“…where q is the vector of joint positions (arm and base), J AG is the Analytic Generalized Jacobian (Umetani, 1989;Menon, 2003), J AG + is its pseudoinverse, e is error of the end-effector (the difference between desired and real position), x d is the desired position and orientation of working space, K p and K c are gain matrixes (K c is positive definite), I is the identity matrix, J c is the Jacobian associated to the constraint task error, e c is the error of the constraint task defined as e c =q bd -q b where q b is a generalized position variable of the free floating base of the robot that is constrained to the desired trajectory q bd . The matrix (I − J AG + J AG ) projects the joint velocity contribution into the null space of the generalized Jacobian in order to separate the constraint and endeffector tasks.…”