The only degrees of freedom needed to represent this assembly are the ones needed to represent the movement of a rigid body. In 2D this is simply two in-plane translations, and the rotation around the z-axis.

In 3D the situation is more complex. Six degrees of freedom, usually selected as three translations and three parameters for the rotation, are necessary. For finite rotations, however, any choice of three rotation parameters is singular at some specific set of angles. For this reason, a four-parameter quaternion representation is used for the rotations in COMSOL Multiphysics. Thus, each rigid connector in 3D actually has seven degrees of freedom, three for the translation and four for the rotation. The quaternion parameters are called a, b, c, and d, respectively. These four parameters are not independent, so an extra equation stating that the following relation is added:

The connection between the quaternion parameters and a rotation matrix R is

Under pure rotation, a vector from the center of rotation (Xc) of the rigid connector to a point X on the undeformed object is rotated into

where x is the new position of the point originally at X. The displacement is by definition

where I is the unit matrix.

When the center of rotation of the rigid connector also has a translation uc, then the complete expression for the rigid body displacements is

The parameter a can be considered as measuring the rotation, while b, c, and d can be interpreted as the orientation of the rotation vector. For small rotations, this relation simplifies to

The rotation vector is available as the variables thx_tag, thy_tag, and thz_tag. Here tag is the tag of the Rigid Connector node in the Model Builder tree.