Most Efficient way to solve for Euler Angles

• Robconway
In summary, the conversation is discussing the task of finding the most efficient way to solve for the euler angles in order to sync a laser with a fixed target point in 3 dimensional space. The values needed to solve this problem are the direct distance from the laser to the target, the distance from the laser to the initial point, and the distance from the initial point to the fixed target. However, more or different information may be needed to fully solve the problem, such as the cartesian coordinates or yaw, pitch, and roll of the laser and target. The conversation also provides a formula for calculating the length and unit vector of a direction in 3 dimensional space, and explains that the Euler angles can be found by taking the arccosines
Robconway

Essentially, a laser is pointing at a certain point in 3 dimensional space. There is a fixed target that the laser is specifically supposed to point to. My job is to find the most efficient way to solve for the euler angles(yaw pitch and roll) in order to sync the laser with the target point. What is the best way to do this? The values known are the direct distance from the laser to to the target, the distance from the laser to the initial point, and the distance from the initial point to the fixed target. Thank you guys, it would help a lot!

You need more or different info. One set of info that would be sufficient is the cartesian coordinates of the initial point and the target with respect to the laser plus the yaw, pitch, and roll of the laser in that coordinate system. Another sufficient set is yaw, pitch, and roll of the laser plus the yaw and pitch of the target position with respect to some fixed coordinate system. It's not clear exactly what your particular situation is, but the info you've given is not sufficient to solve the problem.

A vector pointing from $(x_0, y_0, z_0)$ to $(x_1, y_1, z_1)$ can be written as $\vec{v}= (x_1- x_0)\vec{i}+ (y_1- y_0)\vec{j}+ (z_1- z_0)\vec{j}$. Its length is
$$|\vec{v}|= r= \sqrt{(x_1-x_0)^2+ (y_1- y_0)^2+ (z_1- z_0)2}$$

Finally, the unit vector in that direction is
$$\frac{\vec{v}}{|\vec{v}|}= \frac{x_1- x_0}{r}\vec{i}+ \frac{y_1- y_0}{r}\vec{j}+ \frac{z_1- z_0}{r}\vec{k}$$

So the Euler angles are given by
$$cos(\theta_x)= \frac{x_1- x_0}{r}$$
$$cos(\theta_y)= \frac{y_1- y_0}{r}$$
$$cos(\theta_z)= \frac{z_1- z_0}{r}$$

That is, the Euler angles for a given direction are the arccosines of the components of a unit vector in that direction.

1. How do you define Euler angles?

Euler angles are a set of three rotations that describe the orientation of an object in three-dimensional space. Each rotation is performed around one of the object's axes, usually the x, y, and z axes.

2. What is the most efficient way to solve for Euler angles?

The most efficient way to solve for Euler angles depends on the specific problem and context. In general, it is important to have a clear understanding of the coordinate system and the order in which the rotations are applied. Various mathematical methods, such as matrix transformations or quaternion rotations, can also be used to solve for Euler angles.

3. What are some common applications of Euler angles?

Euler angles are commonly used in robotics, aerospace engineering, and computer graphics to describe the orientation of objects in three-dimensional space. They are also used in physics to describe the motion of rigid bodies.

4. How do you convert between Euler angles and other rotation representations?

There are several methods for converting between Euler angles and other rotation representations, such as quaternions or rotation matrices. These methods involve complex mathematical calculations and may vary depending on the specific representation and coordinate system used.

5. Can Euler angles lead to gimbal lock?

Yes, Euler angles can lead to gimbal lock, which is a situation where one of the rotational axes aligns with another, causing a loss of one degree of freedom. This can occur when the rotations are performed in a specific order and can be avoided by using alternative rotation representations, such as quaternions.

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