Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 9 potx

Figure 8 shows examples of PPM configurations in which the reciprocal figure is connected and the manipulator is in a singular configuration.. Two examples of singular configurations and

236 A Degani and A Wolf

3.3 The Singularity/Self Stress Connection

Once the MLG of a PPM and a reciprocal figure are constructed, one

can use them for the singularity analysis of PPMs Maxwell’s theory

(section 2.2) presents a connection between the existence of a reciprocal

figure and self stress in a framework We will now analyze a self-stressed

MLG (Fig 7) in order to demonstrate the connection between self-stress

and singularity Based on the definition of self stress framework, when a

bar-joint framework is in self stress, the sum of the forces of the bars

connected to a joint is equal to zero Three equations corresponding to the

sum of forces in the three vertices 1,2,3 can be written as:

These three equations are vector summations We arbitrarily assign the

direction of the forces and consistently add the forces Therefore, some of

the forces in Eq 2-4 are negated Summing Eq 2, 3, and 4:

Figure 7 “Singular” (self stress) configuration of MLG

Equation 5 confirms a linear dependency of the three forces a b, and c

These three forces are the forces corresponding to the lines of action of

the three limbs The meaning of this dependency is that these three

limbs cannot generate instantaneous work (virtual work (Hunt, 1978)) on

the end-effector while it is moving in an instantaneous twist deformation

resulting from an external wrench applied on it Therefore, self stress in

a framework is equivalent to a type-II singularity of a PPM It is now

evident that the existence of a reciprocal figure indicates a self stress

framework, and in a similar way indicates a singular configuration in a

mechanism

3.4 Locating the Singular Configurations

To find the configurations where there exists a reciprocal figure to a

particular PPM, and therefore it is in a singular configuration, one

should move the manipulator by changing its joint parameters while

tracking for configurations in which the reciprocal figure is visually

.

237

Graphical Singularity Analysis of 3-DOF PPMs

connected, e.g in Fig 6e, vertices p and q merge with p’ and q’ respectively Figure 8 shows examples of PPM configurations in which the reciprocal figure is connected and the manipulator is in a singular configuration

Figure 8 Two examples of singular configurations and the connected reciprocal figures (3-RPR left, 3-RRR right)

So far the search for a singular configuration was carried out by changing the joint parameters of the manipulator and checking for the existence of a connected reciprocal figure If the analysis is constructed the other way around, so that a connected reciprocal figure is first constructed and only then an MLG is constructed to be reciprocal to it,

we can trace the loci of the singular configurations of the manipulator by changing the reciprocal figure while keeping it connected (Fig 9 left) Note that the construction of the reciprocal figure in this case is based on mechanical constraints of the PPM, e.g the fixed shape of the end-effector Moreover, the singular configuration’s loci are traced relative to

a constant orientation of the PPM in order to enable us to plot the loci as

a 2-D graph We refer the readers to (Sefrioui and Gosselin, 1995) to examine the consistency of the results

More examples, including JAVA applets of this method, can be found at: www.cs.cmu.edu/~adegani/graphical/

Figure 9 (Left) Singularity loci of 3-RPR in two different constant orientations

of the effector (Right) A loci plot from six different orientations of the effector (0,5, 10, 15, 20, and 25 degrees)

end-

.

238 A Degani and A Wolf

4 Conclusion and Future Work

References

Bonev, I.A (2002), Geometric analysis of parallel mechanisms Ph.D Thesis,

Laval University, Quebec, QC, Canada

Crapo, H., and Whiteley, W (1993), Plane self stresses and projected polyhedra I:

Gosselin, C., and Angeles, J (1990), Singularity analysis of closed-loop kinematic

chains IEEE Transactions on Robotics and Automation no 3, vol 6, pp

Mechanism and Machine Theory no 4, vol 30, pp 533-551

Tsai, L.W (1998), The Jacobian analysis of a parallel manipulator using

reciprocal screws, Proceedings of the 6th International Symposium on Recent

Advances in Robot Kinematics, Salzburg, Austria

The graphical method which is presented and implemented in this paper results in comparable outcomes to those obtained by other approaches(e.g Sefrioui and Gosselin, 1995), yet avoids some of the complexities in-volved in analytic derivations It is worth mentioning that the method we present can be potentially applied to non-identical limb manipulators and

to other types of mechanisms as well The method makes use of reciprocal screws to represent the lines of action of PPMs’ limbs in a Mechanism’s Line of action Graph (MLG), an insightful graphical representation of the mechanism Maxwell s theory of reciprocal figure and self stress is then’applied to create a dual figure of the MLG Analyzing this dual (reciprocal) the loci o

figure provides us with the singular configurations of the PPM and with

We are currently facing the challenge of expanding this graphicalable to use our relatively simple method to find the singular configura-tions of complex three-dimensional manipulators, such as a 6-DOF Gough-Stewart platform One possible way to achieve this goal is to method to the analysis of three-dimensional manipulators We hope to be

project the spatial lines of action of the limbs on one or more planes (Karger, 2004) We believe a self-stress analysis of these projected graphs,similar to the one done on PPM, will offer insight into the singular confi-guration of these non-planar manipulators

The basic pattern Structural Topology no 1, vol 20, pp 55-78

Hunt, K.H (1978), Kinematic Geometry of Mechanisms, Oxford, Clarendon Press

singular configurations of the manipulator

f

J Lenarcic and

Karger, A (2004), Projective properties of parallel manipulators,

^ ^

DIRECT SINGULARITY CLOSENESS

INDEXES FOR THE HEXA PARALLEL ROBOT

Carlos Bier*, Alexandre Campos, J¨urgen Hesselbach

Institute of Machine Tools and Production Technology - TU Braunschweig

Langer Kamp 19b, 38106 Braunschweig, Germany

* c.bier@tu-bs.de

Abstract Direct kinematic singularities constrain the internal robot workspace

and the proximity to them must be detected online as fast as possible for non deterministic trajectories Direct singularity proximity for the Hexa parallel robot is measured by means of three measure indexes with two different physical bases In this paper a new index based on Grass- mann geometry to measure the singularity closeness is introduced This method and methods based on constraint minimization are applied and validated in the Hexa robot From the results we observe, for instance, that the new index requires less time than the constraint minimization methods but requires a better knowledge of the robot structure.

Keywords: Parallel Manipulator Singularities, Grassmann Geometry, Constrained

Minimization

A measure of the direct singularity closeness for parallel manipulators

is required aiming at a safe operation space For parallel robots as theHexa robot (Fig 3d), workspace is limited by direct kinematic singu-larities as well as by inverse kinematic singularities Direct kinematicsingularities allow the end effector to gain unconstrained movements.Its identification has been studied from different perspectives The van-ishing of the Jacobian determinant has been used for particular parallelrobots However it is a product of factors and thus it suffers from the factthat close to a singularity, where a factor shrinks to zero, other factorsmay be big enough and the determinant does not indicate the singularitycloseness Additionally, the physical meaning of the determinant is notclear

Qualitative conditions, based on Grassmann geometry, are proposed

to detect singularities of triangular simplified symmetric manipulators[Merlet, 2000] Quantitative approaches use a numerical measure todetermine how close a robot position is to a singularity Different mea-

© 2006 Springer Printed in the Netherlands.

-sures have been used for this task, e.g the natural frequency measure

[Voglewede and Ebert-Uphoff, 2004], the power and the stiffness inspiredmeasure [Pottmann et al., 1998] based on a constraint minimization

In this paper a new method for quantitative measures of the directsingularity closeness based on Grassmann geometry is presented Thisnew method as well as the minimization based methods are applied tothe Hexa robot and the results are analyzed

The six DOF Hexa robot is composed by six limbs connecting thebasis to the end effector, see Fig 3d Each limb contains an active

rotative joint A i (for i = 1, · · · , 6) Its axes are fixed to the basis plane,

a passive universal joint B i and a passive spherical joint C i connected to

the end effector, so that all C i joints define the end effector plane The

cranks and the passive links are connected at B i The six limbs of theHexa robot are arranged in three pairs of two active joints with collinear

rotational axes The pairs of active joints are axisymmetrical, i.e 120 o

between each pair

2.

In spatial parallel manipulators the relationship between actuator

co-ordinates q and end effector Cartesian coco-ordinates x can be stated as a function f (q, x) = 0, where 0 is the 6-dimensional null vector Therefore

the differential kinematic relation may be determined as

tor through the passive link, i.e the distal link of each limb

[David-son and Hunt, 2004] Therefore, a static relation may be stated as

the end effector in axis order, τ = [τ1, · · · , τ6] are the input wrench

mag-nitudes and the columns of J T

x are the normalized screws (axial order)

of wrenches acting on the end effector

Singular configurations appear if either J x or J q drops rank If J x issingular, a direct singularity is encountered and the end effector gainsone or more uncontrollable degrees of freedom (DOF), on other hand if

240

matrix algebra

method as well as the condition number [Xu et al., 1994] based on

C Bier, A Campos and J Hesselbach

J q drops rank it looses at least one DOF The direct singularity occurs

in the workspace and is the main aim of this paper

The new method as well as the minimization based methods are troduced and applied to the Hexa robot

in-3.

The constraint minimization method determines closeness to larity through an optimization problem that results in a correspondinggeneralized eigenvalue problem Using this methodology it is possible

singu-to describe the instantaneous behavior of the end effecsingu-tor near ities for parallel manipulators in general [Voglewede and Ebert-Uphoff,

singular-2004, Hesselbach et al., 2005] In this approach, an objective function

F ($ t ) to be optimized is subject to move on a constraint h This is

formulated mathematically as:

M (X) =

min/max F ($ t) = $T

subject to h = $ T t T $ t − c = 0 (2)

where S (positive semidefinite) and T (positive definite) are n × n

sym-metric matrix and c is some positive constant, e.g c = 1 The solution

of Eq (2) gives the closeness measure to a singularity M (X) at a ticular position and orientation X of the manipulator The proposed

par-constrained optimization problem is found with the application of a

La-grange multiplier λ The local extrema of the Lagrangian ζ($, λ) =

solu-tion to exist, the minimizasolu-tion (or maximizasolu-tion) of Lagrangian yields

eigen-value problem The smallest eigeneigen-value λ min will be the minimum value

of the objective function F ($ t), and so it can be utilized as a measurevalue

In general, this minimization problem was formulated based on an

arbitrary quantity for S and T [Voglewede and Ebert-Uphoff, 2004] Taking J = J x , S = I 6x6 and T = diag[000111], then √

λ min is ated to the minimum power [∼ W ] of the system, which indicates the

associ-manipulator singularity closeness

Another possible way is to choose S as the stiffness matrix of the actuators K Act and T as the mass matrix of the manipulator M EE(or for

simplicity the end effector mass matrix, i.e neglecting the limb masses).

Therefore,√

λ min is associated to the ω natural frequency [ ∼ Hz] of the

system (M EE X¨− ω2K EE = 0), indicating the singularity closeness.Both methods are applied in the Hexa robot (Fig 3d) for its singu-larity approximation measure The measure behaviors of the minimumpower of the system through a singularity (Fig 1a) is showed in Fig 1b,

Direct Singularity Closeness Indexes for the Hexa Parallel Robot 241

here the end effector twists θo around the $min (the end effector twistwhich requires minimum power in this singularity) The singularity oc-curs when √

λ min = 0, but a singular range exists due to clearance andcompliance of the system, where the end effector is still unconstrained.The singular range bound is experimentally identified as 0.029∼ W and

upon it the manipulator stiffness is warranted into the whole workspace

q

0 20 40 60 80

q angle [°]

singular range

Grassmann algorithm V5b limit value (55 mm)

singularity V5b

0.04

singular range

power method limit value (0.029 W) ~

singularity V5a

eties of lines, i.e the sets of linear dependent lines to n given

indepen-dent lines, and characterized them geometrically according to their rank

ma-nipulator may be associated with a linear dependent set of lines, also

242

Grassmann (1809–1877) studied the

vari-C Bier, A Campos and J Hesselbach

.

called line based singularities In general, the reciprocal wrenches $r(Fig 1a) to the passive twists of each manipulator leg are associated

to lines in the direction of the forces acting upon the end effector, alsocalled Pl¨ucker vectors Linear dependence among these lines represents

a direct singularity These wrenches compose the J x matrix (Sec 2).Using the Grassmann geometry we recognize that the Hexa robot may

be associated to several varieties Some singularities of the Hexa robot

as well as correspondent varieties are shown in Figs 1a, 1c and 2 Inthe Hexa configurations of Fig 2a, two wrenches are collinear and so $r1and $r2 represent a Grassmann variety 1 (for short V1) Figure 2b showsthat four wrenches ($r1, $r2, $r3 and $r6) are on a flat pencil V2b Giventhat the Hexa robot has six wrenches acting upon the end effector, theconfiguration in Fig 2a may be associated to V5a and the configuration

in Fig 2b to V4d In Fig 2c all wrenches are parallel to each other andthey form a bundle of lines V3b In Fig 2d all wrenches lie in a planewith different intersection points and represent a V3d In Fig 1a allwrenches belong to a linear complex V5a, and Fig 1c shows an examplewhere all wrenches are meeting one given line V5b Each unconstrainedDOF of the end effector is represented by one $min in the Figs 1a, 1cand 2

V1 1

Figure 2. Grassmann variety on the Hexa robot: a) V1; b) V2b; c) V3b; d) V3d

Considering the workspace of the Hexa robot which is limited by theactuated joint angles, the possible Grassmann varieties may be reduced

to two: V5a and V5b Thus only singularities of Fig 1a and 1c may

Direct Singularity Closeness Indexes for the Hexa Parallel Robot 243

.

occur With the help of the Grassmann geometry all possible singularconfigurations on the Hexa robot are known Aiming at quantify thecloseness to the singularities of Fig 1a and Fig 1c, a measure algorithm

is presented in the sequence

A complex is generated by five skew symmetric lines (e.g wrench axes) Let π be a plane tangent to the complex that contains a point

of the line B6C6 (correspondent to the sixth wrench) The distance

between a certain point of this line and π gives a closeness measure to

that singularity It is possible to build a 4×4 skew symmetric matrix G so

that B T

i G C i = 0 where B i and C i(Fig 1c) are in projective coordinates.This linear system has six unknowns and five equations For simplicityand without loss of validity one unknown is set to 1 A pencil of lines of

the complex that contains B6 defines π Mathematically, if a projective point X p = [x y z 1] T = B6 is an element of π, then B T

Considering the vector U = [u1, · · · , u4]T = B T

of plane is u1x + u2y + u3z + u4 = 0 The distance between C6 and π may be interpreted as a measure for the line B6C6 of the complex:

u2

1+ u2

2+ u2 3

(3)

If all the lengths B i C i are the same and no other variety occurs,

Singularity of V5b occurs if all six wrench axes B i C iintersect one line

L This line crosses two wrenches in the points C These points must

belong to legs whose drive axes are collinear and L must be parallel

to these drive axes The maximal distance between L and all the six

wrenches is a measure to a singularity V5b of the Hexa robot

This algorithm is applied in order to measure the closeness of the Hexarobot to a singularity V5b as shown in Fig 1c The resulting distancemeasure to the singularity is presented in Fig 1d, where it linearly fallsdown to zero in the singularity Similarly to the minimization method,

a singular range is observed under the limit of 55 mm

The Grassmann approach as well as the power and frequency ods are experimentally validated in the Hexa robot and investigated foronline singularity detection All three methods allow a safe monitoring

meth-of such positions and present some properties are described next.movement (manually drived) from a rigid position, through a singularityV5b and twice singularity V5a, to a rigid position Figure 3a compares

244

The experiment presented in Figs 3a and 3b shows the end effector

C Bier, A Campos and J Hesselbach

both Grassmann algorithms with the power method and shows that aGrassmann algorithm V5a does not detect a singularity of V5b and viceversa A combined Grassmann index, the lower of the both algorithms,may be used due to that both present the same limit range of 55mm,which is a general property Comparing the combined Grassmann indexwith the power and the frequency method in Fig 3b, it can be observedthat all three methods have an equivalent behavior It is important tonotice that a scale factor is required due to the different physical base

of each method Additionally, it allows the use of a unique singularitylimit value

time [s]

1900 x power [ W] ~ combined Grassmann [mm]

singularity V5a

Figure 3. a) and b) Comparison of the singularity closeness indexes; c) tional time; d) Hexa robot of Collaborative Research Center 562

Computa-For the online application of an approach, the computing time is adecisive factor For the same trajectory of example Fig 3a, a comput-ing time comparison is presented in Fig 3c The combined Grassmannalgorithm is notably faster than the minimization methods Addition-ally, in the frequency approach has been observed that using only the end

effector instead of the whole manipulator mass matrix M EE, the tation time decreases without any loss of measure accuracy Therefore,

compu-it seems plausible to only use a simplified model of M EE

Direct Singularity Closeness Indexes for the Hexa Parallel Robot 245

.

Properties as computational time, number of measured indexes, ical index meaning, method complexity and implementation costs must

phys-to be considered phys-to choose a suitable approach for a particular tion The frequency method is general and can be applied to any type ofparallel manipulator The drawback of this approach is its complexityand that non-kinematic quantities (mass and stiffness) are introduced tothe measure On the other hand the power approach does not presentthis drawback and is not so complex However, it can not detect fi-nite and infinite (pure translation) unconstrained screw movements ofthe end effector with the same index Both minimization approach in-

applica-dexes have a physical meaning in particular conditions (e.g only unitary

wrenches acting upon the end effector), and generally they are only ciated to this physical meaning The Grassmann approach offers a fastand simple method to detect the singularities closeness with a geomet-rical meaning index The main weakness is that more than one index

asso-is required if the robot presents more than one singular variety It not taken for granted that the measure index can be always combinedaiming at automatic singularity avoidance strategies

can-In this paper, three singularity closeness indexes based on differentphysical meanings, are evaluated in the Hexa robot Each index is able

to detect all the direct singularities into the robot workspace The erties of the new Grassmann based index are compared to the odder twoapproaches and conclusions are presented

prop-References

Hesselbach, J., Bier, C., Campos, A., and L¨ owe, H (2005) Direct kinematic

sin-gularity detection of a hexa parallel robot Proceedings - ICRA, pp 3249-3254.

Barcelona.

Merlet, J (2000) Parallel Robots Kluwer Academic Publisher.

Pottmann, H., Peternell, M., and Ravani, B (1998) Approximation in line space–

applications in robot kinematics and surface reconstruction In Andvances in Robot

Kinematics, pp 403-412, Salzburg Kluwer Acad Publ.

Sciavicco, L and Siciliano, B (1996) Modeling and Control of Robot Manipulators.

McGraw-Hill.

Tsai, L (1999) Robot Analysis: the Mechanics of Serial and Parallel Manipulators.

John Wiley & Sons, New York.

Voglewede, P and Ebert-Uphoff, I (2004) Measuring closeness to singularities for

parallel manipulators In Proceedings - ICRA, New Orleans.

Xu, Y X., Kohli, D., and Weng, T C (1994) Direct differential kinematics of

hybrid-chain manipulators including singularity and stability analyses Journal of

Mechan-ical Design, vol 116, no 2, pp 614-621.

C Bier, A Campos and J Hesselbach

246

Davidson, J and Hunt, K (2004) Robots and Screw Theory Oxford University Press.

STEWART-GOUGH PLATFORMS WITH SIMPLE SINGULARITY SURFACE

Adolf Karger

Charles University Praha

Faculty of Mathematics and Physics

Adolf.Karger@mff.cuni.cz

Abstract The singularity surface of a parallel manipulator is a very complicated

algebraic surface of high degree in six-dimensional space of all possible positions of the manipulator (in the six-dimensional group of all space congruences) In this paper we show that for some classes of manipula- tors we can visualize the singularity set for any fixed orientation of the manipulator by a quadric in the space of translations Some properties and examples are given.

Keywords: Stewart-Gough platforms, parallel manipulators, Study representation,

is very complicated, in general it is given by 24 structural parameters

18 spatial coordinates of 6 points in the base and 18 in the platform, 12

of which can be specialized by using twice the congruence group, whichyields 12 parameters This means that the description of the motionleads to complicated equations and in general it is not possible to study

infor-One possibility is to choose a fixed orientation of the platform and forthis orientation describe all singular positions This can be done, but it

is easy to see that in this case we in general obtain a cubic surface in

E3, as the singularity surface is cubic in translations Cubic surfaces inspace are still relatively complicated objects to give a good idea abouttheir shape Using the general equation of the singular set we can showthat in case of S

and base the equation of the singular set becomes only quadratical and

© 2006 Springer Printed in the Netherlands

-G platforms with affinely corresponding platform

therefore in this case the singular set for fixed orientation is given by aquadric, which is much simpler to represent.

If the platform and base are affinely correspondent and planar, we getmuch more specialized situation the singular set factorizes into threefactors either points of the platform lie on a conic section or the orien-tation belongs to an algebraic hypersurface in the space of orientationsand the manipulator is singular for all translations or there is a plane ofsingular positions (depending on the orientation) This describes the sit-uation relatively well in both cases It seems that the described situation

is not the only possibility for which the singular set is quadratical, butthe general solution seems to be difficult for non-planar base or platform.This means to describe all parallel manipulators for which the singularset is quadratical in translations In the planar case the problem is notdifficult to solve, we show one example

parametrized by Euler parameters a ij = a ij (x α) and Study parameters

t i = t i (x α , y β ), i, j = 1, 2, 3, α, β = 0, , 3 in the usual way, see Botema,

Roth, 1990, Husty, 1991, Husty, Karger, 2000, Karger, 2002

We shall define the Study representation of the displacement group D6

of the Euclidean space We consider the 7-dimensional projective space

P7 of the vector space R8 with coordinates x0, , x3, y0, , y3 Points of

P7 are determined by non-trivial 8-tuples (x0, x1, x2, x3, y0, y1, y2, y3) of

real numbers, given up to a nonzero multiple From P7 we remove the

subspace given by equations x0 = 0, x1 = 0, x2 = 0, x3 = 0, let P 

the remaining part Let S be the set in P 

7 which is determined by theequation

S will be called the Study quadric We have a 1-1 correspondence

between points of S and elements of the group D6of space displacements

To simplify computations we can normalize coordinates in S by the

A Karger

248

by telescopic legs which connect six points of the moving space with sixpoints of the fixed space by spherical joints.

Let us suppose that we have chosen a system {O1,  e1,  e2,  e3}

respectively

Let M = (A, B, C) be a point in the fixed space (lower part of the platform), m = (a, b, c) be a point in the moving space (upper part of the

platform)

Any point m of the moving space can be also expressed by coordinates

In this equation we substitute Study parameters and the result is an

equation of degree four in x i , y i This equation simplifies considerably

if we add 4U2 to it Then the factor K factorizes out and we obtain a

Let us suppose that the Stewart-Gough platform is given by six

ar-bitrary points M i = (A i , B i , C i ) in the lower part and six points m i =

(a i , b i , c i ) in the upper part of the platform, r i be also given, i = 1, , 6.

We substitute coordinates of M i , m i in (2) and we obtain 6 equations

very simple Let us suppose that r i are given as functions of time, r i =

r i (t) This generates a motion in the moving space described by a curve

x i = x i (t), y i = y i (t), i = 0, , 3 in P7 We express the velocity operator

for this motion We can suppose that at instant t = t0 the motionpasses through identity (the frame in the moving space is identical withthe frame in the fixed one),

consequence of K = 1, U = 0 The vector (u1, u2, u3) yields the rotational

part of the velocity operator, (v1, v2, v3) yields its translational part

The derivative of (2) at t = t0 yields

r(t0)r  (t

0)/2 = u1(Bc −Cb)+u2(Ca −Ac)

Application of this procedure to (3) yields a linear mapping φ which transforms velocities (u1, u2, u3, v1, v2, v3) of the motion of the upper

part of the platform (end effector) into the linear velocities (r 

1, , r 

6) oftelescopic motions of legs of the manipulator In practical problems we

more often need the inverse φ −1 of this mapping, to express velocities

of the motion from velocities of legs As we have 6 linear equations

for 6 unknowns, it exists iff the matrix of coefficients of φ is regular.

Coefficients are

which are the Pl¨ucker coordinates of the line connecting points m and M Positions where φ is not invertible are called singular positions of the

parallel manipulator, see Botema, Roth, 1990, Karger, 2001, Karger,

2002, Ma, Angeles, 1992, Merlet, 1992

3.

The singular surface is a very complicated hypersurface in the dimensional space of all possible configurations of the parallel manip-ulator It is of degree 10 in Study parameters and it has about 2000