Tham khảo tài liệu 'robot manipulators trends and development 2010 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 272 Robot Manipulators Trends and Development Finally the coupling torque affecting the motor dynamics see Equation 1 is defined as Ecoup -2EIu1 2. Notice that the coupling torque has the same magnitude and different sign to the joint torque 2EIu1 2. This torque can be expressed as a linear function Vcoup 0 - c A - c 2r 4 where C c1 c2 c Ci 1 i n 2 are parameters which do not depend on the concentrated masses along the structure and c 2 -C 1 1 1 t. For example the transfer functions Gc s and Gt s for only one point mass located in the tip mi are as follows Gc s 3EI L s2 0 and Gt s 0 s2 0 5 in which O1 yj3EI L3m1 . This model can be used for flexible robots with a high payload weight ratio. Assumed mode method The dynamic behaviour of an Euler-Bernoulli beam is governed by the following PDE see for example Meirovitch 1996 EIwIV x t pW x t f x t 6 where f x t is a distributed external force w is the elastic deflection measured from the undeformed link. Then from modal analysis of Equation 6 which considers w x t as TO w x t tj x h Ơ 7 i 1 in which ội x are the eigenfunctions and ĩỊi t are the generalized coordinates the system model can be obtained see Belleza et al. 1990 for more details . Multi-link flexible manipulators For these types of manipulators truncated models are also used. Some examples are De Luca Siciliano 1991 for planar manipulators Pedersen Pedersen 1998 for 3 degree of freedom manipulators and Schwertassek et al. 1999 in which the election of shape functions is discussed. The deflections are calculated from the following expression Wi x t T x Ai t 1 i Hl 8 see for example Benosman Vey 2004 in which i means the number of the link nL the number of links ó x is a column vector with the shape functions of the link for each considered mode Ai t 7ii 7Ni T is a column vector that represents the dynamics of each mode in which N is the number of modes considered. Control of Flexible Manipulators. Theory and Practice 273 The dynamics equations of .
