# Centralized Networks to Generate Human Body Motions

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Network Structure

#### 2.1.1. Background on Scale-Free Networks and Centralized Networks

#### 2.1.2. Centralized Networks for Elementary Human Motions

**A**- Harmonic basis. Here we assume that$${\mathsf{\Phi}}_{j}(q,b)=cos\left(bjq\right),$$
**B**- System of radial basis functions.For the case where a motion consists of many segments and we observe sharp transitions between those segments, we can use radial basis functions$${\mathsf{\Phi}}_{j}=\varphi \left(b\right|q-{\overline{q}}^{\left(j\right)}\left|\right),\phantom{\rule{1.em}{0ex}}j=1,\dots ,{N}_{m},$$$$\varphi \left(\right|z\left|\right)={exp(-|z|}^{2}/2).$$
**C**- Polynomial basis.Here we take$${\mathsf{\Phi}}_{j}\left(q\right)={q}^{j-1},\phantom{\rule{1.em}{0ex}}j=1,\dots ,{N}_{m}.$$The basis
**B**has an important advantage: the radial basis functions provide local approximations that are important to approximate complicated motions with sharp transitions.To perform switching in the network, we will also use the sigmoidal functions $\sigma $. They are increasing and smooth (at least twice differentiable) functions such that$$\sigma (-\infty )=0,\phantom{\rule{1.em}{0ex}}\sigma (+\infty )=1,\phantom{\rule{1.em}{0ex}}{\sigma}^{\prime}\left(z\right)>0.$$Typical examples can be given by$$\sigma \left(h\right)=\frac{1}{1+exp(-h)},\phantom{\rule{1.em}{0ex}}\sigma \left(h\right)=\frac{1}{2}\left(\frac{h}{\sqrt{1+{h}^{2}}}+1\right).$$

#### 2.1.3. Centralized Networks Generating a Large Class of Human Body Motions

- an RBF network defined by (2);
- a switching module that is a network with $M+1$ nodes, where M is the number of different motions.

#### 2.2. Switching Module

**Lemma**

**1.**

#### 2.3. Algorithm of Construction of the RBF Network to Generate Human Body Motions

#### 2.3.1. Non-Segmented Motions

#### 2.3.2. Segmented Motions

#### 2.4. Comparison with DMPs

## 3. Results

#### 3.1. Results without Segmentation and Ad Hoc Segmentations

#### 3.2. Results Based on Algorithmic Segmentations as Pre-Processing Steps

#### 3.3. Comparison with Other Approaches

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**This image shows the control of one of the x-coordinates of human body motions by a network consisting of two oscillators $({v}_{1},{v}_{2})$ and a radial basis function (RBF) network with $N=6$ nodes. The graph consists of eight nodes denoted by ${v}_{1},{v}_{2},{w}_{1},{w}_{2},{w}_{3},{w}_{4},{w}_{5},{w}_{6}$. Each node ${w}_{i}$ corresponds to a contribution of a radial basis function $\mathsf{\Phi}(q-{\overline{q}}^{\left(j\right)})$. The nodes ${v}_{1},{v}_{2}$ form the set of centers $\mathcal{C}$ and they affect ${w}_{i}$. In turn, the nodes ${w}_{i}$ determine the output coordinate ${x}_{1}$.

**Figure 2.**Modular architecture. This can be seen as an example of the architecture described in [6]. The switching module consists of the center z and the satellites ${\tilde{w}}_{1},{\tilde{w}}_{2},{\tilde{w}}_{3}$. The generating module consists of the centers ${v}_{1},{v}_{2}$ and the satellites ${w}_{1},\dots ,{w}_{6}$. Note that there is a feedback between z and the satellites ${w}_{i}$; however, there is no feedback of ${w}_{j}$ on ${v}_{l}$.

**Figure 3.**An approximation of x-coordinates by 25 satellites and two centers of motion (CMU 86 Trial 1) consisting of jumping, kicking, and punching (sampled at 120 Hz). (

**a**) Right heel; (

**b**) Left heel; (

**c**) Right wrist, distal; (

**d**) Left wrist, distal. The red curve shows the experimentally observed coordinates and the green curve gives their neural approximations.

**Figure 4.**An approximation of y-coordinates by 25 satellites and two centers of motion (CMU 86 Trial 1; sampled at 120 Hz). (

**a**) Right heel; (

**b**) Left heel; (

**c**) Right wrist, distal; (

**d**) Left wrist, distal. The red curve shows the experimentally observed coordinates and the green curve gives their neural approximations.

**Figure 5.**An approximation of y-coordinates by 50 satellites and two centers of motion (CMU 86 Trial 1; sampled at 120 Hz). (

**a**) Right heel; (

**b**) Left heel; (

**c**) Right wrist, distal; (

**d**) Left wrist, distal. The red curve shows the experimentally observed coordinates and the green curve gives their neural approximations.

**Figure 6.**An approximation of z-coordinates by 100 satellites and two centers of motion (CMU 86 Trial 1; sampled at 120 Hz). (

**a**) Right heel; (

**b**) The left heel; (

**c**) Right wrist, distal; (

**d**) The left wrist, distal. The red curve shows the experimentally observed coordinates and the green curve gives their neural approximations.

**Figure 7.**Three-dimensional plot of marker trajectory of the motion of the right wrist (distal; CMU 86 Trial 1). The red curve shows the experimentally observed coordinates and the green curve gives their neural approximations (with two centers and 100 satellites).

**Figure 8.**An approximation of z (vertical) coordinates by 200 neurons and two centers for a simple non-segmented motion (sampled at 120 Hz). The center frequencies are $0.30\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ and $0.72\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$. (

**a**) z-coordinate for the right heel; (

**b**) The left heel; (

**c**) z-coordinate for the right wrist, distal; (

**d**) The left wrist, distal. The red curves show the experimentally observed coordinates and the blue curves give their neural approximations.

**Figure 9.**Approximations of x (left) and y coordinates (right) by 200 neurons and two centers for a simple non-segmented motion (sampled at 120 Hz). The center frequencies are $0.30\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ and $0.72\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$. (

**a**) Right heel , x-coordinate; (

**b**) Left heel, x-coordinate; (

**c**) Right wrist, distal, x-coordinate; (

**d**) Left wrist, distal, x-coordinate; (

**e**) Right heel , y-coordinate; (

**f**) Left heel, y-coordinate; (

**g**) Right wrist, distal, y-coordinate; (

**h**) Left wrist, distal, y-coordinate. The red curves show the experimentally observed coordinates and the blue curves give their neural approximations. Both curves coincide up to pixel accuracy in many places.

**Figure 10.**Approximations of x coordinates of a complicated motion (CMU 86 Trial 2; sampled at 120 Hz) by a radial basis function (RBF) network with three centers and 100 satellites. The motion was segmented into three intervals $[1,1800]$, $[1800,2500]$, and $[2500,4000]$. (

**a**) Right heel; (

**b**) Left heel; (

**c**) Right wrist, distal; (

**d**) Left wrist, distal.

**Figure 11.**An approximation of y coordinates of a complicated motion (CMU 86 Trial 2; sampled at 120 Hz) by an RBF-network with three centers and 100 satellites. The motion was segmented into three intervals $[1,1800]$, $[1800,2500]$, and $[2500,4000]$. (

**a**) Right ankle; (

**b**) Left ankle; (

**c**) Right heel; (

**d**) Left wrist, distal. The red curves show the experimentally observed coordinates and the blue curves represent their neural approximations.

**Figure 12.**An approximation of the z coordinates of a complicated motion (CMU 86 Trial 2; sampled at 120 Hz) by an RBF-network with three centers and 100 satellites. The motion was segmented into three intervals $[1,1800]$, $[1800,2500]$, and $[2500,4000]$. (

**a**) Right ankle; (

**b**) Left ankle; (

**c**) Right heel; (

**d**) Left wrist, distal. The red curves show the experimentally observed coordinates and the blue curves represent their neural approximations.

**Figure 13.**Akaike information criterion corrected for finite sample sizes (AICc) for CMU 86 Trial 1 with 1, 2, 3, and 4 centers and varying numbers of satellites. The global optimum was reached for three centers and 150 satellites.

**Figure 14.**Absolute error (ErrABS) and relative integral errors (Err) for all 31 markers and all 11 segments of CMU 86 Trial 1 using two centers and 49 satellites.

**Figure 15.**Approximation errors (in mm) for the algorithmically found segments of motions in CMU 86 Trial 1. We give the errors for using 16 and 100 satellites. As a comparison we give the results using rhythmic dynamic movement primitives (DMPs) with 100 basis functions computed with pydmps, and the average approximation error of the Bayesian approach reported in [28] (Table 3). Notice that segments 2, 4, 6, and 8 are short transitional motions between the neighboring segments. The average over all segments is 8.7 mm for the DMPs, 7.9 mm for two centers and 16 satellites, and 6.7 mm for two centers and 100 satellites.

**Figure 16.**Approximation errors (in mm) for the algorithmically found segments of motions in CMU 86 Trial 2. We give the errors for using 16 satellites and 49 satellites. As a comparison we give the results using rhythmic DMPs with 100 basis functions computed with pydmps. The average over all segments is 8.0 mm for the DMPs, 15.6 mm for two centers and 16 satellites, and 6.3 mm for two centers and 49 satellites.

**Table 1.**Integral relative accuracies of approximations by centralized networks of CMU 86 Trial 1 using different numbers of oscillators and satellites. The motion was segmented into four segments at frames $[1,1300,2000,3000,4500]$.

Number of Oscillators | Number of Satellites | Integral Relative Accuracies | |||
---|---|---|---|---|---|

Segment 1 | Segment 2 | Segment 3 | Segment 4 | ||

1 | 100 | 0.2133 | 0.0827 | 0.423 | 0.0749 |

2 | 25 | 0.1207 | 0.0486 | 0.0405 | 0.076 |

2 | 50 | 0.0351 | 0.0089 | 0.0084 | 0.0586 |

2 | 100 | 0.0189 | 0.0068 | 0.0029 | 0.0299 |

3 | 25 | 0.0832 | 0.0253 | 0.0297 | 0.0685 |

3 | 100 | 0.0133 | 0.0063 | 0.0031 | 0.0071 |

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**MDPI and ACS Style**

Vakulenko, S.; Radulescu, O.; Morozov, I.; Weber, A.
Centralized Networks to Generate Human Body Motions. *Sensors* **2017**, *17*, 2907.
https://doi.org/10.3390/s17122907

**AMA Style**

Vakulenko S, Radulescu O, Morozov I, Weber A.
Centralized Networks to Generate Human Body Motions. *Sensors*. 2017; 17(12):2907.
https://doi.org/10.3390/s17122907

**Chicago/Turabian Style**

Vakulenko, Sergei, Ovidiu Radulescu, Ivan Morozov, and Andres Weber.
2017. "Centralized Networks to Generate Human Body Motions" *Sensors* 17, no. 12: 2907.
https://doi.org/10.3390/s17122907