+4 McGill Dept Of Mechanical Engineering MECH572 Introduction to robotics Lecture 2
Lecture 2 MECH572 Introduction To Robotics Dept. Of Mechanical Engineering
Review Overview of fields of robotics Concept of vector Space and Linear Transformation Ax =b linear system of equation m×nnm Column Space(range), Null space Properties: A(ax+By)=a Ax BAy Useful Linear Transformation in 3-dimensional space Projection Reflection Rotation Important- Understand physical meaning
Review • Overview of fields of Robotics • Concept of Vector Space and Linear Transformation Ax = b linear system of equation m×n n m Column Space(range), Null space Properties: A(αx+y) = Ax + Ay • Useful Linear Transformation in 3-dimensional space Projection Reflection Rotation Important - Understand physical meaning
Review Linear Proiection Reflection Rotation Trans (P) (R) Definition P=1-nnTI R=1-2nnt Q=eeT+csp(1-eeTy p +sin E n Properties 2=P,Pn=0 R2=1 QQ=1 R=R Det det(p)=0 det(R) det()=+1 (singular)
Review Linear Trans. Projection (P) Reflection (R ) Rotation (Q ) Definition Properties Det (singular) p n P' n p P" e p P'
Review Linear trans Projection Reflection Rotation (P) (R) (Q) Geometric Interpretation Z Matrix Representation 100 100 010 010 010 000 00-1
Review Linear Trans. Projection (P) Reflection (R ) Rotation (Q ) Geometric interpretation Matrix Representation 0 0 0 0 1 0 1 0 0 0 0 −1 0 1 0 1 0 0 − − 0 0 1 0 1 0 1 0 0 x y z x y z x y z x z y x y z x y z
review Rotation matrix Q=ee+oos(1-ee)+sin E Alternative form Q=1+sindE +(1-Cos)E (254) Canonical form Euler angles a rotation sequence along different axes roll: e is x axis Pitch: e is Y axis Yaw: e is Z axis
Review • Rotation Matrix Alternative form Canonical form – Euler Angles A rotation sequence along different axes. Roll: e is X Axis Pitch: e is Y Axis Yaw: e is Z Axis
Review Example- Rotation about x axis x=1*x+0*y+0* y=0*x'+coso *y-sind*= 二=0*x+sin*y+cosφ* 10 0 y=0 cos p -sin o y 0 sin o coso Q maps p into p
Review • Example – Rotation about x axis x = 1*x' + 0*y' + 0*z' y = 0*x' + cos*y' - sin*z' z = 0*x' + sin *y' + cos*z' p = Q p' Q maps p' into p = − ' ' ' 0 sin cos 0 cos sin 1 0 0 z y x z y x x y z x' y' O z
Mathematical Background Concept of linear Invariants Cartesian decomposition of any 3x matrix A As≡(A+A2) Symmetric Ass≡方(A-A2) Skew-symmetric The vectorof a has the following property axv≡Assv The trace of a is defined as the eigenvalues of as
Mathematical Background • Concept of Linear Invariants Cartesian decomposition of any 3x3 matrix A: The vector of A has the following property: The trace of A is defined as the eigenvalues of As Symmetric Skew-symmetric
Mathematical Background Concept of linear Invariants For 3x3 matrix a defined in a certain coordinate frame Vector vect(A)≡a≡言a13-a 21-12 Trace tr(A)三a11+a22+a33 Properties vect(a)=0 if a is symmetric tr(a)=0 if A is skew-symmetric vect(ab)=-axb tr(ab)=ab
Mathematical Background • Concept of Linear Invariants For 3x3 matrix A defined in a certain coordinate frame: Properties vect(A) = 0 if A is symmetric tr(A) = 0 if A is skew-symmetric Vector Trace
Mathematical Background Linear lnvariant of a rotation matrix Recall: Q=ee+oos p(1-ee)+sin E vect(Q)=vect(sin pE)=sin oe tr(Q)=tr[ee+cos p(1-eeT e e+cos(3 -ee)=1+2cos p tr(Q-1 Define q=sin ge Group into 4-D vector 入≡[,g,g3,o] 三cG l2+⑤≡sin2d+cs32=1 Linear invariant Points on the surface of 4-D |2=q+++=1
Mathematical Background • Linear Invariant of a rotation matrix Recall: Define Group into 4-D vector Linear Invariant Points on the surface of 4-D sphere
Mathematical Background Linear Invariant of a rotation matrix(cont'd) 士√1-G+省+嗡) sin=+llls e=qsin p lall +(1-9 Q aq xx) Q=o1+⊥qq ox Sign problem Whenφ=π, Q is not uniquely defined
Mathematical Background • Linear Invariant of a rotation matrix (cont'd) Sign Problem When = , Q is not uniquely defined
