Chapter 2
Robot Kinematics: Position Analysis
2.1 INTRODUCTION
Forward Kinematics:
to determine where the robot’s hand is?
(If all joint variables are known)
Inverse Kinematics:
to calculate what each joint variable is?
(If we desire that the hand be
located at a particular point)
Chapter 2
Robot Kinematics: Position Analysis
2.2 ROBOTS AS MECHANISM
Multiple type robot have multiple DOF.
(3 Dimensional, open loop, chain mechanisms)
Fig. 2.1 A one-degree-of-freedom closed-loop
four-bar mechanism
Fig. 2.2 (a) Closed-loop versus (b) open-loop mechanism
Chapter 2
Robot Kinematics: Position Analysis
2.3 MATRIX REPRESENTATION
2.3.1 Representation of a Point in Space
A point P in space :
3 coordinates relative to a reference frame
^
^
^
P  ax i  by j  cz k
Fig. 2.3 Representation of a point in space
Chapter 2
Robot Kinematics: Position Analysis
2.3 MATRIX REPRESENTATION
2.3.2 Representation of a Vector in Space
A Vector P in space :
3 coordinates of its tail and of its head
__
^
^
^
P  a x i  by j  c z k
x
 y
__
P 
z
 
 w
Fig. 2.4 Representation of a vector in space
Chapter 2
Robot Kinematics: Position Analysis
2.3 MATRIX REPRESENTATION
2.3.3 Representation of a Frame at the Origin of a Fixed-Reference Frame
Each Unit Vector is mutually perpendicular. :
normal, orientation, approach vector
 nx ox a x 
F  n y o y a y 
 nz oz a z 
Fig. 2.5 Representation of a frame at the origin of the reference frame
Chapter 2
Robot Kinematics: Position Analysis
2.3 MATRIX REPRESENTATION
2.3.4 Representation of a Frame in a Fixed Reference Frame
Each Unit Vector is mutually perpendicular. :
normal, orientation, approach vector
 nx
n
y

F
 nz

0
Fig. 2.6 Representation of a frame in a frame
ox a x Px 
o y a y Py 
oz a z Pz 

0 0 1
Chapter 2
Robot Kinematics: Position Analysis
2.3 MATRIX REPRESENTATION
2.3.5 Representation of a Rigid Body
An object can be represented in space by attaching a frame
to it and representing the frame in space.
 nx
n
y

Fobject 
 nz

0
Fig. 2.8 Representation of an object in space
ox a x Px 
o y a y Py 
oz a z Pz 

0 0 1
Chapter 2
Robot Kinematics: Position Analysis
2.4 HOMOGENEOUS TRANSFORMATION MATRICES
A transformation matrices must be in square form.
•
•
It is much easier to calculate the inverse of square matrices.
To multiply two matrices, their dimensions must match.
 nx
n
y

F
 nz

0
ox a x Px 
o y a y Py 
oz a z Pz 

0 0 1
Chapter 2
Robot Kinematics: Position Analysis
2.5 REPRESENTATION OF TRANSFORMATINS
2.5.1 Representation of a Pure Translation
A transformation is defined as making a movement in space.
• A pure translation.
• A pure rotation about an axis.
• A combination of translation or rotations.
1
0
T 
0

0
Fig. 2.9 Representation of an pure translation in space
0 0 dx 
1 0 d y 
0 1 dz 

0 0 1
Chapter 2
Robot Kinematics: Position Analysis
2.5 REPRESENTATION OF TRANSFORMATINS
2.5.2 Representation of a Pure Rotation about an Axis
Assumption : The frame is at the origin of the reference frame and parallel to it.
Fig. 2.10 Coordinates of a point in a rotating
frame before and after rotation.
Fig. 2.11 Coordinates of a point relative to the reference
frame and rotating frame as viewed from the x-axis.
Chapter 2
Robot Kinematics: Position Analysis
2.5 REPRESENTATION OF TRANSFORMATINS
2.5.3 Representation of Combined Transformations
A number of successive translations and rotations….
Fig. 2.13 Effects of three successive transformations
Fig. 2.14 Changing the order of transformations will
change the final result
Chapter 2
Robot Kinematics: Position Analysis
2.5 REPRESENTATION OF TRANSFORMATINS
2.5.5 Transformations Relative to the Rotating Frame
Example 2.8
Fig. 2.15 Transformations relative to the current frames.
Chapter 2
Robot Kinematics: Position Analysis
2.6 INVERSE OF TRANSFORMATION MATIRICES
Inverse of a matrix calculation steps :
• Calculate the determinant of the matrix.
• Transpose the matrix.
• Replace each element of the transposed matrix by its own minor(adjoint matrix).
• Divide the converted matrix by the determinant.
Fig. 2.16 The Universe, robot, hand, part, and end effecter frames.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
Forward Kinematics Analysis:
• Calculating the position and orientation of the hand of the robot.
• If all robot joint variables are known, one can calculate where the robot is
at any instant.
• Recall Chapter 1.
Fig. 2.17 The hand frame of the robot relative to the reference frame.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.1 Forward and Inverse Kinematics Equations for Position
Forward Kinematics and Inverse Kinematics equation for position analysis :
(a) Cartesian (gantry, rectangular) coordinates.
(b) Cylindrical coordinates.
(c) Spherical coordinates.
(d) Articulated (anthropomorphic, or all-revolute) coordinates.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.1 Forward and Inverse Kinematics Equations for Position
2.7.1(a) Cartesian (Gantry, Rectangular) Coordinates
IBM 7565 robot
• All actuator is linear.
• A gantry robot is a Cartesian robot.
TP  Tcart
R
Fig. 2.18 Cartesian Coordinates.
1
0

0

0
0 0 Px 
1 0 Py 
0 1 Pz 

0 0 1
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.1 Forward and Inverse Kinematics Equations for Position
2.7.1(b) Cylindrical Coordinates
2 Linear translations and 1 rotation
• translation of r along the x-axis
• rotation of  about the z-axis
• translation of l along the z-axis
TP  Tcyl (r , , l )  Trans(0,0, l )Rot( z, )Trans( r ,0,0)
R
C  S 0 rC 
 S C 0 rS 
R

TP  Tcyl  
 0
0 1 l 


0
0
0
1


Fig. 2.19 Cylindrical Coordinates.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.1 Forward and Inverse Kinematics Equations for Position
2.7.1(c) Spherical Coordinates
2 Linear translations and 1 rotation
• translation of r along the z-axis
• rotation of  about the y-axis
• rotation of  along the z-axis
TP  Tsph (r ,  , l )  Rot( z,  )Rot( y,  )Trans( 0,0,  )
R
C  C  S S  C rS  C 
 C  S C S  S rS  S 
R

TP  Tsph  
  S
0
C
rC 


0
0
0
1


Fig. 2.20 Spherical Coordinates.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.1 Forward and Inverse Kinematics Equations for Position
2.7.1(d) Articulated Coordinates
3 rotations -> Denavit-Hartenberg representation
Fig. 2.21 Articulated Coordinates.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.2 Forward and Inverse Kinematics Equations for Orientation



Roll, Pitch, Yaw (RPY) angles
Euler angles
Articulated joints
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.2 Forward and Inverse Kinematics Equations for Orientation
2.7.2(a) Roll, Pitch, Yaw(RPY) Angles
Roll: Rotation of a about a-axis (z-axis of the moving frame)
Pitch: Rotation of o about o -axis (y-axis of the moving frame)
Yaw: Rotation of n about n -axis (x-axis of the moving frame)
Fig. 2.22 RPY rotations about the current axes.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.2 Forward and Inverse Kinematics Equations for Orientation
2.7.2(b) Euler Angles
Rotation of 
Rotation of 
Rotation of 
about
about
about
a-axis (z-axis of the moving frame) followed by
o -axis (y-axis of the moving frame) followed by
a-axis (z-axis of the moving frame).
Fig. 2.24 Euler rotations about the current axes.
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.2 Forward and Inverse Kinematics Equations for Orientation
2.7.2(c) Articulated Joints
Consult again section 2.7.1(d)…….
Chapter 2
Robot Kinematics: Position Analysis
2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS
2.7.3 Forward and Inverse Kinematics Equations for Orientation
 Assumption : Robot is made of a Cartesian and an RPY set of joints.
TH  Tcart ( Px , Py , Pz )  RPY (a , o , n )
R
 Assumption : Robot is made of a Spherical Coordinate and an Euler angle.
TH  Tsph (r ,  ,  )  Euler ( ,  , )
R
Another Combination can be possible……
Denavit-Hartenberg Representation
Chapter 2
Robot Kinematics: Position Analysis
2.8 DENAVIT-HARTENBERG REPRESENTATION OF
FORWARD KINEMATIC EQUATIONS OF ROBOT
 Denavit-Hartenberg Representation :
@ Simple way of modeling robot links and
joints for any robot 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.
Fig. 2.25 A D-H representation of a general-purpose joint-link combination
Chapter 2
Robot Kinematics: Position Analysis
2.8 DENAVIT-HARTENBERG REPRESENTATION OF
FORWARD KINEMATIC EQUATIONS OF ROBOT
 Denavit-Hartenberg Representation procedures:
Start point:
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.
Chapter 2
Robot Kinematics: Position Analysis
2.8 DENAVIT-HARTENBERG REPRESENTATION OF
FORWARD KINEMATIC EQUATIONS OF ROBOT
 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.
‫ ٭‬Parallel z-axes joints make a infinite number of common normal.
‫ ٭‬Intersecting z-axes of two successive joints make no common
normal between them(Length is 0.).
Chapter 2
Robot Kinematics: Position Analysis
2.8 DENAVIT-HARTENBERG REPRESENTATION OF
FORWARD KINEMATIC EQUATIONS OF ROBOT
 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
2.8 DENAVIT-HARTENBERG REPRESENTATION OF
FORWARD KINEMATIC EQUATIONS OF ROBOT
 The necessary motions to transform from one reference
frame to the next.
(I)
Rotate about the zn-axis an able of n+1. (Coplanar)
(II) Translate along zn-axis a distance of dn+1 to make xn and xn+1
colinear.
(III) Translate along the xn-axis a distance of an+1 to bring the origins
of xn+1 together.
(IV) Rotate zn-axis about xn+1 axis an angle of n+1 to align zn-axis
with zn+1-axis.
Chapter 2
Robot Kinematics: Position Analysis
2.9 THE INVERSE KINEMATIC SOLUTION OF ROBOT
 Determine the value of each joint to place the arm at a
desired position and orientation.
TH  A1 A2 A3 A4 A5 A6
R
C1 (C234C5C6  S 234S 6 )
 S S C
 1 5 6
S1 (C234C5C6  S 234S 6 )

  C1S5C6
 S 234C5C6  C234S 6


0
 nx
n
y

 nz

0
ox a x p x 
o y a y p y 
oz a z p z 

0 0 1
C1 (C234C5C6  S 234C6 )

C1 (C234S5 )  S1C5 C1 (C234a4  C23a3  C2 a2 )
 S1S5C6

S1 (C234C5C6  S 234C6 )
S1 (C234S5 )  C1C5 S1 (C234a4  C23a3  C2 a2 ) 

 C1S5C6
 S 234C5C6  C234C6
S 234S5
S 234a4  S 23a3  S 2 a2 


0
0
1
Chapter 2
Robot Kinematics: Position Analysis
2.9 THE INVERSE KINEMATIC SOLUTION OF ROBOT
 nx

1  n y
A1 
 nz

0
ox a x p x 
o y a y p y 
 A11[ RHS ]  A2 A3 A4 A5 A6
oz a z p z 

0 0 1
C1 S1
0 0

 S1  C1

0 0
0
1
0
0  n x
0 n y

0  n z
 
0 1  0
ox a x p x 
o y a y p y 
 A2 A3 A4 A5 A6
oz a z p z 

0 0 1
Chapter 2
Robot Kinematics: Position Analysis
2.9 THE INVERSE KINEMATIC SOLUTION OF ROBOT
 py 

 px 
1  tan 1 
 2  tan 1
(C3a3  a2 )( pz  S 234a4 )  S3a3 ( pxC1  p y S1  C234a4 )
(C3a3  a2 )( pxC1  p y S1  C234a4 )  S3a3 ( Pz  S 234a4 )
 S3 

C
3
 
 3  tan 1 
 4   234   2  3
 5  tan 1
C234 (C1ax  S1a y )  S 234az
S1ax  C1a y
 6  tan 1
 S 234 (C1nx  S1n y )  S 234nz
 S 234 (C1ox  S1o y )  C234oz
Chapter 2
Robot Kinematics: Position Analysis
2.10 INVERSE KINEMATIC PROGRAM OF ROBOTS
 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
2.11 DEGENERACY AND DEXTERITY
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.
Fig. 2.31 An example of a robot in a degenerate position.
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
Fig. 2.31 The frames of the Stanford Arm.
#

d
a

1
1
0
0
-90
2
2
d1
0
90
3
0
d1
0
0
4
4
0
0
-90
5
5
0
0
90
6
6
0
0
0