This document summarizes research on modeling collective transport behavior in desert ants (Aphaenogaster cockerelli). Experiments were conducted observing ants transporting a load and their positions/load trajectory were tracked. A polynomial stochastic hybrid system (pSHS) model was developed with behavioral states of front, back, detached and transitions between them. The model's parameters were fitted to the experimental data and shown to match the averaged data and predict individual trials. Future work is proposed to further validate the model under different conditions and extend it to incorporate heterogeneity, state-dependent rates, and 2D transport.

1. 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

2. 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 & Rescue
http://tiny.cc/pf4yuw
)
Construction
http://tiny.cc/204yuw

3. Motivation for Biologists
Understanding collective transport in certain ant species
Aphaenogaster cockerelli
carrying lexan structure

4. 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

5. 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

6. 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 cell Back
Detached
Front

7. Polynomial Stochastic Hybrid System Model
Front
Back
 State vector 𝐱 = 𝑁𝐹 𝑁 𝐵 𝑁 𝐷 𝑥 𝐿 𝑣 𝐿
𝑇
 Behavioural states: S = 𝐹, 𝐵, 𝐷
 Population counts: 𝑁𝑖∈𝑆
 Dynamical variables: 𝑥 𝐿, 𝑣 𝐿
 Flow equation d𝐱/d𝑡 = 0 0 0 𝑣 𝐿 𝑎 𝐿
𝑇
 6 Transitions: 𝑋𝑖 → 𝑋𝑗, with rate 𝑟𝑖𝑗
 Transition intensity: 𝜆𝑖𝑗 = 𝑟𝑖𝑗 𝑁𝑖
 Reset map: 𝑁𝑖, 𝑁𝑗 ↦ (𝑁𝑖 − 1, 𝑁𝑗 + 1)
Detached
F
BD
𝑟 𝐷𝐵, 𝑟𝐵𝐷
𝑟 𝐷𝐹, 𝑟𝐹𝐷 𝑟𝐹𝐵, 𝑟𝐵𝐹
Back Front
𝑣 𝐿
Detached
↦ 𝑥 𝐿

8. pSHS : Load Dynamics
 Front and back ants lift with net force:
𝐹𝑢𝑝 = 𝑁𝐹 + 𝑁 𝐵 𝐹𝐿
 Normal force: 𝐹𝑛 = 𝑚 𝐿 𝑔 − 𝐹𝑢𝑝
 Front ants pull with velocity regulation
 Proportional gain: 𝐾
 Velocity set point: 𝑣 𝐿
𝑑
 Individual pulling force: 𝐹𝑝 = 𝐾(𝑣 𝐿
𝑑
− 𝑣 𝐿)
LOAD
𝑚 𝐿 𝑔
𝑁 𝐹 𝐹𝑝𝜇𝐹𝑛
𝐹𝑟𝑜𝑛𝑡
𝐹𝑢𝑝 + 𝐹𝑛
𝐵𝑎𝑐𝑘
𝑥 𝐿 = 𝑣 𝐿
𝑚 𝐿 𝑣 𝐿 = 𝑁𝐹 𝐹𝑝 − 𝜇𝐹𝑛

9. Moment Dynamics
 Moments computed using extended generator 𝐿
 Key property allows moment computation for differentiable 𝜓:
𝑑
𝑑𝑡
𝐸 𝜓 . = 𝐸 𝐿𝜓
 Time evolution of expectations
𝑑𝐸 𝑁𝑖
𝑑𝑡
=
𝑖,𝑗∈𝑆,𝑖≠𝑗
(𝑟𝑖𝑗 𝐸 𝑁𝑖 − 𝑟𝑗𝑖 𝐸 𝑁𝑗 )
𝑑𝐸(𝑥 𝐿)
𝑑𝑡
= 𝐸 𝑣 𝐿
𝑑𝐸 𝑣 𝐿
𝑑𝑡
= 𝑐 𝑔 + 𝑐 𝐹 𝐸 𝑁𝐹 + 𝑐 𝐵 𝐸 𝑁 𝐵 + 𝑐 𝐹𝑣 𝐸 𝑁𝐹 𝐸(𝑣 𝐿)
 Note: 𝐸 𝑁𝐹 𝑣 𝐿 ≈ 𝐸 𝑁𝐹 𝐸 𝑣 𝐿
For our pSHS, 𝐿 is defined as:
𝐿𝜓(𝐱) ≔
𝜕𝜓
𝜕𝑥 𝐿
𝑥 𝐿 +
𝜕𝜓
𝜕𝑣 𝐿
𝑣 𝐿
+
𝑖,𝑗∈𝑆,𝑖≠𝑗
𝜓 𝜙𝑖𝑗 𝐱 − 𝜓 𝐱 𝑟𝑖𝑗 𝑁𝑖

10. 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
𝑣 𝐿
𝑑
= 0.3185 cm∙s−1
𝐀𝐧𝐭 𝐩𝐮𝐥𝐥𝐢𝐧𝐠 𝐟𝐨𝐫𝐜𝐞
𝐹𝑝 = 𝐾 𝑣 𝐿
𝑑
− 𝑣 𝐿

11. Model Predictions vs. Averaged Data

12. Model Validation with Individual Trials

13. Summary
We
 Conducted experiments of ants transporting a load
 Devised a pSHS Model of Collective Transport
 Fit the model parameters to empirical data

14. 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

15. 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