.

²²²

Given the controls -1 to 1 for pitch yaw and roll, the following c++ code applies the controls as a torque then limits the angular velocity which eventually changes the 3D rotation of an object.

```c++
constexpr Vec AIR_CONTROL_TORQUE = Vec(130, 95, 400);
constexpr float CAR_MAX_ANG_SPEED = 5.5f;

torque = (controls.pitch * dirPitch_right * AIR_CONTROL_TORQUE.x) + (controls.yaw * dirYaw_up * AIR_CONTROL_TORQUE.y) + (controls.roll * dirRoll_forward * AIR_CONTROL_TORQUE.z);

Body.applyTorque(_rigidBody.m_invInertiaTensorWorld.inverse() * (torque - damping) * CAR_TORQUE_SCALE);

// ...

if (angVel.length2() > (CAR_MAX_ANG_SPEED * CAR_MAX_ANG_SPEED))
	angVel = angVel.normalized() * CAR_MAX_ANG_SPEED;
```

From a starting rotation and angular velocity, determine the sequence of inputs to reach a desired rotation and come to a full stop in the fewest number of steps.

Implement it within the following testing environment:

```python
import RocketSim as rs
import math
import numpy as np
from random import randint, choice, seed


TICK_RATE = 120
TICK_SKIP = 1
AIR_PD_MULTIPLIER = 1.5
THRESHOLD = 0.01
SEED = 2

seed(SEED)

CAR_TORQUE_SCALE = 0.09587 * AIR_PD_MULTIPLIER
CAR_AIR_CONTROL_TORQUE_PITCH = 130
CAR_AIR_CONTROL_TORQUE_YAW = 95
# CAR_AIR_CONTROL_TORQUE_ROLL = 400


def pfs(pitchspeed):
    """time until pitch full stop"""
    return abs(pitchspeed) / (CAR_TORQUE_SCALE * CAR_AIR_CONTROL_TORQUE_PITCH)


def yfs(yawspeed):
    """time until yaw full stop"""
    return abs(yawspeed) / (CAR_TORQUE_SCALE * CAR_AIR_CONTROL_TORQUE_YAW)


def sign(x):
    """Retuns 1 if x > 0 else -1."""
    return 1 if x > 0 else -1


def clamp(x, r=1):
    if x < -r:
        return -r
    elif x > r:
        return r
    else:
        return x


def circle_wrap(a, pi=math.pi):
    """Limits any angle a to [-pi, pi] range"""
    if abs(a) >= 2 * pi:
        a -= abs(a) // (2 * pi) * 2 * pi * sign(a)
    if abs(a) > pi:
        a -= 2 * pi * sign(a)
    return a


def update_controls(arena, car_id):
	"""Placeholder code for a better solution"""
    car = arena.get_car_from_id(car_id)

    car_state = car.get_state()
    ball_state = arena.ball.get_state()

    ball_local_pos = car_state.rot_mat.as_numpy().dot(
        ball_state.pos.as_numpy() - car_state.pos.as_numpy()
    )

    distance = np.linalg.norm(ball_local_pos)
    azimuth = math.atan2(ball_local_pos[1], ball_local_pos[0])
    inclination = -math.acos(ball_local_pos[2] / distance) + math.pi / 2

    car_local_ang_vel = car_state.rot_mat.as_numpy().dot(
        car_state.ang_vel.as_numpy()
    )
    yaw_ang_vel = -car_local_ang_vel[2]
    pitch_ang_vel = car_local_ang_vel[1]

    yaw = math.copysign(1, circle_wrap(azimuth + yfs(yaw_ang_vel) * yaw_ang_vel))
    pitch = math.copysign(
        1, circle_wrap(inclination + pfs(pitch_ang_vel) * pitch_ang_vel)
    )
    
    yaw *= clamp(10 * arena.tick_rate * abs(azimuth + yaw_ang_vel * arena.tick_time), 1)

    pitch *= clamp(10 * arena.tick_rate * abs(inclination + pitch_ang_vel * arena.tick_time), 1)

    if (
        abs(azimuth) + abs(yaw_ang_vel) + abs(inclination) + abs(pitch_ang_vel)
        < THRESHOLD
    ):
        return False

    car.set_controls(rs.CarControls(yaw=yaw, pitch=pitch))

    return True


def main():
    # init rocketsim
    meshes_path = "collision_meshes"
    rs.init(meshes_path)

    # setup rocketsim arena
    arena_config = rs.ArenaConfig()
    arena = rs.Arena(rs.GameMode.SOCCAR, tick_rate=TICK_RATE, config=arena_config)

    # set mutators
    mutator_config = arena.get_mutator_config()
    # mutator_config.boost_used_per_second = 0  # infinite boost
    mutator_config.gravity = rs.Vec()
    arena.set_mutator_config(mutator_config)

    # set goal score callback
    arena.set_goal_score_callback(lambda *args, **kwargs: arena.reset_kickoff(), None)

    # setup rocketsim cars
    team = rs.Team.BLUE
    car = arena.add_car(team, rs.CarConfig(0))

    for i in range(1000):
        if i % 10 == 0:
            print(i)
        car.set_state(
            rs.CarState(
                pos=rs.Vec(
                    x=randint(-1000, 1000),
                    y=randint(-1000, 1000),
                    z=1000,
                ),
                ang_vel=rs.Vec(
                    5.5 * randint(-1, 1),
                    5.5 * randint(-1, 1),
                    5.5 * randint(-1, 1),
                ),
            )
        )

        # setup ball state
        ball = arena.ball
        ball.set_state(
            rs.BallState(pos=rs.Vec(z=choice([randint(100, 800), randint(1200, 2000)])))
        )

        while update_controls(arena, car.id):
            arena.step(TICK_SKIP)

    print(SEED, THRESHOLD, TICK_SKIP, AIR_PD_MULTIPLIER, arena.tick_count)


if __name__ == "__main__":
    main()

```

Can you come up with a better solution?