Fleet management

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Build a fleet with any number of copters.

• [x] Formation to be decided.
• [ ] Algorithm to be decided.
• [ ] All agents should keep the formation if the fleet is moving.

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It will be focused on linear or feedback-linearizable agents, so it will use the linear consensus algorithm. Specific agents used will be different types of quadcopters and hexacopters.

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Movements: rendezvous first, formation control later.

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The agents will be positioned on an imaginary circle with the leader at the center. All followers will have to maintain a certain distance from one another and from the leader itself.

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Example of formation with 6 agents:

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In the end, or at least up until now, the formation is not geometrically precise. That's because all agents have to account for an adjustment due to the collision avoidance calculations during rendezvous.

The main idea still stands though: the leader is at the center of the formation and the followers will be distributed around it.

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The Rendezvous is achieved through recursive iterations of a modified consensus algorithm:

p_{i+1} = \alpha \times p_{des} + (1-\alpha) \times p_i + \beta \times c_a

where:

• $i$: iteration index
• $p_i$: position vector at iteration $i$
• $p_{des}$: desired/target position (rendezvous point)
• $c_a$: collision adjustment vector
• $\alpha \in [0;1]$: numerical factor indicating the influence of the target position on the agent's movement (with higher values of $\alpha$ the agent will prioritize moving towards the target position)
• $\beta \in [0;1]$: numerical factor indicating the influence of the neighboring agents' position on the agent's movement, for collision avoidance purposes (with higher values of $\beta$ the agent will prioritize avoiding collisions with the other agents)

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The Formation algorithms is achieved through recursive iterations of a consensus algorithm, with the hypothesis that each copter's dynamics can be linearized:

\displaylines{p_t+1 = p_t - u_t + \beta \times c_a \\ u_i = k_p \sum_{j \in \mathcal{N}_i} w_{ij} [(p_j - p_i) - (p_{j_{des}} - p_{i_{des}})]}

where:

• $t$: iteration index
• $p_i$: position vector of agent $i$
• $p{i{des}}$: desired position of agents $i$
• $\mathcal{N}_i$ is the set of neighbors of agent $i$
• $k_p > 0$: proportional gain
• $c_a$: collision adjustment vector
• $\beta \in [0;1]$: numerical factor indicating the influence of the neighboring agents' position on the agent's movement, for collision avoidance purposes
• $w{ij}$: weight of the connection between agents $i,j$. From graph theory, $w{ij} > 0 \leftrightarrow j \in \mathcal{N}_i$. All neighbors are connected in our case, with unary weights