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Mechatronics 16 (2006) 405–416

A performance evaluation of an automotive magnetorheological brake design with a sliding mode controller Edward J. Park *, Dilian Stoikov, Luis Falcao da Luz, Afzal Suleman Department of Mechanical Engineering, University of Victoria, 3800 Finnerty Road, P.O. Box 3055 STN CSC, Victoria, BC, Canada V8W 3P6 Received 14 March 2006

Abstract The aim of this work is to develop a magnetorheological brake (MRB) system that has performance advantages over the conventional hydraulic brake system. The proposed brake system consists of rotating disks immersed in a MR fluid and enclosed in an electromagnet, which the yield stress of the fluid varies as a function of the magnetic field applied by the electromagnet. The controllable yield stress causes friction on the rotating disk surfaces, thus generating a retarding brake torque. The braking torque can be precisely controlled by changing the current applied to the electromagnet. In this paper, an optimum MRB design with two rotating disks is proposed based on a design optimization procedure using simulated annealing combined with finite element simulations involving magnetostatic, fluid flow and heat transfer analysis. The performance of the MRB in a vehicle was studied using a quarter vehicle model. A sliding mode controller was designed for an optimal wheel slip control, and the control simulation results show fast anti-lock braking.  2006 Elsevier Ltd. All rights reserved. Keywords: Magnetorheological brake (MRB); Anti-lock brake system (ABS); Finite element analysis; Multidisciplinary design optimization; Sliding mode control

1. Introduction The topic of ‘‘x-by-wire’’ is a focus topic to automotive industries due to its potential to improve vehicle performance, safety and cost. The ‘‘x’’ in x-by-wire is a technological wildcard for automotive systems such as steering and braking, and means replacing conventional mechanical components by electrical ones [1]. Our work is aimed at the development of a brake-by-wire system using an electromechanical brake (EMB) that employs magnetorheological (MR) fluid. Brake-by-wire replaces the mechanical connection between the brake actuator on each wheel and the brake pedal with electrical components. According to [1], in the first generation of brake-of-wire systems, a conventional hydraulic brake (CHB) system can be installed for backup and all the brake control functions are implemented in one *

Corresponding author. Tel.: +1 250 721 7303; fax: +1 250 721 6051. E-mail address: [email protected] (E.J. Park).

0957-4158/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2006.03.004

main electronic control unit (ECU). A hydraulic backup mode is needed for safety reasons to ensure braking in the case of an ECU or electrical failure. In the second generation brake-by-wire systems, however, with the availability of a fault-tolerant system architecture and redundant fail-safe power management systems (e.g., the 42 V system) that are currently being developed by the automotive industries for the next generation hybrid electric vehicles (HEV), the hydraulic backup mode will no longer be necessary. In addition, a sub-ECU can be installed on every wheel, in order to provide a multi-redundant, independent wheel-specific braking. There are many advantages of using pure electronically controlled brake systems. The properties and behavior of the brake will be easy to adapt by simply changing software parameters and electrical outputs instead of adjusting mechanical components. This also allows easier integration of existing and new control features such as anti-lock braking system (ABS), vehicle stability control (VSC), electronic parking brake (EPB), etc., as well as vehicle chassis

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control (VCC) and adaptive cruise control (ACC). Diagnostic features and the elimination of the water polluting brake fluids are additional benefits [1], as well as a small number of components, simplified wiring and generalized optimized layout. In this paper, we propose a MR actuator for the brake in each wheel. The actuator consists of a rotating disk immersed in a MR fluid, enclosed in an electromagnet. In principle, the brake torque can be controlled by changing the DC current applied to the electromagnet. Magnetorheological fluid – a compound containing fine iron particles in suspension – stiffens in the presence of a magnetic field. Two important characteristics of MR fluids are: (i) they exhibit linear response, i.e., the increase in stiffness is directly proportional to the strength of the applied magnetic field and (ii) they provide fast response, i.e., MR fluid changes from a fluid state to a near-solid state within milliseconds of exposing a magnetic field. CHB systems exhibit about 200–300 ms of delay between the time the brake pedal is pressed by the driver and the corresponding brake response is observed at the wheels due to pressure build-up within the hydraulic lines. An electric brake system has the potential to drastically reduce this time delay, consequently bringing a reduction in braking distance. Recently, Delphi introduced an EMB with performance similar to the existing disk brakes, with the brake pads actuated by an electrical motor [2], instead of the hydraulic actuator. While the application of MR fluid in automotive vehicles has been promising for years, it is only recent that MR fluid based electromechanical devices have started to displace all-mechanical or hydraulic counterparts. For instance, General Motors recently introduced the Magnetic Ride Control [3], which is a MR fluid-based suspension control system developed by Delphi, on the Corvette and Cadillac Seville STS and XLR. The significance with these new systems is that the vehicle control is quickly evolving away from the limitations of traditional mechanical components, such as springs, brakes, shocks and steering gear. Instead, real-time sensors and high-speed, direct electric actuation can now adjust all these systems depending on driving conditions [4]. In this regard, a MR actuator is a promising technology for the automotive industry with high commercial values. In the initial stage of our work, two types of controllable fluids were considered: electrorheological (ER) and MR fluids. ER fluid is also a linear viscous liquid whose rheological behavior changes under the influence of an applied

electric field, instead of a magnetic field. Table 1 presents a brief comparison between ER and MR fluids. It can be seen from the above comparisons that MR fluids present a greater potential for the proposed application than ER fluids, both in terms of performance (yield stress, energy dissipation per unit volume) and requirements (tolerance to impurities and compatibility with more common power supplies). The only advantage that ER fluids have over MR fluids is the lower density, which is not a significant issue considering the fact that only a small amount of these fluids would be needed inside the brake. In Section 2, our proposed MRB design concept is described and modelled. Sections 3 and 4 present the finite element analysis and subsequent design optimization of the proposed MRB. Section 5 deals with the derivation of the real vehicle dynamic model. Then, the design of a wheel slip controller based on the sliding mode control technique is discussed in Section 6. Section 7 presents control simulations of the MRB system, demonstrating its fast response. Section 8 concludes the paper. 2. Proposed MR brake design Fig. 1 is a conceptual, three-dimensional illustration of our proposed MR brake (MRB) actuator, which consists of a disk rotating within MR fluid enclosed in a static casing. A cut has been made to highlight the cross-section that was modelled and analyzed via finite element method. The legend in the figure indicates the various components of the MRB with the exception of the MR fluid, which is located in the narrow channel (part no. 7) surrounding the rotating disk (no. 3) and the stator (no. 5). 2.1. MR brake modelling MR fluids are created by adding micron-sized iron particles to an appropriate carrier fluid such as oil, water or silicon. Their rheological behaviour is nearly the same as that

Table 1 ER versus MR fluids [5]

Yield stress Operating environment Density Energy density Power supply

ER fluids

MR fluids

2–5 kPa 25 C to +125 C Do not tolerate impurities 1–2 g/cm3 0.001 J/cm3 2–5 kV, 1–10 mA

50–100 kPa 40 C to +150 C Tolerate impurities 3–4 g/cm3 0.1 J/cm3 2–25 V, 1–2 A

Fig. 1. Basic configuration of the proposed MR brake.

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of the carrier fluid when no external magnetic field is present. However, when exposed to a magnetic field, the iron particles acquire a dipole moment aligned with the applied magnetic field to form linear chains parallel to the field [6]. This reversibly changes the free flowing liquid to semi-solids that have a controllable yield strength, which depends on the magnitude of the applied magnetic field. Although MR fluids have been known for decades, they had been experiencing stability and longevity issues for commercial applications. Recently, however, these problems have been solved and commercial applications are starting to appear, most notably as controllable dampers in the afore-mentioned car suspensions [4] and in civil engineering applications for seismic response control [6]. In the literature, it is found that the essential magnetic field dependent fluid characteristics of MR fluids can be described by a simple Bingham plastic model [7]. In this model, the total shear stress s is given by s ¼ sH þ lp c_

ð1Þ

where sH is the yield stress due to the applied magnetic field H, lp is the constant plastic viscosity, which is considered equal to the no-field viscosity of the fluid, and c_ is the shear strain rate. Here, the plastic viscosity is defined as the slope between the shear stress and shear stress rate, which is the traditional relationship for Newtonian fluids. While other researchers have tried more elaborate models such as the Herschel–Bulkely model [8,9] to accommodate the shear strain rate dependent shear thinning and shear thickening phenomena in the fluid, the simpler Bingham model is still very effective, especially in the initial design phase [6]. In addition, the Lord Corporation’s MRF-132AD fluid, which was chosen for the MRB design presented in this paper, has a nearly linear experimental stress shear rate curve that is well approximated by the Bingham model. The choice of the hydrocarbon-based MRF-132AD fluid over other types of MR fluids, such as the Lord Corporation’s water-based MRF-241ES, was mainly due to its higher temperature resistance characteristics [10]. Based on Eq. (1) and the given geometrical configuration shown in Fig. 1, the retarding or braking torque – which is caused by the friction on the interfaces between the MR fluid and the solid surfaces within the MR brake – can be written as [11]: Z rz Z rz T b ¼ 2pN sr2 dr ¼ 2pN ðlp c_ þ sH Þr2 dr ð2Þ rw

rw

where N is the number of surfaces of the brake disk(s) in contact with the MR fluid (e.g., 2 for one disk with MR fluid covering the both sides, 4 for two disks, etc.); rz and rw are the outer and inner radii of the brake disk, respectively; and rx and sH ¼ kH b c_ ¼ h where x is the angular velocity of the rotating disk, h is the thickness of the MR fluid gap, H is the magnetic field inten-

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sity, and k and b are constant parameters that approximate the relationship between the magnetic field intensity and the yield stress for the MR fluid. Then, Eq. (2) can be rewritten as Z rz   rx þ kH b r2 dr lp ð3Þ T b ¼ 2pN h rw Eq. (3) is a more accurate form than that of the Lord Corporation’s low torque MRB used in [12], because it can take into account non-constant magnetic field distributions. This improvement is necessary in order to use a greater amount of MR fluid (which causes greater variations in the magnetic field intensity) than that of [12], which was used for AC induction motor braking. Eq. (3) provides some insight into the dynamics of the MRB and shows possible ways to improve the braking torque, including the use of multiple disk surfaces (increasing N) or fluids with high yield stresses (increasing k and/or b). Improving the braking torque by amplifying the first term in the integral, i.e., increasing the plastic viscosity lp or decreasing the gap thickness h, is not desired as this would lead to a greater residual torque (increasing the drag even without the brakes applied). Eq. (3) indicates that, while carrying a one-disk configuration (hence, N = 2) would be ideal in terms of the simplicity of the design, manufacturing and weight of the MRB, having multiple disks generates more braking torque. Hence, in this work, two different geometry configurations, consisting of one disk and two disks, were selected for a detailed analysis, involving the Lord Corporation’s MRF-132AD fluid. Given the number of disk surfaces, additional parameters that influences the performance of the MRB are the physical dimensions of its components. This is addressed in the subsequent section. The applied magnetic field H can be produced within the MRB when current i is supplied to the electromagnet encircling the MR fluid, i.e., H ¼ ai

ð4Þ

where a is a proportional gain. Then, the two contributions of the resulting braking torque, TH due to the yield stress induced by the applied magnetic field and Tl due to the friction and viscosity of the MR fluid, can be derived by performing the integration in Eq. (3) and substituting Eq. (1), i.e., 2p Nkaðr3z  r3w Þi ¼ T i i 3 p T l ¼ N lp ðr4z  r4w Þh_ ¼ T v h_ 2h

TH ¼

ð5Þ ð6Þ

where h_ is the rotational speed of the disk(s). 2.2. MR brake dimensional parameters The dimensions of the proposed MRB shown in Fig. 1 can be optimized for performance and weight. However,

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Fig. 2. MR brake dimensional design parameters.

its overall dimensions must be restricted so that the brake can be fitted inside a wheel rim as the typical CHB does. For example, considering the fact that the general recommended minimum clearance between the wheel rim and the brake is 3 mm, the maximum acceptable MRB radius for a 1600 wheel is about 20 cm [11]. In Section 4, the various dimensional parameters represented in Fig. 2 are optimized using a multidisciplinary design optimization (MDO) procedure.

and loads are all consistent along the tangential direction, only the cross-section was modelled. This way, the solution becomes that of a two-dimensional problem, allowing the use of ANSYS’ plane elements (i.e., the PLANE13 elements for the magnetostatics modelling and the FLUID141 elements for the CFD modelling) with axisymmetric formulation, and thus greatly reducing the computational cost of each simulation. For the magnetostatics simulation, the B–H (magnetic flux density versus applied magnetic field) curve for the MR fluid was obtained from the manufacturer’s specifications and the B–H curve for the steel element (SAE 1010 steel) that makes up the casing and disk(s) were obtained from the ANSYS material library. The current in the coil was applied as an area load. For heat transfer analysis of the CFD model, the velocity of the moving disks was specified, as well as the heat generated by the current flow in the coil (so-called the Joule effect). The heat generated by the friction between the fluid and solid surfaces was computed by the CFD solver. Since the brake is cooled by the flow of outside air around the casing, the convection coefficient was also determined, from empirical relations based on the Nusselt number. Figs. 3 and 4 present the preliminary results of these simulations. Fig. 3 shows the magnetic flux density (B) distribution in the one-disk configuration, where the arrows that represent the direction of the magnetic flux density follow the intended path around the coil and passing along the casing. Fig. 4(a) presents the distribution of the magnetic field intensity. Fig. 4(b) illustrates the linear relationship between the applied magnetic field (H) and the resulting yield stress (sH) for H up to about 130 kAmp/ m, where the saturation effect starts to take place. Hence, it would be important for the MRB to operate within this linear operating range.

3. Finite element modelling A finite element model (FEM) of the MR brake was developed using ANSYS to accurately characterize the brake’s behaviour. This model was a multiphysics model that accounted for magnetostatics, MR fluid flow, heat transfer, structural response within the MRB. ANSYS is capable of dealing with such a multiphysics problem, and also provides the feature of adding user-programmed routines to extend its built-in capabilities. Our finite element analysis procedure consisted of a magnetostatics study followed by a computational fluid dynamics (CFD) simulation in ANSYS. The former gives the magnetic field distribution throughout the MR brake, which allows the determination of the yield stress sH. The magnetic field distribution is then supplied to the CFD model, which computes the wall shear stresses – the friction exerted on the walls and disk surfaces – and the temperature distribution within the MRB. The first step in the finite element modelling was to define the brake geometry. Since our problem is axisymmetric, meaning that the geometry, material properties

Fig. 3. Magnetic flux density distribution in one-disk configuration.

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Fig. 4. Magnetic field intensity in one-disk configuration. (a) Magnetic field intensity and (b) magnetic field intensity versus yield stress.

Table 2 Design space for each variable

4. Design optimization Following the development of the finite element models describing the behaviour of the MRB, an optimization routine was written to obtain the best possible design. For successful employment of the MRB into passenger vehicles, a factor requiring the most improvement is its weight, given that the steel components of the MRB are heavy and may add excessive weight to the vehicle. The braking torque is also an important parameter but, as long as a minimum torque requirement is met, it was considered of less important than the weight. Hence, the objective function for the optimization was defined so that a much greater importance is given to the weight than to the braking torque, by assigning a greater scalar weighting factor (0.9–0.1). The minimum acceptable value for the braking torque and the maximum acceptable value for the brake weight were chosen as 1010 N m and 65 kg, respectively. These numbers are the constraints of the optimization problem. In this initial MRB design phase, while the minimum braking torque value corresponds to that of typical CHBs, the value for the maximum weight was greatly relaxed such that it would allow the optimization procedure’s search for a wider design space. In addition, each MRB can potentially have more weight than a comparable on-wheel CHB as it would no longer have the extra weight carried by the CHB’s hydraulic components: the master cylinder, brake fluid lines, and pump. The optimization problem is expressed by Minimize subject to

Tb W f ðxÞ ¼ 100  0:1 þ 0:9 W ref T ref T b P 1010 N m and W 6 65 kg with xmin 6 x 6 xmax

ð7Þ

The above is the objective function for the optimization, subject to the two constraints, with x containing the design variables which are the dimensional design parameters expressed in Fig. 2 and Table 2. Tref = 1200 N m and

Design variables, x

Allowed values, xmin  xmax (cm)

th_disk

1.0–5.0 (one disk) 0.5–2.5 (two disks) 13.0–18.5 0.25–2.5 1–5 (one disk) 0.5–2.5 (two disks) 0.25–2.5 3.0–8.0 0.1

rad_disk rad_th_coil rad_th_casing ax_th_casing length_disk fl_gap

Wref = 30 kg were chosen as the reference values for the torque and weight, respectively, and xmin and xmax represent the chosen minimum and maximum values for each design variable. Table 2 shows the allowed ranges of these values. A custom-programmed optimization procedure was applied to the above problem using simulated annealing [13]. This is a random-search method that can find a global minimum for the objective function f(x) in Eq. (7). See [13] for the theory behind simulated annealing. Fig. 5 outlines the implemented simulated annealing procedure in our work. The details of this procedure and the results, including the actual values of f(x) and the optimization convergence plots, can be found in [10,11]. 4.1. Optimal MRB design The resulting optimal MRB design and its parameters, employing the Lord Corporation’s hydrocarbon-based MRF-132AD fluid, are given in Table 3. The computation time taken to obtain the solution was around 100 h. As shown in Fig. 6, the two-disk configuration (i.e., N = 4), with a stator between the disks, was the optimal brake design that minimized the objective function. This design yielded a maximum braking torque of 1013 N and a brake

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Fig. 5. Implemented simulated annealing procedure. Table 3 Optimal MRB design parameters Design variables

Optimal values

Number of disks Maximum current Number of wire turns th_disk rad_disk rad_th_coil Rad_th_casing ax_th_casing length_disk Fl_gap (h)

2 12 A Approx. 80 1.2 cm 16.8 cm 1.4 cm 0.5 cm 1.6 cm 5.5 cm 0.1 cm

weight of 27.9 kg, which by itself (without considering the overall brake system) is twice as heavy as that of a comparable performance CHB. Table 4 lists the remaining design parameters that was used or obtained by the simulation. Fig. 7 shows the steady-state temperature distribution in the two-disk MRB to verify that the maximum heat build-up exhibited in the brake is within the operating

Fig. 6. Optimal two-disk MRB configuration.

temperature range of the chosen MR fluid, 40 C to +130 C. Note that in [10,11], a transient finite element analysis was also performed to verify the effects of the

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Table 4 Other MRB design parameters Number of contact surfaces, N Outer radius of brake disk, rz Inner radius of brake disk, rw MR fluid viscosity, lp MR fluid thickness, h Electric constant, k Proportional gain, a Total inertia of brake disks, Iy Brake mass, mb

FL 4 0.168 m 0.118 m 0.09 Pa s 1 · 103 m 0.269 Pa m/A 12.5 · 103 m1 2.5 · 102 kg m2 27.9 kg

Tb

x

Rw mt g

Fr

Fn



Ff

Fig. 8. Forces acting on the driving wheel.

1 mt ¼ mv þ mw 4

ð8Þ

where mv is the mass of the vehicle, and mw is the mass of the wheel. The effective mass moment of inertia I is given by 1 I ¼ I y þ I w þ u2 I e ¼ I y þ I t 2

ð9Þ

where Iy is the total inertia of the brake disks (given in Table 4), Iw is the inertia of the wheel, u is the gear ratio, Ie is the moment of inertia of the engine. The factor 1/2 in Eq. (9) accounts for the distribution of the inertia of the engine to each of the two driving wheels. The rolling resistance Fr is a function of the vehicle’s linear velocity x_ [14]: 2:5

Fig. 7. Temperature distribution in two-disk configuration (in K).

repeated cycles of pressing and releasing the brake pedal on the heat build-up. In addition, [11] showed that the use of the afore-mentioned water-based MRF-241ES fluid yielded much better results in terms of the brake weight (18 kg), but this MR fluid had an unsuitable operating temperature range for the automotive brake application, 10 C to +70 C. 5. Dynamic model of the vehicle This section outlines the development of a dynamic model for a quarter vehicle used to simulate and evaluate the MRB’s performance. This model is comprised of the MRB dynamics, which was obtained in Section 2.1 and the dynamics of a driving wheel, which supports a quarter of the vehicle’s mass, as shown in Fig. 8. Naturally, this model is referred to as a quarter vehicle model. During braking, a braking torque Tb in Eq. (3) is applied to the wheel by the brake that decreases the wheel speed. With reference to Fig. 8, € h is the angular acceleration of the wheel with radius Rw, x is the distance traveled by the vehicle, Fr is the rolling resistance force, Ff is the friction force, Fn is the normal force and FL is the transfer of weight when braking. The total mass of the quarter vehicle, mt is

F r ¼ f0 þ 3:24f s ðK v x_ Þ

ð10Þ

where the constant Kv is a conversion factor from m/s to miles per hour (mph), f0 and fs are the curve fit parameters. The friction force Ff is written as F f ¼ lF n

ð11Þ

with the normal load of the quarter of the vehicle given by F n ¼ mt g 

mv hCE €x ¼ mt g  F L l

ð12Þ

where l is the wheel base and hCE is the center of gravity height. The braking force coefficient l in Eq. (11) is a function of the slip ratio, sr, defined as x_  Rw h_ ð13Þ x_ where h_ is the angular velocity of the wheel. The functional dependence between l and sr for different road conditions is given in Fig. 9. From Newton’s law, the equations of motion for the quarter vehicle are given by

sr ¼

mv hCE €x l I €h ¼ T b þ Rw F f  Rw F r ¼ T b þ lRw F n  Rw F r

mt€x ¼ F f ¼ lmt g þ l

ð14Þ ð15Þ

To verify the performance of the proposed MRB, parameter values for the quarter vehicle model were taken from [15] and listed in Table 5. These values are used to carry the control simulations presented in Section 6.

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In this section, an approach similar to [16] is taken to develop a sliding mode controller that regulates the slip ratio around 0.2, and consequently achieving a minimum braking distance. Using the relations in Eqs. (5) and (6), Eqs. (14) and (15) can be rewritten as lmt g lg €x ¼ ¼ ð16Þ mt þ lmv hCE =l 1 þ lml h€ ¼ si u  sv h_ þ lsn  sr ð17Þ where si = Ti/I, sv = Tv/I, sn = RwFn/I, sr = RwFr/I and u = i. Then, using the above equations, the time derivative of the slip ratio defined in Eq. (13) is given by   Rw lg _ _ h þ x_ ðsv h  lsn þ sr Þ þ x_ si u s_ r ¼ 2  ð18Þ 1 þ lml x_

Fig. 9. Braking force coefficient as a function of the slip ratio for various road conditions.

While it is desired to design a controller which is robust to different road conditions, the exact dynamics of l cannot ^, be exactly known. However, we can estimate it using l with the estimation error assumed to be bound by a known value l*, i.e., ^ j 6 l jl  l

Table 5 Parameters for the quarter vehicle model Wheel radius, Rw Wheel base, l Center of gravity height, hcg Wheel mass, mw 1/4 of vehicle’s mass, 1/4 mv Total moment of inertia of wheel and engine, It Basic coefficient, f0 Speed effect coefficient, fs Scaling constant, Kv

0.326 m 2.5 m 0.5 m 40 kg 415 kg 1.75 kg m2 1 · 102 5 · 103 2.237

6. Design of a sliding mode controller A MRB is naturally a pure electronically controlled brake system (without the hydraulics) that can perform existing and new braking control features such as an anti-lock braking system (ABS). An ABS is designed to prevent the wheel lockup, to minimize the vehicle’s stopping distance, and to enhance stability. One of the objectives of the ABS is to regulate a desired slip ratio so that the maximum braking force coefficient (or road friction coefficient) is maintained during the braking process. This in turn leads to minimization of the braking distance. However, the slip ratio is dependent on the road surface condition (e.g., dry versus wet). Fig. 9 suggests that this slip ratio corresponds to approximately 0.2, regardless of the road conditions. Hence, it is also necessary to develop a robust controller that provides an optimal braking torque by maintaining the desired slip ratio in different road conditions. To achieve this, the sliding mode control technique, which is a nonlinear control method, was employed in our work. The performance of the proposed MRB is investigated through simulations carried out with the dynamic model of a quarter vehicle developed in the previous section.

ð19Þ

The sliding surface, S(t), which will allow the slip ratio sr to track the desired slip ratio srd, can be defined by the scalar equation s(sr, t) = 0, where  n1 d ~sr þk ð20Þ sðsr ; tÞ ¼ dt with ~sr ¼ srd  sr and k a strictly positive constant [17]. Since Eq. (17) has an order of one, n = 1, Eq. (20) becomes s ¼ srd  sr

ð21Þ

The sliding condition which keeps the scalar s at zero is defined by [15]: s  s_ 6 gjsj

ð22Þ

where g is a strictly positive number. Now Eq. (21) needs to be differentiated once in order for the control input u to appear [17]. The derivative of the sliding surface is given by " # Rw lgh_ s_ ¼ _sr ¼ 2  x_ ðsv h_  lsn þ sr Þ  x_ si u ð23Þ x_ 1 þ lml and the dynamics while in sliding mode can be written as s_ ¼ 0

ð24Þ

Solving the above equation for the control input, the best approximation for u is given by ^u ¼

^gh_ l 1 ^ sn þ sr Þ  ðsv h_  l ^ml Þ_xsi si ð1 þ l

ð25Þ

where ^u can be interpreted as the best estimate of the control input that would maintain s_ ¼ 0 if the dynamics of l were exactly known. In order to satisfy the sliding condition in Eq. (22), despite the uncertainties on the dynamics of l, a term u, which is discontinuous across the surface s = 0, must be added to ^u. Inserting Eq. (23) into Eq. (22) yields:

E.J. Park et al. / Mechatronics 16 (2006) 405–416

" # Rw lgh_ _ s  s_ ¼ s  2  x_ ðsv h  lsn þ sr Þ  x_ si u x_ 1 þ lml

ð26Þ

Now, let gþb  u¼ sgnðsÞ ð27Þ si where the term b is defined below. Since u ¼ ^u þ u, Eq. (26) can be rewritten as follows: " ^Þgh_ Rw ðl  l ^Þ_xsn þ ðl  l s  s_ ¼ s  2 ^ ml Þ x_ ð1 þ lml Þð1 þ l #  x_ ðg þ bÞ sgnðsÞ

ð28Þ

If s P 0, Eq. (28) can be written as " ^Þgh_ Rw ðl  l s  s_ ¼ s  2 ^ml Þ x_ ð1 þ lml Þð1 þ l # ^Þ_xsn  x_ b  g_x þ ðl  l Rw ^Þðgh_ þ x_ sn Þ  x_ b  g_x ½ðl  l ð29Þ x_ 2 To satisfy the sliding condition of Eq. (22), b can be determined from Eq. (29) as l b ¼ ðgh_ þ x_ sn Þ ð30Þ x_ Eq. (30) also satisfies the sliding condition of Eq. (23) for s 6 0. Therefore by inserting Eq. (30) into Eq. (27), u is obtained as follows: u ¼ ^u þ  u 6s

¼

^gh_ 1 l _ þ ð^ lsn  sr  sv hÞ ^ml Þ_xsi si ð1 þ l    1 l  _ þ gh þ x_ sn sgnðsÞ gþ si x_

ð31Þ

413

The control input u obtained in Eq. (31) can produce a steady-state error caused by the parametric uncertainties. The steady-state error can be reduced by including an integral term in the sliding surface definition [9]:  Z t d ~sr dr ðk P 0Þ þk ð32Þ sðsr ; tÞ ¼ dt 0 Then the resulting control law is derived following a similar procedure as the above _ ^gh_ 1 l _ þ kh þ x_ k ðsrd  1Þ u¼ þ ð^ lsn  sr  sv hÞ ^ml Þ_xsi si si R w si ð1 þ l    1 l _ þ g þ ðgh þ x_ sn Þ sgnðsÞ ð33Þ si x_ The control law in Eq. (33) satisfies the sliding condition in Eq. (22) and leads to perfect tracking in the presence of the model uncertainties. However, the generated control signal is discontinuous across the surface S(t) [17] and thus can causes control chattering. To eliminate chattering, a thin boundary layer of thickness w neighbouring the switching surface can be introduced [18]. Then the hard switching function sgn(s) can be replaced by the sat(s/w) function. 7. Simulation results Fig. 10 shows the Matlab Simulink model used to investigate the performance of the proposed MRB using the sliding mode control. The vehicle velocity x_ and the wheel angular velocity h_ are obtained by integrating Eqs. (16) and (17) respectively. Then the slip ratio error, ~sr , is calculated by subtracting the current slip ratio from the desired slip ratio. The simulations are for the dry and ice road conditions and the braking force coefficient is obtained from a look-up table containing information on the slip ratio versus braking force coefficient only profile.

Fig. 10. MRB Simulink model.

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It should be noted that the look-up table is used only to simulate the road conditions. Instead, it is Eq. (19) that deals with the model uncertainties due to the changing road _ ~sr , the controller conditions. Based on the values of x_ , h, generates the input signal for the MRB, which in turn produces the braking torque required to maintain the desired slip ratio. Fig. 11 shows the braking simulation results on a dry road for the MRB with design parameters given in Tables 3–5. The vehicle is braked from an initial velocity of

70 km/h and comes to a complete stop in approximately 2.3 s (Fig. 11(a)). It can be observed from Fig. 11(b) and (d) that the braking torque is proportional to the current supplied to the electromagnet. This is an expected result as the braking torque is formed mainly by the TH contribution in Eq. (5). Note that TMRE is a function of the applied current i. As seen in Fig. 11(c), the slip ratio is regulated around the desired optimal value of 0.2. This allows for the wheel speed to closely follow the vehicle speed without locking up (Fig. 11(a)), which suggests that the sliding

Fig. 11. Anti-lock braking simulation results: dry road.

Fig. 12. Anti-lock braking simulation results: ice road.

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Fig. 13. Simulated performance of a hydraulic brake: dry road.

mode controller succeeds in providing an optimal wheel slip control of the MRB. Fig. 12 shows the corresponding braking simulation results on an ice road. Next, the performance of the proposed MRB was compared to the simulated performance of a CHB system. The simulations are carried out with the same quarter vehicle dynamic model and parameters described in Section 5 and a bang–bang controller implemented in the ABS control example of Simulink [19]. The simulation results shown in Fig. 13 reveal fluctuations in the slip ratio which are due to the delay in the response of the hydraulic actuator. This fluctuation leads to a braking response (3 s) which is slower that that of the MRB, even without the consideration of the additional 200–300 ms of delay coming from the initial pressure build-up in the hydraulic line.

In addition, the proposed MRB can further improved in terms of braking torque, structural weight, and heat dissipation by introducing slots/holes in the rotating disks. For instance, as discussed in Section 4.1, the brake weight can be significantly reduced by employing a water-based MR fluid (e.g., Lord Corporation’s MRF-241ES), but then the structural design has to be changed to improve heat dissipation capabilities, unless a MR fluid with better temperature properties becomes available. Acknowledgement This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. References

8. Conclusions Based on a commercially available magnetorheological fluid, a new automotive brake system was proposed. The mathematical model of the proposed MRB was presented, and a finite element model was created to analyze the magnetic, fluid flow and heat transfer phenomena within the system. Using a design optimization procedure centered around the finite element analysis, an optimum MRB design with two rotating disk was proposed. The proposed MRB is naturally a pure electronically controlled brake system that uses the use of ‘‘bytes and amperes instead of bars and compressed brake fluid.’’ This allows easy implementation of advanced braking control features with a smaller number of components, simplified wiring, improved braking response and generally optimized layout. The control performance of the MRB in a vehicle was studied using a quarter vehicle model. A sliding mode controller was designed for an optimal wheel slip control, which is a feature of an ABS. The controller maintains the slip ratio at the desired value which assures a maximum braking force coefficient and provides a minimum stopping distance. The simulation results show the potential of the proposed MRB system to provide fast anti-lock braking. Future work must focus on life-cycle tests to assess the reliability and longevity of the system and to ensure that it can effectively replace the existing hydraulic brake technology.

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