+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