# An Exploration-Exploitation Approach to Anti-lock Brake Systems

Benjamin Sullivan, Jingjing Jiang, Georgios Mavros, Wen-Hua Chen  This work
was supported by the UK Engineering and Physical Sciences Research Council
(EPSRC) Established Career Fellowship “Goal-Oriented Control Systems:
Disturbance, Uncertainty and Constarints” under the grant number EP/T005734/1.
The authors are with the Department of Aeronautical and Automotive
Engineering, Loughborough University, LE11 3TU, UK. Correspondence;
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>This work has been submitted to the IEEE for possible
publication. Copyright may be transferred without notice, after which this
version may no longer be accessible.

###### Abstract

Anti-lock Brake System (ABS) is a mandatory active safety feature on road
vehicles with analogous systems for aircraft and locomotives. This feature
aims to prevent locking of the wheels when braking and to improve the handling
performance, as well as reduce stopping distance of the vehicle. Estimation
uncertainties in the vehicle state and environment (road surface) are often
neglected or handled separately from the ABS controller, leading to sub-
optimal braking. In this paper, a Dual Control for Exploration-Exploitation
(DCEE) approach is taken toward the ABS problem which achieves both accurate
state (and environment) estimation and superior braking performance. Compared
with popular Extremum Seeking methods, improvements of up to $15\%$ and
$8.5\%$ are shown in stopping time and stopping distance, respectively. A
Regularized Particle Filter with Markov Chain Monte Carlo step is used to
estimate vehicle states and parameters of the Magic Formula tyre model that
includes the peak friction coefficient for the environment. The effectiveness
of the DCEE approach is demonstrated across a range of driving scenarios such
as low and high speeds; snow, wet and dry roads and changing road surfaces.

###### Index Terms:

Anti-lock Brake System, Dual Control for Exploration-Exploitation, Particle
Filter

##  I Introduction

Active safety systems, such as Electronic Stability Control, Traction Control
and Adaptive Cruise Control, are safety features found on many ground vehicles
due to the ability to actively improve controllability and safety of the
vehicle. This is either through autonomously activating the vehicle controls
(Advanced Driver Assistance Systems in ground vehicles), or automatically
activating inline with direct driver operation, _e.g._ pressing the
accelerator may also activate Traction Control. This paper focuses on arguably
the most critical active safety feature, the Anti-lock Brake System (ABS). The
performance of ABS becomes limited when either the state of the vehicle
(_i.e._ tyre slip) is poorly estimated or the quality of the road surface is
unknown: too much braking causes the wheels to lock or too little braking
leads to an increased chance of collision. The relationship between friction
coefficient and the vehicle state (wheel slip ratio) is shown in Fig. 1
[1][2]. The aim of our design is to achieve the greatest braking performance,
_i.e._ , the maximum friction force from the road surface. Thus the ABS
problem is twofold: estimate the vehicle states and parameters, as well as
control the vehicle to achieve its maximum retardation.

Figure 1: The relationships between wheel slip and friction coefficient (top),
as well as between wheel slip and normalised wheel acceleration (bottom).

Traditionally, two control methodologies exist for ABS. They are Wheel Slip
Control (WSC) and Wheel Deceleration Control (WDC). In the latter class of
methods, wheel deceleration control is achieved using rule based heuristics
often employing hybrid automata to regulate the wheel deceleration [3, 4, 5].
This methodology is sub-optimal and in fact attempt to bound the vehicle
within the optimal acceleration setpoint using a set of acceleration
thresholds. For example, as shown in Fig. 1, a hypothetical set-point of wheel
acceleration $\eta_{1}$ leads to an operating point that is only suitable for
snow covered roads, all the other road types lead to the vehicle either
sliding significantly or under-braking. WSC methods however, regulate the
wheel slip to its largest value before the vehicle begins sliding, achieving
the greatest retardation forces. Such methods are beneficial since they have a
clear mathematical background and work even if there is no well defined maxima
[6]. [7, 8, 9] proposed control designs using a priori knowledge of the
optimum slip ratio and neglecting measurement noise, while [10, 11, 12]
estimates the slip value of the vehicle for changing road surfaces. Sliding
Mode Control has traditionally been used for ABS controllers [13, 14, 15] and
extended to uncertain systems in [16] where the optimum slip ratio has been
estimated in an online fashion using Extremum Seeking [17, 18, 19]. Despite
this, Sliding Mode Control is prone to high frequency oscillations
(chattering) as the controller attempts to hold onto the surface and can cause
the vehicle to slide [12].

In this paper, we develop a chatter free solution, namely the Dual Control for
Exploration and Exploitation (DCEE) approach which aims to predicatively
estimate and achieve the best control action on any road surface. This problem
is fundamentally an exploration-exploitation of the road surface friction thus
the DCEE is qualified. The DCEE methodology further improves on the current
state of the art by informing the action selection by predicted measurements.
Ultimately, this method predicts the future outcome in a cost function that
minimizes the error between the predicted tyre force and the maximum available
tyre force for any given road surface. It is able to deal with time-varying
road conditions, which is different from other works that only consider a
time-invariant setpoint _e.g._ [20].

Kalman Filtering (KF) and subsequent extensions (e.g Extended [21, 22, 23] or
Unscented [24, 25, 26]) have been used to solve the estimation problems (e.g.
wheel speed and vehicle velocity), yet do not adequately handle the highly
non-linear dynamics of braking systems and non-Gaussian noises. Limited cases
exist for this problem using the Sequential Monte Carlo (SMC) approach [24].
However, such approach cannot deal with higher-dimensional static parameter
estimation in this work. A GPF approach was implemented to identify a tyre
model along with the state of the vehicle [27]. However, this required
measurements of lateral and longitudinal velocities, accelerations and wheel
speed. Hence, the implementation cost is higher than that of typical WSC or
WDC solutions that measure wheel speed and vehicle longitudinal velocity only.
Furthermore a unique piece-wise tyre force model is identified during this
study but lacks usefulness for other systems since it makes assumptions about
linearity, saturation and introduces scaling gains. Bayesian Inference has
shown to perform well in vehicle localization and target tracking for non-
linear systems and/or non-Guassian observations for state estimation
_e.g._[22, 28] but remains challenging when applied for parameter estimation.
We show that a Regularized Particle Filter (RPF) approach for joint state and
parameter estimation of unknown vehicle states and tyre model parameters is
suitable for the ABS problem. The application of the Particle Filter when used
for vehicular active safety is under-examined, particularly for dynamic
parameters as is the case for changing road surfaces which we demonstrate our
method for.

A novel control scheme based on exploitation and exploration has been proposed
and verified with simulation, results of which shows an improvement of up to
$15\%$ in stopping time and $8.5\%$ in stopping distance by comparing with
benchmarks. We demonstrate that the RPF can estimate both vehicle states and
all the parameters of the Magic formula tyre model simultaneously.
Furthermore, a novel resampling method is introduced, namely Retrogressive
resampling that significantly improves the estimation and control result in
the presence of step changes to parameters. These phenomena are typical of
braking on changing road surfaces e.g. traversing newly paved road to an old
road surface, onto untreated icy roads or over ’potholes’.

The structure of the rest of the paper is as follows with the system
architecture shown in Fig. 2. The system model and slip-friction relationship
are described in Section II, while the joint state and parameter estimation
using Particle Filtering is presented in Section III with an effective
resampling method for parameter step changes, namely Retrogressive resampling.
Then a controller is presented using the Dual Control for Exploration and
Exploitation method in Section IV. Simulation results are presented in Section
V, demonstrating the effectiveness of the proposed method. Finally,
conclusions and discussion on future progression are conducted in Section VI.

Figure 2: DCEE for ABS system architecture diagram with plant model.

##  II System Dynamics

This work focuses on the longitudinal only behaviour of the vehicle during
braking as one considers the lateral vehicle stability to be outside the remit
of an ABS feature. Thus we target the 7 degree of freedom (DOF) vehicle model
for demonstration of this work.

###  II-A Vehicle Dynamics: 7 DOF Model

The 7 DOF vehicle dynamics model is shown in Fig. 3. From first principles,
the summation of all forces and moments are found using the center of mass
$[x_{G},y_{G},z_{G}]$ and align with the SAE frame of reference [29]:

$\sum^{4}_{j=1}F_{x,j}=m\left(\frac{dU}{dt}+Wq\right)-m\left(x_{G}q^{2}-z_{G}\frac{dq}{dt}\right)$
(1)
$\sum^{4}_{j=1}F_{z,j}=m\left(\frac{dW}{dt}-Uq\right)-m\left(z_{G}q^{2}+x_{G}\frac{dq}{dt}\right)$
(2)
$\sum^{4}_{j=1}M_{y}=I_{yy}\frac{dq}{dt}+mz_{G}\left(\frac{dU}{dt}+Wq\right)-mx_{G}\left(\frac{dW}{dt}-Uq\right)$
(3)

where $F_{x,j}$ and $F_{z,j}$ denotes the longitudinal and vertical tyre force
at wheel $j\in\left[1,2,3,4\right]$ as shown in Fig. 3 respectively, while $q$
represents the pitch velocity about the centre of gravity. In addition, $m$ is
the total vehicle mass and $U$ and $W$ are the longitudinal (forward) and
vertical vehicle body centred velocities. $I_{yy}$ is the vehicle body
products of inertia about the $yy$ axis, while $M_{y}$ denotes the pitching
moment.

Figure 3: The 7 DOF vehicle dynamics model.

We model the suspension using a spring/damper, with spring stiffness $k$ and
damping coefficient $c$, connecting each wheel to the rigid chassis as:

$F_{z,f}=k_{\mathrm{f}}\left(z-|a|\phi\right)+c_{\mathrm{f}}\left(W-|a|q\right)$
(4)
$F_{z,r}=k_{\mathrm{r}}\left(z+|b|\phi\right)+c_{\mathrm{r}}\left(W+|b|q\right)$
(5)

where $\phi$ is the pitch angle, $a$ and $b$ are the distances to the centre
of gravity from the front and rear axles respectively, $z$ is the vertical
displacement of the centre of gravity. Note that subscripts $f$ and $r$ are
used to denote the front and rear axles.

Finally, the remaining four degrees of freedom are from the wheel
accelerations, $\dot{\omega}$, given by the following relationship:

$I_{w}\dot{\omega}_{j}=T_{j}-RF_{T,j}$ (6)

where $T_{j}$ denotes the sum of the brake or driving action (control input)
applied to the centre of $j$th wheel for all $j\in\\{1,2,3,4\\}$. Note that
$T_{3}=T_{4}=0$ as we only apply braking and driving torques to the front
wheels in this study. Note that all wheels are assumed identical, hence the
wheel inertia $I_{w}$ and the effective tyre radius $R$ are the same between
all wheels.

###  II-B Steady State Tyre Dynamics

Tyre force models such as the non-linear Magic formula provide an accurate
representation of the available road surface friction across a range of
operating conditions, _e.g._ , dry, wet or asphalt, cobblestone as shown in
Fig. 1. The non-sliding (slip ratio smaller than the optimum value) and
sliding (slip ratio larger than the optimum value) regions of the road surface
are captured where the peak friction coefficient leads to the greatest tyre
force and retardation of the vehicle. Thus finding the optimal wheel slip is
the target of this work. Simply, the tyre force $F_{x,j}$ can be calculated
using the normal load $F_{z,j}$ on the tyre:

$F_{x,j}=\mu(\kappa)F_{z,j},$ (7)

where $\mu(\kappa)$ represents the friction coefficient for a certain road and
it is a function of slip ratio $\kappa$. According to the tyre model (_i.e._ ,
Magic formula), it can be found by the following function [30]:

$\mu_{j}=D\sin[C\arctan({B\kappa_{j}-E(B\kappa_{j}-\arctan B\kappa_{j})})]$
(8)

where $B,C,D$ and $E$ are the stiffness factor, the shape factor, the peak
factor (determining the peak friction coefficient) and the curvature factor,
respectively. In addition, the vehicle slip ratio $\kappa_{j}$ is calculated
as:

$\kappa_{j}=\frac{\omega_{j}R-U}{U}$ (9)

Furthermore, the practical slip ratio definition is used here giving negative
force under braking as is usual practice when using the Magic formula.

##  III State and Parameter Estimation

Our work aims to find the maximum friction force and control the vehicle to
generate such force so as to be able to stop as soon as possible. This
requires the identification of all the parameters of the magic formula (_i.e._
, $\mathbf{\Theta}=[B,C,D,E]^{T}$), as well as the states of the ego vehicle,
such as body velocity and wheel speeds. The $7$DOF vehicle dynamics
represented by (1)-(9) can be reduced to the nonlinear discrete-time double
corner model with 3DOF, _e.g._ [2], as

$\displaystyle\mathbf{x}_{k}$
$\displaystyle=f\left(\mathbf{x}_{k-1},{\mathbf{u}}_{k-1},\mathbf{\Theta}_{k-1}\right)$
(10) $\displaystyle\mathbf{y}_{k}$
$\displaystyle=\mathbf{x}_{k}+\mathbf{V}_{k}$

where $\mathbf{u}_{k}$ is the torque control input used to brake the front two
wheels, _i.e._ , $\mathbf{u}_{k}=\left[T_{1,k},T_{2,k}\right]$, and
$\mathbf{x}_{k}$ represents the state of the vehicle, including longitudinal
body velocity and wheel speeds, _i.e._ ,
$\mathbf{x}_{k}=\left[U_{k},\omega_{f,k},\omega_{r,k}\right]$. This is a
simplified version of the previous system model whereby only the longitudinal
and wheel rotational degrees of freedom are maintained. The wheels of each
individual axle are assumed to have the same rotational velocity, hence this
is a single-track model that only distinguished between front and rear slip,
but not left and right. Note that the parameter $\mathbf{\Theta}$ is assumed
to be static, but may exhibit sudden step changes to match the operating
condition. Even though the model does not include pitch and heave motions, it
permits differential vertical loading conditions over the front and rear axles
that can be significant during braking. In addition, we assume that all states
are measurable but with additive noise hence matching that of the sensor, thus
the system output is $\mathbf{y}_{k}\in\mathbb{R}^{3\times 1}$.

Note that we assume that all braking torques are applied to front wheels only
as we achieve the majority of braking here. To estimate states and parameters
at the same time, we create a single augmented vector variable
$\mathbf{\chi}_{k}$ as:

$\mathbf{\chi}_{k}=\left[U_{k},\omega_{f,k},\omega_{r,k},B_{k},C_{k},D_{k},E_{k}\right]^{T},$

whose dynamics are represented by

$\mathbf{\chi}_{k}=\begin{bmatrix}\mathbf{x}_{k}\\\
\mathbf{\Theta}_{k}\end{bmatrix}=\begin{bmatrix}f\left(\mathbf{x}_{k-1},\mathbf{u}_{k-1},\mathbf{\Theta}_{k-1}\right)\\\
\mathbf{\Theta}_{k-1}\end{bmatrix}=\begin{bmatrix}f\left(\mathbf{\chi}_{k-1},\mathbf{u}_{k-1}\right)\\\
\mathbf{\Theta}_{k-1}\end{bmatrix}$ (11)

###### Remark 1

One may take $\mathbf{\Theta}_{k}=\mathbf{\Theta}_{k-1}$ since we assume that
all the parameters in the Magic Formula are piece-wise constants despite the
changing road conditions.

###### Remark 2

Here lies the challenge associated with the tyre model identification during
an ABS scenario. Our system is required to quickly determine the correct
parameters whilst operating robustly to an uncertain environment. Hence the
best braking action is require to be found as quickly as possible to avoid
collision.

###  III-A Estimation Using Bayesian Inference

In having a history of observations _i.e._
$\mathbf{Y}_{k-1}=\\{y_{0},y_{1}:y_{k-1}\\}$ the objective of an observer is
to estimate the current state $\mathbf{\chi}_{k}$ using all the information
available and recursively update the priori belief
$p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k-1}\right)$ as new information
becomes available. Given we can predict the state value using our system model
(11), one may use the Chapman-Kolmogorov equation to calculate the prior
belief as:

$p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k-1}\right)=\int
p\left(\mathbf{\chi}_{k}\mid\mathbf{\chi}_{k-1}\right)p\left(\mathbf{\chi}_{k-1}\mid\mathbf{Y}_{k-1}\right)\mathrm{d}\mathbf{\chi}_{k-1}$
(12)

From a new measurement at the current time step, $y_{k}$, the posterior
likelihood $p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k}\right)$ can be updated
using Bayes theorem based on the likelihood
$p\left(\mathbf{y}_{k}\mid\mathbf{\chi}_{k}\right)$, prior
$p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k-1}\right)$ and marginal likelihood
$p\left(\mathbf{y}_{k}\mid\mathbf{Y}_{k-1}\right)$:

$p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k}\right)=\frac{p\left(\mathbf{y}_{k}\mid\mathbf{\chi}_{k}\right)p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k-1}\right)}{p\left(\mathbf{y}_{k}\mid\mathbf{Y}_{k-1}\right)}$
(13)

where the normalising constant is:

$p\left(\mathbf{y}_{k}\mid\mathbf{Y}_{k-1}\right)=\int
p\left(\mathbf{y}_{k}\mid\mathbf{\chi}_{k}\right)p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k-1}\right)d\mathbf{\chi}_{k}$
(14)

###### Remark 3

We attempt to identify parameters of a widely accepted tyre model which, to
the best of the authors knowledge, is the first complete identification effort
of the Magic Formula using a Bayesian approach.

###  III-B Regularized Particle Filter

Determining the solution to (13) is generally intractable as it requires the
entire observation history $\mathbf{Y}_{k}$ that is costly for real online
systems. In the case of linear finite dimension systems that exhibit Gaussian
noise characteristics, the Kalman Filter provides the optimal recursive
Bayesian solution. However, to better approximate real systems including
system non-linearities and non-Gaussian noises, a Sequential Monte Carlo (SMC)
approach is more appropriate. Due to these reasons, the optimal state
estimation is achieved by using a Bayesian solution [31] that is approximated
by the sum of weighted samples, $w^{i}$ associated with the Dirac function
$\delta$, as the following using the number of particles, $n_{i}$:

$p\left(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k}\right)\approx\sum_{i=1}^{n_{i}}w_{k}^{i}\delta\left(\mathbf{\chi}_{k}-\mathbf{\chi}_{k}^{i}\right)$
(15)

where $w_{k}^{i}$ is found using the importance density $q$:

$w_{k}^{i}\propto
w_{k-1}^{i}\frac{p\left(\mathbf{y}_{k}\mid\mathbf{\chi}_{k}^{i}\right)p\left(\mathbf{\chi}_{k}^{i}\mid\mathbf{\chi}_{k-1}^{i}\right)}{q\left(\mathbf{\chi}_{k}^{i}\mid\mathbf{\chi}_{k-1}^{i},\mathbf{y}_{k}\right)}$
(16)

The expected value is calculated from our particles using the standard form
for finding the weighted mean:

$\mathbb{E}\left[\mathbf{\chi}_{k}\right]=\sum_{i=1}^{n_{i}}W_{k}^{i}\mathbf{\chi}_{k}^{i}$
(17)

where $W^{i}_{k}$ is the normalised particle weight _i.e._
$W^{i}_{k}=w^{i}_{k}/\sum_{i=1}^{n_{i}}w^{i}_{k}$ such that the sum of the
normalised particle weights equals 1. The inference problem so far has been
restricted to Sequential Importance Sampling which forms the basic mechanic
for most particle filtering methods. There are however a number of limitations
to this method including degeneracy and impoverishment which must be
addressed.

The degeneracy problem exists when the weight of a single particle is updated
with increasing likelihood until it reaches nearly 100% likely where the
weight, $w_{k}^{i}$, equals 1. In this extreme scenario, computation of this
single particle leads to the expected state value from the filter since all
other weights are insignificant, hence the efficiency is extremely poor. Once
degeneracy is observed in the weight values, the estimates will begin to
diverge and result in unacceptably poor errors. To mitigate this issue,
particle resampling is utilised to redistribute those with high weightings to
multiple particles of the same value and remove those with low weightings.
This in turn increases the number of active particles. Resampling methods
include systematic, stratified and branch-kill with further discussions found
in [32]. The stratified resampling method is used in this work for its
superior computational efficiency. Furthermore one may decide to selectively
re-sample only when the level of degeneracy is below a threshold. The
effective sample size, $N_{eff}$, is one way to measure this [33] and is
calculated by

${N}_{eff}=\frac{1}{\sum_{i=1}^{n_{i}}\left(w_{k}^{i}\right)^{2}}$ (18)

Resampling is active if $N_{eff}\leq K_{o}n_{i}$, where $K_{o}$ is introduced
to control the resampling frequency as in practice large value of this
threshold alone leads to overly frequent resampling that can be detrimental to
the estimation effort. Here we chose to define this using a linear
relationship, dependant on the expected peak friction.

###### Remark 4

It is clear to see from Fig. 1 that the parameter identification problem,
particularly the estimation of peak friction parameter $D$, is more
challenging on snow or icy surfaces since the peak is less defined than that
for dry or wet roads. Therefore, one would expect a higher resampling
frequency were the peak is harder to identify, ensuring an adequate
distribution of particles.

In this paper, $K_{o}$ is chosen as a time-varying parameter subject to the
expectation of the road condition. At time step $k$, it is defined as

$K_{o,k}=-\frac{3}{20}\mathbb{E}\left[D_{k}\right]+0.295$ (19)

According to (19) there is no need to do manual adaptation among operating
conditions and the observer is automatically adapted to the tyre model
estimation of the peak friction.

Sample impoverishment is an undesirable quality of particle filtering where
the particles move closer together yielding the particle population to have no
diversity and is a side effect of high frequency resampling. When all
particles occupy the same value, leading to poor representation of the
posterior density, this is often referred to as particle collapse. It can be
seen that frequent resampling would instigate this phenomenon, hence
impoverishment and degeneracy are interlinked.

To introduce particle diversity, two methods may be employed to move a
particle to a new value and could be used in unison along with the resampling
threshold, namely a regularization step (Regularized Particle Filter (RPF)
[34]) and a Markov Chain Monte Carlo (MCMC) step. By using the RPF, jittering
of the particle population is achieved. The regularization step may be
approximated to the following using the Gaussian Kernel. Note that the
characteristics of the kernel are zero mean subject to small variance (_i.e._
, $\epsilon_{k}\sim\mathcal{N}\left(0,1\right)$). The regularization step,
using the re-sampled particle population, is conducted as follows:

$\mathbf{\chi}^{i\star}_{k}=\mathbf{\chi}^{i}_{k}+h_{opt}\mathbf{D}_{k}\mathbf{\epsilon}^{i}_{k}$
(20)

where $\mathbf{\chi}^{i\star}_{k}$ represents the new particle value.
$h_{opt}$ is the bandwidth calculated by

$h_{opt}=An_{i}^{\frac{-1}{n_{x}+4}}$ (21)

where $A=\left[4/\left(n_{x}+2\right)\right]^{\frac{1}{n_{x}+4}}$, $n_{x}$
equals to the length of the augmented state and is selected to minimize the
mean integrated square error (MISE). Finally, $\mathbf{D}_{k}$ is the solution
to the equation $\mathbf{S}_{k}=\mathbf{D}_{k}\mathbf{D}_{k}^{T}$, where
$\mathbf{S}_{k}$ denotes the weighted covariance and is calculated by

$\mathbf{S}_{k}=\frac{n_{i}}{n_{i}-1}\frac{1}{\sum_{i}^{n_{i}}W_{i,k}}\sum_{i=1}^{n_{i}}W_{i,k}\left(\mathbf{\chi}_{i,k}-\mathbf{\mu}_{k}\right)^{2}$
(22)

The purpose of the MCMC step is similar to the regularization step. However,
this method ensures that a resampled particle population (or regularized
sample) is free from sample degeneracy and impoverishment [33]. The aim is to
draw more samples from areas with high probability by an acceptance
probability, using the posterior distribution following resampling
$p(\mathbf{\chi}_{k}\mid\mathbf{Y}_{k})$, and a proposal distribution
$p(\mathbf{y}_{k}\mid\mathbf{\chi}_{k}^{i\star})$. MCMC methods randomly draw
a sample (Monte Carlo) that is dependant on the likelyhood of the prior sample
_i.e._ Markov Chain.A common MCMC method utilised in Particle Filters is the
Metropolis-Hastings Step, originally presented in [35] with the replication in
[36] utilized for this work.

###### Remark 5

The Regularization has the additional benefit of introducing artificial
dynamics to the augmented state matrix. If an unlikely scenario, for example,
of extreme tyre wear, very low or high road surface friction coefficients or
very high vehicle speeds are observed, the particle values are able to expand
to handle this.

The complete regularized particle filter with MCMC step is shown in Algorithm
1.

Algorithm 1 Regularized Particle Filter with MCMC Step

Input: $\mathbf{\chi}_{k-1}^{i},w_{k-1}^{i},\mathbf{y}_{k}$
Output: $\mathbf{\chi}_{k}^{i},w_{k}^{i}$

1: Propagate system state according to (11)

2: Determine likelihood by (16)

3: Use (18) to calculate effective ratio

4: Compute the resampling gain $K_{o,k}$ by (19)

5: if $N_{eff}\leq K_{o,k}n_{i}$ then

6: Systematic Resampling

7: Regularization

8: MCMC Step

9: end if

###  III-C Parameter Step Change: Retrogressive Resampling

Particle filtering is well suited to slowly changing states or parameters,
such as vehicle’s velocity which cannot instantaneously change from $0$ to
$20m/s$ due to mechanical constraints, instead it would be accelerating over a
period of time. However, change of road conditions is different. In fact, the
changes in the road surface parameters can be best described as step changes,
for example, traversing from an old road surface to a newly paved surface,
from a dry road to a wet road, or over potholes. The performance of estimating
a changing road surface is shown in Fig. 4 where an RPF estimator first
converges successfully to the first road surface, but fails to quickly
converge to the second one. It takes nearly $1.5s$ to diversify the particle
population and re-converge to the new ground truth. The maximum and minimum
particle values show the diversity and clearly indicate how the filter
immediately becomes impoverished and cannot escape this effect in a timely
manner. In fact the filter is impoverished for over a second whereby the
particle population range doesn’t include the true value.

Figure 4: Estimation result on a dry road to a wet surface using the
Regularized Particle Filter without retrogressive resampling. The mean of the
estimation result for Parameter $D$ is shown with its maximum and minimum
estimations.

To manage such sudden change in operating conditions, a simple yet effective
method for resampling the particle population is introduced, namely
Retrogressive resampling. In the case that the states of the vehicle are
continuously tracked regardless of the road condition but the road surface
parameters are not, a trigger to reset these parameters to their initial
conditions using the current uncertainty of the system is required. If we
observe high estimation uncertainties, a reset is triggered, leading to a
normally distributed particle population over the initial range. The
weightings are also reverted to their initial values thus giving an equal
likelihood over the entire parameter space. The Retrogresive resampling method
is presented in Algorithm 2 where we use the uncertainty of the predicted
maximum tyre force, quantified by its error variance matrix, ${P}_{k+1\mid k}$
described in Section IV. This step is applied at the beginning of every new
measurement.

Algorithm 2 Retrogressive Resampling

Input: ${P}_{k+1\mid
k},{\mathbf{x}}^{i}_{k},\mathbf{\Theta}_{0}^{i},\mathbf{\Theta}_{k}^{i},w^{i}_{0},P_{0},w_{k}^{i}$
Output: ${\mathbf{x}}^{i}_{k},\mathbf{\Theta}_{k}^{i},w_{k}^{i}$

1: if $\max{{P}_{k+1\mid k}}\geq P_{0}$ then

2: $\mathbf{\Theta}_{k}^{i}\leftarrow\mathbf{\Theta}_{0}^{i}$

3: ${\mathbf{x}}_{k}^{i}\leftarrow\mathbf{x}^{i}_{k}$

4: ${w}_{k}^{i}\leftarrow w^{i}_{0}$

5: else

6: $\mathbf{\Theta}_{k}^{i}\leftarrow\mathbf{\Theta}_{k}^{i}$

7: ${\mathbf{x}}_{k}^{i}\leftarrow\mathbf{x}^{i}_{k}$

8: ${w}_{k}^{i}\leftarrow w^{i}_{k}$

9: end if

###### Remark 6

There is no need to tune the Retrogressive resampling process since $P_{0}$,
the initial value of $\max{{P}_{k+1\mid k}}$, is determined by how the initial
conditions are set. Following the initialisation, if at any point during the
manoeuvre the uncertainty exceeds that of the initial value then we resample
accordingly.

##  IV Dual Control for Exploration-Exploitation

The ABS problem is both a stochastic and adaptive control problem since we do
not know the current state of the vehicle and parameters associated with the
vehicle (tyre model) that we may exploit to achieve optimal braking. The
Particle Filter is used to find the current road surface friction $\mu$, the
maximum road surface friction $D$, the wheel speed and body velocity. Thus the
brake torque is selected as the action that produces the smallest tracking
error between the current and the estimated maximum tyre forces. However, this
results in exploitation of the estimation solely through experience of the
recursive RPF observer rather than prediction since the observer can only
determine the correct parameters when it has experienced the maxima. This
maxima is found by driving the vehicle to begin sliding without prediction. It
is clear that altering the brake force will change how the vehicle behaves,
affecting the future state and consequently the future measurements of our
vehicle state. To this effect, the DCEE framework incorporates information
from a one step prediction into our control action selection using predicted
observations, propagated from the RPF observer and a discrete-time
approximation of the vehicle dynamics. In this way, our system intelligently
distinguishes those control actions that do not drive the vehicle to reach the
minumum stopping distance whilst considering the uncertainty of the maximum
tyre force estimation.

###  IV-A DCEE Formulation

The objective function is selected to minimize the error between the predicted
tyre force $F_{k+1\mid k}$ and the maximum tyre force $F_{k}^{*}$, hence is
defined as:

$\displaystyle\min_{{u}_{k}}J\left({u}_{k}\right)$
$\displaystyle=\min_{{u}_{k}}\mathbb{E}_{\mathbf{\Theta}_{k}}\left[\mathbb{E}_{y_{k+1|k}}\left[\big{|}F_{k+1\mid
k}-F_{k}^{*}\big{|}\mid\bar{\mathbf{\chi}}_{k},\mathbf{y}_{k}\right]\right]$
(23a) subject to $\displaystyle{u}_{k}$ $\displaystyle={u}_{k-1}+{\tau}_{k}$
(23b) $\displaystyle{\tau}_{k}\in\mathcal{T}$ (23c)

where $\mathbf{y}_{k}$ and denotes the current measurements. $\mathcal{T}$ is
a set of control actions and $u_{k}$ is the current torque which is assumed to
be deterministic. Different from traditional control schemes, the reference is
unknown and the maximum tyre force must be estimated in an online fashion
according to the changes in road condition. Furthermore, one may expect the
tyre properties to change as they are replaced due to wear during its
lifespan. Thus an adaptive control scheme is necessary to control the vehicles
braking whilst including the typical proprioceptive and exteroceptive
operating dynamics of the vehicle and environment. According to [37], the cost
function (23) is equivalent to the following one using the variance of the
error between the predicted tyre force and its maximum value, ${P}_{k+1\mid
k}$ :

$J({u}_{k})=\big{|}\bar{{F}}_{k+1|k}-\bar{F}_{k+1\mid
k}^{*}\big{|}+{P}_{k+1|k},$ (24)

where ${F}^{*}_{k}$ is the expectation of the predicted estimation of the
total available tyre force using the peak friction coefficient $D$, _i.e._ ,

$\bar{F}^{*}_{k+1\mid k}=\mathbb{E}\left[F_{k+1\mid
k}^{*}\mid\mathcal{Y}_{k+1\mid k}\right].$ (25)

and similarly for the predicted state given a potential control action from
$\mathcal{T}$ and the estimated augmented state matrix where $\bar{{F}}$
denotes the tyre force vector using the admissible action set $\mathcal{T}$:

$\bar{{F}}_{k+1\mid k}=\mathbb{E}\left[{F}_{k+1\mid k}\mid\mathcal{Y}_{k+1\mid
k}\right]$ (26)

Cost function (24) shows the exploration-exploitation effect by not only
incorporating the dynamic nature of the maximum tyre force but also the
likelihood of achieving it, captured in the variance of the tyre force
tracking error ${P}_{k+1\mid k}$. Then, $P_{k+1\mid k}$ is found as the
variance of the predicted tracking error:

$P_{k+1\mid k}=Var\left({\bar{{F}}}_{k+1|k}-\bar{F}_{k+1\mid k}^{*}\right)$
(27)

The first right hand term of (24) exploits the current information gathered by
the observer as per traditional control theory, while the second right hand
term quantifies the uncertainty of the vehicle reaching the predicted maximum
tyre force.

###### Remark 7

ABS is fundamentally an exploration-exploitation problem since the maximum
friction of a road surface is unknown and cannot be measured. Thus the control
system should aim to understand the environment by means of the available
actions (_exploration_) observing the response, then taking the best action
that results in the shortest stopping distance (_exploitation_) in a recursive
manner. Most existing methodologies do not acknowledge this problem instead,
computing the most appropriate action from the posterior belief of the road
(exploitation only). SMC, for example, only considers the reference surface
rather than exploring the system to find it.

###  IV-B Measurement Prediction

The goal of the measurement prediction is to determine the actions that
achieve the highest brake force, before taking them. Since we have already
generated a belief in the current state of the vehicle one may predict what
the next tyre force measurement is, given a selected control input. In
practice, physical measurement of the tyre force represents a significant
challenge since unconventional sensors, such as Load Sensor Bearings [38] or
other Smart Tyre systems [39], are used. However this work does not require
such systems to estimate the tyre force or predict it.

###### Remark 8

This work does not require sophisticated sensing, instead estimating the
current and predicted tyre force from low cost conventional sensors.

The predicted tyre force for $i^{th}$ particle, ${{F}}_{k+1\mid k}^{i}$, is
found using a discrete time approximation of the half car dynamics using the
posterior estimate of the augmented state matrix, $\mathbf{\chi}_{k}^{i}$ from
the RPF and the vertical load on the wheel:

${{F}}_{k+1\mid k}^{i}=\mu_{k+1|k}^{i}F_{z}$ (28)

where $\mu_{k+1\mid k}^{i}$ is the predicted friction coefficient found using
the posterior estimation of the augmented state _i.e._
$\bar{\mathbf{\chi}}_{k}$. Then similarly for the predicted maximum tyre force
using peak factor $D$ where $D_{k+1\mid k}^{i}=D_{k}^{i}$ instead of the
predicted friction coefficient.

This however leads to $n_{i}$ predicted tyre forces for each action which is
unnecessarily large since the performance gain of $n_{i}$ particles
(potentially thousands of measurements for each action) does not justify the
computational cost. Therefore one may ‘intelligently’ select the most relevant
observations using the stratified resampling technique [32] with the posterior
weighting. This facilitates the redistribution of the particle population into
strata or within sub-populations that are drawn randomly from within these
groups. Considering that the prediction is inherently uncertain, we determine
that only $40$ predicted observations is required for ABS. Performing
resampling on the predicted particle population also has the additional
benefit of ensuring that the observations are non-degenerate thus ensuring a
set of unique measurements. Finally, making the assumption that the tyre force
observations follow a Gaussian distribution with variance $\sigma_{f}=250$,
one determines the likelihood of the predicted tyre force. Ensuring that when
approximating the tyre forces by Eqs. (25)-(26), we use the down sampled set
of predicted tyre force measurements when determining the weights.

##  V Simulation

The effectiveness of the aforementioned closed loop control scheme is
demonstrated in simulation using the system architecture shown in Fig. 2 with
the 7DOF vehicle model (_i.e._ , (1)-(9)) parameterised using a Jaguar XJ
vehicle as shown in Table I.

TABLE I: Vehicle parameters. Parameter |  Value |  Units
---|---|---
Front Suspension Stiffness, $k_{f}$ |  20090 |  $N/m$
Rear Suspension Stiffness, $k_{r}$ |  22700 |  $N/m$
Front Damping Coefficient, $c_{f}$ |  2000 |  $N/m^{2}$
Rear Damping Coefficient, $c_{r}$ |  2260 |  $N/m^{2}$
Rolling Radius, $R$ |  0.29 |  $m$
Vehicle Body Inertia, $I_{yy}$ |  3983 |  $kg/m^{3}$
Center of gravity, $x_{G}$ |  0 |  $m$
Center of gravity, $y_{G}$ |  0 |  $m$
Center of gravity, $z_{G}$ |  -0.4427 |  $m$
Distance to front axle, $a$ |  1.455 |  $m$
Distance to rear axle, $b$ |  -1.575 |  $m$
Vehicle Total Mass, $m_{T}$ |  1838.35 |  $kg$
Wheel Inertia, $I_{w}$ |  1.25 |  $kg/m^{3}$

Online estimation of the augmented state vector $\mathbf{\chi}_{k}$ is done in
all scenarios where the road surface friction is modelled using the Magic
formula with the parameters of the road surface shown in Table II. It is
acknowledged that the tyre stiffness changes between road types demonstrating
the insensitivity of this method to different tyres thus further showing the
advantages of the DCEE based solution. Measurements of the ego body velocity
and wheel speed $[U,\omega_{f},\omega_{r}]^{T}$ are corrupted by the additive
noise $\mathbf{V}_{k}$ such that
$\mathbf{V}_{k}\sim\mathcal{N}\left(\left[0,0,0\right]^{T},diag\left(0.2,0.5,0.5\right)\right)$.

TABLE II: Magic formula coefficients and optimal slip ratio for three road surfaces. Parameter |  Dry Road |  Wet Road |  Snowy Road
---|---|---|---
$B$ |  5 |  10.695 |  20
$C$ |  1.4601 |  1.4 |  1.5354
$D$ |  1.3 |  0.8 |  0.3
$E$ |  -10.3522 |  -3.5 |  0.8525
Optimal Slip |  -0.15 |  -0.104 |  -0.18

In real world scenarios and in the context of ABS, the ego vehicle begins in
an unknown environment, particularly if there has been rain or snow for
example. Therefore the particles are initialised with a set of parameter
values that encompass all road types and scenarios mentioned here. These are
represented as uniform distributions: $B\in\mathcal{U}\left[4,21\right]$,
$C\in\mathcal{U}\left[1.2601,1.601\right]$,
$D\in\mathcal{U}\left[0.2,1.6\right]$ and $E\in\mathcal{U}\left[-12,2\right]$.
Additionally, since we assume the vehicle to be in an unknown environment we
also assume proprioceptive uncertainty where we find the initial states to
found from any low cost sensor then uniformly distributed as per the quality
of the sensor. Although some sensors exhibit Gaussian noise, we do not need to
make such assumption given our observer design. In the case of body velocity
and wheel speed, they are within [$v_{0}-2m/s$, $v_{0}+2m/s$] of the first GPS
velocity measurement and [$\omega_{0}-3.5rad/s$, $\omega_{0}+3.5rad/s$] from
the wheel speed sensor.

###  V-A Change In Initial Speed

In reality, the ABS should work in a wide range, from high to low speeds,
including those above the normal driving speeds _i.e._ above the typical speed
limit. Therefore, we tested the proposed DCEE based solution for a range of
initial velocities, from $10mph$ to $100mph$, and aimed to stop the vehicle as
soon as possible. Simulation results are shown in Fig. 5. Beneficial to this
technique is that no gain tuning is required between operating points _i.e._ ,
the same model, as well as the same parameter values, are used for each
starting velocity. No locking of the wheel occurs and the vehicle uniformly
decelerates over all velocities. One notices the oscillation in the wheel
speed at low body velocities when staring from $30$ and $10mph$. This may be
attributed to the low number of steps the controller has taken where the PF
does not converge as well therefore not holding as steady wheel deceleration.
Although this is only for a very short period of $0.2s$. Furthermore, initial
variation, particularly in the $100mph$ test case, is a result of the the
controller uncertainty in the location of the maximum tyre force. This period
is short since the controller can quickly identify and reach the appropriate
tyre force. Further oscillations in the wheel speed are due to the observer
resampling which is always greater in the initial period as the filter
converges to the true state/parameter, but exist throughout the manoeuvre.
This is required to prevent degradation in the filter and improve the
identification of less sensitive parameters.

Very quick convergence of the ego states is achieved as is shown on Figure 5.
In fact less than 0.5% error is observed in all vehicle states after 13 time
steps however the tyre model parameters requires a longer time and is aided
throughout the scenario by the excitation caused through the explorative
effort of the DCEE controller.

Figure 5: Simulation results with different starting velocity, braking to
stationary. The body velocity and wheel speed estimation result is also shown
for the 30mph braking scenario.

###  V-B Adverse Road Conditions

Road conditions vary a lot, from dry to muddy and from wet to snow.
Furthermore, the road surface does not always remain the same. For example,
water may collect on parts of the road but drain away on other sections of the
road, thus ABS should be able to work efficiently and adapt to these changes.
The robustness of our ABS solution to both static and changing road conditions
is tested in this sub-section.

####  V-B1 Static Road Surface Estimation

The DCEE method is well suited to the online identification of many road
surfaces with the estimation result of dry, wet and snow covered road surfaces
during an ABS manoeuvre from $20m/s$ shown in Fig. 6. The road surfaces are
identified correctly, estimating the initial non-sliding elastic region and
defining the maxima appropriately. Small errors found at higher slips where
the vehicle has had limited experience for, _i.e._ , driving the vehicle to a
condition when the slip ratio equals to 0.4 would cause significant sliding
thus the vehicle never reaches this. Therefore due to the lower exploration
effort, the identification performance becomes degraded. However, as the goal
of the ABS is to keep the vehicle working at the condition when the friction
coefficient achieves the maximum value, the results are more than acceptable.

Figure 6: Parameter estimation result during three separate ABS braking
manoeuvres with time-invariant road conditions and same initial conditions.

####  V-B2 Dynamic Road Surface

The RPF estimation results of individual parameters for a time-varying road
surface (the road surface changes from dry to wet at $0.5s$) can be found in
Fig. 7 where the sensitivity of the parameters is made clear. Parameter $D$
converges almost immediately and tracks the step change very well. Comparing
this to the original result (that does not use Retrogressive resampling) in
Fig. 4, one can clearly conclude the importance of the Retrogressive
resampling approach. The remaining parameters have mixed tracking performance
with parameter E showing to be insensitive to changes throughout the surface
but without any detrimental impact on the controller or vehicle since this
parameter governs the sliding region of the tyre model which the ABS prevents
the vehicle from operating in. More importantly, one may look to the product
of the parameters $B$, $C$ and $D$ since this defines the gradient of the
slip-$\mu$ curve and by taking the arcsine of this product one finds the angle
of the slope. Shown in Fig. 7, one can observe that this estimation is slower
than that for parameter $D$, eventually converging by the end of the
manoeuvre.

Figure 7: Estimation result of tyre model parameters.

The vehicle behaviour during the braking procedure can be assessed by tyre
forces, as shown in Fig. 8. The maximum available tyre force is time-varying.
Thus, the weight transfer during the manoeuvre is also time-varying. However,
the DCEE method is able to continually drive the vehicle to this varying
operating condition without lock of any wheels. Estimation of the current
time-varying friction coefficient at the wheels closely tracks the true
friction (see Fig. 8) along with holding a steady state error of less than
0.02 (_i.e._ , $\leq 3\%$) with maximum available.

Figure 8: Brake manoeuvre when road surface changes from dry to wet road at
$0.5$ seconds.

It can be shown that by including the Retrogressive resampling step one not
only improves the augmented state estimation but also reduces the uncertainty
of the location of the maximum available tyre force. Shown in Fig. 9 is a
comparison between the same changing road surface scenario with and without
the resampling step. By using this method, one reduces the estimation
uncertainty (especially after $0.5s$ when the vehicle is driven on a wet road)
and provides a quicker (_i.e._ , within $0.1s$) convergence to the steady
state.

Figure 9: Effect of retrogressive resampling on control system uncertainty.

###  V-C Comparison with Extremum Seeking Control

To demonstrate the superiority of our DCEE method above existing schemes, we
make comparison with two Extremum Seeking Control (ESC) for the ABS problem,
referred to as Continuous Sine Perturbation (CSP) [40] and Bisection method
(Bisection) [1]. Having identified the optimal slip ratio we utilise the
P-controller and reachability based control schemes also presented in these
works. The P-controller uses the gain $c=500$, chosen to match the performance
for our vehicle whereas all other parameters for the CSP model remain the
same. The sampling time is, $\Delta T=0.001$ and the initial conditions for
the Bisection method are: $a=0.1$, $b=0.2$ and tolerance is $l_{d}=0.001$.
Measurements of the ego velocity and wheel speed are made in all systems
whereby a low pass filter with cutoff frequencies of $20Hz$ and $100Hz$ for
ego body and wheel speed sensors respectively are used to remove the noise
except in the DCEE approach where the aforementioned Particle Filter is used.
Simulation results are displayed in Figs. 10 and 11.

In this example, the DCEE approach is compared to both ESC methods, stopping
the vehicle in $2.274s$ which is $15.0\%$ faster than the CSP method and
$11.7\%$ faster than the bisection method. This leads to an overall stopping
distance of $24.45m$ that is $2.09m$ shorter than that of CSP and $1.56m$ than
that of the bisection method. Noticeably, the reachability controller used in
the bisection method is unable to maintain a stable wheel speed and tends away
from constant vehicle deceleration. Furthermore, one may notice slightly
slower deceleration of the wheel speed which can be attributed to the DCEE
being primarily driven by the exploration of the state and parameter space.
This however is beneficial since the wheel speed in our method begins a steady
deceleration much faster than either of the ESC methods and with no chatter.
Similarly the other methods also go through the explorative phase although are
only identifying the optimal slip, not the road condition as and optimal tyre
force as we are. The bisection method has large oscillations, undershooting
the correct wheel speed and the CSP method undershoots but at steady state,
reaching a slower deceleration thus causing higher wheel speeds.

Figure 10: Comparison between the DCEE method and two Extremum seeking
methods, stopping from 20m/s (45mph). Figure 11: Total vehicle tyre force
comparison during braking manoeuvre from $20m/s$.

Noticeably the vehicle with DCEE is able to achieve a greater total tyre force
given the transient loading condition and is highlighted in Fig. 11. The
inclusion of the suspension dynamics is imperative for simulation and shows
the limitation for classical adaptive control, such as the CSP, fails to
maintain as large tyre force throughout the transient loading behaviour of the
vehicle. Conversely the Bisection method is able to track this change although
with significant instability, extending into the steady state part of the
manoeuvre. Our DCEE method achieves and maintains a greater total tyre force
with superior stability even while identifying the tyre model parameters which
the other methods do not.

##  VI Conclusion

The DCEE method has been demonstrated as an effective ABS solution. Estimation
of vehicle states from noisy sensors and identification of Magic formula tyre
model parameters are done so within this framework in an online manner. We
have extended the use of the particle filtering method to parameter
identification in the presence of sudden changes in operation using the
Retrogressive resampling method which incorporates the estimation uncertainty
of the maximum available tyre force in a holistic, goal-orientated way. This
work shows a performance margin of up to $15\%$ in stopping time and $8.5\%$
in stopping distance compared to existing ABS methods while achieving a higher
steady state braking force.

Future work may include introducing a longer prediction horizon to the DCEE
methodology to bring greater stability in the control action although would
undoubtedly introduce higher computational cost. Whilst this method considers
unknown vehicle states and unknown vehicle parameters, extending this
framework to identify the transient tyre behaviour would also be beneficial.

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