Affine Body Revolute Joint

Literature References:

A unified newton barrier method for multibody dynamics

#18 AffineBodyRevoluteJoint

The Affine Body Revolute Joint constitution constrains two affine bodies to rotate relative to each other about a shared axis. This joint allows for rotational motion while restricting translational movement between the two bodies.

We assume 2 affine body indices \(i\) and \(j\), each with their own state vector (to be concerete, transform) \(\mathbf{q}_i\) and \(\mathbf{q}_j\) as defined in the Affine Body constitution.

The joint axis in world space is defined by 2 points \(\mathbf{x}^0\) and \(\mathbf{x}^1\). At the beginning of the simulation, the relationships are kept:

\[ \mathbf{x}^0 = \mathbf{J}^0_i \mathbf{q}_i = \mathbf{J}^0_j \mathbf{q}_j, \]

and,

\[ \mathbf{x}^1 = \mathbf{J}^1_i \mathbf{q}_i = \mathbf{J}^1_j \mathbf{q}_j, \]

intuitively \(\mathbf{J}^0\) and \(\mathbf{J}^1\) tell the local coordinates of the two points on affine bodies \(i\) and \(j\).

The energy function for the Affine Body Revolute Joint is defined as:

\[ E = K \cdot \left( \| \mathbf{J}^0_i \mathbf{q}_i - \mathbf{J}^0_j \mathbf{q}_j \|^2 + \| \mathbf{J}^1_i \mathbf{q}_i - \mathbf{J}^1_j \mathbf{q}_j \|^2 \right), \]

where \(K\) is the stiffness constant of the joint, we choose \(K=\gamma (m_i + m_j)\), where \(\gamma\) is a user defined strength_ratio parameter, and \(m_i\) and \(m_j\) are the masses of the two affine bodies.