Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 32–40___________________________________________________________________________________________________________

An Estimation Method for Environmental Friction Based on Body Dynamic Model of
Caenorhabditis elegans
Zu SohDepartment of System Cybernetics, Hiroshima University, 1-4-1 Kagamiyama
Higashi-Hiroshima, Hiroshima 739-8527, Japan
Michiyo Suzuki

Department of Radiation-Applied Biology, National Institutes for Quantum and Radiological Science and Technology,
1233 Watanuki, Takasaki, Gunma 370-1292, Japan
Toshio Tsuji

Department of System Cybernetics, Hiroshima University, 1-4-1 Kagamiyama
Higashi-Hiroshima, Hiroshima 739-8527, Japan
E-mail: sozu@bsys.hiroshima-u.ac.jp

Abstract
Caenorhabditis elegans is a small worm which is approximately 1 mm in length. The present study proposes anestimation method for frictional force using locomotion information obtained from video analysis of actual worms.
The results indicate that the body model driven by the estimated frictional force can trace the locomotion of theworm within a low error level of 4% of the body length. The proposed method can be applied to analyze therelationship between friction and gait control.
Keywords: Caenorhabditis elegans, frictional force, dynamics model, video analysis, locomotion

become a favored model organism for the analysis ofneuronalmechanisms
underlying
information
processing, especially since White et al. revealed aconnective structure between its neurons and muscle
cells.1
In this context, the mechanism of gait control is aninteresting target among various life phenomena
because commands generated by neural networks can beeasily observed through locomotion. Accordingly, themechanisms of information processing in C. elegans can

1. Introduction
Caenorhabditis elegans is a small soil-dwelling
worm with a slender translucent body composed ofaround 1,000 cells. It is approximately 1 mm in lengthand about 5.0 µg in weight (Figure 1(a)). Its neuralnetwork, which is composed of only 302 neurons,
allows the processing of environmental informationsuch as temperature and chemical gradients so that theworm can act in response to its surroundings. Thanks tothe small scale of its neural network, C. elegans has

Copyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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environmental conditions is therefore needed to support
estimation of the mechanical stimuli affecting on C.
elegans.
In this paper, the authors propose a body dynamics
model incorporating the considerations of dynamic andviscous friction. An estimation method for friction
coefficients based on the model is then proposed. As
this method requires only locomotion information
obtained from video analysis of worms, it enables us toestimate frictions regardless of the environment and theworm's behavior. Chapter 2 describes the body
dynamics model of C. elegans and the estimation
method for environmental frictions. Chapter 3 covers
verification of the proposed algorithm and reports on the
results of environmental friction estimation. Chapter 4
discusses the relationship between translational force
and friction force, and Chapter 5 details the conclusion
and outlines future works.

Fig 1. Body dynamics model of C.elegans
be investigated by recording its behavior and examining
the internal states of the neural network. Even better,recently, dopamine was identified as a key chemical2intermediating the distinctive gait changes observed
when the worm swims in water and crawls on agar3,4. It
was also revealed that dopamine-related changes are
induced by mechanical stimuli2. However, for thereason of both experimental conditions andtechnological limits, it has been difficult to measure theamount of forces between the body and environments
without considering of effects of its behavior. As aresult, the input for its neural network system remains
unclear.
Computer simulation provides a potential solution tothis problem because the worm's thrust is generated byreaction forces based on the frictions between its bodyand environments in the absence of other external
forces, and can be calculated using motion equations.
The body dynamics model of C. elegans, which at firsthas been proposed by Niebur and Erdos5, was able tosimulate the worm's undulatory locomotion so that amodel worm could travel as fast as a real one. As
friction cannot be measured from an actual worm, thefriction coefficients included in the model were
estimated from the friction measured between the glass
fiber and agar6. Recently, more precise frictioncoefficients in the gelatin solution and water were
calculated by Berri et al. and Sznitman et al. based on
the principles of fluid dynamics7~9. However, previous
methods have limitations in terms of the range ofenvironments in which they can be applied to. An
algorithm to determine friction independent of

2. Materials and Methods
When C. elegans moves on a solid surface such as
an agar plate or swims in liquid, it is exposed todifferent environmental drag forces. As these forces are
the basis of propulsion, different drag characteristics orstrengths produce different propulsive movements. Dragforce can therefore be derived by observing the motionof worms to solve dynamic problems. The following
sections describe the materials and methods used torecord worm motions, and highlights the algorithms
used to estimate drag force.
2.1. Strains and culture
The C. elegans wild-type Bristol N210 and the
Escherichia coli OP50 strain were obtained from the
Caenorhabditis Genetics Center. Using standard
methods10 worms were grown at 20°C on 6-cm nontreated petri plates (IWAKI 60 mm/ Non Treated Dish,
AGC Techno Glass Co., Ltd., Shizuoka, Japan)
containing 10 ml of nematode growth medium (NGM)
agar spread with E. coli (food). Well-fed worm at the
young adult stage were used in all assays.
2.2. Sample preparation
A 6-cm plates containing 4 ml of NGM agar forassay were prepared on the same day of experiments.
The plate for the crawling assay (plate A) was
uncovered and placed on a clean bench for33

Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 32–40___________________________________________________________________________________________________________𝐼𝑔 (𝒒)𝒓̈ 𝑔 + ℎ𝑔 (𝒒, 𝒒̇ ) + 𝑔𝑔 (𝒒) = � 𝐹𝑗 ,
𝑗
𝑁−1

(2)

where 𝒒 = (𝑞1 , 𝑞2 , … 𝑞𝑁−1 )T ∈ ℝ
is a vector of
local bending angles, 𝒓 = (𝑥, 𝑦)T and 𝑞0 denotes the tip
position and its posture angle for the head link related to
the global coordinate as shown in Figure 1. The terms
𝐼(𝒒) , 𝐼𝑔 (𝒒) on the left-hand side are inertial forces,ℎ(𝒒, 𝒒̇ ) , ℎ𝑔 (𝒒, 𝒒̇ ) are Coriolis forces and centrifugal
forces, and 𝑔(𝒒) , 𝑔𝑔 (𝒒) are gravity forces. The firstterm 𝝉 ∈ ℝ𝑁−1 on the right-hand side is the torquedriving each link, and the second term represents
friction, where 𝑱𝑇𝑗 is the Jacobean matrix of the j-thTjoint with a representative point. 𝑭𝑗 = �𝑓𝑗,𝑡 , 𝑓𝑗,𝑛 �
represents the drag force acting on the gravity point ofeach joint, and can be described in arbitrary form
depending on the surrounding environment. In thisstudy, drag force was modeled using the properties of
Coulomb friction and viscous drag in a Newtonian fluid
as follows:
𝑣𝑗,𝑑𝑓𝑗,𝑑 = −𝜇𝑑− 𝜂𝑑 𝑣𝑗,𝑑 ,(3)
�𝑣𝑗,𝑑 �where 𝑑 ∈ (𝑡, 𝑛) denotes the tangential and normal
directions related to the link, 𝑣𝑗,𝑑 are the velocities at thegravity point of link j, 𝜇𝑑 is the coefficient of Coulomb
friction, and 𝜂𝑑 is the viscous drag coefficient.
Equation (2) indicates that the drag coefficients andtorque 𝝉 driving each joint can be derived from local
bending angles and the related velocities andaccelerations (𝒒, 𝒒̇ , 𝒒̈ ) and translational acceleration 𝒓̈ 𝑔 ,which can be determined from video analysis. Based on
this equation, an algorithm for friction coefficientestimation is proposed below.

Fig. 2. Image processing
approximately 1h to dry up excess fluid from thesurface. The plate for the swimming assay (plate B) was
not dried up and was added 300 µl of S basal buffer11. A
worm was picked up from a culture plate and washedtwice in a few drops of S basal buffer. Immediately after
wash, a worm was transferred to an assay plate A or B.
2.3. Rigid link modelTo estimate the drag forces acting between C.
elegans and its environment, the worm's body was first
approximated using the N rigid link model shown in
Figure 1 (c), which also illustrates its coordinate
configuration. The motion of the model is restricted tothe two-dimensional x-y plane. The links l2 - lN-1 wereconnected with adjacent links via rotational joints toallow rotation around the z-axis, and the head link l1 andtail link lN had one free end and a joint respectively
connected to links l2 and lN-1. Using this model, therelationship between drag and motion can be derived
from Newton-Euler-type motion equations as shown
below.
𝑞0̈
0
𝐼(𝒒) � 𝒒̈ � + ℎ(𝒒, 𝒒̇ ) + 𝑔(𝒒) = � � − � 𝑱𝑇𝑗 𝑭𝑗 , (1)𝝉

2.4. Friction estimation algorithm
The motion equations (1) and (2) can be solved fordrag coefficients given the motion information (𝒒, 𝒒̇ , 𝒒̈ ,𝒓̈ 𝑔 ). Among these values, the local bending angle 𝒒 andthe position 𝒓𝑔 can be determined by recording the
motion of a worm and analyzing the resulting video, andthe related time derivations can be used to determine
velocity and acceleration. The information obtained canthen be used to solve the motion equations for the dragcoefficients in relation to each video. However, there
are two problems in the implementation of thiscalculation.
Firstly, it is an ill-posed problem because only N+2
coupled equations can be derived from Equations (1)𝒋34Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 32–40___________________________________________________________________________________________________________

acquisition of the x- and y-coordinates of each point on
the body, and (vii) calculation of values concerning withthe body form such as the relative angle between
adjacent dividing points. Video analyses describedabove were carried out using the following software:
Wriggle Tracker (Library Co. Ltd., Tokyo, Japan) forprocedures (i)-(iv) and Move-tr/2D (Library Co. Ltd.,Tokyo, Japan) for procedures (v)-(vii).

and (2), while there are four unknown drag coefficientsplus N-1 unknown torque values (𝝉 ∈ ℝ𝑁−1 ). To solve
this problem, the proposed algorithm involves
simultaneous analysis of two different motions of aworm placed in the same environment in which
identical drag coefficients are assumed. In this manner,
the number of unique coupled equations can be doubled
without increasing the number of unknown dragcoefficients.
Secondly, the drag coefficients calculated from only
one sampling can be significantly affected by video
analysis error. To reduce this impact, the proposed
algorithm involves the solution of a motion equation for𝒓̈ 𝑔 and 𝑞0 for the time duration T with different dragcoefficients. The coefficients that generate the paths
closest to those of the actual worm are then explored byminimizing the following evaluation function:
𝐸=

12𝑙(𝑡max + 1)
𝑗

𝑇

𝑙

2

𝑗

𝑗

� � � ||𝒙𝑖 (𝑡) − 𝒓𝑖 (𝑡)|| ,𝑡=0 𝑖=1 𝑗=1

Block 2: Selection of drag coefficient
In this block, a set of coefficients is selected in threephases as described below.
[Phase 1]
In Phase 1, drag coefficients are selected from theset (𝜇𝑛 , 𝜇𝑡 , 𝜂𝑛 , 𝜂𝑡 ) ∈ {(𝑁1 Δ𝜇, 𝑁2 Δ𝜇, 𝑁3 𝜂, 𝑁4 𝜂)|
𝑁1 , 𝑁2 , 𝑁3 , 𝑁4 = 0, 1, ⋯ , 𝑁 max } based on iterative
alteration of 𝑁1 , 𝑁2 , 𝑁3 , 𝑁4 .
After the search for all combinations is complete,
𝐸𝐸
𝐸
𝐸
d1 % of the drag coefficients �𝜇𝑛1,𝑎 , 𝜇𝑡 1,𝑎 , 𝜂𝑛1,𝑎 , 𝜂𝑡 1,𝑎 �
with smaller evaluation values (Equation (4) areselected, and the iteration proceeds to Phase 2, where𝑎 = 1, 2, … , 𝑑1 (𝑁 max + 1)4 /100.
[Phase 2]
In Phase 2, better drag coefficients are sought fromthe set selected in Phase 1. The search is carried out(4)

Where 𝒙𝑖 represents the position of the i-th evaluation𝑗point on the worm, 𝒓𝑖 represents the corresponding
position on the rigid-link body model, and subscript j
denotes an individual worm.
The proposed algorithm is applied to estimate thedrag coefficients for a worm swimming in liquid andcrawling on an agar surface. The proposed algorithm
consists of four blocks as outlined below.

𝐸

Δ𝜇

𝐸

Δ𝜇

with the set of (𝜇𝑛1,𝑎 + 𝑁1 � � , 𝜇𝑡 1,𝑎 + 𝑁1 � � ,𝐸Δ𝜂𝜂𝑡 1,𝑎 + 𝑁1 � � ,2

𝐸Δ𝜂𝜂𝑡 1,𝑎 + 𝑁1 � �)
2

2

where

2

𝑁1~4 ={−1,0,1} . After the search for all combinations iscomplete,
𝑑2 %
of
the
drag
coefficients

Block 1: Video analysis
To analyze the body form of C. elegans in crawlingand swimming, the worm on an assay plate (A for
crawling and B for swimming) was video-recorded
using a digital camera EXILIM EX-F1 (CASIO
Computer Co., Ltd., Tokyo, Japan) mounted on astereomicroscope for approximately 10 seconds at 300
frames per second. The resolution of each frame was
512 x 384 pixels. Distinct worms picked up from aculture plate were used for crawling assays andswimming assays. The body form in crawling andswimming of C. elegans was analyzed off-line based on
the previous method12 as shown in Figure 2. Briefly,
each frame of the video was processed using thefollowing procedures: (i) binarization, (ii) denoising,(iii) skeletonizing, and (iv) division of body line into the
13 parts. After image processing, time-series data
concerning with the body form were obtained using thefollowing procedures: (v) length-scale calibration, (vi)

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

�𝜇𝑛2,𝑏 , 𝜇𝑡 2,𝑏 , 𝜂𝑛2,𝑏 , 𝜂𝑡 2,𝑏 � with smaller evaluation values(Equation (4)) are selected, and the iteration proceeds to
Phase 3, where 𝑏 = 1, 2, … , 3𝑑1 𝑑2 (𝑁 max + 1)4 /10000.
[Phase 3]
In this phase, the generalized reduced gradientmethod13 is applied for the selection of drag coefficients𝐸using �𝜇𝑛2,𝑏 , 𝜇𝑡 2,𝑏 , 𝜂𝑛2,𝑏 , 𝜂𝑡 2,𝑏 � as initial values so thatthe friction coefficients with the smallest evaluation
values are calculated.
Block 3: Calculation of Motion Equations

In this block, calculation for the path of two rigidlink body models in the time span T s is performed withthe friction coefficients selected in Block 2. Calculation
to solve Equations (1) and (2) for q0, r is carried out

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 32–40___________________________________________________________________________________________________________

Table 1 Parameters used in simulation5s
Δ𝜇1.0× 10−13Δ𝜂
1.0× 10−3
3.84× 10−7 g
𝑑1
1.0 %
10
𝑑2
0.2 %

tmax
l
m
Nmax

True value
Estimated value

1.0

0.01 10

[％]

η

µ
0

µn

ηn

µt

ηt

0

0

(a) Result of F1

ηn

µn

µt

ηt

av

(b) Percentage errors

Fig. 3. Estimated results of friction coefficients

Fig. 5. Locomotion of C. elegans
The paths determined from this simulation were thenused with the proposed algorithm for friction coefficientestimation. Finally, the errors observed between thepreset and estimated friction coefficients were
calculated to verify the performance of the algorithm.
Here, time-dependent joint angles with the following
four patterns were chosen: 𝑞𝑖+1 − 𝑞𝑖 = 0.4sin(5𝑡 −0.6𝑖), 0.6 sin(5𝑡 − 0.6𝑖) , 0.4 sin(5𝑡 − 0.6𝑖) + 0.1,
0.6 sin(5𝑡 − 0.6𝑖) + 0.1 which are respectively denoted
as Locomotion 1~4. In addition, 10 sets of friction
coefficients (F1~F10) were chosen as uniform random
numbers in the range of [0,1] for dynamic friction
coefficients and [0, 0.01] for viscous friction
coefficients. The locomotion pair of 1 and 2 wassimulated with the friction coefficients of F1~F5 and theother pair (Locomotion 3 and 4) was simulated with thefriction coefficients of F5~F10. The parameters of the
body dynamics model and the proposed estimation
algorithm are shown in Table 1.
Figure 3 shows the friction coefficient estimation
results and the related percentage errors against thepreset values. Figure 3(b) shows that the average
percentage errors were less than 2% with a standard
deviation of 2%. Figure 4 shows an example result forthe paths of Joint 7 as well as average path errors for all
simulations performed. It can be seen that the average
error was 1.9 × 104 ± 1.1 × 10−4 mm, which indicatesthat the proposed algorithm is capable of generating

Fig. 4. Estimated paths
using ADAMS (MSC Software Corporation, Tokyo,
Japan) multi-body dynamics simulator.
Block 4: Evaluation of Drag Coefficients
In Block 4, evaluation is performed for dragcoefficients by substituting the paths calculated in Block3 into an evaluation function (Equation (4)), and theiteration returns to Block 2.
3. Simulation Experiments
As a way of verifying the performance of the
proposed body dynamics model and the estimation
algorithm for friction coefficients, testing was
performed to determine whether the algorithm enabled
estimation of artificially preset friction coefficients.
Such coefficients were then estimated for worms
crawling on agar and swimming in a drop of water.
3.1. Verification
The locomotion of the body model was simulatedusing Equations (1) and (2) with artificially preset
friction coefficients and time-dependent joint angles.

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 32–40___________________________________________________________________________________________________________

analysis times for crawling and swimming wererespectively set as tmax=4 s, 1 s, and the other parameters
were the same as those used in the verification
simulation.
Figures 6 and 7 show the values of evaluation
function (Equation (4)) acquired from swimming andcrawling simulation, respectively. Here, the evaluation
function represents the errors between the paths of themodels and that of the actual worms. The average patherror of crawling simulation was about 0.02 mm (2 % of
the worm's length), and the average path error ofswimming simulation was 0.05 mm (4 % of the worm’s
length). Figure 8 shows the paths of seventh joint incrawling and swimming simulation. These outcomesindicate that each of the simulated path fitted worm
paths well. Figures 9 and 10 show friction coefficients
estimated from eight video images for crawling andeight for swimming, with the numbers under each
picture distinguishing individual worms. It can be seenthat the estimated friction coefficients differed greatly
even for the same environments. In particular, in Figure9(A1) and (A3) representing situations in which friction
coefficients were estimated from the same worm
crawling on the same agar plate, the viscous friction
coefficient for the normal direction shown in (A1) wasmuch larger than that shown in (A3). The next section
discusses this friction coefficient variability.

Fig. 6. Resulted value of evaluation function of
crawling simulation

Fig. 7. Resulted value of evaluation function ofswimming simulationStarty [mm]

0

worm

End
0

End
Start
x [mm](a) Paths of crawling

2.0

2.0

End

y [mm]

model

1.5

Start
End

Start
0

0

x [mm]

2.0(b) Paths of swimming

Fig. 8. Paths of seventh joint

locomotion values with errors smaller than 2.9 ×
10−2 % in relation to the worm’s length.
3.2. Estimation of friction coefficients of actual
worms

4.

Friction coefficients for actual worms were
estimated using video of their movement on agar and ina drop of water (Figures 5). Each of the time-dependent
joint angles determined from image processing wassmoothed using a second-order Butterworth low-passfilter (high cutoff frequency: 5 Hz) for denoising. The

Discussion

This section discusses the cause of variances foundin the estimated friction coefficients A1, A2 and A3(Fig. 10) or the same worm on the same agar plate.

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 32–40___________________________________________________________________________________________________________

Fig. 10. Estimated friction coefficients (water)

Fig. 9. Estimated friction coefficients (agar)

to the propulsion forces, this result indicates that the
proposed algorithm with an evaluation function definedby path errors can be used to estimate propulsion force.
In the previous studies regarding the dynamic
modeling of C. elegans, it was reported that a worm's
path can be determined from the ratio between thenormal and tangential friction coefficient.5,7 Indeed, thefriction coefficient ratio is a dominant factor in
determining worm paths. However, according to thesimulation performed here, identical locomotion cannot
be generated if the same ratio but different friction
coefficients are adapted. Figure 13 shows locomotion
paths observed when the dynamic friction coefficientswere kept at a constant ratio (2µt = µn) and the viscous
friction coefficient was set to 0 (ηt= ηn =0). The results
demonstrate that there were path differences dependingon the value of µn, even though the dynamic friction
coefficients were kept at a constant ratio. This suggeststhat if image processing on actual worms can be carriedout with a high degree of accuracy, friction coefficients

To compare the effects of different friction
coefficients adopted for the same joint motion, the timedependent joint angles qA1(t) extracted from thelocomotion of A1 were substituted into Equations (1)and (2), and paths with the friction coefficients of A2and A3 were calculated. Figure 11 shows paths resulting
from simulation under the time-dependent joint angles
qA1(t) and the friction coefficients A2 and A3. The
errors observed between the actual and simulated
worm's paths are also shown in Figure 12(b). These
results indicate that the average path error was about 4%of the worm's length even when different friction
coefficients (A1, A2 and A3) were used. Figure 12(a)highlights the friction force at each link accumulated at
the point of the head tip. The right-hand side ofEquation (2) shows this calculation. From the figure, it
can be seen that the total friction generated from friction
coefficient A1 showed a strong correlation and similarvalues to that generated from friction coefficients A2
and A3. As the total friction reactive forces correspond38

Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 32–40___________________________________________________________________________________________________________

Fig. 13. Paths under the same friction coefficient
ratio

Fig. 11. Comparison of trajectories (agar)

Fig. 12. Comparison of friction force
can be uniquely determined because verification
simulation under artificially preset configurations as
described in Section 3.1 enabled the estimation offriction coefficients close to actual values with anacceptable level of accuracy.
Finally, the effect of friction in different
environments was quantified. Figure 14 shows theaverage RMS value of the total friction forces ascalculated from the crawling and swimming simulation
results. The result of a t-test to determine the RMS
between crawling and swimming showed a significant
difference between the two (p<0.001). Fig. 15 indicatesthat worms experience stronger friction forces whenswimming than when crawling. Using results obtained
from the proposed estimation algorithm may allow toclarify the underlying mechanisms of gait control8 at thefriction level.

Fig. 14. Average RMS
5. Conclusion and Future Works
This paper proposed a method for estimating friction
coefficients based on a body dynamics model and video
images of C. elegans. The study verified that the
proposed algorithm can be used to estimate friction
coefficients under artificially preset configurations, andthe results of experiments using actual worms
confirmed that the model can track worm paths withsmall errors. The algorithm can be used to estimate true
friction coefficient values under ideal noise-free
conditions. Although estimation results were affected bynoise contamination from image processing, thepropulsion forces affecting locomotion paths could be
estimated.

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In future works, the authors plan to improve therobustness of the proposed method to support thecalculation of time-dependent friction changes acting on
worms, and will analyze the underlying mechanisms ofgait control against environmental factors.

5.
6.

Acknowledgements

7.

The authors appreciate to Dr. Y. Hattori and Mr. S.
Nishikawa for valuable discussions. This work was
supported by JSPS KAKENHI Grant Numbers
15H03950 and 20115010.

8.

9.

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