Manipulator Dynamics (1)Read Chapter 6 (chapter 5 helps too)What is dynamics?• Study the force (torque) required to cause the motion of robots – just like engine power required to drive a automobile– Most familiar formula: f = ma– f: force, m: mass, and a: acceleration• Manipulator will be more complicated since each link can move and rotate, not just a point mass.fAcceleration of a rigid body is we need to calculate manipulator dynamics– (1) linear acceleration   • Linear velocity of a vector P in frame {B} can be expressed in frame {A} as• Because• Acceleration can be expressed as• When the origin of {B} is moving• Complicated? due to: a. rigid body, and b. both translational and rotational motions of {B} and P.󰇛󰇜󰇛󰇜{B}{A}󰇗󰇗2     󰇗   󰇛   )󰇗 󰇗󰇗2     󰇗   󰇛   ){B}• If there are three frames {A}, {B} and {C}, one may have:• Then we obtain by taking the derivative of both sides with respect to time:which turns out to be• Angular acceleration in one frame can be expressed in another frame in an iterative way – but could be complicated. That will be used in calculating manipulator dynamicsAcceleration of a rigid body – (2) angular acceleration  󰇗󰇗󰇛󰇜 󰇗󰇗󰇗+   (3) Mass distribution• For rigid body we have to consider both mass and moment of inertia since a rigid body is free to move in the space with translational and rotational motions.• Different configurations need different torques to achieve accelerations: consider 1below• The moment of inertia is related to the mass distribution of the links and its motions in particular coordinate framesXeZeD1D2D0X0Z0X1Z1Z2X21Inertia tensor• Using inertia to describe mass distribution with respect to a coordinate• Inertia tensor in {A} can be expressed as• ,, are called mass moments of inertia• , ,are called mass products of inertia• ρ is the density of the material• If we chose the frame in such a way that the products of inertias are all zero, the axes are called principal axes, and mass moments are called the principal moments of inertia=∭󰇛+)ρdv,=∭󰇛+)ρdv, =∭󰇛+)ρdv=∭󰇛󰇜ρdvi, j = x, y, or z; i≠jdistance × massInertia tensor examples (1) Which one has greater ,, or ?• Calculate the inertia tensor for the following object and the attached frame. Assuming density to be evenInertia tensor examples (2) = 󰇛+)ρdxdydz=󰇛+)ρwdydz=󰇛+)ρwdz=(hw +lw)ρ=󰇛l+4h)= ρdxdydz= • What if we remove the frame to a new place as shown below?• How can we make the mass products of inertia all