Stiffness matrix of manipulators with passive joints: computational aspects

Ecole des Mines de Nantes, 4 rue Alfred-Kastler, Nantes 44307, France
Institut de Recherches en Communications et en Cybernetique de Nantes, 1 rue de la Noe, 44321 Nantes, France

(Submmited to "IEEE Transactions on Robotics", 2011)

Previous page  |  Outline  |  Next page

Appendix B: Recursive computation of the stiffness matrix Kc

Let us assume that the Jacobian  of size  is decomposed into two sub-matrices  of sizes  and  corresponding to non-intersected subsets of passive joints and the derived expression for the stiffness matrix is applied recursively, using sequentially the Jacobians  :

( B1)

To evaluate the obtained matrix, let us substitute the first expression to the second one and perform some equivalent transformations using notations

(B2)

 This allows converting the original bulky expression

(B3)

into a more compact form

(B4)

that allows a matrix presentation

(B5)

Further, using Frobenius formula for the blockwise matrix inverse

(B6)

the derived expression can be presented in the form

(B7)

or

(B8)

that exactly coincide with the expression for the stiffness matrix corresponding to the aggregated Jacobian . Hence, the desired stiffness matrix of the kinematic chain with passive joints can be computed recursively, using arbitrary partitioning of the Jacobian . Obviously, it is more convenient to apply column-wise petitioning that allow sequential modification of the matrix  taking into account geometry of each passive joint separately (and sequentially reducing the rang of the Cartesian stiffness matrix).

Previous page  |  Outline  |  Next page