Taylor expansion for matrix functions of vector variable using the Kronecker product

Nguyen Van Khang, Dinh Cong Dat, Nguyen Thai Minh Tuan


Taylor expansion is one of the many mathematical tools that is applied in Mechanics and Engineering. In this paper, using the partial derivative of a matrix with respect to a vector and the Kronecker product, the formulae of Taylor series of vector variable for scalar functions, vector functions and matrix functions are built and demonstrated. An example regarding the linearization of the differential equations of an elastic manipulator is presented using Taylor expansion.


Taylor expansion; Kronecker product; the partial derivative of a matrix with respect to a vector; elastic manipulator; linearization

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W. Schiehlen and P. Eberhard. Applied dynamics. Springer, Berlin, (2014).

H. Dresig and F. Holzweissig. Dynamics of machinery. Springer, Berlin, (2010).

F. Amirouche. Fundamentals of multibody dynamics/ Theory and applications. Birkhuser, Boston, (2006).

A. A. Shabana. Dynamics of multibody systems. Cambridge University Press, 3rd edition, (2005).

I. N. Bronshtein, K. A. Semendyayer, G. Musiol, and H. Muehlig. Handbook of mathematics. Springer, Berlin, 5th edition, (2007).

A. Graham. Kronecker products and matrix calculus, with applications. John Wiley & Sons, New York, (1981).

N. V. Khang. Partial derivative of matrix functions with respect to a vector variable. Vietnam Journal of Mechanics, 30, (4), (2008), pp. 269–279. https://doi.org/10.15625/0866-7136/30/4/5632.

N. V. Khang. Consistent definition of partial derivatives of matrix functions in dynamics of mechanical systems. Mechanism and Machine Theory, 45, (7), (2010), pp. 981–988. https://doi.org/10.1016/j.mechmachtheory.2010.03.005.

N. V. Khang. Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems. Mechanics Research Communications, 38, (4), (2011), pp. 294–299. https://doi.org/10.1016/j.mechrescom.2011.04.004.

N. T. M. Tuan, P. T. Chung, D. D. Khoa, and P. D. Phong. Kinematic and dynamic analysis of multibody systems using the Kronecker product. Vietnam Journal of Science and Technology, 57, (1), (2019), pp. 112–127. https://doi.org/10.15625/2525-2518/57/1/12285.

N. V. Khang. Dynamics of multibody systems. Hanoi Science and Technics Publishing House, 2nd edition, (2017). (in Vietnamese).

DOI: https://doi.org/10.15625/0866-7136/14196 Display counter: Abstract : 295 views. PDF : 17 views.


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