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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 · · · ·

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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

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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

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MMS I, Lecture 54 Example

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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

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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

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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

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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

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

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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

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

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

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

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MMS I, Lecture 514 TCP SCARA robot

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