Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 3 ppt

Lu & Kota, 2003 introduced a more generalapproach using finite element analysis and a genetic algorithm to deter-mine an optimized compliant mechanism’s topology and dimensions.The presen

d.o.f of the chains It is also possible to obtain chains that change Gˆ but not the number of d.o.f Using again the scheme of Fig 2, the results From these results the schemes of Fig 5 are obtained Other similar configurations can be obtained through suitable permutations of kinematic pairs and groups

Table 2 Modified groups and pairs in Fig 1

Case G 1 = G 2 KP a = KP b Displacements between a 1 and b 2

d E (planar)  (Schoenflies) X From E to a subset of X with 3 d.o.f

e Y (translating

screw)

X (Schoenflies) From Y to a subset of X with 3 d.o.f

Figure 6 shows the kinematic chains resulting from the two cases d

separates the two branches of positions belonging to different groups

Figure 6 Robots that change displacement group but not No of d.o.f

53

reported in Table 2 can be achieved (see Fanghella and Galletti, 1994)

and e of Table 2 The chains are drawn in their singular position that

Figure 5 Robots with E, X, Y, and R groups

Parallel Robots that Change their Group of Motion

.

For example, in the case d, starting from the position drawn and

rotating the revolutes with horizontal axes, the robot acts as a standard planar platform, with 3 d.o.f Starting again from the singular position,

translations and 1 rotation) Then, the group of displacement is changed, but the number of d.o.f is preserved

An analogous situation applies to case e

A slightly different case can be derived from a further interesting intersection group Two Schoenflies groups X can give a group Gˆ = X or

a group Gˆ = U (three-dimensional translation), depending on the relative positions of their rotation directions (see Fanghella and Galletti,

Case G 1 = G 2 KP a = KP b Displacements between a 1 and b 2

(Schoenflies) R

From the original X (Schoenflies)

to an X (Schoenflies) with the axis parallel to the axis of R

Since a group X with 4 d.o.f is obtained in both branches, the platform must have 4 legs in order to apply one driver to each leg, according to the scheme of Fig 7

1994) Therefore, according to Fig 2, the following chain can be derived

Figure 7 Scheme of a 4-legs robot

by moving the revolutes with vertical axes, the platform of the robothas a displacement that is a subset of the group X, with 3 d.o.f (2

group of displacement is not changed, but its invariant property (rotation axis) is changed

pairs Therefore, many different robot structures can be obtained starting from the schemes of Figs 3, 5, and 7

From a practical point of view, in order to control the motion of a kinematotropic chain in a branch it is necessary to provide a number of drivers equal to the number of degrees of freedom of the chain in that branch For a complete control of the chain in all branches, it is necessary to provide a set of drivers equal to the union of the drivers used to control each branch In each branch, the chain is actuated only by the drivers associated with that branch, while other drivers become driven; when passing through a singular position (where the number of infinitesimal degrees of freedom grows), all drivers must act either to maintain their position or to drive the chain to a specific branch

Finally, it is worth noting that, in some cases, starting from the

direction orthogonal to the drawing plane leads to a branch in which the

55

Joint Modifications, Actuators and Branches

be reached For example, for the mechanism in Fig 6-d, a translation in the singular positions in Figs 4, 6 and 8, more than two branches may

Figure 8 Robot that changes the invariant of its displacement group Parallel Robots that Change their Group of Motion

parallel prismatic pairs and one revolute, by 3 parallel revolutes and

1 prismatic pair not normal to them, and so on Moreover, the revolutes

KP in the chains can be substituted, in several circumstances by helical

allowed relative motion between the frame and the platform is a pure planar translation In the paper, for each case, the discussion is limited

to the two branches with the highest number of degrees of freedom

References

Angeles J (1988), Rational Kinematics, Springer

Fanghella P and Galletti C (1994), Mobility Analysis of Single-Loop Kinematic Chains: An Algorithmic Approach Based on Displacement Groups,

Mechanism and Machine Theory, Vol 29, pp 1187-1204

Galletti C and Fanghella P (2001), Single-Loop Kinematotropic Mechanisms,

Mechanism and Machine Theory, Vol 36, pp 743-761

Gogu G (2005), Mobility Criterion and Overconstraints of Parallel Manipulators,

Proc of CK2005 Int Workshop on Computational Kinematics, Cassino, Paper

22-CK2005, pp 1-16

Hervé J (1978), Analyse Structurelle des Mécanismes par Groupe des

Déplacements, Mechanism and Machine Theory, Vol 13, pp 437-450

Kong X and Gosselin C (2004), Type Synthesis of 3T1R 4-DOF Parallel

Manipulators Based on Screw Theory, IEEE Transactions on Robotics and Automation, Vol 20, pp 181-190

Kong X and Gosselin C (2005), Type Synthesis of 3-DOF PPR-Equivalent Parallel Manipulators Based on Screw Theory and the Concept of Virtual

Chain, ASME J of Mechanical Design, Vol 127, pp 1113-1121

Wohlhart K (1996), Kinematotropic Mechanisms, Recent Advances in Robot Kinematics, (J Lenarcic and V Parenti Castelli, Eds.), Kluwer , pp 359-368

This work has been developed under a grant of Italian MIUR.

by one branch to another, are the basic components we have usedfor synthesizing a particular type of parallel robots Three different

Acknowledgement

APPROXIMATING PLANAR, MORPHING CURVES WITH RIGID-BODY LINKAGES

Andrew P Murray

University of Dayton, Department of Mechanical & Aerospace Engineering

Dayton, OH USA

murray@udayton.edu

Brian M Korte and James P Schmiedeler

The Ohio State University, Department of Mechanical Engineering

Columbus, OH USA

korte.16@osu.edu & schmiedeler.2@osu.edu

Abstract This paper presents a procedure to synthesize planar linkages, composed

of rigid links and revolute joints, that approximate a shape change fined by a set of curves These “morphing curves” differ from each other by a combination of rigid-body displacement and shape change Rigid link geometry is determined through analysis of piecewise linear curves, and increasing the number of links improves the shape-change approximation The framework is applied to an open-chain example.

de-Keywords: Shape change, morphing structures, planar synthesis

1 Introduction

For a mechanical system whose function depends on its geometricshape, the controlled ability to change that shape can enhance per-formance or expand applications Examples of adaptive or morphingstructures include antenna reflectors (Washington, 1996) and airfoils(Bart-Smith & Risseeuw, 2003) proposed to include many smart mater-ial actuators Compliant mechanisms also provide a means of achievingshape changes Saggere & Kota, 2001 developed a synthesis procedurefor compliant four-bars that guide their flexible couplers through dis-crete prescribed “precision shapes” that involve both shape change andrigid-body displacement Lu & Kota, 2003 introduced a more generalapproach using finite element analysis and a genetic algorithm to deter-mine an optimized compliant mechanism’s topology and dimensions.The present work introduces synthesis techniques for planar, rigid-body mechanisms that approximate a desired shape change defined by

an arbitrary number of curves, one morphing into another Higher

load-57

carrying capacity makes rigid-body mechanisms better suited than

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 57 64

© 2006 Springer Printed in the Netherlands −

rigid-body mechanisms would likely require fewer actuators acting inparallel, such as along an airfoil with changing camber Furthermore,actuation is not an additional development need because existing tech-nology rather than, for example, smart material technology, is typicallyused to actuate rigid-body mechanisms With rigid links, synthesis can

a

can typically achieve larger displacements, enabling more dramatic shapechanges This paper details a methodology for designing rigid links thatcan be joined together in a chain by revolute joints to approximate theshapes of a set of morphing curves The methodology is applicable toboth open and closed chains, and an open-chain example is presented

2 Rigid Link Geometry

linkage involves converting the desired curves, denoted as “design files”, into “target profiles” that are readily manipulated and compared.The target profiles are divided into segments, and corresponding seg-ments from all of the target profiles are used to generate the rigid links.The key is to divide the target profiles and then generate the rigid links

pro-so as to reduce the error in approximating the design profiles

Design Profiles and Target Profiles. A design profile is a curvedefined such that an ordered set of points on the curve and the arc lengthbetween any two such points can be determined The piecewise linearcurve (solid line) in Fig 1 is a simple example of a design profile A

set of p design profiles defines a shape change problem Because the

change will be approximated with a rigid-body linkage, the error in the

approximation is generally smaller if all p profiles have roughly equal arc

length, though this is not an explicit requirement of the methodology

A target profile is formed by distributing n points, separated by equal

arc lengths, along a design profile Thus, a target profile is a piecewiselinear curve composed of the line segments connecting the ordered set ofpoints, and any design profile can be represented by a target profile of

two or more points In Fig 1, five (x, y) points generate a target profile from the design profile defined by three (a, b) points The target profile

includes the dashed line and does not pass through the design profile’ssecond point In this case, three points could be used to exactly representthe design profile, but the approach is more generally applicable to any

design profile The motivation is to convert a set of p design profiles into target profiles all defined by n points such that corresponding points can

compliant mechanisms for applications with large applied loads Similarly,

priori knowledge of exact external loads Finally, rigid-body mechanisms

be purely kinematic, so the system can be modeled precisely without

The procedure for generating rigid links that compose a shape-changing

Figure 1 Three-point ( a, b) design profile and five-point (x, y) target profile.

be found on each target profile For a closed curve design profile, anypoint can be deemed the first/last point, yielding a closed target profile.Important characteristics of a target profile include the fact that itsarc length is always shorter than the design profile it represents Themost significant loss of shape information occurs where the curvature islargest for a continuous design profile or where the angle at a vertex issmallest in magnitude for a piecewise linear design profile Since points

on the target profile are separated by equal arc lengths along the designprofile, they are not at equidistant intervals along the target profile

Large values of n produce smaller variations between the design profile

and target profile and in the distances between consecutive points on the

target profile A useful heuristic is selection of n such that the target

profile arc length is greater than 99% of the design profile arc length

Shifted Profiles. The j th target profile is defined by, z j i ={x j i y j i } T,

i=1, n A rigid-body transformation in the plane,

will relocate the profile preserving the respective distances between points

in it Any profile relocated in this fashion is called a shifted profile get and mean profiles (described below) are both shifted to performuseful design operations without altering the original design problem

Tar-The “distance” between target profiles j and k is defined to be,

defined metric To determine the rigid-body transformation that shifts

59

Approximating Planar, Morphing Curves

target profile j to the location that minimizes D with respect to target profile k, one must find θ and d such that ∂D ∂θ = ∂d ∂D

profile is formed by shifting target profiles 2 through p to minimize

their respective distances relative to reference target profile 1 A new

piecewise linear curve defined by n points, each the geometric center

of the set of p corresponding points in the shifted target profiles, is

generated For example, two target profiles in Fig 2a are shifted in Fig.profile Fig 2c shows the mean profile that approximates the targetprofiles when regarded as rigid bodies In Fig 2d, this mean profile

is shifted to approximate the shape and location of the target profiles.The described procedure could convert a shape-changing problem to arigid-body guidance problem, as the three locations of the mean profile

in Fig 2d define three finitely separated positions of a moving lamina

A chain of two or more rigid links connected by revolute joints canbetter approximate a shape change than can a single rigid body with theshape of a mean profile The procedure for generating a mean profilemay be applied to any segment of the target profiles To generate a

linkage composed of s rigid links, an initial solution divides the target profiles into s segments of roughly equal numbers of points, the last

point of a segment being the first of the next segment A mean profile

is generated for each set of segments For example, given target profiles

51, 51-76, and 76-102 The first three segments and their correspondingmean profiles each have 26 points, and the last has 27 Once generated,each mean profile can be shifted individually to the location relative to

its corresponding segment in each target profile that minimizes D The positions of the s mean profiles relative to each other will differ as they

are superimposed on each target profile The end points of the segments

in general will not coincide in any of the positions at this stage

Error Reducing Segmentation. Non-uniform target profile mentation can reduce the error in approximating a shape change byshortening segments on the profile where shape change is most dramatic

seg-of n = 102 points, if s = 4, the segments are composed seg-of points 1-26,

26-2b to their respective distance minimizing positions relative to the first

) $0: <):/-< 8:7
4-; * #01.<-, <):/-< 8:7
4-; <7 516151B-
 + -)6 8:7
4- ;741, 416- , -)6 8:7
4- ;01.<-, <7 516151B-
7: -)+0 <):/-< 8:7
4-

-85 49CD1>35  9C 1 @??B C57=5>D1D9?> =5DB93 2531EC5 9D 45@5>4C ?> 1C57=5>DC >E=25B ?6 @?9>DC  25DD5B =5DB93 D85 5BB?B  9C 45
>54 1C6?<<?GC ?B =51> @B?
<5  D85 5BB?B  1CC?391D54 G9D8 =1D389>7 D85DG55> 1>I DG? 3?BB5C@?>49>7 @?9>DC ?> D85 DG? @B?
<5C G85> D85 =51>75D @B?
<5 C57=5>D -85 5BB?B  1CC?391D54 G9D8 D89C =51> @B?
<5 9CD85 =1H9=E= F1<E5 ?6 

5BB?B  9C D85 =1H9=E= F1<E5 ?6  6?B 1<< =51> @B?
<5C    -? B54E35  D85 C57=5>D1D9?> <?31D9?>C ?> D85 D1B75D @B?
<5C 1B5

8 7 6 5 4 3 2 1

c)

l, except the last segment s, is increased by one if E l < ¯ E and decreased

by one if E l > ¯ E, where ¯ E is the average of the E l ’s Segments 1 and s change by one point, and the others by two E sdoes not explicitly deter-

mine whether segment s increases or decreases in length, but its effect on

E and ¯ E does so indirectly With the target profile segments redefined,

a new mean profile for each set is generated, the error E recomputed, and the process repeated until E ceases to decrease To avoid local min- ima, the process continues for several iterations after E increases, and each E is compared to several previous iterations instead of just the im- mediate predecessor The segmentation providing the smallest E is the

error reducing segmentation, and the corresponding mean profiles definethe geometry of the rigid links that compose the linkage Because thetarget profiles typically contain thousands of points, altering segments

by two points is a modest change, and exhaustive approaches involvingsingle-point alterations are unlikely to offer significant benefit

An alternative approach for initial segmentation is to specify an

ac-ceptable error E ainstead of a number of segments, and “grow” segments,

starting with 1, point by point until the error E l of the corresponding

mean profile exceeds E a This generates an unknown number of ments, the last of which generally has the smallest error

sege-3 Example

The three design profiles used to generate the target profiles in Fig 2aare defined by the points listed in Tb 1, and their arc lengths are 6.72,6.78, and 6.76, respectively The target profiles contain 1800 points,

as does the mean profile in Fig 2c The subset of points from themean profile listed in Tb 1 are key points that mark the locations of

Table 1. Defining points of design profiles and key points of mean profile in Fig 2 Mean profile points are in two columns, each ordered top to bottom.

Design Profile 1 Design Profile 2 Design Profile 3 Mean Profile

(2.3,7.6) (7.6,4.3) (4.7,6.4) (2.52,7.23) (0.78,4.39) (1.4,6.5) (7.4,5.1) (4.0,6.2) (1.94,6.85) (0.85,4.01) (1.0,5.5) (6.9,5.8) (3.1,5.6) (1.87,6.80) (0.90,3.83) (1.0,4.0) (6.4,6.4) (2.7,5.0) (1.46,6.38) (0.94,3.73) (1.3,2.8) (5.7,7.0) (2.7,3.9) (1.32,6.19) (1.29,3.08) (1.9,2.1) (4.8,7.3) (3.0,3.4) (1.24,6.10) (1.38,2.93) (2.3,1.7) (4.0,7.3) (3.7,2.9) (0.93,5.54) (1.74,2.53)

(3.3,7.3) (4.4,2.6) (0.91,5.49) (1.76,2.52) (2.5,6.8) (5.3,2.4) (0.90,5.48) (2.04,2.32)

(0.79,4.64) (2.52,2.02)

significant change in slope along the mean profile Figure 3 plots the error

Figure 3. Error in matching target profiles shown in Fig 2a as a function of number

of segments Inset shows 4-segment solution superimposed on target profiles Solution segments correspond to unassembled rigid links of a shape-changing linkage.

E in matching the target profiles as a function of the number of segments,

with a curve fit to the data to more clearly illustrate the trend The datapoint for 1 segment represents the solution shown in Fig 2d, for whichthe error clearly is defined by the top end point of the middle targetprofile In Fig 3, increasing the number of segments beyond 4 offersnoticably diminishing returns in terms of reduced error The plot inset

in Fig 3 contains the 4-segment solution superimposed on the targetprofiles with the segments shown in alternating shades of gray

4 Mechanization

Once the geometry of the rigid links is determined, the links arejoined together at their end points with revolute joints to form a link-age This increases the error since it requires movement of the linksfrom their distance-minimizing positions to bring together the generally

63

non-coincident adjacent endpoints Furthermore, the relative motion

Approximating Planar, Morphing Curves

sitions is more general than that allowed by revolute joints Still, if theerror prior to connecting the links is small, the linkage approximateswell the desired shape change With the links joined, it is often desir-able to add additional links that constrain the linkage to have a reduced

number of degrees of freedom To constrain an s-link open chain to

be a 1-DOF mechanism, s + 1 binary links must be added If five or

fewer design profiles are involved, circle and center points for additionalbinary links can be found exactly, but for six or more design profiles,

1973 are required The details of mechanization are beyond the scope

of this paper, but each additional link further constrains the motion ofthe shape-change-approximating links, thereby increasing the error

5 Conclusions

This work introduces a systematic procedure to determine the etry of rigid links that can be assembled together with revolute joints tocompose a linkage that approximates a desired shape change defined by

geom-an arbitrary number of morphing curves The procedure involves paring piecewise linear curves to reduce the error in the shape changeapproximation, and increasing the number of links generally reducesthat error Mechanizing the generated chains of rigid links presents anumber of challenges, but rigid-body mechanisms have great potential

com-as morphing structures, particularly in heavy load applications

shapes, Journal of Intelligent Material Systems & Structures, vol 14, pp 379–391.

Saggere, L., & Kota, S (2001), Synthesis of planar, compliant four-bar mechanisms

for compliant-segment motion generation, ASME Journal of Mechanical Design,

vol 123, no 4, pp 535–541.

with the least-square approximation of a given motion, ASME Journal of neering for Industry, vol 95, no 2, pp 503–510.

Engi-Structures, vol 5, no 6, pp 801–805.

quired between adjacent links to achieve their distance-minimizing re

po-least-square approximations such as those developed by Sarkisyan, et al.,

Lu, K.J., & Kota, S (2003), Design of compliant mechanisms for morphing structural

Sarkisyan, Y L., Gupta, K.C., & Roth, B (1973), Kinematic geometry associated

Washington, G.N (1996), Smart aperture antennas, Journal of Smart Materials &

Matteo Zoppia, Dimiter Zlatanovb, Rezia Molfinoa

aUniversit`a di Genova, via Opera Pia 15A, 16145, Genova, Italia

bD´epartement de G´enie M´ecanique, Universit´e Laval, Qu´ebec, QC, Canada

[zoppi,molfino]@dimec.unige.it, zlatanov@gmc.ulaval.ca

Abstract In parallel mechanisms (PMs), the passive joint velocities can be

elimi-nated from the velocity equations by a standard screw-theory method, obtaining a system of linear input-output equations A general method for the elimination of the passive joint velocities in non purely paral- lel mechanisms is not yet known The paper addresses the problem by studying the instantaneous kinematics of two non-parallel closed-chain 4-dof mechanisms derived from a 5-dof PM With some modifications and appropriate geometric reasoning the PM methodology can be suc- cessfully applied to the analysis of non-parallel mechanisms.

Keywords: Velocity analysis, parallel mechanisms, closed chain mechanisms

1 Introduction

Parallel mechanisms (PMs) are composed of an end-effector connected

to the base by separate serial leg chains, Fig 1 Most published closedspatial kinematic chains are PMs, but occasionally authors describe as

“parallel” kinematic chains that do not strictly belong to this class

A relatively simple generalization of a parallel (or serial) mechanism iswhen the kinematic chain is a two-terminal series-parallel graph connect-ing the base to the end-effector Starting with a parallel (or serial) chain,substitute individual joints with parallel subchains; a mostly parallel (orserial) series-parallel (S-P) chain (and mechanism, S-PM) is the result(Fig 2) More complex chains can be obtained from a mostly parallelS-P connection when subchains (with at least one joint) are added be-tween links belonging to different leg chains Such mechanisms can bereferred to as interconnected chains (IC) mechanisms (ICMs) (Fig 3)

In a PM, out of singularities, the input-output velocity equations

(re-lating the output twist, ξ = (ω, v), or (ωT|vT)T as a column vector,

and the actuated joint velocities, ˙ q) are obtained in the form: Zξ = Λ ˙ q For PMs, Z and Λ are computed by a screw-theory based method

that can be considered standard It is relatively easy (ignoring unusual

65

© 2006 Springer Printed in the Netherlands –

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 65 72

NON-PARALLEL CLOSED CHAIN

MECHANISMS

ON THE VELOCITY ANALYSIS OF

EE

P P

R R R R

R R R R

R R R R

P

Figure 1.

method cannot be used, without changes, for ICMs In the general case,one deals with the velocity loop equations (rather than linear expressions

of ξ in terms of the leg’s joint screws) Analogously, Ohm’s laws suffice

when an electrical network is series-parallel; otherwise the more generalKirchhoff laws are needed (Davies, 1981)

As we have shown (Zoppi et al., 2006), the ideas of PM velocityanalysis can be applied successfully to ICMs The present paper illus-trates this further by studying two new non-PMs We modify a 5-dof

PM and its analysis to obtain and solve first a 4-dof S-PM and then a4-dof ICM

con-ϕz a vertical force at O The actuated constraints (reciprocal to the leg

passive joints) are VL= Span (ϕz, ϕL), with force ϕL along π23L ∩ πL

45

Constraint and Mobility Analysis

5-dof PM: architecture with leg screws (left) and graph (right).

singularities such as RPM or IIM singularities, (Zlatanov et al., 1994)) togeneralize the passive-velocity elimination for series-parallel chains The

The combined constraint systems are: W = LWL = Span (ϕz);

V = LVL = W + Span (ϕA, , ϕE) So the platform has full tational capability about its point O, which can translate horizontally.Out of singularity, dimV = 6 and the mechanism can be controlled byactuating the five P joints

ro-2.2 Jacobian

The screw-theoretical method for the velocity analysis of PMs was

(Hunt, 1978); (Mohamed and Duffy, 1985);(Kumar, 1992); (Agrawal, 1990); (Zlatanov et al., 1994); (Zlatanov et al.,2002); (Joshi and Tsai, 2002) We provide a detailed general formulation

in (Zoppi et al., 2006)

For each leg, a non-unique actuation system, UL, is identified, VL =

WL⊕ UL, for this PM we use UL= Span (ϕL) The reciprocal product

of the actuations (any basis ofUL) eliminates the passive joint velocities

from the leg twist equation, here ξ = ˙q1LξL1 +5

i=2ωLi ξLi

To obtain an equation Zξ = Λ ˙q with coefficients in terms of the

PM’s geometry, we need symbolic expressions for the actuation screws

ϕL= (fL, mL) We use a moving frame Oijk, Oz always vertical Since

ϕL, a pure force, and the origin are in πL45, mL = rLnL

45, where nL

45

is the unit normal to π45L and rL is the distance of ϕL from O Since

the intensity is irrelevant, fL = nL

fAx fAy

fBrBnB

45 T

fBx fBy

fCrCnC

45 T

The PM of Fig 1 has instantaneous end-effector motions spanned by

2 translations and 3 rotations, all independent Mobility types allowing

67

Analysis

developed in works like

Velocity Analysis of Non-parallel Closed Chain Mechanisms

(3. 3,7 .3) (4.4,2.6)... connect-ing the base to the end-effector Starting with a parallel (or serial) chain,substitute individual joints with parallel subchains; a mostly parallel (orserial) series-parallel (S-P) chain (and. .. present paper illus-trates this further by studying two new non-PMs We modify a 5-dof

PM and its analysis to obtain and solve first a 4-dof S-PM and then a4-dof ICM

con-ϕz