The kinematic modeling of multi-loop mechanisms requires a systematic representation of the kinematic topology, i.e. the arrangement of links and joints. A linear graph, called the topological graph, is used to this end. Various forms of this graph have been introduced for application in mechanism kinematics and multibody dynamics aiming at matrix formulations of the governing equations. For the (higher-order) kinematic analysis of mechanisms a simple yet stringent representation of the topological information is often sufficient. This paper proposes a simple concept and notation for use in kinematic analysis. Upon a topological graph, an order relation of links and joints is introduced allowing for recursive computation of the mechanism configuration. An ordering is also introduced on the topologically independent fundamental cycles. The latter is indispensable for formulating generically independent loop closure constraints. These are presented for linkages with only lower pairs, as well as for mechanisms with one higher kinematic pair per fundamental cycle. The corresponding formulation is known as cut-body and cut-joint approach, respectively.

The

Various types of topological graphs have been proposed in the literature.
They have been an important aspect for modeling of complex MBS.

For kinematical investigations of mechanisms a graph representation of the
kinematic topology has been proposed by

Topological graphs have further interesting features related to generic
properties of the mechanism. The essential kinematic properties (of generic
realizations) were investigated by

Still, topological information are rarely exploited for kinematic analysis. One consequence often observed is that redundant loop constraints are imposed for multi-loop mechanisms. This becomes critical in particular if higher-order analyses are pursued. Furthermore, despite the vast literature on graph modeling of mechanisms and MBS topology, there is no established approach and notation used in mechanism theory. The aim of this paper is to summarize the basic concept of graph representation and of formulations of loop constraints for multi-loop mechanisms in a way appropriate for the higher-order kinematic analysis.

In this paper, the graph representation of the kinematic topology is
recalled, and the essential relations necessary for introducing loop
constraints are derived. Throughout the paper relative coordinates (joint
coordinates) are used to parameterize the configuration allowing for a
recursive evaluation of the mechanism kinematics. The essential topological
relations are:

an order relation to define predecessors of bodies and joints,

an indicator of the direction in which a relative joint motion is defined, and

an order relation defining predecessors of bodies and joints within topologically independent loops.

The first and second allow for recursive determination of configurations of
bodies, and the third for a recursive formulation of topologically
independent loop constraints. To this end, (1) an oriented spanning tree

These topological relations provide the basis for kinematic investigations, in particular the higher-order kinematic analysis. Matrix representations of topological relations (incidence, adjacency, etc.) are omitted as they are of little help for higher-order constraints.

The constituent structural elements of a mechanism are the bodies (links,
members) and the joints between them. The topological graph is an undirected
graph

An edge is an unordered pair of vertices denoted

Figure

The joint

A kinematic chain can be evaluated recursively by starting from an initial
body. For mechanisms with kinematic loops there is a priori no unique chain
between two bodies. Such can be introduced with help of a spanning tree of

The recursive evaluation of the kinematics further requires an order relation that assigns to each body and joint its direct predecessor. Such a relation is induced by directing the spanning tree.

A

Joint

Denote with

The tree in Fig.

The edges of

Cotree

A rigid body is kinematically represented by a body-fixed reference frame.
The configuration of

The joint motion is interpreted according to the direction of the joint. Let
the tree-joint

The majority of technical joints can be modeled as combination of lower
kinematic pairs

Successive combination of the relative configurations of tree-joints in the
spanning tree allows to determine the configuration of all bodies in the
mechanism. This requires taking into account the assigned directions of the
tree-joints. To this end, an indicator function is introduced as

For the manipulator example with

The relative configuration of body

The joint variable

With the relative configuration Eq. (

the oriented topological graph

the root-directed tree

The tree-topology mechanism obtained after removing the cut-joints

For each of the

For the

For the

This method is used for kinematics modeling in computational MBS dynamics.
Consider the FC

The cut-joint formulation is also advantageous for the kinematic analysis
when only one higher kinematic pair is present in a FC. Then the
configuration, velocity, and acceleration, etc. of the two open chains with
terminal

The higher pair

In summary, the cut-joint formulation requires introduction of

the oriented topological graph

the root-directed tree

the FCs

This method is used for kinematic analysis of linkages, i.e. closed
kinematic chains comprising only lower pairs. Instead of eliminating the
cotree-joint

The orientation of

In the manipulator example, the two FCs in Fig.

Two oriented FC for the oriented topological graph

Successive combination of the relative configurations of all joints in the
FC leads to the closure condition for

Consider the manipulator example in Fig.

In summary, the cut-body formulation requires introduction of

the oriented topological graph

the spanning tree

an orientation of the FCs.

The kinematic analysis of a mechanism requires evaluation of the motion of its members, and formulation of a system of generically independent loop closure constraints. Any recursive evaluation of the motion of a mechanism rests on an ordering of bodies and joints. The configuration of a body is given in terms of the configurations of its predecessors that form a kinematic chain to the ground (reference body). For a multi-loop mechanism this chain is no unique. The spanning tree of the topological graph gives rise to a unique predecessor relation. This is introduced in this paper making use of a root-directed spanning tree (a tree such that there is an oriented path from any body to the ground). If the mechanism comprises lower pair joints only, the configuration is then recursively expressible by the product of exponentials (POE) in terms of joint screw coordinates.

The recursive formulation of loop closure constraints also requires an ordering, now within the loop. Here it is important that constraints are formulated for fundamental cycles (FC), i.e. for topologically independent kinematic loops. To this end, fundamental cycles are introduced on the topological graph together with an orientation. Two different constraint formulations are considered: cut-joint and cut-body formulation. The cut-joint formulation allows for a higher kinematic pair in a FC, whereas the cut-body formulation is tailored to linkages with lower pairs.

The basic difference of the proposed topology description compared to the various graph representations is that it does not involve matrix representations. Moreover, the presented notation provides the basis for a systematic higher-order analysis of the mechanism kinematics. This will be reported in forthcoming paper.

The author acknowledges that this work has been partially supported by the Austrian COMET-K2 program of the Linz Center of Mechatronics (LCM). Edited by: L. Birglen Reviewed by: four anonymous referees