Stability regions of periodic trajectories of the manipulator motion
Introduction
In the case of stationary robots, the designed trajectories of a manipulator gripping device motion that result from the technological processes being carried out exhibit most often a character of periodic motion cycles. From the viewpoint of safety, an analysis of the behaviour of the dynamical system such as a manipulator that performs a periodic motion and is subjected to motion perturbations is an interesting issue. To this effect, the motion stability is understood as insensitivity of the trajectory of the dynamical system motion to motion perturbations. In this paper, an investigation of dynamical characteristics in the vicinity of the periodical trajectory, based on Poincare maps of variational equations, has been proposed. A Poincare map is understood as a discrete mapping in the form of (n−1) of the dimensional space that divides the phase space into two subspaces and where (n) is a dimension of the dynamical system. In order to achieve this goal, the equations of Poincare maps [12] have been written and they have been employed in nonlinear equations of perturbations. In the case of manipulators, an application of a Poincare map is convenient due to the fact that the position of the boundary cycle in the phase space is known. Because of this, it is relatively easy to define a Poincare map in the phase space. An analysis of stability regions makes the linearization of maps possible and constitutes an initial stage of an analysis of manipulator vibrations, that is to say, of manipulator bifurcation types. Owing to a possibility of occurrence of unpredictable vibrations due to a stability loss, this issue seems to be very important from the point of view of work safety. Sample stability regions of the manipulator, conditions under which a stability loss takes place and ways in which a stability loss of the manipulator periodic motion due to perturbations occurs have been presented.
Problems involved in an analysis of stability of periodic trajectories of dynamical systems are based on the Floquet and Lapunov theories [3], [5], [7], [9], [12]. The Floquet theory deals with linear differential equations with periodic coefficients. In turn, using the Lapunov theorem, we can state the stability of periodic solutions to the nonlinear equation on the basis of an analysis of the linear variational equation. The Lapunov theorem does not, however, inform what new solutions appear or disappear and what their stability is like, i.e., it does not characterise a bifurcation type. It follows from the fact that this theorem employs an analysis of the linearized equation. In order to determine a type of bifurcation, thus it is indispensable to investigate the nonlinear issue in the vicinity of the parameter value that corresponds to the boundary value of Floquet multiplicators. In typical cases, the investigation of behaviour of Floquet multiplicators that are obtained from the linear analysis allows for determination of a type of bifurcation of the periodic orbit. In other words, in order to identify a bifurcation type, it is sufficient to consider the forms of solutions to the linear variational equation that correspond to the transition of Floquet multiplicators through an elementary circle. As an example, methods of mechanical systems stability analysis based on different conditions can be found in [1], [6], [10].
Our task is to develop a model of the manipulator, a model of the electric and mechanical drive and to determine critical values of parameters of the nonlinear model for which a change in stability, i.e., bifurcation, takes place. Both the regions of stability and kinds of vibrations that can occur during a perturbation of the manipulator periodic motion are interesting and important. On the basis of the manipulator variational equation and employing the theory of Poincare maps, it will be possible to determine the regions of stability of a periodic trajectory of the manipulator gripping device motion. An analysis of stability regions performed for an MAR manipulator is presented in Section 4.
Section snippets
Model of the manipulator and of electric and mechanical driving systems
To simplify the motion equations and to clarify the usage of the method, a model of the manipulator (Fig. 1) with rigid links has been assumed. In the mathematical model of the manipulator with the vector n of generalised coordinates of links, the potential energy has been written as a sum of the potential energy of links, an object being manipulated and the flexibility of kinematic pairs. The potential energy of flexibility has been written as a function of resultant flexibility of rolling,
Stability of periodic trajectories of the manipulator
The issue of stability of a periodic motion becomes important when the gripping device motion becomes unstable for some parameters of the manipulator model. Let us assume that the vector of generalised coordinates is a periodic solution to the equation of the manipulator motion and let us perturb this solution. A perturbation of the manipulator periodic solution is to be understood as a perturbation of the ith generalised coordinate/coordinates that is/are determined
Determination of stability regions of the MAR manipulator
The MAR robot manipulator (Fig. 1) whose main data are presented in Table 1, has been subjected to a sample numerical analysis in order to determine stability regions. The eigenvalues are calculated for varying parameters of velocity of the gripping device motion along the motion trajectory and for control parameters. The stability of matrix (17) is determined by absolute values of eigenvalues of Eq. (22), described with respect to 1 [5], [12]. Thus, matrix (17) is asymptotically stable when
Conclusions
The analysis of stability regions of the periodic motion trajectory, has been carried out on the basis of Poincare maps for non-autonomous, nonlinear systems of differential equations. The presented analysis of the periodic trajectory stability makes it possible to analyse the regions of parameters in which the system operation is advantageous. A phenomenon of instability of motion of manipulator links shows a possibility of occurrence of unpredictable vibrations of individual robot links that
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