3 dimensional torque from angular velocity


by drfrev
Tags: 3-space, angular velocity, torque
drfrev
drfrev is offline
#1
Jan23-13, 10:18 PM
P: 7
Hello,

I am currently making a video game and have run into a bit of a physics problem. I have taken two semesters of physics in college (wasn't too great at it) so I know how to do simple things. I am currently trying to turn an angular velocity into a 3D torque. The angle is a Euler vector ex. (X,Y,Z) using a XYZ order, and I need the equivalent torque in 3 space. I only need the torque direction as the magnitude isn't that important. I have been trying to figure it out but with no luck.

Thanks for any help
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Hassan2
Hassan2 is offline
#2
Jan23-13, 11:27 PM
P: 404
It seems you want to find the rotation axis, but then do you have the rotation matrix? if yes, one of its eigenvectors corresponding the eigenvalue of 1.0 is the rotation axis.
drfrev
drfrev is offline
#3
Jan23-13, 11:43 PM
P: 7
I will be a bit more descriptive to flush out my questions better. I have two "rods" in three space. One rod is fixed, while the other is not. I need to apply a torque to the unfixed rod to make it become parallel to the fixed rod.

Okay, I am able to get the rotation matrix from each of the "rods", is there any way to "subtract" them to get the matrix rotation that separates them, because then I could use what you said. Also how would I go about calculating the eigenvalue of a rotation matrix. (I am not used to rotation matrices which is why I was using angular velocity above. After making this reply I will be looking up eigenvalue on google in case you don't want to go into that)

Hassan2
Hassan2 is offline
#4
Jan24-13, 12:15 AM
P: 404

3 dimensional torque from angular velocity


Each bar is represented by a vector. If you have the rotation matrix to multiply it by the vector of unfixed bar to become parallel the fix one, isn't your problem solve then? If so, I will explain how to construct the matrix from the two vectors.
drfrev
drfrev is offline
#5
Jan24-13, 12:24 AM
P: 7
I don't quite know what you mean but each bar is represented by a vector as the center and a rotation matrix representing orientation.
Hassan2
Hassan2 is offline
#6
Jan24-13, 12:45 AM
P: 404
I hope you are familiar with the following programming language. The calculation is based on Euler–Rodrigues formula.
The matrix rotMat rotates every point of the object in 3D space by alpha radian around rotAx ( rotation axis). It can make the bars parallel.

public static Mat rotMat(Vect rotAx,double alpha)
{
double e1,e2,e3,e4;

e1=rotAx.el[0]*sin(alpha/2);
e2=rotAx.el[1]*sin(alpha/2);
e3=rotAx.el[2]*sin(alpha/2);
e4=cos(alpha/2);

Mat M=new Mat(3,3);
M.el[0][0]=pow(e1,2)-pow(e2,2)-pow(e3,2)+pow(e4,2);
M.el[0][1]=2*(e1*e2-e3*e4);
M.el[0][2]=2*(e1*e3+e2*e4);
M.el[1][0]=2*(e1*e2+e3*e4);
M.el[1][1]=-pow(e1,2)+pow(e2,2)-pow(e3,2)+pow(e4,2);
M.el[1][2]=2*(e2*e3-e1*e4);
M.el[2][0]=2*(e1*e3-e2*e4);
M.el[2][1]=2*(e2*e3+e1*e4);
M.el[2][2]=-pow(e1,2)-pow(e2,2)+pow(e3,2)+pow(e4,2);
return M;
}
Hassan2
Hassan2 is offline
#7
Jan24-13, 12:50 AM
P: 404
I use it for the following problem:

I need to draw arrows showing the magnetic field in the space. Using Java3D, I can draw a 3D arrow in z direction and the arrow center is the origin but I need to draw the arrow at point P and in direction given by the field at point P. Using the matrix, the object is first rotated to become parallel with the field direction, then by a translation, moves to the point. I think it is kind of similar to your problem.
drfrev
drfrev is offline
#8
Jan24-13, 01:02 AM
P: 7
I need to use a torque on the object so that the physics engine can handle collisions. The engine will handle the actual vertex rotation. I have a function which can apply the torque to the object for the next physics tick. It can apply a local or global torque. (Though I do understand that language)
Hassan2
Hassan2 is offline
#9
Jan24-13, 01:28 AM
P: 404
Sorry what i suggested was not physics by linear algebra. Out of curiosity, could you explain about the Euler angle you said. What are X,Y and Z ? I have heard of Euler angles but not Euler vector.
drfrev
drfrev is offline
#10
Jan24-13, 01:37 AM
P: 7
It's the same thing, (x,y,z) being the rotation acording to that axis. XYZ order means that it rotates on the x axis first, then the y axis, then the z axis.
Hassan2
Hassan2 is offline
#11
Jan24-13, 01:51 AM
P: 404
So you have the three angles and can construct the rotation matrix. From this matrix, you can obtain the rotation axis to apply torque to. This axis is the eigenvector of the matrix. see http://en.wikipedia.org/wiki/Rotation_matrix

Eigenvectors of a 3x3 matrix is, to my knowledge, found by iterative methods. However you can use a library for that purpose.
drfrev
drfrev is offline
#12
Jan24-13, 01:57 AM
P: 7
Thanks, I will try it out tomorrow. Thank you for all the help, I really do appreciate it.


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