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Journal ArticleDOI

A calibration procedure for reconfigurable Gough-Stewart manipulators

01 Oct 2020-Mechanism and Machine Theory (Pergamon)-Vol. 152, pp 103920

TL;DR: A calibration procedure for the identification of the geometrical parameters of a reconfigurable Gough-Stewart parallel manipulator is introduced and a numeric algorithm for an efficient solution to the problem is proposed.

AbstractThis paper introduces a calibration procedure for the identification of the geometrical parameters of a reconfigurable Gough-Stewart parallel manipulator. By using the proposed method, the geometry of a general Gough-Stewart platform can be evaluated through the measurement of the distance between couples of points on the base and mobile platform, repeated for a given set of different poses of the manipulator. The mathematical modelling of the problem is described and a numeric algorithm for an efficient solution to the problem is proposed. Furthermore, an application of the proposed method is discussed with a numerical example, and the behaviour of the calibration procedure is analysed as a function of the number of acquisitions and the number of poses.

Summary (2 min read)

1 Introduction

  • Parallel robots are closed-loop mechanisms that are characterized by high stiffness, payload capability and repeatability [1].
  • Furthermore, the estimation of some parameters might not be available at all.
  • In his book, Merlet [1] identifies three main calibration methods for parallel kinematic machines: external calibrations, which are based on measurements with external devices; constrained calibrations, which analyse the motion of the robot in a constrained configuration; auto-calibrations, that only rely on the internal sensors of the robot.
  • While most of the works of the 1990s are focused on practical calibration methods, in the early 2000s several papers on calibration modelling were published.
  • The new decade was also characterized by the rise of new technologies, such as vision-based metrology.

2 Mechanism description

  • The Gough-Stewart mechanism, often called hexapod, is based on a 6-UPS parallel architecture with six identical limbs of varying length, which are controlled by linear motors.
  • The position of each joint on the base platform is expressed by position vector fi, while the relative position of each joint on the moving platform with respect to centre point H is expressed by position vector mi.
  • With reference to Fig. 1, the following parameters are used to define the geometry of the calibration system: .
  • The location of the jth distance sensor on the moving platform is defined by point Sj.
  • A simple iterative procedure based on the Newton-Raphson method with the steps in Fig. 2 is used to solve forward kinematics.

3 Calibration procedure

  • This section presents the mathematical modelling of a calibration procedure that identifies the geometry of a reconfigurable Gough-Stewart platform, which is characterized by a variable position of the joints of the fixed and mobile platform, defined by vectors fi and mi.
  • By assuming perfect passive joints, a general Gough-Stewart platform is characterized by 42 identifiable parameters, namely the xyz coordinates of the mobile joints (18) and fixed joints (18) and the limb offsets (6).
  • A priori estimates are available for the full set of parameters.
  • To compensate errors due to sensor positioning and assembly, the xyz coordinates of sensors (3nr) and of measurement targets (3nr) can be calibrated, for a total of 6nr additional parameters.

4 Calibration in unknown environments

  • The previous section assumes a known coordinate system for the identification of the position of the joints of the base platform.
  • This reference system can then be used to calibrate and identify the geometry of the fixed base and the position of the measuring targets.

5 Experimental validation

  • The proposed calibration procedure is applied to the Free-Hex robot, a reconfigurable Gough-Stewart machining tool, in order to identify the position of its passive joints.
  • Free-Hex, as explained in [33], is a parallel machine tool that is characterized by a mobile platform with fixed geometry and a reconfigurable base platform, with loose magnetic feet at the end of each limb.
  • A second partial calibration has been performed by including the location of all the passive joints as parameters.
  • When compared to the reference geometry of Table 1, the average correction is equal to 1.94 mm, with an average relative correction of 1.03% and a maximum relative correction of 1.63%.
  • A smaller number of poses does not increase the number of iterations to convergence, with 30 to 90 iterations needed for convergence with different subsets.

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A Calibration Procedure for Reconfigurable Gough-Stewart
Manipulators
Matteo Russo
1
, Xin Dong
1
1
Department of Mechanical, Materials and Manufacturing Engineering
University of Nottingham, Nottingham, UK
matteo.russo@nottingham.ac.uk
Abstract. This paper introduces a calibration procedure for the identification of the geometrical
parameters of a reconfigurable Gough-Stewart parallel manipulator. By using the proposed method,
the geometry of a general Gough-Stewart platform can be evaluated through the measurement of
the distance between couples of points on the base and mobile platform, repeated for a given set of
different poses of the manipulator. The mathematical modelling of the problem is described and a
numeric algorithm for an efficient solution to the problem is proposed. Furthermore, an application
of the proposed method is discussed with a numerical example, and the behaviour of the calibration
procedure is analysed as a function of the number of acquisitions and the number of poses.
Keywords: Calibration; Parallel Manipulators; Gough-Stewart; Hexapod; Robotics; Kinematics.
Nomenclature
Var
Description
Var
Description
B
Jacobian matrix of readings wrt pose
o
i
Offset of the i
th
limb (minimum distance between F
i
and M
i
)
C
Calibration matrix
p
Parameter vector to calibrate
e
j
Reading error evaluated as difference between r
j
and ρ
j
R
Rotation matrix of the mobile platform wrt the base platform
e
max
Maximum admitted calibration error
r
j
Distance between S
j
and T
j
as acquired by the j
th
sensor
F
i
Centre point of the i
th
joint on the base platform
r
j
Position vector associated to r
j
f
i
Absolute position vector of point F
i
S
j
Location of the j
th
distance sensor on the mobile platform
H
Centre point of the mobile platform
s
i
Position vector of point S
i
h
Absolute position vector of point H
S
Jacobian matrix of limb lengths wrt pose
i
Limb index
T
j
Location of the j
th
measurement target on the base platform
j
Sensor index
t
i
Absolute position vector of point T
i
k
Pose index
u
i
Unit vector in the direction of the i
th
limb
l
i
Stroke of the linear motor of the i
th
limb
v
j
Unit vector in the direction of the j
th
distance acquisition
l
i
Limb vector of the i
th
limb, from F
i
to M
i
x
First position coordinate of point H (along X-axis)
M
Jacobian matrix of limb lengths wrt parameters m
j
and f
j
y
Second position coordinate of point H (along Y-axis)
M
i
Centre point of the i
th
joint on the mobile platform
z
Third position coordinate of point H (along Z-axis)
m
i
Position vector of point F
i
in the mobile platform frame
ρ
j
Distance between S
j
and T
j
as estimated from kinematics
N
Jacobian matrix of readings wrt parameters s
j
and t
j
α
First orientation coordinate of point H (around X-axis)
n
p
Number of poses
β
Second orientation coordinate of point H (around Y-axis)
n
r
Number of sensors
γ
Third orientation coordinate of point H (around Z-axis)
1 Introduction
Parallel robots are closed-loop mechanisms that are characterized by high stiffness, payload
capability and repeatability [1]. However, the knowledge of their geometrical parameters is needed to
obtain a good accuracy for precision tasks, such as machining. Position control requires the location of
the centres of the joints and the offsets of the links. Estimates of these parameters are usually available,
but deviations due to manufacturing and assembly tolerances can alter significantly their real values.

2
Furthermore, the estimation of some parameters might not be available at all. Thus, the identification
of the geometry of a parallel robot is essential to its proper functioning.
In his book, Merlet [1] identifies three main calibration methods for parallel kinematic machines:
external calibrations, which are based on measurements with external devices; constrained calibrations,
which analyse the motion of the robot in a constrained configuration; auto-calibrations, that only rely
on the internal sensors of the robot. These methods have been successfully used in the last decades, as
proved by the wide literature available [1]. Historically, interest in parallel robot calibration rose in the
1990s with the increasingly common usage of the Gough-Stewart platform in industry [2] and the
invention of the Delta Robot [3]. Both self-calibration and external calibration methods can be found:
in [4], for example, an implicit-loop method is proposed to calibrate a Gough-Stewart platform with
Inverse Kinematics through internal sensors on the spherical and universal joint of one of the parallel
limbs; in [5], a constrained calibration is described. Another constrained calibration method is
introduced in [6], who proposed a self-calibration of a Gough-Stewart manipulator without external
sensors. The same authors also proposed a calibration procedure with two inclinometers in [7]. In [8],
a calibration with a redundant leg is presented.
While most of the works of the 1990s are focused on practical calibration methods, in the early 2000s
several papers on calibration modelling were published. The research in [9] presents a method to
determine all the identifiable parameters of parallel robots, again with a focus on the Gough-Stewart
platform. A complete description of the Gough-Steward platform is also given in [10]. The new decade
was also characterized by the rise of new technologies, such as vision-based metrology. While most of
the methods of the 1990s focus on reducing the number of sensors or simplifying the data acquisition
phase, most of the calibration techniques in the 2000s are based either on laser trackers [11-12] or
cameras [13-17]. Research on alternative procedures, however, went on, as reported in [18-21]. The
most recent works on parallel robot calibration are very wide in scope, with papers on mechanism
synthesis and design [22-23], calibration methods [24-25], non-geometric calibration [26], application
to innovative designs [27-30] and error models [31-34].
Calibration methods for the Gough-Stewart manipulator usually assume a fixed configuration, where
an estimate is available for the parameters and only small errors due to manufacturing and assembly
tolerances need to be evaluated. Thus, most of the standard calibration methods fail to converge when
some of the parameters are unknown or show a large deviation from the initial estimated value. In [33],
an innovative hexapod design is presented as based on the Gough-Stewart architecture with a
reconfigurable geometry of the base platform. Since the position of the fixed joints of the machine can
change from installation to installation, an onboard calibration procedure with external sensors (three
double ball-bars) is manually performed before each operation in order to identify the robot geometry.
A further evolution of the design in [33] is described in [34], which introduces a camera-based self-
calibration method to identify the position of the fixed joints on the ground. The method is detailed for
a three-camera vision system with a previous calibration of the other geometrical parameters through
cameras, laser trackers and additional sensors. The calibration methods in [33-34] are tailored for their
specific applications, by modelling a Gough-Stewart mechanism with reconfigurable base platform and
the specific distance sensor that are selected for the application. Thus, they cannot be used in a general
configuration that is characterized by a different kind of distance sensors or by a reconfigurable
geometry of the mobile platform (in addition to a reconfigurable fixed platform).
To overcome this limitation, this paper expands the mathematical model introduced in [34] with a
general approach for the identification of the geometry of a reconfigurable Gough-Stewart parallel
manipulator with no a-priori knowledge of the location of any passive joint (including the joints on the
mobile platform). The proposed calibration procedure requires distance sensors to measure the distance
between a point of the moving platform and a target on the base platform. The calibration problem is
defined for a general setup, which does not rely on the kind and number of sensors and can be adapted
to a wide range of applications. First, the geometry of the problem is described, and the kinematics of
the Gough-Stewart platform are detailed. Then, the algorithm for the geometrical identification is
explained. Finally, a numerical example is reported in order to validate the proposed method and to
analyse the influence of the calibration parameters on the results.

3
2 Mechanism description
The Gough-Stewart mechanism, often called hexapod, is based on a 6-UPS parallel architecture with
six identical limbs of varying length, which are controlled by linear motors. The limbs are connected to
the moving platform with universal joints and to the base platform through spherical joints. With
reference to Fig. 1, in this paper the following nomenclature is used to describe the geometry of the
Gough-Stewart manipulator:
The location of the centre of the joints on the base platform is defined by point F
i
, for i = 1…6, while
the corresponding joint on the moving platform is defined by point M
i
.
The position of each joint on the base platform is expressed by position vector f
i
, while the relative
position of each joint on the moving platform with respect to centre point H is expressed by position
vector m
i
.
The location of point H can be expressed by position vector h (x y z) and orientation (α β γ), by
assuming the rotation matrix R of the moving platform being composed by a rotation by γ around
the Z-axis first, then by α around the X-axis and finally by β around the Y-axis.
Each limb is modelled as a rigid link with length equal to the sum of a fixed offset o
i
and a variable
length controlled by the motor, which is measured by the motor encoder as reading l
i
.
Limb vector l
i
, going from F
i
to M
i
, can be written as (l
i
+ o
i
) u
i
, where u
i
is a unit vector in the
direction of the i
th
limb.
Fig. 1. Kinematic scheme of a Gough-Stewart platform.
With reference to Fig. 1, the following parameters are used to define the geometry of the calibration
system:
The location of the j
th
distance sensor on the moving platform is defined by point S
j
. The
corresponding measurement target on the base platform is point T
j
.
The position of each target on the base platform is expressed by position vector t
i
, while the relative
position of each sensor on the moving platform with respect to centre point H is expressed by
position vector s
i
.
Each sensor can acquire the distance between point S
j
and point T
j
, which is equal to sensor reading
r
j
, with an associated reading vector r
j
, equal to r
j
v
j
.
A total of n
r
acquisitions can be obtained for each pose of the Gough-Steward platform. Each
acquisition is defined by index j.
The total number of poses used in a calibration is expressed by n
p
, while each pose is defined by
index k.
In order to define a calibration procedure, the inverse kinematic problem (IKP) of the hexapod is
mathematically defined by writing loop-closure equations for the i
th
limb as:

4

(2.1)
The solution of inverse kinematics requires an expression for the i
th
limb length as a function of the
position of the moving platform, given by h and R. Thus, Eq. 2.1 can be rewritten as
󰇛
󰇜
󰇛

󰇜
󰇛

󰇜
(2.2)
When the pose of the moving platform is known, the inverse kinematic formulation can be used to
evaluate a theoretical reading for the j
th
distance sensor. In particular, Eq. 2.2 can be written to express
a reading of the j
th
distance sensor as a function of the pose, as


(2.3)
Even if the inverse kinematics of the hexapod are easy to express in closed form, the forward
kinematic problem (FKP) leads to multiple solutions and is usually evaluated in a discrete way [1]. In
this paper, a simple iterative procedure based on the Newton-Raphson method with the steps in Fig. 2
is used to solve forward kinematics.
Fig. 2. Algorithm for the solution of forward kinematics.
The inputs for the algorithm in Fig. 2 are the parameters of the manipulator (position of the mobile
and fixed joints, offsets of the limbs) and leg displacements. The algorithm starts by defining a tentative
pose of the moving platform. With this pose, the inverse kinematic problem is used to evaluate a
theoretical limb displacement. The error between the theoretical limb displacement and the input one is
evaluated, and a correction of the pose is estimated by using matrix S, which is the 6x6 matrix of the
partial derivatives of the leg length with respect to the pose, given by













(2.4)
Matrix S can be used to relate a small displacement in limb length to a small displacement of the
pose, as













(2.5)

5
and it is used in the algorithm in Fig. 1 as inverse of S to evaluate the pose correction from the error in
limb displacement. The derivation of Eq. (2.5) can be found in Appendix A. When the maximum error
obtained in the iterative process is lower than the desired accuracy e
max
, the solution is found.
3 Calibration procedure
This section presents the mathematical modelling of a calibration procedure that identifies the
geometry of a reconfigurable Gough-Stewart platform, which is characterized by a variable position of
the joints of the fixed and mobile platform, defined by vectors f
i
and m
i
. The calibration is achieved by
measuring the distance between points of the moving platform and targets on the base platform, which
can be acquired by any kind of distance sensor.
By assuming perfect passive joints, a general Gough-Stewart platform is characterized by 42
identifiable parameters, namely the xyz coordinates of the mobile joints (18) and fixed joints (18) and
the limb offsets (6). However, a priori estimates are available for the full set of parameters. In a
reconfigurable platform, a priori knowledge can be used only for a small subset of 6 parameters,
corresponding to the limb offsets, while the others are unknown. Furthermore, the parameters of the
calibration system require identification too. To compensate errors due to sensor positioning and
assembly, the xyz coordinates of sensors (3n
r
) and of measurement targets (3n
r
) can be calibrated, for a
total of 6n
r
additional parameters. Thus, the number of parameters to be calibrated is equal to 42 + 6n
r
if the offsets are included in the calibration, or to 36 + 6n
r
for a simplified model that does not include
them. For each pose of the moving platform, 6 + n
r
measurements can be obtained, respectively by the
encoders of the linear motors and the distance sensors. By acquiring data in n
p
different poses, it is
possible to increase the number of samples available, thus improving the calibration results. The
constraint functions derive from the kinematic model of the robot in Eqs. 2.1-3, and relate the acquired
measurements to the calibrated parameters. In particular, for a given pose k, a theoretical reading ρ of
the j
th
sensor can be evaluated as a function of the pose by using Eq. 2.3 as

(3.1)
This theoretical value can be compared to the real one, which is acquired through the j
th
sensor in
pose k, to obtain error e, which is defined as



(3.2)
In order to calibrate the robot, the influence of both pose and calibration parameters on the reading
must be studied by differentiating Eq. 3.1. With an approach similar to the FKP solution one in the
previous section, matrix B
j,k
can be defined as the matrix of the partial derivatives of the reading with
respect to the pose, and matrix N
j,k
as the matrix of the partial derivatives of the reading with respect to
parameters s
j
and t
j
. In particular, matrix B
j,k
is a 1x6 matrix that expresses the following relation:









(3.3)
By differentiating Eq. 3.1, B
j,k
is obtained as a 1x6 matrix given by









(3.4)
Matrix N
j,k
expresses the following relation:



󰇧


󰇨
(3.5)
By differentiating Eq. 3.1, N
j,k
is obtained as a 1x6 matrix given by




(3.6)
Equations 3.3-6 established a relation between a small variation of the reading of a sensor and a small
variation in pose and calibration parameters. However, to perform a full calibration, the relation between
the pose and the robot parameters must be defined. By using inverse kinematics, it is possible to evaluate
matrix S of Eq. 2.4 for the current pose and reading, to express

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Cites background from "A calibration procedure for reconfi..."

  • ...Because its design and research involve a series of high and new technology fields such as machinery [1], hydraulic [2–4], control [5], computer, signal [6], and sensor [7], it has integrated the knowledge of multiple disciplines such as electromechanical and hydraulic and has been attached great importance by the academic circles [8]....

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References
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Journal ArticleDOI
01 Jun 1997
TL;DR: It is shown that by installing a number of redundant sensors on the Stewart platform, the system is able to perform self-calibration and the approach provides a tool for rapid and autonomous calibration of the parallel mechanism.
Abstract: Self-calibration has the potential of: 1) removing the dependence on any external pose sensing information; 2) producing high accuracy measurement data over the entire workspace of the system with an extremely fast measurement rate; 3) being automated and completely noninvasive; 4) facilitating on-line accuracy compensation; and 5) being cost effective. A general framework is introduced in this paper for the self-calibration of parallel manipulators. The concept of creating forward and inverse measurement residuals by exploring conflicting information provided with redundant sensing is proposed. Some of these ideas have been widely used for robot calibration when robot end-effector poses are available. By this treatment, many existing kinematic parameter estimation techniques can be applied for the self-calibration of parallel mechanisms. It is illustrated through a case study, i.e. calibration of the Stewart platform, that with this framework the design of a suitable self-calibration system and the formulation of the relevant mathematical model become more systematic. A few principles important to the system self-calibration are also demonstrated through the case study. It is shown that by installing a number of redundant sensors on the Stewart platform, the system is able to perform self-calibration. The approach provides a tool for rapid and autonomous calibration of the parallel mechanism.

221 citations


Journal ArticleDOI
01 Oct 1995
TL;DR: A unified formulation for the calibration of both serial-link robots and robotic mechanisms having kinematic closed-loops is presented and applied experimentally to two 6-degree-of-freedom devices: the RSI 6-DOF hand controller and the MEL "modified Stewart platform".
Abstract: A unified formulation for the calibration of both serial-link robots and robotic mechanisms having kinematic closed-loops is presented and applied experimentally to two 6-degree-of-freedom devices: the RSI 6-DOF hand controller and the MEL "modified Stewart platform". The unification is based on an equivalence between end-effector measurements and constraints imposed by the closure of kinematic loops. Errors are allocated to the joints such that the loop equations are satisfied exactly, which eliminates the issue of equation scaling and simplifies the treatment of multi-loop mechanisms. For the experiments reported here, no external measuring devices are used; instead we rely on measurements of displacements in some of the passive joints of the devices. Using a priori estimates of the statistics of the measurement errors and the parameter errors, the method estimates the parameters and their accuracy, and tests for unmodeled factors. >

171 citations


Journal ArticleDOI
01 Dec 1999
TL;DR: The calibration makes use of the motorized prismatic joint positions corresponding to some sets of configurations where in each set either a passive universal joint or a passive spherical joint is fixed using a lock mechanism.
Abstract: Presents a method for the autonomous calibration of six degrees-of-freedom parallel robots. The calibration makes use of the motorized prismatic joint positions corresponding to some sets of configurations where in each set either a passive universal joint or a passive spherical joint is fixed using a lock mechanism. Simulations give us an idea about the number of sets that must be used, the number of configurations by set and the effect of noise on the calibration accuracy. The main advantage of this method is that it can be executed rapidly without need to external sensors to measure the position or the orientation of the mobile platform.

151 citations


Journal ArticleDOI
01 Mar 1998-Robotica
TL;DR: This article deals with the kinematic calibration of the Delta robot and a measurement set-up is presented which allows to determine the end-effector's position and orientation with respect to the base.
Abstract: This article deals with the kinematic calibration of the Delta robot. Two different calibration models are introduced: The first one takes into account deviations of all mechanical parts except the spherical joints, which are assumed to be perfect (“model 54”), the second model considers only deviations which affect the position of the end-effector, but not its orientation, assuming that the “spatial parallelogram” remains perfect (“model 24”). A measurement set-up is presented which allows to determine the end-effector's position and orientation with respect to the base. The measurement points are later be used to identify the parameters of the two calibration model resulting in an accuracy improvement of a factor of 12.3 for the position and a factor of 3.7 for the prediction of the orientation.

117 citations


Proceedings ArticleDOI
21 May 2001
TL;DR: A numerical method for the determination of the identifiable parameters of parallel robots based on QR decomposition of the observation matrix of the calibration system is presented.
Abstract: Presents a numerical method for the determination of the identifiable parameters of parallel robots. The special case of Stewart-Gough 6 degrees-of-freedom parallel robots is studied for classical and self calibration methods, but this method can be generalized to any kind of parallel robot. The method is based on QR decomposition of the observation matrix of the calibration system. Numerical relations between the parameters which are identified and those which are not identifiable can be obtained for each method.

107 citations


Frequently Asked Questions (2)
Q1. What contributions have the authors mentioned in the paper "A calibration procedure for reconfigurable gough-stewart manipulators" ?

This paper introduces a calibration procedure for the identification of the geometrical parameters of a reconfigurable Gough-Stewart parallel manipulator. Furthermore, an application of the proposed method is discussed with a numerical example, and the behaviour of the calibration procedure is analysed as a function of the number of acquisitions and the number of poses. 

Since the procedure follows a linear approximation with the assumption of small parameter variation, it is possible to study the dependency of limb length on position and geometry independently. A direct derivation of the total differential of Eq. ( 2. 2 ) yields the same result without decoupling the system and can be obtained by expanding Eq. ( A. 5 ) without applying conditions ( A. 6 ) or ( A. 9 ).