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A Unified Notation for Serial, Parallel, and Hybrid Kinematic

Structures

U. Thomas, I. Maciuszek, and F. M. Wahl

Institute for Robotics and Process Control, Technical University of BraunschweigMuehlenpfordtstrasse 23, 38106 Braunschweig, Germany, email [email protected]

Abstract

This paper proposes a new notation for kinematic structures

which allows a unified description of serial, parallel, and hy-

brid robots or articulated machine tools. During the past

decades, the well-known Denavit-Hartenberg parameters have

been used widely to describe serial kinematics of robots in sci-

ence and industry. Till now, such a common notation for par-

allel manipulators has not yet been accepted. This paper tries

to fill this gap by presenting a new notation, which is based on

the graph representation known from gear trains. In parallel

manipulators, spherical and cardan joints are widely used. In

order to describe these kinds of joints, the well- known DH-

parameter notation has been extended, so that, to each joint

as many joint variables can be assigned as degrees of freedom

exist. The notation is not only very useful for design, program-

ming, and simulation of parallel robots; it also can be applied

as a convention to refer to parallel or hybrid kinematic struc-

tures elsewhere.

Keywords: Parallel Manipulators, Kinematic Notation.

1 Introduction

During the past decade, parallel manipulators became more

popular. Today many of them are employed in industrial ap-

plications, for instance for high speed pick and place oper-

ations. Currently, many software/hardware tools for parallel

robots are under development. E.g. for graphical simulation,

motion planning and controlling the kinematic structures of

such manipulators need to be described in an appropriate man-

ner. Many scientists use their individual notation for assigning

the corresponding joint parameters to different parallel ma-nipulators depending on the problems they face, and on the

structures that they describe. For a notation of serial chain

segments inside hybrid kinematic structures, they often apply

Denavit-Hartenberg parameters [1]. In parallel and hybrid ma-

nipulators, frequently spherical and cardan joints are involved.

Describing these joint types by DH-parameters leads to a non-

unique representation. Furthermore, notating e.g. spherical

joints by DH-parameters conveniently results in 3 necessary

4-parameter sets. Moreover, many software tools for parallel

manipulators currently are under development. Usually many

engineers from various fields are involved in this process,

hence denoting a particular joint and its associated joint vari-

ables should be defined well. By now, no widely used com-

mon notation simultaneously suitable for serial, parallel, and

hybrid manipulators has been established. Thus, there exists

a strong demand for a unified notation for parallel and hybrid

kinematic structures, which might be as useful as the Denavit-

Hartenberg notation for serial manipulators. This paper pro-

poses a notation by extending the well-known DH-parameters

for denoting spherical and cardan joints in the DH-sense as

well. It also provides a solution for representing and referringto the kinematic structure of parallel and hybrid manipulators

in a unique manner. By specifying such a notation, the follow-

ing requirements have to be taken into account:

COMPLETENESS: The specification for a notationshould provide all kinematic and geometric data of a

given structure;

UNIQUENESS: An instantiated notation for a particularkinematic structure should refer only to this structure;

NON-R EDUNDANCY: Multiple notations of a particularproperty should be avoided. Each property has to be

represented once in the specification;

COMPREHENSIVENESS: The notation provided hereshould be expandable, for instance to describe dynamic

parameters, as well.

In the following, we propose a notation fulfilling these require-

ments while simultaneously reducing the number of parame-

ters necessary to describe spherical or cardan joints. The nota-

tion introduced in this paper is used in a big research effort on

parallel manipulators in which many mechanical and electrical

engineers as well as computer scientists are involved [2].

2 Related Work

Mayer and Gosselin [3] use a notation to describe a general

Gough-Stewart platform in which they refer to joints by their

layers. For describing spherical joints, they employ RPY-

parameters. Merlet and Gosselin [3, 4] classify parallel kine-

matic structures into SSM (Simplified Symmetric Manipula-

tors), TSSM (Triangular Simplified Symmetric Manipulators),


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and MSSM (Minimal Simplified Symmetric Manipulators).

In [5], a closed forward position solution is given for a PPSP-

manipulator. They describe spherical joints by three sets of

DH-parameters. Merlet [4] gives a notation for spherical joints

by three coplanar rotational joints. The notation proposed by

us is based on the approach by Belfirore and Benedetto [6],

who apply graphs to represent serial, parallel, and hybrid re-

dundant kinematic structures with one degree of freedom for

each joint. Graphs as shown there have also been applied inthe field of mechanisms and gear trains [7, 8]. For denoting

spherical joints we augment the DH-parameters, which has

also been mentioned by Veitschegger et al. [9], but just for

calibration purposes. In contrast to our proposal, Veitschegger

uses a rotation around the y-axis concatenated backwards to

the four DH-transformations as extension.

3 Description of Joints with more than one Degree of

Freedom

Joints with one degree of freedom can be described in a veryintuitive manner by DH-parameters (Fig. 1) whereby, to each

robot link, one transformation i1Ti is assigned with:

i1Ti =Rot(zi1,i) Trans(zi1, di) Trans(xi, ai) Rot(xi,i)

Figure 1: The Denavit-Hartenberg parameters

A cylindrical joint possessing two degrees of freedom can also

be denoted by this equation with i and di being the joint vari-ables. Other joints with more than one degree of freedom can

either be described by

several sets of DH-parameters,

or by a more complex transformation.

The advantages of the second approach are a convenient kind

of formulation, a reduction of redundancies and for each joint

as many joint variables are available as DoFs exist. Therefore,

we propose to describe joints with more than one degree of

freedom by augmenting the DH-parameters.

3.1 Notation of Spherical Joints

Spherical joints are often described by RPY-parameters, or by

three rotational coplanar joints as mentioned in [4]. Using this

description, it is possible to obtain a vector with arbitrary ori-

entation in space, i.e. without the orientation around its own

axis, Fig.2.

Figure 2: Notation of spherical joints by Merlet [4]

In our approach we extend the DH-parameters so that it is

possible to obtain three joint variables for a single spherical

joint. For control purposes, it is not essential to specify passive

spherical joints by three joint variables, but e.g. for simulation

we have to deal with all three degrees of freedom explicitly.

Consider the three linked joints depicted in Fig. 3:

Figure 3: The definition of axis for spherical joints

Each spherical joint is characterized by its center point.

Furthermore, one can connect these points virtually to obtain

normal vectors between the spherical joints. The cross prod-

uct ni ni1 of the succeeding and preceding joint normalsis assigned to be the z-axis; the normal itself represents the

x-axis and the y-axis is obtained by applying the right hand

rule. The x-axis points toward the succeeding joint. Using

this convention, it is possible to assign coordinate systems

to spherical joints in a non-ambiguous manner. Therefore,

a single extended transformation i1Ti for a spherical joint

consists of five elementary transformations defined by five


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parameters respectively:

i1Ti = Rot(zi1,i) Trans(zi1, di) Rot(yi1,i)

Trans(xi, ai) Rot(xi,i)

The DH-parameters have to be augmented by a rotation around

the y-axis. Fig. 4 shows a RSR-chain, where each transforma-

tion step is depicted for the spherical joint.

Figure 4: The extended DH-parameters for a spherical joint insidea RSR-chain

The first parameter i describes a rotation around the z-axis,di in this case is zero and the third parameter i is the rotationaround the y i1-axis, so that the new x-axis lies collinear to

the normal vector of the link. The translation along the x-axis

with parameter ai is the distance between the origins of the

corresponding coordinate systems. The last rotation around

the x-axis determines the third joint variable i . With this

transformation the coordinate system which corresponds to therobot linki moves around the sphere with radius ai. Using this

definition we are able to describe kinematic chains which con-

sist of a combination of rotational, prismatic or/and spherical

joints by a maximum of five parameters for each robot link.

For rotational and prismatic joints we set bi = 0 and obtain thefamiliar DH-parameters. In order to describe cylindrical joints

i and di are the joint variables.

3.2 Cardan Joints

Cardan joints can also be described by the notation introduced

above. The two degrees of freedom are represented by joint

variables i and i. The x-axis of the corresponding coordi-nate system lies collinear to the normal of the link and points

toward higher indexed joints (see Fig. 5).

3.3 Conventions for a Unique Extended DH-Definition

In order to define the coordinate systems uniquely, some con-

ventions are necessary:

The origin of the coordinate system i corresponding tolink i is located in the joint i+1.

The z-axis of the coordinate system i corresponds to the

Figure 5: Two cardan joints and their corresponding coordinate sys-tems following the extended DH-definition

rotational or prismatic joint axis of joint i+1. For spher-ical joints, the z-axis is assigned to the cross product of

the joint normal vectors ni

and ni+1

.

The x-axis of linki lies collinear to the joint normal andpoints toward higher indexed joints.

The rotational degrees of freedom must be instantiatedfrom the left side in the transform equation above, i.e.

if a joint has only one degree of freedom, i is the jointvariable, if it has two degrees of freedom i and i arethe variables. For spherical joints, i, i and, i with0 i 180

are the joint variables.

3.4 Examples of the Extended DH-Notation

In Fig. 6, Fig. 7, and Fig. 8 some examples of kinematicchains and their extended DH-parameter notations are shown,

which are typically used in the design of parallel robots.

Joint d a i1 var d i1 0 ai1 90

i var 0 var ai var

i+1 var 0 var ai+1 var

Figure 6: A RSS-Chain and its extended DH-parameter sets


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Joint d a i1 var 0 var 0 vari 90 var 90 0 0


Figure 7: A SPS-Chain and its extended DH-parameter sets

4 Notation of a Universal Robot

A notation of kinematic structures should be applicable to se-

rial, parallel, and hybrid robots. Fig. 9 left shows a Hexa-robot

and its kinematic structure (Fig. 9 middle) . The graph shown

in Fig. 9 right can be obtained, if we map robot links to nodes

of the graph and robot joints to graph edges. By this way,

links with multiple connections are represented by nodes in a

non-ambiguous manner. If we map joints to nodes and links

to edges, we would obtain an ambiguous definition of the tool

and the base platform.

Formally, a kinematic graph or a kinematic net of a universal

robot U R can be described by a 4-tuple

U R :=< N,E,BF, T F >

where N is the set of nodes, E is the set of edges, BF and T F

are used for specifying the robot tool and the base frame. The

kinematic graph is attributed by the following items:

N :=< n1, . . . ,nn > is a set of nodes. A node repre-sents a physical connection between at least two joints.

It corresponds to a robot link and consequently to itsgeometric entity. A node is defined by a tuple ni := . Slinkmeans one con-nection between two joints, Mlink represents multiple

connections to more than two joints and Base and Tool

are used for the base and tool platform respectively.

CSGi is the constructive solid geometry model or any

other CAD-representation for the robot linki.

E :=< e1, . . . ,em > is a set of edges. Each edge is uni-directional and represents a joint which connects two

robot links. Such an edge exists for each joint.

Joint d a i1 var 0 var 0 0

i 90 var 90 0 0


Figure 8: A CPS-Chain and its extended DH-parameter sets

The edges ei :=< Npre Npost, Tfix,1, Tparams, Tfix,2,Type, Range, Flag > are attributed by the following

items:

Npre Npost are pre nodes and post nodes.

Tfix,i 44 are fixed homogeneous transforma-

tions. They are used, for example, to transform

the robot base frame into the frame attached to the

first link of a single chain (see Fig. 10).

Tparams :=< , d,, a, > describes the parame-ters for a transformation i1Ti respective to the ex-

tended DH-parameter notation introduced above.The product Tfix,1

i1 Ti Tfix,2 describes the trans-formation of a single edge. For the transformation

from the base frame to the first joint, Tfix,2 is equal

to the identity matrix I and for the transformation

from the last joint to the tool Tfix,1 is set to I. In all

other cases Tfix,1 and Tfix,2 are the identity matrix

I.

Type=< rotational|prismatic|spherical| cardan|linear| ... > describes the type of joint.

Range defines the range of each joint.

Flag :=< passive, active > is used to mark an ac-tive or passive joint.

BF :=< F, ni > with F 44 as base frame attached

to the base platform represented by the node ni.

T F :=< F, nj > with F 44 as tool frame attached

to the tool platform represented by node nj.

With the notation introduced here, we are able to describe se-

rial, parallel, and hybrid kinematic structures in a systematic

and intuitive way. In any case we obtain a directed graph.


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!

Figure 9: The Hexa-robot its layout and its corresponding graph representation

!

" # $ %

' ( 0 2 4

6 7 9 7 B D

Figure 10: The three homogenous transformation for one edge

Fig. 11 shows the kinematic layout of the Eclipse [5] and its

corresponding directed graph. Fig. 12 demonstrates the use ofthis representation for the Portys-Robot [2]. As shown, we are

able to deal with all kinds of kinematic structures using one

common representation.

4.1 An Algorithm for the Enumeration of Nodes

In the case when different modules, for example a simulator,

a motion planner or a control module, have to exchange data,

a systematic way is desired to refer to links and joints in the

kinematic graph. The following algorithm assigns indexes to

nodes, such that each node obtains a unique number.

1. Unmark all nodes.

2. Start with the node to which the base frame is attached.

3. Assign the current index to the visited node.

4. Sort all unmarked neighbors such that the neighbor with

the geometrically shortest distance d is the first item in

the list etc. The distance between two adjacent link co-

ordinate systems and in their home position is de-fined by

d= (xx,yy,zz)T

or in case of zero by

d= (xx,yy,zz)T.

and represent the transformation annotated byTfix,1

i1 Ti Tfix,2 .

5. Increment the index.

6. For all unmarked successors continue with step 3.

7. Stop if all nodes have been visited.

This algorithm searches the nodes in depth first order and as-

signs indexes according to the criteria above.

5 Conclusion

The notation suggested here yields a representation which is

applicable to serial, parallel, and hybrid kinematic structures.

With the extended DH-parameters, we are able to describe

spherical, cardan, cylindrical, rotational, and prismatic joints

following the well-known intuitive notation by Denavit and

Hartenberg. When applying the proposed notation, it is pos-

sible to describe spherical or cardan joints with only 5 param-

eters instead of 3 4 parameters necessary when using thetraditional DH-notation. With the suggested graph representa-

tion, we can refer any element of any kind of kinematic struc-

ture uniquely. The notation suggested here has been used in a

kinematical simulation system able to deal with various typesof kinematic structures. For storing inertial parameters only

the node attributes of the graph representation need to be aug-

mented.

Acknowledgment

We would like to thank M. Krefft from the Institute of Machine

Tools and Production Techniques of the Technical University

of Braunschweig for the drafts of the robot pictures in Fig.9

left and Fig.12 left.


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Figure 11: The Eclipse-robot [5] its layout and its corresponding graph representation

Figure 12: The Portys-robot [2] its layout and its corresponding graph representation

References

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[3] B. Mayer St.-Onge and C.M. Gosselin: Singularity

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[7] A. G. Erdmann and G. N. Sandor: Mechanism De-

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