# 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

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**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 $c\left(\right)open="("\; close=")">f(s)$ that depends on the value of the B-spline function $f\left(\right)open="("\; close=")">s$ 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