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Accurate and reliable estimation of the kinematic state of a six degrees-of-freedom Stewart platform is a problem of interest in various engineering disciplines. Particularly so in the area of flight simulation, where the Stewart platform is in widespread use for the generation of motion similar to that experienced in actual flight. Accurate measurements of Stewart platform kinematic states are crucial for the application of advanced motion control algorithms and are highly valued in quantitative assessments of simulator motion fidelity. In the current work, a novel method for the reconstruction of the kinematic state of a Stewart platform is proposed. This method relies on an Unscented Kalman Filter (UKF) for a tight coupling of on-platform inertial sensors with measurements of the six actuator positions. The proposed algorithm is shown to be superior to conventional iterative methods in two main areas. First, more accurate estimates of motion platform velocity are obtained and, second, the algorithm is robust to inherent measurement uncertainties like noise and bias. The results were validated on the SIMONA Research Simulator (SRS) at TU Delft. To this end, an efficient implementation of the algorithm was driven, in real time, by actual sensor measurements from two representative motion profiles.
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Improved Stewart Platform State Estimation using Inertial and Actuator
Position Measurements
I. Miletovi´ca,,D.M.Pool
a,O.Stroosma
a,M.M.vanPaassen
a,Q.P.Chu
a
aDelft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands
Abstract
Accurate and reliable estimation of the kinematic state of a six degrees-of-freedom Stewart platform is
a problem of interest in various engineering disciplines. Particularly so in the area of flight simulation,
where the Stewart platform is in widespread use for the generation of motion similar to that experienced in
actual flight. Accurate measurements of Stewart platform kinematic states are crucial for the application of
advanced motion control algorithms and are highly valued in quantitative assessments of simulator motion
fidelity. In the current work, a novel method for the reconstruction of the kinematic state of a Stewart
platform is proposed. This method relies on an Unscented Kalman Filter (UKF) for a tight coupling of
on-platform inertial sensors with measurements of the six actuator positions. The proposed algorithm is
shown to be superior to conventional iterative methods in twomainareas. First,moreaccurateestimates
of motion platform velocity are obtained and, second, the algorithm is robust to inherent measurement
uncertainties like noise and bias. The results were validated on the SIMONA Research Simulator (SRS)
at TU Delft. To this end, an ecient implementation of the algorithm was driven, in real time, by actual
sensor measurements from two representative motion profiles.
Keywords: Sensor integration, State estimation, Unscented Kalman Filter, Parallel robotics, Stewart
platform
1. Introduction
The Stewart platform [1] (see Figure 1) is a six degrees-of-freedom (DOF) parallel manipulator that is in
widespread use throughout various robotic discliplines. The main reasons for this are its relatively low mass,
high accuracy, rigidity and support for heavier loads as compared to conventional serial manipulators. Some
common applications include manufacturing [2], surgery [3,4]andmedicalrehabilitation[5,6]. Perhapsthe5
most popular application is that of vehicular motion simulation, e.g., aeronautical flight simulation [7].
Corresponding author
Email addresses: I.Miletovic@tudelft.nl (I. Miletovi´c), D.M.Pool@tudelft.nl (D.M. Pool), O.Stroosma@tudelft.nl
(O. Stroosma), M.M.vanPaassen@tudelft.nl (M.M. van Paassen), Q.P.Chu@tudelft.nl (Q.P. Chu)
Preprint submitted to Control Engineering Practice March 3, 2017
Published as:
I. Miletovic, D. M. Pool, O. Stroosma, M. M. van Paassen, Q. P. Chu, “Improved Stewart
Platform State Estimation using Inertial and Actuator Position Measurements”, Control
Engineering Practice, Vol. 62 (May 2017), pp. 102-115.
https://doi.org/10.1016/j.conengprac.2017.03.006
Given this wide range of applications, obtaining an accurateestimateofaStewartplatformskinematic
state is often of interest. A typical example is the application of advanced control techniques to Stewart
platforms, e.g., [8, 9]. An increasingly relevant application is also that of flight simulator motion fidelity
assessment. It is well known that humans perceive inertial self-motion predominantly by means of the10
vestibular system, which is sensitive to both specific force and angular acceleration [10]. Current eorts to
quantify motion fidelity therefore focus on the development and standardization of frequency-domain system
identification methods to capture and specify the motion transfer characteristics of contemporary motion
cueing systems [11, 12]. Estimation of the kinematic state of a Stewart platform, however, is inhibited
by two particular diculties. The first is the inference of a Stewart platform’s position and attitude from15
measurements of its six actuator positions (the so-called forward kinematics problem). The second is the
subsequent inference of platform (angular) velocity and acceleration, as required by many of the typical
applications listed here.
The forward kinematics of the Stewart platform have been a subject of study for many decades [13].
Over the years, several authors have shown, for numerous possible platform geometries, that the forward20
kinematics of a Stewart platform has up to 40 possible solutions [14, 15, 16]. As already acknowledged by
Merlet, [17, 18], however, a closed-form algebraic method toobtainasingleunique solution for the platform
position and attitude based on only six actuator length measurements does not exist. As a result, a variety
of numerical methods to solve the forward kinematics have been developed [19, 18, 20]. A common limitation
of these eorts to solve the forward kinematics of a Stewart platform, however, is that they are limited to25
inference of platform position and attitude only. Neither addresses the second problem, namely that of
inference of platform (angular) velocity and acceleration.Whilenumericaldierentiationofpositionand
attitude is applied to obtain the latter, the presence of measurement noise in physical sensors clearly renders
such an approach suboptimal. Nonlinear state observers have therefore also been relied upon to directly
evaluate the platform pose and velocity [21, 22]. These typically require an explicit model of both platform30
dynamics and, especially for hydraulically driven platforms, actuator dynamics. While these methods oer
some robustness to model uncertainties, they lack an inherent mechanism to account for sensor uncertainties,
e.g., noise and bias. In light of the growing availability of both aordable and accurate inertial sensors [23],
this paper presents a novel approach to estimate the kinematic state of a Stewart platform.
The proposed approach relies on the fusion of on-platform inertial sensors, encapsulated in an Inertial35
Measurement Unit (IMU), with the six available actuator position sensors. This is accomplished by using an
extension of the Kalman Filter (KF) [24] to nonlinear systems. Through this approach, the need for explicit
knowledge of motion platform and actuator dynamics is eliminated, while at the same time incorporating
robustness to measurement inaccuracies. The idea of using a Kalman filter-based method for such an
application is not new and has been widely demonstrated in, e.g., robotics [25, 26], aerospace [27, 28],40
biomedical engineering [29] and power plant control [30]. Applications to Stewart platforms, however,
2
remain limited to only a small number of DOFs [31, 32]. More recently, the Iterated Extended Kalman
Filter (IEKF) [33] was applied to extend the sensor fusion scheme to all six DOFs of the Stewart platform
[34]. The current work applies the more advanced Unscented Kalman Filter (UKF) [35], suitable for more
nonlinear problems, and presents both the simulation and real-time experimental validation of the proposed45
state reconstruction algorithm.
The paper is structured as follows. First, a brief introduction to the Stewart platform and its kinematics
is provided. Then, the proposed state reconstruction algorithm is introduced, followed by a verification of
the algorithm on the basis of computer simulations. Subsequently, the validation of the proposed algorithm
on the SIMONA Research Simulator (SRS), using a real-time implementation driven by actual sensor mea-50
surements, is presented. This validation also includes a comparison to an iterative scheme commonly applied
to estimate the kinematic state of Stewart platforms. Finally, a brief discussion is provided and the paper
is concluded.
Figure 1: The SIMONA Research Simulator, using a Stewart platform as the motion providing mechanism [36].
2. Stewart platform kinematics
The kinematics of a Stewart platform are defined by its geometry, shown in Figure 2. The coordinates of55
the joints on the lower (Figure 2b) and upper (Figure 2c) platform, in turn, determine the geometry of the
platform. The locations of these joints are conveniently specified with respect to the centroids of the joints
on the lower and upper platform. These centroids are typically referred to as the Lower Gimbal Point (LGP)
and Upper Gimbal Point (UGP), respectively. The origins of two right-handed reference frames Eaand Eb
are attached to the UGP and LGP, respectively. All joints on the lower and upper platforms lie equidistantly60
spaced with a certain distance (i.e., daor db) in pairs on a circle with a given radius (i.e., raor rb), such
that the coordinates of each joint (i.e., aiand bi)canbederivedfrombasictrigonometry. Notethateven
though this specific configuration of the Stewart is used in thecurrentwork,theauthorsforeseenoissues
in applying the proposed methodology to any non-singular platform geometry.
3
Ea
Eb
1
2
3
4
5
6a1
b1
l1
(a) Perspective
ξb
θb
b1
b2
b3
b4
b5
b6
rb
xb
yb
db
(b) Lower platform
ξa
θa
a1
a2
a3
a4
a5
a6
ra
xa
ya
da
(c) Upper platform
Figure 2: The geometry of a Stewart platform.
The length of each of the connecting elements between the joints then follows from the geometry as:
li(c,Φ)=c+Tba(Φ)aa
ibb
i∥∀i[1,...,6] (1)
The vector cdefines the position of the origin of reference frame Eawith respect to Eband is expressed in
Cartesian coordinates as:
c=!xyz
"!
(2)
Tba is the transformation matrix that describes the transformation aa
iab
iand therefore Equation (1)
depends on the attitude of frame Eawith respect to Eb. Here, attitude is represented using the Euler-
Rodrigues quaternion formulation [37, 38] for numerical eciency. As such, the attitude vector is:
Φ=!e0exeyez"!
(3)
and the transformation matrices Tab and Tba can subsequently be defined as [37]:
Tab =T!
ba =
e2
x+e2
0
e2
ye2
z2(exey+eze0)2(exezeye0)
2(exeyeze0)e2
y+e2
0
e2
xe2
z2(eyez+exe0)
2(exez+eye0)2(eyezexe0)e2
z+e2
0
e2
xe2
y
(4)
Equation (1) specifies the so-called inverse kinematics of the Stewart platform. Inference of the platform
pose from knowledge of the length of each actuator is known as the forward kinematics of the motion
4
platform. While the inverse kinematics are trivial to compute, the forward kinematics cannot be expressed
in a closed algebraic form [17, 18]. To compute the forward kinematics of the platform iterative schemes
such as the Newton-Raphson (NR) or Gauss-Newton (GN) method [19] are therefore commonly applied.
Such methods iteratively compute the position and attitude of the motion platform, denoted collectively as
the platform pose vector p, from the measured actuator position vector lm:
pj+1 =pj+J1
lp (pj)(lmlj)(5)
until satisfactory convergence is reached (i.e., pj+1 pj∥≤egn) or until a fixed number of iterations (i.e.,65
jNgn)havebeenperformed. Jlp is the Jacobian of the actuator position vector lwith respect to the
platform pose vector p.
These iterative numerical methods are limited in that they lack inherent robustness to measurement
inaccuracies in the actuator position sensors and, moreover, provide information on platform pose only. As
knowledge of velocity and acceleration is also of interest, a novel approach based on a tightly-coupled fusion70
of on-platform inertial sensors with actuator position sensors is discussed and evaluated in the remainder of
this paper.
3. Stewart platform state reconstruction using sensor fusion
In [32], a novel sensor fusion scheme was proposed and evaluated for the reconstruction of the partial
(three DOF) kinematic state of a Stewart platform. This method relies on a tightly-coupled fusion of on-75
platform inertial sensors and actuator position sensors. Extensions of the well-known KF [24] to nonlinear
systems, namely the Extended Kalman Filter (EKF) and IEKF [33], were evaluated. The EKF uses di-
rect linearization of the nonlinear system in order to propagate the stochastic properties of the estimated
system state. The IEKF, in turn, iteratively corrects the estimated system state based on a consecutively
re-linearized observation model until satisfactory convergence of the estimated system state is attained. Sys-80
tems which are characterized by a highly nonlinear observation model typically benefit from the application
of the IEKF. In case of the Stewart platform, it was demonstrated that, even for the reconstruction of the
partial kinematic state, the IEKF was required to attain convergence of the filter to the true system state.
In [34], the IEKF-based sensor fusion scheme from [32] was extended to all six DOF of the Stewart platform.
In the current work, the more advanced UKF [35], suitable for more nonlinear problems, is applied. The85
UKF, in contrast to the IEKF, does not rely on direct linearization and oers enhanced robustness to system
nonlinearities.
This section will first introduce the stochastic system model of the Stewart platform that forms the basis
of the proposed state reconstruction algorithm. The working principle of the KF and, specifically, the UKF90
is also briefly recapitulated.
5
3.1. Stochastic system model of the Stewart platform
The basis of a typical sensor fusion scheme is formed by a stochastic system model that relates the
various availa ble physical sensor measurements to the parameters that determine the state of the dynamic
system under consideration. Such a model has the following general structure in the nonlinear case [33]:
˙
x(t)=f(x(t),u(t),t)+G(x(t),t)w(t)
y(t)=h(x(t),u(t),t)+v(t)
(6)
Here, the notation !is used to signify stochastic varia bles. xis the stochastic system state vector, u
is the input vector and yis the output or observation vector. The quantities wand vare the process
and observation noise vectors, respectively. These are commonly assumed to be white and Gaussian noise
processes:
v(t)N(0,R(t)) and E{v(t)v!(t+τ)}=R(t)δ(t)
w(t)N(0,Q(t)) and E{w(t)w!(t+τ)}=Q(t)δ(t)
(7)
The vector function f(·)andmatrixG(·)describetheevolutionofthestochasticprocessstateintime as
a function of the current value of the state, input and process noise vector, respectively. The equation in
which these terms appear is therefore referred to as the prediction model. Similarly, the vector function95
h(·) relates the value of the stochastic process state to the available observation vector and the equation it
appears in is therefore known as the observation model.
In case of the sensor fusion scheme proposed in [32, 34] for thereconstructionofthekinematicstateofa
Stewart platform, the input vector that drives the prediction of the system state consists of measurements
from an IMU. IMUs typically contain three accelerometers and three rate gyros:
u=!f!ω!"!
(8)
where fand ωare the specific force and angular rate vector, respectively, defined in the right-handed
reference frame Eashown in Figure 2:
f=!fxfyfz"!
and ω=!pqr
"!
(9)
Based on this definition of the input vector, the stochastic system state is defined as:
x=!c!V!q!λ!"!
(10)
where cis defined as in Equation (2). qis the vector part of the quaternion defined in Equation (3), that is:
q=!exeyez"!
(11)
6
The inclusion of the scalar part of the quaternion, e0,aspartofthestatevectorisnotrequiredbecauseof
the imposed unit norm constraint:
e2
0+e2
x+e2
y+e2
z=1 (12)
Vin Equation (10) is the inertial velocity of the reference point attached to the upper platform expressed
in reference frame Ea:
V=!uvw
"!
(13)
Finally, λis a vector that contains the six constant biases typically assumed inherent in measurements of
the inertial sensors (see, e.g., [27]):
λ=!λ!
fλ!
ω"!
=!λxλyλzλpλqλr"!
(14)
Note that the inclusion of the inertial sensor biases as part of the state vector allows for their estimation
in conjunction with the kinematic state of the platform. The observation vector is defined to contain the
measured lengths of each of the six actuators:
y=!l1l2l3l4l5l6"!
(15)
Given these definitions in combination with knowledge on the geometry of the Stewart platform as shown
in Figure 2, it is possible to formulate expressions for the vector functions f(·)andh(·)aswellasthematrix
G(·) appearing in Equation (6). From basic kinematics, it follows:
f(x,u)=
˙
c
˙
V
˙
q
˙
λ
=
Tba(Φ)V
(fλf)+Tab(Φ)g(×Λ)V
q(ωλω)
0
(16)
where:
×=
0rq
r0p
qp 0
,Λ=
0λrλq
λr0λp
λqλp0
(17)
and where Tab and Tba are transformation matrices. Furthermore, gis the gravitational acceleration and
qis defined as [37]:
q=1
2
e0ezey
eze0ex
eyexe0
(18)
7
Similarly, G(·)canbederivedas:
G(x)=
03×6
I3v
04×3q
06×6
,with v=
0wv
w0u
vu0
(19)
where I3is the three-dimensional identity matrix and qis given by Equation (18). Finally, the observation
model contains the inverse kinematics of the motion platform:
h(x)=!l1(x)l2(x)l3(x)l4(x)l5(x)l6(x)"!
(20)
where li(x)followsfromthegeometrydepictedinFigure2asgivenbyEquation (1). Notice that, in
contrast to methods that rely on nonlinear state observers [21, 22], the kinematic system model presented
here requires no knowledge of motion platform or actuator dynamics. Moreover, note that Equation (20)100
contains information on the platform position and attitude for any non-singular geometry (which, for the
SRS, is the case [39]). Consequently, it can be shown that the stochastic system is fully observable, including
rate gyro and accelerometer bias [27].
3.2. The Unscented Kalman Filter
The stochastic system model defined by Equation (6) can be used in combination with a nonlinear
recursive filtering algorithm to reconstruct the stochastic system state in real-time. Extensions of the well-
known Kalman Filter (KF) [24, 33] to nonlinear systems are themostwidelyappliedclassoffiltering
algorithms used for this purpose. The KF is a minimum-variance estimator that minimizes the posterior
variance of the stochastic process state at any discrete timetk:
minˆ
x(+)
k!trace )P(+)
k*" (21)
where the superscript (+) denotes posterior quantities and where Pkis the covariance of the stochastic105
process state at time tk. This optimization constitutes two distinct phases, as illustrated in Figure 3.
In the first phase, the prediction,thestochasticsystemstateanditscovariancearepropagated in time
using the system input ukand the prediction model included within the stochastic system model given by
Equation (6). The second phase, the correction, uses the most recent observation vector ykin combination
with the observation model to correct the predicted quantities. This is done by means of the so-called110
Kalman gain matrix Kk.Forthecomputationofthismatrix,thestatisticalproperties of the process and
observation noises are explicitly taken into account through the noise covariance matrices Qkand Rk.
In implementations of the KF for nonlinear systems, the propagation of these statistical properties
becomes problematic and typically results in a suboptimal filter [40]. The fundamental reason for this is
8
Prediction
Correction
0
12
3
4
Kk
Qk
Rk
ˆ
x(+)
k
ˆ
x()
k
ˆ
x()
k+1
P()
k
P()
k+1
P(+)
k
ˆ
x0
P0
yk
uk
z1
Figure 3: Schematic of the Kalman filter [24], illustrating the distinctive prediction and correction phases.
that a Gaussian process which undergoes a nonlinear transformation is, in general, no longer Gaussian115
[41]. Consequently, knowledge of the mean and covariance as propagated by the KF is insucient to
fully capture the probability distribution of the stochastic process. Nonetheless, the KF can be applied to
nonlinear systems by approximating the posterior mean and covariance of the stochastic system state. Such
approximations are typically still classified as optimal filters from a practical point of view. A common
approach is to use local linearization on the stochastic system model given by Equation (6), which results in120
the well-known Extended Kalman Filter (EKF) and Iterated Extended Kalman Filter (IEKF) [33]. A better
alternative to direct linearization of the stochastic system model is provided by the Unscented Kalman
Filter (UKF) [35]. The UKF approximates the nonlinear transformations of the stochastic system state
and its covariance using a technique known as the Scaled Unscented Transformation (SUT) [42]. Given a
general nonlinear function g(·), this technique entails the transformation of a deterministically calculated125
and symmetric set of samples Xaround the mean of a general random variable x,asshowninFigure4,to
obtain the transformed set of samples Y.Theposteriormeanandcovariance,¯
yand Pyy, can subsequently
be approximated as illustrated in the figure. This method is also referred to as statistical linearization and
its merits for nonlinear state reconstruction problems havebeendemonstratedinvariousapplicationsover
the years, e.g., [30, 29, 28].130
S={w(m),w
(c),X}
g(X)
Pxx
¯
x
S={w(m),w
(c),Y}
Pyy
¯
y
Figure 4: Illustration of the SUT inspired by [35].
As outlined in [35, 42], the accuracy and numerical stabilityoftheUKFaremainlyinuencedbya
9
total of three scalar parameters: κ,αand β. These parameters aectthescalarsetofweightsw(m)and
w(c). Together with these set of weights, the collections Xand Yform the so-called sigma-points Sand
S(see Figure 4), respectively, used to approximate the probability distributions of the state and output.
The parameters κ,αand βdetermine the distribution of sigma-points and can also be used to incorporate135
skewness or kurtosis of the probability distribution. In [43], several guidelines can be found for selecting
appropriate values for κ,αand β.
Using computer simulations, the next section will outline the verification of the proposed UKF-based
sensor fusion scheme to the reconstruction of the kinematic state of a Stewart platform.
4. Verification using computer simulations140
Before applying the proposed sensor fusion scheme to an actual Stewart platform, the method is evaluated
using computer simulations in combination with a representative kinematic model of the Stewart platform
of the SRS at TU Delft (see Figure 1). The parameters of the SRS’s platform geometry are summarized in
Table 1. The gravitatio n a l a c c eleratio n i s a s s u med to b e 9 . 8 0665 m/s2.
Table 1: Constants related to the geometry of the SRS (see Figure 2).
Constant Value Unit
ra1.60 m
rb1.65 m
da0.20 m
db0.60 m
lmin 2.08 m
lmax 3.33 m
These parameters, in combination with artificially generated and realistic motion profiles, are used to145
create simulated IMU and actuator position measurements that will be polluted using artificially generated
measurement noises. The noise-polluted signals are subsequently passed to the UKF for evaluation. All
computer simulations are performed using Python, with the open-source scientific libraries Numpy, Scipy
and Matplotlib [44, 45]. The simulation data is sampled at a rate of 100 Hz and, where necessary, use is
made of the forward Euler method for numerical integration.150
4.1. Motion profiles
The simulated sensor measurements are generated from two distinct motion profiles, both illustrated in
Figures 5 and 6. Tables 2 and 3 show the corresponding positions and attitudes of the motion platform
poses depicted. The first motion profile, denoted as Motion Profile 1 (MP1), is selected because it activates
10
most of the six DOFs of the Stewart platform with relatively large amplitudes and therefore excites the155
highly nonlinear kinematics of the platform. After a lead-inperiodof10s,theUGPoftheplatformtraces
acircularpathwitharadiusof0.5 m in the horizontal plane with a period of five seconds. Superimposed
on this circular motion is a periodic roll and pitch motion with an amplitude of 10 deg/s and a period of
2.5s. Thisresultsinamaximumactuatortravelofapproximately 30 cm with respect to the neutral length
of 2.705 m.160
The second motion profile, denoted as Motion Profile 2 (MP2), ismorerepresentativefortypicalex-
periments performed on the SRS. This motion profile has been recorded as part of a human-in-the-loop
(manual tracking) experiment [46] and features a relativelyhigh-frequencycombinedrollandpitchmotion
of small amplitude (max. of approximately 7 deg/s). In this case, the maximum actuator travel from the
neutral position is in the order of 10 cm. Several representative time traces of specific force, angular rate165
and actuator position corresponding to both motion profiles are also shown in Figure 7.
4.2. Simulated sensor data
The selected motion profiles are superimposed with simulated zero-mean Gaussian noise signals to obtain
the artificial sensor measurements used for verification of the proposed algorithm. These noise signals
are fully defined by the standard deviation of the Gaussian distribution used to sample the simulated170
measurements. The inertial sensors are assumed to be polluted by a constant bias as well. Estimates of
appropriate standard deviations and bias levels for the available inertial and actuator position sensors are
obtained from static measurements of the available measurement equipment for the SRS. The measurement
equipment on the SRS comprises an ISIS R
IMU Rev. C manufactured by the Inertial Science Inc. R
and
six Temposonics R
R-series linear position sensors manufactured by the MTS R
Systems [47].175
Figure 8 shows stationary measurements from a representative selection of sensors. Even though the
actual sensor noise is evidently non-Gaussian, for the simulated sensor measurements Gaussian sequences
are assumed. This allows for an evaluation of the theoretical accuracy attainable by the algorithm. The
static IMU and actuator position measurements are used to obtain estimates of noise standard deviations
and bias values, a summary of which can be found in Tables 4 and 5, respectively. The values shown in180
these tables, together with the available motion profile data, are finally used to generate the simulated IMU
and actuator position measurements. A selection of these simulated signals is shown right in Figure 8.
4.3. Method
The performance of the UKF-based algorithm is assessed on the basis of the accuracy and robustness
of the algorithm. Accuracy reflects the error in the estimated quantities, while robustness pertains to the185
resilience of the algorithm to nonlinearities and model errors. Because the true values of estimated quantities
are often unknown in practice, accuracy and robustness are commonly evaluated on the basis of the so-called
11
(a) t=20.00 s (b) t=21.25 s
(c) t=22.50 s (d) t=23.75 s
Figure 5: Impression of motion platform pose during MP1.
Table 2: Motion platform coordinates during MP1.
t 20.00 21.25 22.50 23.75 s
x-0.07 0.49 0.06 -0.49 m
y-0.49 -0.07 0.49 0.07 m
z-2.39 -2.39 -2.39 -2.39 m
φ-4.76 2.71 -4.76 2.71 deg
θ-4.75 2.70 -4.75 2.70 deg
ψ0.19 0.06 0.20 0.06 deg
(e) t=11.50 s (f) t=16.80 s
(g) t=55.30 s (h) t=69.20 s
Figure 6: Impression of motion platform pose during MP2.
Table 3: Motion platform coordinates during MP2.
t 11.50 16.80 55.30 69.20 s
x0.00 0.00 0.00 0.00 m
y0.01 -0.02 0.00 0.01 m
z-2.38 -2.41 -2.39 -2.38 m
φ-2.59 3.86 -0.38 -3.57 deg
θ1.30 -1.59 1.30 -0.64 deg
ψ-0.14 -0.23 -0.67 -0.82 deg
12
020406080100
t(s)
1.0
0.5
0.0
0.5
1.0
1.5
fx(m/s2)
0102030405060708090
t(s)
5.0
2.5
0.0
2.5
5.0
fx(m/s2)
×101
020406080100
t(s)
1.6
0.8
0.0
0.8
1.6
p(rad/s)
×101
0102030405060708090
t(s)
1.2
0.6
0.0
0.6
1.2
p(rad/s)
×101
020406080100
t(s)
2.40
2.55
2.70
2.85
3.00
l1(m)
0102030405060708090
t(s)
2.64
2.68
2.72
2.76
2.80
l1(m)
020406080100
t(s)
2.40
2.55
2.70
2.85
3.00
l3(m)
0102030405060708090
t(s)
2.64
2.68
2.72
2.76
2.80
l3(m)
020406080100
t(s)
2.55
2.70
2.85
3.00
l5(m)
0102030405060708090
t(s)
2.60
2.65
2.70
2.75
2.80
2.85
l5(m)
Figure 7: Representative time traces corresponding to MP1 (left) and MP2(right).
Table 4: Estimates of standard deviations of measurement noises in IMU and actuator position measurements.
Parameter Value Unit Parameter Value Unit
σx6.680 ·104m/s2σl11.093 ·105m
σy8.578 ·104m/s2σl21.132 ·105m
σz6.915 ·104m/s2σl35.741 ·106m
σp1.545 ·104rad/s σl41.482 ·105m
σp1.770 ·104rad/s σl51.054 ·105m
σq1.791 ·104rad/s σl69.770 ·106m
Table 5: Estimates of IMU measurement biases.
Parameter Value Unit
λx4.898 ·101m/s2
λy-9.023 ·103m/s2
λz-1.894 ·101m/s2
λp-1.822 ·102rad/s
λp-5.067 ·103rad/s
λq-2.159 ·102rad/s
13
020406080100
t(s)
4.86
4.88
4.90
4.92
4.94
fx(m/s2)
×101
Mean Standard deviation
020406080100
t(s)
4.86
4.88
4.90
4.92
4.94
fx(m/s2)
×101
020406080100
t(s)
1.90
1.85
1.80
1.75
p(rad/s)
×102
020406080100
t(s)
1.90
1.85
1.80
1.75
p(rad/s)
×102
020406080
t(s)
4
2
0
2
4
l1(m)
×105
020406080100
t(s)
4
2
0
2
4
l1(m)
×105
020406080
t(s)
4
2
0
2
4
l3(m)
×105
020406080100
t(s)
4
2
0
2
4
l3(m)
×105
Figure 8: Selection of actual (left) and simulated (right) IMU and actuator position measurements for stationary motion base.
innovation sequence.TheinnovationappearsinthecorrectionstepoftheUKFasϵk=)ykˆ
y()
k*.Itcan
therefore be interpreted as the dierence between the measured actuator positions, yk, and the reconstructed
measurements obtained from the estimated kinematic state of the Stewart platform. Similarly, the innovation190
covariance appears in the computation of the Kalman gain as Pϵ,k =Pyy,k , i.e., the estimated covariance of
the reconstructed measurements.
In case the process and observation noises wand vare both zero-mean Gaussian white noise, the
innovation sequences should also be zero-mean, Gaussian andwhiteforoptimalfilterperformance[48]. In
practice, this condition is most commonly verified by the observation that approximately 95 percent of the195
innovation sequence is within ±2σϵ,whereσϵis the standard deviation of the innovation obtained from the
diagonal elements of Pϵ[48]. The rate of convergence and similarity of the innovation to such an “optimal”
sequence is a measure of robustness. Since the current section deals with verification of the algorithm using
simulated sensor measurements, however, additional information can be derived from the estimation errors
in the estimated states.200
4.4. Tuning of the UKF
Before the proposed algorithm can be applied to simulated sensor measurements, several parameters need
to be configured. The UKF needs an initial estimate of the stochastic process state and its corresponding
covariance as well as information about the process and observation noise covariances. In order to evaluate
the algorithm in the most general sense, the initial process state estimate and auto-covariance are chosen
14
arbitrarily far away from the true initial process state as:
ˆ
x0=[0 0 1000000000000]
!
P0=diag+12,12,12,12,12,12,0.12
0.12,0.12,12,12,12,0.12,0.12,0.12,
(22)
The process and observation noise covariances selected are based on the values in Tables 4 and 5 as:
Qk=1.1·diag(σ2
x,σ
2
y,σ
2
z,σ
2
p,σ
2
q,σ
2
r)
Rk=1.1·diag(σ2
l1,σ
2
l2,σ
2
l3,σ
2
l4,σ
2
l5,σ
2
l6)
(23)
The ten percent increase in the magnitude of these covariances ensures some robustness against numerical
round-oerrors. In the current work, the system and observation noise covariance are assumed constant.
Other parameters that influence the UKF are κ,αand β,whichareassignedvaluesof0,0.1and2,
respectively. These settings are in line with recommendations given in [43]. Except for α,whichifchosen205
too small degrades numerical stability, these parameters were not found to critically aect the obtained
results.
4.5. Results
The proposed algorithm is subsequently applied to both MP1 and MP2. A representative selection of
the obtained results is shown in Figures 9, 10 and 11. It is evident that the UKF shows quick convergence210
and optimal filter performance for both motion profiles, as witnessed by the seemingly white and Gaussian
innovation sequences. In addition, both the platform statesaswellastheIMUbiasesareestimatedwitha
high accuracy. For most of the estimated quantities, the obtained accuracy is equivalent regardless of the
motion profile evaluated. However, a periodic trend in the estimation errors of λz,wand zfor the case of
MP1 can observed. Such fluctuations are typically associated with sub-optimal filter performance, although215
in this particular case the fluctuations do not visibly propagate through to the innovation sequences.
The pronounced fluctuations in λz,wand zfor MP1 can be explained by the physical limitations of
the SUT inherent in the UKF. The SUT still uses only the mean and covariance to describe the posterior
probability distribution of the estimated quantities. Higher order moments of the posterior probability
distributions are not taken into account, even though they may be important for fully capturing the eect220
of the highly nonlinear platform kinematics for large motions.
As such, it can be expected that the more violent the platform motion, i.e., the more pronounced the
nonlinear platform kinematics, the more the performance of the UKF will degrade. Nonetheless, even for
the case of MP1, which is considered extreme for all practicalintentsandpurposes,positionisestimatedto
an accuracy of O(105)m,velocitytoO(104) m/s, attitude to O(106) rad and the IMU biases to O(105)225
m/s2and rad/s. In flight simulation, accuracy requirements are closely related to human motion perception
15
thresholds. These are in the order of O(102)m/s
2for translations and O(103)rad/s
2for rotations [49].
Therefore, the accuracy attainable by the proposed algorithm is considered sucient by a large margin for
the purpose of flight simulator motion fidelity assessment. However, requirements for other applications
may be far more stringent. An example is precision surgery, where a positional accuracy of the platform in230
the order of micrometers is required [3].
020406080100
t(s)
5.0
2.5
0.0
2.5
5.0
l1(m)
×105
Innovation (ϵ)Bounds (±2σϵ)
0102030405060708090
t(s)
5.0
2.5
0.0
2.5
5.0
l1(m)
×105
020406080100
t(s)
4
2
0
2
4
l3(m)
×105
0102030405060708090
t(s)
2
0
2
4
l3(m)
×105
020406080100
t(s)
6
3
0
3
6
l5(m)
×105
0102030405060708090
t(s)
5.0
2.5
0.0
2.5
5.0
l5(m)
×105
Figure 9: Selection of innovation sequences for MP1 (left) and MP2 (right).
5. Validation on the SRS
The proposed state reconstruction algorithm is subsequently implemented and validated on the SIMONA
Research Simulator (SRS) at TU Delft [36]. First, a real-time implementation of the algorithm proposed
in Section 3 is discussed, after which the method of validation will be presented. Then, the tuning of the235
algorithm for application on the SRS will be discussed, followed by a presentation of the obtained results.
Finally, the validation is concluded with a comparison to a conventional iterative method (e.g., [19]) used
to reconstruct the kinematic state of a Stewart platform.
5.1. Real-time implementation
In order to be able to execute the proposed state reconstruction algorithm on the SRS in real-time240
and using physical sensor measurements, the algorithm was implemented in the software architecture of
the SRS, namely the Delft University Environment for Communication and Activation (DUECA) [50].
DUECA is a middleware software layer that takes care of many ofthecomplexissuesthatarisefrom
developing software that is suitable for execution on an operational full-flight simulator in real-time. Typical
functionality includes the synchronization of processes with wall clock time, inter-process communication245
within a distributed computing environment and communication with simulator hardware (e.g., sensors,
controls, displays, etc.).
16
020406080100
t(s)
2
1
0
1
2
x(m)
×105
Estimation error (e) Standard deviation (σ)
0102030405060708090
t(s)
2
1
0
1
2
x(m)
×105
020406080100
t(s)
2
1
0
1
2
z(m)
×105
0102030405060708090
t(s)
0.8
0.4
0.0
0.4
0.8
z(m)
×105
020406080100
t(s)
1.2
0.6
0.0
0.6
1.2
u(m/s)
×104
0102030405060708090
t(s)
0.8
0.0
0.8
1.6
u(m/s)
×104
020406080100
t(s)
3.0
1.5
0.0
1.5
3.0
w(m/s)
×104
0102030405060708090
t(s)
1.0
0.5
0.0
0.5
1.0
w(m/s)
×104
020406080100
t(s)
5.0
2.5
0.0
2.5
5.0
ex(-)
×106
0102030405060708090
t(s)
6
3
0
3
6
ex(-)
×106
020406080100
t(s)
5.0
2.5
0.0
2.5
5.0
ey(-)
×106
0102030405060708090
t(s)
5.0
2.5
0.0
2.5
5.0
ey(-)
×106
Figure 10: Selection of errors in estimated platform states for MP1 (left) and the MP2 (right).
020406080100
t(s)
3.2
2.4
1.6
0.8
0.0
λx(m/s2)
×104
Estimation error (e) Standard deviation (σ)
0102030405060708090
t(s)
3.2
2.4
1.6
0.8
0.0
λx(m/s2)
×104
020406080100
t(s)
1.5
1.0
0.5
0.0
λz(m/s2)
×104
0102030405060708090
t(s)
4
2
0
2
4
λz(m/s2)
×105
020406080100
t(s)
1.2
0.6
0.0
0.6
λp(rad/s)
×105
0102030405060708090
t(s)
0.8
0.4
0.0
0.4
0.8
λp(rad/s)
×105
020406080100
t(s)
0.8
0.4
0.0
0.4
0.8
λq(rad/s)
×105
0102030405060708090
t(s)
0.6
0.0
0.6
1.2
1.8
λq(rad/s)
×105
Figure 11: Selection of errors in estimated IMU biases for MP1 (left) and the MP2 (right).
17
The majority of the code base of DUECA is written in the C and C++computerlanguages,toallow
for high performance and favourable real-time characteristics. Software developed for the SRS is typically
encapsulated in so-called DUECA modules, which use the services provided by DUECA to ensure adherence250
to the various computational requirements and constraints.Theproposedalgorithmwasdevelopedasa
separate module in DUECA and thus in the C++ computer language. To support the complex matrix
algebra inherent in the UKF-based algorithm, use was made of the open-source Eigen library [51]. This
implementation was found to require an average computational time in the order of a few tenths of a
millisecond per measurement sample on modern desktop and server hardware. It is therefore considered255
more than adequate for execution on the SRS in real time at a nominal rate of 100 Hz.
5.2. Method
The method employed to validate the proposed algorithm is identical to the one used for the verification
outlined in Section 4. That is, the same motion profiles (MP1 and MP2) will be used and the same criteria
to assess the performance of the algorithm will be applied. There are, however, a number of additional260
limitations and practical issues to consider in a real world application. One important limitation is the
lack of knowledge of the true kinematic state of the platform. In case of simulated sensor measurements,
the true state of the platform is known, such that the estimation error in each of the estimated states can
be evaluated (e.g., see Figures 10 and 11). In practice, the true states are of course not available and the
innovation sequences must be largely relied upon to assess the performance of the algorithm. However, in265
Section 5.5, a comparison to a traditional method which relies only on the actuator positions is presented.
Also, as shown in Figure 8, the actual sensor noise may be non-Gaussian, such that a degraded performance
can be expected.
An important issue to also take into consideration is the mounting location of the IMU. Until this point,
it was assumed that the the IMU was located in the UGP of the platform, i.e., the origin of reference frame
Eain Figure 2. In reality, this may not be the case for various practical reasons. As such, the algorithm
needs to be adapted to account for a generic IMU location. To illustrate the issue, the dierence between
the specific forces in the UGP (or any arbitrary point on the platform) and at the IMU location can be
written as:
fimu fugp =c!
imu )˙
×+
2
×*(24)
where cimu is the location of the IMU with respect to the UGP and where ×is given by Equation (17).
This equation shows that there is a mismatch between the specific forces in any two points on the moving
platform, which arises from the rotation of the platform. An elegant way to correct for this mismatch in the
proposed algorithm is to adapt the definition of the joint coordinates aiappearing in Equation (1) as:
aimu
i=aicimu i[1,...,6] (25)
18
This eectively moves the origin of frame Eain Figure 2 to the IMU location. The consequence of this
approach is that the complete estimate of the stochastic process state inferred by the algorithm corresponds270
to the IMU location. The state vector can, however, be re-calculated with respect to the UGP or any
arbitrary location on the platform using Equation (24) and its equivalent for platform velocity and position.
Other issues that aect the performance of the proposed algorithm include: actuator position bias, time-
varying IMU biases, measurement time synchronization errors and IMU misalignment. Care was taken to
calibrate the actuator position sensors using laser-based distance measuring equipment so as to ensure the275
biases are no more than a few tenths of a millimetre in magnitude. For the brief motion profiles considered in
the current work, time-dependent variations in the IMU biases are negligible. It should be noted, however,
that for more prolonged measurements variations in the IMU biases can be accounted for by introducing
so-called random walks (e.g., [43]). This entails modeling the evolution of the biases in the stochastic
system model as artificial noises. Time delays between the inertial sensors and generic auxilliary position280
measurements are known to degrade accuracy, e.g., [52], and methods to account for such delays have been
proposed in the literature, e.g., [43, 53]. Their application, however, would entail a considerable increase
in the complexity of the proposed algorithm and are thereforebeyondthescopeofthecurrentwork. IMU
misalignment causes couplings between measurements from the rate gyros in the IMU. This eect can be
minimized by careful alignment of the IMU with the orthogonalplatformaxes,butonecouldalsoattempt285
to incorporate rate gyro misalignment in the stochastic system model. This too is beyond the scope of the
current work, however.
5.3. Tuning of the UKF
Before the proposed algorithm was executed on the SRS in real-time, actual IMU and actuator position
measurements corresponding to MP1 and MP2 were collected foro-lineanalysisandtuning. Inthisanalysis,
the sensitivity of the algorithm to changes in the governing parameters of the UKF-based algorithm was
investigated. It was found that all but one of the UKF tuning parameters could remain unchanged with
respect to the settings presented in Section 4. This parameter turned out to be Qk,i.e.,thesystemnoise
covariance matrix. The magnitude of Qkhad to be increased by a factor of a thousand in order to obtain
favourable characteristics of the innovation sequences (see Section 4):
Qk=1000·diag(σ2
x,σ
2
y,σ
2
z,σ
2
p,σ
2
q,σ
2
r)(26)
where all the standard deviations used are as given in Table 4.Changesinthemagnitudeofthismatrixwere
found to have by far the largest impact on the results, whereas the state reconstruction algorithm seemed290
less sensitive to moderate changes in the magnitude of Rk. It is also not desirable to increase the magnitude
of Rkin this case, since the actuator position measurements are the sole basis for estimation of the biases
inherent in the IMU measurements. Moreover, as mentioned in Section 5.2, some deficiencies in the IMU
19
measurements are currently unaccounted for, e.g., IMU misalignment and time delays between IMU and
actuator position measurements. A more detailed analysis into the magnitude and eect of time delays295
between the two sets of measurements was performed. This analysis revealed that the IMU measurements
are delayed by 10 to 50 ms with respect to the actuator position measurements. The delays were furthermore
found to be axis dependent and fluctuating in time. An explanation for the delays on the software level (i.e.,
DUECA) could not be found and the issue is therefore likely caused by the IMU itself. Regardless of the
exact cause, it seems evident that from the perspective of theUKF,theIMUmeasurementsarelessreliable300
than the actuator position measurements. This is reflected in the significantly higher magnitude of Qkwith
respect to Rkrequired to obtain favourable filter performance.
As discussed in Section 5.2, the last parameter required by the algorithm for application on the SRS is
the mounting location of the IMU. In the SRS, the IMU is located0.305behindand0.0105metersabove
the UGP, such that:
cimu =!0.305 0 0.0105"!
(27)
5.4. Results
The obtained results from real-time execution on the SRS are included in Figures 12, 13 and 14, which
show a selection of innovation sequences, state estimates and IMU bias estimates corresponding to both305
MP1 and MP2. Several conclusions can be drawn from these results.
First, from the innovation sequences, it can be seen that rapid convergence of the algorithm to zero-mean
values o ccurs for both mot ion profiles. Also, the i nnovation sequences are largely within the band that is
bounded by 2σϵ.Theamplitudeofthisbandisintherangeofonlyseveraltenths of a millimeter, indicating
high accuracy of the estimated states. However, the innovation sequences are evidently not characterized as310
white and are therefore indicative of sub-optimal filter performance [48]. This could be expected, given the
remaining (unmodelled) measurement uncertainties and the fact that the actual measurement noise is non-
Gaussian. It can be seen that the innovation sequences corresponding to MP1 clearly exhibit a component
that resembles the periodicity of the motion profile. This eect is less pronounced in case of MP2, which is
explained by the smaller amplitude of the motion in this profile relative to the motion in MP1. In addition,315
anumberofshortperiodswithanelevatedvalueoftheinnovation sequences appear during both MP1 and
MP2. The cause of these “periodic beats” was traced to the actuator position measurements, which were
found to exhibit an identical periodic elevation in noise level. This increase in noise level was found to occur
synchronously over all six actuator position measurements and therefore could not be isolated to a single
(faulty) sensor. Consequently, the problem is likely situated in a databus that is aected by some external320
disturbance. As it does not impact the validity of the presented results, however, a further consideration of
the issue is beyond the scope of the current work.
20
Regardless of all practical issues encountered, the state estimates shown in Figure 13 are highly accurate
nonetheless, as also illustrated by the generally small standard deviations calculated from the diagonal
elements of the covariance matrix Pk.Theestimatesoftheinertialsensorbiases,showninFigure14,show325
atrendofsomewhatslowerconvergencesimilarasobservedin the results from the computer simulations
presented in Section 4. A comparison to the static sensor measurements shown in Figure 8, however,
reveals that their values are also estimated accurately after a brief period of convergence. Note that a
minor mismatch may exist between the estimated IMU biases and the static measurements. This is because
the static measurements were recorded at a dierent date and time than the validation experiments were330
performed. As such, temperature variations may slightly influence the true values of the inertial sensor
biases and small variations in their estimated values may therefore be observed when the experiments are
repeated at dierent times or for an extended amount of time.
The promising results obtained from the validation experiments motivate a comparison of the proposed
state reconstruction algorithm to a traditional method to estimate the kinematic state of a Stewart platform.335
This comparison is the subject of the next section.
020406080100
t(s)
1.2
0.8
0.4
0.0
0.4
0.8
l1(m)
×103
Innovation (ϵ)Bounds (±2σϵ)
020406080
t(s)
6
3
0
3
6
l1(m)
×104
020406080100
t(s)
0.5
0.0
0.5
1.0
l3(m)
×103
020406080
t(s)
4
2
0
2
4
l3(m)
×104
020406080100
t(s)
6
3
0
3
6
l5(m)
×104
020406080
t(s)
5.0
2.5
0.0
2.5
5.0
l5(m)
×104
Figure 12: Selection of innovation sequences from real-time sensor data for MP1 (left) and MP2 (right).
5.5. Comparison with an iterative method
In Section 5, it was shown that the proposed UKF-based state reconstruction algorithm seems to con-
verge to accurate estimates of the kinematic state of the SRS and is also able to infer inherent IMU biases.
This brings about two distinct advantages over contemporaryiterativeschemescommonlyappliedtosolve340
the forward kinematics of a Stewart platform (e.g., [19]), which only provide direct estimates of position
and attitude. In such traditional methods, to obtain (angular) velocity and acceleration, numerical dier-
entiation is required. As demonstrated, the proposed UKF-based algorithm does not rely on any numerical
dierentiation to infer velocity and translational acceleration. Translational velocity is an element of the
21
020406080100
t(s)
1.00
0.75
0.50
0.25
0.00
0.25
x(m)
Estimated value Standard deviation (σ)
020406080
t(s)
3.28
3.20
3.12
3.04
2.96
x(m)
×101
020406080100
t(s)
5.0
2.5
0.0
2.5
5.0
y(m)
×101
020406080
t(s)
2
1
0
1
2
y(m)
×102
020406080100
t(s)
9
6
3
0
3
6
9
u(m/s)
×101
020406080
t(s)
4
2
0
2
4
u(m/s)
×102
020406080100
t(s)
1.0
0.5
0.0
0.5
w(m/s)
×101
020406080
t(s)
6
3
0
3
6
w(m/s)
×102
020406080100
t(s)
4
2
0
2
ex(-)
×102
020406080
t(s)
4
2
0
2
4
ex(-)
×102
020406080100
t(s)
4
2
0
2
4
ey(-)
×102
020406080
t(s)
4.5
3.0
1.5
0.0
1.5
ey(-)
×102
Figure 13: Selection of estimated platform states from real-time sensor data for MP1 (left) and MP2 (right).
020406080100
t(s)
4.80
4.83
4.86
4.89
4.92
λx(m/s2)
×101
Estimated value Standard deviation (σ)
020406080
t(s)
4.905
4.920
4.935
4.950
λx(m/s2)
×101
020406080100
t(s)
1.84
1.76
1.68
1.60
1.52
λz(m/s2)
×101
020406080
t(s)
1.59
1.56
1.53
1.50
1.47
λz(m/s2)
×101
020406080100
t(s)
1.88
1.86
1.84
1.82
1.80
λp(rad/s)
×102
020406080
t(s)
1.92
1.88
1.84
1.80
λp(rad/s)
×102
020406080100
t(s)
0.25
0.00
0.25
0.50
0.75
λq(rad/s)
×102
020406080
t(s)
0.3
0.0
0.3
0.6
0.9
λq(rad/s)
×102
Figure 14: Selection of estimated IMU biases from real-time sensor data for MP1 (left) and MP2 (right).
22
stochastic state vector (see Equation (10)) and is therefore inferred directly. Estimates of translational345
acceleration and angular velocity can be obtained from raw IMU measurements, corrected for bias using
real-time estimates from the proposed UKF-based algorithm.
In order to fully validate the proposed UKF-based state reconstruction algorithm, it is compared to a
traditional iterative scheme commonly applied to solve for the forward kinematics of Stewart platforms. The
algorithm selected is one that is currently used within the motion control system of the SRS [54], namely the350
Gauss-Newton (GN) method given by Equation (5). In order to compare the two algorithms, the error with
respect to set-points is computed for both MP1 and MP2. Note that this includes al l errors, i.e., control
system errors, transport delays and estimation errors. Some representative results are included in Figure 15.
This figure shows excerpts of timetraces of the errors in position, attitude and velocity estimated using both
algorithms for both motion profiles.355
The GN-based algorithm does not provide a direct estimate of platform velocity. For a fair comparison,
use is therefore made of a dierentiator combined with a second-order Butterworth filter to compute velocity
from estimated position [55]:
B2(s)=Y(s)
U(s)=s
τ2
ds2+2τds+1 (28)
The cut-ofrequency of the filter is set to 25 Hz (i.e., τdof 0.04 s). Also note that, as the UKF-based state
estimates are inferred with respect to the location of the IMU, they must be translated to the platform’s
UGP for comparison to the GN-based estimates. This is done using the following kinematic entities:
ˆ
cugp =ˆ
cTba(ˆ
Φ)cimu
ˆ
Vugp =ˆ
V(׈
Λ)!cimu
(29)
where ×and Λare as defined in Equation (17) and cimu is given by Equation (27). The notation ˆ
!
signifies estimates that are derived from the UKF-based algorithm. Also note that ×contains raw angular
rate measurements from the IMU.
From insp e c t i on of Figure 15, it s e e m s t h at the UKF-base d a l g orithm generally attains better accuracy
than the GN-based algorithm. The dierence is most apparent for MP1, where the error due to the phase360
lag incurred by the dierentiating filter is clearly visible in the GN-based velocity estimate. This frequency-
dependent phase lag can be reduced by increasing the cut-ofrequency of the filter, but this results in more
noise in the GN-based velocity estimates.
To conclude the comparison between the two algorithms, the corresponding standard deviations of the
errors in the estimates are illustrated in Figures 16 and 17 for MP1 and MP2, respectively. It can be seen365
that the UKF-based algorithm provides slightly improved estimates of position and attitude for both MP1
and MP2. However, by far the largest dierence in favor of the UKF-based algorithm is observed in the
estimates of platform velocity for MP1. Here, the uand vestimates are found to be approximately twice as
accurate for the UKF-based algorithm. For MP2, the dierence between the two methods is less pronounced.
23
This can be explained by the dierent frequency spectrum of MP2 as compared to MP1. In this case, this370
means that the phase lag induced by the dierentiating filter in the GN-based method has a smaller impact
on the observed error for MP2 than for MP1. As a final note, it canbeseenthattheaccuracyofboth
algorithms is equivalent for those kinematic states that are barely excited, i.e., z,ψand wfor MP1 and y,
ψand vfor MP2.
05101520
t(s)
2
1
0
1
2
x(m)
×102
GN UKF
05101520
t(s)
1.6
0.8
0.0
0.8
1.6
x(m)
×103
05101520
t(s)
6
3
0
3
6
φ(rad)
×103
05101520
t(s)
3.0
1.5
0.0
1.5
3.0
φ(rad)
×103
05101520
t(s)
6
3
0
3
6
u(m)
×102
05101520
t(s)
1.2
0.6
0.0
0.6
1.2
u(m)
×102
Figure 15: Selection of estimation errors for UKF- and GN-based method to compute platform states in case of MP1 (left) and
MP2 (right).
0.00.81.6
σUKF (m) ×102
0.0
0.8
1.6
σGN (m)
×102
σx
σy
σz
0.02.55.0
σUKF (rad) ×103
0.0
2.5
5.0
σGN (rad)
×103
σφ
σθ
σψ
024
σUKF (m/s) ×102
0
2
4
σGN (m/s)
×102
σu
σv
σw
Figure 16: Standard deviations of errors in UKF- and GN-based estimates for MP1.
0.00.40.81.2
σUKF (m) ×103
0.0
0.4
0.8
1.2
σGN (m)
×103
σx
σy
σz
0.00.81.62.4
σUKF (rad) ×103
0.0
0.8
1.6
2.4
σGN (rad)
×103
σφ
σθ
σψ
0.00.30.60.91.2
σUKF (m/s) ×102
0.0
0.3
0.6
0.9
1.2
σGN (m/s)
×102
σu
σv
σw
Figure 17: Standard deviations of errors in UKF- and GN-based estimates for MP2.
24
6. Discussion375
In this paper, a novel algorithm to reconstruct the kinematicstateofaStewartplatforminanoperational
environment, namely that of the SIMONA Research Simulator (SRS) at TU Delft, was investigated. The
proposed algorithm relies on the fusion of inertial and actuator position sensors, realised using the Unscented
Kalman Filter (UKF) in conjunction with a purely kinematic model of a Stewart platform. This method
enables direct estimation of platform position, attitude and translational velocity. By means of this so-called380
tightly-coupled sensor fusion, it also becomes possible to account for inherent measurement uncertainties,
such as noise and inertial sensor bias. Furthermore, becauseoftherecursivenatureofthealgorithm,itis
possible to execute the algorithm in real-time.
The main contribution of this work is twofold. First, a rigorous verification of the proposed algorithm,
using o-line simulations with simulated sensor measurements, was presented. This verification has demon-385
strated, using two representative motion profiles, the theoretically attainable performance of the algorithm
in terms of accuracy and robustness. Some minor signs of sub-optimal filter performance could be observed
due to the strongly nonlinear kinematics of the Stewart platform. However, the overall performance of
the algorithm is remarkable, yielding highly accurate estimates and exhibiting a rapid rate of convergence
without a need for careful tuning of configuration parameters and initial conditions. These results served390
as the basis for the second contribution of this work, namely a real-time implementation and validation
of the novel algorithm on the SRS. To this end, an ecient implementation within DUECA, the software
framework of the SRS, was conceived. This implementation was found to easily meet the requirements for
real-time execution on the SRS.
During the tuning of the algorithm, a number of practical issues arising from the application in an395
operational environment were encountered. While some of these issues could be resolved in a relatively
straightforward manner, others were deemed beyond the scope of the current work. These issues relate to
measurement time synchronization, IMU misalignment and non-stationary noise levels in measurements.
Nonetheless, the overall results obtained from the validation were satisfying. That is, a direct comparison
of the results to those obtained from a traditional iterativescheme,commonlyappliedtosolvetheforward400
kinematics of Stewart platforms, were in favor of the proposed UKF-based algorithm.
There are a number of areas in which the proposed approach can be extended and improved. First,
potential issues pertaining to the sensory equipment and data acquisition system should be investigated
and resolved. These include minimizing measurement time synchronization errors, IMU misalignment, and
removing any external disturbances in the data acquisition system that may aect measurement noise405
levels. Therefore, even though the present results already highlight the potential of the proposed algorithm
as an improved estimator for the kinematic state of a Stewart platform, the UKF-based algorithm can be
improved even further by, e.g., the use of a more modern and accurate IMU. Second, the robustness of the
25
state reconstruction algorithm itself could be improved as well. In the current work, the UKF system and
observation noise covariances were held constant, however, it could be investigated if adaptive modification410
of these matrices yields improved results (see, e.g., [56]).Finally,adistinctadvantageoftheproposed
algorithm is its extendability with additional sensor measurements. For example, given angular acceleration
measurements [57], the kinematic model of the Stewart platform at the heart of the state reconstruction
algorithm could easily be modified to enable direct inferenceofangularvelocityandunbiasedangular
acceleration.415
7. Conclusion
In this paper, a novel method to reconstruct the kinematic state of a Stewart platform was presented
and validated using real-time sensor measurements on the SIMONA Research Simulator (SRS) at TU Delft.
The proposed method relies on an implementation of the Unscented Kalman Filter (UKF) to enable a
tighly-coupled fusion of on-platform inertial sensor data with measurements of actuator position. In a direct420
comparison, the approach is shown to have two main advantagesoverconventionaliterativemethodsfor
estimating the state of Stewart platforms in real time, enabling direct and accurate estimation of translational
platform velocity, as well as an inherent robustness to measurement uncertainties like sensor noise and bias.
The current work therefore demonstrates the eectiveness oftheproposedmethodforStewartplatform
state estimation and its potential for supporting improved motion control and simulator motion fidelity425
verification.
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