['regression: rotation angle function (partial repair)', [200, 10, 190, 55],
[44.561, 'tracking']],
['control 1', [60, 0, 20, 45], [0.0, 'night']],
['control 2', [340, 60, 170, 45], [5.725, 'tracking']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [0, 10, 190, 60], [44.561, 'tracking']],
['regression: rotation angle function (partial repair)', [300, 45, 20, 45],
[-44.561, 'tracking']],
['control 1', [135, 0, 0, 60], [0.0, 'night']],
['control 2', [60, 20, 170, 55], [-55.0, 'limited']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [45, 10, 20, 45], [45.0, 'limited']],
['regression: rotation angle function (partial repair)', [150, 45, 170, 55],
[-18.882, 'tracking']],
['control 1', [90, 20, 0, 60], [60.0, 'limited']],
['control 2', [30, 3, 190, 45], [-45.0, 'limited']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [30, 20, 170, 45], [-45.0, 'limited']],
['regression: rotation angle function (partial repair)', [340, 3, 20, 55], [-55.0, 'limited']],
['control 1', [100, 3, 190, 60], [-60.0, 'limited']],
['control 2', [90, 75, 170, 55], [-14.782, 'tracking']]]]
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: sun on the horizon | [0.0, 'night'] | [0.0, 'night'] | Passed |
| normal: solar noon on N-S axis | [60.0, 'limited'] | [0.0, 'tracking'] | Failed |
| boundary: low morning sun hits east limit | [60.0, 'limited'] | [-60.0, 'limited'] | Failed |
| regression: rotation angle function | [55.0, 'limited'] | [-55.0, 'limited'] | Failed |
| regression: rotation angle function (partial repair) | [45.0, 'limited'] | [-13.65, 'tracking'] | Failed |
| control 1 | [45.0, 'limited'] | [-45.0, 'limited'] | Failed |
| control 2 | [55.0, 'limited'] | [0.0, 'tracking'] | Failed |
SHA-256 / 78b87274dc87a07e5d9f69053fb07240cd34c2d4bac4a2674e39a6450c4ac917
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, axis_az, max_angle):
if sun_el <= 0:
return [0.0, 'night']
az = math.radians(sun_az)
el = math.radians(sun_el)
e = math.cos(el) * math.sin(az)
n = math.cos(el) * math.cos(az)
u = math.sin(el)
pa = math.radians(axis_az + 90)
perp = e * math.sin(pa) + n * math.cos(pa)
rot = math.degrees(math.asin(perp))
status = 'tracking'
if abs(rot) > max_angle:
rot = math.copysign(max_angle, rot)
status = 'limited'
return [round(rot, 3), status]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [30, 10, 180, 55], [-55.0, 'limited']],
['regression: rotation angle function (partial repair)', [315, 75, 20, 45], [-13.65, 'tracking']],
['control 1', [150, 10, 190, 45], [-45.0, 'limited']],
['control 2', [180, 89, 0, 55], [0.0, 'tracking']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [100, 3, 20, 60], [60.0, 'limited']],
['regression: rotation angle function (partial repair)', [200, 10, 190, 55],
[44.561, 'tracking']],
['control 1', [60, 0, 20, 45], [0.0, 'night']],
['control 2', [340, 60, 170, 45], [5.725, 'tracking']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [0, 10, 190, 60], [44.561, 'tracking']],
['regression: rotation angle function (partial repair)', [300, 45, 20, 45],
[-44.561, 'tracking']],
['control 1', [135, 0, 0, 60], [0.0, 'night']],
['control 2', [60, 20, 170, 55], [-55.0, 'limited']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [45, 10, 20, 45], [45.0, 'limited']],
['regression: rotation angle function (partial repair)', [150, 45, 170, 55],
[-18.882, 'tracking']],
['control 1', [90, 20, 0, 60], [60.0, 'limited']],
['control 2', [30, 3, 190, 45], [-45.0, 'limited']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [30, 20, 170, 45], [-45.0, 'limited']],
['regression: rotation angle function (partial repair)', [340, 3, 20, 55], [-55.0, 'limited']],
['control 1', [100, 3, 190, 60], [-60.0, 'limited']],
['control 2', [90, 75, 170, 55], [-14.782, 'tracking']]]]
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: sun on the horizon | [0.0, 'night'] | [0.0, 'night'] | Passed |
| normal: solar noon on N-S axis | [0.0, 'tracking'] | [0.0, 'tracking'] | Passed |
| boundary: low morning sun hits east limit | [-60.0, 'limited'] | [-60.0, 'limited'] | Passed |
| regression: rotation angle function | [-29.499, 'tracking'] | [-55.0, 'limited'] | Failed |
| regression: rotation angle function (partial repair) | [-13.566, 'tracking'] | [-13.65, 'tracking'] | Failed |
| control 1 | [-39.273, 'tracking'] | [-45.0, 'limited'] | Failed |
| control 2 | [0.0, 'tracking'] | [0.0, 'tracking'] | Passed |
SHA-256 / 72a2a4cdb7a6e6d6ed7063924ef875496f8583adb1d6fc7418db47568be06d23
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, axis_az, max_angle):
if sun_el <= 0:
return [0.0, 'night']
az = math.radians(sun_az)
el = math.radians(sun_el)
e = math.cos(el) * math.sin(az)
n = math.cos(el) * math.cos(az)
u = math.sin(el)
pa = math.radians(axis_az + 90)
perp = e * math.sin(pa) + n * math.cos(pa)
rot = math.degrees(math.atan2(perp, u))
status = 'tracking'
if abs(rot) > max_angle:
rot = math.copysign(max_angle, rot)
status = 'limited'
return [round(rot, 3), status]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [30, 10, 180, 55], [-55.0, 'limited']],
['regression: rotation angle function (partial repair)', [315, 75, 20, 45], [-13.65, 'tracking']],
['control 1', [150, 10, 190, 45], [-45.0, 'limited']],
['control 2', [180, 89, 0, 55], [0.0, 'tracking']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [100, 3, 20, 60], [60.0, 'limited']],
['regression: rotation angle function (partial repair)', [200, 10, 190, 55],
[44.561, 'tracking']],
['control 1', [60, 0, 20, 45], [0.0, 'night']],
['control 2', [340, 60, 170, 45], [5.725, 'tracking']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [0, 10, 190, 60], [44.561, 'tracking']],
['regression: rotation angle function (partial repair)', [300, 45, 20, 45],
[-44.561, 'tracking']],
['control 1', [135, 0, 0, 60], [0.0, 'night']],
['control 2', [60, 20, 170, 55], [-55.0, 'limited']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [45, 10, 20, 45], [45.0, 'limited']],
['regression: rotation angle function (partial repair)', [150, 45, 170, 55],
[-18.882, 'tracking']],
['control 1', [90, 20, 0, 60], [60.0, 'limited']],
['control 2', [30, 3, 190, 45], [-45.0, 'limited']]],
[['boundary: sun on the horizon', [90, 0, 180, 60], [0.0, 'night']],
['normal: solar noon on N-S axis', [180, 60, 180, 60], [0.0, 'tracking']],
['boundary: low morning sun hits east limit', [90, 10, 180, 60], [-60.0, 'limited']],
['regression: rotation angle function', [30, 20, 170, 45], [-45.0, 'limited']],
['regression: rotation angle function (partial repair)', [340, 3, 20, 55], [-55.0, 'limited']],
['control 1', [100, 3, 190, 60], [-60.0, 'limited']],
['control 2', [90, 75, 170, 55], [-14.782, 'tracking']]]]
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: sun on the horizon | [0.0, 'night'] | [0.0, 'night'] | Passed |
| normal: solar noon on N-S axis | [0.0, 'tracking'] | [0.0, 'tracking'] | Passed |
| boundary: low morning sun hits east limit | [-60.0, 'limited'] | [-60.0, 'limited'] | Passed |
| regression: rotation angle function | [-55.0, 'limited'] | [-55.0, 'limited'] | Passed |
| regression: rotation angle function (partial repair) | [-13.65, 'tracking'] | [-13.65, 'tracking'] | Passed |
| control 1 | [-45.0, 'limited'] | [-45.0, 'limited'] | Passed |
| control 2 | [0.0, 'tracking'] | [0.0, 'tracking'] | Passed |
SHA-256 / 2fe4f4362df9ccdcf425f580e112068acaef8cfbd47cf9a1544918c30012c5d7
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 2026-09-29T14:51:54.349919+00:00.
Case digest / dd3ebacd22b90b1d533e157ddea73daa8e585c7a5880e5e2906a7bad08c5ae61