# Estimation of the trajectory and attitude of railway vehicles using inertial
sensors with application to track geometry measurement

J. González-Carbajal 11footnotemark: 1 Pedro Urda 22footnotemark: 2 Sergio
Muñoz 33footnotemark: 3 José L. Escalona44footnotemark: 4 Department of
Mechanical and Manufacturing Engineering, University of Seville, Spain
Department of Materials and Transportation Engineering, University of Seville,
Spain

###### Abstract

This paper describes a novel method for the estimation of the trajectory curve
and orientation of a rigid body moving along a railway track. Compared to
other recent developments in the literature, the presented approach has the
significant advantage of tracking the position and orientation of a railway
vehicle using inertial sensors only (a 3D gyroscope and a 3D accelerometer),
excluding global position sensors (GNSS or total station) and also excluding
global orientation sensors (magnetometers or inclinometers). The algorithm is
based on a kinematic model of the relative motion of the body with respect to
the track. This kinematic model is used as the system equations of a Kalman
filter algorithm that includes in the state vector the coordinates used to
define the position and orientation of the body. Two different Kalman filter
approaches are described. In the first one, the position and orientation are
calculated independently. In the second one, both position and orientation are
calculated as a coupled problem. Crucial to the success of the results of the
Kalman filters is the use of the correct covariance matrices associated with
the system process and the measurements. Again, two approaches are used for
the estimation of the covariance matrices. One is based on the use of
experimental results with a known output. The other one is based on
constrained maximum likelihood estimation. The calculated trajectory and
orientation are applied in this research to the problem of track geometry
measurement. This is a very demanding application because the measurement of
track irregularities requires millimetric accuracy. A scale track with known
design geometry and irregularities is used to conduct experiments for tuning
and evaluating the quality of the output of the algorithm. Results show that
the developed algorithm is accurate enough for this application. They also
show that, using either of the two proposed Kalman filtering approaches, the
constrained maximum likelihood method for the estimation of the covariance
matrices performs similarly to the known-output method. This is very
convenient because it allows a straightforward application of the observation
model in different scenarios.

###### keywords:

rail vehicles , track irregularities , multibody dynamics , inertial sensors ,
computer vision.

##  1 Introduction

The problem of finding the trajectory of a body in space using inertial
sensors is a classic problem that has been treated by many scientists in the
fields of inertial navigation of vehicles, robotics, or biomechanics. In fact,
the first application of the Kalman filter in the 60’s of the last century was
the inertial navigation of spacecrafts in the Apolo project. The inherent
difficulty of the estimation of a trajectory is that inertial sensors provide
signals that are functions of the time-derivatives (first and second
derivatives) of the variables needed to calculate it. Due to the existence of
noise in the sensors’ signals, the calculated trajectory necessarily drifts
unbounded with respect to the real trajectory. This problem can be fixed
including new sensors that provide signals proportional of the trajectory
coordinates themselves, not their derivatives. To this end, the most popular
technology is the Global Navigation Satellite System, GNSS.

Nowadays there exist numerous algorithms for inertial navigation that work
reasonably well in many applications. The use of state observers aims to
determine the state of the system, which may include the sensors biases [1, 2,
3] or IMU errors parameters [4]. The fusion algorithms are based on combining
the information of two or more sensors measuring different variables of the
same system, leading to estimated trajectory and orientation without the drift
problem. These fusion algorithms can be based on Linear Kalman Filter [5],
Non-linear Extended Kalman Filter [6, 7], Unscented [8], Cubature Kalman
Filter [9]), Complementary Filter [10] or Unknown Input Filter [11]. The
optimization algorithms are based on minimizing the error function between the
real orientation (unknown) and the estimations based on measured data. This
minimization can be approached by different algorithms such as the Levenberg-
Marquardt Algorithm (LMA) [12], Gauss-Newton Algorithm (GNA), Gradient Descent
Algorithm (GDA) [13, 14] and Control Loop [15, 16]. All these algorithms are
successfully applied to the trajectory estimation of aircrafts, spacecrafts,
cars, robots or humans. The estimation of the trajectory of an aircraft in
space is much more difficult than the estimation of the trajectory of a
vehicle along a rail track. The obvious reason is that the aircraft moves
freely in an unbounded space while the rail vehicle follows more or less
closely the track centerline 3D curve. In other words, the trajectory of the
rail vehicle is approximately known beforehand. Only slight deviations about
the reference trajectory are expected. However, the estimation of the
trajectory of a rail vehicle has other difficulties. In order to get advantage
of the knowledge of the approximate solution (track centerline), the use of
track-relative kinematics is recommended. Track-relative kinematic equations
are much more involved than the kinematics of a free body in space. Besides,
it requires the use of the design geometry of the track and an odometry
algorithm that estimates the position of the vehicle along the track. When the
estimation of trajectory of the rail vehicle is used for the estimation of the
real track geometry (rail geometry measurement), the accuracy that is needed
is millimetric. This is much more than the accuracy needed for the inertial
navigation of other vehicles. This paper treats the problem of the trajectory
estimation of railways with application to track geometry measurement.

In the railway industry, track geometry measurement is a fundamental phase of
track maintenance. Two different types of equipment are used for this task:
man-driven rail track trolleys (RTT) and track recording vehicles (TRV). On
one hand, the technology behind the RTT is simple and accurate. The relative
irregularities (track gauge and cross-level) are directly measured with a
distance sensor, like an LVDT and an inclinometer, respectively. The absolute
irregularities (alignment and vertical profile) require and absolute
positioning system, like a total station or a very accurate GNSS. Despite of
their precision and good performance, in addition to its low cost, the main
handicap of RTT devices is their slowness when measuring the track. On the
other hand, the technology used in TRV is more varied. Essentially, there are
two technologies [17]: versine measurement systems (VMS), and inertial
measurement systems (IMS). The VMS, also called chord method, are based on
simple kinematics that only requires the measurements of distance sensors,
obtaining both horizontal and vertical measurement of rails [18]. The IMS are
based on the use of inertial sensors (accelerometers and gyroscopes), the most
serious problem being the need to integrate the sensor signals in time to get
the irregularities. The main drawback of the TRV methods is that are based on
the use of expensive and dedicated vehicles.

The alternative to both measuring methods (RTT and TRV) is the use of
inexpensive measuring systems, mounted on in-service vehicles for continuous
monitoring of track conditions. That way, the frequency of the track
inspection is increased and the time-evolution of the irregularities can be
followed closely, all this at a very low cost. An extensive review on the use
of in-service vehicles for the monitoring of railway tracks can be found in
[19]. Weston et al. published a set of papers[20, 21], showing the measurement
of vertical and lateral track irregularities using accelerometers and
gyroscopes mounted in the bogie frame and a very simple kinematic model of the
vehicle motion. Lee et al. [22] presented a Kalman filter data fusion approach
based on accelerometer signals mounted on the body frame and the axle-box. In
this work, the Kalman filter is used as a kind of integrator of lateral
acceleration of the wheelset to obtain lateral displacements and,
subsequently, a set of compensation filters are used to predict lateral
irregularities. Wei et al. [23] proposed a method to estimate the alignment of
the track through a double integration of the acceleration measured by several
accelerometers placed on the vehicle. Tsai et al. [24, 25] presented a fast
inspection technique based on the Hilbert–Huang Transform, that is a very
useful time–frequency analysis technique, applied to the signal of an
accelerometer mounted on the axle-box of an in-service car. Both methods that
are based on axle-box-mounted accelerometers are only valid to measure
vertical track irregularities. Tsumashima et al. [26] estimated the vertical
track geometry using the accelerations measured in the car-body of the
Japanese Shinkansen using a Kalman filter based on a very basic car model.
More recently, De Rosa et. al. [27] proposed three different model-based
methods to estimate both lateral track alignment and cross-level
irregularities: 1) pseudo-inversion of the vehicles frequency response
function (FRF) matrix, 2) unknown input estimation using a deterministic
observer and 3) unknown input estimation using a linear Kalman filter as a
stochastic observer. In [28] Muñoz et. al. proposed a Kalman filter model-
based methodology for the estimation of lateral track irregularities from
measurements from different sensors mounted on an in- service vehicle. The
proposed methodology was experimentally validated through an experimental
campaign carried out in a 1:10 scaled vehicle, obtaining promising results. In
the work of Escalona et al. [29], a track measurement system that can be
installed on in-service vehicles and combines a kinematic model, a
computational vision system and inertial measurement is presented.

Finally, it can be concluded that, many of the different methodologies for the
track geometry measurement require the estimation of the trajectory for the
measurement of the absolute irregularities. Consequently, it would be very
convenient to develop an accurate method for the calculation of the trajectory
of the vehicles using only inertial measurements. In the present work, a new
method for the estimation of the trajectory and orientation of a rigid body is
presented, with application to track geometry measurement.

This paper is organized as follows. Section 2 explains the kinematic
description of the geometry of a track, of a vehicle running on the track, and
their relationship with the measurements of inertial sensors. Section 3
explains the different Kalman filters used to estimate the trajectory and
attitude of a body moving along the track. Section 4 shows two different
methods that can be used to find the covariance matrices needed for the
application of the Kalman filters. Section 5 describes the experiments used
for the application of the developed estimation techniques to the geometry
measurement of a scale track. Section 6 shows and discusses the validity of
the estimation techniques based on the comparison with the experimental
measurements. Finally, summary and conclusions of this work are given in
Section 7.

##  2 Kinematic models

This section shows the kinematic models used in the estimation of the
trajectory and orientation of a body moving along a rail track. It is a
summary of the more detailed kinematic description given in [29]. This section
describes the kinematics of the track design geometry, the kinematics of the
body based on track-relative kinematics and the relationships between the
signals measured with the inertial sensors and the body and track kinematics.

Figure 1: Kinematics of a body with track-relative coordinates

Figure 1 shows an arbitrary rigid body $b$ that moves along a rail track. The
track is simply represented by its centerline. The figure shows three frames:
(1) an inertial and global frame (GF) $<X,Y,Z>$ that is fixed in space, (2) a
track frame (TF) $<X^{t},Y^{t},Z^{t}>$ whose position and orientation are
known functions of the arc-length coordinate along the track $s$ and (3) the
body frame (BF) $<X^{b},Y^{b},Z^{b}>$ that is rigidly attached to the body.

###  2.1 Kinematics of the track centerline

Describing the kinematics of the track centerline is equivalent to providing
the functions that give the position and orientation of the TF with respect to
the GF. The definition of the TF is such that the $X^{t}$ axis is tangent to
the track design centerline, the $Y^{t}$ axis is perpendicular to $X^{t}$ and
connects the left rail centerline and the right rail centerline and the
$Z^{t}$ axis is perpendicular to both $X^{t}$ and $Y^{t}$. The components of
the absolute position vector of an arbitrary point on the design track
centerline with respect to the GF is a function of the arc-length $s$, as
follows:

${{\bf{R}}^{t}}\left(s\right)=\left[{\begin{array}[]{*{20}{c}}{R_{x}^{t}}\\\
{R_{y}^{t}}\\\ {R_{z}^{t}}\end{array}}\right]$ (1)

And the rotation matrix from the TF to the GF can be approximated to:

${{\bf{A}}^{t}}\left(s\right)\simeq\left[{\begin{array}[]{*{20}{c}}{{\rm{c}}{\psi^{t}}}&{-{\rm{s}}{\psi^{t}}}&{{\varphi^{t}}{\rm{s}}{\psi^{t}}+{\theta^{t}}{\mathop{\rm
c}\nolimits}{\psi^{t}}}\\\ {{\rm{s}}{\psi^{t}}}&{{\mathop{\rm
c}\nolimits}{\psi^{t}}}&{{\theta^{t}}{\rm{s}}{\psi^{t}}-{\varphi^{t}}{\rm{c}}{\psi^{t}}}\\\
{-{\theta^{t}}}&{{\varphi^{t}}}&1\end{array}}\right]$ (2)

where the Euler angles ${\psi^{t}}$ ($azimut$ or $heading$ angle),
${\theta^{\,t}}$ ($verticalslope$) and ${\varphi^{\,t}}$ ($cant$ or
$superelevation$ $angle$) that can be observed in Fig. 1 have been used. In
this equation, “s” means “sine” and “c” means “cosine” of the angle. Note that
the vertical slope and cant angles are assumed to be small and the rotation
matrix is linearized using the small-angle assumption.

The six functions $R_{x}^{t}\left(s\right)$, $R_{y}^{t}\left(s\right)$,
$R_{z}^{t}\left(s\right)$, ${\psi^{t}\left(s\right)}$,
${\theta^{\,t}\left(s\right)}$, and ${\varphi^{\,t}\left(s\right)}$ are
tabulated in the railway industry and they represent the design geometry of
the track centerline. In this paper, it is assumed that these functions are
given.

Each body $b$ moving along the track has an associated TF at each instant of
time. Its position and orientation is obtained substituting the position of
the body along the track, $s^{b}(t)$, in the functions
${{\bf{R}}^{t}}\left(s\right)$ and ${{\bf{A}}^{t}}\left(s\right)$.

###  2.2 Kinematics of a body moving along the track

The coordinates used to describe the position and orientation of the arbitrary
body $b$ shown in Fig. 1 moving along the track are:

${{\bf{q}}^{b}}={\left[{\begin{array}[]{*{20}{c}}{{s^{b}}}&{r_{y}^{t,b}}&{r_{z}^{t,b}}&{{\varphi^{t,b}}}&{{\theta^{t,b}}}&{{\psi^{t,b}}}\end{array}}\right]^{T}}$
(3)

where $s^{b}$ is the arc-length along the track of the position of the body,
$r_{y}^{t,b}$ and $r_{z}^{t,b}$ are the non-zero components of the position
vector $\vec{r}^{t,b}$ of the BF with respect to the TF, this is
${{\bf{\bar{r}}}^{t,b}}={\left[{\begin{array}[]{*{20}{c}}0&{r_{y}^{t,b}}&{r_{z}^{t,b}}\end{array}}\right]^{T}}$
, and ${\varphi^{t,b}},\,\,{\theta^{t,b}}\,\,{\rm{and}}\,\,{\psi^{t,b}}$ are
three Euler angles (roll, pitch and yaw, respectively) that define the
orientation of the BF with respect to the TF.

The linearized transformation matrix from the BF to the TF is given by:

${{\bf{A}}^{t,b}}\simeq\left[{\begin{array}[]{*{20}{c}}1&{-{\psi^{t,b}}}&{{\theta^{t,b}}}\\\
{{\psi^{t,b}}}&1&{-{\varphi^{t,b}}}\\\
{-{\theta^{t,b}}}&{{\varphi^{t,b}}}&1\end{array}}\right]$ (4)

The absolute position vector and the absolute orientation matrix of body $b$
are given by:

${\bf R}^{b}={\bf R}^{t}+{\bf A}^{t}{{\bf{\bar{r}}}^{t,b}},\;{\bf A}^{b}={\bf
A}^{t}{\bf{A}}^{t,b}$ (5)

The time-derivatives of these expressions are used to find the absolute
acceleration and angular velocity of body $b$. See more details in [29].

###  2.3 Kinematics of the gyroscope signals

In the remainder of this paper it will be assumed that an inertial sensor or
IMU is installed on the moving body. The IMU is assumed to be installed at the
origin of the BF of the moving body with the sensor frame parallel to the BF.
The IMU measures the 3 components of the absolute angular velocity and the the
three components of the absolute acceleration vectors in the sensor frame. The
three signals measured with the gyroscope can be interpreted as:

${{\boldsymbol{\omega}}^{imu}}=\hat{\boldsymbol{\omega}}^{b}$ (6)

where ${{\boldsymbol{\omega}}^{imu}}$ is the $3\times 1$ array of the
gyroscope signals and $\hat{\boldsymbol{\omega}}^{b}$ contains the 3
components of the absolute angular velocity in the BF. As explained in [29],
the time derivative of the absolute Euler angles of the body $b$ are related
to the gyroscope signals as follows:

$\left[{\begin{array}[]{*{20}{c}}{\dot{\varphi}^{b}}\\\ {\dot{\theta}^{b}}\\\
{\dot{\psi}^{b}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}1&{\frac{{\sin\varphi^{b}\sin\theta^{b}}}{{\cos\theta^{b}}}}&{\frac{{\cos\varphi^{b}\sin\theta^{b}}}{{\cos\theta^{b}}}}\\\
0&{\cos\varphi^{b}}&{-\sin\varphi^{b}}\\\
0&{\frac{{\sin\varphi^{b}}}{{\cos\theta^{b}}}}&{\frac{{\cos\varphi^{b}}}{{\cos\theta^{b}}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{{\hat{\omega}}_{x}}}\\\
{{{\hat{\omega}}_{y}}}\\\ {{{\hat{\omega}}_{z}}}\end{array}}\right]$ (7)

where
${\bf{\Phi}}^{b}={[\begin{array}[]{*{20}{c}}{\varphi^{b}}&{\theta^{b}}&{\psi^{b}}\end{array}]^{T}}$
is the set of absolute angles (with respect to the GF) of the body $b$. Note
that these are not the same angles given in Eq. 3 which are BF to TF relative
angles. This expression is non-linear in terms of absolute Euler angles, but
linear in terms of their time-derivatives. Using again the small angles
assumption of the roll and pitch angles, the following linearization can be
adopted:

$\begin{array}[]{l}\dot{\varphi}^{b}\approx{{\hat{\omega}}_{x}}+\theta^{b}{{\hat{\omega}}_{z}}\\\
\dot{\theta}^{b}\approx{{\hat{\omega}}_{y}}-\varphi^{b}{{\hat{\omega}}_{z}}\\\
\dot{\psi}^{b}\approx\varphi^{b}{{\hat{\omega}}_{y}}+{{\hat{\omega}}_{z}}\end{array}$
(8)

###  2.4 Kinematics of the accelerometer signals

Nowadays, most IMU’s used in vehicle dynamics have MEMS-type accelerometers.
In contrast to piezoelectric accelerometers, the measured signals include the
effect of gravity, as follows:

${{\bf{a}}^{imu}}=\hat{\ddot{\bf{R}}}^{b}+{\left({{{\bf{A}}^{b}}}\right)^{T}}{\left[{\begin{array}[]{*{20}{c}}0&0&g\end{array}}\right]^{T}}$
(9)

where ${{\bf{a}}^{imu}}$ is the $3\times 1$ array of the accelerometer
signals, $\hat{\ddot{\bf R}}^{b}$ contains the 3 components of the absolute
acceleration of the IMU in the sensor frame, ${\bf{A}}^{b}$ is the absolute
rotation matrix of the IMU and $g$ is the acceleration of gravity that is
assumed to act in the $Z$ direction. As shown in [29], developing this
equation the following approximate relationship between the accelerometer
signals and the body coordinates are obtained:

$\displaystyle\begin{bmatrix}0\\\ \ddot{r}_{y}^{t,b}\\\
\ddot{r}_{z}^{t,b}\end{bmatrix}+\begin{bmatrix}-2\rho_{h}V&2\rho_{v}V\\\
0&-2\rho_{tw}V\\\
2\rho_{tw}V&0\end{bmatrix}\begin{bmatrix}\dot{r}_{y}^{t,b}\\\
\dot{r}_{z}^{t,b}\end{bmatrix}+$ (10)
$\displaystyle+\begin{bmatrix}V^{2}(\rho_{tw}\rho_{v}-\rho^{\prime}_{h})-\rho_{h}\dot{V}&\rho_{v}\dot{V}+\rho_{tw}\rho_{h}V^{2}\\\
-V^{2}(\rho_{tw}^{2}+\rho_{h}^{2})&\rho_{v}\rho_{h}V^{2}-\rho_{tw}\dot{V}\\\
\rho_{tw}\dot{V}+\rho_{v}\rho_{h}V^{2}&-V^{2}(\rho_{tw}^{2}+\rho_{h}^{2})\end{bmatrix}\begin{bmatrix}r_{y}^{t,b}\\\
r_{z}^{t,b}\end{bmatrix}=$
$\displaystyle=\begin{bmatrix}a_{x}^{imu}+a_{z}^{imu}\theta^{t,b}-a_{y}^{imu}\psi^{t,b}+g\theta^{t}-\dot{V}\\\
a_{y}^{imu}+a_{x}^{imu}{\psi^{t,b}}-a_{z}^{imu}{\varphi^{t,b}}-g{\varphi^{t}}-{\rho_{h}}{V^{2}}\\\
a_{z}^{imu}-a_{x}^{imu}{\theta^{t,b}}+a_{y}^{imu}{\varphi^{t,b}}-g+{\rho_{v}}{V^{2}}\end{bmatrix}$

where $V$ is the forward velocity of the body and $\rho_{h}$, $\rho_{v}$, and
$\rho_{tw}$ are the horizontal, vertical and twist curvatures, respectively,
of the track centerline.

A further simplification that is also obtained in [29] relates the ”corrected”
accelerometer signals with the absolute Euler angles of the body $b$, as
follows:

${\bf{a}}^{imu}_{corr}={\bf{a}}^{imu}_{filt}-\left[{\begin{array}[]{*{20}{c}}{\dot{V}}\\\
{{\rho_{h}}{V^{2}}}\\\
{-{\rho_{v}}{V^{2}}}\end{array}}\right]\approx\it{g}\left[{\begin{array}[]{*{20}{c}}{-\theta^{b}}\\\
\varphi^{b}\\\ 1\end{array}}\right]$ (11)

where ${\bf{a}}^{imu}_{corr}$ includes the ”corrected” accelerometer signals.
The correction is made to find a set of signals that are only due to the
effect of gravity, eliminating to some extent the effect of the body
acceleration. This correction is the result of low-pass filtering the signals
to find ${\bf{a}}^{imu}_{filt}$ and subtracting and approximate value of the
acceleration. See [29] for more details.

##  3 Kalman filters

Two different algorithms have been developed to obtain the trajectory and
attitude of the moving body from the IMU measurements.

  * •

Two consecutive Kalman filters. The first one obtains the absolute orientation
of the body $b$, while the second one provides its position relative to the
TF.

  * •

One Kalman filter that includes the coupled orientation + position kinematic
model.

Please note that the term Kalman filter is used in a broad sense throughout
the paper, since the procedure applied to obtain the position and orientation
of the moving body is actually a Kalman smoother, which yields more accurate
results [30].

The chosen nomenclature for the state space models is as follows:

$\left\\{\begin{aligned} \textbf{x}_{k+1}=\textbf{Fx}_{k}+\textbf{q}_{k}\\\
\textbf{z}_{k}=\textbf{H}_{k}\textbf{x}_{k}+\textbf{r}_{k}\end{aligned}\right\\},\quad\textrm{with}\quad\left\\{\begin{aligned}
\textbf{q}_{k}\sim\textbf{N}(\textbf{0},\textbf{Q})\\\
\textbf{r}_{k}\sim\textbf{N}(\textbf{0},\textbf{R})\end{aligned}\right\\}.$
(12)

  * •

$\textbf{x}_{k}$: state vector of the system at time instant $k$.

  * •

$\textbf{z}_{k}$: measurements vector at time instant $k$.

  * •

F: transition matrix of the system.

  * •

$\textbf{H}_{k}$: measurements model matrix at time instant $k$.

  * •

Q: Process noise covariance matrix.

  * •

R: Measurements noise covariance matrix.

  * •

$\textbf{N}(\textbf{M},\textbf{S})$: Multivariate normal random variable with
mean M and covariance matrix S.

In the next subsections, all vectors and matrices in Eq. (12) will be
specified for the different proposed algorithms, based upon the kinematic
models described in Section 2. Once these vectors and matrices are defined,
the equations required to implement a Kalman filter or Kalman smoother
algorithm can be found in any text on the subject, such as [30, 31].

###  3.1 Two consecutive Kalman filters

####  3.1.1 Kalman filter for the orientation

This filter estimates the time history of angles
$\left\\{\varphi^{b},\theta^{b},\psi^{b}\right\\}$. It uses the kinematic
model defined by equations (8) and (11).

It should be noted that the yaw angle $\psi^{b}$ does not directly appear in
(8), (11), but only its time derivative $\dot{\psi}^{b}$. This means that the
numerical integration that allows obtaining $\psi^{b}$ from $\dot{\psi}^{b}$,
which is part of the Kalman filter algorithm, may produce a drift effect by
which, after certain time, the values of $\psi^{b}$ are totally erroneous. To
solve this problem, it is helpful to include in the Kalman filter the
additional equation

$\psi^{b}=\psi^{t},$ (13)

which represents a fictitious measurement. It might seem that the introduction
of Eq. (13) will yield incorrect results, since it imposes the absolute yaw
angle of the body to be exactly equal to that of the track. However, it should
be kept in mind that all measurement equations have an associated noise–see
$\textbf{r}_{k}$ in (12). As a consequence, the actual effect of Eq. (13) will
be to keep $\psi^{b}$ close to $\psi^{t}$.

The state vector for this filter is defined as

$\textbf{x}=\left[\begin{array}[]{*{20}{c}}\varphi^{b}&\dot{\varphi}^{b}&\theta^{b}&\dot{\theta}^{b}&\psi^{b}&\dot{\psi}^{b}\end{array}\right]^{T}.$
(14)

Using (8), (11), (13) and (14), all matrices required to build the state space
model (12) can be obtained. It is convenient to make some comments about these
matrices, which are shown in Appendix A:

  * •

$\mathrm{\Delta}t$ represents the time interval between measurements.

  * •

The diagonal form of matrix R corresponds to the assumption that the noises of
the different measurements are statistically independent.

  * •

The form of matrix Q represents the statistical assumption that variables
$\dot{\varphi}^{b}(t)$, $\dot{\theta}^{b}(t)$ and $\dot{\psi}^{b}(t)$ evolve,
between each pair of instants $k$ and $k+1$, following Wiener processes. This
is usually referred to in the literature as a CWNA model (Continuous White
Noise Acceleration) [32, 30].

  * •

In order to define matrices Q and R, the values of 7 parameters need to be
specified. We collect these parameters in a vector p:

$\textbf{p}=\left[\begin{array}[]{*{20}{c}}q_{\varphi}&q_{\theta}&q_{\psi}&R_{\omega}&R_{ax}&R_{ay}&R_{\psi}\end{array}\right]^{T}.$
(15)

####  3.1.2 Kalman filter for the trajectory

This filter estimates the time history of the body position with respect to
the track, given by variables $\left\\{r_{y}^{t,b},r_{z}^{t,b}\right\\}$. It
uses the kinematic model defined by the last two equations in Eq. (10).

It is assumed here that the orientation of the body along time has already
been estimated (see Section 3.1.1). Note that the Kalman filter for the
orientation provides the absolute angles
$\left\\{\varphi^{b},\theta^{b},\psi^{b}\right\\}$, while the relative angles
$\left\\{\varphi^{t,b},\theta^{t,b},\psi^{t,b}\right\\}$ are actually more
convenient to use in the Kalman filter for the trajectory (see Eq. (10)). The
transformation from one set of angles to the other can be easily performed
thanks to the following relations:

$\displaystyle\varphi^{t,b}\approx\varphi^{b}-\varphi^{t}$ (16)
$\displaystyle\theta^{t,b}\approx\theta^{b}-\theta^{t}$
$\displaystyle\psi^{t,b}\approx\psi^{b}-\psi^{t}$

where the assumption that
$\left\\{\varphi^{t,b},\theta^{t,b},\psi^{t,b}\right\\}$ are small angles has
been used.

As done in the Kalman filter for the orientation–see Eq. (13)–virtual sensors
are introduced to avoid drift:

$\displaystyle r_{y}^{t,b}=0$ (17) $\displaystyle r_{z}^{t,b}=\delta$

where $\delta$ represents the vertical distance at rest between the IMU and
the track design centerline.

The state vector is defined in this case as

$\textbf{x}=\left[\begin{array}[]{*{20}{c}}r_{y}^{t,b}&\dot{r}_{y}^{t,b}&\ddot{r}_{y}^{t,b}&r_{z}^{t,b}&\dot{r}_{z}^{t,b}&\ddot{r}_{z}^{t,b}\end{array}\right]^{T}.$
(18)

Using (10), (16), (17) and (18), all matrices required to define the state
space model (12) can be obtained. Let us make some remarks about these
matrices, which are presented in Appendix B:

  * •

The diagonal form of matrix R represents the assumption that the noises of the
different measurements are statistically independent.

  * •

The form of matrix Q corresponds to the statistical assumption that variables
$\ddot{r}_{y}^{t,b}$ and $\ddot{r}_{z}^{t,b}$ follow Wiener processes between
each pair of instants $k$ and $k+1$. This is usually known in the literature
as a CWPA model (Continuous Wiener Process Acceleration) [32, 30].

  * •

The 6 scalar parameters needed to define matrices Q and R are collected in a
vector p:

$\textbf{p}=\left[\begin{array}[]{*{20}{c}}q_{y}&q_{z}&R_{y1}&R_{z1}&R_{y2}&R_{z2}\end{array}\right]^{T}.$
(19)

###  3.2 One Kalman filter

This filter simultaneously estimates the absolute orientation of the moving
body and its position relative to the track, given by variables
$\\{\varphi^{b},\theta^{b},\psi^{b},r_{y}^{t,b},r_{z}^{t,b}\\}$. The algorithm
exploits the kinematic model given by (8), (10) and (16). Virtual measurements
(17) are also used here to avoid drift problems.

The state vector is defined as

$\textbf{x}=\left[\begin{array}[]{*{20}{c}}\varphi^{b}&\dot{\varphi}^{b}&\theta^{b}&\dot{\theta}^{b}&\psi^{b}&\dot{\psi}^{b}&r_{y}^{t,b}&\dot{r}_{y}^{t,b}&\ddot{r}_{y}^{t,b}&r_{z}^{t,b}&\dot{r}_{z}^{t,b}&\ddot{r}_{z}^{t,b}\end{array}\right]^{T}.$
(20)

Using (8), (10), (16), (17) and (20), all matrices required to formulate the
state space model (12) can be obtained. It is pertinent to make some
clarifications about these matrices, which are shown in Appendix C:

  * •

The diagonal form of matrix R represents the assumption of statistically
independent noises for the different measurements.

  * •

The form of matrix Q corresponds to the statistical assumption that variables
$\dot{\varphi}^{b}(t)$,
$\dot{\theta}^{b}(t),\dot{\psi}^{b}(t),\ddot{r}_{y}^{t,b}$ and
$\ddot{r}_{z}^{t,b}$ evolve, between each pair of instants $k$ and $k+1$,
following Wiener processes.

  * •

Eleven parameters are required to fully define covariance matrices Q and R .
We gather these parameters in a vector p:

$\textbf{p}=\left[\begin{array}[]{*{20}{c}}q_{\varphi}&q_{\theta}&q_{\psi}&q_{y}&q_{z}&R_{\omega}&R_{x}&R_{y1}&R_{z1}&R_{y2}&R_{z2}\end{array}\right]^{T}.$
(21)

Before concluding Section 3, it is relevant to note that we also tested
augmented versions of the 3 presented Kalman filters that incorporate
additional state variables to model sensor biases. The corresponding equations
have not been included here because the results were not satisfactory, as
discussed in Section 6.4.

##  4 Methods for the estimation of covariance matrices

In the last section, two different Kalman filtering approaches have been
presented with the aim of estimating the trajectory and attitude of a generic
body $b$ moving along a railway track. To be able to use these algorithms, the
first step is to fully define covariance matrices Q and R in Eq.(12).

We use
$\textbf{p}=[\begin{array}[]{*{20}{c}}p_{1}&p_{2}&...&p_{M}\end{array}]^{T}$
to denote a vector containing the scalar parameters that determine matrices Q
and R. The number of elements $M$ in p depends on the specific Kalman Filter
under consideration–see Eqs. (15), (19) and (21).

Before proceeding to describe the proposed strategies to estimate vector p,
some clarifications are appropriate. The developments presented in Sections 2
and 3 are completely general, in the sense that they can be used in any
application that requires estimating the position and orientation of a vehicle
moving along a railway track. On the other hand, these techniques have been
applied in the current research to the specific problem of measuring track
irregularities. A Track Geometry Measuring System (TGMS) has been developed by
the authors, combining computer vision with the estimation of the trajectory
and attitude of the vehicle using inertial sensors. The details of this TGMS
technology can be found in [29] and are summarized in Section 5. The present
section expounds two alternative procedures to estimate vector p. The first
one is specific to the TGMS application, while the second one is as general as
Sections 2 and 3:

  * •

Known-output method. The covariance matrices are tuned through a comparison
between the track irregularities obtained with the TGMS and a set of reference
irregularities.

  * •

Constrained maximum likelihood estimation. This technique uses the IMU
measurements to obtain the most likely vector p.

In both cases, a global optimization of certain objective function is
conducted, with the components of vector p varying within a specified range.
In order to choose appropriate range limits, consider first that every
parameter $p_{i}$ must be positive due to their physical meaning–see Section
3. On the other hand, the range should be wide enough to encompass a
meaningful optimum of the objective function. After several trials, the
following range has been found to produce accurate results:

$10^{-4}\leq p_{i}\leq 10^{4}\qquad\textrm{for}\ \ i=1,...,M$ (22)

with each $p_{i}$ measured in SI.

After testing different global optimization techniques included in the MATLAB
Optimization Toolbox, the method that was found to work best for this
application was the Surrogate Optimization [33]. This scheme does not require
an initial point for the optimum search, but only the ranges where the
parameters can vary and, optionally, one or more conditions that they must
fulfill.

###  4.1 Known-output method

In order to apply the Known-Output Method (KOM), the vehicle with the
operating TGMS has to travel along a track whose irregularities have already
been obtained through some other reliable methodology (reference
irregularities). For the present work, these reference irregularities were
obtained by using an LVDT, an inclinometer and a Total Station, as described
in detail in [34].

It is convenient at this point to recall that we are interested in the 4
irregularities that are usually controlled in the railway industry:

  * •

Alignment and Vertical Profile (absolute irregularities)

  * •

Gauge Variation and Cross Level (relative irregularities)

See [29] for precise definitions of these terms.

The optimization algorithm searches for the vector p that minimizes the RMS of
the difference between the TGMS irregularities and the reference
irregularities. Thus, the objective function (OF) for the optimization is
defined as

$\displaystyle\textrm{OF}=\textrm{RMS}[\textrm{TGMS
Alignment}(s)-\textrm{Reference Alignment}(s)]+$ (23)
$\displaystyle\textrm{RMS}[\textrm{TGMS Vertical Profile}(s)-\textrm{Reference
Vertical Profile}(s)]+$ $\displaystyle\textrm{RMS}[\textrm{TGMS Cross
Level}(s)-\textrm{Reference Cross Level}(s)].$

Note that the gauge variation is not included in (23). The reason is that this
irregularity has been found to depend almost exclusively on the computer
vision of the TGMS, with a negligible influence of the vehicle trajectory and
attitude.

Let us make some remarks about the described strategy:

  * •

From a pragmatic point of view, this approach has a clear disadvantage: when
installing the TGMS on a new commercial vehicle, it would need to travel along
a track whose irregularities were reliably known beforehand, which is not easy
to find in practice. Furthermore, the appeal of the TGMS technology could get
limited by the need of a complex initial calibration.

  * •

This procedure relies on the assumption that, once the covariance matrices
have been estimated for a certain track, with certain levels of irregularity
and a vehicle circulating at certain speed, the Kalman filters will still
produce accurate results for other tracks, with different irregularities and
different vehicle speeds. The validity of this hypothesis is not evident. In
fact, the results presented in Section 6 suggest that the mentioned assumption
does not always hold. In other words, the covariance matrices obtained for a
specific railway track could be ineffective to estimate the irregularities of
other tracks.

###  4.2 Constrained maximum likelihood estimation

Maximum Likelihood Estimation (MLE) is a very common approach for tuning the
parameters of statistical models–the sense of the word constrained in the
title will be seen at the end of the section. The basic idea, for those
readers who are not familiar with this kind of methods, is the following.
Consider a generic multivariate random variable y whose probability density
function, $p(\textbf{y}|\textbf{p})$, depends on a set of parameters p. Assume
these parameters are unknown and we are interested in estimating them.
According to the MLE strategy, the estimation includes 2 steps:

  * •

Carry out the experiment that is modelled by the probability density function
$p(\textbf{y}|\textbf{p})$. Let $\textbf{y}_{0}$ denote the obtained result
(observation).

  * •

Estimate p as the set of parameters that maximizes
$p(\textbf{y}_{0}|\textbf{p})$. That is to say, we choose the parameters that
are most likely to generate the observed data.

For a more rigorous and complete explanation of the MLE concept, see [35].

The generic scheme outlined above can be directly particularized to the
trajectory and attitude estimation that is being investigated. While the
vector of parameters p is given in Eqs. (15), (19) and (21), the set of
observed data $\textbf{y}_{0}$ represents the sequence of accelerations,
angular velocities, forward speeds and positions along the track registered
during a specific ride of the moving body $b$ (Fig. 1). Finally, the
probability density of the observations $p(\textbf{y}|\textbf{p})$ can be
obtained from the model equations (12), as shown in [36].

In summary, given a set of observations $\textbf{y}_{0}$, the proposed global
optimization algorithm will explore the space of parameters p within the range
specified in Eq. (22), searching for those parameter values that maximize
$p(\textbf{y}_{0}|\textbf{p})$. The specific equations that need to be
appended to the Kalman Filter algorithm in order to calculate
$p(\textbf{y}_{0}|\textbf{p})$ can be found in [36].

Some comments about this approach are pertinent:

  * •

It is usual in the Kalman filtering literature to conduct the likelihood
maximization through a specific procedure called Expectation Maximization
algorithm (EM algorithm) [37, 38]. This method has been implemented by the
authors for the trajectory and attitude estimation, but the attained results
have not been satisfactory. This is why the presented global optimization
scheme has been chosen instead. In any case, it should be noted that the EM
algorithm intends to find a local maximum of the likelihood function, which in
principle makes a global maximization technique preferable.

  * •

When the described methodology is used for the TGMS application, it has a
plain advantage over the one set forth in Section 4.1, since in this case
there is no need of a previously known set of irregularities to tune the
covariance matrices. Consider further the following benefit: assume that a
railway vehicle, with an operating TGMS, performs two rides on two very
different tracks, with different irregularity levels and moving at different
forward speeds. As was commented in Section 4.1, it is not obvious that one
only pair of matrices Q and R could provide accurate results for both tracks.
The MLE would take this automatically into account by obtaining one set of
most likely covariance matrices for each of the rides.

####  4.2.1 Constraints imposed for the optimization

The experimental results obtained during this research have shown that the
MLE, as has been described, can produce inaccurate outcomes in this
application. That is to say, the most likely covariance matrices do not
necessarily provide a good position and attitude estimation. This problem has
been solved by slightly guiding the search for the optimum considering some
conditions on the parameters. In particular, accurate results have been
found–as will be shown in Section 6–thanks to the following constraints:

  * •

Two consecutive Kalman Filters

– Kalman Filter for the Orientation

$R_{ay}\leq q_{\varphi}/10$ (24)

– Kalman Filter for the Trajectory

$R_{y1}\leq R_{y2}/10$ (25)

  * •

One Kalman Filter

$R_{y1}\leq R_{y2}/10$ (26)

with all parameters measured in SI.

The interpretation of the above inequalities can be readily seen by recalling
the physical meaning of each of the parameters. For example, in the case of
Eq. (26), $R_{y1}$ represents the variance of the noise associated with the
accelerometer signal in $y$ direction, while $R_{y2}$ is the variance of the
noise associated with the virtual measurement $r_{y}^{t,b}=0$, as can be found
in Section 3.2. Considering that the noise variance for each sensor represents
the uncertainty of the corresponding measurement, the meaning of Eq. (26)
could be stated as follows: measuring all parameters in SI, the Kalman filter
must have a considerably higher confidence in the accelerometer measurements
in $y$ direction than it has in the virtual measurements $r_{y}^{t,b}=0$. This
way, we avoid the optimization process from leading to a solution of the type
$r_{y}^{t,b}(t)\approx 0$ due to an excessive weight of the fictitious
position measurements in the calculations.

It can be said after the above considerations that, instead of a plain MLE,
the proposed approach falls within the category of Constrained Maximum
Likelihood estimation (CML estimation) [39, 40].

##  5 TGMS method and experimental setup

The TGMS is a technology developed by the authors for the measurement of the
irregularities of rail tracks. It combines inertial sensors with computer
vision. The sensors include an IMU, two video cameras, two laser projectors
and an encoder for the odometry and calculation of the forward velocity. This
equipment is schematically represented in Fig 2. This technology is explained
in detail in [29]. Results shown in this paper are obtained with a 1:10 scale
experimental facility installed at the School of Engineering of the University
of Seville, Spain. Figure 3 shows the 90 m-scale track that has been designed
to create an arbitrary distribution of irregularities. To this end, the
sleepers have been substituted with 4-dof mechanisms that allow the in-plane
motion of both rail cross-sections. Figure 4 shows the scale vehicle used in
the experiments that incorporates the TGMS. The vehicle has a classical
structure of 4 rigid wheelsets, two bogie frames and one carbody, with primary
and secondary suspensions. In this figure, two video cameras and one of the
laser projectors can be distinguished in the central part of the carbody. The
IMU cannot be seen in the figure, but it is located also in the central part
of the carbody. Therefore in this case, body $b$, whose trajectory and
orientation will be estimated, is the vehicle carbody. It is important to
emphasize that the TGMS allows the installation of the sensors in any body of
the vehicle.

The algorithms developed in [29] for the estimation of irregularities with the
TGMS require the evaluation of the position and orientation of the IMU. While
the vehicle trajectory has a strong influence on the estimation of alignment
and vertical profile, the vehicle orientation (particularly the roll angle) is
very relevant for the estimation of alignment and cross level–the effect of
both trajectory and orientation on the gauge estimation is negligible, as
commented in Section 4.1. Thanks to the improvements in the trajectory and
attitude estimation described in this paper, the obtained alignment, vertical
profile and cross level will be shown to be notably more accurate than those
presented in [29].

Figure 2: Track geometry measuring system Figure 3: Scale track: (a) Aerial
view, (b) detail of track supports Figure 4: Scale vehicle with TGMS

##  6 Results

This section presents the experimental results that have been obtained when
applying the procedures developed in this paper to the problem of track
geometry measurement, as described in Section 5. For the details on how the
trajectory and attitude estimation is used within the TGMS technology to
obtain track irregularities, see [29].

All shown results correspond to one ride of the instrumented vehicle along the
scale track, with the forward velocity shown in Fig. 5.

Figure 5: Forward velocity of the scale vehicle during the test

Some of the irregularity profiles displayed throughout this section are
filtered. Whether or not each particular irregularity has been filtered for
the representation is specified in the corresponding figure caption (Figs.
6-17). Filtered irregularities only preserve wavelengths between
$0.3\text{\,}\mathrm{m}\text{/}$ and $7\text{\,}\mathrm{m}\text{/}$. This is
in accordance with European Standard [41], which states that irregularity
wavelengths between $3\text{\,}\mathrm{m}\text{/}$ and
$70\text{\,}\mathrm{m}\text{/}$ are the ones directly associated with railway
vehicles safety–note that the 1:10 scale of the track has been applied to the
wavelength range.

###  6.1 Two consecutive Kalman filters

The irregularities shown in Figs. 6-9 correspond to the application of 2
consecutive Kalman filters for the trajectory and attitude estimation, as
presented in Section 3.1. We compare the accuracy of the CML estimation for
covariance matrices (Section 4.2) with that of the KOM (Section 4.1).

Figure 6: Track Gauge obtained with 2 consecutive Kalman filters, compared to
the Reference Gauge. Top: Unfiltered. Bottom: Filtered.

Figure 7: Cross Level obtained with 2 consecutive Kalman filters, compared to
the Reference Cross Level. Top: Unfiltered. Bottom: Filtered. Figure 8:
Alignment obtained with 2 consecutive Kalman filters, compared to the
Reference Alignment (all signals are filtered). Figure 9: Vertical Profile
obtained with 2 consecutive Kalman filters, compared to the Reference Vertical
Profile (all signals are filtered).

###  6.2 One Kalman filter

Figs. 10-13 represent the irregularities obtained when using one Kalman filter
for the trajectory and attitude estimation, as described in Section 3.2. The
precision of the CML estimation for covariance matrices (Section 4.2) is
compared to that of the KOM (Section 4.1).

Figure 10: Track Gauge obtained with one Kalman filter, compared to the
Reference Gauge. Top: Unfiltered. Bottom: Filtered.

Figure 11: Cross Level obtained with one Kalman filter, compared to the
Reference Cross Level. Top: Unfiltered. Bottom: Filtered. Figure 12: Alignment
obtained with one Kalman filter, compared to the Reference Alignment (all
signals are filtered). Figure 13: Vertical Profile obtained with one Kalman
filter, compared to the Reference Vertical Profile (all signals are filtered).

###  6.3 Robustness of the known-output method

It was mentioned in Section 4.1 that the KOM used to estimate the covariance
matrices is based upon the hypothesis that, once the required parameters have
been obtained for a specific track, with specific irregularities and vehicle
speed, they will remain valid in other conditions. The present subsection
intends to test the validity of this assumption.

One way to verify this would be to to repeat the test with a different
distribution of irregularities–recall that the scale track rests upon a set of
mechanisms that allow for the generation of any desired profile of
irregularities. Although this will be surely done in the near future, a
simpler approach that does not require any additional experimental tests has
been adopted for this paper. The idea consists in mentally dividing the
vehicle ride in 2 parts and checking if the covariance matrices obtained for
one of the parts are able to give a good irregularity estimation for the whole
test. The results attained, using one Kalman filter, are displayed in Figs.
14-17.

Figure 14: Track Gauge obtained with one Kalman filter and the KOM
(optimization conducted for a partial section of the track), compared to the
Reference Gauge. Top: Unfiltered. Bottom: Filtered.

Figure 15: Cross Level obtained with one Kalman filter and the KOM
(optimization conducted for a partial section of the track), compared to the
Reference Cross Level. Top: Unfiltered. Bottom: Filtered. Figure 16: Alignment
obtained with one Kalman filter and the KOM (optimization conducted for a
partial section of the track), compared to the Reference Alignment (all
signals are filtered). Figure 17: Vertical Profile obtained with one Kalman
filter and the KOM (optimization conducted for a partial section of the
track), compared to the Reference Vertical Profile (all signals are filtered).

###  6.4 Discussion

The results shown in Figs. 6-13 clearly display a remarkably accurate
estimation of the irregularities with all 4 combinations of 1 / 2 Kalman
filters, CML estimation / KOM, as evidenced by the similarity between the
reference irregularities and those yielded by the TGMS.

As for the robustness of the KOM, evaluated in Figs. 14-17, reasonably good
results can be found for gauge, cross level and vertical profile. However, the
alignment graph does manifest a detrimental effect of estimating the
covariance parameters based on only a portion of the track. When optimizing
between $0.8\text{\,}\mathrm{m}\text{/}$ and $45\text{\,}\mathrm{m}\text{/}$,
the 3 higher alignment peaks are not properly captured, while the optimization
between $45\text{\,}\mathrm{m}\text{/}$ and $65\text{\,}\mathrm{m}\text{/}$
produces large errors at the start of the test. These results suggest a
limited robustness of the KOM to estimate the covariance matrices. However, if
the optimization was carried out over a sufficiently long track with a highly
varied frequency content for all irregularities, it might be the case that the
obtained covariance parameters yielded good estimations for a wide range of
tracks and irregularity levels. In any case, it should be remembered that
these issues do not exist when the CML estimation is used, because the
algorithm estimates specific covariance matrices for each different ride of
the vehicle.

Taking the considerations of the last paragraph into account, it is reasonable
to propose the CML estimation for covariance matrices as the most promising
algorithm for the implementation of the TGMS technology on commercial
vehicles. The KOM may still be useful in order to see how accurate the TGMS
results can be when covariance matrices are optimally tuned.

Focusing on the results found with CML estimation, the obtained irregularities
are notably precise using either 1 or 2 Kalman filters, with the exception of
a spurious alignment peak at the beginning of the test (Figs. 8 and 12). This
initial peak is found to be smaller when using 2 consecutive Kalman filters.
On the other hand, a close look to Figs. 7 and 11 reveals that the unfiltered
cross level is slightly more accurate with one Kalman Filter. Nevertheless,
more experimental results would be needed in order to confirm these small
differences. What is apparent from the presented results is that both
approaches (CML estimation with 1 Kalman filter / CML estimation with 2 Kalman
filters) give rise to sufficiently accurate results for the current
application.

Finally, it is appropriate to note that we also tested variants of all
presented Kalman filters that include additional state variables to model
sensor biases. This was done by assuming the bias variables to evolve in time
as Wiener processes, as is common in the literature [42, 43]. Although, as
expected, this proved to slightly improve the irregularity estimation when
using the KOM, the CML estimation strategy was not able to properly tune the
parameters associated with the bias variables. For this reason, the results
reached by using these augmented Kalman filters are not shown in the paper.

##  7 Summary and conclusions

This paper is about the estimation of the trajectory and attitude of a body
moving along a railroad track. Compared with the equivalent, and well
developed in the literature, estimation problems used for inertial navigation
of air or road vehicles, this estimation problem has two specific properties:
(1) the resulting trajectory and attitude are very similar to the known
trajectory and attitude of the track centerline, and (2) the required accuracy
is very demanding for applications like track geometry measurement. The
developed estimation techniques are based on the kinematics of an arbitrary
body moving along an arbitrary track and the exact relationship between the
description of this type of motion and the measurements of inertial sensors.

The estimations are based on the simple discrete Kalman filter. This basic
method can be used thanks to an accurate linearization of the kinematics due
to the small body to track relative motion. The estimation of trajectory and
attitude can be done independently, using a different Kalman filter for each,
or as a coupled problem, using a single Kalman filter. Anyway, crucial to the
success of the estimation is the accurate calculation of the parameters of the
covariance matrices used in the Kalman filters. Two methods are used in this
work to calculate these parameters: the KOM and the CML estimation. The KOM,
which is specific for the application of track geometry measurement, requires
the use of a railroad track with known irregular geometry. The CML estimation
is much more general and does not rely upon the previous knowledge of any
irregularity profiles. Both techniques are explained in detail.

The application of the estimation techniques to the track geometry measurement
of a scale track shows the accuracy of the four combinations of techniques
that have been presented. In general, results show a remarkably accurate
estimation of the track irregularities with all four combinations. Therefore,
it is obvious to propose the use of the much simpler and less costly CML
method for the calculation of the covariance matrices. This is very good news
for the application of the track geometry measurement method proposed by the
authors. Regarding the use of one or two Kalman filters, results are not
concluding. Thanks to the developed estimation techniques, the accuracy of the
measurement of the track irregularities has improved significantly with
respect to the results presented in [29].

## Acknowledgements

This research was supported by the Spanish Ministerio de Ciencia e Innovación,
under the program ”Proyectos I+D+I–2020”, with project reference
PID2020-117614RB-I00. This support is gratefully acknowledged.

## References

  * [1] I. Ahmad, A. El Hadri, L. Benziane, A. Benallegue, Globally asymptotic attitude estimation for accelerated aerial vehicles, Aerospace Science and Technology 84 (2019) 1175–1181.  doi:10.1016/j.ast.2018.06.008.
URL https://doi.org/10.1016/j.ast.2018.06.008

  * [2] T. H. Bryne, R. H. Rogne, T. I. Fossen, T. A. Johansen, Attitude and Heave Estimation for Ships using MEMS-based Inertial Measurements, IFAC-PapersOnLine 49 (23) (2016) 568–575.  doi:10.1016/j.ifacol.2016.10.496.
URL http://dx.doi.org/10.1016/j.ifacol.2016.10.496

  * [3] M. D. Hua, Attitude estimation for accelerated vehicles using GPS/INS measurements, Control Engineering Practice 18 (7) (2010) 723–732.  doi:10.1016/j.conengprac.2010.01.016.
URL http://dx.doi.org/10.1016/j.conengprac.2010.01.016

  * [4] W. J. Scholte, V. Rodrigo Marco, H. Nijmeijer, Experimental Validation of Vehicle Velocity, Attitude and IMU Bias Estimation, IFAC-PapersOnLine 52 (8) (2019) 118–123.  doi:10.1016/j.ifacol.2019.08.058.
URL https://doi.org/10.1016/j.ifacol.2019.08.058

  * [5] M. Aldimirov, R. Arnaudov, Method for automated reconstruction of a car’s path during crash from GPS/INS data using a Kalman filter, Advances in Engineering Software 115 (November 2017) (2018) 386–390.  doi:10.1016/j.advengsoft.2017.10.009.
URL https://doi.org/10.1016/j.advengsoft.2017.10.009

  * [6] J. Vaganay, M. J. Aldon, Attitude estimation for a vehicle using inertial sensors, Control Engineering Practice 2 (2) (1994) 281–287.  doi:10.1016/0967-0661(94)90209-7.
  * [7] M. Anderle, S. Čelikovský, Sensor Fusion for simple walking robot using low-level implementation of Extended Kalman Filter, IFAC-PapersOnLine 51 (13) (2018) 43–48.  doi:10.1016/j.ifacol.2018.07.252.
  * [8] B. Kada, K. Munawar, M. S. Shaikh, M. A. Hussaini, U. M. Al-Saggaf, UAV Attitude Estimation Using Nonlinear Filtering and Low-Cost Mems Sensors, IFAC-PapersOnLine 49 (21) (2016) 521–528.  doi:10.1016/j.ifacol.2016.10.655.
URL http://dx.doi.org/10.1016/j.ifacol.2016.10.655

  * [9] H. Benzerrouk, A. Nebylov, H. Salhi, Quadrotor UAV state estimation based on High-Degree Cubature Kalman filter, IFAC-PapersOnLine 49 (17) (2016) 349–354.  doi:10.1016/j.ifacol.2016.09.060.
URL http://dx.doi.org/10.1016/j.ifacol.2016.09.060

  * [10] P. Huang, Z. Zhang, X. Luo, J. Zhang, P. Huang, Attitude Estimation of Agricultural Implements Based on Quaternion and Complementary Filter, IFAC-PapersOnLine 51 (17) (2018) 837–842.  doi:10.1016/j.ifacol.2018.08.090.
URL https://doi.org/10.1016/j.ifacol.2018.08.090

  * [11] M. Abolhasani, M. Rahmani, Robust deterministic least-squares filtering for uncertain time-varying nonlinear systems with unknown inputs, Systems and Control Letters 122 (2018) 1–11.  doi:10.1016/j.sysconle.2018.09.005.
URL https://doi.org/10.1016/j.sysconle.2018.09.005

  * [12] X. Li, W. Chen, C. Chan, B. Li, X. Song, Multi-sensor fusion methodology for enhanced land vehicle positioning, Information Fusion 46 (March 2017) (2019) 51–62.  doi:10.1016/j.inffus.2018.04.006.
  * [13] S. O. Madgwick, An efficient orientation filter for inertial and inertial/magnetic sensor arrays, Report x-io and University of Bristol (2010) 32doi:10.1109/ICORR.2011.5975346.
  * [14] S. O. Madgwick, A. J. Harrison, R. Vaidyanathan, Estimation of IMU and MARG orientation using a gradient descent algorithm, IEEE International Conference on Rehabilitation Robotics (2011) 1–7doi:10.1109/ICORR.2011.5975346.
  * [15] S. Mansoor, U. I. Bhatti, A. I. Bhatti, S. M. D. Ali, Improved attitude determination by compensation of gyroscopic drift by use of accelerometers and magnetometers, Measurement: Journal of the International Measurement Confederation 131 (2019) 582–589.  doi:10.1016/j.measurement.2018.08.067.
URL https://doi.org/10.1016/j.measurement.2018.08.067

  * [16] W. Alqaisi, J. Ghommam, A. Alazzam, M. Saad, V. Nerguizian, Three-loop uncertainties compensator and sliding mode quadrotor control, Computers and Electrical Engineering 81 (2020) 106507.  doi:10.1016/j.compeleceng.2019.106507.
URL https://doi.org/10.1016/j.compeleceng.2019.106507

  * [17] S. L. Grassie, Measurement of railhead longitudinal profiles: a comparison of different techniques, Wear 191 (1) (1996) 245–251, 4th International Conference on Contact Mechanics and Wear of Rail-Wheel Systems.  doi:https://doi.org/10.1016/0043-1648(95)06732-9.
URL https://www.sciencedirect.com/science/article/pii/0043164895067329

  * [18] S.-B. Chiou, J.-Y. Yen, Precise railway alignment measurements of the horizontal circular curves and the vertical parabolic curves using the chord method, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 233 (5) (2019) 537–549.  doi:10.1177/0954409718800527.
  * [19] P. Weston, C. Roberts, G. Yeo, E. Stewart, Perspectives on railway track geometry condition monitoring from in-service railway vehicles, Vehicle System Dynamics 53 (2015) 1063–1091.
  * [20] P. Weston, C. S. Ling, C. Roberts, C. J. Goodman, P. Li, R. M. Goodall, Monitoring vertical track irregularity from in-service railway vehicles, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 221 (1) (2007) 75–88.
  * [21] P. Weston, C. S. Ling, C. J. Goodman, C. Roberts, P. Li, R. M. Goodall, Monitoring lateral track irregularity from in-service railway vehicles, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 221 (1) (2007) 89–100.
  * [22] J. Seok Lee, S. Choi, S.-S. Kim, C. Park, Y. Guk Kim, A mixed filtering approach for track condition monitoring using accelerometers on the axle box and bogie, IEEE T. Instrumentation and Measurement 61 (2012) 749–758.
  * [23] X. Wei, F. Liu, L. Jia, Urban rail track condition monitoring based on in-service vehicle acceleration measurements, Measurement 80 (2016) 217–228.  doi:https://doi.org/10.1016/j.measurement.2015.11.033.
URL https://www.sciencedirect.com/science/article/pii/S0263224115006193

  * [24] H. C. Tsai, C. Y. Wang, N. E. Huang, Fast inspection and identification techniques for track irregularities based on hht analysis, Advances in Adaptive Data Analysis 04 (01n02) (2012) 1250016.  arXiv:https://doi.org/10.1142/S1793536912500161, doi:10.1142/S1793536912500161.
URL https://doi.org/10.1142/S1793536912500161

  * [25] H.-C. Tsai, C.-Y. Wang, N. Huang, T.-W. Kuo, W.-h. Chieng, Railway track inspection based on the vibration response to a scheduled train and the hilbert-huang transform, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 229 (01 2014).  doi:10.1177/0954409714527930.
  * [26] H. Tsunashima, Y. Naganuma, T. Kobayashi, Track geometry estimation from car-body vibration, Vehicle System Dynamics 52 (2014) 207–219.
  * [27] A. De Rosa, S. Alfi, S. Bruni, Estimation of lateral and cross alignment in a railway track based on vehicle dynamics measurements, Mechanical Systems and Signal Processing 116 (2019) 606–623.
  * [28] S. Muñoz, J. Ros, P. Urda, J. L. Escalona, Estimation of lateral track irregularity using a kalman filter. experimental validation, Journal of Sound and Vibration 504 (2021) 116122.  doi:https://doi.org/10.1016/j.jsv.2021.116122.
URL https://www.sciencedirect.com/science/article/pii/S0022460X21001942

  * [29] J. L. Escalona, P. Urda, S. Muñoz, A track geometry measuring system based on multibody kinematics, inertial sensors and computer vision, Sensors 21 (3) (2021) 683.
  * [30] J. Hartikainen, A. Solin, S. Särkkä, Optimal filtering with kalman filters and smoothers, Department of biomedical engineering and computational sciences, Aalto University School of Science, 16th August (2011).
  * [31] S. Särkkä, Bayesian filtering and smoothing, no. 3, Cambridge University Press, 2013.
  * [32] Y. Bar-Shalom, X. R. Li, T. Kirubarajan, Estimation with applications to tracking and navigation: theory algorithms and software, John Wiley & Sons, 2004\.
  * [33] N. V. Queipo, R. T. Haftka, W. Shyy, T. Goel, R. Vaidyanathan, P. K. Tucker, Surrogate-based analysis and optimization, Progress in aerospace sciences 41 (1) (2005) 1–28.
  * [34] P. Urda, S. Muñoz, J. F. Aceituno, J. L. Escalona, Application and experimental validation of a multibody model with weakly coupled lateral and vertical dynamics to a scaled railway vehicle, Sensors 20 (13) (2020) 3700.
  * [35] K. Miura, An introduction to maximum likelihood estimation and information geometry, Interdisciplinary Information Sciences 17 (3) (2011) 155–174.
  * [36] P. Abbeel, CS294-40 Learning for Robotics and Control. Kalman Filtering, EKF, Unscented KF, Smoother, EM (Lecture Notes), University of California, Berkeley, 2008.
URL https://inst.eecs.berkeley.edu/~cs294-40/fa08/scribes/lecture14.pdf

  * [37] J. van den Berg, Lecture Slides on EM-Algorithm, Algorithmic Robotics Lab, The University of Utah.
URL http://arl.cs.utah.edu/resources/EM%20Algorithm.pdf

  * [38] N. Sammaknejad, Y. Zhao, B. Huang, A review of the expectation maximization algorithm in data-driven process identification, Journal of Process Control 73 (2019) 123–136.  doi:https://doi.org/10.1016/j.jprocont.2018.12.010.
URL https://www.sciencedirect.com/science/article/pii/S0959152418305614

  * [39] R. J. Hathaway, A constrained formulation of maximum-likelihood estimation for normal mixture distributions, The Annals of Statistics 13 (2) (1985) 795–800.
  * [40] N. Chatterjee, Y.-H. Chen, P. Maas, R. J. Carroll, Constrained maximum likelihood estimation for model calibration using summary-level information from external big data sources, Journal of the American Statistical Association 111 (513) (2016) 107–117.
  * [41] EN 13848-5 (2008). Railway applications - Track - Track geometry quality - Part 5: Geometric quality levels.
  * [42] J.-Y. Keller, M. Darouach, Optimal two-stage kalman filter in the presence of random bias, Automatica 33 (9) (1997) 1745–1748.
  * [43] R. Anderson, D. M. Bevly, Using gps with a model-based estimator to estimate critical vehicle states, Vehicle System Dynamics 48 (12) (2010) 1413–1438.

## Appendix A
Kalman filter for the orientation: matrices of the state space model

$\textbf{z}=\left[\begin{array}[]{*{20}{c}}\hat{\omega}_{x}&\hat{\omega}_{y}&\hat{\omega}_{z}&{a_{imu}^{corr}|}_{x}&{a_{imu}^{corr}|}_{y}&\psi^{t}\end{array}\right]^{T}$

$\textbf{F}=\begin{bmatrix}\textbf{F}_{1}&\textbf{0}&\textbf{0}\\\
\textbf{0}&\textbf{F}_{1}&\textbf{0}\\\
\textbf{0}&\textbf{0}&\textbf{F}_{1}\end{bmatrix},\quad\textrm{with}\quad\textbf{F}_{1}=\begin{bmatrix}1&\mathrm{\Delta}t\\\
0&1\end{bmatrix}$

$\textbf{H}=\begin{bmatrix}0&1&-\hat{\omega}_{z}&0&0&0\\\
\hat{\omega}_{z}&0&0&1&0&0\\\ -\hat{\omega}_{y}&0&0&0&0&1\\\ 0&0&-g&0&0&0\\\
g&0&0&0&0&0\\\ 0&0&0&0&1&0\end{bmatrix}$

$\textbf{Q}=\begin{bmatrix}q_{\varphi}\textbf{Q}_{1}&\textbf{0}&\textbf{0}\\\
\textbf{0}&q_{\theta}\textbf{Q}_{1}&\textbf{0}\\\
\textbf{0}&\textbf{0}&q_{\psi}\textbf{Q}_{1}\end{bmatrix},\quad\textrm{with}\quad\textbf{Q}_{1}=\begin{bmatrix}\mathrm{\Delta}t^{3}/3&\mathrm{\Delta}t^{2}/2\\\
\mathrm{\Delta}t^{2}/2&\mathrm{\Delta}t\end{bmatrix}$

$\textbf{R}=\begin{bmatrix}R_{\omega}&0&0&0&0&0\\\ 0&R_{\omega}&0&0&0&0\\\
0&0&R_{\omega}&0&0&0\\\ 0&0&0&R_{ax}&0&0\\\ 0&0&0&0&R_{ay}&0\\\
0&0&0&0&0&R_{\psi}\\\ \end{bmatrix}$

## Appendix B
Kalman filter for the trajectory: matrices of the state space model

$\textbf{z}=\begin{bmatrix}a_{y}^{imu}+a_{x}^{imu}{\psi^{t,b}}-a_{z}^{imu}{\varphi^{t,b}}-g{\varphi^{t}}-{\rho_{h}}{V^{2}}\\\
a_{z}^{imu}-a_{x}^{imu}{\theta^{t,b}}+a_{y}^{imu}{\varphi^{t,b}}-g+{\rho_{v}}{V^{2}}\\\
0\\\ \delta\end{bmatrix}$

$\textbf{F}=\begin{bmatrix}\textbf{F}_{1}&\textbf{0}\\\
\textbf{0}&\textbf{F}_{1}\\\
\end{bmatrix},\quad\textrm{with}\quad\textbf{F}_{1}=\begin{bmatrix}1&\mathrm{\Delta}t&\mathrm{\Delta}t^{2}/2\\\
0&1&\mathrm{\Delta}t\\\ 0&0&1\end{bmatrix}$

$\textbf{H}=\begin{bmatrix}\textbf{H}_{1}&\textbf{H}_{2}\end{bmatrix}$

$\textbf{H}_{1}=\begin{bmatrix}-V^{2}(\rho_{tw}^{2}+\rho_{h}^{2})&0&1\\\
\rho_{v}\rho_{h}V^{2}+\rho_{tw}\dot{V}&2\rho_{tw}V&0\\\ 1&0&0\\\
0&0&0\end{bmatrix}$

$\textbf{H}_{2}=\begin{bmatrix}\rho_{v}\rho_{h}V^{2}-\rho_{tw}\dot{V}&-2\rho_{tw}V&0\\\
-V^{2}(\rho_{tw}^{2}+\rho_{h}^{2})&0&1\\\ 0&0&0\\\ 1&0&0\end{bmatrix}$

$\textbf{Q}=\begin{bmatrix}q_{y}\textbf{Q}_{1}&\textbf{0}\\\
\textbf{0}&q_{z}\textbf{Q}_{1}\\\
\end{bmatrix},\quad\textrm{with}\quad\textbf{Q}_{1}=\begin{bmatrix}\mathrm{\Delta}t^{5}/20&\mathrm{\Delta}t^{4}/8&\mathrm{\Delta}t^{3}/6\\\
\mathrm{\Delta}t^{4}/8&\mathrm{\Delta}t^{3}/3&\mathrm{\Delta}t^{2}/2\\\
\mathrm{\Delta}t^{3}/6&\mathrm{\Delta}t^{2}/2&\mathrm{\Delta}t\end{bmatrix}$

$\textbf{R}=\begin{bmatrix}R_{y1}&0&0&0\\\ 0&R_{z1}&0&0\\\ 0&0&R_{y2}&0\\\
0&0&0&R_{z2}\end{bmatrix}$

## Appendix C
Kalman filter for the orientation and trajectory: matrices of the state space
model

$\textbf{z}=\begin{bmatrix}\hat{\omega}_{x}\\\ \hat{\omega}_{y}\\\
\hat{\omega}_{z}\\\
a_{x}^{imu}+\theta^{t}(g-a_{z}^{imu})+a_{y}^{imu}\psi^{t}-\dot{V}\\\
a_{y}^{imu}+{\varphi^{t}}(a_{z}^{imu}-g)-a_{x}^{imu}{\psi^{t}}-{\rho_{h}}{V^{2}}\\\
a_{z}^{imu}-g-a_{y}^{imu}{\varphi^{t}}+a_{x}^{imu}{\theta^{t}}+{\rho_{v}}{V^{2}}\\\
0\\\ \delta\end{bmatrix}$

$\textbf{F}=\begin{bmatrix}\textbf{F}_{1}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}\\\
\textbf{0}&\textbf{F}_{1}&\textbf{0}&\textbf{0}&\textbf{0}\\\
\textbf{0}&\textbf{0}&\textbf{F}_{1}&\textbf{0}&\textbf{0}\\\
\textbf{0}&\textbf{0}&\textbf{0}&\textbf{F}_{2}&\textbf{0}\\\
\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{F}_{2}\\\ \end{bmatrix}$

$\textbf{F}_{1}=\begin{bmatrix}1&\mathrm{\Delta}t\\\
0&1\end{bmatrix},\qquad\textbf{F}_{2}=\begin{bmatrix}1&\mathrm{\Delta}t&\mathrm{\Delta}t^{2}/2\\\
0&1&\mathrm{\Delta}t\\\ 0&0&1\end{bmatrix}$

$\textbf{H}=\begin{bmatrix}\textbf{H}_{1}&\textbf{H}_{2}&\textbf{H}_{3}\end{bmatrix}$

$\textbf{H}_{1}=\begin{bmatrix}0&1&-\hat{\omega}_{z}&0&0&0\\\
\hat{\omega}_{z}&0&0&1&0&0\\\ -\hat{\omega}_{y}&0&0&0&0&1\\\
0&0&-a_{z}^{imu}&0&a_{y}^{imu}&0\\\ a_{z}^{imu}&0&0&0&-a_{x}^{imu}&0\\\
-a_{y}^{imu}&0&a_{x}^{imu}&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
\end{bmatrix}$

$\textbf{H}_{2}=\begin{bmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\\\
V^{2}(\rho_{tw}\rho_{v}-\rho^{\prime}_{h})-\rho_{h}\dot{V}&-2\rho_{h}V&0\\\
-V^{2}(\rho_{tw}^{2}+\rho_{h}^{2})&0&1\\\
\rho_{v}\rho_{h}V^{2}+\rho_{tw}\dot{V}&2\rho_{tw}V&0\\\ 1&0&0\\\
0&0&0\end{bmatrix}$

$\textbf{H}_{3}=\begin{bmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\\\
\rho_{v}\dot{V}+\rho_{tw}\rho_{h}V^{2}&2\rho_{v}V&0\\\
\rho_{v}\rho_{h}V^{2}-\rho_{tw}\dot{V}&-2\rho_{tw}V&0\\\
-V^{2}(\rho_{tw}^{2}+\rho_{h}^{2})&0&1\\\ 0&0&0\\\ 1&0&0\end{bmatrix}$

$\textbf{Q}=\begin{bmatrix}q_{\varphi}\textbf{Q}_{1}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}\\\
\textbf{0}&q_{\theta}\textbf{Q}_{1}&\textbf{0}&\textbf{0}&\textbf{0}\\\
\textbf{0}&\textbf{0}&q_{\psi}\textbf{Q}_{1}&\textbf{0}&\textbf{0}\\\
\textbf{0}&\textbf{0}&\textbf{0}&q_{y}\textbf{Q}_{2}&\textbf{0}\\\
\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&q_{z}\textbf{Q}_{2}\end{bmatrix}$

$\textbf{Q}_{1}=\begin{bmatrix}\mathrm{\Delta}t^{3}/3&\mathrm{\Delta}t^{2}/2\\\
\mathrm{\Delta}t^{2}/2&\mathrm{\Delta}t\end{bmatrix},\qquad\textbf{Q}_{2}=\begin{bmatrix}\mathrm{\Delta}t^{5}/20&\mathrm{\Delta}t^{4}/8&\mathrm{\Delta}t^{3}/6\\\
\mathrm{\Delta}t^{4}/8&\mathrm{\Delta}t^{3}/3&\mathrm{\Delta}t^{2}/2\\\
\mathrm{\Delta}t^{3}/6&\mathrm{\Delta}t^{2}/2&\mathrm{\Delta}t\end{bmatrix}$

$\textbf{R}=\begin{bmatrix}R_{\omega}&0&0&0&0&0&0&0\\\
0&R_{\omega}&0&0&0&0&0&0\\\ 0&0&R_{\omega}&0&0&0&0&0\\\ 0&0&0&R_{x}&0&0&0&0\\\
0&0&0&0&R_{y1}&0&0&0\\\ 0&0&0&0&0&R_{z1}&0&0\\\ 0&0&0&0&0&0&R_{y2}&0\\\
0&0&0&0&0&0&0&R_{z2}\end{bmatrix}$


# On the spin content of the classical massless Rarita–Schwinger system

Mauricio Valenzuela1,2 and Jorge Zanelli1,2
1Centro de Estudios Científicos (CECs), Arturo Prat 514, Valdivia, Chile
and
2Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián,
Valdivia, Chile

###### Abstract

We analyze the Rarita–Schwinger (RS) massless theory in the Lagrangian and
Hamiltonian approaches. At the Lagrangian level, the standard gamma-trace
gauge fixing constraint leaves a spin–$\tfrac{1}{2}$ and a spin–$\tfrac{3}{2}$
propagating Poincaré group helicities. At the Hamiltonian level, the result
depends on whether the Dirac conjecture–that all first class constraints
generate gauge symmetries–is assumed or not. In the affirmative case, a
secondary first class constraint is added to the total Hamiltonian and a
corresponding gauge fixing condition must be imposed, completely removing the
spin–$\tfrac{1}{2}$ sector. In the opposite case, the spin–$\tfrac{1}{2}$
field propagates and the Hamilton field equations match the Euler-Lagrange
equations.

##  1 Introduction

In 1939, Markus Fierz and Wolfgang Pauli discussed the obstacles in the
attempt to quantize fields of arbitrary spin $\geq 1$ interacting with photons
[1]. Two years later, William Rarita and Julian Schwinger simplified the
Fierz-Pauli treatment, writing down a set of field equations describing
fermions of arbitrary spin $\geq 3/2$ [2]. The Rarita-Schwinger system (RS)
describes a field of spin $k+1/2$ as a tensor-spinor of rank $k$,
$\psi^{\alpha}_{\mu_{1}\cdots\mu_{k}}$, symmetric in its tensor indices
$\mu_{1}\cdots\mu_{k}$, satisfying a Dirac-like field equation with mass,

$(\not{\partial}+M)\psi_{\mu_{1}\cdots\mu_{k}}=0\;,\qquad\gamma^{\mu}\,\psi_{\mu\mu_{2}\cdots\mu_{k}}=0\,.$
(1)

The subsidiary conditions $\partial^{\mu}\,\psi_{\mu\mu_{2}\cdots\mu_{k}}=0$
(transverse) and $\psi^{\mu}{}_{\mu\mu_{2}\cdots\mu_{k}}=0$ (traceless),
appear as consequence of (1) for $M\neq 0$. In the spin–$\tfrac{3}{2}$ case,
of a vector-spinor $\psi^{\alpha}_{\mu}$, Rarita and Schwinger also noted that
there is a class of Lagrangians parametrized by the mass ($M$) and a
dimensionless coefficient ($A$) that gives rise to the equations (1) (see e.g.
[3, 4, 5]). Then they chose some $A$ “which permits a relatively simple
expression of the equations of motion in the presence of electromagnetic
fields”.

The description of spin–$\tfrac{3}{2}$ particles adopted in supergravity,
however, traditionally referred to as the Rarita–Schwinger Lagrangian [6, 7]
(see also [8] and references therein) corresponds to a different choice of $A$
which in the massless limit gives the Lagrangian111Here $\gamma_{\mu}$,
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\eta_{\mu\nu}$, are Dirac matrices,
$\eta_{\mu\nu}=\texttt{diag}\,(-1,1,\dots)$ and
$\gamma_{\mu\cdots\nu}=\gamma_{[\mu}\cdots\gamma_{\nu]}$ are completely
antisymmetric products. We assume the Majorana reality condition
$\psi^{\dagger}=\psi$, $\bar{\psi}=\psi^{t}C$, were $C^{t}=-C$.

$\mathcal{L}:=-\frac{i}{2}\bar{\psi}_{\mu}\gamma{}^{\mu\nu\lambda}\partial_{\nu}\psi_{\lambda}\,,$
(2)

whose corresponding field equations are (see [10, 9] for the chiral spinor
version of these equations)

$\gamma{}^{\mu\nu\lambda}\partial_{\nu}\psi_{\lambda}=0\,.$ (3)

We emphasize that (2) is not equivalent to massless limit of the original RS
action since, as shown below, the $\gamma$-trace condition arises as a gauge
choice and not as consequence of the field equations.

The action changes by a boundary term and (3) is invariant under the gauge
transformation

$\delta\psi_{\mu}=\partial_{\mu}\epsilon\,.$ (4)

Eq. (3) can also be written as

$\not{\partial}\psi_{\mu}-\partial_{\mu}\,\gamma\cdot\psi=0\;,\qquad\partial\cdot\psi-\not{\partial}\gamma\cdot\psi=0\,,$
(5)

and, in the $\gamma^{\mu}\psi_{\mu}=0$ gauge, as

$\not{\partial}\psi_{\mu}=0\,,\qquad\partial^{\mu}\psi_{\mu}=0\,,\qquad\gamma^{\mu}\psi_{\mu}\overset{\scriptscriptstyle
gf}{=}0\,.$ (6)

The third equation corresponds to the gauge choice that fixes the freedom (4),
where the symbol $\overset{\scriptscriptstyle gf}{=}$ reflects this. In the
massive RS system, $\partial^{\mu}\psi_{\mu}=0$ is a consistency condition of
the field equations, hence (6) can be obtained from the massless limit of the
original RS equations.222This is analogous to the transverse condition
$\partial^{\mu}A_{\mu}=0$ that is required by consistency of the Proca
equations but is only a gauge option in Maxwell’s theory., however (6) can not
be obtained by direct variation of the massless action.

The RS field $\psi_{\mu}$ belongs to the reducible representation
$\frac{3}{2}\oplus\frac{1}{2}$ of the Lorentz group and it can be split into
its irreducible parts as $\psi_{\mu}:=\rho_{\mu}+\gamma_{\mu}\kappa$, where
the spin-3/2 part $\rho_{\mu}$ is gamma-traceless, $\gamma^{\mu}\rho_{\mu}=0$,
and $\kappa$ represents the spin-1/2. Then the Euler-Lagrange equations (5)
read

$(D-1)\not{\partial}\kappa-\partial^{\mu}\rho_{\mu}=0\;,\quad\quad\not{\partial}\rho_{\mu}-\gamma_{\mu}\not{\partial}\kappa-(D-2)\partial_{\mu}\kappa=0\,.$
(7)

Using the gauge freedom (4) it is possible to make $\rho_{\mu}$, not only
gamma-traceless but also divergence-free. Hence, the first equation reduces to
the massless Dirac equation for the spin–$\tfrac{1}{2}$ field $\kappa$, while
the second becomes a Dirac equation for the massless spin–$\tfrac{3}{2}$ with
source $\partial_{\mu}\kappa$. This shows that at least in the gauge
$\partial^{\mu}\rho_{\mu}=0$ both spin sectors seem to propagate.

Another way to see that the RS may propagate is the fact that in the vacuum of
the spin–$\tfrac{3}{2}$ field $\rho_{\mu}=0$ the RS action produces the Dirac
action,

$\mathcal{L}:=i\frac{(D-1)(D-2)}{2}\;\bar{\kappa}\not{\partial}\kappa\,.$ (8)

where $D$ is the number of spacetime dimensions.

Though this result might seem straightforward, it should be unexpected if the
spin–$\tfrac{1}{2}$ is pure gauge. One would expect a trivial action
principle, as it happens in standard gauge theories. In this paper we look for
an explanation to this problem.

By excellence, Hamiltonian analysis is the standard framework for elucidating
what are the degrees of freedom of gauge systems. We shall see that either
result can be obtained depending on whether or not the validness of the Dirac
conjecture—which says that all first class constraints are gauge generators—is
assumed. This is a technical observation: if the Dirac conjecture is not
assumed, and a gauge fixing condition is impossed only for the primary first
class constraint, then the spin–$\tfrac{1}{2}$ field propagates. Otherwise,
the sum of the secondary first class constraint to the extended Hamiltonian
introduce a new arbitrary function of time (the corresponding Lagrange
multiplier) which, in order to produce a deterministic system of equations,
requires an additional gauge fixing condition, which removes the
spin–$\tfrac{1}{2}$ field, in agreement with [8, 14, 15, 13, 16].

We shall see exactly in which step of the Dirac algorithm the two branches are
generated: the one in which the spin–$\tfrac{1}{2}$ sector remains and the
other where it is removed.

##  2 Space and time splitting

In terms of Poincaré group the vector-spinor $\psi^{\alpha}_{\mu}$ can be
decomposed in irreducible representation of spins $(1\oplus
0)\otimes\frac{1}{2}=\frac{3}{2}\oplus\frac{1}{2}\oplus\frac{1}{2}$ [8]. Hence
the RS action should be regarded has a spin–$\tfrac{1}{2}$ and
spin–$\tfrac{3}{2}$ particle systems. The Poincaré spin split can be achieved
explicitly in terms of Ogievetsky-Sokatchev projectors [11, 8], which involve
nonlocal operators. These projectors reveals the gauge invariant components of
the RS action and it produces a decoupled system of spin–$\tfrac{3}{2}$ and
spin–$\tfrac{1}{2}$ governed by Dirac kinetic terms, in agreement with the
discussion following (7) (for further details see [17]). The second
spin–$\tfrac{1}{2}$ mode is the pure gauge mode, as it belongs to the kernel
of the RS kinetic operator off-shell. The analogous treatment of the Maxwell
theory would introduce the transverse (spin-one) and the longitudinal (spin-0)
projectors, of which only the transverse mode would propagate.

Non-locality along time directions, however, may be problematic: they might be
incompatible with the integration of the field equations under initial
conditions on a Cauchy surface. Thus another method is necessary to analyze
the problem. Spatial nonlocality, on the other hand, is compatible with the
Cauchy data because it leaves the initial surface intact and therefore gauge
transformations or field redefinitions involving nonlocal spatial operators
such as $\not{\nabla}^{-1}:=(\gamma^{i}\,\partial_{i})^{-1}$, should not lead
to inconsistencies.

We first split the vector-spinor $\psi_{\mu}$ into $\psi_{0}$ and $\psi_{i}$,
which is in turn split into three more pieces: the spatial divergence
$\partial^{i}\psi_{i}$, the $\gamma$–trace $\gamma^{i}\psi_{i}$, and a spatial
$\gamma$–traceless and divergenceless vector-spinor $\xi_{i}$
($\gamma^{i}\xi_{i}=0=\partial^{i}\xi_{i}$). Thus we have one
spin–$\tfrac{3}{2}$ field $\xi_{i}$, and three spin–$\tfrac{1}{2}$
representations of the spatial rotation group, $\psi_{0}$,
$\partial^{i}\psi_{i}$ and $\gamma^{i}\psi_{i}$. We shall see that the
$\gamma$–traceless and divergenceless conditions (6) remove one
spin–$\tfrac{1}{2}$ representations of the rotation group each, whilst the
third is a propagating spin–$\tfrac{1}{2}$ irreducible representation of the
Poincaré group.

Consider the decomposition of the identity of vector-spinors in three
orthogonal projectors, $\mathds{1}=P^{{\perp\\!\\!\\!\perp}}+P^{N}+P^{L}$,

$(P^{N}{})_{ij}:=\frac{1}{D-2}N_{i}N_{j}\,,\qquad(P^{L}{})_{ij}:=L_{i}L_{j},\qquad
P^{{\perp\\!\\!\\!\perp}}=\mathds{1}-{P}^{N}-P^{L}\,,$ (9)

where $N_{i}:=\gamma_{i}-L_{i}$ and $L_{i}:=\not{\nabla}^{-1}\partial_{i}$.
These are space-like Ogievetsky–Sokatchev projectors that decompose the
spatial vector-spinor as

$\psi_{i}=\xi_{i}+N_{i}\zeta+L_{i}\lambda,\quad\mbox{where}\quad\xi_{i}=P^{{\perp\\!\\!\\!\perp}}{}_{i}{}^{j}\psi_{j}\,,\quad\zeta=\frac{1}{D-2}N^{i}\psi_{i}\,,\quad\lambda=L^{i}\psi_{i}\,,$
(10)

which can be verified with the help of the identities $N_{i}\,N^{i}=D-2$,
$L_{i}\,L^{i}=1$, $N_{i}\,L^{i}=0$.

The gamma traceless relation is equivalent to

$\psi_{0}=-\gamma_{0}\gamma^{i}\psi_{i}=-\gamma_{0}((D-2)\zeta+\lambda)\,,$
(11)

which, together with the decomposition (10), reduce (6) to

$\not{\partial}\xi_{i}=0\,,\qquad\not{\partial}\tilde{\lambda}=0\,,\qquad\dot{\zeta}=0\,,\qquad\not{\nabla}\zeta=0\,,$
(12)

where $\tilde{\lambda}=\gamma^{0}\lambda$. Thus the explicit solution of (6)
containing the spin–$\tfrac{1}{2}$ and spin–$\tfrac{3}{2}$ sector is,

$\psi_{0}=\gamma^{0}\lambda\,,\qquad\psi_{i}=\xi_{i}+\partial_{i}\,\not{\nabla}^{-1}\lambda\,,$
(13)

given in terms of standard solutions of the Dirac equations (12), restricted
by double-transverse condition $\gamma^{i}\xi_{i}=\partial^{i}\xi_{i}=0$ [12,
13]. It follows that fields $\xi_{i}$ and $\lambda$ propagate, whilst $\zeta$
is a constant spinor –and must therefore vanish on-shell–, and $\psi_{0}$ is
not an independent field. Hence, the massless RS equations in the gauge
$\gamma^{\mu}\psi_{\mu}=0$ eliminates two spin–$\tfrac{1}{2}$ modes,
$\psi_{0}$ and $\zeta$, whilst one spin–$\tfrac{1}{2}$ and one
spin–$\tfrac{3}{2}$ field propagate. Thus, there are no arbitrary functions of
time left in the system: the dynamical equations (12) completely determine the
evolution of the fields, provided initial data is given on a Cauchy surface.

The number of degrees of freedom in the system is defined by one half the
number of functions in the set
$\\{\xi^{\alpha}_{i}(t_{0},\vec{x}),\lambda^{\alpha}(t_{0},\vec{x})\\}$
necessary to specify the evolution. These functions are: $k\times(D-1)-2k$
components of $\xi_{i}^{\alpha}$, $\alpha=1,\dots,k$, and $k=2^{[D/2]}$
components in $\lambda^{\alpha}$; in both cases the Dirac equation restricts
half of them. In total, we are left with $k(D-2)/2$ degrees of freedom which
are equivalent to two massless states of spin–$\tfrac{1}{2}$ and
spin–$\tfrac{3}{2}$ , respectively.

It is relevant now to discuss the picture in Dirac’s time honored Hamiltonian
analysis [18].

##  3 Hamiltonian analysis

Splitting $\psi_{\mu}$ as in (10) one obtains, up to boundary terms,

$\mathcal{L}=-i\bar{\psi}_{0}\gamma^{0ij}\partial_{i}\psi_{j}+\frac{i}{2}\bar{\psi}_{i}\gamma^{0ij}\dot{\psi}_{j}-\frac{i}{2}\bar{\psi}_{i}\gamma^{ijk}\partial_{j}\psi_{k}\,.$
(14)

The definition of momenta,
$\pi^{\mu}:=\partial\mathcal{L}/\partial\dot{\psi}_{\mu}$, yields the primary
constraints

$\displaystyle\pi^{0}\approx 0\,,$ (15)
$\displaystyle\chi^{i}:=\pi^{i}-\frac{i}{2}\mathcal{C}^{ij}{\psi}_{j}\,\approx
0\,,$ (16)

where
$\mathcal{C}^{ij}_{\alpha\beta}:=-(C\gamma^{0ij})_{\alpha\beta}=\mathcal{C}^{ji}_{\beta\alpha}$
is invertible,
$\mathcal{C}^{ij}_{\alpha\beta}(\mathcal{C}^{-1})_{jm}^{\beta\kappa}:=\delta^{i}_{m}\delta_{\alpha}^{\kappa}$,

$(\mathcal{C}^{-1})_{ij}^{\alpha\beta}=\Big{(}-\frac{1}{(D-2)}\gamma_{i}\gamma_{j}\gamma_{0}C^{-1}+\delta_{ij}\gamma_{0}C^{-1}\Big{)}^{\alpha\beta}\,.$
(17)

The constraint (15) states that $\psi_{0}$ is a Lagrange multiplier and (16)
is a consequence of the first order character of the system. The Hamiltonian,
including a linear combination of the primary constraints is

$\displaystyle H=\int
d^{\scriptscriptstyle{D-1}}x\,\Big{(}i\bar{\psi}_{0}\gamma^{0ij}\partial_{i}\psi_{j}+\frac{i}{2}\bar{\psi}_{i}\gamma^{ijk}\partial_{j}\psi_{k}+\chi^{i}_{\alpha}\mu_{i}^{\alpha}+\pi^{0}_{\alpha}\mu_{0}^{\alpha}\Big{)}\,,$
(18)

where $\mu_{i}$ and $\mu_{0}$ are arbitrary spinorial Lagrange multipliers.
Preservation in time of the primary constraint $\pi^{0}\approx 0$ yields a
secondary constraint,

$\dot{\pi}^{0}=\\{\pi^{0},H\\}=-\frac{\delta
H}{\delta\psi_{0}}=-iC\gamma^{0ij}\partial_{i}\psi_{j}\approx
0\quad\Leftrightarrow\quad\varphi:=-i\mathcal{C}^{ij}\partial_{i}\psi_{j}\approx
0\,,$ (19)

which is equivalent to the equation of motion obtained varying (14) with
respect to $\psi_{0}$. Preservation in time of the other primary constraints,

$\dot{\chi}^{i}=-(C\gamma^{i}\gamma_{0}C^{-1})\varphi+i\mathcal{C}^{ij}\partial_{j}\psi_{0}^{\beta}+iC\partial^{i}\gamma^{j}\psi_{j}-iC\not{\nabla}\psi^{i}+i\mathcal{C}^{ij}\mu_{j}\approx
0\,,$ (20)

and of the secondary one,

$\dot{\varphi}=i\mathcal{C}^{ij}\partial_{i}\mu_{j}\approx 0\,,$ (21)

yield conditions that determine the Lagrange multipliers $\mu_{i}$ in terms of
the phase space fields, and there are no further constraints. It is easily
checked that $\chi^{i}$ is second class, while $\pi^{0}$ and the linear
combination of constraints

$\tilde{\varphi}:=\varphi+i\partial_{i}\chi^{i}\approx 0\,,$ (22)

are first class. The second class constraint $\chi^{i}$ will be eventually
dropped, leaving $\pi^{0}\approx 0$ and $\varphi\approx 0$ as the only
remaining first class constraints.

It should be noticed that the secondary constraint $\varphi$ is not purely
first class. In particular (21) fixes part of the Lagrange multipliers of the
system, whilst $\tilde{\varphi}$ mixes primary and secondary constraints. This
will be relevant for the discussion of the Dirac conjecture.

The system can be reduced to the surface of the second class constraints (16)
by strongly setting $\chi^{i}=0$ and replacing Poisson by Dirac brackets,
$\\{f\,,g\\}_{D}:=\\{f\,,g\\}-\\{f\,,\chi^{i}_{\alpha}\\}\,\mathcal{C}^{-1}{}_{ij}^{\alpha\beta}\\{\chi^{j}_{\beta}\,,g\\}$,
which in the variables $\psi_{0}$, $\pi^{0}$, $\xi$, $\zeta$ and $\lambda$,
reads

$\displaystyle\\{f\,,g\\}_{D}$ $\displaystyle=$ $\displaystyle(-1)^{f}\int
d^{\scriptscriptstyle{D-1}}z\left[-i\frac{\delta
f}{\delta\xi_{i}}\;P^{{\perp\\!\\!\\!\perp}}_{ij}\gamma_{0}C^{-1}\frac{\delta
g}{\delta\xi_{j}}-i\frac{D-3}{D-2}\frac{\delta
f}{\delta\lambda}\gamma_{0}C^{-1}\frac{\delta g}{\delta\lambda}\right.$ (23)
$\displaystyle\left.+i\frac{1}{D-2}\Big{(}\frac{\delta
f}{\delta\lambda}\gamma_{0}C^{-1}\frac{\delta g}{\delta\zeta}+\frac{\delta
f}{\delta\zeta}\gamma_{0}C^{-1}\frac{\delta
g}{\delta\lambda}\Big{)}+\Big{(}\frac{\delta
f}{\delta\psi_{0}^{\alpha}}\frac{\delta
g}{\delta\pi^{0}_{\alpha}}+\frac{\delta f}{\delta\pi^{0}_{\alpha}}\frac{\delta
g}{\delta\psi_{0}^{\alpha}}\Big{)}\right]\,,$

and the first class Hamiltonian (18) reduces to

$\displaystyle H_{1}=\int
d^{\scriptscriptstyle{D-1}}x\,\Big{(}i(D-2)\bar{\psi}_{0}\gamma^{0}\not{\nabla}\zeta-\frac{i\displaystyle{(D-2)(D-3)}}{2}\bar{\zeta}\not{\nabla}\zeta+\frac{i}{2}\bar{\xi}^{i}\not{\nabla}\xi_{i}+\pi^{0}_{\alpha}\mu_{0}^{\alpha}\Big{)}\,,$
(24)

where the first class secondary constraint $\varphi\approx 0$ is equivalent to
$\not{\nabla}\zeta\approx 0$.

The question now is whether one should add this secondary first class
constraint to the Hamiltonian as an independent gauge generator. This is
equivalent to asking whether the Dirac conjecture (DC) holds in this case,
namely, whether all secondary first class constraints generate gauge
transformations. If the conjecture is valid, the gauge transformations
generated by $\varphi$ would require gauge fixing; if that is not the case,
$\varphi$ does not generate gauge transformations, it should not be included
in the Hamiltonian and no gauge fixing would be required.

One can examine the effect of adding $\varphi$ to the Hamiltonian (24) with a
Lagrange multiplier. The time evolution defined by
$\dot{f}=\\{f\,,H^{\prime}\\}$, with respect to the extended Hamiltonian
$H^{\prime}:=H_{1}+\tau^{\alpha}\varphi_{\alpha}$, is

$\displaystyle\dot{\xi}_{i}$ $\displaystyle=$
$\displaystyle-\gamma_{0}\not{\nabla}\xi_{i}\,,\qquad\dot{\lambda}=-(D-3)\gamma_{0}\not{\nabla}\zeta+\not{\nabla}\psi_{0}+\not{\nabla}\tau\,,$
(25) $\displaystyle\dot{\psi}_{0}$ $\displaystyle=$
$\displaystyle-\mu_{0}\,,\qquad\pi^{0}=0\,,\qquad\dot{\zeta}=0,\qquad\not{\nabla}\zeta=0\,.$
(26)

The gauge symmetry generated by $\pi^{0}$ is fixed by specifying $\psi_{0}$,
which can be chosen to implement the standard $\gamma$-traceless condition in
(6) as $\psi_{0}+\gamma_{0}\gamma^{i}\psi_{i}\approx 0$. This, together with
$\pi^{0}\approx 0$, form a pair of second class constraints that can be
readily eliminated from the phase space. This gauge choice is accessible since
$\pi^{0}$ generates arbitrary shifts in $\psi_{0}$ and, in particular, the
shift $\delta\psi_{0}=-(\psi_{0}+\gamma_{0}\gamma^{i}\psi_{i})$, renders
$\psi_{\mu}^{\prime}=\psi_{\mu}+\delta\psi_{\mu}$ $\gamma$-traceless. Thus in
the phase space spanned by $\xi_{i},\zeta,\lambda$, using (11) reduces the
system (25) to

$\dot{\xi}_{i}=-\gamma_{0}\not{\nabla}\xi_{i}\,,\qquad\dot{\lambda}=\gamma_{0}\not{\nabla}\lambda+\not{\nabla}\tau\,,\qquad\dot{\zeta}=0=\not{\nabla}\zeta\,.$
(27)

Assuming the DC as valid, would imply that the evolution of $\lambda$ is
indeterminate, from the presence of the arbitrary function of time $\tau$, and
therefore an external gauge condition in convolution with the constraint
$\varphi\approx 0$ would be necessary. Choosing the gauge condition
$\lambda\approx 0$, the stationary condition $\dot{\lambda}\approx 0$
determines the Lagrange multipliers, $\tau=0$. This removes the
spin–$\tfrac{1}{2}$ sector and the only propagating field is $\xi_{i}$. In
this case, the Hamilton approach, with the Dirac conjecture assumed to be
valid, do not match the Euler-Lagrange equations (12).

The framework that matches the Lagrangian approach is the one where the DC is
not assumed. Then $\varphi$ would not be regarded as a gauge generator, and it
should not be added to $H_{1}$. This is equivalent to setting $\tau=0$ in
(27), and we reproduce the Euler-Lagrange equations (12) and (7) in the gauge
$\partial^{\mu}\rho_{\mu}=0$, whose solution is given by the
spin–$\tfrac{1}{2}$ –spin–$\tfrac{3}{2}$ system (13).

The two scenarios presented above are consistent. Although in the first case
the resulting Hamiltonian evolution is not equivalent to the Lagrangian
dynamics, it yields a physical subsystem. There are Lagrangian models whose
Hamiltonian formulation leads to secondary first class constraints that do not
generate gauge transformations [19, 20, 21]. For those counterexamples to the
DC it is still possible to postulate the validity of the conjecture without
running into inconsistencies. Moreover it has been argued that not adopting
the Dirac conjecture might lead to problems in the quantization, which
supports the idea that it would be safer to assume the validity of the DC in
general [21].

On the other hand, it seems unnecessary to postulate the DC in our case; the
resulting system is still consistent and in agreement with the Lagrangian
description, and the Dirac bracket (23) does not lead to quantization problems
of the sort found in the counterexample of the DC in [21]: the Dirac field can
be quantized. In addition, in Chapter 3 of [21] the DC is shown to follow from
Dirac’s constrained Hamiltonian analysis for dynamical systems in which first
and second class constraints do not mix in the process. As noted above (22),
this condition does not hold here since the secondary constraint
$\\{H,\pi^{0}\\}=\varphi=\tilde{\varphi}-i\partial_{i}\chi^{i}$ is a linear
combination of first class and second class constraints. Furthermore, the
constraint $\tilde{\varphi}$ (22) is a mixture of a secondary first class
constraint and a second class one. Since it mixes both types, it does not have
the form required by the proof of the DC presented in [21]. For a critical
discussion on the DC see [22, 23].

If the DC is not valid because some secondary first class constraints do not
generate gauge transformations, there is no need to provide a gauge condition
for those constraints, and the standard formula for the counting of degrees of
freedom [25, 24, 21] generalizes as

2$\times\left[\begin{tabular}[]{ c }{\footnotesize Number of}\\\
{\footnotesize d.o.f.}\end{tabular}\right]$ =$\left[\mbox{\begin{tabular}[]{ c
}{\footnotesize Dimension of}\\\ {\footnotesize phase
space}\end{tabular}}\right]$ -$\left[\mbox{\begin{tabular}[]{ c
}{\footnotesize 2nd class}\\\ {\footnotesize
constraints}\end{tabular}}\right]$ -2$\times\left[\mbox{\begin{tabular}[]{ c
}{\footnotesize 1st class}\\\\[-3.0pt] {\footnotesize gauge}\\\\[-3.0pt]
{\footnotesize generators}\end{tabular}}\right]$
-$\left[\mbox{\begin{tabular}[]{ c }{\footnotesize 1st class}\\\\[-3.0pt]
{\footnotesize non-gauge}\\\\[-3.0pt] {\footnotesize
generators}\end{tabular}}\right]$ .

Note that the last term on the right hand side could be odd, leading to a
paradoxical (possibly inconsistent) quantum scenario. However in systems of
spinors, first class constraints have an even number of components and
therefore not necessarily inconsistent. For the RS system in 4 dimensions,
this counting gives $(16\times 2-12-2\times 4-4)/2=4$ degrees of freedom,
which correspond to two spin–$\tfrac{3}{2}$ helicities plus two
spin–$\tfrac{1}{2}$ helicities. In references [8, 14, 15, 13, 16], on the
other hand, the DC is assumed to be valid, concluding that there are only 2
degrees of freedom, those of a massless spin–$\tfrac{3}{2}$ field.

##  4 Conclusions

The apparent presence of a propagating spin–$\tfrac{1}{2}$ mode in the RS
system contradicts the expectation that the spin–$\tfrac{1}{2}$ field is a
pure gauge mode. A dynamical spin–$\tfrac{1}{2}$ mode in RS sounds similar to
the claim that there is a propagating spin-0 field in the Maxwell theory.
However, in contrast to what happens in gauge theories like Maxwell, Yang-
Mills or Chern-Simons when evaluated on a pure gauge configuration like
$A_{\mu}=\Lambda^{-1}\partial_{\mu}\,\Lambda$, the RS action neither vanishes
nor reduces to a boundary term when evaluated on
$\psi_{\mu}=\gamma_{\mu}\zeta$ for a generic $\zeta$. This means that
configurations $\psi_{\mu}=\gamma_{\mu}\zeta$ are not zero-modes of the
action, unlike what happens in gauge theories for pure gauge configurations.
The reduction $\psi_{\mu}=\gamma_{\mu}\zeta$ is precisely what is done in
unconventional supersymmetry [26, 27, 28, 29, 30], while in supergravity the
complementary option is selected by imposing $\gamma^{\mu}\psi_{\mu}=0$ [8].

As for quantization issues, the spin–$\tfrac{3}{2}$ sector of the massless RS
field has been quantized in various approaches [14, 15, 16]. In all of them,
both spin–$\tfrac{1}{2}$ sectors of the Poincaré group decomposition are
factored out. Following reference [31]—where it is shown that the massless RS
field decomposes in a spin–$\tfrac{1}{2}$ (pure gauge) sector with 0-norm, and
spin–$\tfrac{1}{2}$ and spin–$\tfrac{3}{2}$ sectors of positive norm—the
massless RS can be quantized à la Gupta-Bleuler factoring out only the zero
norm state.

So far we have assumed a flat spacetime, although the generalization to a
curved background is straightforward. In the light of these results, it would
be interesting to consider supergravity theories without enforcing the
validity of the Dirac conjecture, which must contain a spin–$\tfrac{1}{2}$
excitation along with the gravitino. The spin–$\tfrac{1}{2}$ sector will
inherit the gravity and gauge interactions of the vector spinor, which would
generate new supergravity phenomenology.

## Acknowledgements

We warmly thank discussions with L. Andrianopoli, G. Bossard, N. Boulanger, C.
Bunster, A. Marrani, R. Matrecano, R. Noris, M. Rausch de Traubenberg, P.
Sundell, M. Trigiante for their challenging questions, comments and
suggestions. This work was partially funded by grant FONDECYT 1220862 and USS-
VRID project VRID-INTER22/10.

## References

  * [1] M. Fierz and W. Pauli. On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. Roy. Soc. Lond. A173 211–232 (1939).
  * [2] W. Rarita and J. Schwinger. On a theory of particles with half-integral spin, Phys.Rev. 60 61 (1941).
  * [3] K. Johnson and E. C. G. Sudarshan. Inconsistency of the local field theory of charged spin 3/2 particles, Annals Phys. 13 126–145 (1961).
  * [4] L. M. Nath, B. Etemadi, and J. D. Kimel. Uniqueness of the interaction involving spin 3/2 particles. Phys. Rev. D3 2153–2161 (1971).
  * [5] M. Benmerrouche, R. M. Davidson, and N. C. Mukhopadhyay. Problems of Describing Spin 3/2 Baryon Resonances in the Effective Lagrangian Theory. Phys. Rev. C39 2339–2348 (1989).
  * [6] D. Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara. Progress toward a theory of supergravity, Phys. Rev. D13 3214–3218 (1976).
  * [7] S. Deser and B. Zumino. Consistent supergravity, Phys. Lett. B62 335 (1976).
  * [8] P. van Nieuwenhuizen. Supergravity, Phys. Rept. 68 189 (1981).
  * [9] M. Rausch de Traubenberg and M. Valenzuela. A Supergravity Primer: From Geometrical Principles to the Final Lagrangian, World Scientific Press, 2020 ISBN 978-981-12-1051-8, 978-981-12-1053-2 doi10.1142/11557
  * [10] R. Ducrocq, M. Rausch De Traubenberg and M. Valenzuela, A pedagogical discussion of $N=1$ four-dimensional supergravity in superspace, Mod. Phys. Lett. A 36 (2021) no.16, 2130015 doi10.1142/S0217732321300159 [arXiv:2104.06671 [hep-th]].
  * [11] V. I. Ogievetsky and E. S. Sokatchev. Equations of Motion for Superfields. In 4th International Conference on Nonlocal Quantum Field Theory 183–203 (1976).
  * [12] A. K. Das and D. Z. Freedman, Gauge Quantization for Spin 3/2 Fields, Nucl. Phys. B 114 (1976), 271-296 doi10.1016/0550-3213(76)90589-7
  * [13] S. Deser, J. H. Kay, and K. S. Stelle, Hamiltonian Formulation of Supergravity, Phys. Rev. D16 2448 (1977).
  * [14] G. Senjanovic, Hamiltonian Formulation and Quantization of the Spin 3/2 Field, Phys. Rev. D16 307 (1977).
  * [15] M. Pilati, The Canonical Formulation of Supergravity, Nucl. Phys. B132 138–154 (1978).
  * [16] E. S. Fradkin and Mikhail A. Vasiliev, Hamiltonian Formalism, Quantization and S Matrix for Supergravity, Phys. Lett. B72 70–74 (1977).
  * [17] M. Valenzuela and J. Zanelli, On the propagating modes of the Rarita–Schwinger system, in preparation.
  * [18] P. A. M. Dirac, Lectures on Quantum Mechanics, Dover, New York, 2001.
  * [19] R. Cawley, Determination of the Hamiltonian in the Presence of Constraints, Phys.Rev.Lett. 42 (1979) 413.
  * [20] M. J. Gotay, On the validity of Dirac’s conjecture regarding first-class secondary constraints, J.Phys. A16 (1983) L141.
  * [21] M. Henneaux and C. Teitelboim, Quantization of gauge systems. Princeton University Press, New Jersey, 1992.
  * [22] J.M. Pons, On Dirac’s incomplete analysis of gauge transformations, Stud. Hist. Phil. Sci.. B36 491–518 (2005).
  * [23] J. Earman, Tracking down gauge: An ode to the constrained Hamiltonian formalism. In Katherine Brading & Elena Castellani (eds.), Symmetries in Physics: Philosophical Reflections. Cambridge University Press. 140–62 (2003).
  * [24] A. J. Hanson, T. Regge, and Claudio T. Constrained Hamiltonian Systems. Accademia Nazionale dei Lincei, Rome, 1976.
  * [25] M. Henneaux, C. Teitelboim, and J. Zanelli, Gauge invariance and degree of freedom count, Nucl. Phys. B332 169 (1990).
  * [26] P. D. Alvarez, M. Valenzuela, and J. Zanelli, Supersymmetry of a different kind, JHEP 04 058 (2012).
  * [27] P. D. Alvarez, M. Valenzuela, and J. Zanelli, Role of gravity in particle physics: A unified approach, Int. J. Mod. Phys. D29 2041012 (2020).
  * [28] P. D. Alvarez, M. Valenzuela, and J. Zanelli, Chiral gauge theory and gravity from unconventional supersymmetry, JHEP 07 205 (2020).
  * [29] P. D. Alvarez, L. Delage, M. Valenzuela, and J. Zanelli, $\mathcal{N}$ = 2 extended MacDowell-Mansouri supergravity, JHEP 07 176 (2021).
  * [30] P. D. Alvarez, L. Delage, M. Valenzuela, and J. Zanelli, Unconventional SUSY and Conventional Physics: A Pedagogical Review, Symmetry, 13 628 (2021).
  * [31] W. Heidenreich, On solutions spaces of massless field equations with arbitrary spin, J. Math. Phys. 27 (1986), 2154-2159 doi:10.1063/1.527037