MASTER THESIS
DEVELOPMENT OF A CAD/CAE TOOL – ROBOKINE
(ROBOtic KINEmatics)–FOR WORKSPACE, INVERSE KINEMATICS
AND TRAJECTORY PLANNING
BY
MUKUND V. NARASIMHAN
SUPERVISING PROFESSOR: Dr. T.C. YIH
COMMITTEE MEMBERS : Dr. K. L. LAWRENCE
Dr. B. P. WANG
MECHANICAL ENGINEERING DEPARTMENT
THE UNIVERSITY OF TEXAS AT ARLINGTON
November 19th 2002
ROBOT
A reprogrammable, multifunctional manipulator
designed to move material, parts or tools through various
programmed motions for the performance of a variety of
tasks.
Robots were used mostly in the automobile industry but
now a days they can be seen in hospitals, laboratories,
energy plants, warehouses etc.
By 2003 there will be nearly 900,000 multi-purpose
robots in use worldwide compared with 750,000 that are
currently available. According to “World Robotics 2000”, a
survey published by the united nations economic
commission for Europe in co-operation with the
international federation of robotics.
REVIEW OF C-B NOTATION
This notation is based on the homogeneous cylindrical
coordinates and bryant angles transformations matrices and
hence termed c-b notation.
The homogeneous transformation matrix is given by
Ti (i, hi, ri, i, i) =Tci(i, hi, ri) Tbi(i, i)
Zi-1
Oi-
ezi
i
Zi
Oi+
ri
1
1
i
Zi+1
exi
Xi+1
hi
i
i-1
Oi
i
Xi
ex(i1)
CYLINDRICAL COORDINATES
Tc(, h, r)= Tr(Z, ) Tt(Z, h) Tt(X, r) =
 cθ
 sθ

0
0

 sθ
cθ
0
0
0
0
1
0
rcθ 
rsθ 

h 
1 

Tr(Z, ) represents rotation  about Z-axis
Tt(Z, h) represents translation h along Z-axis
Tt(X, r) represents translation r along X-axis
BRYANT ANGLES CONVENTION
Tb(, , ) = Tr(X, ) Tr(Y, ) Tr(Z, ) =
 cβcγ
 cβsγ
sβ

cαsγ  sαsβcγ cαcγ  sαsβsγ  sαcβ


sαsγ  cαsβcγ sαcγ  cαsβsγ cαcβ

0
0
0

Tr(X, ) represents rotation about X-axis
Tr(Y, ) represents rotation about Y-axis
Tr(Z, ) represents rotation about Z-axis
0
0
0
1






HOMOGENEOUS SHAPE MATRIX
T(, h, r, , ) = Tc(, h, r) Tb(, , 0) =
cc  sss  sc cs  ssc rc 


sc  css cc ss  csc rs 


s
cc
h 
  cs

0
0
0
1 


Y
Z
Z
X
l
Y
h=l
r=0
Z
 = 90
X
Y
=0
l
Y
X
Z
X
h = 0 r = l  = 90  = 0
ROBOKINE
Written in java and java3d.
Java used for generating the homogeneous matrices
Java3d for generating 3-dimensional features and simulations.
Contains 18 classes, 350+ methods and numerous variables
Advanced features include
Automatic C-B notation table generation (24 X 24 X 2)
Workspace generation (solid and wire frame modes)
Dynamic slice of the workspace in real time
Single slice of the workspace
Solving inverse kinematic problems
Trajectory planning
FEATURE 1
DISPLACEMENT ANALYSIS AND WORKSPACE GENERATION
Generate workspace profiles for even 3-dimensional robots and
also for robots which does not have adjacent axes parallel or
perpendicular to each other.
Can validate the generated profiles with the use of “Dials”.
Kinematic analysis consists of position or displacement, velocity
and acceleration analyses.
The workspace / work volume of a robotic manipulator is defined
as the set of all 3-dimensional points that can be accessed by the
manipulator.
One of the design criteria for design of robots.
Study of robotic workspaces is important in arranging the
associated flexible manufacturing cell of a robot and assessing its
efficiency in a manufacturing line.
DISPLACEMENT ANALYSIS
The general homogeneous characteristic matrix, ti, for different
kinematic lower pairs are given by

cici  sisisi  sici cisi  sisici rici 


Ti =
s

ic i  cisis i
c

ici
s

is i  cisic i risi


  cisi
s i
cici
hi 


0
0
0
1 

The position analysis of the end effector or manipulator can
be obtained from this resultant matrix h given below
 Di
H =  Ti  
i 1
0 0
n
Pi 
0 1 
Where D is direction cosine matrix and P is position vector and these specify the
orientation and position respectively.
FEATURE 2
VISUALIZATION BASED INVERSE KINEMATICS SOLVER
A graphical, visualization based technique is developed to solve inverse
kinematic problems without resorting to the numerical procedures.
Given the target point, finding the values of the joint parameters to reach
that desired point.
The difficulty in solving inverse kinematic problems is because of its
non-linear nature. the difficulties associated with this non-linearity are
Multiple to infinite solutions
No solutions because of divergence
The convergent set of solutions obtained may not be a desirable solution
to the problem
FEATURE 3
VISUALIZATION BASED TRAJECTORY PLANNER
A visualization based trajectory planner is developed to
accomplish this task.
Drastic reduction in time for planning trajectories and ease of
use.
Obstacle avoidance.
Robots can be pre-programmed in either point-to-point mode or
continuous path mode.
Point-to-point mode is used for tasks such as spot welding,
inspection and moving parts.
Continuous path mode is used for tasks such as spray painting
and arc welding.
ROBOKINE
USER INPUTS
NUMBER OF JOINTS AND MAXIMUM CO-ORDINATE
VALUES OF SKETCHING
USER INPUTS FOR GENERATING
THE HOMOGENEOUS SHAPE
MATRICES FOR EACH OF THE
JOINTS
DATA SAVED
INTO AN
OUTPUT FILE
UNIMATE 2000
SPHERICAL ROBOT
ALGORITHM GENERATES SHAPE
MATRICES FROM C-B NOTATION
TABLE, FOR EACH OF THE JOINTS IN
THE CONFIGURATION
UPJ ROBOT
1- PLOT OPTION
2- SLICE OPTION
EXIT
3- REAL TIME OPTION
NUMERICAL EXAMPLES
CINCINNATI MILACRON T3 ROBOT
BENEDIX AA/CNC INDUSTRIAL ROBOT
UNIMATE 2000 SPHERICAL ROBOT
KR 60 P/2 ROBOT
UPJ ROBOT
CINCINNATI MILACRON T3 ROBOT (RRR/RRR)
C-B NOTATION TABLE
WORKSPACE
INVERSE KINEMATICS
BENDIX AA/CNC INDUSTRIAL ROBOT (RRP/RRR)
C-B NOTATION TABLE
WORKSPACE
INVERSE KINEMATICS
UNIMATE 2000 SPHERICAL ROBOT (SP/RRR)
C-B NOTATION TABLE
WORKSPACE
INVERSE KINEMATICS
KR 60 P/2 (6R)
C-B NOTATION TABLE
WORKSPACE
INVERSE KINEMATICS
UPJ ROBOT
C-B NOTATION TABLE
WORKSPACE
INVERSE KINEMATICS
LIMITATIONS
The dials take in integer values
The orientations of the links have to be either horizontal and vertical while sketching
the links at home position
Range of the input parameters of the joints should pass through zero or contain zero
If one of the joints in the configuration is helical other than the base then the
workspace is not generated
Workspaces generated in certain cases may not be 100% accurate because of the
following reasons
1) Joints with no range of motion
2) Algorithm considers maximum reach position when all the links are set to zero which
may not be the case in certain configurations
Workspace generated is based on the projection of the profile on the sagittal plane
The results and plots generated can be saved nor printed
The inputs have to be provided each time the application is started
Software limitations
Hardware limitations
FUTURE WORK
Velocity and acceleration spaces
Dynamic analysis
Closed chain and combined open and closed chain analyses