Given an angular velocity represented by the vector [x, y, z] where the magnitude of the vector represents the magnitude of the angular velocity in radians/s and the direction of the vector represents the axis in which the rotation is applied, how do I convert this vector into body-fixed euler angular velocity (roll_d, pitch_d, yaw_d)?
For infinitesimally small timesteps body-fixed angular velocity and "Euler angular velocity" are the same.
To convert from world-frame angular velocity to body-fixed angular velocity you need to rotate the vector from the world frame to the body frame, using the current orientation of your vehicle.
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I have a given 3x3 rotation matrix and I want to calculate the rotation angle around z axis. How do I get there?
For example, in this case below, how did they calculated the "-30deg rotation around the x axis"? Or how did they get to the "-74deg" value around that axis?
This is my original matrix:
Thank you!
It is simple if the rotation matrix is just a rotation matrix and there is no scaling. Here is a site that explains in more pretty terms then I am willing to diagram here. Basically the rotation matrix is composed of sinf(x) and cosf(x) of euler angles (well you can think of it like that at least). You can therefore use values within it to back calculate the euler angles.
http://nghiaho.com/?page_id=846
If you have scaling involved you will need to normalize each row of the matrix first. Then apply the above method.
I have a rigid body whose centre of mass is located at position p relative to origin of fixed global reference frame. ux,uy and uz are three orthogonal unit vectors relative to origin of the fixed global reference frame. These unit vectors represent the local coordinate frame of the rigid body. The local coordinate frame follows linear and rotational motion of the rigid body.
How can I build an initial quaternion for this rigid body in its current orientation ?
"ux,uy and uz are three orthogonal unit vectors relative to origin of the fixed global reference frame."
It is exactly the 3x3 rotation matrix. You can just convert from this 3x3 matrix to quaternion, using good known implementation.
I would like to find a solution for taking a rotation represented as a matrix and then resetting one of it's components. Basically I want to be able to multiply a vector by this matrix and get a direction that is rotation around x and z axis and be constant along the y axis (up). I want to take object rotation and get the vector that represents gravity but in object local space and disregarding the yaw. So I want to reset the yaw.
I don't want to convert this to euler angles. I would prefer using a quaternion or doing some sequence of operations on the rotation matrix directly in order to avoid possible bugs with certain angles.
Ok, so I have the follwoing:
btTransform t;
mBody->getMotionState()->getWorldTransform(t);
btMatrix3x3 trans = t.getBasis().inverse();
btVector3 up = (trans * btVector3(0, 1, 0));
I realized that if I used quaternion then I got completely wrong results (why?). Now I'm getting a vector in object space that represents up vector in world space. BUT I want to rotate this vector so that it represents global up vector in object space WHEN MODEL HAS ZERO rotation around Y axis. So I have to somehow rotate this vector back. How?
You can use quaternion swing twist decomposition with passed "Y" axis. It will decompose quaternion to rotation around Y axis and rotation around axis that is perpendicular to Y.
It is described here, in my answer.
Component of a quaternion rotation around an axis
I have a 3d object which is free to rotate along x,y and z axis and it is then saved as a transform matrix. In a case where the sequence of rotation is not known and the object is rotated for more than 3 times (eg :-if i rotate the object x-60degress, y-30 degrees, z-45 degrees then again x->30 degrees), is it possible to extract the angles rotated from the transform matrix?.I know that it is possible to get angles if the sequence of rotation is known, but if I have only the final transform matrix with me and nothing else, is it possible to get the angles rotated(x,y,and z) from the transform matrix ?
Euler angle conversion is a pretty well known topic. Just normalize the matrix orientation vectors and then use something like this c source code.
The matrix is the current state of things it has no knowledge of what the transformation has been in the past. It does not know how the matrix was built. You can just take the matrix into and decompose it into any pieces you like, as long as:
The data do not overlap. For example:Two X turns after each other is indistinguishable form each other (no way to know if its 1 2 or three different rotations summed).
The sequence order is known
A decomposition can be built out of the data (for example scale can be measured)
I'm trying to make a rubik cube game in webgl using three.js (you can try it here).
And I have problems to detect on witch axis I have to rotate my cube according the rotation of the cube. For instance, if the cube is in original position/rotation, if I want to rotate the left layer from down to up, I must make a rotation on the Y axis. But I rotate my cube 90 degrees on Y, I will have to rotate It on the Z axis to rotate my left layer from down to up.
I'm trying to find a way to get the correct rotation axis according the orientation of the cube.
For the moment I check witch vector of the axis of the rotation matrix of the cube is most parallel with the vector(0,1,0) if I want to move a front layer from down to up. But it do not works in edge cases like this for instance :
I guess there is some simple way to do that, but I'm not good enough in matrix and mathematical stuff :)
An AxisHelper can show the aixs of the scene which you could determine the orientation with.
var axishelper = new THREE.AxisHelper(40);
axishelper.position.y = 300;
scene.add(axishelper);
You could also log your cube and check the position and rotation properties with Chrome Developer Tools or Firebug.
You can store the orientation of each cube in its own 4x4 matrix (i.e. a "model" matrix) that tells you how to get from the cube's local coordinates to the world's coordinates. Now, since you want to rotate the cube around to an axis (i.e. vector) in world coordinates, you need to translate the axis into cube coordinates. This is exactly what the inverse of the model matrix yields.