Flight Testing of Multiple-Spacecraft Control on SPHERES During Close-Proximity Operations

A, B, C = state-space matrices a = acceleration due to linear-quadratic-regulatorand artificial-potential-field-determined control effort aAPF = acceleration due to artificial-potential-fielddetermined control effort aLQR = acceleration due to linear-quadratic-regulatordetermined control effort am = maximum acceleration aobs = acceleration of chaser spacecraft toward an obstacle ax;y;z = acceleration due to the control effort Do = obstacle region of influence da = goal acceleration decay constant dg = goal exponential decay constant do = stopping distance constant JLQR = linear quadratic regulator cost function KLQR = linear quadratic regulator state feedback gain ka = acceleration shaping parameter kd = docking safety parameter kg = velocity shaping function ko = obstacle function ks = safety function kv = velocity shaping parameter Lo = obstacle exterior surface N = linear quadratic regulator gain matrix Q = linear quadratic regulator state gain matrix R = linear quadratic regulator control effort gain matrix r = Euclidean norm distance or relative range r = relative distance vector rc = position vector of the chaser spacecraft rg = position vector of the chaser spacecraft from the goal rinit = initial distance of the chaser spacecraft from the goal rm = maximum allowable distance of the chaser spacecraft from the goal ro = position vector of the chaser spacecraft from the obstacle rt = position vector of the target spacecraft with respect to the Earth S = solution of the Riccati equation u = control effort vector V = potential function Vg = goal potential function Vo = obstacle potential function vm = maximum relative velocity vo = desired velocity of chaser spacecraft toward an obstacle vobs = velocity of chaser spacecraft toward an obstacle x = state vector x, y, z = positions, or states, along the Cartesian axis Q = linear quadratic regulator state performance gain R = linear quadratic regulator control effort gain t = time increment = standard deviation for the obstacle’s region of influence ! = orbital angular velocity