Abstract
Many flying robots can be regarded as rigid bodies propelled by a thrust force whose direction is constant from the viewpoint of the robot, but whose magnitude can be controlled. Such robots are endowed with a mechanism to induce torques about three mutually orthogonal directions, allowing one to control the direction of the thrust force. Quadrocopters are examples of robots in this class. This chapter presents a mathematical model of such flying robots and it revisits the flocking problem for this model. In the special case when the robots fly in a horizontal plane, the solutions of the flocking problem for unicycles and flying robots turn out to be very similar. The chapter closes with a discussion about rendezvous of flying robots, an exciting open problem whose solution may lead to distributed coordination algorithms for formation control.
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Notes
- 1.
In the derivation of this model we have ignored drag and other aerodynamic effects. We have also ignored the dynamics inherent in the propulsion mechanism.
- 2.
This choice of notation creates a minor inconsistency with the previous section, where we have used, for instance, \(\omega _1\) to denote the first component of the angular velocity vector \(\omega \). From now on, \(\omega _i\) will denote instead the angular velocity vector of robot i.
- 3.
The LaSalle invariance principle requires solutions to be bounded. The speeds \(\dot{\theta }_i\) are bounded because of the damping term \(-b_i \dot{\theta }_i\) in (6.9). As for the angles \(\theta _i\), we view them as points of a unit circle, a compact set.
- 4.
This result is analogous to Theorem 4.2, which covers the case of unit weights, \(a_{lk}=1\) for all l, k.
- 5.
In our formulation of the flocking problem, nothing prevents the robots from crashing to the ground. A more meaningful problem statement would require \(v_{ss}\) to be parallel to the ground, but this problem is to date open and significantly harder than the one considered in this section.
- 6.
As in the proof of Theorem 6.1, we may apply the LaSalle invariance principle because all solutions of (6.12) with control (6.15) are bounded. The boundedness of \(\omega _i\) follows from the presence of the dissipation term-\(B_i \omega _i\) in the \(\tau _i\). The matrices \(R_i\) have unit norm columns so they are bounded as well.
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Francis, B.A., Maggiore, M. (2016). Introduction to Flying Robots. In: Flocking and Rendezvous in Distributed Robotics. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-24729-8_6
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DOI: https://doi.org/10.1007/978-3-319-24729-8_6
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