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Coordinated gradient descent: a case study of Lagrangian dynamics with
projected gradient information

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Luc Moreau, Ralf Bachmayer and Naomi Ehrich Leonard

*Proceedings, 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods
for Nonlinear Control*, Seville, Spain, April 2003
The paper studies gradient descent algorithms for vehicle networks.
Each vehicle within the network is modeled as a double integrator in the
plane. For each individual vehicle, the control input enabling coordinated
gradient descent consists of a gradient descent control term and additional
inter-vehicle forcing terms. When each vehicle has enough sensors to measure
the full gradient at its current position, then the closed-loop system
becomes Lagrangian. We focus in the present paper upon the more practical
situation where each vehicle has only one sensor with which to sample the
environment. We take this into account by replacing the full gradient in
the closed-loop equations by its projection on the direction of motion
for each individual vehicle. This gives rise to a differential equation
with *discontinuous* right-hand side. In order to avoid the (practical
and theoretical) complications that arise as a consequence of these discontinuities,
we modify the inter-vehicle forcing terms and represent the velocity of
each vehicle by a magnitude and an angle, resulting in a set of *smooth*
differential equations. We demonstrate our approach with simulations.

(6 pages pdf)
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