Consider a system on N agents moving in the plane with the following dynamics where r, is the position of agent i in the plane (,9), v; its linear velocity, and a; its acceleration, which is also its control input. Assume that the each agent can communicate with every other agent, namely that the resulting graph is complete. Let the relative positions between agents i and j be denoted by r = T - Tj. 2 V t = a; i = 1, ...,N 9 Ti 08. ;a; 0 -V, Vi Figure 5: Forces acting on agent i. Figure 6: Edge tension (Energy function) between agents i.and j The objective is to steer each agent via its acceleration input in order to achieve the Boid action, namely to achieve cohesion, collision avoidance, and alignment. Consider the following energy as displayed in Figure 6, 1 Vij (|| Tij ||) = + log || 1 || which leads to the energy V; for node i and V, for the complete network given below, V = V = Let a control law be given by, do; N V (Tij j=1j#i N (V+0x + ) i=1 1,N 12 ( V4 +00+0) a + ao; U = ar; = -V, Vi N = 0-0 9 (2 x U) Note that 2 x v; denotes the cross product between velocity v; and 2 (refer to Figure 5). 1. (3 points). Explain (in words) what Vij is intended to achieve. (6) (8) (9) (10) 2. (3 points). Explain (in words) what the control laws a,, and ag, are intended to achieve. 3. (5 points). Consider a set of N = 5 identical agents and let the initial positions and velocities be random within a ball of radius 10 (m or m/s) and let each agent radius of communication be 15 m. Simulate the trajectory of the ten agents using the control law in equation (10) and describe the resulting behavior for 3 random initial conditions. 4. (4 points). Next, let the radius of communication be 8 m. Simulate the trajectory of the ten agents using the control law in equation (10) and describe the resulting behavior for 3 random initial conditions. NAME 10