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APPROACHES TO THE DESIGN OF A PLANAR PARALLEL MANIPULATOR.

1. INTRODUCTION

Manipulator is an important part of a whole robot assembly, forming the mechanical infrastructure of a mechatronic system. Selection of the manipulator affects a broad area extending from modelling to design, from control to operation and furthermore from accuracy to its economy. Manipulator can be constructed by bringing together rotary and/or sliding elements or a combination of these in a suitable manner. Within this context, open or closed kinematic chains formed as such will result in the so-called serial or parallel robotic structures, (Duffy, 1996).

The most fundamental manipulator types like cartesian, cylindrical, spherical, articulated arm and scara are the most widespread examples of serial manipulator. Most classical works rely upon the serial manipulator, which is based on the open chain, (Koren, 1987; Stadler, 1995; Fu et al., 1987, Groover et al., 1986). Although serial manipulators have a high maneuverability within a large workspace, they are subject to significant limitations. First of all, they have limited load carrying capacities due to their highly deformable structures, which are prone to vibrations under large velocities. Additionally, it is probable that the fact that an actuator is needed at each joint, for a serial manipulator having many joints in an open chain, might lead to a rise in the initial and operating costs of the robotic assembly. On the other hand, most of the issues mentioned above are solved by using parallel manipulators facing only the limitation of having small workspaces. Thus, such advantages have invited the research attention on parallel manipulators in recent years, (Innocenti and Castelli, 1990; Bernier et al., 1995, Harris, 1995; Liu, 1995).

Since the workspace of a parallel manipulator is limited, the problem of overlapping the actual space in which the physical tasks like welding, cutting conveying etc. Are to be fulfilled with that of the manipulator becomes significant. The solution of the problem passes through the accurate positioning of the manipulator. Thus, determining optimum values of all the adjustable parameters in a robot assembly constitutes a design task.

Here in this work, approaches to the design of a parallel planar manipulator with two degrees of freedom have been shown. The design of the manipulator has been reduced to the design of modules which considerably dissolve the complexity of the original assembly, mathematically and physically, thus always assuring closed-form solution. Then how actuators may be utilized to operate the assembly and to form an analog robot have been demonstrated.

2. THEORY

The kinematic scheme of the parallel manipulator in consideration is drawn in Fig. 1.

It can be seen that the parallel manipulator in question can be constituted by bringing together two of the basic module shown in Fig. 2.

The fundamental problem here is to determine the most appropriate values of the module parameters involved such that the end point C of the module follow a desired trajectory y (x) within [x0, xn] domain. To this end, the following can be written from Fig. 2:

x = [r.sub.1] + (a -- d)cos[theta] + ssin [theta] (1)

y = [r.sub.2] + (a -- d)sin[theta] -- s cos[theta] (2)

If (a-d) is designated by b and s is eliminated from the above equations, then a displacement function G(x, y, [theta]) characterizing the motion of the module on the trajectory is obtained:

G(x, y,[theta]) = ysin[theta] + x cos[theta] -- [r.sub.1] cos[theta] -- [r.sub.1]cos[theta] - [r.sub.2] sin[theta] -- b = 0 (3)

Considering that the robot arm will rotate about O according to the [theta]' motion variable starting from an initial position [theta]0, the following relationships can be written to meet the motion co-ordination requirements:

[theta] = [theta]0 + [theta]' (4)

[theta]' = [r.sub.x] (x -- x0) (5)

[r.sub.x] = [DELTA][theta]/[DELTA]x, [DELTA][theta] = [DELTA]n - [DELTA]0, [DELTA]x = xn -- x0 (6)

where [theta]n is the final position of the robot arm corresponding to the point (xn) of the given trajectory. Evaluating the above relationships together with the trigonometric identities and rearranging will yield the following expression:

[mathematical expression not reproducible] (7)

Now, it is possible to obtain velocity relationships by taking the first derivative of the displacement function G with respect to time:

[mathematical expression not reproducible] (8)

If the velocities of the end point of the manipulator in the x and y directions are represented by Vx and Vy, respectively, and the angular speed of the rotating arm is designated by Z, then the following will come out of (8):

[mathematical expression not reproducible] (9)

D A where

[mathematical expression not reproducible] (10)

Examination of the basic displacement and velocity functions (7) and (9) will reveal that there are four available parameters (r1 , r2 , [theta]0, b) for formulating a design. One approach for a formulation of manipulator design is to require that the end point of the manipulator fit to specified two-position and two-velocity values. In such a context, if Precision-Point or Accuracy-Point (Hartenberg and Denavit, 1964) Subdomain (Akcali and Dittrich, 1989a) and Galerkin (Akcali and Dittrich, 1989b) methods are applied to displacement and velocity functions, then the following will result as the basic design equations:

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

where:

[mathematical expression not reproducible] (13)

The coefficients contained in (13) are defined is accordance with each method as follows: Accuracy-Point method takes into account points xi i =1,2:

[mathematical expression not reproducible] (14)

Subdomain Method considers subintervals [xi-1, xi] i = 1,2:

[mathematical expression not reproducible] (15)

The coefficients appearing in (13) are evaluated by Galerkin method, with reference to selected weighting functions wi i = 1,2 as shown below:

[mathematical expression not reproducible] (16)

[mathematical expression not reproducible]

In the solution phase of design formulation, which consists of four non-linear equations, b, r2 and r1 are eliminated, reducing the set (11)-(12) to the following:

[mathematical expression not reproducible] (17)

In the general case, there are two solutions given by:

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

[mathematical expression not reproducible] (21)

Since for a given tangent value, there exist two angles separated by 180[degrees], four possible angles might satisfy equation (17). Thus, in order to decide on the technically meaningful ones as well as on the quality of outcome, a motion analysis should be carried out.

If the actuation of the manipulator is based upon ([theta] , s) which are computed by (4)-(6) and (22) given below, then position error e is evaluated by means of (1), (2), (22) and (23), where xth, yth, xac, yac are theoretical and actual co-ordinates, respectively,

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible] (23)

In order to determine error in velocity, first theoretical velocities Vxth, Vyth and Vth, then actual velocities Vxac, Vyac and Vac are calculated with reference to (24)-(26), finally ending in (27).

[mathematical expression not reproducible] (24)

[mathematical expression not reproducible] (25)

[mathematical expression not reproducible] (26)

[mathematical expression not reproducible] (27)

3. SIMPLIFIED APPROACH

By letting a = d or b = 0 in Fig. 2, and by requiring that three point-positions on the specified trajectory be satisfied in the sense of Accuracy-Point, Subdomain and Galerkin methods, by the end point of the robot arm, a simplified approach can be made to the problem. In that case, the displacement function G becomes:

[mathematical expression not reproducible] (28)

Then the design equations take the form of a set of three linear equations as shown below:

[mathematical expression not reproducible] (29)

where:

Coefficients in Accuracy-Point Method are as follows:

[mathematical expression not reproducible] (31)

These in Subdomain Method are:

[mathematical expression not reproducible] (32)

In Galerkin Method, the coefficients are defined as such:

[mathematical expression not reproducible] (33)

Solution of the design equations yields the sought set (r1, r2, T[theta]) as given below:

[mathematical expression not reproducible] (34)

where:

[mathematical expression not reproducible] (35)

[mathematical expression not reproducible] (36)

[mathematical expression not reproducible] (37)

To secure a pair of solutions needed in the manipulator, the solution process is implemented twice by a change in method or some input parameters like the amount of arm rotation or the sense of rotation, if necessary.

4. ACTUATION POSSIBILITIES

The basic module is, in fact, a serial manipulator, Fig. 2, while the manipulator constructed out of two or possibly more basic modules will be of parallel type, securing more stiffness by its mechanical structure. One disadvantage of the serial manipulator is that one actuator should be available at each joint to the loss of payload capacities of the robot assembly. Thus, the advantageous feature of the parallel robot, namely having less actuators than the number of joints, offers possibilities of using analogously programmable actuators attached to the ground. Since actuation of the manipulator under consideration depends on slider displacement (s) and arm rotation ([theta]), possible analogously programmable actuators are either a function generating four-bar or an inverted slider-crank mechanism, the dimensions of which are continuously adjustable according to the design of the generator, which changes by trajectory (y), Fig. 3 (a), (b).

The functions of the 4-bar OKML in Fig. 3(a) and the inverted slider-crank QOD in Fig. 3(b) are to provide the necessary rotation [theta] required about M and the needed slider displacement (s) along AB at the right time for the fulfilment and control of the trajectory task, just like the supply of right voltages at the right time in the case of electrical drive. In other words, two-degree-of-freedom manipulator receives one rotary actuation ([theta]) from the motor at ground pivot O, and the other actuation either at ground pivot M through OKML 4-bar ([theta]) angle generator, instead of a motor, Fig. 3(a), or along motion direction AB through QOD slider-crank (s) displacement generator instead of a hydraulic or pneumatic drive on the moving arm, Fig. 3(b). In this manner, the dimensions of the 4-bar or the slider-crank function generators can be viewed as an information storage medium transforming input data like trajectory, manipulator design values and motor rotation into new actuation variables like [theta] or s in accordance with the following mathematical relationships

[mathematical expression not reproducible] (38)

[mathematical expression not reproducible] (39)

[mathematical expression not reproducible] (40)

where (r3, r4, h) are the parameters of the second serial manipulator like (r1, r2, b) in the first serial manipulator.

From the discussion above, it is to be understood that next to the rotary actuator at O, if the two-degree-offreedom robot assembly is to be brought into motion to follow a trajectory y by means of a programmed motion of the slider, then the slider-crank is designed with a suitable method like (Ak9ali, 1987) such that functional relationship (38) is generated between the reverse rotations (-[theta]) of the motor at O and slider translation (s). If two-rotary actuators are to be used for the same purpose, in that case a 4-bar is designed by means of (Akcali and Dittrich, 1989b) to generate (39) function between [theta] and [theta] in place of a motor at M in addition to the one located at O. Of course, in construction of the 4-bar or the inverted slider-crank, adjustability of dimensions is foreseen.

One advantage of these analogous designs is that the sense of actuation ([theta], [theta]) will be kept same within long intervals as opposed to possibly frequent changes in the sign of actuation in digital applications. In the direct kinematic analysis, corresponding to ([theta], [theta]) actuation, the following (x,y) trajectory co-ordinates will be generated:

[mathematical expression not reproducible] (41)

[mathematical expression not reproducible] (42)

5. ON APPLICATIONS

The design and operation of the manipulators proposed here are primarily based on analogous variables. As is well known, analogous variables are continuous in contrast with the discrete nature of digital variables, (raven 1987). Hence there are no zigzags in the functioning of actuators in driving the manipulators considered here. This aspect is consistently taken into considerations when applying the techniques presented here. Take, for instance, the problem of transporting an article from the point with (x0, y0) co-ordinates on a conveyor moving with velocity V0 to the point with (xn, yn) co-ordinates on another conveyor with a linear velocity of Vn. In order to associate this problem with both general theory and simplified approach, it is sufficient to find a trajectory function y(x) satisfying the velocity and the given end points. For instance, the cubic polynomial

[mathematical expression not reproducible] (43)

with the following computed coefficients (a, b, c, d) will be an answer to the requirements.

[mathematical expression not reproducible] (44)

[mathematical expression not reproducible] (45)

[mathematical expression not reproducible] (46)

[mathematical expression not reproducible] (47)

where rx, [omega] are selected as required in the techniques.

6. NUMERICAL RESULTS AND DISCUSSION

Analytical thoughts developed have been transformed into computer programs under Fortran 77 coding. Utilizing these programs, comprehensive illustrations are presented here to review the numerical results and to discuss their significance.

Example 1 A trajectory described by y = 0.5 x+ 0.5 0[less than or equal to] x [less than or equal to] 1 is to be followed under a uniform velocity requirement matching a 10s. travel on the part of a two-arm manipulator with the sliding directions passing through the fixed revolute joints.

The solution lies in the implementation of the simplified approach twice. Data for the first implementation are in case of Precision-Point Method xi, i=1,2,3 being [0.2;0.6;1.0] in case of Subdomain Method subintervals being [xi-1,xi] i = 1,2,3 [0.0;0.40],[0.40;0.75], [0.75;1.00] and finally for Galerkin Method weighting functions wi i =1,2,3 being [x,x2,x3]. The amount of arm rotation [DELTA][theta] is taken to be 45[degrees], for every method. The numerical results are displayed in Table 1, indicating also max absolute error, emax.

Input for the second implementation are in Precision-Point Method xi = 1, 2, 3 being [0.03;0.40;0.62] in Subdomain Method subintervals being [0.00;0.35], [0.35; 0.40], [0.40; 1.00]; in Galerkin Method weighting functions wi = 1, 2, 3 being [x, x2, x3]. Arm rotation angle ([DELTA][theta]) has been taken as 45o in Precision-Point and Subdomain Methods but in Galerkin Method as 42[degrees]. The outcome has been presented in Table 2. When the two arms are put together to form the parallel manipulator in question, then the maximum absolute (emax) error turns out to be 0.0034 in Precision-Point, 0.0056 in Subdomain and 0.0489 in Galerkin methods. Theoretical velocity on the trajectory is supposed to be 0.1118, constant throughout the motion. Maximum velocity errors in Precision-Point, Subdomain and Galerkin designs become 0.0152; 0.0183 and 0.0165, respectively.

Example 2 Requirements of Example 1 are to be met by the general parallel manipulator of Fig. 1.

General theory is applied once, here. When data for precision points xi, i=1,2, [0.4,0.7], for subdomains, [xi-1,xi], [0.2;0.6],[0.6;0.9] for weighting functions wi,i=1,2;[sin x,cos x] and [DELTA][theta] = 45[degrees] are taken into account, the results shown in Table 3 are obtained.

Two significantly different module designs are brought together for the construction of the parallel manipulator. After this process, b values are slightly affected, leading to 1.4439, 1.3496 and 1.0902, in Precision-Point, Subdomain and Galerkin methods, respectively, leaving all others same. Maximum errors in the resulting parallel manipulators become 0.0026, 0.0026, 0.0007 in the aforementioned methods, respectively. Maximum deviations from the constant 0.1118 velocity value in Precision-Point, Subdomain and Galerkin methods turn out to be 0.0023, 0.0024, 0.0013, respectively.

If the results of two examples above are compared, it will be seen that the chances of getting more refined designs are always much more in the general theory against simplified approach. While more than one application is needed in the simplified approach for the formation of a parallel manipulator, only one implementation of general theory is sufficient. Another advantage of the general theory is that the mirror-image manipulator (with [theta]+) is a good alternative when space for the manipulator (with [theta].) is not appropriate, since they both produce the same trajectories with the same end velocities.

The robotic assembly designed in this work differs from the classical constrained-motion mechanisms in that it has a flexible structure. While a classical one-degree of frecdom mechanism generates only one and constant curve, the robotic assembly under consideration can produce as many curves as the intervals of adjustability of parameters permit. If a change in the positions of fixed pivots of the two manipulators is not seen practical, then following the design of manipulators, the dimensions of the function-generating four-bar relating [theta]'c rotations of the first manipulator to the [theta]'c rotations of the second one can easily be made adjustable. In that case, the design of the function-generating 4-bar is realised with respect to a transformed function ([theta]'c, [theta]'c) corresponding to trajectory yc (xc) in a co-ordinate system xc - yc located at the ground pivot (O) of the first manipulator in the following way : First, [theta]M = tan-1[r4 - r2 ) /(r3 - r1)] is computed, then co-ordinate transformations xc=(x-r1)cos[theta]M+(y-r2)sin[theta]M, yc= -(x-r1)sin[theta]M+(y-r2)cos[theta]M [theta]c' = [theta]c - [theta]c, [theta]c = [theta]c - [theta]c ,where [theta]c = [theta]0 - [theta]M, [theta]c = [theta]0 - [theta]M and [mathematical expression not reproducible], [mathematical expression not reproducible] OM = [(r3 - r1 )2 + (r - r)2]1/2 are carried out. To demonstrate the practical nature of the proposition, an experimental model based on the simplified approach has been constructed, Fig. 4. A synchronous motor with 1 rpm located at O and controlled by a timer is used together with light aluminium arms. The pen attached to the generating-point (C) draws desirable curves with insignificant, unnoticeable errors always remaining inside the thickness of the line. By the presence of a mechanical drive, 4-bar, a motor is saved from the second manipulator. Hence, the assembly is suitably termed as analog robot. Two illustrations are shown in Fig. 5, in which straight lines between points having co-ordinates (10, 30) and (24.5, 18) in the first one and between (10, 24) and (24, 28.5) co-ordinates in the second one are traced. Relevant adjustable parameters (x1, x2, x3, x6) turn out to be (8.5161, 29.9633, 16.0945, 35.0000) in the first design and (33.1220, 11.3563, 14.9201, 35.0000) in the second one.

Conclusively, all numerical results and experimental work indicate that methods of design for a parallel robotic assembly work very well, leading to optimum results.

REFERENCES

Akcali, I. D. and Dittrich, G. (1989a) "Path Generation by Subdomain Method", Mech. Mach. Theory, Vol. 24, No.1, pp.45-52,

Akcali, I. D. and Dittrich, G. (1989b) "Function Generation by Galerkin's Method" Mech. mach. theory, Vol. 24, No.1, pp. 39-43

Akcali, I. D. (1987) "Design of Slider-Crank Mechanism for Function Generation" Proc. 7 World Congress on Theory of Machines and Mechanism, Sevilla, pp. 119-124.

Bernier, D., Castelian, V. and Li, X. (1995) "A New Parallel Structure with 6 Degrees of Freedom" 9 World Congress on the Theory of Machines and Mechanisms, Proc. Milano, pp. 8-12.

Duffy, J. (1996) Statics and Kinematics with Applications to Robotics, Cambridge University Press, UK.

Fu, K. S., Gonzales, R.C. and Lee, C. S. G. (1987) Robotics, Control, Sensing, Vision and Intelligence, McGraw-Hill Book Co., USA.

Hartenberg, R. S. and Denavit, J. (1964) Kinematic Synthesis of Linkages, McGraw-Hill Book Co, USA.

Harris, D. M. J. (1965). "Parallel-Linkage Robot Coordinate Transformation through Screw Theory" 9th World Congress on the theory of Machines and Mechanisms, Milano, pp. 1565-1568.

Groover, M. P., Weiss, Nagel, R. N. and Odrey, N. G. (1986). Industrial Robotics Technology Programming and Applications, McGraw-Hill, Singapore.

Koren, Y. (1987) Robotics for Engineers, McGraw-Hill, Singapore.

Liu, A. (1995) "Configuration Analysis of a Class of Parallel Structures Using Improved Continuation", 9th World Congress on the Theory of Machines and Mechanisms, Milano, pp. 155-158.

Innocenti, C. and Parenti-Castelli, V. (1990). "Direct Position Analysis of the Stewart Platform Mechanism" Mech. Mach. Theory, Vol. 25, No. 6, pp. 611- 621.

Stadler, W. (1995). Analytical Robotics and Mechatronics, McGraw-Hill Inc., Usa.

Raven, (1987). Automatic Control Engineering, 4th Ed., Mc Graw-Hill Book Co., Usa.

Huseyin Mutlu (*1) and Iskender Ozkul (2)

(1) Mersin University, Engineering Faculty, Mechanical Engineering Department, Mersin, Turkey ORCID ID 0000-0002-4770-2873 huseyinmutlu@yahoo.com

(2) Mersin University, Engineering Faculty, Department of Mechanical Engineering, Mersin, Turkey ORCID ID 0000-0003-4255-0564 iskender@mersin.edu.tr

(*) Corresponding Author

Received: 20/10/2017 Accepted: 20/12/2017

DOI: 10.31127/tuje.345351
Table 1. First Robot Arm for Trajectory y = 0.5 x+ 0.5
0 [less than or equal to] x [less than or equal to] 1

Method           r1       r2      [theta]0([degrees])  Emax

Precision-Point  -0.0155  2.0311  -90.44               0.0035
Subdomain        -0.0768  2.0064  -87.74               0.0060
Galerkin         -0.0414  2.0306  -89.47               0.0042

Table 2. Second Robot Arm for Trajectory y = 0.5 x+ 0.5
0[less than or equal to]x[less than or equal to]1

Method           [r.sub.3]  [r.sub.4]  [theta]0([degrees])  emax

Precision-Point  -0.3934    1.8573     -74.90               0.0681
Subdomain        -0.1794    1.9583     -83.66               0.0163
Galerkin         -0.0891    2.1232     -87.82               0.0036

Table 3. Modules For Trajectory y = 0.5 x+ 0.5
0[less than or equal to]x[less than or equal to]1

Method           r1       r2      [theta]0([degrees])  b       emax

Precision-Point  -0.7203  3.3156  -88.18               2.8405  0.0035
                 -0.8205  2.0423  -24.75               1.3698  0.0653
Subdomain        -0.7082  3.3087  -88.49               2.8290  0.0026
                 -0.8216  2.0405  -25.50               1.3870  0.0514
Galerkin         -0.7529  3.2559  -85.94               2.8022  0.0006
                 -0.5659  1.9826  -22.50               1.0708  0.2228
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Author:Mutlu, Huseyin; Ozkul, Iskender
Publication:Turkish Journal of Engineering (TUJE)
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Date:May 1, 2018
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