Metric Relations and Displacement Groups in Mechanism and Robot Kinematics

Abstract

This paper presents a systematic theory for metric relations between the invariant properties of displacement groups, and shows this theory application to mechanism kinematics. Displacement groups, their invariant properties and operations are briefly described. Kinematic constraints are then introduced as tools for relating abstract group properties to actual mechanism constraints. Criteria and operating rules to employ metric relations for the generation of a meaningful set of closure equations for kinematic chains are detailed.

Kinematic synthesis using tree topologies

“…The exceptional kinematic chain loop can be substituted for the subalgebra corresponding to the intersection which may not always have a simple equivalence. Fanghella and Galletti [43] find many equivalences of intersections using group theory, and Rico and Ravani [42] do the same using Lie algebras. However, finding the equivalent open kinematic chain is not always a simple task, depending on the mobility class of the loop.…”

“…(43), that the PRR chain needs m R = 5 positions, the PRP chain needs m R = 2 position and the entire system would also need m R = 2 positions. In the case of the PRP chain, the exact synthesis is limited by m R to only two positions with a rotational component.…”

Structural synthesis of two-layer and two-loop spatial mechanisms with coupling chains

“…5(b). With the existing methods [5,23], it is easily obtained that the initial DOF of the end platform is a translations, along the intersecting line formed by the normal plane of R c4 3 and the one of R c5 3 . Thus, M 0 = 1, the initial twist system is Table 2 Representations of geometrical relations between two joint axes.…”

“…The motion pattern of the end platform associated with the initial mobility is obtained directly through the existing methods for parallel mechanisms [5,23]. The key problem is to determine the motion pattern of the end platform in the equivalent subparallel mechanism without consideration of kinematics analysis.…”

“…Kong and Gosselin [2], and Fang and Tsai [3] also studied structural synthesis of parallel mechanisms based on constraint screws. Herve [4], Fanghella and Galletti [5], and Rico et al [6] proposed the structural synthesis method for spatial parallel mechanisms based on group theory. Later, Li et al [7] and Meng et al [8] further extended the relevant research based on differential manifold.…”