Algebras of Stepping Motor Programs

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1 Applied Mathematical Scieces, Vol. 5, 2011, o. 4, Algebras of Steppig Motor Programs Aa Gorbeko Departmet of Mathematics ad Mechaics Ural State Uiversity Ekateriburg, Russia Alexader Lutov Departmet of Mathematics ad Mechaics Ural State Uiversity Ekateriburg, Russia Maxim Morev Departmet of Mathematics ad Mechaics Ural State Uiversity Ekateriburg, Russia Vladimir Popov Departmet of Mathematics ad Mechaics Ural State Uiversity Ekateriburg, Russia Abstract I this paper we cosider algebraic models for steppig motors. We itroduce the otio of the status of a fiite groupoid ad cosider this otio for groupoids of steppig motor programs. Mathematics Subject Classificatio: 08A99 Keywords: groupoids, steppig motor programs

2 1680 A. Gorbeko, A. Lutov, M. Morev ad V. Popov 1 Itroductio Steppig motors are widely used as digital actuators i automatio equipmets such as robots, office automatio istrumets ad productive facilities. May studies o the dyamic characteristics of steppig motors have bee carried out. For example, the studies cocerig the developmet of the mathematical model of magetic steppig motors [1], the steppig motor failure model [2], the aalysis of the vibratio pheomeo of steppig motors [], ad the dyamic characteristics of steppig motor system i cosideratio of elasticity of rotatig shaft [4] have bee reported. The, the microstep drive of steppig motors were developed for steppig motors to improve the positio resolutio ad solve the problems of overshoot ad resoace, [5], [6]. O the other had, desk robots ad semicoductor wafer trasfer robots usig steppig motors are i practical use [7], [8]. Furthermore, the studies about the dyamic characteristics of two-lik robot arm drive by steppig motors ad the trajectory trackig cotrol of three-lik semicoductor wafer trasfer robot arm drive by steppig motors have bee reported [9], [10]. I this paper we cosider algebraic models for steppig motors. 2 Algebraic models Followig a commo approach i reasoig about actios, dyamic systems are modeled i terms of state evolutios caused by actios. A state is a complete descriptio of a situatio the system ca be i. Actios cause state trasitios, makig the system evolve from the curret state to the ext oe. I priciple we could represet the behavior of a system (i.e. all its possible evolutios) as a trasitio graph, where: Each ode represets a state, ad is labeled with the properties that characterize the state. Each arc represets a state trasitio, ad is labeled by the actio that causes the trasitio. We ca suppose that for some states of the steppig motor form the followig set of agles: S { 2 + k, k {0, 1,..., 2}}. 2 Actios ca be represeted by the followig set of agles: A { + k, k {0, 1,..., 4}}. 2

3 Algebras of steppig motor programs 1681 Correspodetly, ay program for a steppig motor ca be represeted as some sequece of this actios. For ay state ϕ 1 ad ay actio ψ a steppig motor evolves to the state ϕ 2 where ϕ 2 ϕ 1 + ψ if ϕ 1 + ψ [ 2, 2 ]; ϕ 2 2 ϕ 2 2 if ϕ 1 + ψ > 2 ; if ϕ 1 + ψ < 2. Usig this rule we ca cosider a partial groupoid P A, o the set A such that x y z if ad oly if actio y cause the state trasitio, makig the steppig motor evolve from state x to state z; x y udefied i all other cases. Note that i the partial groupoid P a same elemet ca have two differet iterpretatios. For example, a elemet x y i term z (x y) at first cosidered as a state of a steppig motor ad after this cosidered as a actio which cause the state trasitio, makig the steppig motor evolve from state z to state z (x y). Clearly, if y > 0 the x y (x (y 2 )) 2. Correspodetly, if y < 0 the x y (x (y + 2 )) ( 2 ). Therefore, ay term x y, x S, y A, ca be represeted as some term which elemets oly from S. So, it is sufficiet to cosider oly elemets from S. Cosider a subgroupoid G of partial groupoid P geerated by S. Theorem 2.1 G S. Proof. By defiitio, ay elemet of G ca be cosidered as a state of the steppig motor. Sice the steppig motor have the fiite set of states, G have a fiite umber of elemets. I particular, G S. Ay elemet of G ca be cosidered as some state of the steppig motor. Ay term ca be cosidered as a sequece of actios which cause state trasitios. Therefore, we ca cosider terms from G as programs for the steppig motor. We say that G is a groupoid of a steppig motor programs.

4 1682 A. Gorbeko, A. Lutov, M. Morev ad V. Popov From algebraic poit of view groupoid G has uatural defiitio. So, it is of iterest to fid a presetatio of G by a set of geerators ad a set of relatios. For example, we ca cosider G as a groupoid give by the set of geerators S ad the followig set of relatios: R {τ ϕ ψ τ ϕ + ψ, ϕ + ψ [ 2, ], ϕ S, ψ S} 2 { 2 ϕ ψ ϕ + ψ >, ϕ S, ψ S} 2 { 2 ϕ ψ ϕ + ψ <, ϕ S, ψ S}. 2 Clearly, i this case S 2 + 1, R (2 + 1) 2. Directly from defiitios of the steppig motor ad operatio we obtai followig relatios: 2 ( 2 ) 0, ((( 2 ) ( 2 ))... ) ( 2 ) 2 + ( r), r {1, 2,..., }, 2 (( 2 2 )... ) }{{ 2} 2 ( r), r {1, 2,..., }. 2 Therefore, we ca cosider elemets from the set S\{ 2, 2 } as otatios of correspodet terms. I this case we ca suppose that G 2, 2 R. Robot joit agles ad steppig motor agles Note that assumptio that a steppig motor allow us to obtai some fixed rotatioal agle ca be used oly at some level of abstractio. Usually each motor has some omial voltage ad ca provide some amout of torque cotiuously with a matchig curret cosumptio ad a maximum permissible speed. So,

Algebras of steppig motor programs 168 Figure 1: A mobile robot with robotic arm Figure 2: Robotic arm with a cushioig sprig to achieve required levels of abstractio we eed to create a correspodece

Oe drawback of such approach is that ay error i the measuremet of correspodece betwee a amout of torque ad real rotatioal agle accumulates over time, causig the robots positio to become less ad less

5 Algebras of steppig motor programs 168 Figure 1: A mobile robot with robotic arm Figure 2: Robotic arm with a cushioig sprig to achieve required levels of abstractio we eed to create a correspodece betwee a amout of the twistig force that teds to cause rotatio ad real rotatioal agle. You ca move the robot alog a pre-determied path by specifyig amout of torque. Oe drawback of such approach is that ay error i the measuremet of correspodece betwee a amout of torque ad real rotatioal agle accumulates over time, causig the robots positio to become less ad less accurate. Moreover, for practical problems we eed to cosider a robot joit agle istead of a steppig motor agle. At first, we eed to calibrate our robots motors. After this we eed to calibrate our robots joits to esure that it ca drive correct. For differet robots a same steppig motor ca demostrate differet correspodeces betwee a amout of the twistig force ad real rotatioal agle. Similarly, for differet robots a same joit ca demostrate differet correspodeces betwee a amout of the twistig force ad real rotatioal agle (see figures 1 ad

vertical collisio with the floor Figure 5: A horizotal collisio

6 1684 A. Gorbeko, A. Lutov, M. Morev ad V. Popov Figure : Robotic arm without cushioig sprigs Figure 4: A vertical collisio with the floor Figure 5: A horizotal collisio with aother object 2). Some robotic joits have differet cushioig sprigs (see figures 2 ad ).

Algebras of steppig motor programs 1685 Figure 6: Iitial image of the robotic fork Robots ca maipulate with objects with differet weights ad geometrical properties (see figures ad 4).

7 Algebras of steppig motor programs 1685 Figure 6: Iitial image of the robotic fork Robots ca maipulate with objects with differet weights ad geometrical properties (see figures ad 4). Durig maipulatios robot may come ito some vertical or horizotal collisios with other objects or with the floor (see figures 4 ad 5). So, we eed take ito cosideratio basic properties of robot, effects of cushioig sprigs, properties of objects of maipulatios, differet collisios, attritio wear, egie wear ad so o. Eve startup calibratio of robot joit agles require sigificat efforts (see [11], [12]). The performace of a robot depeds heavily o the performace of its joit servo. I order to desig a better robot joit servo system, the performace of the system eeds to be moitored i a real-time eviromet such that the servo parameters ca be adjusted to achieve the optimum performace. Therefore, a robot joit servo developmet system with real-time moitorig capability is eeded. As to the desig of such system, sice computatioal requiremet is imperative to the performace of a robot system such that the choice of the microprocessor for the digital cotrol of a robot servo system is a crucial factor. For example, the detailed desig of a robot servo developmet system based o a microprocessor with architectures optimized for high speed umeric computatios preseted i [1]. Based o this system, the servo parameters for the desiged servo system ca be optimized by moitorig the servo performace. I our laboratory we develop a visual system for real-time calibratio of joits. This system uses eural etworks. It creates some test sequeces for each joit of the robot, rus this sequeces, ad captures pictures of test sequeces. It aalyzes images of this test sequeces for each step of a joit agle chage. If it is possible our system geerates ew correspodece. Otherwise it creates some ew test sequeces ad so o. Periodically we use a global test of joits. I this case we obtai images of etire sequeces of joi states (approximately 2000 of images per sequece). For example, see figures 6 ad 7. First eural etwork used for joit detectio. Secod eural etwork eeded to estimate correspodeces betwee

1686 A. Gorbeko, A. Lutov, M. Morev ad V. Popov Figure 7: Iitial image of third joit of robotic arm Figure 8: Detectio of the robotic fork a amout of the twistig force ad real rotatioal agle.

But eve this first eural etwork require approximately 1 Gb per sequece ad approximately 800 6000 hours if oly oe processor used (Widows XP, Petium IV 2.40GHz).

8 1686 A. Gorbeko, A. Lutov, M. Morev ad V. Popov Figure 7: Iitial image of third joit of robotic arm Figure 8: Detectio of the robotic fork a amout of the twistig force ad real rotatioal agle. First eural etwork allow us to obtai oly relatively rough detectio (see figures 8 ad 9). But eve this first eural etwork require approximately 1 Gb per sequece ad approximately hours if oly oe processor used (Widows XP, Petium IV 2.40GHz). We use heterogeeous cluster based o three clusters (Cluster USU, Liux, 8 calculatio odes, Itel Petium IV 2.40GHz processors; umt, Liux, 256 calculatio odes, Xeo.00GHz processors; um64, Liux, 124 calculatio odes, AMD Optero 2.6GHz bi-processors). Eve i this case first eural etwork require approximately approximately 4 0 hours. To miimize such periodic activity it is of iterest to cosider geeratig sets with relatively small umber of elemets. From other had, the legth of commad sequeces is a crucial factor for a robotic system performace. So, we eed to cosider geeratig sets such that every elemet is expressible as a product of relatively small umber of elemets.

9 Algebras of steppig motor programs 1687 Figure 9: Detectio of third joit of robotic arm 4 The status of a fiite groupoid The otio of the status of a fiite semigroup was itroduced i [15]. Similarly, we ca itroduce the otio of the status of a fiite groupoid. For each geeratig set A of a fiite groupoid T the iteger (A) is defied as the least for which every elemet of T is expressible as a product of at most elemets of A. The status Stat(T ) of T is defied as the least value of A (A). Sice G 2 + 1, it is clear that Stat(G) It is easy to see that for geeratig set {, } every elemet of G is expressible as a product of 2 2 at most elemets of this set. Therefore, Stat(G) 2. Cosider a set It is easy to see that M { 2 + 2, 2, 2, 2 }. 2 ((( 2 + ) ( 2 2 ))... ) ( 2 ) ( r), ((( 2 + ) 2 2 )... ) }{{ 2} ( + r), ((( 2 ) ( 2 ))... ) ( 2 ) 2 r,

10 1688 A. Gorbeko, A. Lutov, M. Morev ad V. Popov ((( 2 2 ((( 2 2 ) 2 )... ) }{{ 2} ( + r), ) ( 2 ))... ) ( 2 ) 2 2 ( + r), (( 2 2 )... ) }{{ 2} 2 r, r {1, 2,..., }. Note that Therefore, S {0, ( r), ( + r), 2 r, ( + r), 2 2 ( + r), 2 r r {1, 2,..., }}. ( ) ( ) Stat(G) M Similarly, we ca cosider a set Clearly, M k {a p, b p, 2, 2 0 p k 1}, a p b p 2 2 (1 + 2p), (1 + 2p). M k 2k + 2, (M k ) + 1.

11 Algebras of steppig motor programs 1689 Theorem 4.1 Stat(G) (2k + 2) ( Now let 1+2k + 1 ). Suppose that The If the N k {c p, d p, 2, 2 1 p k}, 2p c p d p p, p. r (2p 1). 2 r ((c p ( 2 ))... ) ( 2 ), 2p 1+2k 2 r ((d p ( 2 ))... ) ( 2 ). 2p 1+2k (2p 1) r 2(p 1) 2 r ((c p 1 ( 2 ))... ) ( 2 ), r 2(p 1) 1+2k times 2 r ((d p 1 ( 2 ))... ) ( 2 ). r 2(p 1) 1+2k times Suppose that a() + b where b <. The c k k (a + 1)k a() + b 2ak + 2k 2 a + 2ak + b a + 2ak + b + 2k a b a + 2ak + b 2k + a + b a + 2ak + b 2 + ( 2k + a + b). 2

12 1690 A. Gorbeko, A. Lutov, M. Morev ad V. Popov Therefore, d k k (a + 1)k a() + b 2ak + 2k 2 a + 2ak + b a + 2ak + b + 2k a b 2 a + 2ak + b 2 2k + a + b 2 a + 2ak + b 2 ( 2k + a + b). 2 Sice b <, it is easy to see that Therefore, Notice that 2 ((c k ( 2 ))... ) ( 2 ), 2k+a+b times 2 ((d k ( 2 ))... ) ( 2 ). 2k+a+b times 2k + a + b a (N k ) + 1. c p c 1 c p d 1 d p d 1 d p c 1 p. (p + 1) c p+1, p + (p 1) c p 1, p + (p + 1) d p+1, p (p 1) d p 1.

13 Algebras of steppig motor programs 1691 Theorem 4.2 Stat(G) (2 ( k + 4) 1+2k + 2 ). ACKNOWLEDGEMENTS. The work partially supported by Grat of Presidet of the Russia Federatio MD ad Aalytical Departmetal Program Developig the scietific potetial of high school 2.1.1/1775. Refereces [1] D.J. Robiso ad C.K. Taft. A Dyamic Aalysis of Magetic Steppig Motors. IEEE Trasactios o Idustrial Electroics ad Cotrol Istrumetatio IECI-16 (2) (1969) [2] C.K. Taft ad R.G. Gauthier. Steppig Motor Failure Model. IEEE Trasactios o Idustrial Electroics ad Cotrol Istrumetatio IECI-22 () (1975) [] T. Egami, K. Shimizu, ad K. Abe. Aalysis of Vibratio Pheomeo of Steppig Motors. IEEJ Trasactios o Idustry Applicatios 12-D (2) (200) [4] H. Kojima ad S. Ikeda. Aalysis of Dyamic Characteristics of Steppig Motor System i Cosideratio of Elasticity of Rotatig Shaft. The Trasactios of the Istitute of Electrical Egieers of Japa 104-B (4) (1987) [5] M.F. Rahma, A.N. Poo, ad C.S.Chag. Approaches to Desig of Miisteppig Step Motor Cotrollers ad Their Accuracy Cosideratios. IEEE Trasactios o Idustrial Electroics IE-2 () (1985) [6] M.F. Rahma ad A.N. Poo. A Applicatio Orieted Test Procedure for Desigig Microsteppig Step Motor Cotrollers. IEEE Trasactios o Idustrial Electroics IE-5 (4) (1988) [7] idustrial/ jp/ products/ desktop robot/ idex.html [8] products/ advaced/ robot/ robot 0.html [9] H. Kojima, S. Ikeda, ad T. Tabata. Dyamic Characteristics of Horizotal Two-Lik Robot Drive by Step Motors. Trasactios of the Society of Istrumet ad Cotrol Egieers 26 (4) (1990) [10] H. Kojima, H. Chigira, Y. Kuwao, K. Abe, ad N. Kikuchi. Trajectory Trackig Cotrol ad Error Aalysis of Three-Lik Semicoductor Wafer Trasfer Robot Arm Drive by Steppig Motors. Trasactios of

14 1692 A. Gorbeko, A. Lutov, M. Morev ad V. Popov the Japa Society of Mechaical Egieers C-72 (719) (2006) [11] US Patet Robot joit agle detectig system. [12] S. Guarsso, M. Norrlof, E. Rahic, M. Ozbek. Iterative learig cotrol of a flexible robot arm usig accelerometers. I Proceedigs of the 2004 IEEE Iteratioal Coferece o Cotrol Applicatios, 2 (2004) [1] C.-H. Wu, P.N. Koch. Desig of robot joit servo developmet system. I Proceedigs of 24th IEEE Coferece o Decisio ad Cotrol, 24 (1985) [14] parallel.ura.ru/ mvc ow/ hardware/ supercomp.htm [15] A. Cherubii, J. Howie, B. Piochi. Rak ad status i semigroup theory. Commuicatios i Algebra 2 (2004) Received: November, 2010

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