2004b, Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations, European Journal of Mechanics– A/Solids, no.. The presented technique
Trang 1Figure 2 Example of fully-isotropic hexapod: Isoglide6-E1(a) and its associated
Figure 3 Example of fully-isotropic hexapod: Isoglide6-E2 (a) and its
graph (b)
associated graph (b)
Trang 2joints or by introducing some new joints with idle mobilities to obtain non overconstrained (isostatic) solutions Due to space limitations, we have reduced our presentation in this paper to fully-isotropic overconstrained solutions without idle mobilities integrating just revolute and prismatic pairs in the legs A, B and C
B and C of type P ɠ R ɠ R ɠ R (Fig 2) and complex legs A, B and C of type
P ɠ R ɠ Rb1 ɠ R (Fig 3) The workspace of these solutions must be correlated
with the angular capability of the homokinetic joints and translational capability of the telescopic shafts
3 Conclusions
of 2197 fully-isotropic hexapods with six degrees of mobility called Isoglide6-E Special legs were conceived to achieve fully-isotropic conditions The Jacobian matrix mapping the joint and the operational vector spaces of the fully-isotropic hexapods presented in this paper is realize a one-to-one mapping between the actuated joint velocity space and the operational velocity space The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission Moreover, the solutions of fully-isotropic hexapods presented in this paper have all actuators mounted directly on the fixed base As far as we are aware, this paper presents for the first time fully-isotropic parallel manipulators with six degrees of freedom and a method for their structural synthesis
4 Acknowledgement
This work was sustained by CNRS (The French National Council of Scientific Research) in the frame of the projects ROBEA-MAX (2002-2003) and ROBEA-MP2 (2004-2006)
References
Angeles, J (1987), Fundamentals of Robotic Mechanical Systems : Theory,
Methods, and Algorithms, New York, Springer
Angeles, J (2004), The qualitative synthesis of parallel manipulators, Trans
ASME Journal Mech Design, vol 126, pp 617-624
the 6 × 6 identity matrix throughout the entire workspace These solutions
An approach has been proposed for structural synthesis of a family The two examples presented in Figs 2 and 3 have elementary legs A,
Trang 3Carricato, M and Parenti-Castelli, V (2002), Singularity-free fully-isotropic
translational parallel mechanisms, Int Journal of Robotics Research, no 2,
vol 22, pp 161-174
Caricato, M (2005), Fully-isotropic four degrees-of-freedom parallel mechanisms
for Schoenflies motion, Int Journal of Robotics Research, no 5, vol 24, pp
397-414
articulate, Braúov, Ed Orientul Latin
Braúov, Ed Trisedes Press
Fang, Y., and Tsai, L.-W (2002), Structural synthesis of a class of 4-dof and 5-dof
parallel manipulators with identical limb structures Int Journal of Robotics
Research, no 9, vol 21, pp 799-810
Fassi, I., Legnani, G and Tosi, D (2005), Geometrical conditions for the design of
partial or full isotropic hexapods, Journal of Robotic Systems, no 10, vol 22,
pp 505-518
manipulators Int Journal of Robotics Research, no 9, vol 21, pp 811-824
Frisoli, A., Checcacci, D., Salsedo F and Bergamasco, M (2000), Synthesis by screw algebra of translating in-parallel actuated mechanisms, in: Lenar ĀiĀ, J Academic Publishers, pp 433-440
Gogu, G (2004a), Fully-isotropic T3R1-type parallel manipulators, in: LenarĀiĀ,
J., Galletti, C (Eds.), On Advances in Robot Kinematics, Dordrecht, Kluwer
Academic Publishers, pp 265-274
Gogu, G (2004b), Structural synthesis of fully-isotropic translational parallel
robots via theory of linear transformations, European Journal of Mechanics–
A/Solids, no 6, vol 23, pp 1021-1039
Gogu, G (2004c), Fully-isotropic over-constrained planar parallel manipulators,
Proceedings of IEEE/RSJ Int Conf on Intelligent Robots and Systems (IROS 2004), Sendai, pp 3519-3520
Gogu, G (2005a), Singularity-free fully-isotropic parallel manipulators with
Schönflies motions, Proceedings of 12th International Conference on Advanced
Robotics (ICAR 2005), Seattle, pp 194-201
Gogu, G (2005b), Fully-isotropic parallel robots with four degrees of freedom
T2R2-type, Proceedings of IEEE/RSJ International Conference on Intelligent Robots
and Systems (IROS 2005), Edmonton, pp 1190-1195
Gogu, G (2005c), Fully-isotropic T1R2-type parallel manipulators with three
degrees of freedom, Proceedings of International Design Engineering Technical
Conferences & Computers and Information in Engineering Conference (IDETC/CIE 2005), Long Beach, Paper DETC2005-84313
Gogu, G (2005d), Fully-isotropic over-constrained parallel wrists with two degrees
of freedom, Proceedings of IEEE International Conference on Robotics and
Automation (ICRA 2005), Barcelona, pp 4025-4030
Gogu, G (2005e), Mobility and spatiality of parallel robots revisited via theory of
linear transformations, European Journal of Mechanics / A –Solids, vol 24, pp
690-711
Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms, Milan, pp 2079-2082
DudiĦӽ, F., Diaconescu, D., Jaliu, C., Bârsan, A and Neagoe, M (2001a), Cuplaje mobile
DudiĦӽ, F., Diaconescu, D., Lateú, M and Neagoe, M (2001b), Cuplaje mobile podomorfe ,
Fattah, A and Hasan Ghasemi A.M (2002), Isotropic design of spatial parallel
and Stanišiþ, M.M (Eds), Advances in robot kinematics, Dordrecht, Kluwer
Hervé, J.M (1995), Design of parallel manipulators via the displacement group ,
Trang 4Lie-groups: Y-STAR and H-ROBOT, Proceedings of IEEE Intl Workshop on
Advanced Robotics, Tsukuba, pp 75-80
parallelogram, Proceedings of the 11 th World Congress in Mechanism and Machine Science, vol 4, China Machine Press, pp 1599-1603
symmetrical lower-mobility parallel manipulators and several novel
manipulators, Int Journal of Robotics Research, no 2, vol 21, pp 131-145 parallel mechanisms using the constraint-synthesis method, Int Journal of
Robotics Research, no 1, vol 22, pp 59-79
Hunt, K H ( 1973), Constant-velocity shaft couplings: a general the ory, Trans
ASME Journal of Eng Industry, vol 95B, pp 455-464
Trans ASME Journal Mech Design, vol 105, pp 705-712
manipulator, in: J LenarĀiĀ and F Thomas, eds., Advances in Robot
Kinematics, Dordrecht, Kluwer Academic Publishers, pp 21-28
translational degrees of freedom based on screw theory, Proceedings of
CCToMM Symposium on Mechanisms, Machines and Mechatronics, Montreal
parallel manipulators, in: J LenarĀiĀ and F Thomas, eds., Advances in Robot
Kinematics, Dordrecht, Kluwer Academic Publishers, pp 453-462
manipulators based on screw theory, IEEE Trans Robotics and Automation,
no 2, vol 20, pp 181-190
parallel manipulators, Proceedings of the 11 th World Congress in Mechanism and Machine Science, vol 4, China Machine Press, pp 1642-1646
parallel manipulators based on screw theory, Trans ASME Journal of
Mechanical Design, vol 126, pp 83-92
mechanisms using the Lie group of displacements IEEE Trans Robotics and
Automation, no 2, vol 20, pp 173-180
e
Tsai, L.-W (2000), Mechanism Design: Enumeration of kinematic structures
according to function CRC Press
Hervé, J.M and Sparacino, F (1993), Synthesis of parallel manipulators using
Hervé, J.M (2004), New translational parallel manipulators with extensible
Huang, Z and Li, Q.C (2002), General methodology for type synthesis of
Huang, Z and Li, Q.C (2003), Type synthesis of symmetrical lower-mobility
Hunt, K.H (1978), Kinematic Geometry of Mechanisms, Oxford University Press
Hunt, K.H (1983), Structural kinematics of in-parallel-actuated robot arms, Kim, H.S and Tsai, L.-W (2002), Evaluation of a Cartesian parallel
Kong, X., Gosselin, C.M (2001), Generation of parallel manipulators with three
Kong, X and Gosselin, C.M (2002), Type synthesis of linear translational
Kong, X and Gosselin, C.M (2004a) Type sysnthesis of 3T1R parallel
Kong, X and Gosselin, C.M (2004b), Type synthesis of analytic translational
Kong, X and Gosselin, C.M (2004c), Type synthesis of 3-dof translational
Li, Q., Huang, Z and Hervé, J M (2004 ), Type synthesis of 3R2T 5-DOF parallel
, 2 édition, Paris, Hermès
Merlet, J.P (1997), Les robots parallèls
Merlet, J.P (2000), Parallel Robots, Dordrecht, Kluwer Academic Publishers
Trang 5A CLASS OF PARALLEL MECHANISMS
Philipp Last
Institute of Machine Tools and Production Technology
Langer Kamp 19b, 38106 Braunschweig, Germany
p.last@tu-bs.de
J¨urgen Hesselbach
Institute of Machine Tools and Production Technology
Langer Kamp 19b, 38106 Braunschweig, Germany
j.hesselbach@tu-bs.de
Abstract Geometric calibration has been proven to be an efficient way to
en-hance absolute accuracy of robotic systems The idea is to identify the geometric parameters of the kinematic model matching the real robots’ geometry Basically calibration is performed by analyzing the differ- ence between conflicting information gained by the kinematic model and corresponding redundant information In all existing robot calibra- tion approaches required redundancy is achieved either by extra sensors
or by special constraint devices This paper for the first time proposes
a calibration method that does not rely on any extra device, thus being very economical The presented technique which only holds for parallel robots is based on a method that allows passing singularities of type two By means of simulation studies using a FiveBar-structure as an example the approach is verified.
Keywords: Parallel Kinematics, Calibration, Singularities
Although automated robot programming is a well engineered ogy, most robotic systems are still programmed by using the teach-inapproach This is due to an insufficient absolute accuracy offered bymost industrial robots Positioning errors are mainly caused by a devia-tion between the controller model and the real robots’ geometry effected
technol-by thermal influences, manufacturing and assembly tolerances [Mooring
et al., 2005] Geometric calibration has been shown to be a suitablemethod to overcome that drawback It is a process by which the pa-rameters of the kinematic model are estimated in a way that best fitsthe real robot Parametric calibration requires redundant measurementinformation that is usually obtained by additional internal or external
© 2006 Springer Printed in the Netherlands.
331
J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 331–338.
Trang 6measurement systems such as lasertracker-devices, theodolites, camerasystems or passive joint sensors Alternatively the robots’ degree offreedom (dof) may be restrained by passive devices In that case theactuator encoders of the system deliver enough information allowing forparameter identification Various calibration technique of both cate-gories are compared in [Hollerbach and Wampler, 1996].
This contribution for the first time presents a calibration strategywhich does not require any calibration equipment Due to the aban-donment of external measurement systems or constraint devices theproposed calibration approach is very inexpensive compared to othertechniques Furthermore it belongs to the class of self-calibration meth-ods [Maurine et al., 2005] and can thus be completely automated andrepeated whenever necessary Redundancy is achieved by special knowl-edge about singular configurations of type two which need to be passed
in order to identify the kinematic parameters of a parallel kinematic nipulator Our approach is therefore based on a technique that allows
ma-to savely guide a parallel mechanism through singularities of type two,introduced in [Helm, 2003] Without a loss of generality the new cali-bration approach will be explained and validated by means of a simple2-dof planar parallel structure, theFiveBar-robot [Sachau et al., 2002]
2.
As mentioned in the previous section our new calibration approach lies on passing singularities of type two Because these constitute struc-ture configurations where two solutions of the direct kinematic problem(DKP) coincide, they are also called direct kinematic singularities It
re-is well known that a robot-structure re-is uncontrollable in thre-is kind ofconfigurations [Hesselbach et al., 2005] and hence particular strategiesneed to be applied to savely guide a manipulator through singularities
of type two With the intention of workspace enlargement Helm sented a technique to pass direct kinematic singularities, which has been
pre-approach has been extended to spatial parallel structures in
tuate the robot system during passing the singular configuration and touse an additional driving force to guide the structure through the directkinematic singularity By means of the planar FiveBar-structure theconfiguration (a) the structure is underactuated by releasing one actua-tor (b) While the second actuator is kept at a constant motor-positionapproach is exemplarily summarized in Fig 1 In a pose near the singular
experimentally proven at a planar robot-structure [Helm, 2003] The
et al., 2005] Both methods rely on the basic idea to temporarily
underac-[Budde
Idea of the Calibration cheme
the endeffector-point C passes the singularity (c) driven by gravity
S
Trang 7Figure 1 FiveBar -structure: Kinematic design (a) and basic steps while passing
leased actuator can be activated again Instead of exploiting gravity asthe driving force which has been also done in [Budde et al., 2005], struc-ture inertia may be used to pass the singularity as described in [Helm,2003]
Mathematically singular configurations of type two can be detectedby
det(J) = 0 (1)
with the Jacobian matrix J = ∂q
∂X relating endeffector velocities ˙X and
actuator velocities ˙q [Gosseling and Angeles, 1990]
˙q = J ˙ X (2)Following the above mentioned strategy to pass the singularity one actu-ator needs to be released while all others remain at a constant position
In that case the systems’ dof is one and J reduces to a scalar expression
Trang 8re-a direct kinemre-atic singulre-arity is re-according to Eq 1 rere-ached under thecondition
˙
q = 0 ∧ ˙X = 0 (5)Thus, in a singular structure configuration the velocity of the releasedactuator ˙qreleasedis zero while the endeffector is still in motion This cor-responds to the structure in Fig 1 where the released actuator changesits direction of movement exactly in the singular configuration (indicated
by the dashed line in a,b,c) Consequently by observing the movement ofthe released actuator by its own encoder it is possible to identify and savethe actuator coordinate ˆqreleasedsing that corresponds to a singular config-uration Furthermore, since particular geometric conditions need to befulfilled at a singular configuration of type two, it is possible to computethe actuator coordinate qsreleaseding (k) from the kinematic model including the kinematic parameters k Comparing both information leads to a
residual
r(k) = ˆqsreleaseding − qs ing
released(k) (6)Passing the direct kinematic singularity at different locations allows for
a formulation of different residual functions These may be assembled
real robot-structure, then r(k) = 0 Since we assume parameter errors, mathematical optimization methods may be applied to find k such that
r(k) is minimized.
3.
In order to validate the presented approach the FiveBar-structurefrom Fig 1 will be calibrated Its kinematic model is defined by onlyfive parameters These are (Fig 1a):
1 parameter L0 defining the distance A1A2 between the two ator base points
actu-1 parameter Li1 for each kinematic chain i = 1, 2 describing thelength of the crank AiBi
1 parameter Li2 for each kinematic chain i = 1, 2 specifying therod length BiC
While typical kinematic problems are concernd with relating fector and actuator coordinates the calibration approach presented hererequires to determine the actuator coordinate qreleasedsing from arbitrary
endef-given fixed actuator coordinates qf ixed and a vector of kinematic
pa-rameters k
qreleasedsing = fSKP(k, qf ixed) (7)
in a vector r(k) Ideally, if the kinematic model exactly matches the
FiveBar- obotR K inematics
Trang 9We refer to this problem as the Singular Kinematic Problem (SKP).TheFiveBar-structures’ SKP can be solved analytically For brevityindex f is introduced for parameters of the chain with the fixed actuatorand index r for parameters of the chain whose actuator is released We
assume that rAf = [xAf, yAf]T and rAr = [xAr, yAr]T pointing from thebase coordinate system to point Af and Ar respectively, are given withthe restriction that|xAr − xAf| = L0 and consequently yAr = yAf By
known vector rBr = [xBr, yBr]T, qreleased can be solved to
qreleased= qr= atan2(yBr− yAr, xBr − xAr) (8)The structure is in a singular configuration under the geometric con-
dition that the two rods of the robot build a common line Hence rBr
can be computed by intersecting a circle KI with its center in Bf andradius RI = L12+ L22 and a second circle KII around Ar with radius
RII = Lr1 With the substitutes
The technique presented above is a very promising strategy to brate parallel mechanisms and thus to enhance their absolute accuracy.There are however several limitations:
cali-As there is a risk of damaging a robot-structure it is usually avoided
to approach direct kinematic singularities Due to this several allel structures are dimensioned and designed in a way that no sin-gular configuration of type two exist in their workspace Obviouslythese manipulators cannot be calibrated by means of the proposedcalibration scheme
par-L
Trang 10Each time a singularity is passed one redundant information can
be gathered In order to identify n parameters by the calibrationprocess at least n independent informations need to be determined.This requires that different singular configurations exist, which isfor example not the case for theParaplacer-structure presented
in [Helm, 2003]
If only angular measurements are used for parameter calibration
no unique parameter-set can be identified as each scaled version
of the robot defines a possible solution to the calibration problem.Consequently one metric parameter needs to be known in advanceand serves as a reference-dimension during calibration, meaning
problem is the same for all calibration strategies)
Since direct kinematic singularities for the FiveBar-robot occurunder the geometric condition of the two rods building one line, thecalibration process cannot differ between a parameter deviation of
L12and one in L22 This however results from the particular designfeature that both kinematic chains are directly connected to eachother in one joint Most parallel structures, especially those with
a dof > 2 are designed in a way that the chains are not directlyconnected to each other but to an additonal passive platform link
In that case this problem does not occur
The last two items limit parameter identification for the robot under consideration In order to circumvent the scaling problem,
FiveBar-it is assumed that L0 is exactly known and will serve as the referenceparameter during calibration Furthermore, as it cannot be differedbetween parameter-deviations in L12 and L22, L12+ L22 will be handled
as one parameter of the calibration procedure In summary only three
of the five parameters describing the system can be identified
In order to validate the proposed calibration approach various
sim-tor kreal containing the actual robot geometry-parameters is generatedwhich adds random values in the range [±1mm] to the nominal values
of the three kinematic parameters L11, L21, L12+ L22 that are supposed
to be identified by the calibration process Nominal parameters knom as
well as typical real robot parameters kreal
Gathering of redundant information is simulated by application ofˆ
qsing= fSKP(kreal, q1,j), where j indicates a specific configuration This
that this parameter remains constant during calibration (This
S
ulation studies have been performed For simulation purposes a
vec-are given in Table 1
Trang 11result is compared to qs2,jing = fSKP(k, q1,j) in a residual rj(k) according
to Eq 6, with k the current parameter set In order to ensure that
m ≥ 3 different residuals rj exist, q1,j is set to j = 1 m different ues All rj(k) are then assembled in r(k) = [r1, , rj]T Starting with
val-k = val-knom, the idea is to find an optimal parameter set kopt that best
fits the real robot-structure by minimization of r(k) For mathematical convenience a function F is defined as F = rTr Minimization of F
is then performed by the Levenberg Marquard algorithm [Scales, 1985]that has been proven to be successful in various calibration approaches,e.g [Zhuang, 1997]
2 shows a typical result of our calibration studies which
corre-real values within a small number of iteration steps indicating that thealgorithm works successful
L 11
L 22
L +L 12 22
1 2 3 4 5 6
Figure 2.
sponds to the parameters from Table 1 The parameters converge to the
presented calibration approach.
Nominal and exemplary real dimensions for the simulation studies.
Trang 12-6 Conclusion
For the first time a robot calibration approach has been presentedthat does exclusively rely on the information delivered by the robot-system itself Hence, as neither additional sensors nor special constraintdevices are required in order to apply parameter identification methods,the proposed technique is very economical The basic idea of the newcalibration scheme has been explained and validated by means of simpleplanar parallel structure Simulation results emphasize the promisingpotential of the approach Future work will focus on application of themethod to more complex parallel structures In addition the effect ofsensor noise in the actuator encoders will be analyzed
The research work reported here was supported by the German ach Foundation (DFG) within the scope of the Collaborative ResearchCenter SFB 562
Reser-References
Budde, C., Last, P., Hesselbach, J (2005), Workspace Enlargement of a Triglide Robot by Changing Working and Assembly Mode, Proc of IASTED International Gosselin, C., Angeles, J (1990), Singularity Analysis of Closed-Loop Kinematic Chains, Helm, M (2003), Durchschlagende Mechanismen f¨ ur Parallelroboter, Dissertation, TU Hesselbach, J., Bier, C., Campos, A., L¨ owe, H (2005), Direct Kinematic Singularity Detection of a Hexa Parallel Robot, Proc of the IEEE International Conference Hollerbach, J M., Wampler, C W (1999), The Calibration Index and Taxonomy for Robot Kinematic Calibration Methods, International Journal of Robotics Research, Maurine, P., Liu, D M., Uchiyama, M (1998), Self Calibration of a New Hexa Parallel Robot, Proc of the 4th Japan-France Congress and 2nd Asia-Europe Congress on Mooring, B W., Roth, Z S., Driels, M R (1991), Fundamentals of Manipulator Calibration, John Wiley and Sons INC.
Sachau, D., Breitbach, E., Rose, M., Keimer, E (2001), An Adaptronic Solution to Increase Efficiency of High Speed Parallel Robots, Proc of the 12th Int Conference Scales, L E (1985), Introduction to Non-Linear Optimization, Macmillan Publishers Ltd.
Zhuang, H (1997), Self-Calibration of Parallel Mechanisms with a Case study on
Braunschweig.
IEEE Trans on Robotics and Automation, 6(3), pp 281-290.
Conference on Robotics and Applications, Cambridge, USA, pp 244-248.
on Robotics and Automation, ICRA, Barcelona, Spain, pp 3507-3512.
Mechatronics, Kitakyushu, Japan, pp 290-295.
on Adaptive Structures and Technologies, College Park, Maryland, USA.
pp 387-397.
vol 15.
Stewart Platforms, IEEE Transaction on Robotics and Automation, vol 13, no 3,
Trang 13Abstract It is well known that parallel kinematic machines (PKM) have a lot
of significant advantages In recent years several machines have been developed and presented on international fairs But up to now only a handful of machines have been transferred to industrial applications Reasons for this lack of technical transfer are structural drawbacks and
an awful design Therefore, this paper presents a new optimization proach for the optimal design of PKM according to given application requirements The main idea of this concept is the separation of the transmission matrices concerning translational and rotational behavior Hence the resulting criteria allow an exact physical analysis of the kine- matic and dynamic properties To assure the practicability our approach
ap-is demonstrated by means of 6-dof Stewart-Gough platform.
Keywords:
Due to the fact that the tool center point (TCP) is supported by veral guiding chains, the main advantages of PKM are high stiffness,process forces and accuracy In addition the symmetric layout allows
se-a modulse-ar design bse-ased on stse-andse-ard mse-achine elements (Merlet, 2000).Beside the evolution of increasingly powerful machine control and sophi-sticated new components, the structure synthesis and optimization is one
of the most important defiance’s for the developing engineer This is thestarting point for several researchers dealing with the analysis and opti-mization of the kinematic structure The main focus of these approaches
is the optimization of two characteristics: workspace and the Jacobianmatrix In general, these first optimization strategies utilized currentlylook at the kinematics and dynamics of the mechanism in a successivemanner Moreover, the dynamic properties of PKM are optimized after
Parallel kinematics, Jacobian matrices, optimization
© 2006 Springer Printed in the Netherlands
J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 339–348
Institute of Machine Tools and Production Technology (IWF)
Technical University Braunschweig, Langer Kamp 19b, 38106 Braunschweig, Germany
Technical University Braunschweig, Langer Kamp 19b, 38106 Braunschweig, Germany
Trang 14the kinematics is defined This is a typical approach because the tools forkinematic and dynamic analysis are often not the same and the dynamicanalysis is usually very difficult to perform In this paper we will discussnew performance criteria By integrating the kinematics and dynamics
in the design process enhanced PKM can be designed
The kinematic and dynamic analysis of PKM is strongly associated
with the linear transmission of the drive velocities by the Jacobian Ja:
˙
In literature a large number of approaches discussing this transmissionbehavior can be found, e.g Ma and Angeles, 1991 Thus the standardized
drive velocity ˙ q, which describes the surface of a F -dimensional hyper
sphere in the joint space, are transmitted on a hyper ellipsoid in theCartesian space representing the time derivates of the endeffector pose.This transmission is characterized by the extension/amplification anddeformation of the hyper sphere and the phase shifting of the output
vector ˙ X according to the input vector ˙ q The maximum amplification
of an input drive velocity is given by the spectral norm of the Jacobianmatrix:
ξ max(JTJa) is the maximum semi half axis of
the hyper ellipsoid and σ min(Ja) ≡ 0ξ min(JTJa ) = σ −1
max
J−1 a
is theminimum semi half axis Thus the deformation of the hyper ellipsoid can
be described by the condition number κ of the Jacobian matrix:
Using the inverse of the condition number the range of values can be
shifted from [1, ∞] to the interval [0, 1] Salisbury and Craig, 1982 Even
though the inverse condition number is a local value, which depends onthe pose of the endeffector, a lot of authors, e.g Ma and Angeles, 1991,use this value for the optimization of parallel robots The aim of this
Trang 15approach is a uniform velocity transmission in all directions If κ −1= 1,
the structure is calledisotropic, irrespective of other poses Furthermore,
the general opinion that κ −1 = 1 means that the PKM can reach the
same velocities in all directions, is completely wrong from a physicalpoint of view This fact has serious consequences for the use of singularvalues Considering Eq 1, the difference quotient for accuracy analysiscan be derived:
Fig 1 shows the Biglide-structure based on the PRRRP kinematicchain in a kinematic isotropic position (P = prismatic joint, R = rotationaljoint) However, considering the maximum errors of each drive, the maxi-
1 The reason for this lies in the fact that the input vector ∆q has to
be described by a hyper square (index) with the side length 2∆q max
If ∆q 1,max = ∆q 2,max, the ratio of maximum and minimum Cartesian
errors is κ x =∆X max 2∆X min −12 = 1, 41, which is the same as the ratio of the input drive errors κ q =∆q max 2∆q min −12 = 1, 41 = κ x thus κ = κ x /κ q
conditions for the PRRRP-structure as shown in Fig 1 Thus, the
sphere 1.00 1.00 0.70 1.77 1.00 2.55 2.55 square 1.00 1.41 0.92 2.51 1.41 2.74 2.40
mum and minimum endeffector displacements differ as depicted in Table
me Cartesian errors can be calculated by
Figure 1. Exact Modeling of Independent Input Parameters
Table 1. Error Transmission of the PRRRP-Structure
= 1 Table 1 shows the spectral norm of the errors and
.