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Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 7

Tham khảo tài liệu 'multi-arm cooperating robots- dynamics and control - zivanovic and vukobratovic part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Mathematical Models of Cooperative Systems 107 are possible in all directions irrespective of whether they are positive or negative. On the other hand the manipulator tips and manipulated object form one whole as if they were glued so that the elasticity model introduced holds either for the rigid manipulator and elastic object or for a rigid manipulator with an elastic tip and rigid object or for an elastic manipulator tip and elastic object. The simplest case is when the tips of the manipulators are elastic as then all the model parameters can be easily determined. The system manipulators-manipulated object is decomposed into the elastic system and rigid manipulators. The elastic system is first described approximately by considering the discontinual structure with 6m 6 DOFs of motion which communicates with the manipulators via force and position. Practically an elastic spatial grid is formed with rigid objects at the nodes which is a relatively rough picture of reality but a very practical one for engineering applications and sufficiently correct provided the influential Maxwell s coefficients are constant. If the grid is elastic there will be a unique relationship between the grid position and forces acting on the grid. For the selected model of elastic system only the grid nodes are under the influence of forces equal to the sum of inertial damping gravitational and contact forces. The acting forces are balanced by elastic forces both under the static and dynamic conditions. A description of this property has the same form when the unloaded state is either at rest or in the state of motion and is given by the relations Fe y Ky K const e R m m . rankK 6m 185 9y Fe Y 8rh 7Y K Y Y K Y e R 6m 6 x 6m 6 rankK Y 6m. dY From the Lagrange equations 93 for the immobile and mobile unloaded state we have sn dy I 1 d dT I r-TT dt dy dT d D dy dy I Q 1 dna 1 1 1 I d dTa dTa d Da 1Ỹ 0Y I 1 Qa dY 1 1 1 1 1 dt dY 1 d ý 1 I I I AAA -Ộ ý I 1 G F ý y Y a Fe Fd Fd G F . 186 .