tailieunhanh - Kinematic and dynamic analysis of multibody systems using the kronecker product

Using the Kronecker product, a much larger size matrix can be formed from two matrix operands; therefore, the capability of matrix algebra in analyzing the kinematics and dynamics of multibody systems are extended. This paper employs 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. | Kinematic and dynamic analysis of multibody systems using the kronecker product Vietnam Journal of Science and Technology 57 (1) (2019) 112-127 doi: KINEMATIC AND DYNAMIC ANALYSIS OF MULTIBODY SYSTEMS USING THE KRONECKER PRODUCT Nguyen Thai Minh Tuan1, *, Pham Thanh Chung1, Do Dang Khoa1, Phan Dang Phong2 1 Hanoi University of Science and Technology, No. 1, Dai Co Viet, Hai Ba Trung, Ha Noi 2 National Research Institute of Mechanical Engineering, Pham Van Dong, Cau Giay, Ha Noi * Email: Received: 23 May 2018; Accepted for publication: 6 December 2018 Abstract. Using the Kronecker product, a much larger size matrix can be formed from two matrix operands; therefore, the capability of matrix algebra in analyzing the kinematics and dynamics of multibody systems are extended. This paper employs 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. Keywords: Jacobian matrix, Hessian matrix, Kronecker product, velocity-free Coriolis matrix, matrix form of Lagrange’s equations. Classification numbers: 1. INTRODUCTION Matrix operations are commonly used in the field of multibody system due to the convenience of writing generalized formulas. However, basic operations such as matrix multiplication and addition are not enough in certain cases. For instance, while rotational and translational Jacobian matrices are popular, their partial derivatives, Hessian matrices, .