SISOM 2006, Bucharest 17-19 May
A CONTROL APPROACH OF A 3-RRR PLANAR PARALLEL MINIROBOT
Radu BĂLAN*, Vistrian MĂTIEŞ*, Sergiu STAN**
**
*
Department of Mechatronics, Faculty of Mechanics, Technical University of Cluj-Napoca, Cluj-Napoca 400641, Romania
Department of Mechanics and Programming, Faculty of Machine Building, Technical University of Cluj-Napoca, Cluj-Napoca
400641, Romania
Corresponding author: [email protected]
The paper presents an approach and concept to control a mechatronic system. Robots may be
considered as typical mechatronic systems. The control is applied on a planar parallel robot with 3
DOF. The control was achieved based on SimMechanics toolbox of Matlab/Simulink. A method for
determining the inverse kinematics of the 3-RRR planar parallel minirobot is presented. Planar
parallel robots are good candidates for microminiaturization into a microdevice. Planar parallel robots
have great potential for microminiaturization using processing technology developed for
microsystems, because their components move on parallel planes and are connected by simple joints.
Key words: mechatronics; control; parallel robot.
1. INTRODUCTION
A mechatronic system is an integrated electro-mechanical one. Mechatronic is a highly
interdisciplinary domain, and robots may be considered as typical mechatronic systems. Robots are good
examples of mechatronic artifacts, as are DVD- players, cars, copying machines, cameras, printers, etc.
Figure 1. Basic structure of a mechatronic system (VDI 2206)
A mechatronic system has a multi-disciplinary nature; it encompasses several components: a
mechanical structure, actuators (motors), sensors, a control computer, and a programming or human/machine
interface (Fig. 1).
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Some applications for nonlinear processes of a predictive control algorithm
It simultaneously combines several scientific/engineering disciplines, such as: machine design,
structural dynamics, control engineering, real-time software engineering, actuator and sensor technology,
optics, etc., in a synergetic way.
A parallel robot is a closed-loop mechanism in which the mobile platform is connected to the base by
at least two serial kinematic chains (legs) [1] [2]. Applications of this type of robots can be found in the
motion platform for the pilot training simulators and the positioning device for high precision surgical tools
because of the high force loading and very fine motion characteristics of the closed-loop mechanism.
The advantages of parallel robots as compared to serial ones are:
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higher payload-to-weight ratio since the payload is carried by several links in parallel,
higher accuracy due to non-cumulative joint error,
higher structural rigidity, since the load is usually carried by several links in parallel and in some
structures in compression-traction mode only,
location of motors at or close to the base,
simpler solution of the inverse kinematics equations.
One of the latest tendencies in parallel kinematics is miniaturization. In the case of parallel robots
miniaturization proved to be possible and a lot of resources are spend in order to build new parallel
minirobots. Planar parallel robots have great potential for microminiaturization using processing technology
developed for microsystems, because their components move on parallel planes and are connected by simple
joints.
2. 3-RRR PLANAR PARALLEL MINIROBOT
A. General description of the parallel minirobot
The kinematics of planar, 3-RRR parallel robot, actuated with DC motors, will be developed in this
section. Electric motors require minimal auxiliary support devices, which are typically a power supply and
motor drive. These support devices are small and fairly portable. Secondly, the motor mass is fixed to the
base frame and not part of the moving linkage mass, keeping the manipulator inertia lower. Additionally, the
rotary type of actuator does not require a large base assembly and therefore the overall dimensions of the
manipulator can be kept small. For these reasons, the electric motor was considered the best actuator for this
parallel manipulator.
The schematic of the planar 3RRR parallel minirobot is presented in Fig. 2.
Figure 2. Planar parallel robot with 3 DOF
Radu BĂLAN, Vistrian MĂTIEŞ, Sergiu STAN
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B. Inverse Kinematics Problem (IKP) of the parallel minirobot
Inverse Kinematic Problem (IKP) is defined as the inverse problem of finding the joint variables in
terms of the end-effector position and orientation of a manipulator. Here it is summarized the method of
solving the IKP for the 3-RRR parallel manipulator.
The inverse kinematics of planar parallel robots can be found by evaluating the loop closure equations
that describe the closed kinematic chains of the manipulator. The inverse kinematics of many manipulators
can be found in literature. The kinematics of the 3-RRR manipulator can be found in Tsai's book [3].
Figure 3 and figure 4 show the 3-RRR planar minirobot and its parameters. The robot in Figure 3 is
actuated by three revolute actuators, located at M1, M2 and M3. Two links for each leg, labeled Link A and
Link B, are connected to the End Effector by passive joints at the points labeled A and B. The parameters are
described here without the numerical subscripts, however, when they appear with a numerical subscript (or i)
the subscript denotes the leg number. The position of the end effector is given by P; this is an arbitrary point
on the end effector, typically taken to be the center of mass of the moving platform. P is also the x-y
coordinates of the manipulator in global coordinates. The orientation of the end effector is given by the
angleψ , measured from the x-axis. la is the length of Link A, lb is the length of Link B, and lci is the distance
from point P to Bi. θ i is the joint angle measured from the x-axis. The angle given by ψ i is the angle of
Link B measured from the x-axis. γ i is the angle of the line from point P to the joint at Bi - it is measured
from the line connecting B1 and B2 on the moving platform (see Fig. 4).
Figure 3. Kinematic scheme of the planar parallel robot with 3 DOF
The derivation of the inverse kinematics comes from the loop closure equation for the 3-RRR parallel
robot:
OP = OM i + M i Ai + Ai Bi + Bi P
Writing the equations in component form:
x P = l a cos(θ 1) + l b cos(ψ 1 ) + l c cos(φ + γ 1 ) + x M1
y P = l a sin(θ 1) + lb sin(ψ 1 ) + l c sin(φ + γ 1 ) + y M1
(1)
(2)
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Some applications for nonlinear processes of a predictive control algorithm
Rewriting equations 1 and 2:
l b cos(ψ 1 ) = x P − l a cos(θ 1 ) − l c cos(φ + γ 1 ) − x M − y M
1
1
l b sin(ψ 1 ) = y P − l a sin(θ 1 ) − l c sin(φ + γ 1 ) − y M 1
(3)
(4)
Figure 4. Close-up of End Effector for 3-RRR planar parallel minirobot
Now squaring equation 3 and 4 then summing the two:
2
2
lb2 = x P2 + y P2 + l a2 + l c2 + x Mi
+ y Mi
− 2 x P l a cosθ i − 2 x P lc cos(φ + γ i )
− 2 x P x Mi + 2l a lc cos(θ i ) cos(φ + γ i )
+ 2l a x M i cos(θ i ) + 2l c x Mi cos(φ + γ i )
(5)
− 2 y P l a sin θ i − 2 y P lc sin(φ + γ i )
− 2 y P y Mi + 2l a lc sin(θ i ) sin(φ + γ i )
+ 2l a y M i sin(θ i ) + 2l c y Mi sin(φ + γ i )
Now combining terms in equation 5 so that all terms with
sin θ i are grouped together, and similarly, terms with
cosθ i are grouped.
e1 sin θ i + e2 cosθ i + e3 = 0
(6)
2
2
e3 = x P2 + y P2 + l a2 − lb2 + l c2 + x Mi
+ y Mi
− 2 x P lc cos(φ + γ i ) − 2 x P x Mi
+ 2l c x Mi cos(φ + γ i )
(7)
− 2 y P lc sin(φ + γ i ) − 2 y P y Mi
+ 2l c y Mi sin(φ + γ i )
e2 = −2 x P l a + 2l a lc cos(φ + γ i ) + 2l a x Mi
(8)
Radu BĂLAN, Vistrian MĂTIEŞ, Sergiu STAN
308
e1 = −2 y P l a + 2l a lc sin(φ + γ i ) + 2l a y Mi
(9)
Rather than the solution presented in Tsai’s book the solution of equation (8) will use the cosine-sine
solution method suggested in Lipkin and Duffy [4][5]. Although the tangent half-angle solution used by Tsai
is the most commonly employed method, Lipkin and Duffy point put that there are algebraic indeterminacies
which occur when e3 − e2 = 0 . In fact, numerical roundoff can cause significant error even when
e3 − e2 ≅ 0 .
Dividing equation (6) by
e12 + e22 :
e1 sin θ i
e12 + e22
+
e2 cosθ i
e12 + e22
e3
+
e12 + e22
.
(10)
Now, rearranging equation (7) and using the trigonometric identity:
cos(α − β ) = cos α cos β + sin α sin β
(11)
to yield,
cos(θ − ρ ) = −
e3
(12)
e12 + e22
where,
cos ρ =
e1
e12 + e22
, sin ρ =
e2
e12 + e22
(13)
2
2
Using the identity cos α + sin β = 1, and eq. (12) gives:
sin(θ − ρ ) = ±
e12 + e22 − e32
e12 + e22
.
(14)
By defining σ such that:
θ± =σ± + ρ
(15)
cosθ ± = cosσ cos ρ + sin σ sin ρ
(16)
and applying the identity of equation (12)
Now,
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Some applications for nonlinear processes of a predictive control algorithm
cosθ =
±
− e1e2 ∓ e2 e12 + e22 − e32
e12 + e22
(17)
and
sin θ ± =
− e2 e3 ± e1 e12 + e22 − e32
e12 + e22
(18)
Now this equation can easily be solved for any number of legs by simply changing the subscript i in
equations (17) and (18). For the particular robot there are two possible solutions given by the solutions to
equations (17) and (18). The two solutions are called elbow left and elbow right corresponding to θ + and
θ − . In practice, the atan2 command is used to find the solution in the proper quadrant. When the roots are
imaginary (i.e. there is no real solution) for some given pose, it means that the position/orientation of that
configuration is outside the reachable workspace.
Figure 5. Configuration of the 3-RRR planar parallel minirobot for x=0.2, y=0.15, φ=45°
C. Simulation results for the 3-RRR planar parallel minirobot
Simulation of the parallel robot was made in SimMechanics. SimMechanics is a toolbox from
Matlab/Simulink.
Figure 6. SimMechanics control scheme of the parallel robot 3-RRR
Radu BĂLAN, Vistrian MĂTIEŞ, Sergiu STAN
310
Control scheme describes the mechanical structure of the robot together three controllers of type PID
(Fig. 6). Driving joints are actuated by rotation actuators. To each joint it is attached a position sensor.
Figure 7. The blocks SimMechanics Joint Actuator, Joint Sensor and Scope
Figure 8. 3D visualization of the 3-RRR parallel robot in SimMechanics
Figure 9. Simulation results for independent PID control of 3-RRR planar parallel minirobot
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Some applications for nonlinear processes of a predictive control algorithm
Numerous control techniques are described in literature for serial manipulators. They include joint
space control, task space control, impedance control, and model based control.
Because parallel manipulators result in a loss of full constraint at singular configurations, any control
applied to a parallel manipulator must avoid such configurations.
Simple independent joint control is a practical approach to robot control. Linear PID compensators are
commercially available; this provides a reliable source for industrial applications, where availability of
replacement parts and operational reliability is important.
The results of the simulation for 3-RRR parallel robot for PID controllers is presented in fig.9. The
simulation results are for Kp = 2500, Kd = 35, and Ki = 1500.
4. CONCLUSIONS
In the paper was presented an approach and concept to control a mechatronic system. Robots may be
considered as typical mechatronic systems. The control was applied on a planar parallel robot with 3 DOF.
The control was achieved based on SimMechanics toolbox of Matlab/Simulink. A method for determining
the inverse kinematics of the 3-RRR planar parallel minirobot was presented. Planar parallel robots have
great potential for microminiaturization using processing technology developed for microsystems, because
their components move on parallel planes and are connected by simple joints.
REFERENCES
1. MERLET, J-P., The parallel robots, Kluwer Academic Publ., The Netherland, 2000
2. STAN, S., Diplomarbeit, Analyse und Optimierung der strukturellen Abmessungen von Werkzeugmaschinen mit Parallelstruktur,
IWF-TU Braunschweig, 2003, Germany.
3. TSAI, L.-W. Robot Analysis. The Mechanics of Serial and Parallel Manipulators. John Wiley & Sons, 1st edition, 1999.
4. LIPKIN, H., DUFFY, J. “A vector analysis of robot manipulators“. In G. Beni and S. Hackwood, editors, Recent Advances in
Robotics, pages 175-241. John Wiley & Sons, New York, 1985.
5. CHAN, V. K., “Singularity Analysis and Redundant Actuation of Parallel Manipulators“, PhD Thesis, Georgia Institute of
Technology, March 2001.
6. MATLAB DOCUMENTATION, www.mathworks.com