1. MMS I, Lecture 51 Short repetition of mm4 Motions of links –Jacobians short –Acceleration of riged bobyes Linear F = mv c Angular N = I c ω + ω x I c ω –Newtons equations –Eulers equations –Iterative Newton-Euler dynamic formulation Outward iteration to get (v c, ω, v c, ω) Inward iteration to get N Content MM5 · · · ·

2. MMS I, Lecture 52 Denavit-Hartenberg Frame Attachment Frame attachment 1. Identify joint axis 2. Identify common perpendicular 3. Assign z i pointing along i-th joint axis 4. Assign x i pointing along common perpendicular 5. Assign y i to complete frame 6. Assign frame {0} (base) to match {1} Link parameters a i =dist(z i, z i+1 ) along x i  i =ang(z i, z i+1 ) about x i d i =dist(x i-1, x i ) along z i θ i =ang(x i-1, x i ) along z i

3. MMS I, Lecture 53 Denavit-Hartenberg Link Parameters a i =dist(z i, z i+1 ) along x i  i =ang(z i, z i+1 ) about x i d i =dist(x i-1, x i ) along z i θ i =ang(x i-1, x i ) along z i

4. MMS I, Lecture 54 Example

5. MMS I, Lecture 55 Example on the blackboard Get recursive angular velocity i ω i+1 and i+1 ω i+1 and Linear velocity i v i+1 and i+1 v i+1 for i = 0, 1, 2

6. MMS I, Lecture 56 Jacobian for examble · x = l 1 c 1 + l 2 c 12 y = l 1 c 1 + l 2 s 12 x = -l 1 s 1 θ 1 - l 2 s 12 θ 1 - l 2 s 12 θ 2 y = l 1 c 1 θ 1 + l 2 c 12 θ 1 + l 2 c 12 θ 2 · · · · · · · · · · x = -l 1 s 1 - l 2 s 12 - l 2 s 12 θ 1 y = l 1 c 1 + l 2 c 12 + l 2 c 12 θ 2 · · · · · x = J( θ ) θ or θ = J( θ ) x

7. MMS I, Lecture 57 Iterative Newton-Euler dynamics - 1 1. Compute angular and linear velocities and accelerations outward from {0}-{N} by iteration 2. Compute forces and torques acting on each link 3. Compute forces and torques from {N}-{0} by iteration

8. MMS I, Lecture 58 1) Angular and linear velocities and accelerations Outwards propagation: 1. Angular velocity and acceleration: 2. Linear acceleration of frames 3. Linear acceleration of link CoM

9. MMS I, Lecture 59 2) Force and torque on each link

10. MMS I, Lecture 510 3a) Forces and torques F i : force on link i by link i-1 N i : torque on link i by i-1

11. MMS I, Lecture 511 3b) Forces and torques iteration Force and torque equilibrium Iteration:

12. MMS I, Lecture 512 Combined Newton-Euler Dynamics - 1 1) Outwards iterations, i: 0-5

13. MMS I, Lecture 513 Combined Newton-Euler Dynamics -2 2) Inward iterations: i: 6-1

14. MMS I, Lecture 514 TCP SCARA robot