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This paper presents a novel technique for the kinematic analysis of planar and spatial open chain systems, called the Incidence and Transfer Method (IT), which is based on the incidence matrices associated with the edge-oriented graph attached to the mechanism and the transfer joints. Relative to such joints, a set of independent equations can be automatically generated for the efficient computation of manipulator's joint positions and velocities. Kinematics of open chain manipulators with common joints such as revolute (R), prismatic (P), cylindrical (C), and helical (H) can be solved by using the sparse matrices derived from the manipulator's graph model. Complete kinematic equations are obtained in matrix form using a base of circuits from a cycle matroid. The proposed method has general applicability and can be employed for systems with any number of links and degrees of freedom, as illustrated by the numerical example presented.

Proper valve angles and concentric valve seats are critical to performance of an engine. If the valve seat were not right, the valve is not going to seat properly resulting in reduced power output. Although the performance of CNC machines is accurate, unavoidable human errors at the part loading position have serious repercussions on engine performance. A solution algorithm presented in this paper employs the principles of inverse kinematics wherein a faulty compound-hole angle axis in space caused by the translational and rotational errors at the part loading position is identified with an imaginary true axis in space by enforcing identity through a modified machine axes.