# Experimental Evaluation of the Pitch Angle Righting Capabilities of a High Speed Terrestrial Vehicle during Ballistic Phases

Marc Davis, Philippe Vaslin, Jean-Christophe Fauroux, Christophe Gouinaud, Liang Ju

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

## To cite this version:

Marc Davis, Philippe Vaslin, Jean-Christophe Fauroux, Christophe Gouinaud, Liang Ju. Experimental Evaluation of the Pitch Angle Righting Capabilities of a High Speed Terrestrial Vehicle during Ballistic Phases. Applied Mechanics and Materials, 2012, 162, pp.57-66. 10.4028/www.scientific.net/AMM.162.57. hal-02059500

HAL Id: hal-02059500

https://uca.hal.science/hal-02059500v1

Submitted on 28 Mar 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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
# Experimental evaluation of the pitch angle righting capabilities of a high speed terrestrial vehicle during ballistic phases

Marc Davis<fnref n="1" />,<fnref n="2" />,ᵃ, Philippe Vaslin<fnref n="1" />,ᵇ, Jean-Christophe Fauroux<fnref n="2" />,³, Christophe Gouinaud<fnref n="1" />,⁴, Liang Ju<fnref n="2" />,⁵

**Keywords:** High speed ballistic robot, pitch angle control, angular momentum, moment of inertia, high speed camera, noise filtering

**Abstract.** We present a study of the feasibility of controlling the pitch angle of a high velocity ground vehicle in ballistic phase by accelerating or decelerating its wheels. The vehicle was filmed during several jumps, and various wheel acceleration commands were sent to it during that time. Images were recorded using a high-speed video camera, and the trajectories of markers placed on the vehicle were analyzed in order to measure the relationship between the changes in angular velocities of the chassis and the wheels. It appears that it is possible to achieve pitch righting through the modification of the wheels' angular velocities with adequate performance, thus opening the road towards automated control of pitch, then later yaw and roll angles.

## Introduction

High speed terrestrial vehicles can be used in a variety of situations such as mapping and reconnaissance in unknown or hazardous environments, surveillance of vast installations, search and rescue operations, agricultural monitoring, and delivery of critical supplies in humanitarian missions. Even though maneuverability on unstructured terrains at high velocities can be maintained through the use of optimized suspension systems [1], terrain discontinuities and impacts with obstacles can cause ballistic phases which require aerial self-righting if the vehicles are to land with their suspensions facing the ground. In some applications [2], [3], ballistic phases might be voluntary, but still require attitude correction for providing safe landing conditions, and in order to provide real-time image readability for a remote operator using an on-board camera for first person view control.

The means for providing aerial self-righting can be derived from techniques evolved in living beings [4], and lizard-inspired solutions [5] have been explored, for example by using a robotic tail to provide pitch righting similarly to techniques used by other reptiles [6]. Traditional self-righting means used in aerospace mainly include the use of thrusters, spin stabilization, reaction wheels, or control moment gyroscopes [7].

Previously, we studied this pitch correction maneuver in the case of motocross jumps. A kinematic analysis on video footage taken for this purpose was used to identify the key parameters of this phenomenon. A mechanical model was constructed, and numerical analysis in a custom-written 2D Java simulator was conducted so as to study the influence of the initial conditions of jumps, as well as the influence of the vehicle's characteristics [8].

This study focuses on evaluating the controllability of the pitch angle of an airborne vehicle in ballistic phase through the exchange of partial angular momentums between its chassis and its wheels. After having described the experimental setup and results, the latter will be discussed before concluding this study.

<footnote n="1">¹Clermont University, Blaise Pascal University (UBP), UMR CNRS 6158, Computer Science Research Group (LIMOS), Campus des Cézeaux, 63177 Aubière Cedex, France</footnote>

<footnote n="2">²Clermont University, French Institute for Advanced Mechanics (IFMA), EA3867, FR TIMS / CNRS 2856, Mechanical Engineering Research Group (LaMI), BP 10448, F-63000</footnote>

<footnote>ªmarc.davis@isima.fr, ᵇphilippe.vaslin@isima.fr, ᶜjean-christophe.fauroux@ifma.fr,</footnote>

<footnote>dchristophe.gouinaud@isima.fr, eLIang.ju@ifma.fr</footnote>
Experimental Setup

The vehicle used for this study is a 1:8 scale radio-controlled truck from Traxxas, the E-Maxx Brushless Model 3908 depicted in Fig. 1a. The vehicle features a four wheel drive powered by a brushless motor, and provides a suspension system adapted for recovering from jumps and falls. The fact that all four wheels are actuated makes this vehicle a good candidate for wheel-based pitch righting, since approximately 20% of the truck’s mass is concentrated in the wheels, and the studied maneuver relies on the exchange of partial angular momentums between the vehicle and its wheels.

<figure><img src="image_3.png" /><figcaption>Fig. 1a: The E-Maxx Brushless,
with its local reference frame.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 1b: The ramp at the end of the concrete track, with cones for calibration, and the vehicle performing a jump.</figcaption></figure>

The experiment consisted in having the vehicle perform three sorts of jumps, with different actions on the throttle, thus on the changes of angular velocity of the wheels. During “normal” jumps, no commands were sent to the throttle in mid-air; during “accelerated” jumps, the throttle trigger was manually depressed during the aerial phases; during “braked” jumps, the throttle was used to brake the wheels.

A jumping ramp was installed at the end of a concrete track on a javelin throw field, providing the vehicle with a smooth surface for accelerating towards the said ramp. After jumping, the vehicle would then land on smooth grass, as shown in Fig. 1b. Cones were placed at regular intervals in the plane of motion, used for calibrating the horizontal axis in the images to be filmed; the vertical axis was calibrated with a tall object of known dimensions.

The vehicle's remote controller was modified so as to provide repeatable input during the acceleration phase on the runway: the throttle trigger was attached to a linkage controlled by a motor, which itself was driven by a Lego Mindstorms NXT control brick (Fig. 2a). Pressing a single on/off button connected to the control brick ensured better throttle repeatability than driving the system by hand.

Images were captured with a resolution of 1280x1024 pixels at 500 frames per second using an S-MOTION high speed camera from AOS Technologies [9]. A Pentax K-7 camera was used for filming at 30 frames per second the ramp itself, providing means for measuring the vehicle's velocity upon leaving the ramp (Fig. 2b), and a Pentax K-5 camera was used for filming isometric images.

<figure><img src="image_5.png" /><figcaption>Fig. 2a: The automated vehicle remote control.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2b: Measuring take-off velocity
with measuring tape on the ramp.</figcaption></figure>
The first set of images captured by the high speed camera was used for spatial calibration of the scene by capturing objects of known dimensions. The scenes' coordinate system was defined with the x axis oriented in the vehicle's general horizontal direction of travel (towards the left in Fig. 4), and the z axis towards the sky. All angles measured around the pitching axis (the y axis) are thus defined as being increasing in the images' counter-clockwise direction.

The image sequences captured with the high speed camera were imported in Vannier-Photolec's MotionTrack video analysis software for tracking the key points on the vehicle. The vehicle attitude was manually tracked as the segment joining the center of both left wheels, and the angular positions of the wheels were automatically tracked by a contrast following algorithm.

The angular positions of the wheels and chassis for the “normal” and “accelerated” jumps were interpolated using sixth degree polynomials, so as to provide angular velocity data in the form of fifth degree curves after differentiation. The angular positions for the “braked” jump had to be interpolated through piecewise functions of degrees one and two.

The moments of inertia of the vehicle and of its wheels were measured using the pendulum method, following the protocol described in [10]. Images were captured at 30 frames using a Panasonic DMC-LX2 digital camera, and the markers visible in Fig. 3a and 3b were tracked in the video analysis software mentioned previously.

<figure><img src="image_7.png" /><figcaption>Fig. 3a: Measurement of the moment of inertia of the vehicle around its y axis.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 3b: Measurement of the moment of inertia of a wheel around its y axis.</figcaption></figure>

The measured moments of inertia were used to compute the total angular momentum of the vehicle in ballistic phase, with the expectation of it being constant during that entire time. Eq. 1 gives the expression of the vehicle's total moment of inertia [8]:

$$ \vec{\sigma}^i = J_{G\Sigma y} \vec{\omega}_{\Sigma} + \sum_{k=0}^{4} J_{GW_k y} \vec{\omega}_k. \quad (1) $$

In Eq. 1, $\vec{\sigma}^i$ is the total angular momentum of the system, $J_{G\Sigma y}$ is the moment of inertia of the complete vehicle around the pitching axis and $\vec{\omega}_{\Sigma}$ is the angular velocity of the vehicle. $J_{GW_k y}$ is the moment of inertia of wheel number k around its axis of rotation, and $\vec{\omega}_k$ is the angular velocity relative to the local reference frame of wheel number k.

## Results

In Fig. 4a (“normal” jump), it appears that the vehicle’s attitude varies very little between positions 1 and 3, and that the vehicle pitches very slightly in the negative direction (clockwise) before landing safely in position 7. The throttle of the vehicle could have been slightly actuated, resulting in the observed motion. This jump is thus a very slightly accelerated jump.

In Fig. 4b (“accelerated” jump), the vehicle’s pitch angle varies very little until position 2, then abruptly changes as the throttle is pressed and the wheels accelerate. It is to be noted that the tires seem to have bulged due to the centrifugal effect in positions 6 and 7 to a diameter larger than in positions 1 and 2.

In Fig. 4c (“braked” jump), the vehicle seems to be in pure translation at positions 2, 3 and 4, then the brakes are applied shortly after position 4, and the vehicle pitches forward in the positive direction.
<figure><img src="image_9.png" /><figcaption>b)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>c)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 4: A “normal” jump (a), an “accelerated” jump (b) and a “braked” jump (c).</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 5: Angular positions (left column) and angular velocities (right column) of the wheels and chassis during a “normal” jump (a), an “accelerated” jump (b) and a “braked” jump (c).</figcaption></figure>
The "normal" jump exhibits a slight angular acceleration throughout the ballistic phase. The remote control's throttle was manually set close to zero, but it appears that a slight acceleration command was still being sent. However, the acceleration seems constant, and while the wheels follow a positive acceleration, the vehicle's nose pitches up as expected.

During the "accelerated" jump, it appears that an acceleration command was sent around t=0.452s, and stopped at t=0.686s. While the throttle was set to its maximum value gradually (not in an abrupt manner), the vehicle's angular velocity went decreased by 5.18 rad/s in 0.234s, yielding an angular acceleration of -22.15 rad/s². At that rate of acceleration, the vehicle could be capable of modifying its pitch angle by half a turn in 0.53s.

During the "braked" jump, the brakes were applied gradually from t=0.472s to t=568s. The angular velocity increased by 4.82 rad/s in 0.96s, yielding an acceleration of 50.2 rad/s². While this angular acceleration is only available until the wheels come to a stop, it would have been possible to lengthen this forward pitching maneuver by accelerating the wheels in reverse gear after the braking was completed.

<figure><img src="image_13.png" /><figcaption>Fig. 6: Total angular momentum of the "normal" (a), "accelerated" (b) and "braked" (c) jumps.</figcaption></figure>

Overall, it appears that using the vehicle's wheels for pitch righting maneuvers is completely feasible, and that the potential for executing these maneuvers is present for even the most extreme cases and initial conditions.

For the sake of verifying the data against the mechanical model, the moments of inertia of the vehicle and its wheels were measured and used in Eq. 1 along with the relevant angular velocities for each jump in order to compute the system's total angular momentum. While the calculated values were found to be close to constant (Fig. 6) during the ballistic phases, variations were still observed, which violates the law of conservation of angular momentum. These variations could be caused by a number of factors, which will be discussed in the next section.

## Discussion

The results of this study are subject to measurement errors emanating from multiple sources, and which can be divided into two categories [11]: random errors and systematic errors. Random errors fluctuate in an unpredictable manner, and can be diminished by repeating the same measurement in the exact same conditions multiple times. Systematic errors are those introduced by the methods of measurement, the operators, the hardware used, the environment and the objects being measured. Many factors were identified as being possible sources of error, and they were found to be almost exclusively of systematic nature.

**Random Errors.** They have a multitude of possible causes, which can include camera sensor noise, transient changes in illumination of the objects, camera vibrations, dust particles passing in front of the lens, errors in the video analysis software's contrast tracking algorithm and various round-off errors.

In order to quantify these random errors, a stationary marker in the scenes was filmed: the marker was chosen to be at the bottom part of a loudspeaker at the corner of a building (Fig. 4, position "A"). This marker was chosen because of the similarity in contrast and size to the markers to be followed on the vehicle. In future studies, markers identical to those to be tracked on the truck will be placed on stationary surfaces for random error measurement and for scene calibration. The marker was automatically tracked by the video analysis software during a whole jumping sequence.
The raw **x** (horizontal) and **y** (vertical) pixel coordinates of the marker were extracted (Fig. 7), and it appears that the **x** coordinates fluctuate in a range of 0.6 pixels (with a standard deviation of 0.1 pixels), while the **y** coordinates vary in a range of 1.4 pixels, with a standard deviation of 0.2 pixels. It is interesting to note that the random errors are not equal in their vertical and horizontal distributions, which could be caused by the wide and short shape of the tracked object. These errors were found to be negligible relatively to the other sources of uncertainty in this study.

<figure><img src="image_14.png" /><figcaption>Fig. 7: Raw **x** (bottom curve, left scale) and **y** (top curve, right scale) pixel coordinates of a stationary marker.</figcaption></figure>

**Tracking of Markers on the Wheels.** Markers on the wheels were materialized by a square piece of white tape placed on the tire, and these markers were automatically followed by the video analysis software. Initial measurements of the wheel and chassis angular velocities proved to be undulating around the expected values, with a frequency equal to that of the wheels’ rotation.

It was decided to measure the evolution of a supposed constant, the radius of the wheels, defined as the distance between the markers in the centers of the wheels and the markers at their peripheries. It was found that instead of being constant, this distance was measured as varying significantly. While the average value of the radius was measured at 55mm, the actual values oscillated in an 18mm range around that average (Fig. 7a).

<figure><img src="image_15.png" /><figcaption>Fig. 7a: Measured wheel radius variation.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 7b: Variation in wheel marker size.</figcaption></figure>

This change in measured radius being cyclical and depending essentially on the wheel's angular position, it could have several causes. Firstly, the background of the markers being the tire itself, the contrast between the marker and the rubber varied with the angular position relative to incident sunlight. Secondly, the marker being square, its shape varied continuously in terms of discrete pixel representation, thus leading the automatic contrast following algorithm to introduce uncertainties on the marker size and geometric center. Thirdly, the marker having been placed on a curved shape not perpendicular to the camera's axis, the projected and pixel-discrete size of the marker in the camera's focal plane varied with each rotation as well. All these variations led the video analysis software to consider the marker to be varying in size (Fig. 7b), and introduced large errors which propagated undulating errors in every subsequent calculation. In future work it will be necessary to use markers of appropriate shape, size, contrast, placement, background and orientation relative to
the camera in order to reduce this behavior. This source of error called for the use of polynomial interpolation of the angular positions of the wheels and chassis.

**Polynomial Interpolation.** Because the main source of error in the measurements was by nature of the same frequency than that of one of the phenomena to be measured, it was impossible to isolate and filter out that frequency without severely distorting the data from its actual meaning. It was decided to use polynomial interpolation as a temporary solution for succeeding in observing the relationship between the chassis and wheel angular velocities. The angular positions were interpolated, and the obtained polynomials were differentiated in order to obtain angular velocities. This solution enabled for the extraction of adequate tendencies concerning the phenomenon, but proved to have its limits. Using too high of a degree for the polynomials tended to introduce parasite undulation on the interpolated data, but too low of a degree induced severe loss of definition on the tendencies obtained through differentiation of the polynomials. In the case of the braked jump, curve fitting was simply not acceptable, and the obtained angular velocities lost all meaning, thus leading for a radical use of first and second degree curves for tendency extraction. In future work it is expected to obtain data with much less of the cyclical noise measured in this work, enabling the use for more traditional and refined filtering methods.

**Marker Placement.** It was chosen to measure the vehicle's attitude using the wheel centers, since it is a plane tangent to the wheels that is to be made parallel to the ground before landing. However, it appears that the vehicle's suspensions are excited during the jumps, as visible on the angular position curves of the “accelerated” and “braked” jumps (Fig. 5b, c). In order to be able to measure the actual angular momentum of the vehicle, it is necessary to measure the wheel position relative to the chassis, thus introducing the need for markers on the chassis itself.

**Moments of Inertia.** While experimental conditions for measuring an object's moment of inertia while minimizing measurement errors are known [12], they are not always easily applicable: the localization of a usable pivot axis is not always trivial, and the measurement of the distance of that axis to the vehicle's center of mass is a potential source for error. Moreover, it was observed that the wheels change in shape and diameter as a function of their angular velocity (Fig. 8). This necessarily introduces changes in their moments of inertia, which must be either modeled and taken into account, or mechanically forbidden.

<figure><img src="image_17.png" /><figcaption>Fig. 8a: The wheels bulging at high velocity</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 8b: The wheels at rest</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 8c: Left tires spinning and bulging while right tires adhere</figcaption></figure>

**Scene Lighting.** The Sun was the primary light source during the tests. While before each jump the high speed camera's exposure settings were adjusted, changes in illumination conditions took place between the end of the adjustment and the filming of the sequences. While the human eye could hardly notice the subtle changes in lighting due to the changing thickness of the cloud cover, the camera's sensor eventually produced some ill-exposed images. In some sequences the sensor was saturated, producing images beyond recovery by adjustment of brightness and contrast. As an effect, contrast following proved difficult in some captured sequences. Subsequent manipulations should be either artificially-lit, or camera exposure should be set when the sky is clear and it is certain that no clouds could block the Sun at the time of capture.
**Jump Initial Conditions.** While the vehicle's throttle was automatically actuated as it approached the ramp, steering was done manually and could have introduced some uncertainty on the jumps' initial conditions. As a result, the parabolic motion of the center of mass of the vehicle could have been in a different plane than that used for calibrating the scene, thus inducing errors in distance measurements. Moreover, slight steering commands could have been sent to the vehicle while it was on the ramp, causing it to have initial yawing and rolling motions during the jumps. Fig. 9 displays various positions of a vehicle initially meant to perform a "normal" jump. In this case, one of the wheels was not actually on the ramp during takeoff, introducing a great amount of yaw on the vehicle during the ballistic phase.

<figure><img src="image_20.png" /><figcaption>Fig. 9: Strong yawing initial conditions</figcaption></figure>

The "normal" jump (Fig. 4) appears to exhibit a slight amount of yawing motion in the negative direction, and the "braked" jump shows some rolling in the positive direction. In order to eliminate this problem it could be necessary to automate to some extent the vehicle's steering, by example following a painted line, or by taking readings from a compass or magnetometer. Another solution would be to use a second camera for stereo vision, enabling the reconstruction of actual trajectories in 3D, as well as the tracking of not only pitching, but also rolling and/or yawing motions.

**Control Law during Ballistic Phases.** The throttle and brakes were manually actuated during the jumps, leaving no indication of what the actual commands were, and of their timing. While the commands can be inferred from the measurements of the wheels' velocities, they are still not explicitly known. Moreover, it could have been interesting to have the commands be automatically applied directly after takeoff, thus lengthening the duration of their application.

## Conclusion and Future Work

In this work, we demonstrated that it was feasible to modify the pitch angle of a four-wheeled vehicle around its center of mass during an expected aerial phase using wheel acceleration or braking. The acceleration/braking were only applied during a short time frame with respect to the total ballistic phase but their effect was quite pronounced with a standard brushless engine. The forward pitching motion during the braked jump was only achieved through engine braking and could be intensified by reverse acceleration of the wheels.

A high speed camera was successfully applied for the measurement of the vehicle pitch angle and wheels angular position during the flight. A certain amount of noise was observed on the angular data, requiring the careful choice of filtering methods for eliminating it without deforming the actual trajectories. For subsequent tests, specular reflections have to be suppressed and markers of appropriate size and contrast must be placed on a flat surface of the rim, thus avoiding problems from tire deformation, cyclic changes in illumination conditions and partial cyclic occlusion of the target. Vehicle attitude should also be measured using markers on the chassis rather than at the center of each wheel. Moreover, tire bulging due to the centrifugal force must either be modeled or be inhibited. These improvements will hopefully allow us to observe the expected constant nature of the total angular momentum.

Future work will focus on implementing the automatic control of the pitch motion, using proprioceptive sensors and a very fast embedded control loop. As the jumps can last well under a second (typically 0,2 s), the control loop is expected to be able to run at least 1 kHz. A second task will be to correct also the yaw and roll angles, with the same constraints. The vehicle will then be
capable of actual 3D self-righting, requiring the use of high-speed stereo vision for the experimental aspects.

The previous ANR FAST project allowed to design and control the suspensions of a vehicle to maintain its stability at high speed on irregular terrains. The current RobCat project extends these capabilities also to ballistic phases. Combining both will allow the design and control of a fully autonomous agile robot capable to move and jump at high speeds over unstructured terrains for surveillance, mapping, agricultural, search and rescue, and humanitarian tasks.

## Acknowledgements

This work was financed by Région Auvergne through the "Innov@Pôle" project. It was supported by the "Fédération de Recherche Technologies de l'Information de la Mobilité et de la Sûreté" (FR TIMS, CNRS 2856), and its "Véhicules et Infrastructures Intelligents" (V2I) project (for the "RobCat" project). This federation is supported by the French Ministry for Higher Education and Research, the Agence Nationale de la Recherche (for the ANR FAST project), Région Auvergne, and the "Europe en Auvergne" FEDER (Fonds Européen de DÉveloppement Régional) program.

The authors wish to thank Issam Rajib and Samir Belazouz, who were of great help in conducting the experiments. Lastly, we wish to thank Christine Dulac-Rougerie, Deputy Mayor of Clermont-Ferrand in charge of Sports, Outdoor Leisure and Vacation Centers, David Rocton, director of the Stadium Jean Pellez in Aubière, as well as the whole staff of the Stadium for having authorized and enabled the use of their sports installations for our tests.

## References

[1] J. C. Fauroux, J. Dakhlallah, and B. C. Bouzgarrou, "A New Concept of FAST Mobile Rover with Improved Stability on Rough Terrain," May 2010.

[2] D. Hurst, "Sandia hopping robots to bolster troop capabilities," *Sandia Labs News Releases*, 11-Sep-2009. [Online]. Available: http://goo.gl/YuzlO.

[3] M. Hoffman, "Jumping robot to get tested in Afghanistan," *ArmyTimes*, 07-Nov-2011. [Online]. Available: http://goo.gl/gvhCt.

[4] M. Davis, C. Gouinaud, J. C. Fauroux, and P. Vaslin, "A Review of Self-Righting Techniques for Terrestrial Animals."

[5] E. Chang-Siu, T. Libby, M. Tomizuka, and others, "A lizard-inspired active tail enables rapid maneuvers and dynamic stabilization in a terrestrial robot," in *Intelligent Robots and Systems (IROS), 2011 IEEE/RSJ International Conference on*, 2011, pp. 1887-1894.

[6] A. Jusufi, D. I. Goldman, S. Revzen, and R. J. Full, "Active tails enhance arboreal acrobatics in geckos," *Proceedings of the National Academy of Sciences*, vol. 105, no. 11, pp. 4215-4219, Mar. 2008.

[7] C. Kaplan, "Leo satellites: attitude determination and control components; some linear attitude control techniques," Middle East Technical University, 2006.

[8] P. Vaslin, V. Pouzols, C. Gouinaud, J.-C. Fauroux, S. Deleplanque, and M. Davis, "Contrôle du tangage d'un véhicule en phase balistique," in *20ème Congrès Français de Mécanique*, Besançon, 2011.

[9] AOS Technologies AG, "S-MOTION High Speed Camera Technical Specifications." [Online]. Available: http://goo.gl/P3qJX.

[10] V. Pouzols, "Modélisation des facteurs influençant la trajectoire aérienne du système {pilote + moto} lors d'un saut en moto-cross. Validation expérimentale du modèle." 2006.

[11] Mouvement Français pour la Qualité. Besançon. MFQ Franche Compté, "Guide pour la détermination des incertitudes de mesure," 1995.

[12] J. J. Dowling, J. L. Durkin, and D. M. Andrews, "The uncertainty of the pendulum method for the determination of the moment of inertia," *Medical engineering & physics*, vol. 28, no. 8, pp. 837-841, 2006.