The finite-position dimensional synthesis of parallel robots is not sufficient to ensure the trajectory of the parallel system. Circuit or branch defect problems cannot be controlled; in the most extreme cases, it may yield a system with negative mobility, that can be assembled at each task positions but cannot be driven from task position to task position.

Our approach is to relate a finite set of position to a complete description of the workspace. Considering the workspace of the linkage as a set of finite screws corresponding to finite displacements of the end-effector, the workspace takes a simple expression in some cases.

It is sometimes possible to start with the joint-parameterized version of the workspace, namely the forward kinematics equations of the robot, and perform implicitization to obtain an algebraic description of the workspace.

The images and documentation below correspond to the application of this method to the closed RC-CR linkage, which is mobile when R and C joints are parallel. The method has been applied so far to the closed RCCR, RPRP and RRRR linkages.

The RC-CR linkage has a constant-orientation workspace, hence the workspace of relative displacements is a translation-only workspace. The algebraic expression for the relative point workspace of an RC chain is a circular cylinder passing through zero. The intersection of two of the four possible cylinders passing through the four points creates the 1-dof trajectory of the linkage.

Selecting two of the cylinders allows us to control the shape of the workspace. Finding the linkage dimensions from the cylinder expression is relatively simple. The movie below shows the linkage obtained selecting cylinders 1 and 3, with a single circuit containing all the points.

## Resources

The draft paper below shows the application of the method for the RCCR linkage. The Mathematica file contains all the instructions in order to synthesize an RCCR linkage.

Draft: Exact Workspace Design for the RCCR Linkage

Implementation: Mathematica files for the synthesis of the RCCR linkage .

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