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Kalman濾波模型的建立
Kalman濾波是在時域內以信號的一、二階統計特性已知為前提、以均方誤差極小為判據，能自動跟蹤信號統計性質的非平穩變化，具有遞歸性質的一種算法[5]。Kalman濾波的狀態方程和預測方程如式(2)所示：
其中，Xk+1為系統的狀態向量，Φk+1，k為狀態轉移矩陣，Tk+1，k為控制系數矩陣，Uk為系統控制輸入，Γk+1為噪聲系數矩陣，Wk為系統的過程噪聲序列，Zk+1為系統的觀測序列，Hk+1為觀測矩陣，Vk+1為觀測噪聲序列。
使用陀螺儀數據建立線性模型，如式(3)所示：
其中，θk為飛行姿態角，βk為陀螺儀輸出的誤差，Uk是陀螺儀輸出的角速度數據，ΔT是采樣時間間隔。因為陀螺儀輸出的誤差整體來看變化不大，可認為βk+1=βk。因Wk對系統影響甚小，這里規定Wk=0。
由以上建立狀態矩陣方程，如式(4)所示：
陀螺儀和加速度計測量姿態角時各有優缺點，為獲得比較可靠的姿態角數據，必須對兩者的數據優勢互補。以上算法用加速度計數據得到不同階段的測量噪聲協方差，并帶入解算過程中，大大減少了姿態角的測量誤差。
3 Kalman硬件實現
目前Kalman濾波器的實現方式主要有PC機和DSP兩種方式，PC機可以同時滿足計算精度和實時性的要求，但是PC機體積大、質量重、成本高；DSP實現的Kalman濾波器雖然體積小，質量輕，但是因其指令順序執行的CPU架構，在系統復雜時無法滿足系統的實時性要求[6-7]。
因此，Kalman濾波器在工程應用中的實現遇到了系統體積、重量、成本與系統精度、速度等性能不能兼顧的問題。FPGA是解決這一問題的有效途徑，FPGA具有極高并行度的信號處理能力、高度的靈活性、現場可以編程、較短的開發周期和較低的成本等優勢，在數字信號處理領域已經得到廣泛的應用[8]。
本系統使用的FPGA芯片是Altera公司Cyclone II系列中的EP2C35F672C6，它共有33 216個邏輯單元、483 840個存儲單元、70個乘法器單元（9 bit）。雖然Altera公司提供了很多功能強大的IP核，其中就包括加減乘除單精度浮點數運算，但是提供的IP核占用資源過多，所以使用自己編寫的運算模塊。下面以除法模塊為例說明浮點數運算的硬件實現。
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