**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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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.

**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 $[1,1300,2000,3000,4500]$.

**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 |