ISSN 0005-1179, Automation and Remote Control, 2018, Vol. 79, No. 3, pp. 554–570.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
S.A. Krasnova, A.S. Antipov, 2016, published in Problemy Upravleniya, 2016, No. 6, pp. 10–21.
Hierarchical Design of Sigmoidal Generalized Moments
of Manipulator under Uncertainty
S. A. Krasnova
and A. S. Antipov
Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
Bauman Moscow State Technical University (National Research University), Moscow, Russia
Received January 18, 2016
Abstract—For the orientation control system of the manipulator’s end eﬀector with electrical
actuators, we develop a decomposition pr ocedure of feedback law design to track given trajec-
tories in the end eﬀector’s coordinate system. Owing to the S-shaped smooth sigma-functions
used as the local feedback laws and corrections of the state observer, the tracking system is
invariant with a given accuracy with respect to existing uncertainties under constraints imposed
on the variables of the mechanical subsystem. The suggested approach does not involve the
solution of the inverse kinematics and dynamics problems and also relaxes the requirements to
the volume of a priori information about the plant and external perturbations.
Keywords: manipulator, tracking, invariance, state variable constraints, sigma-function, state
In this paper, we consider an electromechanical system that includes a dynamical model of
a manipulator (the mechanical subsystem) operating under uncertainty and incomplete measure-
ments and also a reduced model of electrical actuators. The problem under study is to track given
trajectories in the end eﬀector’s coordinate system of the manipulator. The standard methods of
manipulator planning and orientation control in the generalized coordinate space are inapplicable
to this problem, as they require the real-time solution of the inverse kinematics and dynamics
problems, which have a unique analytical solution merely in some cases .
Below we suggest a direct method for the block design of the mechanical subsystem written
in the tracking errors that does not involve the solution of the inverse kinematics problems. The
generalized moments gained by the actuators act as ﬁctitious controls in the mechanical subsystem.
The novelty of our approach lies in the use of sigmoidal feedback laws for solving the problems of
control and observation under uncertainty [2, 3]. It is shown that the smooth bounded actions in the
control and observation loops lead to similar properties as in the discontinuous control systems that
operate in sliding modes, namely, the decomposition of joint motion into diﬀerent-pace components
and the ε-invariance with respect to external perturbations.
The ﬁctitious controls generated in the mechanical subsystem are the reference signals for the
actuators. Therefore, the decomposition principle is implemented in the course of tracking system
design. Moreover, at the stage of electromechanical system design, it is now possible to choose
diﬀerent complete actuators for a guaranteed execution of diﬀerent jobs by the manipulator under