+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
Rigid-Body mechanics Description of a Rigid-Body motion Rigid-body motion preserves distance Q Q(p +Q(p d A Define da= P三p-p lp=a-p+Q(p-a p a-p+Q(p-a)+ dA+(Q-1(p-a) edp=e dA+e(Q-1)(p-a)o P-Arbitrary Reference Left multiply QT. Take transpose Q e=e Q-1)=0 d dA三do Displacement of any point projected onto the rotation axis are same
Q O P P' A p a p' a' P – Arbitrary A - Reference Rigid-Body Mechanics • Description of a Rigid-Body Motion Rigid-Body motion preserves distance Define Left multiply Take transpose = 0 A' Displacement of any point projected onto the rotation axis are same
Rigid-Body mechanics General rigid-Body motion o 3.2.1 The component of the dis aoerments of all the points of a rigid body undergoing a general motion along the aris of the underlying rotation is a constant. e p dA三do A F Theorem 3.2.2(Mowzi, 1763; Chasles, 1830) Given a rigid body un- dergoing a general motion, a set of its points located on a line l undergo identical displacements of minimun magnitude. Moreover, line c and the minimum-magnitude displacement are parallel to the aris of the rotation nvolved
Rigid-Body Mechanics • General Rigid-Body Motion e
Rigid-Body mechanics General Rigid-body motion Geometric Interpretation /A A B Pitch 2-D case i 3-D case Any rigid body motion in 2-D can be Any rigid body motion in 3-D can be regarded as a pure rotation around regarded as a screw-like motion along one point O an axis. e
Rigid-Body Mechanics • General Rigid-Body Motion Geometric Interpretation A B A' B' O 2-D case 3-D case e Pitch p0 Any rigid body motion in 2-D can be regarded as a pure rotation around one point O Any rigid body motion in 3-D can be regarded as a Screw-like motion along an axis, e
Rigid-Body mechanics Screw Motion of a rigid-Body a rigid body can attain any configuration from its original to arbitrary position following a screw-like motion defined by Screw axis l Pitch P d o dpe or do-minimum displacement φ- Magnitude Pitch P
Rigid-Body Mechanics • Screw Motion of a Rigid-Body A rigid body can attain any configuration from its original to arbitrary position following a screw-like motion defined by – Screw Axis L – Pitch p L Pitch p d0 – minimum displacement - Magnitude
Rigid-Body mechanics Screw of rigid-body motion Question to answer how to define the screw axis of rigid-body motion? Known Q, a and a (direction+reference point Seek po(a point to define the screw axis C do=dA+(Q-1)(po-a P dA+(Q-1)(p0-a)=eedA (Q-1)p=(Q-1)a-(1-eedA (310b) p=0 39) Rearrange(3. 10b)&(3.9) Apo=b A=Q1 b=(Q-1)a-(1-eedai Linear sy stems of equation (overdetermined)
Rigid-Body Mechanics • Screw of Rigid-Body Motion Question to answer – how to define the screw axis of rigid-body motion? Known Q, a and a' (direction+reference point) Seek p0 (a point to define the screw axis) Rearrange (3.10b) & (3.9) e O A P0 a p0 Linear systems of equation (overdetermined)