<figure><img src="image_1.png" /></figure>

HAL
open science

# Optimal estimation of complex aerial movements using dynamic optimisation

André Venne, François Bailly, Eve Charbonneau, Jennifer Dowling-Medley,

Mickaël Begon

► To cite this version:

André Venne, François Bailly, Eve Charbonneau, Jennifer Dowling-Medley, Mickaël Begon. Optimal estimation of complex aerial movements using dynamic optimisation. Sports Biomechanics, 2022, pp.1-16. 10.1080/14763141.2022.2066015. hal-03689547

HAL Id: hal-03689547

https://inria.hal.science/hal-03689547v1

Submitted on 24 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Optimal estimation of complex aerial movements using dynamic
optimization

André Venne a*, François Bailly b, Eve Charbonneau a, Jennifer Dowling-Medley a and Mickaël Begon a

ªLaboratoire de Simulation et Modélisation du Mouvement, Université de Montréal, Montréal, QC, Canada

bNational Institute for Research in Computer Science and Automation, Camin Team, Montpellier, France

*andre.venne@umontreal.ca
Optimal estimation of complex aerial movements using dynamic
optimization

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

When estimating full-body motion from experimental data, inverse kinematics
followed by inverse dynamics does not guarantee dynamical consistency of the
resulting motion, especially in movements where the trajectory depends heavily on
the initial state, such as in free-fall. Our objective was to estimate dynamically
consistent joint kinematics and kinetics of complex aerial movements. A 42-
degrees-of-freedom model with 95 markers was personalized for five elite
trampoline athletes performing various backward and forward twisting
somersaults. Using dynamic optimization, our algorithm estimated joint angles,
velocities and torques by tracking the recorded marker positions. Kinematics,
kinetics, angular and linear momenta, and marker tracking difference were
compared to results of an Extended Kalman Filter (EKF) followed by inverse
dynamics. Angular momentum and horizontal linear momentum were conserved
throughout the estimated motion, as per free-fall dynamics. Marker tracking
difference went from 17±4 mm for the EKF to 36±11 mm with dynamic
optimization tracking the experimental markers, and to 49±9 mm with tracking of
EKF joint angles. Joint angles from the dynamic optimizations were similar to
those of the EKF, and joint torques were smoother. This approach satisfies the
dynamics of complex aerial rigid-body movements while remaining close to the
experimental 3D marker dataset.

Keywords: dynamic optimization; optimal control; inverse dynamics; aerial
acrobatics; initial conditions

**Introduction**

23
24
25
26
27
28

Computing joint kinematics and kinetics from the time history of 3D point clouds of markers is a common problem in biomechanics. Inverse kinematics methods such as nonlinear least-square (Cahouët et al., 2002; Kuo, 1998) or extended Kalman filter (EKF) (Begon et al., 2018) followed by inverse dynamics do not guarantee that the resulting motion will respect the equations of motion (Dumas et al., 2017; Silva & Ambrósio, 2002). Inverse dynamics is subject to high-frequency noise amplified by the numerical
differentiation (Silva & Ambrósio, 2002). Inverse kinematics is sensitive to underdetermined systems, due to missing or occluded markers, causing discontinuous joint angles (Buss, 2004). This is an issue for twisting somersaults since marker self-occlusions are likely to occur (Crépeau Rousseau, 2014). EKF relies on local linearization of the system's dynamics, and thus does not accurately account for nonlinear dynamics. Yet, it is a computationally light and fast state-of-the-art method that gives reasonable estimates. However, in motions with no contact forces (e.g., aerial phases), the parabolic trajectory of the centre of mass depends on the take-off state. Since these methods do not enforce the system dynamics equation continuously over time, they cannot ensure accurate final state, nor momenta conservation.

Techniques have been developed to improve kinematic estimation (Bailly et al., 2019; Begon et al., 2018; Silva & Ambrósio, 2002, 2004) and inverse dynamics consistency and accuracy (Bailly, Carpentier, et al., 2021; Cahouët et al., 2002; Faber et al., 2018; Forner-Cordero et al., 2006). However, they all require scenarios with contact forces to constrain the dynamics, and hence they have never been validated for aerial movements. Another method to improve the estimation is dynamic optimization, often solved using optimal control. In gait, it improves joint moment estimations by reducing the effect of kinematic measurement errors and gets rid of residual forces that appear in inverse dynamics (Remy & Thelen, 2009). Dynamic optimization guarantees that the dynamics will be continuously consistent throughout the movement, providing an accurate estimate of the final state in a forward dynamics simulation (Remy & Thelen, 2009). Dynamic optimization has been mainly used in gait kinematics estimation, by tracking joint angles previously calculated using inverse kinematics or motion capture markers.
Tracking the joint angles from inverse kinematics has been typical since the pioneering work of Chao & Rim (1973). Compared to marker tracking, it has a lower number of variables, which should make the optimization problem easier to solve. It allows for more complex objective function and constraints (Felis et al., 2015). Ambrósio & Kecskeméthy (2007) used a 3D 44-degrees-of-freedom (DoF) model for walking, and Nitschke et al. (2020) used a 3D 33-DoF model for running with changes in direction. Using a simplified 2D 6-DoF model, Koh & Jennings (2003) estimated a gymnastics movement with an aerial phase, for an athlete, by tracking the centre of mass along with the joint angles. Errors were up to 8 cm between the optimally estimated centre of mass trajectory and the one computed from digitized coordinates. Joint tracking does not necessarily provide the best marker fit, especially at the end of long kinematic chains where errors accumulate.

Tracking the experimental marker positions is more accurate, however, it is more computationally demanding and sensitive to model and experimental marker placement errors. Reinbolt et al. (2005) optimized subject-specific parameters of a 3D 12-DoF lower-body model while tracking the markers for gait. Remy & Thelen (2009) and Ausejo et al. (2011) presented marker tracking dynamic optimizations of detailed full-body models for gait, reaching and steering movements. However, studies based on 3D models with several DoF were done on movements with relatively low velocities and accelerations (Mentiplay et al., 2018; Umberger & Martin, 2007), and thus their conclusions cannot be generalized to high-speed gymnastics skills. These movements also included contact forces, breaking the relation to the initial state. This does not apply to contactless environments such as free-fall. Hence, our objective was to estimate dynamically consistent joint kinematics and kinetics of highly rotational aerial movements. It was hypothesized that dynamic optimization, especially marker tracking,
78

would provide an accurate estimation of kinematics and kinetics of high velocity twisting
somersaults.

79

80

**Methods**

81

82

83

84

85

86

87

88

Five elite trampoline athletes were equipped with 95 reflective markers (Fig. 1) and performed various backward and forward twisting somersaults in straight and pike positions. Four movements were selected: 44/, 821<, 822/ and 833/ (see Appendix A for an explanation of notations). The aerial phases duration (mean ± standard deviation) was 1.4±0.2 s. Marker positions were captured using 18 optoelectronic cameras (Vicon T40S and T20S) at 200 Hz. The protocol (No. CERC-19-002-D) was approved by the Université de Montréal Research Ethics Committee, and participants signed written informed consents.

89

90

91

92

93

94

To obtain the participants’ inertial parameters, 95 anthropometric measurements (Yeadon, 1990) were taken along with their total height and weight (Appendix B), personalizing a 17-segment 42-DoF model, generated with Biorbd (Michaud & Begon, 2021). The pelvis was defined as a floating root segment with 6-DoFs. The joints were inelastic and their centre of rotation were located using the SCoRE algorithm (Ehrig et al., 2006) from a set of pre-experimental calibration trials.

95

96

97

98

99

100

The dynamic optimization problem was solved using a direct multiple-shooting scheme in Bioptim (Michaud et al., 2021) connected to the nonlinear programming solver IPOPT (Wächter & Biegler, 2006) on four cores of a 2.9 GHz Intel Core i5-9400 processor. Direct multiple-shooting was chosen over direct collocation because it seems more suited to problems where the final state depends strongly on the initial conditions, such as in free-fall conditions (van den Bogert et al., 2011). The shooting nodes were placed every third frame of the experimental marker dataset, resulting in an estimation

101
frequency of 66.6 Hz (200/3 Hz). Kinematics and kinetics limits were enforced as path constraints to avoid non-physiological movements (Boone & Azen, 1979; Günal et al., 1996), as well as limits to avoid Gimbal locks at the pelvis (root segment), torso, head, glenohumeral joint and hip joint (Appendix C). The dynamics was defined by:

$$ \tau = M(q) \ddot{q} + N(q, \dot{q}) + G(q), \qquad (1) $$

where $\tau$ is the torques vector, and $q, \dot{q}$ and $\ddot{q}$ are the joint positions, velocities and accelerations vectors in generalized coordinates. $M(q)$ is the mass matrix, $N(q, \dot{q})$ embodies the Coriolis and centrifugal forces, and $G(q)$ represents gravity effects. By choosing the state of the system as $X = [q(t), \dot{q}(t)]$, and the control as $U = [\tau]$, the forward dynamics solves $\dot{X} = f(X, U)$. This first-order differential equation was numerically integrated between nodes with four steps of a 4th order Runge-Kutta algorithm.

An EKF with constant acceleration for the prediction was applied to the recorded marker trajectories to obtain a first estimate of the states. In the first frame, the EKF was initialized using a least-square L-BFGS-B with joint bounds (Appendix C). Joint torques were calculated using inverse dynamics (Eq. [1]). This provided favourable initial guesses to initialize the dynamic optimization and comparative values of joint angles and torques.

The optimal control problem was formulated to retrieve a dynamically consistent full-body motion. The objective function was the sum of a tracking term $J_{track}$ and a regularization term $J_{reg}$ at each node. $J_{track}$ aimed to either minimize the Euclidean distance between the 95 experimental and estimated markers, subsequently referred to as marker tracking (MT), or to minimize the difference between the 42 joint angles estimated by EKF and the dynamic optimization, subsequently referred to as joint angle tracking (QT). $J_{reg}$, which included torques, and their derivatives when required, sought to avoid
non-physiological joint torques (Appendix D) and non-physiological variations between
nodes caused by noise in the data.

Marker occlusions were ignored in the objective function. On average, 6±3% of
the experimental markers were occluded throughout the trials. Consequently, the
Jacobians of the objective function ($J_{track}$) with respect to joint angles were of full rank
throughout the whole motion in only 13 out of the 26 trials, regardless of marker
redundancy. Both dynamic optimizations used the output of the EKF as initial guesses in
the optimal control problem (Fig. 2). To assess EKF dynamics consistency, residual
torques on the root segment and root mean square error (RMSE) of the angular and linear
momenta were reported. To evaluate the performance of the dynamic optimizations
compared to EKF, root mean square difference (RMSD) of markers, convergence time,
RMSD of joint angles and joint torques, and mean power frequency of joint torques were
determined. Joint torque RMSD and torque mean power frequency excluded the root
segment. To further assess the impact of the main segment groups on each movement,
markers RMSD per movement for the trunk, arms and legs were also reported for EKF,
MT and QT. To evaluate error propagation along kinematic chains, markers RMSD per
topological distance from the root (pelvis) segment were calculated. The topological
distance is an integer describing how many segments separate each segment from the
root, the hands being the farthest.

## Results

All 26 trials reached an optimal solution (e.g., Supplementary Material Figures – Kinematics and Kinetics) in 6.1±0.7 s for EKF, 2085±1313 s (54±29 iterations) for MT and 2249±1324 s (60±32 iterations) for QT. Torque derivatives were included in $J_{reg}$ in 9 out of 26 trials for MT, and in 4 trials for QT, to ensure convergence.
150 Markers RMSD was 17±4, 36±11 and 49±9 mm in EKF, MT and QT respectively.
151 For MT, the trunk segment seemed to consistently have the highest markers RMSD
152 (Fig. 3). The mean RMSD in translation of the root and in joint angles between EKF and
153 MT was respectively 24±5 mm and 8±1°, and QT was 18±4 mm and 0.5±0.2°. The root's
154 velocity in translation was up to 8.8 m/s for EKF, 9.4 m/s for MT and 9.5 m/s for QT, and
155 in rotation, up to 1275°/s, 1173°/s and 1236°/s, respectively. The mean RMSD in joint
156 velocities between EKF and MT was 176±66°/s, and QT was 181±88°/s.

157 Using EKF, the angular momentum was not conserved with peak-to-peak
158 variations up to 30 kgm²/s and a mean RMSE of 2.6±1.6 kgm²/s, for mean absolute values
159 of 68±22, 8±7 and 4±3 kgm²/s for the somersault, tilt and twist components. Linear
160 horizontal momentum was not constant either, with a mean RMSE of 9.6±5.2 kgm/s, for
161 mean absolute values of 20±12 and 7±6 kgm/s, in the anteroposterior and mediolateral
162 components. This resulted in mean RMSE residual torques of 473±144 N in translation
163 and 114±48 Nm in rotation on the root segment, with peak residuals of 5.8 kN and
164 2.4 kNm in straight position along the longitudinal axis, although these residual torques
165 are expected to be null in free fall. The mean RMSD in joint torques between EKF and
166 MT was 7.7±4.2 Nm, and between EKF and QT was 5.8±2.8 Nm. This represented joint
167 torque values for EKF that were 59% and 39% larger, respectively. Min-max torques for
168 each method are listed in Appendix D. Torque fluctuations were smaller than those of
169 EKF (Fig. 4) with mean power frequency of 10.5±2.3 Hz for EKF versus 2.4±0.7 Hz and
170 2.9±0.9 Hz respectively for MT and QT.

171 **Discussion and implications**

172 Our main objective was to estimate dynamically consistent joint kinematics and
173 kinetics of a model with several DoF, performing highly rotational aerial movements (up
174 to 9.5 m/s and 1275°/s). As expected, joint kinematics resulting from the EKF was not
175 dynamically consistent, contrary to marker (MT) or joint angle (QT) tracking-based
176 dynamic optimizations. MT and QT had similar convergence time, which was
177 counterintuitive since MT had 6.8 times more variables to track. As anticipated, MT led
178 to smaller markers RMSD than QT, especially at the hands and feet.

179
*Dynamic consistency and computational time*

180 EKF and dynamic optimization-based tracking are robust estimators when data
181 are noisy or partially missing. Despite the use of 18 cameras, some markers were occluded
182 during our data collection, especially due to the pike position and other self-occlusions
183 caused by the extremely complex nature of the studied motions. By examining the rank
184 of the Jacobian of the objective function (J<sub>track</sub>) with respect to joint angles, we noted
185 that half of the trials were not of full rank throughout the whole motion, leading to
186 underdetermined estimation problems that cannot be solved using a frame-by-frame
187 nonlinear least square algorithm, therefore requiring a predictive approach. In the present
188 paper, we used an EKF with a constant acceleration for the prediction, which provided a
189 good estimate of the kinematics in previous studies (kayaking (Fohanno et al., 2010);
190 running (Bonnet et al., 2017)). Unfortunately, since our EKF was not bounded, it resulted
191 in non-anatomical local minima (such as elbow overextension) in some trials, requiring
192 multiple initialization approaches. Moreover, EKF does not ensure dynamic consistency:
193 the angular momentum and horizontal linear momentum were not preserved throughout
194 the motion (mean RMSE of 2.6 kgm²/s and 9.6 kgm/s, respectively). This lack of
195 consistency brought unrealistic residual forces and torques applied on the root segment,
196 up to 5.8 kN and 2.4 kNm. Even so, these residual torques were instantaneous, meaning
197 that they were unlikely to largely affect the kinematics.
When using an optimal estimator enforcing the dynamics through constraints, momenta were consistent with freefall equations, but markers RMSD increased from 17 mm in EKF to 36 mm with MT and to 49 mm with QT. Convergence times were also larger, going from 6.1 s for EKF to about 35 min for MT and QT. From our experience, this convergence time and the number of iterations highlight a complex problem. The movement duration (e.g., 1.55 s with 106 shooting nodes for the longest trials) and the number of variables (12,444) and constraints (8,652) led to a large, yet sparse, Jacobian of the objective function. Nevertheless, the complexity may also arise from the dependency between the state at take-off (which determines the angular momentum and the centre of mass trajectory over the whole aerial phase) and all subsequent states. Fortunately, with direct multiple shooting, the nonlinear effects of the continuity conditions are distributed over the whole-time horizon. In contrast, the dependency on the initial state is weakened for movements with contact forces, because external forces are added to the system during the movement. This enables the solver to balance fluctuations in joint forces with those of the contact point, diminishing the need for accurate joint force distribution. Therefore, the present study was necessary to show that dynamic optimization with no contact forces could accurately estimate complex acrobatic movements.

The regularization term in the dynamic optimization (least torques and least torque derivatives for some trials) tended to smooth the joint torques. In musculoskeletal modelling, the various models of activation dynamics (Buchanan et al., 2005; Thelen & Anderson, 2006) act as low-pass filters, muscle activations having lower mean power frequencies than muscle excitations. Our regularization term on joint torques and their derivatives seemed relevant in the absence of activation dynamics. Furthermore, marker occlusions may explain large or uncontrolled changes in joint angle values (Buss, 2004),
and demonstrate the necessity of using a method such as dynamic optimization, mitigating this effect to obtain realistic kinematics and kinetics values. The Bioptim library (Michaud et al., 2021) allows for easy implementations of such problems.

We used the robust interior point algorithm IPOPT to solve the nonlinear program, but in the future we should explore other sequential quadratic programming based solvers such as SNOPT (Gill et al., 2005) or Acados (Verschueren et al., 2020) which are more efficient for direct multiple shooting problems when the initial guess is close to the optimal solution. For example, in simple biomechanical problems, Acados converged 50 times faster than IPOPT (Bailly, Ceglia, et al., 2021; Michaud et al., 2021). In the present study, robustness was prioritized over computational time.

While several types of Kalman filters exist, adding constraints (e.g. constant angular momentum) and considering the whole movement is not trivial (Bonnet et al., 2016). Mohammadi et al. (2020) proposed an original optimally tuned EKF with a simple musculoskeletal arm model. However, it needs to be extended to a 3D full-body model and validated for high-speed movements with large range of motion. Furthermore, in a long duration aerial phase, noisy initial state values may still compromise the convergence of a constrained EKF. Another option would be a Kalman smoother, which moves backward in time through the data. While it can improve the kinematics (De Groote et al., 2008), it still does not guarantee dynamical consistency. Optimal control is a more relevant tool to estimate full-body joint kinematics and dynamics when the focus is about consistency.

## Marker tracking differences and models' uncertainties

The kinematics of airborne acrobatic movements can require extremely complex strategies described by interdependent variables and highly nonlinear equations (Bailly,
247 Charbonneau, et al., 2021). To perform aerial twist and control body's angular velocity
248 using transfer of angular momentum, arm and hip kinematics are key (Frohlich, 1980),
249 though every segment can affect the movement. In the present study, we considered that
250 the kinematic and dynamic models were free of error and that the only source of error
251 came from the measurement of the marker trajectories and soft tissue artifacts. The larger
252 markers RMSD in dynamic optimization-based solutions are certainly due to inaccuracies
253 in the kinematic and dynamic models, namely position of joint centers, DoF modeling,
254 inertial parameters, etc. In MT, the trunk segment had consistently higher markers
255 RMSD, which highlights that the kinematic (and probably dynamic) model could be
256 improved. In particular, the spine was composed of only one segment and therefore
257 cannot reproduce rounding positions. In future studies, our algorithms could be
258 augmented to also identify some key inertial parameters or the gravity orientation
259 according to sensitivity analyses and the ability of the movements to excite the parameters
260 (Ayusawa et al., 2014).

261 QT is often recommended due to a lower number of tracking variables than in
262 MT, which makes the optimization problem easier to solve. While RMSD between EKF
263 and QT kinematics was small, the error propagation led to larger errors at the hands and
264 feet. This is reflected on the arms and legs in Fig. 3 and by the increasing error at the end
265 of the kinematic chain as the topological distance from the root segment increases in
266 Fig. 4. On the other hand, MT resulted in smaller markers RMSD than QT, but estimated
267 joint kinematics different from EKF by 8±1°, which nevertheless reproduced a similar
268 movement. However, it was harder with MT to reach an optimal solution. Small changes
269 in the weighting and/or the regularization term in the objective function could jeopardize
270 the convergence. Almost a third of the trials required the extra regularization term on the
torque derivative. In models with several DoF, we recommend using MT when marker fit is a priority, and QT when ease of estimation is preferred.

## 273 Physiological joint torques

In gait for example, joint torques strongly depend on ground reaction forces, which can balance residual torque, potentially helping convergence. In aerial motions, joint torques mainly depend on joint acceleration and nonlinear effects due to full-body rotations (Bailly, Charbonneau, et al., 2021; Yeadon & Hiley, 2014). Consequently, the large difference between EKF and QT in terms of joint velocities meant that EKF required joint torque values 39% larger than QT to reproduce the same movement. The contrast was even greater with MT which required 59% less joint torque than EKF with a better markers RMSD than QT. However, the difference in joint kinematics was bigger with a similar difference in joint velocities. The mean power frequency of joint torque of EKF was 10.5±2.3 Hz, which is physiologically improbable. In another gymnastic discipline, namely hand balancing, Yeadon & Trewartha (2003) found reaction frequencies at the wrists of up to 8.3 Hz. Also, physiological tremor is associated with frequencies of 8 to 12 Hz (Bigland-Ritchie et al., 1992). Therefore, EKF followed by inverse dynamics did not estimate realistic joint torques in our study (Appendix D).

Furthermore, tracking only the experimental markers without torque regularization does not necessarily provide a physiologically plausible joint dynamics, since experimental errors (e.g., from marker occlusion) add large fluctuations to the joint torques. Hence the regularization term $J_{reg}$ to dampen this effect. Although the EKF did not regulate the joint torques, the predictive method helped to mitigate the impact of marker occlusions.
Our model did not include muscles to ensure physiological joint torques. Indeed, it has been shown that generic musculoskeletal models are weak especially for the upper-limbs (Martelli et al., 2008), inaccurate at high speed (Happee & Van der Helm, 1995) and not realistic in terms of muscle trajectories (Hoffmann et al., 2017). Moreover, muscle parameter personalization remains an ill-conditioned problem and may not be sufficient to provide accurate muscle forces (Żuk et al., 2018). As an alternative in skeletal models, joint torques may be bounded as a function of joint position and angular velocity using a series of maximal voluntary contractions on an isokinetic dynamometer (Forrester et al., 2011). Such bounds are relevant when performance is maximized (e.g., minimum time, maximum speed). We believe that in optimal estimation of twisting somersaults, the challenge is not about bounding the joint torques, but controlling their fluctuations.

## Conclusion

Full-body kinematics and kinetics of highly rotational aerial movements can be estimated with accuracy using dynamic optimization. Furthermore, marker tracking provides a better estimate than joint angle tracking, though convergence is easier to achieve with joint angle tracking.

## Acknowledgements

This work was supported by Mitacs (Acceleration program) under Grant #IT11976 and by the Institut National du Sport du Québec.

## Disclosure statement

No potential conflict of interest was reported by the authors.
316 References

317 Ambrósio, J. A. C., & Kecskeméthy, A. (2007). Multibody Dynamics of Biomechanical

318 Models for Human Motion via Optimization. In J. C. García Orden, J. M. Goicolea, & J.

319 Cuadrado (Eds.), *Multibody Dynamics* (Vol. 4, pp. 245–272). Springer Netherlands.

320 https://doi.org/10.1007/978-1-4020-5684-0_12

321 Ausejo, S., Suescun, Á., & Celigüeta, J. (2011). An optimization method for
322 overdetermined kinematic problems formulated with natural coordinates. *Multibody

323 *System Dynamics*, 26(4), 397–410. https://doi.org/10.1007/s11044-011-9263-x

324 Ayusawa, K., Venture, G., & Nakamura, Y. (2014). Identifiability and identification of
325 inertial parameters using the underactuated base-link dynamics for legged multibody
326 systems. *The International Journal of Robotics Research*, 33(3), 446–468.

327 https://doi.org/10.1177/0278364913495932

328 Bailly, F., Carpentier, J., Benallegue, M., Watier, B., & Souéres, P. (2019). Estimating
329 the Center of Mass and the Angular Momentum Derivative for Legged Locomotion—A

330 Recursive Approach. *IEEE Robotics and Automation Letters*, 4(4), 4155–4162.

331 https://doi.org/10.1109/LRA.2019.2931200

332 Bailly, F., Carpentier, J., & Souères, P. (2021, May). Optimal Estimation of the
333 Centroidal Dynamics of Legged Robots. *ICRA 2021 - IEEE International Conference
334 on Robotics and Automation*. https://hal.archives-ouvertes.fr/hal-03193940

335 Bailly, F., Ceglia, A., Michaud, B., Rouleau, D. M., & Begon, M. (2021). Real-Time
336 and Dynamically Consistent Estimation of Muscle Forces Using a Moving Horizon
337 EMG-Marker Tracking Algorithm—Application to Upper Limb Biomechanics.

338 *Frontiers in Bioengineering and Biotechnology*, 9.

339 https://doi.org/10.3389/fbioe.2021.642742
Bailly, F., Charbonneau, E., Danès, L., & Begon, M. (2021). Optimal 3D arm strategies for maximizing twist rotation during somersault of a rigid-body model. *Multibody System Dynamics*, 52(2), 193–209. https://doi.org/10.1007/s11044-020-09759-5

Begon, M., Andersen, M. S., & Dumas, R. (2018). Multibody Kinematics Optimization for the Estimation of Upper and Lower Limb Human Joint Kinematics: A Systematized Methodological Review. *Journal of Biomechanical Engineering*, 140(3), 030801.
https://doi.org/10.1115/1.4038741

Bigland-Ritchie, B. R., Furbush, F. H., Gandevia, S. C., & Thomas, C. K. (1992).
Voluntary discharge frequencies of human motoneurons at different muscle lengths.
*Muscle & Nerve*, 15(2), 130–137. https://doi.org/10.1002/mus.880150203

Bonnet, V., Daune, G., Joukov, V., Dumas, R., Fraisse, P., Kulić, D., Seilles, A.,
Andary, S., & Venture, G. (2016). A constrained Extended Kalman Filter for
dynamically consistent inverse kinematics and inertial parameters identification. *2016 6th IEEE International Conference on Biomedical Robotics and Biomechatronics (BioRob)*, 944–949. https://doi.org/10.1109/BIOROB.2016.7523749

Bonnet, V., Richard, V., Camomilla, V., Venture, G., Cappozzo, A., & Dumas, R. (2017). Joint kinematics estimation using a multi-body kinematics optimisation and an extended Kalman filter, and embedding a soft tissue artefact model. *Journal of Biomechanics*, 62, 148–155. https://doi.org/10.1016/j.jbiomech.2017.04.033

Boone, D. C., & Azen, S. P. (1979). Normal range of motion of joints in male subjects.
*The Journal of Bone and Joint Surgery American Volume*, 61(5), 756–759.

Buchanan, T. S., Lloyd, D. G., Manal, K., & Besier, T. F. (2005). Estimation of Muscle Forces and Joint Moments Using a Forward-Inverse Dynamics Model. *Medicine &
363 *Science in Sports & Exercise*, 37(11), 1911–1916.

364 https://doi.org/10.1249/01.mss.0000176684.24008.6f

365 Buss, S. R. (2004). Introduction to Inverse Kinematics with Jacobian Transpose,

366 Pseudoinverse and Damped Least Squares methods. *IEEE Journal of Robotics and Automation*, 17, 19.

368 Cahouët, V., Luc, M., & David, A. (2002). Static optimal estimation of joint

369 accelerations for inverse dynamics problem solution. *Journal of Biomechanics*, 35(11),
370 1507–1513. https://doi.org/10.1016/S0021-9290(02)00176-8

371 Chao, E. Y.-S., & Rim, K. (1973). Application of optimization principles in determining
372 the applied moments in human leg joints during gait. *Journal of Biomechanics*, 6(5),
373 497–510. https://doi.org/10.1016/0021-9290(73)90008-0

374 Crépeau Rousseau, A. (2014). *Développement et validation d'un modèle de simulation numérique personnalisé à une athlète de plongeon*.

376 https://papyrus.bib.umontreal.ca/xmlui/handle/1866/10402

377 De Groote, F., De Laet, T., Jonkers, I., & De Schutter, J. (2008). Kalman smoothing
378 improves the estimation of joint kinematics and kinetics in marker-based human gait
379 analysis. *Journal of Biomechanics*, 41(16), 3390–3398.

380 https://doi.org/10.1016/j.jbiomech.2008.09.035

381 Dumas, R., Brånemark, R., & Frossard, L. (2017). Gait Analysis of Transfemoral

382 Amputees: Errors in Inverse Dynamics Are Substantial and Depend on Prosthetic

383 Design. *IEEE Transactions on Neural Systems and Rehabilitation Engineering*, 25(6),
384 679–685. https://doi.org/10.1109/TNSRE.2016.2601378
385 Ehrig, R. M., Taylor, W. R., Duda, G. N., & Heller, M. O. (2006). A survey of formal
386 methods for determining the centre of rotation of ball joints. *Journal of Biomechanics*,
387 39(15), 2798–2809. https://doi.org/10.1016/j.jbiomech.2005.10.002

388 Faber, H., van Soest, A. J., & Kistemaker, D. A. (2018). Inverse dynamics of
389 mechanical multibody systems: An improved algorithm that ensures consistency
390 between kinematics and external forces. *PLOS ONE*, 13(9), e0204575.
391 https://doi.org/10.1371/journal.pone.0204575

392 Felis, M. L., Mombaur, K., & Berthoz, A. (2015). An optimal control approach to
393 reconstruct human gait dynamics from kinematic data. *2015 IEEE-RAS 15th
394 International Conference on Humanoid Robots (Humanoids)*, 1044–1051.
395 https://doi.org/10.1109/HUMANOIDS.2015.7363490

396 FIG executive committee (2021). 2022–2024 Code of points. (2020). FÉDÉRATION
397 INTERNATIONALE DE GYMNASTIQUE.

398 https://www.gymnastics.sport/publicdir/rules/files/en_ACRO%20CoP%202022-
399 2024.pdf

400 Fohanno, V., Colloud, F., Begon, M., & Lacouture, P. (2010). Estimation of the 3D
401 kinematics in kayak using an extended Kalman filter algorithm: A pilot study.

402 *Computer Methods in Biomechanics and Biomedical Engineering*, 13(sup1), 55–56.
403 https://doi.org/10.1080/10255842.2010.491958

404 Forner-Cordero, A., Koopman, H. J. F. M., & van der Helm, F. C. T. (2006). Inverse
dynamics calculations during gait with restricted ground reaction force information
405 from pressure insoles. *Gait & Posture*, 23(2), 189–199.

407 https://doi.org/10.1016/j.gaitpost.2005.02.002
408 Forrester, S. E., Yeadon, M. R., King, M. A., & Pain, M. T. G. (2011). Comparing

409 different approaches for determining joint torque parameters from isovelocity

410 dynamometer measurements. *Journal of Biomechanics*, 44(5), 955–961.

411 https://doi.org/10.1016/j.jbiomech.2010.11.024

412 Frohlich, C. (1980). The Physics of Somersaulting and Twisting. *Scientific American*,

413 242(3), 154–165. JSTOR.

414 Gill, P. E., Murray, W., & Saunders, M. A. (2005). SNOPT: An SQP Algorithm for

415 Large-Scale Constrained Optimization. *SIAM Review*, 47(1), 99–131.

416 https://doi.org/10.1137/S0036144504446096

417 Günal, I., Köse, N., Erdogan, O., Göktürk, E., & Seber, S. (1996). Normal Range of

418 Motion of the Joints of the Upper Extremity in Male Subjects, with Special Reference to

419 Side*. *JBJS*, 78(9), 1401.

420 Happee, R., & Van der Helm, F. C. T. (1995). The control of shoulder muscles during

421 goal directed movements, an inverse dynamic analysis. *Journal of Biomechanics*,

422 28(10), 1179–1191. https://doi.org/10.1016/0021-9290(94)00181-3

423 Hoffmann, M., Haering, D., & Begon, M. (2017). Comparison between line and surface

424 mesh models to represent the rotator cuff muscle geometry in musculoskeletal models.

425 *Computer Methods in Biomechanics and Biomedical Engineering*, 20(11), 1175–1181.

426 https://doi.org/10.1080/10255842.2017.1340463

427 Koh, M. T. H., & Jennings, L. S. (2003). Dynamic optimization: Inverse analysis for the

428 Yurchenko layout vault in women’s artistic gymnastics. *Journal of Biomechanics*,

429 36(8), 1177–1183. https://doi.org/10.1016/S0021-9290(03)00085-X
430 Kuo, A. D. (1998). A Least-Squares Estimation Approach to Improving the Precision of
431 Inverse Dynamics Computations. *Journal of Biomechanical Engineering*, 120(1), 148–
432 159. https://doi.org/10.1115/1.2834295

433 Martelli, S., Veeger, H. E., & Van der Helm, F. C. (2008). Scaling of a shoulder
434 musculoskeletal model does not lead to significant improvements. *Proceedings of the
435 7th Conference of the International Shoulder Group*.

436 Mentiplay, B. F., Banky, M., Clark, R. A., Kahn, M. B., & Williams, G. (2018). Lower
437 limb angular velocity during walking at various speeds. *Gait & Posture*, 65, 190–196.
438 https://doi.org/10.1016/j.gaitpost.2018.06.162

439 Michaud, B., Bailly, F., Charbonneau, E., Ceglia, A., Sanchez, L., & Begon, M. (2021).
440 Bioptim, a Python framework for Musculoskeletal Optimal Control in Biomechanics.
441 *BioRxiv*, 2021.02.27.432868. https://doi.org/10.1101/2021.02.27.432868

442 Michaud, B., & Begon, M. (2021). biorbd: A C++, Python and MATLAB library to
443 analyze and simulate the human body biomechanics. *Journal of Open Source Software*,
444 6(57), 2562. https://doi.org/10.21105/joss.02562

445 Mohammadi, H., Khademi, G., Simon, D., van den Bogert, A. J., & Richter, H. (2020).
446 Upper body estimation of muscle forces, muscle states, and joint motion using an
447 extended Kalman filter. *IET Control Theory & Applications*, 14(19), 3204–3216.
448 https://doi.org/10.1049/iet-cta.2020.0321

449 Nitschke, M., Dorschky, E., Heinrich, D., Schlarb, H., Eskofier, B. M., Koelewijn, A.
450 D., & van den Bogert, A. J. (2020). Efficient trajectory optimization for curved running
451 using a 3D musculoskeletal model with implicit dynamics. *Scientific Reports*, 10(1),
452 17655. https://doi.org/10.1038/s41598-020-73856-w
Reinbolt, J. A., Schutte, J. F., Fregly, B. J., Koh, B. I., Haftka, R. T., George, A. D., & Mitchell, K. H. (2005). Determination of patient-specific multi-joint kinematic models through two-level optimization. *Journal of Biomechanics*, 38(3), 621–626.
https://doi.org/10.1016/j.jbiomech.2004.03.031

Remy, C. D., & Thelen, D. G. (2009). Optimal Estimation of Dynamically Consistent Kinematics and Kinetics for Forward Dynamic Simulation of Gait. *Journal of Biomechanical Engineering*, 131(3). https://doi.org/10.1115/1.3005148

Silva, M. P. T., & Ambrósio, J. A. C. (2002). Kinematic Data Consistency in the Inverse Dynamic Analysis of Biomechanical Systems. *Multibody System Dynamics*, 8(2), 219–239. https://doi.org/10.1023/A:1019545530737

Silva, M. P. T., & Ambrósio, J. A. C. (2004). Sensitivity of the results produced by the inverse dynamic analysis of a human stride to perturbed input data. *Gait & Posture*, 19(1), 35–49. https://doi.org/10.1016/S0966-6362(03)00013-4

Thelen, D. G., & Anderson, F. C. (2006). Using computed muscle control to generate forward dynamic simulations of human walking from experimental data. *Journal of Biomechanics*, 39(6), 1107–1115. https://doi.org/10.1016/j.jbiomech.2005.02.010

Umberger, B. R., & Martin, P. E. (2007). Mechanical power and efficiency of level walking with different stride rates. *Journal of Experimental Biology*, 210(18), 3255–3265. https://doi.org/10.1242/jeb.000950

van den Bogert, A. J., Blana, D., & Heinrich, D. (2011). Implicit methods for efficient musculoskeletal simulation and optimal control. *Procedia IUTAM*, 2, 297–316.
https://doi.org/10.1016/j.piutam.2011.04.027
475 Verschueren, R., Frison, G., Kouzoupis, D., Frey, J., van Duijkeren, N., Zanelli, A.,

476 Novoselnik, B., Albin, T., Quirynen, R., & Diehl, M. (2020). *acados: A modular open-

477 source framework for fast embedded optimal control. *ArXiv:1910.13753 [Math]*.

478 http://arxiv.org/abs/1910.13753

479 Wächter, A., & Biegler, L. T. (2006). *On the implementation of an interior-point filter*

480 line-search algorithm for large-scale nonlinear programming. *Mathematical*

481 *Programming*, 106(1), 25–57. https://doi.org/10.1007/s10107-004-0559-y

482 Yeadon, M. R. (1990). *The simulation of aerial movement—I. The determination of*

483 orientation angles from film data. *Journal of Biomechanics*, 23(1), 59–66.

484 https://doi.org/10.1016/0021-9290(90)90369-E

485 Yeadon, M. R., & Hiley, M. J. (2014). *The control of twisting somersaults. Journal of*

486 *Biomechanics*, 47(6), 1340–1347. https://doi.org/10.1016/j.jbiomech.2014.02.006

487 Yeadon, M. R., & Trewartha, G. (2003). *Control Strategy for a Hand Balance. Motor*

488 *Control*, 7(4), 421–442. https://doi.org/10.1123/mcj.7.4.421

489 Żuk, M., Syczewska, M., & Pezowicz, C. (2018). *Influence of Uncertainty in Selected*

490 Musculoskeletal Model Parameters on Muscle Forces Estimated in Inverse Dynamics-

491 Based Static Optimization and Hybrid Approach. *Journal of Biomechanical*

492 *Engineering*, 140(12). https://doi.org/10.1115/1.4040943
# Appendices

## Appendix A

The FIG notation was used to identify the trampoline movements studied (see Supplementary Material Videos – Twisting Somersaults). The first digit represents the number of ¼ somersaults, while the second and third represent the number of ½ twists in each somersault. The trailing symbol represents the position in which the movement is executed: / is for straight, < is for piked and o is for tucked. For example, FIG 821< has two somersaults (8 x ¼), one twist in the first somersault (2 x ½), half a twist on the second somersault (1 x ½) and is performed in a piked position (<) (FIG Executive Committee (2021). 2022 - 2024 Code of Points, 2020).
Appendix B

The 95 anthropometric measurements (Yeadon, 1990) that were taken to obtain the participants' inertial parameters, personalizing a 17-segment 42-DoF model. The height and mass were measured to adjust the densities used for each body segment. The body parts that are strikethrough were not measured but estimated. The length of the midarm was estimated by halving the length from shoulder to elbow. The length of the midthigh and the foot arch were estimated by averaging the length of the two adjacent segments (i.e., average of length from hip to crotch and hip to knee for midthigh, and average of length from ankle to heel and ankle to ball for foot arch).

<table><thead><tr><th></th><th></th><th>Length From</th><th>To</th><th>Perimeter</th><th>Width</th><th>Depth</th></tr></thead><tbody><tr><td rowspan="4">Torso</td><td rowspan="4">Hip</td><td>Umbilicus</td><td>Hip</td><td>Hip</td><td>Hip</td><td>Shoulder</td></tr><tr><td>Ribcage</td><td>Umbilicus</td><td>Umbilicus</td><td>Umbilicus</td><td></td></tr><tr><td>Nipple</td><td>Ribcage</td><td>Ribcage</td><td>Ribcage</td><td></td></tr><tr><td>Shoulder</td><td>Nipple</td><td>Nipple</td><td>Nipple</td><td></td></tr><tr><td rowspan="3">Head</td><td rowspan="3">Neck</td><td>Neck</td><td>Neck</td><td>Neck</td><td>Shoulder</td><td></td></tr><tr><td>Nose</td><td>Nose</td><td>Nose</td><td>Nose</td><td></td></tr><tr><td>Ear</td><td>Ear</td><td>Ear</td><td>Ear</td><td></td></tr><tr><td rowspan="4">Left/Right Arm</td><td rowspan="4">Shoulder</td><td>Top</td><td></td><td></td><td></td><td></td></tr><tr><td>Midarm</td><td>Shoulder</td><td>Shoulder</td><td>Wrist</td><td></td></tr><tr><td>Elbow</td><td>Midarm</td><td>Elbow</td><td>Elbow</td><td></td></tr><tr><td>Forearm</td><td>Elbow</td><td>Forearm</td><td>Forearm</td><td></td></tr><tr><td rowspan="4">Left/Right Hand</td><td rowspan="4">Wrist</td><td>Wrist</td><td></td><td></td><td></td><td></td></tr><tr><td>Thumb</td><td>Thumb</td><td>Thumb</td><td>Thumb</td><td></td></tr><tr><td>Knuckle</td><td>Knuckle</td><td>Knuckle</td><td>Knuckle</td><td></td></tr><tr><td>Nails</td><td>Nails</td><td>Nails</td><td>Nails</td><td></td></tr><tr><td rowspan="5">Left/Right Leg</td><td rowspan="5">Hip</td><td>Crotch</td><td>Crotch</td><td>Crotch</td><td>Crotch</td><td>Crotch</td></tr><tr><td>Midthigh</td><td>Midthigh</td><td>Midthigh</td><td>Midthigh</td><td>Midthigh</td></tr><tr><td>Knee</td><td>Knee</td><td>Knee</td><td>Knee</td><td>Knee</td></tr><tr><td>Calf</td><td>Calf</td><td>Calf</td><td>Calf</td><td>Calf</td></tr><tr><td>Ankle</td><td>Ankle</td><td>Ankle</td><td>Ankle</td><td>Ankle</td></tr><tr><td rowspan="4">Left/Right Foot</td><td rowspan="4">Ankle</td><td>Heel</td><td>Heel</td><td>Heel</td><td>Ball</td><td>Heel</td></tr><tr><td>Arch</td><td>Arch</td><td>Arch</td><td>Arch</td><td>Arch</td></tr><tr><td>Ball</td><td>Ball</td><td>Ball</td><td>Ball</td><td>Ball</td></tr><tr><td>Nails</td><td>Nails</td><td>Nails</td><td>Nails</td><td>Nails</td></tr></tbody></table>
Appendix C

Bounds and limits (metres in translation and radian in rotation) to avoid non-physiological movements and Gimbal locks on the 17 segments, encompassing the 42-DoF.

<table><thead><tr><th>Segments</th><th>x</th><th>y</th><th>z</th></tr></thead><tbody><tr><td>Pelvis (translation)</td><td>-∞, ∞</td><td>-∞, ∞</td><td>-∞, ∞</td></tr><tr><td>Pelvis (rotation)</td><td>-∞, ∞</td><td>-π/2, π/2</td><td>-∞, ∞</td></tr><tr><td>Thorax</td><td>-π/2, π/2</td><td>-π/2, π/2</td><td>-π/2, π/2</td></tr><tr><td>Head</td><td>-π/2, π/2</td><td>-π/2, π/2</td><td>-π/2, π/2</td></tr><tr><td>Right shoulder</td><td>-</td><td>-π/2, π/2</td><td>-π/2, π/2</td></tr><tr><td>Right arm</td><td>-π, π</td><td>-π/2, π/2</td><td>-π, π</td></tr><tr><td>Right forearm</td><td>0, π</td><td>-</td><td>-π/2, π/2</td></tr><tr><td>Right wrist</td><td>-π/2, π/2</td><td>-π/2, π/2</td><td>-</td></tr><tr><td>Left shoulder</td><td>-</td><td>-π/2, π/2</td><td>-π/2, π/2</td></tr><tr><td>Left arm</td><td>-π, π</td><td>-π/2, π/2</td><td>-π, π</td></tr><tr><td>Left forearm</td><td>0, π</td><td>-</td><td>-π/2, π/2</td></tr><tr><td>Left wrist</td><td>-π/2, π/2</td><td>-π/2, π/2</td><td>-</td></tr><tr><td>Right thigh</td><td>-π, π</td><td>-π/2, π/2</td><td>-π/2, π/2</td></tr><tr><td>Right leg</td><td>-π, 0</td><td>-</td><td>-</td></tr><tr><td>Right foot</td><td>-π/2, π/2</td><td>-</td><td>-π/2, π/2</td></tr><tr><td>Left thigh</td><td>-π, π</td><td>-π/2, π/2</td><td>-π/2, π/2</td></tr><tr><td>Left leg</td><td>-π, 0</td><td>-</td><td>-</td></tr><tr><td>Left foot</td><td>-π/2, π/2</td><td>-</td><td>-π/2, π/2</td></tr></tbody></table>
Appendix D

Min-max pelvis (root segment) translation forces (N) and joint torques (Nm) of the 42-DoF for all 26 trials, distributed between 17 segments, for the Extended Kalman Filter (EKF), marker tracking (MT) and joint angle tracking (QT).

<table><thead><tr><th>Segments</th><th>Method</th><th>x</th><th>y</th><th>z</th></tr></thead><tbody><tr><td rowspan="3">Pelvis (translation)</td><td>EKF</td><td>-5754, 3578</td><td>-4524, 3171</td><td>-5324, 4056</td></tr><tr><td>MT</td><td>0, 0</td><td>0, 0</td><td>0, 0</td></tr><tr><td>QT</td><td>0, 0</td><td>0, 0</td><td>0, 0</td></tr><tr><td rowspan="3">Pelvis (rotation)</td><td>EKF</td><td>-1533, 2448</td><td>-1490, 1995</td><td>-1087, 538</td></tr><tr><td>MT</td><td>0, 0</td><td>0, 0</td><td>0, 0</td></tr><tr><td>QT</td><td>0, 0</td><td>0, 0</td><td>0, 0</td></tr><tr><td rowspan="3">Thorax</td><td>EKF</td><td>-821, 914</td><td>-1074, 857</td><td>-308, 416</td></tr><tr><td>MT</td><td>-190, 100</td><td>-98, 91</td><td>-65, 50</td></tr><tr><td>QT</td><td>-161, 127</td><td>-126, 150</td><td>-92, 101</td></tr><tr><td rowspan="3">Head</td><td>EKF</td><td>-136, 112</td><td>-123, 130</td><td>-28, 22</td></tr><tr><td>MT</td><td>-41, 25</td><td>-32, 39</td><td>-12, 25</td></tr><tr><td>QT</td><td>-190, 80</td><td>-124, 158</td><td>-51, 65</td></tr><tr><td rowspan="3">Right shoulder</td><td>EKF</td><td>-</td><td>-265, 307</td><td>-402, 230</td></tr><tr><td>MT</td><td>-</td><td>-41, 71</td><td>-36, 29</td></tr><tr><td>QT</td><td>-</td><td>-70, 101</td><td>-61, 54</td></tr><tr><td rowspan="3">Right arm</td><td>EKF</td><td>-130, 271</td><td>-352, 234</td><td>-40, 57</td></tr><tr><td>MT</td><td>-40, 70</td><td>-24, 88</td><td>-10, 19</td></tr><tr><td>QT</td><td>-56, 79</td><td>-34, 99</td><td>-24, 30</td></tr><tr><td rowspan="3">Right forearm</td><td>EKF</td><td>-114, 101</td><td>-</td><td>-5, 5</td></tr><tr><td>MT</td><td>-18, 23</td><td>-</td><td>-3, 4</td></tr><tr><td>QT</td><td>-22, 37</td><td>-</td><td>-5, 7</td></tr><tr><td rowspan="3">Right wrist</td><td>EKF</td><td>-12, 8</td><td>-6, 3</td><td>-</td></tr><tr><td>MT</td><td>-6, 5</td><td>-5, 5</td><td>-</td></tr><tr><td>QT</td><td>-7, 7</td><td>-7, 8</td><td>-</td></tr><tr><td rowspan="3">Left shoulder</td><td>EKF</td><td>-</td><td>-266, 320</td><td>-209, 290</td></tr><tr><td>MT</td><td>-</td><td>-87, 52</td><td>-41, 32</td></tr><tr><td>QT</td><td>-</td><td>-264, 108</td><td>-70, 126</td></tr><tr><td rowspan="3">Left arm</td><td>EKF</td><td>-192, 204</td><td>-305, 263</td><td>-105, 86</td></tr><tr><td>MT</td><td>-39, 65</td><td>-81, 30</td><td>-36, 9</td></tr><tr><td>QT</td><td>-83, 115</td><td>-87, 166</td><td>-50, 35</td></tr><tr><td rowspan="3">Left forearm</td><td>EKF</td><td>-79, 67</td><td>-</td><td>-7, 3</td></tr><tr><td>MT</td><td>-31, 42</td><td>-</td><td>-9, 7</td></tr><tr><td>QT</td><td>-87, 128</td><td>-</td><td>-19, 17</td></tr><tr><td rowspan="3">Left wrist</td><td>EKF</td><td>-11, 7</td><td>-4, 9</td><td>-</td></tr><tr><td>MT</td><td>-10, 8</td><td>-5, 9</td><td>-</td></tr><tr><td>QT</td><td>-17, 13</td><td>-12, 15</td><td>-</td></tr><tr><td rowspan="3">Right thigh</td><td>EKF</td><td>-937, 575</td><td>-700, 448</td><td>-100, 109</td></tr><tr><td>MT</td><td>-93, 122</td><td>-52, 162</td><td>-24, 24</td></tr><tr><td>QT</td><td>-125, 100</td><td>-73, 159</td><td>-34, 33</td></tr><tr><td rowspan="3">Right leg</td><td>EKF</td><td>-133, 258</td><td>-</td><td>-</td></tr><tr><td>MT</td><td>-44, 35</td><td>-</td><td>-</td></tr><tr><td>QT</td><td>-75, 54</td><td>-</td><td>-</td></tr><tr><td rowspan="3">Right foot</td><td>EKF</td><td>-11, 12</td><td>-</td><td>-1, 0</td></tr><tr><td>MT</td><td>-25, 28</td><td>-</td><td>-5, 5</td></tr><tr><td>QT</td><td>-54, 50</td><td>-</td><td>-8, 8</td></tr><tr><td rowspan="3">Left thigh</td><td>EKF</td><td>-831, 279</td><td>-858, 470</td><td>-71, 126</td></tr><tr><td>MT</td><td>-78, 133</td><td>-179, 22</td><td>-20, 30</td></tr><tr><td>QT</td><td>-88, 136</td><td>-167, 20</td><td>-27, 34</td></tr><tr><td rowspan="3">Left leg</td><td>EKF</td><td>-196, 219</td><td>-</td><td>-</td></tr><tr><td>MT</td><td>-45, 46</td><td>-</td><td>-</td></tr><tr><td>QT</td><td>-70, 54</td><td>-</td><td>-</td></tr><tr><td rowspan="3">Left foot</td><td>EKF</td><td>-19, 9</td><td>-</td><td>-1, 1</td></tr><tr><td>MT</td><td>-15, 14</td><td>-</td><td>-4, 4</td></tr><tr><td>QT</td><td>-34, 29</td><td>-</td><td>-6, 6</td></tr></tbody></table>
Figures

<figure><img src="image_3.png" /><figcaption>Figure 1</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3</figcaption></figure>

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4</figcaption></figure>

**Figure captions**

* **Figure 1:** Kinematic model with 42-degrees-of-freedom and 95 reflective markers as blue dots. The orthogonal frames are the coordinate systems of each joint: x-axis in red, y-axis in green and z-axis in blue. A generic humanoid silhouette is displayed in the background for reference.

* **Figure 2:** Optimal estimation workflow. From the marker dataset, states $X(q, \dot{q})$ were first estimated by an Extended Kalman Filter (EKF), followed by inverse dynamics (InvDyn) to obtain the joint torques, i.e., the controls U. The following dynamic optimizations were initialized by the EKF results. It tracked the experimental markers from the EKF (in A) and the joint angles in a least-squares sense (in B) every third frame while enforcing the dynamics.

* **Figure 3:** Mean RMSD between estimated and experimental markers per movement for each method, for segments trunk in red, arms in purple and legs in brown. The black lines are the standard deviation, while the black dots are the RMSD of each segment for each trial.
* **Figure 4:** On the left, mean power frequency of joint torque for EKF, marker tracking and joint angle tracking, excluding torques on the root segment which were null for the dynamic optimizations, but not for EKF. On the right, mean markers RMSD with standard deviation for segments according to their topological distance from the root (pelvis).