Nguyen Thai Minh Tuan, Chung Thanh Pham, Do Dang Khoa, Phan Dang Phong


This paper employ Khang’s definition of the partial derivative of a matrix with respect to a vector and the Kronecker product to define translational and rotational Hessian matrices. With these definitions, the generalized velocities in the expression of a linear acceleration or an angular acceleration are collected into a quadratic term. The relations of Jacobian and Hessian matrices in relative motion are then established. A new matrix form of Lagrange’s equations showing clearly the quadratic term of generalized velocities is also introduced.


Jacobian matrix; Hessian matrix; Kronecker product; velocity-free Coriolis matrix; matrix form of Lagrange’s equations

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Zhao Tieshi, et al. - Kinematics and Dynamics Hessian Matrices of Manipulators Based on Screw Theory, Chin. J. Mech. Eng., 28 (2015) 226-235.

Wen-Hua D., et al. - Control-orientated dynamic modeling of forging manipulators with multi-closed kinematic chains, Robotics and Computer-Integrated Manufacturing, 30 (5) (2014) 421-431.

Nguyen Van Khang - Consistent definition of partial derivatives of matrix functions in dynamics of mechanical systems, Mechanism and Machine Theory, 45 (2010) 981-988.

Nguyen Van Khang - Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems, Mechanics Research Communications, 38 (2011) 294-299.

Fuzhen Z. - Matrix Theory: Basic Results and Techniques, Springer New York, 2011, pp. 399.

Brewer J. - Kronecker products and matrix calculus in system theory, IEEE Transactions on Circuits and Systems, 25 (9) (1978) 772-781.

Laub A.J. - Matrix Analysis for Scientists and Engineers, SIAM, 2005, pp. 157.

Nguyen Van Khang - Dynamics of Multibody Systems, (second printing with some corrections and additions) (in Vietnamese). Hanoi Science and Technics Publishing House, 2017.

Schiehlen W. and Eberhard P. - Applied Dynamics (Vol. 57), Springer, 2014, pp. 215.

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