A Stochastic Hybrid System Model of Collective Transport in the Desert Ant Aphaenogaster cockerelli
GANESH P KUMAR1, AURรLIE BUFFIN2, THEODORE P PAVLIC2,
STEPHEN C PRATT2, SPRING M BERMAN1
1FULTON SCHOOLS OF ENGINEERING / 2SCHOOL OF LIFE SCIENCES
ARIZONA STATE UNIVERSITY
Motivation for Engineers
Developing robust strategies for Multi-Robot Collective Transport No prior information about load or obstacles Applications: Construction, Search & Rescue, Manufacturing Swarms in nature inspire swarm robot control strategies
Khepera III Robots (K-Team) Search & Rescuehttp://tiny.cc/pf4yuw
)
Constructionhttp://tiny.cc/204yuw
Motivation for Biologists
Understanding collective transport in certain ant species
Aphaenogaster cockerellicarrying lexan structure
Prior Work
Collective Transport in Ants
Berman et.al. Proc. IEEE, Sep 2011
Czaczkes and Ratnieks, Myrmecol. News, 2013
Polynomial Stochastic Hybrid Systems (pSHS)
Hespanha and Singh, Intl. J. Robust Nonlinear Control, Oct 2005
pSHS Models of Multi-Robot Systems
Mather and Hsieh, Proc. RSS, June 2011
Napp et.al, Proc. RSS, June 2009
Experiments: Ants Transporting Load
17 Video-recorded trials of ants carrying foam-mounted dime Segments spanning 145s extracted from each video Ant positions and load trajectory tracked using ImageJ and Mtrack plugin
Observations
Load trajectory was typically almost straight Random switches among 3 states: Front, Back,
Detached Ants lift load with force ๐น๐ฟโ2.65 mN, measured
with load cellBack
Detached
Front
Polynomial Stochastic Hybrid System Model
Front
Back
State vector ๐ฑ = ๐๐น ๐๐ต ๐๐ท ๐ฅ๐ฟ ๐ฃ๐ฟ๐
Behavioural states: S = ๐น, ๐ต, ๐ท Population counts: ๐๐โ๐ Dynamical variables: ๐ฅ๐ฟ , ๐ฃ๐ฟ
Flow equation d๐ฑ/d๐ก = 0 0 0 ๐ฃ๐ฟ ๐๐ฟ๐
6 Transitions: ๐๐ โ ๐๐, with rate ๐๐๐ Transition intensity: ๐๐๐ = ๐๐๐๐๐ Reset map: ๐๐ , ๐๐ โฆ (๐๐ โ 1,๐๐ + 1)
Detached
F
BD
๐๐ท๐ต , ๐๐ต๐ท
๐๐ท๐น , ๐๐น๐ท ๐๐น๐ต , ๐๐ต๐น
Back Front
๐ฃ๐ฟ
Detached
โฆ ๐ฅ๐ฟ
pSHS : Load Dynamics
Front and back ants lift with net force: ๐น๐ข๐ = ๐๐น +๐๐ต ๐น๐ฟ
Normal force: ๐น๐ = ๐๐ฟ๐ โ ๐น๐ข๐ Front ants pull with velocity regulation
Proportional gain: ๐พ Velocity set point: ๐ฃ๐ฟ
๐
Individual pulling force: ๐น๐ = ๐พ(๐ฃ๐ฟ๐ โ ๐ฃ๐ฟ)
LOAD
๐๐ฟ๐
๐๐น๐น๐๐๐น๐
๐น๐๐๐๐ก
๐น๐ข๐ + ๐น๐
๐ต๐๐๐
๐ฅ๐ฟ = ๐ฃ๐ฟ๐๐ฟ ๐ฃ๐ฟ = ๐๐น๐น๐ โ ๐๐น๐
Moment Dynamics
Moments computed using extended generator ๐ฟ
Key property allows moment computation for differentiable ๐:๐
๐๐ก๐ธ ๐ . = ๐ธ ๐ฟ๐
Time evolution of expectations๐๐ธ ๐๐๐๐ก
=
๐,๐โ๐,๐โ ๐
(๐๐๐๐ธ ๐๐ โ ๐๐๐๐ธ ๐๐ )
๐๐ธ(๐ฅ๐ฟ)
๐๐ก= ๐ธ ๐ฃ๐ฟ
๐๐ธ ๐ฃ๐ฟ๐๐ก
= ๐๐ + ๐๐น๐ธ ๐๐น + ๐๐ต๐ธ ๐๐ต + ๐๐น๐ฃ๐ธ ๐๐น ๐ธ(๐ฃ๐ฟ)
Note: ๐ธ ๐๐น ๐ฃ๐ฟ โ ๐ธ ๐๐น ๐ธ ๐ฃ๐ฟ
For our pSHS, ๐ฟ is defined as:
๐ฟ๐(๐ฑ) โ๐๐
๐๐ฅ๐ฟ ๐ฅ๐ฟ +
๐๐
๐๐ฃ๐ฟ ๐ฃ๐ฟ
+
๐,๐โ๐,๐โ ๐
๐ ๐๐๐ ๐ฑ โ ๐ ๐ฑ ๐๐๐๐๐
Fitting Model Parameters
Rates, units of ๐ฌโ๐
๐๐ท๐ต = 0.0197, ๐๐ต๐ท= 0.0205
๐๐ท๐น = 0, ๐๐น๐ท = 0
๐๐น๐ต = 0.0301, ๐๐ต๐น= 0.0184
Proportional gain
๐พ = 0.0035 N โ cmโ1โsโ1
Velocity set point
๐ฃ๐ฟ๐ = 0.3185 cmโsโ1
๐๐ง๐ญ ๐ฉ๐ฎ๐ฅ๐ฅ๐ข๐ง๐ ๐๐จ๐ซ๐๐
๐น๐ = ๐พ ๐ฃ๐ฟ๐ โ ๐ฃ๐ฟ
Model Predictions vs. Averaged Data
Model Validation with Individual Trials
Summary
We
Conducted experiments of ants transporting a load
Devised a pSHS Model of Collective Transport
Fit the model parameters to empirical data
Future Work
Further validate the model, by Varying the load mass and coefficient of friction
Fitting second and higher-order moments to data statistics
Compare ant transport with optimal strategies Criteria: minimize load path variance, transit time, team size
Extend the model, by incorporating Heterogeneity in ants
State-dependent transition rates
Two-dimensional load transport
Acknowledgements
ONR, Wallonie-Bruxelles International: for funding
Jessica Ebie, Ti Ericksson, Kevin Haight (ASU): for ant collection and care
Denise Wong, Vijay Kumar (UPenn): for measurement of ant forces
Sean Wilson (ASU): for valuable feedback on paper and presentation