Introduction to Vectors and Frames

Introduction to Vectors and Frames
CIS Russell Taylor Sarah Graham ; Copyright © 1999, 2000, 2001 rht+sg

Information Flow Diagram
Model the Patient Plan the Procedure The representation of information flow we are using contains three primary components: modeling planning, and execution. The arrows represent information flowing between these components as well as into and out of the system (to & from the real world). Modeling is defined as the process that takes raw and non-patient specific information and creates representations of the specific patient within the computer. Planning is the process of creating an intended course of action, based on those models. Execution is the process of carrying out that plan. Notice, though, that the real world (patient) might have changed between the modeling and the execution, so, as part of the execution, the system must relocalize or Register the patient (as well as any other sensing or manipulation tools) to the models and plans created previously. Execute the Plan Real World ; Copyright © 1999, 2000, 2001 rht+sg

x Femur Planned hole Pins CT image Tool path
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x Tool path Planned hole Pin 2 Pin 3 Pin 1 CT image Femur Assume equal
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Can calibrate (assume known for now) Tool path
Want these to be equal Tool tip Tool holder Can calibrate (assume known for now) Tool path Can control Pin 2 Planned hole Pin 3 Base of robot Pin 1 CT image Femur Assume equal ; Copyright © 1999, 2000, 2001 rht+sg

Tool tip Tool holder Target Base of robot
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Tool path Planned hole Pin 2 Pin 3 Pin 1 CT image Femur Assume equal
; Copyright © 1999, 2000, 2001 rht+sg

Tool tip Tool holder Tool path Pin 2 Pin 3 Base of robot Pin 1
CT image ; Copyright © 1999, 2000, 2001 rht+sg

… Let’s review some math
But: We must find FCT … Let’s review some math Tool tip Tool holder Tool path Pin 2 Base of robot Pin 3 Pin 1 CT image ; Copyright © 1999, 2000, 2001 rht+sg

Coordinate Frame Transformation
x1 y1 z1 x0 y0 z0 F ; Copyright © 1999, 2000, 2001 rht+sg

Coordinate Frames b F = [R,p]
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Forward and Inverse Frame Transformations
This [R,p] is useful, even without the 4x4 representation. These formulae can be derived from the Homogenous form, but don’t require it. Note: the negative in the translation component of the inverse of F comes from the fact that the inverse of p is -p. ; Copyright © 1999, 2000, 2001 rht+sg

Composition Assume Then So
Two successive motions can be converted into a single motion by composing the corresponding frames representing them The order of composition is important In graphics world, the matrices are column major and vectors are rows. So all operations appear transposed. We can ignore this computation for our purposes. To see that the graphics terminology is essentially the same, one can check: So ; Copyright © 1999, 2000, 2001 rht+sg

Vectors v w u = vxw v•w z y x
Vectors are the basic representation of points or directions in space. In general, a vector can either be a row or column of numbers, though for the rest of the course we’re going to use only the column representation. The most basic measure of a vector is its length, which, from the Pythagorean Theorem is where  means “defined as”, so means “length of v” The dot product is an operator that takes two vectors and returns a single value. This value can be interpreted in several ways: The length of either of the vectors “projected onto” the other The product of the lengths of the vectors and the cosine of the angle between them The cross product takes two vectors and returns another vector. This vector has two interesting properties. Its direction is perpendicular to both of the original vectors Its length is the product of the vectors’ lengths and the sine of the angle between them ; Copyright © 1999, 2000, 2001 rht+sg

Vectors as Displacements
z w v v+w w y x v v-w w y x ; Copyright © 1999, 2000, 2001 rht+sg

Vectors as Displacements Between Parallel Frames
x1 y1 z1 x0 y0 z0 v1 v0 w Vectors can not only represent points in space, but displacements as well. After adding two vectors, the result can be seen as the arrow produced by placing the input vectors head to tail. Negating a vector simply reverses its arrow, thus subtraction can be seen as adding the first to the negative of the second. Interesting point: You can also a displacement as transforming the “coordinate space” from which you measure points. If you consider two “coordinate spaces”, where the origin of the second is the origin of the first plus some difference vector (w), when you measure the same point with respect to these two origins, the values will differ by -w. ; Copyright © 1999, 2000, 2001 rht+sg

Rotations: Some Notation
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Rotations: A few useful facts
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Properties of Rotation Matrices
Inverse of a Rotation Matrix equals its transpose: R-1 = RT RT R=R RT = I The Determinant of a Rotation matrix is equal to +1: det(R)= +1 Notice that the inverse of a rotation around a single axis is simple the rotation by the negative of the angle. However, if the rotation is more complex you may not know the axis of rotation. However, the transpose always inverts the rotation. Notice also that if you invert a rotation which has been built of component rotations, not only do you need to invert the individual rotations, but you need to swap their order as well (because (AB)T=BTAT ) Any Rotation can be described by consecutive rotations about the three primary axes, x, y, and z: R = Rz, Ry, Rx, ; Copyright © 1999, 2000, 2001 rht+sg

Canonical 3D Rotation Matrices Note: Right-Handed Coordinate System
R2 is a rotation matrix. This 2-D case is actually the 3-D rotation of coordinates axes 1 (x1 ,y1 and z1), about the z axis, Rz, . Once again, the matrix for the original axes is given by R1. The 3-D rotation matrix for rotating a set of coordinate axes, or a frame, about the z axis by an angle  is then given by R2 . Around y: Around x: ; Copyright © 1999, 2000, 2001 rht+sg

Homogeneous Coordinates
Widely used in graphics, geometric calculations Represent 3D vector as 4D quantity For our purposes, we will keep the “scale” s = 1 ; Copyright © 1999, 2000, 2001 rht+sg

Representing Frame Transformations as Matrices
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Frame transformation from 3 point pairs
x x x ; Copyright © 1999, 2000, 2001 rht+sg

Rotation from multiple vector pairs
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Introduction to Vectors and Frames

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