# A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion

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## Abstract

**:**

## 1. A Survey of Some Geometrical Concepts

#### 1.1. Riemannian Manifolds

#### 1.2. Matrix Lie Groups

#### 1.3. Special Euclidean Group

## 2. Proposed Method for Change-Point Detection

#### 2.1. Design for Change-Point Detection Method

#### 2.2. Method for Change-Point Detection

**Theorem**

**1.**

**Proof.**

Algorithm 1 Algorithm for calculating the series of geodesic distances |

Inputs: Points ${X}_{1},\dots ,{X}_{n}$ on $SE(3)$ and window size N.Output: The approximation of the series ${(d({X}_{i},{\mathcal{M}}_{f}({X}_{i})))}_{i}$.For $i=N+1,\dots ,n-N.$Initialization: Set ${\widehat{X}}_{i}:=exp\left(\frac{1}{2N+1}{\sum}_{j=i-N}^{i+N}log({X}_{j})\right).$Main loop:$d{X}_{j}={\widehat{X}}_{i}^{-1}{X}_{j},j=i-N,\dots ,i,\dots ,i+N$. $d{\widehat{X}}_{i}=exp\left(\frac{1}{2N+1}{\sum}_{j=i-N}^{i+N}log(d{X}_{j})\right)$. ${\widehat{X}}_{i}={\widehat{X}}_{i}d{\widehat{X}}_{i}$. If $\parallel {\sum}_{j=i-N}^{i+N}log(d{X}_{j}){\parallel}_{F}$ is sufficiently small, the output is $d({X}_{i},{\mathcal{M}}_{f}({X}_{i}))\approx {\parallel log({X}_{i})-log({\widehat{X}}_{i})\parallel}_{F}$. Otherwise continue looping.Endfor. |

## 3. Simulations

#### 3.1. Change-Point Detection for the Motion of a Cylinder

#### 3.2. Change-Point Detection for the Motion of the STAUBERT TX90XL Manipulator

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**When a change-point occurs, the series ${(d({X}_{i},{\mathcal{M}}_{f}({X}_{i})))}_{i}$ displays a local maximum.

**Figure 2.**(

**a**) Blue cylinders denote identified change-points; (

**b**) Detection results with different window sizes.

**Figure 3.**(

**a**) Scanning on the surface of a rectangular workpiece with rounded corners by a STAUBLI TX90XL manipulator; (

**b**) Change points of the motion trajectory.

**Figure 4.**Comparison of the original velocity without change-point detection to the correction velocity for each joint axis.

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**MDPI and ACS Style**

Duan, X.; Sun, H.; Zhao, X.
A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion. *Entropy* **2019**, *21*, 531.
https://doi.org/10.3390/e21050531

**AMA Style**

Duan X, Sun H, Zhao X.
A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion. *Entropy*. 2019; 21(5):531.
https://doi.org/10.3390/e21050531

**Chicago/Turabian Style**

Duan, Xiaomin, Huafei Sun, and Xinyu Zhao.
2019. "A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion" *Entropy* 21, no. 5: 531.
https://doi.org/10.3390/e21050531