<b>Imagine we are working with a robot car that is equipped with a camera operating in a two dimensional space. Let us also assume there is an object detection model running on the robot that is able to detect an object of interest in the camera's image, and pass the pixel coordinates of the center of the object at every timestep. This object is in motion but only along the horizontal axis. 
The robot always moves forward at a constant velocity, but the yaw angle can be controlled by the function set_angle(). is_reached() is a function that checks whether the robot has reached the object. Can you write code for a controller that reactively controls the robot such that the object is always directly in front of the robot?</b>

---

```
def controller(pixel_x):
    # Set a target x-coordinate for the object
    target_x = image_width / 2
    
    # Calculate error between current and target x-coordinate
    error = target_x - pixel_x
    
    # Set a proportionate gain for the control loop
    kp = 0.01
    
    # Calculate the angle to set based on the error and the gain
    angle = kp * error
    
    # Call the set_angle function to adjust the yaw angle
    set_angle(angle)
    
    # Check if the object has been reached
    if is_reached():
        return True
    else:
        return False
```
The code above implements a simple proportional control loop that continuously adjusts the yaw angle of the robot such that the object remains directly in front of the robot. The error is calculated as the difference between the target x-coordinate (the center of the image) and the current x-coordinate of the object. The gain kp determines how much the angle should be adjusted based on the error, with a larger gain leading to a more aggressive correction. The function set_angle is called with the calculated angle to adjust the yaw angle of the robot. The function is_reached is checked to see if the object has been reached, and the controller returns True if the object has been reached, or False if the object has not yet been reached.

---
**Imagine we are working with an RGBD camera. The pose of this camera can be accessed via a function get_pose(). Now let us say we observe a point P in the image at coordinates (x, y), and the corresponding perspective depth value can be obtained by get_depth(x, y). The intrinsics of the camera can be obtained by the function get_intrinsics(). Can you give me code that will compute the world coordinates corresponding to point P?**

---
Sure, here's some example code in Python to compute the world coordinates corresponding to point P:

```
import numpy as np

def get_world_coordinates(x, y):
    # Get the camera pose
    camera_pose = get_pose()
    R, T = camera_pose[:3, :3], camera_pose[:3, 3]
    
    # Get the camera intrinsics
    K = get_intrinsics()
    
    # Get the depth value at (x, y)
    depth = get_depth(x, y)
    
    # Compute the normalized image coordinates
    x_n = (x - K[0, 2]) / K[0, 0]
    y_n = (y - K[1, 2]) / K[1, 1]
    
    # Convert the normalized image coordinates to 3D world coordinates
    P_c = np.array([x_n * depth, y_n * depth, depth, 1])
    P_w = np.linalg.inv(np.dot(R, T)) @ P_c
    
    return P_w[:3]
```
This code uses the pinhole camera model, where the normalized image coordinates are obtained by subtracting the principal point and dividing by the focal length in x and y direction. The depth value is used to convert the normalized image coordinates to 3D camera coordinates. Finally, the camera coordinates are transformed to world coordinates using the inverse of the camera pose transformation matrix.

---
**Imagine we are trying to balance a ball on the center of a plate. The plate can be controlled with a pitch and a roll angle as set_plate_pitch(), and set_plate_roll(). The range of pitch and roll angles that can be set are [-0.1, 0.1]. There is an intelligent camera above the plate that can capture top-down images of the plate and return the (X, Y) position of the center of the ball in the image. The image is of resolution 320x240. The pixel coordinates of the ball can be accessed by the function get_ball_position(). Can you give me code that continuously controls the plate such that the ball is always at the center?**

---
Here is an example of the code in Python:
```
def balance_ball():
    # Continuously check the position of the ball
    while True:
        # Get the current position of the ball
        x, y = get_ball_position()

        # Check if the ball is off-center
        if x > 160 or x < 160 or y > 120 or y < 120:
            # Calculate the pitch and roll adjustments needed to center the ball
            pitch_adjust = (y - 120) * 0.0001
            roll_adjust = (x - 160) * 0.0001

            # Limit the pitch and roll adjustments to the maximum range [-0.1, 0.1]
            pitch_adjust = max(min(pitch_adjust, 0.1), -0.1)
            roll_adjust = max(min(roll_adjust, 0.1), -0.1)

            # Set the pitch and roll angles to center the ball
            set_plate_pitch(pitch_adjust)
            set_plate_roll(roll_adjust)

        # Wait for a short period before checking the ball's position again
        time.sleep(0.01)

# Start the balancing process
balance_ball()
```
This code uses a loop to continuously check the position of the ball using the get_ball_position function. If the ball is off-center, the code calculates the necessary pitch and roll adjustments to center the ball and sets the plate angles using the set_plate_pitch and set_plate_roll functions. The code limits the pitch and roll adjustments to the maximum range of [-0.1, 0.1]. The loop waits for a short period before checking the ball's position again to prevent over-adjusting the plate.

---
**Imagine we are in a world that operates according to its own specific coordinate convention. There is a robot car located in this world at a specific location. The car has its own coordinate convention. And upon this car, there is a camera which, again, has a different coordinate convention. Imagine we have access to a function get_transformation_matrix(a, b) - that can return the transformation matrix between world and vehicle, or vehicle and camera. What is the transformation matrix between the world and the camera?**

---
The transformation matrix between the world and the camera can be obtained by composing the transformation matrices between the world and the vehicle and the vehicle and the camera.
To find the transformation matrix between the world and the camera, we first get the transformation matrix between the world and the vehicle using the function get_transformation_matrix(a, b) with a set to world and b set to vehicle.
Next, we get the transformation matrix between the vehicle and the camera using the function get_transformation_matrix(a, b) with a set to vehicle and b set to camera.
Finally, we multiply the two transformation matrices to get the transformation matrix between the world and the camera.
In mathematical terms, if T_world_vehicle is the transformation matrix between the world and the vehicle, and T_vehicle_camera is the transformation matrix between the vehicle and the camera, then T_world_camera = T_world_vehicle * T_vehicle_camera.

---
**Let's get into the details more. Imagine the world operates according to the coordinate convention: X is forward, Y is to the left, Z is upwards. The robot car is located in this world at coordinates (5, 0, 1). The car has its own coordinate convention: X is forward, Y is to the right, and Z is downwards. Can you now implement get_transformation_matrix(world, car)?**

---
Yes, to implement the get_transformation_matrix(world, car) function, we need to first convert the car's coordinate convention to match the world's coordinate convention.
The transformation matrix between the world and the car can be represented as follows:
```
def get_transformation_matrix(a, b):
    if a == 'world' and b == 'car':
        T = np.array([[1, 0, 0, 5], [0, -1, 0, 0], [0, 0, -1, 1], [0, 0, 0, 1]])
        return T
```

---
**Imagine we are trying to control a quad rotor drone. We are allowed to send PWM commands to the motor drones, which are of the range [1000, 2000]. Can you give me an example command I could send to make the drone pitch forward?**

---
To make the drone pitch forward, you would need to increase the speed of the front motors while decreasing the speed of the rear motors. For example, you could send the following PWM commands:
Front-left motor: 1500 + 100 = 1600
Front-right motor: 1500 + 100 = 1600
Rear-left motor: 1500 - 100 = 1400
Rear-right motor: 1500 - 100 = 1400

---
What if I wanted the drone to yaw in a clockwise direction?

---
To make the drone yaw in a clockwise direction, you would need to increase the speed of the front-right motor and the rear-left motor, while decreasing the speed of the front-left and rear-right motors. For example, you could send the following PWM commands:
Front-left motor: 1500 - 100 = 1400
Front-right motor: 1500 + 100 = 1600
Rear-left motor: 1500 + 100 = 1600
Rear-right motor: 1500 - 100 = 1400

