Extension of Usable Workspace of Rotational Axes in Robot Planning

Singularity of a robot manipulator is one of the obstacles that influences its capabilities. This paper discusses constrained and allowable rotational motion resulting from lost translational freedom when the robot is singular. A convenient method and simple and clear expression to determine the allowable rotational axes and the subspace that they form, under Jacobian singularity, is analyzed and presented. Different configurations, reciprocal screws, and subspaces of allowable-rotational-axes are derived in a case study involving a classic robot. The result is useful in applications involving robot path planning in task space as it extends the usable workspace of rotational axes. INTRODUCTION Robotic manipulators have been used in industry, but their utilization is sometimes limited by the singularity of the manipulators. As a result, its workspace decreases and its kinematic and dynamic properties deteriorate. This problem has been studied extensively [1-9], and most of the studies are based on the screw theory [1 0-13]. Baker analyzed the relative motion of two-degree-of-freedom mechanisms and determined the special configuration using common reciprocal screws, which intersects with all the kinematic pair axes of the mechanisms [1 ,2]. Sugimoto reported that a robot end effector can not move along the direction of a reciprocal screw [3]. Waldron derived three singular conditions of some industrial robot manipulators and their corresponding reciprocal screws [4]. Hunt solved the reciprocal screw of some robot manipulators. He pointed out that there are infinite' twists in space belonging to five-system, and determined the twists with different pitch [5,6]. Liu in [7] indicated the method of pseudo inverse of Jacobian matrix and developed a differential motion approach based on the singularity-robust inverse to deal with the singularity problem. Using the method of pseudo inverse sometimes results in discontinuities in joint space variables. 1 Visiting professor from Yanshan University, China. Based on the screw theory and the work done by Waldron [4] and Hunt [5,6], the first named author has shown that the constrained rotation of a body does not affect its translation while the constrained translation of an object does influence its rotation [14, 15]. Specifically, when the translational freedom is constrained, the rotational freedom is also partially constrained. As a result, the selection of rotational axes during path planning reduces, (i.e., the body may still have rotational freedom) but it can not rotate about any arbitrarily chosen axis. Its allowable rotational axes form only a subspace of the original space. The practical implication is that at singular points, trajectory planning of a manipulator in the task space may still proceed, but with certain restrictions. This effectively extends the robot workspace. This paper further discusses the allowable rotational freedom when translational freedom is constrained, and demonstrates how to conveniently determine the allowable rotation under singularity and further a usable subspace of all rotational axes using a classical robot as an example. The results presented in this paper can be implemented in manipulator path planning software for industrial applications. POSSIBLE MOVEMENT UNDER CONSTRAINTS If screws $1(s 1 ; S 01 ) , i = 1 ,2,3, ... ,n. s 1•s 1 = 1 and S;•S 01 = 0, represent the axes of the kinematic joints of the robot, and those screws are linearly independent. There exist (6 n) reciprocal screws, $',that are reciprocal to each screw of the screw system. The reciprocal screw denoting constrain is defined as: