MEAM 520. Homogenous Transformations

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1 MEAM 520 Homogenous Transformations Katherine J. Kuchenbecker, Ph.D. General Robotics, Automation, Sensing, and Perception Lab (GRASP) MEAM Department, SEAS, University of Pennsylvania Lecture 3: September 13, 2012

5 Office Hours Monday 1-2 pm :: Denise Wong, GRASP Conference Room Tuesday 1:30-2:30 pm :: Katherine Kuchenbecker, Towne 224 Tuesday 2:30-3:30 pm :: Denise Wong, GRASP Conference Room Wednesday 5-6 pm :: Philip Dames, GRASP Conference Room Thursday am :: Philip Dames, GRASP Conference Room Thursday 3-4 pm :: Katherine Kuchenbecker, Towne 224

6 New Course Website

7 Rotation Matrices Slides created by Jonathan Fiene

8 Three-Dimensional Rotation Matrix y 0 R 0 1 = x 1 x 0 y 1 x 0 z 1 x 0 x 1 y 0 y 1 y 0 z 1 y 0 x 1 z 0 y 1 z 0 z 1 z 0 y 1 x 1 z 1 x 0 SO(3) Special Orthogonal group of order 3 z 0

9 Rotation Matrices y 0 Rotation matrices serve three purposes (p. 47 in SHV): p 0 b 1. Coordinate transformation θ =90 x 0 relating the coordinates of a p 0 a point p in two different y 0 z 0 frames y 1 x1 2. Orientation of a transformed coordinate frame with respect to a fixed frame z 1 x 0 y 0 p 3. Operator taking a vector z 0 y 1 x1 and rotating it to yield a new vector in the same x 0 coordinate frame z 1 z 0

10 Composite Rotations

11 Composition of Rotations with Respect to the Current Frame

12 Composition of Rotations with Respect to the Current Frame the result of a successive rotation about the current (intermediate) frame can be found by post-multiplying by the corresponding rotation matrix R 0 2 = R 0 1 R 1 2

13 Composition of Rotations with Respect to a Fixed Frame

14 Composition of Rotations with Respect to a Fixed Frame the result of a successive rotation about a fixed frame can be found by pre-multiplying by the corresponding rotation matrix R 0 2 = RR 0 1 Note that R is a rotation about the original frame

15 You should read the book and think about this.

16 Parameterizing Rotations

17 Parameterization of Rotations R 0 1 = x 1 x 0 y 1 x 0 z 1 x 0 x 1 y 0 y 1 y 0 z 1 y 0 x 1 z 0 y 1 z 0 z 1 z 0 in three dimensions, no more than 3 values are needed to specify an arbitrary rotation which means that the 9-element rotation matrix has at least 6 redundancies numerous methods have been developed to represent rotation/orientation with less redundancy Euler Angles Roll, Pitch, Yaw Angles Axis/Angle Representation Conventions vary, so always check definitions!

18 Euler Angles Define a set of three intermediate angles, φ, θ, ψ, to go from 0 3 z 0 z 3 y 3 x 3 y 0 x 0

19 Euler Angles step 1: rotate by z 1 z 0 φ about z 0 y 1 y 0 x 0 x 1 φ

20 Euler Angles step 2: rotate by θ about y 1 z 1 z 2 y y 2 1 x 2 θ x 1

21 Euler Angles step 3: rotate by ψ about z 2 z 3 z 2 y 3 y 2 x 2 x 3 ψ

22 Euler Angles to Rotation Matrices (post-multiply using the basic rotation matrices) R = R z,φ R y,θ R z,ψ = c φ s φ 0 s φ c φ c θ 0 s θ c ψ s ψ 0 s ψ c ψ s θ 0 c θ = c φc θ c ψ s φ s ψ c φ c θ s ψ s φ c ψ c φ s θ s φ c θ c ψ + c φ s ψ s φ c θ s ψ + c φ c ψ s φ s θ s θ c ψ s θ s ψ c θ

23 Roll, Pitch, Yaw Angles defined as a set of three angles about a fixed reference z 0 φ roll y 0 x 0 ψ yaw θ pitch

24 Roll, Pitch, Yaw Angles to Rotation Matrices (pre-multiply using the basic rotation matrices) R = R z,φ R y,θ R x,ψ = c φ s φ 0 s φ c φ c θ 0 s θ c ψ s ψ s θ 0 c θ 0 s ψ c ψ = c φc θ c φ s θ s ψ s φ c ψ s φ s ψ + c φ s θ c ψ s φ c θ s φ s θ s ψ + c φ c ψ s φ s θ c ψ c φ s ψ s θ c θ s ψ c θ c ψ

25 Axis/Angle Representation rotation by an angle about an axis in space z 0 θ k k = k x k y k z y 0 x 0

26 Axis/Angle Representation z 0 any rotation matrix can be represented this way! k θ y 0 x 0

27 Talk to the person next to you. Explain one of the three parameterization approaches to your partner, then switch. Talk about the third one together. z 1 z 10 z 0 z 0 z 3 z 2 φ k y 3 y y 2 12 θ x 0 φ θ x 2 x 1 x 3 ψ y 0 x 0 ψ θ y 0 x 0 y 0

28 Homogeneous Transformations

29 Rigid Motion a rigid motion couples pure translation with pure rotation z 1 z 0 p v 1 p y 1 o 1 x 1 o 0 d 0 1 y 0 x 0 rigid motions can be expressed as v 0 p = R 0 1v 1 p + d 0 1

30 Homogeneous Transforms a homogeneous transformation is a matrix representation of rigid motion, defined as [ ] R d H = 0 1 where R is the 3x3 rotation matrix, and d is the 3x1 translation vector H = n x s x a x d x n y s y a y d y n z s z a z d z

31 Homogeneous Transformations the homogeneous representation of a vector is formed by concatenating the original vector with a unit scalar P = [ p 1 ] where p is the 3x1 vector P = p x p y p z 1

32 Homogeneous Transformations rigid body transformations are accomplished by pre-multiplying by the homogenous transform P 0 = H 0 1 P 1 {example with quadrotor model}

33 P 0 = H 0 1 P 1 z 0 o 0 y 0 R z,θ = cos θ sin θ 0 sin θ cos θ x 0

34 z 0 o 0 y 0 x 0

35 Homogeneous Transformations composition of multiple transforms is the same as for rotation matrices: post-multiply when successive rotations are relative to intermediate frames H 0 2 = H 0 1 H 1 2 pre-multiply when successive rotations are relative to the first fixed frame H 0 2 = HH 0 1

36 Homogeneous Transformations Composition (intermediate frame) H 0 2 = H 0 1 H 1 2 = [ R 0 1 d ][ R 1 2 d ] = [ R 0 2 R 0 1d d ] Inverse Transform H 1 0 = [ R 1 0 d = 0 1 ] [ (R 0 1 ) (R 0 1) d 0 1 ]

37 Homework 1 Homework 1: Rigid Motions and Homogeneous Transformations MEAM 520, University of Pennsylvania Katherine J. Kuchenbecker, Ph.D. September 11, 2012 This assignment is due on Tuesday, September 18, by 5:00 p.m. sharp. You should aim to turn the paper part in during class that day. If you don t finish until later in the day, you can turn it in to Professor Kuchenbecker s office, Towne 224. The code must be ed according to the instructions at the end of this document. Late submissions of either or both parts will be accepted until 5:00 p.m. on Wednesday, but they will be penalized by 25%. After that deadline, no further assignmentsmaybesubmitted. You may talk with other students about this assignment, ask the teaching team questions, use a calculator and other tools, and consult outside sources such as the Internet. To help you actually learn the material, what you write down should be your own work, not copied from a peer or a solution manual. Book Problems (30 points) The first set of problems is from the textbook, Robot Modeling and Control by Spong, Hutchinson, and Vidyasagar (SHV). Please follow the extra clarifications and instructions when provided. Write in pencil, show your work clearly, box your answers,andstapletogetherallpagesofyourassignment. 1. SHV 2-10, page 66 Sequence of Rotations (5 points) Please specify each element of each matrix in symbolic form and show the order in which the matrices should be multiplied; as stated in the problem, you do not need toperformthematrixmultiplication. 2. SHV 2-14, page 67 Rotating a Coordinate Frame (5 points) Sketch the initial, intermediate, and final frames by reading thetextintheproblem. ThenfindR in two ways: by inspection of your sketch and by calculation. Check your solutions against one another. 3. SHV 2-23, page 68 Axis/Angle Representation (10 points) 4. SHV 2-39, page 70 Homogeneous Transformations (10 points) Treat frame o 2 x 2 y 2 z 2 as being located at the center of the cube s bottom surface (as drawninfigure 2.14), not at the center of the cube (as stated in the problem). MATLAB Programming (30 points) This class will use MATLAB to analyze and simulate robotic systems and also to control real robots. While Professor Kuchenbecker loves MATLAB, she recognizes that it canbedifficulttouseatthestart. Evenif you don t like MATLAB now, please give it a chance, and come to office hours or contact the teaching team if you feel lost or frustrated. Your task for this question is to update a provided MATLAB script so that it animates the movement of rectangular block that was moved in a specific way. The motion was captured on video, and the positions and orientations of the block were recorded over time using a Ascension TrakStar magnetic motion tracking system that includes a sensor located inside the block. 1

40 z 2 z 0 the flying box y 0 y 2 frame 0 - camera x 2 frame 2 - sensor y 1 x 1 x 0 frame 1 - transmitter z 1

42 Flying Box by PUT YOUR NAME HERE z0 (in.) y0 (in.) x0 (in.) 0

43 Programming Homework Tips: 1. Write out your approach before sitting down to program. 2. Start early to give yourself time to figure things out.

44 Questions?

MEAM 520. More Velocity Kinematics

MEAM 520. More Velocity Kinematics MEAM 520 More Velocity Kinematics Katherine J. Kuchenbecker, Ph.D. General Robotics, Automation, Sensing, and Perception Lab (GRASP) MEAM Department, SEAS, University of Pennsylvania Lecture 12: October