Synthesis of an adaptive gripper

Applied Mathematical Modelling 38 (2014) 3175–3181

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Synthesis of an adaptive gripper V. Galabov a, Ya. Stoyanova b,⇑, G. Slavov c a b c

Department of Theory of Mechanisms and Machines, Technical University of Sofia, 8, Kl. Ohridski Blvd., 1797 Sofia, Bulgaria Department of Theory of Mechanisms and Machines, Faculty of Industrial Technology, Technical University of Sofia, 8, Kl. Ohridski Blvd., 1797 Sofia, Bulgaria Department of Mechanics, Todor Kableshkov University of Transport, 158, Geo Milev Str., Sofia 1113, Bulgaria

a r t i c l e

i n f o

Article history: Received 2 November 2012 Received in revised form 20 June 2013 Accepted 29 November 2013 Available online 19 December 2013 Keywords: Adaptive gripper Structure Synthesis Analysis

a b s t r a c t The problem about synthesis of an original adaptive gripper with simplified structure is formulated and solved. The synthesized gripper is mainly oriented to automatic manipulation devices, which are designed for carrying, and assembly work-pieces under production of appliances and electrical-industrial articles. A developed example illustrates the application of the compound mathematical models for gripper synthesis and results evaluation. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Grasping devices interact with the work-piece and thus facilitating interaction between robotic systems and complexes and their working environment [1]. The grasping can be done mechanically (force, geometric, mixed), by keeping (vacuum, magnetic, electrostatic), adhesion (using the chemical element), and by means of grasping penetration into the surface of the work-piece (with needle and hook etc.) [2]. The information about grippers with different functions ranges albums [3], monographs [2,4–6], and papers devoted to structural–functional problems of the grippers [7,8], different approaches in their analysis and synthesis [9–14]. Gripper mechanisms as end-effectors of different technical means (including industrial robots and manipulators) [15] convert input motion and force into necessary output motion and grasping force. Therefore some authors call them converting mechanisms [7,9]. The known mechanical grippers with two symmetric situated slider-crank mechanisms do not possess necessary adaptability at grasping of manipulation work-pieces that are inaccurately situated [4,8,16]. There are known grippers possessing adaptability, which is achieved at the expense of their complicated structure [5]. The purpose of the work is to be synthesized an original adaptive gripper with simplified structure [17,18], which is orientated to automatic manipulation devices designed for carrying and assembly work-pieces under production of appliances and electrical-industrial articles. 2. Structure and working principle of the gripper The adaptive gripper (Fig. 1) consists of carrier 1 with fixed to it pneumatic cylinder 2 and stem 3, ending with roller 4, which presses flat spring 5, that is jointly connected to grasping links 6 and 7, which are connected to the carrier ⇑ Corresponding author. Tel.: +359 2 965 37 95. E-mail address: yast@tu-sofia.bg (Y. Stoyanova). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.11.038

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Fig. 1. Constructional scheme of the gripper.

1 by bearings. Two grasping links end with fixed to them removable jaws 8 and 9 for grip on the manipulated work-pieces 10. The piston motion is transformed into rotational motion of the grasping links by the follower and a flat spring. When the piston moves forward, the grasping links begin to close and grasp the manipulated work-piece, and opposite – when the piston moves backward the grasping links start opening under the action of the elastic force of the flat spring. If the manipulated work-piece is inaccurately situated, one of the jaws reaches it. Then that jaw stops moving but due to the centrode pair between the follower and flat spring and the action of the piston force, the follower rolls about the spring and by stretching it additionally sets in additional rotation another grasping link while its jaw touches manipulated work-piece. On this way, a passive gripper adaptation toward inaccurately situated grasping work-pieces is achieved. The grasping links restore their symmetric position after realizing of the work-piece from its socket, due to the recovery of the equilibrium of the inner about gripper static forces. 3. Force and geometrical characteristics For the synthesis purpose, the gripper‘s force characteristics are transformed into geometrical using the virtual power (work) principle. By the equality of the griper force power Fr (Fig. 2) and piston force power FS = 0.25gCD2p of the leading cylinder, the necessary for the synthesis first transfer function is obtained:

S00 ¼ dS=du ¼ cS F r ;

cS ¼ 2H=F S :

ð1Þ

Here cS = const, if the working pressure p of the leading cylinder having diameter D of the piston and efficiency gC, is accepted constant. By H is denoted the length of the grasping links, which are rotated toward an initial position (Fig. 2a), that is defined by free parameter (varying) parameter u = u0 + Du to angles Du = d, Du = 0, Du = d, where d is given angle. These three angles are determined by corresponding three values of the diameter U of the cylindrical part of the grasping workpiece along which the grasping is realized. The initial position when u = u0, respectively U = UN and the grasping links are parallel, is also called nominal position. In fact, the angle Du is an angle of the opening of the grasping links. The function S = S(u) determines the position function, where S determines the piston shifting (Fig. 2a), S00 ¼ ddSu determines first transfer function, its second derivative S00 ¼ ddSu – second transfer function, and so on. For the synthesis purpose, we use the following transfer functions 2

ð2Þ

3

ð3Þ

S00 ¼ d S=du2 ¼ cS F 0r ; S000 ¼ d S=du3 ¼ cS F 00r :

The synthesis task includes three values (Fr(Du = d), Fr(Du = 0), Fr(Du = d), d > 0), of the necessary grasping force that are conformable with the mass and inertial loading of the grasping work-piece as the diameter U. The grasping force can be described by the parabolic function

F r ¼ F r ðDuÞ ¼ a0 þ a1 Du þ a2 Du2

ð4Þ

where the coefficients

a0 ¼ F r ðDu ¼ 0Þ;

ð5Þ

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Fig. 2. Kinematic scheme of the gripper with introducing 3th order kinematic invariants.

a1 ¼

a2 ¼

F r ðDu ¼ dÞ  F r ðDu ¼ dÞ ; 2d F r ðDu ¼ dÞ þ a1 d  a0 d2

;

ð6Þ

ð7Þ

are defined by the system of equations

  F r ðDu ¼ 0Þ ¼ a0    F r ðDu ¼ dÞ ¼ a0  a1 d þ a2 d2   F r ðDu ¼ dÞ ¼ a0 þ a1 d þ a2 d2 To define necessary for gripper synthesis transfer functions we substitute the grasping force (4) and its derivatives

F 00r ¼ a1 þ 2a2 Du;

ð8Þ

F 00r ¼ 2a2 ;

ð9Þ

into equalities (1)–(3). 4. Synthesis of the gripper The synthesis problem can be reduced to synthesis in infinitely close relative positions after transforming of the synthesis force conditions into geometrical. Two synthesis variants are possible. 4.1. Variant I Synthesis in definite values S00 ðDu ¼ 0Þ; S00 ðDu ¼ 0Þ and the angle c of motion transmission towards grasping links since the reaction forces in kinematics pairs directly depend on its values, respectively dimensioning and durability of the gripper. The angle u, determining the position of the grasping link, is varying parameter and that allows finding a kinematic scheme suitable for constructive development under definite conditions. It is expedient that parameter to be varied in the range (60°, 90°). If it is necessary, the angle c can be varied in the same range. The distance a and length H of the grasping links are taken accordingly nominal diameter UN of the cylindrical surface in which the grasping of the work-piece becomes. The gripper is synthesized at the nominal position (Du = 0) in the introduced coordinate system (see Fig. 2) and the following are determined consecutively:

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– The slope of the straight line on which the centre A of the rotating kinematic pair between grasping link and flat spring (line 1 in Fig. 2b) is sought:

ka ¼ tan u;

ð10Þ

– The slope of the straight line on which the centre A and the last contact point B1 of the flat spring and the roller (line 2 in Fig. 2b) is sought:

ðm ¼ j arctan km jÞ;

km ¼ tanðu þ cÞ;

ð11Þ

– The position of the relative instantaneous centre of velocity P:

xP ¼ S00 ðyP ¼ 0Þ;

ð12Þ

– The coordinates of the centre A, interception point of the straight lines 1 and 2 (Fig. 2b):

xA ¼

km xP ; y ¼ k a xA ; km  ka A

ð13Þ

– The length of the arm OA:

lOA ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2A þ y2A ;

ð14Þ

– The tangent of the angle l, that collineation axis q encloses with the straight line determining position of the right branch of the spring outside the roller (line 2 in Fig. 2b), and the slope of the collineation axis:

tan l ¼ S0 =S00 ; kq ¼

ð15Þ

km þ tan l ; 1  km tan l

ð16Þ

– The coordinates of the absolute instantaneous centre of velocity Q as interception point of the axis q and straight line 1 (Fig. 2b):

xQ ¼

kq xP ; kq  ka

yQ ¼ ka xQ ;

ð17Þ

– The coordinates of point B1 as the interception point of the straight line PA and a line passing through Q and parallel to the x-axis:

xB1 ¼ xP þ

yQ ; km

yB1 ¼ yQ ;

ð18Þ

– The length of the segment B1A:

lB1 A ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxB1  xA Þ2 þ ðyB1  yA Þ2 ;

ð19Þ

– The coordinates of the centre B of the roller which are obtained by interception of the straight line passing through B1 and perpendicular to PA, and the symmetry axis of the gripper:

a xB ¼  ; 2

yB ¼ yB1 þ

1 ð0:5a þ xB1 Þ; km

ð20Þ

– The radius of the roller:

r¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxB1  xB Þ2 þ ðyB1  yB Þ2 ;

ð21Þ

– The working length of the flat spring

l ¼ 2ðlB1 A þ r mÞ:

ð22Þ

On that way, the parameters of the gripper’s kinematic scheme are determined. 4.2. Variant II After transforming the gripper‘s force characteristics into geometrical the task can be reduced to synthesis of functiongenerating mechanism in definite values S00 ðDu ¼ 0Þ; S00 ðDu ¼ 0Þ; S000 ðDu ¼ 0Þ of the transfer functions (1)–(3), i.e. to synthesis in four infinitely closed relative positions. The task can be solved on the easiest way by the formula of Freudenstein [19] and circle of Carter-Hall [20]. They are transformed about cases in which translation is transformed into rotation [21]. In the case, utilizing of the Carter-Hall circle is not quite correct since the real mechanism is equivalent to replacing slider-crank

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mechanism only up to second derivatives. Despite all it is interesting to be specified how much the incorrectness influences over obtained results. The angle u that defines the position of the grasping link remains free (varying) parameter. In nominal position (Du = 0), what the gripper is synthesized, the following are determined consecutively: – The slope ka = tan u determined by (10); – The position of the relative instantaneous centre of velocity P determined by (12); – The directional segment,

rc ¼ 3

S002 þ S002 ; S00 þ S000

which defines the diameter dC = lPC = |rC| of the Carter-Hall circle and the abscissa

xC ¼ xP þ r C ; of the diametrically opposite point P of the circle c; – The abscissa of the centre K of the circle c:

xK ¼ 0:5ðxP þ xC Þ ðyK ¼ 0Þ; – The coordinates of the absolute instantaneous centre of velocity Q as interception point of the straight line with slope ka = tanu and the circle c:

xQ ¼

xK þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 x2K þ ð1 þ ka Þðdc =4  x2K Þ 2

1 þ ka

;

y Q ¼ k a xQ ;

– The slope of the collineation axis q  PQ:

kq ¼

yQ ; xQ  xP

– The tangent of the angle l which is determined by (15) and the slope of that line:

km ¼

kq  tan l ; ðm ¼ j arctan km jÞ; 1 þ kq tan l

– The coordinates of the centre A as interception point of the straight line with slopes ka and km:

xA ¼ – – – – –

The The The The The

km xP ; km  ka

yA ¼ ka xA ;

length lOA of the arm OA determined by (14); coordinates of the point B1 determined by (18); length lB1 A of the spring determined by (19); coordinates of the roller centre and radius determined by (20) and (21); working area l of the flat spring that is determined by (22).

On that way the parameters of the gripper’s kinematic scheme are determined. 5. Estimate of the results The estimate presumes determining of the values F~r ¼ ðDu ¼ dÞ, F~r ¼ ðDu ¼ dÞ of the generated by gripper force F~r , since the value F~r ðDu ¼ 0Þ ¼ F r ðDu ¼ 0Þ is realized a priori. Because of the character of the adaptive gripper the method of kinematicly equivalent four-bar mechanisms is applied for determining of the first transfer function, respectively the grasping force values. For this purpose positions of the relative instantaneous centre of the velocity P are defined by mathematical model which is adequate to corresponding geometrical constructions. After determining the coordinates

xA ¼ lOA cosðu þ DuÞ;

yA ¼ lOA sinðu þ DuÞ;

ð23Þ

the ordinate of the centre B is calculated using Taylor series

yB ¼ yB ðDu ¼ 0Þ þ S0 ðDu ¼ 0ÞDu þ

S00 ðDu ¼ 0ÞDu2 S000 ðDu ¼ 0ÞDu3 þ ; 2 6

ðxB ¼ 0:5aÞ;

ð24Þ

Bounded to known third order derivatives. The following slope is determined

kAB ¼

yB  yA xB  xA

ð25Þ

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as well as the length

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxB  xA Þ2 þ ðyB  yA Þ2 :

lBA ¼

ð26Þ

According the last can be defined

lB1 A ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lBA  r 2 ;

tan n ¼ r=lB1 A ;

ð27Þ ð28Þ

and the slope

km ¼

kAB  tan n : 1 þ kAB tan n

ð29Þ

The position of the relative instantaneous centre of velocity P is determined by intercepting of the x-axis and a straight line passing through A and possessing slope determined by (28):

xP ¼ xA 

yA : km

ð30Þ

According to (13) ~ S00 ¼ xP . Then by virtue of (1) the grasping force is obtained

F~r ¼ ~S0 =cS :

ð31Þ

At the end, the relative errors of the defined by (31) grasping forces of the given values Fr = (Du = d) Fr = (Du = d) can be calculated. The calculations show if the solution is final or synthesis can be continued with another value of the free parameter u. If the synthesis is done according to variant II then the value of the transmission angle has to be taken into consideration:

 

c ¼ arctan

 km  ka  : 1 þ km ka 

ð32Þ

6. Example Let the adaptive gripper be synthesized in giving values Fr(Du = 0) = 300 N, Fr(Du = - 0.1) = 200 N, and Fr(Du = 0.1) = 500 N. A force cylinder with FS = 1100 N is chosen, the length of the grasping links is H = 55 mm, and a = 34 mm. The coefficients a0 = 300, a1 = 1500, a2 = 5000 are determined by (5)–(7). Substituting these coefficients into (4), (8), and (9) leads to Fr(Du = 0) = a0 = 300 N, F 00r ðDu ¼ 0Þ ¼ a1 ¼ 1500 N and F 00r ðDu ¼ 0Þ ¼ 2a2 ¼ 10000 N. The last values are substitute into (1)–(3). Then S00 ðDu ¼ 0Þ ¼ 30 mm, S00 ðDu ¼ 0Þ ¼ 150 mm and S000 ðDu ¼ 0Þ ¼ 1000 mm are calculated. Solution according to variant I. At chosen values u = (0.5p  0.1) rad = 84.2704° and c = 65° by equalities (10)–(23) the following are determined consequently: ka = 9.9667; km = 0.5945; xP = 30 mm; xA = 1.689 mm; yA = 16.830 mm; lOA = 16.914 mm; tan l = 0.2; kq = 0.9017; xQ = 2.489 mm; yQ = 24.806 mm; xB1 ¼ 11:728 mm; yB1 ¼ 24:806 mm; lB1 A ¼ 15:608 mm; xB = 17 mm; yB = 15.937 mm; r = 10.316 mm; l = 42.283 mm. Results estimate when Du = 0.1. By utilizing of the equalities (23)–(31) the following are determined consequently: xA = 0; yA = 16.916 mm; xB = 17 mm; yB = 18.354 mm; kAB = 0.0846; lAB = 17.061 mm; lB1 A ¼ 13:588 mm; tann = 0.7593; km = 0.5945; xP = 18.7589 mm; F~r ¼ 187:59 N. At the given force Fr = 200 N the relative error is 6.2% which is acceptable result keeping in mind the value 1.3 of the safety coefficient that usually is used to be increased necessary grasping force. Results estimate when Du = 0.1. The equalities (23)–(31) are applied. Then: xA = 3.361 mm; yA = 16.579 mm; xB = 17 mm; yB = 12.020 mm; kAB = 0.2239; lAB = 20.865 mm; lB1 A ¼ 18:136 mm; tann = 0.5689; km = 0.5945; xP = 57.536 mm; F~r ¼ 575:36 N. At the given force Fr = 500 N the relative error is 15%, which also can be considered as an acceptable result. Otherwise it is necessary the value of the free parameter u to be varied up to finding an acceptable error of the given grasping force. Solution according to variant II. By means of the mathematical model about this variant the following are determined: ka = 9.9667; xP = 30 mm; rc = 68.155 mm; xC = 38.155 mm; xK = 4.078 mm; xQ = 3.337 mm; yQ = 33.261 mm; tanl =  0.2; kq = 1.2475; km = 0.8383; m = 39,974°, xA = 2.328 mm yA = 23.198 mm; lOA = 23.315 mm; xB1 ¼ 9:676 mm, yB1 ¼ 33:261 mm; lB1 A ¼ 15:664 mm; xB = 17 mm; yB = 24.524 mm; r = 11.4 mm; l = 47.235 mm. The motion transmission angle c = 55.756° is determined by (32). Results estimate when Du = 0.1. By utilizing of the equalities (23)–(31) the following are defined consequently: xA = 0; yA = 20.868 mm; xB = 17 mm; yB = 32.621 mm; kAB = 0.6914; lAB = 20.667 mm; lB1 A ¼ 20:906 mm; tann = 0.1825; km = 1.0001; xP = 20.868 mm; Ffr ¼ 208:68 N.

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At the given force Fr = 200 N the relative error is 4.3% which is an acceptable result. Results estimate when Du = 0.1. The equalities (23)–(31) are applied. Then: xA = 4.146 mm; yA = 20.452 mm; xB = 17 mm; yB = 26.287 mm; kAB = 0.3216; lAB = 21.936 mm; lB1 A ¼ 21:602 mm; tann = 0.1766 km = 0.5282; xP = 42.866 mm. At the given force Fr = 500 N the relative error is 14%. A new solution can be sought with smaller and positive error because of the obtained solution the grasping force is vastly less than the given grasping force. 7. Conclusion The problem about synthesis of an original adaptive gripper with simplified structure is formulated and solved. The synthesized gripper is mainly oriented to automatic manipulation devices which are designed for carrying and assembly workpieces under production of appliances and electrical-industrial articles. Two mathematical models are developed. These models treat synthesis of a gripper with wide range varying grasping force accordingly in varying diameter of the cylindrical surface on which the grasp is done. For purposes of the synthesis, the gripper’s force characteristics are transformed into geometrical using the virtual power (work) principle. On that way the kinetostatic problem is reduced to synthesis by infinitely closed relative positions and taking into consideration the motion transmission angle towards grasping links. The method of kinematicly equivalent four-bar mechanisms is applied for determining of the first transfer function, respectively the grasping force values. For this purpose positions of the relative instantaneous centre of the velocity P are defined by mathematical model which is adequate to corresponding geometrical constructions. A developed example illustrates the application of the compound mathematical models for gripper synthesis and results evaluation. Acknowledgments The research is partially related to the project No BG051PO001-3.3.06-0046 ‘‘Development support of PhD students, postdoctoral researchers and young scientists in the field of virtual engineering and industrial technologies’’. The project is implemented with the financial support of the Operational Programme Human Resources Development, co-financed by the European Union through the European Social Fund. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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