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Q. 1 If the robots are fast or heavy, it is no longer enough to consider velocities. Instead, one needs to define the controllers at the level of accelerations, i.e., we can assume that the agents are double integrators, i.e. ; = ui. Assume that we can measure the relative displacement as well as the relative velocity displacement, and we let our control algorithm be ((: -1)+(4; - ;)), MEN where we assume the graph to be static, undirected, and connected. a Letting Gi ER, we can set 21 IN 2 = (1 : ON Using this notation, find a matrix M such that = M2. = b Let 11,...,be the eigenvalues of the graph Laplacian. After a bunch of algebra, one can show that M's eigenvalues M1+41-, ...,HN+, HN- (there are 2N eigenvalues) are -70k V7212 - 41k Mkt = 2 For what values of is consensus achieved both with respect to the positions, and with respect to the 7 velocities? Q. 1 If the robots are fast or heavy, it is no longer enough to consider velocities. Instead, one needs to define the controllers at the level of accelerations, i.e., we can assume that the agents are double integrators, i.e. ; = ui. Assume that we can measure the relative displacement as well as the relative velocity displacement, and we let our control algorithm be ((: -1)+(4; - ;)), MEN where we assume the graph to be static, undirected, and connected. a Letting Gi ER, we can set 21 IN 2 = (1 : ON Using this notation, find a matrix M such that = M2. = b Let 11,...,be the eigenvalues of the graph Laplacian. After a bunch of algebra, one can show that M's eigenvalues M1+41-, ...,HN+, HN- (there are 2N eigenvalues) are -70k V7212 - 41k Mkt = 2 For what values of is consensus achieved both with respect to the positions, and with respect to the 7 velocities