Manipulator Dynamics (2)Read Chapter 6Dynamic equations for each link• For each link there are two equations to describe the effects of force and torque to the motion:• Newton equation• Euler’s equation{Ci} has its origin at the center of mass of the link and has the same orientation as the link frame {i}12󰇗  󰇗 xc1x1Dynamic equations for all the links• Use the iterative Newton-Euler algorithm• Between links there are action and reaction forces (torques)• For link n, we have n+1 f = 0 and n +1n = 0; therefore the equations can start from the last link and go inwardsxizizi+1i Ni Fi nifi +1ni +1fxi+1• Look the top figure in the previous slide, one has:• From the above, one can obtain the following iterative equations:• Note that n+1 f = 0 and n +1n = 0; so we can start the iteration from link n.• Since every joint has only one torque applied, we haveQuestion: how the above equation should be changed for a prismatic joint?Compute forces and torques                  󰇛󰇜 The iterative Newton-Euler dynamics algorithm (1)• The iterative algorithm takes two steps- First step: perform outward iterations to compute velocities and accelerations- Second step: perform inward iterations to computer forces and torques• First step:- Angular acceleration- Linear acceleration- For the center of the link: 󰇛󰇗󰇜   󰇗󰇗󰇗+   󰇗󰇗󰇗󰆹+ 󰇘󰆹 󰇗󰇗2     󰇗   󰇛   )󰇗[ 󰇗( 󰇜  󰇠󰇗󰇗󰇗( )+ 󰇗• Second step:- Draw the Newton-Euler equations- Inward iteration from link n to link 1:The iterative Newton-Euler dynamics algorithm (2) 󰇗  󰇗        󰇛󰇜 Iterative algorithm example• Use the OSU hexapod as an example which is similar to the manipulator shown below – to simplify the problem, let m1and m2be point mass and located at the tip of the linkXeZeD1D2D0X0Z0X1Z1Z2X21m1m2Use the two iterations• Determine the values needed in the algorithm- The position vectors of the center of mass- The inertia tensor at the center of mass- Rotational matrices00 00001 00010• Outward iteration- For link 1󰇗󰇘󰆹 00󰇘ω󰇗󰆹 00󰇗󰇗 00001000= 00󰇗0󰇘0󰇛󰇗󰇜00󰇗 󰇛󰇗󰇜󰇘󰇛󰇗󰇜󰇘= 󰇛󰇗󰇜󰇘000- For link 2󰇗  󰇗󰇗󰇘󰇗󰇗󰇘󰇘ω 󰇗󰆹󰇗󰆹 󰇗󰇗000󰇗󰇗󰇗󰇗󰇗2󰇗1󰇗22󰇘12󰇗1󰇗22󰇘1󰇘2200+2󰇗12󰇗1󰇗22󰇗12󰇗1󰇗2200󰇛󰇗󰇜󰇛󰇗󰇜󰇘+ 0= 󰇛󰇗󰇜󰇛󰇗󰇜󰇘󰇛󰇗󰇜󰇘222󰇗122󰇗2221󰇗1222󰇘2222󰇗1221󰇗122222󰇗1󰇗222󰇘11󰇘1000󰇗00010󰇛󰇗󰇜󰇘0󰇛󰇗󰇜󰇛󰇗󰇜󰇘󰇗22󰇗2222 222󰇗2󰇗0󰇘󰇗󰇗󰇘222󰇗122󰇗22222󰇗1222󰇗1󰇗2+ 21󰇗12221󰇛󰇗1󰇜221󰇘1= 222󰇗122󰇗2221󰇗1222󰇘2222󰇗1221󰇗122222󰇗1󰇗222󰇘11󰇘1• Inward iteration- For link 2- For link 1200 02󰇛222󰇗1󰇗222󰇘11󰇘1󰇜2󰇛2󰇘2222󰇗1221󰇗122󰇜 