McGill Dept. of Mechanical Engineering MECH572 Introduction to robotics Lecture 10
MECH572 Introduction To Robotics Lecture 10 Dept. Of Mechanical Engineering
Review Fundamentals of multibody dynamics Newton-Euler equation I;=-×工+n+nF mici=fh Rely on free-body diagram. Constraint force involved Compact form M t i=-WiMti+w+ Working Constraint rench Wrench M;三 IO|,wa≡oO o mi
Review • Fundamentals of Multibody Dynamics Newton-Euler Equation Rely on free-body diagram. Constraint force involved. Compact form: Working Wrench Constraint Wrench
Review Fundamentals of Multibody dynamics Euler-Lagarange Equation d(a0)-80= or d/0L\_aL=pn d/at aT d(6/-80 L≡T-V Kinetic energy T=∑=∑M41=M= atTu In terms of twist T=50 I(0)0 In terms of generalized coordinates and inertia Alternative form ofE-L equation I+(b-1[ab+a=中n
Review • Fundamentals of Multibody Dynamics Euler-Lagarange Equation Kinetic energy: Alternative form of E-L equation or In terms of twist In terms of generalized coordinates and inertia
Review Fundamentals of multibody dynamics Summary Newton-Euler Equation Element(body ) level formulation All forces/moments involved(active, constraints,.) Reference point -mass centre Euler-Lagarange Equation System level formulation System kinetic/potential energy involved Generalized coordinates
Review • Fundamentals of Multibody Dynamics Summary: Newton-Euler Equation - Element (body) level formulation - All forces/moments involved (active, constraints,…) - Reference point – mass centre Euler-Lagarange Equation - System level formulation - System kinetic/potential energy involved - Generalized coordinates
Inverse Dynamics Overview of Recursive Algorithm Inverse Dynamics: Known time history of joint position, rate and acceleration, compute joint torque Recursive algorithm: The problem is formulated as a recursive process in such a way that the computation can be proceed from one link to the next Z Kinematics Computation 0,0 ,,b,.0 Cc Dynamics Computation
Y Z 1 1 1 , , 2 2 2 , , n n n , , … 1 1 c , c 2 2 c , c 3 3 c , c 1 1 ω ,ω 2 2 ω ,ω 3 3 ω ,ω Kinematics Computation Dynamics Computation f Inverse Dynamics • Overview of Recursive Algorithm Inverse Dynamics: Known time history of joint position, rate and acceleration, compute joint torque Recursive Algorithm: The problem is formulated as a recursive process in such a way that the computation can be proceed from one link to the next. X n
Recursive lnverse dynamics Procedure Summary (1) Kinematic Computation (Outward) Known: 0, 0, 0 y Compute t, t 62,62,62 (t1t1) Link 1 Link 2
Recursive Inverse Dynamics • Procedure Summary (1) Kinematic Computation (Outward) Known: Compute θ,θ,θ t t , 1 1 1 , , 2 2 2 , , 1 1 c , c 2 2 c , c 1 1 ω ,ω 2 2 ω ,ω ( , ) 1 1 t t ( , ) 2 2 t t … Link 1 Link 2
Recursive lnverse dynamics Procedure Summary (2) Dynamic Computation(Inward) Known: Kinematic quantities of each link(from outward recursion) Compute: Joint wrench and external wrench NE WEE(EE, nEE)+Wn(fn, nn)-+Wnl(fn1, nn1)
Recursive Inverse Dynamics • Procedure Summary (2) Dynamic Computation (Inward) Known: Kinematic quantities of each link (from outward recursion) Compute: Joint wrench and external wrench WEE (fEE, nEE) Wn (fn, nn) Wn-1 (fn-1, nn-1) … N.E
Recursive lnverse dynamics Outward Recursions- Kinematics Computation (i Angular velocity and acceleration swi-1+ei, if the ith joint is R (625a) w-1 if the ith joint is P Wi-1+wi-1 Biei +eiei, if the ith joint is R Wi-1, if the ith joint is P(6.25b) Expressed in(i+1) frame QT(wi-1+0iei), if the ith joint isR if the ith joint is P (626a) Q wi Qi-1+Wi-1 x e:+0,ei), if the ith joint isR Q-1 the讪 oint is103b Initial conditions o=0
Recursive Inverse Dynamics • Outward Recursions - Kinematics Computation (i) Angular velocity and acceleration Expressed in (i+1) frame Initial conditions
Recursive lnverse dynamics Outward Recursions - Kinematics Computation Computational complexity for angular velocity and acceleration Coordinate Transformation [rl+1=Qrl cos e sin a 01 71c06+72sin6 [r]+1=-A@i A cos e: Air2 uisin ]i -Hi cos 0i Ai Lr3 4r+入3 h三c0ap≡aina 7三T1sin-72co6 The extra term in wi computation ei [wi-1 x0 eli 6 Also [ojei] 000
Recursive Inverse Dynamics • Outward Recursions - Kinematics Computation Computational complexity for angular velocity and acceleration Coordinate Transformation The extra term in computation: Also
Recursive lnverse dynamics Outward Recursions-Kinematics Computation Complexity 8M 5A R) IOM 7A (R 8M 4A(P) 8M 4A(P)
Recursive Inverse Dynamics • Outward Recursions - Kinematics Computation Complexity: 8M 5A (R) 8M 4A (P) 10M 7A (R) 8M 4A (P)
