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Tham khảo tài liệu 'automation and robotics part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Enhanced Motion Control Concepts on Parallel Robots 19 serial manipulators representing differential kinematic relation x Jq and static relation T JTf are used for deduction. The second step - deduction of an exact model for a given structure - can be done via Lagrange-D Alembert-Formulation d dL ỔL T l - -T T JT fpxt d t l ổq J ổq 1 with L T - V representing Lagrange function T kinetic energy V potential energy q vector of joint space variables T actuator torques and J G T serial manipulator Jacobian on which external forces fext are applied. Computing energy functions 1 q T 2qTMq q q V fnq q dq 2 qo leads to a differential equation in joint space coordinates Mq q q Cq q q q nq q T JTfext 3 Its elements can be calculated considering a discrete model the main idea is based upon discrete point masses mi. Starting with the simple case of planar structures each link can be replaced by a combination of at least three single point masses without neglecting and disturbing properties concerning mass center of mass and moment of inertia thus guaranteeing correct dynamical behavior Dizioglu 1966 . Without loss of generality this concept can be transferred to more complex structures. With growing complexity in structure the number of discrete elements increases resulting in the finite element method. The concept of discrete point masses leads to Mq 2 mi JT Ji diag ỉm ỉm i dM 1 d q X Cqt 4 nq 2 mi JTg i with drive inertia ỉm and g being vector of gravity. All Jacobians Ji can be described by a linear combination of endeffector- and passive joints Jacobians. The choice of Coriolis-Matrix is not unique Using Christoffel-Symbols and following the notation of Vetter 1973 and Weinmann 1991 with discussion in Bohn 2000 leads to Cq I qT 0 Irq - X T 1 ổMq 1 17 TJ ổMq ì 0 q ổq 7p1 nq 0 q ổq J 5 20 Automation and Robotics where denotes the Kronecker-product nq is the number of degrees of freedom of the parallel structure and Mq m J JJ t 0JT 6 Illi _ Ji V- na Ji V ỡq ỡq ỡq q ỡq A basic
