在智慧製造技術快速發展的時代，許多重複性高與高精度的動作已逐漸由機器人代為執行。欲使機器人達到任務精度要求，精準的控制尤為重要，而精確的速度與加速度的估測更是其中不可或缺的關鍵。本論文針對自適應卡爾曼濾波器提出了改良策略。首先，由於應用於卡爾曼濾波器之Selective-Scaling自適應策略在估測過程中缺少速度及加速度兩者的實際狀態，因此本論文透過卡爾曼濾波器滿足最佳無偏估測器之特性，結合迴歸模型方法，建立線上模型並預測出可能之實際速度及加速度值並用於過程雜訊協方差矩陣Q之更新，此為本論文提出之「改良型自適應卡爾曼濾波器」。本論文發現，隨著疊代過程，估測值會更準確，使得誤差減少導致協方差P收斂。因此本論文提出「自適應策略停止之判斷方法」，根據P之跡值的斜率變化判斷是否達到閾值，並決定是否停止執行自適應策略。最後本論文將所提方法與類神經網路結合，在估測時可以避免參數與迴歸模型之選擇，並減少Q值之自適應時間。最後針對三種模擬函數以及兩組機械手臂進行速度及加速度估測實驗。結果顯示本論文所提出方法確實有效。

As smart manufacturing technologies rapidly advance, many highly repetitive and precise tasks can be performed by robots. As a result, feedback control in robotic applications has become important, where accurate estimation of velocity and acceleration is essential. This study proposes a modified strategy for Adaptive Kalman Filter by integrating four regression modeling approaches. Furthermore, the proposed method employed a neural networks to learn adaptive strategy. First, this study adopts a Selective-Scaling adaptive strategy within the Kalman Filter, since the true states of velocity and acceleration are unavailable during the estimation process. As Kalman Filter satisfies the Best Linear Unbiased Estimator property, the proposed method integrated with regression model and constructed online, then predicted the possible true states. This process used to update process noise covariance Q, and is proposed as "Modified Adaptive Kalman Filter ." It is observed that, as the iterative process progresses, the estimation becomes more accurate, leading to a gradual decrease in error covariance P. Based on this observation, this study proposes a "Stopping Criterion for the Adaptive Strategy, " which determines whether to terminate the adaptative strategy process by evaluating the slope of the trace of P and checking whether it' s below the threshold. The proposed method is further integrated with a neural network to eliminate the need for selecting parameters and regression models during the estimation, while reducing the required adaptation time for updating Q. Finally, through the subsequent experiments, the proposed approach is evaluated using three simulated functions and two robotic arm datasets, , which demonstrates the effectiveness of the proposed algorithms based on the analysis of velocity and acceleration estimation results.

[1] R. H. Brown, S. C. Schneider, and M. G. Mulligan, “Analysis of algorithms for velocity estimation from discrete position versus time data,” IEEE Transactions on Industrial Electronics, vol. 39, no. 1, pp. 11–19, Jan. 1992.
[2] N. S.-H. Lee and N. J.-B. Song, “Acceleration estimator for low-velocity and low-acceleration regions based on encoder position data,” IEEE/ASME Transactions on Mechatronics, vol. 6, no. 1, pp. 58–64, Mar. 2001.
[3] L. J. Puglisi, “On the Velocity and Acceleration Estimation from Discrete Time-Position Signal of Linear Encoders,” Journal of Control Engineering and Applied Informatics, vol. 17, no. 3, pp. 30–40, Sep. 2015.
[4] R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of Basic Engineering, vol. 82, no. 1, pp. 35–45, Mar. 1960.
[5] G. L. Smith, S. F. Schmidt, and L. A. McGee, Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle. 1962.
[6] M. Sever, T. Y. Erkeç, and C. Hajiyev, “Comparison of adaptive Kalman filters in aircraft state estimation,” WSEAS TRANSACTIONS ON SIGNAL PROCESSING, vol. 19, pp. 128–138, Oct. 2023.
[7] L. Xiang, X. Shi, F. Zheng, S. Luo, Z. Chen, L. Zhu, B. Yu, and H, Luo, “Junction Temperature Estimation of IGBT Based on Adaptive Kalman Filter,” in Proc. 2024 CPSS & IEEE International Symposium on Energy Storage and Conversion (ISESC), Xi'an, China, 2024, pp. 212-217.
[8] S. Akhlaghi, N. Zhou and Z. Huang, “Adaptive adjustment of noise covariance in Kalman filter for dynamic state estimation,” in Proc. 2017 IEEE Power & Energy Society General Meeting, Chicago, IL, USA, 2017, pp. 1-5.
[9] L. Chu, Y. Shi, Y. Zhang, H. Liu and M. Xu, “Vehicle lateral and longitudinal velocity estimation based on Adaptive Kalman Filter,” in Proc. 2010 3rd International Conference on Advanced Computer Theory and Engineering(ICACTE), Chengdu, 2010, pp. V3-325-V3-329.
[10] B. Zheng, P. Fu, B. Li, and X. Yuan, “A Robust Adaptive Unscented Kalman Filter for Nonlinear Estimation with Uncertain Noise Covariance,” Sensors, vol. 18, no. 3, p. 808, Mar. 2018.
[11] R. Volkmar, K. Thormann, K. Soal, Y. Govers, M. Böswald, and M. Baum, “Adaptive Kalman filter tracking for instantaneous aircraft flutter monitoring,” in Proc. 2022 25th International Conference on Information Fusion (FUSION), pp. 1–8, Jun. 2023.
[12] T. Kruse, T. Griebel, and K. Graichen, “Adaptive Kalman Filtering: Measurement and process noise covariance estimation using Kalman smoothing,” IEEE Access, p. 1, Jan. 2025.
[13] A. Almagbile, J. Wang, and W. Ding, “Evaluating the performances of adaptive Kalman filter methods in GPS/INS integration,” Journal of Global Positioning Systems, vol. 9, no. 1, pp. 33–40, Jun. 2010.
[14] A. H. Mohamed and K. P. Schwarz, “Adaptive Kalman filtering for INS/GPS,” Journal of Geodesy, vol. 73, no. 4, pp. 193–203, May 1999.
[15] S. A. B. Birjandi, H. Khurana, A. Billard, and S. Haddadin, “A stable adaptive extended Kalman filter for estimating robot manipulators link velocity and acceleration,” in Proc. 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), vol. 200, pp. 346–353, Oct. 2023.
[16] L. He, Y. Wang, Y. Wei, M. Wang, X. Hu, and Q. Shi, “An adaptive central difference Kalman filter approach for state of charge estimation by fractional order model of lithium-ion battery,” Energy, vol. 244, p. 122627, Nov. 2021.
[17] C. Hajiyev, S. Y. Vural and U. Hajiyeva, “Adaptive Fading Kalman Filter with Q-adaptation for estimation of AUV dynamics,” in Proc. 2012 20th Mediterranean Conference on Control & Automation (MED), Barcelona, Spain, 2012, pp. 697-702.
[18] Q. Xia, M. Rao, Y. Ying, and X. Shen, “Adaptive fading Kalman filter with an application,” Automatica, vol. 30, no. 8, pp. 1333–1338, Aug. 1994.
[19] X. Luan, W. Xue, S. Zhao, and F. Liu, “A Kalman & Fading Memory Co-Filter for Uncertain Systems Based on Self-Perception Mechanism,” IEEE Transactions on Automatic Control, pp. 1–15, Jan. 2025.
[20] T.-S. Lou, Z.-H. Wang, M.-L. Xiao, and H.-M. Fu, “Multiple adaptive fading Schmidt-Kalman filter for unknown bias,” Mathematical Problems in Engineering, vol. 2014, pp. 1–8, Jan. 2014.
[21] E. Zerdali, R. Yildiz, L. Ozbek, and M. Barut, “Adaptive EKF algorithm with novel fading memory: Design and validation,” Computers & Electrical Engineering, vol. 123, p. 110130, Feb. 2025.
[22] C. Hu, W. Chen, Y. Chen, and D. Liu, “Adaptive Kalman filtering for vehicle navigation,” Journal of Global Positioning Systems, vol. 2, no. 1, pp. 42–47, Jun. 2003.
[23] G. Ozkurt and E. Zerdali, “Design and implementation of hybrid adaptive extended Kalman filter for state estimation of induction motor,” IEEE Transactions on Instrumentation and Measurement, vol. 71, pp. 1–12, Jan. 2022.
[24] C. Hide, T. Moore and M. Smith, “Adaptive Kalman filtering algorithms for integrating GPS and low cost INS,” in Proc. of PLANS 2004. Position Location and Navigation Symposium (IEEE Cat. No.04CH37556), Monterey, CA, USA, 2004, pp. 227-233.
[25] J. Kim, D. Lee, B. Kiss, and D. Lee, “An adaptive unscented Kalman filtering approach using selective scaling,” in Proc. 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC), Budapest, 2016, pp. 000784-000789.
[26] J. Kim, D. Lee, B. Kiss, and D. Kim, “An adaptive Unscented Kalman filter with selective scaling (AUKF-SS) for overhead cranes,” IEEE Transactions on Industrial Electronics, vol. 68, no. 7, pp. 6131–6140, May 2020.
[27] J. Kim, B. Kiss, D. Kim, and D. Lee, “Tracking control of overhead crane using output feedback with adaptive unscented Kalman filter and Condition-Based selective scaling,” IEEE Access, vol. 9, pp. 108628–108639, Jan. 2021.
[28] J. Kim, M. Song, D. Kim, and D. Lee, “An adaptive Kalman Filter-Based Condition-Monitoring technique for induction motors,” IEEE Access, vol. 11, pp. 46373–46381, Jan. 2023.
[29] C. Wang, R. Li, Y. Cao, and M. Li, “A hybrid model for state of charge estimation of lithium-ion batteries utilizing improved adaptive extended Kalman filter and long short-term memory neural network,” Journal of Power Sources, vol. 620, p. 235272, Aug. 2024.
[30] W. Fang, W. Cai, B. Fan, J. Yan and T. Zhou, “Kalman-LSTM Model for Short-term Traffic Flow Forecasting,” in Proc. 2021 IEEE 5th Advanced Information Technology, Electronic and Automation Control Conference (IAEAC), Chongqing, China, 2021, pp. 1604-1608.
[31] F. Song, Y. Li, W. Cheng, L. Dong, M. Li, and J. Li, “An improved Kalman filter based on Long Short-Memory recurrent neural network for nonlinear radar target tracking,” Wireless Communications and Mobile Computing, vol. 2022, pp. 1–10, Aug. 2022.
[32] T. Wen, J. Liu, B. Cai, and C. Roberts, “High-Precision state estimator design for the state of Gaussian linear systems based on deep neural network Kalman Filter,” IEEE Sensors Journal, vol. 23, no. 24, pp. 31337–31344, Nov. 2023.
[33] Z. Huo et al., “Optimal compensation of MEMS Gyroscope noise Kalman filter based on ConV-DAE and MultiTCN-Attention model in static base environment,” Sensors, vol. 22, no. 19, p. 7249, Sep. 2022.
[34] 黃毅瑄，應用於卡爾曼濾波器汽車導航的共變異數矩陣更新策略，碩士論文，國立清華大學，通訊工程研究所，2023
[35] Q. Ge, Y. Li, Y. Wang, X. Hu, H. Li, and C. Sun, “Adaptive Kalman filtering based on model parameter ratios,” IEEE Transactions on Automatic Control, vol. 69, no. 9, pp. 6230–6237, Mar. 2024.
[36] L. Zhang, D. Sidoti, A. Bienkowski, K. R. Pattipati, Y. Bar-Shalom, and D. L. Kleinman, “On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-Old Problem,” IEEE Access, vol. 8, pp. 59362–59388, Jan. 2020.
[37] Y. Chen, W. Li, and Y. Wang, “Online adaptive Kalman filter for target tracking with unknown noise statistics,” IEEE Sensors Letters, vol. 5, no. 3, pp. 1–4, Feb. 2021.
[38] Y. Huang, Y. Zhang, Z. Wu, N. Li, and J. Chambers, “A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices,” IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 594–601, Sep. 2017.
[39] H. Zhu, G. Zhang, Y. Li, and H. Leung, “An adaptive Kalman filter with inaccurate noise covariances in the presence of outliers,” IEEE Transactions on Automatic Control, vol. 67, no. 1, pp. 374–381, Feb. 2021.
[40] X. Lyu, B. Hu, K. Li, and L. Chang, “An adaptive and robust UKF approach based on Gaussian Process Regression-Aided Variational Bayesian,” IEEE Sensors Journal, vol. 21, no. 7, pp. 9500–9514, Feb. 2021.
[41] J. Meng, M. Yue, and D. Diallo, “A Degradation Empirical-Model-Free Battery End-of-Life Prediction Framework based on Gaussian Process regression and Kalman Filter,” IEEE Transactions on Transportation Electrification, vol. 9, no. 4, pp. 4898–4908, Sep. 2022.
[42] H. D. Hesar and M. Mohebbi, “An Adaptive Kalman Filter Bank for ECG Denoising,” IEEE Journal of Biomedical and Health Informatics, vol. 25, no. 1, pp. 13–21, Mar. 2020.
[43] H. L. Wah and A. M. T. Sin, “An Adaptive Kalman Filtering Algorithm Without Using Kinematic Models,” ASEAN Engineering Journal, vol. 13, no. 4, pp. 53–59, Oct. 2023.
[44] R. H. Brown, S. C. Schneider, and M. G. Mulligan, “Analysis of algorithms for velocity estimation from discrete position versus time data,” IEEE Transactions on Industrial Electronics, vol. 39, no. 1, pp. 11–19, Jan. 1992.
[45] D. Simon, Optimal state estimation. 2006.
[46] C. H. Lim, L. Y. Ong, T. S. Lim, and V. C. Koo, “Kalman Filtering and its Real‐Time applications,” InTech eBooks, 2016.
[47] A. P. Sage and G. W. Husa, "Algorithms for sequential adaptive estimation of prior statistics," in Proc. 1969 IEEE Symposium on Adaptive Processes (8th) Decision and Control, University Park, PA, USA, 1969, pp. 61-61.
[48] T. Kariya and H. Kurata, Generalized least squares. John Wiley & Sons, 2004.
[49] L. Piegl and W. Tiller, The NURBS Book. Springer Science & Business Media, 2012.
[50] C. K. Williams and C. E. Rasmussen, Gaussian Processes for Machine Learning, vol. 2, no. 3. Cambridge, MA, USA: MIT Press, 2006.
[51] J. L. Elman, “Finding structure in time,” Cognitive Science, vol. 14, no. 2, pp. 179–211, Mar. 1990.
[52] S. Hochreiter and J. Schmidhuber, “Long Short-Term memory,” Neural Computation, vol. 9, no. 8, pp. 1735–1780, Nov. 1997.
[53] J. Chung, C. Gulcehre, K . Cho,, and Y. Bengio, “Empirical evaluation of gated recurrent neural networks on sequence modeling,” arXiv preprint arXiv:1412.3555, 2014.
[54] T. Wen, J. Liu, B. Cai, and C. Roberts, “High-Precision state estimator design for the state of Gaussian linear systems based on deep neural network Kalman Filter,” IEEE Sensors Journal, vol. 23, no. 24, pp. 31337–31344, Nov. 2023.
[55] F. Song, Y. Li, W. Cheng, L. Dong, M. Li, and J. Li, “An improved Kalman filter based on Long Short-Memory recurrent neural network for nonlinear radar target tracking,” Wireless Communications and Mobile Computing, vol. 2022, pp. 1–10, Aug. 2022.
[56] C. Lea, R. Vidal, A. Reiter, and G. D. Hager, “Temporal Convolutional Networks: a unified approach to action segmentation,” in Lecture notes in computer science, 2016, pp. 47–54.
[57] S. Bai, J. Z. Kolter, and V. Koltun, “An empirical evaluation of generic convolutional and recurrent networks for sequence modeling,” arXiv preprint arXiv:1803.01271, 2018.
[58] Z. Huo et al., “Optimal compensation of MEMS Gyroscope noise Kalman filter based on ConV-DAE and MultiTCN-Attention model in static base environment,” Sensors, vol. 22, no. 19, p. 7249, Sep. 2022.
[59] H. Maghfiroh, M. Nizam, M. Anwar, and A. Ma’Arif, “Improved LQR control using PSO optimization and Kalman Filter Estimator,” IEEE Access, vol. 10, pp. 18330–18337, Jan. 2022.
[60] S. Jouaber, S. Bonnabel, S. Velasco-Forero and M. Pilté, "NNAKF: A Neural Network Adapted Kalman Filter for Target Tracking," in Proc. of ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Toronto, ON, Canada, 2021, pp. 4075-4079.
[61] C. R. Henderson, “Best Linear Unbiased Estimation and Prediction under a Selection Model,” Biometrics, vol. 31, no. 2, p. 423, Jun. 1975.