A planar 2-DOF serial robot is shown in Fig. 1.1, with usual notations. Assume that the...

 

 

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A planar 2-DOF serial robot is shown in Fig. 1.1, with usual notations. Assume that the mass centre for each link is located at the centre of the link, and that the payload mass centre is located at the tip of link 2. The payload is assumed to be a particle with a mass of Mp. Note that quantities in the figure are all variables. (a) Discuss the relations between the lengths of the two links (4 and 4) to determine the condition that the payload mass centre can reach the fixed joint O. (b) Derive equations of the payload mass centre's coordinates, velocities, and accelerations along the X- and Y-axes, in terms of the two input angles. [5 marks] [11 marks] (c) Determine the dynamic equations of the robot using the Lagrange approach (Eq.(1.1)). Note that it is sufficient to obtain the kinetic energy equation, potential energy equation, generalized coordinates and generalized forces for the Lagrange equation. L=T-U Kinetic energy Potential energy d OL OL dt oq, oq, 1 Generalized coordinate joint angle) T Generalized force (torque) Eq. 1.1 Lagrange equation Ti, L, Mi, and I; (i=1 or 2) denote torque, length, mass, and moment of inertia, respectively.," Motor Y Torque T, X 0 Payload Link1: L, M and h Mp Link 2: L2, M and 12 Motor Torque T Fig. 1.1 A planar 2-DOF serial robot A planar 2-DOF serial robot is shown in Fig. 1.1, with usual notations. Assume that the mass centre for each link is located at the centre of the link, and that the payload mass centre is located at the tip of link 2. The payload is assumed to be a particle with a mass of Mp. Note that quantities in the figure are all variables. (a) Discuss the relations between the lengths of the two links (4 and 4) to determine the condition that the payload mass centre can reach the fixed joint O. (b) Derive equations of the payload mass centre's coordinates, velocities, and accelerations along the X- and Y-axes, in terms of the two input angles. [5 marks] [11 marks] (c) Determine the dynamic equations of the robot using the Lagrange approach (Eq.(1.1)). Note that it is sufficient to obtain the kinetic energy equation, potential energy equation, generalized coordinates and generalized forces for the Lagrange equation. L=T-U Kinetic energy Potential energy d OL OL dt oq, oq, 1 Generalized coordinate joint angle) T Generalized force (torque) Eq. 1.1 Lagrange equation Ti, L, Mi, and I; (i=1 or 2) denote torque, length, mass, and moment of inertia, respectively.," Motor Y Torque T, X 0 Payload Link1: L, M and h Mp Link 2: L2, M and 12 Motor Torque T Fig. 1.1 A planar 2-DOF serial robot