MATLAB: How to rotate ANY vector to a given angle in the space

MATLABrotation vector angle 3d space

I would like to rotate the source vector to be a dot from upper view (xy plane). How is it possible in Matlab?

E.g.: There is a vector: X = [0 4]; Y = [0 4]; Z = [0 0];

This is a 45° line in the xy plane. If a rotation is made around the z-axis with 45°, then 90° around the y-axis, it gives the result, what I would like to achive in ANY vector.

Due to compatibility with other softwares, it's important to rotate only around the Y and Z axis.

Can anybody help?

Best Answer

I am assuming that what you want to know are the rotation angles that would be used to perform a Z-Y rotation sequence in order to align a general vector with the Z axis. I will assume that the general vector is defined by it's tail point, P1, and it's head point P2. This vector points from P1 to P2.

The first thing that needs to happen is to translate this vector to the origin. This is done by subtracting the tail from the head, i.e. V = P2 - P1.

We now have a vector V with it's tail at the origin. We will perform a Z rotation through the angle psi which will place the vector in the X-Z plane, as in the first figure, below. The angle of rotation is computed from the X and Y components of the vector using Tan(psi) = -Vy / Vx. Notice in the figure that using a positive value for Vy and the right-hand rule for vector rotation, the rotation angle in the figure is negative, therefore, we put a negative sign on Vy.

With the vector rotated to position (2), it is now in the X-Z plane. Now we want to rotate through the angle theta to align it with the Z axis. The angle theta is calculated according to cos(theta) = -Vz / mag(V). Once again, using the right-hand rule, this is a negative rotation angle.

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MATLAB: Accelerometer Coordinate system rotation

In general, you can construct a 3-dimensional transformation matrix using a sequence of three 2-dimensional planar rotations. First, we have to define our terms.

We define Phi as the rotation angle about the X-axis

Theta is the rotation angle about the Y axis.

Psi is the rotation angle about the Z axis.

All rotations are performed according to the right-hand rule (positive rotation is a positive right-hand curl)

Reference frame A is fixed and used to observe the motion of frame B, so frame B is said to be displaced from frame A. Let's assume frame A is the inertial frame and B is the body frame.

(NOTE that the angles are defined as the displacement of B from A . Think about it as if B starts out in alignment with A, and the three rotation angles are used to describe how B moves from A)

Now you can define the 2 dimensional transformations that will rotatre a set of coordinates from the displaced frame (B) to the fixed reference frame (A).

ROLL  = [1, 0, 0; 0, cos(Phi),-sin(Phi); 0, sin(Phi), cos(Phi)];
PITCH = [cos(Theta), 0, sin(Theta); 0, 1, 0; -sin(Theta), 0, cos(Theta)];
YAW   = [cos(Psi), -sin(Psi), 0; sin(Psi), cos(Psi) 0; 0, 0, 1];

Now the three dimentional transformation matrix may be constructed by providing the appropriate sequence of the three planar rotations. Note that the order of the rotations is not commutative, so the problem geometry will dictate what order must be used. In aerospace systems, we usually construct the attitude of an aircraft using the Z-Y-X or YAW-PITCH-ROLL sequence. In this case, the transformation matrix T = YAW * PITCH * ROLL. This is usually accompanied by a frame where Z is the "down" axis. Then any vector in the displaced frame (V_measured) can be rotated into the Inertial frame by multiplying it by T:

V_inertial = T * V_measured.

If your system is using Y as the "down" axis, you might need to alter the rotation sequence. I can't tell without being able to visualize how you move the sensor to align with the inertial frame.

Also, to rotate a vector from the inertial frame to the sensor frame (i.e. the inverse rotation) simply use the transpose of T:

V_measured = T' * V_inertial

MATLAB: What is the rotation sequence for the Euler angles in the Quaternions to Euler Angles block in the Aerospace Blockset

Documentation on the Quaternions to Euler Angles block (for the default sequence of rotation for the Euler angles) is missing from the Aerospace Blockset manual.

Here is additional information on the Quaternions to Euler Angles block:

The sequence of rotation for the Euler angles is 3-2-1. The angles are output as Roll, Pitch, Yaw (X, Y, Z).

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MATLAB: Accelerometer Coordinate system rotation

MATLAB: What is the rotation sequence for the Euler angles in the Quaternions to Euler Angles block in the Aerospace Blockset

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