# Multi-Fidelity Information Fusion to Model the Position-Dependent Modal Properties of Milling Robots

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

**Physics-based methods**

- conventional rigid-body models [9],
- rigid-body models with additional virtual joints [10] or
- finite element models [11].

**Data-driven algorithms**

#### 1.2. Motivation

#### 1.3. Scope and Approach

## 2. Methods

- the natural frequencies ${\omega}_{i}\left(\mathit{x}\right)$ for each vibration mode $i=1,\dots ,M$ of the robot structure,
- the damping ratios ${\xi}_{i}\left(\mathit{x}\right)$ for each vibration mode $i=1,\dots ,M$ of the robotstructure and
- the residues ${\mathcal{R}}_{i,d}\left(\mathit{x}\right)$ for each vibration mode $i=1,\dots ,M$ and in each mode direction $d\in \{\mathrm{xx},\mathrm{xy},\mathrm{xz},\mathrm{yx},\mathrm{yy},\mathrm{yz},\mathrm{zx},\mathrm{zy},\mathrm{zz}\}$.

**Data generation:**first, the modal parameters are derived from the analytical model and gathered experimentally at the robot (see Section 2.1).**Data preparation:**the data sets are divided into training and testing data sets (see Section 2.2).**Model setup and training:**the spatial behavior of the vibrational features is modeled as follows (see Section 2.3):- (a)
- The primary features (natural frequencies) are modeled using multi-fidelity information fusion approaches (see Section 2.3.1).
- (b)
- The secondary features (damping ratios and residues) are modeled using conventional Gaussian process regression techniques (see Section 2.3.2 and Section 2.3.3).

#### 2.1. Data Generation: Spatial Modal Parameter Identification

#### 2.2. Data Preparation: Sampling Methodology

- The testing data set ${\mathcal{D}}_{\mathrm{test}}$ remains the same for all investigations.
- The actually used training data points ${\mathcal{D}}_{\mathrm{train},\phantom{\rule{4.pt}{0ex}}\mathrm{used}}$ are subsampled from the original training data set ${\mathcal{D}}_{\mathrm{train}}$.

#### 2.3. Model Setup and Training: Data Driven Modeling of the Modal Properties

- natural frequencies (Section 2.3.1),
- damping ratios (Section 2.3.2) and
- residues (Section 2.3.3).

#### 2.3.1. Primary Feature: Natural Frequencies

- a linear kernel: a linear kernel makes it possible to incorporate a linear spatial model using the variance $\sigma $:$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathrm{linear}}(\mathit{x},{\mathit{x}}^{\prime})& ={\sigma}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathit{x}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{x}}^{\prime}.\hfill \end{array}$$
- a quadratic kernel: a quadratic kernel allows more flexibility than a simple linear kernel and is represented by$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathrm{quadratic}}(\mathit{x},{\mathit{x}}^{\prime})& ={\mathit{K}}_{\mathrm{linear}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathrm{linear}}.\hfill \end{array}$$
- a cubic kernel: similar to the generation of a quadratic kernel, the idea of a polynomial kernel can be extended to a cubic kernel, given by$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathrm{cubic}}(\mathit{x},{\mathit{x}}^{\prime})& ={\mathit{K}}_{\mathrm{linear}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathrm{linear}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathrm{linear}}.\hfill \end{array}$$
- a radial basis function (RBF) kernel: the conventional RBF kernel is a very popular kernel, as no assumption on the data’s structure is incorporated. However, an RBF kernel may be prone to overfitting. The kernel is defined as$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathrm{RBF}}(\mathit{x},{\mathit{x}}^{\prime})& ={\sigma}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}exp\left(-\frac{\parallel \mathit{x}-{\mathit{x}}^{\prime}{\parallel}^{2}}{2{l}^{2}}\right),\hfill \end{array}$$

#### 2.3.2. Secondary Feature: Damping Ratios

#### 2.3.3. Secondary Feature: Residues

## 3. Results

#### 3.1. Primary Feature: Natural Frequencies

#### 3.1.1. Validity of the Approach

#### 3.1.2. Accuracy Using an Increasing Number of Training Data Points

#### 3.2. Secondary Feature: Damping Ratios

#### 3.3. Secondary Feature: Residues

#### 3.4. Reconstruction of Frequency Response Functions

#### 3.5. Implementation Details

## 4. Discussion

## 5. Conclusions

- First, an information fusion approach to model the robot’s position-dependent natural frequencies improves the prediction accuracy significantly in comparison to that of conventional Gaussian process regression techniques, especially in scenarios with only a very small number of training data points. In those cases, the prediction uncertainty of conventional Gaussian processes is unreliable, whereas the uncertainty estimation of the linear information fusion scheme is reliable.
- Second, a detailed study was conducted to evaluate different kernel design choices for modeling the robot’s damping ratios and mode residues using conventional Gaussian process regression methods. The data analysis of the position-dependent damping ratios motivates the use of cubic kernels, whereas an RBF kernel is best suited for modeling the residues.
- Third, the position-dependent models can be used to estimate the position-dependent directional dynamics of the robot accurately and quantify the combined model uncertainty using a Monte Carlo algorithm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Measurement Locations

x in m | y in m | z in m |
---|---|---|

1.175 | −0.250 | 1.240 |

1.175 | −0.500 | 1.240 |

1.175 | 0.0 | 1.240 |

1.175 | 1.000 | 1.240 |

1.175 | 0.250 | 1.240 |

1.175 | 0.500 | 1.240 |

1.175 | 0.750 | 1.240 |

1.400 | −0.250 | 1.240 |

1.400 | −0.500 | 1.240 |

1.400 | 0.0 | 1.240 |

1.400 | 1.000 | 1.240 |

1.400 | 250 | 1.240 |

1.400 | 500 | 1.240 |

1.400 | 750 | 1.240 |

1.625 | −250 | 1.240 |

1.625 | −500 | 1.240 |

1.625 | 0.0 | 1.240 |

1.625 | 1.000 | 1.240 |

1.625 | 0.250 | 1.240 |

1.625 | 0.500 | 1.240 |

1.625 | 0.750 | 1.240 |

1.850 | −0.250 | 1.240 |

1.850 | −0.500 | 1.240 |

1.850 | 0.0 | 1.240 |

1.850 | 1.000 | 1.240 |

1.850 | 0.250 | 1.240 |

1.850 | 0.500 | 1.240 |

1.850 | 0.750 | 1.240 |

0.950 | −0.250 | 1.240 |

0.950 | −0.500 | 1.240 |

0.950 | 0.0 | 1.240 |

0.950 | 1.000 | 1.240 |

0.950 | 0.250 | 1.240 |

0.950 | 0.500 | 1.240 |

0.950 | 0.750 | 1.240 |

## Appendix B. Cross Residuals R_{i,xy} and R_{i,yx}

## Appendix C. Rigid Body Model of the Milling Robot

**Figure A2.**Rigid body model of the robot. The orientation of the six local body coordinate systems are illustrated in colored lines ( = x, = y, = z.)

## Appendix D. Mode Shape Visualization

**Figure A3.**The first four mode shapes (first mode 1 in

**a**), second mode in

**b**) third mode in

**c**), fourth mode in

**d**)), simulated with the rigid body model at ${\mathit{x}}_{T}$.

## Appendix E. R^{2} results of R_{i,xx} and R_{i,yy}

**Figure A4.**Benchmarks of different kernel designs for modeling spatial behavior of ${\mathcal{R}}_{i,xx}$ and ${\mathcal{R}}_{i,yy}$ based on ${R}^{2}$ (${R}^{2}$ values lower than 0.1 are not displayed to their full extend for better comprehensibility).

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**Figure 3.**Measured directional frequency response functions and synthesized estimation by Least-Squares Complex Frequency Domain (LSCF) algorithm.

**Figure 5.**Training and testing data points for sampling strategies ${\mathrm{B}}_{3}$, ${\mathrm{B}}_{5}$, ${\mathrm{B}}_{7}$, ${\mathrm{B}}_{10}$ and ${\mathrm{B}}_{15}$ (A).

**Figure 6.**Spatial behavior of first natural frequency as measured (column

**a**)) and predicted by simulation (column

**b**)). There is a linear dependency between simulation results and measurements (column

**c**)).

**Figure 8.**Benchmark of conventional Gaussian process regression with an RBF kernel and the two multi-fidelity schemes AR1 and NARGP using iterative sampling of training data points.

**Figure 9.**Comparison of prediction accuracy on testing data of ${\omega}_{2}$ using sampling strategy ${\mathrm{B}}_{3}$, including model’s predictive uncertainty in form of the 95% prediction interval.

**Figure 11.**Benchmarks of different kernel designs for modeling spatial behavior of ${\mathcal{R}}_{i,xx}$ and ${\mathcal{R}}_{i,yy}$.

**Figure 12.**Measured directional frequency response functions and model predictions at testing point ${\mathit{x}}_{T}={[1.175\mathrm{m},1.0\mathrm{m},1.24\mathrm{m}]}^{T}$ using sampling strategy ${B}_{5}$ (for simplicity, only 300 of 10,000 Monte Carlo samples are illustrated).

Purpose | Package | Version | Language |
---|---|---|---|

Data acquisition | Data acquisition toolbox | R2021a | Matlab |

Generation of frequency domain data | Signal processing toolbox | R2021a | Matlab |

Experimental modal analysis | pyEMA [29] | 0.23 | Python |

(Maximin) LHS sampling | scikit-optimize | 0.8.1 | Python |

Rigid body model | RBDL (ORB Version) [30] | 3.0.0 | C++/Python |

HFGP models | GPy [31] | 1.9.9 | Python |

AR1 model | emukit [32] | 0.4.8 | Python |

NARGP model | emukit [32] | 0.4.8 | Python |

Monte Carlo simulation | Uncertainpy [28] | 1.2.3 | Python |

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**MDPI and ACS Style**

Busch, M.; Zaeh, M.F.
Multi-Fidelity Information Fusion to Model the Position-Dependent Modal Properties of Milling Robots. *Robotics* **2022**, *11*, 17.
https://doi.org/10.3390/robotics11010017

**AMA Style**

Busch M, Zaeh MF.
Multi-Fidelity Information Fusion to Model the Position-Dependent Modal Properties of Milling Robots. *Robotics*. 2022; 11(1):17.
https://doi.org/10.3390/robotics11010017

**Chicago/Turabian Style**

Busch, Maximilian, and Michael F. Zaeh.
2022. "Multi-Fidelity Information Fusion to Model the Position-Dependent Modal Properties of Milling Robots" *Robotics* 11, no. 1: 17.
https://doi.org/10.3390/robotics11010017