Matrix Methods in Kinematics Vector Methods with Matrix Notation Rigid Body Rotation Matrices Rigid Body – points have same relative position Displacement = Rotation and Translation Angular rotations described about 1. Right hand Cartesian axes (x,y,z) 2. Arbitrary Axis 3. Euler Angles

Matrix Methods in Kinematics Holonomic Systems Holonomic Constraints Relations between co-ordinates and possibly time) Rigid body xyz displacement is not order dependent

Matrix Methods in Kinematics Rigid body rotations Rotations not commutative (order dependent)

Matrix Methods in Kinematics 1. Rotate about Z (α) Components in the fixed system x-y Rotation matrix

Matrix Methods in Kinematics Rotation about three Cartesian axes y x z X,Y,Z axes fixed in space

Matrix Methods in Kinematics Plane rotation (2D) (rotation about z) Spatial Rotation – sequence z,y,x

Matrix Methods in Kinematics

Matrix Methods in Kinematics y u 2. Rotation about axis u (φ) Decompose to rotations about x, y, z Rotate u about z and then back again +/- show directions x z

Matrix Methods in Kinematics y x z u

Matrix Methods in Kinematics

Matrix Methods in Kinematics 3. Euler Angles Displacement of a rigid body in terms of three relative displacement angles Each rotation about an axis whose location depends on preceding rotation 1 2 3 4 sets of axes initially coincident xyz fixed in the body

Matrix Methods in Kinematics

Matrix Methods in Kinematics

Matrix Methods in Kinematics 12 different possible rotation sets z-x-z x-y-x y-z-y z-y-z x-z-x y-x-y x-y-z y-z-x z-x-y x-z-y z-y-x y-x-z

Matrix Methods in Kinematics

Matrix Methods in Kinematics

Matrix Methods in Kinematics Euler Angles Angular Velocity Vectors 3 1 Rigid body 3 1 2 2

Matrix Methods in Kinematics Rotation of Rigid Body in 2d Cartesian Space p1 q1 Fixed x-y Vector form

Matrix Methods in Kinematics 2 1

Matrix Methods in Kinematics 3x3 D = plane displacement matrix

Matrix Methods in Kinematics Spatial (3D) Rigid Body Displacement Replace Cartesian u Axis Euler Using Cartesian new original 4x4 matrix (3D)

Matrix Methods in Kinematics Screw displacement matrix y q u q1 p=p1+su s p1 z x Screw displacement matrix

Matrix Methods in Kinematics 12 non constant elements 3D space= 6 DOF, 6 elements are dependent To Define D: euler angles, dir cos, points on body, etc…

Matrix Methods in Kinematics Example Displacement of a point Moving with a rigid body

Matrix Methods in Kinematics 3x3 Displacement matrix

Matrix Methods in Kinematics Finite Rotation Pole – plane rotation about p0 po With new position vectors p1=p2=p0

Original displacement matrix Matrix Methods in Kinematics Previously p1 and p2 in Displacement matrix D now written as: q2 is the same point Original displacement matrix

Matrix Methods in Kinematics

Matrix Methods in Kinematics HW #5 Salute Y X q shoulder 30◦ p Z p1=elbow q1=tip of finger Use Cartesian angles to find [D] Treat as 3 independent rotations

Matrix Methods in Kinematics HW #5 Salute Y Y Changed axes notation X X shoulder 2 Z p 1 Z q p1=elbow move origins for 2,3 rotations Y q1=tip of finger X 30◦ z-y-z rotation 3 Z

Matrix Methods in Kinematics Finding the Displacement Matrix by Inversion y B1 Known points C1 C2 A1 A2 B2 x

Matrix Methods in Kinematics Displacement Matrix by Inversion

Matrix Methods in Kinematics q q1 d a x y 2D Planar motion z D=d*a-1

Matrix Methods in Kinematics Finding the inverse of A (by hand – the long way) adj - adjoint ith row=ith column α=co factor Mij= minor of A

Matrix Methods in Kinematics Using MATLAB inv(a) ans = 0.5000 -0.5000 2.0000 0.5000 0.5000 -3.0000 -1.0000 0 2.0000 d = 5 7 6 1 1 2 1 1 1 a = 2 2 1 4 6 5 1 1 1 0 1 1 -1 0 3 0 0 1 D=d*a-1 Displacement matrix

Matrix Methods in Kinematics HW 5 by Inversion Using My arm

Matrix Methods in Kinematics By Inverse method – position 1 2 -0.0500 0.0500 1.7500 0 -0.0500 -0.7500 0.0500 0 0 D = 0 1.0000 0 -1.0000 0 0 0 0 1.0000

Matrix Methods in Kinematics By Inverse method – position 2 3 0.0500 0.0500 1.7500 -0.0500 0 -0.7500 0 -0.0500 0 -0.8660 0.5000 -27.9900 -0.5000 -0.8660 -7.5000 0 0 1.0000

Matrix Methods in Kinematics