FA-93701 / Solar tracker geometry / Open access

Tilted single-axis tracker frame: axis vector tilt · case 01

Incidence angles on tilted-axis rows are underestimated in the morning.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The axis vector used for incidence ignores the axis tilt.

THE FAILURE

The axis vector used for incidence ignores the axis tilt.

Unsuccessful approach: Tilting the axis downward reverses the along-axis component.

Case contract

The axis points toward axis_az rising by axis_tilt: A = (sin a cos b, cos a cos b, sin b). The zero-rotation panel normal is N0 = (-sin a sin b, -cos a sin b, cos b) and positive rotation turns toward R = (cos a, -sin a, 0). With unit sun vector s, rotation = atan2(s.R, s.N0) and the incidence at ideal tracking is asin(|s.A|). Return [rotation, incidence] rounded to 3, or 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(sun_az, sun_el, axis_az, axis_tilt):
    if sun_el <= 0:
        return None
    az = math.radians(sun_az)
    el = math.radians(sun_el)
    a = math.radians(axis_az)
    b = math.radians(axis_tilt)
    s = [math.cos(el) * math.sin(az), math.cos(el) * math.cos(az), math.sin(el)]
    A = [math.sin(a), math.cos(a), 0.0]
    N0 = [-math.sin(a) * math.sin(b), -math.cos(a) * math.sin(b), math.cos(b)]
    R = [math.cos(a), -math.sin(a), 0.0]
    dot = lambda u, v: sum(p * q for p, q in zip(u, v))
    theta = math.degrees(math.atan2(dot(s, R), dot(s, N0)))
    aoi = math.degrees(math.asin(min(1.0, abs(dot(s, A)))))
    return [round(theta, 3), round(aoi, 3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [0, 89, 180, 5], [-0.0, 4.0]],
  ['regression: axis vector tilt (partial repair)', [90, 75, 170, 10], [-15.118, 12.239]],
  ['control 1', [300, 0, 180, 0], None], ['control 2', [225, 3, 180, 20], [105.236, 42.957]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [225, 89, 180, 5], [0.711, 5.707]],
  ['regression: axis vector tilt (partial repair)', [135, 3, 0, 20], [67.625, 40.215]],
  ['control 1', [45, 60, 170, 10], [-24.406, 7.588]],
  ['control 2', [135, 75, 170, 20], [-10.079, 31.978]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [225, 30, 180, 10], [57.771, 43.622]],
  ['regression: axis vector tilt (partial repair)', [315, 10, 170, 10], [61.157, 49.845]],
  ['control 1', [340, 45, 170, 20], [7.747, 24.363]],
  ['control 2', [340, 10, 180, 20], [35.076, 54.117]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [180, 10, 180, 10], [180.0, 90.0]],
  ['regression: axis vector tilt (partial repair)', [180, 45, 180, 20], [0.0, 65.0]],
  ['control 1', [340, 20, 180, 0], [43.219, 62.009]],
  ['control 2', [30, 3, 0, 20], [116.285, 56.159]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [135, 20, 170, 5], [-63.084, 52.81]],
  ['regression: axis vector tilt (partial repair)', [225, 10, 0, 10], [-67.255, 40.968]],
  ['control 1', [100, 30, 0, 5], [59.062, 6.098]],
  ['control 2', [135, 75, 180, 10], [-11.257, 20.363]]]]
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: horizontal axis at noon	`[0.0, 30.0]`	`[0.0, 30.0]`	Passed
normal: tilted axis morning	`[-71.564, 9.391]`	`[-71.564, 12.714]`	Failed
boundary: night	`None`	`None`	Passed
regression: axis vector tilt	`[-0.0, 1.0]`	`[-0.0, 4.0]`	Failed
regression: axis vector tilt (partial repair)	`[-15.118, 2.576]`	`[-15.118, 12.239]`	Failed
control 1	`None`	`None`	Passed
control 2	`[105.236, 44.922]`	`[105.236, 42.957]`	Failed

SHA-256 / 498898516afe85b6322f4c985f28dc6ed84239e9d0bd55e7b5060f82a795e050

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, axis_tilt):
    if sun_el <= 0:
        return None
    az = math.radians(sun_az)
    el = math.radians(sun_el)
    a = math.radians(axis_az)
    b = math.radians(axis_tilt)
    s = [math.cos(el) * math.sin(az), math.cos(el) * math.cos(az), math.sin(el)]
    A = [math.sin(a) * math.cos(b), math.cos(a) * math.cos(b), -math.sin(b)]
    N0 = [-math.sin(a) * math.sin(b), -math.cos(a) * math.sin(b), math.cos(b)]
    R = [math.cos(a), -math.sin(a), 0.0]
    dot = lambda u, v: sum(p * q for p, q in zip(u, v))
    theta = math.degrees(math.atan2(dot(s, R), dot(s, N0)))
    aoi = math.degrees(math.asin(min(1.0, abs(dot(s, A)))))
    return [round(theta, 3), round(aoi, 3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [0, 89, 180, 5], [-0.0, 4.0]],
  ['regression: axis vector tilt (partial repair)', [90, 75, 170, 10], [-15.118, 12.239]],
  ['control 1', [300, 0, 180, 0], None], ['control 2', [225, 3, 180, 20], [105.236, 42.957]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [225, 89, 180, 5], [0.711, 5.707]],
  ['regression: axis vector tilt (partial repair)', [135, 3, 0, 20], [67.625, 40.215]],
  ['control 1', [45, 60, 170, 10], [-24.406, 7.588]],
  ['control 2', [135, 75, 170, 20], [-10.079, 31.978]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [225, 30, 180, 10], [57.771, 43.622]],
  ['regression: axis vector tilt (partial repair)', [315, 10, 170, 10], [61.157, 49.845]],
  ['control 1', [340, 45, 170, 20], [7.747, 24.363]],
  ['control 2', [340, 10, 180, 20], [35.076, 54.117]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [180, 10, 180, 10], [180.0, 90.0]],
  ['regression: axis vector tilt (partial repair)', [180, 45, 180, 20], [0.0, 65.0]],
  ['control 1', [340, 20, 180, 0], [43.219, 62.009]],
  ['control 2', [30, 3, 0, 20], [116.285, 56.159]]],
 [['normal: horizontal axis at noon', [180, 60, 180, 0], [0.0, 30.0]],
  ['normal: tilted axis morning', [100, 20, 180, 10], [-71.564, 12.714]],
  ['boundary: night', [270, 0, 180, 10], None],
  ['regression: axis vector tilt', [135, 20, 170, 5], [-63.084, 52.81]],
  ['regression: axis vector tilt (partial repair)', [225, 10, 0, 10], [-67.255, 40.968]],
  ['control 1', [100, 30, 0, 5], [59.062, 6.098]],
  ['control 2', [135, 75, 180, 10], [-11.257, 20.363]]]]
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: horizontal axis at noon	`[0.0, 30.0]`	`[0.0, 30.0]`	Passed
normal: tilted axis morning	`[-71.564, 5.814]`	`[-71.564, 12.714]`	Failed
boundary: night	`None`	`None`	Passed
regression: axis vector tilt	`[-0.0, 6.0]`	`[-0.0, 4.0]`	Failed
regression: axis vector tilt (partial repair)	`[-15.118, 7.092]`	`[-15.118, 12.239]`	Failed
control 1	`None`	`None`	Passed
control 2	`[105.236, 40.215]`	`[105.236, 42.957]`	Failed

SHA-256 / ddb4b0d2894c34d53963f9bad0edf13a02e949cf42f1ac29ef62a102f4272cc1

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:51:57.500491+00:00.

Case digest / 5ec7475ec98805221c9cb679e50961e17701a330c723ae4bbaa7732a0780333b