Chapter 1 Intro duution
c. / Mathematical Modeling of Robots
1. 1 /
.
Symbolic Representation of Manipulators (Robots)
Manipulators have links that are connected to joints to form a kinematic chain
Joints →
usually rotary or linear
" "
Revolute Joint → Allows relative rotation between two links (R)
"
Prismatic Joint → Allows a linear relative motion between two links (P)
"
A 3- link arm with 3
rotary joints would be RRB
Joint Variables → ④ for Revolute Joint (d) for Prismatic Joint
I. 1.2 Configuration Space of Manipulator
configuration → the location of
point on the manipulator
every
configuration Space → set of all possible configurations
A configuration is denoted by 9i where joint Variables can be set to % % % , ,
. . .
.
In
qi=⊖i for revolute joint Ii =D ± for Prismatic Joint where each % is a vector
1. 1.3 State Space
State of a
manipulator →
contains a set of Variables that describes the manipulator 's dynamics and
"
Important for determine future state !
'
input .
state space → the set of all possible states I=(9 IF I
,
contains joint velocities .
Dimension of the state
space
is Zn it
system has a D0F
1. 1.4 The Workspace
Workspace
→ Max space or dimensions of Motion of a manipulator 's end effector
Points that can be reached by the end effector .
Workspace ( w ) PEW Piseud effector Position
Joint space
Joint space → set of all Possible postures
of C- Q 9 is individual postures (9,4 %) . . .
•
Joint space ! Workspace
Forward
kinematics B→
Joint Space Workspace
9. c- Q PEW
%
Inverse kinematics
" "
For g. there is up → I→ P PLAY £ ) C- w Forward kinematics
every ,
be[o◦r•]q" →
"
For any P there
may p→q
, I. 1 Extra Info on Workspaces and Joint spaces
Voids
Voids are areas or dimensions that are unreachable by the end effector
☆ Unreachable by the end effector in any pose -
Dexterous workspace
" "
that be reached with all
possible orientations
set of all positions can
☐
Workspaces of Different configurations
THPP → cylindrical Arm RRP →
Spherical Arm PPP → Cartesian
widely used for
printers
Factors influencing shape and size of Workspace
1) # of 1) of
2) Type of Joints
3) Order of Arrange Joint
4) Link Lengths
5) Constrained Joints 1? )
, 1. 2 Robots as Mechanical Devices
I. 2.1 Classification of Robotic Manipulators
criteria of a manipulator → Power source geometry application
, ,
area
,
method of control
.
Method of Control : Servo → closed to determine their motion
°
loop
-
comp control .
Non -
servo → The earliest robots and are
open
-
loop .
No sensor to determine movement
Geometry →
articulated (RRR ) , Spherical (RRP) ,
SCARA (RRP ) , cylindrical (RPP ), Cartesian (PPP
articulated →
9=[0,0-203]
c. / Mathematical Modeling of Robots
1. 1 /
.
Symbolic Representation of Manipulators (Robots)
Manipulators have links that are connected to joints to form a kinematic chain
Joints →
usually rotary or linear
" "
Revolute Joint → Allows relative rotation between two links (R)
"
Prismatic Joint → Allows a linear relative motion between two links (P)
"
A 3- link arm with 3
rotary joints would be RRB
Joint Variables → ④ for Revolute Joint (d) for Prismatic Joint
I. 1.2 Configuration Space of Manipulator
configuration → the location of
point on the manipulator
every
configuration Space → set of all possible configurations
A configuration is denoted by 9i where joint Variables can be set to % % % , ,
. . .
.
In
qi=⊖i for revolute joint Ii =D ± for Prismatic Joint where each % is a vector
1. 1.3 State Space
State of a
manipulator →
contains a set of Variables that describes the manipulator 's dynamics and
"
Important for determine future state !
'
input .
state space → the set of all possible states I=(9 IF I
,
contains joint velocities .
Dimension of the state
space
is Zn it
system has a D0F
1. 1.4 The Workspace
Workspace
→ Max space or dimensions of Motion of a manipulator 's end effector
Points that can be reached by the end effector .
Workspace ( w ) PEW Piseud effector Position
Joint space
Joint space → set of all Possible postures
of C- Q 9 is individual postures (9,4 %) . . .
•
Joint space ! Workspace
Forward
kinematics B→
Joint Space Workspace
9. c- Q PEW
%
Inverse kinematics
" "
For g. there is up → I→ P PLAY £ ) C- w Forward kinematics
every ,
be[o◦r•]q" →
"
For any P there
may p→q
, I. 1 Extra Info on Workspaces and Joint spaces
Voids
Voids are areas or dimensions that are unreachable by the end effector
☆ Unreachable by the end effector in any pose -
Dexterous workspace
" "
that be reached with all
possible orientations
set of all positions can
☐
Workspaces of Different configurations
THPP → cylindrical Arm RRP →
Spherical Arm PPP → Cartesian
widely used for
printers
Factors influencing shape and size of Workspace
1) # of 1) of
2) Type of Joints
3) Order of Arrange Joint
4) Link Lengths
5) Constrained Joints 1? )
, 1. 2 Robots as Mechanical Devices
I. 2.1 Classification of Robotic Manipulators
criteria of a manipulator → Power source geometry application
, ,
area
,
method of control
.
Method of Control : Servo → closed to determine their motion
°
loop
-
comp control .
Non -
servo → The earliest robots and are
open
-
loop .
No sensor to determine movement
Geometry →
articulated (RRR ) , Spherical (RRP) ,
SCARA (RRP ) , cylindrical (RPP ), Cartesian (PPP
articulated →
9=[0,0-203]
