Exact computation and analysis of the workspace volume of robot manipulators is very important for its optimum design and application. A numerical method based on random probability is used to generate the workspaces of 3D robot

according to the kinematics mapping relationship from the joint space to the workspace. With deeply studying the differential geometry relationship between 2D and 3D figure

the 3D workspace is addressed by enveloping boundary curves of each plane which are obtained by partitioning 3D space into various layers along Z-axis. The numerical integration is used to calculate the volume of the workspace

and one example is illustrated to show the process of computation in detail.