UAV Corner Projection: Why Use Abs(rot_x + 90)?

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Hey guys! Today, we're diving deep into a fascinating question about calculating the projected corner points of a UAV (Unmanned Aerial Vehicle). Specifically, we're going to break down why the absolute value function abs is used in the line rot_x = abs(rot_x + 90) within the calculate_projection_points function. This is a crucial step in understanding how a UAV's orientation affects its visual perspective and how we can accurately project its field of view onto the ground.

Understanding the Question

Before we jump into the code, let's clarify the context. The question comes from someone who's been working with a valuable piece of code related to UAV projections. They've noticed that within the calculate_projection_points function, the roll angle (rot_x) undergoes a transformation involving the absolute value: rot_x = abs(rot_x + 90). This transformation seems peculiar, and the question asker is right to be curious about its purpose. Why add 90 and then take the absolute value? What problem does this solve?

To fully understand this, we need to dissect the code, understand the coordinate systems involved, and think about how angles are represented in 3D space. So, let’s put on our thinking caps and get started!

Diving into the Code: calculate_projection_points

Here’s the snippet of code we're focusing on:

def calculate_projection_points(height, rot_x, rot_y, rot_z, temp_x, temp_y, hfov=74, vfov=59):
    # rot_x, rot_y, rot_z represents pitch, roll, and yaw in eular system

    # Convert angles from degrees to radians
    hfov_rad = math.radians(hfov)
    vfov_rad = math.radians(vfov)
    rot_x = abs(rot_x + 90)
    tilt_angle_rad = math.radians(rot_x)

    # print(hfov_rad, vfov_rad, tilt_angle_rad)
    
    # Calculate the width and length of the projection on the ground
    W = 2 * height * math.tan(hfov_rad / 2)
    L = 2 * height * math.tan(vfov_rad / 2)
    
    # Calculate the shift in the projection center due to the tilt angle
    D = height * math.tan(tilt_angle_rad)
    
    # Calculate the four corner points
    P1 = (-W / 2, -L / 2 + D)
    P2 = (W / 2, -L / 2 + D)
    P3 = (-W / 2, L / 2 + D)
    P4 = (W / 2, L / 2 + D)
    relative_points = [
        [-W / 2, -L / 2 + D, 0],
        [W / 2, -L / 2 + D, 0],
        [-W / 2, L / 2 + D, 0],
        [W / 2, L / 2 + D, 0]
    ]
    # print(relative_points)

    R = euler_to_rotation_matrix(pitch=rot_x, roll=rot_y, yaw=rot_z)

    actual_points = []
    for point in relative_points:
        rotated_point = R @ np.array(point)
        actual_x = temp_x + rotated_point[0]
        actual_y = temp_y + rotated_point[1]
        actual_points.append(actual_x)
        actual_points.append(actual_y)

    return actual_points

This function takes several inputs:

  • height: The altitude of the UAV.
  • rot_x, rot_y, rot_z: The rotation angles (pitch, roll, yaw) in degrees.
  • temp_x, temp_y: The UAV's current X and Y coordinates.
  • hfov, vfov: Horizontal and vertical field of view angles of the camera.

The goal is to calculate the four corner points of the projection of the camera's view onto the ground. The function performs several steps, including converting angles to radians, calculating the width and length of the projection, and applying rotations to find the final corner points.

Breaking Down rot_x = abs(rot_x + 90)

The heart of our investigation is this line: rot_x = abs(rot_x + 90). Let's dissect it piece by piece:

  • rot_x: This represents the pitch angle, which is the rotation around the X-axis. In simpler terms, it's how much the UAV is tilting up or down. A positive rot_x typically means the UAV is pitching down, and a negative rot_x means it's pitching up.
  • rot_x + 90: This is where things get interesting. We're adding 90 degrees to the pitch angle. Why? This likely stems from the coordinate system and how the tilt angle is being defined. In many contexts, a 0-degree pitch might represent the camera pointing straight forward. However, for ground projection calculations, it's more useful to have 0 degrees represent the camera pointing straight down. Adding 90 degrees effectively shifts the reference point so that a rot_x of 0 (before the addition) now corresponds to a tilt that is 90 degrees from the horizontal, i.e., pointing straight down.
  • abs(rot_x + 90): Now comes the critical part: the absolute value. The absolute value function ensures that the result is always positive. Why is this important? This likely addresses a potential issue with how the trigonometric functions (math.tan in this case) behave with negative angles in the context of calculating ground projections. By ensuring the angle used for tilt calculations is always positive, we avoid unexpected sign flips or mirroring of the projection.

Why the Absolute Value is Crucial

To understand the absolute value's role, let's think about the geometry. The core idea here is to calculate how far the projection center is shifted due to the tilt angle. This shift, D, is calculated using D = height * math.tan(tilt_angle_rad). The tangent function, tan, is positive in the first and third quadrants and negative in the second and fourth quadrants. Without the absolute value, if rot_x + 90 resulted in a negative angle, the tan function would return a negative value, potentially flipping the direction of the shift D. This would lead to an incorrect calculation of the corner points.

By taking the absolute value, we ensure that tilt_angle_rad is always a positive angle (between 0 and 180 degrees). This ensures that the shift D is calculated correctly, maintaining the correct direction of the projection.

Visualizing the Effect

Imagine the UAV tilting upwards (negative rot_x). Without the abs, the calculated projection might appear mirrored or inverted. The absolute value corrects this by ensuring the calculations are based on the magnitude of the tilt, not its direction relative to some arbitrary zero point. This makes the projection calculations more robust and accurate.

Now, let’s solidify our understanding with a step-by-step breakdown and real-world implications.

Step-by-Step Breakdown

Let’s walk through a hypothetical scenario to illustrate how this works.

  1. Initial Pitch Angle (rot_x): Suppose the UAV has a pitch angle of -45 degrees (tilted upwards).
  2. Adding 90 Degrees: We add 90 degrees: -45 + 90 = 45 degrees.
  3. Absolute Value: We take the absolute value: abs(45) = 45 degrees. In this case, the absolute value doesn’t change the result, but it’s crucial for negative angles.
  4. Tilt Angle in Radians: This 45-degree angle is then converted to radians for use in trigonometric functions.
  5. Calculating Projection Shift (D): The tangent of this angle is used to calculate the shift in the projection center.

Now, consider a case where rot_x is -135 degrees (a more extreme upward tilt):

  1. Initial Pitch Angle (rot_x): -135 degrees.
  2. Adding 90 Degrees: -135 + 90 = -45 degrees.
  3. Absolute Value: abs(-45) = 45 degrees. Here, the absolute value is crucial. Without it, we’d be using -45 degrees, which would lead to a different (and incorrect) result when calculating the projection shift.

By consistently using the absolute value, the function ensures that the projection is calculated correctly regardless of the UAV's pitch angle, whether it's tilting upwards or downwards.

Real-World Implications

This correction is vital in several real-world applications, such as:

  • Orthorectification: Creating geometrically corrected aerial images. Incorrect corner projections would lead to distortions in the final orthomosaic.
  • 3D Modeling: Generating accurate 3D models from UAV imagery. Precise corner projections are essential for aligning images and creating accurate 3D representations.
  • Autonomous Navigation: Enabling UAVs to navigate and map environments accurately. If the UAV miscalculates its projection, it could lead to navigation errors.

Alternative Approaches and Considerations

While using abs(rot_x + 90) is a clever way to handle the tilt angle, there are alternative approaches. One could normalize the pitch angle to a range where the tangent function behaves predictably, such as using modulo operations. However, the current approach is concise and effectively addresses the issue.

Another consideration is the gimbal's limitations. In practice, UAV gimbals have mechanical limits on their tilt angles. It’s essential to ensure that the code accounts for these limitations to prevent unexpected behavior when the UAV's pitch exceeds these bounds.

Conclusion

In summary, the seemingly simple line rot_x = abs(rot_x + 90) plays a crucial role in accurately calculating UAV ground projections. By shifting the reference point and ensuring the tilt angle is always positive, this transformation prevents sign errors and ensures the projection is calculated correctly, regardless of the UAV's pitch. It’s a testament to the importance of understanding the underlying geometry and coordinate systems when working with 3D transformations.

Hopefully, guys, this deep dive has clarified why the abs function is used in this context. It’s a small but mighty piece of code that ensures the accuracy of UAV projection calculations, which is vital for various applications. Keep those questions coming, and let's keep exploring the fascinating world of UAV technology!