Study on Mathematical Modeling and Experiment of Inductive Magnetic-gas Mixed Suspension Spherical Joint

The multi-degree-of-freedom inductive spherical driving joint owns high mechanical integrity and has advantages such as control and trajectory planning. Because of its limit of rotor pose, instant stepping angle and inability to spin around any axis through center of sphere, the suspension and rotation precision of spherical reluctance driving joint isn’t high and the response speed can’t reach the best. Based on mechanism of rotation and suspension, due to distribution of air gap magnetic induction intensity, a novel inductive maglev spherical driving joint was presented, and electromagnetic suspending force and electromagnetic torque were established. Analyze the magnetic induction intensity, electromagnetic suspending force and electromagnetic torque through finite element simulation, build a bench test of maglev spherical driving joint, and do testing research on displacement characters and suspension and rotation characters of joint rotor.


Introduction
Robots and mechanical arms are usually driven by multiple single-degree of freedom devices and complex transmission mechanism.They gain three-dimensional motion, which lead to a serious of problems (1,2) , such as the increase of complexity, big volume, serious wear of friction surface of each joint, low efficiency, slow response and poor dynamic performance.As an entry point of research, the study of the system of magnetic levitation spherical driving joint can simplify the mechanism, realize the aim of no friction between rotor and stator and no wear of suspending supporting, improve the positional accuracy of joint and response speed and reduce the volume of mechanism to obtain high response speed.In this research field, magnetic levitation spherical motor and driving joint were first introduced in the world and a number of national patents were applied (3,4) .The research work has a study of mechanism of spherical surface reluctance motor's maglev spherical reluctance driving joint (5) , builds magnetic field flux and electromagnetic suspending force and electromagnetic torque model of joint (6,7) , and establishes joint ideal spherical rotor three-dimensional dynamics nonlinear coupling model and linearization of decoupling control is performed (8,9) .With the limit of rotor pose, instant stepping angle and inability to spin around any axis through center of sphere, its suspension and rotation precision of spherical reluctance driving joint is insufficient.In order to make response speed does not reach the fastest, the final pose of end effectors of joint have to be determined according to the planned track in advance.Therefore, a novel inductive maglev spherical driving joint was presented.
Study mechanism of rotation and suspension and establish a mathematical model of joint system.Analyze the suspension and rotation characters through finite element simulation and testing.The stator is stacked by 50W470 silicon steel sheet.There is a groove, with concentrated distribution winding method, in the middle of stator, embedded with three-phase winding whose connection mode is Y shape.When there is electricity, each phase winding not only generated electromagnetic torque that can drive rotor revolve but also support radial maglev force for rotor.Spherical rotor, compacted on axle by round nut and shaft shoulder, is stacked also by 50W470 silicon steel sheet.On the surface of rotor, there is a mutually orthogonal groove.In the orthogonal point along radial direction to the centre of sphere, there is a hole.The Cu or Al is injected into groove and hole and then the armature structure of emanate orthogonal spherical cage shape will be formed in the core of centre of sphere short sub and intersect.The magnetic potential is generated by each phase current of three-phase stator winding of maglev spherical driving joint in the way of sine or cosine distribution.By ignoring ultra harmonics component, groove leakage inductance, end region leakage inductance and magnetic saturation.A three-phase fundamental wave combining magnetic induction intensity (10) came over three-phase winding magnetic potential of a stator, whose air gap of magnetic field between stator and rotor is figured out.That is defined as ( , ) cos( )

Maglev Spherical Joint Mathematical Model
Where θ is the (tangential) angle of rolling stator axle, Bm is magnetic induction intensity amplitude, which depend on winding incoming current and air gap length and p is magnetic field number of pole pairs generated by joint winding.
The magnetic flow of inductive magnetic levitation driving joint stator winding cut rotor spherical cage armature winding in space and induct ac electromotive force and electricity in the rotor.The rotating flux is produced in air gap as ac electricity pass rotor.Its magnetic induction intensity is defined as ( , ) cos( ) Where K is inductance coefficient of joint stator and rotor, α is phase (including rotor rotating angular displacement) difference between rotor winding and stator winding rotating magnetic field.K and α all depend on structure and slip ratio of inductive magnetic levitation driving joint stator, rotor.The total magnetic induction intensity generated by stator winding and rotor winding in peripheral air gap is defined as As shown in Eq.( 3), distribution regularities of magnetic induction intensity of inductive magnetic levitation spherical driving joint air gap are relating to magnetic field number of pole pairs generated by stator winding, tangential angular position rotating stator axle and phase difference between rotor and stator rotating magnetic field.
(2) Distribution regularities of joint air gap magnetic energy The envelope radius of the sphere of stator salient pole is R, the envelope radius of the sphere of rotor salient pole is R r , the air gap between inner diameter of stator salient pole and external diameter of rotor salient pole is g =R-R r ,if rotor centre of sphere of magnetic levitation spherical joint has no deviation.A rectangular coordinate system f(X，Y， Z) and coordinate system F(R，φ，θ) (see Fig. 3) are built by adopting a stator axle coincides z axle.If rotating virtual displacement generated by rotor rotates around stator axle is Δα, radial virtual displacement is e=ix+ jy+ kz, that is to say rotor centre of sphere moves from O(0,0,0) point to O 1 (x, y, x) point, as R r >>x, R r >>y, R r >>z, the change rules of radial air gap length between stator and rotor salient pole envelope surface on optional position of stator magnetic pole envelope annularly spherical surface can be expressed by： 0 ( ) cos sin sin sin cos The plane between stator and rotor salient pole is regarded as parallel plane, because the position deviated from equilibrium position is very small.The reluctance of working air gap is only considered in the whole magnetic circuit system by ignoring iron core, reluctance and eddy-current loss.The enveloping spherical crown on the stator pole, when air gap is g c (φ), take a tiny sphere, surround stator axis, which width is Rdφ, length is Rcosφdθ If joint stator parameter is φ 1 =0, φ 2 =π/4, then air gap total magnetic energy is defined as: (3)The distribution regularities of joint electromagnetic suspending force and electromagnetic torque According to principle of virtual displacement and Eq.( 6), radial virtual displacement z and rotating virtual displacement Δα are in derivative respectively.Consequently, electromagnetic suspending force along direction of stator axle and electromagnetic torque rotated around stator axle, generated by a stator of magnetic levitation spherical driving joint, are got.That is defined as As Eq.( 7) is illustrated, magnetic induction intensity is defined as Where N is joint stator number of turns of winding, I is current amplitude in flowed into stator winding and g is air gap length between spherical rotor and certain joint stator.
As Eq. ( 7) is shown, the distribution regularities of (2 )(1 2 cos ) 16 Where M and J respectively are spherical rotor mass and rotational inertia, When rotors are interfered and their location are changed, winding current changes with that but magnetic induction intensity remain unchanged.Therefore, stability of joint electromagnetic torque nearly doesn't be influenced.Geometric mesh model of stator and rotor of magnetic levitation spherical driving joint can be built (see Fig. 4) by Table 1 parameters.It border subdivision picture is magnified, as Fig. 5 shows.

3. 1
Joint suspending force and electromagnetic torque model (1) Joint air gap magnetic induction intensity's regularities of distribution

Fig. 3 .
Fig.3.Schematic diagram of change of rotor air gap According to the theory, magnetic field permeability of vacuum is μ0 then the total stored energy of joint air gap magnetic field is defined as

( 4 )
electromagnetic suspending force and electromagnetic torque generated by inductive magnetic levitation spherical driving joint is cosine and sine along direction of rotating.The amplitude of electromagnetic suspending force and electromagnetic torque has the direct ratio to square of magnetic induction intensity Bm (have the inverse ratio to square of air gap length g).Joint rotor dynamics equation According to Newton's second law, a kinetic equation of suspension and rotating of spherical rotor driven by a stator can be defined as F d and M d separately are disturbing force and moment besides spherical rotors are under electromagnetic suspending force/ torque.If spherical rotor deviation is z, under spherical rotors are interfered.In order to recover spherical rotor in balance state and keep air gap length is g , controller should be adjusted to current of stator winding is magnified or decreased I=I 0 ±ΔI.Then, magnetic induction intensity of air gap in equation mentioned is defined as

4. 1 .( 1 )
Air gap magnetic field character of magnetic levitation spherical driving joint Finite element analysis model Structure parameters of magnetic levitation spherical driving joint are given in

5. 2 Fig. 15 .( 2 )Fig. 18 .
Fig.15 shows the oscillogram of rotor when it is floating.There is on load on the driving joint.The designed air gap between rotor and stator is 0.6mm.The primary clearance of rotor's offset downward is 0.1mm.With the action of suspending control, spherical rotor will move to the center of stator, and achieve stable suspension.Rotor's maximum fluctuating peak value is 8μm when stably suspending.
Fig.18 shows the changing law of spherical rotor suspending force affected by rotor displacement.Use eddy current displacement sensor to detect rotor displacement and rotor and the rotor's rotating position is detected by two sensors.Voltage source controlled by digital microprocessor is direct voltage of 36V that is offered by a 3-phase pulse width modulation inverter, and the inverter's switching frequency is 100 kHz.The sampling frequency of suspending control and current control is 300μs and 100μs, separately.From the picture, when the spherical rotor deviates from the balance point with a distance of under 0.3mm, it is usually linear.
Papers submitted to ICISIP must describe original and unpublished researches.They should not have been submitted to other conferences or journals.Any paper which is under review elsewhere will not be considered for publication.Once an eligible paper is accepted, it will be published under a Creative Commons Attribution 3.07.ConclusionsThis paper introduces the fundamental composition and structure features of inductive maglev spherical driving joint.Use the principle of virtual displacement to establish the model of electromagnetic levitation force and electromagnetic torque, based on the distribution law of air gap magnetic field of inductive maglev spherical driving joint.By finite element simulation analysis, verify the distribution of air gap magnetic induction intensity, electromagnetic levitation force and electromagnetic torque of inductive maglev spherical driving joint.Put forward the control circuit of inductive maglev spherical driving joint.Study the characteristics of rotor displacement, electromagnetic levitation force and electromagnetic torque of inductive maglev spherical driving joint.

Table 1 .
1. Structure parameters of magnetic levitation spherical driving joint