# Kinematic modeling at the ant scale: propagation of model parameter uncertainties

Santiago Arroyave-Tobón, Jordan Drapin, Anton Kaniewski, Jean-Marc Linares, Pierre Moretto

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Santiago Arroyave-Tobón, Jordan Drapin, Anton Kaniewski, Jean-Marc Linares, Pierre Moretto. Kinematic modeling at the ant scale: propagation of model parameter uncertainties. Frontiers in Bioengineering and Biotechnology, 2022, 10, 10.3389/fbioe.2022.767914. [hal-03592115](https://hal.science/hal-03592115)

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# Kinematic modeling at the ant scale:
propagation of model parameter uncertainties

Santiago Arroyave-Tobon <fnref n="1" />,†,*, Jordan Drapin <fnref n="2" />,†, Anton Kaniewski <fnref n="1" />,
Jean-Marc Linares <fnref n="1" />, Pierre Moretto <fnref n="2" />

Correspondence*:

Corresponding Author

santiago.arroyave-tobon@univ-amu.fr

†These authors have contributed equally to this work and share first authorship

## 2 ABSTRACT

Quadrupeds and hexapods are known by their ability to adapt their locomotive patterns in function of the environment. Computational modeling of animal movement can help to better understand the emergence of locomotive patterns and their body dynamics. Although considerable progress has been made in this subject in recent years, the strengths and limitations of kinematic simulations at the scale of small moving animals are not well understood. In response to this, this work evaluated the effects of modeling uncertainties on kinematic simulations at small scale. In order to do so, a multibody model of a *Messor barbarus* ant was developed. The model was built from 3D scans coming from X-ray micro-computed tomography. Joint geometrical parameters were estimated from the articular surfaces of the exoskeleton. Kinematic data of a free walking ant was acquired using high-speed synchronized video cameras. Spatial coordinates of 49 virtual markers were used to run inverse kinematics simulations using OpenSim software. The sensitivity of the model's predictions to joint geometrical parameters and marker position uncertainties was evaluated by means of two Monte Carlo simulations. The developed model was four times more sensitive to perturbations on marker position than those of the joint geometrical parameters. These results are of interest for locomotion studies of small quadrupeds, octopods and other multi-legged animals.

19

20 **Keywords:** multibody, inverse kinematics, ant, motion capture, uncertainty

## 1 INTRODUCTION

Legged locomotion is the most common form of terrestrial animal movement (Christensen et al., 2021).

Even if quadrupedal and hexapodal locomotion have evolved independently (Blickhan and Full, 1987),

they present similarities. Both quadrupeds and hexapods can adapt their locomotive patterns according to

their objective (Hoyt and Taylor, 1981; Nirody, 2021). Like quadrupeds, hexapods exhibit a wide variety

of locomotor strategies (Nirody, 2021), e.g. walking, running, jumping (Musthak Ali et al., 1992) or even

swimming (Schultheiss and Guénard, 2021) and gliding (Yanoviak et al., 2005). As some quadrupeds

do, insects change smoothly the inter-leg coordination patterns based on their locomotion speed (Ambe

et al., 2018). In the metachronous gait (or direct wave gait), hexapods propagate swinging movements

from the hind legs to the forelegs, similarly as quadrupeds do in the walking gait (Ambe et al., 2018). In

<footnote n="1">¹Aix Marseille Univ, CNRS, ISM, Marseille, France</footnote>

<footnote n="2">²Centre de Recherches sur la Cognition Animale (CRCA), Centre de Biologie
Intégrative (CBI), Université de Toulouse, CNRS, UPS, Toulouse, France</footnote>
tripod gait, hexapods move their diagonal legs in phase, as quadrupeds do in the trotting gait (Ambe et al., 2018). These equivalences in the locomotion mechanics generate similar ground reaction force patterns in quadrupeds and hexapods, as demonstrated experimentally by Full et al. (1991). In that study, the authors demonstrated that at constant average speed, cockroaches function as a spring-mass system in which three legs add up to function as one leg of a biped or two legs of a quadruped.

As opposed to bipedal and quadrupedal locomotion, hexapodal locomotion is characterized by its plasticity. For instance, hexapods can adopt quadrupedal or bipedal gaits to increase speed, as has been shown in cockroaches (Full et al., 1991). The bipodal posture adopted when the insect stands up, allows for a longer stride length while maintaining the same stride frequency, thus raising the speed. In stick insects, the coordination of the middle legs and hindlegs is similar to typical regular gaits of quadrupeds Grabowska et al. (2012a). The emergence of quadrudepal gaits on hexapod robots has also been demonstrated when a sudden fault event occurs to one leg (Yang and Kim, 1998). However, these adaptations deserve further analysis to better understand the plasticity and dynamics of multi-legged gait.

The hexapodal gait has been first described as an alternative tripod gait that ensures high static stability (Hughes, 1952) regardless of the support. Yet, studies estimating ground reaction forces demonstrate different functions of the rear, median and front legs (sustain, propel, push or drag) (Cruse, 1976; Full et al., 1991; Grabowska et al., 2012b; Reinhardt and Blickhan, 2014; Wöhrl et al., 2017). Other studies, dedicated to the effects of the ground substrates or load carried, demonstrated the plasticity of the tripod gait in response to mechanical constraints (Bernadou et al., 2011; Pfeffer et al., 2019; Merienne et al., 2020). These studies suggest that hexapodal gait is more complex than a mere alternating tripod one. Furthermore, the small scale and lack of a precise description of the architecture of the musculoskeletal system could explain why the hexapodal gait is less documented than the quadrudepal or bipedal gaits.

Learning how insects adapt their locomotion strategies to their environment (motor and neural control), how each body segment moves for a given locomotion strategy (kinematics), and how forces are generated (muscle actuation) and transmitted (joint dynamics), could help answer biological questions, and develop engineering applications. For instance, kinematic, dynamic and motor control data regarding animal locomotion proves indispensable for bio-inspired robotics development. Particularly, some examples of applications include: bio-inspired robot architecture (Lu et al., 2018), bio-inspired control strategies for legged robots (Dupeyroux et al., 2019; Ouyang et al., 2021), and bio-inspired actuation systems (Ahn et al., 2019), among others.

Computational modeling of animal movement can help to better understand the emergence locomotive patterns and their mechanics by means of the musculoskeletal models. A musculoskeletal model is comprised of a kinematic model coupled to a dynamic model. The kinematic model, which represents the skeletal system, is a set of body segments connected by joints (i.e. a multibody system). A dynamic model, which represents the muscular system, is a set of actuators attached to the skeletal system.

The proper development of the kinematic model is essential for predicting later muscle and joint forces (Dunne et al., 2021). In kinematic modeling, constrained inverse kinematics, as opposed to with unconstrained inverse kinematics, leads to a more realistic prediction of joint kinematics. Conversely, unconstrained inverse kinematics, which permits a fast exploitation of experimental data using stick models, can generate unrealistic behaviors, such as a model's body segments changing length (Dunne et al., 2021). This kind of behavior is unsuitable for musculoskeletal simulations. In constrained kinematic modeling, which is conducted using multibody models, the position and orientation of each segment of the kinematic chain is derived from the trajectories of experimental markers. This is done by optimizing procedures
that minimize the weighted least-squares distance between experimental markers and the corresponding
markers placed on the kinematic model (Lu and O'Connor, 1999). The position and orientation of each
segment of the kinematic chain, together with their first order derivatives, can be used for further muscle
and joint force estimation.

In the case of vertebrates, the development and use of musculoskeletal models are mainly motivated by
medical applications (REFS). In the case of insects, motivations are mostly related to biology, ecology
and evolution. Ramdya et al. (2017) developed a multibody model of drosophila to study fast locomotor
gaits. Guo et al. (2018) proposed a neuro-musculo-skeletal model for insects to study control strategies
in gait patterns. David et al. (2016) and Blanke et al. (2017) developed musculoskeletal models of the
dragonfly's mandible to study bite forces. A kinematic model of stick insects was developed by Theunissen
and Dürr (2013). In the case of ants, locomotion studies mostly focus on experimental procedures. For
example: video-based kinematic analysis (Weihmann and Blickhan, 2009; Moll et al., 2010; Pfeffer et al.,
2019), stepping pattern analysis (Zollikofer, 1994), center of mass tracking (Reinhardt and Blickhan, 2014;
Merienne et al., 2020, 2021), quantification of ground reaction forces (Reinhardt et al., 2009; Wöhrl et al.,
2017), and mandible forces (Zhang et al., 2020), among others.

Despite the aforementioned examples, the use of musculoskeletal models at the insect-scale is not yet
widespread; probably due to the technological barriers to acquire experimental data (kinematic, dynamic
and morphometric data). When comparing the relative resolution of motion capture systems vs. the subject
size, it can be argued that motion capture at the human scale is far more accurate than at the insect-scale.
In human motion analysis using reflective markers, the measuring uncertainty can reach 0.33 mm in a
volume of 5.5 x 1.2 x 2.0 m³ (Eichelberger et al., 2016) (0.0275% in the smallest dimension). Motion
analysis by means of physical markers is not easy in small insects. A pattern matching procedure based on
video films is a feasible solution for the moment. Using this technique at small scale, our setup reached, on
average, 3% resolution in each dimension of the calibrated volume (including tracking errors and pattern
recognition errors). The difficulty with small scales lies in keeping the depth of field of the camera at a
reasonable size when zooming in to get a clear whole-body image. This problem is not encountered in
larger subjects because the lenses are far from the objective. Similar difficulties are faced in morphometric
data acquisition in small insects, which is required for the definition of joint locations in musculoskeletal
modeling. This implies that the effect of uncertainties in musculoskeletal modeling at the insect-scale must
be considered and evaluated to understand the limits of this tool in locomotion analysis. Estimation of
uncertainties in kinematic modeling has been widely addressed at the human scale (see for example (Groen
et al., 2012; El Habachi et al., 2015; Martelli et al., 2015)). At the insect-scale, however, it is unclear how
modeling assumptions affect predicted results in kinematic modeling.

The present work therefore evaluated the effects of modeling assumptions in kinematic analysis at the
small insect-scale, particularly on a *Messor barbarus* ant. To achieve this objective, i) a whole-body
kinematic model of the *Messor barbarus* ant was developed (Section 2.1), ii) an inverse kinematics
simulation of the ant gait was reproduced using the developed model and experimental kinematic data
(Section 2.6), and iii) the sensitivity of the predicted results regarding model parameter uncertainties was
evaluated (Section 2.7).

## 2 METHODS

The global research methodology followed in this work is illustrated in Figure 1. Specimens 1 and 2 belong
to the medium sized caste of the *Messor barbarus* species (more details in Section 2.1). Specimen 1 was
used to build a 3D model from micro-computed tomography (Section 2.2). 3D models of body segments
115 were used to extract joint geometrical parameters and to create a multibody model (Sections 2.3 and 2.4).
116 Specimen 2 was used to acquire experimental kinematic data and to extract marker trajectories (Section 2.5).
117 Experimental kinematic data was used to scale the multibody model and to run an inverse kinematics simulation (Section 2.6).
118 To evaluate the impact of the propagation of model parameter uncertainties on joint angles, two Monte Carlo (MC) simulations were conducted (Section 2.7).
119 Model parameters subjected to uncertainty are represented by a Gaussian distribution icon in Figure 1.
120

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Followed research methodology. Gaussian distribution icon indicates model parameters subjected to uncertainty.</figcaption></figure>

## 2.1 Experimental model

122 We used workers from a colony of *Messor barbarus* collected in April 2018 in Saint Hipolyte (42° 78' North; 2° 97' East, Pyrénées Orientales, France). *Messor barbarus* is a seed-collecting ant whose mature colonies can harbor tens of thousands of individuals (Hölldobler and Wilson, 1990). The body mass of the scanned subject was 8.92 mg.

126 The main colony was kept in a box (L: 50 x W: 30 x H: 15 cm) with walls coated with Fluon® to prevent ants from escaping. The ants could shelter inside nests formed with test tubes (length: 20 cm; diameter: 2.5 cm) covered with opaque paper. They had access to water and a mixture of bird seeds. The experimental room was maintained at a constant temperature of 26 °C (thermometer: TFA Dostmann / Wertheim) and under an artificial photoperiod regime 12h: 12h (light: dark).

## 2.2 Micro-Computed Tomography

132 Following the procedure used by Peeters et al. (2020), specimen 1 was stored in 90% ethanol, then stained in a 2 M iodine solution for a minimum of 24 h and transferred into micro tubes filled with 99% ethanol. It was then transferred to Okinawa Institute of Science and Technology Graduate University (OIST, Japan) to be scanned using micro-Computed Tomography (μ-CT). This was performed using a Zeiss Xradia 510 Versa 3D X-ray microscope operated by the Zeiss Scout-and-Scan Control System software (version 11.1).
135 A vertical stitching enabled a 3 times scanning along a head-trunk-gaster axis each with a resolution of 933
138 × 1013 × 988 pixels (providing a voxel of 5.7 µm). These scans were compiled to increase the resolution
139 of the whole ant body to 3159 × 1013 × 988 pixels. The DICOM images of the µ-CT scan were used to
140 build the 3D models of the body segments. A segmentation was done using ITK-SNAP (version 3.6.0)
141 (Yushkevich et al., 2006) to differentiate the body segments as follows: head, thorax, abdomen, coxa,
142 trochanter, femur, tibia, metatarsus and tarsus. The four tarsal segments were lumped all into a unique rigid
143 segment called tarsus in this work.

## 2.3 Extraction of joint geometrical parameters

145 Defining the types of joints was done from both literature and morphometric data (Liu et al., 2019).
146 From the 3D models of the body segments, joint geometrical parameters were estimated from the articular
147 surfaces of the exoskeleton using a CAD software (3D EXPERIENCE, Dassault Systèmes, France). For
148 ball-and-socket joints, the center of a sphere fitted to the articular surface was considered as the center of
149 rotation of the joint (see Figure 2 (b,f)). For hinge joints, the rotation axis was defined as the line passing
150 through the center of two spheres fitted to the condyles of the joint (see Figure 2 (c,d,e)). The procedure to
151 determine joint geometrical parameter was also used in insect biomechanical modeling by Blanke et al.
152 (2017). Because of low perceived motion, and to facilitate the convergence of the inverse kinematics
153 algorithm, the internal rotation of the metatarsus of each leg was not considered (it was assumed as a
154 blocked degree of freedom (DOF)).

## 2.4 Creation of the multibody model

155 A multibody model was created representing the whole-body locomotor system of the *Messor barbarus*.
156 According to the recommendations of the ISB (Wu et al., 2002, 2005), a coordinate system was defined
157 for each body segment and for the ground. All coordinate systems were defined as right-handed and
158 orthogonal, as follows (see Figure 2):
159

160 • Definition of the sagittal plane: plane perpendicular to the line passing through the center of two spheres fitted to the propodeal spiracles and containing the point *P*<sub>SP</sub>. Point *P*<sub>SP</sub> was defined as the mid-point of the line segment defined by the two propodeal spiracles, see Figure 2b.

163 • Global coordinate system (*x*<sub>g</sub>, *y*<sub>g</sub>, *z*<sub>g</sub>): The *z*<sub>g</sub>-axis points upward, parallel to the field of gravity. The *x*<sub>g</sub>-axis points in the direction opposite to the direction of travel. The *y*<sub>g</sub> axis was defined as the common axis perpendicular to *x*<sub>g</sub>- and *z*<sub>g</sub>-axis.

166 • Thorax coordinate system (*x*<sub>t</sub>, *y*<sub>t</sub>, *z*<sub>t</sub>): the origin of this coordinate system was defined as the mid-point of the line segment passing through the center of the spheres fitted to the thorax/neck joint and thorax/abdomen joints, points *P*<sub>*h*</sub> and *P*<sub>*a*</sub> in Figure 2b respectively. The *y*<sub>*t*</sub>-axis was defined parallel to the line segment *P*<sub>*h*</sub>*P*<sub>*a*</sub> and pointing anteriorly. The *x*<sub>*t*</sub>-axis was defined as the common axis perpendicular to the normal vector of the sagittal plane and to *y*<sub>*t*</sub>. The *z*<sub>*t*</sub>-axis was defined as the common axis perpendicular to *x*<sub>*t*</sub>- and *y*<sub>*t*</sub>-axis.

172 • For hinge joints, the origin of the coordinate system was chosen as the mid-point of the line segment
173 representing the rotation axis (for example: points *P*<sub>ml_to</sub> and *P*<sub>ml_fe</sub> in Figure 2c, *P*<sub>ml_ti</sub> in Figure 2d
174 and *P*<sub>ml_mt</sub> in Figure 2e). The *z*-axis was defined parallel to the rotation axis and pointing medially.
175 The *y*-axis was defined perpendicular to the *z*-axis and pointing to the origin of the coordinate system
176 of the previous segment. The *x*-axis was defined as the common axis perpendicular to *y*- and *z*-axis.

177 • For ball-and-socket joints, the origin of the coordinate system was chosen as the center of the sphere
178 fitted to the articular surface (for example: points *P*<sub>ml_co</sub> and *P*<sub>ml_ta</sub> in Figure 2b and 2f, respectively).
179 The *y*-axis was defined parallel to the line passing through the origin of the coordinate system and
180 the origin of the coordinate system of the previous segment, and pointing proximally. The *x*-axis was
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Definition of the joint geometrical parameters and coordinate systems. For all coordinate systems, the *x*-axis is represented in red, the *y*-axis in green, and the *z*-axis in black. For ball-and-socket joints, a sphere fitted to the articular surface was considered as the center of the rotation of the joint. For hinge joints, the rotation axis was defined as the line passing through the center of two spheres fitted to the condyles of the joint. Fitted spheres are represented in red and rotation axes are represented in yellow. (a) Representation of the sagittal plane. (b) Geometrical elements used to define the sagittal plane, the coordinate system of the thorax (*x*<sub>t</sub>, *y*<sub>t</sub>, *z*<sub>t</sub>) and the coordinate system of the middle left coxa. The point *P*<sub>SP</sub> was defined as the mid-point of the line segment passing through the center of the two propodeal spiracles. The points *P*<sub>h</sub> and *P*<sub>a</sub> correspond to the center of the spheres fitted to the thorax/head and thorax/abdomen joints, respectively. The point *P*<sub>ml_co</sub> corresponds to the center of the sphere fitted to the articular surface of the middle left thorax/coxa joint. (c, d, e) Geometrical elements used to define the rotation axes and the coordinate systems of the coxa/trochanter, trochanter/femur, femur/tibia and tibia/metatarsus joints. In respective order, the points *P*<sub>ml_to</sub>, *P*<sub>ml_fem</sub>, *P*<sub>fe ti</sub> and *P*<sub>ti mt</sub> were defined as the mid-points of the line segments representing the rotation axis of the joints. (f) Geometrical elements used the rotation center and the coordinate systems of the metatarsus/tarsus joint. The point *P*<sub>ml ta</sub> corresponds to the center of the sphere fitted to the articular surface of the metatarsus/tarsus joint.</figcaption></figure>

181 defined as the common axis perpendicular to the normal vector of the sagittal plane and to *y*. The

182 *z*-axis was defined as the common axis perpendicular to *x*- and *y*-axis.
According to the previous definitions of the coordinate systems, the following convention for rotations was adopted. Abduction: positive rotation about the *x*-axis. Adduction: negative rotation about the *x*-axis. Internal rotation: positive rotation about the *y*-axis. External rotation: negative rotation about the *y*-axis. Flexion: negative rotation about the *z*-axis. Extension: positive rotation about the *z*-axis.

The model was composed of 39 segments and 65 DOF. Segments were considered as rigid bodies and joints were considered without clearance. Half of the kinematic chain of this model is represented in Figure 3. 47 virtual markers were placed on the model according to the tracked anatomical landmarks (see Figure 4). The model was created using the software tool NSM Builder (version 2.1) (Valente et al., 2017) and finally exported in an OpenSim format. Range of motion of the joints were constrained to feasible values to aid the convergence of the inverse kinematics algorithm. These values were determined in OpenSim by articulating each DOF of the model until some structures of the joint segments touch each other. Obtained values are presented in Tables 1 and 2.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Kinematic chain representing half of the ant locomotor system. Anatomy: th (thorax), pet (petiole), abd (abdomen), cox (coxa), tro (trochanter), fe (femur), ti (tibia), mt (metatarsus), ta (tarsus). Type of joint: hinge (example: mt/ta) or ball-and-socket (example: head/thorax).</figcaption></figure>

## 2.5 Kinematic data acquisition and treatment

Kinematic data of a free walking ant (mean speed over the length of the calibrated walkway: 3.4 mm s⁻¹) was acquired using high-speed synchronized video cameras (AI GO-5000M-PMCL). The experimental setup was composed of a wide walkway where the ant walked through, 5 cameras (1 of the top and 2 for each side or the walkway) and 3 infra-red spots (see Figure 5). The shutter time was 1/3333 s, the acquisition time was set to 10 s with a sampling frequency of 300 Hz. The infrared spots were added to compensate this short shutter time. The resolution of the camera sensor was 2560 × 2048 pixels. Using Hiris software of R&D Vision (version 5.2.0), the active sensor window was adjusted to the ant size in a 2000 × 418 pixels rectangular area. The average field of vision of the cameras was 15.8 x 4.9 x 7.8 mm that gives a spatial resolution of 0.0096 mm/pixel. Obtained raw videos are available from the project repository.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Position and denomination of the markers on the model. The following abbreviations are used for the denomination of the markers. Prefix: f (front), m (middle) or r (rear); Position: l (left), r (right). Anatomy and articulation: th (thorax), pet (petiole), abd (abdomen), cox (coxa), tro (trochanter), fe (femur), ti (tibia), mt (metatarsus), ta (tarsus).</figcaption></figure>

<table><caption>**Table 1.** Maximum range of motion allowable for each degree of freedom of trunk joints. Values are presented per leg: front, middle, rear. The same values were used for left and right legs. These values were determined in OpenSim by articulating each degree of freedom of the model until some structures of the joint segments touch each other.</caption><thead><tr><th>degree of freedom</th><th>maximum<br>allowable range<br>of motion (deg)</th></tr></thead><tbody><tr><td>thorax/head adduction</td><td>75</td></tr><tr><td>thorax/head internal rotation</td><td>40</td></tr><tr><td>thorax/head flexion</td><td>120</td></tr><tr><td>thorax/abdomen adduction</td><td>25</td></tr><tr><td>thorax/abdomen internal rotation</td><td>60</td></tr><tr><td>thorax/abdomen flexion</td><td>100</td></tr></tbody></table>

Following a similar protocol than Merienne et al. (2020), the filming procedure was: (1) the ant was randomly collected from the colony and left in a box for 15 minutes in order to reduce the stress of the capture. (2) The ant was located at the beginning of the walkway and the recording started when it entered in the calibrated volume. The temperature of the room was 26± 0.2° during the filming procedure. Only one gait cycle was studied to avoid the variability of the motor control during different gait cycles (change of the walking speed, balance management and change of movement direction).

Video recordings were processed afterwards with the Vicon Peakmotus (version 10) software tool.

Segment extremities were tracked semi-automatically during a gait cycle using a pattern matching technique.
<table><caption>**Table 2.** Maximum range of motion allowable for each degree of freedom of the leg joints. Values are presented per leg: front, middle, rear. The same values were used for left and right legs. These values were determined in OpenSim by articulating each degree of freedom of the model until some structures of the joint segment touch. Nonallocated values (NA) correspond to blocked degrees of freedom</caption><thead><tr><th>degree of freedom</th><th>front legs (deg)</th><th>middle legs (deg)</th><th>rear legs (deg)</th></tr></thead><tbody><tr><td>thorax/cox abduction</td><td>70</td><td>80</td><td>110</td></tr><tr><td>thorax/cox internal</td><td>40</td><td>55</td><td>105</td></tr><tr><td>thorax/cox flexion</td><td>80</td><td>100</td><td>105</td></tr><tr><td>cox/tro abduction</td><td>120</td><td>NA</td><td>NA</td></tr><tr><td>cox/tro internal</td><td>165</td><td>NA</td><td>NA</td></tr><tr><td>cox/tro flexion</td><td>180</td><td>120</td><td>130</td></tr><tr><td>tro/fe flexion</td><td>130</td><td>180</td><td>120</td></tr><tr><td>fe/ti flexion</td><td>160</td><td>165</td><td>190</td></tr><tr><td>ti/mt flexion</td><td>190</td><td>200</td><td>175</td></tr><tr><td>mt/ta flexion</td><td>200</td><td>200</td><td>240</td></tr></tbody></table>

214 The gait cycle was defined when the left middle leg leaves the ground and lifts, and it ends when that same leg leaves the ground again. Kinematic data was filtered with 4th order Butterworth low-pass filters with a cut-off frequency of 5 Hz. It was then resampled from 300 to 100 Hz to decrease computation time. Spatial coordinates of the anatomical landmarks (those represented in Figure 4) were exported on a c3d format file. 215 216 217 This file is available from the project repository.

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Video acquisition system. The experimental setup was composed of a wide walkway where the ant walked through, and was captured by 5 cameras. C: cameras, IR: infrared spots, P: 250x20 mm wide walkway.</figcaption></figure>
## 2.6 Model scaling and inverse kinematics analysis

Spatial coordinates of the anatomical landmarks were used to scale the multibody model and to run inverse kinematics simulations. A scaling procedure was carried out to fit the model (originally created from the morphology of specimen 1) to the morphology of specimen 2. This was performed using the open source software tool OpenSim (version 4.0) (Seth et al., 2018). Using the scaled model, inverse kinematics simulations were also performed in OpenSim. Joint angles as well as root mean square errors (RMSE) were obtained from these simulations.

## 2.7 Propagation of model parameter uncertainties

In order to evaluate the sensitivity of the calculated kinematic data to model parameter uncertainties, two MC simulations were conducted. A similar procedure was used by Martelli et al. (2015) and Myers et al. (2015).

In the first MC simulation, position of model markers was randomly perturbed according to their uncertainty. Random values were assumed to have a uniform distribution (i.e. all outcomes were considered as equally likely). Variations were assumed to be the same in all directions of the measurement volume. Therefore, the uncertainty zone for the model markers was assumed to be spherical. The radius of these spherical uncertainty zones was chosen as a common residual value for the camera calibration process for the used experimental setup: 0.4 mm.

In the second MC simulation, joint geometrical parameters (location and orientation) were randomly disturbed. The uncertainty in location and orientation of joints is mainly related with operator-dependent variability of the treatment and identification of the articular surfaces. In order to define perturbation magnitude (translation and rotation) introduced to the joint geometrical parameters, several procedures of identification of articular surfaces were carried out. Cylindrical uncertainty zones were assumed for hinge joints while spherical uncertainty zones for ball-and-socket joints. The radius of the cylindrical and spherical uncertainty zones was considered to be the same for all the joints and equal to: 0.2 mm.

These MC simulations were implemented and run by means of the OpenSim API. 1000 iterations were carried out for each MC simulation, which were enough to guarantee a stabilization of average values. Average values of joint angles at each time step were calculated from the obtained results. Coverage intervals were defined as twice the standard deviation. The sensitivity of the kinematic results regarding model parameter uncertainties was defined as the signal-to-noise ratio (SNR) of the joint angles during the gait. The SNR was calculated as the maximum amplitude of the signal (also called power of the signal, $P_s$) divided by the maximum coverage interval (also called power of the noise, $P_n$) of the joint angle during the gait. Therefore, a SNR value was obtained per degree of freedom for the analyzed gait cycle.

# 3 RESULTS

In order to determine how modeling assumptions affect inverse kinematics results at the ant scale, a multibody model of the *Messor barbarus* was developed together with a simulation framework to evaluate its sensitivity. Both the model and the simulation framework are freely available on the SimTK repository: https://simtk.org/projects/barbarus. From the experimental kinematic data, an inverse kinematics simulation was conducted. The results of this simulation, representing a gait cycle of free locomotion of the *Messor barbarus*, are summarized in Tables 3 and 4. A video of the simulated kinematics is available from the project repository.

These results correspond to the range of motion of the joint angles. The whole set of results is available from the project repository and can also be reproduced from the model and the experimental kinematic
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Representation of the uncertainty of the computed kinematics of the ant gait. The uncertainty Δθ is represented for the femur/tibia angle (θ<sub>fe/ti</sub>) of the middle left leg for: joint axis perturbation Δθ<sub>a</sub> (black shaded confidence intervals) and marker position perturbation Δθ<sub>m</sub> (green shaded confidence intervals).</figcaption></figure>

260 data. It can be noticed that the trochanter/femur (tr/fe) joint is the one with the wider range of motion, while
261 the thorax/coxa joints exhibit the smallest one. The average RMSE of the inverse kinematics simulation
262 was 0.21 mm, which corresponds to 3.2% of specimen size.

263 The sensitivity of the kinematic results regarding model parameter uncertainties was evaluated by means
264 of the SNR. These results are summarized per set of joints, from marker perturbation, as well as from axis
265 perturbation, in Table 5. High SNR values indicate that the power of the signal (computed joint angle) is
266 representative with respect to the power of the noise (confidence intervals). SNR values near or lower than
267 1 indicate that the dynamics of the signal of interest might be hidden by noise. It can be noticed that the
268 computed kinematics is more sensitive to marker perturbation compared to joint axis perturbation (Table
269 5). The perturbation applied to the markers generated a SNR of 2.2 in average for all the joints. This means
270 that the dynamics of the studied signal (computed joint angles) can be observed despite possible variations
<table><caption>**Table 3.** Results of the inverse kinematics simulation for the trunk joints. Reported values represent the range of motion in degrees of the joint angles.</caption><thead><tr><th>degree of freedom</th><th>range of motion (deg)</th></tr></thead><tbody><tr><td>thorax/head abduction</td><td>19.5</td></tr><tr><td>thorax/head internal rotation</td><td>9.0</td></tr><tr><td>thorax/head flexion</td><td>13.3</td></tr><tr><td>thorax/abdomen abduction</td><td>11.9</td></tr><tr><td>thorax/abdomen internal rotation</td><td>14.2</td></tr><tr><td>thorax/abdomen flexion</td><td>13.8</td></tr></tbody></table>

<table><caption>**Table 4.** Results of the inverse kinematics simulation for the leg joints. Reported values represent the range of motion in degrees of the joint angles. Nonallocated values (NA) correspond to blocked degrees of freedom.</caption><thead><tr><th>degree of freedom</th><th>front right leg (deg)</th><th>middle right leg (deg)</th><th>rear right leg (deg)</th><th>front left leg (deg)</th><th>middle left leg (deg)</th><th>rear left leg (deg)</th></tr></thead><tbody><tr><td>thorax/cox abduction</td><td>40.7</td><td>32.0</td><td>16.5</td><td>19.1</td><td>23.4</td><td>27.2</td></tr><tr><td>thorax/cox internal rotation</td><td>29.3</td><td>54.0</td><td>35.7</td><td>28.1</td><td>26.0</td><td>16.3</td></tr><tr><td>thorax/cox flexion</td><td>26.5</td><td>25.9</td><td>90.6</td><td>22.1</td><td>39.4</td><td>14.9</td></tr><tr><td>cox/tro abduction</td><td>78.2</td><td>NA</td><td>NA</td><td>74.6</td><td>NA</td><td>NA</td></tr><tr><td>cox/tro internal rotation</td><td>76.3</td><td>NA</td><td>NA</td><td>47.3</td><td>NA</td><td>NA</td></tr><tr><td>cox/tro flexion</td><td>90.0</td><td>30.6</td><td>30.2</td><td>43.9</td><td>48.3</td><td>42.4</td></tr><tr><td>tro/fe flexion</td><td>120.7</td><td>53.1</td><td>103.9</td><td>78.6</td><td>27.2</td><td>15.6</td></tr><tr><td>fe/ti flexion</td><td>61.6</td><td>73.6</td><td>66.7</td><td>39.3</td><td>38.2</td><td>60.2</td></tr><tr><td>ti/mt flexion</td><td>53.8</td><td>41.0</td><td>29.0</td><td>63.4</td><td>17.7</td><td>17.2</td></tr><tr><td>mt/ta flexion</td><td>56.9</td><td>36.2</td><td>43.5</td><td>57.7</td><td>NA</td><td>23.4</td></tr></tbody></table>

271 during the motion analysis process. The SNR from axis perturbation was almost four times higher than that 272 from markers perturbation. No significant differences in sensitivity were found between the joints of the 273 legs on the right side of the body with respect to those on the left side. No tendency can be inferred from 274 the sensitivity of the joints with respect to their anterior-posterior position: front, middle and rear. The joint 275 that showed the highest SNR values (consequently a lower sensitivity) was the fe/ti joint, and this was the 276 case for both marker and axis perturbation.

277 Figure 7 illustrates kinematic results obtained from the simulation of the ant model and the experimental 278 kinematic data for joints of the middle right leg ((a) thorax/cox, (b) cox/tro, (c) tro/fe, (d) fe/ti, (e) ti/mt 279 and (f) mt/ta flexion angles). Mean values (in solid lines) from both MC simulations (marker and axis 280 perturbation) are shown with their corresponding confidence intervals (shaded regions). The green line 281 and shaded region represent the results from the marker perturbation, and the black line and shaded region 282 represent the results from the axis perturbation. The SNR of the thorax/cox flexion angle obtained from 283 the axis perturbation simulation is illustrated in Figure 7 (a). From these results, it can be noticed that the 284 confidence intervals of the joint angles when disturbing the axis location and orientation were smaller than 285 the confidence intervals obtained from the marker position perturbation.

## 4 DISCUSSION AND CONCLUSIONS

286 In this paper, the propagation of parameter uncertainties in kinematic modelling has been evaluated at the 287 small-scale. This work demonstrates the feasibility of using biomechanical models to study locomotion in 288 relatively small animals. Because of their scale, motion analysis techniques for hexapods are less developed
<table><caption>**Table 5.** Results of the sensitivity analysis. These results represent the average of signal-to-noise ratio per joint obtained from: marker perturbation (first column) and axis perturbation (second column). In the case of cox/tro, tro/fe, fe/ti, ti/mt, and mt/ta joints, averages were calculated from the signal-to-noise ratio of the six legs.</caption><thead><tr><th>joint</th><th>signal-to-noise<br>ratio from marker<br>perturbation</th><th>signal-to-noise<br>ratio from axis<br>perturbation</th></tr></thead><tbody><tr><td>all joints</td><td>2.20</td><td>8.10</td></tr><tr><td>right-hand side joints</td><td>2.29</td><td>7.96</td></tr><tr><td>left-hand side joints</td><td>2.27</td><td>8.92</td></tr><tr><td>front legs joints</td><td>1.54</td><td>9.13</td></tr><tr><td>middle legs joints</td><td>2.82</td><td>8.05</td></tr><tr><td>rear legs joints</td><td>2.69</td><td>7.97</td></tr><tr><td>thorax/head</td><td>1.71</td><td>5.08</td></tr><tr><td>thorax/abdomen</td><td>1.38</td><td>4.97</td></tr><tr><td>thorax/cox</td><td>1.58</td><td>4.65</td></tr><tr><td>cox/tro</td><td>2.73</td><td>7.25</td></tr><tr><td>tro/fe</td><td>1.57</td><td>5.52</td></tr><tr><td>fe/ti</td><td>6.87</td><td>27.63</td></tr><tr><td>ti/mt</td><td>1.70</td><td>8.62</td></tr><tr><td>mt/ta</td><td>1.19</td><td>7.14</td></tr></tbody></table>

compared to those for quadrupeds and bipeds. In relatively big animals, the use of several reflective markers per segment allows a good precision of the kinematic data. However, the use of physical markers is not easy in motion analysis in small insects. This implies that the capabilities of the small-scale biomechanical modeling techniques must be well evaluated.

To do so, a multibody model of a *Messor barbarus* ant was developed. It is available in open source from the project repository and can be used and enhanced by the scientific community. Besides, the model could allow biologists to study function/structure relationships of *Messor barbarus*. The whole set of experimental and simulated kinematic data is also available from the project repository.

In spite of the differences in morphology of the studied species, the obtained joint angles were in the same order of magnitude as those reported in the literature about ant kinematics (see Table 6). The difference between angle range of left and right legs comes from the fact that the ant did not walk perfectly straight. Obtained kinematic data are valuable for roboticians to implement bio-inspired gaits in robots (see Ouyang et al. (2021) for example).

A possible error source in the conducted kinematic simulation could be linked to the use of two different specimens for acquiring experimental data (one for the geometrical 3D model and one for the experimental kinematic data). When using two subjects to perform a constrained kinematics simulation, a scaling procedure is required, which is naturally an additional source of errors. This might be one of the main reasons for the obtained RMSE values. In comparison to human locomotion simulations, the obtained normalized RMSE values for ant locomotion simulation were greater. In human simulations, it is recommended not to exceed 0.6% relative RMSE regarding body size (in contrast to a normalized RMSE of 3.2% obtained in this work). This difference can also be related to the fact that the ant body is composed of more segments than the human one.

Thanks to the developed model, the impact of the propagation of model parameter uncertainties in inverse kinematics simulations at insect-scale was evaluated. Obtained SNR values indicate that the geometric and kinematic measurement techniques used are feasible for the development of multibody models at
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 7.** Kinematic results obtained from the simulation of the ant model and the experimental kinematic data for flexion angle of the middle right leg of (a) thorax/cox; (b) cox/tro; (c) tro/fe; (d) fe/ti; (e) ti/mt and (f) mt/ta. The recorded and simulated gait cycle lasted 1.39 s. These results are a sample of the whole set of results available from the project repository. Solid lines indicate the mean values from the Monte Carlo simulations from the marker perturbation (green) and from the axis perturbation (black). For marker and axis perturbation respectively, the green and black shaded region represent the confidence interval (calculated as twice the standard deviation). The dashed vertical lines (37% and 87%) indicate when legs of both tripods were on the ground. The SNR of the thorax/cox flexion angle obtained from the axis perturbation simulation is illustrated in 7 (a). $P_s$ (standing for power of the signal) corresponds to the peak-to-peak amplitude of the signal. $P_n$ (standing for power of the noise) corresponds to the maximal coverage interval of the joint angle during the gait.</figcaption></figure>

the ant-scale. The fact that the model is more sensitive to marker perturbations, indicates that efforts in kinematic modeling at the ant-scale must be centered around the kinematic acquisition (marker definition, placement, tracking, etc.) rather than geometric acquisition (μ-CT, segmentation, joint parameter definition, etc.). The fact of experiencing lower sensitivity at the fe/ti joint can be explained by the large range of motion of this joint and, also, because it is composed of the two longest segments of the limb. Long
<table><caption>**Table 6.** Summary of studies investigating ant kinematics.</caption><thead><tr><th>Study</th><th>Analyzed angle</th><th>Specie</th><th>Methods</th><th>Range of motion (deg)</th><th>Corresponding range of motion from this study (deg)</th></tr></thead><tbody><tr><td>Weihmann and Blickhan (2009)</td><td>thorax/head flexion</td><td>Cataglyphis fortis</td><td>video-based analysis (250 Hz, 480x480 pixels of camera resolution)</td><td>5</td><td>13</td></tr><tr><td>Weihmann and Blickhan (2009)</td><td>thorax/head flexion</td><td>Formica pratensis</td><td>video-based analysis (250 Hz, 480x480 pixels of camera resolution)</td><td>5</td><td>13</td></tr><tr><td>Reinhardt and Blickhan (2014)</td><td>thorax/head flexion</td><td>Formica polyctena</td><td>video-based analysis (500 Hz, 768x512 pixels of camera resolution)</td><td>10</td><td>13</td></tr><tr><td>Weihmann and Blickhan (2009)</td><td>thorax/abdomen flexion</td><td>Cataglyphis fortis</td><td>video-based analysis (250 Hz, 480x480 pixels of camera resolution)</td><td>10</td><td>14</td></tr><tr><td>Weihmann and Blickhan (2009)</td><td>thorax/abdomen flexion</td><td>Formica pratensis</td><td>video-based analysis (250 Hz, 480x480 pixels of camera resolution)</td><td>14</td><td>14</td></tr><tr><td>Reinhardt and Blickhan (2014)</td><td>thorax/abdomen flexion</td><td>Formica polyctena</td><td>video-based analysis (500 Hz, 768x512 pixels of camera resolution)</td><td>10</td><td>14</td></tr><tr><td>Guo et al. (2018)</td><td>thorax/cx flexion angle on middle left leg</td><td>Cataglyphis fortis</td><td>video-based analysis (500 Hz, 480x480 pixels of camera resolution)</td><td>63</td><td>39</td></tr><tr><td>Guo et al. (2018)</td><td>cx/fe flexion angle on middle left leg</td><td>Cataglyphis fortis</td><td>video-based analysis (500 Hz, 480x480 pixels of camera resolution)</td><td>37</td><td>27</td></tr><tr><td>Guo et al. (2018)</td><td>fe/tb flexion angle on middle left leg</td><td>Cataglyphis fortis</td><td>video-based analysis (500 Hz, 480x480 pixels of camera resolution)</td><td>83</td><td>38</td></tr></tbody></table>

319 segments are easier to track, plus the perturbation of the measurement process has lower impact than in the case of short segments. The fact of having no significant differences in sensitivity between the joints of the legs on the right side of the body compared to those on the left side can be associated to the symmetry of the video acquisition system regarding the walkway.

323 This study presents several limitations, however. From an experimental point of view, the following aspects can be improved. Each body segment was tracked by only two markers. The number of tracked markers per segments could be increased to improve the quality of the simulation. Additionally, emerging automatic tracking techniques (i.e. deep learning powered motion-tracking) must be explored as an alternative to reduce tracking time and to increase the number of tracked points per segment. Finally, the
328 four tarsal segments were all lumped into a unique rigid segment. This was due to the configuration and the 329 capacity of the experimental setup (camera resolution, number of cameras, camera position, etc.) which 330 did not provide enough resolution to track the tarsals segments individually. On the other hand, from a 331 modeling point of view, the segments of the ant were considered as rigid bodies because of the complexity 332 of taking body deformation into consideration. This assumption merits a profound analysis in order to 333 determine the effects of segment compliance in insect locomotion, which seems to play an important role 334 (Blickhan et al., 2021).

335 Finally, future work is required to develop a dynamic model of the ant gait. This requires determining 336 muscle parameters (geometrical and force-generating parameters), segment mass and inertia properties, as 337 well as ground reaction forces. This study contributes to the construction of a musculoskeletal model of 338 ants which can be useful in the study of evolution, neural control, and bio-mimetic applications.

FUNDING

339 This work was partially subsidized by the CNRS AO MITI Biomim founding. Work by JD was supported 340 by a scholarship from the *Collectivité Territoriale de Martinique*.

ACKNOWLEDGEMENTS

341 The authors would like to thank Pr. Evan Okonomo and Dr. Francisco Hita Garcia (Biodiversity and 342 Biocomplexity Unit, Okinawa Institute of Science and Technology, Japan) who performed the scans that 343 enabled to determine segment and joint geometries. They would also like to acknowledge Moran Le 344 Gleau's contribution to the treatment of the μ-CT under the supervision of Adam Khalife, as well as Tanguy 345 Puluhen for his help during the development of the multibody model. Thanks also to the CNRS GDR 2088 346 Biomim who encouraged this work.

CONFLICT OF INTEREST STATEMENT

347 The authors declare that the research was conducted in the absence of any commercial or financial 348 relationship that could be construed as a potential conflict of interest.

SUPPLEMENTAL DATA

349 The ant model and the simulation framework are freely available at the SimTK repository: https: 350 //simtk.org/projects/barbarus

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