The planar mechanism know as Pantograph is shown in Figure 1. Pantograph has five links: the...

Transcribed Image Text:

The planar mechanism know as Pantograph is shown in Figure 1. Pantograph has five links: the fixed link N (with zero length for the depicted system) and symmetric links A, D and B, C (lA = ID and lB = lc). The links A and D are actuated with torques TA and TD. The end effector is located at the intersection of B and C and is marked as point Q. A particle E of mass me is attached to point Q. Variables 2 and denote the position of the end effector with respect to the center of the workspace CW. External forces Fr and F, act on the system at point Q. y 08 y B 00 I cw Oc R Figure 1: Schematic representation of the Pantograph Link lengths of the device are set as la = lp = 144 mm, l = lc = 195 mm. The center of the workspace CW is located at x=y=0 mm and the point O is located at x = 0 mm and y = -255 mm. When the end effector is located at the point CW, then the orientations of the links can be calculated as qa = 40.5416, qB = 124.1376, qc= 55.86241, and qp = 139.4584, respectively. (These values are required for proper initializations. Make sure to properly fill in the initial conditions of integrator blocks in your model.) a) Symbolically derive the configuration level forward and inverse kinematics of the mechanism using the cross product method. b) Symbolically determine the kinematic Jacobian of the mechanism using analytical vector methods. c) Determine the largest square workspace that lies within the reachable workspace of the pantograph mechanism. d) Determine configurations of the end-effector when the kinematic Jacobian becomes rank deficient. Provide a schematic drawing of the robot in these configurations. e) Symbolically derive the equations of motion governing the dynamics of the Pantograph using Newton- Euler Method. Make sure to provide Free Body Diagrams and Equivalent Force-Torque Diagrams for each link. The planar mechanism know as Pantograph is shown in Figure 1. Pantograph has five links: the fixed link N (with zero length for the depicted system) and symmetric links A, D and B, C (lA = ID and lB = lc). The links A and D are actuated with torques TA and TD. The end effector is located at the intersection of B and C and is marked as point Q. A particle E of mass me is attached to point Q. Variables 2 and denote the position of the end effector with respect to the center of the workspace CW. External forces Fr and F, act on the system at point Q. y 08 y B 00 I cw Oc R Figure 1: Schematic representation of the Pantograph Link lengths of the device are set as la = lp = 144 mm, l = lc = 195 mm. The center of the workspace CW is located at x=y=0 mm and the point O is located at x = 0 mm and y = -255 mm. When the end effector is located at the point CW, then the orientations of the links can be calculated as qa = 40.5416, qB = 124.1376, qc= 55.86241, and qp = 139.4584, respectively. (These values are required for proper initializations. Make sure to properly fill in the initial conditions of integrator blocks in your model.) a) Symbolically derive the configuration level forward and inverse kinematics of the mechanism using the cross product method. b) Symbolically determine the kinematic Jacobian of the mechanism using analytical vector methods. c) Determine the largest square workspace that lies within the reachable workspace of the pantograph mechanism. d) Determine configurations of the end-effector when the kinematic Jacobian becomes rank deficient. Provide a schematic drawing of the robot in these configurations. e) Symbolically derive the equations of motion governing the dynamics of the Pantograph using Newton- Euler Method. Make sure to provide Free Body Diagrams and Equivalent Force-Torque Diagrams for each link.