robot kinematics
DESCRIPTION
Robot Kinematics. Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems. Robot Kinematics: what. Kinematics: study of the relationship between any two displacement variables in a dynamic system. - PowerPoint PPT PresentationTRANSCRIPT
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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
Logics of presentation:
Kinematics: what
Coordinate system: way to describe motion
Relation between two coordinate systems
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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).
Robot Kinematics: coordinate system
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
Robot Kinematics: coordinate system
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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
Robot Kinematics: 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
Robot Kinematics: coordinate system
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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
Logics of presentation:
Kinematics: what
Coordinate system: way to describe motion
Relation between two coordinate systems
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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 ?
To be given in the classroom
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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)
To be given in the classroom
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:
To be given in the classroom
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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[
To be given in the classroom
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XB
XA
YBYA
5
10
30o
P (3, 7, 0)
Fig. 2-20
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To be given in the classroom
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