On local approximation-based adaptive control with applications to robotic manipulators and biped robots

The adaptive approximation control is a powerful tool for controlling robotic systems with unmodeled dynamics. The local (partitioned) approximation-based adaptive control includes representation of the uncertain matrices and vectors in the robot model as finite combinations of basis functions. Update laws for the weighting matrices are obtained by the Lyapunov-like design. However, one of the inherent limitations of this category of approximation is curse dimensionality associated with the approximation of uncertain matrix. There are three possible representations for the approximation of the uncertain matrix: Kronecker product, sparse matrices, and GL operator. Both Kronecker product and sparse matrices can grow exponentially with the dimension of the target matrix, whereas GL operator can grow linearly but without the use of conventional operations of matrices. In light of the above, this paper proposes a simple representation for the approximation of the uncertain matrix. The proposed representation is directly linear with respect to the dimension of the target matrix using the conventional operations of matrices. A comparative study is performed to all previously investigated representations, including the proposed representation in view of control law, adaptive law and the order of approximation for robotic manipulators in free space. Two case studies are simulated which are: two-link manipulator and 6-link biped robots during the single support phase. The results show that for low dimension robotic manipulators, all representations can be conducted equivalently. There is no large difference in view of simulation time, whereas for higher degrees of freedom robots the Gl operator and the proposed representation are superior in view of simulation time. © 2017, Springer-Verlag Berlin Heidelberg.