Torque Calculation Of Permanent Magnet Spherical Motor Based On Virtual Work Method

In recent years,modern industryhas put forward more and more stringent performance requirements for the mechanical device capable of multi-degree-freedom operation,such as industrial robots,artificial satellites,etc.A system that relies on cascade of multiple motor to achieve multi-degree-freedom operationgradually shows its drawbacks.The permanent magnet spherical motor(PMSM)that can realize three-degree-freedom operation with single jointhas the advantages of small size,simple structure,convenient control,which is more suitable for the mechanical device with multi-degree-freedom operation.However,the PMSM brings more complicated electromagnetic relationship.Torque analysis is a connect point of the structure design and control strategy,and it is especially important for the research and development of the PMSM.Therefore,it is necessary and important to establish the torque model of the PMSM.In this paper,based on the mapping relationship between the rotor magnetic pole distribution of the PMSM and its corresponding magnetic field distribution,the rotor magnetic poles are placed at the extreme position of its corresponding magnetic field expressed by the spherical harmonic.In order for the stator energized coil to generate the same magnetic field distribution as the rotor to produce a constant torque,it is proposed to place the coil in the same spatial position as the rotor pole.In addition,a non-contact support scheme for the PMSM is proposed by a gas-static bearing with a curved surface embedding in the stator hollow coil.Secondly,under the premise of considering only the fundamental magnetic field of the stator pole and rotor pole,the torque calculation model of the single pair of stator pole and rotor pole is derived by the virtual work method.In view of the fact that the obtained torque model is mainly related to the spatial angle of the stator pole and the rotor pole,the calculation method of spatial angle for any pair of stator pole and rotor pole of a single-layer octapole PMSMis derived by the spatial geometry method.In view of the three-degree-of-freedom running characteristics of the PMSM,the torque vector is decomposed into the stator coordinate system and the calculation method of the direction angle ofthree torque components is derived.The influence of the interaction between one pair of poles and another pole on the use of the superposition theorem is quantified,and then torque model of the single-layer PMSM is obtained.Thirdly,considering the computational complexity of the spatial angle and the direction angle,the calculation method of these two factors is derived by the space vector method for the double-layer PMSM.And then the normalized torque model of the double-layer PMSM is obtained by the superposition theorem.Finally,the finite element simulation model is established to verify the correctness of the obtained torque model.A prototype of double-layer permanent magnet spherical motor was fabricated and a three-degree-of-freedom torque measurement platform was designed to further verify the torque model.In addition,a relative error formula is used to measure the accuracy of the obtained torque model.