SimScape multibody rotation matrix

Hello everyone,

I am currently working with simscape multibody and trying to determine a rotation matrix for a rigid transform. Let's assume I have an extruded solid along its z axis and the coordinate of the director vector of the rotation axis of the revolute joint is known QP=[-7 32.7 94.2]. (New origin has to be M=[129.9984 145.1076 353.5218].)

Now I want to calculate the rotation matrix needed to transform the wolrd frame's Z axis so it matches the joint's axis. For that, I caculate first the angle around the X axis, by calculating the angle between the projection of the vector on the YZ plane and the Z axis (-19.1436°), then caculate the angle around the Y axis calculating the angle between the projection of the vector on the XZ plane and the Z axis (-4.2498°).

After that, I determine rotx(theta1) and roty(theta2), and determine the final rotation matrix R=rotx*roty.

To verify my calculation, I multiply Z axis coordinate by the rotational matrix and get QP' [-7.4106 32.7035 94.2102].

Element by element division QP/QP' gives me [0.9446 0.9999 0.9999], which means the rotation matrix is incorrect.

When keeping the same method for the X axis rotation and calculating the angle between QP (coordinate in the new base after the X rotation) and Z axis, I obtain an angle of -4.0156°. The new QP/QP' gives [-1.0581 -0.9971 0.9996], which is still wrong.

Would you have an idea on what is wrong? Thank you for your answers.

rS5x=Zvector(1,2)*rQP(1,3)-Zvector(1,3)*rQP(1,2);  %get the sign of the angles  (Xa-Yb)*(Ya-Xb)
rS5y=Zvector(1,3)*rQP(1,1)-Zvector(1,1)*rQP(1,3);

rtheta5x=sign(rS5x)*acos(dot([0 rQP(1,2) rQP(1,3)],Zvector)/(norm(Zvector)*norm([0 rQP(1,2) rQP(1,3)])))*180/PI;  %angle around X axis
rR5x=rotx(rtheta5x); %rot matrix X

rQPprime=rQP*rR5x; %rQP coordinate in new base

rtheta5y=sign(rS5y)*acos(dot([rQPprime(1,1) 0 rQPprime(1,3)],Zvector)/(norm(Zvector)*norm([rQPprime(1,1) 0 rQPprime(1,3)])))*180/PI; %angle around Y axis
rR5y=roty(rtheta5y);  %rot matrix Y

rR5=rR5x*rR5y; % global rot matrix
VerifQP=Zvector*rR5*100; %QP calculated from Z axis and rot matrix
AB=rQP./VerifQP; %element by element division

[James Tursa]

Can you post an image of how things are related?

[Killian Hyron]

Here you can see points Q and P and all the suspension mechanism. X Axis is along the car (from right to left on the picture), Z is vertical and Y horizontal (from center to the left wheel here)

[Killian Hyron]

Hi James,

Little update : By calculating the axis of rotation between Zvector and QP, and deducing the rotation matrix from it, I get QP/QP' = [-0.9996 -0.9996 0.9996], which is way better (but unfortunately not entirely precise)!

I am still wondering what is wrong in my first method.

The 2 different matrixes are :

Rot5 =

0.9975 0.0118 -0.0700

0.0118 0.9449 0.3271

0.0700 -0.3271 0.9424 Second method

rR5 =

0.9975 0 -0.0700 First method

0.0230 0.9447 0.3271

0.0662 -0.3279 0.9424

As we can see, Rot5 is almost antisymmetric, and rR5 isn't at all. Do you have an idea of the problem of the first method?