Performed the experiments: TL KS. Analyzed the data: TL HN FK RR. Contributed reagents/materials/analysis tools: RR. Wrote the paper: TL.
The authors have declared that no competing interests exist.
The honeybee dance “language” is one of the most popular examples of information transfer in the animal world. Today, more than 60 years after its discovery it still remains unknown how follower bees decode the information contained in the dance. In order to build a robotic honeybee that allows a deeper investigation of the communication process we have recorded hundreds of videos of waggle dances. In this paper we analyze the statistics of visually captured highprecision dance trajectories of European honeybees (
After returning from a valuable food source honeybee foragers move vigorously, in a highly stereotypical pattern, on the comb surface. Intriguingly, an increasing number of nestmates can be observed visiting the feeding site only a few minutes past that event (
A tailwagging forager moves on the vertical comb in an approximate figure of eight. In the central part  the so called waggle run  it throws its body from side to side in a pendulum like motion at a frequency of about 13 Hz. Throughout that run the dancer holds tight to the comb moving forward in a rather straight line. Each wagglephase is followed by a returnphase, in which the dancer circles back to the approximate starting point of the previous waggle run, alternatingly performed clockwise and counterclockwise. Interestingly, dance parameters reflect feeding site properties. In the waggle phase, the body angle with respect to gravity approximates the direction to the feeder relative to the sun's azimuth. The length and duration of the waggle run correlate highly with the distance to the target location (
The bees actively pursuing the dancer, commonly called follower bees, are most likely to be recruited after joining several dance periods. In that process they tend to remain in close contact with the dancer and detect a variety of stimuli. Antennal contacts with the body of the dancer are frequently observable and likely transmit information about the dancer's body orientation (
Moreover, the dance communication involves highly complex cognitive tasks. A dancer bee extracts and translates field site properties from the memory formed on her foraging trips  the dance is no instantaneous or spontaneous response nor a signal for an immediate action. Follower bees “read” the dance, translate their sensory input into the remote target location and find the feeder even with large detours around obstacles like high buildings or hills. The bee dance community has gathered an amazing amount of knowledge on navigation, memory and communication in honeybees (
To tackle a part of these questions we are developing a robotic honeybee that can emulate all known stimuli (
In order to mimic the dance as realistic as possible we captured the dance motion via an automatic tracking program from video recordings of natural dances. This paper covers the analysis of the resulting trajectory data, i.e. body position and orientation over time and targets two goals. First, we would like to understand the general nature of the dance motion, i.e. what are frame properties like maximum velocities, angular precision and the size of the dance area. These global parameters define the target properties of the robot's mechanics and actuators. Secondly, we would like to learn from the observations which motion features are invariant throughout a large set of dances and are therefor likely to be of significance in the communication process. These key motion features are then implemented in the honeybee robot in order to generate realistic waggle dances.
We present a comprehensive list of dance characteristics, discuss interesting aspects of natural dances and show the trajectories generated by the final robotic dance model. This analysis gives a thorough description of the variability of the body motion of waggle dancers and thus might serve as the base of future analyses of the dance. The video recordings, the trajectory data, the matlab code for the model and all results of this analysis will be made available to the scientific community through a video and metadata management system on
Video recordings (100 fps, VGA resolution) of waggle dances of European Honeybees (
The position of the dancer can be marked with a rectangular box by the user for a starting frame. The position of its center and the angle of the box are found and stored for consecutive frames automatically.
Any tracked dance motion might be divided into a (hidden) dance model component and secondary, nondance behaviors (e.g. evasive maneuvers, trophallaxis). We assume that the observed dance motion is a mixture of the dance motion and other motion that is superimposed or interspersed depending on external and internal stimuli. Thus, for the robotic dance it makes sense to make use of a model rather than playing back previously tracked dances. By analyzing many dance trajectories we expect that the influence of nondance behaviour will cancel out statistically. To assess a minimal set of parameters with which to model a realistic dance we first made a list of numerous candidate properties of the trajectory data. We then discarded parameters showing high variance. Some of the remaining parameters were redundant. The parameter with lower variance was selected for the dance model. The trajectory data allows a vast variety of statistical queries. To retain order, we divide the parameters in sections “Global Dance Parameters”, “Waggle Parameters”, “Return Parameters” and “IntraWaggle Parameters”.
Parameter  Unit/Ann. 



Global Parameters  
dance duration  s  5.24  0.85  0.16 
dance radius  mm    
x    7.6    
y    7.1    
wagglereturn    0.22  0.10  0.44 
duration ratio  
Waggle Parameters  
duration  s  0.44  0.16  0.36 
waggle length  mm  6.32  2.36  0.37 
waggle orientation 

−0.03  28.06   
waggle direction 

1.24  24.85   
waggle drift  mm  
forward  0.27  4.57    
sideward  −0.01  4.62    
divergence A 


orientation  31.98      
direction  22.11      
divergence B 


orientation  33.12  19  0.57  
direction  22.76  29.86  1.31  
Return Parameters  
return duration  s  2.13  0.47  0.22 
return velocity  mm s 
20.10  4.27  0.21 
IntraWaggle Parameters  
orientation 

13.89  8.33  0.60 
amplitude  
displacement  mm  2.64  1  0.38 
amplitude  
waggle frequency  Hz  12.67  1.89  0.15 
waggle velocity  mm s 
15.04  5.00  0.33 
waggle steps  mm  
forward  1  0.9    
sideward  0.04  0.9   
This section encloses parameters specific to a dance period, i.e. the sequence wagglereturnwagglereturn (see
Top: Plot of planar positions of the dancer bee's center. Two consecutive waggle runs are depicted. Return runs are dashdotted. Start and end are marked with an asterisk and plus sign, respectively. On the left waggle run we show the linear least squares fit (dashed line) of the smoothed waggle (light grey solid curve) and the mean orientation line (solid line). In the right waggle run we marked the left and right turning points of the waggle oscillation with a diamond and square sign, respectively. Bottom: Orientation of the bounding box over time. Return run orientation is dashdotted.
The waggle run class is specific to parameters describing a waggle run or relations between waggle runs. These are:
In this section parameters describing return run properties are clustered:
All parameters that describe intrinsic properties of waggle runs are specified in this class: The
A number of parameters were previously identified to be highly correlated to particular properties of the feeding location.
E.g. the angles
Besides the properties known from the literature we describe the
The program MATLAB® is used for the whole analysis. We manually reviewed all automatically obtained results. First we run a script that identifies the waggle runs to obtain the separation needed for the classspecific analysis. Therefor we Fourier transform the onedimensional derivatives of the x and y coordinates using a sliding window of 0.2 seconds width. A Fourier transform decomposes a signal into its constituent frequencies. If we find a spectral activity of the body motion at 12 Hz higher than a threshold the respective window was selected to contain tailwagging. The resulting binary data was postprocessed using dilution and erosion operations, known from image processing, to close gaps in some waggle runs. A few angular measures in the waggle phase were computed using two different methods. The “orientation method” is the angular mean of all stored orientations of the bounding box throughout the waggle phase. The “direction method” is the angle of the linear least squares fit of the 2D trajectory of a waggle run. The x,y positions were smoothed before the fit. This was necessary for very short waggle phases because these were “shorter than wide”, i.e. the fit would express the lateral motion rather than the forward motion. Smoothing the trajectory helps to extract the direction of the waggle, but might be a source of additional error and will be discussed later. Some authors have used one or the other method to measure the waggle angle (
To compare the variability of parameters of different scales and units we compute the coefficient of variation (CoV) which is defined as the standard deviation divided by the mean. The twodimensional test of difference of mean is performed using Hotelling's Tsquare statistic. Difference of means of onedimensional data is done with a Student's ttest. Test for uniformity of circular data is performed with Rao's spacing test.
The CoV of the
The
The
The average velocities are drawn dashdotted, the solid line depicts the polynomial fit. The top plot shows the forward (top) and sideward (bottom) velocities, the second plot depicts the angular velocities.
The
The former corresponds to wagging on the spot, the latter reflects the forward motion through one waggle movement.
We include the following parameters in the dance model:
The solid line shows the trajectory of the center of the robot's body. The dashdotted line depicts the curve of the 2D plotter that carries the motor used for the rotation and the lateral wagging motion. Bottom: Body orientation over time.
We propose dance parameters for the construction of a waggle dance model which produces trajectories closely resembling real ones. In this study we limited our analysis to obtain the parameters for the description of honeybee waggle dances advertising a fixed distance of 230 m. Future work will include recording, tracking and analyzing dances for a discrete set of distances.
The proposed low variance parameters indicate the significance of particular body pose or motion properties in the communication process. The way how the dancer's body moves and how this motion modulates other dancerelated stimuli can, however, plausibly be allocated to and modeled with these particular properties. Furthermore, our evaluation allows some interesting inferences.
The mean angle of all positions throughout the dance (the
We found that the two angular measures yield a significant difference in the divergence angle. This angle is either assessed by comparing the means of left and right waggle runs (method A). Or it is measured “sequentially”, i.e. only consecutive waggle runs are used to collect the angular differences which, in the end, are averaged (method B). Both divergences differ with respect to the angular measure used. The orientation measure (based on the average body orientation) always yields larger divergences than the direction measure (that refers to the direction of the waggle path). This might be explained by the transitions of the waggle to return run or vice versa. While waggling, the dancer bee often turns her body into the return run's direction but keeps the body's trajectory straight. To this might add that at the beginning of the waggle the dancer bee might have turned not entirely into the right angle. To prove these assumptions we recalculated the divergences discarding the first and last 10%, 20% and 30% of the waggle run. Leaving out the first portions, the difference of the two divergences gets even larger, entirely on the account of the direction measure. By discarding the end of the waggle the difference of the two angular measures drops to 4
Looking at real dance trajectories, the figure eight that is used commonly to describe the dance shape is observed rather infrequently. This is on the one hand due to the followers vigorous physical efforts to keep close body contact to the dancer and, by doing so, being obstacles in her way. On the other hand the dance floor is usually no free space and also the followers can not move freely. If the dancer's path is occupied she continues the turn on the spot or executes evasive maneuvers. Yet very effectively, a group of 2–3 followers usually clears the area for the dancer's subsequent waggle run with every turn they follow. We superimposed waggle periods (wagglereturnwagglereturn) and created a two dimensional histogram of the body coordinates (
Dark regions denote a high frequency of presence. The trajectories were centered periodwise, i.e. we subtracted the mean from each of the WRWRpaths.
Applying Peirce's definition, communication codes can be of three kinds: iconic, indexical and symbolic (
So far, the robotic dance motion alone has not yet recruited followers to an unknown feeder. In a yet unpublished experiment, adding wing beats to the pure dance motion increases the rate with which foragers visit a known but depleted, unscented feeding site. Until now, we haven't yet been able to prove that the robot is communicating the direction to a feeder. This is closely related to the fact that followers have to sample a high number of waggle runs in order to gain a feasible estimate of the direction (
This document contains calculations of different divergence measures, the Matlab code for the dance model and figures showing the distributions of most of the dance properties.
(PDF)
We are indebted to the reviewers and the editor for numerous valuable comments. The first author also likes to thanks Dr. Rodrigo J. De Marco for discussing an early version of this work.