Shape functions, also called generalized velocities, are the projections of the velocity of material point M_k attached to body k on the kinematic variables of the model. Generalized velocities of a material point are depicted in the figure below:

Figure 5.7: Generalized Velocities of a Material Point

Because of the choice of relative degrees of freedom, the velocity of M_k is a function of kinematic variables of the joint located between body k and its parent body L(k), as well as those of the joint between L(k) and L(L(k)), continuing until the ground is reached.

To understand how these generalized velocities are formed, it helps to first focus on the contribution of the first joint of the chain (pictured below). This joint is located between body k and its parent, L(k).

Figure 5.8: Contribution of the Parent Joint to the Generalized Velocities

Because body k is rigid, the velocity of point M_k with respect to the ground 0 can be expressed from the velocity of point O_k . Point O_k is the material point on the mobile coordinate system of the joint between body k and its parent, L(k). This is expressed as follows:

(5–8)

The angular velocity of body k with respect to the ground can be expressed as the angular velocity of its parent, plus the contribution of the joints linking body k and its parent, L(k). This is expressed as follows:

(5–9)

Similarly, can be expressed using point R_k , which is the reference coordinate system of the joint between body k and its parent, L(k). Note that R_k is a material point on body L(k). This is expressed as follows:

(5–10)

where is the joint relative velocity, i.e. the translational velocity between body k and its parent, L(k).

It is important to realize that the vector has an angular velocity of . Joints can have translational degrees of freedom, and rotational degrees of freedom. The translation is expressed in the reference coordinate system, while the rotation center is the moving coordinate system. In other words, the joint translation is applied first, and the rotation is applied after the coordinate system is updated with the results of the joint translation. The decomposition of the Model Topology graph into a tree results in an oriented parent-child relationship. When the joint has both translational and rotational degrees of freedom and its reference coordinate system is on the child side, the joint must be split into a rotational joint linked to the parent side, and a translational joint linked to the child side, with a fictitious mass-less body between these two joints. While this is an internal representation of that "reverted" joint (that is, a joint that has both translational and rotational degrees of freedom and a link to the ground on the mobile coordinate system side), results are reported on the original user-defined joint.

Because R_k is a material point of body L(k), the same methodology can be used to decompose the velocity into the contribution of the parent joint located between L(k) and L(L(k)) and the contribution of the parent.

Two important quantities have been introduced in this process:

The concept of recursive calculation of the generalized velocities has also been introduced. The generalized velocities on body k can be computed by adding the contribution of the parent joint to the generalized velocities of body L(k).

The contribution of each joint in the chain between body k and the ground can be found and expressed as:

(5–11)

(5–12)

Vector , which is associated with the kinematic variable q_i , is the "partial velocity" of the variable expressed at point M_k . It is configuration dependent, that is, it varies with the geometric variables of the joints located between body k and the ground.

The translational and accelerations can similarly be derived to obtain:

(5–13)

(5–14)