Theoretical studies on parallel mechanisms

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In contrast to the case of serial manipulators, the analysis and design of parallel manipulators is much less intuitive. For example the boundaries of the workspace are relatively easy to determine for a serial robot — one simply extends the arm of the robot — whereas they are much more complex for a parallel robot.

Thus, algorithms must be developed for the analysis of the different properties of parallel robots, such as:

• the direct kinematic problem: the calculation of the Cartesian coordinates for given joint coordinates;

• the inverse kinematic problem: the calculation of the joint coordinates for given Cartesian coordinates;

• the workspace: the determination of the range of poses that can be attained by the platform given the architecture of the manipulator;

• the dexterity: the graphical representation of the ability of a manipulator to execute fine movements with precision;

• the singularity analysis: the determination of the locus of singularities — the configurations in which one loses control of the mechanism — in the Cartesian workspace.

These problems have been addressed in numerous research studies in the Robotics laboratory and algorithms have been developed for various types of parallel manipulators: planar, spherical and spatial. Three graphical examples of solutions developed are shown below.

Fig. 1: Representation of the dexterity and the singularity locus for the Gough-Stewart platform. The singularity locus, represented by dark black lines, has been plotted using an analytical expression while the dexterity, represented by the grey scale, was obtained through the analysis of the Jacobian matrix.

Fig. 2: Representation of the workspace for the Gough-Stewart platform. The boundary of the workspace was determined analytically using a geometric analysis. The calculation takes but a few milliseconds.

Fig. 3: Superposition of the workspace (blue curves) on the singularity locus (red curves) for a 6-DOF spatial manipulator. This superposition indicates that singularities are unfortunately present within the workspace. Tools have been developed in the laboratory to analytically detect the presence of singularities within the workspace.