## 3 Control Strategy  
### Coupling Error Control Model  
In the previous study, the distribution control method was adopted for MMS of the FC. The command signal was directly assigned to each OSM according to the desired angle of each mode. The steering angle tracking control of each OSM was implemented based on bridge circuit, but the coupling motion of four OSMs was not considered. This methodology is simple and easy to implement when the four OSMs are steering independently without restraining each other. In this study, a coupling control strategy is proposed. The CE of an off-center arm is obtained by comparing the angular velocity between two adjacent OSMs. Then, the CE of each OSM is compensated using the coupling control algorithm. Any changes in the angular velocity of an OSM will provide feedback to an adjacent OSM. All adjacent steering mechanisms are coupled in pairs, eventually forming a coupling loop. This study observes the effect of the proposed coupling control by comparing it with the distribution control method. In the MMS process, the expected angular velocity of an off-center arm, \(\omega_{d}\), is the only input signal. However, the angular velocities of the off-center arms are different under different motion modes. In a motion mode, the proportional relationship between the angular velocities of all off-center arms is expressed as:  
\[\frac{\omega_{1}}{\eta_{1}}=\frac{\omega_{2}}{\eta_{2}}=\cdots=\frac{\omega_{ n}}{\eta_{n}}, \tag{14}\]  
where \(\eta_{p}\) is proportionality coefficient (\(p=1\), \(2\),..., \(n\)). The \(k\)th off-center arm with the worst control performance is selected as a reference, and its proportionality coefficient is \(\eta_{k}\). For convenience of synchronization error derivation, a normalized proportional coefficient \(\eta_{p}^{*}\) is introduced:  
\[\eta_{p}^{*}=\frac{\eta_{p}}{\eta_{k}}=\frac{\omega_{p}}{\omega_{k}}. \tag{15}\]  
In a control system with \(n\) OSMs, the angular velocity tracking error of OSM \(p\) is defined as  
\[e_{p}=\omega_{p}^{*}-\omega_{p}, \tag{16}\]  
where \(\omega_{p}^{*}=\eta_{p}^{*}\cdot\omega_{d}\), and \(\omega_{d}\) represents the demand angular velocity. The synchronization error between any two adjacent OSMs is expressed using Equation (17). Synchronization control of all OSMs is achieved when \(\varepsilon_{p}\) = 0:  
\[\varepsilon_{p}=\frac{\omega_{p}}{\eta_{p}^{*}}-\frac{\omega_{p+1}}{\eta_{p+1}^{* }}. \tag{17}\]  
The dynamic characteristics of the OSM are described using:  
\[\dot{\omega}_{p}=\frac{1}{I_{o}}(T_{e}-T_{s})-(\frac{B_{o}}{J_{o}})\omega_{p}, \tag{18}\]  
where \(T_{s}\) is the load torque (N-m); \(B_{o}\) is the viscous friction damping coefficient of the OSM (N-m); and \(J_{o}\) denotes the OSM moment of inertia (kg:m\({}^{2}\)). Equation (18) can be simplified as:  
\[\dot{\omega}_{p}=f-ba\omega_{p}, \tag{19}\]  
where \(f=(T_{e}-T_{s})/J_{o}\), \(b=B_{o}/J_{o}\). The CE of the \(p\)th OSM after compensation in coupling control is given as:  
\[E_{p}=e_{p}-\lambda_{p}\varepsilon_{p}, \tag{20}\]  
where \(\lambda_{p}\) represents the synchronization error coefficient, \(e_{p}\) is the OSM tracking error, and \(\varepsilon_{p}\) is the OSM coupling error. To achieve coupling control, \(E_{p}\) must be maintained at zero. According to [PERSON]'s direct method, a Lyapunov function can be constructed for a nonlinear differential equation to study its stabilization in a control system. To make \(E_{1}\) zero, the Lyapunov function is used to judge the system stability.