Implementation of a control system for an adaptive damper in a quarter car context

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NTNU Norwegian University of Science and Technology Faculty of Information Technology and Electrical Engineering Department of Engineering Cybernetics

Marius Hamre Nordrik

Implementation of a control system for an adaptive damper in a quarter car context

Master’s thesis in Cybernetics and Robotics Supervisor: Ole Morten Aamo

June 2021

Master ’s thesis

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Marius Hamre Nordrik

Implementation of a control system for an adaptive damper in a quarter car context

Master’s thesis in Cybernetics and Robotics Supervisor: Ole Morten Aamo

June 2021

Norwegian University of Science and Technology

Faculty of Information Technology and Electrical Engineering

Department of Engineering Cybernetics

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Abstract

This work presents a complete system for the control of a magnetorheological adaptive damper in a quarter car environment. The system implements a current controller, force controller, and force reference generator on an embedded microcontroller. A Kalman filter is also implemented to provide estimates of unmeasured states which are necessary for the control of the damper. The entire system is implemented on a printed circuit board, and with components ready to put it on a Formula Student type racecar. All of the components are tested together on a physical quarter car rig, to give a realistic representation of how the force controller tracks the references. It is shown that the overall system behaves ade- quately, but that the system has a significant error in force reference tracking. Several areas of improvement, and further work are disclosed, like evaluation of the reference generator, improvements to the control systems, and improvements to the mechanical systems which can improve the performance of the force tracking.

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Preface

Towards the end of the work towards this thesis, I started to reflect on my past years as a student at NTNU. A topic I strongly regret thinking about right before I was supposed to fall asleep.

However, I want to start by acknowledging the opportunity that NTNU has given me to meet like-minded peers. The ability to learn through practical assignments and to choose topics I find interesting.

Next, I want to emphasize the role which Revolve NTNU has played in my education.

Instead of making small things on my own, I am able to build amazing racecars together with others. I started in Revolve as a Group leader for the Vehicle dynamics & control systems group. I learned so much about how to interact with other people, especially in the context of technological development. When I applied for the position, I would never have guessed how much more I would get the opportunity to learn.

I have been exposed to students with impressive skills, dedication, and willpower. Several of which have inspired me, and I believe, will continue to do so.

I have always been a curious person, and during my time in Revolve, I have taken nearly every possibility to learn something new. I will list some of the learning opportunities I have gotten, and things I have worked on.

• Lead a group of awesome people through conceiving, developing, designing, pro- ducing, and testing amazing technological systems. Be it a torque vectoring system, adaptive dampers, slam, or autonomous control.

• Design the kinematics of a race car suspension.

• Contribute on torque vectoring, and motor control.

• Designing and manufacturing a printed circuit board.

• Designing an adaptive damper, using FEM and optimization techniques to maximize the performance.

• Analyzing telemetry data from the racecar during the summer, for the purpose of debugging, tuning, and getting the most out of the car.

• Experience the emotional roller coaster of testing a racecar, during both day shifts and night shifts. Having the car experience more and less severe breakdowns, and using all of our skills and willpower to get it back up and running.

• Feeling the enormous team spirit through this emotional roller coaster, and always trying to get the best out of each other.

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• Celebrate that we came second in one of the world’s biggest engineering competi- tions.

and so much more.

I listed these, not for others to see what I have done, but for them to understand what they can do. I hope that future students at NTNU will see the unique opportunity that they are given by the technical student organizations.

I think that student organizations add something which is hard for a university to provide, teaching students to be independent. During primary school, high school, and university there is always a clear expectation. You have your learning goals, deadlines, etc.. In student organizations nothing is clear, but you learn to figure it out.

Now that my stay at NTNU comes to an end, I want to thank my study buddies and my team members at Revolve. I want to thank my supervisor Ole Morten Aamo for the support and for asking questions that make me feel stupid and over-complicating. I would also thank Lars Cornelis Marion van der Lee, not because he attended every meeting during the project- and master thesis, but because he has been my friend since the first weeks at NTNU, has kept up with me through three years at Revolve, and continues to be a solid friend.

The last years have been some of my best, and my hardest years. I have changed and developed more than I could have imagined. I am utterly grateful for both the friends I have gained through the last years and painfully sorry for the ones that have passed.

I hope that this thesis is of help for the future development of the adaptive damper project and that it might be an interesting read.

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List of Tables

3.1 Table of the test runs used for parameter estimation . . . 24

3.2 Table of quarter car rig sensors . . . 25

3.3 Table of the standard deviation of measurements . . . 32

4.1 Table of the current controller gains . . . 40

A.1 Table of the system parameters of the quarter car rig and simulation environment . . . 65

A.2 Table of Bingham model parameters . . . 65

A.3 Table of the modified Dahl model parameters . . . 66

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List of Figures

1.1 Atmos (left) and Nova (right), Revolve NTNU’s latest Driverless and Elec- tric vehicle respectively, depicted in front of Hovedbygget NTNU Trond- heim. . . 1 1.2 Schematics of the hierarchical control system model . . . 6 2.1 An overview of damper categories and their properties, from Savaresi et al.

[16, Chapter. 2]. . . 10 2.2 A cross section view of the damper used in this work. . . 11 2.3 Graph of the analytical(calculated) and linearly approximated accumulator

chamber forces in the operating range of the damper. From Nordrik [1] . . 12 2.4 Relation between yield strength and magnetic field strength for Lord MR-

122 from the corresponding datasheet [18] . . . 13 2.5 The colored area shows an example of the forces an MR damper is able to

produce. . . 13 2.6 Coarse illustration of a magnetization curve for a steel. From [24] . . . . 15 2.7 Illustration of the magnetic circuit inside the damper piston . . . 15 2.8 Example of force velocity response of the Bingham Damper Model. From

Nordrik [1]. . . 16 2.9 Illustration of a quarter car model. From Abid et al. [31]. . . 18 3.1 The quarter car test rig. Some sensors are missing in this illustration, like

damper load cell and tire deflection. . . 24 3.2 A 3D view of the Damper Control Unit (DCU) . . . 25

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3.3 The tire forces implemented in Simulink. . . 31

3.4 An overview of the road profile module. . . 31

3.5 The sensor mocking of the damper extension in Simulink . . . 32

3.6 An simplified overview of the control system structure. . . 33

3.7 Current controller in Simulink . . . 33

3.8 An overview of the force controller structure. . . 34

3.9 The current required to obtain a desired damper force given a damper velocity. The unit of the currents in the contour plot is Ampere. . . 36

3.10 An overview of the force reference generator structure. . . 36

4.1 Current step response with constant duty cycle. One step from 0.0 Ato 0.5 Aand from0.0 Ato2.5 A . . . 40

4.2 (a) shows the relation between steady state duty cycle and current, together with a linear fit. (b) shows the error between the measured values and the fit, relative to the maximum duty cycle, which was 0.05. . . 41

4.3 Isolated current controller tests. (a) shows a current controller step response. Steps between0.3 Aand1.3 A. (b) shows the current controller tracking a sinusoidal signal with an amplitude of0.5 Aand a DC offset of 0.8 A. . . 42

4.4 The damper current and its reference during the use of the Force controller, with a linear damper reference (a) and with the ground hook method (b). . 42

4.5 The estimated damper extension and damper velocity plotted together with their true values, in Figure 4.5a and Figure 4.5b respectively. Results from simulation. . . 43

4.6 The estimated damper extension error and damper velocity error, in Figure 4.6a and Figure 4.6b respectively. Results from simulation. . . 44

4.7 The estimated bias plotted together with the true bias. (a) shows the initial response while (b) shows how the bias estimate behaves when it has converged. Results from simulation. . . 44

4.8 The damper velocity as estimated by the Kalman filter and the smoothing spline. Detail view in (b) . . . 45

4.9 Damper velocity as estimated by the Kalman filter, while the accelerometers were bandwidth limited), and the smoothing spline(Post-process velocity). . . 46

4.10 A plot of the estimated relationship between damper force and damper velocity at different measurement delays. The input is a sine excitation of the damper and the delays are given as a percentage of the period. . . 46

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4.11 (a) shows a run with no current, thus defining the minimum force the damper can exert. (b) shows the same run with an approximate controllable region drawn onto the graph. The notes show the approximate for needed to overcome the static friction and hysteresis in the damper. . . 47 4.12 The force measured directly by the load cell force plotted against the

damper extension. The marked regions show where the damper force is zero, despite movement in the damper. . . 48 4.13 The damping force plotted against the damper velocity, after the pressure

in the damper is increased. . . 48 4.14 The damper force and reference plotted against damper velocity. The fig-

ures show results where the reference has a damper coefficient of1500 N s m⁻¹, 2500 N s m⁻¹,4000 N s m⁻¹and6000 N s m⁻¹, in that order. . . 49 4.15 Root Mean Square Error (RMSE) of the force controller following linear

damper references at different damping coefficients. . . 50 4.16 Damper current and damper force plotted with their references. Following

a linear damper reference with a damper coefficient of4000 N s m⁻¹. . . 51 4.17 Results fro running the damper map controller. (a) and (b) show the force

reference and the measured damper force for a damping coefficient of 1500 N s m⁻¹ and2500 N s m⁻¹, respectively. In both cases the initial force was60 N. . . 52 A.1 Inputs used to tune the Kalman filter. Detail views to the right . . . 64

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Acronyms

ADC Analog-to-Digital Converter. 24, 26, 32 CAN Controller Area Network. 25

DCU Damper Control Unit. vi, xi, 4, 6, 24–26, 31, 37, 50, 53 ECS Electronically Controlled Suspension. v, 2, 3

FEB Frequency Estimation-Based Controller. 3 LPV Linear parameter-varying. 3

MPC Model Predictive Control. 55

MR Magnetorheological. xi, 2, 4, 13, 15, 16, 36 MRF Magnetorheological fluid. 3, 12, 15, 16 PCB Printed Circuit Board. 24

PI Proportional Integral. 33

PID Proportional Integral Derivative. 33, 54, 55 PWM Pulse Width Modulation. 24

RMSE Root Mean Square Error. vi, xiii, 18, 29, 43, 45, 50–52

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Chapter 1 Introduction

This thesis is written as a part of the Formula Student team, Revolve NTNU. Formula Stu- dent is an engineering competition where student teams from all over the world compete not only to build the fastest car but also to learn how to be better engineers. Each year Re- volve NTNU develops a four-wheel-drive electric racecar to compete in Formula Student, with a goal to be among the top three teams. With teams from highly ranked universities, the competition is fierce, and the cars feature advanced technologies. One of the technologies being developed at Revolve NTNU is an electronically controlled suspension using semi-active dampers. [1]

Figure 1.1: Atmos (left) and Nova (right), Revolve NTNU’s latest Driverless and Electric vehicle respectively, depicted in front of Hovedbygget NTNU Trondheim.

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The semi-active dampers are based on the use of Magnetorheological (MR) fluids. MR fluids are types of fluids that change properties when exposed to magnetic fields. The fluid used in the damper increases its apparent viscosity when exposed to an increasing magnetic field. This allows for the force of the damper to be controllable. There are several uses for such fluids which are already proven, automotive clutches, seat dampers, prosthetic, and shock absorbers. The use of Magnetorheological (MR) fluids is however not widespread and is not seen in many commercial vehicles. [2]

1.1 Motivation

Traditional suspension systems suffer great compromises between handling and comfort, and between straight-line performance against cornering performance [3, Chapter 22].

Electronically Controlled Suspension (ECS) can mitigate these tradeoffs and in addition offer benefits like driver aid by changing their properties in a real-time manner.

The motivation behind the design of a semi-active system is to get the performance gain of an ECS without the complexity and cost of a fully active suspension. The semi-active damper is implemented using a magnetorheological fluid which offers responsiveness and low price but suffers from nonlinearities and hysteresis compared to other alternatives.

These effects have to be taken into account for efficient control of the dampers [4, 5].

1.2 State of the art

1.2.1 State of the art on force control

This section is adapted from the prequel to this work [1].

The control of adaptive dampers is a field that has had several publications in later years, the applications range from stabilizing buildings during seismic events to control of vehicle suspension or driver seats. To effectively use the dampers a form of control system needs to be implemented. The control system can be either totally reactive or use some kind of model for predicting the required input.

Most literature found focuses on using models to predict the required input, this has the benefit of low response time and also the fact that the number of measurements can be reduced if feedback measurements can be omitted. Several articles have been released regarding the modeling and parameter estimation in adaptive dampers. Most of them focus on using models known for their capability in hysteretic modeling like the Bouc–Wen model of hysteresis and modifying them to allow varying hysteretic behavior. Spencer et al. [4] is one of the works showing the application of known hysteretic models and how they can be adapted to model the output force based on input current and velocity measurements. It is also shown that simpler models like the Bingham model are able to model the damper forces, albeit without considering the hysteretic behavior. Other articles also propose the use of black box models instead of grey box models, Savaresi et al. [6]

and Jin et al. [7] are examples of this. The former uses an NARX model with a neural network as the parametric function and compares it to grey box models like the Bingham

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and Bouc Wen model, while the latter uses a wavelet shrinking algorithm. A comparison of several dynamic algebraic models can be found in S¸ahin et al. [8].

The input-output relationship is useful for simulating the system but does not directly synthesize a controller for the damper. Several methods for creating controllers for adaptive dampers are found in the literature, with varying structure and complexity from simple feed-forward schemes to robust full car models used as reference models. De Jesus Lozoya-Santos et al. [9] presents a linear parameter varying approach to controlling the damper, assuming a quarter car model to describe the system dynamics, thus being an example of a holistic approach. An example of an approach where only the damper and its behavior is looked at is presented in publications like Kasprzyk et al. [10], where an input-output model is fitted to test data and the inverted algebraically to give a relationship from desired force to input current.

1.2.2 State of the art force references

As mentioned in the previous section the approach of modeling the damper as a part of a bigger system is described in De Jesus Lozoya-Santos et al. [9], and also Lozoya-Santos et al. [11]. The work lists several reasons for combining the damper model with the model of the suspension and use one controller. Mainly the benefit is that the mapping from desired force to current is removed, and the desired current is found directly. They also show how a Linear parameter-varying (LPV) controller and a Frequency Estimation-Based Controller (FEB) controller can be used to attain the desired suspension behavior.

Approaches also use different suspension models, ranging from simple linear quarter car models to more complex and nonlinear full car models [5]. Simpler state of the art control strategies like Ground hook, skyhook, and a combination of these, hybrid hook has been demonstrated in several works [5, 12, 13]. An interesting concept is also the one given by Krauze and Kasprzyk [12], where a skyhook control strategy is used for the front suspensions, the front suspension is also used to sense the road, allowing the rear suspension controller to use this information for preview control, without the need for any preview sensor.

1.2.3 Revolve NTNU’s Electronically Controlled Suspension (ECS) research

Revolve NTNUs development of an Electronically Controlled Suspension (ECS) started with the development of a concept using servos to adjust the valve of a commercial damper.

A master’s thesis, Devold [14], was written on the implementation of the embedded control system for this damper. The concept suffered from slow actuation and was thus only feasible for setting static damper presets. The next year the concept was iterated on to use solenoids to actuate the damper valves, but the solution was deemed unfeasible due to electromechanical limitations.

A new concept was developed during the 2019 season, utilizing a damper based on the usage of a Magnetorheological fluid (MRF). A prototype damper was produced and dynamometer tests were performed to evaluate the adaptive damper with constant currents.

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The main focus was to validate the mechanical design of the damper. A quarter car jig was also created, and used to test the effects of traditional dampers on suspension dynamics.

This work is presented in Vigerust [15]. However, the adaptive damper was not tested in the quarter car rig, and no control of the adaptive damper was performed. A inital version of the Damper Control Unit (DCU) was also designed, but due to prioritizations within the organization the control unit was untested.

During last season the author of this work, together with Christian Trandem, designed a new MR damper based on the results and experiences from the year before. This damper is currently being produced and is not finished yet. Therefore the damper used in this work is the one from 2019. A new damper control unit was also designed and tested.

During the first half of this season, when writing the project thesis, the MR damper was mounted on the quarter car rig together with a functioning DCU and initial tests of the damper were performed, mainly with constant currents. A methodology was developed for gathering data from the quarter car rig and use it to parameterize models of the adaptive damper. These models were used to simulate a simple adaptive damper and to synthesize controllers to control the damper force.

1.3 Scope

The goal of this work is to implement and test a control system for following a desired damper force in the context of a quarter car system. In the prequel to this work, Nordrik [1] a methodology was developed for estimating parameters for several damper models, and a control system for tracking the damper force was tested in simulations. This work will continue off that basis by adapting the approach to work in the real world, and a virtual, quarter car rig and implement the necessary components needed.

The initial goal was to focus on the higher-level control systems responsible for finding force references for the dampers. Such force references can have different purposes, like optimizing grip, comfort, or a combination of these. However, the goal was modified to the one stated earlier due to two main factors. Firstly, it was deemed more important for the future research and development of Revolve NTNU’s adaptive dampers, that the low-level control systems work satisfactorily. The reason for this being that it should let the next season’s team focus on the force reference generator, rather than being forced to improve both the force controller and the reference generator.

Secondly, as there often is, there was a greater workload to be done to get the force controller working than first expected. The damper is Revolve NTNU’s first prototype of a MR damper and has surpassed its lifetime. A new revision has been designed, but several factors, including the ongoing pandemic, have delayed the production. Thus the initial prototype had to be reused. As expected for an early prototype, the damper has several design and production issues. The damper has been partly disassembled and remounted several times. In this process, a new leak has been caused, which given the current design, can’t be repaired. As shown in the results, other quirks of the damper also surface after more testing.

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The chosen control system hierarchy is given in Figure 1.2, the previous work implements the Force Controller and tests it in a simulated damper dynamometer. This work will test the force controller on a physical and a simulated quarter car rig. This work will also implement the current controller and the force reference generator parts of the control system. A state estimation module is also implemented to get the states that can not be measured directly on the quarter car rig.

The control system hierarchy and methodology in this work is, like mentioned in Nor- drik[1], greatly influenced by Revolve NTNU’s needs and project management. The organization has a short project span as a new car is developed each season (a season is approx one year long), the season starts with technical members starting during August.

The members then have until May the following year, approximately, to develop, produce and test their systems before they will be put onto the car to drive during May to July. The car is then competing during July and August. In addition to the short development time, the organization also has a big turnover of members, more than half of the members are new each year. The new members have a limited time to learn their systems before starting the development. The short development time together with the amount of new members calls for solutions that are modular and simple to test. To simplify the development the control system hierarchy is modularized, this allows for development in one module at a time, while keeping the others unchanged. The simulation environment presented, further enhances the feedback cycle during development, allowing for rapid testing of new solutions. During the writing of this work, it also allowed the development of the control systems during periods of maintenance and development of the mechanical and electrical system, which turned out to be the majority of the semester.

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Damper Current Controller

Force Controller Force Reference generator

Force Voltage

Current Reference Force Reference

Figure 1.2:Schematics of the hierarchical control system model

1.4 Limitations

The adaptive damper system consists of several subsystems, including but not restricted to 1. Mechanical and fluid dynamical design of the adaptive damper

2. Design of the physical quarter car rig

3. Synthesis of the control and state estimation system 4. Electrical design of the Damper Control Unit (DCU) 5. Low-level code and drivers for the DCU

This work is mainly concerned with item 3 and does not take credit for the other points.

However, all calculations, results, and discussions presented are new work, unless noted otherwise.

1.5 Outline

Chapter 2 will present the relevant theory for understanding the underlying adaptive damper, the control system components, and the quarter car model in which the system is tested.

Chapter 3 described the methodology for developing and testing the control system both in a simulated and physical quarter car rig.

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Chapter 4 presents the results together with a description and explanation of the main find- ings. The results are presented in a bottom up manner, starting with the current controller, and moving up through the hierarchy presented in Figure 1.2. Where relevant the data presented in the results will show both the data from the physical testing and the data from the simulation environment.

Lastly, chapter 5 offers a review of the results, a conclusion of the work, and suggestions for further work.

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Chapter 2 Theory

2.1 Adaptive dampers

The damper used and described in this work is an adaptive or semi-active damper. The definition of semi-active dampers is not a very rigid definition, but generally adaptive dampers are passive dampers that can vary their properties and semi-active dampers are adaptive dampers that can vary their properties at a high rate. Unlike active dampers the adaptive and semi-active dampers cannot add energy to the system, they can only store energy, like a spring, or directly dissipate the energy. Figure 2.1 gives an overview of the different types of suspension systems and their properties. [16, Chapter. 2] The adaptive dampers in this work are of the telescopic type, with an inline accumulator chamber. The dampers being telescopic simply implies that they feature a linear motion. Other possibilities would be rotary dampers. The inline accumulator chamber is an air-filled chamber with a floating piston, the chamber is compressed when the damper is compressed in order to make up for the volume displaced by the rod entering the damper. The pressure from the chamber also mitigates hysteretic effects, described in Section 2.1.2. The main damping force comes from the piston movement forcing the fluid in the damper to pass through channels in the damper piston. An electromagnetic coil is placed inside the damper and the damper piston is designed to direct the electromagnetic field through the damper fluid. The damper fluid’s properties change due to the magnetization and this is the main principle of operating the damper. An illustration of the damper internals is given in Figure 2.2.

2.1.1 Damper forces

There are several factors which contribute to the forces created by a physical damper. In the ideal case dampers are often described simply as exerting a force proportional to the damper velocity like this,

F=cvvd. (2.1)

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Figure 2.1:An overview of damper categories and their properties, from Savaresi et al. [16, Chap- ter. 2].

Here,F, is the force developed by the damper,c_v, is the viscous damping coefficient and the damper velocity is given asvd. This idealized version of a damper assumes that there is only linear viscous damping. In the real world additional effect like friction, turbulence, inertia and spring forces also play a role. [17]

Viscous friction

Viscous friction is the friction arising from viscous forces in a fluid counteracting the relative motion inside the fluid. In dampers this can generally be described by the formula given in Equation 2.1. The viscous friction is the main damping component in modern hydraulic dampers and is considered the desired form of damping. The viscous friction is also the friction that is controllable in the damper described in this work, this will be described in Section 2.1.2. [17, Chapter .1]

Fluid dynamic friction

Fluid dynamic arises from the energy dissipation associated with turbulence and is proportional to the flow rate squared. Fluid dynamic friction is generally undesired as it gives too large forces at high velocities, and too small at low velocities. [17, Chapter .1]

Dry solid friction

The dry solid friction in dampers arises from the relative movement of components like, guide rings and seals. The dry solid friction can be divided into two main components

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Accumulator Chamber

Oil reservoir

Floating Piston Damper coil Fluid Gap

Figure 2.2:A cross section view of the damper used in this work.

static friction (stiction) and kinetic friction. The static friction will counteract any relative movement up to a given force and can be modeled as,

F ≤µsFn (2.2)

, where ,F, is the force exerted,µ_s, is the friction coefficient between the surfaces andF_n is the normal force. After the static friction is overcome the components will start to move and the kinetic friction comes in to play, this can be written as,

F = sgn(v)µ_kF_n (2.3)

, whereµ_k is the kinetic coefficient of friction. Thesgn(v)term ensures that the friction always opposes the direction of movement.

Both static and kinetic dry solid friction is generally minimized in modern suspensions as they have negative effects on the handling of the vehicle [17, Chapter .1]. As these friction components are fixed by the mechanical design of the damper they also aren’t controllable by the adaptive damper and thus set the lower boundary of the force the damper can exert.

[17, Chapter .7]

Accumulator chamber

As mentioned earlier the damper features an accumulator chamber, which compresses to compensate for the volume displaced by the damper rod which enters the fluid reservoir, and to pressurize the damper fluid to avoid cavitation [17, Chapter .1]. Due to the compression of the gas in the accumulator chamber a force will be exerted on the damper rod.

By modeling the gas as ideal, it can be shown that the force can be expressed as, F =Arod(pc0

Vc0

Vc0−Arodxd

−pa) (2.4)

[1].A_rodis the cross-sectional area of the damper rod.p_c0V_c0and are the uncompressed pressure and volume in the accumulator chamber, respectively. p_a is the ambient pressure andxd is the damper extension. For a sufficiently largeVc0 the force will increase almost linearly in for small changes inx. The exact solution together with such a linear

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approximation is given in Figure 2.3. The largest deviation between the analytical and approximate approach is less than2 N. The linear approximation is given as

F =F0+kxd (2.5)

, whereF₀is the approximate force with an uncompressed chamber andkis the linearly approximated spring constant.

-50 -40 -30 -20 -10 0

Damper Extension [mm]

70 75 80 85 90 95 100 105 110

Force [N]

Calculated Force Linearily Approximated Force

Figure 2.3: Graph of the analytical(calculated) and linearly approximated accumulator chamber forces in the operating range of the damper. From Nordrik [1]

2.1.2 Magnetorheological dampers

As explained in Nordrik [1], the damper in this work is made adaptive by the use of a Mag- netorheological fluid (MRF). An MRF is a fluid consisting of a carrier fluid, like mineral oil, and suspended iron particles. When a magnetic fluid is applied to the fluid the fluid changes its apparent viscosity, which in turn increases the fluid dynamical friction. More specifically, the fluid behaves like a Bingham plastic. A Bingham plastic is a material that behaves like a solid at low stresses and like a fluid at high stresses. The stress required to go from the solid behavior to the fluid behavior, the yield stress, is increased by increasing the magnetic field in the fluid. The fluid used in this damper isLord MR-122and the relation between the yield stress and the magnetic field strength from the datasheet is depicted in Figure 2.4.

There are several constraints to the realizable damper forces. During a positive damper velocity the lower bound of the force the damper can produce is defined by the dry solid friction, combined with the minimum viscous friction. The upper force limit is constrained by how much the yield stress in the fluid is able to rise under maximum magnetization.

The controllable region of the damper is between those limits, for negative velocities the argument is the same but the upper limit becomes the lower and vice versa. An example of such a controllable region is given in Figure 2.5.

Hysteresis

As defined by the Oxford dictionary, hysteresis is [19];

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Figure 2.4:Relation between yield strength and magnetic field strength for Lord MR-122 from the corresponding datasheet [18]

-200 -150 -100 -50 0 50 100 150 200 Velocity [mm/s]

-1500 -1000 -500 0 500 1000 1500

Force [N]

Figure 2.5:The colored area shows an example of the forces an MR damper is able to produce.

The phenomenon in which the value of a physical property lags behind changes in the effect causing it, as for instance when magnetic induction lags behind the magnetizing force.

The definition is repeated here, because it is important for later discussion and for clarity that hysteresis is caused by some lag, in other words, the history of the system. Hysteresis allows a model to have different outputs, given the same inputs, due to some internal history.

There are several sources of hysteresis in the relation between damper velocity and damper force, with varying degrees of effect. The ones described below are some of the noticeable effects, and which are deemed relevant for this work. This is not meant to be an exhaustive list.

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Aeration and compressibility are affects which cause hysteresis in dampers. Compressibil- ity comes from the fact that the increasing pressured during damper operation compress parts of the damper, like the fluid, suspended gases, and seals. This leads to the damper force being dependent of earlier velocities, as they dictate how much the components are compressed. It can also be shown that the force due to compressibility, is dependent on the damper acceleration [17]. Aeration similarly comes from the pressure decrease under damper operation, causing dissolved gasses to form bubbles[20]. When the damper pressure increases again, the bubbles collapse.

If the pressure in the damper is low enough, cavitation might also occur. Cavitation is the sudden boiling of fluids, due to a sudden reduction in pressure. Small bubbles arise, which suddenly collapse when the pressure increases. This leads to shockwaves in the damper, which reduce the lifetime of components in the damper and lead to unwanted vibrations.

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2.1.3 Electromagnetic considerations

The electromagnet implemented inside the damper consists of an iron core and two air gaps. An illustration of the damper piston and how it is compared to a traditional iron core is given in Figure 2.7.

For modeling the current and magnetization of the damper the circuit of the coil is modeled as an RL-circuit. In the linear case, the differential equation is given as

Ldi

dt+Ri= 0, (2.6)

whereRis the circuit resistance,Lis the inductance andiis the current. [21, p. 236] In this case the time constant for the system isτ=L/R. However, this approach is based on a constant inductance. Due to the iron core in the damper not behaving in a linear fashion the above mentioned simplification is not exact. The exact magnetization curve for the steel is not known exactly as it depends on the annihilation procedure used [22]. Typical magnetization curves for magnetic steel have a steep linear region, then a intermediate region where the curve flattens, and lastly a saturated region where the permeability is close to the vacuum permeability. An example for such a curve is given in Figure 2.6. The inductance of a coil with varying permeability can be written as

L=∂λ

∂i (2.7)

, whereLis the inductance,λis the flux linkage andiis the inductor current [23, p. 65].

The flux linkage is given as

λ= Z

S

BdS (2.8)

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, whereSis the domain enclosed by the conductor andBis the magnetic flux density.

Since the magnetic flux density, B, varies non-linearly with the magnetization current as described earlier, the inductance is thus non-constant. Further, since the inductance is the ratio between current and the increase in magnetic flux density the inductance will decrease as the current increases and the magnetic core saturates.

Figure 2.6:Coarse illustration of a magnetization curve for a steel. From [24]

Figure 2.7:Illustration of the magnetic circuit inside the damper piston

2.2 Adaptive damper models

2.2.1 Bingham model

This description of the bingham model is partly taken from Nordrik [1], but extended upon.

The Magnetorheological fluid (MRF) can be observed to behave like a viscoelastic Bing- ham Plastic and models have thus been proposed which model the MR damper as having

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a yield force depending on current as well as a viscous term. [25–28] The name originates from the fact that the model assumes that the damper fluid behaves like a viscoelastic Bingham plastic. The model is however not able to represent any hysteric behavior and thus does not model the MRF well at low velocities [29]. The model equations presented below are closely related to how they are formulated in by Soltane et al. [25].

F= sgn(vd)Fy+vdc+k0(xd−x0) (2.9)

Fy=Fy_aid+Fy_b (2.10)

c=c_ai_d+c_b (2.11)

Fy models the yield force of the MRF, that is, the force required to make the bingham plastic behave like a fluid.Fyis assumed to be a linear function of the damper currentid, as given in Equation 2.10.FyaandFybare arbitrary constants, to fit the model. Thesgn(vd) gives the yield force the same sign as the velocity as it will always oppose the movement of the damper. vdcmodels the viscous damping of the damper whilek0(x−x0)models the spring effects in the damper.k0is the spring constant, andx0is the initial compression of the spring. The viscous damping coefficient,c, is also assumed to be a linear function of the damper current as given in Equation 2.11, withcaandcbbeing constants to describe the relation to the current. The model does not include any terms which allow the modeling of hysteretic behavior and thus neglects this effect. A representative parameterization of the model is shown in the force vs velocity curve given in Figure 2.8.

Figure 2.8:Example of force velocity response of the Bingham Damper Model. From Nordrik [1].

2.2.2 The modified Dahl model

The Dahl model is a dynamic model and was proposed by Dahl [30] and has shown to be able to represent several types of friction. The model has later been reformulated by Zhou and Qu to include the changing properties experienced in MR dampers, called the modifed Dahl model.

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The modified Dahl model is given as

F =K₀x+C₀x˙+F_dz−f₀ (2.12)

˙

z=σx(1˙ −zsgn( ˙x)) (2.13)

C₀=C_0s+C_0du (2.14)

Fd=Fds+Fddu (2.15)

˙

u=−η(u−V), (2.16)

like it is stated by S¸ahin et al. [8]. Equation 2.12, models the damper force,F.K₀is the linear spring stiffness,C0is the viscous damping coefficient,Fdis the Coulomb force, and f0is a constant offset in force, added by seals and the pressure chamber. To include the magnetorheological effects,C0andFdare assumed to be a linear function in current. This results in Equation 2.14, whereC0sandC0d are arbitrary constants, and equation 2.16, whereFdsandFddare also arbitrary constants. Equation 2.16 models the response from input voltage,V to damper parameter change as a first order filter, with filter constantη.

As in Nordrik the damper current is measured instead of the voltage. This subtracts the time constant of the electromagnet in Equation 2.16. The model is thus given as

F =K0x+C0x˙+Fdz−f0 (2.17)

˙

z=σx(1˙ −zsgn( ˙x)) (2.18)

C₀=C_0s+C_0du (2.19)

Fd=Fds+Fddu (2.20)

˙

u=−η(u−i_d). (2.21)

2.3 Quarter car model

The quarter car model is a simplified suspension system model where only one corner of the suspension is modeled. The model consists of two main components, the sprung mass, and the unsprung mass. The unsprung mass is connected to the ground by a model of the tire, and to the sprung mass by a damper and a spring. An illustration of the model is shown in Figure 2.9. As the names indicate the sprung mass is called so, since it is sprung by the spring in the suspension system, while the unsprung mass is not. The unsprung mass does however have some spring effect and even damping due to the spring and damping in the tire itself.

The accelerations of the sprung and unsprung masses,asandau, respectively, are given by,

as= 1 ms

(−ks(zs−zu)−cs(vs−vu)−g) (2.22) au= 1

mu

(ks(zs−zu) +cs(vs−vu)−kt(zu−zr)−ct(vu−vr)−g). (2.23) Note that the velocity of the damper, vd can be expressed as the difference between the velocity of the sprung mass,vs, and the unsprung mass,vu. And similarly also the damper

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Figure 2.9:Illustration of a quarter car model. From Abid et al. [31].

extensionxd

xd=zs−zu (2.24)

vd=vs−vu (2.25)

2.4 Performance metrics

2.4.1 Root Mean Square Error (RMSE)

The Root Mean Square Error (RMSE) between a predicted signal,y, and the true signal,ˆ y, is given as

RMSE= s

PN

k=1(y−y)ˆ ²

N (2.26)

2.5 Smoothing splines

Smoothing splines construct a function estimate,s(x), for a set of noisy measurements,yi. To be precise, the smoothing splines used in this work are cubic smoothing splines which minimize the following cost function,

pX

i

wi(yi−s(xi))²+ (1−p)

Z d²s dx²

²

dx. (2.27)

Wherew_iis the weight of data pointi,y_iis the measurement,x_iis the time of the measurement, andpis a smoothing parameter. The smoothing parameter can be chosen to be betweenp= 0, which produces a least-square straight line fit, andp= 1, which gives a cubic spline interpolant. [32]

The cubic spline, which is minimized, is a piece-wise polynomial,s(x), where each poly- nomialsk(x)is defined for x < xk+1 andx > xk. Each of the polynomials is of the

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form

sk(x) =ak+bk(x−xk) +ck(x−xk)²+dk(x−xk)³. (2.28)

2.6 Control strategies

2.6.1 Skyhook

In skyhook control the main principle is to dampen the motions of the chassis by emulating the existence of a damper between the chassis and a fixed reference(the sky). Due to the passivity constraint of the adaptive dampers the adaptive damper can only emulate the skyhhok when the relative movement between the sprung- and unsprung masses is in the same direction as the movement of the chassis.

The two state skyhook has either an on or off state, either it applies a maximum damping coefficient, c_max, or a minimum damping coefficent c_min. The law that describes the desired damping coefficient is thus given as

cin=

(cmin ifz˙z˙def ≤0

c_max ifz˙z˙_def >0 (2.29)

There is also a linear version of the skyhook strategy. Linear in this context being that the method interpolates a desired damping coefficient betweencmin andcmaxinstead of switching discretely between them. The desired damping coefficient is given as

cin=

(cmin ifz˙z˙def ≤0 satc_in∈[cmin;c_max]

_αc

maxz˙def+(1−α)cmaxz˙

˙ zdef

ifz˙z˙_def >0[16, Chapter.6]

(2.30)

2.6.2 Groundhook

Similar to the skyhook the ground hook emulates the existence of a damper, not between chassis and the sky but but rather between the unsprung mass and the ground.

c_in=

(cmin if −z˙tz˙def ≤0

cmax if −z˙tz˙def >0 (2.31) And similarly, a linear version can be formulated

cin=

(cmin if −z˙tz˙def ≤0 sat_c_in_∈[c_min_;c_max_]_αc

maxz˙def+(1−α)cmaxz˙t

˙ z_def

if −z˙tz˙def >0 (2.32) [16, Chapter. 6].

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2.6.3 Hybrid hook 2.6.4 Damper map

Dampers can generally be described by their force vs. velocity curve,F(v), whereF is the damper force andv is the damper velocity. The curve is usually given as forces at given velocities. Adjustable dampers allow the modification of these curves, which are often given for different valve settings. The curves shown in Figure 2.8, to demonstrate the Bingham model, are an example of such damper curves.

2.7 State estimation

2.7.1 Discrete Kalman filter

For a discrete-time system model given as

x(k+ 1) =Φx(k) +∆u(k) +Γw(k) (2.33) y(k) =Hx(k) +v(k) (2.34) the Kalman filter is defined as given below [33, p. 296].

Initially the state a priori, x(0), and the error covariance a priori¯ P¯(0), are set to their inital values.

x(0) =¯ x0 (2.35)

P¯(0) =P0 (2.36)

Then the Kalman gain matrix,K(k)is calculated K(k) = ¯P(k)H^T(k)

H(k) ¯P(k)H^T(k) +R(k)⁻¹

(2.37) The state estimate,x(k)ˆ is calculated

ˆ

x(k) = ¯x(k) +K(k) [y(k)−H(k) ¯x(k)] (2.38) and the error covariance,Pˆ(k), is updated

Pˆ(k) = [I−K(k)H(k)] ¯P(k) [I−K(k)H(k)]^T+K(k)R(k)K^T(k) (2.39) Then the a priori for next state, x(k¯ + 1), and the next error covariance, P¯(k+ 1), is calculated

¯

x(k+ 1) =Φ(k) ˆx(k) +∆(k)u(k) (2.40) P¯(k+ 1) =Φ(k) ˆP(k)Φ^T(k) +Γ(k)Q(k)Γ^T(k) (2.41) The design matrices

Q(k) =Q^T(k)>0 (2.42) R(k) =R^T(k)>0 (2.43) (2.44)

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are the process noise covariance matrix, and measurement noise covariance matrix, respectively.

Discretization

Given a continuous system of the form

˙

x=Ax+Bu+Ew (2.45)

y=Hx+v (2.46)

a discrete-time model of the form given in Equations 2.33 - 2.34 can be obtained from the continuous system model by

Φ=exp(Ah)≈I+Ah (2.47)

∆=A⁻¹(Φ−I)B (2.48)

Γ=A⁻¹(Φ−I)E (2.49)

wherehis the time step.

The discretized model can now be used in the discrete Kalman filter.

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Chapter 3 Method

This chapter will present the method used to create the simulation environment as well as the physical test rig. Further, the method for implementing the different control strategies will be shown.

3.1 The quarter car test rig

The quarter car test rig in its physical form was set up by Vigerust [15]. This work has been focused on the electrical and control system implementation and on adding the sensors needed to evaluate the control systems.

A short description of the physical layout of the test rig will be given before a more thor- ough description of the most important aspects.

The physical model is shown in Figure 3.1 and the main components are highlighted in the figure. The quarter car model is excited by a rotary AC machine driving a pulley connected to a cam and follower system at the bottom of the road plate. The cam and follower systems generate a sinusoidal waveform. The movement of the road is transferred through a load cell up to a plate at the bottom of the tire. The wheel is rigidly mounted to the rest of the unsprung mass. The unsprung mass and the sprung mass are connected by the damper and springs and the displacement between them is measured. The damper also features an in-line load cell to precisely measure the damper forces. The sprung mass is53.0 kgand the unsprung mass is15.8 kg. The masses are chosen to resemble a quarter of the masses of a formula student type racecar.

The wheel mounted on the rig is a rim from the 2019 car from Revolve NTNU with a Continental Formula Student C19 tire (item 43328). The tire was used on Revolve NTNU’s cars from 2017 to 2019 but will be substituted this year for a Hoosier Racing Tire (item 43075).

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Figure 3.1:The quarter car test rig. Some sensors are missing in this illustration, like damper load cell and tire deflection.

During the evaluation of the controllers presented in the results of this work, the quarter car rig is exited by frequencies according to Table 3.1 and with an amplitude of2.5 mm.

Each of the input frequencies is applied for approximately4 s.

Table 3.1:Table of the test runs used for parameter estimation Motor Velocity[RPM] Input Frequency [Hz]

700 5.1

900 6.6

1100 8.0

1300 9.5

3.1.1 Damper Control Unit (DCU)

The Damper Control Unit (DCU) is a Printed Circuit Board (PCB) featuring a microcontroller, measurement circuitry and an H-bridge to drive the damper current. An illustration of the PCB is given in Figure 3.2. The circuit board is used as both a hub for the sensors and as the driving circuit for the dampers. The circuit board is developed and manufac- tured by Revolve NTNU with help from its sponsors and is purpose-built for the quarter car jig and the prototype racecar. The microcontroller is a Microchip ATSAME70N21 running at300 MHzand features built-in Analog-to-Digital Converter (ADC) functional- ity used to sample the sensor measurements in addition to Pulse Width Modulation (PWM) controllers used to control the H-bridge [34]. The microcontroller is also responsible for running all of the code necessary to control the damper and transmit measurements for logging.

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The control system, which is described later, is implemented in Simulink and is trans- formed into C code using Matlab’s embedded coder. The code is combined with the application code and executed on the DCU.

The sensor data and signals from the control system are sent from the DCU onto a Con- troller Area Network (CAN). Due to a limited capacity on the network, only a subset of the messages are sent for each test, and the frequencies are varied depending on the need.

When testing high-frequency components, like the current controller or state estimators, the messages are sent at 10 kHz. For tests with slower dynamics, like testing the force controller, the messages are sent at 2 kHz, allowing more measurements to be sent over the bus.

A

B

C

D

E

1 1

2 2

3 3

4 4

Realistic View

Figure 3.2:A 3D view of the Damper Control Unit (DCU)

3.1.2 Sensors

The sensors used on the quarter car rig are given in Table 3.2.

Table 3.2:Table of quarter car rig sensors

Measurement Model Manufacturer

Sprung mass acceleration ADXL335 Analog Devices Unsprung mass acceleration ADXL335 Analog Devicesr

Damper extension MLP-75 TE Connectivity

Damper force LCM200 Futek

Road force HBM

Tire deflection HG-C1100-P Panasonic

Mass accelerations

The mass acceleration measurements are used both for evaluating the performance metrics based on the acceleration of the sprung body and used to estimate the vertical velocities of the bodies. The vertical velocities of the bodies are used in some of the force reference generators. The acceleration measurements are filtered by the built-in filter in the

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accelerometers, set to a cut-off frequency of1600 Hz. In addition to a filter on the Analog- to-Digital Converter (ADC) of the DCU of with a cut-off frequency of500 Hz.

Damper Extension

The damper extension is measured by an MLP-75 and sampled by an ADC on the DCU at a frequency of10 kHz. The damper extension measurement is used to estimate the spring force associated with compressing the damper. The measurement is also used to estimate the damper velocity used for calculating the damper forces.

Damper Force

The damper force is measured directly by a load cell mounted in line with the damper.

It should be noted that the load cell will measure all forces exerted by the damper, be it the viscous damping, magnetorheological damping, or the forces from the accumulator pressure. Since the springs are mounted in parallel with the damper the spring forces will not be measured by the load cell.

The damper force measurement is mainly used for evaluating the performance of the force controller, but may also be used as feedback in the force controller.

Road force

The road force is measured using a load cell mounted under the plate on which the tire is resting. The load cell is moving together with the plate as the cam is oscillating and the load cell will thus measure the forces used to accelerate both itself and the plate, in addition to measuring the road force acting on the wheel. The road force is used to evaluate how the tire load changes when different force reference strategies are applied.

Tire deflection

The tire deflection is measured using a laser measurement. The measurement is used to evaluate the performance of the control schemes in how well they mitigate tire deflection and can also be used to estimate the velocity of the sprung mass together with the acceleration measurement,

3.2 Damper velocity estimation

With the current configuration of sensors, the damper velocity is not measured directly, but the damper velocity is however used to a great extent in the control of the damper. Several relevant measurements are available to estimate the damper velocity, among them,

• Damper extension

• Sprung acceleration

• Unsprung acceleration

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• Damper force

• Road force

• Tire extension

The chosen approach was to create a Kalman filter to utilize the damper extension measurement, together with the two acceleration measurements, to estimate the damper velocity.

As will be shown this results in simple kinematic equations. The damper velocity estimate of the proposed Kalman filter should have little drift due to the correction offered by the damper extension measurement. The filter should also offer little phase shift and smoothing by utilizing the acceleration measurements. The structure of the filter is inspired by Brekke (Accelerometer with slowly varying bias) [35, Chapter .4]. The filter is however extended to accommodate two acceleration measurements.

System equations

Firstly the dynamics of the system are modeled. Starting with the damper extension,xd, we have that the derivative of the damper extension is simply the damper velocity,vd.

˙

xd=vd (3.1)

The damper extension is measured and the measurement is assumed to be affected by Gaussian noise. The measurement,xmdis thus given as,

xmd=xd+vx, (3.2)

wherevxis the Gaussian measurement noise. The derivative of the damper velocity,vd, is given as the difference of the sprung and unsprung accelerations

˙

vd=as−au. (3.3)

a_sanda_uare the sprung mass and unsprung mass accelerations, respectively.

Each of the acceleration measurements is assumed to be a combination of the real acceleration, some Gaussian noise, and a bias.

First, the bias of the two accelerometers is expressed as the sum of a common mode bias and the difference in bias between the two. The bias for the sprung mass,abs, and unsprung massabuare thus written as

a_bs=a_bc+abd

2 (3.4)

abu=abc−abd

2 , (3.5)

whereabcis the common bias andabd is the difference in bias between the two measurements. The two acceleration measurements can then be written as

ams=as+ws+abc+abd

2 (3.6)

amu=au+wu+abc−abd

2 , (3.7)

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which can be rewritten to get the true accelerations, as=ams−ws−abc−abd

2 (3.8)

au=amu+wu−wu−abc+a_bd

2 . (3.9)

Inserting Equation 3.8 - 3.9 into Equation 3.3 gives,

˙

vd =ams−ws−amu−wu−abd. (3.10) Note that the common bias cancels out and only the difference in bias is left.

The bias of each of the accelerometers is assumed to vary slowly and so the difference between,abd, should vary slowly too. The difference in bias is thus modeled as a first- order filter given by,

˙

abd=−cabd+wb, (3.11)

wherew_bis Gaussian noise andcis a tuning parameter for the time constant of the bias.

State space model

Based on the dynamic equations, the state-space model as given in Equation 2.45 - 2.46 will be constructed. The following state vector is chosen,

x=



 xd

vd

abd



, (3.12)

and the input is chosen to be the two acceleration measurements.

u= a_ms

a_mu

, (3.13)

The process noise is,

w=



 w_s wu

wb



 (3.14)

And measurement vector is chosen as, y=

x_md

(3.15) together with the measurement noise,

v= vx

(3.16)

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The model matrices are then constructed from the dynamic Equations, Eq. 3.1, Eq. 3.10, and Eq. 3.11, together with the measurement given in Equation 3.2. The matrices are

A=





0 1 0 0 0 1 0 0 −c



 (3.17)

B=



 0 0 1 −1 0 0



 (3.18)

E=





0 0 0

−1 −1 0

0 0 1



 (3.19)

H=

1 0 0

(3.20) (3.21) Kalman filter tuning and evaluation

To create a benchmark for the Kalman filter an offline approach is used to estimate the damper velocity. The approach chosen is to fit a smoothing spline to the damper extension and then differentiate this spline to obtain the damper velocity. The smoothing spline method used is described in Section 2.5. The smoothing parameter is set to,p= 1−5· 10⁻⁸, the parameter is chosen based on a reduction of noise and oscillations in the output and their derivatives, while still giving a sufficient fit to the data.

The smoothing spline input data, the inputs to the Kalman filter, and the measurements for the Kalman filter were gathered at the frequency that the Kalman filter is running at, 10 kHz.

The Kalman filter was then tuned to the gathered data and evaluated by calculating the Root Mean Square Error (RMSE) values for several errors. The errors were, the error between the velocity of the Kalman filter and the smoothing spline, the error between the Kalman filters predicted measurement and the actual measurement. While tuning the filter, it is important to note that the smoothing spline is not a proper ground truth as it simply a curve fit to the measurements. Having good acceleration measurements, the Kalman filter should also be able to outperform the smoothing spline by merging the measurements.

The smoothing spline is also a parametric fit and prone to the wrongful selection of the smoothing parameter.

3.3 Simulation environment

A simulation model of the quarter car model is implemented to facilitate rapid testing and prototyping of the control system in a safe and risk-free environment and to be able to isolate effects. The simulation model is implemented using Simulink.

The simulation model implements the main physics involved in the system, the damper, the sprung and unsprung masses, and the tire.

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3.3.1 Sprung and unsprung dynamics

The dynamics of the sprung and unsprung masses are modeled similarly to the linear quarter car as described in 2.3. The sprung acceleration,a_sis modeled as

as= 1 ms

(−Fs−Fd−msg), (3.22)

wheremsis the sprung mass,Fsis the force from the suspension spring,Fd is the force from the damper, and g is the gravitational acceleration. The sprung velocity, vs, and vertical position,zs, is obtained by integration. Similarly for the unsprung acceleration, au

au= 1

m_u(Fs+Fd+Ft−mug), (3.23) whereFtis the force from the tire andmuis the unsprung mass. Otherwise, the symbols are the same as for the sprung dynamics.

3.3.2 Tire dynamics

Only the vertical dynamics of the tire are considered as the quarter car system is only exited in a vertical manner. The tire is modeled as a damper and spring but limited to only exert a force when the tire is lifted from the ground.

The tire spring force,F_tsis modeled as Fts=

(0 if(zu−zr)>0

k_t(z_u−z_r) otherwise (3.24)

and the tire damping force,F_tdis modeled as Fts=

(0 if(zu−zr)>0

c_t(v_u−v_r) otherwise (3.25)

The implementation in Simulink is written somewhat differently to decrease the run time of the solver. The implementation is shown in Figure 3.3. The saturation block limits the extension used for the spring force calculation to be negative so that the spring force from the wheel will not be able to pull the wheel towards the ground. Similarly, the absolute value block and sign value block will result in the damper velocity of the tire being multiplied by zero if the extension is zero or positive and multiplied by unity otherwise.

3.3.3 Road profile

Two road profiles are implemented in the simulation, as can be seen in Figure 3.4. The first method is a simple sinusoidal road profile, where both the amplitude and frequency can be changed. This approach resembles the input given by the cam follower system on the quarter car rig. The other approach is a road profile regenerated from ISO 8606 classifications of roads using the sinusoidal approximation described in Tyan et al. [36].

The output of the road profile is the profile height, and profile velocity, corresponding to zr, andvrin the tire dynamics.

Implementation of a control system for an adaptive damper in a quarter car context

Marius Hamre Nordrik

Implementation of a control system for an adaptive damper in a quarter car context

Master ’s thesis

Marius Hamre Nordrik

Implementation of a control system for an adaptive damper in a quarter car context

Master’s thesis in Cybernetics and Robotics Supervisor: Ole Morten Aamo

June 2021

Norwegian University of Science and Technology

Faculty of Information Technology and Electrical Engineering

Department of Engineering Cybernetics

Abstract

Preface

Contents

List of Tables

List of Figures

Acronyms

Chapter 1

Introduction

1.1 Motivation

1.2 State of the art

1.2.1 State of the art on force control

1.2.2 State of the art force references

1.2.3 Revolve NTNU’s Electronically Controlled Suspension (ECS) research

1.3 Scope

1.4 Limitations

1.5 Outline

Chapter 2

Theory

2.1 Adaptive dampers

2.1.1 Damper forces

2.1.2 Magnetorheological dampers

2.1.3 Electromagnetic considerations

2.2 Adaptive damper models

2.2.1 Bingham model

2.2.2 The modified Dahl model

2.3 Quarter car model

2.4 Performance metrics

2.4.1 Root Mean Square Error (RMSE)

2.5 Smoothing splines

2.6 Control strategies

2.6.1 Skyhook

2.6.2 Groundhook

2.6.3 Hybrid hook 2.6.4 Damper map

2.7 State estimation

2.7.1 Discrete Kalman filter

Chapter 3

Method

3.1 The quarter car test rig

3.1.1 Damper Control Unit (DCU)

3.1.2 Sensors

3.2 Damper velocity estimation

3.3 Simulation environment

3.3.1 Sprung and unsprung dynamics

3.3.2 Tire dynamics

3.3.3 Road profile