stochastic hybrid system model for collective transport in desert ant a.cockerelli
TRANSCRIPT
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