Study on the Suspension Model and Control of Spherical Driving Joint with Magnetic Levitation

The traditional machine joints has friction and wear on the mechanical support when they are working, so these make the moving parts hot, and cause the static and dynamic performance worse. Based on the integration of motor, magnetic levitation and robot technology, a new kind of mult i degree freedom magnetic levitation active ball joint without mechanical friction, wear and lubrication is presented. The model of electromagnetic levitation fo rce and the electromagnetic torque for the dynamic the jo int is established, which is based on the state reconstruction principle and is used to realize the state synchronous suspension control and the self-detection feedback control without sensors. Then the simulation is done, which shows that the system has good dynamic response and anti-disturbance.


Introduction
Robots and mechanical arms are usually driven by mu ltip le single-degrees of freedo m devices and comp lex transmission mechanism.They gain three-d imensional motion, which lead to a serious of problems [1,2] , such as the increase of co mp lexity, b ig volu me, 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 simp lify the mechanis m, 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 volu me of mechanis m 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 jo int [5] , builds magnetic field flu x and electro magnetic 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][10] .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 driv ing jo int 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.There-fore, a novel inductive mag lev spherical driving joint was presented.Study mechanis m of rotation and suspension and establish a mathematical model of joint system.Analy ze the suspension and rotation characters through finite element simulation and testing.

Maglev Spherical Joint Basic Composition
The 3-D structural drawing of stator and rotor of magnetic lev itation spherical driv ing jo int is shown in Fig. 1.
Stator 1 and 3 are spherically symmetric and keep coaxial, and they are with functions of driving rotor spin around stator axle and generating maglev force wh ich is able to make rotor along the direction of stator axle.Stator 2 and 4 are spherically sy mmetric and keep coaxial, and their roles are driving rotor spin around its axle and producing mag lev force wh ich can make rotor stable suspending along the direction of rotor axle.Stator 5 was arranged on the bottom of spherical rotor, and its effects are driving spherical rotor spin around stator axle and producing maglev force wh ich can make rotor stably suspend along the direction of stator axle.Every stator axle is mutually orthogonal, which produce an intersection that is center of sphere composed by every stator salient pole spherical enveloping surface.The structural drawing of magnetic levitation spherical driving jo int's rotor and stator is shown in Fig. 2. The stator is stacked by 50W 470 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 electro magnetic torque that can drive rotor revolve but also support radial mag lev force fo r 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 o rthogonal groove.In the orthogonal point along radial d irect ion to the centre of sphere, there is a hole.The Cu or Al is in jected into groove and hole and then the armature structure of emanant orthogonal spherical cage shape will be formed in the core of center of sphere short sub and intersect.

Suspension Control Model of Spherical Reluctance Driving Joint with Magnetic Levitation
Fro m foregoing, the magnetic levitation sphere driving joint is the driving joint wh ich is supported and driven by Multi-Degree-of-Freedo m spherical reluctance motor.The research for magnetic lev itation s phere driving joint can start at the air-gap magnetic energy between stator and rotor of the motor fo llo wing the method to study the magnetic reluctance motor.It means that establish the transformational relat ion of mechanic and electric energy according to the relevance theory of air-gap magnetic energy, and then obtain the relationship between electro magnetic torque of driving rotor and electro magnetic levitation force which can keep the rotor suspend.In order to brief the analysis, assumes that: 1) three phase stator winding distribute balanced in the space, the magnetic potential generated by each phase current is sinusoidal distribution(or cosine distribution)in air-gap space, and neglect it's higher harmonic component; 2) neglect the effection of short range co mponent group and distribute winding for air-gap magnetic potential, it means that the value of winding factor is 1; 3) neglect groove leakage inductance, end leakage inductance and magnetic saturation; 4) neglect the magnetic resistance of iron core and eddy current loss, only the magnetic resistance of operating air-gap is considered in the whole magnetic circuit system.
With the rotation of spherical rotor of the magnetic levitation sphere driving joint, the magnetic torque is created by the magnetic pulling force wh ich is generated while one surface of salient pole of rotor overlapped with the surface of salient pole of stator partly.Moreover, this magnetic pulling force is the magnetic levitation force which keeps the rotor suspended.Therefore, the variat ion of air-gap length which be passed by the flu x wh ile the flu x generated by magnetic field of stator flows to spherical rotor is co mp lexity.And it is a function of  and , it is obtained as ( , ) As a result, the magnetic conductance is the function of  and  which can be defined as ( , )   .According to the method of dealing with the magnetic conductance of the maglev spherical reluctance motor driven by the levitation force in axial plane in document, the unit-area magnetic conductance of the mag lev spherical reluctance motor d riven by the levitation force in axial p lane in this paper is inversely proportional to the gap e which between the stator and the rotor while their magnetic pole of salient pole aligns with each other.Furthermore, it is cosine distribution shown as where K 0 and K are magnetic conductance constant which have the relation with the structure of stator and rotor; ω is the angular speed of the rotor relat ives to the stator; ψ is the phase difference between one stator and the rotor wh ile the stator drives the rotor rotating around the coordinate axis of the stator; e is the radial air-gap between the magnetic salient pole of stator and the rotor.Due to the width of the magnetic pole enveloping spherical surface ring on the stator is small, it means the variation range of φ (φ1<φ<φ2) is small.It is considered that the surface of magnetic po le on stator is parallel with the surface of magnetic pole on the rotor which aligned with the stator, thus e could be shown as where  0 is the angle between the normal line at the position of centroid of stator magnetic poleand Z-axial.Thus, the magnetic conductance when flu x crosses the micro-area (3) Assuming that magnetic potential(magnetic pressure) h s of stator winding generated in air-gap d istributes as sinusoidal distribution or cosine distribution [5] .It is cos( ) where H s is magnetic potential amp litude, the product of the current I and number of turns of stator N, that is H s = I• N. The micro-magnetic-energy generated in air-gap between stator and rotor is 2 1 ( , ) 2 Taking the range of integration(φ1<φ<φ2, 0<θ<2π), and getting the integral of above equation, the air-gap total magnetic energy is obtained as The equation 6 is the relat ion of the mechanic and electric energy transformation of magnetic levitat ion sphere driving jo int.It is used to obtain the derivation of the difference of phase between stator and rotor and the displacement in radial d irection.Thus, it can obtain the electro magnetic torque and maglev force of magnetic levitation sphere driving joint.The mag lev force along Z-axial direct ion generated by the magnetic levitation sphere driving joint is where Where K z is displacement stiffness; K i is current stiffness.
In terms of the Newton's Second Law, the equation of the spherical rotor's movement along Z-axial direction is Where m is the mass of spherical rotor; F d is disturbance except maglev levitation force.
Substituting equation 8 to 10, the equation of the spherical rotor's movement along Z-axial direct ion is obtained as According to the law of electro magnetic induction, the relationship of current i and voltage u in winding is obtained as where . The state structure of the system is shown as Fig. 4 Fig. 4 State structure drawing of the stator sub-system In accordance with the state equation of sub-system (equation 13), the characteristic root of the system is positive real nu mber or positive real part co mplex nu mber.So, the system is an open loop unstable system wh ich should be took the closed loop integrated correction to keep the suspension system of magnetic levitation sphere driving joint to content the performance requirement.

Design of Synchronous Suspension Control System Based on State Reconstitution
The structure of synchronous suspension control system with feedback [4,5] with observer constituted by the sub-system (Fig. 4) is shown in Fig. 5.In this drawing, sub-system 1 designs the value of the feedback matrix of output error G on the principles of state reconstruction, and makes it 's state ́ following the state x̀ of sub-system 2.Moreover, the follow error of the stable state is zero.The state observer is another sub -system which accomp lished by computer or analogue circuit with the selfsame structure with sub-system 1.The purpose of this design is getting the s tate feedback control of the spherical rotor obtains self-detected displacement at one direction without sensor, as well as realizing the position feedback, speed feedback and the integrated correction control of current feedback.
According to the equation 13, controllability matrix and observability matrix o f the system is full ran k.It means that the system is co mpletely controllable and observable.
Thus, it can be state reconstructed and state feedback controlled.In order to driv ing the spherical rotor to move in synchronism along right about when the two systems come to steady state, and achieving self-detect state feedback control without sensor by the state observer, the output error feedback matrix can be designed according to the principle of state reconstruction to get the two stator sub-systems and the state observer with selfsame state variable.
Assuming the output error feedback matrix between the sub-system and state observer is state equation of the two stator sub-system from Fig. 5 is The equation of state observer is ∫ = (  −  )  +  + ̀ (15) Their closed loop characteristic polynomial is According to the pole point assigned by the performance requirement of the system, the value of the state synchronous error feedback regulate matrix G between the two stator sub-systems and the observer can be confirmed by the above equation.Substituting the definite parameters in matrix A and C of the equation (13) to above equation, the characteristic polynomial is obtained as In accordance with the indices of performance of the system, if pole point assigned at 1,2 40 s  (key pole point), 3 400 s  , the expected closed loop characteristic polynomial of the system is 2 ( ) ( 40) ( 400) Substituting the structure parameter of the system to the equation 17 and 18, and co mparing them, the state reconstruction matrix can be obtained as Assuming the state feedback matrix is   After the state reconstruction is achieved, the state variable of each sub-system and obsever and output variable tend to equality when the system comes to steady state.So, the closed loop state equation of the sub-system state feedback is shown as The closed loop characteristic polynomial is Similarly, according to the pole point assigned by the performance requirement of the system, the value of the system's state feedback matrix K can be determined by above equation.Substituting the definite parameter of the matrix A and b in 13 to equation 17 and 18, characteristic polynomial is obtained as Substituting the structure parameter of the system to equation 22 and 18, and co mparing them, the state feedback matrix can be obtained as 102322.80130.24 96.7468 in Fig. 7.It can be seen that, the response speed of the system is rapid.And the adjustment time is less than 0.2s.

Suspension Control Characters of Spherical Reluctance Driving Joint with Magnetic Levitation
The motion trial of the hot core of spherical rotor deviates fro m equilib riu m position along Z-axial direction while the input signal is sine AC signal with frequency as 50 Hz in Fig. 8.With the effect of the two stator sub-system's synchronous control, it follows that the deviant of the rotor deviates from equilibriu m position is stochastic and quite small which less than4×10 -18 m.After the balance of the rotor, at the t ime s t 3  , adding step interference signal at z d irection and observe the displacement at z direction in Fig. 9.It uses state observer and state feedback control in Fig. 9 (a) and uses PID control in Fig. 9

Fig. 1
Fig.1 Schematic diagram of maglev spherical driving joint

Fig. 3
Fig.3 Schematic diagram of the change of rotor air gapWith the condition that there is no excursion for the center of sphere of rotor of the magnetic levitation sphere driving joint, the radius of spherical surface of stator salient pole is R, the radius of spherical surface of rotor s alient pole is R r , and the gap value between the inner diameter of stator salient pole and the external diameter of rotor salient pole is g 0 = R -R r .Taking the symmetry axis coincides the Z-axial.Establishing the coordinate frame  (  , ,  ) = (, , ).If the center of sphere of rotor moves fro m point O to point O 1 under the disturbance of the driving jo int, shown in Fig.3, the excursion along with Z-axial direction is z.Taking a micro -spherical-surface in the enveloping spherical surface ring on the six stators, and giving the value of the broad is Rd and the length is rd , thus it's aera is ( ) ( ) A rd Rd is the current influx in coil; N is the number of turns of stator coil.Thus it can be seen, the lev itation force of magnetic levitation sphere driving joint along Z-axial d irection is inversely proportional to the square of the gap ( between the salient pole of stator and rotor, and p roportional to the square of the magnetic potential Hs generated by coil.Consequently, changing the magnetic potential (or current in the coil) could control the displacement z at Z-axial d irection of spherical rotor, and keeping the sphere driving joint rotor suspended stably.As the displacement z of rotor generated, decreasing the current i flows into coils to regulate the position of the rotor and get it returns to the equilibriu m position.The current can be shown as I = I 0i.When the variation of the displacement is small ( induction of windings when the rotor at the equilibrium position.The equation 8 and 12 is the open loop suspension control model of stator sub-system of the magnetic levitation sphere driving joint.The state variable is assumed without disturbance as 1 x z  ，  2 = ， 3 x i  .And taking the winding current i as output to get the result that the displacement of rotor is detected without sensor.Thus, according to the equation 11 and 12, the state equation and output equation of open loop is obtained as

Fig. 5
Fig.5 Structure drawing of the synchronous suspension closed loop control based on state reconstruction In order to driv ing the spherical rotor to move in synchronism along right about when the two systems come

Fig. 6 Fig. 7
Fig.6 Control system maglev spherical driving join As shown in Fig 6, design a control-driven system of joint based on the verification of the characteristics of rotation and suspension of inductive maglev spherical driving joint.The system makes a differential d rive of spherical rotor through two stators up and down, which makes the rotor to suspend and rotate.The control of rotor's rotating and suspending around z-axes is based on the closed-loop control of motor control system: Input the

Fig. 8 Fig. 9
Fig.8 Rator trajectory of departure from balance position on inputting 50Hz AC signal (b).It can be seen that the two both could get the system back to steady state soon.Using state observer and state feedback control can back to steady state pass by one oscillation.And the biggest deviant is 3.8×10 -4 m.Using PID control can back to steady state pass by two oscillat ions.And its biggest deviant is 3.1×10 -4 m.It is thus clear that basing on the use of state observer and state feedback controller; it has better capacity of resis ting disturbance and robustness.