# An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Nonlinear Weighted Least Squares Data Approximation

#### 1.2. Trajectory Optimization

#### 1.3. Bayesian Filters

#### 1.4. Contribution

#### 1.5. Structure of the Data Set

#### 1.6. Outline

## 2. Methods

#### 2.1. B-Spline Function Representation

#### 2.2. Marginalized Particle Filter

Algorithm 1: The marginalized particle filter derived from References [44,46] |

#### 2.3. Nonlinear Recursive B-Spline Approximation

#### 2.3.1. Initialization

#### 2.3.2. Measurement Update

Algorithm 2: Nonlinear recursive B-spline approximation |

#### 2.3.3. Time Update with Shift Operation

#### 2.3.4. Effect of the Shift Operation

## 3. Numerical Experiments

#### 3.1. General Experimental Setup

#### 3.2. Effect of Weighting and Nonlinear Measurement Function

_{t,1}of the data points (s

_{t}, y

_{t}). For a better visualization of the approximating functions, only two representative data dots per spline interval are displayed. For f (s) = 30, the deviation between the value of c and its target value y

_{t,4}= 0 has a local maximum (c.f. Figure 2). In NRBA

^{N}and LM

^{N}, this deviation is penalized strongly; hence, these solutions avoid f (s) = 30. In contrast, NRBA

^{L}and LM

^{L}approximate data with y

_{t,1}= 30 closely because the nonlinear criterion is weighted only to a negligible extent.

^{L}and LM

^{N}in Figure 3 reflect this symmetry.

_{n}, y

_{n}) is taken into account.

^{L}and NRBA

^{N}are both asymmetrical and mostly delayed with respect to LM

^{L}and LM

^{N}. However, with NRBA

^{N}, the asymmetry is less distinct. The reason for this is that, in the nonlinear problem, the PF removes states with a high delay more quickly from the particle set because they create a larger error. Additionally, the range of values in NRBA

^{N}is smaller than in NRBA

^{L}so that a present lag is less obvious.

^{med}and NRMSE

^{max}differ only slightly. This suggests that, for the investigated settings, P = 6561 suffices for a convergence of NRBA solutions.

#### 3.3. Effect of Interval Count

^{max}differs more from NRMSE

^{med}and shows larger oscillation amplitudes between s = 130 and s = 170 than for I = 3. This suggests that P = 625 is not sufficient for a convergence of NRBA for I = 1. Although we use only 625 particles for I = 1, the required increase to P = 15,625 for I = 3 is quite strong. This illustrates that keeping the sampling density constant quickly becomes infeasible, especially if computation time constraints are present [44]. Figure 5 shows the results for the nonlinear approximation problem and supports the previously drawn conclusions. Additionally, we see for s ≤ 20, that the conflicting target criteria in the nonlinear approximation problem cause a larger period for stabilization.

^{max}solution to the corresponding NRMSE

^{med}solution, we notice that they differ much more for I = 3. This indicates that more particles are needed for convergence for I = 3. Especially, we notice that, with I = 3, these differences are much larger for NRBA

^{N}than for NRBA

^{L}.

^{med}solutions in both figures. Figure 6 also shows that NRBA

^{N}temporarily decreases below f (s) = 30, the position of the maximum of c (c.f. Figure 2). This illustrates how the sequential data processing of filter-based methods can lead to solutions that differ from those of a batch method.

#### 3.4. Effect of Particle Count on Convergence

#### 3.5. Mean and Standard Deviation of NRBA Error

## 4. Trajectory Optimization

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zhao, X.; Kargoll, B.; Omidalizarandi, M.; Xu, X.; Alkhatib, H. Model Selection for Parametric Surfaces Approximating 3D Point Clouds for Deformation Analysis. Remote Sens.
**2018**, 10, 634. [Google Scholar] [CrossRef] - Jiang, Z. A New Approximation Method with High Order Accuracy. Math. Comput. Appl.
**2017**, 22, 11. [Google Scholar] [CrossRef] - Majid Amirfakhrian, S.D. Approximation of Parametric Functions by Bicubic B-spline Functions. J. Am. Sci.
**2013**, 9. [Google Scholar] - Du, M.; Mei, T.; Liang, H.; Chen, J.; Huang, R.; Zhao, P. Drivers’ Visual Behavior-Guided RRT Motion Planner for Autonomous On-Road Driving. Sensors
**2016**, 16, 102. [Google Scholar] [CrossRef] - Elbanhawi, M.; Simic, M.; Jazar, R.N. Continuous Path Smoothing for Car-Like Robots Using B-Spline Curves. J. Intell. Robot. Syst.
**2015**, 80, 23–56. [Google Scholar] [CrossRef] - Shih, C.L.; Lin, L.C. Trajectory Planning and Tracking Control of a Differential-Drive Mobile Robot in a Picture Drawing Application. Robotics
**2017**, 6, 17. [Google Scholar] [CrossRef] - Liu, H.; Lai, X.; Wu, W. Time-optimal and jerk-continuous trajectory planning for robot manipulators with kinematic constraints. Robot. Comput.-Integr. Manuf.
**2013**, 29, 309–317. [Google Scholar] [CrossRef] - Zhao, D.; Guo, H. A Trajectory Planning Method for Polishing Optical Elements Based on a Non-Uniform Rational B-Spline Curve. Appl. Sci.
**2018**, 8, 1355. [Google Scholar] [CrossRef] - Kineri, Y.; Wang, M.; Lin, H.; Maekawa, T. B-spline surface fitting by iterative geometric interpolation/approximation algorithms. Comput.-Aided Des.
**2012**, 44, 697–708. [Google Scholar] [CrossRef] - Yunbao Huang, X.Q. Dynamic B-spline surface reconstruction: Closing the Sensing-and-modeling loop in 3D digitization. Comput.-Aided Des.
**2007**, 39, 987–1002. [Google Scholar] [CrossRef] - Monir, A.; Mraoui, H. Spline approximations of the Lambert W function and application to simulate generalized Gaussian noise with exponent α = 1/2. Digit. Signal Process.
**2014**, 33, 34–41. [Google Scholar] [CrossRef] - Rebollo-Neira, L.; Xu, Z. Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval. Signal Process.
**2010**, 90, 2308–2313. [Google Scholar] [CrossRef] - Reis, M.J.; Ferreira, P.J.; Soares, S.F. Linear combinations of B-splines as generating functions for signal approximation. Digit. Signal Process.
**2005**, 15, 226–236. [Google Scholar] [CrossRef] - Panda, R.; Chatterji, B. Least squares generalized B-spline signal and image processing. Signal Process.
**2001**, 81, 2005–2017. [Google Scholar] [CrossRef] - Roark, R.M.; Escabi, M.A. B-spline design of maximally flat and prolate spheroidal-type FIR filters. IEEE Trans. Signal Process.
**1999**, 47, 701–716. [Google Scholar] [CrossRef] - Izadian, J.; Farahbakhsh, N. Solving Nonlinear Least Squares Problems with B-Spline Functions. Appl. Math. Sci.
**2012**, 6, 1667–1676. [Google Scholar] - Dahmen, W.; Reusken, A. (Eds.) Numerik für Ingenieure und Naturwissenschaftler, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Ruhe, A.; Wedin, P.A. Algorithms for Separable Nonlinear Least Squares Problems. SIAM Rev.
**1980**, 22, 318–337. [Google Scholar] [CrossRef] - Haupt, G.T.; Kasdin, N.J.; Keiser, G.M.; Parkinson, B.W. Optimal recursive iterative algorithm for discrete nonlinear least-squares estimation. J. Guid. Control Dyn.
**1996**, 19, 643–649. [Google Scholar] [CrossRef] - Zhao, K.; Ling, F.; Lev-Ari, H.; Proakis, J.G. Sliding window order-recursive least-squares algorithms. IEEE Trans. Signal Process.
**1994**, 42, 1961–1972. [Google Scholar] [CrossRef] - Dias, F.M.; Antunes, A.; Vieira, J.; Mota, A. A sliding window solution for the on-line implementation of the Levenberg-Marquardt algorithm. Eng. Appl. Arti. Intell.
**2006**, 19, 1–7. [Google Scholar] [CrossRef] - Gao, Z.; Yan, W.; Hu, H.; Li, H. Human-centered headway control for adaptive cruise-controlled vehicles. Adv. Mech. Eng.
**2015**, 7. [Google Scholar] [CrossRef] - Radke, T. Energieoptimale Längsführung von Kraftfahrzeugen durch Einsatz vorausschauender Fahrstrategien. Ph.D. Thesis, KIT Scientific Publishing, Karlsruhe, Germany, 2013. [Google Scholar] [CrossRef]
- Wahl, H.G. Optimale Regelung eines prädiktiven Energiemanagements von Hybridfahrzeugen. Ph.D. Thesis, KIT Scientific Publishing, Karlsruhe, Germany, 2015. [Google Scholar] [CrossRef]
- Zhang, S.; Xiong, R. Adaptive energy management of a plug-in hybrid electric vehicle based on driving pattern recognition and dynamic programming. Appl. Energy
**2015**, 155, 68–78. [Google Scholar] [CrossRef] - Winner, H.; Hakuli, S.; Lotz, F.; Singer, C. Handbook of Driver Assistance Systems—Basic Information, Components and Systems for Active Safety and Comfort; Springer: Cham, Switzerland, 2016. [Google Scholar] [CrossRef]
- Passenberg, B. Theory and Algorithms for Indirect Methods in Optimal Control of Hybrid Systems. Ph.D. Thesis, Technische Universität München, Munich, Germany, 2012. [Google Scholar]
- Giron-Sierra, J.M. Kalman Filter, Particle Filter and Other Bayesian Filters. In Digital Signal Processing with Matlab Examples; Springer: Singapore, 2017; Volume 3, pp. 3–148. [Google Scholar] [CrossRef]
- Wu, Z.; Li, J.; Zuo, J.; Li, S. Path Planning of UAVs Based on Collision Probability and Kalman Filter. IEEE Access
**2018**, 6, 34237–34245. [Google Scholar] [CrossRef] - Arulampalam, M.S.; Maskell, S.; Gordon, N.; Clapp, T. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process.
**2002**, 50, 174–188. [Google Scholar] [CrossRef] - Cappe, O.; Godsill, S.J.; Moulines, E. An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo. Proc. IEEE
**2007**, 95, 899–924. [Google Scholar] [CrossRef] - Alessandri, A.; Cuneo, M.; Pagnan, S.; Sanguineti, M. A recursive algorithm for nonlinear least-squares problems. Comput. Optim. Appl.
**2007**, 38, 195–216. [Google Scholar] [CrossRef] - Deng, F.; Chen, J.; Chen, C. Adaptive unscented Kalman filter for parameter and state estimation of nonlinear high-speed objects. J. Syst. Eng. Electron.
**2013**, 24, 655–665. [Google Scholar] [CrossRef] - Ha, X.V.; Ha, C.; Lee, J. Trajectory Estimation of a Tracked Mobile Robot Using the Sigma-Point Kalman Filter with an IMU and Optical Encoder. In Intelligent Computing Technology; Huang, D.S., Jiang, C., Bevilacqua, V., Figueroa, J.C., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; pp. 415–422. [Google Scholar]
- Nieto, M.; Cortés, A.; Otaegui, O.; Arróspide, J.; Salgado, L. Real-time lane tracking using Rao-Blackwellized particle filter. J. Real-Time Image Process.
**2016**, 11, 179–191. [Google Scholar] [CrossRef] - Lee, Y. Optimization of Moving Objects Trajectory Using Particle Filter. In Intelligent Computing Theory; Huang, D.S., Bevilacqua, V., Premaratne, P., Eds.; Springer International Publishing: Cham, Switzerlan, 2014; pp. 55–60. [Google Scholar]
- Xin, L.; Hailong, P.; Jianqiang, L. Trajectory prediction based on particle filter application in mobile robot system. In Proceedings of the 2008 27th Chinese Control Conference, Kunming, China, 16–18 July 2008; pp. 389–393. [Google Scholar] [CrossRef]
- Zhou, F.; He, W.J.; Fan, X.Y. Marginalized Particle Filter for Maneuvering Target Tracking Application. In Advances in Grid and Pervasive Computing; Bellavista, P., Chang, R.S., Chao, H.C., Lin, S.F., Sloot, P.M.A., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 542–551. [Google Scholar]
- Qian, K.; Ma, X.; Dai, X.; Fang, F. Improved Rao-Blackwellized particle filter for simultaneous robot localization and person-tracking with single mobile sensor. J. Control Theory Appl.
**2011**, 9, 472–478. [Google Scholar] [CrossRef] - Yatim, N.M.; Buniyamin, N. Development of Rao-Blackwellized Particle Filter (RBPF) SLAM Algorithm Using Low Proximity Infrared Sensors. In Proceedings of the 9th International Conference on Robotic, Vision, Signal Processing and Power Applications, Penang, Malaysia, 2–3 February 2017; Ibrahim, H., Iqbal, S., Teoh, S.S., Mustaffa, M.T., Eds.; Springer: Singapore, 2017; pp. 395–405. [Google Scholar]
- Liu, J.; Wang, Z.; Xu, M. A Kalman Estimation Based Rao-Blackwellized Particle Filtering for Radar Tracking. IEEE Access
**2017**, 5, 8162–8174. [Google Scholar] [CrossRef] - Skoglar, P.; Orguner, U.; Tornqvist, D.; Gustafsson, F. Road target tracking with an approximative Rao-Blackwellized Particle Filter. In Proceedings of the 2009 12th International Conference on Information Fusion, Seattle, WA, USA, 6–9 July 2009; pp. 17–24. [Google Scholar]
- Jauch, J.; Bleimund, F.; Rhode, S.; Gauterin, F. Recursive B-spline approximation using the Kalman filter. Eng. Sci. Technol. Int. J.
**2017**, 20, 28–34. [Google Scholar] [CrossRef] - Hendeby, G.; Karlsson, R.; Gustafsson, F. A New Formulation of the Rao-Blackwellized Particle Filter. In Proceedings of the 2007 IEEE/SP 14th Workshop on Statistical Signal Processing, Madison, WI, USA, 26–29 August 2007; pp. 84–88. [Google Scholar] [CrossRef]
- Lyche, T.; Mørken, K. Spline Methods Draft; Department of Informatics, Centre of Mathematics for Applications, University of Oslo: Oslo, Norway, 2008. [Google Scholar]
- Schön, T.; Gustafsson, F.; Nordlund, P.J. Marginalized particle filters for mixed linear/nonlinear state-space models. IEEE Trans. Signal Process.
**2005**, 53, 2279–2289. [Google Scholar] [CrossRef] - Schön, T. Rao-Blackwellized Particle Filter—MATLAB Code. 2011. Available online: http://user.it.uu.se/~thosc112/research/rao-blackwellized-particle.html (accessed on 16 January 2018).
- Arasaratnam, I.; Haykin, S. Cubature Kalman Filters. IEEE Trans. Autom. Control
**2009**, 54, 1254–1269. [Google Scholar] [CrossRef] - Jauch, J. NRBA Matlab Files. 2019. Available online: http://github.com/JensJauch/nrba (accessed on 23 March 2019).
- Marquardt, D.W. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math.
**1963**, 11, 431–441. [Google Scholar] [CrossRef] - Hendeby, G.; Karlsson, R.; Gustafsson (EURASIPMember), F. The Rao-Blackwellized Particle Filter: A Filter Bank Implementation. EURASIP J. Adv. Signal Process.
**2010**, 2010, 724087. [Google Scholar] [CrossRef] - Bargende, M.; Reuss, H.; Wiedemann, J. 14. Internationales Stuttgarter Symposium: Automobil- und Motorentechnik; Springer: Wiesbaden, Germany, 2014. [Google Scholar]
- Bender, S.; Chodura, H.; Groß, M.; Kühn, T.; Watteroth, V. e-generation—Ein Forschungsprojekt mit positiver Bilanz. Porsche Eng. Mag.
**2015**, 2, 22–27. [Google Scholar] - Zimmer, M. Durchgängiger Simulationsprozess zur Effizienzsteigerung und Reifegraderhöhung von Konzeptbewertungen in der Frühen Phase der Produktentstehung; Wissenschaftliche Reihe Fahrzeugtechnik Universität Stuttgart, Springer: Wiesbaden, Germany, 2015. [Google Scholar]
- Vaillant, M. Design Space Exploration zur multikriteriellen Optimierung elektrischer Sportwagenantriebsstränge. Ph.D. Thesis, KIT Scientific Publishing, Karlsruhe, Germany, 2016. [Google Scholar] [CrossRef]
- Zhao, Z.S.; Feng, X.; Lin, Y.y.; Wei, F.; Wang, S.K.; Xiao, T.L.; Cao, M.Y.; Hou, Z.G.; Tan, M. Improved Rao-Blackwellized Particle Filter by Particle Swarm Optimization. J. Appl. Math.
**2013**, 2013, 302170. [Google Scholar] [CrossRef] - Yin, J.; Zhang, J.; Mike, K. The Marginal Rao-Blackwellized Particle Filter for Mixed Linear/Nonlinear State Space Models. Chin. J. Aeron.
**2007**, 20, 346–352. [Google Scholar] [CrossRef]

**Figure 1.**The allocation of the available data points and computed estimates $\widehat{\mathit{x}}$ to KF iterations in RBA versus MPF iterations in NRBA: The arrows indicate the needed information for computing the estimates. By definition, we use $(s{}_{1},\mathit{y}{}_{1})$ for computing ${\widehat{\mathit{x}}}_{0}^{+}$ and $s{}_{n}$ for ${\widehat{\mathit{x}}}_{n+1}^{-}$ as indicated by the dashed arrows.

**Figure 2.**The nonlinear measurement function $c\left(f(s)\right)$ that depends on the value of the B-spline function $f\left(s\right)$ that approximates the data. c is itself a B-spline function.

**Figure 3.**Approximating the B-spline function f determined by NRBA with a number of spline intervals $I=1$ and particle count $P=6561={9}^{4}$ in comparison to the LM solution: ${\mathrm{NRBA}}^{\mathrm{L}}$ and ${\mathrm{LM}}^{\mathrm{L}}$ denote solutions of the algorithms for the quasi-linear problem whereas ${\mathrm{NRBA}}^{\mathrm{N}}$ and ${\mathrm{LM}}^{\mathrm{N}}$ refer to solutions for the nonlinear problem. ${\mathrm{NRMSE}}^{med}$ and ${\mathrm{NRMSE}}^{max}$ denote the NRBA solution with the median or maximum normalized root mean square error (NRMSE) compared to the LM solution with the same weighting. Forty of the 400 data points $(s{}_{t},y{}_{t,1})$ and the knots $\kappa =0,5,\dots ,200$ are shown. The arrow indicates the range in which NRBA can adapt $f(s)$, while the data in the interval $[190,200)$ is processed.

**Figure 4.**Approximating B-spline function, f is determined by NRBA for various numbers of spline intervals $I$ and various particle counts $P$ in comparison to the LM solution. ${\mathrm{NRBA}}^{\mathrm{L}}$ and ${\mathrm{LM}}^{\mathrm{L}}$ denote solutions of the corresponding algorithm for the quasi-linear approximation problem. ${\mathrm{NRMSE}}^{\mathrm{med}}$ and ${\mathrm{NRMSE}}^{\mathrm{max}}$ denote the NRBA solution with the median or maximum normalized root mean square error (NRMSE) compared to the LM solution with the same weighting. Forty of the 400 data points $(s{}_{t},y{}_{t,1})$ and the knots $\kappa =0,5,\dots ,200$ are shown. The arrows indicate the range in which NRBA can adapt $f(s)$, while the data in the interval $[190,200)$ is processed.

**Figure 5.**Approximating B-spline function, f is determined by NRBA for various numbers of spline intervals $I$ and various particle counts $P$ in comparison to the LM solution. ${\mathrm{NRBA}}^{\mathrm{N}}$ and ${\mathrm{LM}}^{\mathrm{N}}$ denote solutions of the corresponding algorithm for the nonlinear approximation problem. ${\mathrm{NRMSE}}^{\mathrm{med}}$ and ${\mathrm{NRMSE}}^{\mathrm{max}}$ denote the NRBA solution with the median or maximum normalized root mean square error (NRMSE) compared to the LM solution with the same weighting. Forty of the 400 data points $(s{}_{t},y{}_{t,1})$ and the knots $\kappa =0,5,\dots ,200$ are shown. The arrows indicate the range in which NRBA can adapt $f(s)$, while the data in the interval $[190,200)$ is processed.

**Figure 6.**Approximating B-spline function, f is determined by NRBA with the number of spline intervals $I=3$ and particle count $P={9}^{4}=6561$ in comparison to the LM solution. ${\mathrm{NRBA}}^{\mathrm{L}}$ and ${\mathrm{LM}}^{\mathrm{L}}$ denote solutions of the algorithms for the quasi-linear problem whereas ${\mathrm{NRBA}}^{\mathrm{N}}$ and ${\mathrm{LM}}^{\mathrm{N}}$ refer to solutions for the nonlinear problem. ${\mathrm{NRMSE}}^{med}$ and ${\mathrm{NRMSE}}^{max}$ denote the NRBA solution with the median or maximum normalized root mean square error (NRMSE) compared to the LM solution with the same weighting. Forty of the 400 data points $(s{}_{t},y{}_{t,1})$ and the knots $\kappa =0,5,\dots ,200$ are shown. The arrow indicates the range in which NRBA can adapt $f(s)$, while the data in the interval $[190,200)$ is processed.

**Figure 7.**The convergence of NRBA: Normalized root mean square error (NRMSE) of NRBA versus the particle count $P$. ${\mathrm{NRMSE}}^{\mathrm{min}}$, ${\mathrm{NRMSE}}^{\mathrm{med}}$, and ${\mathrm{NRMSE}}^{\mathrm{max}}$ denote the nonlinear recursive B-spline approximation (NRBA) solution with the minimum, median, or maximum NRMSE compared to the LM solution in the Monte Carlo analysis. L and N denote the quasi-linear and nonlinear weighting and I, the number of spline intervals.

**Figure 8.**First diagram: Velocity v versus time $\tau $ according to the velocity set points ${v}_{\mathrm{Set}}$ of the reference and three trajectories ${\mathrm{NRBA}}^{1}$, ${\mathrm{NRBA}}^{2}$, and ${\mathrm{NRBA}}^{3}$ optimized by NRBA that differ in the variance of power measurement. Second diagram: Estimated electric traction power ${\widehat{\mathrm{P}}}_{\mathrm{elec}}$ according to mathematical model. Third diagram: Traction power loss ${\mathrm{P}}_{\mathrm{loss}}$. Fourth diagram: Traction energy E.

**Table 1.**The mean and standard deviation of error vector of function values between NRBA and LM over all 50 Monte Carlo runs with a quasi-linear approximation problem L, nonlinear approximation problem N, and number of spline intervals $I$.

Mean of Error Vector | Standard Deviation of Error Vector | |||||||
---|---|---|---|---|---|---|---|---|

Particle Count | L, I = 1 | L, I = 3 | N, I = 1 | N, I = 3 | L, I = 1 | L, I = 3 | N, I = 1 | N, I = 3 |

256 | 0.0041 | −0.0088 | −0.2268 | −0.5820 | 0.8738 | 0.9143 | 0.6225 | 1.2525 |

625 | 0.0150 | 0.0072 | −0.0979 | −0.4386 | 0.7692 | 0.8224 | 0.4030 | 1.0902 |

729 | −0.0064 | −0.0176 | −0.0930 | −0.4350 | 0.8231 | 0.7988 | 0.3904 | 1.0975 |

1296 | −0.0028 | −0.0156 | −0.0611 | −0.2248 | 0.7294 | 0.7365 | 0.3361 | 0.6988 |

2401 | 0.0005 | −0.0009 | −0.0445 | −0.1851 | 0.6480 | 0.6851 | 0.2965 | 0.6454 |

4096 | 0.0014 | 0.0050 | −0.0296 | −0.2189 | 0.6436 | 0.6538 | 0.2583 | 0.7106 |

6561 | 0.0011 | −0.0069 | −0.0334 | −0.1340 | 0.5930 | 0.6084 | 0.2498 | 0.5673 |

15,625 | 0.0056 | −0.0015 | −0.0204 | −0.0512 | 0.5502 | 0.5715 | 0.2216 | 0.3124 |

**Table 2.**The mean and standard deviation of error vector of coefficient values between NRBA and LM over all 50 Monte Carlo runs with a quasi-linear approximation problem L, a nonlinear approximation problem N, and number of spline intervals $I$.

Mean of Error Vector | Standard Deviation of Error Vector | |||||||
---|---|---|---|---|---|---|---|---|

Particle Count | L, I = 1 | L, I = 3 | N, I = 1 | N, I = 3 | L, I = 1 | L, I = 3 | N, I = 1 | N, I = 3 |

256 | 0.0041 | −0.0098 | −0.3036 | −0.5902 | 1.2019 | 1.3057 | 1.2664 | 1.7251 |

625 | 0.0133 | 0.0044 | −0.1640 | −0.4562 | 1.0725 | 1.1953 | 0.8450 | 1.4869 |

729 | −0.0040 | −0.0055 | −0.1507 | −0.4742 | 1.1971 | 1.1239 | 0.8092 | 1.5128 |

1296 | −0.0024 | −0.0171 | −0.1212 | −0.2695 | 1.0045 | 1.0828 | 0.7122 | 1.1541 |

2401 | 0.0005 | 0.0001 | −0.1099 | −0.2271 | 0.9309 | 0.9916 | 0.6654 | 1.0487 |

4096 | 0.0019 | 0.0072 | −0.0725 | −0.2511 | 0.9334 | 0.9402 | 0.5621 | 1.0439 |

6561 | 0.0013 | −0.0053 | −0.0738 | −0.1889 | 0.8614 | 0.9000 | 0.5201 | 0.9672 |

15,625 | 0.0054 | −0.0008 | −0.0506 | −0.1064 | 0.8211 | 0.8329 | 0.4605 | 0.6524 |

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**MDPI and ACS Style**

Jauch, J.; Bleimund, F.; Frey, M.; Gauterin, F. An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization. *Mathematics* **2019**, *7*, 355.
https://doi.org/10.3390/math7040355

**AMA Style**

Jauch J, Bleimund F, Frey M, Gauterin F. An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization. *Mathematics*. 2019; 7(4):355.
https://doi.org/10.3390/math7040355

**Chicago/Turabian Style**

Jauch, Jens, Felix Bleimund, Michael Frey, and Frank Gauterin. 2019. "An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization" *Mathematics* 7, no. 4: 355.
https://doi.org/10.3390/math7040355