Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 14 pps

3.3 Generation of Mechanisms Equations 7-8 provide the geometric conditions for legs of US-type to fit within the desired spherical motion between platform and base.. That is, in a feasi

3.2 Synthesis Procedure

The synthesis of a US-PM amounts to finding the location of the U and

S joints on the base and on the platform respectively, and the lengths of

the connecting legs That is, if for each leg k, we introduce the column

array of components p k >p k ; q k ; r k@, with respect to S1, of the point Pk which

is the center of either the U or the S joint on the platform, the column

array of components B k >A k ; B k ; C k@, with respect to a frame parallel to S0

but centered in C, of the point Bk which is the center of either the S joint

or the U joint on the base, and the length l k= _Pk  Bk_ of the connecting

leg between Pk and Bk , we have to search for N unknown arrays of

geometric parameters, i.e Q k > p k ; B k ; l k @ for k 1, ,N Use of the method

M1 or M2 makes it possible to find the conditions for Q k so that the leg

PkBk fits within the desired 2-dof spherical motion between platform and

base

Method M1: Assuming the description and the parameterization of the

orientation introduced in Section 3.1, the length of the k-th US-leg is

given by

P B

which, according to Eq 1 is a function of the motion parameters D and E

However, for the k-th US-leg to comply within the desired motion of the

platform, the length l k must not depend on D and E This means that if

one considers the expression

2 2

all the coefficients in cD, sD, cE and sE of the polynomial f D,E must

vanish, and the length of the k-th leg must become

k k

B q , B r k k 0, C q k k 0, C r k k 0, (6.2)

to be satisfied simultaneously for each leg PkBk

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It turns out that the only non trivial solutions, i.e l k z 0, are obtained

for the following two sets of parameters

1Q i A i; B i; C p i; i 0;q i 0;r i 0;l i A i2 B i2 C i2 , (7)

2Q j A j 0;B j 0; C j; p q j; j 0;r j 0;l j p2j C2j (8)

Method M2: Since for a general motion of the US-PM the parameters

D and E are functions of time, based on the time derivative of the

where D and E are the first time derivatives of the motion parameters

D and E, respectively Since the legs PkBk have constant length l k, it must

hold that, during the motion, the components of the velocity v k in the

direction of the leg axis must be zero, i.e

 , k ks k kc c k kc s k kc k ks c k ks

g D E p A D q A D Er A D Ep B D q B D E r B D E, (12)

 , k kc s k kc c k ks s k ks c k kc k k

h D E q B D E r B D E q A D E r A D E q C E r C sE (13)

Since this condition must hold for every configuration and motion of the

platform (that is it should not depend on D, E, D and E), all the

coefficients in cD, sD, cE and sE of the two polynomials g D,E and h D,E

must vanish

Vanishing of the coefficients of Eqs 12-13 leads to the same conditions

expressed by Eq 6 and, therefore, to the non-trivial solutions

3.3 Generation of Mechanisms

Equations 7-8 provide the geometric conditions for legs of US-type to fit within the desired spherical motion between platform and base The conditions identify two types of legs Legs of type-1 have one joint located

at point C in the platform and the other joint located anywhere in the base Legs of type-2 have the joint in the based located on kk0 axis and the joint in the platform located on ii1 axis

Generation of mechanisms amounts to combining a proper number I

of legs of type-1 and a proper number J of legs of type-2 Of course, the

choice of I and J clearly affects the mechanism architecture and its feasibility In particular, certain conditions on I and J must be satisfied First, in order for the mechanism to have 2-dof, the axes of the legs in the set ^1 1 2 2

1, , I, 1, , J

Q Q Q Q ` must belong to a linear variety of lines with rank 4 (Merlet, 1989), usually referred to as linear line congruence Therefore, at least four legs with linear independent axes are needed, i.e Here, the axis of the k-th leg is defined as the

line through point P

4

IJt

k to point Bk Second, since the legs (of type-1) in the set ^1 1

1, , I

Q Q ` pass through the common point C, i.e the center of the reference frame 1, while the legs (of type-2) in the set ^

S

1, , J

Q Q ` lie in the plane defined by the

vectors k0 and i1, the axes of the legs within each type form, at most, a linear variety of lines with rank 3 (Merlet, 1989) Indeed, the axes of the legs within the family of type-1 generate at most a bundle of lines, while the axes of the legs within the family of type-2 generate at most a plane of lines Therefore, in order for the set of geometric parameters

1, , I, 1, , J

Q Q Q Q ` to define a linear variety with rank 4, at least one leg for each type is needed, i.e It 1 and Jt 1

That is, in a feasible 2-dof spherical fully parallel mechanism with legs

of US-type, the axes of all the legs must define a degenerate congruence,

i.e the variety of lines which lie in the plane defined by unit vectors k0and i1 or pass through the point C of that plane

In practice, depending on the varieties of lines spanned by the axes of the legs within a type, three families of mechanism architectures can be identified:

x Family-1 (Fig 2): The axes of the legs in the set ^1

1 , ,1 I

, ,

` define a linear variety with rank 1, i.e a single line passing through C but

with direction different to k0, and the axes of the legs in the set

^2 1 2

J

, ,

` define a linear variety of lines with rank 3, i.e a plane of

lines defined by k0 and i1

x Family-2 (Fig 3): The axes of the legs in the set ^1

1 1 I

Q Q ` define a linear variety with rank 2, i.e a planar pencil of lines with center in

R Vertechy and V Parenti-Castelli

390

C but which does not contain the line through k0, and the axes of the

legs in the set ^2Q1, , 2Q J

, ,

` define a linear variety of lines with rank

2, i.e a planar pencil of lines in the plane defined by k0 and i1

x Family-3 (Fig 4): The axes of the legs in the set ^1

1 1 I

, ,

` defines a linear variety of lines with rank 3, i.e a bundle of lines centered in C, and the axes of the legs in the set ^2 1 2

J

Q Q ` define a linear variety of lines with rank 1, i.e a single line in the plane defined by the unit

vectors k0 and i1 but which does not pass through C

Figures 2-4 depicted the basic (non-over-constrained) mechanisms which are the generators of the three families For convenience, in the figures, both U and S joints are represented by circles

Moreover, addition of legs of type-1 and/or of type-2 to such basic PMs does not alter the mechanism kinematics but renders the systems redundant and with self-motion

US-Examples of over-constrained US-PM with five and six US-legs are depicted in Fig 5 For ease of understanding, the redundant legs are drawn in long-dash-dot lines Note that over-constrained architectures have several advantages with respect to the basic ones Indeed, the former make it possible to augment the mechanism stiffness-to-weight

391

Figure 2 Family-1 Figure 3 Family-2 Figure 4 Family-3

Figure 5 Over-constrained mechanisms

Synthesis of 2-DOF Spherical Fully Parallel Mechanisms

and encumbrance ratios, the mechanism strengthweight and encumbrance ratios, allow the mechanism to be preloaded so as to reduce system backlash, and allow the system to be built through simpler elements such as rafters and wireropes As an example, the mechanism depicted in Fig 5.a can be made by means of one rafter (leg drawn in long-dash-dot line) and by four wireropes (legs drawn in solid line) Besides, over-constrained architectures further limit the range of motion

-to-of the mechanism and render its assemblage more complex

From a kinematic standpoint, note that many of the U and S joints of the mechanisms depicted in Figs 2-5 may, in practice, be suppressed and/or replaced by simpler pairs Indeed, joints which are not placed

along the axes k0 and i1 are idle; the joints which are placed either on k0

or i1 work as simpler revolute joint with rotation axis along k0 or i1,

respectively; and the joints which are placed on both k0 and i1 work as U

joints with rotation axes along k0 and i1 However, from the kinetostatic standpoint, suppression of the idle degrees of freedom of the U and S joints makes the legs to bear consistent flexional loads which may cause the mechanism to be oversized Replacement of the idle pairs with elastic hinges introduces much smaller flectional loads and, therefore, may be the most effective way to implement the mechanism

4 Actuation

The mechanisms presented in Section 3.3 are inherently suited to be actuated in-parallel by motors with linear motion In practice, the addition of two UPS-legs (P stands for actuated prismatic P joint), each connecting the base and the platform by means of a U joint and an S joint, provides a very simple means to fully control the mechanism throughout the desired range of motion

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Figure 6 Actuated mechanism of family-3

.

decouple the motion about the axes k0 and i1 Indeed, since a force is not able to generate moments about the lines it crosses, it is clear that every UPS-leg whose connecting joint on the base is centered in a point Bk,

which lies on k0, makes it possible to control rotations about i1 only, while every UPS-leg whose connecting joint on the platform is centered in a point Pk, which lays on i1, makes it possible to control rotations about k0

only

A decoupled actuated manipulator obtained from a US-PM of family-3

is represented in Fig 6 (UPS-legs are drawn as telescopic legs) The actuated UPS-leg, P5B5, controls the rotation about the axis k0 only, while the actuated UPS-leg, P6B6, controls the rotation about the axis i1 only Note that the manipulator obtained from the mechanisms of family-3 coincides with the fully parallel spherical wrist with the P actuator on the leg P4B4 locked

5 Kinematic, Workspace and Singularity Analyses Due to the decoupled actuation of the rotations of the mechanism

about the k0 and i1 axes, the direct kinematic, workspace and singularity analyses are very straightforward Indeed, these problems are reduced to the study of two spatial Whitworth’s quick-return mechanisms Solutions

of these problems can be accomplished as in Di Gregorio and Sinatra,

2002, Di Gregorio, 2002, and Carricato and Parenti-Castelli, 2004

6 Conclusions

This paper presented the synthesis of 2-dof spherical fully parallel mechanisms In particular, by means of two analytical methods derived from the literature, three families of 2-dof spherical mechanisms have

analyses have been addressed which show that such mechanisms are very easy to analyze and control

7 Acknowledgements

Support of this work by the Advanced Concept Team (ACT) of

The collaboration and discussions with Dr Carlo Menon of the ACT are acknowledged and appreciated

393

In practice, proper placements of the actuators make it possible to

been synthesized These families comprise over-constrained mecha- nisms too Actuation issues and kinematic, workspace and singularity

European Space Agency (ESA) under ESTEC/Contract No 18911/05/NL/MV is gratefully acknowledged

Synthesis of 2-DOF Spherical Fully Parallel Mechanisms

.

References

Shönflies, A (1886), Geometrie der Bewegung in Sinthetischer Darstellung , Lipzieg

Bricard, R (1906), Memoire sur les Displacements a Trajectoires Spheriques,

Borel, E (1908), Memoire sur les Displacements a Trajectories Spheriques,

Grassmann Geometry, The Int J of Robotic Resear h s c , vol 8, pp 45-56

Husty, M.L and Zsombor-Murray, P (1994), A Special Type of Singular Stewart Gough Platform, Advances in Robots Kinematics , pp 439-449

Baumann, R., Maeder, W Glauser, D and Clavel, R (1997), The PantoScope: a Spherical

Remote-Center-of-Motion Parallel Manipulator for Force Reflexion, Proc IEEE Int Conf

Robotics and Automation, pp 718-723

Gosselin, C.M and Caron F (1998), Two Degree-Of-Freedom Spherical Orienting Device, US Patent #5,966,991.

Karger, A and Husty, M (1998), Classification of all Self-Motions of the Original Stewart-Gough Platform, Computer-Aided Design , vol 30, pp 205-215

Roshel O and Mick S (1998), Charachterization of Architecturally Shaky Platforms, Advances in Robots Kinematics , pp 465-474

Kong X (1998), Generation of Singular 6-SPS Parallel Manipulators, Proc of

1998 ASME Design Techni al Conferences , 98DETC/MECH-5952

Dunlop, G.R and Johnes, T.P (1999), Position Analysis of a two DOF Parallel Mechanism – the Canterbury Tracker, Mechanism and Machine Theory , vol

34, pp 599-614

Wiitala, J.M and Stanisic, M.M (2000), Design of an Overconstrained Dextrous Spherical

Wrist, Journal of Mechanical Design, vol 122, pp 347-353

Bauer, J.R (2002), Kinematics and Dynamics of a Double-Gimballed Control Moment

Gyroscope, Mechanism and Machine Theory, vol 37, pp 1513-1529

Husty, M.L and Karger, A (2002), Self Motions of the Stewart-Gough-Platforms,

an overview, Proceedings of the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanism and Manipula ors , Quebec, Canada, pp 131-141

Di Gregorio, R and Sinatra, R (2002), Singulariy Curves of a Parallel Pointing System, MECCANICA , vol 37, pp 255-268

Di Gregorio, R (2002), Analytic Determination of Workspace and Singularities in

a Parallel Pointing System, Journal of Robotics Systems , vol 18, pp 37-43 Wohlhart, K (2003), Mobile 6-SPS Parallel Manipulators, J urnal of Robotics Systems , vol 20, pp 509-516

Carricato, M and Parenti-Castelli, V (2004), A Novel Fully Decoupled Degrees-of-Freedom Parallel Wrist, The Int J of Robotics Resear h , vol 23,

Two-pp 661-667

Gogu, G (2005), Fully-Isotropic Over-Constrained Parallel Wrists with Two Degrees of

Freedom, Proc IEEE Int Conf Robotics and Automation, pp 4025-4030

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Journal de l’Ecole Polytechnique , vol 11, no 2, pp 1-96

Merlet, J.P (1989), Singular Configurations of Parallel Manipulators and

Memoires Presentes par Divers Savants , Paris, vol 33, no 2, pp 1-128

Machine Theory, vol 28, pp 553 561

-CONSTRAINT SYNTHESIS

Gim Song Soh

gsoh@uci.edu

J Michael McCarthy

jmmccart@uci.edu

Robotics and Automation Laboratory

University of California, Irvine

Irvine, CA 92697

Abstract In this paper, we control the joints of a planar nR chain mechanically

using RR dyads and obtain a one degree-of-freedom system that guides the end-effector smoothly through five specified task positions To solve this problem, we specify the nR chain and determine it configurations when its end-effector is positioned in each of the five task positions This yields a set of RR chain synthesis problems that constrain alternating links in a way that ensures that the relative joint angles required by the task positions are attained In general, we cannot guaranteed that the resulting assembly will move smoothly between the task positions without jamming, however we present a strategy based on enforcing symmetry of the nR chain that yields useful solutions Examples of constrained 3R, 4R, 5R and 6R are discussed The procedure is general and can be applied to arbitrarily long serial chains.

In this paper, we consider the addition of n-1 RR chains to a planar

nR serial chain robot in order to mechanically prescribe its movementthrough five task positions In the case of a planar 3R robot this isequivalent to the synthesis of a Watt I six-bar linkage For a planar4R, 5R and 6R serial chains, we obtain planar eight-bar, 10-bar and

12-bar linkages In general, our synthesis results transform an n

degree-of-freedom nR chain into a single degree-degree-of-freedom 2n-bar linkage that

Our design equations only ensure that the solution linkage can beassembled in each task position, not that it can move smoothly betweenthe positions Therefore, it can happen that a resulting linkage hastask positions reachable from different assemblies This is known as a

circuit defect ? A fundamental challenge in linkage synthesis is finding

© 2006 Springer Printed in the Netherlands

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J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 395–4 02

N

solutions that have the task positions on the same single circuit In whatfollows, we present a synthesis strategy that yields successful constrained

nR chains that have the five task positions on the same circuit

This work is inspired by ?, who derived synthesis equations for planar

nR planar serial chains in which the n joints are constrained by a cable

drive They obtained a “single degree-of-freedom coupled serial chain”that they use to design an assistive device

? formulated and solved the design equations for six-bar linkages, and

? extended this to eight-bar linkages Our approach is simpler in that

we do not attempt to design the entire 2n-bar linkage, rather we assumethe nR serial chain is given, and use standard dyad synthesis theory to

solve for individual RR constraints, see ?.

Once a linkage has been designed, we analyze it to determine its figuration for given values of the input crank, in order to simulate its

con-movement ? presents an analysis methodology for general planar

link-ages using complex number loop equations and the Dixon determinant.However, the our synthesis approach yields linkages that are a series offour-bar loops and are easily analyzed individually

Let the configuration of an nR serial chain be defined by the

Introduce a world frame G and task frame H so the kinematics

equa-tions of the nR chain are given by

of the base of the chain relative to the world frame, and [H] locates the task frame relative to the end-effector frame The matrix [D] defines the coordinate transformation from the world frame G to the task frame H.

Figure 1. This shows the kinematic structure of mechanically constrained serial

chains The linkage graph is on the left, which has each link as a node and each R joint as an edge The contracted graph on the right shows only links with three or four edges as nodes This shows that the structure extends to any length of nR robot.

chain, we can solve the equations

Be-cause there are three independent constraints in this equation, we have

free parameters when n > 3 In what follows, we show how to use these

free parameters to facilitate the design of mechanical constraints on thejoint movement so the end-effector moves smoothly through the taskpositions

397

Constraint Synthesis for Planar N- R Robots

-Figure 2. This six-bar chain is obtained by designing two RR chains to constrain the joint movement of a 3R robot, so its end-effector passes through the five specified positions.

Assume that we have solved the inverse kinematics problem that

([V k,j]Pk − B k)· ([V k,j]Pk − B k ) = R k , j = 1, , 5. (3)These are the well-known constraint equations for an RR chain, whichcan be solved algebraically to determine as many as four solutions for

satisfy these design equations, which guarantees the presence of secondreal solution

G.S Soh and J.M McCarthy

398

The RR design equations allow us to constrain an nR chain to reach

five task positions Figure ?? lists the planar linkages that this procedure

allows us to design Notice that in each the chain is a sequence of

four-bar linkages extending from the base frame F to the moving frame M

Furthermore, while the base and moving frames are binary links having

the remaining links are quaternary

Figure 3. This eight-bar chain is obtained by constraining the joints of a 4R serial chain to pass through the same five specified task positions.

In the design or serial chain robots it is convenient to have near-equallength links to reduce the size of workspace holes For this reason, weassume that the link dimensions of our nR chain satisfy the relationship

This reduces the specification of the nR chain to the location of the base

If the serial chain has three joints, then the inverse kinematics tions are completely prescribed and the synthesis of two RR chains yields

equa-the constrained serial chain Figure ?? shows equa-the results of this synequa-the-

synthe-sis In this case, though we obtained a solution that passes smoothlythrough the task positions, we are not able to guarantee that this willoccur

399

Constraint Synthesis for Planar N- R Robots

-Figure 4. This 10-bar chain is obtained from a 5R that reaches the same five task positions.

If the serial chain has n > 3 joints, then we impose a symmetry

condition

in order to obtain a unique solution to the inverse kinematics equations

existing links of the nR chain In this case the design equations havefour simultaneous roots

In order to obtain useful RR constraints, we perturb the condition

(??), so these angles are close in value but not equal The result is a

set of solutions to the design equations that are near the existing links.While this process does not guarantee a solution that does not have acircuit defect, we have been successful in finding 4R, 5R and 6R linkages

that move smoothly through the five task positions, see Figures ??, ?? and ??.

In order to evaluate and animate the movement of these linkage tems we must analyze the system to determine its configuration for each

G.S Soh and J.M McCarthy

400

Figure 5. This 12-bar chain is a mechanically constrained 6R chain that guides the end-effector through the same five task positions.

of a network of interconnected four-bar linkages

entire system can be analyzed as a sequence of four-bar linkage analysisproblems

We begin the analysis with the results of the inverse kinematics ysis of the nR chain at each of the five task positions Our approach

anal-uses the analysis procedure of 4 bar linkage from ? We analyze each

401

This means that the

Constraint Synthesis for Planar N- R Robots

-Starting from the first four-bar linkage in frame K, we solve for P1 for a

n − 1st four-bar linkage in frame A n−1

The result is a complete analysis of the mechanically constrained nRserial chain

Our synthesis of a mechanically constrained nR chain yields a onethrough five task positions We set the size of each of the links of thechain to the same value, and specify the coordinates of its attachment

to ground and to the end-effector The inverse kinematics of the nRchain in each of the five task positions provides relative positions that

combine with the nR chain back-bone to form a network of four-bar ages which are easily analyzed to simulate the movement of the chain

link-A strategy of perturbation of a singular solution has yielded linkage 4R,5R and 6R linkage systems that have all of the task positions on thesame circuit

References

anisms,” ASME Journal of Mechanical Design, 115(2):214-222.

Krovi, V., Ananthasuresh, G K., and Kumar, V., 2002, “Kinematic and Kinetostatic

Synthesis of Planar Coupled Serial Chain Mechanisms,” ASME Journal of

Me-chanical Design, 124(2):301-312.

Positions,” Mechanism and Machine Theory, 22:411-419.

Subbian, T., and Flugrad, D R., 1994, “6 and 7 Position Triad Synthesis using

Con-tinuation Methods,” Journal of Mechanical Design, 116(2):660-665.

Sandor, G N., and Erdman, A G., 1984, Advanced Mechanism Design: Analysis and

Synthesis, Vol 2 Prentice-Hall, Englewood Cliffs, NJ.

Perez, A., and McCarthy, J M., 2005, “Clifford Algebra Exponentials and Planar

Linkage Synthesis Equations,” ASME Journal of Mechanical Design,

127(5):931-940, September.

nant and a Complex Plane Formulation”, ASME Journal of Mechanical Design,

McCarthy, J M., 2000, Geometric Design of Linkages, Springer-Verlag, New York.

G.S Soh and J.M McCarthy

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degree-of-freedom system that guides the end-effector of the chain smoothly

J A.,

Mirth, and Chase, T R., 1993, “Circuit Analysis of Watt Chain Six-Bar

Mech-Lin, C.-S., and Erdman, A G., 1987, “Dimensional Synthesis of Planar Triads for Six

US-Examples of over-constrained US-PM with five and. .. planar 3R robot this isequivalent to the synthesis of a Watt I six-bar linkage For a planar4R, 5R and 6R serial chains, we obtain planar eight-bar, 10-bar and

12-bar linkages In general,...

degree-of-freedom nR chain into a single degree-degree-of-freedom 2n-bar linkage that

Our design equations only ensure that the solution linkage can beassembled in each task position,