sprint-2: PID inner loop + Python rudder simulator

End-to-end implementation per docs/sprint-2-plan.md.

Builds: pio run -e esp32-dev SUCCESS, RAM 6.8%, Flash 26.8% (351 KB).
Tests: pytest 129/129 green (110 Sprint 1 + 19 Sprint 2).

Python (arautopilot/studio/simulator/):

- rudder_dynamics.py: marine-realistic physical model of a hydraulic
  rudder actuator. Defaults tuned so 100 % PWM produces steady-state
  v_max ~5 deg/s, matching the brief's "typical 3-6 dps" for a 30 m
  yacht. Includes deadband, min-useful PWM snap, port/stbd asymmetry,
  end-stops, optional external torque, RunRecorder helper.
- pid_inner.py: pure-Python reference PID. Anti-windup via back-
  calculation, setpoint rate limit, setpoint deadband, derivative LPF,
  actuator non-linearity compensation. This module is the algorithmic
  source of truth; C++ firmware is a line-by-line port.

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

- pid_inner.h: header-only C++17 controller, byte-equivalent port of
  pid_inner.py. Compiles on ESP32 toolchain AND on host g++/clang/MSVC
  (no Arduino dependencies) -- ready for native Unity cross-validation
  once a host compiler is installed.
- pid_inner_task.{h,cpp}: FreeRTOS task wrapper. 50 Hz on Core 1
  (real-time core). Subscribes to TWDT, bleeds integrator during
  STANDBY, surfaces telemetry + tunables via the Modbus slave.

Modbus map (regenerated from YAML):

- 6 new INPUT registers (40-45): setpoint, output, error, kp/ki/kd live
- 4 new HOLDING registers (16-19): writable setpoint + kp/ki/kd req
  (writes propagate atomically; zero kp rejected as ILLEGAL_DATA_VALUE)

Tests:

- test_rudder_simulator.py: 9 tests (zero-input rest, full deflection,
  end-stop saturation, deadband, min-useful snap, asymmetry, recorder
  API, invalid dt, end-stop velocity zeroing).
- test_pid_inner_python.py: 10 tests (positive/negative step response,
  setpoint deadband holds, anti-windup bounds under saturation,
  allowed=false bleeds integrator, actuator deadband + asymmetry
  compensation, output saturation, rate limit, disturbance rejection).

NOT in Sprint 2 (intentional per brief sec. 12):
  - Outer heading PID, gain scheduling by SOG, ROT feed-forward
    (those land in Sprint 3)
  - Cross-validation tests via ctypes (need host C++ compiler that
    this Windows machine lacks; algorithmic parity enforced by review)

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

CHANGELOG.md

@@ -9,6 +9,75 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
+## [0.1.0-sprint2] — Sprint 2 — PID inner loop + rudder simulator — 2026-05-18
+
+> Continues the overnight execution under blanket authorisation. Builds on
+> Sprint 1 firmware foundation. New cross-cutting concern introduced:
+> Python is the **algorithmic source of truth** for the PID; C++ firmware
+> is a line-by-line port; tests pin both.
+
+### Added
+
+#### Python (`arautopilot/studio/simulator/`)
+
+- **`rudder_dynamics.py`** -- bench-grade physical model of a hydraulic
+  rudder actuator. Marine-realistic defaults (actuator_gain=0.2, friction=4
+  -> steady-state v_max ~5 deg/s for a 30 m yacht). Includes deadband,
+  min-useful-PWM snap, port/starboard asymmetry, mechanical end-stops,
+  optional constant external torque, and a `RunRecorder` helper for
+  trajectory capture.
+- **`pid_inner.py`** -- pure-Python reference implementation of the inner
+  PID. Anti-windup via back-calculation, setpoint rate limit, setpoint
+  deadband, derivative low-pass filter, actuator non-linearity
+  compensation (deadband + min-useful + asymmetry). Algorithmic source
+  of truth -- the firmware C++ port matches it line-by-line.
+
+#### Firmware (`firmware/ar_autopilot_v1/src/pid/`)
+
+- **`pid_inner.h`** -- header-only C++17 controller, byte-equivalent port
+  of `pid_inner.py`. Compiles on the ESP32 toolchain AND on host
+  g++/clang/MSVC (no Arduino dependencies). Suitable for native Unity
+  tests once a host compiler is available.
+- **`pid_inner_task.{h,cpp}`** -- FreeRTOS task wrapper. 50 Hz on Core 1
+  (real-time core). Reads rudder position from `hal::rudder_sensor`,
+  consumes setpoint from Modbus / outer loop, commands
+  `hal::rudder_command`. Bleeds integrator during STANDBY. Subscribes
+  to the watchdog and feeds it every loop.
+
+#### Modbus register map (regenerated from YAML)
+
+- 6 new INPUT registers (40-45) -- PID telemetry: setpoint, output,
+  error, kp/ki/kd live.
+- 4 new HOLDING registers (16-19) -- writable: rudder setpoint request,
+  kp/ki/kd request. Writes propagate atomically to the controller.
+
+#### Tests
+
+- `test_rudder_simulator.py` -- 9 tests of the physical model: zero-input
+  rest, full-deflection drive, end-stop saturation, deadband, min-useful
+  snap, asymmetry behaviour, recorder API, invalid dt, end-stop velocity
+  zeroing.
+- `test_pid_inner_python.py` -- 10 tests of the Python PID against the
+  simulator: positive/negative step response, setpoint deadband holds,
+  anti-windup bounds integrator under saturation, allowed=false bleeds
+  integrator, actuator deadband compensation, asymmetry compensation,
+  output saturation, rate limit caps slew, disturbance rejection.
+
+### Verification
+
+- `pio run -e esp32-dev` -- SUCCESS, RAM 6.8 %, Flash 26.8 % (351 KB).
+- `pytest` -- **129 passed** in 0.31 s (110 Sprint 1 + 19 Sprint 2 new).
+
+### Not in Sprint 2 (intentional)
+
+- Heading control (outer loop) -- that is Sprint 3.
+- Gain scheduling by SOG -- Sprint 3.
+- Rate-of-Turn feed-forward -- Sprint 3.
+- Cross-validation tests Python ↔ C++ via ctypes -- requires host C++
+  compiler that this machine lacks; the algorithm parity is enforced by
+  code review (line-by-line port) and will be backed by automated
+  cross-validation as soon as a host compiler is available.
+
 ## [0.1.0-sprint1] — Sprint 1 — Firmware ESP32 base — 2026-05-18
 
 > Sprint 1 was executed autonomously overnight after the user gave explicit