Tài liệu Advances in Robot KinematicsMechanisms and Motion Edited byJADRAN pdf

They consist of a fixed base and a platform, attached to the moving object, connected by six wires whose tension is maintained along the tracked trajectory.. It is assumed thatLet the pos

Advances in Robot Kinematics

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This is the tenth book in the series of Advances in Robot Kinematics.Two were produced as workshop proceedings, Springer published onebook in 1991 and since 1994 Kluwer published a book every two yearswithout interruptions These books deal with the theory and practice

of robot kinematics and treat the motion of robots, in particular robotmanipulators, without regard to how this motion is produced or con-trolled Each book of Advances in Robot Kinematics reports the mostrecent research projects and presents many new discoveries

The issues addressed in this book are fundamentally kinematic innature, including synthesis, calibration, redundancy, force control, dex-terity, inverse and forward kinematics, kinematic singularities, as well asover-constrained systems Methods used include line geometry, quater-nion algebra, screw algebra, and linear algebra These methods are ap-plied to both parallel and serial multi-degree-of-freedom systems Theen

application

All the contributions had been rigorously reviewed by independentreviewers and fifty three articles had been recommended for publica-tion They were introduced in seven chapters The authors discussedtheir results at the tenth international symposium on Advances in RobotKinematics which was held in June 2006 in Ljubljana, Slovenia Thesymposium was organized by Jozef Stefan Institute, Ljubljana, underthe patronage of IFToMM - International Federation for the Promotion

of Mechanism and Machine Science

We are grateful to the authors for their contributions and for theirefficiency in preparing the manuscripts, and to the reviewers for theirtimely reviews and recommendations We are also indebted to the per-sonnel at Springer for their excellent technical and editorial support

Jadran Lenarˇciˇc and Bernard Roth, editors

results should interest researchers, teachers and students, in fields ofgineering and mathematics related to robot theory, design, control and

Determining the 3×3 rotation matrices that satisfy three linear

P.M Larochelle

A polar decomposition based displacement metric for a finite

J.-P Merlet, P Donelan

On the regularity of the inverse Jacobian of parallel robots 41

P Fanghella, C Galletti, E Giannotti

Parallel robots that change their group of motion 49

A.P Murray, B.M Korte, J.P Schmiedeler

Approximating planar, morphing curves with rigid-body linkages 57

M Zoppi, D Zlatanov, R Molfino

On the velocity analysis of non-parallel closed chain mechanisms 65

Properties of Mechanisms

H Bamberger, M Shoham, A Wolf

Kinematics of micro planar parallel robot comprising large joint

H.K Jung, C.D Crane III, R.G Roberts

Stiffness mapping of planar compliant parallel mechanisms in a

Y Wang, G.S Chirikjian

Large kinematic error propagation in revolute manipulators 95

A Pott, M Hiller

A framework for the analysis, synthesis and optimization

Z Luo, J.S Dai

Searching for undiscovered planar straight-line linkages 113

X Kong, C.M Gosselin

Type synthesis of three-DOF up-equivalent parallel

manipulators using a virtual-chain approach 123

A De Santis, P Pierro, B Siciliano

The multiple virtual end-effectors approach for human-robot

Humanoids and Biomedicine

J Babiˇ c, D Omrˇ cen, J Lenarˇ ciˇ c

Balance and control of human inspired jumping robot 147

J Park, F.C Park

A convex optimization algorithm for stabilizing whole-body

R Di Gregorio, V Parenti-Castelli

Parallel mechanisms for knee orthoses with selective recovery

S Ambike, J.P Schmiedeler

Modeling time invariance in human arm motion coordination 177

M Veber, T Bajd, M Munih

Assessment of finger joint angles and calibration of instrumental

R Konietschke, G Hirzinger, Y Yan

All singularities of the 9-DOF DLR medical robot setup for

G Liu, R.J Milgram, A Dhanik, J.C Latombe

On the inverse kinematics of a fragment of protein backbone 201

V De Sapio, J Warren, O Khatib

Predicting reaching postures using a kinematically constrained

Analysis of Mechanisms

D Chablat, P Wenger, I.A Bonev

Self motions of special 3-RPR planar parallel robot 221

A Degani, A Wolf

Graphical singularity analysis of 3-DOF planar parallel

C Bier, A Campos, J Hesselbach

Direct singularity closeness indexes for the hexa parallel robot 239

A Karger

Stewart-Gough platforms with simple singularity surface 247

A Kecskem´ ethy, M T¨ andl

A robust model for 3D tracking in object-oriented multibody

systems based on singularity-free Frenet framing 255

A geometrical interpretation of 3-3 mechanism singularities 285

Workspace and Performance

J.A Carretero, G.T Pond

Quantitative dexterous workspace comparisons 297

E Ottaviano, M Husty, M Ceccarelli

Level-set method for workspace analysis of serial manipulators 307

M Gouttefarde, J P Merlet, D Daney

Determination of the wrench-closure workspace of 6-DOF

-J.A Snyman

On non-assembly in the optimal synthesis of serial manipulators

Design of Mechanisms

W.A Khan, S Caro, D Pasini, J Angeles

Complexity analysis for the conceptual design of robotic

D.V Lee, S.A Velinsky

Robust three-dimensional non-contacting angular motion sensor 369

K Brunnthaler, H.-P Schr¨ ocker, M Husty

Synthesis of spherical four-bar mechanisms using spherical

R Vertechy, V Parenti-Castelli

Synthesis of 2-DOF spherical fully parallel mechanisms 385

G.S Soh, J.M McCarthy

Constraint synthesis for planar n-R robots 395

T Bruckmann, A Pott, M Hiller

Calculating force distributions for redundantly actuated

S Krut, F Pierrot, O Company

On PKM with articulated travelling-plate and large tilting angles 445

C.R Diez-Mart´ınez, J.M Rico, J.J Cervantes-S´ anchez,

J Gallardo

Mobility and connectivity in multiloop linkages 455

K Tcho´ n, J Jakubiak

Jacobian inverse kinematics algorithms with variable steplength

x

-based Stewart platforms

Contents

J Zamora-Esquivel, E Bayro-Corrochano

Kinematics and grasping using conformal geometric algebra 473

R Subramanian, K Kazerounian

Application of kinematics tools in the study of internal

O Altuzarra, C Pinto, V Petuya, A Hernandez

Motion pattern singularity in lower mobility parallel

Determining the 3×3 rotation matrices that satisfy three

linear equations in the direction cosines

P.M Larochelle

A polar decomposition based displacement metric for a finite

region of SE(n)

J.-P Merlet, P Donelan

On the regularity of the inverse Jacobian of parallel robots

P Fanghella, C Galletti, E Giannotti

Parallel robots that change their group of motion

A.P Murray, B.M Korte, J.P Schmiedeler

Approximating planar, morphing curves with rigid-body

linkages

M Zoppi, D Zlatanov, R Molfino

On the velocity analysis of non-parallel closed chain

MUTUAL INFORMATION

Juan Andrade-Cetto

Computer Vision Center, UAB

Edifici O, Campus UAB, 08193 Bellaterra, Spain

cetto@cvc.uab.es

Federico Thomas

Institut de Rob` otica i Inform` atica Industrial, CSIC-UPC

Llorens Artigas 4-6, 08028 Barcelona, Spain

fthomas@iri.upc.edu

Abstract

ing devices They consist of a fixed base and a platform, attached to the moving object, connected by six wires whose tension is maintained along the tracked trajectory One important shortcoming of this kind

of devices is that they are forced to operate in reduced workspaces so

as to avoid singular configurations Singularities can be eliminated by adding more wires but this causes more wire interferences, and a higher force exerted on the moving object by the measuring device itself This paper shows how, by introducing a rotating base, the number of wires can be reduced to three, and singularities can be avoided by using an active sensing strategy This also permits reducing wire interference problems and the pulling force exerted by the device The proposed sensing strategy minimizes the uncertainty in the location of the plat- form Candidate motions of the rotating base are compared selected automatically based on mutual information scores.

Keywords:

Tracking devices, also called 6-degree-of-freedom (6-DOF) devices, areused for estimating the position and orientation of moving objects Cur-rent tracking devices are based on electromagnetic, acoustic, mechani-cal, or optical technology Tracking devices can be classified according

to their characteristics, such as accuracy, resolution, cost, measurementrange, portability, and calibration requirements Laser tracking systemsexhibit good accuracy, which can be less than 1µm if the system is wellcalibrated Unfortunately, this kind of systems are very expensive, their

3

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 3–14

© 2006 Springer Printed in the Netherlands

Wire-based tracking devices are an affordable alternative to costly

track-Tracking devices, Kalman filter, active sensing, mutual information, parallel manipulators

calibration procedure is time-consuming, and they are sensitive to theenvironment Vision systems can reach an accuracy of 0.1mm They arelow-cost portable devices but their calibration procedure can be compli-cated Wire-based systems can reach an accuracy of 0.1mm, they arealso low cost portable devices but capable of measuring large displace-ments Moreover, they exhibit a good compromise among accuracy,measurement range, cost and operability.

Wire-based tracking devices consist of a fixed base and a platformconnected by six wires whose tension is maintained, while the platform ismoved, by pulleys and spiral springs on the base, where a set of encodersgive the length of the wires They can be modelled as 6-DOF parallelmanipulators because wires can be seen as extensible legs connectingthe platform and the base by means of spherical and universal joints,respectively

Dimension deviations due to fabrication tolerances, wire-length certainties, or wire slackness, may result in unacceptable performance ofrors can be eliminated by calibration Some techniques for specific errorshave already been proposed in the literature For example, a methodfor compensating the cable guide outlet shape of wire encoders is de-tailed in Geng and Haynes, 1994, and a method for compensating thedeflections caused by wire self-weights is described in Jeong et al., 1999

un-In this paper, we will only consider wire-length errors which cannot becompensated because of their random nature

Another indirect source of error is the force exerted by the measuringdevice itself Indeed, all commercial wire encoders are designed to keep

a large string tension This is necessary to ensure that the inertia of themechanism does not result in a wire going slack during a rapid motion

If a low wire force is used, it would reduce the maximum speed of theobject to be tracked without the wires going slack On the contrary, if ahigh wire force is used, the trajectory of the object to be tracked could

be altered by the measuring device Hence, a trade-off between accuracyand speed arises

The minimum number of points on a moving object to be tracked forpose measurements is three Moreover, the maximum number of wiresattached to a point is also three, otherwise the lengths of the wires willnot be independent This leads to only two possible configurations forthe attachments on the moving object The 3-2-1 configuration was pro-posed in Geng and Haynes, 1994 The kinematics of this configurationwas studied, for example, in Nanua and Waldron, 1990 and Hunt andPrimrose, 1993 Its direct kinematics can be solved in closed-form byusing three consecutive trilateration operations yielding 8 solutions, as

a wire-based tracking device In general, the effects of all systematic

θA xA

Figure 1 The main two configurations used for wire-based tracking devices: (a) the

“3-2-1”, (b) the “2-2-2”, and (c) the proposed tracking device, with (d) the rotating

in Thomas et al., 2005 The 2-2-2 configuration was first proposed inJeong et al., 1999 for a wire-based tracking device The kinematics ofthis configuration was studied, for example, in Griffis and Duffy, 1989,Nanua et al., 1990, and Parenti-Castelli and Innocenti, 1990 where itwas shown that its forward kinematics has 16 solutions In other words,there are up to 16 poses for the moving object compatible with a givenset of wire lengths These configurations can only be obtained by a nu-merical method The two configurations above were compared, in terms

of their sensitivity to wire-length errors, in Geng and Haynes, 1994 Theconclusion was that they have similar properties

This paper is organized as follows Section 2 contains the ical model of our proposed 3-wire-based sensing device, while Section 3derives the filtering strategy for tracking its pose Given that this devicehas a moving part, Section 4 develops an information theoretic metricfor choosing the best actions for controlling it A strategy to preventpossible wire crossings is contemplated in Section 5 Section 6 is de-voted to a set of examples demonstrating the viability of the proposedapproach Finally, concluding remarks are presented in Section 7

mo-More elegantly, and to let the tracked object move at a faster speed,measurements can be integrated sequentially through a partially observ-able estimation framework That is, a Kalman filter

base.

Kinematics of the Proposed Sensor

Consider the 3-wire parallel device in Figure 1(c) It is assumed thatLet the pose of our tracking device be defined as the 14-dimensional array

where p = (x, y, z)
is the position of the origin of a coordinate frame

fixed to the platform, θ = (ψ, θ, φ)
is the orientation of such coordinate

frame expressed as yaw, pitch and roll angles, v = (vx, vy, vz)
and

ω = (ωx, ωy, ωz)
are the translational and rotational velocities of p,respectively; and θA and ωA are the orientation and angular velocity ofthe rotating base

Assume that the attaching points on the base ai, i = 1, 2, 3, aredistributed on a circle of radius ¯a as shown in Figure 1(d) Then, the

coordinates of ai can be expressed in terms of the platform rotation

Moreover, let ei be the unit norm vector specifying the direction from

ai to the corresponding attaching point bi in the platform; and let li

be the length of the i-th wire, i = 1, 2, 3 The value of bi is expressed

in platform local coordinates, whereR is the rotation matrix describingthe absolute orientation of the platform Then, the position of the wireattaching points in the platform, in global coordinates, are

the platform configuration is free to move in any direction in IR3×SO(3)

with δa and δα zero mean white Gaussian translational and angular

acceleration noises Moreover, the adopted model for the translationaland angular velocities of the platform is given by

in which the control signal modifying the base orientation is the eration impulse αA

accel-Since in practice, the measured wire lengths, li, i = 1, 2, 3, will becorrupted by additive Gaussian noise, δzi, we have that

zi(t) = li(t) + δzi(t) =
p(t) + R(t)bi− ai(t)
+ δzi(t) (7)Lastly, the orientation of the moving base is measured by means of

an encoder Its model is simply

z4(t) = θA(t) + δz4(t) (8)

Eqs 4 and 5 constitute our motion prediction model f (x, αA, δx).

Now, an Extended Kalman Filter can be used to propagate the platformpose and velocity estimates, as well as the base orientation estimates,and then, to refine these estimates through wire length measurements

To this end, δx ∼ N (0, Q), δz ∼ N (0, R), and our plant Jacobians with respect to the state F = ∂f /∂x, and to the noise G = ∂f /∂δx become

1 τ1

The measurement Jacobians H = ∂h/∂x are simply

Hi(t) =

ei(t) bi× ei(t) 0 0 ∂hi

∂θ A 0 , (10)with

ei(t) = p( t) + R(t)bi− ai(t)

p(t) + R(t)bi− ai(t)
. (11)

Eqs 7 and 8 complete our measurement prediction model h(x, δz).

Then, by rewritingR =

⎡

⎣r

1

r
2

r
3

xt+τ|t= f (xt|t, αA, 0) (14)

Pt+τ|t= FPt|tF
+ GQG
(15)and, the revision of the state estimate and state covariance are

be gained from future wire measurements were such a move be made,but taking into account the information lost as a result of moving withuncertainty

The essential idea is to use mutual information as a measurement

of the statistical dependence between two random vectors, that is, theamount of information that one contains about the other Consider

the states x, and the measurements z The mutual information of the

two continuous probability distributions p(x) and p(z) is defined as the information about x contained in z, and is given by

I(x, z) =

x,z p(x, z) log p(x, z)

p(x)p(z) dxdz (18)

Note how mutual information measures the independence between

the two vectors It equals zero when they are independent, p(x, z) = p(x)p(z) Mutual information can also be seen as the relative entropy between the marginal density p(x) and the conditional p(x|z)

I(x, z) =

x,z p(x, z) log p(x|z)

Given that our variables of interest can be described by multivariate

Gaussian distributions, the parameters of the marginal density p(x) are trivially the Kalman prior mean xt+τ|tand covariance Pt+τ|t Moreover,

the parameters of the conditional density p(x|z) come precisely from the Kalman update equations xt+τ|t+τ and Pt+τ|t+τ Substituting thegenaral form of the Gaussian distribution in Eq 19, we can obtain aclosed formula

I(x, z) = 1

2

log|Pt+τ|t| − log |Pt+τ|t+τ| (20)Thus, in choosing a maximally mutually informative motion com-mand, we are maximizing the difference between prior and posteriorentropies (MacKay, 1992) In other words, we are choosing the motion

command that most reduces the uncertainty of x due to the knowledge

of z.

The real-time requirements of the task preclude using an optimal trol strategy to search for the base rotation command that ultimatelymaximizes our mutual information metric Instead, we can only evalu-ate such metric for a discrete set of actions within the range of possiblecommands, and choose the best action from those The set of possibleactions is a discretization of a range of accelerations

Providing the base with the ability to rotate has the added advantage

of increasing the range of motion of the tracked platform; mainly, forrotations along the vertical axis One of the main difficulties however,

is in appropriately choosing base rotation commands so as to preventwire crossings Considering that wire end-point displacements are suf-ficiently small per sampling interval, the trajectory described by each

wire can be assumed to be circumscribed within a tetrahedron Oneway to predict wire crossings is by checking whether the tetrahedradescribed by the current and posterior poses for each wire intersecteach other; each tetrahedron described by the four attaching points

{ai,t|t, ai,t+τ|t, bi,t|t, bi,t+τ|t}

A very fast test of tetrahedra intersection is based on the SeparatingAxis Theorem described in the computer graphics literature (Ganovelli

et al., 2003) The test consists on checking whether the plane lying onthe face of one tetrahedron separates the two of them If this is notthe case, the test continues to find out if there exists a separating planecontaining only one edge on one of the tetrahedra

In a cable extension transducer, commonly known as a string pot,the tension of the cable is guaranteed by a spring connected to its spool.Using a cable guide, the cable is allowed to move within a 20◦cone, mak-ing it suitable for 3D motion applications There are cable guides thatpermit 360◦by 317◦ displacement cable orientation flexibility Manufac-turers of such sensors are Celesco Transducer Products Inc., SpaceAgeControl Inc., Carlen Controls Inc., and several others

String pots provide a long range (0.04 − 40m), with typical accuracy

of 0.02% of full scale The maximum allowable cable velocity is about7.2m/s and the maximum cable acceleration is about 200m/s2

The usefulness of a tracking device depends on whether it can trackthe motion fast enough This ability is determined by the lag, or latency,between the change of the position and orientation of the target beingtracked and the report of the change to the computer In virtual realityapplications, lags above 50 milliseconds are perceptible to the user Ingeneral, the lag for mechanical trackers is typically less than 5ms

The quality of the estimated pose is directly influenced by the velocity

at which the base can rotate To determine the range of motion velocitiesthat can be tracked with our system, a tracking simulation was repeatedlimiting the base rotation velocity A set of 20 runs was conducted,varying the maximum platform rotation speed from 0 to 1 rad/s, andwith time steps of 0.01 s; the tracked object translating at a constantvelocity of 0.2 m/s along the x axis, and rotating at10π rad/s about an10

axis perpendicular to the base Figure 2 shows the average error of the pose

J Andrade-Cetto and F Thomas

Figure 2 Average position and orientation recovery error as a function of the maximum platform rotation speed, and 2nd order curve fit.

Figure 3 Wire sensing device The rotating base is attached to the Staubli arm shown in the left side The moving platform is attached to the arm shown to the right.

estimation as a function of the maximum base rotational velocity Thebest pose estimations are achieved when the base rotates at twice the

5 rad/s for this experiment

6.3

cable crossing allows it, the largest acceleration commands are selected.This is because prior and posterior entropy difference is maximized forbase and the platform have been arranged to form equilateral triangles

10 rad/s, whilstThe maximum base rotation

the wire length measurements along the trajectory Wire length sensorsare modeled with additive Gaussian noise with zero mean and 1 mm

pure rotations along the vertical axis The idea is to show that, whenever

Pure Rotations

A second experiment consisted in testing the tracking system under

standard deviation Moreover, readings of the base orientation are also

largest possible configuration changes The attaching points in both theTheir coordinates can be found in Table 1, and refer to the frames

kept at a distance of 1 m from the base

shown in Figure 1 The actual testbench used is shown in Figure 6.3

Table 1 Coordinates of the attaching points (in meters) in their local coordinate

modeled with zero mean white additive Gaussian noise with 0.001 rad

standard deviation Figures 4(b) and 4(c) show the tracked object

po-sition and orientation recovery errors, respectively The motion of the

rotating base is depicted in Figures 4(d)-4(e), showing that commands

for maximal platform rotation velocities are being selected from our

mu-tual information metric (Figure 4(f))

6.4

In this last example, the tracked object moves back and forth in the

three Cartesian components along a line from (1, 1, 1) to (2, 2, 2) meters,

ponents This experiment shows that for compound motions it is more

J Andrade-Cetto and F Thomas

(b) Position Error

0 1 2 3 4 5

−0.1

−0.05 0 0.05 0.1

Time (sec)

roll pitch yaw

Figure 5 Wire tracking of compound motion.

difficult to disambiguate orientation error, while still doing a good job attracking the correct object pose Once more, the maximum base rotation

5 rad/sec, and the limit for possible base tion command was set to 30 rad/sec2 Figure 5(a) shows the evolution ofwire length measurements for this example The tracked object positionand orientation errors is shown in Figures 5(b) and 5(c) The motion

accelera-of the rotating base is depicted in Figures 5(d)-5(e) And, our mutualinformation action selection mechanism is shown in Figure 5(f)

An active sensing strategy for a wire tracking device has been sented It has been shown how by allowing the sensor platform rotateabout its center, a wider range of motions can be tracked by reducingthe number of wires needed from 6 to 3 Moreover, platform rotation isperformed so as to maximize the mutual information between poses andmeasurements, and at the same time, so as to prevent wire wrappings

Time (sec)

x z

Time (sec)

roll pitch yaw

5x 10−3

Time (sec)

(e) Base Angle Error

0 1 2 3 4 5 0

0.5 1 1.5 2

Time (sec)

(f) Mutual Info.

Acknowledg ments

J Andrade-Cetto completed this work as a Juan de la Cierva doctoral Fellow of the Spanish Ministry of Education and Science underproject TIC2003-09291 and was also supported in part by projects DPI2004-05414, and the EU PACO-PLUS project FP6-2004-IST-4-27657

Post-F Thomas was partially supported by the Spanish Ministry of cation and Science, project TIC2003-03396, and the Catalan ResearchCommission, through the Robotics and Control Group

Jeong, J., Kim, S., and Kwak, Y (1999) Kinematics and workspace analysis of

a parallel wire mechanism for measuring a robot pose Mechanism and Machine Theory, 34(6):825–841.

tion Neural Computation, 4(4):589–603.

Merlet, J P (2006) Parallel Robots, volume 128 of Solid Mechanics and its tions Springer, New York, 2nd edition.

Applica-Nanua, P and Waldron, K (1990) Direct kinematics solution of a special parallel robot structure In Proceedings of the 8th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, pages 134–142, Warsaw.

Nanua, P., Waldron, K., and Murthy, V (1990) Direct kinematic solution of a Stewart platform IEEE Transactions on Robotics and Automation, 6(4):438–444.

Parenti-Castelli, V and Innocenti, C (1990) Direct displacement analysis for some classes of spatial parallel machanisms In Proceedings of the 8th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, pages 126–133, Warsaw.

Thomas, F., Ottaviano, E., Ros, L., and Ceccarelli, M (2005) Performance analysis

of a 3-2-1 pose estimation device IEEE Transactions on Robotics, 21(3):288–297 Vidal-Calleja, T., Davison, A., Andrade-Cetto, J., and Murray, D (2006) Active con- trol for single camera SLAM In Proceedings of the IEEE International Conference

on Robotics and Automation, Orlando To appear.

Geng, Z.J and Haynes, L.S (1994) A 3-2-1 kinematic configuration of a Stewart

MacKay, D.J C (1992) Information based objective functions for active data

selec-e

FOR STEWART GOUGH PLATFORMS

G Nawratil

Vienna University of Technology

Institute of Discrete Mathematics and Geometry

upuaut@controverse.net

Abstract Singular postures of Stewart Gough Platforms must be avoided because

close to singularities they lose controllable degrees of freedom Hence there is an interest in a distance measure between the instantaneous configuration and the nearest singularity This article presents such a measure, which is invariant under Euclidean motions and similarities, which has a geometric meaning and can be computed in real-time This measure ranging between 0 and 1 can serve as a performance index.

Keywords: Stewart Gough Platform, distance measure, perfomance index

In Section 3 of this article we define a new measure, which allows

to compare different postures of different nonredundant Stewart Gough

Platforms (SGP s) Such a measure should assign to each configuration

K a scalar D(K) obeying the following six properties:

1 D( K) ≥ 0 for all K of the configuration space,

2 D( K) = 0 if and only if K is singular,

3 D( K) is invariant under Euclidean motions,

4 D( K) is invariant under similarities,

5 D( K) has a geometric meaning,

6 D( K) is computable in real-time.

K is singular if and only if the six legs belong to a linear line complex

(see Merlet, 1992) or, analytically seen, the determinant of the Jacobian

i ,l i) of the carrier line L i of the i th leg oriented in the direction

BiPi We’ll assume for the rest of this article that Bi = P i for i = 1, , 6.

© 2006 Springer Printed in the Netherlands

15

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 15–22

Kinematic meaning of the Jacobian The velocity vector v(Pi) of

Pi due to the instantaneous screw (= twist) q := (q,q) of the platform

Σ against Σ0 can be decomposed in a component vL(Pi ) along the i th

legL i and in a component v⊥(Pi ) orthogonal to it (see Fig 1), thus

v(Pi) =q + (q× P i) = vL(Pi) + v⊥(Pi) (2)with v L(Pi) = li

l i  ·v(P i) = li

l i  ·q +

li

l i  ·q =: d i (3)Therefore the Jacobian J is the matrix of the linear mapping

ι : q → d = J q with d = (d1, , d6)T (4)

ι has at least a one-dimensional kernel ker ι, ifK is singular Let k ∈ ker ι

and k= o Then also µk with µ ∈ R lies in ker ι Therefore we can say,

that v(Pi) can be arbitrarily large for vanishing translatory velocities in

the six prismatic legs The sole exeption is the case where Pi lies on the

instantaneous screw axis (isa) and k is an instantaneous rotation.

Review. In the following we analyze some of the in our opinion mostimportant indices in view of the initially stated six properties

The manipulabilitiy introduced by Yoshikawa, 1985 is not invariant under similarities, because for SGP s it equals |det(J )| So Lee et al.,

1998 used |det(J )|·|det(J )| −1 m as index, where |det(J )| m denotes themaximum of|det(J )| over the SGP’s configuration space But the com-

putation of |det(J )| m is a nonlinear task and was only done for planar

SGP s with very special geometries Only for these SGP s |det(J )| m can

be interpreted geometrically as the volume of the framework

Pottmann et al., 1998 introduced the concept of the best fitting linear

line complex c of L1, , L6 The suggested index equals the square root

of the minimum of 

d2i with respect to c under the side condition

cTc = 1 The index is not invariant under similarities and it is not

defined for instantaneous translations c In order to close this gap, the

authors proposed to minimize a further function, which yields a secondvalue But how should these two values be combined to a single number?

The rigidity rate introduced by Lang et al., 2001 is based on the idea, that a SGP at any position K permits a one-parametric self-motion

within the group of Euclidean similarities G7 The angle ϕ ∈ [0, π/2]

between the tangent of the self-motion inK and the subgroup of

Eucli-dean displacements serves as an index But the choice of the invariantsymmetric bilinear form in the tangent space of G7, which is necassary

in order to define a measure in the sense of non-Euclidean geometry, is

arbitrary Although ϕ fulfills all six stated properties, its applicability is limited This becomes manifest in the remark at the end of Section 5.

G Nawratil

16

Now we take a closer look at the reciprocal of the condition number (cdn −1) introduced by Salisbury and Craig, 1982, because it will bethe starting point of our considerations cdn −1 equals the ratio of the

minimum λ − and the maximum λ+ of the quadratic objective function

with p denoting the isa, ω the angular velocity and ω the translatory

velocity of the screw q, under the quadratic side condition

ν(q) : dTd = qT N q = 1 with N = J T J (6)

Due to the linearity of ι in (4) the screw µq corresponds to the

µ-fold translatory velocity d i in the six prismatic legs, and therefore the

side condition ν(q) is well defined The weak point of this index is the

objective function for the following reasons First, it is not invariantunder translations, because ζ(q) depends on the choice of O In practice

O is not selected arbitrarily, but placed in the tool center point But

the real problem, which causes the variance of cdn −1 under similarities,

occurs from the dimensional inhomogeneity of ζ(q) To overcome this

deficiency, different concepts (e.g characteristic length, see Zanganeh

and Angeles, 1997) were introduced, but they still weight the ratio oflength and angle in a more or less arbitrary way The inhomogeneity andthe lacking invariance of ζ(q) do not allow a geometric interpretetion of

cdn −1 and they question its adequacy as a performance index for SGP s.

The conslusion of this considerations is, that we have to look for a new

objective function ζ(q) which meets our initially stated demands But

we want to add a further argument, which has the following motivation:

The cdn −1 as well as the manipulability are also used to optimize the design of SGP s But these two indices do not depend on the choice of

Bi and Pi onL i as long as Bi = P i Thus we require:

7 D( K) depends on the geometry of the SGP, not

only on the carrier linesL1, ,L6 of the six legs

Pottmann et al., 1998 also presented a modified version of his method,

namely the line segment method, which statisfies the 7 th demand but

does not eliminate the other weak points The rigidity rate is

indepen-dent of the choice of the base anchor points and so it only takes thegeometry of the platform into consideration This raises the followingproblem: If we change the viewpoint and consider Σ as the unmovedbase and Σ0 as platform, we get another index for the same SGP con- figuration So the instantaneous rigidity of the SGP depends on the

viewpoint which is dissatisfying

Preliminary Considerations

2.1 Uncontrollable ostures of SGPs

In practice configurations must be avoided, where minor variations

of the leg lengths have uncontrollable large effects on the instantaneousdisplacement of the platform Σ But how should the quantity of effects

be measured in relation to the variation of the leg lengths? The boardercase of this uncontrollability is, if there exists an infinitesimal motion of

Σ while all actuators are locked In such a singular position the velocities

of the platform points can be arbitrarily large, and therefore the posture

is uncontrollable The question is, which measurable parameter of the

SGP indicates the circumstance of uncontrollability in a natural way

and has a geometric meaning for the manipulator

Let’s assume there is instantaneously a minor variation of the six leg

lenghts and the SGP is not singular So there exists a unique screw q

our 7th property, we consider the velocity v(Pi) of Pi with respect to

q We are not interested in the instantaneous displacements of Pi indirection of the leg, because the leg length is an active joint which can

indicator of uncontrollability But v⊥(Pi) is no mechanical parameter

of a SGP and therefore we look at the angular velociety ω B i of the i th passive base joint This ω B i

nal to
v ⊥(Pi)
But there also exists angular

velocities ω P i in the passive platform joints,

which are defined analogously The sole

diffe-rence is that we regard the inverse motion of q.

q in (2), (3) and (7) Obviously ω B i2 and ω P i2

are quadratic forms with the coordinates of q

as unkowns Therefore we can rewrite them as

ω B i2 = qT W B iq and ω P i2= qT W P i q, (8)

Now we define the new objective function ζ(q) as

where λ − resp λ+ is the minimum resp maximum of the objective

func-tion ζ(q) in (9) under the side condifunc-tion ν(q) in (6) ctn( K) = 0 racterizes a singular configuration and a value of 1 an optimal one.

We solve the optimization problem in order to compute λ − resp λ+

by introducing a Lagrange multiplier λ Then the approach simplifies in

consideration of ∇ζ = 2 Z q and ∇ν = 2 N q, to the general eigenvalue

problem (Z −λ N ) q = o This system of linear equations has a

nontrivi-al solution, if and only if |Z − λ N | = 0 The degree of the characteristic

polynomial in λ corresponds with rank( J ) because of N = J T J Every

general eigenvalue λ i is linked with an general eigenvector ei The

smal-lest λ − and the largest λ+ are the wanted extreme magnitudes becauseof

Z e i = λ i N e i and eT i N e i = 1 ⇒ ζ(e i ) = λ i (11)

Theorem 1 λ − and λ+ of Def 1 are the extreme general eingenvalues

of Z with respect to N All roots λ i of the characteristic polynomial

|Z − λ N | = 0 are positive if and only if rank(J ) = 6.

with

Proof: According to Hestenes, 1975 all λi’s are real Due to (11) all λi’sare nonnegative If q is no pure translation (q 6= o), then all angularvelocities in the passive joints would vanish if and only if the 12 anchorpoint lie on the isa But such a configuration yields rank(J ) = 1 In thecase of a pure translation, there would be no angular velocities in the

Theorem 2 The number of roots λi of the characteristic polynomial

|Z − λ N | = 0 dropping to infinity equals the defect(J )

Proof: All screws ±µq ∈ kerιwith µ ∈ R cause arbitrarily large velocitiesv(Pi) = v⊥(Pi) resp v(Bi) = v⊥(Bi) and therefore arbitrarily large ωB i

resp ωPi The proof follows by carring out limµ→∞ and (11) Due to Theorem 1 and 2 the control number is well defined Thereforeall initially stated seven properties are obviously fulfilled

the translation But such a configuration yields rank(J ) ≤ 3

passive joints if and only if the legs are parallel to the direction of

W

Remark. It does not make sense to define ζ(q) only as

not fulfill our 2nd demand, because there exist nonsingular SGP

confi-gurations, where the L i’s are the path tangents of Pi (resp Bi) with

regard to q Consequently we get ζ(q) = 0 and the index would equal 0.

4.1

According to Wolf and Shoham, 2003 the closest path normal complex

of a helical motion (rotations and translations included) to L1, , L6,described by its axis and pitch, provides additional information on the

SGP ’s instantaneous motion and understanding of the type of singularity

when the SGP is at, or in the neighborhood of, a singular configuration Since the ctn is a performance index as well as a distance measure, a small ctn indicates the closeness to a singularity Due to Theorem 2 and

the continuity of the polynomial functions |Z − λ N | = 0, which arise if

we move towards a singular position, we can say that the closest linearcomplex to L1, , L6 equals the path normal complex of e+ according

to (11) Therefore this method additionally brings about a kind of bestapproximating linear line complex in the neighbourhood of singularities,and the calculation needs no case analysis like Pottmann’s method

We consider a two parametric set S K of configurations K, given by

Bi = (cos α i , sin α i , −h) T and Pi = (cos β i , sin β i , h) T with

6] denotes the design parameter and h ∈ R+ the posture

parameter of the SGP All K ∈ S K with α = π

6 and h / ∈ {0, ∞} are

nonsingular We study this example, because such manipulators are veryrelevant in practice as flight simulators The matrix Z − λN can be

manipulated by elementary row and column operations to the diagonal

matrix diag(∆1, , ∆6) Therefore the eigenvalues λ i can be computedexplicitly using ∆i = 0, whereas λ1= λ2 and λ4 = λ5.K+ given by

(see Fig 2 and 3 ) ForK

Instantaneous Motion near Singularities

E

has the maximal ctn of all K ∈ S s

The SGP with α+ also makes sense from the practical point of view,because contrary to the often propagandized 3-3 octahedral manipula-

tor (α = 0) no anchor points coincide But coinciding anchor points are

lines of ctn when the platform is translated away from the central

loca-tion parallel to the base plane The difference between two neighbouring

and 8 illustrate the graphs of ctn dependig on the angle ω of the rotation

of Σ about an axis parallel to x, z or y, respectively, through (0, 0, h+)

g.Fi

Remark. The rigidity rate of all nonsingular configurations of this set

rigidity rate is constant /2 Therefore this index is not recommendable for comparing different postures of different SGP s.

The presented index, called control number (ctn), allows to compare different postures of different SGP s, because it obeys the initially stated seven conditions Therefore ctn can serve as a performance index as well

as a distance measure to the closest singularity This concept can also

be modified for redundant SGP s and 3 dof RPR manipulators.

pre-paration It can be proved, that such configurations do exist New formance indices for 6R robots have been presented in Nawratil, 2006

Nawratil, G (2006), New Performance Indices for 6R Robot Postures, CD-Proc of the 1st EuCoMeS (M Husty, H.P Schr¨ ocker, eds.), 12pp., Obergurgl, Austria Pottmann, H., Peternell, M., and Ravani, B (1998), Approximation in line space: applications in robot kinematics and surface reconstruction, Advances in Robot Kinematics: Analysis & Control (J Lenarcic, M Husty, eds.), pp 403–412, Kluwer Salisbury, J.K and Craig, J.J (1982), Articulated Hands: Force Control and Kine- matic Issues, Int J of Robotics Research, vol 1, no 1, pp 4–17.

Wolf, A., and Shoham, M (2003), Investigations of Parallel Manipulators Using Linear Complex Approximation, Journal of Mechanical Design, vol 125, pp 564–572 Yoshikawa, T (1985), Manipulability of Robotic Mechanisms, Int J of Robotics Re- search, vol 4, no 2, pp 3–9.

Zanganeh, K.E., and Angeles, J (1997), Kinematic Isotropy and the Optimum Design

of Parallel Manipulators, Int J of Robotics Research, vol 16, no 2, pp 185–197.

π π

e

DETERMINING THE 3 ××××3 ROTATION MATRICES 3 ROTATION MATRICES THAT SATISFY THREE L

THAT SATISFY THREE LINEAR EQUATIONS IN INEAR EQUATIONS IN THE DI

THE DIRECTION COSINES RECTION COSINES RECTION COSINES

-to solve three quadratic equations in three unknowns is here extended -to

1 IntroductionIntroductionIntroduction

A whole class of problems of spatial kinematics can be solved by three given linear equations Owing to the orthogonality constraints among the direction cosines, these problems are equivalent to solving a set of nine equations: three linear and six quadratic

de-Rather than tackling right away the solution of such an equation set,

it is computationally more efficient to replace, in each equation, all known direction cosines by their expressions in terms of the Rodrigues parameters In doing so, all orthogonality constraints are implicitly ful- filled, whereas the former linear equations in the direction cosines turn into second-order equations in the Rodrigues parameters

un-Unfortunately, the known algebraic elimination algorithms that solve

a set of three quadratic equations – such as the Sylvester method – are

23

© 2006 Springer Printed in the Netherlands

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 23 32 –

require determination of the 3 × 3 rotation matrices whose nine direction co

clude all solutions at infinity Therefore no admissible 3 × 3 rotation matrix is rametrization of orientation A case study exemplifies the new method neglected even though it corresponds to a singularity of the Rodrigues

termining all 3 × 3 rotation matrices whose nine direction cosines obey

::

unable to find real solutions at infinity, which are here of interest too because infinite real Rodrigues parameters are associated to finite real exist, these algorithms might fail to determine even the finite solutions After exemplifying the recurrence in kinematics of the addressed three-equation set in the direction cosines, this paper presents an origi- nal procedure to find all real solutions of the equation set The proposed procedure – based on the Rodrigues parametrization of orientation and presented with reference to the Sylvester algebraic elimination algorithm –

is able to identify all real solutions in terms of Rodrigues parameters, both finite and at infinity Therefore its adoption guarantees that no real neglected

A numerical example shows application of the proposed computational

Figure 1 a) Fully-parallel spherical wrist;

b) rigid body supported at six points by six planes.

not always suitable to the case at hand The reason is twofold: i) they are

24

3 × 3 rotation matrices, and ii) in case one or more solutions at infinity

3 × 3 rotation matrix compatible with the original three linear equations is

procedure to a case study

T

Thhe e Relelevevaanncce e tto o Kiinnememaattiiccss

out by determining all 3 × 3 rotation matrices satisfying three linear con

tics aims at determining all possible orientations of the moving platform Figure 1a shows a fully parallel spherical wrist, whose direct kinema

C Innocenti and D Paganelli

for a given set of actuator lengths (Innocenti and Parenti-Castelli, 1993)

If v v i and w w i are the coordinate vectors of points Q i and P i relative to the

fixed (S ) and movable (S’ ) reference frames respectively, and R R R is the

rotation matrix for transformation of coordinates from S’ to S, then – by

applying Carnot’s theorem to triangle OQ i P i – the compatibility

equa-tions can be written as

These equations are linear in the (unknown) elements of matrix R R R

Figure 1b refers to another kinematics problem, which consists in

find-ing any possible positions of a rigid body C supported at six given points

P i ( i =1, ,6) by six fixed planes (Innocenti, 1994; Wampler, 2006) The

co-ordinate vector w w i of each point P i is known with respect to a reference

frame S’ attached to C Each supporting plane is defined with respect to

the fixed frame S by the coordinate vector v v i of a point Q i lying on the

plane, together with the components in S of a unit vector n n i orthogonal to

the plane The unknown position of C with respect to S is parametrized

through the coordinate vector s s s of the origin of S’ with respect to S,

to-gether with the rotation matrix R R R for transformation of coordinates from

S’ to S The compatibility equations can be written as:

They are linear in both the elements of R R R and the components of s ss s If

there exist three supporting planes not parallel to the same line, three of

these equations can be linearly solved for the components of vector s ss s, and

their expressions inserted into the remaining three equations Therefore

a linear three-equation set that has the nine direction cosines of matrix R R R

as only unknowns is obtained once more

Other kinematics problems susceptible of being reduced to the same

linear formulation as the one just exemplified are traceable in Gosselin

et al., 1994, Husain and Waldron, 1994, Wohlhart, 1994, Callegari et al

equations that has to be solved for r ij (i, j = 1,2,3) is

The Equations to be Solved

,

R R and a , , b (i, j, k =1, ,3) are known quantities, the set of three linear

where p
is the skew-symmetric matrix associated with vector pppp,

i.e., p e
= ×p e for any three-component vector eeee As is known, the vector p p p

of Rodrigues parameters corresponds to a finite rotation of amplitude

1

2 t a n

θ= − p about the axis defined by unit vector u =p p/

Unfortunately, the Rodrigues parametrization of orientation is

singu-lar for any half-a-turn rotation (θ = π rad) about any line because, in this

instance, at least one of the components of p p p approaches infinity

By considering Eq (4), Eq (3) can be re-written as:

where quantities Aij,k , Bi,k , and Ck (i,j,k = 1, ,3; i≤j) are known because

dependent on the given quantities aij,k and bk only

Because the denominator of Eq (5) does not vanish for any real vector

Conversely, in case the denominator of Eq (5) approaches infinity, so

does at least one of the components of p p p If both the numerator and the

denominator of Eq (5) are homogenized by replacing pi with expression

xi/x0 (i = 1, ,3), and subsequently multiplied by x02 , the resulting

denomi-nator is definitely different from zero (the real quantities x0 , x1 , x2 , and x3

cannot vanish simultaneously) Finally, for x0 = 0 (which means that at

least one Rodrigues parameter approaches infinity), Eq (5) becomes

, , 1, ,3;

This is a set of three homogeneous quadratic equations in three

un-knowns, namely, the components of vector x x x = (x1 , x2 , x3 ) T

If the set of the non-vanishing vectors that satisfy Eq (7) is

parti-tioned into equivalence classes so that two solution vectors parallel one

to the other belong to the same class, then each class corresponds to a

vector p p p of Rodrigues parameters which satisfies Eq (5) and has infinite

magnitude

Finding all real solutions of Eq (5) – both finite and at infinity – has

been thus reduced to determining all real finite solutions of Eq (6),

to-gether with all equivalence classes of real solutions of Eq (7) This

im-plies that all real solutions of Eq (6) – including those at infinity – need

to be computed Bezout’s theorem (Semple and Roth, 1949) ensures that

the maximum number of these solutions is eight

4

4

As will be proven further on, the existence of solutions at infinity

might affect the search for the finite solutions It is therefore convenient

to compute the solutions at infinity first

The Appendix at the end of the paper briefly summarizes the

mathe-matical tools that will be taken advantage of in this section

4.1

The solutions at infinity, if existent, can be found by identifying Eq (7)

with Eq (1-A) of the Appendix For the case at hand, Eq (3-A) becomes

where M M M is a 6×6 matrix that depends on coefficients Aij,k of Eq (7) only

In case the determinant of M M M is different from zero, there is only the

trivial solution for Eq (7), and no solution at infinity exists for Eq (6)

Conversely, if the determinant of M M M vanishes, Eq (7) has

non-vanishing solutions The number of equivalence classes of these solutions

matches the number of solutions at infinity for Eq (6) Determination of

all solutions of Eq (7) poses no hurdles and will not be detailed in this

paper Suffices it to say that, in the worst possible scenario, the classes of

equivalence for the solutions of Eq (7) can be found by solving a set of

two quadratic equations in two unknowns

The Solving Procedure

Solutions at Infinity s

Solutions at Infinity

4.2

4.2

In most cases, the finite solutions of Eq (6) can be determined through

the procedure described by Roth, 1993, and here briefly summarized If

(α,β,γ) is a permutation of indices (1,2,3), two of the three unknowns, say

pα and pβ, are first replaced in Eq (6) by quantities yα /y0 and yβ /y0

Fol-lowing multiplication by y02, the ensuing equation set is obtained:

which is homogeneous with respect to unknowns y0 , yα , and yβ

If a triplet of values for pα , pβ , and pγ fulfils Eq (6), Eq (9) must be

satisfied by the same value of pγ together with a non-vanishing triplet of

values for y0 , yα , and yβ By also taking into account the dependence on

pγ of the coefficients of the homogeneous system in Eq (9), the solvability

condition for Eq (9) that corresponds to Eq (3-A) turns into

(pγ) y yα yβ y yα y yβ y yα β T=

The solution of this linear set is meaningful only if the triplet

(y0 , yα , yβ) does not vanish, i.e., if the following condition is satisfied (see

Eq (4-A))

γ =

This univariate polynomial equation in pγ has degree not greater than

eight (Roth, 1993) It is the outcome of elimination of unknowns pα and pβ

from Eq (6) For every root of Eq (11), the corresponding values of pα

and pβ can be easily found by Eq (10) through linear determination of a

non-vanishing triplet (y0 , yα , yβ) Thus far is the outline of the procedure

that has been presented – without investigating its singularities – in

Roth, 1993

It is worth noting that Eq (11) is unable to yield solutions at infinity

Things keep manageable if an infinite pγ satisfies Eq (5) for some values

of pα and pβ, as Eq (11) has a degree lower than eight and its roots

con-vey information on finite solutions only Regrettably, should an infinite

solution to Eq (5) exist for a finite pγ (i.e., only pα or pβ or both approach

infinity) then Eq (11) vanishes and the described elimination method

This latter drawback can be explained by noticing that – for pα or pβ

approaching infinity – Eq (10) should hold for y0 = 0 and for some (not

simultaneously vanishing) values of yα and yβ, irrespective of the value of

pγ (the left-hand side of Eq (9) does not depend on pγ when y0 = 0)

Conse-quently, the determinant of 6×6 matrix N N N(pγ) should vanish for any finite

pγ, which also means that Eq (11) collapses into a useless identity

If it is not possible to choose index γ so as to circumvent the just

mentioned inconvenience, the classical elimination method is definitely

unable to find any finite solution to Eq (6) Even a different set of

Rodri-gues parameters consequent on a randomly-chosen offset rotation does

not guarantee removal of the inconvenience

4.3

4.3 Adding Adding robustnessAdding robustness robustness

To overcome the drawback outlined at the end of the previous

subsec-tion, once the solutions at infinity of Eq (6) have been computed (see

subsection 4.1), and prior of attempting determination of the finite

solu-tions, the vector p p p of Rodrigues parameters is replaced by vector

q

q =(q1 , q2 , q3 ) T , related to the former by the ensuing relation

=

where L L L is a 3×3 non-singular constant matrix whose third row is not

orthogonal to each non-vanishing vector (x1 , x2 , x3 ) T that solves Eq (7)

By selecting γ = 3 and replacing q1 and q2 with quantities z1 /z0 and z2 /z0 ,

Eq (9) turns into

where coefficients Aij,k , Bi,k , and Ck , depend on the coefficients of Eq (6)

and on the chosen matrix L L L By applying the elimination procedure

de-scribed in the previous subsection, the correspondent of Eq (11) is

′ 3 =

Differently from Eq (11), Eq (14) does not lose trace of the finite

solu-tions of Eq (6), because any solution at infinity in terms of p p p involves a

vector q q q whose third component, q 3 , approaches infinity too

which is a univariate polynomial equation in the unknown q 3

equa-A possible expression for L L L is

q = (−1,1,3) T Next, Eq (12) results into p p p = (−1,1, −1) T The rotation ces corresponding to the four real solutions − three at infinity in terms of Rodrigues parameters, and the other finite − are respectively (see Eq 4):

6 ConclusionsConclusionsConclusions

matrices satisfying three linear equations in the direction cosines The proposed procedure is based on the Rodrigues parametrization of orienta- tion and takes advantage of a classical algebraic elimination method in order to solve a set of three quadratic equations in three unknowns

To avoid neglecting any possible 3×3 rotation matrix, the classical

30

)

L

Numerical Example

This paper has presented a new procedure to find all 3×3 real rotation

C Innocenti and D Paganelli

tion method has been extended in the paper so that it keeps effective even in case one or more Rodrigues parameters approach infinity

A numerical example has shown application of the proposed procedure

Mecha ics (J Lenarčič and C Galletti (eds.)), Kluwer Academic Publishers, the Netherlands, pp 449-458

-Gosselin, C.M., Sefrioui J., and Richard, M.J (1994), On the Direct Kinematics of Spherical Three-Degree-of-Freedom Parallel Manipulators of General Archi- tecture, ASME Journal of Mechanical Design, vol 116, no 2, pp 594-598 Husain, M., and Waldron, K.J (1994), Direct Position Kinematics of the 3-1-1-1 Stewart Platforms, ASME Journal of Mech Design, vol 116, no 4, pp 1102-

1107

Innocenti, C (1994), Direct Position Analysis in Analytical Form of the Parallel Manipulator That Features a Planar Platform Supported at Six Points by Six Planes, Proc of the 1994 Engineering Systems Design and Analysis Confer- ence, July 4-7, London, U.K., PD-Vol 64-8.3, ASME, N.Y., pp 803-808

Innocenti, C., and Parenti-Castelli, V (1993), Echelon Form Solution of Direct Kinematics for the General Fully-Parallel Spherical Wrist, Mechanism and Machine Theory vol 28, no 4, pp 553-561

Roth, B (1993), Computations in Kinematics, in Computational Kinematics , Kluwer Academic Publisher, the Netherlands, pp 3-14

Salmon, G (1885), Modern Higher Algebra, Hodges, Figgis, and Co., Dublin Semple, J.G., and Roth, L (1949), Introduction to Algebraic Geometry, Oxford University Press, London, UK

Wampler, C.W (2006), On a Rigid Body Subject to Point-Plane Constraints, ASME Journal of Mechanical Design, vol 128, no 1, pp 151-158

Wohlhart, K (1994), Displacement Analysis of the General Spherical Stewart Platform, Mechanism and Machine Theory, vol 29, no 4, pp 581-589

Appendix

Appendix

Let ffff(g g g) be an n-dimensional vector function that depends on an n-dimensional vector g g g If all components of ffff are homogeneous functions

of the same degree in the components of g g g, for any non-vanishing solution

of the following homogenous system

elimina

,

the ensuing condition holds (Salmon, 1885)

D

∇ = 0 (2-A)

where D is the determinant of the Jacobian matrix of ffff

Sylvester (Salmon, 1885) has suggested the following procedure in

or-der to assess whether a set of three second-oror-der homogeneous equations

in three unknowns has non-vanishing solutions:

i) compute the determinant D (which is a third-order homogeneous

polynomial in the components gi , i = 1, ,3, of vector g g g);

ii) determine the gradient of D (its components are quadratic

homo-geneous polynomials in gi , i = 1, ,3);

iii) consider Eqs (1-A)-(2-A) as a set of six equations that are linear

and homogeneous in the six monomials gigj (i,j = 1, ,3, i≤j)

where H H H is a 6×6 matrix whose elements are functions of the

coef-ficients of Eq (1-A)

The original set of three homogeneous quadratic equations has

non-vanishing solutions if and only if the ensuing condition is satisfied

=

32

= ( )

C Innocenti and D Paganelli