Consistent definition of partial derivatives of matrix functions in dynamics of mechanical systems

Dynamic Model With a New Formulation of Coriolis/Centrifugal Matrix for Robot Manipulators

“…The detailed proofs of both theorems are presented clearly in [11]. In the following, the matrix form (29) can be built based on the Euler-Lagrange equations (16) by using the above definitions and theorems.…”

“…From (46), the requirement is to extract matrix C that satisfies the skew-symmetric property of matrixṀ − 2C. There is a kind of matrix C taken from (46) (presented in [11,12]); But it does not assure the mentioned property. To achieve this objective, we give a lemma as follows.…”

“…Several types of the Coriolis/centrifugal matrix developed directly in matrix-based manner are studied to preserve the skew symmetry property; One is derived from the Lie group based recursive Newton-Euler algorithm by replacing the original Coriolis matrix of the motion equations for each body level with a modified Coriolis matrix [17]; One is obtained by dropping the term that does not contribute to the Coriolis/centrifugal force [9]; One is taken after removing a zero term from the Coriolis/centrifugal vector [3]; One is extended from [3] for geared manipulators with ideal joints [2]; And another can be simplified, after being split into a skew-symmetric matrix and a symmetric matrix, by omitting the skew-symmetric part in the case that this part is trivial in compared to the other part [18]. Some other studies also offer a new Coriolis/centrifugal matrix, but it does not satisfy the skew symmetry property [11,12,15].…”

“…In our paper, taking advantages of the definitions and theorems for partial derivatives of a matrix, a product of two matrices with respect to a vector, and the time derivative of a matrix [11] using Kronecker product [4,10,22], we build the matrix form of dynamic equations of robot manipulators based on the Euler-Lagrange equations. Moreover, a new formulation of the Coriolis/centrifugal matrix is established directly in matrix-based manner and guarantees the skew symmetry property.…”

A comparison study of some control methods for delta spatial parallel robot

“…The equations of motion of a parallel robot can be obtained by either sub-structural method, Newton-Euler equations [1][2][3][4] or Lagrangian multiplier equation [5,6,7]. These equations are written in matrix form as follows:…”