Multi Degrees of Freedom Maglev System with Permanent Magnet Motion Control

Transcription

1 Multi Degrees of Freedo Maglev Syste with Peranent Magnet Motion Control Cui Tianshi A dissertation subitted to Kochi University of Technology in partial fulfillent of the requireents for the degree of Doctor of Philosophy (Special Course for International Students) Departent of Intelligent Mechanical Engineering Graduate School of Engineering Kochi University of Technology Kochi, Japan March 6

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3 Abstract As agnetic suspension syste is supporting syste without echanical contacts, there is a growing deand for agnetic suspension, especially in the field of high accuracy ulti-diensional positioning. Our otivation is to develop a peranent agnet suspension syste with novel agnetic suspension technologies. It can realize noncontact positioning and icroanipulation. It can be applied in the fields in which ultra-clean environent is needed to avoid saple containation, such as seiconductor processing, biotechnology experients, especially, when the object is oved with icro displaceents. There are any inds of aglev syste, such as superconducting aglev syste, electroagnetic aglev syste, peranent aglev syste and etc. we propose a novel active aglev syste with peranent agnets and a otion control echanis in place of electroagnets and a current control echanis. Using peranent agnets, the features of this syste are effective for saving energy, avoiding heat generation, no echanical wearing and dust free. Multi-DOF (degrees of freedo) peranent agnet suspension echaniss that anipulate the object in the vertical plane and horizontal plane have been developed. The control ethod is that the position of the suspended object is controlled by eans of adjusting the reluctance in a agnetic circuit of the suspension echanis. In this research, the concrete ethod for control reluctance is to adjust the air gap between the peranent agnet and the suspended object. As the reluctance is proportional to the air gap in the agnetic circuit, if the air gap is adjusted with the linear actuator, the reluctance in the agnetic circuit can be actively adjusted. As the first step of ulti-dof icroanipulation, the suspension echanis of the -DOF aglev syste is investigated. The principle of suspension is that the suspended object is suspended by an attractive force of a peranent agnet which is driven by an actuator and positioned above the suspended object. The direction of levitation is vertical (both of the agnet and suspension object are only oved in this direction). The suspended object is ferroagnetic aterial. The equilibriu position of the ferroagnetic body is deterined by eans of a balance between the gravity force and the attractive force of agnet. The attractive force can be adjusted with adjusting the air gap between the agnet and the ferroagnetic body. The nonlinear attractive force of agnet is linearized and calculated. The odel of the suspension syste is created and the feasibility of suspension echanis is analyzed theoretically. The controller of the suspension syste is calculated with the LQR (linear quadratic regulator) control law, which ensures the stability of close loop syste, and the syste is exained by eans of nuerical siulation and experiental exaination. i

4 Secondly, a -DOF peranent aglev syste, in which three peranent agnets are used to anipulate the object in the vertical plane, is investigated. Based on the suspension principle, the prototype has been built. In this prototype, the suspended object is an iron ball and agnets are driven by actuators to control the position of iron ball. One agnet is located on the upper to suspend the iron ball, and the other two agnets are located on the two sides of the iron ball to control its otion on the horizontal direction. left agnet N S d d suspension agnet right agnet S N f f N S iron ball Fig.. Outline of -DOF aglev syste Fig.. shows the outline of -DOF agleve syste. There are three peranent agnets in this suspension syste and each agnet has two poles, N and S, so there are soe different arrangeents which have different flux distribution. The pole arrangeent is naed in ters of the agnets polarity, for instance, if the agnet polarities which are facing the iron ball are S, S and S, the nae of pole arrangeent is called SSS. The poles order of arrangeent is the top agnet, left agnet and right agnet. Therefore, the pole arrangeent showed in Fig., is called SNN. In the -DOF aglev syste, the total nuber of pole arrangeents is 6 which are SSS, SSN, SNN, NNN, NNS, and NSS. Fro the perforance of agnet field we now that the SSS and NNN have the sae flux distribution, the SSN and NNS have the sae flux distribution, and the SNN and NSS have the sae flux distribution, so we choose the forer three arrangeents for analysis. The agnet pole arrangeent is studied with FEM (finite eleents ethod) analysis. As the result of FEM analysis, the SNN (or NSS) arrangeent is used in this syste, because this arrangeent can provide sufficient force acting on the suspended object and the equilibriu position can be assued to the center of the agnets. Next, the horizontal otion odel of the syste is created, in which the influence of the vertical attractive force is neglected. There are three situations of this syste considering the input and output of syste, two forces input syste, sae force input syste, and two agnets connecting syste. The controllability and observablity of these odels are analyzed by the linear control theory. Consequently, only the two forces input syste can be controlled and observed, so it is applied as the otion odel of the suspension syste. The otion equations are obtained based on the free body diagra of the odel. In this odel, the ii

5 agnet force is nonlinear with the air gap length, so these equations are nonlinear. First, these equations are linearized and then the state space equation can be used to express the otion odel. In ters of the LQR control law, an optial controller of suspension syste is obtained by choosing the state weighting atrix Q and input weighting atrix R. The syste is exained with this controller. However, the dynaic characteristic of step response of the siulation result is different fro that of the experiental result. This phenoenon would be caused by the vertical agnet attractive force that had been neglected when the otion odel is created. So an iproved odel is created in which the vertical agnet attractive force is considered. Based on the iproved odel, a new controller is obtained by eans of LQR control law. With the new controller, the siulation result is consistant with the experiental result and the dynaic characteristic verifies that the syste is stable. The displaceents of the iron ball in the horizontal direction were easured about reference input signals. The relationship between displaceent and reference input is linear within a range of.6() fro left to right of the equilibriu position, so the iron ball can be positioned at any position within this range in the horizontal direction. Thirdly, a 4-DOF peranent aglev syste is developed, in which the iron ball can be anipulated in the horizontal plane, i.e. the iron ball spins along the suspension axis and can be positioned near the center of echanis. In this syste, five peranent agnets are used. One is used to suspend the iron ball and the other four agnets are uniforly distributed in the horizontal plane of the suspended object to ae the iron ball rotate and ove in horizontal plane. The first step is to ae the iron ball rotate along the suspension axis, and the second step is to ae the iron ball ove in the horizontal plane. In the 4-DOF syste, the arrangeent of the polarity of the agnet is SNNNN or (NSSSS) as defined above. The suspended object is an iron ball also, which is a ferreoagnetic substance. It ust be agnetized so that there is reanent agnetis on the surface of the iron ball. When the iron ball is suspended, the influence of the strongest reanent agnetis decides the vertical direction of suspension. When a agnet located in horizontal plane approaches the iron ball, there is a agnetized point on the surface of iron ball facing the approaching agnet. And when this agnet withdraws and the next agnet approaches, the approaching agnet will attractive the agnetized point. The four agnets alternatively approaching to the iron ball ae the iron ball rotate along the suspension axis. In the rotation echanis, a laser sensor is applied to easure the rotation velocity. The four peranent agnets are driven by four linear actuators oving forward or bacward for adjusting the air gap between the peranent agnet and the iron ball to control the torque. The otions ae the iron ball rotate. The rotation odel has been created and the otion equations have been obtained. The experiental exainations are perfored. However, the result indicates that the rotation velocity is not unifor. This proble can be iproved by increasing the nuber of the agnets which are located in the horizontal plane. iii

6 With the sae construction of the rotation syste, the iron ball can also be located at any position in the horizontal plane near the central position of the syste. The principle is that through controlling the otion of four agnets which are located in the horizontal plane we can adjust the air gaps between the iron ball and agnet, the attractive forces of four agnets are changed, so the iron ball can be positioned at arbitrary position in the horizontal plane within a sall range near the original point. The otion odel is created and analyzed. The syste can be controlled and observed, therefore, the LQR control law can be used to design the controller of this syste. The syste exaination is perfored with this controller. The feasibility of the otion is confired in ters of the exaination results. The presented wor deonstrates the feasibility of anipulating the suspended object in the vertical plane and horizontal plane. Based on the outcoe of experients, it is concluded that i. The suspended object is suspended successfully and stably. ii. The suspended object can be anipulated in the vertical plane. iii. The suspended object can be rotated and anipulated in the horizontal plane. Finally, several suggestions are given to continue this research for iproving the rotate quality and the robustness of suspension syste. iv

7 Contents General introduction. Overview... Definition and Requireents for suspension echanis..... Perforance of agnetic suspension echanis.... Classification and application of agnetic suspension echanis..... Classification..... Application Disadvantage of EMS syste Goals and otivation of the investigation 3.3. Motivation Goals Control ethod Structure of thesis... 5 Principle of peranent agnetic suspension 7. Principle of peranent agnetic suspension..7.. Fundaental for of suspension echanis Perforance of peranent agnet suspension echanis..8. Theoretical analysis and exaination of feasibility 9.. Attractive force between agnet and suspended object... Theoretical analysis... Model analysis Calculation of feedbac gain Exaination for peranent agnet suspension syste Prototype of suspension syste Measureent of the peranent agnet attractive force Nuerical siulation exaination Experiental exaination....3 Discussion.. 3 Magnetic suspension echanis with the horizontal otion control Principle of -DOF aglev syste FEM analysis and pole location.6 v

8 3.3 Prototype of -DOF peranent aglev syste Model analysis Siple odel of -DOF peranent aglev syste Description of siple odel Linearization of the otion odel Six situations according the input and output of syste Two forces input syste Sae force input syste Two agnets connecting syste Three situations according the input and output of syste Accuracy odel of -DOF peranent aglev syste Description of accuracy odel Linearization of the otion odel Controllability and observability Controller design Controller design ethod Choice of weighting atrix Controller design for siple odel Controller design for accuracy odel Syste exaination Syste exaination for siple odel Siulation exaination for siple odel Step response Frequency response Step response when the white noise is added Experiental exaination for siple odel Step response Frequency response Syste exaination for accuracy odel Siulation exaination for accuracy odel Step response Frequency response Experiental exaination for accuracy odel Step response Frequency response Measureents of the otion of the iron ball and agnets vi

9 3.8 Discussion Magnetic suspension echanis with the rotation otion control Construction of echanis Suspension Principle Model analysis Attractive force exaination Suspension odel Controller design Exaination Siulation exaination Experiental exaination Rotation Rotation Principle Rotation odel and control Exaination Siulation exaination Experiental exaination Locating the iron ball in horizontal plane Principle Model analysis Controller design Exaination Discussion Conclusion and recoendation Conclusion One-DOF suspension syste Two-DOF suspension syste Four-DOF suspension syste Zero power control Recoendation.. Appendix. 3 Bibliography... 5 Relevant papers of this research... 8 Acnowledgeent. vii

10 Abstract i Contents v viii

11 Chapter Generation introduction. Overview.. Definition and Requireent for suspension echanis DEFINITION Within an orthogonal coordinate syste, every object or body possesses a linear position of three coordinates as well as an orientation of three rotations relative to another body. Generally, six independent variables are necessary and sufficient to define the spatial position fully in any given coordinate syste. These are called degrees of freedo (DOF). Suspension or bearing is restriction/confineent of soe DOF towards zero deviation, whilst, other DOF are non-restricted as good as possible. This coprised both static load carrying capacity as well as the dynaic copensation of disturbances. REQUIREMENT The ost coon custoarily used bearings are rolling eleent bearings, and ost of the support echanis just lie the rolling echanis and the slipping echanis are supported with the echanical contact force. It is inevitably to cause friction in these contact for echaniss, so appropriate lubrication is necessary. Moreover, because a echanical contact part becoes the source of dust, special consideration is needed for using in a clean environent... Perforance of agnetic suspension echanis Magnetic suspension echanis is one ind of the support echanis in which the object is supported by eans of agnetic force, and there is no echanical contact force when the agnetic suspension echanis is woring. Contactless operation leads to eliination friction, sticiness, wear, and dissipation, hence ore accurate, noise free, reliable, long life, high speed, and clean operation at low operating costs even under severe conditions lie extree teperatures, containation free environents, and Ultra High Vacuu []-[6].

12 . Classification and application of agnetic suspension echanis.. Classification of agnetic suspension syste There is diversifor in agnetic suspension echanis []-[9]. As we now that there are three inds of aterial can generate agnetic force, i.e. a peranent agnet, a usual-state conductivity electroagnet and a superconductivity electroagnet, and there are three inds of aterial is easy to let current and flux pass, i.e. a superconductor, a etal conductor, and a ferroagnetic substance. Cobined these six inds aterial, various agnetic suspension syste can be constructed. According to the generation principle of the suspension force, agnetic suspension syste is classified as () Levitation using forces of repulsion between peranent agnets. () Levitation using forces of repulsion diaagnetic aterials. (3) Levitation using superconducting agnets. (4) Levitation by forces of repulsion due to eddy currents induced in a conduction surface or body. (5) Levitation using the force which acts on a current carrying linear conductor in a agnetic field. (6) Suspension using a tuned LCR circuit and the electrostatic force of attraction between two plates. (7) Suspension using a tuned LCR circuit and the agnetic force of attraction between an electroagnet and a ferroagnetic body. (8) Suspension using controlled DC electroagnets and the force of attraction between agnetized bodies. (9) Suspension using the force of attraction between a peranent agnet and a ferroagnetic body. () Suspension using the force of attraction between an electroagnet and a ferroagnetic body... Application of agnetic suspension syste So far, any applications of agnetic suspension echanis are coercially available already, e.g. Magnetic levitation conveyance syste using forces of repulsion between peranent agnets [6]; the support syste of aglev trains using forces of repulsion due to eddy currents induced in a conduction surface or body, such as transforer RAPIDDO of Gerany and Shanghai [5]-[6]; the agnetic surfacing experient line of JR using superconducting agnets []-[4]; the agnetic bearing using the force of attraction between an electroagnet and a ferroagnetic body, which is

13 call the EMS syste [7]-[4];the agnetic bearing using the force of attraction between an electroagnet and peranent agnet [], [], and the superconductivity repulsion type thrust bearing using the force of repulsion between superconducting agnet and peranent agnet [3], etc. The ost widely used agnetic suspension syste is the electroagnet suspension (EMS) syste...3 Disadvantages of the EMS syste Because the Lorentz force is wea, the lightweight thing is needed to choose as a suspended object. Although, a large suspension force can be acquired through ae an electric wire into the shape of a coil, it becoes ipossible to disregard the generation of heat when pass the current. Moreover, since the structure becoes coplicated it is not easy to constitute a inute suspension echanis with an EMS syste. In a coon EMS syste, a bias current is required in order to support the object prudence when the current passed in the coil of an electroagnet. Therefore, the proble of generation of heat by the EMS syste cannot be generally disregarded. Moreover, because of the inductance ingredient of the coil of an electroagnet, the iron loss will be generated and there is delay of exciting force. It is difficult to control the EMS syste in a high frequency doain. On the other hands, the volue of the coil is a proble when the iniaturization of a syste is considered. In addition, it is restricted to eep away a coil fro a suspended object if the lea of flux fro a agnetic circuit is taen into consideration. This has been the obstacle of a iniaturization of an EMS syste In this research, a peranent agnet is used to replace the electroagnet, i.e. the force of attraction between an electroagnet and a ferroagnetic body is used to ae the suspension echanis. Although, it is not a solution of the proble for the iniaturization of a suspension syste, it can ae the coil eep away fro the suspended object, itigate a power loss and eliinate the influence of generated heat..3 Motivation, goals and control ethod.3. Motivation The purpose of this research is to develop a new active peranent agnetic suspension echanis in which the suspended object can be anipulated in ulti DOF. It can realize noncontact reote control and icroanipulation. It can be applied in such fields in which ultra-clean environent is needed to avoid saple containation, such as seiconductor processing, 3

14 biotechnology experients, especially, when the object is oved with icro displaceent..3. Goals At first, applied the linear control theory exaines the feasibility of the peranent agnet suspension syste. Based on the considerations of above, the research project will ai at aing the suspended object ove in ulti DOF, i.e. ove in vertical plane and in horizontal plane. Furtherore, the accuracy of the otion of the suspended object is considered..3.3 Control ethod Adjusting the reluctance in a agnetic circuit of the suspension echanis is considered to control the position of the suspended object. As a ater of fact, in this research, the concrete ethod is to adjust the air gap between the peranent agnet and the suspended object. That is to adjust the attractive force with peranent agnet. As the reluctance is proportional to the air gap in the agnetic circuit, if the air gap is adjusted with the linear actuator, the reluctance in the agnetic circuit can be actively adjusted. Based on adjusting the air gap with the actuator to copose a peranent agnet suspension echanis, the stabilization of the suspension syste has to be considered. Due to that the attractive force wors strongly if the peranent agnet is approaching to the suspended object, whereas the attractive force wors wealy if the peranent agnet is departing fro the suspended object, it is unstable in a peranent agnet and ferroagnetic. If a peranent agnet can be driven ore quicly copared with the oveent of the suspended body, the restoration force to the suspended body can be gained. In a word, in such a syste, the control object is not the size of the woring force but the oveent of the peranent agnet and the suspended object. It is the feature of this suspension syste. If it is possible that the suspension echanis which is coposed by using this ethod, generally, sees to be ipossible, the siilarity suspension echanis can be coposed by using various generation forces. For instance, the suspension echanis that uses the interolecular force and the atoic force, etc. can be expected to be coposed. A concrete target of this research is to confir the possibility of the suspension echanis of the for to adjust the attractive force by controlling the oveent of a peranent agnet with the actuator, and adjusting the air gap with the suspended body as described above. 4

15 .4 Structure of thesis This chapter that deals with the bacground and the perforance of agnetic suspension echanis and the goal and otivation of this investigation is the first part of this thesis. The succeeding chapter, chapter, treats the principle of peranent agnetic suspension syste. The possibility of the peranent agnetic suspension syste is verified theoretically and the exaination is fulfilled. It is the second party of this thesis. The third part of this thesis, chapter 3, describes a -DOF peranent agnetic suspension syste in which the suspended object is controlled to ove in the vertical plane. The arrangeent of the agnet poles is analyzed with the FEM (finite eleent ethod) in this syste. The controller of this syste is designed with LQR (linear quadratic regulator) control law. The nuerical siulation and experiental exaination is carried out. In chapter 4, a 4-DOF peranent agnet suspension syste in which the suspended object is controlled to ove not only in the vertical but the horizontal plane. It is designed based on the -DOF peranent agnetic suspension syste. As usual, this thesis will end with a listing of the ain conclusions in chapter 5, and the well eant advice and unfinished aspects that are recoended as topics for further research. 5

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17 Chapter Peranent agnetic suspension echanis In chapter, a new type agnetic suspension syste in which the peranent agnet is used was proposed. In this chapter, the principle of this suspension syste is described and the prototype of this suspension syste is constructed based on the principle. Considering the feature of peranent agnet, the suspension feasibility and the experient of suspension are exained. The attractive force of peranent agnet and the ethod to adjust the attractive force is analyzed theoretically.. Peranent agnetic suspension echanis.. Fundaental for of suspension echanis An outline of proposed suspension syste with a peranent agnet and linear actuator is shown in Fig... In this syste, the suspended object is a ferroagnetic body. It is suspended by an attractive force of a peranent agnet which is driven by an actuator and positioned above the suspended object. The direction of levitation is vertical (the agnet and the object ove only in this direction). The equilibriu position of the ferroagnetic body is deterined by eans of a balance between the gravity force and the agnet force. The attractive force can be adjusted through adjust the air gap between the agnet and the ferroagnetic body. If the actuator does not actively control the agnet s position, the levitated object will either fall or adhere to the agnet. However servo-control of the actuator can ae this syste stable. Because there is a saller attractive force for a larger air gap between the peranent agnet and object, the actuator drives the agnet upwards in response to object oveent fro its equilibriu position towards the agnet. Siilarly, the actuator drives the agnet downwards in response to object oveent away fro the agnet. In this way, the object can be stably suspended without contact. In 7

18 coparison to the electrical control ethod of electroagnetic suspension systes, this syste is a echanical control aglev syste [7]. Actuator Peranent Magnet attractive Air Gap force Suspended Object g direction of suspension direction of actuator Fig.. Outline of suspension echanis with air gap control.. Perforance of peranent agnet suspension echanis The perforance of peranent agnet suspension syste shown in Fig.. is: (i) (ii) (iii) (iv) Suspension is realized by echanical control Since no electroagnet is used, it is not necessary to tae into consideration the influence of heat generated with a coil. Since the control point and the suspension point are separated, no power loss in suspension point Besides agnetic force, electrostatic power, interolecular force and the force between atos can be used to constitute the suspension echanis In the perforance (i), since the suspension echanis is stabilized by echanical control, it can only be consisted of a peranent agnet when the equipent has a predictive position control such as a robot anipulator. When the noncontact conveyance echanis is constituted according to the principle of this echanis, the wiring for a coil can be oitted copared with the thing using an electroagnet. 8

19 In the perforance (ii), as we now, if a suspension echanis is constructed with an electroagnet, it is necessary to feed a passing current continually for supporting the gravity of the suspended object and a bias current for linearization in the coil. Thus, the heat is generated by the resistance of the coil. In this suspension echanis since no coil is used, the influence of the heat can be disregarded. In the perforance (iii), when the distance between a coil and a suspended object is long, even if the agnetic resistance of the iron core was disregarded for analysis, the influence of the agnetic resistance of the iron core and lea of flux can not be disregarded in practice, thus sufficient suspension power ay not be acquired actually. When the suspension syste is operated with peranent agnet otion control, if there is a distance between the control part and the suspended object, the agnet can be connected with a long stic to the control part, and the suspension syste has no power loss. In the perforance (iv), the suspension echanis shown in Fig.3. enables a non-contact suspension because of the attractive force between suspension echanis and the suspended object. A oveent echanis is used for stabilization of the suspension syste. Therefore, this echanis does not need to control the generating power itself. Any suspension echaniss can be theoretically constructed by using a attractive force. Such as static electricity force, interolecular force and the force between atos can be applied to construct a sall suspension echanis without liit the suspended object to ferroagnetic substance [8].. Theoretical exaination of feasibility This section, the suspension echanis shown in Fig.3. is exained theoretically. The suspension echanis is odeled and a oveent equation is drawn. The attractive force near the equilibriu point is linearized and a linear odel of suspension syste is obtained. The feasibility of suspension echanis is exained by eans of linear control theory. The following two odels are considered by whether the influence of the attractive force is taen into consideration to oveent of the peranent agnet. Just lie that shown in Fig.., when a peranent agnet is driven an actuator, an attractive force fro the suspended object wors in addition to the driving force fro an actuator. Copared with the ass of the suspended object, if the ass of the peranent agnet is large, the force fro the suspended object can be disregarded during the otion of the peranent agnet. At this tie, the oveent of the peranent agnet can be considered independently with the oveent of the suspended object. It can be considered that position control of the oveent of the peranent agnet is carried out by the actuator. The input of a suspension syste serves as a peranent agnet position. 9

20 Since the attractive force equals the weight of the suspended object at the equilibriu position, if the ass of the suspended object becoes large, the attractive force will becoe large. And when the peranent agnet is driven directly by a otor, it is necessary to tae the attractive force into consideration to the oveent of a peranent agnet. At this tie, it can be considered that oveent of a peranent agnet is a power control with the actuator. The input of the suspension syste is the forces applied to a peranent agnet, and an output is the position of the suspended object. For analysis Fig. is used to express the odel shown in Fig.. ξ f a g f z z f d gap g Fig.. Model of suspension syste with peranent agnet otion control The sign used In Fig.. is explained to below. z : suspended object position z : peranent agnet position : ass of suspended object : ass of peranent agnet f a : force of the actuator f : attractive force between peranent agnet and suspended object d gap : air gap between peranent agnet and suspended object : constant of the peranent agnet attractive force : linearization constant of the peranent agnet attractive force : spring constant of the peranent agnet support part

21 ξ : daping coefficient of a peranent agnet support part Moreover, a inute deviation fro an equilibriu position is expressed as... Attractive force between agnet and suspended object For analyzing the suspension syste, an attractive force between the suspended object and the peranent agnet is exained. The attractive force is proportional to the square of the flux which passes the air gap, and the flux is inverse proportional to the length of the air gap. Therefore the attractive fore is shown in equation (..). f = (..) d gap Assue that at the equilibriu point, the attractive force is f and the air gap is d gap. d gap is the inute change of the air gap, so when there is a inute change the air gap becoes to d gap + d gap. At this tie, the attractive force is f = ( d gap + d gap ) (..) = d gap + d gap d gap + d gap = ( d gap + d gap ( d d gap gap d + d gap gap d )( d gap gap + d d gap gap ) d gap + d gap ) = ( d d 4 gap gap d (d gap gap d d gap gap + d + d gap gap ) ) (..3) Since d gap is very sall copared with d gap, the second order ter can be neglected. Then f = d gap = f d gap = f 3 d gap d gap f (..4) If only the inute deviation is considered f = d (..5) gap Usually the spring coefficient at the right side of equation is positive, however, in the agnetic suspension syste it is negative ( >). On the other words, in a agnetic suspension syste, the

22 suspended object is supported by a negative spring constant... Theoretical analysis... Model analysis Fro Fig.. the otion equations of the suspended object and the peranent agnet can be obtained respectively. The otion equation of the suspended object is & z = f g (..6) And the otion equation of the peranent agnet is & z = ξ z& z f + f g (..7) a Moreover, d gap = z z (..8) Linearization is perfored near the equilibriu point and a linear odel is obtained. Equations (..6) and (..7) are linearized to be the equations (..9) and (..) respectively. And then the linear control law can be applied in the analysis of suspension odel. & z = ( z ) (..9) z & z = ξ z z ) + (..) & ( z z f a The state variable of this suspension syste is x ( z z z & ) & (..) = z The force f a which acts on the peranent agnet is the input u of syste and the position of the suspended object z is the output y of the syste. The state space equation is Where & (..) x = Ax + bu y = cx (..3) b ( / ) =

23 3 = A ξ ( ) = c By eans of the linear control theory, the controllability and observability can be deterined. In this syste, the controllability C and the observability O and their deterinant are [ ] A b b A Ab b C 3 = (..4) [ ] = c A c A A c c O 3 ) ( ) ( (..5) 4 ) det( C = (..6) ) det( O = (..7) It is verified that this suspension syste can be controlled and observed [6].... Calculation of feedbac gain In ters of linear control theory, due to that the syste is controlled and observed, the LQR (linear quadratic regulator) full state feedbac control law can be used to design the controller of this syste. Z Z Plant Controller ref u Controller Fig..3 Control syste bloc diagra

24 The control syste bloc diagra is shown in Fig..3. In this diagra ref the reference input of syste u the input of syste z, z the iron ball s position and the peranent agnet s position respectively. Fro this diagra and otion equations we can see that the iron ball s position can be controlled via control the peranent agnet position. The reference signal and z is added to control z through the controller. The cost function is: J = t t t [ x Qx u Ru] + dt (..8) In this equation, Q is the state weighting atrix and R is the input weighting atrix. K=R - b T P (..9) P is the solution of Riccati Equation. After choose the states weighting atrix Q and input weighting atrix R and solve the Riccati Equation, the feedbac gain of syste K can be gotten [5]. Choosing weighting atrixes Q and R appropriately, the feedbac gain K can be obtained. Using LQR control law to design the controller ensures the stability of close loop of syste. So this syste ust be stable under such feedbac gain [7]...3 Exaination for peranent agnet suspension syste..3. Prototype of suspension syste Based on the principle of suspension syste, a prototype is constructed. It is shown in Fig..4. It is a prototype of -DOF peranent aglev syste in which we only consider the vertical direction, that is a suspension echanis. In this echanis, the suspended object is an iron ball which has a diaeter of.5() and a weight of.63(g), i.e. =.63(g). The peranent agnets have cylindrical shape with a diaeter of.8() and a length of.(). The oving part with the agnet has a weight of.375(g), i.e. =.375(g). The actuator, VCM (voice coil otor), has a stoe length of.5() and a propulsive force of (N) at a coil current of (A). The gap sensor senses the peranent agnet position. The peranent agnet is driven by the VCM and oved up and down to adjust the air gap between the peranent agnet and the iron ball, i.e. adjust the attractive force of the iron ball to balance the gravity of the iron ball. 4

25 eddy current sensor peranent agnet iron ball Fig..4 Prototype of suspension syste..3. Measureent of the peranent agnet attractive force The value of the attractive force between the peranent agnet and the iron ball object was easured by the anufactured easureent equipent shown in Fig.5. A Lode Cell whose easureent range is 5 g is used for easureent. The easureent ethod is that the iron ball is attached on the Load Cell and through oving the peranent agnet to adjust the air gap between the iron ball and the peranent agnet to easure the attractive force under different air gap. In that case, the interval is.5() in the range fro.5() to.5(). The results is shown in the Fig..6. Fro the equation (..), we now that the attractive force between the peranent agnet and the iron ball is inverse proportion to the square of the air gap length. Therefore, fro the relationship curve of the attractive force and the air gap shown in Fig..6, the constant of the peranent agnet attractive force can be nown,.9-5 (N ). Moreover, it turned out that the agnet used for an experient can tae out sufficient support force to support the iron ball during the suspension tie. 5

26 peranent agnet iron ball load cell Fig..5 Photograph of experiental set up Fig..6 Relationship between attractive force and air gap 6

27 A/D Controller D/A Aplifier Fig..7 Configuration of the suspension echanis..3.3 Nuerical siulation exaination The syste configuration is shown by Fig..7. The gap sensor sensing the peranent agnet oveent, which is located behind the actuator, is an eddy current type. It has a sensing range of.() and a resolution of (µ). The oveent of the iron ball is sensed by axes photo sensor and has a sensing range of.() and a resolution of 5(µ). Controller is a digital DSP controller with bit resolution A/D converters and 6 bits D/A converter. Through A/D converters, sensors signals are converted to digital value and input to the DSP Controller. Controller coputes the current for VCM and through D/A converter the currents are converted to analog value and input to the current aplifier to control the oveent of the peranent agnet. Applied the ethod described in section..., the feedbac gain K is obtained. K= [ ] Applied the feedbac gain K in the control syste, the nuerical siulation of step response is obtained. Fig..8 shows the nuerical siulation exaination of step response of the linear odel, and Fig..9 shows the nuerical siulation exaination of step response of the nonlinear odel. The step value is.(). It is added at the tie at second. After the step disturbance is added, the results had been recorded for second about the iron ball position and the peranent agnet position. In these two figures, the positive direction is the upward otion direction of the peranent agnet and the iron ball. When the step signal is added, the peranent agnet is oved downward and the air gap length between the peranent agnet and the iron ball becoes sall so the 7

28 attractive force becoes large to attract the iron ball ove upward. And then the peranent agnet is oved upward also, i.e. they are converged. After the transient tie the peranent agnet and the iron ball is balanced at the new position. The step disturbance is eliinated. The peranent agnet position is perforing an operation for stabilization, after oving in the opposite direction. Moreover, copared the transient tie of the response showed in Fig..8 and Fig..9, it can be checed that nonlinearity is in a syste. Copared with these two figures, we can see that the transient tie of the Fig..8 is shorter than that of the Fig..9 slightly, i.e. the convergence tie is late in the Fig..9. That is the difference between the linear odel and nonlinear odel in the step response. Furtherore, when the step value is increased, which the siulation results are shown in Fig.. and Fig.., in the linear odel, the response is stable, but in the nonlinear odel, the syste becoes unstable. It turns out that the air gap of a peranent agnet and a suspended object disappeared, and the attractive force will becoe infinite. Therefore, the data is no eaning. Fro these results we now that when a big step disturbance is applied the nonlinear control technique have to be considered. The siulation paraeter is shown in table... Table. values and sybols of -DOF peranent aglev syste Mass of the iron ball Mass of suspension agnet VCM spring constant VCM daping constant Magnet field constant of left agnet VCM force constant ξ a 63.7g 373g 6.63* -5 N 4.6N/A 8

29 Fig..8 Nuerical siulation of step response of linear odel (step value is.()) Fig..9 Nuerical siulation of step response of the nonlinear odel (step value is.()) 9

30 Fig.. step response of linear odel (step value is.() Fig.. step response of nonlinear odel (step value is.())

31 ..3.4 Experiental exaination Applied the controller in the experiental echanis, the suspension is succeeded. Fig.. is the photograph of the peranent agnet and the iron ball during the suspension. The step response and ipulse response of the experiental exaination are shown in the Fig..3 and Fig..4 respectively Fig. Photograph during suspension Fig..3 Experiental exaination of step response

32 And we also do the experient when the ass of iron ball is varied under the sae controller, the iron ball can be suspended. Table. ass of different iron ball Mass of the iron ball Mass of the iron ball Mass of the iron ball 3 Mass of the iron ball 4 Mass of the iron ball g 3.4g 49.8g 7.9g.g.3 Discussion In this syste, the suspension is success. It is controlled by eans of adjusting the air gap between the peranent agnet and the suspended object. It can be said that this echanis in which the ain coponents of the attractive force of a peranent agnet is instable is stabled by eans of echanic control. For this reason, the feature of this syste is no heat generation copared with the suspension echanis using an electroagnet. The odel was ade about this echanis and the feasibility of this suspension syste was considered fro a viewpoint of linear control theory. Modeling was perfored and it is checed that the odel can be controlled and observed. The nuerical siulation and experiental exaination is perfored. The results verified that the proposed suspension syste can be realized.

33 Chapter 3 Magnetic suspension echanis with the horizontal otion control This chapter will describe a agnetic suspension echanis in which the suspended object is anipulated in the vertical plane. It is a -DOF agnetic suspension echanis and is ade based on the suspension syste described in the chapter. In this chapter, the otion principle is explained and the arrangeent of agnetic polarity is analyzed with FEM ethod. A siple otion odel and an accuracy otion odel are ade respectively. Based on the otion odel the controller of control syste is obtained with LQR optial control ethod. The siulation and experiental exaination results verifies that the odel is correct. Actuator Peranent Magnet attractive Air Gap force Suspended Object g direction of suspension direction of actuator Fig.3. Outline of suspension echanis 3

34 3. Principle of -DOF aglev syste The -DOF agnetic suspension echanis principle showed with Fig.3. can be siplified to be that shown with Fig.3.. In Fig.3., the actuator showed in Fig.3. is oitted, i.e. a peranent agnet is used to represent the peranent agnet and the actuator. The different color eans the different polarity of a peranent agnet. We assued that the red color represents the S polarity of a peranent agnet and the blac color is N polarity. The principle of -DOF peranent aglev syste in which the suspended object can be oved in the vertical direction has been introduced in chapter. Here, we want to ove the suspended object in the horizontal direction when it is suspended. That is the suspended object can be oved in two directions, vertical and horizontal, so this syste is a -DOF peranent aglev syste. For this reason, three peranent agnets are used. One is used to suspend the object, and the other two agnets are used to ove the object oving in horizontal direction. The outline of -DOF peranent aglev echanis is shown in Fig.3.3. Due to that the suspended object is anipulated in the horizontal direction the attractive force of suspension agnet can be neglected. The syste shown in Fig.3.3 can be siplified to be that shown with Fig.3.4 peranent agnet and actuator iron ball Fig.3. Siplified Outline of suspension echanis 4

35 suspension agnet and actuator left agnet and actuator right agnet and actuator d d f f iron ball Fig.3.3 Outline of the -DOF peranent aglev syste left agnet and actuator iron ball d d right agnet and actuator f f Fig.3.4 Outline of the horizontal otion echanis In Fig.3.4, d is the air gap between the left agnet and the iron ball and d is the air gap between the right agnet and the iron ball. The attractive force of left agnet is f and that of right agnet is f. During the suspension, through adjust the air gap d and d ae the suspended object ove in the horizontal direction. The concrete ethod is to adjusting the reluctance in a agnetic circuit of the suspension echanis to control the position of the suspended object. It has been described in chapter. 5

36 3. FEM analysis and pole location A agnet has two poles, N and S and different poles arrangeent will have different flux distribution. Fro Fig.3.3 we now that there three peranent agnets in this syste, so it is required to decide agnetic polarity of three peranent agnets which face to the iron ball for stabilization suspension. In the -DOF peranent aglev syste the total nuber of polarity cobination is 6, SSS, SSN, SNN, NNN, NNS and NSS. The poles order of cobination is of suspension agnet, left agnet and right agnet. For instance, if the surface of the iron ball counters S pole at the vertical direction and both N poles at horizontal direction it is called SNN arrangeent, i.e. SNN expresses that the polarity of the agnet which faces to the iron ball is that the suspension agnetic pole is S, and left agnetic pole and right agnetic pole are N. Fro the perforance of agnets we now that the forer three cobinations and later three cobinations have the sae flux distribution, so we only choose the forer three cobinations as the object of the analysis. The agnetic field under different agnetic pole arrangeent is studied with FEM (finite eleents ethod). In this research, it is three diensions and two diensions analysis to the agnetic field in the suspended object circuference. For SSS arrangeent, SNN arrangeent and SSN arrangeent, the Magnetic field analysis results are shown with the figures fro Fig.3.5 to Fig.3.. Fig.3.5, Fig.3.6 and Fig.3.7 show the three diensions agnetic field analysis results about the pole arrangeent of SSS, SSN and SNN respectively. In every figure, the upper part shows the agnetis of peranent agnet acting on the suspended object, and the lower part shows the agnetis of the iron ball which is agnetized by the peranent agnet. Different color expresses different strength of the agnetis. The strongest agnetis is expressed with the red color. In the figure of a suspended object, a contour line is used to express agnetis. Furtherore, Fig.3.8, Fig.3.9 and Fig.3. show the two diensions agnetic field analysis results about the pole arrangeent of SSS, SSN and SNN respectively. In every figure, the upper part shows the agnetis and the lower part shows the flux linage which is expressed with the line, the white part is a agnet, and the round shape near a center is the suspended object. In these figures, the air gap length between the agnet and iron ball are all.5() as an initial value. The suspended object was arbitrarily oved within ±.3() in horizontal direction. Magnetic field analysis was perfored. Fro the analysis results, we can see that near the center of a suspended object either the passing flux oppose or suit for suspension. 6

37 Fig.3.5 Result of agnetic force 3D analysis (SSS) 7

38 Fig.3.6 Result of agnetic force 3-D analysis (SSN) 8

39 Fig.3.7 Result of agnetic force 3D analysis (SNN) 9

40 Fig.3.8 Result of agnetic force and flux diagra -Danalysis (SSS) 3

41 Fig.3.9 Result of agnetic force and flux diagra -Danalysis (SSN) 3

42 Fig.3. Result of agnetic force and flux diagra -Danalysis (SNN) 3

43 force (N) (SSS) virtual wor (SSS) axwell stress tensor (SNN) virtual wor (SNN) axwell stress tensor (SSN) virtual wor (SSN) axwell stress tensor displaceent() Fig.3. Result of agnetic field horizontal analysis force (N) SSS) virtual wor SSS) axwell stress tensor SNN) virtual wor SNN) axwell stress tensor SSN) virtual wor SSN) axwell stress tensor displaceent() Fig.3. Result of agnetic field vertical analysis 33

44 Fig.3. shows the analysis results of agnetis in horizontal direction which is coposed of virtual wor and Maxwell stress tensor under different arrangeent, whilst, Fig.3. shows the analysis results of agnetis in vertical direction. Fro a horizontal analysis result, we can see that the arrangeent SNN and SSS have alost the sae results which are syetric with the original point, but the results of arrangeent SSN is different which is not syetric with the original point. It turns out that leftward force is acting on the suspended object near the original point in the SSN arrangeent. Fro the analysis result of the vertical direction, we can see that the difference is rearable. The agnetis can be checed near the initial value. Furtherore, the agnitude order of the agnetis acted on the suspended object is SNN arrangeent, SSN arrangeent and SSS arrangeent. This is considered to be the difference of agnetic circuit in each agneto arrangeent. Moreover, for explain the choice of the pole arrangeent and obtain the relationship between the vertical force and displaceent and the relationship between the horizontal force and displaceent, the agnetic field analysis is perfored. The results are shown fro Fig.3.3 to Fig.3.6. Fig.3.3 shows the agnetic flux distribution of the opposite polarity. Fig.3.4 shows the agnetic flux of the sae polarity. The relationship between the vertical force and displaceent of the suspended object is shown in Fig.3.5, and the relationship between the horizontal force and displaceent of the suspended object is shown in Fig.3.6. Consequently, the arrangeent SNN (or NSS) is chosen in this research, because in this arrangeent the vertical force is largest and the horizontal force is syetric. That eans using this arrangeent the ost weighting suspended object can be suspended and it is easy to control the oveent of the suspended object in the horizontal direction. 34

45 Fig.3.3 agnetic flux diagra (opposite poles face to object) Fig.3.4 agnetic flux diagra (sae poles face to object) 35

46 Fig.3.5 Result of agnetic field for vertical analysis Fig.3.6 Result of agnetic field for horizontal analysis 36

47 3.3 Prototype of -DOF peranent aglev syste According to the principle of the -DOF peranent agnet suspension echanis and the polarity arrangeent, a prototype of this syste is set up. It is shown in Fig.3.7. In this echanis, the suspended object is an iron ball which has a diaeter of.5() and a weight of.63(g), i.e. =.63(g). The peranent agnets have cylindrical shape with a diaeter of.8() and a length of.(). The oving part with the agnet has a weight of.375(g), i.e. =.375(g). The actuator, VCM (voice coil otor), has a stoe length of.5() and a propulsive force of (N) at a coil current of (A). The gap sensor senses the peranent agnet position. The peranent agnet is driven by the VCM and oved up and down to adjust the air gap between the peranent agnet and the iron ball, i.e. adjust the attractive force of the iron ball to balance the gravity of the iron ball. Fro the picture we can see that there are three voice coil actuators which drive the three peranent agnets respectively. Motion of the agnet is sensed by eddy current sensor. An iron ball is the suspended object. When the iron ball is suspended, through control the otion of two horizontal agnets we can ae the iron ball ove along horizontal direction. So there are two otion directions in this syste, vertical otion and horizontal otion, i.e. this syste is a -DOF peranent agnetic suspension syste. eddy current sensor peranent agnet ironball voice coil otor Fig.3.7 Prototype of peranent aglev syste 37

48 3.4 Model analysis Fro the prototype and principle of -DOF peranent aglev syste, the otion odel of iron ball and peranent agnets were created. At first, a siple odel in which the influence of the attractive force of the suspension agnet was created. For verify the correction of siple odel, an accuracy odel was created. They are introduced as following Siple odel of -DOF peranent aglev syste Description of siple odel Fro Fig.3-7 we can see that the otions of the iron ball and the agnets divide into two directions, vertical and horizontal, oveent. It is consued that two otions are independent of each other. The analysis of vertical otion has been already investigated in chapter. Here, the horizontal otion, which involves the otions of an iron ball and two peranent agnets driven by actuators, are ainly investigated. Horizontal otion odel is shown in Fig.3. z z ξ ξ z d d f f f a f a Fig.3.8 Siple odel of horizontal otion As can be seen, the sybols used in the following study are z : displaceent of the iron ball z : displaceents of the left peranent agnet z : displaceents of the right peranent agnet : ass of the iron ball : ass of the left agnet : ass of the right agnet f a : generating forces of the left actuator f a : generating forces of the right actuator 38

49 f : attractive force between the ball and left agnet f : attractive force between the ball and right agnet d : the left side initial air gap length between the iron ball and left agnet d : the right side initial air gap length between the iron ball and right agnet d : the left side air gap length between the iron ball and left agnet d : the right side air gap length between the iron ball and right agnet : the coefficient of the left agnetic field : the coefficient of the right agnetic field : the linear coefficient of the left agnetic field : the linear coefficient of the right agnetic field : the spring coefficient of the left actuator : the spring coefficient of the right actuator ξ : the daping coefficient of the left actuator ξ : the daping coefficient of the right actuator The displaceents and forces are considered positive when they act in the right side direction. And it is consued that the friction of the guide of the actuator and viscosity of the air are negligible. The equations of the otion of the suspended object and the agnets are shown with equations (), () and (3). & z = f f (3.4.) & z = ξ z& z + f + f (3.4.) a & z = ξ z& z f + f (3.4.3) a Where f = and f =. Due to the relationship between agnetic force and the d d air gap is nonlinear, the equations (3.), (3.) and (3.3) are nonlinear equations. The syste is nonlinear syste. We will use the linear control theory to analyze the syste, so the nonlinear equations ust be linearized. f = = (3.4.4) d ( ) d + z z f = = (3.4.5) d ( ) d + z z Q z (3.4.6) = z z z = z z 39

50 4 ( ) z d f + = (3.4.7) ( ) z d f + = (3.4.8) Linearization of the otion odel According the linearization principle [5], equation (3.7) can be linearized to be ( ) ) ( 3 z z f z d d z d d z f z z d d f z d f + = + = = = + = = = + = (3.4.9) And the equation (3.8) can be linearized to be ( ) ) ( 3 z z f z d d z d d z f z z d d f z d f + = + = = = + = = = + = (3.4.) So the otion equations (3.), (3.) and (3.3) are changed to be ) ( ) ( z z z z f f z + = & & (3.4.) a f z z f z z z = ) ( & & & ξ (3.4.)

51 & z = ξ z z f ) + f (3.4.3) & Q f and f f o are constants, and ( z z ( z z ) ( z ) f << z & z = z z ) ( z ) (3.4.4) ( z & z ξ z& z + z z + f (3.4.5) = ( ) a & z ξ z& z z z + f (3.4.6) = ( ) a a The equations (3.4), (3.5) and (3.6) are linearized otion equations. After linearizing, the horizontal otions can be expressed by the state space odel. It is shown with equations (3.7) and (3.8). x & = Ax + bu (3.4.7) y = cx (3.4.8) Where x is the state variable, y is the output, A is the state atrix, b is the input atrix and c is the output atrix. Applied the linear control law, we have to chec the controllability and observability of this syste. Considering the input of this syste there are three situations [6] Three situations according the input and output of syste In ters of the linear control law, the controllability atrix C and observability atrix O are expressed with equations (3.9) and (3.) respectively [ b Ab A b A b A b A b] C = (3.4.9) [ c A c A c A c A c A ] ) ( ) ( ) ( ) O = ( c (3.4.) According to the ran of the controllability atrix and observability atrix we can now the controllability and observability of this syste under different situation. [6]. Theore : if and only if the ran of controllability atrix is equal to the nuber of state variable, the syste is controllable. Theore : if and only if the ran of observability atrix is equal to the nuber of state variable, 4

52 the syste is observable. When the controllability atrix is square atrix, there are another two theores to chec the controllability and observability of the syste Theore 3: if and only if the deterinant of controllability atrix is not equal to, the syste is contollable. Theore 4: if and only if the deterinant of observability atrix is not equal to, the syste is observable Two forces input syste In the odel shown in Fig.3., the two agnets are driven by eans of different VCM and the current of VCM is fed with different aplifier. That is the otion of two agnets is independent. The propulsion force f a and f a are the syste inputs, so this syste is called Two Forces Input Syste. z z ξ ξ z d d f f f a f a Fig.3. Model of two forces input syste When the output of this syste is y=z, in the state space odel, where x ( z& z z& z z& ) = z b = / / 4

53 43 = d d d d d d d d A ξ ξ ( ) = c ( ) = a f a f u In this odel, the controllability atrix is C and the observability atrix is O. The Ran of C is 6, and the Ran of O is 4, so this odel is controllable but observable. However, when the output y=[z z z ], the output atrix becoes = c At this tie, the controllability atrix is C, and the observability atrix is O. The Ran of C and O are all equal to 6, so the syste is controllable and observable. d d f f f a f a z z z ξ ξ Fig.3. Model of the sae force input syste

54 Sae force input syste In the odel shown with Fig.3., the agnets are driven by two VCM respectively, but the current of VCM is fed only with one aplifier. That eans the sae force is generated with two VCM, i.e. f a =f a =f a, the syste has only one input f a. We call this syste Sae Force Input Syste. When the output of this syste is y=z, in the state space odel, where ( ) = z z z z z z x & & & = d d d d d d d d A ξ ξ ( ) = b ( ) = c f a u = In this odel, the controllability atrix is C 3, and the observability atrix is O 3. The Ran of C 3 and O 3 are all equal to 4 this syste is uncontrollable and unobservable. However, when the output y= [z z z ], the output atrix becoes = c The controllability atrix is C 4, and the observability atrix is O 4. The Ran of C 4 and O 4 are all equal to 6. But when =, = and d=d, the ran of C 4 is 4 and the ran of O 4 is 6. Therefore, this syste is uncontrollable but observable.

55 z z z d d ξ f f f a Fig.3. Model of two agnets connecting syste Two agnets connecting syste In the odel shown with Fig.3., the two agnets are dependent, i.e. they are connected together and driven by eans of one VCM. Such a syste is called Two Magnets Connecting Syste. in this syste = =, z =z. When the output of this syste is y=z, in the state space odel, where x ( z& z z& z z& ) = z A = 3 d d d 3 d 3 3 ξ 3 d 3 d ξ 3 d 3 d ( ) b = c = ( ) u = f a In this case, the deterinant of the controllability atrix C 5 and observability atrix O 5 are all equal to, so the syste is uncontrollable and unobservable. 45

56 However, when the output y=[z z z ], the output atrix becoes c = The ran of controllability atrix C 6 is 4, and the ran of observability atrix O 6 is 6. This syste is uncontrollable but observable Deterination the control plant of -DOFaglev syste Table 3. controllability and observability of different plant input output controllability observability Two forces input One output controllable unobservable sytee Three out put controllable observable Sae forces input One output uncontrollable unobservable syste Three out put uncontrollable observable Two agnets One output uncontrollable unobservable connecting syste Three out put uncontrollable observable Table 3. shows the controllability and observability of the -DOF peranent aglev syste under different input and output situations. According the linear control theory we have to choose one situation of the as a control plant which ust be controllable and observable. Fro the table 3., we can see that only one situation, two forces input and three outputs, is controllable and observable. Thus, this situation is chosen as the plant in the research. 46

57 agnet initial position f d f f d d iron ball g final position f f v d f α f z d ' z d ' z f h g Fig.3.3 Iron ball s initial position and final position free body diagra ξ ξ ξ z z z f d d f f f a f a Fig.3.4 Accuracy odel of horizontal otion 47

58 3.4. Accuracy odel of -DOF peranent aglev syste In the last section, a siple odel in which the influence of suspension attractive force is neglected is introduced. For exaining this influence, an accuracy odel is created. a free body diagra of the iron ball is drawn when the iron ball is at the center position and is deviated to the right (it is syetric with the free body diagra when the iron ball is oved to the left ). It is shown in Fig.3.3. The upper part of this figure is the iron ball free body diagra at the center position, initial position. The lower part is the iron ball free body diagra when the iron ball is oved to the right direction Description of accuracy odel In the Fig.3.3 and Fig.3.4 z : displaceent of the iron ball z : displaceents of the left peranent agnet z : displaceents of the right peranent agnet : ass of the iron ball : ass of the left agnet : ass of the right agnet f a : generating forces of the left actuator f a : generating forces of the right actuator f : attractive force between the ball and left agnet f : attractive force between the ball and right agnet d : the left side initial air gap length between the iron ball and left agnet d : the right side initial air gap length between the iron ball and right agnet : the coefficient of the left agnetic field : the coefficient of the right agnetic field : the linear coefficient of the left agnetic field : the linear coefficient of the right agnetic field : the spring coefficient of the left actuator : the spring coefficient of the right actuator ξ : the daping coefficient of the left actuator ξ : the daping coefficient of the right actuator f : the attractive force of suspension agnet. f v : the vertical coponent of f. f h : the horizontal coponent of f. d : the initial air gap between suspension agnet and iron ball. 48

59 α: the angle between f and f v. g: the gravity force of iron ball d : left air gap of the finial position between the iron ball and the left agnet d : right air gap of the finial position between the iron ball and the right agnet tg f f f z d h α = (3.4.) v g h = z = (3.4.) ' ' d d d The otion equations of the aglev syste are written as equation (3.3), (3.4) and (3.5). & f (3.4.3) z & = f f h & z = ξ z& z + f + f (3.4.4) a & z = ξ z& z f + f (3.4.5) a Linearization of the otion odel As described in the last section, the equation (3.3), (3.4) and (3.5) are nonlinear equations. We have to linearize the. The linearized otion equation are shown with equation (3.6), (3.7) and (3.8). g & (3.4.6) z = ( z z ) ( z z) z d & z ξ z& z + z z + f (3.4.7) = ( ) a & z ξ z& z z z + f (3.4.8) = ( ) a Controllability and observability After linearized the otion equations, the accuracy odel can be expressed with state space odel shown as equations (3.7) and (3.8). Where x ( z& z z& z z& ) = z 49

60 5 = d g A ξ ξ = / / b = c ( ) = a f a f u In the accuracy odel, we only consider controllability and observability under the two forces input and three outputs situation. In this case, the ran of controllability atrix C 7 and observability atrix O 7 are all equal to 6, so the syste can be controlled and observed. Z Z Z Plant Controller Controller ref u u Controller Fig.3.5 Control syste bloc diagra

61 3.5 Controller design For realize the -DOF peranent agnet suspension syste, the controller of this syste have to be designed. The control syste bloc diagra is shown in Fig.3.5. In this diagra Ref is the reference input of syste u, u are the inputs of syste z, is the iron ball s displaceent. z, is the left agnet s displaceent z, is the right agnet s displaceent The relationship aong the iron ball s position and the two peranent agnets position is coplicated. Fro this diagra and otion equations we can see that the iron ball s position can be controlled via control the two peranent agnets position. Two agnets are coupled that eans these two peranent agnets position effect each other. The reference signal and z are divided into two parts to control z and z through the controller. The paraeter of this syste is shown in table.3.. Table 3. values and sybols of -DOF peranent aglev syste Mass of the iron ball Mass of the left agnet Mass of the right agnet Left VCM spring constant Right VCM spring constant Left VCM daping constant Right VCM daping constant Magnet field constant of left agnet Magnet field constant of right agnet VCM force constant ξ ξ a 63.7g.5g.5g 6.63* -5 N 6.63* -5 N 4.6N/A 3.5. Controller design ethod There any ethods to design the controller for a linear syste. In ters of linear control theory, due to this syste is controllable and observable, the LQR (linear quadratic regulator) full state feedbac control law can be used to design the controller of this syste. The cost function is J = t t t [ x Qx u Ru] + dt (3.5.) 5

62 In this equation, Q is the state weighting atrix and R is the input weighting atrix. T K = R b P (3.5.) P is the solution of Riccati Equation. After choose the states weighting atrix Q and input weighting atrix R and solve the Riccati Equation, the feedbac gains of syste K can be gotten [6]. Using LQR control law to design the controller ensures the stability of close loop of syste. So this syste ust be stable under such feedbac gain [7] Choice of weighting atrix Fro the Fig.3., we now that z, z and z are the displaceent, therefore, z&, z& and z& are the velocity of horizontal agnets and iron ball. Based on the state space equation, we choose the weighting atrix Q and R. Q = diag R = diag( ) ( ) Using MATLAB software, we calculate the feedbac gain K. i =, K = ij (3.5.3) j =,,, 3, 4, Controller design for siple odel It is assued that the initial left and right air gap length are.8(). For siple odel of -DOF peranent aglev syste, the feedbac gain K is K = Controller design for accuracy odel The initial air gap length is also.8(). For accuracy odel of -DOF peranent aglev syste the feedbac gain K is K 7.76 =

63 A/D A/D Controller D/A Aplifier position sensor optical sensor VCM vertical horizontal iron ball Fig.3.6. Configuration of -DOF peranent aglev syste 3.6 Syste exaination Controller is a digital DSP controller with bit resolution A/D converters and 6 bits D/A converter. Through A/D converters, four sensors signal are converted to digital value and input to the DSP Controller. Controller coputes the current for VCM and through D/A converter the currents are converted to analog value and input to the current aplifier to control the oveent of agnets. Using the feedbac gain, which is gotten with the LQR linear control ethod, the syste is exained with nuerical siulation and the experiental exaination Syste exaination for siple odel The nuerical siulation and experiental exaination have been done for siple odel. It includes step response, frequency response and white noise response which is only applied in the nuerical siulation. 53

64 3.6.. Siulation exaination for siple odel The step response, frequency response and the step response in which the white noise is added have been done for the siple odel. They are introduced in this section Step response The initial air gap d =d =.8() and the step input of.() is added as the reference signal. The nuerical siulation result is shown in Fig.3.7. displaceent() displaceent() displaceent() right agnet left agnet Fig.3.7. Step response of nuerical siulation In this figure, the right oveent direction is positive. There is no overshot, and the syste is converged in the.(s). Fro the figure, we can also see that when the step input is added, two horizontal agnets are oved towards the left. The right air gap becoes saller and the left air gap becoes larger, i.e. the attractive force of right agnet is larger than that of left agnet. The iron ball is oved to the right, at the sae tie two horizontal agnets are oved to the right. At last, the iron ball and two agnets are balanced in the new position. At the new position, the left air gap length equals the right air gap length, i.e. the attractive force of the right agnet is equal to that of the left agnet. Thus through control the otion of two peranent agnets, we can control the otion of the iron ball in the horizontal direction. 54

65 Frequency response The frequency response is shown in the Fig.3.8. The frequency response shows the gain argin and phase argin of this syste. In this syste, the gain argin is 5 (db), and the phase argin is 5 degrees. Syste has no resonant point. Fig.3.8 Frequency response of siulation for siple odel 55

66 Step response when the white noise is added The easureent noise and disturbance inputs are often odeled as white noise when evaluating the control syste perforance. Thus, a band-liited white noise disturbance is added when the step signal is a reference input. The result is shown in Fig The paraeter of white noise is shown in table.3.3. Table.3.3 Paraeter of white noise disturbance Noise power Saple tie Seed.W.s 334 Fig.3.9 Step response when white noise is added in siulation for siple odel Fro Fig.3.9, we can see that when the disturbance is added, the dynaic characteristic of this syste is not influenced. Only the position of the iron ball and two peranent agnets is influenced. It turns out that there are soe fluctuations of the position of the iron ball and peranent agnets caused by the white noise. This phenoenon can be controlled by ultiplying by the proportionality gain. It is better to perfor intensive control so that a gain can be freely set up in the deflection of the suspension object. 56

67 3.6.. Experiental exaination for siple odel Experiental exaination was carried out on the syste and the -DOF agnetic levitation was succeeded. The Fig.3.3 shows the -DOF peranent aglev syste during the suspension. The step response and frequency response is introduced. Fig.3.3 Photo of -DOF peranent aglev syste during operation Fig.3.3. Step response of experiental exaination for siple odel 57

68 Step response The initial position is that the ball is located in the center of agnets. The air gap between horizontal agnets is set to.8() through adjust the position of the horizontal agnets. A step input of.() was added as the reference signal of the iron ball. The experiental exaination of step response is shown in Fig.3. In this figure, the response curves show that after the step input is applied, when the syste reach the stable state, the iron ball is oved to the right and two horizontal agnets are oved to the left copared with their initial position. So at the new position, the left and right air gap length is different with that at initial position. It is different with the siulation results. And in the transient process, the overshoot is large copared that with the siulation results Besides the different in dynaic characteristic, the final position of the iron ball and two peranent agnets are different. Fig.3.3 shows the otion of iron ball and two peranent agnets in the siulation and experiental exaination. In this figure, (a) is about the otion in the siulation of step response, and (b) is about the otion in the experiental of step response. When a positive step input is added, the two agnets are all oved to the left to ae the iron ball ove to right, and then they are all oved to the right. At the final position, in the siulation exaination, the air gap is not changed but changed in the experiental exaination. initial position initial position d d d d final position final position d ' d ' d ' d ' (a). otion in the siulation (b). otion in the experient Fig.3.3 Motion of iron ball and two agnets in siulation and experiental exaination 58

69 This phenoenon is ainly caused by the attractive force of suspension agnet. This can be seen fro the free body diagra of the iron ball when the iron ball is diverted fro the center of the syste. It is shown in the Fig agnet initial position f d f f d d iron ball g final position f f v d f α f d ' z d ' f h g Fig Free body diagra of the iron ball when the iron ball is located at the center position and offset fro the center position Fro this figure, we can now that when the iron ball is located at the center of the syste, the attractive force of the suspension agnet is equal to the gravity of the iron ball. When the iron ball is offset fro the center position, the direction attractive force is not opposite to the gravity. There are an angle between the attractive force and the opposite direction of gravity. Therefore, the attractive force will generate two coponent forces, one is in the opposite direction of gravity and another is in the horizontal direction. If the iron ball is oved to right, the direction of the horizontal coponent is left, vice versa. Therefore, we create an accuracy odel of this syste to eliinate this phenoenon. The experiental exaination will be introduced in the following section. 59

70 Frequency response Fig.3.34 shows the frequency response of experiental exaination for siple odel. Fro it, the resonant point and bandwidth can be seen. In this syste, the resonant point is at π (rad/sec), and the bandwidth is 4 (rad/sec). Fig.3.34 Frequency response of experiental exaination for siple odel 3.6. Syste exaination for accuracy odel The exaination results of the siple odel shows a difference between the siulation exaination and experiental exaination. For explain this difference, an accuracy odel has been created and analyzed. Hence, the exaination for the accuracy odel is perfored. The exaination includes two aspects, siulation exaination and experiental exaination Siulation exaination for accuracy odel In this section, the step response and frequency response of the accuracy odel is introduced Step response A feedbac gain K for the accuracy odel has been obtained by eans of LQR control law in the section 3.5. The siulation exaination of the accuracy odel is carried out with the feedbac gain and K. The step responses of siulation are shown in Fig.3.35 and Fig.3.36 respectively. 6

71 At first, the feedbac gain K, the feedbac gain for siple odel, was used to chec the accuracy odel. The step response of siulation exaination is shown in Fig In this figure, when the iron ball is oved to the right, at the equilibriu position, the two agnets are oved the left copared with that at the initial position. it is the sae with the step response of the experiental exaination for the siple odel. Copared this figure with Fig.3.3, the step response of the experiental exaination for siple odel, these two figures confir that the siple odel is correct. Secondly, the feedbac gain K is applied in the accuracy odel. The step response is shown in Fig Fro this figure, we can see that the displaceent of the iron ball is lager than the displaceent of the agnets. Therefore, the right air gap is slightly saller than the left one. This is caused by eans of the attractive force of the suspension agnet. Moreover, when the iron ball is oved to the right, at the equilibriu position, the two agnets are oved to the right copared with that at the initial position. 6

72 Fig.3.35 Siulation exaination of accuracy odel with K right agnet iron ball left agnet Fig.3.36 Siulation step response of accuracy odel with K 6

73 Frequency response The frequency response of the accuracy odel is shown in fig In this figure, there is a resonant point at 34.6 (rad/sec), and the bandwidth is 8.9 (rad/sec). Fig.3.37 Frequency response of siulation for accuracy odel with K 63

74 3.6.. Experiental exaination for accuracy odel In this section, the step response and frequency response of the experiental exaination are perfored for the accuracy odel to exaine characteristic of the accuracy odel Step response Using the feedbac gain K, the experiental step response for the accuracy odel was obtained under the sae condition with that for the siple odel. The result is shown in Fig Copared this with the Fig.3.36, the siulation result, we can find that there are soe fluctuation when the syste converged, whilst, in the experiental exaination, the ipact of step input on the syste is larger than that in the siulation exaination. This phenoenon is caused by the linearization, i.e. this syste is nonlinear syste but for analyzing this syste we linearized the syste. The other aspects of experiental step response for accuracy odel confir to that of the siulation step response for accuracy odel. It is verified that the accuracy odel is correct and the siple odel is enough for anipulate the suspended object to ove in the horizontal direction. right agnet iron ball left agnet Fig.3.38 Experiental step response of accuracy odel with K 64

75 Frequency response Fig.3.34 shows the frequency response of experiental exaination for siple odel. Fro it, the resonant point and bandwidth can be seen. In this syste, the resonant point is at.9 (rad/sec), and the bandwidth is.3 (rad/sec). Fig.3.39 Experiental frequency response of accuracy odel with K 65

76 3.7 Measureents of the otion of the iron ball and agnets The -DOF peranent agnet suspension syste is built for anipulate the suspended object in the horizontal direction, so the easureents of the otion of the iron ball and agnets when the iron ball is oved in the horizontal direction are perfored with the siple odel of syste. They are shown in the Fig.3.4 and Fig.3.4. Fig.3.4 shows the relationship between the displaceent of the iron ball and two agnets under different reference input. right agnet iron ball-x left agnet displaceent () reference input () Fig.3.4 Relationship of displaceent of iron ball and horizontal agnets with reference input In this figure, the origin is the initial position. About the Y axis, the positive direction is the right direction and the negative direction is the left direction of oveent. Fro this figure we can see that, when the reference input value is positive, the iron ball oves to the right and two horizontal agnets ove to the left at the equilibriu position. The left air gap length is larger than the right air gap length, i.e. the attractive force of the right agnet is larger than that of the left agnet. When the reference input value is negative, the iron ball oves to the left and two horizontal agnets ove to the right at the equilibriu position. The right air gap length is larger than the left air gap length, i.e. the attractive force of left agnet is larger than that of the right agnets. It confirs to the experiental exaination results for siple odel. Fro this figure, we can also find that within the range of -3() to 3() of the reference input, the otion of the iron ball in horizontal direction is 66

77 linear with the reference input, so the iron ball can be located at any position in the horizontal direction within that range. The anipulation is successful. The photos in which the iron ball is anipulated to deviate fro the center position are shown with Fig.3.4 and Fig Furtherore, as we now, while the iron ball deviates fro the central position, the attractive force of the vertical agnet will generate a horizontal coponent, which acts on the iron ball to ae a balance between the left and right attractive force. Fro Fig.3.4, we can get the coponent of the attractive force of the suspension agnet. In this syste, = 6.63* -6 (N/). The horizontal coponent force of the vertical agnet attractive force can be calculated with the data shown in Fig.3.4 (here the calculation is oitted). It is verified that the equation (3) is correct. The otion of the iron ball and the suspension agnet in the vertical direction is easured while the iron ball is oved in the horizontal direction. The relationship between the vertical displaceent of the iron ball and suspension agnet and reference input is shown in Fig.3.4. In this figure, however, the displaceent in the vertical direction is very sall copare with that in the horizontal direction. We thin that it is another reason for the fluctuation in dynaic characteristic of the experiental step response. iron ball vertical agnet displaceent () reference input () Fig.3.4 Relationship of displaceent of iron ball and suspension agnet with reference input 67

78 Fig.3.4 Photo of the syste when the iron ball is anipulated to the right direction Fig.3.43 Photo of the syste when the iron ball is anipulated to the left direction 68

79 3.8 Discussion In this chapter, a -DOF peranent agnet suspension echanis is created based on the -DOF syste which is introduced in the last chapter. In this syste, the suspended object, the iron ball is anipulated in the horizontal direction while it is suspended. The syste is consisted of three peranent agnets controlled by VCM respectively and the suspended object. The position of agnets is sensed by eddy current sensors and that of the suspended object is sensed by an optical sensor. The syste is controlled by eans of adjusting the air gap between the peranent agnet and the suspended object. For anipulate the suspended object to ove in the horizontal direction, firstly, the arrangeent of the polarity of three agnets, SNN (or NSS), is chosen with the FEM analysis fro six situations according the agnetic field distribution. Fro this arrangeent, the ost large support force for supporting the suspended object will be generated and the force acted on the suspended object in horizontal direction is syetrical. It is clear that this arrangeent is easy for control the otion of the suspended object in vertical and horizontal direction. Secondly, the prototype is constructed according the arrangeent of polarity and principle. Based on the prototype a siple odel in which the suspension agnet attractive force is neglected is created. Consequently, fro the analysis result of linear control theory, two inputs and three outputs syste is chosen as the control plant which can be controlled and observed for this syste. The controller of this syste is designed according the LQR full state feedbac control law. Due to that there is soe difference between the siulation exaination and experiental exaination for siple odel, an accuracy odel is created and exained. Fro the exaining results for the accuracy odel, we can say that the siple odel is correct and enough for anipulation. Finally this syste is operated successfully. The suspended object can be anipulated in vertical and horizontal direction. Within a range of -3() to 3() of the reference input, the relationship of the displaceent of the suspended object and two horizontal agnets with the reference input is linear. Therefore, the suspended object can be located at any position in horizontal direction within the range of the reference input. However, during the suspended object is oved in horizontal direction, there is a tendency to be unstable in the DOF that is perpendicular to the otion direction in the horizontal plane. For solving this proble and ae the suspended object ove in other DOF, a new peranent aglev syste has been developed. It is introduced in the following chapter. 69

80 7

81 Chapter 4 4-DOF Magnetic suspension echanis with peranent agnet otion control So far, the suspended object can be anipulated in the vertical plane via the -DOF peranent agnet suspension echanis that is described in the chapter 3. In that syste, there is a tendency to be unstable in the other DOF when the suspended object is oved in the horizontal direction. For solving this proble, a new agnetic suspension echanis in which the suspended object can be anipulated not only in the vertical plane but in the horizontal plane is developed. Firstly, a new peranent agnet suspension echanis is built in which the suspended object is suspended and rotated along the suspension axis. The principle of this echanis is introduced. The rotation odel is created and the experiental exaination verified that the feasibility of this echanis. Secondly, using this echanis, the suspended objec can be located in the horizontal plane within a sall range while it is suspended. Therefore, this syste is a four-dof peranent agnetic suspension syste. The principle is introduced and the odel is created. The siulation result verified the feasibility of this syste. 4. Construction of echanis A new peranent aglev syste is developed. The construction of this syste is shown with Fig.4. and Fig.4.. Fig.4. is an illustration of this echanis in which the suspended object is an iron ball, and five peranent agnets are used to anipulate the suspended object. One of the is the suspension agnet located at the above of the iron ball, and the others four agnets are uniforly located in the horizontal plane around the iron ball. The iron ball is suspended in the center of horizontal plane of this echanis, and the agnets are driven by eans of the VCM actuators. The VCM actuators are fixed at the point that has the sae distance to the center of echanis. Fig.4. is the photo of this prototype in which the iron ball is oitted. 7

82 peranent agnet VCM iron ball eddy current sensor Fig.4. Illustration of echanis eddy current sensor Fig4.. Photo of Prototype 7

83 Fro these figures we can see that there are five VCM actuators that drive the five peranent agnets respectively. Positions of the agnets and iron ball are sensed by eans of eddy current sensors. When the iron ball is suspended, through control the otion of the upper agnet the height of iron ball can be adjusted to ae the iron ball and four horizontal agnets in the sae horizontal plane. At the sae tie, through control the otion of the horizontal agnets to ae the iron ball rotate along suspension axis and position the iron ball in the horizontal plane within a range. In the following sections, the principle, odel analysis and experiental exaination of suspension, rotation and position will be introduced. 4. Suspension The rotation of the suspended object and positioning the suspended object in the horizontal plane are perfored while the suspended object is suspended. Therefore, the first step is to ae the iron ball suspend. Actuator Peranent Magnet attractive Air Gap force Suspended Object g direction of suspension direction of actuator Fig4.3. Outline of Suspension 4.. Principle The suspension echanis of this syste has the sae construction with the -DOF peranent aglev syste which has been introduced in chapter. Therefore, they have the sae principle. An outline of proposed suspension syste with a peranent agnet and linear actuator is shown in Fig.4.3. A ferroagnetic body is suspended by an attractive force fro a peranent agnet, which is driven by an actuator positioned above it. The direction of levitation is vertical; the agnet and the object ove only in this direction. The equilibriu position of the ferroagnetic body is deterined by eans of a balance between the gravity force and the agnet force. If the actuator does not 73

84 actively control the agnet s position, the levitated object will either fall or adhere to the agnet. However servo-control of the actuator can ae this syste stable. Because there is a saller attractive force for a larger air gap between the peranent agnet and object, the actuator drives the agnet upwards in response to object oveent fro its equilibriu position towards the agnet. Siilarly, the actuator drives the agnet downwards in response to object oveent away fro the agnet. In this way, the object can be stably suspended without contact. In coparison to the electrical control ethod of electroagnetic suspension systes, this syste is a echanical control aglev syste [7]. 4.. Model analysis Although the principle of this syste is the sae with that of -DOF peranent aglev syste, different peranent agnets ae the attractive force between the suspended object and the agnet be different. 4 attactive force [N] gap [] Fig4.4. Relationship between suspension attractive force and air gap 4...Attractive force exaination In this suspension syste, cylindrical neodyiu peranent agnets are used for control the attitude of the suspended object. The diaeter of agnet is.8 and its height is.8 also. The suspended object is an iron ball whose diaeter is.5 and its ass is.63 g. The attractive force between the agnet and iron ball is easured under different air gap. It is shown in Fig.4.4, fro which we can see that the attractive force, f is inverse proportional with the square of air gap. It is nonlinear force. 74

85 f = (4..) d gap Where, K is the constant of attractive force. It can be obtained fro the Fig.. In this case, K =6.9-6 N. When the suspension syste is balanced, i.e. the attractive force is equal to the ass of the iron-ball, the air gap is.5. so far, used such a agnet, the feasibility of suspension is confired. ξ f a g f z z f d gap g Fig.4.5 Model of suspension syste 4... Suspension odel Fig.4.5 shows the odel of suspension echanis. Where, z : displaceent of the iron ball z : displaceent of peranent agnet : ass of the iron ball : ass of the agnet f a : generating force of the actuator f : attractive force between iron ball and agnet d gap : air gap length between iron ball and agnet : the coefficient of agnetic field. : the spring coefficient of the actuator ξ : the daping coefficient of the actuator. Thus the otion equations of iron ball and agnet during suspension are: 75

86 & z = f g (4..) & z = ξ z& z f + f g (4..3) a Equations (4.) and (4.3) are linearized to be equations (4.4) and (4.5) respectively. And then the linear control law can be applied in the analysis of suspension odel. & z = z ) (4..4) ( z & z = ξ z z z z ) + (4..5) & ( f a In this research, a LQR full state feedbac control law is applied. The state space equation of suspension odel is shown with equation (6). Where & (4..6) x = Ax + bu y = cx (4..7) x ( z z& z & ), b ( / ) = z = A = ξ, c = ( ) u = f a The deterinant of Syste controllability atrix C and observability atrix O are shown with equations (7) and (8). det( C) = (4..8) det( O) = (4..9) Thus the suspension syste can be controlled and observed. 76

87 Controller ref Controller u Plant Z Z Fig.4.6 Control bloc diagra of suspension syste 4..3 Controller design Based on the controllability and observablility of this suspension echanis the LQR control law is applied to design the controller of it. The control bloc diagra is shown in Fig.4.6. The cost function of syste is shown with equation (4.). J = t t t [ x Qx u Ru] + dt (4..) State weighting atrix Q = diag Input weighting atrix R = () The feedbac gain ( ) K s = ( ) 4..4 Exaination A step disturbance is applied to exa the suspension echanis with siulation and experiental ethod. The step value is.(). Table 4. shows the sybols and values of this suspension echanis. 77

88 Table 4. values and sybols of suspension echanis Mass of the iron ball Mass of suspension agnet VCM spring constant VCM daping constant Magnet field constant of left agnet VCM force constant ξ a 63.7g.5g.69* -5 N 5N/A Siulation exaination Fig. 4.7 Step response of siulation exaination Fig.4.7 shows the step response of siulation exaination result. Fro this figure we can see that the step disturbance is added at (s). Converge tie is.(s) and there is no overshoot in the dynaic characteristic Experiental exaination Fig.4.8 shows the configuration diagra of suspension syste. Fro it, we can see that the gap sensor, which is located upper of the actuator and senses the peranent agnet oveent, is an eddy current type. It has a sensing range of () and a resolution of (µ). The oveent of the iron ball is sensed also via an eddy current sensor and its resolution is 5(µ). Controller is a digital DSP controller with bit resolution A/D converters and 6 bits D/A converter. Through A/D 78

89 converters, two sensors signal are converted to digital value and input into DSP Controller. Controller coputes the current for VCM. Through D/A converter, the currents is converted to analog value and input into current aplifier to control the oveent of agnet. Position of Iron Ball Fig.4.8 Configuration of suspension syste When the feedbac gain is applied, the iron ball is suspended successfully. The result is shown in Fig.4.9. This figure is a tie history of the displaceent of iron ball and agnet in which the saple tie is.5(s), and the interval is (s) Fig.4.9 Step response of experiental exaination 79

90 Fig.4. Photo of suspension 4.3 Rotation Through control the otion of agnets which located in the horizontal plane the suspended object can be rotated along the suspension axis during suspension. This section describes the rotation principle, rotation odel and exaination. a d b c Fig.4. Iage of spinning echanis 8

91 4.3. Rotation principle One of the purposes of this investigation is ae a rotation control echanis in which the suspended object is rotated along the suspension axis when it is suspended. The iage of rotation is shown in Fig.4.. For siplification, only the agnets that are located in the horizontal plane and iron ball are shown in it. This figure is a planfor of the syste in which the suspension agnet is oitted. Fro it we can see the location of four agnets and iron ball. The iron ball is located in the center position, and four agnets are located perpendicularly each other around the iron ball in the horizontal plane. When the agnet approaches the iron ball alternatively, the iron ball can be rotated along the suspension axis. Due to the suspended object is an iron ball, which is a ferroagnetic substance, there is reanent agnetis on the surface of it [Appendix]. However, when the iron ball is suspended, the influence of reanent agnetis is ostly in the vertical direction of suspension. It eeps the iron ball suspended. When the agnet in horizontal approaches the iron ball, there is a agnetization point on the surface of iron ball facing the approaching agnet. And when this agnet withdraws and the next agnet approaches, the approaching agnet will attractive the agnetization point, so the iron ball can be rotated. In Fig.4., the approaching order of agnets is agnet a, b, c and d. It is one circle Rotation odel and control z d 4 d 4 ' d R α γ θ λ O z 4 z d ' A β rotation direction d ' d d 3 ' Y d 3 X z 3 a. Configuration of the agnets and reanent point 8

92 z f '' d ' d f d ' a θ f ' f ' f f '' z θ R b d b. Attractive forces between two agnets and reanent point Fig.4. Rotation odel Fro the rotation principle and prototype of this syste, a rotation odel is created. It is shown in Fig.4.. There are two parts in this figure, figure a shows the configuration of the agnets and reanent point fro which we can get the relationship between the air gap and the distance fro the agnets to the reanent point, and figure b shows the relationship between the attractive forces and rotation angle θ, where A: the reanent point θ: the rotation angle of the iron ball α: the angle between d and OA β x : the angle between d and OA λ: the angle between d 3 and OA γ: the angle between and d 4 OA z : displaceents of the peranent agnet z : displaceents of the peranent agnet z 3 : displaceents of the peranent agnet 3 z 4 : displaceents of the peranent agnet 4 : ass of the iron ball : ass of the agnet : ass of the agnet 3 : ass of the agnet 3 8

93 4 : ass of the agnet 4 f a : generating forces of the actuator f a : generating forces of the actuator f a3 : generating forces of the actuator 3 f a4 : generating forces of the actuator 4 f : attractive force between the reanent point and agnet f : attractive force between the reanent point and agnet f 3 : attractive force between the reanent point and agnet 3 f 4 : attractive force between the reanent point and agnet 4 d : the initial air gap length between the iron ball and agnets r: the axiu value of the displaceent of the agnets d : the air gap length between the iron ball and agnet d : the air gap length between the iron ball and agnet d 3 : the air gap length between the iron ball and agnet 3 d 4 : the air gap length between the iron ball and agnet 4 d : the distance between the reanent point and agnet d : the distance between the reanent point and agnet d 3 : the distance between the reanent point and agnet 3 d 4 : the distance between the reanent point and agnet 4 : the coefficient of the agnetic field of agnet : the coefficient of the agnetic field of agnet 3 : the coefficient of the agnetic field of agnet 3 4 : the coefficient of the agnetic field of agnet 4 f : the coponent of f along the tangent direction of reanent point f : the coponent of f along the tangent direction of reanent point f 3 : the coponent of f 3 along the tangent direction of reanent point f 4 : the coponent of f 4 along the tangent direction of reanent point f : the coponent of f in the Y direction f : the coponent of f in the X direction f 3 : the coponent of f 3 in the Y direction f 4 : the coponent of f 4 in the X direction Fro this figure, the rotation equation of the iron ball and peranent agnet can be obtained. I & θ = f f + f f R (4.3.) ( 4 3 ) I = R (4.3.) 83

94 f = f π cos( α ) = f sinα = f d + R sin( d θ ) (4.3.3) π d + R f = f cos( β ) = f sin β = f cos( θ ) (4.3.4) d f = f π cos( γ ) = f sin γ = f d3 + R sin( d θ 3 ) (4.3.5) π d 4 + R f 4 = f 4 cos( λ ) = f 4 sin λ = f 4 cos( θ ) (4.3.6) d Exaination Siulation exaination In this investigation, the open loop control with which the requireent of rotation can be satisfied is applied. Therefore, it is only needed to input a current to ae the agnets ove to and fro. Table 4. shows the sybols and values of this suspension echanis. 4 Table 4. values and sybols of suspension echanis Mass of the iron ball Mass of suspension agnet Initial air gap length Maxiu displaceent of agnet Radiu of iron ball VCM spring constant VCM daping constant Magnet field constant of left agnet VCM force constant d r R ξ a 63.7g.5g * -5 N 5N/A Fig.4.3 shows the siulation results of the iron ball rotation and agnets oveent. We assued that the direction is positive direction when the agnet oves towards the iron ball. Fro this figure, we can see that when the agnets oved periodically the iron ball can be rotated along the suspension axis. 84

95 Fig.4.3 Siulation exaination of the rotation velocity Experiental exaination Fig.4.4 shows the configuration of spinning echanis. Only one horizontal agnet otion control is shown in this figure. The other three agnets otion has the sae control syste with this figure. The agnet position is sensed by an eddy current sensor. The sensor s signal is converted to digital signal through A/D converter and then is input into DSP controller. Controller coputes the current for VCM and the signal is converted into analog value through D/A converter, and then the value is input a current aplifier to drive the VCM. The agnet is driven by VCM actuator to ove straightly towards or away to the iron ball. In this investigation, the suspension and rotation control are independent each other. 85

96 Fig.4.4 Spin echanis configuration Fig.4.5 Spin echanis control diagra Fig.4.5 shows the control syste of the experient. In this syste, four input signals have the sae period and the input signal phase difference of each VCM is π/ (rad) that eeps the approaching order of agnets. 86

97 Rotational Speed [rps] Tie [s] Fig.4.6 Result about rotation velocity Rotational Speed [rps] Tie [s] Fig.4.7 Result about rotation velocity The experient is carried out with the control syste. The iron ball is rotated successfully. The frequency of reference input signal is Hz, the rotation velocity tie history is shown in Fig.4.6. The rotation velocity is easured by eans of a laser feed onitor. Fig.4.7 shows the rotation velocity at the first second of the Fig.4.6. The average rotation velocity of the iron ball is about r/s Fro Fig.4.6 and Fig.4.7, we can see that the velocity of rotation is not unifor. That is caused 87

98 by open loop control as well as wea reanent on the surface of the iron ball. The cycle of agnet oveent is difficult to select when the rotation is started. A feed bac control of the iron ball angle is necessary for iproving the rotation. 4.4 Position the suspended object in horizontal plane With the sae construction and arrangeent of agnets polarity, the suspended object can be anipulated in the horizontal plane while it is suspended. It can be positioned at an arbitrary position in horizontal plane within a sall range. In this section, the principle, otion odel analysis, controller design and exaination are described. Y 4 R R eddy currenct sensor R X 3 eddy currenct sensor Fig.4.8 Planfor of syste in which the suspension agnet is oitted 4.4. Principle Fig.4.8 is the planfor of this syste in which the suspension agnet is oitted. In this figure, Y axis is perpendicular to X axis, so they can be regarded as the coordinate of this figure. The gravity point of the iron ball is located at the original point. Y axis and X axis divided the horizontal plane into four equal parts, part, part, part and part. Magnet and 3 can be oved along the Y axis whilst agnet and 4 can be oved along X axis. Through control the otion of four agnets to adjust the air gap between the iron ball and agnet, the iron ball can be positioned at 88

99 arbitrary position in the horizontal plane within a sall range near the original point. For instance, if the air gap between agnet and the iron ball and the air gap between agnet and the iron ball are adjust to saller, and at the sae tie, the air gap between agnet 3 and the iron ball and the air gap between the agnet 4 and the iron ball are adjusted to be larger, the iron ball can be anipulated to ove into part. As the sae reason, the iron ball can be positioned in the part, part and part in the horizontal plane. The eddy current sensor and sense the position of the iron ball in the horizontal plane. If the iron ball is oved along the X axis or Y axis this syste can be regarded as a -DOF syste, however, this syste can solve the proble which has been discussed in chapter 3. That eans when the iron ball is ove along the X axis, it is stable in the Y axis, vice versa. Y ξ f a z y z z x z ξ 4 z 4 f 4 f d d z ξ d 4 4 d 3 f X f 3 f a4 z 3 f a ξ 3 3 f a3 Fig.4.9 odel of positioning the suspended object in horizontal plane 4.4. Model analysis Fig.4.9 shows the otion odel of the iron ball and agnets when the iron ball is positioned in the horizontal plane. z : displaceent of the iron ball 89

100 z x : coponent of displaceent of the iron ball in X direction z y : coponent of displaceent of the iron ball in Y direction z : displaceents of the peranent agnet z : displaceents of the peranent agnet z 3 : displaceents of the peranent agnet 3 z 4 : displaceents of the peranent agnet 4 : ass of the iron ball : ass of the agnet : ass of the agnet 3 : ass of the agnet 3 4 : ass of the agnet 4 f a : generating forces of the actuator f a : generating forces of the actuator f a3 : generating forces of the actuator 3 f a4 : generating forces of the actuator 4 f : attractive force between the ball and agnet f : attractive force between the ball and agnet f 3 : attractive force between the ball and agnet 3 f 4 : attractive force between the ball and agnet 4 d : the initial air gap length between the iron ball and agnet d : the initial air gap length between the iron ball and agnet d 3 : the initial air gap length between the iron ball and agnet 3 d 4 : the initial air gap length between the iron ball and agnet 4 : the coefficient of the agnetic field of agnet : the coefficient of the agnetic field of agnet 3 : the coefficient of the agnetic field of agnet 3 4 : the coefficient of the agnetic field of agnet 4 : the linear coefficient of the agnetic field of agnet : the linear coefficient of the agnetic field of agnet 3 : the linear coefficient of the agnetic field of agnet 4 : the linear coefficient of the agnetic field of agnet : the spring coefficient of the actuator : the spring coefficient of the actuator 3 : the spring coefficient of the actuator 4 : the spring coefficient of the actuator ξ : the daping coefficient of the actuator 9

101 ξ : the daping coefficient of the actuator ξ 3 : the daping coefficient of the actuator ξ 4 : the daping coefficient of the actuator f x : the coponent of f in the X direction f y : the coponent of f in the Y direction f 3x : the coponent of f 3 in the X direction f 4y : the coponent of f 4 in the Y direction f : attractive force between the agnet and the iron ball at the initial position f : attractive force between the agnet and the iron ball at the initial position f 3 : attractive force between the agnet 3 and the iron ball at the initial position f 4 : attractive force between the agnet 4 and the iron ball at the initial position Fro Fig.4.9, the otion equations of the iron ball and agnets can be obtained. & (4.4.) zx = f f 4 f x f 3x & (4.4.) z y = f f 3 f y f 4 y & z = ξ z& z + f + f (4.4.3) a & z = ξ z& z + f + f (4.4.4) a & z = ξ z& z f + f (4.4.5) a3 & z = ξ z& z f + f (4.4.6) a4 The linearized otion equations are: & (4.4.7) zx = 4 ( z4 zx ) ( zx z ) ( f / d + f 3 / d 3 ) zx & (4.4.8) z y = 3 ( z3 z y ) ( z y z) ( f / d + f 4 / d 4 ) z y & z ξ z& z + z z + f (4.4.9) = ( y ) a & z ξ z& z + z z + f (4.4.) = ( x ) a & z ξ z& z z z + f (4.4.) 3 3 = ( 3 y ) a3 & z ξ z& z z z + f (4.4.) 4 4 = ( 4 x ) a4 9

102 9 The state space equation is bu Ax x + = & (4.4.3) cx y = (4.4.4) Where [ ] = z z z z z z z z z z z z x y y x x & & & & & & + + = ) ( ) ( d f d f d f d f A ξ ξ ξ ξ = 4 3 / / / / b = c = 4 3 a a a a f f f f u The ran of controllability atrix and observability atrix of this state space equation is, so the syste is controllable and observable Controller design The LQR (linear quadratic regulator) full state feedbac control law is used to design the

103 controller of this syste. The cost function is: J = t t t [ x Qx u Ru] + dt (4.4.5) Q = diag R = diag ( ) ( ) It is assued that the initial air gap length is.8(). The feedbac gain K is K = Exaination A step disturbance is applied to exa the suspension echanis with siulation ethod. The step value is.(). Table 4.3 shows the sybols and values of this echanis. Table 4.3 values and sybols of suspension echanis Mass of the iron ball Mass of suspension agnet VCM spring constant VCM daping constant Magnet field constant of left agnet VCM force constant ξ a 63.7g.5g.69* -5 N 5N/A Siulation exaination Fig.4. shows the step response of siulation exaination result. Fro this figure we can see that the step disturbance is added at (s). Converge tie is.(s) and there is no overshoot in the dynaic characteristic. 93

104 Fig.4. Step response of siulation 4.5 Discussion In this chapter, as the one step of the developent of the noncontact anipulation echanis, a noncontact spinning echanis with linear actuators and peranent agnets has been proposed. The echanis consists of five linear actuated agnets. And the reanent agnetization of the iron ball is utilized. In this echanis, the suspended object is suspended successfully; it can be rotated along the suspended axis when it is suspending; and it is verified theoretically that the suspended object can be positioned at the horizontal plane near the center position of the echanis. This echanis is a 4-DOF peranent aglev syste. As further wors, the angle control of the iron ball aybe realized as shown in Fig.4.. The adjustents of the positioning of the peranent agnets ae the iron ball to be positioned in arbitrary angle. 94

105 Fig.4. Angle control diagra Fig.4. Photo of the device 95