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This paper presents a novel framework for studying the statics and the instantaneous kinematics of robot manipulators based on the Grassmann-Cayley algebra. This algebra provides a complete mathematical interpretation of screw theory, in which twist and wrench spaces are represented by means of the concept of extensor, and the reciprocity condition between twist and wrench spaces of partially constrained rigid bodies is reflected by its inherent duality. Kinestatic analysis of robot manipulators entails computing sums and intersections of the twist and wrench spaces of the composing kinematic chains which are carried out by means of the operators join and meet of this algebra. The Grassmann-Cayley algebra permits us to work at the symbolic level, that is, in a coordinate-free manner. Moreover, it has an explicit formula for the meet operator that gives closed-form expressions of twist and wrench spaces of robot manipulators. Besides being computationally advantageous, the resulting formalism is conceptually much closer to the way humans think about kinestatics than geometric and coordinate-dependent methods, and therefore provides a deeper insight into the kinestatics of robot manipulators.