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