## 4 The Complementary Filter Method and Performance Analysis  
### The Fusion Acceleration Based on the Modified Complementary Filter Method  
According to Equation (5), considering the CCD's delay \(e^{-\tau s}\), we can get the transfer function of the virtual accelerometer as follows. \(H_{CCD}\) respects the transfer characteristic of the CCD without delay, \(H_{ACC}\) is the transfer characteristic of the MEMS accelerometer. \[ \begin{array}{lll}\frac{a_{fusion}}{a_{real}}&=&\frac{C(s)}{1+C(s)}\cdot s^{2 }e^{-\tau s}H_{CCD}+\frac{1}{1+C(s)}H_{ACC}\\ &=&B_{closed}\cdot s^{2}e^{-\tau s}H_{CCD}+B_{restrain}\cdot H_{ACC}\\ &=&\frac{f_{s}(s)}{G_{a}(s)}\end{array}  \tag{16}\]  
When in very low frequency, \(B_{restrain}\) approaches to 0, \(a_{fusion}\) is equal to the acceleration got through the CCD with no drift. Similarly, when in relatively high frequency, \(B_{closed-loop}\) approaches to 0, the \(a_{fusion}\) is equal to the acceleration got by the MEMS accelerometers with nearly no delay. In medium frequency, \(a_{fusion}\) contains both the CCD's and MEMS accelerometers' signals. However, in fact, although the original complementary filter method can eliminate the low-frequency drift, it still cannot get a very satisfied acceleration because of the MEMS accelerometer's high-frequency noise. Assume \(R_{CCD}\) and \(R_{ACC}\) are respectively the CCD's signal without delay and the MEMS accelerometer's signal and they contain both useful signal and interference signal. According to Equation (16), we can deduce the fusion acceleration as follows. \[ \begin{array}{lll}a_{fusion}(s)&=&C_{low-pass}(s)s^{2}e^{-\tau s}R_{CCD}+C_{ high-pass}(s)R_{ACC}\\ &=&C_{low-pass}(s)s^{2}e^{-\tau s}R_{CCD}+[1-C_{low-pass}(s)]R_{ACC}\\ &=&R_{ACC}+C_{low-pass}(s)[s^{2}e^{-\tau s}R_{CCD}-R_{ACC}]\end{array}  \tag{17}\]  
The new fusion acceleration contains two items, the \(R_{ACC}\) and the low-frequency data filtered by \(C_{low-pass}\). When given a low-frequency input, \(R_{ACC}\) not only contains a zero drift but also contains high-frequency noise. The second item of Equation (17) could help eliminate the drift of \(R_{ACC}\) but cannot deal with the high-frequency noise signal, which will decrease the accuracy of the last acceleration. Therefore, in this case the basic complementary filter cannot work well. In order to prevent the pollution of the high-frequency noise, we change \(C_{high-pass}\) to be a band-pass filter \(C_{band-pass}\). Considering that the high-order low-pass filter would bring a large phase lag, we choose a first-order low-pass filter as \(C_{low-pass}\). Equation (18) presents the new fusion acceleration. \[ \begin{array}{lll}a_{fusion}(s)&=C_{low-pass}(s)R_{CCD}s^{2}e^{-\tau s}+C_{ band-pass}(s)R_{ACC}\\ &=&\frac{1}{1+T_{1}s}R_{CCD}s^{2}e^{-\tau s}+\frac{T_{1}s}{1+T_{1}s}\frac{1}{1+T _{2}s}R_{ACC}\\ &(T_{2}<<\tau<<T_{1})\end{array}  \tag{18}\]  
In Equation (18), although the double differential of \(R_{CCD}\) to get the acceleration would amplify the impact of the noise, since the signal of noise's differential mainly distribute on high-frequency domain, \(C_{low-pass}\) will nearly eliminate the effect of the noise. Similarly, the band-pass filter could cut down the low-frequency drift and decrease the pollution of the high-frequency noise. Hence, now we could ignore the drift and noise and get \(R_{\text{CCD}}s^{2}\approx R_{\text{ACC}}\approx a_{real}\), then  
\[a_{fusion}(s)\ \approx(\frac{1}{1+T_{1}s}e^{-\tau s}+\frac{T_{1}s}{1+T_{1}s} \frac{1}{1+T_{2}s})a_{real}\ =\ \frac{T_{1}s+(1+T_{2}s)e^{-\tau s}}{(1+T_{1}s)(1+T_{2}s)}a_{real} \tag{19}\]  
In Equation (19), the form of the acceleration transfer function has been fixed. The rest is to determine the handover frequency \(1/T_{1}\). Theoretically, whatever the value of \(1/T_{1}\) is, the final acceleration model could be identified by spectrum fitting. Nevertheless, in fact, \(1/T_{1}\) will affect the linearity of the object, which will determine the difficulty of the controller design.