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Journal ArticleDOI

Elasto-geometrical modeling and calibration of redundantly actuated PKMs

01 May 2010-Mechanism and Machine Theory (Pergamon)-Vol. 45, Iss: 5, pp 795-810

AbstractRedundantly actuated parallel kinematic machines (PKMs) offer a number of advantages compared to classical non-redundant PKMs. Particularly, they show a better stiffness thanks to singularity avoidance and they have an improved repeatability due to a better behavior against backlashes. The main problem with the calibration of these machines is that the redundancy leads to some mechanical strains in their structure. This makes it difficult to identify the geometrical errors of their structure without taking into account the effects of the elastic deformations. The main originality of this work is to propose an efficient elasto-geometrical and calibration method that allows the identification of both the geometrical and stiffness parameters of redundantly actuated parallel mechanisms with slender links. The first part of the paper explains the proposed method through its application on a simple redundant planar mechanism. The second part deals with its experimental application to the redundant Scissors Kinematics machine.

Topics: Kinematics (50%)

Summary (6 min read)

1 Introduction

  • Redundantly actuated Parallel Kinematic Machines (PKMs) have recently attracted interest of researchers because they allow the reduction of some ∗ Corresponding author.
  • Whereas for classical PKMs am insufficient knowledge of the mechanism geometrical properties, such as link length or joint position/orientation, leads exclusively to Cartesian position inaccuracies at the Tool Center Point (TCP) [9], in the case of redundant PKMs, such errors also lead to internal constraints.
  • First, the proposed methodology that is used to derive the elasto-geometrical models of parallel mechanisms with one or more actuated redundant chains is described for planar mechanisms.
  • Then, the calibration strategy that has been used to perform the geometrical and stiffness parameter identification of the obtained models is explained.

2.1 Method description

  • The high dynamics of PKMs suppose low moving masses, i.e., slender elements and light joints [11] [12], which are then subject to elastic deforma- tions.
  • For calibration purposes, these elastic deformations that depend on the PKM configuration [13] have to be calculated in order to be compensated.
  • For this purpose, an analytical finite-element modeling using beam elements is proposed to describe redundant PKMs.
  • The analytical finite-element method allows a more accurate calculation of the platform situation because all possible deformation effects are taken into account [18],[19].
  • Calculation of all stiffness matrices into the global reference frame of the structure.

2.2 Elasto-geometrical modeling of redundant planar mechanisms

  • In this section the description of the method is presented for planar mechanisms.
  • For this purpose, the values of the geometrical parameters are considered as nominal, that is to say without any errors.
  • As a result, the position and orientation of the platform associated frame is identical and can be calculated with any non-redundant subsystems of the structure.

2.2.2 Modeling of the structure joints

  • The solution that is used here to describe a structure with joints is to consider them as beam elements with coincident nodes.
  • The parameters kx and ky stand respectively for the radial stiffness coefficients along x and y-axes.
  • A passive revolute joint of axis z would be described with a beam element with a very small stiffness value for the rotation around the z-axis (krz ≈ 0) and a high stiffness value along the other directions.
  • The small and high stiffness coefficients must be chosen so that they are as far as possible from the other stiffness coefficients and that the numerical accuracy is still valid.
  • Its advantages reside in the ease of implementation and in particular the fact that its associated matrix is positive definite.

2.2.3 Mapping and assembly of all stiffness matrices

  • During this step, the stiffness contributions of all links and joints are integrated into a global stiffness matrix that describes the whole structure stiffness.
  • For this purpose, the local stiffness matrices of all the elements have to be expressed within the global reference frame gRij 03 03 gRij where gRij is the rotation matrix from the local frame.
  • It is to be noted that the elastic deformations that are induced by the structure’s own weight are considered by reporting the weight of each link to its two associated nodes and then by merging the resulting equivalent efforts to the vector of external actions F g.

2.3.5 Calculation of all nodal displacements

  • The TCP displacement due to elastic deformations is ∆Xe = dx1 dy1.
  • The node rotational displacement around the z-axis is not considered.
  • The final TCP position is calculated through the resulting forward elastogeometrical model X = fgm(qnr, ξ) + ∆Xe.

2.4 Generalization of the proposed method to three-dimensional mechanisms

  • The extension of the elasto-geometrical modeling method is presented for three-dimensional mechanisms.
  • For this purpose, the nodal force and displacement vectors are modified to consider the six degree-of-freedom as follows: Fi = (fx,i fy,i fz,i mx,i my,i mz,i) T Ui = (dx,i dy,i dz,i rx,i ry,i rz,i) T (16) where fz,i is the force along z and mx,i and my,i are the moments around x and y, in the local beam axis.
  • The displacements are dx,i, dy,i, dz,i, for the local displacements along x, y and z respectively, and rx,i, ry,i and rz,i, for the local rotations around these same axes.
  • For the modeling of the structure bodies, the 12× 12 stiffness matrix that is used to describe a 2-node link between the nodes i and j is: Kij =.
  • The difference with the modeling of planar mechanisms is that the submatrix Kdkl must be extended as Kdkl = diag(kx, ky, kz, krx, kry, krz).

3 Elasto-geometrical calibration of redundant PKMs

  • In the first part of this section the elasto-geometrical modeling method that has been proposed previously is modified to be used for calibration.
  • This leads to the error model that will be involved for the identification of both geometrical and stiffness parameters of the PKM structure.
  • In the second part of the section, the whole calibration methodology will be illustrated by using again the Redundant Triglide.
  • Qset are the actuator set values, ∆X is the vector of measurement errors obtained by the difference between Xmod and Xmeas, J ∗ is the pseudo-inverse of the Jacobian matrix of the calibration and ξcal is the vector of calibrated parameters.

3.1 Error model for the calibration of redundant PKMs

  • In order to derive the calibration error model, the situation of the PKM platform has to be calculated with both nominal and real parameters, respectively ξnom and ξ = ξnom+∆ξ.
  • The consideration of some geometrical errors ∆ξ leads to the fact that the nodes of some joints involved in the elastic modeling are not coincident anymore and the proposed elasto-geometrical method has to be modified.
  • To illustrate this problem with the Redundant Triglide, the authors first consider that the position of all actuators is calculated for a given TCP position with the nominal geometrical parameters.
  • If some geometrical errors exist, the distance between the two nodes of some joints is non-null before application of the external forces and they will behave as beam elements with an initial length δ0.
  • Their node displacements calculated through the resolution of (7) will not fit with reality.

3.1.1 Method of joint internal forces

  • The proposed solution to solve the problem of non-coincident joints consists in the following steps: (1) Calculating the platform position/orientation with the forward geometrical model Xnr = fgm(qnr, ξ) of any non-redundant substructure of the machine where qnr is the actuator position vector of the non-redundant subsystem.
  • (2) Calculating the position of the nodes of the redundant link(s).
  • The nominal geometrical parameters ξnom have to be used for this calculation and the distance between two nodes of a joint has to be minimal.
  • For step 3, the joint stiffness is given by (5) and, since the distance δ0 between the two nodes 8 and 9 is non null (Fig. 9), an internal force proportional to the distance has to be applied and added to the right-hand side of the equation system as follows: F ′89 = F89 + K89.
  • For step 4, the system (7) is solved to derive all nodal displacements and then the position of the Triglide’s TCP (Fig. 10).

3.2 Global Jacobian matrix for the calibration

  • In order to perform the calibration, the local Jacobian matrix Ji that gives for a configuration i the relationship between the variations of the geometrical/stiffness parameters and the variations of the PKM platform situation is calculated as follows: Ji =.
  • Since the finite-element method that is used to derive this forward elasto-geometrical model (fegm) requires a numerical solution of the equation system (7), those partial derivatives are calculated as a finite-difference of (11).
  • Then, the 6m× p global Jacobian matrix J that will be involved further for the sensitivity and observability analyses is obtained as the concatenation of the local Jacobian matrices Ji calculated for the m configurations of measurement.

3.3 Sensitivity and observability analyses

  • The range in which each of those parameters can vary can be set based on considerations related to the machine part manufacturing and assembly.
  • Js from which only the columns of the selected parameters are kept.
  • In order to calculate the global Jacobian matrix J and to perform the sensitivity and observability studies, 231 measurement points are taken over the entire workspace.
  • The angle γ is the most influent parameter for both redundant and non-redundant mechanisms.
  • The calibration of the non-redundant PKM will then tend to be more stable and accurate.

3.4 Identification

  • Simulations were then carried out for the Redundant Triglide.
  • The simulated measured TCP positions along the x and y-directions were obtained through the forward elasto-geometrical model with the real geometrical/stiffness parameters ξ. Figure 12(a) shows the mean final parameter error for the calibration of both mechanisms with respect to the number of measurement points (a Gaussian noise with a standard deviation of 10µm is added to each measurement point).
  • The mean TCP positioning accuracy was calculated for a set of 10 points taken within the workspace.

4.1 Description and elasto-geometrical modeling of the Scissors-Kinematics machine

  • The method that has been proposed for the elasto-geometrical modeling of redundant PKMs has been tested on the Scissors-Kinematics machine developed at the Fraunhofer Institute for Machine Tools and Forming Technology IWU in Chemnitz for tool and die machining [27].
  • The elasto-geometrical modeling method that has been developed so far involves mechanisms with one redundant branch.
  • Its application is also suitable for mechanisms with a higher order of redundancy and, therefore, it has been applied to the Scissors-Kinematics architecture.
  • The Scissors-Kinematics redundant parallel structure includes a moving plat- form, four linear actuators along the y-axis and five fixed-length rods (Fig. 14(a)).

4.2 Elasto-Geometrical Modeling Validation

  • For several TCP positions along the x-axis a variation was applied on one of the parameters, the new actuator positions were calculated with this modified parameter set and all actuators were driven to these positions.
  • The scanning head is mounted directly in the spindle and the plate is fixed on the machine table.
  • The TCP displacements are too small to be measurable.
  • The average deviation between the measured TCP displacements and those obtained by the elasto-geometrical model for all parameters and over the whole x-movement is 14 %.
  • For the redundant part, it is 100 % because these parameters are not taken into account in the model.

4.3 Elasto-geometrical calibration validation

  • The calibration of the Scissors-Kinematics was carried out using the gridencoder to measure 80 TCP positions over the whole workspace.
  • They do not appear in the list of parameters to be identified because of the column normalization of the Jacobian matrix used during the optimization process.
  • The mean position difference for the redundant actuators between the results of the forward geometrical model with the calibrated parameters and the measurements are ∆q2 = 0.873 mm and ∆q3 = 0.305 mm for actuators 2 and 3, respectively.
  • The measurement data are then excluded as being the cause of the discrepancy between the results of the elasto-geometrical calibration and the real mechanism.

5 Conclusion

  • The external forces and the internal constraints linked to the actuation redundancy.
  • The method uses a partly analytical finiteelement analysis based on beam elements, so that it is quick enough for an on-line implementation.
  • This method can also be applied for the study of the influence of machine parameter errors on the TCP, thus it is adapted for the calibration of such mechanisms.
  • The calibration simulations revealed the complementary facts that redundant PKMs are more robust to parameter errors and that for this reason they are more difficult to calibrate than the classical non-redundant PKMs.
  • A self-calibration strategy could be used where the redundant actuators would act as extra measuring systems or extra sensors could be used on passive joints.

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Elasto-geometrical modeling and calibration of
redundantly actuated PKMs
Gaël Ecorchard, Reimund Neugebauer, Patrick Maurine
To cite this version:
Gaël Ecorchard, Reimund Neugebauer, Patrick Maurine. Elasto-geometrical modeling and calibration
of redundantly actuated PKMs. Mechanism and Machine Theory, Elsevier, 2010, 45 (5), pp.795-810.
�10.1016/j.mechmachtheory.2009.12.008�. �hal-00755943�

Elasto-geometrical modeling and cali bration of
redundantly actuated PKMs
Ga¨el Ecorchard
a,
, Reimund Neugeba uer
b
, Patrick Maurine
c
a
IWP Institute, Chemnitz University of Technology, Reichenhainer Straße 70,
09126, Germany.
b
Fraunhofer Institute f or Machine Tools and Forming Technology IWU,
Reichenhainer Straße 88, 09126 Chemnitz, Germany.
c
Laboratory GCGM, Institut National des Sciences Appliqu´ees, 20 Av. des Buttes
de Co¨esmes 35043 Rennes, France.
Abstract
Redundantly actuated Parallel Kinematic Machines (PKMs) offer a numb er of ad-
vantages compared to classical non-redundant PKMs. Particularly, they show a bet-
ter stiffness thanks to singularity avoidance and they have an improved repeatability
due to a better behavior against backlashes. The main problem with the calibra-
tion of these machines is that the redund an cy leads to some mechanical strains in
their structure. This makes it difficult to identify the geometrical errors of their
structure without taking into account the effects of the elastic deformations. T he
main originality of this work is to propose an efficient elasto-geometrical and cali-
bration method that allows the identifi cation of both the geometrical and stiffness
parameters of redundantly actuated parallel mechanisms with slender links. The
first part of the paper exp lains the proposed method through its application on a
simple redundant planar mechanism. The second part deals with its experimental
application to the redundant Scissors Kinematic Machine.
Key words: Calibration, Parallel Mechanisms, Redundant Actuation
1 Introduction
Redundantly actuated Parallel Kinematic Machines (PKMs) have recently
attracted interest of researchers because they allow the reduction of some
Corresponding author. Tel.: +49 371 53 97 14 10.
Email address: gael.ecorchard@s2001.tu-chemnitz.de (Ga¨el Ecorchard).
Preprint submitted to Elsevier December 10, 2009
*Manuscript
Click here to view linked References

drawbacks of classical, non-redundant PKMs [1]. The presence of one or more
redundant actuated chains in the structure allows the avoidance of mechanism
singularities [2] [3] a nd the reduction of joint backlash effects using a contro l
on the internal force directions [4] [5] [6]. Those redundant chains can also be
used to perfor m the autonomous calibration of these mechanisms [7] [8].
Whereas for classical PKMs am insufficient knowledge of the mechanism ge-
ometrical properties, such as link length or joint position/orientation, leads
exclusively to Cartesian position inaccuracies at the Tool Center Point (TCP)
[9], in the case of redundant PKMs, such errors also lead to internal con-
straints. These mechanical strains in the structure can in turn cause early part
wearing and loss of energy in the actuators [5] [1 0]. These internal constraints
make it difficult, not to say impossible, to identify the errors of the geometri-
cal parameters involved in the control model without taking into account the
resulting structure elastic deformations. The purpose of this paper is to pro-
vide a computation metho d of the platform situation (position/orientation)
of redundantly actuated PKMs and t o show how it can be involved in their
calibration. By using this method, one can improve the calibration quality of
redundant PKMs by carrying out backlash-free measurements while taking
into the effect of the internal constraints used to reduce this backlash.
This paper is organized as follows. F irst, t he proposed methodology tha t is
used to derive the elasto-geometrical models of parallel mechanisms with one
or more actuated redundant chains is described for planar mechanisms. The
development o f the method is then illustrated on the Redundant Triglide, a
simple r edundant planar mechanism. Explanations are then given to show how
this modeling method can be easily extended to three dimensional redundant
parallel mechanisms. Then, the calibration strategy that has been used to
perform the geometrical and stiffness parameter identification of the obtained
models is explained. Sensitivity and observability analyses as well as the re-
sults of calibration simulations show the efficiency of the proposed approach.
The last section deals with the experimental application of the method for the
elasto-geometrical modeling and calibration of the redundant Scissors Kine-
matics Machine developed at the Fraunhofer Institute of Machine Tools and
Forming Technology IWU in Chemnitz, Germany.
2 Elasto-Geometrical Modeling of Redundant PKMs
2.1 Method description
The high dynamics of PKMs suppose low moving masses, i.e., slender ele-
ments and light joints [11] [12], which are then subj ect to elastic deforma-
2

tions. For calibration purposes, these elastic deformations that depend on the
PKM configuration [13] have to be calculated in order to be compensated.
For this purpose, an a na lytical finite-element modeling using beam elements
is proposed to describe redundant PKMs. This approach enables a reduction
in the calculation times as well as the number of parameters in comparison
to a CAD finite-element method with numerous surfacic or volumic elements.
This makes it compatible with calibration issues [14]. However, the calculation
of the platform situation for redundant PKMs cannot be achieved in a similar
manner as for non-redundant PKMs because it involves an over-determined
equation system [15]. The number of loop-closure equations is greater than the
platform’s degree of freedom. Some authors proposed some purely geometrical
methods [16] and some methods based on lumped models [5], [17]. However,
the analytical finite-element method allows a more accurate calculation of the
platform situation because all possible deformation effects are taken into ac-
count [18],[19]. Thus, the final aim of the modeling is to o btain the platform
position/orientation X = (P
T
φ
T
)
T
under actuation redundancy.
The modeling method is based on the following steps:
(1) Calculation of the platform position/orient ation X
nr
= (P
T
nr
φ
T
nr
)
T
thanks
to the forward geometrical model (fgm) of a non-redundant substructure
of the redundant mecha nism: X
nr
= fgm(q
nr
, ξ), where q
nr
is the vector
of actuator positions of the non-redundant subsystem and ξ, the vector
of the geometrical and stiffness parameters.
(2) Calculation of the platform displacements induced by the structure elastic
deformations due to its own weight and the applied external forces. This
is done through a forward elastic model (fem) X
e
= fem(q, ξ, F ), where
q is the vector of all actuator positions and F the wrench including the
external forces acting on the structure and the structure’s own weight.
(3) Calculation of the final platform position/orientation through the re-
sulting forward elasto-geometrical model f egm: X = fegm(q, ξ, F ) =
X
nr
+ X
e
.
In order to derive the forward elastic model (fem) involved in step (2), the
following finite-element approach is used:
Determination of the stiffness matrices of all links and joints of the mecha-
nism within a local frame a t t ached t o each link and joint.
Calculation of all stiffness matrices into t he global reference frame of the
structure.
Mapping and assembly of all the resulting stiffness matrices to derive the
global stiffness matrix of the structure.
Calculation with the global stiffness matrix of the structure of the displace-
ment of all nodes and in particular of the node tha t corresp onds to the
TCP.
In next section, the proposed method is detailed for redundant planar parallel
3

mechanism and then expanded for three dimensional structures.
2.2 Elasto-geome trical modeling of redund ant planar mechanisms
In this section the description of t he method is presented for planar mecha-
nisms. For this purpose, the values of the geometrical parameters a r e consid-
ered as nominal, t ha t is to say without any errors. As a result, the position
and orient ation of the platform associated frame is identical and can be calcu-
lated with any non-redundant subsystems of the structure. A non-redundant
subsystem is defined here as a part of the mechanism that contains as many
actuated chains as the platform’s degree of freedom and that constitutes a
viable mechanism.
2.2.1 Modeling of the structure link s
The slender links of the planar structures are considered as rods that can be
described by planar 2-node beams. For each beam, a local reference frame
ij
is defined in such a way that its origin is at node i, its x-axis g oes through
nodes i and j and its z-axis is the same as t he z-axis o f the global frame that
is denoted
g
(Fig. 1). The stiffness of each two-node beam is first expressed
in the beam local reference frame
ij
as [20]:
K
ij
=
K
a
ij
K
b
ij
T
K
b
ij
K
c
ij
(1)
with:
K
a
ij
=
ES
L
0 0
0
12EI
z
L
3
6EI
z
L
0
6EI
z
L
4EI
z
L
, K
b
ij
=
ES
L
0 0
0
12EI
z
L
3
6EI
z
L
0
6EI
z
L
2EI
z
L
, (2)
K
c
ij
=
ES
L
0 0
0
12EI
z
L
3
6EI
z
L
0
6EI
z
L
4EI
z
L
. (3)
L denotes the element’s length, S its cross-section area, I
z
its quadratic mo-
ment along the z-axis and E the Young’s modulus of its material. For a beam
4

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01 Jun 1974
TL;DR: Since the lm function provides a lot of features it is rather complicated so it is going to instead use the function lsfit as a model, which computes only the coefficient estimates and the residuals.
Abstract: Since the lm function provides a lot of features it is rather complicated. So we are going to instead use the function lsfit as a model. It computes only the coefficient estimates and the residuals. Now would be a good time to read the help file for lsfit. Note that lsfit supports the fitting of multiple least squares models and weighted least squares. Our function will not, hence we can omit the arguments wt, weights and yname. Also, changing tolerances is a little advanced so we will trust the default values and omit the argument tolerance as well.

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  • ...It can be calculated according to the relation U r = Kr−1F r by inverting the matrix Kr but it can be more efficiently obtained by solving the system through its singular value decomposition (SVD) [23]....

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Abstract: It has been shown that redundant actuation provides an effective means for eliminating singularities of a parallel manipulator, thereby improving its performance such as Cartesian stiffness and homogeneous output forces. Based on this concept, several high-performance parallel manipulator prototypes have been designed. A major difficulty that prevents application of the vast control literature developed for the serial counterparts to redundantly actuated parallel manipulators is the lack of an efficient dynamical model for real-time control. In this paper, using the Lagrange-D'Alembert formulation, we propose a simple scheme for computing the inverse dynamics of a redundantly actuated parallel manipulator. Based on this approach, four basic control algorithms, a joint-space proportional derivative (PD) control, a PD control in generalized coordinates, an augmented PD control, and a computed-torque control, are formulated. A two-degrees-of-freedom redundantly acutated parallel manipulator designed for a high-speed assembly task is used to verify the simplicity of the proposed approach and to evaulate the performance of the four control algorithms.

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Journal ArticleDOI
TL;DR: A new stiffness modeling method for overconstrained parallel manipulators with flexible links and compliant actuating joints is presented, based on a multidimensional lumped-parameter model that replaces the link flexibility by localized 6-dof virtual springs that describe both translational/rotational compliance and the coupling between them.
Abstract: The paper presents a new stiffness modeling method for overconstrained parallel manipulators with flexible links and compliant actuating joints. It is based on a multidimensional lumped-parameter model that replaces the link flexibility by localized 6-dof virtual springs that describe both translational/rotational compliance and the coupling between them. In contrast to other works, the method involves a FEA-based link stiffness evaluation and employs a new solution strategy of the kinetostatic equations for the unloaded manipulator configuration, which allows computing the stiffness matrix for the overconstrained architectures, including singular manipulator postures. The advantages of the developed technique are confirmed by application examples, which deal with comparative stiffness analysis of two translational parallel manipulators of 3-PUU and 3-PRPaR architectures. Accuracy of the proposed approach was evaluated for a case study, which focuses on stiffness analysis of Orthoglide parallel manipulator.

238 citations


"Elasto-geometrical modeling and cal..." refers methods in this paper

  • ...Some authors proposed some purely geometrical methods [16] and some methods based on lumped models [5], [17]....

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Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "Elasto-geometrical modeling and calibration of redundantly actuated pkms" ?

The main originality of this work is to propose an efficient elasto-geometrical and calibration method that allows the identification of both the geometrical and stiffness parameters of redundantly actuated parallel mechanisms with slender links. The first part of the paper explains the proposed method through its application on a simple redundant planar mechanism.