RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR

Transcription

1 RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR Yuanjie Fang *, Bing Xu, Chunhu Sun College of Mechanical and Electical Engineeing Chaohu Univesity, Hefei, China s: Submitted: Aug.30, 2016 Accepted: Oct.3, 2016 Published: Dec.1, 2016 Abstact- The attitude detection of 3-DOF actuato is an impotant technology of 3-DOF spheical actuato contol. Fo the space attitude change of the stato and the taditional detection method is invalid, the elative coodinate tansfomation is poposed in this pape, though the analysis of the coodinate systems of the spheical actuato. The tansfomed elative coodinate data is poved by using MATLAB platfom. The taditional coodinate tansfomation and the elative coodinate tansfomation ae compaed in this pape. Using the elative coodinate tansfomation algoithm, the elative position detection system based on the MEMS six-axis senso is designed. Using two sensos to detect both the attitude of the stato and the oto, the elative position detection system is ealized. The data eceived fom senso tansfomed by this method is used to veify the feasibility. Index tems: Relative Coodinate Tansfomation; Spheical Actuato; MEMS Six-Axis Senso; Position Detection System 1710

2 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR I. INTRODUCTION As a multiple degees of feedom mechanism, spheical actuato can ealize moe than 1 DOF movement. The mechanical stuctue can be simplified by using spheical actuato. Fo the diffeent stuctue of the of the actuato, the diffeent contol algoithm should be designed.[1-7] Single DOF sevo moto is studied deeply,moe and moe contol stategy and contol algoithms ae poposed[8] and the contol theoy is been developed vigoously.[9] Since the space scalability of the spheical actuato, the contol method should be based on the attitude detection algoithm. Now, the eseach of the spheical actuato detection algoithm has been developed widely. Hefei Univesity of Technology developed a detection algoithm which using the visual senso to detect the position of the oto that the suface is spayed.[10-11] A passive ball joint is designed as detection mechanism by Beihang univesity. The inclination angle senso and optical encode ae built-in the passive ball joint.[12-13]. Since it can only calculate the elative displacement between the oto and stato, the optical encode can t detemine the attitude of the oto. The detection method based on inclination angle senso can achieve the absolute attitude of the oto, the absolute position detection of the spheical actuato is difficult to apply to the space joint. The attitude detection system based on the attitude detection senso is elative to the geodetic coodinate system. Since the geodetic coodinate system is static, the stato of spheical actuato always assumed absolute est. The spheical actuato s contol algoithm should base on the elative position between the stato and oto. The actuato may be used in space motion to achieve the geate degees of feedom,. The elative position detection between the stato and oto is needed, the elative coodinate tansfomation and detection system is imminent. II. THE COORDINATE SYSTEMS OF 3-DOF SPHERICAL ACTUATOR The 3-DOF spheical actuato, same as convention moto, is composed of stato and oto. Since the convention moto has a fixed output shaft, by contast, the spheical actuato didn t have a fixed shaft, the otation axis is vitual that doesn t exist. The output shaft is used to poduce toque. The 3-DOF spheical actuato s stuctue is shown in Fig 1. When the spheical actuato otates aound the expected axis, the elative position between the oto and the stato is change. The contol algoithm of the 3-DOF spheical actuato will be 1711

3 influenced by the change of the elative position. Due to the stuctue of the 3-DOF spheical actuato, the tadition detection method is invalid when the spheical actuato acted in the space. In fact, 3-DOF spheical actuato geneally composed of 3 basic coodinate systems, shown as Fig 1, the stato coodinate system R s, the oto coodinate system R and the geodetic coodinate system R 0. In the taditional detection method, the stato coodinate system Rs and geodetic coodinate system R0 ae egaded as the same coodinate system. Fig. 1 The coodinate systems of 3-DOF spheical actuato a. Geodetic Coodinate System R 0 The geodetic coodinate system R 0 is Descates coodinate system; it s a fixed coodinate system. It meets with ight-handed scew ule; consist of thee othogonal coodinate axis X, axis Y, axis Z. The coodinate system is based on the geodetic, setting as efeence coodinate system. The diection cosine matix whose paamete can be diect measued is elated to this coodinate system. b. Stato Coodinate System R s The stato coodinate system R s, attached to the spheical actuato s stato, has 3 quateed axis x s,y s,z s and the cente of the stato sphee is set as oiginal point. The axis zs is pointed to the noth of the stato sphee. The stato is static elate to the oto. Since the stato coodinate system otates with stato in the space movement, the stato coodinate system is not a static coodinate system. c. Roto Coodinate System R 1712

4 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR The oto coodinate system R, attached to the spheical actuato s oto, has 3 quateed axis x,y,z and the cente of the oto sphee is set as oiginal point. The axis z is pointed to the noth of the oto sphee. The output toque, Loenz foce and space voltage vecto should be conveted in this coodinate system. Since the oto coodinate system is otated with oto, the oto coodinate system is a moving coodinate system. III. SPATIAL COORDINATE TRANSFORM The coodinate tansfomation is vey impotant in vaiable fequency closed loop system fo the single axis AC moto. The paametic convesion based on the position detection space coodinate tansfomation play an impotant ole in the 3 DOF spheical actuato detection systems. 3.1 Taditional Coodinate Tansfomation The coodinate tansfomation is widely used in the stapdown inetial navigation system. The quatenion [14] and the diection cosine matix methods often adopted in coodinate tansfomation method. The diection cosine matix is discussed in this pape. The elative position between oto and stato can be detemined by a slide fame measuing system. [15] Thee ae 3 optical encodes in the slide fame to detect position signals. The attitude of the oto can t be detected, and the detection system leading to inceasing of the fiction. The detection system is shown in the Fig2. Fig 2 Position Detection System with thee Encode Devices and Slide Holde 1713

5 The spheical actuato taditional coodinate tansfomation often contains two detection elements, the attitude of the oto and the elative position between stato and oto. The stato coodinate system is assumed fixed in taditional coodinate tansfomation, and the oto coodinate system is assumed otated by Eula angles. In this tansfomation method, the attitude of the oto can be detemined, but the elative position between oto and stato is leak measued. A senso is set to detect the oto attitude shown in fig 3, and the convesion of coodinates is shown in fig 4. [11] When the stato is otate in the space, the taditional coodinate tansfomation is invalid. Fig 3 Roto Oientation Measuement System Fig 4 Convesion of Coodinate Thee is a coodinate system a in the space, and the coodinate system is otated aound axis X α degees, otate aound axis Y β degees, otate axis Z γ degees. The coodinate system b is tansfomed. The otation of the 3-DOF spheical actuato is shown in the Fig 4.The coodinate tansfom diection cosine matix A b a fom coodinate system a to coodinate system b is shown below. 1714

6 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR b A a cos βcosγ cos βsin γ sin β = cosαsinγ sinαsin βcosγ cosαcosγ sinαsin βsinγ sinαcos β + + sinαsin γ + cosαsin βcosγ sinαcosγ + cosαsin βsin γ cosαcos β (1) The tansfomation matix can tansfom the coodinate system xyz to the coodinate dqp shown in the Fig 4. Assuming a vecto a 0 =[x a,y a,z a ] T in coodinate system a, it s tansfomed coodinate in coodinate b is b 0 =[x b,y b,z b ] T. The tansfomed new coodinate data should obey the fomula below: b = Aa b 0 a 0 (2) Using diection cosine matix can achieve any vecto coodinate data afte otation. This method is used in the 3 DOF spheical actuato position detection systems. Though measuing the otated angle, the cosine matix could be obtained [11]. 3.2 Relative Coodinate Tansfomation Accoding to the analysis of the coodinate system of the spheical actuato based on the font section, the spheical actuato s coodinate system should include the geodetic coodinate system R 0, stato coodinate system R s, oto coodinate system R. When the stato s attitude elate to the geodetic coodinate system have changed, the stato coodinate system will change with it. In this situation, the taditional position detection method is invalid. The 3-DOF spheical actuato s elative coodinate tansfomation includes two tansfomations, the stato-oto tansfomation (the tansfomation of stato coodinate system to oto coodinate system) and the oto-stato tansfomation (the tansfomation of oto coodinate system to stato coodinate system). The tansfomation of stato-oto could convet physical quantity in stato coodinate system into oto coodinate system. Relatively the oto-stato tansfomation could convet physical quantity in oto coodinate system into stato coodinate Roto-Stato Coodinate Tansfomation Assuming the spheical actuato wok in the space, the stato coodinate system R s is otated α,β,γ degees aound axis X,Y,Z accoding the otation ode x-y-z. The angles otated ae the 1715

7 stato coodinate system elate to the geodetic coodinate system. The two coodinate systems should obey: Rs = AR s 0 0 (3) A s 0 = Rx( α) Ry( β) Rz( γ) Whee is the diection cosine matix otation 3 times, satisfy the fomula (1). The oto coodinate system R is otated θ,φ,ψ degees aound axis X,Y,Z accoding the otation ode x-y-z. The angles otated ae the oto coodinate system elate to the geodetic coodinate system. Two coodinate systems should obey: R = AR 0 0 (4) A0 = Rx( θ) Ry( ϕ) Rz( ψ) Whee s is the diection cosine matix otation 3 times, A0 and A0 satisfy the fomula (1). Accoding to the diection cosine matix s popety, the matix s deteminant equals 1. Since s 1 A s 0 is exist, 0 ( ) A is evesible, and it can be solved fomula (5). A cos βcosγ cosαsin γ + sinαsin βcosγ sinαsin γ + cosαsin βcosγ = β γ α γ α β γ α γ α β γ + + sin β sinαcos β cosαcos β (5) s 1 ( 0 ) cos sin cos cos sin sin sin sin cos cos sin sin And fomula (3) can be ewitten in fomula (6). = s 1 R0 ( A0) R s (6) Take fomula(6) into fomula(4), it can be achieve fomula(7). R = A A R 1 ( s 0( 0) ) s (7) Rewite 1 A 0( s s = A A0), fomula(8) obtained. R = AR (8) s s As is the elative coodinate tansfomation matix. 1716

8 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR As is the stato-oto elative tansfomation matix. Accoding to fomula (1), the As is a special case of the stato otate the given axis 0. Using As tansfomation matix, abitay coodinate data in the stato coodinate system can be tansfomed into oto coodinate system. Only the otation angles α,β,γ, which acquied by the stato coodinate system elate to the geodetic coodinate system, and the otation angles θ,φ,ψ, which acquied by the oto coodinate system elate to the geodetic coodinate system should be measued to calculate the tansfomation matix elative position detection can be solved by the elative coodinate tansfomation. A s. The Stato-Roto Coodinate Tansfomation A s = 1 Accoding to, the elative tansfomation matix is invetible. The following fomula will be obtained. 1 Rs = ( As) R (9) s 1 That is, A = ( As) The fomula shows the data in the oto coodinate system can be conveted to the stato s A coodinate system. is the coodinate tansfomation matix. The matix of the stato to oto coodinate tansfomation and the matix of the oto to stato coodinate tansfomation ae too complicated will not shown in the pape. 3.3 Data Simulation Expeiment Taditional Coodinate Tansfomation Expeiment The data simulation expeiment is taken to test the taditional coodinate tansfom. Assume a point (0,0,1) on the sphee in the geodetic coodinate system, the oto coodinate system otate 1 ~20 aound x axis. The stato coodinate system doesn t otate and coincide with the geodetic coodinate system. These tansfomed coodinate data in the stato coodinate system ae shown in the Fig 5. These data equals the points in space coodinate otate 1 ~20 aound the x axis. The taditional detection is based on the algoithm. 1717

9 Fig. 5 Taditional Coodinate Tansfomation Simulation Result Stato-Roto Coodinate Tansfomation Expeiment 20 points on the spheical suface of stato ae taken, which coodinate data is (0, sin1,cos1 )~ (0, sin20,cos20 ), Assume the oto coodinate system otate 20 degees aound the Y axis, the stato otate 10 degees aound X axis,otate 30 degees aound Y axis, otate 0 degee aound Z axis. The coodinate data ae conveted to oto coodinate system by using the matix A s. The conveted coodinate and the oiginal coodinate ae shown in the same coodinate system. The display esult is shown in the Fig 6. The X point in the gaph is the point afte the tansfomation of the coodinate system, and the point of the O point is the font of the coodinate tansfomation. Fig. 6 Stato-Roto Coodinate Tansfomation Simulation 1718

10 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR Roto-Stato Coodinate Tansfomation Expeiment Similaly, 20 points on the spheical suface of the oto ae taken, which coodinate data is (0, sin1,cos1 )~ (0, sin20,cos20 ), Assume the oto coodinate system otate 20 degees aound the Y axis, the stato otate 10 degees aound X axis,otate 30 degees aound Y axis, otate 0 degee aound Z axis. The coodinate data ae conveted to stato coodinate system by using the matix s A.The conveted coodinate and the oiginal coodinate ae shown in the same coodinate system. The display esult is shown in the Fig 7. The X point in the gaph is the point afte the tansfomation of the coodinate system, and the point of the O point is the font of the coodinate tansfomation. The data points ae all on the spheical suface. Accodingly, the 20 sample data will otate 20 degees accoding to the Y axis, and elative to the stato coodinate evese otating 10 degees aound the X axis, evese otate 30 degees aound Y axis sample data ae pesented to show the otation befoe and afte the otation of the point. Fig. 7 Roto-Stato Coodinate Tansfomation Simulation The Compaison of Taditional and Relative Coodinate Tansfomation The compaison expeiment is taken to shown the diffeence of tansfomation method. Assuming a point, which coodinate is (0,0,1), on the spheical suface. The coodinate data is calculated in the situation of oto otate and the oto and stato otate both otate. The oto coodinate system elate to the geodetic coodinate system otate 1 ~20 aound X axis, otate 20 aound Y axis. The stato coodinate system is not otated elative to the geodetic coodinate system. The tansfomed coodinate system is calculated, the data achieved shown in the Table

11 Table 1: Tansfomation of Roto Coodinate System Rotation Situation (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) (0, , ) Relate to the geodetic coodinate system, the oto coodinate system otate 20 aound Y axis, the stato coodinate system otate 1 ~20 aound X axis, otate 20 aound Y axis. The oto coodinate system data ae tansfomed into stato coodinate system, the tansfomed data ae shown in the Table 2. Table 2: Tansfomation of the Stato and Roto Coodinate System Rotation Situation ( , ,0.9998) ( , ,0.9994) ( , ,0.9986) ( , ,0.9976) ( , ,0.9962) ( , ,0.9945) ( , ,0.9925) ( , ,0.9903) ( , ,0.9877) ( , ,0.9848) ( , ,0.9816) ( , ,0.9781) ( , ,0.9744) ( , ,0.9703) ( , ,0.9659) ( , ,0.9613) ( , ,0.9563) ( , ,0.9511) ( , ,0.9455) ( , ,0.9397) 1720

12 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR Fig 8 Compaative Simulation Diagam of Relative Coodinate Tansfomation It is seen fom the figue; the taditional coodinate tansfomation and oto-stato coodinate tansfomation data in diffeent coodinate systems diffeent esults. If the stato coodinate system is otated 45, using the esults of the taditional coodinate tansfomation is calculated fom the stato coodinate system is not otating coodinate system, that is geodetic coodinate system, which coesponds to the stato coodinate system is otated 45 incompatible with the actual situation. Taditional coodinate tansfomation is invalid. The esults of elative coodinate tansfomation shows the oiginal data of the oto coodinate system is otated aound the Z axis 45. It matches the actual situation. The elative coodinate tansfomation is suitable fo all kinds of situations. Taditional coodinate tansfomation can only applied to the stato coodinate system and the geodetic coodinates ovelap situation. Relative coodinate tansfomation is moe univesal significance. IV. DESIGN OF DETECTION PLATFORM 4.1 MEMS Six Axis Senso Attitude Matix Solution The attitude matix solution has been widely used in stapdown inetial navigation system. In the stapdown inetial navigation system, the aicaft geneally has a pitch, oll, yaw motion, which is consistent with the thee degee of feedom motion of the spheical moto. Using MEMS six-axis senso, the space otation between the caie coodinate system and navigation coodinate system can be detemined. The detected otation angles ae geneally used to measue to fill the attitude matix of the stapdown inetial navigation system. The stapdown inetial navigation system pinciple diagam is shown in Fig 9.[14-16] 1721

13 Fig 9. The Stapdown Inetial Navigation System Pinciple Diagam The attitude matix is used to obtain the solution of the spatial attitude, which is obtained fom the output of acceleomete and gyo senso. Attitude solution esult is often epesented as the pitch angle θ, oll angle ψ, yaw angle γ. The data obtained by MEMS six-axis senso include acceleation, angula velocity, attitude angle data and the output data of tempeatue compensation. The data solved in spatial otation ae deliveed to host compute. The data tansmitted by senso shown in Fig 10 by using MATLAB platfom. Fig. 10 Six-axis Senso Data Recod Chat 1722

14 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR 4.2 Relative Position Detection System of Spheical Actuato The spheical actuato elative position detection system should detect position of oto coodinate elative to the geodetic efeence coodinate system and the position of stato coodinate system elative to the geodetic efeence coodinate system. Two six-axis sensos ae used to detect both the stato attitude and the oto attitude. MPU6050 gyoscope is used to detect the attitude in the system. The stato gyoscope is mounted in the stato sphee south, and the oto gyoscope is mounted in the oto sphee south. The installation diagam is shown in the Fig 11. Fig. 11 Relative Position Detection System Installation Diagam of Spheical Actuato The otation angles ae detected by the gyoscope MPU6050. The output of MPU6050 is solved by attitude solution is thee Eule angles, which obey the otation ode x-y-z. Thee Eule angles is tansmitted to the USB to TTL cicuit by seial communication. The maste compute using Labview softwae to access 2 COM pots to obtain stato and oto attitude solution of the two gyoscopes. The system hadwae block diagam is shown in Fig

15 Fig. 12 System Hadwae Block Diagam The maste compute softwae is based on the Labview softwae in the designed elative position detection system. The diection cosine matix A0 s detected by the oto detection gyoscope. The diection cosine matix A0 is established by the Eule angles is established by the Eule angles detected by the stato detection gyoscope. The Eule angles obeyed the x-y-z otation ode. The data fom stato gyoscope and oto gyoscope ae accessed though COM inteface by Labview. Accoding to the communication potocol, the seial communication baud ate is 9600, 1 bit stop bit. The data obtained fom sensos ae filteed by median filteing method, the data ae shown in the chat 1 and chat 2 of Fig 13. In each display inteface, the thee cuves show the otation angle of the stato and oto data. Using the MATLAB scipts of Labview, the elative coodinate tansfomation matix is calculated. When the oto and stato otated in the space, the oto coodinate point (0,0,1) is tansfomed into stato coodinate system. Thee cuves in chat 3 show the tansfomed data. The system flow chat is shown in the Fig 14. s A 1724

16 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR Fig. 13 Maste Compute Panel Fig. 14 Maste Compute Softwae Flow Chat The physical pictue of the entie design platfom is shown in Fig 15. The annotation in gaphs is shown in Table

17 Fig. 15 Physical Relative Position Detection Platfom Table 3. Spheical Actuato Relative Position Detection System Numbe Name Roto Stato Output Shaft Stato Senso Roto Senso USB to TTL Cicuit Host Compute V. SENSOR DATA TRANSFORMATION When the oto otates, the otation angle of the senso is ead continuously by the oto attitude detection senso. The point (1,0,0) in the oto coodinate, in diffeent situation of stato otates 0 aound Y axis, stato otates 15 aound Y axis, stato otates 30 aound Y axis, stato otates 45 aound Y axis. The locus of the point is show in the Fig 16 annotated in. 1726

18 Yuanjie Fang, Bing Xu, Chunhu Sun, RESEARCH ON THE RELATIVE COORDINATE TRANSFORMATION AND RELATIVE POSITION DETECTION SYSTEM OF 3-DOF SPHERICAL ACTUATOR Obviously, due to the oto otation movement, the point data tansfomed in the oto coodinate system is shown as a seies of tace. That is, the point is otated equivalent. Whateve the stato attitude change, the tajectoy shape of the fou sub gaphs should be consistent. The simulation esult shows the chaacteistic. Accoding to the sub gaph 1-4, the sub gaph 2 to 4 is equivalent to the 1 sub gaph on the oiginal basis aound the Y axis once again otated 15 degees, 30 degees and 45 degees. Fig 16 Simulation Result of Roto Senso Data VI. PHYSICAL SIGNIFICANCE OF RELATIVE TRANSFORMATION MATRIX In the field of spheical actuato, the stato coodinate system Rs is consideed as the geodetic coodinate system R0. The paticulaity of the spheical moto makes it necessay to obtain the elative position of the stato and oto. In the situation of stato doesn t move, the stato coodinate system Rs is equivalent to the geodetic coodinate system R0, as a special case. When using Loentz foce to contol the spheical actuato, the elative position between the coil of the stato and the pemanent magnet should be obtained. Though detecting the otation angles the stato coodinate system Rs elative to the geodetic coodinate system R0, s the diection cosine matix A0 can be obtained. A0 As can be obtained in the same method. Then, the elative tansfomation matix is acquied. Geneally extended to thee degees of feedom of the moto Loenz foce vecto, magnetic vecto, stato and oto coodinates can complete the stato to oto and oto to the stato coodinate tansfomation. 1727

19 VII. CONCLUSION Though the analysis of the simulation esult, the taditional absolute position detection method of spheical actuato is impoved. The taditional coodinate tansfomation is set as a special case of the elative coodinate tansfomation. Using the elative coodinate tansfomation also can solve the taditional coodinate tansfomation. The actuato can be pefomed any attitude in space, i.e., applied to the joints of the obot am. Setting both oto and stato MEMS six-axis sensos, the stato and the oto attitude can be detected. The elative position detection can be achieved by using the elative coodinate tansfomation. The elative coodinate tansfomation has physical meaning. Relative coodinate tansfomation applies fo 3 DOF spheical actuato and is moe scientific and univesal. ACKNOWLEDGMENT This wok is patially suppoted by Povincial Natual Science Reseach Poject of Anhui Povince (KJ2014A173) and Suppoted by the Open Reseach Fund of Anhui Key Laboatoy of Electic Dive and Contol, AnHui Polytechnic Univesity. The authos also gatefully acknowledge the helpful comments and suggestions of the eviewes, which have impoved the pesentation. REFERENCES [1]Leopoldo Rossini;Emmanuel Onillon;Olivie Chételat, Closed-loop magnetic beaing and angula velocity contol of a eaction sphee actuato, Mechatonics, Vo. 30, 2015, pp [2]Liang Yan, Zewu Wu, Zongxia Jiao, Equivalent enegized coil model fo magnetic field of pemanent-magnet spheical actuatos, Sensos and Actuatos A: Physical, 2015, 229,68-76; [3]Z. Li, Q. Lun, D. Xing, and P. Gao, Analysis and Implementation of a 3-DOF Deflection-Type PM Moto, IEEE TRANSACTIONS ON MAGNETICS.VOL. 51, NOVEMBER 2015 [4].MASAHITO TSUKANO, KATSUHIRO HIRATA, SHUHEI MAEDA, Development of Contol Method fo Oute-Roto Spheical Actuato, Electical Engineeing in Japan, Vol.194, 1728

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