Abstract
This chapter proposes a distributed approach for the resolution of a multi-agent problem under collision and obstacle avoidance conditions. Using hyperplane arrangements and mixed integer programming, we provide an efficient description of the feasible region verifying the avoidance constraints. We exploit geometric properties of hyperplane arrangements and adapt this description to the distributed scheme in order to provide an efficient Model Predictive Control (MPC) solution. Furthermore, we prove constraint validation for a hierarchical ordering of the agents.
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Notes
- 1.
We have made the simplifying convention that all the half-spaces appearing in (17.3) are of form \(\mathcal R^+(\cdot )\).
- 2.
Sometimes this construction is called in the literature the “big M” formulation.
- 3.
Whenever the time instant is clear we abuse the notation and denote the current state, \(x(k)_i\), as \(x_i\). The same simplified notation is applied to the input.
- 4.
Note that in the hierarchical implementation the neighborhoods are disjoint, see also the definition of neighborhoods \(\mathcal N_i\) in Sect. 17.3.
- 5.
Not necessarily true when the dynamics describe systems which have a minimal velocity—unmanned aerial vehicles (UAVs) for example.
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Acknowledgments
The research of Ionela Prodan is financially supported by the EADS Corporate Foundation (091-AO09-1006).
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Prodan, I., Stoican, F., Olaru, S., Stoica, C., Niculescu, SI. (2014). Mixed-Integer Programming Techniques in Distributed MPC Problems. In: Maestre, J., Negenborn, R. (eds) Distributed Model Predictive Control Made Easy. Intelligent Systems, Control and Automation: Science and Engineering, vol 69. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7006-5_17
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