Rotation Matrices and Quaternion Siamak Faal 2/2 07/02/2016 1 Reviews from Last Presentation 2 Application Analytical Navigation calculations Visual representations Automation 3 Representation of Orientation The same shape in three different orientations 4 Exercise Q) We have a vector that is defined as: = 4 + 3 1 ⃗ What are the coordinates ̂ of̂ in coordinate frame {0}? ⃗ 0 A) 1 = 4 cos 30 3 sin 30 0 = 4 sin 30 + 3 cos 30 0 ⋅ ̂ − cos30⋅ ̂ sin 30 4 1.964 1 = sin 30 cos 30 3 4.598 − 30° 0 5 Integrating Angular Velocities Forces Dynamic Linear Accelerations Moments Equations Angular Accelerations Lets go over two simple examples in 2D and 3D spaces 6 2-D Example Outcomes of kinetics / kinematics computation: degree = = 30 s Model 2 where:1 2 1 = Initial Initial conditions: configuration = = 0 1 2 Q) What is the final configuration of Final the mechanism at t = 3 seconds? configuration 7 3-D Example degree = = = 30 s Q) at t = 3 seconds which one of the configurations represent the orientation of the object? A) None! We cant be sure! 8 Quaternion and Orientation Puzzle Orientation 9 Background Information 10 Complex Numbers A short review: Im = + = 1 where − Re Euler’s formula = cos + sin 11 Angle-Axis Any orientation (or a rotation operation) can be defined as a rotation about a defined axis: Euler Parameters = sin 2 1 = sin Orientation can 2 visualized as point 푧 2 on a 4D sphere = sin 2 ′ 3 = cos 2 where: 1 ′ = 1 2 = 1 2 12 � Quaternion and Orientation Puzzle Orientation Angle-Axis 13 Quaternions Theory Sir William Rowan Hamilton August 4, 1805, Dublin, Republic of Ireland September 2, 1865, Dublin, Republic of Ireland 14 History Broom Bridge Dublin, Irland + + = = 1 2 2 2 15 − Quaternions • Quaternions are a number system which extends the complex numbers + + + • Quaternions are noncommutative푎 : • Hamilton defined a quaternion1 as2 the≠ quotient21 of two directed lines in a 3D space = • The algebra of quaternions is often denoted by H (for Hamilton) 16 Quaternions • As a set, the quaternions may be identified with • H has three operations: 4 ℝ • Addition • Scalar multiplication • Quaternion multiplication 17 Quaternions Addition The sum of two elements of is defined to be their sum as elements of 4 ℝ Example: Let = + + + and = + + + , then: 1 1 1 1 1 2 2 2 2 2 + = + + + + + + + 1 2 1 2 1 2 1 2 1 2 18 Quaternions Scalar Multiplication The product of an element of by a real number is defined to be the same as the product by a scalar in 4 ℝ Example: Let = + + + and , then: 1 1 1 1 = + +∈ ℝ + 1 1 1 1 19 Quaternion Multiplication The identities are: = = = = 1 2 2 2 The above identities can be used to derive multiplication − of basis elements Example: Multiplying both sides of the above equation with yields = = 1 = = 2 All the possible products are: − − ⇒ = = = = − = = − 20 − Quaternion Multiplication Let = + + + and = + + + , then: 1 1 1 1 1 2 2 2 2 2 =12 + + + + + + + 2 + 1+2 12 12 12 12 12 12 12 2 Using⋯ the basis12 element multiplication rules: = 1+2 1+2 − 1+2 − 12 − 12 + 12 12 + 12 +− 12 + 12 −+ 12 12 + 12 1 2 1 2 1 2 1 2 − 21 Conjugation Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition Let = + + + , the conjugate of is the quaternion: = ∗ Conjugation is an involution (it is its own − inverse) − − = ∗ ∗ The conjugate of a product of two quaternions is the product of the conjugates in the reverse order = ∗ ∗ ∗ 12 21 22 Norm = + + + Scalar part Vector part The scalar part of is + /2 and the vector part of is /2 ∗ ∗ The norm of H is defined as: − = ∗ = = The unit quaternion is defined1 as:2 =1/ 2 23 Reciprocal Using conjugation and the norm makes it possible to define the reciprocal of a non-zero quaternion. = ∗ −1 2 The division between two quaternions can be defined as their multiplication by ones reciprocal: = 1 −1 1 2 2 24 Quaternion and Orientation Puzzle Orientation Angle-Axis Quaternions Theory 25 Quaternions As a definition for orientation 26 Introduction • Notations: • : The orientation of frame B relative to frame A • � : Vector described in frame A Quaternion̂ as a rotation operator / orientation: = = cos sin sin sin 2 2 2 2 1 2 3 4 � − − − is the conjugate of � =� = ∗ � � 1 −2 −3 −4 Compound Orientations and Mappings A compound orientation can be defined as multiplication of two quaternions = � �� A vector mapping using quaternions is defined as: = where: ∗ � � = 0 = 0 1 2 3 1 2 3 Rotation matrix from quaternions The rotation matrix based on quaternions is defined as: 2 1 + 2 2 + 2 = 2 2 2 2 1 + 2 2 + 1 − 2 23 14 24 − 13 2 + 2 2 2 2 1 + 2 2 3 1 4 1 3 3 4 1 2 − − 2 2 The z-y-x Euler angles24 based13 on quaternions34 − 1are2: 1 − 4 = 2 , 2 + 1 2 2 푎= sin23 −214 + 21 2 − −1 = 2− 24 , 2 1+3 1 2 2 푎 34 − 12 1 4 − Angular velocity and rate of change of quaternions If the angular velocity in the body coordinate frame defined as: = 0 The quaternion derivative describing the rate of change of orientation of the earth frame relative to the body frame is defined as: 1 = 2 ̇ � Quaternion and Orientation Puzzle Orientation Angle-Axis Quaternions Theory Quaternions for Orientation 31.