# 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 $N\in \mathcal{N}=\left\{295\dots 305\right\}\subset \mathbb{N}$.
- 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 | ${363}^{\xb0}$ | ${1057}^{\xb0}$ | ${1087}^{\xb0}$ | ${1124}^{\xb0}$ | ${1180}^{\xb0}$ | ${1764}^{\xb0}$ |

## References

- Frohlich, C. The physics of somersaulting and twisting. Sci. Am.
**1980**, 242, 154–165. [Google Scholar] [CrossRef] [PubMed] - Sanders, R.H. Effect of ability on twisting techniques in forward somersaults on the trampoline. J. Appl. Biomech.
**1995**, 11, 267–287. [Google Scholar] [CrossRef] - Yeadon, M.F. Learning how to twist fast. In Applied Proceedings of the XVIIth International Symposium on Biomechanics in Sports–Acrobatics; School of Biomedical and Sport Sciences, Edith Cowan University: Perth, Australia, 1999; pp. 37–47. [Google Scholar]
- Dullin, H.R.; Tong, W. Twisting somersault. SIAM J. Appl. Dyn. Syst.
**2016**, 15, 1806–1822. [Google Scholar] [CrossRef][Green Version] - Yeadon, M.R. The biomechanics of twisting somersaults Part IV: Partitioning performances using the tilt angle. J. Sports Sci.
**1993**, 11, 219–225. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yeadon, M.R. The limits of aerial twisting techniques in the aerials event of freestyle skiing. J. Biomech.
**2013**, 46, 1008–1013. [Google Scholar] [CrossRef][Green Version] - Bailly, F.; Charbonneau, E.; Danès, L.; Begon, M. Optimal 3D arm strategies for maximizing twist rotation during somersault of a rigid-body model. Multibody Syst. Dyn.
**2020**, 1–17. [Google Scholar] [CrossRef] - Yeadon, M.R.; Mikulcik, E. The control of non-twisting somersaults using configuration changes. J. Biomech.
**1996**, 29, 1341–1348. [Google Scholar] [CrossRef][Green Version] - Huchez, A.; Haering, D.; Holvoet, P.; Barbier, F.; Begon, M. Differences Between Expert and Novice Gymnasts Performance of a Counter Movement Forward in Flight on Uneven Bars. Sci. Gymnast. J.
**2016**, 8, 31–41. [Google Scholar] - Sayyah, M.; Yeadon, M.; Hiley, M.J.; King, M.A. Adjustment in the Flight Phase of 1m Springboard Forward Pike Dives. ISBS Proc. Arch.
**2017**, 35, 16. [Google Scholar] - Heinen, T.; Walter, N.; Hennig, L.; Jeraj, D. Spatial Perception of Whole-body Orientation Depends on Gymnasts’ Expertise. Sci. Gymnast. J.
**2018**, 10, 5–15. [Google Scholar] - Balter, S.G.; Stokroos, R.J.; Akkermans, E.; Kingma, H. Habituation to galvanic vestibular stimulation for analysis of postural control abilities in gymnasts. Neurosci. Lett.
**2004**, 366, 71–75. [Google Scholar] [CrossRef] - Hiley, M.J.; Yeadon, M.R. Investigating optimal technique in a noisy environment: Application to the upstart on uneven bars. Hum. Mov. Sci.
**2013**, 32, 181–191. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mombaur, K. Performing Open-Loop Stable Flip-Flops—An Example for Stability Optimization and Robustness Analysis of Fast Periodic Motions. In Fast Motions in Biomechanics and Robotics; Springer: Berlin/Heidelberg, Germany, 2006; pp. 253–275. [Google Scholar]
- Ashby, B.M.; Delp, S.L. Optimal control simulations reveal mechanisms by which arm movement improves standing long jump performance. J. Biomech.
**2006**, 39, 1726–1734. [Google Scholar] [CrossRef] [PubMed] - Spägele, T.; Kistner, A.; Gollhofer, A. A multi-phase optimal control technique for the simulation of a human vertical jump. J. Biomech.
**1999**, 32, 87–91. [Google Scholar] [CrossRef] - Yeadon, M.R.; Hiley, M.J. The mechanics of the backward giant circle on the high bar. Hum. Mov. Sci.
**2000**, 19, 153–173. [Google Scholar] [CrossRef][Green Version] - Bharadwaj, S.; Duignan, N.; Dullin, H.R.; Leung, K.; Tong, W. The diver with a rotor. Indag. Math.
**2016**, 27, 1147–1161. [Google Scholar] [CrossRef][Green Version] - Yeadon, M.R. Airborne Movements: Somersaults and Twists. In Handbook of Human Motion; Springer: Cham, Switzerland, 2017; pp. 1–19. [Google Scholar]
- Yeadon, M.R.; Hiley, M.J. Twist limits for late twisting double somersaults on trampoline. J. Biomech.
**2017**, 58, 174–178. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yeadon, M.R. Twisting Somersaults; SB & MC: Loughborough, UK, 2015. [Google Scholar]
- Featherstone, R. Rigid Body Dynamics Algorithms; Springer: Cham, Switzerland, 2014. [Google Scholar]
- Yeadon, M.R. The simulation of aerial movement—II. A mathematical inertia model of the human body. J. Biomech.
**1990**, 23, 67–74. [Google Scholar] [CrossRef][Green Version] - Namdari, S.; Yagnik, G.; Ebaugh, D.D.; Nagda, S.; Ramsey, M.L.; Williams, G.R., Jr.; Mehta, S. Defining functional shoulder range of motion for activities of daily living. J. Shoulder Elb. Surg.
**2012**, 21, 1177–1183. [Google Scholar] [CrossRef] - Wächter, A.; Biegler, L.T. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program.
**2006**, 106, 25–57. [Google Scholar] [CrossRef] - Andersson, J.A.E.; Gillis, J.; Horn, G.; Rawlings, J.B.; Diehl, M. CasADi—A software framework for nonlinear optimization and optimal control. Math. Program. Comput.
**2019**, 11, 1–36. [Google Scholar] [CrossRef] - Huchez, A.; Haering, D.; Holvoët, P.; Barbier, F.; Begon, M. Local versus global optimal sports techniques in a group of athletes. Comput. Methods Biomech. Biomed. Eng.
**2015**, 18, 829–838. [Google Scholar] [CrossRef] [PubMed] - Decatoire, A. Analyse Tri-Dimensionnelle de La Gestion des Mouvements Vrillés en Gymnastique: La Simulation: Vers un Outil de Formation des Entraîneurs en Activités Acrobatiques. Ph.D. Thesis, Université de Poitiers, Poitiers, France, 2004. [Google Scholar]
- Hiley, M.J.; Yeadon, M.R. Optimisation of high bar circling technique for consistent performance of a triple piked somersault dismount. J. Biomech.
**2008**, 41, 1730–1735. [Google Scholar] [CrossRef] [PubMed][Green Version] - Grapton, X.; Lion, A.; Gauchard, G.C.; Barrault, D.; Perrin, P.P. Specific injuries induced by the practice of trampoline, tumbling and acrobatic gymnastics. Knee Surg. Sports Traumatol. Arthrosc.
**2013**, 21, 494–499. [Google Scholar] [CrossRef] [PubMed] - Yang, Y.; Gu, Y.; Fan, X.; Cheng, H. Multi-Objective Optimization of Virtual Human Motion Posture for Assembly Operation Simulation. In Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Charlotte, NC, USA, 21–24 August 2016; American Society of Mechanical Engineers: New York, NY, USA; Volume 50077, p. V01AT02A056. [Google Scholar]
- Mombaur, K.; Clever, D. Inverse optimal control as a tool to understand human movement. In Geometric and Numerical Foundations of Movements; Springer: Cham, Switzerland, 2017; pp. 163–186. [Google Scholar]
- Tong, C.; Wolpert, D.M.; Flanagan, J.R. Kinematics and dynamics are not represented independently in motor working memory: Evidence from an interference study. J. Neurosci.
**2002**, 22, 1108–1113. [Google Scholar] [CrossRef]

**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