ALGORITHMIC SUPPORT OF PROBLEMS OF ELECTRONIC KINEMATICS

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1 VOL. 3 NO. 5 MARCH 8 ISSN Asian Research Publishing Networ (ARPN. All rights reserved. ALGORITHMIC SUPPORT OF PROBLEMS OF ELECTRONIC KINEMATICS Inga Niolaevna Bulatniova and Natala Niolaevna Gershunina Department of Applied Mathematics Kuban State Technological Universit Mosovsaa Krasnodar Russia inras@andex.ru ABSTRACT This article discusses digital simulation and possible implementation of inematic mechanisms and units on the basis of simple computational means (microprocessors. The proposed digital simulation is supported b geometrical plotting methods developed in descriptive geometr and successfull applied in engineering graphics. These methods aided b compass ruler protractor and triangular ruler mae it possible to plot an traector for instance inematic one. Technological basis of this simulation is a set of rapid algorithms based on integer arithmetic carried out b RISC microprocessors. Possibilities of this approach to designing of algorithmic support are exemplified b evolvent of circle. The obtained set of algorithms does not contain operations of multiplication and division; nevertheless quite complex traectories are obtained without trigonometric computations. The efficienc of the proposed procedure of algorithm designing is based on their fast operation low cost of designing and simplicit of software implementation using microprocessors. Kewords: mechatronics integer algorithms microprocessors machiner designs.. INTRODUCTION Academician Frolov a prominent engineer introduced the term "electronic inematics". Its essence is comprised of provision of preset snchronous motions of woring tools not b means of conventional mechanic inematic tools (oints rods cranshafts cams and so on but b power drives (usuall electric snchronousl controlled b computers []. This simplifies designing and manufacturing of complex mechanic sstems reducing it to snthesis of existing algorithms and their implementation b computers. Taing into consideration widespread application of computers we recommend to use microprocessors. In this regard an urgent issue is efficient algorithmic support of microprocessors which are used for solution of electron inematic problems. Herewith it is required to account for microprocessor architecture which differs significantl from regular PC architecture. In our wors we proved the efficienc of microprocessor information technologies based on integer arithmetic [ 3 4]. Such integer algorithms are ver fast and accurate maing them efficient upon implementation b microprocessors for instance of RISC architecture. Since all units of mechanisms are distributed and move in space then it is important to proceed from static algebraic computational models to dnamic geometric models. Moreover it should be taen into account that man algebraic problems are solved better b geometrical methods. Such approach would provide sharp increase in processing rate of control information not onl due to internal cloc rate (in mhz but mainl due to algorithm operation speed. Development of such algorithms is aided b geometrical plotting methods developed in descriptive geometr and widel applied in engineering graphics. Thus on the basis of integer algorithms of digital linear interpolation (ruler and circular interpolation (compass algorithms of measurement of angular motions (protractor determination of perpendicular to straight line (triangular ruler it is possible proceed to integer arithmetic and hence to relieve sharpl requirements of architecture of microprocessors which carr out such integer algorithms. The most important requirement to relieve requirements to microprocessors is elimination of multiplication and division in their sstem of commands. This increases their operation speed and data processing accurac in solving problems of electron inematics.. MAIN COMPUTATIONAL METHODS ON THE BASIS OF INTEGER ARITHMETIC Tools of integer data processing for electronic inematic problems are listed below.. Difference iterative methods Difference iterative methods are non-analtical computational methods eliminating multiplication and division [] required for solution of electronic inematic problems. For instance the function x can be z ( computed using such algorithm (for x : if Zi ; qi signzi if Zi ; if Zi stop ; ( 833

2 VOL. 3 NO. 5 MARCH 8 ISSN Asian Research Publishing Networ (ARPN. All rights reserved. i Z x Zi Zi qi Zn ; Y i ( x x Yi Yi qi x Yn. Here and below iis the number of iteration i...n ; and. nis the binar digits of operands x. Vector pseudo-rotation methods These methods were proposed in [345]. The "digit-b-digit" methods can be exemplified as follows (at x : if Yi ; i signyi ( if Yi ; ( i Y Yi Yi i Xi Yn ; ( i X x Xi Xi i Yi Xn K x ; i i i arctg ( i Q n arctg x where K is the coefficient: n m K. m Algorithm for rotation of coordinates b angle is available similar to algorithm (..3 Digital linear interpolation There are numerous algorithms of digital interpolation of straight lines. Some of them for the sae of simplicit permit absolute error equaling to one step of interpolation (. We developed the so-called algorithm of digital linear interpolation with twice as lower error and double operation speed in comparison with non-optimized algorithms [6]. This provides more accurate geometrical simulation of inematic sstems upon their designing..4 Digital circular interpolation The situation with circular interpolation is the same as with linear one: there are numerous nonoptimized algorithms with absolute error up to. We substantiated and proposed optimum algorithm of digital circular interpolation with absolute error equaling to. Operation speed is also doubled [6]. This is achieved b that the algorithm estimates deviation (evaluation function in two adacent alternative interpolation nodes and selects the node with minimum absolute value of evaluation function. The evaluation function is adusted b means of integer expressions such as: Fi xi uponmovement along x axis Fi Fi xi i uponmovement along diagonal Fi i uponmovement along axis; where x i and i are the coordinates of current node of circle interpolation. Algorithm of digital circular interpolation is readil modified for angular rotations in the range of 36 as well as for reverse motion along arc of circle (clocwise or counterclocwise..5 Angular rotation of radius vector Using integer algorithm of measurement of actual rotation of radius vector digital circular interpolation determines angles of rotation of moving parts. This is required for instance upon simulation of complex inematic curves (ccloid epi- and hpoccloid. This algorithm is based on consideration for interrelation between the value of angular sector and its area. The latter is calculated b increments at each step of digital interpolation using simple equation (such as area of rectangular triangle with unit base interpolation step Δ [7].5 i upon movement along x axis Si.5( xi i upon movement along diagonal.5 i upon movement along axis..6 Alignment of perpendicular to straight line Equation of perpendicular to this straight line in predefined point ( x (or passing via predefined point is based on nown expressions of analtical geometr. Their angular coefficients are correlated as and (where is the angular coefficient of preset (3 (4 straight line and perpendicular offset b is calculated as follows: x b (5 Since simulation of perpendicular as straight line requires onl for expression of its evaluation function then X and Y are important (parameters of digital linear interpolation which are correlated with the angular coefficient as follows: X Y. (6 834

3 VOL. 3 NO. 5 MARCH 8 ISSN Asian Research Publishing Networ (ARPN. All rights reserved. This correlation is used for instance upon determination of parameters of digital interpolation of tangent to circle..7 Dnamic multiplication of variables This procedure maes it possible to multipl variables in integer format. This multiplication is considered as dnamic because each variable varies in time (step-b-step but with condition that in each step it varies discretel (integer arithmetic b or does not var (zero increment. Such variables can be for instance coordinates of successive node of digital interpolation of this or that line (straight line circle and so on. This procedure is carried out according to the recursive equation: i i xi ( t i xi ( t i i i where i is the step number i= 3 ; x i (t x i (t are the current values of the first and the second variables; i i are the increments of the first and the second variables in the i-th step equaling to the elements of set { }. (7 At the initial moment ( i the variable x ( t x ( t is a parameter (constant of integer algorithm..8 Concept of geometric simulation of inematic sstems Geometric simulation is based on the thesis: man algebraic problems are solved simpler b geometrical method. Well-nown plotting methods of descriptive geometr are successfull applied in engineering graphics and mae it possible to determine complex configurations and curves. Four maor graphic tools are used for this purpose: ruler compass protractor triangular ruler. All these tools can be substituted b integer algorithms of digital linear circular interpolation algorithm of measurement of angles and algorithm of determination of parameters of digital interpolation of perpendicular as a straight line. Some other algorithms are also used including logical algorithms. Advantage of integer algorithms is based on their operation speed due to implementation b efficient computing means: microprocessors for instance of RISC architecture. Another advantage of such algorithms is their improved accurac due to eliminated errors. This is achieved b non-use of floating point operations as well as multiplication and division. Using these algorithms it is possible to carr out traectories of separate elements of inematic sstems and essentiall to control positions of required movable elements (for instance manipulator gripper. This is one of electronic inematic problems declared at the beginning of this article []. 3. CASE STUDY OF EVOLVENT DIGITAL INTERPOLATION In the case stud of this curve - evolvent of circle - we will show how using integer algorithms and model properties of this curve to achieve motion of woring element (for instance cutting tool along the evolvent traector. This is based on algorithms of digital circular interpolation [6] and generalized algorithm of digital interpolation of arbitrar curves [8]. The essential concept of the algorithm is illustrated in Figure-. The control of evolvent steps is comprised of determination of the fact that the current point C of evolvent reaches the tangential AC after one or several steps of digital interpolation of evolvent (arc BC. This is determined b the sign of evaluation function for the tangential AC passing via the point A (x i i with angular coefficient x i i. Evaluation function of the tangential is as follows: F( x xb whereb i xi (8 Figure-. Algorithm of evolvent interpolation. In the points above the tangential it is positive and below the tangential it is negative. On the tangential it is zero. This zero crossing denotes the end of steps of digital interpolation after each step of digital interpolation of circle. Them the next step of circular interpolation is performed and the process is repeated. Herewith this is accompanied b variation of the coordinates of point A (not more than b ± the angular coefficient and respectivel the value of evaluation function F(x of new tangential AC. 835

4 VOL. 3 NO. 5 MARCH 8 ISSN Asian Research Publishing Networ (ARPN. All rights reserved. Digital interpolation of evolvent is performed according to generalized algorithm [8] with varing X and Y equaling to xi and i respectivel. Finall the evaluation function of tangential for the first quadrant is as follows: F( x i xi x ( xi i. (9 The sum in bracets Equation (9 equals approximatel to a where a is the circle radius. It exactl equals to a for the nodes of circular interpolations on the circle and for remaining points it is: ( xi i fi a ( where fi is the excess of ( xi i over a due to that the i-th node is outside the circle. In this case f i if it is inside then f i. In the proposed algorithm fi equaling to zero at the beginning for other nodes is calculated according to iterative integer expression and subtracted from the evaluation function F ( x. The algorithm of adustment is as follows. If the step prior to the i-th one was coordinate ( x i x i or i i then fi will equal to ( x i or ( i and if it x i x i and i i fi ( xi i was diagonal ( then. These equations follow from Eq. (.3. Let us temporaril omit fi from the considerations. Then we have in simplified form: F( x i xi x a. ( Zero of Eq. ( follows from equation of circle with the radius a for all points of the tangential AC. In order to avoid misunderstanding in further conclusions let us denote coordinates of evolvent as w (abscissa and u (ordinate for all its current points. Then the coordinates of the next point (interpolation node will be as follows: w w dw u u du ( where dw and du are the possible increments of evolvent coordinates upon interpolation equaling to or at the -th step of interpolation according to generalized algorithm for arbitrar curves. Let us determine the increment F ( w u as difference: F ( w u F( w u F w u du x ( i i dw (3 Multiplication operations in Equation (3 b or are substituted with additions (if b.then the evaluation function is adusted for the last point (node on evolvent according to the expression similar to Equation (3 using increment: F ( x ui d wi dx (4 where d и dx are the increments of coordinates of current node of circular interpolation ( or ±. Equation (4 also does not contain operations of multiplication. Initial value of evaluation function F( x in the point B (origin of exponent is zero. Therefore the algorithm of digital interpolation of evolvent is separated into two parts: athe driving algorithm is that of circular interpolation (exactl one step is performed and the second part is activated; bthe driven algorithm of digital interpolation (one or several steps are performed until the evaluation function of tangential AC F ( xi i varies its sign. Herewith parameters for the second part of algorithm are provided b the first part in the form of successive point ( x i i of circular interpolation. After change of sign of evaluation function of tangential as a consequence of action of the second part the action of the first part is transferred. 4. RESULTS AND DISCUSSIONS Finall completel integer algorithm (without multiplication and division is obtained oriented at rapid implementation b microprocessors for instance of RISC architecture. The obtained algorithm of digital interpolation of evolvent was approbated on PC using Delphi b emulation of command sstem of modern RISC microprocessors. Quantitative results (maximum absolute errors of digital interpolation of evolvent are summarized in Table-. 836

5 VOL. 3 NO. 5 MARCH 8 ISSN Asian Research Publishing Networ (ARPN. All rights reserved. Table-. Error of digital interpolation of evolvent (in interpolation steps Δ =. mm. a= a=5 a=5 a= a= The developed integer algorithm of digital interpolation of evolvent is recommended for application in numerical control machines automated sstems of designing and simulation of complex inematic sstems. 5. CONCLUSIONS Conversion to integer algorithmization of information technologies of data processing for solution of problems of electronic inemics is ver promising. On the basis of simple technical aids (microprocessors and using rapid integer algorithms it permits to implement not onl simulation but phsical replacement of elements of complex inematic sstems [7 ]. REFERENCES [] K.V. Frolov and V.I. Babitsii Mehaniaiisusstvoonstruirovania v epohu EVM. [Engineering and designing in PC age] Izobretatel` iratsionalizator. : 6-7. [] A.M. Oransii Apparatnemetod v tsifrovoivchislitel`noitehnie [Hardware in digital computing technique]. (BGU Mins. [3] I.N. Bulatniova and V.I. Clucho.. Informatsionnetehnologii s ispol`zovaniemtselochislennoiarifmetii [Information technologies using integer arithmetic]. Geoengineering. NIPI InzhGeo.: microprocessors of local automation (functional transformation and data processing]: Doctoral thesis specialt Kiev. [7] Morris Coordinate transformations and programming for small revolute - coordinate robots. Microprocessors and Microsstems. 9(6: [8] I.N. Bulatniova.. Tsifroveinterpolatorrivolineinhtraetorii. [Digital interpolators of curved paths ] Izv.vuzov. Sev.-Kavazsii region. Ser. Tehnichesienaui. : 6-9. [9] N.S. Anishin. 3. Modelirovanieinematiiplosihmehanizmovnabazet selochislennhalgoritmov. [Simulation of inematics of plain mechanisms on the basis of integer algorithms] Izv.vuzov. Sev.-Kav. region. Ser. Tehnichesienaui. 4: -5. [] K.P. Corwell et al Computers complexit and controvers. IEEE Computer. 8(4: 8-9. [] E.Vitrant Canudas-De-Vit C. D. Georges 4. Alamir Remote stabilization via time - varing communication networ delas. IEEE Conf. in Control Applications Taiwan. Sept -4. [] R.E. Carlle 985. Risc business. Datamation. 8(4: 8-9. [4] V.D. Baiov and V.B. Smolov Spetsializirovanneprotsessor: iteratsionnealgoritmistrutur [Specialized processors: iterative algorithms and structures]. (Radio isvaz` Moscow. [5] Adir E. Almog L. Fournier E. Marcus M. Rimon M. Vinov A. Ziv. 4. Geness-Pro: Innovations in Test. Program Generation for Functional Processor Vertification. Design and Test. [6] N.S. Anishin. 99. Metodologiaproetirovaniaalgoritmichesogoobesp echeniamiroprotsessornhustroistvloal`noiavtom atii (funtsional`noepreobrazovanieiobrabotainformatsii [Designing procedures of algorithmic support of 837