# 無跡卡曼濾波器

## Sigma點

${\displaystyle \{\mathbf {s} _{0},\dots ,\mathbf {s} _{N}\}={\bigl \{}{\begin{pmatrix}s_{0,1}&s_{0,2}&\ldots &s_{0,L}\end{pmatrix}},\dots ,{\begin{pmatrix}s_{N,1}&s_{N,2}&\ldots &s_{N,L}\end{pmatrix}}{\bigr \}}}$

• 一階權重${\displaystyle W_{0}^{a},\dots ,W_{N}^{a}}$ 滿足
1. ${\displaystyle \sum _{j=0}^{N}W_{j}^{a}=1}$
2. 對所有啲${\displaystyle i=1,\dots ,L}$ 有：${\displaystyle E[x_{i}]=\sum _{j=0}^{N}W_{j}^{a}s_{j,i}}$
• 二階權重${\displaystyle W_{0}^{c},\dots ,W_{N}^{c}}$ 滿足
1. ${\displaystyle \sum _{j=0}^{N}W_{j}^{c}=1}$
2. 對所有啲𠵿${\displaystyle (i,l)\in \{1,\dots ,L\}^{2}:E[x_{i}x_{l}]=\sum _{j=0}^{N}W_{j}^{c}s_{j,i}s_{j,l}}$

UKF演算法入便一種揀啲sigma點同埋權重畀${\displaystyle \mathbf {x} _{k-1\mid k-1}}$ 嘅簡單方式係：

{\displaystyle {\begin{aligned}\mathbf {s} _{0}&={\hat {\mathbf {x} }}_{k-1\mid k-1}\\-1&

{\displaystyle {\begin{aligned}\mathbf {s} _{0}&={\hat {\mathbf {x} }}_{k-1\mid k-1}\\W_{0}^{a}&={\frac {\alpha ^{2}\kappa -L}{\alpha ^{2}\kappa }}\\W_{0}^{c}&=W_{0}^{a}+1-\alpha ^{2}+\beta \\\mathbf {s} _{j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}+\alpha {\sqrt {\kappa }}\mathbf {A} _{j},\quad j=1,\dots ,L\\\mathbf {s} _{L+j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}-\alpha {\sqrt {\kappa }}\mathbf {A} _{j},\quad j=1,\dots ,L\\W_{j}^{a}&=W_{j}^{c}={\frac {1}{2\alpha ^{2}\kappa }},\quad j=1,\dots ,2L.\end{aligned}}}

${\displaystyle \alpha }$ ${\displaystyle \kappa }$ 啲sigma點嘅擴散。${\displaystyle \beta }$ ${\displaystyle x}$ 嘅分佈有關。

## 過程

### 預測

${\displaystyle \mathbf {x} _{j}=f\left(\mathbf {s} _{j}\right)\quad j=0,\dots ,2L}$ .

{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid k-1}&=\sum _{j=0}^{2L}W_{j}^{a}\mathbf {x} _{j}\\\mathbf {P} _{k\mid k-1}&=\sum _{j=0}^{2L}W_{j}^{c}\left(\mathbf {x} _{j}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)\left(\mathbf {x} _{j}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)^{\textsf {T}}+\mathbf {Q} _{k}\end{aligned}}}

### 更新

${\displaystyle \mathbf {z} _{j}=h(\mathbf {s} _{j}),\,\,j=0,1,\dots ,2L}$ .

{\displaystyle {\begin{aligned}{\hat {\mathbf {z} }}&=\sum _{j=0}^{2L}W_{j}^{a}\mathbf {z} _{j}\\[6pt]{\hat {\mathbf {S} }}_{k}&=\sum _{j=0}^{2L}W_{j}^{c}(\mathbf {z} _{j}-{\hat {\mathbf {z} }})(\mathbf {z} _{j}-{\hat {\mathbf {z} }})^{\textsf {T}}+\mathbf {R} _{k}\end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {C_{sz}} &=\sum _{j=0}^{2L}W_{j}^{c}(\mathbf {s} _{j}-{\hat {\mathbf {x} }}_{k|k-1})(\mathbf {z} _{j}-{\hat {\mathbf {z} }})^{\textsf {T}}\end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {K} _{k}=\mathbf {C_{sz}} {\hat {\mathbf {S} }}_{k}^{-1}.\end{aligned}}}

{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid k}&={\hat {\mathbf {x} }}_{k|k-1}+\mathbf {K} _{k}(\mathbf {z} _{k}-{\hat {\mathbf {z} }})\\\mathbf {P} _{k\mid k}&=\mathbf {P} _{k\mid k-1}-\mathbf {K} _{k}{\hat {\mathbf {S} }}_{k}\mathbf {K} _{k}^{\textsf {T}}.\end{aligned}}}

## 考

1. Julier, Simon J.; Uhlmann, Jeffrey K. (1997). "New extension of the Kalman filter to nonlinear systems" (PDF). 出自 Kadar, Ivan (編). Signal Processing, Sensor Fusion, and Target Recognition VI. Proceedings of SPIE.第3卷. pp. 182–193. Bibcode:1997SPIE.3068..182J. CiteSeerX 10.1.1.5.2891. doi:10.1117/12.280797. S2CID 7937456. 喺2008-05-03搵到.
2. Menegaz, H. M. T.; Ishihara, J. Y.; Borges, G. A.; Vargas, A. N. (October 2015). "A Systematization of the Unscented Kalman Filter Theory". IEEE Transactions on Automatic Control. 60 (10): 2583–2598. doi:10.1109/tac.2015.2404511. hdl:20.500.11824/251. ISSN 0018-9286. S2CID 12606055.
3. Gustafsson, Fredrik; Hendeby, Gustaf (2012). "Some Relations Between Extended and Unscented Kalman Filters". IEEE Transactions on Signal Processing. 60 (2): 545–555. Bibcode:2012ITSP...60..545G. doi:10.1109/tsp.2011.2172431. S2CID 17876531.
4. Bitzer, S. (2016). "The UKF exposed: How it works, when it works and when it's better to sample". doi:10.5281/zenodo.44386. {{cite journal}}: Cite journal requires |journal= (help)
5. Wan, E.A.; Van Der Merwe, R. (2000). "The unscented Kalman filter for nonlinear estimation" (PDF). Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373). p. 153. CiteSeerX 10.1.1.361.9373. doi:10.1109/ASSPCC.2000.882463. ISBN 978-0-7803-5800-3. S2CID 13992571. 原著 (PDF)喺2012年3月3號歸檔. 喺2021年7月5號搵到.
6. Sarkka, Simo (September 2007). "On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems". IEEE Transactions on Automatic Control. 52 (9): 1631–1641. doi:10.1109/TAC.2007.904453.