## 2. Commutation and Power Loss Equalizing Degree of Coil Array
### Commutation of Coil Array
This matrix characterizes the force/torque coeficients of MLPM between the coil currents and the electromagnetic forces/torques, just as the torque constant of the rotating motor. However, this coeficient matrix is not a constant matrix and it is still dependent on the position and attitude vector \(\mathbf{q}\) that the moving coordinate system _o-xyz_ locates in the fixed coordinate system _O-XYZ_. Herein assuming coils of \(n\) number are commutated by current vector \(\mathbf{i}\):  
\[\mathbf{i}=\left(i_{1},\ i_{2},\ \ldots,\ i_{n}\right)^{\mathrm{T}} \tag{5}\]  
and the mover consisting of coils of \(n\) number as a rigid body, the \(x\)-, \(y\)-, and \(z\)-direction forces \(F_{x}\left(\mathbf{q}\right)\), \(F_{y}\) (\(q\)), and \(F_{z}\left(\mathbf{q}\right)\) and \(x\)-, \(y\)-, \(z\)-axis torques \(T_{x}\left(\mathbf{q}\right)\), \(T_{y}\left(\mathbf{q}\right)\), and \(T_{z}\left(\mathbf{q}\right)\), which act on the mover, can be added up by using the superposition principle as:  
\[\mathbf{W}(\mathbf{q})=\left[F_{x}(\mathbf{q}),\ F_{y}(\mathbf{q}),\ F_{z}(\mathbf{q}),\ T_{x}(\bm {q}),\ T_{y}(\mathbf{q}),\ T_{z}(\mathbf{q})\right]^{\mathrm{T}}=\mathbf{K}(\mathbf{q})\ i \tag{6}\]  
Conversely, if the desired electromagnetic forces and torques of the real-time movements in position and attitude \(\mathbf{q}\) are:  
\[\mathbf{W}_{\mathrm{d}}=\left(F_{\mathrm{xd}},F_{\mathrm{yd}},F_{\mathrm{zd}},T_{ \mathrm{xd}},T_{\mathrm{yd}},T_{\mathrm{zd}}\right)^{\mathrm{T}} \tag{7}\]  
then the commutated coil currents \(\mathbf{i}_{\mathrm{c}}\) should satisfy  
\[\mathbf{W}_{\mathrm{d}}=\mathbf{K}(\mathbf{q})\mathbf{i}_{\mathrm{c}} \tag{8}\]  
In light of Equation (8), the commutated coil currents \(\mathbf{i}_{\mathrm{c}}\) can be derived by:  
\[\mathbf{i}_{\mathrm{c}}=\left[\mathbf{K}(\mathbf{q})\right]^{-}\mathbf{W}_{\mathrm{d}} \tag{9}\]  
where \(\left[\mathbf{K}(\mathbf{q})\right]^{-}\) represents an inverse computation about force/torque coeficient matrix \(\mathbf{K}(\mathbf{q})\), because MLPM is usually over-actuated so as to provide adequate energy for moving in six-DOF, which implies the number of coils \(n\) is larger than the number of DOF: \(n>6\), \(\mathbf{K}(\mathbf{q})\) is a rectangular matrix. Hence, \(\left[\mathbf{K}(\mathbf{q})\right]^{-}\) can be solved analytically by the pseudo inverse of \(\mathbf{K}(\mathbf{q})\):  
\[\left[\mathbf{K}(\mathbf{q})\right]^{-}=\left[\mathbf{K}(\mathbf{q})\right]^{\mathrm{T}}\left[ \mathbf{K}(\mathbf{q})\left[\mathbf{K}(\mathbf{q})\right]^{\mathrm{T}}\right]^{-1} \tag{10}\]  
and the uniquely analytical solution of commutated coil currents \(\mathbf{i}_{\mathrm{c}}\) can be solved by  
\[\mathbf{i}_{\mathrm{c}}=\left[\mathbf{K}(\mathbf{q})\right]^{\mathrm{T}}\left[\mathbf{K}(\mathbf{q })\left[\mathbf{K}(\mathbf{q})\right]^{\mathrm{T}}\right]^{-1}\mathbf{W}_{\mathrm{d}} \tag{11}\]  
Equation (11) implies that the coil currents are commutated in sense of the minimum square sum of all coil currents, but that does not mean that the current squares are evenly distributed among all coils [11; 19].