Singularity And Its Dynamic Stability For Parallel Manipulators

It is well known that singular configurations are inherent to parallel manipulators. At the singular point, the motion of the manipulator is uncertain. The workspace of manipulators has been divided into several independent singularity-free regions by the singularity distribution hypersurface. For enhancing the motion stability and certainty while the manipulator passes through the singular points, in this paper, considering the application background of the axisymmetric vectoring exhaust nozzle (AVEN), the singularity and the dynamic stability of the parallel manipulators at the singular points investigated, which are supported by Chinese national science foundation committee (project number:50375111/50675188).Firstly, with the aid of the homotopy tracing method, the three-dimensional pre-assembly approach, and the expended equation method for figuring out singular points accurately, the configuration bifurcation characteristics going with the input parameters, the assembly configurations at singular points, and the reasons to cause singularities are analyzed. The research reveals that the dimensional-utmost singularity is the only singular type under the single input parameter, and with increasing of the input number, the dimensional-utmost singularity, the line vectors correlation singularity and the Jacobian matrix correlation singularity can occur individually or jointly.For overcoming the disadvantages that the configuration transform error of the linear disturbance function method for avoiding singularity of parallel manipulators is relative large and the motion uncertainty exists during passing through the singular points, through analyzing the distances from the disturbed moving trace points to the corresponding disturbed singular points, using the optimization method to determine the increments for each configuration component, and taking the Lagrange interpolation polynomial to construct the nonlinear disturbance function, the modified nonlinear disturbance function method for the parallel manipulator passing through the singularity hypersurface with controllable motion is presented based on the concept of the maximum loss control domain.In order to prevent the parallel manipulator falls into the singularity working regions in the design and executing processes, taking the 3-RPS parallel manipulator as an example, the analytical distribution equation of the singular points going with the input parameters is figured out. Based on this analytical distribution equation, the singularity of the parallel manipulator can be avoided through checking and controlling the lengths of the input parameters. Then, the singularity distribution curves of the driving parallel manipulator of axisymmetric vectoring exhaust nozzle (AVEN),3-SPS/3-PRS, and the singularity-free input parameters zones are presented. While the lengths of the input parameters locate in the singularity-free input parameters zones, the motion of the parallel manipulator is certain.For enhancing the motion stability of the parallel manipulator at the singular points, with the aid of the analytical mechanics method, the geometry constraint equations are treated as the restriction forces through introducing Lagrange operators, then the type-I analytical Lagrange dynamic equation of 3-RPS parallel manipulator is set up. In the equation, the impacts of the velocities and accelerations of input parameters, the initial velocities of the moving points, the external forces, and the gravity on the dynamic responses are revealed. Utilizing the center flow theorem and the first approximate Liyapunov method, which are the principles to investigate the stability of dynamic systems at the singular points for n dimensional nonlinear system, the effects of same dynamic parameters, such as the velocities and accelerations of the input parameters, the initial velocities of moving points, and loads, on the dynamic stability of the parallel manipulator at the singular points are studied. The distributing surfaces of these dynamic parameters corresponding to the steady motion of the parallel manipulator at the singular points are figured out. Based on analyzing the relations between the eigenvalue Gerschgorin distribution of the first order linear approximate system of the dynamic system and the some dynamic parameters, the distributions and the parameters regions corresponding to the stable motion at the singular points for some dynamic parameters, such as the initial velocities of moving points, are figured out.