sprint-3: PID outer + Heading Hold + ROT feed-forward + gain scheduling

End-to-end implementation per docs/sprint-3-plan.md.

Closes the cascade: outer loop (heading control, 10 Hz on Core 1) drives
the inner loop (rudder position control, 50 Hz from Sprint 2). First real
mode other than STANDBY is now activable: HEADING_HOLD.

Builds: pio run -e esp32-dev SUCCESS, RAM 6.8%, Flash 27.1% (355 KB).
Tests: pytest 258/258 green (231 Sprint 2.5 + 27 Sprint 3 new).

Python (arautopilot/studio/simulator/):

- vessel_heading.py: first-order yaw model. ROT responds to
  rudder*speed; damping returns ROT to zero when rudder is centred.
  Defaults tuned so 5 deg rudder @ 10 kn -> ~3 dps steady-state ROT.
  Includes heading_error_deg() shortest-arc helper.
- pid_outer.py: pure-Python outer heading PID. Anti-windup via back-
  calculation, gain scheduling by SOG, deadband, derivative LPF,
  output saturation, ROT feed-forward (brief sec. 6 -- the term that
  distinguishes a premium autopilot from a basic one), rate limit on
  produced rudder setpoint, shortest-arc heading wrap-around.

Firmware (firmware/ar_autopilot_v1/src/pid/):

- pid_outer.h: header-only C++17 port. Same algorithm, same variables,
  same numerics. Fixed-capacity gain schedule (up to 8 points).
- pid_outer_task.{h,cpp}: 10 Hz FreeRTOS task on Core 1. Subscribes to
  TWDT. Reads heading + ROT from the NMEA 2000 snapshot. Uses
  operator-configurable SOG (default 15 kn until PGN 129026 wiring in
  Sprint 5). Pushes rudder setpoint into the inner loop only when
  current_mode == HEADING_HOLD.

Modes (firmware/ar_autopilot_v1/src/modes/standby.cpp):

- HEADING_HOLD activable via request_mode(). Pre-conditions:
    * NMEA 2000 heading sensor valid (fresh PGN 127250)
    * Rudder sensor valid (median filter filled)
  On success, captures current heading as initial setpoint so the
  operator doesn't get a sudden swing toward an old setpoint.

Modbus (regenerated from YAML):

- 7 new INPUTs (50-56): outer heading setpoint, produced rudder
  setpoint, error, current SOG, live kp/ki/kd.
- 5 new HOLDINGs (24-28): writable heading setpoint, SOG override,
  outer base gains. Writing any of kp/ki/kd disables the built-in
  3-point gain schedule (operator override).

Tests:

- test_vessel_heading_simulator.py: 6 dynamics tests + 9 parameterised
  heading_error_deg edge cases (wrap-around).
- test_pid_outer_python.py: 12 tests covering gain interpolation,
  per-tick PID behaviour (deadband, sign, ROT feed-forward,
  saturation, rate limit, allowed=false), and three end-to-end cascade
  tests (positive step, negative step, wrap-around 360->10).

Cascade verification: outer + inner + rudder dynamics + vessel-heading
simulator settles a 30 deg step within +-2 deg in 60 s.

NOT in Sprint 3 (intentional):
  - True Course / Track Keeping / Dodge -- Sprint 5
  - Off-course alarms + auto-disengage on sensor loss -- Sprint 6
  - COG / SOG / Position via N2K PGN 129025/9/6 -- Sprint 5

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

arautopilot/tests/test_vessel_heading_simulator.py

@@ -0,0 +1,85 @@
+"""Tests for the vessel-heading simulator."""
+
+from __future__ import annotations
+
+import pytest
+
+from arautopilot.studio.simulator.vessel_heading import (
+    VesselHeadingConfig,
+    VesselHeadingSimulator,
+    heading_error_deg,
+)
+
+
+def test_zero_rudder_holds_heading() -> None:
+    sim = VesselHeadingSimulator()
+    sim.reset(heading_deg=42.0)
+    for _ in range(2000):
+        sim.step(dt=0.01, rudder_deg=0.0)
+    assert sim.state.heading_deg == pytest.approx(42.0, abs=1e-3)
+
+
+def test_positive_rudder_turns_starboard() -> None:
+    sim = VesselHeadingSimulator()
+    sim.reset(heading_deg=0.0)
+    for _ in range(2000):
+        sim.step(dt=0.01, rudder_deg=5.0)
+    # After 20 s with +5 deg of rudder, heading should advance (mod 360).
+    assert sim.state.rate_of_turn_dps > 0.0
+    assert sim.state.heading_deg != 0.0
+
+
+def test_negative_rudder_turns_port() -> None:
+    sim = VesselHeadingSimulator()
+    sim.reset(heading_deg=0.0)
+    for _ in range(2000):
+        sim.step(dt=0.01, rudder_deg=-5.0)
+    assert sim.state.rate_of_turn_dps < 0.0
+
+
+def test_speed_increases_yaw_response() -> None:
+    sim_slow = VesselHeadingSimulator(VesselHeadingConfig(speed_kn=5.0))
+    sim_fast = VesselHeadingSimulator(VesselHeadingConfig(speed_kn=20.0))
+    sim_slow.reset()
+    sim_fast.reset()
+    for _ in range(2000):
+        sim_slow.step(dt=0.01, rudder_deg=5.0)
+        sim_fast.step(dt=0.01, rudder_deg=5.0)
+    # Fast vessel turns farther in the same time.
+    assert abs(sim_fast.state.rate_of_turn_dps) > abs(sim_slow.state.rate_of_turn_dps)
+
+
+def test_heading_wraps_at_360() -> None:
+    sim = VesselHeadingSimulator()
+    sim.reset(heading_deg=359.0, rate_of_turn_dps=10.0)
+    for _ in range(20):
+        sim.step(dt=0.1, rudder_deg=0.0)
+    # heading must remain in [0, 360)
+    assert 0.0 <= sim.state.heading_deg < 360.0
+
+
+def test_invalid_dt() -> None:
+    sim = VesselHeadingSimulator()
+    sim.reset()
+    with pytest.raises(ValueError):
+        sim.step(dt=0.0, rudder_deg=5.0)
+
+
+# ----------------------------------------------------------------------------
+# heading_error_deg
+# ----------------------------------------------------------------------------
+
+
+@pytest.mark.parametrize("sp, meas, expected", [
+    (90.0, 80.0, 10.0),
+    (80.0, 90.0, -10.0),
+    (0.0, 0.0, 0.0),
+    (0.0, 359.0, 1.0),         # crossing 0 going stbd
+    (359.0, 0.0, -1.0),
+    (180.0, 0.0, 180.0),
+    (0.0, 180.0, 180.0),       # ambiguity at 180 -- convention is +180
+    (10.0, 350.0, 20.0),
+    (350.0, 10.0, -20.0),
+])
+def test_heading_error_shortest_arc(sp: float, meas: float, expected: float) -> None:
+    assert heading_error_deg(sp, meas) == pytest.approx(expected, abs=1e-9)