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Representation of the kinematic topology of mechanisms for kinematic analysis

Abstract and Figures

The kinematic modeling of multi-loop mechanisms requires a systematic representation of the kine- matictopology,i.e.thearrangementoflinksandjoints.Alineargraph,calledthetopologicalgraph,isusedtothis 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 formulat- ing 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.
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Manuscript prepared for Mechanical Sciences
with version 2014/09/16 7.15 Copernicus papers of the L
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T
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X class copernicusAM.cls.
Date: 13 March 2017
Representation of the Kinematic Topology of
Mechanisms for Kinematic Analysis
Andreas Müller
Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria
Correspondence to: A. Müller (a.mueller@jku.at)
Abstract. 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 topo-
logical 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 represen-5
tation 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 with10
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.
Keywords: Kinematic topology, linear graphs, loop constraints, higher pairs, cut-joint, cut-body,
kinematic analysis, Lie groups
1 Introduction15
The kinematic topology of a mechanism refers to the existence and the arrangement of links and
joints, i.e. kinematic constraints. It is hence an adjacency relation that can be represented by an
undirected linear graph, referred to as the topological graph denoted with Γ. This graph provides the
basis for a systematic treatment of the kinematics of mechanisms and multibody systems (MBS).
Various types of topological graphs have been proposed in the literature. They have been an im-20
portant aspect for modeling of complex MBS. Wittenburg (1977, 2008) introduced a linear graph to
represent general interconnections of rigid bodies within a MBS. This linear graph includes not only
1
Corrected preprint of the paper published in Mechanical Sciences
doi:10.5194/ms-6-137-2015
Last revision: March 13, 2017
kinematic interconnections but also physical interactions like springs and dampers. This concept
was taken up by Arczewski (1992a, b, c). An exhaustive overview of graph representations in MBS
dynamics was presented by Jain (2011a, b). All these formulations can be used to derive compact25
matrix formulations of the MBS motion equations.
For kinematical investigations of mechanisms a graph representation of the kinematic topology
has been proposed by Davis (1981, 2015). In particular the velocity constraints were considered as
a variant of Kirchoff’s law for electric circuits, and it was concluded that the principle concepts and
results available for electric networks can be adopted to the velocity analysis of mechanisms. This30
approach addresses the velocity analysis leading to systems of linear equations, and thus adjacency
matrices, incidence matrices, etc. could be used to manipulate the governing equations. Wohlhart
(2004), for example, used the topological graph to derive the first-order constraints for topologically
independent loops, and to deduced the connectivity of links.
Topological graphs have further interesting features related to generic properties of the mecha-35
nism. The essential kinematic properties (of generic realizations) were investigated by Simoni et
al. (2013) using topological graphs. It was shown that properties like mobility and connectivity are
preserved by any automorphism of the graph. This may be important for topological synthesis.
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 becomes40
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 repre-
sentation and of formulations of loop constraints for multi-loop mechanisms in a way appropriate
for the higher-order kinematic analysis.45
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 co-
ordinates (joint coordinates) are used to parameterize the configuration allowing for a recursive
evaluation of the mechanism kinematics. The essential topological relations are:
1. an order relation to define predecessors of bodies and joints,50
2. an indicator of the direction in which a relative joint motion is defined, and
3. 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 oriented55
spanning tree ~
G, 2.) an oriented topological graph ~
Γ, and 3.) an oriented system of fundamental
cycles (FC) are introduced.
2
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.60
2 Graph Representation of Mechanism Topology
2.1 Topological Graphs
The constituent structural elements of a mechanism are the bodies (links, members) and the joints
between them. The topological graph is an undirected graph Γ=(B, J), where Bis the set of
vertices (representing bodies) and Jis the set of edges (representing joints). The graph Γis simple,65
i.e. two vertices are connected by no more than one edge. The number of joints and bodies is denoted
with N:= |J|and M:= |B|, respectively. Bodies are indexed with greek letters α= 0,...,M 1.
The index 0 is often used to refer to a ’fixed member’ or to the ’ground’. Joints are indexed with
Latin characters i= 1,...,N.
An edge is an unordered pair of vertices denoted Ji= (Bα,Bβ)J. This indicates that body Bα
70
and Bβare connected by joint Ji. For the sake of simplicity, the shorthand notations Ji= (α, β)J
and (α,β)Jare used.
Fig. 1b) shows the topological graph for the parallel mechanism in fig. 1a) that was reported by
Carricato & Parenti-Castelli (2002); Kim & Tsai (2002); Kong & Gosselin (2002). The mechanism
contains M= 11 bodies and N= 12 joints. Notice that for sake of simplicity, in all figures through-75
out the paper, edges are denoted simply with the index αinstead of Bα.
3
2
1
3
2
1
0
6
5
6
5
4
10
4
11 10 12
9
8
7
7
9
8
G
0
1
2
3
10 9
8
7
6
5
4
2
1
4
5
6
3
10
11
9
8
7
12
Ground
b)a)
Figure 1. a) 3 DOF parallel manipulator consisting of 10 moving bodies (labeled with circles) connected by 12
joints (labeled with rectangles). b) Topological graph Γfor the manipulator.
3
2.2 Joint Orientations and Oriented Topological Graphs
The joint Ji= (Bα,Bβ)constrains the relative motion of the two bodies Bβand Bα. That is, it can
be considered to determine the relative configuration of Bαw.r.t. Bβor that of Bβw.r.t. Bα. The
definition of this ’joint direction’ is an indispensable step within the kinematics modeling that further80
includes introduction of certain joint variables (angles, displacements) to describe the joint motion.
This is not revealed by the undirected edge JiΓ. For example in fig. 2a), the joint angle of the
revolute joint J2measures the rotation of B2relative to B1, whereas the angle of joint J4measures
the rotation of B1w.r.t. B4, so that the joint angle has the opposite meaning. In order to represent the
directions in which the joint motions are to be interpreted, an oriented graph ~
Γis introduced. This85
is obtained by considering the edges of Γas ordered pairs of vertices. That is, if (Bα,Bβ)~
Γ, then
(Bβ,Bα)/~
Γ. The vertex Bβis the source (or tail) and Bαis the target (or head) of the edge. This is
graphically indicated by an arrow. Then Ji= (Bβ, Bα)~
Γmeans that joint Jiis assumed to define
the motion of Bαw.r.t. Bβ. Edges of ~
Γare called arcs. Fig 2b) shows the oriented topological graph
for the mechanism in fig. 2a).90
Remark 1 Frequently the joint orientations are introduced upon algorithmic considerations. In par-
ticular, directions are often assigned so to form a root-directed spanning [Jain (2011a); Wittenburg
(2008)]. This limits the generality, however. Moreover, in an MBS modeling environment, joints are
introduced with well-defined and prescribed orientations. These orientations are in general different
from those introduced for an oriented spanning tree, which is merely an algorithm construct (see95
next section and section 3.3).
B1
B
B
2
3
B0
B4
B5
3
5
1
0
1
4
5
2
3
1
2
3 5
4
2
4
1
5
3
b)a)
Figure 2. a) A linkage with 5 revolute joints. The direction of positive joint angles is shown. b) Edges in the
oriented topological graph ~
Γare directed according to the positive joint angles.
4
2.3 Spanning Trees and Predecessor Relations
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 Γ. A spanning tree, denoted G=B, JG, is an acyclic subgraph of100
Γ, i.e. there is exactly one path between any two vertices. The spanning tree of a graph is not unique.
Fig. 3b) shows a spanning tree for the manipulator in fig. 1.
3
2
1
3
2
1
0
6
5
6
5
4
10
4
10
9
8
7
7
9
8
3
2
1
3
22
1
0
6
5
6
5
4
10
4
10
9
8
7
7
9
8
b)a)
Figure 3. a) A spanning tree Gfor the topological graph in fig. 1. b) The corresponding unique root-directed
tree ~
G0.
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.
Aroot-directed tree, denoted ~
G0, is introduced such that there is a directed path in ~
G0from B0
105
(the ground) to every vertex of Γ.B0is the root of the tree. Edges of ~
G0are called arcs. Fig. 3b)
shows the root-directed spanning tree for the PKM example in fig. 1.
Remark 2 Jain (2011a) called ~
G0the ’standard digraph’ associated to a mechanism. Moreover, this
has been used to actually define directions of joints. Consequently, the original allocation of joint
directions would be changed according to the particular (but not unique) directed spanning tree (see110
remark 1).
5
The predecessor of a body can now be defined relative to the root-directed spanning tree. Body Bβ
is the direct predecessor of Bα, if they are connected by an arc, i.e. (Bβ,Bα)~
G0. This is denoted
as Bβ=Bα1(for short β=α1).
Joint Jjis the direct predecessor of Ji, if the target of Jjis the source of Ji, i.e. Jj= (·)~
G0
115
and Jj= (α,·)~
G0. This is denoted as Jj=Ji1(for short j=i1). Joint Jjis a predecessor of
Ji, if there is a finite k, such that Jj=Ji11...1(ktimes). This is expressed as Jj=Jik.
Being a predecessor is indicated by Jj< Ji. Because of the tree topology it is possible that two
bodies have the same predecessor, i.e. α1 = β1, and analogously for joints.
Denote with J(Bα)the tree-joint that connects Bαwith its predecessor, i.e. J(Bα)=(β,α)∈ G,120
and with Jroot (Bα)the joint that connects to the root B0in the path from Bαwithin G.
The tree in fig. 3b), induces the following predecessor relations
J1=J21J2=J31J3=J10 1J4=J51
J5=J61J7=J81J8=J91
B0=B11B0=B41B0=B71B1=B21B2=B31
B3=B10 1B4=B51B5=B61B7=B81B8=B91
and the following assignment of tree-joints connecting the bodies
Jroot (Bα) = J1,α = 1,2,3,10 Jroot (Bα) = J4,α = 4,5,6Jroot (Bα) = J7= 7,8,9
J(Bα) = Ji,α =i.
125
2.4 Cotrees and Fundamental Cycles
The edges of Γthat are not in Gconstitute the cotree, denoted H=B,JH, with JH:= J/JG.
The cotree edges (or cut-edges) are the chords of the spanning tree. The topological graph of a
mechanism, where all restrains are due to kinematic couplings, consists of exactly one connected
component. Such a graph possesses γ=NM+1 independent fundamental cycles (FC), also called130
fundamental loops. The integer γis called the cyclomatic number (or Euler number) of Γ. A FC of
Γis a closed path (i.e. a sequence of edges) without repeated edges or vertices that contains exactly
one cotree-edge. Thus Hpossesses γedges. The FCs are denoted with Λl, where lis the index of
the cotree-edge in the FC. The FCs are not unique. The manipulator in fig. 1a) has 12 11 + 1 = 2
FCs, thus Hcomprises 2 edges. Fig. 4 shows the cotree Hto the spanning tree in fig. 3a) and the135
corresponding two FCs.
3 Recursive Determination of Mechanism Configurations
3.1 Rigid Body Configurations
A rigid body is kinematically represented by a body-fixed reference frame. The configuration of
Bαw.r.t. to a global reference frame can be represented by a homogenous transformation matrix140
6
3
2
1
3
2
1
0
6
5
6
5
4
10
4
11 10 12
9
8
7
7
9
8
L11 L12
Figure 4. Cotree Hto the spanning tree Gin fig. 2a). Λ11 and Λ12 are two FCs according to H. Here the
notation (α, β)is used instead of (Bα, Bβ).
[Murray et al. (1994); Selig (2005)]
Cα=
Rαrα
01
SE (3) (1)
where RαSO (3) is the rotation matrix transforming coordinates from the body-fixed reference
frame to the global frame, and rαR3is the position vector to the origin of the body-fixed reference
frame expressed in the world frame.145
3.2 Relative Joint Motions – Lower Pair Joints
The joint motion is interpreted according to the direction of the joint. Let the tree-joint Jibe con-
necting body Bαand Bβ. According to its direction Ji= (β, α)~
Γit determines the relative con-
figuration of Bαw.r.t. Bβ, which is given as
Di:= C1
βCαSE (3) .(2)150
This relation follows immediately, since Cαand Cβis the configuration of body αand βw.r.t. the
global frame. That is, Cαtransforms from body-fixed reference frame on Bαto the global frame,
and C1
βtransforms from global frame to the body-fixed frame on Bβ. Hence, Ditransforms from
body-fixed frame on Bαto body-fixed frame on Bβ.
7
The majority of technical joints can be modeled as combination of lower kinematic pairs [Uicker et155
al. (2013)]. Moreover, their motion can be expressed as combination of 1-DOF screw motions, with
pure rotations and translation as special cases, which is a traditional approach in mechanisms and
MBS modeling. The relative configuration of joint Jiis then (with appropriate choice of reference
frames) determined with the exponential mapping on SE (3) as
Di= exp(qiYi)(3)160
where YiR6is the screw coordinate vector and qithe joint variable [Selig (2005)]. This is a basic
result in space kinematics [Angeles (2003); McCarthy (1990)]. Details are omitted here as this is
beyond the scope of this paper.
3.3 Mechanism Configuration
Successive combination of the relative configurations of tree-joints in the spanning tree allows to165
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
σ(Ji) =
1, Bβis direct predecessor of Bα
1, Bαis direct predecessor of Bβ
0, Jiis not a tree-joint, for Ji= (β, α).
The short hand notation σ(i)is used for simplicity. More precisely, σ(i)=1, if (Bβ, Bα)~
G0
(Bβ,Bα)~
Γ; it is σ(i) = 1, if (Bβ,Bα)~
G0(Bα,Bβ)~
Γ; and σ(i)=0, if (Bβ,Bα)/∈ G.170
For the manipulator example with Γin fig. 1b), a joint orientation is chosen according to ~
Γin fig.
5, which also shows the indicator function.
The relative configuration of body Bαw.r.t. its predecessor Bβ=Bα1due to the tree-joint
Ji= (β, α)is then Dσ(i)
i, and with (3) this is expressed as Dσ(i)
i= exp(σ(i)qiYi). That is, the
transformation is reversed, if Jiis directed (i.e. measured) opposite to the root-directed tree.175
The joint variable qiis interpreted according to the joint direction. For a revolute joint, (3) is
the transformation matrix (1) in terms of the rotation angle qi. The meaning of the joint direction
is apparent. If the joint angle is measured from body Bαto Bβ, i.e., if it is directed opposite to
the root-directed tree, then σ(i) = 1, so that its meaning is reversed, and Dσ(i)
i= exp(qiYi).
For instance, the rotation angle of the revolute joint J4as defined in fig. 2a) is reversed in order to180
determine the motion of B4w.r.t. B1.
With the relative configuration (2) of the tree-joints, the configuration of an arbitrary body Bαis
given as
Cα=Dσ(r)
r·...·Dσ(i2)
i2·Dσ(i1)
i1·Dσ(i)
i, r =Jroot (α),i =J(α)(4)
8
3
2
1
3
2
1
0
6
5
6
5
4
10
4
11 10 12
9
8
7
7
9
8
Figure 5. a) Oriented topological graph ~
Γfor Γin fig. 1b). b) The function σ(i)indicates the direction of joint
Jirelative to the root-directed tree ~
Gin fig. 3b.
and using the expression (3) for lower pair joints yields185
Cα= exp(σ(r)qrYr)·...·exp(σ(i1) qi1Yi1)·exp(σ(i)qiYi),(5)
with r=Jroot (α),i =J(α).
For example, the configuration of B10 of the mechanism in fig. 1, according to the oriented tree in
fig. 3b) and the oriented topological graph in fig. 5, is
C10 =D1·D1
2·D3·D1
10
190
= exp(q1Y1)·exp(q2Y2)·exp(q3Y3)·exp(q10Y10 ).
For the mechanism in fig. 2a) the configuration of B5is (with σ(1) = σ(5) = 1, σ (4) = 1)
C5=D1·D1
4·D5
= exp(q1Y1)·exp(q4Y4)·exp(q5Y5).
The interpretation of the recursive relations (4) and (5) is straightforward. The configuration of Bαis195
the combination of the joint configurations when traversing the kinematic chain in ~
G0starting from
B0to Bαwhile noting the joint directions encoded in ~
Γ.
Remark 3 The formulation (5) is referred to as the product of exponentials (POE) formula [Brockett
(1984)] that gave rise to very compact formulations for the mechanism kinematics and algorithms
9
for the dynamics of MBS [Ploen & Park (1997); Park (1994)] employing the Lie group SE (3). An200
important aspect is that it leads to simple explicit algebraic relations for velocities, accelerations,
and higher-order derivatives of any order [Müller (2014b, a)]. Furthermore, this provides the basis
for higher-order kinematic analysis of mechanisms. This has been pursued in [Rico et al. (1999,
2008); Müller & Rico (2008)] for single-loop mechanisms. The extension to multi-loop mechanisms
requires a systematic yet simple description of the mechanism topology. This is the aim of the present205
paper.
In summary, starting from the basic topological information encoded in Γ, the determination of
the configurations of the bodies in a mechanism requires introduction of
the oriented topological graph ~
Γin order to represent the joint orientations as assigned in the
kinematics modeling, and210
the root-directed tree ~
G0in order to define an ordering that determines a unique predecessor
for each body.
4 Kinematic Loop Constraints
For each of the γFCs (and only for these) a system of kinematic constraints is introduced. Such
loop constraints can be formulated in two basically different ways: the cut-joint and the cut-body215
formulation.
For the cut-body formulation, a body within the loop, called the ’cut-body’, is virtually cut open
so to obtain an open kinematic chain comprising all joints in the loop. The loop closure constraints
require the cut-body to be (re)connected.
For the cut-joint approach, a joint within the loop, called the ’cut-joint’, is eliminated (cut open)220
from the kinematic model leading to an open kinematic chain comprising all joints in the loop except
the cut-joint. The loop closure constraints restrict the relative motion of the two bodies connected by
the cut-joint according to its mobility.
4.1 Cut-Joint Approach
This method is used for kinematics modeling in computational MBS dynamics. Consider the FC Λl
225
with cotree-edge Jl= (Bβ,Bα). The joint Jlis used as cut-joint, and removed from the FC. This
leaves two open kinematic chains with the respective terminal bodies Bβand Bαof the spanning
tree. Their configurations are determined by the tree-joint configurations via (4). A system of cut-
joint constraints is then formulated for Jlof the form
hl(Cα,Cβ) = 0.(6)230
The constraints (6) restrict the relative displacement and orientation, i.e. the configuration, of Bβ
and Bαaccording to the mobility of joint Ji. These are well-known for various joint types [Uicker
10
et al. (2013); Wittenburg (2008)]. Cotree edges are the chords of the spanning tree, which bears an
obvious kinematic meaning: a cut-joint reconnects the two terminal vertices of the spanning tree that
are linked by a chord so to close the FC.235
Remark 4 The constraints (6) only involve the joint variables of the tree-joints since the cut-joint
is removed from the kinematic model. The cut-joint approach is used in recursive MBS dynamics
algorithms. The main reason is that the dynamic motion equations of the tree topology system (the
mechanism defined by G) can be derived and evaluated, possibly with low-order algorithms, and the
constraints be imposed. As example, the tree-topology system for the manipulator in fig. 1 according240
to the tree in fig. 3 is shown in fig. 6. The cut-joints J11 and J12 are removed.
0
1
2
3
10
9
8
7
6
5
4
2
1
4
5
6
3
10
9
8
7
Ground
Figure 6. The tree-topology mechanism obtained after removing the cut-joints J11 and J12.
Remark 5 The overall system of loop constraints is possibly redundant due to the particular mech-
anism geometry. The computational treatment of such situations is a topic of ongoing research
[Arabyan & Wu (1998); Meijaard (1993); Wojtyra & Fraczek (2013)]. It should be remarked that the
POE formulation and its underlying Lie group concept allows to deduce redundant loop constraints245
and eventually to determine a reduced non-redundant set of constraints [Müller (2011, 2014c)].
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 Bβand Bαcan be expressed by the POE (5) since all other tree-joints
are lower pairs. For instance, the mechanism in fig. 7a) comprises a higher kinematic pair: the pin-250
in-slot joint J2. An oriented topological graph is shown in fig. 7b). The spanning tree in fig. 8a) is
introduced so that J2is in the cotree. Fig. 8b) shows the corresponding FCs.
11
a)
Ground
2
1
0
4
5
3
1
2
6
7
5
4
3
b)
0
61
4
33
5
7
54
1
22
Figure 7. a) Linkage with a higher kinematic pair J2(pin-in-slot joint). b) An oriented topological graph ~
Γfor
this linkage.
6L
3
7
5
2
0
1
4
33
54
1
2
0
1
4
2
3
L5
54
1
2
6
7
b)a)
Figure 8. a) A root-directed spanning tree ~
G0for the linkage in fig. 5. b) Oriented FCs Λ2and Λ5.
The higher pair J2= (B2,B1)is the cut-joint in Λ2. It imposes two rotational constraints and two
translational constraints on the relative motion of body B2and B1[Uicker et al. (2013)] that are
summarized as h2(C1,C2) = 0. The configurations of B2and B1are determined by the lower pairs255
of the tree topology system. With the orientation of tree-joints defined by ~
Γand ~
G0, these are
C1=D1
1= exp(q1Y1)
C2=D3D6= exp(q3Y3)exp(q6Y6).
12
The loop constraints h5(C4,C5) = 0for Λ5due to the revolute (lower pair) joint J5are derived
with260
C4=D4= exp(q4Y4)
C5=D3D7= exp(q3Y3)exp(q7Y7).
In summary, the cut-joint formulation requires introduction of
the oriented topological graph ~
Γrepresenting the joint orientations,
the root-directed tree ~
G0in order to determine the configuration of the bodies connected by265
the cut-joint, and
the FCs Λldefining the kinematic loops for which closure constraints are introduced.
4.2 Cut-Body Approach
This method is used for kinematic analysis of linkages, i.e. closed kinematic chains comprising only
lower pairs. Instead of eliminating the cotree-joint Jl= (Bβ,Bα), it is regarded as part of the closed270
kinematic chain defined by the FC Λl. This requires taking into account the orientation of the joints
within the FC. To this end, an orientation of the FC Λlis introduced such that it is aligned with the
cotree-edge Jl. The orientations of edges relative to Λlare indicated by the cycle incidence function
σl(Ji) =
1,(Bβ,Bα)~
Γis aligned with Λl
1,(Bβ,Bα)~
Γis directed opposite to Λl
0,(Bβ,Bα)/Λl,for Ji= (β, α).
Notice that σl(l)=1. These numbers are commonly arranged in the cycle incidence matrix of the275
oriented graph.
The orientation of Λlinduces an order relation in the FC. Jjis considered as predecessor of Jiin
Λl, if it is met after Jiwhen traversing the FC Λlstarting from Jl. This is denoted with Jj<lJi.
Clearly Ji<lJlfor all i6=l. Joint Jland the last joint in the FC connect to the same body Bα, i.e.
Ji= (·,Bα)and Jl= (Bα,·).280
In the manipulator example, the two FCs in fig. 4 can be oriented as in fig. 9 that also shows the
cycle incidence function. The ordering in Λ11, for instance, is such that 10 <11 3<11 2<11 1<11
4<11 5<11 6<11 11.
Successive combination of the relative configurations of all joints in the FC leads to the closure
condition for Λl
285
fl=I(7)
with
fl:= Dσl(i)
i·Dσl(j)
j·...·Dσl(k)
k·Dl,(8)
for i <lj <l...k <ll, and Ji,Jj,...,JlΛl
13
10
10
11 12
L11 L12
3
2
1
3
2
1
0
6
5
6
5
4
4
9
8
7
7
9
8
Figure 9. Two oriented FC for the oriented topological graph ~
Γin fig. 4). The cycle incidence function σl(Ji)
indicates whether joint Jiis oriented along or oposite to the FC Λl.
where Iis the 4×4identity matrix. If all joints are 1-DOF lower pairs, this can be written as290
fl:= exp(σl(i)qiYi)·exp(σl(j)qjYj)·...·exp(σl(k)qkYk)·exp(qlYl).(9)
The expression (8) and (9) can be interpreted as the configuration of the terminal body of a kinematic
chain comprising the joints i <l... <ll. The closure condition then requires this terminal body to
remain fixed. For this reason this approach is also known as the cut-body method [Samin & Fisette
(2003)]. The body connecting joints Jiand Jl(the cut-body) is virtually cut, and one half serves as295
terminal body of the chain. Merging the two halves then leads to the above constraints.
Consider the manipulator example in fig. 1a) with oriented topological graph and FCs in fig. 9.
The system of loop constraints for the FC Λ11 and Λ12 is respectively f11 =Iand f12 =Iwith
f11 := D10 ·D3·D1
2·D1·D4·D5·D1
6·D11
f12 := D10 ·D3·D1
2·D1·D7·D8·D1
9·D12
300
In summary, the cut-body formulation requires introduction of
the oriented topological graph ~
Γrepresenting the joint orientations,
the spanning tree Gdefining the FCs Λl, i.e. the kinematic loops for which closure constraints
are introduced, and
an orientation of the FCs.305
Remark 6 The cut-body formulation involves the configurations, thus the joint variables, of all
joints in the FC. The expression (9) has a major importance for the kinematic analysis of linkages
14
with lower pair joints. Firstly because it is determined solely in terms of the joint screw coordinate
vectors Yj, but secondly and most importantly, because its derivatives of any order can be deter-
mined by simple algebraic operation, namely the screw products (Lie brackets) of the instantaneous310
joint screws [Rico et al. (1999); Selig (2005); Müller (2014a)]. This is the basis for any higher-order
kinematic analysis of mechanisms.
Remark 7 The constraints (9) define the variety of admissible configurations of the kinematic loop
Λlas Vl:= {qVn|fl(q) = I}, and thus the configuration space of the linkage as
V:= \
l∈H
Vl.(10)315
This configuration space variety is the chief subject in the mobility analysis of mechanisms. Clearly,
a systematic method for multi-loop mechanisms shall rest on the identification of topologically inde-
pendent FCs. This has been introduced by Davis (1981, 2015) adopting the principles of Kirchoffs
circuit law for electric networks. It is not yet been used widely, however. This frequently leads to the
introduction of redundant constraints when topologically redundant loops are considered.320
Remark 8 The kinematic topology is inextricably connected to the (generic) structural mobility, i.e.
the mobility that a generic realization of a mechanism with a given number of bodies and joints
possesses. Structural mobility criteria hence estimate a lower bound on the mobility of a particular
mechanism. They only require structural information but no information about the topology. It is
instructive though, to note how topological information enters these criteria. The best-known mobil-325
ity criterion is the Chebyshev-Kutzbach-Grübler formula δgen =g(M1) PJiΓ(gδi)where
δiis the DOF of joint Ji, and gcharacterizes the ’motion type’ of the mechanism. For instance,
g= 3 for planar and spherical, and g= 6 for spatial mechanisms [Angeles (2003)]. The number g
can be specified without investigating the particular geometry if the motion of the loops form a mo-
tion subgroup of SE (3). Then the generic mobility, i.e. for generic geometries, is determined with330
g= 1,2,3,4,6. Now with PJiΓg=Ng this reads
δgen =X
iJ
δi. (11)
In other words for each FC a system of gconstraints is imposed. Hence, the generic DOF is deter-
mined once the number FCs is known. It was shown in Müller (2009) that this is in fact the correct
mobility for generic realizations.335
5 Conclusions
The kinematic analysis of a mechanism requires evaluation of the motion of its members, and for-
mulation 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
15
is given in terms of the configurations of its predecessors that form a kinematic chain to the ground340
(reference body). For a multi-loop mechanism this chain is no unique. The spanning tree of the topo-
logical 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.345
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 topo-
logically independent kinematic loops. To this end, fundamental cycles are introduced on the topo-
logical 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 a350
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 represen-
tations 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.355
Acknowledgements. The author acknowledges that this work has been partially supported by the Austrian
COMET-K2 program of the Linz Center of Mechatronics (LCM).
16
Appendix A: Nomenclature
N- number of joints in a mechanism
M- number of bodies in a mechanism
Γ,G,H- topological graph, spanning tree, and cotree in Γ
Bα- vertex representing body α= 0,...,M 1
Ji= (Bβ,Bα)- edge representing joint i= 1,...,N between bodies Bβand Bα
~
Γ- oriented topological graph indicating assigned direction of joint transformations
~
G0- root-oriented spanning tree, so that there is a unique directed path from the root B0to any Bα
σ(i)- function indicating the direction of joint irelative to the root-oriented tree
γ- number fundamental cycles of Γ
Λl- fundamental cycle of Γassigned to co-tree edge l∈ H
qi- joint variable of 1-DOF lower pair joint i
qVN- vector comprising all joint variables
Yi- screw coordinate vector of joint iin the zero reference configuration q=0
SE (3) - matrix representation of the Lie group of rigid body motions
δi- DOF of joint i
δgen - generic DOF of a mechanism
17
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19
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A Mathematical Introduction to Robotic Manipulation presents a mathematical formulation of the kinematics, dynamics, and control of robot manipulators. It uses an elegant set of mathematical tools that emphasizes the geometry of robot motion and allows a large class of robotic manipulation problems to be analyzed within a unified framework. The foundation of the book is a derivation of robot kinematics using the product of the exponentials formula. The authors explore the kinematics of open-chain manipulators and multifingered robot hands, present an analysis of the dynamics and control of robot systems, discuss the specification and control of internal forces and internal motions, and address the implications of the nonholonomic nature of rolling contact are addressed, as well. The wealth of information, numerous examples, and exercises make A Mathematical Introduction to Robotic Manipulation valuable as both a reference for robotics researchers and a text for students in advanced robotics courses.
Book
I Theory.- 1 Fundamental Mechanics.- 2 Dynamics of rigid bodies.- 3 Tree-like multibody structures.- 4 Complex multibody structures.- 5 Symbolic generation.- II Special topics.- 6 Road vehicles: wheel/ground model.- 7 Railway vehicles: wheel/rail model.- 8 Mechanisms: cam/follower model.- 9 Multibody systems with flexible beams.- 10 Time integration of flexible MBS.- III Tutorial.- 11 Introduction.- 12 Problems.
Chapter
In this paper a procedure is decribed which allows determining the relative screw spaces between any two links in a multilooped linkage most efficiently. The leading role is played by the null space of the general constraint matrix of the linkage, which is generated quasi automatically by a planar graph with edges oriented according to the joint screws. The null space of the constraint matrix provides a computer program numerically. With the joint screws on an arbitrary path between any two vertices in the graph a path matrix can be composed which, multiplied by the null matrix gives the screws which constitute the screw space between the two links the path connects in the graph. This space is independent of the chosen path. The rank of the null space matrix equals the overall degrees of freedoms of the linkage and the ranks of the relative screw spaces between any two links yield their relative degrees of freedom, i.e. their connectivity. All connectivities are finally compiled in a symmetric connectivity matrix. As the procedure takes into account the dimensions of the linkages right from the beginning, paradoxical linkages are not excluded.
Chapter
This paper describes a new 3-DOF translational parallel manipulator named a Cartesian Parallel Manipulator (CPM). The machine behaves like a conventional X-Y-Z Cartesian machine due to the orthogonal arrangement of the three supporting limbs. It is shown that there exists a one-to-one correspondence between the input and output of the manipulator. Three possible arrangements of the Z actuator are evaluated by stiffness mapping. A method for compensating actuator misalignment is described.
Article
This paper introduces a 3-DOF translational parallel manipulator called Cartesian Parallel Manipulator (CPM). The manipulator consists of a moving platform that is connected to a fixed base by three limbs. Each limb is made up of one prismatic and three revolute joints and all joint axes are parallel to one another In this way, each limb provides two rotational constraints to the moving platform and the combined effects of the three limbs lead to an over-constrained mechanism with three translational degrees of freedom. The manipulator behaves like a conventional X-Y-Z Cartesian machine due to the orthogonal arrangement of the three limbs. Two actuation methods are analyzed. However the rotary actuation method is discarded because of the existence of singularities within the workspace. For the linear actuation method, there exists a one-to-one correspondence between the input and output displacements of the manipulator The effects of misalignment of linear actuators on the motion of the moving platform are discussed. Each limb structure is exposed to a bending moment induced by external forces exerted on the moving platform. In order to minimize the deflection at the joints caused by the bending moment, a method to maximize the stiffness is suggested. A numerical example of the optimal design is presented.
Chapter
The configuration of a kinematic chain can be uniquely expressed in terms of the joint screws via the product of exponentials. Twists on the other hand can be represented in various forms. The particular representation is determined by the reference frame in which the velocity is measured and the reference frame in which this velocity is expressed. For kinematic analyses the spatial twists are commonly used. Analytical mechanism dynamics, on the other hand, uses body-fixed twists. The body-fixed twist of a moving body is the velocity of a body-attached frame relative to the spatial frame expressed in the body-attached moving frame. Accordingly the spatial and body-fixed twists are expressed in terms of spatial and body-fixed instantaneous joint screw coordinates, respectively. Crucial for analytical kinematics and dynamics are the derivatives of twists, and thus of the mechanism's screw system. Whereas higher-order derivatives of screw systems in spatial representation have been a subject of intensive research, the body-fixed representation has not yet been addressed systematically. In this paper a closed form expression for higher-order partial derivatives of the screw system of a kinematic chain w.r.t. the joint variables is reported. The final expression is a nested Lie bracket of the body-fixed instantaneous joint screws. It resembles the previously presented results for the spatial representation.