## 1 Introduction  
Magnetically levitated planar motors (MLPM) show promising applications in driving advanced manufacturing equipment (extreme-ultraviolet lithography, nano-imprint lithography, and other nano/micro manufacturing equipment) that are used in high-vacuum conditions, because these motors can move without any mechanical friction and wear [1; 2; 3]. They are mainly constituted of two typical parts: stator and mover. The stator and the mover can be collocated in two ways: the mover consists of a coil array and the stator has a permanent magnet array (MLPM with moving coils) [2; 3; 4; 5; 6], and the mover consists of a permanent magnet array and the stator consists of a coil array (MLPM with moving magnets) [1; 7; 8]. No matter how MLPM are configured, they can all be classified as permanent magnet synchronous motors. For these permanent magnet synchronous motors, either planar or rotary, the commutation of windings or coils is always the key link. Once these motors have been physically configured, the magnetic field distribution cannot be changed; thus, the regulation of coil or winding currents, i.e., the commutation becomes the only alternative to control these motors [7; 9; 10]. These coils or windings have to be commutated to provide adequate energy for six degrees of freedom (DOF) movements in three-dimension Euclidean space [11]. Moreover, these coils or windings still have to be accurately commutated so that the six-DOF movements can obtain better decoupling and controlling performances [12; 13]. Figure 1 shows the basic control structure of MLPM, in which \(\mathbf{q_{n\!f}}\) and \(\mathbf{q}\) denote reference input vector and output vector, respectively; \(\mathbf{W_{\mathrm{c}}}\) and \(\mathbf{W_{\mathrm{d}}}\) denote control forces/torques and desired forces/torques, respectively, and \(\mathbf{i}\) denotes the commutated current vector. Different commutation methods will result in different distributions of currents and ohmic losses (refer to Joule losses or dissipated power of coils, hereinafter named power loss) among all the coils, although these currents can generate identical electromagnetic forces and torques. In the literature [10; 14], a commutation method called direct wrench-current decoupling was proposed, which includes smooth switching of coil currents for MLPM with moving magnets. Using this method, wherein the coil currents are solved by the pseudo inversion of the electromagnetic force/torque coefficient matrix, the coils are commutated optimally in the sense of minimum power loss of the coil array. On the other hand, peak current is not taken into account and power loss distribution among coils usually shows disequilibrium. In page 11 of reference [14], the temperature distribution of commutated coils in some situations are measured by thermal imaging and the temperature range from 20 \({}^{\circ}\)C to 37.6 \({}^{\circ}\)C is observed among commutated coils. In reference [15], an apparent temperature distribution difference from 30 \({}^{\circ}\)C to 75 \({}^{\circ}\)C is also observed. Reference [16] illustrates a coil current distribution from 0 A to 3 A when a MLPM with moving magnets is desired to track a designed rectilinear movement. Reference [17] also illustrates the different temperature distributions from 10 \({}^{\circ}\)C to 60 \({}^{\circ}\)C. These studies indicate the obvious disequilibrium with respect to currents and power loss of coil array of MLPM. When the power loss distribution of coil array is under a state of imbalance for a relatively long term, it may result in serious MLPM control problems. One or more coils could break down or even burn out too early because of frequent overheating and thermal shock. It could also cause irreversible demagnetization of the permanent magnets. Thus, the commutation and control precision of the motor could deteriorate or go out of control. In addition, the power loss disequilibrium of the coil array will confuse the capacity selection of the coil power amplifiers.