**無跡卡曼濾波器**（英文：**unscented Kalman filter**，**UKF**）係一種卡曼濾波器，攞嚟解決種問題係擴展卡曼濾波器喺高度非線性系統模型下表現嘸佳噉嘅情況；呢種卡曼濾波器使到一種叫做無跡變換（UT）嘅確定性抽樣技巧，即喺平均值周圍揀一組最細嘅樣本點（喊做sigma點）。然之後，啲sigma點識透過非線性函數傳播（propagated），從中形成到個新估計值畀平均值同埋協方差。噉樣可以避免似喺EKF當中噉經由線性化個潛在嘅非線性模型嚟傳播啲協方差。係噉呢種濾波器嘅具體效能取決於點樣計UT啲轉換統計量同埋使到邊一組採樣到嘅sigma點。需要注意嘅係，啲新嘅UKF始終可以以一致（consistent）嘅方法整出。對於某啲系統，噉樣得到嘅UKF更加準確噉估計得到真實嘅平均值同協方差，個論斷可以透過蒙地卡羅抽樣或者啲後驗統計值個泰勒級數展開嚟驗證到。另外，種技術免走明式噉計算Jacobi矩陣嘅要求：對於啲複雜函數來講，一嘸係啲函數嘸微分得所以冇辦法計得出Jacobi矩陣、就係計算本身好難無論係使分析法（導數複雜）抑或係數值法（計算成本高）都好。

Sigma點
------

對於隨機向量{\displaystyle \mathbf {x} =(x\_{1},\dots ,x\_{L})}，啲Sigma點係任何啲向量嘅集：

{\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 \}}}
有：

* 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&<W\_{0}^{a}=W\_{0}^{c}<1\\\mathbf {s} \_{j}&={\hat {\mathbf {x} }}\_{k-1\mid k-1}+{\sqrt {\frac {L}{1-W\_{0}}}}\mathbf {A} \_{j},\quad j=1,\dots ,L\\\mathbf {s} \_{L+j}&={\hat {\mathbf {x} }}\_{k-1\mid k-1}-{\sqrt {\frac {L}{1-W\_{0}}}}\mathbf {A} \_{j},\quad j=1,\dots ,L\\W\_{j}^{a}&=W\_{j}^{c}={\frac {1-W\_{0}}{2L}},\quad j=1,\dots ,2L\end{aligned}}}
其中{\displaystyle {\hat {\mathbf {x} }}\_{k-1\mid k-1}}係 {\displaystyle \mathbf {x} \_{k-1\mid k-1}}個平均估計。枚向量{\displaystyle \mathbf {A} \_{j}}係{\displaystyle \mathbf {A} }個第*j*棟，其中{\displaystyle \mathbf {A} }有{\displaystyle \mathbf {P} \_{k-1\mid k-1}=\mathbf {AA} ^{\textsf {T}}}。矩陣{\displaystyle \mathbf {A} }應使數值上高效又穩定嘅方法似Cholesky分解嚟計算。平均值嘅權重{\displaystyle W\_{0}}可以任意噉揀。

第個流行嘅參數化（對上述普適化）方式係： 

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

啱嘅值取決於手頭係咩樣嘅問題，而典型建議係揀{\displaystyle \alpha =10^{-3}}，{\displaystyle \kappa =1}同埋{\displaystyle \beta =2}。之不過，大啲嘅值畀{\displaystyle \alpha }（似{\displaystyle \alpha =1}）可能捕捉返個分佈嘅擴散同埋啲可能有嘅非線性會好啲。若果個{\displaystyle x}真實分佈係高斯分佈，係噉揀{\displaystyle \beta =2}係最啱嘅。

過程
--

### 預測

同EKF一樣，UKF預測可以獨立於UKF更新，可以佮線性（或者EKF）更新一齊用或者反過嚟。攞有估計值{\displaystyle {\hat {\mathbf {x} }}\_{k-1\mid k-1}}、{\displaystyle \mathbf {P} \_{k-1\mid k-1}}畀平均值同協方差嘅話，可以攞到{\displaystyle N=2L+1}粒sigma點，似上節所講。啲sigma點係透過轉換函數*f* （transition function）傳播嘅：

{\displaystyle \mathbf {x} \_{j}=f\left(\mathbf {s} \_{j}\right)\quad j=0,\dots ,2L}.
着傳播過嘅 sigma 點又着加權嚟產生啲預測嘅平均值同埋協方差。

{\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}}}
其中{\displaystyle W\_{j}^{a}}同埋{\displaystyle W\_{j}^{c}}分別係一階同埋二階權重喺啲原始sigma點上。矩陣{\displaystyle \mathbf {Q} \_{k}}係轉換噪音{\displaystyle \mathbf {w} \_{k}}嘅協方差。

### 更新

畀有預測估計{\displaystyle {\hat {\mathbf {x} }}\_{k\mid k-1}}同{\displaystyle \mathbf {P} \_{k\mid k-1}}，計得出新一組{\displaystyle N=2L+1}粒sigma點{\displaystyle \mathbf {s} \_{0},\dots ,\mathbf {s} \_{2L}}有相應一階權重{\displaystyle W\_{0}^{a},\dots W\_{2L}^{a}}二階權重{\displaystyle W\_{0}^{c},\dots ,W\_{2L}^{c}}嘅。 啲sigma點通過{\displaystyle h}轉變：

{\displaystyle \mathbf {z} \_{j}=h(\mathbf {s} \_{j}),\,\,j=0,1,\dots ,2L}.
係噉轉變過嘅啲點嘅觀測平均值同埋協方差就噉計出：

{\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}}}
其中{\displaystyle \mathbf {R} \_{k}}係觀測噪音{\displaystyle \mathbf {v} \_{k}}嘅協方差矩陣。另外仲要到交叉協方差（cross covariance ）矩陣：

{\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}}}
其中{\displaystyle \mathbf {s} \_{j}}係攞{\displaystyle {\hat {\mathbf {x} }}\_{k\mid k-1}}、{\displaystyle \mathbf {P} \_{k\mid k-1}}整出嘅啲未轉變過嘅sigma點。

個卡曼增益即係：

{\displaystyle {\begin{aligned}\mathbf {K} \_{k}=\mathbf {C\_{sz}} {\hat {\mathbf {S} }}\_{k}^{-1}.\end{aligned}}}
更新後嘅估計值畀平均值同埋協方差即係：

{\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
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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.