lecture 3

Introduction to Robotics

Lecture 3

Lecture Contents

Spatial Descriptions and Transformations

Descriptions: positions, orientations and frames

Mapping: Changing descriptions from frame to frame

Central Topic

Problem Robotic manipulators, by definition,

implies that parts and tools will be moving around in space by the manipulator mechanism

This requires representation of positions and orientations of the parts, tools and the mechanism itself

Solution Mathematical tools for representing

position and orientation of objects/frames in a 3D space

Coordinate System

Universe Coordinate System

Assume, there is a Universe Coordinate System to which everything can be referenced

All positions and orientations will be described with respect to the Universe Coordinate System or with respect to other Cartesian Coordinate System which are( or could be) defined relative to the Universe Coordinate System

Description of a Position

The location of any point in space can be described as a 3x1 position vector in a reference coordinate system

Coordinate system

Position Vector

Description of a Position

Vectors will be written with a leading superscript indicating the coordinate system to which they are referenced, for example AP

This means that the components of AP have numerical values which indicate distances along the axes of { A }

Each of these distances along an axis can be thought of as the result of projecting the vector onto the corresponding axis

Description of an Orientation

The orientation of a body is described by attaching coordinate system to the body {B} and then defining the relationship between the body and the reference {A} using the rotation matrix

Position : AP

Orientation : AXB , AYB ,

A ZB

Description of Orientation

Thus, positions of points are described with vectors and orientations of bodies are described with an attached coordinate system

One way to describe the body attached to the coordinate system, {B}, is to write the unit vectors of its three principal axes in terms of the coordinate system {A}

The unit vectors giving the principal directions of coordinate system {B} as

XB ,YB , ZB When written in terms of coordinate system {A}, they are

called as AXB ,

AYB , A ZB

Rotation Matrix

It is convenient to stack these three unit vectors together as the columns of a 3X3 matrix, in the order

The matrix is then called as rotation matrix, and because this particular rotation matrix describes {B} relative to {A}, it is named with the notation

Reference Frame

Body Frame

Rotation Matrix

In summary, a set of three vectors may be used to specify an orientation

For convenience, construct a 3X3 matrix that has these three vectors as its columns

Hence, whereas the position of a point is represented with a vector, the orientation of a body is represented with a matrix

The expressions for scalars rij can be computed by noting that components of any vector are simply the projections of that vector onto the unit directions of its reference frame and so can be written as the dot product of a pair of unit vectors

Rotation Matrix

AXB = ARB

BX

AXB = ARB

1

0

0

AYB = ARB

0

1

0

AZB = ARB

0

0

1

Rotation matrix describes {B} relative to {A}

AXB = . . .

Components of XB in {A}

Rotation Matrix

BZAT


Since the dot product of two unit vectors yields the cosine of the angle between them, the components of rotation matrices are often referred to as direction cosines

The rows of the rotation matrix ARB are the unit vectors of {A} expressed in {B}, that is

= BART


Hence, BAR the description of {A} relative to {B} is given by the transpose of ABR , that is

This means that the inverse of a rotation matrix is equal to its transpose


{A}

{B}

ZA YB

XA XB

ZB YA

Example 1 1 0 0 0 0 -1 0 1 0

ABR =

Rotated frames General Notation

Description of a Frame

The information needed to completely specify the whereabouts of the manipulator hand is a position and an orientation

The point on the body whose position is described could be chosen arbitrarily

For convenience, however, the point whose position will be described is chosen as the origin of the body attached frame

The situation of a position and an orientation pair arises so often in robotics that an entity is defined for it called as frame


The frame is a set of four vectors giving position and orientation information

For example, one vector locates the fingertip position and three more describe its orientation

Equivalently, the description of a frame can be thought of as a position vector and a rotation matrix

Thus, frame is a coordinate system, where in addition to the orientation, a position vector is given which locates its origin relative to some other embedding frame


For example, frame {B} is described by ABR and APBORG is the vector that locates the origin of the frame {B}

{A}

{B}

APBORG {B}={ ABR APBORG }

Rotation matrix describing frame {B} relative to frame {A}

The origin of frame {B} relative to frame {A}


There are three frames shown along with the universe coordinate system

Frame{A} and {B} are known relative to universe coordinate system and frame {C} is known relative to frame {A}

Description of a Frame A frame is depicted by 3

arrows representing unit vectors defining the principal axes of the frame

An arrow representing a vector is drawn from one origin to another

This vector represents the position of origin at the head of arrow in terms of the frame at the tail of the arrow


The direction of this locating arrow tells us, for example, that {C} is known relative to {A} and not vice versa

In summary, frame can be used as a description of one coordinate system relative to another

Mappings Mappings refer to changing the description of a point(or vector) in space

from one frame to another

The second frame has three possibilities in reference to first frame:

1. Second-frame is moved away from the first, the axes of both frames remain parallel, respectively. This is a translation of the origin of the second frame from the first frame in space

2. Second-frame is rotated with respect to the first; the origin of both the frames is same. In robotics this referred as changing the orientation

3. Second-frame is rotated with respect to the first and moved away from it, that is, the second frame is translated and its orientation is also changed

Mappings: Translated Frames In this figure, position is

defined by the vector BP with respect to frame {B}

Now, need to express this point in space in terms of frame {A} has the same orientation as frame {B}

In this case, frame {B} differs from frame {A} only by a translation which is given by APBORG , a vector which locates origin of frame {B} relative to frame {A}

Changing the position description of a point P

Mappings: Translated Frames

Note, that only in special case of equivalent orientations may we add vectors which are defined in terms of different frames

This idea of mapping is important concept, quantity itself ( here a point in space) is not changed; only description is changed

Point described by BP is not translated, but remains the same, and instead a new description of the same point was computed, but now with respect to frame {A}

The vector APBORG defines this mapping, since all information needed to perform the change in description is contained in APBORG (along with the knowledge that the frames has equivalent orientation)

Mappings: Rotated Frames

The rotation matrix describes frame {B} relative to frame {A}, it was named with ABR

By definition, the columns of a rotation matrix all have unit magnitude, and further, these unit vectors are orthogonal

A consequence of this is:

Therefore, since the columns of ABR are the unit vectors of {B} written in {A}, then rows of ABR are the unit vectors of {A} written in {B}


If P is given in {B}: BP

AP = BP BP = BAR

AP


Figure shows a frame {B} which is rotated relative to frame {A} about Z-axis by 30 degrees

Here Z is pointing out of the page

Writing the unit vectors of {B} in terms of {A} and stacking them as the columns of the rotation matrix obtain:

ABRZ( ) =


Here, ABR acts as a mapping that is used to describe BP relative to frame

{A} , AP

Important to remember is that, viewed as a mapping, the original vector P is not changed in space

Rather, a new description of the vector is computed relative to another frame

Mappings: General transform

Here, the origin of frame {B} is not coincident with that of frame {A} but has a general vector offset

The vector that locates {B}s origin is called APBORG

Also {B} is rotated with respect to {A} as described by ABR

Given, BP we wish to compute AP

Mappings: General Transform

First change BP to its description relative to an intermediate frame that has the same orientation as {A}, but whose origin is coincident with the origin of {B}

This is done by multiplying by ABR

Then account for the translation between origins by simple vector addition and obtain:

lecture 3

Documents

terms of coordinate

attached coordinate

cartesian coordinate

rotation matrix position

position vectors

zb rotation matrix

3x3 matrix

particular rotation