McGill Dept of Mechanical Engineering MECH572 Introduction to robotics Lecture 11
MECH572 Introduction To Robotics Lecture 11 Dept. Of Mechanical Engineering
Review Recursive Inverse dynamics Inverse Dvnamics-Known joint angles compute joint torques 1)Outward Recursion- Kinematic Computation Known 0, 0, 0 L Compute t, From 0 to n, recursively based on geometrical and differential relationship associated with each link 2)Inward Recursion-Dynamics Computation Compute wrench wi based on wi+l and kinematic quantities obtained from 1) From n+I to 0, recursively using Newton-Euler equation
Review • Recursive Inverse Dynamics Inverse Dynamics – Known joint angles compute joint torques 1) Outward Recursion – Kinematic Computation Known Compute From 0 to n, recursively based on geometrical and differential relationship associated with each link. 2) Inward Recursion – Dynamics Computation Compute wrench wi based on wi+1 and kinematic quantities obtained from 1) From n+1 to 0, recursively using Newton-Euler equation θ,θ,θ t t
Review The Natural Orthogonal Compliment Each link-6-DOF, Within the system-1-DOF 5-DOF constrained Kinematic Constraint equation Kt=0 KT=O T: Natural Orthogonal Complement (Twist Shape Function)
Review • The Natural Orthogonal Compliment Each link – 6-DOF; Within the system – 1-DOF 5-DOF constrained Kinematic Constraint equation T : Natural Orthogonal Complement (Twist Shape Function)
Review Natural Orthogonal Complement(cont'd) Use T in the Newton-Euler Equation, the system equation of motion becomes C6+7+6+Y Where I≡mMT Generalized inertia matrix T≡Tw Active fo d=TWo Dissipative force Gravitational force C(,0三TM+TwMT Vector of Coriolis and fugal fo Consistent with the result obtained from Euler-Lagrange equation
Review • Natural Orthogonal Complement (cont'd) Use T in the Newton-Euler Equation, the system equation of motion becomes: where Consistent with the result obtained from Euler-Lagrange equation Generalized inertia matrix Active force Dissipative force Gravitational force Vector of Coriolis and centrifugal force
Natural Orthogonal Complement Constraint Equations twist-Shape MatriX 1)Angular velocity Constraint e ea×(w1-w-1)=0 E2{;-w-1)=0 (6.63) O Ei: Cross-product matrix of ei OF1 2) Linear Velocity Constraints Ci=Ci1+ Srl+ Pi Differentiate c;-2-1+Pxi+6-1×w-1=0 c;-c-1+R1+D2-1;-1=0 (6.64)
Natural Orthogonal Complement • Constraint Equations & Twist-Shape Matrix 1) Angular velocity Constraint Ei : Cross-product matrix of ei 2) Linear Velocity Constraints ci = ci-1+ i-1 + i Differentiate: Oi-1 Oi O Ci-1 Ci ci-1 c i-1 i Oi+1
Natural Orthogonal Complement Constraint equations twist Shape matrix -R Joint Equations(6.63)and(6. 64) pertaining to the first link E141=0 (665a) 81+R1w1=0 6.65b K1t1=0 6 K;-1t-1+Kt;=0,t=1,:7 666b) Er O K R11 6.67a K;;1≡ ei o (667b) E; O R (6.67
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint Equations (6.63) and (6.64) pertaining to the first link:
Natural Orthogonal Complement Constraint equations twist shape matrix -R Joint Kll O6 O6 6 6 K21K22O6 O K (6.68) O606O Kn-1 ) Oc O6 O6 K K 6n xin matrix O6 denoting the 6 x 6 zero matrix
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint 6n 6n matrix
Natural Orthogonal Complement Constraint equations twist shape matrix -R Joint Define partial Jacobian j.6=t (6.69) 6x n matrix with its element defined as ifj≤ ei xr 0 (670) otherwise Mapping the first i joint rates to ti of the ith link
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint Define partial Jacobian 6 n matrix with its element defined as Mapping the first i joint rates to ti of the ith link
Natural Orthogonal Complement Constraint equations twist shape matrix -R Joint +1 +1 aj a;+a+1+…+a-1+p2ifj< r;;≡〈p ifj=勾 (6.71) 03 otherw
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint
Natural Orthogonal Complement Constraint Equation and twist Shape matrix-R Joint t;=61ti+b2t2+…+6;tn;i=1,…,7 (672) 10 0 0 T三 (6.73) 77 Easy to verify KT=o -11 Oe o O t110 0 K21K22O6 O 0 O6 Oa O6 K O OG O6 O K nn-1 KnnI Tmi t K t 11U1 0 Recall K;-1ta-1+Kt=0,=1,…:7
Natural Orthogonal Complement • Constraint Equation and Twist Shape Matrix – R Joint Easy to verify Recall