Robot Kinematics

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Robot Kinematics

Logics of presentation:

Kinematics: what

Coordinate system: way to describe motion

Relation between two coordinate systems

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Robot Kinematics: what

Kinematics: study of the relationship between any

two displacement variables in a dynamic system.

Robot kinematics: a robot is a dynamic system.

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Robot Kinematics: whatA robot consists of a set of servomotors which drive the end-effector. Therefore we have: (a) the motion of the end-effector, and (b) the motion of the servomotors. These two are related.

Given (b) to find (a): forward kinematics (process 1)

Given (a) to find (b): inverse kinematics (process 2)

1 z

1

x

y 2

(b) (a)

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Robot Kinematics


Kinematics: what



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Robot Kinematics: coordinate system

The general principle to describe motion:

Coordinate systems, as it provides a reference upon

which motion of an object can be quantitatively

described.

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There are two coordinate systems to measure two types of motions (joint level and end-effector level), respectively:

motor or joint coordinate system for joint level motions (see Fig. 2-11).world coordinate system for end effector level motions (see Fig. 2-12).


1

Fig. 2-11

Fig. 2-12

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The relationship of the attached coordinate system with

respect to the world coordinate system completely describes

the position and orientation of that body in the world

coordinate system (Fig. 2-13).

Xw

Fig. 2-13

Mi

PYw


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Attached coordinate system (also called local coordinate system, LCS) replaces the object and represents it with respect to the world coordinate system (WCS) or reference coordinate system (RCS)

The motion of the object with respect to the reference coordinate system reduces to the relation between the local coordinate system with respect to the reference coordinate system


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Denote the relation between A and B as Rel (A-B)

Suppose {A} is the LCS of Object A, and {B} is a reference coordinate system,

Note that by defining {A} on an object, we imply that details of the object are defined, with respect to {A} in this case,

Motion of the object (i.e., A) in the reference coordinate system is thus the relation of {A} with respect to {B}.

The above thinking process shows how the motion of an object becomes the relation between two coordinate systems


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As well, how a point P on Object A should be represented with respect to {B}

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Robot Kinematics


Kinematics: what



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Robot Kinematics: relation between two coordinate systems {A} and {B}

Case 1: two origins are coincident Case 2: two coordinate systems are in parallel

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Remark 1: Motion is also related to velocity and

acceleration. The general idea is that they should be

obtained by the differentiation of the transformation matrix.

Remark 2: coordinate system is also called frame.

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Two origins of the frames are coincident

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Unit vectors giving the principal directions of {B} as

When these vectors are written in terms of {A}, we denote

Stack these three together, and call rotation matrix

= (2-1)

^^^

BBB ZYX

^^^

BA

BA

BA ZYX

RAB

^^^

BA

BA

BA ZYX

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Equation (2-1) can be further written as

(2-2)

The components in equation (2-2) are simply the projections of

that vector onto the axes of its reference frame. Hence, each

component of equation (2-2) can be within as the dot product

of a pair of unit vectors as

To be given in the classroom

(2-3)To be given in the classroom

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RABB with respect to A

How A with respect to B ?


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The inspection of equation (2-3) shows that the rows of the

matrix are the column of the matrix ; as such we have

(2-4)


RABBAR

It can be further verified that the transpose of R matrix is its inverse matrix. As such, we have

1B B T AA A BR R R (2-5)

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When frame A and frame B are not at the same location (see Fig. 2-14), we will consider two steps to get the relationship between {A} and {B}:

Step 1: Consider that {A} and {B} are in parallel first. Then, {B} translates to the location which is denoted as

ABORGP : The origin of {B} in Frame {A}

ABORGP

Step 2: Imagine that {A} and {B} are at the same origin but {B} rotates with respect to {A}. The relation between {A} and {B} in this case is: RAB

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},{}{ BORGAA

B PRCRM

So the total relation between {A} and {B} is:

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Fig. 2-14

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Further, if we have three frames, A, B, C, (Fig. 2-15) then we have a chain rule such that (see the figure in the next slide)

},{ CORGAA

C PR =

},{ BORGAA

B PR },{ CORGBB

C PR

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Fig. 2-15

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Point P at different frames

Fig. 2-16 shows that the same point, P, is expressed in two different frames, A and B.

P

Fig. 2-16

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Case 1: Frame A and Frame B are in parallel but at different locations (see Fig. 2-17)

PFig. 2-17

In this case, we have the following relation

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BORGABA PPP (2-7)

in {A} in {B}

Case 2: A and B are at the same location but with different orientations (see Fig. 2-18).

In this case, we have

PRP BAB

A (2-8)

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Fig. 2-18

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We have:

BORGABA

BA PPRP

We can further write equation (2-9) into a frame-like form, namely a kind of mapping

(2-10)

(2-9)

PTP BAB

A

The matrix T has the following form:

See Fig. 2-16, A and B are both at different locations and with different orientations

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1000BORG

AABA

B

PRT

T matrix is a 4 x 4 matrix, and it make the representation of

P in different frames {A} and {B} a bit convenient, i.e.,

equation (2-10).

For example, for Fig. 2-15, {A}, {B}, {C}, {U}, we have for P

in the space:

PTTP BAB

UA

U PTTTP ABA

CB

UC

U

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PTTTP ABA

CB

UC

U Notation helps to verify the correctness of the expression

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Example 1: Fig. 2-19 shows a frame {B} which is rotated

relative to frame {A} about ^

Z

is an axis perpendicular to the sheet plane^

Z

Please find:

(1) Representation of Frame {B} with respect to Frame {A}

(2)

(3) Representation of P with respect to Frame {A}

BP

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Fig. 2-19

10

30o

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Solution:


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(2-11)

Example 2: Fig.2-20 shows a frame {B} which is rotated relative to frame {A} about Z by 30 degrees, and translated 10 units in XA and 5 units in YA. Find

where PA

TBP ]0.0,0.7,0.3[


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XB

XA

YBYA

5

10

30o

P (3, 7, 0)

Fig. 2-20

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Solution:

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Summary

Forward kinematics versus inverse kinematics.

Motion is measured with respect to coordinate system or frame.

Frame is attached with an object.

Every details of the object is with respect to that frame, local frame.

Relation between two frames are represented by a 4 by 4 matrix, T, in general.

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Summary (continued)

When two frames are in the same location, T is expressed by

10000

0

0

RT

ABA

B

When two frames are in parallel but different locations, T is expressed by

1000000

000

000

BORGA

AB

PT

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Robot Kinematics