Denoting as the locations of the centers of gravity of the j-th and k-th structures respectively, and as the connecting point in the global axes, the vectors between the joint point and the j-th and k-th structures are written as

(8–1)

Let us further denote the translational and rotational movements of these two linked structures as and and the unit vectors of the local articulation axes with respect to the global axes as

(8–2)

For the locked constraint case, the constraint boundary conditions in the local articulation frame are

(8–3)

Introducing the matrix form, the above equations can be expressed as

(8–4)

where

For the hinge constraint case, in which the rotation about the local articulation x-axis is free, the boundary conditions are similarly given by

(8–5)

where

For the universal constraint case in which rotations about the local articulation x- and y-axes are free, the boundary conditions have the same form as Equation 8–5 but

(8–6)

Finally for the ball/socket constraint case, the boundary conditions have the same form as Equation 8–5 but

(8–7)

Equation 8–4 for the locked constraint case can be converted into the same form of Equation 8–5 by simply defining .

From the above discussion, the boundary conditions of all constraint types can be defined by Equation 8–5, differing only in the form of the matrix. Further denoting

(8–8)

Equation 8–5 is rewritten as

(8–9)

Denoting the constraint reaction force/moment matrix acting on the j-th structure at the articulation point in the local articulation axes as , the motion including the reaction forces and moments of the two linked structures can be determined from

(8–10)

where is the total stiffness matrix of these two structures, and and are the forces and moments acting on the j-th and k-th structures respectively (excluding the reaction force component).