Orientation

Description

This object describes the rotation from a frame 1 into a frame 2.An instance of this type should never be directly accessed butonly with the access functions providedin package Modelica.Mechanics.MultiBody.Frames. As a consequence, it is not necessary to knowthe internal representation of this object as described in the next paragraphs.

"Orientation" is defined to be a record consisting of twoelements: "Real T[3,3]", the transformation matrix to rotate frame 1into frame 2 and "Real w[3]", the angular velocity of frame 2 withrespect to frame 1, resolved in frame 2. Element "T"has the following interpretation:

Orientation R;R.T = [ex, ey, ez];    e.g., R.T = [1,0,0; 0,1,0; 0,0,1]

where e_x,e_y,e_zare unit vectors in the direction of the x-axis, y-axis, and z-axisof frame 1, resolved in frame 2, respectively. Therefore, if v₁is vector v resolved in frame 1 and v₂ isvector v resolved in frame 2, the following relationship holds:

v2 = R.T * v1

The inverse orientationR_inv.T = R.T^T describes the rotationfrom frame 2 into frame 1.

Since the orientation is described by 9 variables, there are6 constraints between these variables. These constraintsare defined in function Frames.orientationConstraint.

R.w is the angular velocity of frame 2 with respect to frame 1, resolvedin frame 2. Formally, R.w is defined as:
skew(R.w) = R.T*der(transpose(R.T))with

          |   0   -w[3]  w[2] |skew(w) = |  w[3]   0   -w[1] |          | -w[2]  w[1]     0 |