Position and Orientation

1. Position and Orientation

2. Objectives of the Lecture • Learn to represent position and orientation • Be able to transform between coordinate systems. • Use frames and homogeneous coordinates Reference: Craig, “Introduction to Robotics,” Chapter 2. Handout: Chapter 1 Almost any introductory book to robotics

3. Introduction • Robot manipulation implies movement in space • Coordinate systems are required for describing position/movement • Objective: describe rigid body motion • Starting point: there is a universe/ inertial/ stationary coordinate system, to which any other coordinate system can be referred

4. a coordinate system {A} ZA AP XA Representation of a Position • point = position vector ZB BP YA XB

5. z0 {A} y0 y1  x1 x0 z1 {B} Description of an Orientation Often a point is not enough: need orientation • In the example, a description of {B} with respect to {A} suffices to give orientation • Orientation = System of Coordinates • Directions of {B}: XB, YB & ZB • In {A} coord. system: AXB, AYB and AZB

6. {A} XB aZ aY aX YA XA From {B} to {A} We conclude:

7. Rotation Matrix • Stack three unit vectors to form Rotation Matrix • describes {B} with respect to {A}: • Each vector in can be written as dot product of pair of unit vectors: cosine matrix • Rows of : unit vectors of {A} with respect to {B} • What is ? What is det? • Combination of position and orientation is called “pose” • To describe pose we need a frame

8. z0 {A} y0 y1  x1 x0 z1 {B} Description of a Frame • Frame: set of four vectors giving position + orientation • Description of a frame: position + rotation matrix • Ex.: • position: frame with identity as rotation • orientation: frame with zero position

9. {B} ZB AP {A} ZA BP YB APBORG XB YA XA Mapping: from frame 2 frame Translated Frames • If {A} has same orientation as {B}, then {B} differs from {A} in a translation: APBORG AP = BP + APBORG • Mapping: change of description from one frame to another. The vector APBORG defines the mapping.

10. ZA BP ZB YB YA XA XB Rotated Frames Description of Rotation = Rotation Matrix

11. Rotated Frame (cont.) • The previous expression can be written as • The rotation mapping changes the description of a point from one coordinate system to another • The point does not change! only its description

12. YA y0 YB XB x1 y1 x0 XA Example (2D rotation) 

13. BP AP XB ZA ZB APBORG YB YA XA General Frame Mapping Replace by the more appealing equation: {A} A row added here A “1” added here

14. Homogeneous Coords • Homogeneous coordinates: embedding of 3D vectors into 4D by adding a “1” as the fourth entry • Allows to write a general transformation in linear form: • More generally, the transformation matrix T has the form: … but we will use only the simple form above in this course

15. Operators: Translation, Rotation and General Transformation • Translation Operator:

16. Translation Operator • Translation Operator: • Only one coordinate frame, points are mapped • Equivalent to mapping point to a second frame • Point Forward = Frame Backwards • How does TRANS look in homogeneous coordinates? -Q

17. Operators (cont.) • Rotational Operator Rotation around axis: AP1 AP2

18. Rotation Operator • Rotational Operator The rotation matrix can be seen as rotational operator • Takes AP1and rotates it to AP2=R AP1 • AP2=ROT(K, q)(AP2) • Write ROT for a rotation around K

19. Operators (Cont.) • Transformation Operators • A transformation mapping can be viewed as a transformation operator: map a point to any other in the same frame • Transform that rotates by R and translates by Q is the same a transforming the frame by R & Q

20. Compound Transformation Suppose {C} is known relative to {B}, and {B} is known relative to {A}. Problem: transform P from {C} to {A}: Write down the compound in homog. coords

21. Inverse Transform Write down the inverse transform in HC’s

22. More on Rotations • We saw that a rotation can be represented by a rotation matrix • Matrix has 9 variables and 6+ constraints (which?) • Rotations are far from intuitive: they do not commute! • Rotation matrix can be parameterized in different manners: • Roll, pitch and yaw angles • Euler Angles • Others