Road Condition Predicting
with Kalman Filter
for Magneto-Rheological Damper
in Suspension System
Zuohai Yan
Shuqi Zhao
This thesis is presented as part of Degree of
Master of Science in Electrical Engineering
Blekinge Institute of Technology
July 2012
Blekinge Institute of Technology School of Engineering
Department of Applied Signal Processing Supervisor: Feng Wang
Abstract
This thesis develops a new way to predict the road roughness with Kalman filter. It suggests applying the Kalman filter to predict road condition in suspension system. According to the literature review and to the knowledge of authors, no similar appli-cations of Kalman filter in predicting the road roughness are found at the time of the writing thesis. Most of the prediction nowadays is around the road prediction with GPS. It concentrates on avoiding the road bumps by the operator. This research is brand new in this field. What the authors focus on is to predict the road condition and to pass this information to the control system. By this way, the passenger comfort is improved.
This research is practical in transportation industry. Nowadays the passenger comfort is crucial. This road condition predictor can help the vehicle to improve the passenger comfort. Furthermore, this predictor can be adjusted to different road conditions.
A suspension system is important to improve passenger comfort. Magneto-Rheological(MR) damper, which is a controllable damper, can improve the perfor-mance of the suspension system. This thesis presents a menthod to predict the road condition for MR damper. Firstly, three suspension systems, passive, active and semi-active suspension systems, are evaluated by their costs and performances. The semi-active suspension system has good performance with low cost. This suspension system shows better performance with proper control strategy.
Additionally, two different levels of road roughness are simulated by Harmonic superposition method in time domain. One of the road roughness scenarios is chosen to test the prediction method. The road roughness is predicted by a Kalman filter. The result shows that the Kalman filter can estimate the road condition with a high accuracy. The prediction frequency is high in this method. The control strategy can adjust its coefficient based on the high prediction frequency. Thus, the performance of the suspension system is enhanced and the passenger comfort is also improved.
Acknowledgements
First we would like to express our sincere gratitude to our thesis supervisor, Feng Wang. He not only offered us the opportunity to work on this thesis but also provided valuable orientation and direction in this field.
Contents
Abstract . . . i Acknowledgements . . . iii 1 Introduction 1 1.1 Overview . . . 1 1.2 Motivation . . . 2 1.3 Objective . . . 2 1.4 Outline . . . 3 1.5 Contribution . . . 3 2 Background 4 2.1 Primary Suspension . . . 4 2.2 Passive Suspension . . . 5 2.3 Active Suspension . . . 5 2.4 Semi-active Suspension . . . 82.5 Control Schemes for a 2 Degree of Freedom (2DOF) System . . . 9
2.5.1 Skyhook Control . . . 9 2.5.2 Groundhook Control . . . 10 2.6 Magneto-Rheological Dampers . . . 11 2.6.1 Magneto-Rheological Fluids . . . 11 2.6.2 Modes of MR Operation . . . 12 2.6.3 MR Dampers . . . 14
3 Road Roughness Data Realization 16 3.1 Road Roughness Characterization . . . 16
3.2 The International Roughness Index . . . 16
3.3 Road Roughness Power Spectral Density . . . 17
3.4 Methods of Realizating Road Roughness . . . 18
3.4.1 Harmonic Superposition Method . . . 18
3.4.2 The AR Model . . . 18
3.4.3 The Inverse Fourier Transform Model . . . 20
3.4.4 The Integral White Noise Model . . . 20
4 Experimental Approach and Discussion 23
4.1 Introduction of Kalman Filter . . . 23
4.2 Modelling . . . 23
4.3 Road Condition Predicting Result . . . 25
4.4 Passenger Comfort Comparison . . . 27
5 Conclusion 29
Chapter 1
Introduction
This thesis develops a new way to predict the road roughness with Kalman filter. It suggests applying the Kalman filter to predict road condition in suspension system. According to the literature review and to the knowledge of authors, no similar appli-cations of Kalman filter in predicting the road roughness are found at the time of the writing thesis. Most of the prediction nowadays is around the road prediction with GPS. It concentrates on avoiding the road bumps by the operator. This research is brand new in this field. What the authors focus on is to predict the road condition and to pass this information to the control system. By this way, the passenger comfort is improved.
This research is practical in transportation industry. Nowadays the passenger comfort is crucial. This road condition predictor can help the vehicle to improve the passenger comfort. Furthermore, this predictor can be adjusted to different road conditions.
The purpose of this chapter is to provide an overview of the whole thesis. It briefly discusses the road condition prediction for vehicles equipped with semi-active suspension system.
1.1
Overview
The vibration phenomenon has gained a huge expansion in the past few years. The vibration theory expands from discrete systems to continuous systems, from harmonic vibration analysis to random vibration problems, from linear vibration theory to nonlinear vibration analysis, and from vibration analysis to vibration control system design [1].
With all these control systems and suspension systems, the comfort of passengers has been improved significantly. A good suspension system should achieve three goals:
• Isolating the vehicle from uneven road
• Guaranteeing the stability of vehicle
Suspension systems support the weight of the vehicle and enhance the traction force between the tires and the road surface [2]. They have been developed significantly in recent years.
Many kinds of suspension systems have been implemented. For example, passive, active and semi-active suspension systems have been applied in real vehicle design. The active suspension system is an efficient one with high cost. In order to reduce the cost, semi-active suspension is designed as a substitute. This kind of suspension system costs less when implemented in the vehicle, while the performance is close to that of active suspension system. By implementing semi-active suspension system, the requirement on compromise between the ride comfort and vehicle stability is reduced.
For a semi-active suspension system, a good control strategy is crucial. With a good control strategy, unwanted vibration is isolated. There are many kinds of control strategies. Skyhook control is one of the most commonly used ones. The semi-active suspension system is built with the skyhook control strategy.
The semi-active suspension system is easy to implement with the controllable damper. The Magneto-Rheological (MR) damper is a damper controlled by magnetic field. Nowadays, a lot of researchers are working on Magneto-Rheological fluid for the MR damper.
1.2
Motivation
This thesis concerns the prediction of road condition for the vehicle suspension system. The passive suspension system only controls the damper with one fixed coefficient. Unlike the passive suspension system, the semi-active suspension system controls the damper by the control system. For a semi-active suspension system, the control strategy is fairly important as it affects the performance of the system.
The MR damper is commonly used in a semi-active suspension system. It is a controllable damper controlled by the magnetic field. Thus, the strategy of the coefficient control becomes really important. Nowadays the customers focus more on comfort and safety. A better strategy for the suspension system is able to provide both comfort and safety.
Additionally, MR damper reduces the cost of the system with better performance. The road condition prediction can improve the performance of the semi-active sus-pension system. It helps the system to decide the coefficient of the sussus-pension system. Furthermore, it is important to verify the prediction with true road data. The coeffi-cient of the prediction system determines the quality of experiment result. It should be carefully chosen.
1.3
Objective
• The first one is to simulate the true road data. A model of the road condition is built to provide current data. This data is set as the input of the predictor to test the prediction. Road roughness classification based on the Power Spec-tral Density (PSD) has been proposed by the International Organization for Standardization (ISO). The true road data is built based on this road rough-ness classification. Harmonic superposition method is chosen to build the time domain model.
• The second one is to predict the road condition with measurement data from a road. The measurement data is built based on true road data with random measurement noise. In this thesis, the Kalman filter is chosen to predict the road condition. The prediction filter is then built to get the future data.
1.4
Outline
The next chapter describes the theoretical aspects of suspension system. A primary suspension system is introduced. It gives a fundamental knowledge of the suspension system. There is always compromise between passenger comfort and stability in suspension system. It is impossible to avoid the compromise. A few suspension systems are discussed including semi-active suspension system. This brings in the study of the control strategy and controllable damper.
In chapter 3, simulation of the road roughness is discussed. The harmonic super-position simulation method is chosen to build the road roughness. This simulation method is easy to implement.
Chapter 4 presents the solution to the prediction. The F level road is close to the rough road condition in reality. This kind of roughness is much more common in daily life. A Kalman filter is built to predict this level road roughness. The input of the Kalman filter is the measurement data. The result of the prediction is then evaluated.
Chapter 5 gives out a overview of this thesis. It concludes the main achievement of this thesis and presents recommendations for future study.
1.5
Contribution
This thesis makes the following contributions: • Study the theory of suspension system.
• Simulate the road roughness data based on power spectral density.
• Develop a new way to predict the road roughness with Kalman filter.
Chapter 2
Background
During these years, the automobile industry is growing quickly. Instead of concen-trating on price, customers pay more attention to both comfort and safety than ever. Then, the comfort and safety should be considered in the design of the vehicles. This makes the designing of suspension system becoming trickier.
The suspension system designers have to consider two things. One is the high operability of vehicle. The other is the feeling of smooth ride for passengers. In this chapter, the background of suspension system is provided.
The suspension systems are divided into three categories: passive, active and semi-active suspension systems. And, these three different kinds of suspension systems are discussed and compared.
2.1
Primary Suspension
A conventional suspension system usually consists of two components: a spring and a damper [3]. The spring is chosen depending on the whole weight of vehicle. It is used to pass the force and torsional force between the tire and the vehicle frame. While the damper aims to cancel the impact force from bad road conditions. The primary suspension system isolates the vehicle body from the uneven road. Thus, the vibration is reduced and the safety of the passenger is also guaranteed.
However, it is difficult to decide the constant of the damper in a suspension system. Because it is hard to do the trade-off analysis between comfort and stability, their relationship is shown in Figure 2.1. For an instance, if a higher constant damper is equipped in a suspension system, more stability is achieved but with less comfort and vice versa.
Comfort
Stability
Low Damping High Damping
Figure 2.1: Trade-off between comfort and stability
2.2
Passive Suspension
A passive suspension system, as shown in Figure 2.3, contains two elements, spring and damper [4]. Once the spring and damper are chosen, the character of this sus-pension system is fixed. So a challenge for designers is to make a trade-off between the spring and damper.
The frequency response of a passive suspension system is shown in Figure 2.2. Higher constant of damper equipped in a vehicle, less comfort is felt by passengers while the vehicle passing the uneven road. Most of the energy from the bumpy road is transmitted to the passengers and the cargo in the vehicle. The more stable system is, the more uncomfortable passengers feel. But it is easier to operate the vehicle in this situation
On the contrary, if the designer choses a damper with small coefficient, it results in comfort for passengers. In this case, the stability of the suspension system is conspicuously reduced [5]. With a less stable suspension system, the vehicle behavior is affected, especially when it comes to turns and road condition with side force. The vehicle body roll might happen in some driving situation, for example, accelerating and braking. Designers should try to optimize the character of the suspension system with guidance of the compromise between comfort and stability.
2.3
Active Suspension
0 50 100 150 0 1 2 3 4 5 6 Frequency (Hz) Transmissibility damper constant=0.9 damper constant=0.5 damper constant=0.1
Figure 2.2: Frequency response of primary suspension system
vehicle. The suspension system consists of an electromagnetic actuator, a spring, a piezoelectric accelerometer and an analog control circuit [4]. When the vehicle is running on the road, the piezoelectric accelerometer collects the road condition. It passes the data to the electromagnetic actuator with the analog control circuit. With this data, the actuator can adjust itself according to the road condition. The adjustment of the actuator improves the passenger comfort.
The active suspension system not only dissipates energy, but also gives out force to reduce the vibration. Based on the message given by the accelerometer, the actuator adjusts itself in order to minimize the vibration of vehicle. For instance, it changes the character of actuator and spring to a larger value when the vehicle just starts. Through changing the character, the suspension system dissipates the recoil.
In addition, the system also takes actions when the vehicle reaches certain speed. The system prevents the vehicle from nose-diving by adjusting the spring. Figure 2.4 shows a comparison between a passive system and an active system in two-degree-of-freedom system.
Figure 2.4 is the comparison between passive and active suspension system. The active suspension system has both advantages and disadvantages. The advantage of this system is that it is easy to know how the system works. Within short time, the operator is able to be pretty familiar with the behavior and response to the different situations.
MASS spring Force actuator F MASS spring Damper
Passive Suspention Active Suspension
Figure 2.3: Passive and Active Suspensions [4]
In addition, the active suspension system is not widely used because the cost is high and it requires higher power consumption. And it is commonly used in luxury vehicle.
2.4
Semi-active Suspension
The semi-active suspension system was first proposed by Crosby and Karnopp in the 1970s [8]. This suspension system was applied in the 1980s [9].In an active suspension system, both actuator and spring are controllable. Being similar with the active suspension system, the semi-active suspension system also adjusts the vibration and the height of vehicle frame. The different part is that the system can only adjusts the character of the damper.
The Semi-active Suspension system is simply implemented due to its simple struc-ture [10]. Semi-active suspension system functions with only a little amount of energy from the vehicle. The prospect of applying the system is good.
1 m 2 m t K s K c
Figure 2.5: Semi-active Suspensions
There are two kinds of damper in this system:
• One is continuously variable semi-active suspension system [2]. The coefficient of damper is controlled continuously from the minimum to the maximum value. According to the acceleration, velocity or displacement, the system calculates the corresponding damper coefficient. The system adjusts the damper according to the calculated coefficient. The semi-active suspension system does not need external energy supply device. But the cost of this system is expensive since it requires more sensors than conventional system.
2.5
Control Schemes for a 2 Degree of Freedom
(2DOF) System
In this section, the control schemes for a 2DOF system are introduced. There are three basic ways to control the semi-active suspension system. These control strategies are skyhook control strategy, ground-hook control strategy and hybrid control strategy.
2.5.1
Skyhook Control
Skyhook control is the most commonly used strategy in semi-active suspension sys-tem. The ideal model of skyhook control is shown in Figure 2.6. The damper is connected to an inertial reference in the sky. It is impossible to implement the ideal model in real vehicle. The designer usually uses a controllable damper to reach the similar performance of the idal model in a semi-active suspension system.
M
K
BASE
sky
C
Figure 2.6: Ideal Model of Skyhook Control System
As shown in Figure 2.6, the basic model of skyhook includes a spring with coef-ficient of k and a damper with coefcoef-ficient of C. The velocity of m and base is Vm
and Vb which are upwards. The velocity is defined as positive. For the first situation,
assuming m is separating with a positive upwards. The force given by the skyhook damper should eliminate the vibration. Considering of that, the force should be
F = −C × Vm
V = Vm− Vb > 0
Considering the force should eliminate the vibration, the force given by the con-trollable damper is
Fcon = −Ccon× V
The force given by these two systems should be the same. This means
F = Fcon
Ccon = C
Vm
V
Based on the definition above, Vm and V are positive, then Ccon should equal to
CVm
V . It proves that the semi-active suspension system works as a skyhook system
in this situation. In other situation, the system is separated with a negative velocity. This means both the mass and the base are moving downward. So Vm < 0 ,C > 0,
Ccon < 0, which is physically impossible. The direction is the same as the force
provided by skyhook damper. The semi-active system cannot provide force with negative direction. The ideal solution is to make Ccon = 0. This is also hard to
realize. In reality, the system sets the controllable damper to a minimum coefficient. The capability of system is optimized.
To sum up, the control strategy is
Ccon =
(
high damping, f or V × Vm > 0
low damping, f or V × Vm < 0
This is the simplest control strategy. With this strategy, the skyhook control suspension system is built up with semi-active damper.
2.5.2
Groundhook Control
The damper is connected to the unsprung mass instead of the sprung mass to modify the skyhook control system. This modified system is then defined as ground-hook control system. Although, the system has similar character as skyhook control, the vibration of unsprung mass is reduced and sprung mass is increased. This shows a better capability of isolating the unsprung mass. The structure is shown in Figure 2.7.
And the control strategy is drawn with the same method as skyhook control, which is
Ccon =
(
high damping, f or V × Vb > 0
BASE M b K m K ground C
Figure 2.7: Groundhook Control System
2.6
Magneto-Rheological Dampers
In recent years, Magneto-Rheological (MR) damper has been approved to be used in vibration control applications [11]. MR damper is very popular among all kinds of controllable dampers. It is designed as part of the semi-active suspension system. In this section, the MR damper is introduced as well as the background and application of MR Fluids.
2.6.1
Magneto-Rheological Fluids
MR fluids and Electro-Rheological(ER) fluids are both smart materials. The char-acters of these fluids change due to external reason. They respond to magnetic and electric field separately. ER fluids are discovered before MR fluids. They are highly sensitive to high yield stress. Due to this reason, MR fluids are more widely used than ER fluids [11].
MR was invented by Jacob Rabinow at the US National Bureau of Standards in 1948 [12]. MR fluid is a type of oil which changes state from liquid to semi-solid. Compared to ferrofluids, the MF fluids particles are much bigger. The size of ferrofluids particle is 1 ∼ 2microns and the size of MF fluids particle is 20 ∼ 50microns. There was little research about MR fluids until the early 1990s [13].
The MR fluids are divided into four kinds according to their compositions and capabilities [14]:
• First one is micron magnetic particles with non-magnetic carrier MR fluids. It is a typical MR fluid. The magnetic particles are paramagnetic materials. In this case, magnetic particles are usually carbonyl iron dust. The percentage of magnetic particles in the fluid is from 20% ∼ 40%, even up to 50%. The diameter of particles is usually from 0.1um to 100um. The typical range is from 3um to 5um. The corresponding carrier oils are silicone oil, mineral oil, synthetic oil, water and glycol.
• The second type of MR fluid is Nano-meter magnetic particles with non-magnetic carrier MR fluids. The fluid is composed of Nano-scale magnetic particles with non-magnetic carrier MR fluids.
• The third type is magnetic particles with magnetic carrier MR fluids. The fluid is consisted of micron magnetic particles and magnetic carrier.
• And there is another kind of MR fluid called non-magnetic particles with mag-netic carrier MR fluids. The fluid is composed of Nano-scale non-magmag-netic particle sand magnetic carrier MR fluids.
MR fluids display Newtonian-like behavior when deactivated. Without the mag-netic field, the particles disperse to the fluid without direction. When applied with a powerful magnetic field, the particles align themselves along the line of magnetic flux. It is shown in Figure 2.8. The particles form the particles chains. These chains restrict the movement of the fluid. They lead to an increase of the viscosity, plasticity and yield stress. The chains break with certain amount of shear stress. The shear stress is defines as the yields stress.
However, the reason of viscosity is still unsolved. There are two theories which are phase transformation theory and dipole moment under field theory. One of the theories is widely accepted while the other theory only explains part of the effect of MR fluids. Most of scientists believe that each particle is polarized to magnetic dipole with the fluency of magnetic field. With the magnetic attraction, the dipole attracts each other into a line. The strength of this effect is determined by many factors. This theory is based on that all the particles are in regular shape. In fact, the particles are in irregular shape. This fact is important when analyzing of the motion of the particles.
2.6.2
Modes of MR Operation
It is known that the MR fluids have three application modes which are valve, direct shear and squeeze modes. All of these modes have been applied on MR damper. Each of them is designed for different purpose.
Figure 2.8: MR fluid Ferrous Arrangement when applied in magnetic field [15]
The flow is directed perpendicularly between two magnetic pole plates [16]. The flow of the fluid absorbs the energy. The speed of the flow is determined by the variation of the magnetic field. With the changing of the magnetic field, the MR damper becomes controllable.
Figure 2.9: MR fluid in valve mode with an applied magnetic field [15]
It is widely known that the direct shear mode is also very popular. The shear mode is shown in Figure 2.10. The plates are perpendicular to the magnetic field. The particle chain prevents relative motion of the pole plates. The particles try to prevent the chain from broken. The MR fluid only reduces the relative motion with the normal amount. The operation is deactivated when it is not applied in magnetic field.
While the shear mode does not work in the circumstance that requires high stiff-ness with less displacement, squeeze mode is designed for these circumstances. The squeeze mode utilizes the analogy of a buckling columnar structure.
Figure 2.10: MR fluid in shear mode with an applied magnetic field [15]
After activated, the MR fluid hinders the energy with the power magnetic chain. Normally, the magnetic chain tries to stay in line. The chain breaks when applied with certain amount of displacement. Figure 2.11 gives a clear procedure of the squeeze mode operates.
Figure 2.11: MR fluid in squeeze mode with an applied magnetic field [15]
2.6.3
MR Dampers
Nowadays, MR damper is the most commonly usage of the MR fluids. Its character makes it perfectly suitable for the semi-active suspension system. It changes rheo-logical properties due to the changes of the magnetic field. Since the changing of the rheological properties makes the damper easy to be controlled. The design of control system is based on velocity or acceleration. With this quality, the semi-active system has more advantages than other suspension systems.
model is simple and the design of the magnetic circuit is easy to implement. Figure 2.12 shows the structure of the basic MR damper with valve mode.
Figure 2.12: MR damper operating in valve mode [6]
Chapter 3
Road Roughness Data Realization
3.1
Road Roughness Characterization
Road roughness has influence on passenger comfort, operation of driver, maintenance costs, fuel consuption and fatigue, etc. It is widely known that the cost on fuel is less when the road is smoother [18]. The random process of road roughness char-acterization is commonly assumed as a zero-mean, stationary and ergodic Gaussian process [19].
The purpose of this thesis is to predict the road roughness data in time domain. The prediction is based on an uneven road surface. This chapter describes realization of road roughness data. The road roughness data is set as the input of the predictor which is simulated in time domain in this thesis.
There are two menthods to obtain these data. One is to record the data by measuring the true road. The other one is to simulated road roughness. It is built based on PSD when the measurement data are not provided. Here, road roughness is realized by the simulation.
3.2
The International Roughness Index
Simulation of road roughness is important for analyzing and evaluating the vehicle ride quality. Some organizations have made different classifications of road roughness in the past. The PSD method and International Roughness Index (IRI) method in evaluating road roughness are well-accepted in classifying the road condition.
The IRI is the most popular road roughness index. It has been adopted by most road authorities around the world [20]. The mathematical model of IRI is a quarter-car vehicle model.
3.3
Road Roughness Power Spectral Density
A road roughness classification based on the PSD has been proposed by the Interna-tional Organization for Standardization (ISO).
The general form of fitted PSD is
Gd(n) =Gd(n0) ·(n/n0) −w (3.1) or Gd(Ω) =Gd(Ω0) ·(Ω/Ω0) −w (3.2) where n0 (n0= 0.1cycle/m) is the reference spatial frequency, Ω0 (Ω0= 1 rad/m) is
the reference angular spatial frequency, n is the spatial frequency, w is the exponent of the fitted PSD and Gd(n0) is the displacement PSD.
When the w = 2, the
Gd(n) =Gd(n0) ·(n/n0) −2
(3.3) Considering the influence of velocity v, the spatial frequency spectrum is usually transformed into a temporal frequency spectrum.
Gd(f ) =Gd(n) /v (3.4)
As f = v × n,
Gd(f ) = n20
v
f2Gd(n0) (3.5)
Gd(f ) depends on velocity, temporal frequency and the value of displacement PSD.
According to the value of displacement PSD, the road roughness is divided into eight degrees. The classifications are shown in Table 3.1 [22].
Table 3.1: Road Classification
Road class
Degree of roughness Gd(n0) (10−6m3)
Lower limit Geometric mean Upper limit
3.4
Methods of Realizating Road Roughness
There are many menthods to stimulate stochastic excitation in time domain to form road roughness. For example, Harmonic superposition method, integral white noise model, AR model and inverse Fourier transform model are the commonly used.
3.4.1
Harmonic Superposition Method
The main idea of the Harmonic superposition method is an approximation of a target random process by the Discrete Spectral. In temporal frequency (f 1, f 2), the power spectrum density is Gd(f ). The variance σ2 of this interval is simulated by Gd(f ).
σ2= Z f2
f1
Gd(f ) df (3.6)
In order to derive the discretion formula, the interval (f 1, f 2) is divided into n intervals. The value of middle frequency fmid−i is considered as the power spectrum
of each small interval. Then the function is transformed as
σ2=
n
X
i=1
Gd(fmid−i)·4fi (3.7)
The standard deviation PSD of each small interval is expressed based on sine function as
σ=p2Gd(fmid−i) 4fi·sin(2πfmid−it+θi) (3.8)
Then the stationary-zero mean value-stochastic process is simulated by the sum of the sine function of each small interval.
q (t) =
n
X
i=1
p
2Gd(fmid−i) 4fi·sin(2πfmid−it+θi) (3.9)
where q(t) is the road roughness height and θ is a uniform random number in the interval [0, 2π].As n tends to infinity, the accuracy increases. This means large amount of calculation is needed to obtain an acceptable signal. Z. Yonglin proposed that the operational frequencies range from 0.139 to 39.3Hz in vehicle analysis [23].
3.4.2
The AR Model
The autoregressive-moving-average model is
q (k) = −
m
X
k=1
ak·q (n−k)+w(n) (3.10)
where ak is parameter to be determined, m is the order of AR model and n is the
samples and a white noise process w(n). The mean of the white noise process w(n) is zero. The variance is of w(n) is σ2. After Z-transform, the equation is
H (z) =Q(Z) A(Z)= 1 1+Rk=1m ak·Z−1 = 1 (1−z1Z−1) (1−z2Z−1) · · · (1−zmZ−1) (3.11)
When all the poles z1, z2, . . . , zm are inside the unit circle, the signal of the AR
model is stable. According the property of the autocorrelation function, the value of Rq(m) is Rq(m) =E [q (n) q (n+m)] = E ( q (n) " − m X k=1 ak·q (n−k+m)+w (n+m) #) = − m X k=1 akRq(m−k)+E[q (n) w (n+m) ] (3.12) Then the Rq(m) is Rq(m) = − m X k=1 akRq(m−k)+σ2, if m = 0 (3.13) or Rq(m) = − m X k=1 akRq(m−k), if m 6= 0 (3.14)
If m = 1, 2, . . . , p, the equation is transformed into matrix below. Rq(0) Rq(−1) Rq(−2) · · · Rq(−p) Rq(1) Rq(0) Rq(−1) · · · Rq(−p + 1) Rq(2) Rq(1) Rq(0) · · · Rq(−p + 2) .. . ... ... . .. ... Rq(p) Rq(p − 1) Rq(p − 2) · · · Rq(0) 1 a1 a2 .. . aq = σ2 0 0 .. . 0 (3.15)
The Rq(m) =Rq(−m), the matrix is transformed as
Rq(0) Rq(1) Rq(2) · · · Rq(p) Rq(1) Rq(0) Rq(1) · · · Rq(p − 1) Rq(2) Rq(1) Rq(0) · · · Rq(p − 2) .. . ... ... . .. ... Rq(p) Rq(p − 1) Rq(p − 2) · · · Rq(0) 1 a1 a2 .. . aq = σ2 0 0 .. . 0 (3.16)
This is the Yule—Walke function. If the value of each Rq(q = 1, 2, . . . , p) is known,
the ak can be calculated from the Yule—Walke function. It is solved by
3.4.3
The Inverse Fourier Transform Model
The power spectrum density of road roughness is replaced by the square of Fourier transform of random sequence.
Gd(fk) = 2 N fs |Xk| 2 (3.17)
where k = 0, 1, 2, 3, . . . , N/2, Xkis a discrete Fourier transform of road random signal
in time domain, fkis the frequency of the Fourier transform, fsis the sample frequency
and N is the number of sample points. Therefore,
|Xk| =
r
Gd(fk) N fs
2 (3.18)
|Xk| is the magnitude-frequency characteristic of Xk. For an n-point discrete signal,
the discrete Fourier transform is a complex number. Xk is transformed as
Xk = |Xk| ejϕk (3.19)
where ϕk is a phase of Xk.
The first N/2 + 1 value of ϕk is its phase in the Inverse Fourier Transform model.
To simulate the road roughness based on inverse Fourier transform, |Xk| and ϕk are
combined. Then, xt is obtained by inverse Fourier transform of Xk.
3.4.4
The Integral White Noise Model
The road roughness is considered as the result of a white noise filtered by an appro-priate filter. The mathematical model of a single-point time domain model is
˙
q (t) = w (t) (3.20)
And the variance of w(t) is defined as
σ2w = E[w − E[w]]2 = 4π2Gd(n0) n02v (3.21) So, ˙ q (t) = 2πn0 p Gd(n0) v·w1(t) (3.22)
where w1(t) is the Gaussian white noise with σ2 = 1, q(t) is the road roughness and
v is the velocity of the vehicle. The solution of the differential equation is the road roughness in the vertical direction. There are two the road roughness data. One uses an integrator, and the other uses a forming filter.
3.5
Road Roughness Realization Results
Table 3.2: Roas roughness Simulation Parameters
Road class Parameters
Velocity(m/s) Displacement PSD (×10−6m−3)
C 15 256
F 15 16384
large amount of calculation. But it is widely used in different road situations. It provides good simulation precision. In this thesis, the road roughness is re-constructed by this method.
In this thesis, the C and F degrees of road roughness are simulated. The road simulation parameters are shown in Table 3.2. The simulation results are shown in Figure 3.1 and Figure 3.2.
0 50 100 150 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
The length of the road (m)
Road roughness (m)
Road roughness in time domain
Figure 3.1: C degree road roughness
0 50 100 150 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
The length of the road (m)
Road roughness (m)
Road roughness in time domain
Chapter 4
Experimental Approach and
Discussion
4.1
Introduction of Kalman Filter
The Kalman filter is a remarkable method to predict and estimate the state of a stationary process by minimizing the mean square error [25].
The results of Kalman filter have very small error. The Kalman filter has appli-cations in spacecraft orbit determination, estimation and prediction of target trajec-tories, simultaneous localization and mapping, etc [25].
The discrete Kalman filter cycle is shown in Figure 4.1. It consists of two steps: • Prediction step. In prediction step, the goal is to obtain the predicted state for
next time step by forward projection of the current state and error covariance estimates [26].
• Correction step. In correction step, the aim is to correct the estimate state and error covariance.
4.2
Modelling
In this thesis, the random road roughness is a zero-mean, stationary and ergodic Gaussian process. The road roughness, xk, is measured by the displacement sensor at
time k. The discrete-time process can be expressed as the linear stochastic difference equation:
xk= Axk−1+ wk−1 (4.1)
with measurement data z,
zk= Hxk+ vk (4.2)
Prediction step: project the state and the priori error covariance
Correction step：correct the estimate state and
the priori error covariance
Initial value
Figure 4.1: The prediction and correction steps of Kalman filter
the measurement zk [26]. they are assumed as constant, here. The random variable
wk is a zero-mean white noise with normal distribution which is defined as
wk ∼ N (0, Q)
vk is the noise from measurement. It is a zero-mean white noise with normal
distri-bution independent of wk. It is defined as
vk∼ N (0, R)
Where Q is covariance of the process noise and R is covariance of the measurement noise.
In this model, the road roughness xk is assumed as a linear process. The
dis-placement sensor assembled on the tires is used to measure the disdis-placement. This displacement is assumed as the road roughness over time. z is road roughness mea-surement data with noise. The parameter H = 1 and A = 1 .
x represents the road roughness and z is the measurement value of road roughness. The prediction update step can be expressed as
ˆ
Xk− = A ˆXk−1 (4.3)
Pk− = APk−1AT + Q (4.4)
error covariance. It is used to calculate the Kalman gain in the correction step. The correction update steps are
Kk= Pk−H T(HP− k H T + R)−1 (4.5) ˆ Xk = ˆXk−+ Kk(Zk− HkXˆk−) (4.6) Pk = (I − KkH)Pk− (4.7)
where Kk is the Kalman gain that minimizes the posteriori error covariance, ˆXk is a
posteriori estimated state, Pk is posteriori error covariance and I is unit matrix.
4.3
Road Condition Predicting Result
The F level of road roughness was assumed as the true roads data in the predic-tion experiment predicpredic-tion. The vehicle speed is assumed as 15m/s over an entire distance of 20m. The response time of the Magneto-rheological Fluids is within six milliseconds. So, the prediction frequency is assumed as 100Hz. If the prediction frequency is higher, it requires the MR damper with shorter response time which is now impossible to implement. The parameters are shown in Table 4.1.
Table 4.1: F level load roughness experimental parameters
Parameters Values
Velocity 15m/s
Updated state estimate 0.01s
Q covariance of the process noise 81 − 250mm2 R covariance of the measurement noise 25 − 900mm2
Gd(n0) 16384 × 10−6m3
Figure 4.2 and Figure 4.3 show the performance of Kalman with the varied process noise covariance Q and measurement noise covariance R on predicting the F level road roughness.
In Figure 4.2, it shows that Kalman filter is more accurate with a lower process noise Q on road prediction. According to Figure 4.3, Kalman filter predicts road roughness less accurate in noisy measurement environment(R). The predicted road roughness approaches the actual value quickly. Figures 4.2 and 4.3 also indicate the actual road roughness character.
Table 4.2 shows the mean error with variant measurement noise covariance R and process noise covariance Q in F level road roughness. These coefficients are chosen to test the performance of the predictor.
According to Figures 4.4, Kalman filter performans better on predicting F level road roughness with assumed Q and R.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.3 −0.2 −0.1 0 0.1 R1,Q1 Time [s]
Road roughness (m) Predicted road roughness
True road roughness
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.3 −0.2 −0.1 0 0.1 R1,Q2 Time (s)
Road roughness (m) Predicted road roughness
True road roughness
Figure 4.2: The performance of Kalman with measurement noise covariance R1=25 mm2 and varied process noise covariance Q1=100 mm2,Q2=250 mm2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.2 −0.1 0 0.1 0.2 R3,Q3 Time (s) Road roughness (m)
Predicted road roughness True road roughness
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.2 −0.1 0 0.1 0.2 R4,Q3 Time (s) Road roughness (m)
Predicted road roughness True road roughness
Figure 4.3: The performance of Kalman with process noise covariance Q3=100 mm2
and varied measurement noise covariance R3=100 mm2,R4=900 mm2
convergence very fast with different R values and Q values. It takes no more than 10 iterations. This advantage makes the predictor applicable.
Table 4.2: Mean error with variant R values and Q values R (mm2) Q(mm2) error Mean (m) 25 100 0.0164 25 250 0.0180 100 100 0.0181 900 100 0.0309 25 81 0.0152 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.2 −0.1 0 0.1 0.2 R5,Q5 Time [s] Road roughness [m]
Predicted road roughness True road roughness
Figure 4.4: The performance of Kalman with the process noise covariance Q=25mm2
and measurement noise covariance R=81mm2
4.4
Passenger Comfort Comparison
The vehicle runs through the road at speed of 15m/s. The process noise Q is 81mm2 and the measurement noise R is 25mm2. The passenger comfort comparison is shown
in Figure 4.6.
In Figure 4.6, the red line shows the absolute value of true road roughness. This road has three large bumps and three valleys. The maximum peak is 20cm. The passenger feels uncomfortable on this road.
0 5 10 15 20 25 30 35 40 45 50 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Iteration vaule kalman gain R1,Q1 R1,Q2 R3,Q3 R4,Q3 R5,Q5
Figure 4.5: Kalman gains with different R values and Q values
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (s) Road roughness (m)
absolute value of true road roughness error between predicted and true data
Chapter 5
Conclusion
The implementation of the research is constructed as follows. Firstly, the data of true road and the model of Kalman filter are built. Then Kalman filter is applied to predict the road condition. The result shows the passenger comfort is optimized.
A new solution of road condition prediction is proposed in this thesis. It needs less calculation and only real-time data. The system response is fast. The Kalman filter is also capable of predicting the real-time road condition with accurate response. This leads to a better performance of the suspension system. The system provides more comfort with less trade off to stability.
This thesis realizes the road roughness model to simulate the road condition as the input of road condition predictor. The random process of the road roughness is a stationary random process.
Predictor is built based on Kalman filter. The simulation result is fairly close to the true road data. The prediction of road condition is considered as predicting the roughness of the road.
The advantage of the Kalman filter is that the filter only needs the current data. It gives the Kalman filter an outstanding performance in the real-time prediction. The calculation is fast and easy to implement. This reduces the energy consumption of the control system.
The passenger comfort is proportional to the error. A passenger feels more com-fortable when the error is small. According to the result of prediction, the comfort of passengers is improved significantly with the proposed method.
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