+4 McGill Dept Of Mechanical Engineering MECH572 Introduction To robotics Lecture 3
MECH572 Introduction To Robotics Lecture 3 Dept. Of Mechanical Engineering
Review Rigid-body rotation -Representations Representat Matrix Linear invariant Quadratic Invariant Natural Invariant Ion (E uter para ameters) Definition Q=[el e2 e3]q=sinpe 三sn 90 c08 n=(2) Number of 9 Elements Constraints llell=1,lell=1 ll2+4=1| 2+ro2=1 el‖l ee2=0,e2"e3=0, e3.e Inde pendent 9-6=3 4-1=3 Elements
Review • Rigid-body Rotation - Representations Representat ion Matrix Linear Invariant Quadratic Invariant (Euler Parameters) Natural Invariant Definition Number of Elements 9 4 4 4 Constraints ||e1|| = 1, ||e2|| = 1, ||e3|| = 1 e1•e2= 0, e2•e3= 0, e3•e1= 0 ||e|| = 1 Independent Elements 9 - 6 = 3 4 – 1 = 3 4 – 1 = 3 4 – 1 = 3
Review Alternative form to represent a rotation- Euler Angles a sequence of rotation Q=Q(0)Q(阝)Q(y) a,B, y rotation angles about certain axes Coordinate Transformation General form pA=[bA+[Q]A[T]B Origin offset Homogeneous form pJA=TJAT)E ①TA≡ LA b pM [p] 1
Review • Alternative form to represent a rotation – Euler Angles A sequence of rotation: Q = Q()Q()Q() , , rotation angles about certain axes. • Coordinate Transformation General form Homogeneous form Origin offset
Review Similarity transformations Transformation of matrix entries(compare with vector entries which uses linear transformation) [L]A=[AALLBLA-JA The concept of invariance After transformation between frames, certain quantities are unchanged or frame invariant (inner product, trace, moments, etc
Review • Similarity Transformations - Transformation of matrix entries (compare with vector entries which uses linear transformation) • The concept of invariance After transformation between frames, certain quantities are unchanged or frame invariant (inner product, trace, moments, etc)
Overview of Rigid-Body mechanics Purpose- Lay down foundations of kinetostatics(kinematics t statics) and dynamics of rigid bodies using matrix method Scope Linear and angular displacement velocity and acceleration analysis Static analysIs Mass Inertial properties Equation of motion for single rigid body Useful tools/concepts to be introduced Screw theory Twist renc
Overview of Rigid-Body Mechanics • Purpose – Lay down foundations of kinetostatics (kinematics + statics) and dynamics of rigid bodies using matrix method • Scope – Linear and angular displacement, velocity, and acceleration analysis – Static analysis – Mass & Inertial properties – Equation of motion for single rigid body • Useful tools/concepts to be introduced – Screw theory – Twist – Wrench