http://www.dept.aoe.vt.edu/~cdhall/courses/aoe4140/attde.pdf
Triad algorithm
有两对观测矢量s^,m^" role="presentation" style="position: relative;">s^,m^s^,m^（分别有在惯性系和body系中的值）
为了确定Rib" role="presentation" style="position: relative;">RbiRib，考虑一个中间坐标系t。
t的基矢及在惯性系和body系中的值如下：
可得Rib" role="presentation" style="position: relative;">RbiRib:
注意这里s^" role="presentation" style="position: relative;">s^s^和m^" role="presentation" style="position: relative;">m^m^的地位是不一样的，Ribsi=sb" role="presentation" style="position: relative;">Rbisi=sbRibsi=sb，而Ribmi" role="presentation" style="position: relative;">RbimiRibmi却不一定等于mb" role="presentation" style="position: relative;">mbmb，即认为s^" role="presentation" style="position: relative;">s^s^的测量值更准确。
Orientation Estimation by Error-State Extended Kalman Filter in Quaternion Vector Space
rate random walk是指bias满足b˙=nw" role="presentation" style="position: relative;">b˙=nwb˙=nw
过程模型（连续）
预测步中的期望和方差计算：
其中
注意，由于噪声在过程模型中不是以加项形式存在，而是在f" role="presentation" style="position: relative;">ff中，故有Jacobian M。
测量过程：
rb=(1−E)Rnbrn+n" role="presentation" style="position: relative;">rb=(1?E)Rbnrn+nrb=(1?E)Rnbrn+n
其中(1−E)" role="presentation" style="position: relative;">(1?E)(1?E)表示由error-state（欧拉角）构成的小角旋转矩阵，r" role="presentation" style="position: relative;">rr为重力和地磁。再做EKF。
或者再做点近似，(也就是下面的文章中做的）
δz=rmeasurement−rpredicted=2[r^b×]δq→+n" role="presentation" style="position: relative;">δz=rmeasurement?rpredicted=2[r^b×]δq? +nδz=rmeasurement?rpredicted=2[r^b×]δq→+n
其中rpredicted=r^b=Rnb(qnominal)rn" role="presentation" style="position: relative;">rpredicted=r^b=Rbn(qnominal)rnrpredicted=r^b=Rnb(qnominal)rn
The Use of an Orientation Kalman Filter for the Static Postural Sway Analysis
error-state是δq" role="presentation" style="position: relative;">δqδq的矢量部分和gyro bias误差。用矢量部分而不是整个四元数是为了避免协方差矩阵奇异。
A Sensor Fusion Method for Smart phone Orientation Estimation
θ=(α)∗(θg)+(1−α)∗(θam)" role="presentation" style="position: relative;">θ=(α)?(θg)+(1?α)?(θam)θ=(α)?(θg)+(1?α)?(θam)
其中α=ττ+Ts" role="presentation" style="position: relative;">α=ττ+Tsα=ττ+Ts，Ts" role="presentation" style="position: relative;">TsTs是采样周期，filter的时间常数τ" role="presentation" style="position: relative;">ττ是the relative duration of signal it will act on。
互补滤波的思想：通过加速度计和磁力计算出的θam" role="presentation" style="position: relative;">θamθam没有漂移问题，但噪声较大。而通过陀螺仪算出的θg" role="presentation" style="position: relative;">θgθg短时间内较精确，但有漂移问题。因此对θam" role="presentation" style="position: relative;">θamθam低通滤波，滤除噪声，对θg" role="presentation" style="position: relative;">θgθg高通滤波，保留方位角的快速变化。
Standalone Inertial Pocket Navigation System
magnetic disturbance detector 根据磁场强度和方向
A sensor fusion algorithm for an integrated angular position estimation with inertial measurement units
观测值为加速度信号，预测值值为重力加速度通过变换后在body坐标系中的值。不能修正yaw值
Heading Estimation for Indoor Pedestrian Navigation Using a Smartphone in the Pocket
wkgyro" role="presentation" style="position: relative;">wgyrokwkgyro是陀螺仪的测量误差
process noise covariance matrix
是陀螺仪的测量误差协方差矩阵
measurement model
这里a是没有加速时的值。
adaptive measurement covariance matrix
Jacobian
线性化
3D orientation tracking based on unscented Kalman filtering of accelerometer and magnetometer data
freescale
误差函数：
sk" role="presentation" style="position: relative;">sksk是body系的测量值，rk" role="presentation" style="position: relative;">rkrk是导航系中的参考值，R" role="presentation" style="position: relative;">RR是方向余弦矩阵，αk" role="presentation" style="position: relative;">αkαk是权重
定义
则
考虑归一化的矢量，即
则有
定义
以及四元数
则
引入拉格朗日乘子，
由极值条件
推出
即q" role="presentation" style="position: relative;">qq是κ" role="presentation" style="position: relative;">κκ的本征矢量
由
得q" role="presentation" style="position: relative;">qq对应的是最大本征值的本征矢量。
gyro模型
bk" role="presentation" style="position: relative;">bkbk是zero rate offset
磁场指向水平面下时磁倾角为正