See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/224640406
`
`Low-order modeling of vehicle roll dynamics
`
`Conference Paper  in  Proceedings of the American Control Conference · July 2006
`
`DOI: 10.1109/ACC.2006.1657345 · Source: IEEE Xplore
`
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`Sean N Brennan
`Pennsylvania State University
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`Magna - IPR2020-00777 - Ex. 2006 - 001
`
`

`

`Low-Order Modeling of Vehicle Roll Dynamics
`
`Bridget C. Hamblin, Ryan D. Martini, John T. Cameron, and Sean N. Brennan, Member, IEEE
`
`
`
`Abstract— This work presents results of an ongoing
`investigation into models and control strategies suitable to
`prevent vehicle rollover due to untripped driving maneuvers.
`For use as a design model for controller synthesis, low-order
`models are sought
`that have sufficient complexity
`to
`characterize a vehicle’s roll behavior, yet are not unnecessarily
`complex or nonlinear. To compare different low-order models
`found in literature, this work investigates the validity of several
`roll dynamic models by comparing model prediction to
`experiment in both the time and frequency domains. Discussion
`is also given on methods for parametric fitting of the models
`and areas where significant model error is observed.
`
`
`Index Terms—Vehicle rollover, vehicle dynamics, modeling
`of dynamic systems
`
`A
`
`I. INTRODUCTION
`ccidental death due to motor vehicle accidents claim
`over 1.2 million life-years of unlived life each year, and
`is the largest premature death factor for those under the age
`of 65 [1]. Vehicle accidents are the single largest cause of
`fatalities for males 44 years and under and for females 34
`years and under [2]. These deaths are sudden, and often
`strike when a person is at the peak of both their professional
`and personal/family life.
`Among the myriad causes of vehicle accidents, rollover
`stands out as an area deserving of particular focus. While
`vehicle rollover is involved in only 2.5% of the 11 million
`accidents a year, it accounts for approximately 20% of all
`fatalities [3].
`To study rollover, the National Highway Traffic Safety
`Administration (NHTSA) has developed a number of
`transient maneuvers that are observed to induce untripped
`wheel lift or even untripped vehicle rollover in some vehicle
`models [4, 5]. While
`this experimental approach
`is
`unarguably valid for illustrating shortcomings in vehicle
`behavior, the method does have shortcomings. In particular,
`it is difficult to definitively establish from only a very small
`subset of tests whether or not roll safety is ensured over all
`possible transient maneuvers. Additionally, experimental
`
`
`B. Hamblin and R. Martini are current graduate students at Penn State.
`J. T. Cameron is a recent graduate from Mechanical and Nuclear
`Engineering who is now working with Harris Corp. in Florida.
`S. Brennan is an Assistant Professor in the Mechanical & Nuclear
`Engineering Department, Penn State University, 318 Leonhard Building,
`University Park, PA 16802. He shares a joint appointment with the
`Pennsylvania Transportation Institute. (corresponding author, phone
`(814)863-2430; sbrennan@psu.edu)
`
`results do not directly translate into a vehicle model suitable
`for rollover mitigation through feedback control. Both
`factors highly motivate the development of vehicle roll
`models.
`In considering the choice of which roll dynamic model to
`use, the choice of complexity in the model should match
`well with the intended use of the model. This study is
`focused on finding dynamic models that are well suited to
`the design and
`implementation of online,
`real-time
`controllers to prevent the onset of rollover. A goal of this
`work and particular departure from previous studies is to not
`only understand and validate the linear vehicle dynamics of
`roll behavior, but also to understand the relative impact of
`various assumptions in creating the vehicle models to ensure
`the use of the simplest model possible in later controller
`synthesis.
`The remainder of the paper is organized as follows:
`Section 2 presents preliminaries on nomenclature and model
`formulation. Section 3 presents several models that will be
`compared in this study. Section 4 discusses how the inertial
`parameters of the models were obtained. Section 5 discusses
`experimental fitting of linear tire parameters. Section 6
`details modifications to the linear tire to account for camber.
`Section 7 presents dynamic model fits. Conclusions
`summarize the main points.
`
`II. PRELIMINARIES
`The following notation is used for each of the models
`described in this work:
`U
`
` Longitudinal velocity (body-fixed frame)
` m, ms
` Vehicle mass and sprung mass respectively
` Ixx, Iyy, Izz Inertia about roll (X), pitch (Y),vertical (Z) axis
`Ixz
`
` Inertia product
`lf, lr
`
` Front- and Rear-axle-to-CG distances
`L
`
` Track of vehicle (lf + lr)
`Kφ
`
` Effective roll stiffness of the suspension
` Dφ
` Effective roll damping of the suspension
`h
`
` height from roll axis to CG
`fα , rα Slip angle of the front, rear tires
`
`
`β
`
` Slip angle of the vehicle body
` Cf, Cr
` Front, Rear cornering stiffness
`
`fδ
` Front steering angle
`For ease of comparison model to model, each of the
`models is presented in a compact symbolic notation of the
`form:
`
`Magna - IPR2020-00777 - Ex. 2006 - 002
`
`

`

`(1)
`fuFqKqDqM
`+⋅
`+⋅
`=⋅
`⋅
`&&
`&
`where i denotes the model number (1 to 4 for this study),
`and
`
`between the different models can be found in [6].
`
`
`{
`}T
`
`(2)
`
`q
`y
`φψ=
`which denotes the lateral position, yaw angle, and roll angle
`respectively. The input to the model,
`{
`}T
`
`(3)
`
`F
`F
`u =
`r
`f
`denotes the front and rear lateral tire forces respectively. The
`general MDK form described by Eq. (1) allows for an
`intuitive
`term-by-term comparison between different
`models. Further, this MDK form can be readily transformed
`to the general state-space form of:
`dx
`
`uBxA
`⋅+⋅=
`dt
`
`
`
`(4)
`
`(9)
`
`(10)
`
`
`
`
`
`⎥⎥⎥ ⎦⎤
`
`2
`
`
`
`⎥⎥⎥ ⎦⎤
`
`I
`
`xx
`
`zz
`I
`xz
`mU
`0
`hUm
`s
`
`Figure 1: SAE Coordinate System
`
`A. Model 1 - 3DOF Model Assuming Existence of Sprung
`Mass and No X-Z Planar Symmetry
` The most complex model considered in this work is
`derived by assuming a sprung and unsprung mass, and
`assuming the unsprung mass has a non-symmetric mass
`distribution about the x-z plane. It is motivated by the model
`presented by Mammar et. al. [10]; further details and
`assumptions are listed therein. Following the MDK form
`specified earlier, the MDK matrices are given by:
`m
`hm
`0
`s
`I
`I
`0
`−
`xz
`hm
`hm
`+
`−
`s
`s
`0
`0
`φD
`
`0
`0
`0
`00
`00
`00
`
`⎢⎢⎢ ⎣⎡
`
`⎢⎢⎢ ⎣⎡
`
`M
`
`1
`
`=
`
`D
`1
`
`=
`
`
`
`
`
`(11)
`
`(12)
`
`
`
`⎥⎥⎥ ⎦⎤
`
`0
`0
`ghmK
`−
`s
`φ
`1
`b
`−
`0
`
`⎥⎥⎥ ⎦⎤
`
`
`
`⎢⎢⎢ ⎣⎡
`
`1
`a
`0
`
`⎢⎢⎢ ⎣⎡
`
`F
`1
`
`=
`
`K
`1
`
`=
`
`
`
`
`
`B. Model 2 - 3DOF Model Assuming Existence of Sprung
`Mass and X-Z Planar Symmetry
`A common assumption in the above model is that the
`vehicle is symmetric about the x-z plane, thus making Ixz
`zero and eliminating all cross terms. An example of this is
`presented by Kim and Park in [8]. In presenting this model,
`some authors absorb the suspended mass term, mgh, in the
`(3, 3) element of the K matrix, into the roll stiffness. For
`example, Kim and Park cited above make this assumption.
`Placing the equations of motion into the form specified by
`Eq. (1), the damping, stiffness, and force matrices remain
`the same,
`
` , KDD
`
`=
`2
`1
`2
`while the mass matrix becomes:
`
`=
`
`
`
` , FK
`1
`2
`
`=
`
`
`
`F
`1
`
`(13)
`
`with the state vector,
`]φφ &
`[
`
`(5)
`
`rVx =
`representing lateral velocity, yaw rate, roll angle and roll
`rate respectively. The transformation from MDK form to
`state space is given by the following. Let:
`001
`000
`0000
`⎡
`⎤
`⎡
`⎤
`⎡
`010
`000
`0000
`000
`100
`1000
`100
`000
`0000
`⎣
`and define,
`)
`(
`
`(7)
`
`R
`I
`IMRE
`T +
`⋅
`⋅=
`−
`where
`nI represents the identity matrix of size n. Then the
`state-space matrices A and B are obtained from matrices M,
`D, K and F as:
`(
`)
`EA
`RDR
`=
`⋅−⋅
`⋅
`
`(
`)FR
`EB
`=
`⋅
`⋅
`The state-space form more conveniently allows numerical
`simulation and model comparisons used in later sections.
`
`⎤
`
`
`
`(6)
`
`⎥⎥⎥⎥
`
`⎦
`
`⎢⎢⎢⎢
`
`⎣
`
`,
`
`T
`
`=
`
`⎥⎥⎥⎥
`
`⎦
`
`⎢⎢⎢⎢
`
`⎣
`
`S
`
`=
`
`,
`
`⎥⎥⎥⎥
`
`⎦
`
`⎢⎢⎢⎢
`
`R
`
`=
`
`3
`
`4
`
`T
`
`SKR
`⋅−
`⋅
`
`T
`
`+
`
`T
`
`
`
`(8)
`
`11
`−−
`
`III. VEHICLE MODELS
`A search of recent literature found over two dozen unique
`vehicle models inclusive of roll dynamics, but of these, only
`a few are chosen for further analysis. Considerations used to
`eliminate certain models are detailed in previous work (see
`[6]), but the main criteria are based primarily on model
`complexity, whether or not the model had been validated
`experimentally by the authors of the model, and how easily
`model parameters can be measured or inferred.
`To emphasize the similarity between the models used in
`this study, each is presented in the same coordinate system
`regardless of the coordinate system used in the original
`publication of the model [7-9]. Herein they all follow the
`SAE right-handed sign convention shown in Fig 1.
` For brevity, details of each model derivation have been
`omitted from this work, but further details can be found in
`the original publications [7-9] and in previous work [6]. A
`discussion of
`the notable similarities and differences
`
`Magna - IPR2020-00777 - Ex. 2006 - 003
`
`

`

`
`
`order to find or fit chassis parameters for models 1-3.
`
`IV. OBTAINING VEHICLE INERTIAL PARAMETERS
`To analyze validity of the models to describe vehicle
`chassis behavior, experiments were performed on a 5-door
`1992 Mercury Tracer station wagon available at Penn
`State’s Pennsylvania Transportation Institute test track. A
`significant improvement over previous work is that this
`study was conducted using Novatel’s GPS/INS “SPAN”
`system. This GPS/INS system is based off two Novatel
`OEM4 dual frequency GPS receivers and the Honeywell
`HG1700 military tactical-grade IMU. This combination can
`provide estimates of position, velocity and attitude at rates
`up to 100Hz. In differential carrier phase fixed-integer
`mode and with continuous presence of GPS data, the system
`achieves a position solution with an accuracy of 2 cm.
`Attitude can be estimated with a 1-sigma accuracy of 0.013
`degrees for roll, 0.04 deg for pitch and 0.04 degrees for yaw.
`All velocity errors are 0.007 m/s (one sigma) [12].
`Many of the inertial parameters appearing in models 1-4
`are easily measured or obtained from the National Highway
`Traffic Safety Administration database [13]. The table
`below presents these parameter values, units, and their
`source.
`
`
`
`
`
`
`Table 1: Inertial parameter values
` Variable Value
`Units How obtained
`m
`
`1030
`kg
`Measured
`ms
`
`824
`kg
`Estimated
`Wf
`
`6339
`N
`Measured
`Wr
`
`3781
`N
`Measured
`lf
`
`0.93
`m
`NHTSA*
`lr
`
`1.56
`m
`NHTSA*
`L
`
`2.49
`m
`Calculated
`h
`Measured
`
`0.25
`m
`kg-m2 NHTSA*
`Izz
`
`1850
`kg-m2 NHTSA
`Iyy
`
`1705
`kg-m2 NHTSA
`Ixx
`
`375
`kg-m2 NHTSA
`Ixz
`
`72
`*measurements were also made and these confirmed the NHTSA
`value to within a few percent
`
`Estimates of sprung mass, ms, were obtained by
`approximating the sprung mass as 0.8 times the total mass.
`The CG height was found to be 0.25 meters above the roll
`axis. The roll axis was found by video-taping the vehicle
`undergoing a rocking motion from the front and rear,
`determining the center of rotation at the front and rear axles,
`then using similar triangles to determine the axis of rotation
`at the center-of-gravity of the vehicle. Note that the sprung-
`mass height above the roll-axis is not the height of the CG
`above the road surface reported by NHTSA, 0.52 meters for
`this vehicle.
`
`V. FITTING BICYCLE MODEL TIRE PARAMETERS
`Several model parameters, especially the tire cornering
`stiffnesses,
`require experimental
`fitting and careful
`consideration of the tire’s impact on the model behavior.
`The models presented in this study lump right- and left-side
`
`m
`hm
`0
`s
`I
`0
`0
`zz
`hm
`I
`hm
`0
`+
`s
`s
`xx
`With the assumption of a symmetric mass distribution, the
`yaw dynamics are not directly coupled to the roll and
`sideslip dynamics in the MDK form. In state-space, they can
`only be coupled through the inversion of the mass matrix,
`1−E , and through the tire forces, if these forces are
`dependent on roll.
`C. Model 3 - 3DOF Model Assuming Sprung Mass Only
`In addition to the assumptions given previously, it is
`sometimes assumed that the entire mass of the vehicle is
`concentrated at the sprung mass. The paper by Carlson et. al
`[7] uses this assumption. To modify previous models to
`express this assumption, the unsprung mass is made zero
`and the sprung mass is made equal to the total mass of the
`vehicle. The resulting MDK matrices are:
`m
`hm
`0
`s
`s
`I
`0
`0
`zz
`I
`hm
`hm
`0
`+
`s
`s
`Um
`0
`0
`s
`0
`0
`0
`hUm
`φD
`0
`s
`The F and K matrices are unchanged, e.g.
`
`
`K
` , FK
`
`F
`=
`=
`2
`2
`3
`3
`D. Model 4- 2DOF Model Assuming No Roll Dynamics
`Finally, if one assumes that the sprung mass height is zero,
`the roll dynamics are completely eliminated because there is
`no longer any coupling from yaw or lateral velocity into roll.
`Without this coupling, there is no energy input to the roll
`model other
`than
`initial conditions. This assumption
`produces the well-known “bicycle model” which describes
`the vehicle’s planar dynamics [11].
`m
`0
`0
`0
`0
`0
`
`
`
`
`
`(18)
`
`(19)
`
`⎢⎢⎢ ⎣⎡
`
`⎢⎢⎢ ⎣⎡
`
`M
`
`4
`
`=
`
`D
`4
`
`=
`
`
`
`
`
`(14)
`
`
`
`⎥⎥⎥ ⎦⎤
`
`2
`
`⎢⎢⎢ ⎣⎡
`
`M
`
`2
`
`=
`
`(15)
`
`(16)
`
`(17)
`
`
`
`⎥⎥⎥ ⎦⎤
`
`2
`
`
`
`⎥⎥⎥ ⎦⎤
`
`xx
`
`⎢⎢⎢ ⎣⎡
`
`M
`
`3
`
`=
`
`⎢⎢⎢ ⎣⎡
`
`D
`3
`
`=
`
`
`
`
`
`
`
`0
`0
`0
`0
`0
`0
`
`⎥⎥⎥ ⎦⎤
`
`⎥⎥⎥ ⎦⎤
`
`0
`zzI
`0
`mU
`0
`0
`000
`000
`000
`Again, the F matrix is unchanged:
`(21)
`
`
`4 FF = 3
`
`
`While this model does not include any roll dynamics and
`only exhibits lateral and yaw dynamics, it is included in this
`work because bicycle model parameters are used in all other
`models. It is therefore important to consider this model in
`
`(20)
`
`
`
`⎥⎥⎥ ⎦⎤
`
`⎢⎢⎢ ⎣⎡
`
`4K
`
`=
`
`
`
`Magna - IPR2020-00777 - Ex. 2006 - 004
`
`

`

`
`
`
`Mag (dB)
`
`-100
`
`-200
`
`Phase (deg)
`
`
`
`20
`
`
`
`High Amplitude
`Low Amplitude
`100
`
`w (rad/s)
`
`101
`
`15
`
`10
`
`
`
`05
`
`0
`
`100
`
`101
`
`w (rad/s)
`Figure 1: Frequency Response, Steering Input to
`Lateral Velocity
`
`
`
`lateral tire forces to a single force on the front and rear
`axles,
`fF and rF . This single-wheel representation of a two-
`wheel axle is why the bicycle model is so named. Further, it
`is assumed that the lateral forces acting on each tire are
`directly proportional to the tire slip with proportionality
`constants on the front and rear tires of
`rC
`fC and
`respectively:
`
`(22)
`
`F
`C
`F
`C
`,
`,
`=
`α
`α =
`r
`r
`r
`f
`The slip angles,
`iα, are defined as the angle between the
`tire’s orientation and the velocity vector of the center of the
`tire:
`
`⋅
`
`f
`
`r
`
` (23)
`
`−
`
`δ
`f
`
`r
`
`
`
`(24)
`
`δ
`f
`
`≈
`
`⎠⎞
`
`≈⎟
`
`r
`
`⎠⎞
`
`r
`
`⋅
`
`f
`
`f
`
`⋅
`
`f
`
`r
`
`lV
`+
`U
`
`⎜⎝⎛
`
`−1
`
`⎝⎛
`
`⎜⎜
`
`α
`f
`
`=
`
`−1
`
`tan
`
`
`
`
`
`lV
`+
`−⎟⎟
`U
`lV
`lV
`⋅−
`⋅−
`tanα
`r
`r
`=
`f
`U
`U
`The simplifying assumptions made for Eqs. (23) and (24)
`are that the slip angles are small enough to allow a linear
`approximation and that right- and left-side differences in tire
`forces are negligible. With these assumptions, the tire forces
`can be written as:
`
`
`⎠⎞
`
`δ
`⎟⎟
`
`f
`
`f
`
`−
`
`⎝⎛
`
`f
`
`F
`
`f
`
`=
`
`(26)
`
`
`
`⎟⎠⎞
`
`r
`
`⋅
`
`r
`
`⎜⎝⎛
`
`r
`
`F
`r
`
`=
`
`
`
`lV
`+
`C
`⎜⎜
`U
`lVC
`−
`U
`Longitudinal forces acting upon the tires are assumed to be
`zero, and longitudinal velocity, U, is assumed to be constant.
`The resulting expressions, when substituted into models
`1-4 above, predict linear models. However, it is well known
`that the linearity assumption is violated under aggressive
`maneuvering. Others have noted
`that
`if
`the
`lateral
`acceleration remains below 0.4 g’s, then assumptions of
`linearity appear to hold (many cite [14] as support).
`Therefore, care was taken in all testing to ensure that the
`experiments were conducted at lower accelerations.
`To test whether or not linearity is actually preserved in the
`measured data, two frequency responses were conducted on
`the vehicle: one for steering inputs of small amplitude (1/4
`rotation of the hand wheel) and one for large amplitudes
`(slightly less than 1/2 rotation of the handwheel). The
`resulting Bode plots are overlaid and shown in Figs. 1 and 2
`below for the two states of the bicycle model, yaw rate and
`lateral velocity recorded at a speed of 25 mph. The linearity
`of the models is evident.
`To find the cornering stiffnesses, two methods were used
`that are both based on steady-state data. Steady-state data
`was chosen since these data should be least influenced by
`model-to-model differences in high-order dynamics. The
`first fitting method attempts to match the DC gains of the
`sinusoidal frequency responses of Figs. 1 and 2. The second
`method is based on matching measured responses from
`steady-state turning around a skid-pad circle. Each is
`detailed below.
`
`High Amplitude
`Low Amplitude
`
`101
`
`w (rad/s)
`
`10
`
`Mag (dB)
`
`0
`
`
`100
`
`0
`
`-50
`
`-100
`
`Phase (deg)
`
`100
`
`101
`
`G
`V
`
`=
`
`f
`
`f
`
`2
`
`f
`
` (29)
`
`lC
`⋅
`r
`r
`
`)
`
`w (rad/s)
`Figure 2: Frequency Response, Steering Input to
`Yaw Rate,
`
`From the state-space form of Eq. (4), the DC gains, G , of
`the bicycle model from steering input to state output are
`given by:
`
`(28)
`
`DG
`BCA
`1−
`−
`=
`For lateral velocity, V, this DC gain was parametrically
`solved to be:
`(
`)
` ( ) ( UmlCLlCCU
`
`)
`
`
`+⋅
`⋅
`⋅
`⋅
`⋅
`
`f
`f
`r
`f
`r
`2
`2
`
`
` ( lCUm
`l
`l
`l
` (lCC
`)
`2
`2
`+
`−
`⋅+
`⋅+
`⋅
`⋅
`f
`r
`f
`r
`r
`and for yaw rate, r:
`G
`r
`
`=
`
`2
`
`2
`
`f
`
`
` ( ) LCCU
`
`⋅
`⋅
`f
`r
` (lCC
`l
` ( lCUm
`
`lC
`
`l
`l
`)
`2
`)
`2
`+
`⋅
`⋅+
`⋅+
`−⋅
`⋅
`⋅
`f
`r
`r
`f
`r
`r
`r
`f
`f
`The numerical values of
`VG and
`rG can be read from Figs. 1
`and 2 as 3.804 m/s lateral velocity per radian of steering
`input and 3.599 rad/sec yaw rate per radian of steering input,
`
` (30)
`
`(25)
`
`Magna - IPR2020-00777 - Ex. 2006 - 005
`
`

`

`For the sideslip maneuvers as shown in Fig. 3, the average
`fδ , during the point at which sideslip passed
`steering input,
`through zero were approximately 0.0830 rad of front wheel
`angle.
`This
`gives
`an
`approximate
`value
`of
`.
`C f
`N
`rad
`700,57−=
`
`
`/
`
`
`Figure 3: Using the zero-sideslip condition to measure
`cornering stiffness
`
`=
`
`f
`
`
`
`(36)
`
`
` The understeer gradient can also be used to find the front
`cornering stiffness from Eq. (34):
`CW
`⋅
`
`f
`r
`C
`KCW
`−
`⋅
`r
`r
`us
`Measurements of the understeer gradient can be obtained by
`plotting steering input versus lateral acceleration as shown
`in Fig. 4 below. Using these results, an understeer gradient
`. This gives a front cornering
`was found to be
`.0=usK
`016
`stiffness value of
`which is roughly
`C f
`N
`rad
`,68−=
`400
`/
`15% different from the previous value.
`
`respectively. Rearranging the above equations, one can
`directly solve for cornering stiffnesses after substitution of
`known vehicle parameters:
`GUml
`)
`(
`2
`⋅
`⋅
`⋅
`
`f
`r
`LGlG
`)
`(
`⋅
`⋅−
`r
`v
`r
`lCGmU
`(
`)
`N
`2
`⋅
`⋅
`⋅
`⋅
`−
`r
`r
`r
`( lGmULCGLUC
`
`)
`rad
`2
`2
`−⋅
`⋅
`⋅
`⋅
`−
`⋅
`⋅
`⋅
`r
`r
`r
`r
`f
`These values, when substituted into the bicycle model, did
`appear to match the measured sinewave time responses
`obtained
`in
`conducting
`the
`frequency
`responses.
`Additionally, it appeared to match maneuvers that did not
`excite significant roll, e.g. lane changes and the like.
`However, the model had a very poor fit for vehicle response
`data collected during steady-state turning on the skid pad. In
`an attempt to reconcile steady-turning data, a method was
`sought to determine cornering stiffness values for this data
`alone.
` For steady-state turning around a constant radius circle,
`, measured at the center-of-gravity
`the side-slip,
`UV /=β
`of the vehicle is given by [15]:
`
`
`(32)
`
`
`
`RU
`
`2
`
`C
`r
`
`=
`
`C
`
`f
`
`=
`
`=
`
` N -88,385
`
`
`rad
`
`
`
`(31)
`
`=
`
`-83,014
`
`
`
`+
`
`Rl
`
`r
`
`=β
`
`W
`r
`gC
`⋅
`r
` The side-slip is clearly dependent on speed and radius of
`the turning circle. This therefore suggests a method to
`determine the rear cornering stiffness: measure side slip on
`the vehicle and slowly increase the vehicle speed traversing
`a steady circle to the point where side slip becomes zero. At
`0=β , the above expression gives the cornering stiffness as:
`W
`
`
`(33)
`C
`2U
`r
`−=
`gl
`⋅
`r
`Note that this expression is independent of the radius of the
`turn. Fig. 3 below shows data collected during one of many
`maneuvers to determine the speed at which zero sideslip
`occurred. Repeated measurements showed that zero sideslip
`occurred for this vehicle around 14.1 m/s. Using values
`measured for the vehicle, the calculated rear cornering
`.
`stiffness was found to be
`N
`rad
`Cr
`/
`300,49−=
` The front cornering stiffness can be found using the
`relationship at steady state between steering input and front
`and rear cornering stiffness:
`
`r
`
`
`
`(34)
`
`U
`2
`Rg
`⋅
`
`⎞
`
`⎟⎟⎟⎟ ⎠
`
`ff
`
`CW
`
`−
`
`CW
`
`rr
`
`⎛
`
`⎜⎜⎜⎜ ⎝
`
`+
`
`RL
`
`43421
`Kus
`is the understeer gradient (assuming
`
`
`
`δ
`
`=
`
`⎟⎟⎠⎞
`
`ff
`
`CW
`
`−
`
`CW
`
`rr
`
`⎜⎜⎝⎛
`
`where
`
`K
`
`us
`
`=
`
`negative cornering stiffnesses). So
`−
`
`C
`
`
`
`(35)
`
`CW
`
`rr
`
`W
`f
`Rg
`+⋅
`U
`2
`
`⎟⎠⎞
`RL
`
`⎜⎝⎛
`
`δ
`f
`
`−
`
`=
`
`f
`
`Magna - IPR2020-00777 - Ex. 2006 - 006
`
`

`

`forces are assumed to depend on roll by the following
`relationship:
`
`
`(37)
`
`
`
`F
`C
`C
`α
`φ
`=
`+
`f
`f
`fwf
`f
`φ
`CF
`C
`α
`=
`φ
`+
`r
`r
`r
`r
`wr
`φ
`rwφ are the camber angles of the front and
`fwφ and
`where
`rear tires, and
`fCφ and
`rCφ are the proportionality constants
`representing the change in tire force as a function of roll
`F
`angle,
`, etc. Now define the change in wheel
`f
`C
`φ
`φ
`camber angle as a function of the entire vehicle’s roll angle
`via a proportionality constant, S:
`φ
`φ
`=
`⋅
`
`wf
`v
`φ
`
`φ⋅
`=
`wr
`v
`r
`The steady-state roll angle of the vehicle for a constant
`vφ , is solved by moment
`velocity, constant radius turn,
`balance:
`
`
`(38)
`
`(39)
`
`
`
`f
`
`SS
`
`2
`
`
`
`RU
`
`s
`
`m
`
`⋅
`
`Kh
`
`φ
`v
`
`=
`
`∂∂
`
`=
`
`f
`
` (40)
`
`⎥⎦⎤
`
`RU
`
`2
`
`S
`
`f
`
`⎥⎦⎤
`
`RU
`
`2
`
`S
`
`r
`
`f
`
`−
`
`−
`
`2
`
`⋅
`
`⋅
`
`Ll
`⎢⎣⎡
`
`r
`
`⎢⎣⎡
`
`11
`
`C
`f
`α
`
`=
`
`=
`
`
`
`(41)
`
`RU
`
`−
`
`C
`
`*
`f
`
`⋅
`
`Ll
`⎢⎣⎡
`
`r
`
`=
`
`f
`
`S
`
`f
`
`αα
`
`r
`
`f
`
`
`
`
`
`
`
`KUS=0.016
`
`KUS=0.016
`
`Large CCW
`Large CW
`Small CCW
`Small CW
`
`0.12
`
`0.115
`
`0.11
`
`0.105
`
`0.1
`
`0.095
`
`0.09
`
`0.085
`
`0.08
`
`0.075
`
`Steer Angle, δf (rad)
`
`
`0
`
`0.1
`
`0.7
`
`0.8
`
`
`
`0.6
`0.5
`0.4
`0.3
`0.2
`Lateral Acceleration, U2/R⋅g (g's)
`Figure 4: Steady state steering measurements measured as a function of
`lateral acceleration for two different radii turning circles along
`two different directions, clockwise and counter-clockwise.
`
`
` When comparing the cornering stiffness values obtained
`from the steady-state circle methods to the steady-state circle
`time responses, the agreement was good. However, the same
`cornering stiffness values poorly matched the frequency
`response data. With these observations, it was inferred that
`another mechanism to produce tire force was occurring only
`during steady turning. Therefore, the model for tire force
`generation was revisited.
`
`φ
`A steady-state force balance for a vehicle traversing a
`steady-state turn gives the following slip angles:
`C
`hm
`⋅
`⋅
`mU
`2
`s
`φ
`R
`k
`r
`hmC
`l
`⋅
`⋅
`mU
`r
`s
`f
`φ
`C
`L
`R
`k
`r
`r
`α
`which can be more compactly represented as:
`mU
`2
`2
`R
`mU
`R
`
`⎥⎦⎤
`
`⎥⎦⎤
`
`RU
`
`2
`
`2
`
`−
`
`C
`
`*
`r
`
`⋅
`
`l
`f
`L
`
`⎢⎣⎡
`
`11
`
`S
`
`r
`
`=
`
`αα
`
`r
`
`hmC
`f
`s
`φ
`k
`s
`hmC
`r
`s
`φ
`k
`s
`The steady-state steering input necessary to traverse a
`constant radius turn at constant speed is therefore given by:
`l
`
`(43)
`UC
`mU
`UC
`L
`mU
`1
`1
`2
`2
`f
`−
`+⎥
`R
`R
`L
`C
`R
`R
`CR
`which allows one to solve for the steady-state steering gains
`for a constant velocity, constant radius turn.
`
`
`
` (44)
`
`⎥⎦⎤
`
`RU
`
`2
`
`−
`
`C
`*
`r
`
`2
`
`⋅
`
`mU
`R
`
`Ll
`⎢⎣⎡
`
`f
`
`1
`C
`r
`
`RU
`
`⎦⎤
`
`+⎥
`
`RU
`
`2
`
`−
`
`C
`
`*
`f
`
`⋅
`
`2
`
`mU
`R
`
`Ll
`⎢⎣⎡
`
`r
`
`L
`1
`−
`CR
`
`f
`
`r
`δ
`f
`
`=
`
`circle
`
`and
`
`(42)
`
`S
`
`f
`
`
`
`S
`
`r
`
`==
`
`C
`
`*
`f
`
`C
`
`*
`r
`
`⎥⎦⎤
`
`2
`
`−
`
`*
`r
`
`⋅
`
`⎢⎣⎡
`
`r
`
`⎦⎤
`
`2
`
`−
`
`*
`f
`
`⋅
`
`Ll
`⎢⎣⎡
`
`r
`
`f
`
`with
`
`
`
`δ
`f
`
`=
`
`VI. TIRE MODELS WITH CAMBER
`The previous analysis suggests that there is a significant
`tire force generation mechanism that is dependent on
`whether or not the vehicle is in a steady turn. One possible
`explanation for this behavior is that the tire model is
`dependent on the vehicle’s roll angle, an inference supported
`by the work of others. For example, in the reference [8],
`Kim and Park introduce a incremental change in force on the
`∂ f
`tire model in the form of
`. This effect is commonly
`φ
`
`φα ∂
`
`known as “roll steer” and is usually assumed to be a
`constant value when the amount of tire slip is small.
`According to [8], the magnitude of the coefficient for the
`front tires was 0.2, and -0.2 for the rear tires.
`To find the roll-steer parameters for this study, it appears
`that one must simultaneously solve for four parameters –
`two cornering stiffnesses and two camber coefficients – to
`match the measured steady-state data. However, here it is
`assumed that the cornering stiffnesses obtained by matching
`the frequency responses are not greatly influenced by tire
`camber due to the very low roll angles exhibited at low
`frequencies. Therefore, one only needs to consider the
`turning circle data to measure the influence of tire camber.
`
`To experimentally obtain parameters representative of tire
`camber, the following procedure is utilized. First, the tire
`
`Magna - IPR2020-00777 - Ex. 2006 - 007
`
`

`

`w (rad/s)
`
`101
`
`
`
`Data-Large Amp
`Data-Small Amp
`Bicycle
`Sprung Only
`Sprung+Unsprung
`Asymmetric
`100
`
`15
`
`10
`
`05
`
`
`
`0
`
`Mag (dB)
`
`
`
`-50
`
`-100
`
`Phase (deg)
`
`Data-Large Amp
`Data-Small Amp
`Bicycle
`Sprung Only
`Sprung+Unsprung
`Asymmetric
`100
`101
`w (rad/s)
`Figure 6: Frequency Resp. Steering Input to Yaw Rate
`0
`
`
`
`-10
`
`-20
`
`-30
`
`Mag (dB)
`
` (45)
`
`⎥⎦⎤
`
`RU
`
`2
`
`2
`
`−
`
`C
`
`*
`r
`
`⋅
`
`⎢⎣⎡
`
`⎟⎠⎞
`
`r
`
`+
`
`α
`r
`
`Rl
`⎜⎝⎛
`
`r
`
`U
`
`⋅
`
`⎦⎤
`
`RU
`
`2
`
`2
`
`−
`
`C
`
`*
`f
`
`⋅
`
`Ll
`⎢⎣⎡
`
`r
`
`V
`δ
`f
`
`=
`
`circle
`
`f
`
`*r
`
`*f
`
`l
`mU
`L
`mU
`1
`1
`f
`−
`+⎥
`C
`L
`R
`CR
`R
`The steady-state values of these two transfer functions are
`easily obtained using the turning data. This allows the values
`of
`C to be directly calculated since all other
`C and
`parameters are known.
`
`Data-Large Amp
`Data-Small Amp
`Sprung Only
`Sprung+Unsprung
`Asymmetric
`100
`
`-40
`
`
`
`50
`0
`-50
`-100
`-150
`-200
`-250
`
`
`
`Phase (deg)
`
`VII. DYNAMIC MODEL FITTING
`In previous work [6], examination of the phase lag
`
`observed in the frequency response data showed that a
`model of tire lag was necessary to obtain a reasonable model
`fit. The tire-lag phenomenon is commonly modeled as a
`first-order system with zero steady-state gain [16]. In
`previous work [6], a first-order tire lag model was
`introduced on the front steering input. After consultation
`with the authors of [7] and reviewing [11, 17], the following
`model of tire lag was used which considers tire lag as a
`function of tire slip on the front and rear tires:
`dF
`lV
`r
`+
`⋅
`CU
`⎜⎜
`dt
`U
`σ
`dF
`lVCU
`⋅−
`r
`r
`⎟⎟
`⎜⎜
`dt
`U
`σ
`The best fits were obtained with a tire lag value of sigma =
`0.8 for front and 0.6 for rear, but difference between these
`and an average value of 0.7 were minor. The above tire-lag
`model is used hereafter for all tire force calculations.
`A. Frequency Response Tests – Roll Model Fit
`The two parameters that remained to be estimated for roll
`model fitting were Kφ and Dφ. To accomplish this, these
`parameters were varied manually until the models best
`matched
`the frequency response data. The resulting
`frequency-domain fits are seen in Figs. 5-7, where both
`experimental data and model are shown. The data was
`collected at 25 mph.
`15
`
`⎝⎛
`
`f
`
`⎜⎝⎛
`
`r
`
`⎜⎜⎝⎛
`
`⎝⎛
`
`f
`
`=
`
`=
`
`
`
`
`
`(46)
`
`⎟⎟⎠⎞
`
`F
`
`f
`
`⎠⎞
`
`−⎟⎟
`
`f
`
`−
`
`δ
`f
`
`⎠⎞
`
`F
`r
`
`⎠⎞
`
`−⎟
`
`r
`
`w (rad/s)
`
`101
`
`
`
`Data-Large Amp
`Data-Small Amp
`Bicycle
`Sprung Only
`Sprung+Unsprung
`Asymmetric
`100
`
`10
`
`05
`
`
`
`50
`
`0
`
`Mag (dB)
`
`-50
`
`-100
`
`-150
`
`Data-Large Amp
`Data-Small Amp
`Bicycle
`Sprung Only
`Sprung+Unsprung
`Asymmetric
`101
`100
`w (rad/s)
`Figure 5: Frequency Resp. Steering Input to Lateral
`Velocity
`
`-200
`
`
`
`
`
`Phase (deg)
`
`w (rad/s)
`
`101
`
`
`
`
`
`Data-Large Amp
`Data-Small Amp
`Sprung Only
`Sprung+Unsprung
`Asymmetric
`101
`100
`w (rad/s)
`Figure 7: Frequency Resp. Steering Input to Roll Angle
`Several points are evident from the above plots:
`1) There is little difference between the models. Errors due
`to parameter-fitting seem to be larger than model-to-
`model differences.
`2) The yaw rate and roll responses fit quite well. The lateral
`velocity has relatively large errors in fit, particularly
`around the roll dynamic frequencies.
`3) The inclusion of asymmetry in the inertia appears
`unnecessary and perhaps detrimental to the model fit
`B. Time Response Tests
`In order to obtain a more intuitive understanding of the
`model fit obtained by the frequency response tests, time
`response data were also taken. Shown in Figs. 8-10 are state
`responses during a representative lane-change at 25 mph.
`Both the frequency and time-domain responses showed
`that the predicted yaw response of all of the models is nearly
`identical.
`A noticeable discontinuity is visible in the yaw rate
`response. After inspection of the raw yaw data, delays were
`observed on one-second intervals which correspond to both
`the SPAN system’s internal Kalman filter updates and the
`differential corrections
`from
`the GPS base station.
`Investigation is ongoing to find the exact source and
`solution to this error.
`
`Magna - IPR2020-00777 - Ex. 2006 - 008
`
`

`

`
`
`
`
`7
`
`
`
`significant. In fact, these differences appear less significant
`than errors observed in parameter fitting.
`Further work is currently under way to better verify
`vehicle parameters and to obtain improved models of
`vehicle behavior. One evident shortcoming in the approach
`used thus far is the requirement that all of the models be
`linear. Herein lies a significant difficulty in model-based
`rollover prediction and model-based controller synthesis to
`prevent vehicle rollover: while linearity greatly simplifies
`controller design,
`the
`limit handling maneuvers
`that
`ultimately induce rollover nearly always involve large tire
`forces and tire saturation. However, prior to examining non-
`linear models and control schemes, it is important to fully
`understand and control vehicles dynamics in the linear
`range.
`
`[6]
`
`[1]
`
`[2]
`[3]
`
`REFERENCES
`Injury Statistics Query and Reporting System
`"Web-based
`(WISQARS): Years of Potential Life Lost (YPLL) Reports, 1999 -
`2002," Atlanta, Georgia: The Center for Disease Control (CDC), 2002.
`- "Leading Causes of Death Reports, 1999 - 2002,"
`"Traffic Safety Facts 2003 - Final Report," U.S. Department of
`Transportation: National Highway Traffic and Safety Board 2004.
`[4] USDOT, "An Experimental Examination of Selected Maneuvers That
`May Induce On-Road Untripped, Light Vehicle Rollover - Phase II of
`NHTSA’s 1997-1998 Vehicle Rollover Research Program," NHTSA
`report HS 808 977, July 1999.
`[5] USDOT, "A Comprehensive Experimental Examination of Selected
`Maneuvers That May Induce On-Road, Untripped, Light Vehicle
`Rollover - Phase IV of NHTSA's Light Vehicle Rollover Research
`Program," NHTSA report HS 809 513, October 2002.
`J. Cameron and S. Brennan, "A Comparative, Experimental Study of
`Model Suitability to Describe Vehicle Rollover Dynamics for
`Controller Design," presented at the 2005 ASME IMECE.
`[7] C. R. Carlson and J. C. Gerdes, "Optimal Rollover Prevention with
`Steer-by-Wire and Differential Braking," presented at 2003 ASME
`IMECE.
`[8] H.-J. Kim and Y.-P. Park, "Investigation of robust roll motion control
`considering varying speed and actuator dynamics," Mechatronics,
`2003.
`[9] S. Mammar, "Speed Scheduled Vehicle Lateral Control," presented at
`Proceedings of the 1999 IEEE/IEEJ/JSAI International Conference on
`Intelligent Transportation Systems, 1999.
`[10] S. Mammar, V. B. Baghdassarian, and L. Nouveliere, "Speed
`Scheduled Vehicle Lateral Control," presented at Proceedings of the
`1999
`IEEE/IEEJ/JSAI
`International Conference on
`Intelligent
`Transportation Systems (Cat. No.99TH8383), 1999.
`[11] D. Karnopp, Vehicle Stability. New York: Marcel Dekker, Inc., 2004.
`[12] T. Ford, J. Neumann, M. Bobye, and P. Fenton, "OEM4 Inertial: A
`Tightly Integrated Decentralised Inertial/GPS Navigation System,"
`presented at Proceedings of ION GPS ‘01, Salt Lake City, Utah.,
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