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Chapter 2 Robot Kinematics: Position Analysis Denavit-Hartenberg Representation :
@ Simple way of modeling robot links and joints for any robot configuration, configuration, regardless of its sequence or complexity.
@ Transformations in any coordinates is possible.
@ Any possible combinations of joints and links and all-revolute articulated robots can be represented. represented.
Fig. 2.25 A 2.25 A D-H representation of a general-purpose joint-link combination
Chapter 2 Robot Kinematics: Position Analysis
Chapter 2 Robot Kinematics: Position Analysis Denavit-Hartenberg Representation procedures:
Assign joint number n to the first shown joint. Assign a local reference frame for each and every joint before or after these joints. Y -axis does not used in D-H representation because always perpendicular to both of axes.
Chapter 2 Robot Kinematics: Position Analysis Procedures for assigning a local reference frame to each joint:
All joints are represented by a z -axis. (right-hand rule for rotational joint, linear movement for prismatic joint) The common normal is one line mutually perpendicular to any two skew lines. -axes joints make a infinite number of common normal. Parallel z -axes of two successive joints make no common Intersecting z normal between them.
Common normal
Chapter 2 Robot Kinematics: Position Analysis Symbol Terminologies :
:
A rotation about the z -axis.
d : The distance on the z -axis.
a : The length of each common normal (Joint offset).
: The angle between two successive z -axes (Joint twist)
Only
and
d are joint variables.
Chapter 2 Robot Kinematics: Position Analysis
The necessary motions to transform from one reference frame to the next. (I)
Rotate about the z n -axis an able of
n +1.
(Coplanar)
(II) Translate along z n -axis a distance of d n +1 to make x n and x n+1 colinear.
Chapter 2 Robot Kinematics: Position Analysis
The necessary motions to transform from one reference frame to the next.
(III) Translate along the x n -axis a distance of a n +1 to bring the origins of x n +1 together.
Chapter 2 Robot Kinematics: Position Analysis
The necessary motions to transform from one reference frame to the next.
(IV) Rotate z n -axis about x n+1 axis an angle of
n+1 to
align z n -axis with z n+1 -axis.
Chapter 2 Robot Kinematics: Position Analysis
Chapter 2 Robot Kinematics: Position Analysis
Chapter 2 Robot Kinematics: Position Analysis
Chapter 2 Robot Kinematics: Position Analysis
Chapter 2 Robot Kinematics: Position Analysis
Determine the value of each joint to place the arm at a desired position and orientation.
T H A1 A2 A3 A4 A5 A6
R
C 1 (C 23 4C 5C 6 S 23 4S 6 ) C 1 (C 23 4C 5C 6 S 23 4C 6 ) S S C S 1S 5C 6 1 5 6 S 1 (C 23 4C 5C 6 S 23 4S 6 ) S 1 (C 23 4C 5C 6 S 23 4C 6 ) C 1S 5C 6 C 1S 5C 6 S 23 4C 5C 6 C 23 4S 6 S 23 4C 5C 6 C 23 4C 6 0 0
n x n y n z 0
o x a x p x
o z a z p z 0 0 1
o y a y p y
C 23a3 C 2 a2 ) S 1 (C 23 4a4 C 23 a3 C 2 a2 ) S 23 4a4 S 23a3 S 2 a2 1
C 1 (C 23 4S 5 ) S 1C 5 C 1 (C 23 4a4 S 1 (C 23 4S 5 ) C 1C 5 S 23 4S 5
0
Chapter 2 Robot Kinematics: Position Analysis
n x 1 n y A1 n z 0
o x a x p x
A11[ RHS ] A2 A3 A4 A5 A6 o z a z p z 0 0 1
o y a y p y
C 1 S 1 0 0 n x 0 0 1 0 n y S 1 C 1 0 0 n z 0 0 0 1 0
o x a x p x
A2 A3 A4 A5 A6 o z a z p z 0 0 1
o y a y p y
Chapter 2 Robot Kinematics: Position Analysis
1
p y tan1 p x
2
tan1
3
S 3 tan 1 C 3
4
234 2 3
5
tan 1
6
tan1
(C 3a3 a2 )( p z S 234a4 ) S 3a3 ( p xC 1 p y S 1 C 234a4 ) (C 3a3 a2 )( p xC 1 p y S 1 C 234a4 ) S 3a3 ( P z S 234a4 )
C 234 (C 1a x S 1a y ) S 23 4a z S 1a x C 1a y
S 23 4(C 1n x S 1n y ) S 23 4n z S 23 4(C 1o x S 1o y ) C 23 4o z
Chapter 2 Robot Kinematics: Position Analysis
A robot has a predictable path on a straight line, Or an unpredictable path on a straight line.
A predictable path is necessary to recalculate joint variables. (Between 50 to 200 times a second) To make the robot follow a straight line, it is necessary to break the line into many small sections. All unnecessary computations should be eliminated.
Fig. 2.30 Small sections of movement for straight-line motions
Chapter 2 Robot Kinematics: Position Analysis
Degeneracy : The robot looses a degree of freedom and thus cannot perform as desired.
When the robot s joints reach their physical limits, and as a result, cannot move any further. ’
In the middle point of its workspace if the z -axes of two similar joints becomes colinear.
Dexterity : The volume of points where one can position the robot as desired, but not orientate it.
Chapter 2 Robot Kinematics: Position Analysis 2.12 THE FUNDAMENTAL PROBLEM WITH D-H REPRESENTATION Defect of D-H presentation : D-H cannot represent any motion about the y -axis, because all motions are about the x - and z -axis.
TABLE 2.3 THE PARAMETERS TABLE FOR THE STANFORD ARM
