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Content MM5 Short repetition of mm4 Motions of links – Jacobians short – Acceleration of riged bobyes • Linear • Angular F = mv·c · + ω x Ic ω N = Ic ω – Newtons equations – Eulers equations – Iterative Newton-Euler dynamic formulation · • Outward iteration to get (vc, ω, ·vc, ω) • Inward iteration to get N MMS I, Lecture 5 1 Denavit-Hartenberg Frame Attachment Frame attachment 1. Identify joint axis 2. Identify common perpendicular 3. Assign zi pointing along i-th joint axis 4. Assign xi pointing along common perpendicular 5. Assign yi to complete frame 6. Assign frame {0} (base) to match {1} MMS I, Lecture 5 Link parameters ai=dist(zi, zi+1) along xi ai=ang(zi, zi+1) about xi di=dist(xi-1, xi ) along zi θi=ang(xi-1, xi) along zi 2 Denavit-Hartenberg Link Parameters axis i-1 ai=dist(zi, zi+1) along xi ai=ang(zi, zi+1) about xi axis i link i-1 di=dist(xi-1, xi ) along zi θi=ang(xi-1, xi) along zi link i-2 yi zi-1 zi yi-1 xi-1 ai-1 MMS I, Lecture 5 xi ai-1 di i 3 Example 3 x L L1 y0 y2 y1 L2 3 y x2 3 x1 x0 MMS I, Lecture 5 4 Example on the blackboard Get recursive angular velocity iωi+1 and and Linear velocity ivi+1 and i+1vi+1 i+1ω i+1 for i = 0, 1, 2 MMS I, Lecture 5 5 Jacobian for examble x = l1c1 + l2c12 y = l1c1 + l2s12 · · · x· = -l1s1θ1 - l2s12 θ1 - l2s12 θ2 · · y· = l1c1 θ1 + l2c12 θ1 + l2c12 θ· 2 x· = y· = · -l1s1 - l2s12 - l2s12 θ1 · l1c1 + l2c12 + l2c12 θ2 · x· = J(θ ) θ or -1 · θ = J(θ ) x· MMS I, Lecture 5 6 Iterative Newton-Euler dynamics - 1 1. 2. 3. Compute angular and linear velocities and accelerations outward from {0}-{N} by iteration Compute forces and torques acting on each link Compute forces and torques from {N}-{0} by iteration MMS I, Lecture 5 7 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 MMS I, Lecture 5 8 2) Force and torque on each link MMS I, Lecture 5 9 3a) Forces and torques i+1 i N f i+ 1 i n i+1 i f i F ni Fi: force on link i by link i-1 Ni: torque on link i by i-1 MMS I, Lecture 5 10 3b) Forces and torques iteration Force and torque equilibrium Iteration: MMS I, Lecture 5 11 Combined Newton-Euler Dynamics - 1 1) Outwards iterations, i: 0-5 MMS I, Lecture 5 12 Combined Newton-Euler Dynamics -2 2) Inward iterations: i: 6-1 MMS I, Lecture 5 13 SCARA robot TCP TCP TCP MMS I, Lecture 5 14