A Stochastic Hybrid System Model of Collective Transport in the Desert Ant Aphaenogaster cockerelli

GANESH P KUMAR1, AURÉLIE 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 𝐹𝐿≈2.65 mN, measured

with load cellBack

Detached

Front

Polynomial Stochastic Hybrid System Model

Front

Back

State vector 𝐱 = 𝑁𝐹 𝑁𝐵 𝑁𝐷 𝑥𝐿 𝑣𝐿𝑇

Behavioural states: S = 𝐹, 𝐵, 𝐷 Population counts: 𝑁𝑖∈𝑆 Dynamical variables: 𝑥𝐿 , 𝑣𝐿

Flow equation d𝐱/d𝑡 = 0 0 0 𝑣𝐿 𝑎𝐿𝑇

6 Transitions: 𝑋𝑖 → 𝑋𝑗, with rate 𝑟𝑖𝑗 Transition intensity: 𝜆𝑖𝑗 = 𝑟𝑖𝑗𝑁𝑖 Reset map: 𝑁𝑖 , 𝑁𝑗 ↦ (𝑁𝑖 − 1,𝑁𝑗 + 1)

Detached

F

BD

𝑟𝐷𝐵 , 𝑟𝐵𝐷

𝑟𝐷𝐹 , 𝑟𝐹𝐷 𝑟𝐹𝐵 , 𝑟𝐵𝐹

Back Front

𝑣𝐿

Detached

↦ 𝑥𝐿

pSHS : Load Dynamics

Front and back ants lift with net force: 𝐹𝑢𝑝 = 𝑁𝐹 +𝑁𝐵 𝐹𝐿

Normal force: 𝐹𝑛 = 𝑚𝐿𝑔 − 𝐹𝑢𝑝 Front ants pull with velocity regulation

Proportional gain: 𝐾 Velocity set point: 𝑣𝐿

𝑑

Individual pulling force: 𝐹𝑝 = 𝐾(𝑣𝐿𝑑 − 𝑣𝐿)

LOAD

𝑚𝐿𝑔

𝑁𝐹𝐹𝑝𝜇𝐹𝑛

𝐹𝑟𝑜𝑛𝑡

𝐹𝑢𝑝 + 𝐹𝑛

𝐵𝑎𝑐𝑘

𝑥𝐿 = 𝑣𝐿𝑚𝐿 𝑣𝐿 = 𝑁𝐹𝐹𝑝 − 𝜇𝐹𝑛

Moment Dynamics

Moments computed using extended generator 𝐿

Key property allows moment computation for differentiable 𝜓:𝑑

𝑑𝑡𝐸 𝜓 . = 𝐸 𝐿𝜓

Time evolution of expectations𝑑𝐸 𝑁𝑖𝑑𝑡

=

𝑖,𝑗∈𝑆,𝑖≠𝑗

(𝑟𝑖𝑗𝐸 𝑁𝑖 − 𝑟𝑗𝑖𝐸 𝑁𝑗 )

𝑑𝐸(𝑥𝐿)

𝑑𝑡= 𝐸 𝑣𝐿

𝑑𝐸 𝑣𝐿𝑑𝑡

= 𝑐𝑔 + 𝑐𝐹𝐸 𝑁𝐹 + 𝑐𝐵𝐸 𝑁𝐵 + 𝑐𝐹𝑣𝐸 𝑁𝐹 𝐸(𝑣𝐿)

Note: 𝐸 𝑁𝐹 𝑣𝐿 ≈ 𝐸 𝑁𝐹 𝐸 𝑣𝐿

For our pSHS, 𝐿 is defined as:

𝐿𝜓(𝐱) ≔𝜕𝜓

𝜕𝑥𝐿 𝑥𝐿 +

𝜕𝜓

𝜕𝑣𝐿 𝑣𝐿

+

𝑖,𝑗∈𝑆,𝑖≠𝑗

𝜓 𝜙𝑖𝑗 𝐱 − 𝜓 𝐱 𝑟𝑖𝑗𝑁𝑖

Fitting Model Parameters

Rates, units of 𝐬−𝟏

𝑟𝐷𝐵 = 0.0197, 𝑟𝐵𝐷= 0.0205

𝑟𝐷𝐹 = 0, 𝑟𝐹𝐷 = 0

𝑟𝐹𝐵 = 0.0301, 𝑟𝐵𝐹= 0.0184

Proportional gain

𝐾 = 0.0035 N ∙ cm−1∙s−1

Velocity set point

𝑣𝐿𝑑 = 0.3185 cm∙s−1

𝐀𝐧𝐭 𝐩𝐮𝐥𝐥𝐢𝐧𝐠 𝐟𝐨𝐫𝐜𝐞

𝐹𝑝 = 𝐾 𝑣𝐿𝑑 − 𝑣𝐿

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