Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 16 potx

For the most part, these analyses havebeen focused on single-loop kinematic chains and parallel platforms.Most of the few studies about mobility and connectivity of general mul-tiloop li

has a constant direction with respect to a fixed frame, the second axis is orthogonal to the first one

A detailed kinematic analysis is carried out and leads to geometrical conditions to be verified by the mechanism for proper functioning Then a kinematic modelling illustrates the mechanism simplicity and provides a first evaluation of the machine’s workspace Finally, preliminary information is given regarding practical implementation of this new architecture

The proposed machine is a 6-actuator / 5-dof parallel mechanism In Fig 1, a joint-and-loop graph is depicted: grey boxes represent actuated joints; white boxes passive joints and circles express a kinematic coupling between two joints

Note that, in a general matter, the “spatial-parallelogram” chains

2

parallelogram” made of PR(RR)2R chains (as done on the Orthoglide, see Chablat et al., 2002) adds two constraints on the mechanism (3 translations and 1 rotation remain feasible)

449

chains) only add one constraint on a mechanism (3 translations and 2 rotations remain feasible) while a “spatial-

(that is, the P(SS)

Figure 1 Joint-and-loop graph Figure 2 Kinematics scheme

On PKM with Articulated Travelling-plate

The travelling plate is the one introduced in Krut et al., 2003, with the

I4L robot: while the two sub-parts shift one relatively to the other, a

mechanical device transforms this motion into a rotation Two types of

travelling plates exist (see Fig 3): Type 1 is made of two prismatic joints

and two kinematically coupled rack-and-pinion systems It has a

symmetrical design

3.2

Singularities analysis is often based on the analysis of the standard

Jacobean matrices J x and J representing the input-output velocity q

relationship:

where q and x are respectively the joint velocity vector and the

operational velocity vector

But other kind of singularities can occur (Zlatanov et al., 1998) To

reveal them, a deeper analysis is required At first, we will recall the fact

that “spatial parallelograms” can be seen in two different ways The

realistic case where spherical joints are modelled as 3-DoF joints and not

as 2-DoF joints is considered here Then, two types of modelling will be

given: one suggesting that the linear guide is a cylindrical joint (isostatic

modelling), and another assuming that it is a prismatic joint

(over-constrained modelling) In both cases, geometrical constraints, which

must be fulfilled to get rid of internal singularities, will be derived

According to Hervé’s notations (see Hervé, 1999) about displacement

subgroups, ^ `T stands for the subgroup of spatial translations and

^X( )u` stands for the subgroup of Schoenflies displacements (or Scara

motions), where u is a unitary vector collinear to the rotation axis If a

closed loop mechanism is composed of two chains producing Schoenflies

displacements withv uz , then:

Figure 3 Different possibilities of travelling plate

the parts Type 2 is made up with one part less, but loses Type 1’s

symmetrical design, which is good for balancing the loads among all

A Remark on Singularity Analysis

2easily handled with such a technique since those chains correspond to Schoenflies subgroup

(a) RR(RR)2R (b) R(SS)2

The case of machines with R(SS)2 chains (Fig 4-b) is more complex: each chain provides 5 DoF, 3T-2R, and does not correspond to a group Indeed, it is possible that the union (‰) of two 3T-2R chains generates a 3T-3R motion

This implies the recourse to a more complex analysis when dealing

with mechanisms based on R(SS)2 chains: for lack of space, this is recalled here, but the reader may find relevant information in Company

k  Note that the amplification ratio k1 is chosen equal to

D in order to have same rotation capabilities for T and M (+/-90 degrees for this design) Lengths of rods are: l i 0.9 m, i {1, ! , 6} Actuators limits are: 0 dq id 1.3m Fig 5 presents the domain where the condition number of the normalized Jacobean matrix is smaller than 8 (note that

by the actuators’ stroke)

451

e: this guarantees a large

along the Z direction corresponding to z, the workspace is only limited

(See Fig 2

for geometrical parameters explanations) The travelling plate is of type

Figure 4 Two ways to model “spatial parallelograms”

that is to say that such a mechanism will produce only three trans-

lations The case of machines with RR(RR) R chains (Fig 4-a) is

On PKM with Articulated Travelling-plate

ure

J J W ) < 8 Figure 6 CAD View of the Eureka

It could be interesting, for simplicity purposes, to connect the “single rods” directly to the travelling plate; however, such a practical design faces too many self-collisions The machine depicted in Fig 7 (left) shows such a practical design Another architecture avoiding self-collisions is shown in Fig 7 (right) It involves curved shapes of the single rods in order to avoid self-collisions

A prototype is about to be built The practical design is extremely simple thanks to Linear motors (Fig 6) Dimensions are the ones introduced for computing the workspace Rods and travelling plate are

prototype

Figure 7 Self-collision-free design #1 and #2

made of aluminium Instead of using rack-and-pinion systems, the mobile platform has been equipped with cable-pulley devices This kinematics provides the same displacements as those of the Tricept robot This design is well suited for the manipulation of light objects, but other applications are still possible

A design of a haptic master arm based on this kinematics is proposed

in Fig 8 It uses revolute actuators instead of prismatic ones, so the

footprint is reduced DD motors are used in order to reduce friction The required range for angular displacements is +/ 45 degrees This allows the use of an articulated travelling plate based on a planar parallelogram

provides three Translations plus two Rotations (3T-2R) The missing rotation (to get the complete master arm) is obtained using a carried revolute axis, located directly on the ending stick This is similar to the design of classical master arms, such as the PHANToM (SensAble Technologies)

In this paper, several techniques for reaching high tilting angles have been presented, with a focus on solutions related to articulated travelling plates Even though such results are still at an early stage of

development, they show that it might be possible to use (i) on the one

hand, travelling plates embedding passive joints which allows local

motion amplification, and (ii) on the other hand, actuation redundancy as

453

to provide the desired rotation The translation to rotation trans- formation is then suppressed and friction reduced The Eureka base

Figure 8 CAD view of the Eureka haptic arm

On PKM with Articulated Travelling-plate

–

a way to overcome some singular positions that usually limit the range of motion

References

Angeles, J., Morozov, A., Navarro, O., A novel manipulator architecture for the

production of SCARA motions, Proceddings of IEEE International Conference

on Robotics and Automation, San Francisco, April 24-28, 2000, pp 2370-2375

Chablat, D., and Wenger, P., Design of a Three-Axis Isotropic Parallel

Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, October 3-4, Québec, Québec, Canada, 2002

Clavel, R., Une nouvelle structure de manipulateur parallèle pour la robotique

légère, APII, 23(6), 1985, pp 371-386

Company, O., Krut, S., and Pierrot, F., Internal Singularity Analysis of a Class of

Lower Mobility Parallel Manipulators with Articulated Travelling Plate, IEEE Transactions on Robotics, 2006 (to appear)

Koevermans, W.P., Design and performance of the four dof motion system of the

NLR research flight simulator, Proceedings of AGARD Conference, La Haye,

20-23 October 1975, pp 17-1/17-11

Krut, S., Company, O., Benoit, M., Ota, H., and Pierrot, F., I4: A new parallel

mechanism for Scara motions, Proceedings of IEEE International Conference

on Robotics and Automation, Taipei, Taiwan, September 14-19, 2003

Pierrot, F., and Company, O., H4: a new family of 4-dof parallel robots,

Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Atlanta, Georgia, USA, September 19-22, 1999, pp 508-513

Reboulet, C., Rapport d’avancement projet VAP, thème 7, phase 3 Rapport de Recherche 7743, CNES/DERA, January 1991

Rolland L., The Manta and the Kanuk: Novel 4 dof parallel mechanism for

industrial handling, Proceedings of ASME Dynamic Systems and Control

Ryu, S.J., Kim, J.W., Hwang, J.C., Park C., Ho, H.S., Lee, K., Lee, Y., Cornel U., Park, F.C., and Kim, J., ECLIPSE: An Overactuated Parallel Mechanism for

Rapid Machining, Proceedings of ASME International Mechanical Engineering Congress and Exposition, Vol 8, USA, 1998, pp 681-689

Zlatanov, D., Fenton, R.G., and Benhabib, B., Identification and classification of

the singular configurations of mechanisms, Mechanism and Machine Theory,

Vol 33, No 6, pp 743-760, August 1998

pp 831-844

Division IMECE’99 Conference, Nashville, November 14-19, 1999, vol 67,

Manipulator for Machining Applications: The Orthoglide, Proceedings of

719-730

Hervé, J.M., The Lie group of rigid body displacements, a fundamental tool

for mechanism design, Mechanism and Machine Theory, Vol 34, 1999, pp

MOBILITY AND CONNECTIVITY

Abstract This contribution provides new definitions of infinitesimal mobility and

connectivity of kinematic chains These definitions are straightforwardly connected with accepted definitions of finite mobility and connectivity Further, screw theory is applied to the determination of the infinitesimal mobility and connectivity of multi-loop linkages These results provide

a guide for the determination of the finite mobility and connectivity of general topology of multiloop linkages, one of the important remaining problems in mobility theory.

Keywords:

The last three years have seen a flurry of studies about mobility andconnectivity of kinematic chains For the most part, these analyses havebeen focused on single-loop kinematic chains and parallel platforms.Most of the few studies about mobility and connectivity of general mul-tiloop linkages deal with the mobility and connectivity determined fromtheir velocity analysis It is well known that the information gatheredvia velocity analysis of any class of kinematic chains does not provideconclusive information about their mobility and connectivity In this

© 2006 Springer Printed in the Netherlands.

455

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 455–464.

IN MULTILOOP LINKAGES

contribution, it is shown that higher order analyses, in particular

Mobility, connectivity, multiloop linkages, screw theory

C´esar Real Diez-Mart´ınez

Instituto Tecnol´ogico y de Estudios Superiores de Occidente

Guadalajara, Jal M ´ exico

Facultad de Ingenier a Meca´nica El ´ ectrica y Electr´onica

Universidad de Guanajuato, Salamanca, Gto M ´ exico

Departamento de Ingenier´a Mecanic ´

Instituto Tecnol´ogico de Celaya, Celaya, Gto M ´ exico

i

noted that Wohlhart, 1999 and Wohlhart, 2000, employed higher orderanalyses to shed light into the characteristics of singular positions offully parallel platforms In contrast, in this contribution the authors areinterested in general topology multiloop linkages Further, the higherorder analyses are employed, in addition, as an important aid in thedetermination of their mobility, a problem that remain unsolved in thisgeneral case.

In this section, a review of the concepts of mobility and connectivity

as well as new definitions of infinitesimal mobility and connectivity arepresented

chain The finite mobility of the chain, denoted by M F, in a givenconfiguration is the number, minimum and necessary, of scalar variablesrequired to determine the pose, with respect to a link regarded as refer-ence, of all the remaining links of the kinematic chain

In our approach, the finite mobility depends not only on the kinematicchain, but also of the configuration of the kinematic chain to be analyzed,and it becomes a property of the configuration of the kinematic chainand its neighborhood This definition is motivated by the presence ofkinematotropic chains, Galletti and Fanghella, 2001, and differs from thedefinition, usually presented in undergraduate and graduate textbooks,and adopted by Gogu, 2005

Consider now the velocity analysis equation of the kinematic chain,

in a given configuration, which can be written as follows

where, the Jacobian matrix, J , is a matrix with as many columns as

screws associated with the kinematic pairs of the chain and as many rows

as fundamental circuits and loops of the kinematic chain, multiplied by

6 Additionally,
ω is the vector of joint rates, translational or angular, associated with the screws of the chain, and the vector
0 has the same

number of rows as the Jacobian matrix

Definition 2 Consider a kinematic chain whose velocity analysis

equation is given by Eq 1 Then, the kinematic chain has first order

infinitesimal mobility if there exists a vector
ω1
=
0 that satisfies the

Eq 1 Moreover, the number of independent components of the vector

ω determines the number of first order degrees of freedom, or first

acceleration analysis, can be successfully employed in shedding light onthe mobility and connectivity of general multiloop linkages It should be

order infinitesimal mobility, denoted by M1, of the chain in a givenconfiguration.

Consider now the acceleration analysis equation of the kinematic chain

in the same given configuration, which can be written as follows

Therefore, a vector ˙
ω whose components are numerically equal to those

of any of the vectors
ω1, solution of the Eq 1, is also a solution of Eq

3 Furthermore, a necessary and sufficient condition for the Eq 2 tohave a particular solution is given, Bentley and Cooke, 1971, by

Rank(J ) = Rank(J, $ L ). (4)

If Rank(J ) is less than the number of matrix rows, Eq 4 frequently poses additional conditions over the components of the vector
ω1 Theseadditional conditions require that one or more of the independent com-

im-ponents of
ω1 satisfy additional equations, frequently, these conditions

require that one or more of the independent components of
ω1 be zero

Let
ω2 be the solution of both, the velocity and the acceleration ses equations This result, provides the rationale for defining the secondorder infinitesimal mobility of the chain

analy-Definition 3 Consider a kinematic chain in a given configuration,

such that the velocity and acceleration analyses equations are given byEqs 1, 2 The chain has a second order infinitesimal mobility if there

is a non-zero vector
ω2 that satisfies both equations Furthermore, the

number of independent components of the vector
ω2determines the

num-ber of second order degrees of freedom, or second order infinitesimal

Similarly, it is possible to define similar concepts regarding the nectivity

con-Consider a kinematic chain in a given configuration

The finite connectivity between a pair of links (i, j), in the kinematic

457

see Rico et al., 1999

Mobility and Connectivity

of the non-homogeneous system, Bentley and Cooke, 1971 Thus, the

Definition 4.

chain is the minimum and necessary number of joint – scalar – variablesthat determine the pose of one link with respect to the other, and it if

denoted as C F (i, j).

Consider an arbitrary kinematic chain and assume that (i, j) is an

arbitrary pair of links of the chain Further, assume that the velocity

analysis of the chain has been solved and that the unique velocity state,

i V
j

1, of link j with respect to link i, following all possible paths between links i and j has been determined The velocity state i V
1j depends onindependent variables that solve the velocity analysis solution, contained

in
ω1 These elements are linear or angular velocities associated withthe kinematic pairs of the chain Further, i V
j

1 is a vector space Then

it is possible to define the first order infinitesimal connectivity

Consider a kinematic chain in a given configuration

C1(i, j) = dim( i V
1j ). (5)

Consider a kinematic chain in a given configuration

and let (i, j) be a pair of links Further, assume that the velocity and

acceleration analyses of the chain has been solved The number of

in-dependent variables of the vector
ω2 might be less than the number of

independent variables of the vector
ω1 Assume that the unique velocity

state of link j with respect to link i, using the solution of the velocity and acceleration analyses
ω2, denoted by i V
2j, is also known Then,

their second order infinitesimal connectivity, denoted by C2(i, j),

is defined by

C2(i, j) = dim( i V
2

j

Higher order mobilities and connectivities can be defined accordingly

Linkages

Consider the multiloop spatial kinematic chain shown in Fig 1, posed by Fayet, 1995 and used by Wohlhart, 2004 The chain has twospherical pairs, four cylindrical pairs, a planar pair and three revolutepairs

pro-Definition 6.

Definition 5.

and let (i, j) be a pair of links The first order infinitesimal connec

-z

y

x

Locating the origin of the coordinate system at point A, the screws

associated with the kinematic pairs are given by

A1 5$5a = (1, 0, 0; 0, 0, 0), B1 4$4a = (1, 0, 0; 0, 1, 0), A2 5a$5b = (0, 1, 0; 0, 0, 0), B2 4a$4b = (0, 1, 0; −1, 0, 0), A3 5b$4 = (0, 0, 1; 0, 0, 0), B3 4b$3 = (0, 0, 1; 0, 0, 0), C1 1$1b = (0, 0, 0; 1, 0, 0), D1 2$2a = (0, 0, 0; 0, 1, 0), C2 1b$1c = (0, 0, 0; 0, 1, 0), D2 2a$1 = (0, 1, 0; 0, 0, 1), C3 1c$6 = (0, 0, 1; −1, 0, 0), E1 1$1a = (0, 0, 0; 0, 1, 0),

F 3$2 = (0, 1, 0; −1, 0, 1), E2 1a$5 = (0, 1, 0; 0, 0, 0), G1 6$6a = (0, 0, 0; 0, 1, 0), H 8$7 = (0, 0, 1; 1, 1, 0), G2 6a$4 = (0, 1, 0; −1/2, 0, 0), J1 7$7a = (0, 0, 0; 0, 1, 0),

K 6$8 = (0, 1, 0; −1/2, 1, 0), J2 7a$3 = (0, 1, 0; −1, 0, 0),

where the screws 5$5a, 5a$5b and 5b$4 correspond to the spherical pair

located in point A Similarly, for the remaining kinematic pairs Up to

the velocity analysis, the approach follows that proposed by Wohlhart,chain is shown in Fig 2 The graph associates links with vertex andkinematic pairs with edges

2004, see also Baker, 1981 The graph associated with the kinematic

Figure 1. Kinematic chain proposed by Fayet and used by Wohlhart.

4ω 4a 4a ω 4b 4b ω3

3ω2

2ω 2a 2a ω1

1ω 1a 1a ω5

1ω 1b 1b ω 1c 1c ω6

6ω 6a 6a ω4

6ω8

8ω7

7ω 7a 7a ω3

where $0is a six dimensional vector whose elements are all equal to zero.The solution of the velocity analysis solution, given by Eq 7, is found

to be

5ω 5a = 0, 5a ω 5b= 21ω 1b+ 24b ω3− 1a ω5, 5b ω4 = − 4b ω3, 4ω 4a = 0,

4a ω 4b= − 21ω 1b − 2 4b ω3, 3ω2 = 21ω 1b+ 24b ω3,

2ω 2a = −1ω 1a , 2a ω1 = − 21ω 1b − 2 4b ω3, 1c ω6 = − 4b ω3, 6ω 6a = 1ω 1a − 1b ω 1c ,

6a ω4 = 21ω 1b+ 24b ω3, 8ω7 = 4b ω3,

7ω 7a = 1ω 1a − 1b ω 1c − 4b ω3− 21ω 1b , 7a ω3 = − 21ω 1b ,

6ω8 = 21ω 1b ,

where,1ω 1b,1b ω 1c,1ω 1a,1a ω5and4b ω3can be selected arbitrarily

There-fore, the first order mobility is, M1 = 5 Furthermore, the first orderconnectivity matrix is given by

The acceleration analysis equation has a solution, if and only if, the

augmented matrix, J a, obtained by augmenting the coefficient matrix,

J , with the column given by L S =

$L1 $L2 $L3 T, or

J a=

J L S 

(9)where, $L1, $L2 and $L3 are the Lie screws of the three loops of thekinematic chain, satisfy the condition

Rank(J a ) = Rank(J ). (10)This condition yields

following additional results

Thus, only1ω 1a,1a ω5 and 1b ω 1c can be arbitrarily selected Hence, the

second order infinitesimal mobility is M2= 3 Further, the second orderinfinitesimal mobility matrix is given by

paths, I, II, III, IV , between links 1, regarded as the fixed platform,

and 5, regarded as the moving platform, are the column spaces of thematrices

It is easy to recognize 1V a5 as the subalgebra, of the Lie algebra of the

Euclidean group, se(3), associated with cylindrical displacements along the y axis This result accounts for two of the degrees of freedom, from

the three determined by the second order infinitesimal mobility Theyare the finite displacements associated with the screws 1$1a and 1a$5

translational motion, along the same axis y, and produced by the screws 1b$1c , located in point C, and7$7a , located in point J , while the revolute

joints located between them remain inactive This degree of freedom ispassive when the fixed and moving platforms are links 1 and 5

The conclusion is that the finite mobility of the multiloop linkage is

M F = M2 = 3 Therefore, the finite connectivities among the diferent

links, C F (i, j) are given by the elements of the second order infinitesimal connectivity matrix C II

This contribution has shown that it is possible to provide higher-orderdefinitions of infinitesimal mobility and connectivity that are congruentwith the usual definitions of finite mobility and connectivity They pro-vide a guide for the computation of finite mobility of general multilooplinkages, this is, in the opinion of the authors, the most difficult task inmobility computations The results have been verified using Adamsc

The first author thank Conacyt for the support of his M Sc studies.The authors thank Concyteg for the support of several projects, includ-ing a thesis scholarship for the first author This work is based on his

M Sc thesis at the Instituto Tecnol´ogico de Celaya

References

Baker, J E (1981), On Mobility and Relative Freedoms in Multiloop Linkages and

Structures, Mechanism and Machine Theory, vol 16, no 6, pp 583-597.

Bentley, D L., and Cooke, K L (1971), Linear Algebra with Differential Equations,

New York: Holt, Rinehart and Winston.

Fayet, M (1995), M´ ecanismes Multi-Boucles I D´ etermination des Espaces de Torseurs Cin´ ematiques Dans un M´ecanisme Multi-Boucles Quelconque, Mechanism and Ma- chine Theory, vol 30, no 2, pp 201-217.

Galletti, C and Fanghella, P (2001), Single-Loop Kinematotropic anism and Machine Theory, vol 36, no 6, pp 743-761.

Mechanisms,Mech-Gogu, G (2005), Mobility of Mechanisms: a Critical Review, Mechanism and Machine Theory, vol 40, no 9, pp 1068-1097.

located in point E The remaining degree of freedom is related to the

,