Denavit Hartenberg Algorithm

Denavit Hartenberg Algorithm

1. DENAVIT HARTENBERG
ALGORITHM
Hitesh Mohapatra
https://www.linkedin.com/in/hiteshmohapatra/

2. WHAT IS KINEMATICS ?

3. KINEMATICS
PROPERTY OF MOTION OF AN OBJECT
FORWARD KINEMATICS INVERSE KINEMATICS

4. KINEMATICS
FORWARD KINEMATICS BACKWARD KINEMATICS
VALUES OF JOINT PARAMETERS POSITION OF END EFFECTORS
KINEMATIC EQUATION
POSITION OF END EFFECTORS VALUES OF JOINT PARAMETERS
KINEMATIC EQUATION
PARAMETRIC VALUES

5. PARAMETRIC VALUES
FOR REVOLUTE JOINTS FOR PRISMATIC JOINTS
ANGLES BETWEEN LINKS LINK EXTENSIONS
θ
i
d
i

6. LINKS AND JOINTS

7. IF ‘N+1’ LINKS
THEN
‘N’ JOINTS

9. ANY ARBITRARY FRAME COULD BE ATTACHED WITH EACH LINK
BUT
WE CAN FOLLOW CERTAIN CONVENTION
DENAVIT HARTENBERG CONVENTION
ONE OF SUCH CONVENTIONS IS

10. WHAT IS THE BENEFIT?

11. IF FRAME
0
0x
0y
0z
0
ATTACHED WITH ROBOT’S BASE
INERTIAL FRAME

13. HOMOGENOUS TRANSFORMATION MATRIX
EXPRESSES POSITION AND ORIENTATION
OF
0
ix
iy
iz
i W.R.T0
i-1x
i-1y
i-1z
i-1
REPRESENTED BY
A
i

14. HOMOGENOUS TRANSFORMATION MATRIX
HENCE, IN D-H CONVENTION A
iIS REPRESENTED AS A PRODUCT
OF FOUR BASIC TRANSFORMATIONS

15. BENIFIT
A
i
D-H
6 PARAMETERS USED 4 PARAMETERS USED

16. LINK AND JOINT PARAMETERS
•JOINT ANGLE (θ
i) : THE ANGLE OF ROTATION FROM X
i-1AXIS TO THE Z
i-1 AXIS.
IT IS THE JOINT VARIABLE IF JOINT I IS ROTATORY.
•JOINT DISTANCE (d
i) : THE DISTANCE FROM THE ORIGIN OF (i-1) CO-ORDINATE
SYSTEM TO THE INTERSECTION OF THE Z
I-1AXIS AND X
iAXIS ALONG THE Z
i-1 AXIS.
IT IS THE JOINT VARIABLE IF JOINT I IS PRISMATIC.
•LINK LENGTH (a
I) : THE DISTANCE FROM THE INTERSECTION OF Z
i-1 AXIS AND
THE X
iAXIS TO THE ORIGIN OF THE I
th
CO-ORDINATE SYSTEM ALONG THE X
i.
•LINK TWIST ANGLE (α
i) : THE ANGLE OF ROTATION FROM THE Z
i-1AXIS TO THE
Z
i-1AXIS ABOUT X
i AXIS.

18. EXAMPLE : TWO LINK ELBOW MANIPULATOR

20. D-H CONVENTION ALGORITHM
STEP-1:
LOCATEANDLABELTHEJOINTAXES
Z
0,………..,Z
n-1.
STEP-2:
ESTABLISHTHEBASEFRAME.SETTHE
ORIGINANYWHEREONTHEZ
0AXIS.THE
X
0ANDY
0AXESARECHOSENCONVENIENTLY
TOFORMARIGHTHANDFRAME.

21. STEP-3:
LOCATE THE ORIGIN OI WHERE THE
COMMON NORMAL TO Z
iAND Z
i-1INTERSECTS
Z
i. IF Z
iINTERSECTS Z
i-1LOCATE O
iAT THIS
INTERSECTION. IF Z
iAND Z
i-1ARE PARALLEL,
LOCATE O
iIN ANY CONVENIENT POSITION
ALONG Z
i.
STEP-4:
ESTABLISH T
iALONG THE COMMON
NORMAL BETWEEN Z
i-1AND Z
iTHROUGH O
i, IN
THE DIRECTION NORMAL TO THE Z
i-1–Z
i
PLANE IF Z
i-1AND Z
iINTERSECT.

22. STEP-5:
ESTABLISH Y
iTO COMPLETE A RIGHT
HAND FRAME.

23. STEP-6:
ESTABLISHTHEENDEFFECTORFRAME0
nX
nY
nZ
n.
ASSUMINGTHEN-THJOINTISREVOLUTE,SETZ
n=AALONG
THEDIRECTIONZ
n-1.ESTABLISHTHEORIGIN0N
CONVENIENTLYALONGZ
n,PREFERABLYATTHECENTEROF
GRIPPERORATTHETIPOFANYTOOLTHATTHEMANIPULATOR
MAYBECARRYING.SETY
n=SINTHEDIRECTIONOFGRIPPER
CLOSUREANDSETX
n=NASS×A.IFTHETOOLISNOTA
SIMPLEGRIPPERSETX
nANDY
nCONVENIENTLYTOFORMA
RIGHTHANDFRAME.

24. STEP-7:
CREATEATABLEOFLINKPARAMETERSA
i,D
i,Α
i,Θ
i.
A
i= DISTANCE ALONG X
iFROM 0
iTO THE INTERSECTION OF THE
X
iAND Z
i-1AXES.
D
i= DISTANCE ALONG Z
i-1FROM 0
i-1TO THE INTERSECTION OF
THE X
iAND Z
i-1AXES. D
i IS VARIABLE IF JOINT I IS PRISMATIC.
α
i= THE ANGLE BETWEEN Z
i-1AND Z
iMEASURED ABOUT X
i.

26. LINK COORDINATE FRAMES

27. ASSIGN LINK COORDINATE FRAMES:
TO DESCRIBE THE GEOMETRY OF ROBOT MOTION , WE ASSIGN A
CARTESIAN COORDINATE
FRAME (0
i, X
i, Y
i, Z
i) TO EACH LINK, AS FOLLOWS:
ESTABLISH A RIGHT-HANDED ORTHO-NORMAL COORDINATE FRAME
0
0AT THE SUPPORTING BASE WITH Z
0LYING ALONG JOINT 1 MOTION
AXIS.
THE Z
iAXIS IS DIRECTED ALONG THE AXIS OF MOTION OF JOINT
(I+1) , THAT IS , LINK (I+1) ROTATES ABOUT OR TRANSLATES ALONG Z
i.

28. LOCATE THE ORIGIN OF THE I
TH
COORDINATE AT THE
INTERSECTION OF Z
i& Z
i-1OR AT THE INTERSECTION OF
COMMON NORMAL BETWEEN Z
i& Z
i-1AXES AND THE Z
iAXIS.
THE X
iAXIS LIES ALONG THE COMMON NORMAL FROM THE
Z
i-1AXIS TO THE Z
iAXIS
X
i= ±(Z
i-1×Z
i) / ||Z
i-1×Z
i|| , (IF Z
i-1IS PARALLEL TO Z
i,
THEN X
iIS SPECIFIED ARBITRARILY, SUBJECT ONLY TO X
i
BEING PERPENDICULAR TO Z
i)

29. LINK COORDINATE FRAMES

30. ASSIGN YI = +(Z
i×X
i)/ ||Z
i×X
i|| TO COMPLETE THE RIGHT HANDED
COORDINATE SYSTEM.
THE HAND COORDINATE FRAME IS SPECIFIED BY THE GEOMETRY OF THE
END EFFECTOR.
NORMALLY ,ESTABLISH Z
nALONG Z
n-1AXIS AND POINTING AWAY FROM THE
ROBOT; ESTABLISH X
n
SUCH THAT IT IS NORMAL TO BOTH Z
n-1AND Z
nAXES. ASSIGN Y
nTO
COMPLETE THE RIGHT HANDED COORDINATE SYSTEM.