Extended unified wrench model suitable for the end effect of the ironless permanent magnet planar motor

The ironless permanent magnet (PM) planar motor (IPMPM) has great potential to realise multi-degree of freedom (DoF) motion. The insufficient ability to model the end effect limits its performance. Here, an extended unified wrench model (EUWM) of IPMPM is proposed through a systematic extension of previously proposed unified wrench model (UWM). Based on the concept of periodic extension, the EUWM is established in three steps. First, the side region and the vertex region are defined and extended into periodic pattern and the residual magnetisation vectors of the PM array in each region are expressed in Fourier series form. Then, the problem with unequal periods in the x and y direction is presented. Second, the space is divided into three layers and the magnetic flux density in each layer is solved by scalar potential method. In the last step, the decoupling characteristic of the wrench vector act on the infinitesimal current carrying volume is found. Then, the analytical expression of the EUWM is derived. It is much more general and suitable for the IPMPM with arbitrary-shaped coil and the PM array with unequal periods in the x and y direction. Finally, the EUWM is verified by experiment.