Singulairty Analysis Of A Dissimilar Class Of The Six Degrees Of Freedom Parallel Manipulators

The dissimilar class of the Stewart platform is a typical parallel manipulator with sixdegrees of freedom, whose moving platform and fixed one are two dissimilarsemi-symmetrical hexagons connected by six SPS/SPU branches. The mechanism have beenwidely applied in many high technology fields, such as micro-displacement mechanism,force/torque sensor, virtual-axis machine tool, motion simulator, spacial docking mechanism.Singular configuration seriously affects the performance of the mechanism. Therefore, it isessential to thoroughly explore the regularity of the singularity and then to discuss the methodof the singularity-avoidance.In order to easily obtain the distribution pattern of the singularity loci of the mechanism,the singular configuration is classified into two types: position-singularity for a constantorientation and orientation-singularity for a given position. Based on the force Jacobianmatrix of the mechanism, a general symbolic analytical expression describing the position-singularity loci and a general symbolic analytical expression describing the orientation-singularity loci based on the unit quaternion representation are derived, respectively.For the position-singularity exploration, it is found that the position-singularity locus inevery different characteristic plane (moving plane) is a quadratic curve with obviousgeometric property. Two basic laws, which describe the geometric property of theposition-singularity curve in the characteristic plane, are summarized and proved. Theinstantaneous kinematic property of the mechanism being singular is analyzed by using theGrassmann line geometry and the screw theory.Orientation-singularity of the mechanism is investigated by using the unit quaternion asthe orientation parameters. It is illustrated that a theoretical singularity-free void should existsurrounding the orientation origin and inside the orientation-singularity surface. The minimalinscribed sphere of the orientation-singularity surface, whose center is the orientation origin,is defined as the orientation-singularity-free ball. The radius measuring the volume of thissphere is called orientation-capability which can be regarded as the measurement of thetheoretical singularity-free void size. The influence of the geometry parameters and theposition parameters of the mechanism on the orientation-capability is further discussed.The position-workspace for a constant-orientation and the orientation-workspace for agiven position are explored, respectively. New numerical methods for determining theconstraint position-workspace and singularity-free position-workspace for a constantorientation, the constraint orientation-workspace and singularity-free orientation-workspacefor a given position of the mechanism are developed by synthetically using the step by stepsearch and the bisection method. The minimal inscribed sphere of the practicalorientation-singularity-free workspace boundary is called the practical orientation-capabilitywhich is used as the measurement of the practical orientation-singularity-free size. Theaverage practical orientation-capability of the all positions inside the position-workspace isdefined as the global orientation-capability(GOC), and then the influence of the geometryparameters on the GOC is studied. Base on the singularity analysis of the mechanism, both of the singularity-free positionpath planning for a constant-orientation and the singularity-free orientation path planning fora given position are further investigated, respectively. By using the geometric property of theposition-singularity curve in the moving plane, the singularity-free position path planning inthe moving plane of the mechanism for a constant-orientation is explored. By using the unitquaternion as the orientation parameters, the orientation kinematic equation and theorientation trajectory of a rigid body with a quick twist are constructed based on thequaternion theory, respectively. The method of time optimal singularity-free orientationmotion planning is constructed by analyzing the distribution of the orientation-singularity lociand using the orientation trajectory equation of the rigid body with a twist. It is furtherpresented that the orientation redundant freedom can be optimized with making the conditionnumber of the force Jacobian matrix be minimal by using the genetic algorithm, after whichthe performance of the motion transmission and the force transmission of the mechanism canbe greatly improved.