A Generalized Approach on Design and Control Methods Synthesis of Delta Robot
Trinh Duc Cuong, Tuong Phuoc Tho, Nguyen Truong Thinh
A Generalized Approach On Design And Control Methods
Synthesis Of Delta Robot
Trinh Duc Cuong
Tuong Phuoc Tho
Nguyen Truong Thinh
Department of Mechatronics,
University of Technical Education
Ho Chi Minh City, Viet Nam
+84-903.839.238
Department of Mechatronics,
University of Technical Education
Ho Chi Minh City, Viet Nam
+84-909.160.264
Department of Mechatronics,
University of Technical Education
Ho Chi Minh City, Viet Nam
+84-903.675.673
[email protected]
[email protected]
[email protected]
ABSTRACT
1. INTRODUCTION
This paper will describe the kinematics and dynamics of parallel
robot named Delta with 3 degree of freedom (d.o.f). The use of
dynamics coupled with kinematics for the control of parallel
robot has been gaining increasing popularity in recent years.
Relationship between generalized and articular velocities is
established, hence jacobian and inverse jacobian analyses are
determines. The inverse formulas are generally shown simply
and the direct formulas are also described. Besides, this paper
deal with the direct and inverse dynamics to determine the
relations between the generalized accelerations, velocities,
coordinates of the end-effector and the articular forces based on
simulation and control. Parallel robots have become the
important machines to manufacturing. They are used for various
purposes in industry and life. The dynamic model of parallel
robot with 3 dof is presented, and an adaptive control strategy for
this robot is described. The robustness of the control system with
respect to the nonlinear dynamic behavior and parameter
uncertainties is investigated by computer simulation.
Experiments were implemented to evaluate the responding of
controlling system based on dynamics and kinematics controlling
method for tracking desired trajectories. The results show that
the use of the suitable control system based on dynamics model
can provide the high performance of the robot.
Parallel robots are closed-loop mechanisms presenting very good
performances in terms of accuracy, regidity and abality to
manipulate large loads. Many applications in the field of
production automation, such as assembly and material handling,
require machines capable of very high speeds and accelerations.
The parallel robots are able to work on some tasks with a much
better performance. However, there are still several unanswered
questions and few papers published studying robots with parallel
architectures. This paper introduces a three d.o.f parallel
manipulator dedicated to pick-and-place: Delta Parallel Robot.
First a kinematics model of a Delta parallel robot is obtained
using a generic geometrical formulation then the model is used
for a workspace analysis. Delta robot has many advantages like
operating required accurary, rigidity and manipulation of large
loads. A Parallel Robot is a mechanism that has links that form
closed kinematics chains. Because of this, Parallel mechanisms
have many advantages compared to serial mechanisms, such as
speed and accuracy. Generally, a parallel robot is made up of a
mobile platform (end-effector) with n d.o.f, and a fixed base,
linked together by at least two independent kinematics chains.
Normally, each kinematics chain has a series of links connected
by joints. Manipulators with 3 degrees prove extremely
interesting for pick-and-place operations. Several prototypes have
been suggested. The most famous robot with 3 d.o.f is Delta. All
the kinematic chains of this robot are 3 rotary actuators allowing
to obtain 3 dof in translation. This paper introduces a 3-dof
parallel manipulator architecture Delta dedicated with
kinematics and dynamics analyses to pick-and-place and
developed to perform high speed and acceleration. In this article
we have discussed the inverse and direct kinematics solution as
well as dynamics for the Delta parallel robot. With this
manipulator it is often difficult to determine the kinematics and
dynamics analyses. Thus, this paper includes five seperated
sections. The main properties of parallel robot is described in
section II as well as focusing on kinematics and dynamics
analyses, respectively. Experiment and discussions is established
in Section IV. Finally, in Section V is shown the conclusion.
Categories and Subject Descriptors
B.1.2 [Control Structures And Microprogramming] Control
Structure Performance Analysis and Design Aids -Automatic
synthesis, Formal models, Simulation.
General Terms
Performance, Design, Experimentation, Verification.
Keywords
Delta platform, Design, Dynamics, Delta Robot, Parallel robot,..
2. KINEMATIC AND JACOBIAN
ANALYSES
.
Research Notes in Information Science (RNIS)
Volume13, May 2013
doi:10.4156/rnis.vol13.36
In this section, the description and kinematics of the parallel
robot – 3 dof are shown in Fig.1. Generally, parallel robot is a
179
A Generalized Approach on Design and Control Methods Synthesis of Delta Robot
Trinh Duc Cuong, Tuong Phuoc Tho, Nguyen Truong Thinh
closed loop manipulator is more difficult to calculate the
kinematics. The moving plate always stays parallel to the base
platform and its orientation around the axis perpendicular to the
base plate is constantly zero. Thus, the parallelogram type joints
(forearm) can be substituted by simple rods without changing the
robot kinematic behaviour. The revolute joints (between the base
plate and the upper arms and between the forearms and the
travelling plate) are identically placed on a circle. Thus, the
travelling plate can be replaced by a point P which the three
forearms are connected to.
The modelling of Delta robot has the assumptions like as: 1, 2,
3 are the rotate angle of 3 link, dA is the distance from the center
of the base (origin) to the spin axis of the transmission, F1; F2; F3
are the center of the spindle attached to the transmission, rA is
the distance from the center stand on compared to the projection
axis of the arm to stand on. And L1, L2 are the length of 2 link as
describe in Fig. 2. Because, the inverse kinematics of Delta
parallel robot is more easier than Direct Kinematics (Forward
Kinematics), so firstly the inverse kinematics is shown. The
inverse kinematics of a parallel manipulator determines the i
angle of each actuated revolute joint given the (x,y,z) position of
the travel plate in base-frame.
 z j1
 y F  yJ
 1
1
1  arctan 



Fig.2. Shows model simplification of the Delta parallel robot.
Use of the vector translation of y-axis displacement, we have:
   
OJ1'  OF1  F1 J1  J1 J1'
With a length of the vector, the distance from the original
quadrant to the swivel point of the transmission are:
OF1  OF2  OF3  rA2  d A 2
(1)
Such algebraic simplicity follows from good choice of reference
frame: joint F1J1 moving in YZ plane only, so we can completely
omit X coordinate. To take this advantage for the remaining
angles 2 and 3, we should use the symmetry of delta robot.
First, let's rotate coordinate system in XY plane around Z-axis
through angle of 120o counterclockwise.
 x  x.cos  120   y.sin  120 

 '
o
o
 y0   x.sin  120   y.cos  120 

z0'  z0

o
(4)
Distance from center of the three spheres intersect at the center
base
J 2 J 2 '  J 2 J 2 '  J 3 J 3'  rB
(5)
Radius of the sphere is L2, so:
We've got a new reference frame X'Y'Z', and it this frame we can
find angle 2, 3 using the same algorithm that we used to find
1.
'
0
(3)
F1J1  L2 cos 1
(6)
F2 J 2  L2 cos  2
(7)
F3 J 3  L2 cos 3
(8)
We have:
o
(2)
Now the three joint angles 1, 2 and 3 are given, and we need
to find the coordinates (x0, y0, z0) of end effector point E0.
(9)
r  rA2  d A2  rB
OJ1'  OF1  F1 J1  J1 J1'
(10)
And (x, y, z) is the coordinates of sphere centers J1’, J2’, J3’.
So the coordinate of J1’ is:
0
r  L2 cos 1
T
L2 sin 1  d A    x1
y1
z1 
(11)
Similarly we have the coordinates of J2’ and J3’ as follows:
J 2 '   x2 ; y2 ; z2  
((r  L2 cos  2 ) cos30 0 ; (r  L2 cos  2 ) sin 30 0 ; L2 sin 2  d A )
(12)
'
3
J   x3 ; y3 ; z3 
 ( ( r  L2 cos 3 ) cos 300 ; ( r  L2 cos 3 ) sin 300 ; L2 sin 3  d A )
(13)
So the intersection of 3 sphere here:
Fig.1. Modelling of Delta parallel robot.
180
A Generalized Approach on Design and Control Methods Synthesis of Delta Robot
Trinh Duc Cuong, Tuong Phuoc Tho, Nguyen Truong Thinh
( x  x1 ) 2  ( y  y1 ) 2  ( z  z1 ) 2  L12

2
2
2
2
( x  x2 )  ( y  y2 )  ( z  z2 )  L1

2
2
2
2
( x  x3 )  ( y  y3 )  ( z  z3 )  L1
1
3
1
2


a1 z0  b1
d
(13)

1

1
3

2

2a2  px cos 2  p y sin 2  h  r  sin 12  p z cos 12 
a z b
y0  2 0 2
d
(14)
 3   ma a 2  mb a 2  13   ma  mb  gc a cos 13
b  
2a
(15)
 2   ma a2  mb a 2  12   ma  mb  gc a cos 12
And, we have solutions like as:
x0 
z0 
With help from computer this equation system can be solved.
There will be two solutions that describe the two intersection
points of the three spheres. Then the solution that is within the
robots working area must be chosen. With the base frame {R} in
this case it will lead to the solution with negative z coordinate.
3. DYNAMIC ANALYSIS OF DELTA
ROBOT
k
 L
g
 Qj    i
 

q

qk

1
i
j

(16)
Where L is the Lagrange function, where L = T - V, T is the total
kinetic energy of the body, V is the total potential energy of the
body, q is the kth generalized coordinate, Q is a generalized
external force, λi is the Lagrange multiplier and gi is the
constrain equation. By employing the formula above it is possible
to determine the external forces of a body. However, friction
forces are not constraints even though they play an important role
in the dynamics analysis so they can be treated separately.
The Lagrange multipliers are derived as.
3
px
2  i ( px  h cos i  r cos i  a cos i cos1i )  (mp  3mb ) 
(17)
i 1
3
py
2  i( py  h sin i  r sin i  a sin i cos 1i )  (mp  3mb ) 
(18)
i 1
2  i ( p z  a sin 1i )  ( m p  3mb ) 
p z  (m p  3mb ) g c
(21)

1

1
3

2

2a3  px cos 3  p y sin 3  h  r  sin 13  pz cos 13 
(22)
The analytical inverse dynamics solutions for Delta parallel robot
can be obtained from Eqs.(20-22)
4. EXPERIMENTS AND DISCUSSIONS
To valid the analyses of kinematics and dynamics in previous
section, an experimental setup was built to perform the control of
Delta parallel robot (Fig.3). The specifications of Delta parallel
robot is shown in Table 1.
Table 1. Specifications of Delta parallel robot
One important step in design process of a robot is to understand
the behaviour of the device as it moves around its workspace or
doing a specific task. This behaviour is determined through the
study of the dynamics of the mechanism, where the forces acting
on the elements and torques required by the actuators can be
determined. Consequently, each component must be optimized in
dimensions and material to be used in the manufacturing
processes. In section, the dynamics of Delta parallel robot is
described based on Lagrangian formulation, which is based on
calculus variations, states that a dynamic system can be express
in terms of its kinetic and potential energy leading in an easy
way the solution to the problem. In addition, it is considered a
good option to be used for real-time control for parallel
manipulators [4]. The Lagrange equations can be derived.
d  L

dt  q j
(20)
 2 a 1   p x cos 1  p y sin 1  h  r  sin 11  p z cos 11 
(14)
3


 1   m a a 2  mb a 2  11   m a  mb  g c a cos 11
(19)
i 1
When the Lagrangian multiplies are found the actuator torque
can be determined as.
Parameters
Value
Upper robot arm ma [kg]
1.1
Parallelogram mb [kg]
0.9
Moving platform mp [kg]
0.2
Radius of the fixed base a [mm]
150
Radius of the moving platform b [mm]
100
Upper arm length l1 [mm]
250
Parallelogram length l2 [mm]
480
No. of AC Servo motor
3
Motor power [W]
200
Encoder resolution [ppr]
1000
Maximum load capacity [kg]
Maximum moving platform velocity [m/s]
Position repeatibility [mm]
Workspace
5
5.0
0.2
Diameter [mm]
500
Height [mm]
200
This experimental implementation is built on the PC and Delta
robot. The software for Delta parallel robot is implemented in
Matlab using the kinematics and dynamics analyses from above
solutions to control the moving platform. The proposed analyses
are applied the Delta parallel robot for material cutting and
drawing. The program is used to control the moving platform
with predefined trajectory. We will apply the kinematics, Jacobi
and dynamics to control suitable trajectory of parallel robot based
on positions, velocities. In the section, some experimental results
by kinematics - dynamics control are addressed. To demonstrate
the capability controller, several responses were taken into
account with several various trajectories. In these experiments, a
pen attached to moving platform of Delta parallel robot is
regulated following the several predefined paths including curves
of circle, butterfly, flower, heart.
181
A Generalized Approach on Design and Control Methods Synthesis of Delta Robot
Trinh Duc Cuong, Tuong Phuoc Tho, Nguyen Truong Thinh
Fig.3. Delta parallel robot for experiments.
The first experimental results for controlling the moving platform
with contour of flower are illustrated in Fig.4. A curve has the
shape of a petalled flower and the polar equation of the rose is
follows.
r  a sin  n 
Fig.4. Trajectory of moving platform (a) and responding of 3
motors(b) with flower curve path.
(23)
The drawing on paper or cutting on acrylic reveals that the
analysis results are almost near the desired ones shown in
Fig.4(a). Compared desired contour, we can see that the very
small differences between the desired and experimental values
may be attributed to the following reasons: first, there is error of
mechanical transmission and calculation of kinematics and
dynamics of Delta robot. The improvements will bring better
results for generating trajectories. And responding of three AC
servo motors with time is shown in Fig.4(b) .
Next, other responses for reference commands for butterfly
contour are presented to evaluate the performance of the
controller based kinematics and dynamics. The equation of
butterfly curve is follows.
  2    
r  esin  2 cos  4   sin 5 

 24 
(24)
Fig.5 shows output of responding trajectory and input responses
for contour of butterfly. The control results for butterfly are good
enough to track the perfect shape while moving path of pen has a
little bit error.
Fig.5. Trajectory of moving platform (a) and responding of 3
motors (b) with butterfly curve path.
182
A Generalized Approach on Design and Control Methods Synthesis of Delta Robot
Trinh Duc Cuong, Tuong Phuoc Tho, Nguyen Truong Thinh
5. CONCLUSION
This paper is mainly concerned with kinematic and dynamic
analyses as well as the application of solutions of kinematics and
dynamics to modeling and control of parallel manipulators. A
practical implementation is completed to evaluate the results of
an designed controller for Delta manipulator control system. It
can be said that, excepted results has been achieved for these
cases. The inverse and forward kinematics and velocity equations
have been derived. The results presented in the paper will be
valuable for both the design and development of Delta parallel
robot for various applications. With the aid of computer, these
equations with the design of this robot base on dynamic modeling
and dynamic control in order to improve the behavior of the robot
while reaching high acceleration. By fitting grippers or other
tools to this small platform the delta robot can handle all sorts of
items. Their design enables them to move both rapidly and
accurately, and they are deployed for tasks varying from highspeed packaging to the assembly of miniature products.
6. ACKNOWLEDGMENTS
This study was financially supported Ho Chi Minh city
University of Technical Education, Viet Nam (HCMUTE).
7. REFERENCES
[1]
[2]
Fig.6. Trajectory of moving platform (a) and responding of 3
motors(b) with heart curve path.
Besides, we also generate the trajectory of heart curve with pole
equation like as:
r  2  2sin  
sin  cos 
7
sin  
5
[3]
[4]
(25)
Fig.6(a) shows the actual time response signals and the
command signals of parallel to a heart profile, and the time
history of the controlled position output.
[5]
André Olsson, Modeling and control of a Delta-3
robot, 2009.
Jon Martínez García, Inverse-Forward Kinematics of
a Delta Robot, 2010.
Manuel Napole and Cardona Gutierrez, Kinematics
Analysis of a Delta Parallel Robot, 2011.
S.M.Ha, P.V.B. Ngoc and H.S.Kim, “Dynamics
Analysis of a Delta-type Parallel Robot,” 2011 11th
International Conference on Control, Automation
and System, 2012.
S.M.Ha, P.V.B.Ngoc and H.S.Kim, “Dynamics
Analysis of a Delta-type Parallel Robot”, 2011 11th
International Conference on Control, Automation
and Systems, pp.855-857, 2011.
The movement of moving platform followed the commanded
signals quite well for long time. Present results show that the
analyses of kinematics and dynamics can be successfully applied
to the dynamic tracking of various contour profiles.
183