# Optimal Control as a Tool for Innovation in Aerial Twisting on a Trampoline

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Skeletal Model

#### 2.2. Formulation of the Optimization Problem

#### 2.3. Practical Implementation

#### 2.4. Multi-Start Approach

- Twist time history (${q}_{6}\left(t\right)$) linearly increasing from 0 to $[2,3,4,5]$ rotations.
- Number of shooting nodes $N\in \mathcal{N}=\left(\right)open="\{"\; close="\}">295\dots 305$.
- Random arm elevation (${q}_{7,8}\left(t\right)$).
- Random arm torques (${\tau}_{1-4}\left(t\right)$).

#### 2.5. Robustness Analysis of the Optimal Solutions

#### 2.6. Biomechanical Analysis of the Aerial Twist Strategies

- The arm is moving (${\dot{q}}_{7,8}\left(t\right)>{90}^{\xb0}/s$).
- The arm is not aligned with the body (the angle between the arm and the body is in the range ${[10,170]}^{\xb0}$).

#### 2.7. Biomechanical Analysis of the Cat Twist Strategies

- The twist (i.e., z-axis) component of the arm’s angular momentum in the global frame is non-null ($|{L}_{arm}|0.1$ kg m${}^{2}$ s${}^{-1}$).
- The twist (i.e., z-axis) components of the arm’s and body’s angular momenta are of opposite signs ($|{L}_{arm}|\times |{L}_{body}|0$).
- The twist velocity resulting from the gravity-free simulation is significant (${q}_{6}>{30}^{\xb0}$/s).

## 3. Results

#### 3.1. Distribution of Optimal Solutions

#### 3.2. Analysis of Locally Optimal Selected Solutions

#### 3.3. Robustness of Selected Techniques

#### 3.4. Best Tilting Plane

#### 3.5. Cat Twist Strategies

## 4. Discussion

#### 4.1. A Wide Range of Techniques

#### 4.2. Robustness of Optimal Techniques

#### 4.3. Best Tilting Plane

#### 4.4. Limitations

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

OCP | Optimal control problem |

BTP | Best tilting plane |

DoF | Degree-of-freedom |

PHP | penalized hand path |

UHP | unpenalized hand path |

RMS | root mean square |

## Appendix A. The Best Tilting Plane

**Figure A1.**Illustration of the BTP. (

**a**) 3D representation, with the somersault axis (in black), fixed in an inertial frame and the twist axis (red, blue and green) at different instants ($1,2$ and 3), rotating due to the somersaulting and tilting. At each instant, the BTP is spanned by the somersault and the twisting axes. (

**b**) 2D representation, in a frame fixed to the BTP (rotating around the somersault axis). The twist axis rotates in this plane, due to the tilting of the body.

**Figure A2.**Kinograms of a 2D adduction of the arm (

**a**) and the best-case possible of a 3D adduction of the arm happening in the BTP (

**b**). Both kinograms are presented from the BTP point of view. To the right-hand side of the arrows, the snapshots from the kinograms are superimposed to show arm trajectory.

## Appendix B. Cat and Aerial Contributions

**Figure A3.**Sections of technique V involving cat (

**orange shaded**) and aerial (

**blue shaded**) twist strategies according to criteria previously mentioned in Section 3.3. Regions where arms are not moving are gray shaded and hatched. Arm’s (

**bold orange line**) and body’s (

**thin orange line**) angular momentum expressed in the body’s reference frame, the angle of deviation of the arm movement from the BTP (

**blue line**) and the angle between the body and the arm (

**black line**) as functions of time are also presented. Under the graphs are snapshots of the body configuration seen from the top of the head at corresponding instants.

## Appendix C. Reoptimization vs. Simulation

**Figure A4.**Time histories of the tilt (

**a**) and twist (

**b**) angles for 100 simulations in perturbed conditions for the first half of technique III with (

**pink lines**) and without (

**green lines**) reoptimization of the second half of the skill. The optimal (original technique III) is presented (

**black line**) for reference.

## Appendix D. Performance of Cat Twist Strategies

Technique | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Cat twist | − | − | ${20}^{\xb0}$ | ${25}^{\xb0}$ | ${21}^{\xb0}$ | $-{39}^{\xb0}$ |

Total twist | ${363}^{\xb0}$ | ${1057}^{\xb0}$ | ${1087}^{\xb0}$ | ${1124}^{\xb0}$ | ${1180}^{\xb0}$ | ${1764}^{\xb0}$ |

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**Figure 1.**Model definition with 10 DoFs: translations of the root (${q}_{1-3}$), rotations of the root (${q}_{4-6}$), right arm and left arm abduction/adduction (${q}_{7-8}$), right arm and left arm change in plane of elevation (${q}_{9-10}$). Arm rotation sequence: change in plane of elevation, elevation.

**Figure 2.**Number of twists generated by optimal techniques as a function of the hand path length. The color code represents the mean angle from the arm movement to the best tilting plane (BTP). The size of each circle represents the time ratio wherein the arms are moving. Magenta points represent the selected solutions. Black boxes delimit the most dense areas of solutions. The magenta box delimits a cluster of solutions which seem to be greatly affected by the use of the BTP; an enlargement of this area is presented above the graph. The black dashed line presents the linear regression over all four types of problem.

**Figure 3.**Animations of 2D techniques presented in a frame attached to the root segment in order to emphasize the motions of the arms. All arm motions occur in the frontal plane.

**Figure 4.**Animations of 3D techniques presented in a frame attached to the root segment in order to emphasize the motion of the arms.

**Figure 5.**Sections of technique V involving cat (orange shaded) and aerial (blue shaded) twist strategies according to previously mentioned criteria superimposed to the arm elevation and the twist rate.

Start | Skill | End | |
---|---|---|---|

$\mathit{t}=\mathit{0}$ | $\mathbf{0}<\mathit{t}<\mathit{T}$ | $\mathit{t}=\mathit{T}$ | |

Standing position on the trampoline | Reasonable tilt | Landing position on the trampoline | |

$\left[\mathrm{m}\right]$ | $-1.0e-4\le \phantom{\rule{0.222222em}{0ex}}{q}_{1-3}\le 1.0e-4$ | $-0.1\le \phantom{\rule{0.222222em}{0ex}}{q}_{1-3}\le 0.1$ | |

${[}^{\xb0}]$ | $-6e-3\le \phantom{\rule{0.222222em}{0ex}}{q}_{4-6}\le 6e-3$ | $-45\le \phantom{\rule{0.222222em}{0ex}}{q}_{5}\le 45$ | $350\le \phantom{\rule{0.222222em}{0ex}}{q}_{4}\le 370$ |

$-15\le \phantom{\rule{0.222222em}{0ex}}{q}_{5}\le 15$ | |||

Arms upward | Realist arm amplitudes | Arms upward | |

${[}^{\xb0}]$ | ${q}_{7,8}=160$ | $0\le \phantom{\rule{0.222222em}{0ex}}{q}_{7,8}\le 180$ | $155\le \phantom{\rule{0.222222em}{0ex}}{q}_{7,8}\le 166$ |

$-6e-3\le \phantom{\rule{0.222222em}{0ex}}{q}_{9,10}\le 6e-3$ | $-45\le \phantom{\rule{0.222222em}{0ex}}{q}_{9,10}\le 135$ | ||

Realistic ejection velocities | |||

$[\mathrm{m}/\mathrm{s}]$ | $-10\le \phantom{\rule{0.222222em}{0ex}}{\dot{q}}_{1,2}\le 10$ | ||

$-5.9\le \phantom{\rule{0.222222em}{0ex}}{\dot{q}}_{3}\le 6.1$ | |||

${[}^{\xb0}/\mathrm{s}]$ | $-6e-3\le \phantom{\rule{0.222222em}{0ex}}{\dot{q}}_{5,6}\le 6e-3$ | ||

Null velocity at each arm DoF | Arm velocities | Arm velocities | |

${[}^{\xb0}/\mathrm{s}]$ | $-6e-3\le \phantom{\rule{0.222222em}{0ex}}{\dot{q}}_{7-10}\le 6e-3$ | $-5730\le \phantom{\rule{0.222222em}{0ex}}{\dot{q}}_{7-10}\le 5730$ | $-5730\le \phantom{\rule{0.222222em}{0ex}}{\dot{q}}_{7-10}\le 5730$ |

Arm torques | Arm torques | Arm torques | |

$\left[\mathrm{N}\mathrm{m}\right]$ | $-100\le \phantom{\rule{0.222222em}{0ex}}{\tau}_{1-4}\le 100$ | $-100\le \phantom{\rule{0.222222em}{0ex}}{\tau}_{1-4}\le 100$ | $-100\le \phantom{\rule{0.222222em}{0ex}}{\tau}_{1-4}\le 1004$ |

**Table 2.**Characteristics of selected locally optimal arm techniques, including the number of twists, the mean angle between the BTP and the plane of motion of the arms (column $\varphi $), the length of the hand path (column Complexity), the sum of the joint torques (column Effort), the robustness (column ${\Delta}_{perf}\pm {\sigma}_{perf}$), the reliability of the robustness (column ${\Delta}_{kin}\pm {\sigma}_{kin}$) and the convergence rate of the reoptimization (column Convergence).

# | Type | Twists | $\mathit{\varphi}$ | Complexity | Effort | ${\Delta}_{\mathit{p}\mathit{e}\mathit{r}\mathit{f}}\pm {\mathit{\sigma}}_{\mathit{p}\mathit{e}\mathit{r}\mathit{f}}$ | ${\Delta}_{\mathit{k}\mathit{i}\mathit{n}}\pm {\mathit{\sigma}}_{\mathit{k}\mathit{i}\mathit{n}}$ | Convergence |
---|---|---|---|---|---|---|---|---|

[${}^{\xb0}$] | [m] | [Nm] | [${}^{\xb0}$] | [${}^{\xb0}$] | [%] | |||

I | 2D-PHP | $1.01$ | 56 | $6.92$ | $4.5$ | $2.37\pm 0.06$ | $10\pm 6$ | 87 |

II | 2D-UHP | $2.94$ | 41 | $18.2$ | $151.3$ | $3.11\pm 0.04$ | $12\pm 7$ | 27 |

III | 3D-PHP | $3.02$ | 42 | $16.4$ | $89.3$ | $4.31\pm 0.26$ | $19\pm 5$ | 91 |

IV | 3D-PHP | $3.12$ | 45 | $15.0$ | $64.4$ | $3.77\pm 0.34$ | $17\pm 9$ | 82 |

V | 3D-PHP | $3.28$ | 25 | $10.8$ | $140.9$ | $5.17\pm 0.34$ | $25\pm 17$ | 78 |

VI | 3D-UHP | $4.90$ | 52 | $51.0$ | $512.6$ | $4.85\pm 0.31$ | $37\pm 25$ | 8 |

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## Share and Cite

**MDPI and ACS Style**

Charbonneau, E.; Bailly, F.; Danès, L.; Begon, M.
Optimal Control as a Tool for Innovation in Aerial Twisting on a Trampoline. *Appl. Sci.* **2020**, *10*, 8363.
https://doi.org/10.3390/app10238363

**AMA Style**

Charbonneau E, Bailly F, Danès L, Begon M.
Optimal Control as a Tool for Innovation in Aerial Twisting on a Trampoline. *Applied Sciences*. 2020; 10(23):8363.
https://doi.org/10.3390/app10238363

**Chicago/Turabian Style**

Charbonneau, Eve, François Bailly, Loane Danès, and Mickaël Begon.
2020. "Optimal Control as a Tool for Innovation in Aerial Twisting on a Trampoline" *Applied Sciences* 10, no. 23: 8363.
https://doi.org/10.3390/app10238363