An optimization approach to computing the boundaries of the workspaces of planar manipulators is presented. This numerical method consists of finding a suitable radiating point in the output coordinate space and then determining the points of intersection of a representative pencil of rays, emanating from the radiating point, with the boundary of the accessible set. This is done by application of a novel constrained optimization approach that has the considerable advantage that it may easily be automated. The method is illustrated by its application to two planar mechanisms, namely a planar Stewart platform and a planar redundantly controlled serial manipulator. In addition to the exterior boundaries of the workspace, interior curves that represent configurations at which controllability and mobility may be limited, are also mapped. The optimization methodology, implemented here for the planar case, may readily be extended to spatial Stewart platforms. [S1050-0472(00)00304-4]

SUMMARYA new path-planning interpolation methodology is presented with which the user may analytically specify the desired path to be followed by any planar industrial robot. The user prescribes a set of nodal points along a general curve to be followed by the chosen working point on the end-e ector of the mechanism. Given these speciÿed points along the path and additional prescribed kinematical requirements, Overlapping Cubic Arcs are ÿtted in the Cartesian domain and a cubic Spline interpolation curve is ÿtted in the time-domain. Further user-speciÿed information is used to determine how the end-e ector orientation angle should vary along the speciÿed curve. The proposed trajectory-planning methodology is embodied in a computer-algorithm (OCAS), which outputs continuous graphs for positions, velocities and accelerations in the time-domain. If a varying end-e ector orientation angle is speciÿed, the OCAS-algorithm also generates continuous orientation angle, orientation angular velocity and orientation angular acceleration curves in the time-domain. The trajectory-planning capabilities of the OCAS-algorithm are tested for cases where the prescribed nodal points lie along curves deÿned by analytically known non-linear functions, as well as for nodal points speciÿed along a non-analytical (free-form) test-curve. The proposed trajectory-planner may be implemented as part of kinematic and kinetic simulation software, and it also has the potential application for controlling machine tools in cutting along free-form curves.

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SUMMARYAn optimization approach to the computation of the boundaries of di erent dextrous workspaces of parallel manipulators is presented. A speciÿc dextrous workspace is the region in space in which, at each position of the working point, a manipulator can control the orientation of its upper working platform through a speciÿed range of orientation angles. Here the dextrous workspace is determined from the intersection of suitably chosen ÿxed orientation workspaces, which are found by application of a constrained optimization algorithm. The procedure is simple and has the considerable advantage that it may easily be automated. The method is illustrated by its application to both a planar and spatial Gough-Stewart platform.