FA-93826 / Solar tracker geometry / Open access
Dual-axis pointing error: elevation trig terms · case 01
Azimuth errors near the zenith are reported as huge.
ROOT CAUSE
Elevations are treated as zenith angles in the spherical law of cosines.
VERIFIED REPAIR
Use sin e1 sin e2 + cos e1 cos e2 cos dAz.
Unsuccessful approach: Dropping the cos e factors lets azimuth error dominate at any elevation.
Case contract
Pointing error is the great-circle angle between the commanded direction and the sun: cos(err) = sin e1 sin e2 + cos e1 cos e2 cos(az1 - az2), clamped to [-1, 1]. Return [err degrees rounded 3, err <= tol].
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(cmd_az, cmd_el, sun_az, sun_el, tol):
e1 = math.radians(cmd_el)
e2 = math.radians(sun_el)
d = math.radians(cmd_az - sun_az)
c = math.cos(e1) * math.cos(e2) + math.sin(e1) * math.sin(e2) * math.cos(d)
c = max(-1.0, min(1.0, c))
err = math.degrees(math.acos(c))
return [round(err, 3), err <= tol]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [340, 45, 60, 80, 0.5], [44.136, False]],
['regression: elevation trig terms (partial repair)', [340, 30, 270, 45, 1], [55.737, False]],
['control 1', [90, 10, 225, 30, 0.5], [121.081, False]],
['control 2', [340, 80, 225, 45, 0.5], [49.874, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [200, 60, 60, 45, 5], [70.03, False]],
['regression: elevation trig terms (partial repair)', [180, 60, 0, 60, 0.5], [60.0, False]],
['control 1', [180, 80, 200, 10, 5], [70.628, False]],
['control 2', [45, 60, 60, 80, 1], [20.49, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [135, 60, 45, 45, 5], [52.239, False]],
['regression: elevation trig terms (partial repair)', [270, 45, 340, 60, 5], [42.837, False]],
['control 1', [135, 80, 60, 60, 5], [28.915, False]],
['control 2', [60, 30, 315, 10, 0.5], [97.696, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [200, 30, 60, 80, 0.5], [67.839, False]],
['regression: elevation trig terms (partial repair)', [0, 45, 250, 60, 2], [60.564, False]],
['control 1', [180, 45, 45, 60, 1], [68.754, False]],
['control 2', [300, 10, 135, 45, 5], [123.357, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [270, 30, 250, 30, 5], [17.298, False]],
['regression: elevation trig terms (partial repair)', [300, 30, 135, 60, 1], [89.155, False]],
['control 1', [270, 10, 250, 60, 0.5], [52.187, False]],
['control 2', [300, 30, 180, 30, 2], [97.181, 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 |
|---|---|---|---|
| normal: on target | [0.0, True] | [0.0, True] | Passed |
| boundary: across north | [4.995, False] | [8.658, False] | Failed |
| normal: elevation only | [10.0, False] | [10.0, False] | Passed |
| regression: elevation trig terms | [75.894, False] | [44.136, False] | Failed |
| regression: elevation trig terms (partial repair) | [42.837, False] | [55.737, False] | Failed |
| control 1 | [37.676, False] | [121.081, False] | Failed |
| control 2 | [99.876, False] | [49.874, False] | Failed |
SHA-256 / 12ae00c9fb9b2081c9f2c130adf51368905793ec0046c2f3d1f7936ccf7e2e47
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(cmd_az, cmd_el, sun_az, sun_el, tol):
e1 = math.radians(cmd_el)
e2 = math.radians(sun_el)
d = math.radians(cmd_az - sun_az)
c = math.sin(e1) * math.sin(e2) + math.cos(d)
c = max(-1.0, min(1.0, c))
err = math.degrees(math.acos(c))
return [round(err, 3), err <= tol]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [340, 45, 60, 80, 0.5], [44.136, False]],
['regression: elevation trig terms (partial repair)', [340, 30, 270, 45, 1], [55.737, False]],
['control 1', [90, 10, 225, 30, 0.5], [121.081, False]],
['control 2', [340, 80, 225, 45, 0.5], [49.874, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [200, 60, 60, 45, 5], [70.03, False]],
['regression: elevation trig terms (partial repair)', [180, 60, 0, 60, 0.5], [60.0, False]],
['control 1', [180, 80, 200, 10, 5], [70.628, False]],
['control 2', [45, 60, 60, 80, 1], [20.49, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [135, 60, 45, 45, 5], [52.239, False]],
['regression: elevation trig terms (partial repair)', [270, 45, 340, 60, 5], [42.837, False]],
['control 1', [135, 80, 60, 60, 5], [28.915, False]],
['control 2', [60, 30, 315, 10, 0.5], [97.696, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [200, 30, 60, 80, 0.5], [67.839, False]],
['regression: elevation trig terms (partial repair)', [0, 45, 250, 60, 2], [60.564, False]],
['control 1', [180, 45, 45, 60, 1], [68.754, False]],
['control 2', [300, 10, 135, 45, 5], [123.357, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [270, 30, 250, 30, 5], [17.298, False]],
['regression: elevation trig terms (partial repair)', [300, 30, 135, 60, 1], [89.155, False]],
['control 1', [270, 10, 250, 60, 0.5], [52.187, False]],
['control 2', [300, 30, 180, 30, 2], [97.181, 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 |
|---|---|---|---|
| normal: on target | [0.0, True] | [0.0, True] | Passed |
| boundary: across north | [0.0, True] | [8.658, False] | Failed |
| normal: elevation only | [0.0, True] | [10.0, False] | Failed |
| regression: elevation trig terms | [29.54, False] | [44.136, False] | Failed |
| regression: elevation trig terms (partial repair) | [45.927, False] | [55.737, False] | Failed |
| control 1 | [128.337, False] | [121.081, False] | Failed |
| control 2 | [74.113, False] | [49.874, False] | Failed |
SHA-256 / dc39dc85d73026d175a0a05a7b911992b796f2745f4939e7ed89e1382f44bc12
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(cmd_az, cmd_el, sun_az, sun_el, tol):
e1 = math.radians(cmd_el)
e2 = math.radians(sun_el)
d = math.radians(cmd_az - sun_az)
c = math.sin(e1) * math.sin(e2) + math.cos(e1) * math.cos(e2) * math.cos(d)
c = max(-1.0, min(1.0, c))
err = math.degrees(math.acos(c))
return [round(err, 3), err <= tol]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [340, 45, 60, 80, 0.5], [44.136, False]],
['regression: elevation trig terms (partial repair)', [340, 30, 270, 45, 1], [55.737, False]],
['control 1', [90, 10, 225, 30, 0.5], [121.081, False]],
['control 2', [340, 80, 225, 45, 0.5], [49.874, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [200, 60, 60, 45, 5], [70.03, False]],
['regression: elevation trig terms (partial repair)', [180, 60, 0, 60, 0.5], [60.0, False]],
['control 1', [180, 80, 200, 10, 5], [70.628, False]],
['control 2', [45, 60, 60, 80, 1], [20.49, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [135, 60, 45, 45, 5], [52.239, False]],
['regression: elevation trig terms (partial repair)', [270, 45, 340, 60, 5], [42.837, False]],
['control 1', [135, 80, 60, 60, 5], [28.915, False]],
['control 2', [60, 30, 315, 10, 0.5], [97.696, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [200, 30, 60, 80, 0.5], [67.839, False]],
['regression: elevation trig terms (partial repair)', [0, 45, 250, 60, 2], [60.564, False]],
['control 1', [180, 45, 45, 60, 1], [68.754, False]],
['control 2', [300, 10, 135, 45, 5], [123.357, False]]],
[['normal: on target', [180, 45, 180, 45, 1], [0.0, True]],
['boundary: across north', [355, 30, 5, 30, 1], [8.658, False]],
['normal: elevation only', [90, 30, 90, 40, 5], [10.0, False]],
['regression: elevation trig terms', [270, 30, 250, 30, 5], [17.298, False]],
['regression: elevation trig terms (partial repair)', [300, 30, 135, 60, 1], [89.155, False]],
['control 1', [270, 10, 250, 60, 0.5], [52.187, False]],
['control 2', [300, 30, 180, 30, 2], [97.181, 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 |
|---|---|---|---|
| normal: on target | [0.0, True] | [0.0, True] | Passed |
| boundary: across north | [8.658, False] | [8.658, False] | Passed |
| normal: elevation only | [10.0, False] | [10.0, False] | Passed |
| regression: elevation trig terms | [44.136, False] | [44.136, False] | Passed |
| regression: elevation trig terms (partial repair) | [55.737, False] | [55.737, False] | Passed |
| control 1 | [121.081, False] | [121.081, False] | Passed |
| control 2 | [49.874, False] | [49.874, False] | Passed |
SHA-256 / feed4f40c0ff098eb5470b609ab9c49d8d0f2ea5136e5ddcae0c814ded8ac430
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 2026-09-29T14:51:58.743657+00:00.
Case digest / a8c71cfbb0f59565c2ec045d139f8d158448f284ba5948866c6bd21e69683cda