Patent Publication Number: US-2023139179-A1

Title: Architecture and methodology of adaptive tire force prediction for integrated vehicle motion control

Description:
INTRODUCTION 
     The present disclosure relates to control systems for motor vehicles, and more specifically to systems and methods for accurately modeling automotive tire characteristics. 
     Static and dynamic motor vehicle control systems are increasingly being used to manage a wide variety of static and dynamic motor vehicle performance characteristics. This is especially true with respect to challenging driving scenarios involving tire slip. In many challenging driving scenarios, control actions such as wheel and/or axle torques should be allocated in an optical manner such that tire capacity is fully utilized in longitudinal and lateral directions. Typical tire capacity management is performed within on-board computing platforms or controllers and sensors, including inertial measurement units (IMUs) to measure how the motor vehicle moves in space, referred to as vehicle dynamics. IMUs measure vehicle acceleration in three axes: x (forward/back), y (side-to-side), and z (up/down). IMUs additionally measure how quickly the motor vehicle is turning about the three axes, referred to as the pitch rate (about y), yaw rate (about z), and roll rate (about x). On-board computing platforms or controllers use the measurement data to estimate forces acting on the vehicle. 
     While current systems and methods for tire force prediction in operate for their intended purpose, there is a need for new and improved systems and methods for tire force prediction that are capable of managing motor vehicle control over the lifespan of a tire and under numerous controlled and uncontrolled tire operational conditions while maintaining or reducing cost and complexity, improving simplicity, and while providing increased redundancy and robustness. 
     SUMMARY 
     According to several aspects of the present disclosure a system for adaptive tire force prediction in a motor vehicle includes one or more sensors disposed on the motor vehicle, the plurality of sensors measuring real-time static and dynamic data about the motor vehicle. The system further includes one or more actuators disposed on the motor vehicle, the plurality of actuators altering static and dynamic behavior of the motor vehicle. The system further includes a control module having a processor, a memory, and input/output (I/O) ports, the control module executing program code portions stored in the memory. The program code portions include: a first program code portion that receives, via the I/O ports, the real-time static and dynamic data from the one or more sensors, a second program code portion that models forces at each tire of the motor vehicle at one or more incremental time steps, and a third program code portion that estimates actual forces at each tire of the motor vehicle at each of the one or more incremental time steps. The program code portions further include a fourth program code portion that adaptively predicts tire forces at each tire of the motor vehicle at each of the one or more incremental time steps, a fifth program code portion that generates one or more control commands for the one or more actuators of the motor vehicle, and a sixth code portion that captures discrepancies between real-time force estimations and nominal force calculations at each tire of the motor vehicle, and applies compensation parameters to reduce tracking errors in the one or more control commands to the one or more actuators of the motor vehicle. 
     In another aspect of the present disclosure the first program code portion further receives real-time static and dynamic data from one or more of: inertial measurement units (IMUs) capable of measuring position, orientation, acceleration, and velocity in at least three dimensions, wheel speed sensors capable of measuring angular speeds of wheels of the motor vehicle, and throttle position sensors capable of measuring a throttle position of the motor vehicle. The first program code portion further receives real-time static and dynamic data from accelerator position sensors capable of measuring a position of an accelerator pedal of the motor vehicle, and tire pressure monitoring sensors capable of measuring pressures of tires of the motor vehicle. 
     In still another aspect of the present disclosure the real-time static and dynamic data further includes lateral velocity, longitudinal velocity, yaw rate, wheel angular velocity, and longitudinal, lateral, and normal forces on each tire of the motor vehicle. 
     In still another aspect of the present disclosure the second program code portion further includes a piecewise affine model that generates predictions of longitudinal and lateral forces on each tire of the motor vehicle. 
     In still another aspect of the present disclosure the piecewise affine model further includes a program code portion that calculates a linear approximation of longitudinal forces, lateral forces, self-aligning torques, and coefficients of friction at a contact patch between the tire and a surface such that the linear approximation models tire force behavior in both linear and nonlinear regions at one or more incremental time steps. 
     In still another aspect of the present disclosure the third program code portion further includes utilizing a lookup table to estimate actual forces at each tire of the motor vehicle based on the real-time static and dynamic data from the one or more sensors. 
     In still another aspect of the present disclosure the fourth program code portion adaptively predicts tire forces at each tire of the motor vehicle at each of the one or more incremental time steps to compensate for effects of tire deformation, tire wear, tire temperature, tire inflation pressure, and a coefficient of friction of a surface in contact with the tire at a contact patch. 
     In still another aspect of the present disclosure tire deformation is quantified in terms of longitudinal and lateral slip including a slip angle and slip ratio. 
     In still another aspect of the present disclosure slip angle and slip ratio are defined respectively by: 
     
       
         
           
             
               
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     where actual tire forces are mathematically defined by: 
     
       
         
           
             
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     where y i  represents a force calculation for each of the tires of the motor vehicle, and wherein the coefficients c 1 , c 2 , c 3 , c 4 , c 5  are based on actual tire forces in different slip angles and different normal forces using nonlinear least squares data. 
     In still another aspect of the present disclosure the tire forces are predicted based on predicted state variables and the model of forces on each of the tires of the motor vehicle in X and Y directions is defined as: 
     
       
         
           
             
               
                 
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     and where the tire forces are utilized to calculate state variables such as wheel angular velocities according as J w {dot over (ω)} i,j =T i,j −R e F xi,j . 
     In still another aspect of the present disclosure a method for adaptive tire force prediction in a motor vehicle includes processing static and dynamic motor vehicle information through a control module, the control module having a processor, a memory, and the I/O ports, the control module executing program code portions stored in the memory. The program code portions measure real-time static and dynamic data with one or more sensors disposed on the motor vehicle; utilize one or more actuators disposed on the motor vehicle to alter static and dynamic behavior of the motor vehicle, and receive, via the I/O ports, the real-time static and dynamic data from the one or more sensors. The program code portions further model forces at each tire of the motor vehicle at one or more incremental time steps, estimate estimating actual forces at each tire of the motor vehicle at each of the one or more incremental time steps, and adaptively predict tire forces at each tire of the motor vehicle at each of the one or more incremental time steps. The program code portions further generate one or more control commands for the one or more actuators of the motor vehicle, capture discrepancies between real-time force estimations and nominal force calculations at each tire of the motor vehicle, and apply compensation parameters to reduce tracking errors in the one or more control commands to the one or more actuators of the motor vehicle. 
     In still another aspect of the present disclosure the method further includes program code portions that receive real-time static and dynamic data from one or more of: inertial measurement units (IMUs) capable of measuring position, orientation, acceleration, and velocity in at least three dimensions, wheel speed sensors capable of measuring angular speeds of wheels of the motor vehicle, throttle position sensors capable of measuring a throttle position of the motor vehicle, accelerator position sensors capable of measuring a position of an accelerator pedal of the motor vehicle, and tire pressure monitoring sensors capable of measuring pressures of tires of the motor vehicle. 
     In still another aspect of the present disclosure measuring real-time static and dynamic data further includes measuring lateral velocity, longitudinal velocity, yaw rate, wheel angular velocity, and longitudinal, lateral, and normal forces on each tire of the motor vehicle. 
     In still another aspect of the present disclosure the method further includes program code portions that generate predictions of longitudinal and lateral forces on each tire of the motor vehicle with a piecewise affine model. 
     In still another aspect of the present disclosure generating predictions of longitudinal and lateral forces on each tire of the motor vehicle with a piecewise affine model further includes calculating a linear approximation of longitudinal forces, lateral forces, self-aligning torques, and coefficients of friction at a contact patch between the tire and a surface such that the linear approximation models tire force behavior in both linear and nonlinear regions at one or more incremental time steps. 
     In still another aspect of the present disclosure the method further includes utilizing a lookup table to estimate actual forces at each tire of the motor vehicle based on the real-time static and dynamic data from the one or more sensors. 
     In still another aspect of the present disclosure the method further includes adaptively predicting tire forces at each tire of the motor vehicle at each of the one or more incremental time steps to compensate for effects of tire deformation, tire wear, tire temperature, tire inflation pressure, and a coefficient of friction of a surface in contact with the tire at a contact patch. 
     In still another aspect of the present disclosure the method further includes quantifying tire deformation in terms of longitudinal and lateral slip including a slip angle and a slip ratio, wherein slip angle and slip ratio are defined respectively by: 
     
       
         
           
             
               
                 α 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 
                   tan 
                   
                     - 
                     1 
                   
                 
                 ( 
                 
                   
                     v 
                     
                       y 
                       ij 
                     
                   
                   
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                       ij 
                     
                   
                 
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                       v 
                       
                         x 
                         ij 
                       
                     
                   
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                     x 
                     ij 
                   
                 
               
             
           
         
       
     
     wherein actual tire forces are mathematically defined by: 
     
       
         
           
             
               min 
               
                 
                   c 
                   1 
                 
                 , 
                 
                   c 
                   2 
                 
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                   c 
                   3 
                 
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     where y i  represents a force calculation for each of the tires of the motor vehicle, and wherein the coefficients c 1 , c 2 , c 3 , c 4 , c 5  are based on actual tire forces in different slip angles and different normal forces using nonlinear least squares data. 
     In still another aspect of the present disclosure the method further includes predicting tire forces based on predicted state variables and the model of forces on each of the tires of the motor vehicle in X and Y directions is defined as: 
     
       
         
           
             
               
                 
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     and wherein the tire forces are utilized to calculate state variables such as wheel angular velocities according to the following equation: J w {dot over (ω)} i , =T i,j −R e F xi,j . 
     In still another aspect of the present disclosure a method for adaptive tire force prediction in a motor vehicle includes processing static and dynamic motor vehicle information through a control module, the control module having a processor, a memory, and the I/O ports, the control module executing program code portions stored in the memory. The program code portions measure real-time static and dynamic data with one or more sensors disposed on the motor vehicle, the real-time static and dynamic data including: utilizing one or more actuators disposed on the motor vehicle to alter static and dynamic behavior of the motor vehicle, receiving, via the I/O ports, real-time static and dynamic data from one or more of. inertial measurement units (IMUs) capable of measuring position, orientation, acceleration, and velocity in at least three dimensions, wheel speed sensors capable of measuring angular speeds of wheels of the motor vehicle, throttle position sensors capable of measuring a throttle position of the motor vehicle, accelerator position sensors capable of measuring a position of an accelerator pedal of the motor vehicle, and tire pressure monitoring sensors capable of measuring pressures of tires of the motor vehicle. The method further includes modeling forces at each tire of the motor vehicle at one or more incremental time steps by generating predictions of longitudinal and lateral forces on each tire of the motor vehicle with a piecewise affine model that calculates a linear approximation of longitudinal forces, lateral forces, self-aligning torques, and coefficients of friction at a contact patch between the tire and a surface such that the linear approximation models tire force behavior in both linear and nonlinear regions at one or more incremental time steps. The method further includes estimating actual forces at each tire of the motor vehicle at each of the one or more incremental time steps with a lookup table that provides estimates actual forces at each tire of the motor vehicle based on the real-time static and dynamic data from the one or more sensors. The method further includes adaptively predicting tire forces at each tire of the motor vehicle at each of the one or more incremental time steps to compensate for effects of tire deformation, tire wear, tire temperature, tire inflation pressure, and a coefficient of friction of a surface in contact with the tire at a contact patch. The method further includes generating one or more control commands for the one or more actuators of the motor vehicle, and capturing discrepancies between real-time force estimations and nominal force calculations at each tire of the motor vehicle, and applying compensation parameters to reduce tracking errors in the one or more control commands to the one or more actuators of the motor vehicle. The compensation parameters include: quantifying tire deformation in terms of longitudinal and lateral slip including a slip angle and a slip ratio. Slip angle and slip ratio are defined respectively by: 
     
       
         
           
             
               
                 α 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 
                   tan 
                   
                     - 
                     1 
                   
                 
                 ( 
                 
                   
                     v 
                     
                       y 
                       ij 
                     
                   
                   
                     v 
                     
                       x 
                       ij 
                     
                   
                 
                 ) 
               
             
             , 
             
               
                 κ 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       
                         R 
                         e 
                       
                       ⁢ 
                       
                         ω 
                         
                           i 
                           , 
                           j 
                         
                       
                     
                     - 
                     
                       v 
                       
                         x 
                         ij 
                       
                     
                   
                   ) 
                 
                 
                   v 
                   
                     x 
                     ij 
                   
                 
               
             
           
         
       
     
     where actual tire forces are mathematically defined by: 
     
       
         
           
             
               min 
               
                 
                   c 
                   1 
                 
                 , 
                 
                   c 
                   2 
                 
                 , 
                 
                   c 
                   3 
                 
                 , 
                 
                   c 
                   4 
                 
                 , 
                 
                   c 
                   5 
                 
               
             
             
               
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                         μ 
                         
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                     - 
                     
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                   ) 
                 
                 2 
               
             
           
         
       
     
     where y i  represents a force calculation for each of the tires of the motor vehicle, and the coefficients c 1 , c 2 , c 3 , c 4 , c 5  are based on actual tire forces in different slip angles and different normal forces using nonlinear least squares data; and predicting tire forces based on predicted state variables and the model of forces on each of the tires of the motor vehicle in X and Y directions is defined as: 
     
       
         
           
             
               
                 
                   F 
                   
                     x 
                     
                       i 
                       , 
                       j 
                     
                   
                 
                 = 
                 
                   
                     F 
                     
                       z 
                       
                         i 
                         , 
                         j 
                       
                     
                   
                   ⁢ 
                   
                     μ 
                     
                       x 
                       
                         i 
                         , 
                         j 
                       
                     
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     The tire forces are utilized to calculate state variables such as wheel angular velocities according to the following equation: 
         J   w {dot over (ω)} i,j   =T   i,j   −R   e - F   xi,j .
 
     Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
         FIG.  1    is a schematic diagram of a motor vehicle having a system for adaptive tire force prediction for integrated motion control in a motor vehicle according to an aspect of the present disclosure; 
         FIG.  2   a    is a block diagram of the system of adaptive tire force prediction for integrated motion control in a motor vehicle of  FIG.  1    according to another aspect of the present disclosure; 
         FIG.  2   b    is a block diagram of a portion of the system for adaptive tire force prediction for integrated motion control in a motor vehicle of  FIG.  2   a   , depicting a real-time computation of forces according to another aspect of the present disclosure; 
         FIG.  3    is a partial functional block diagram of a portion of the system for adaptive tire force prediction for integrated motion control in a motor vehicle according to an aspect of the present disclosure; and 
         FIG.  4    is a flowchart depicting a method for adaptive tire force prediction for integrated motion control in a motor vehicle according to an aspect of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
     Referring to  FIGS.  1 - 2     b , a system  10  for adaptive tire force prediction in integrated vehicle motion control for a motor vehicle  12  is shown. The system  10  includes the motor vehicle  12  and one or more controllers  14 . The motor vehicle  12  is shown as a car, but it should be appreciated that the motor vehicle  12  may be a van, bus, tractor-trailer, semi, SUV, truck, tricycle, motorcycle, airplane, amphibious vehicle, or any other such vehicle which makes contact with the ground without departing from the scope or intent of the present disclosure. The motor vehicle  12  includes one or more wheels having tires  18  and a drivetrain  20 . The drivetrain may include a variety of components such as internal combustion engines (ICE)  22  and/or electric motors  24 , and transmissions  26  capable of transmitting motive power developed by the ICEs  22  and/or electric motors  24  to the wheels  27 , and ultimately to the tires  18  affixed thereto. In one example, the motor vehicle  12  may include an ICE  22  acting on a rear axle  28  of the motor vehicle  12  as well as one or more electric motors  24  acting on a front axle  30  of the motor vehicle  12 . It should be appreciated, however, that the motor vehicle  12  may use one or more ICEs  22 , and/or one or more electric motors  24  disposed in other configurations without departing from the scope or intent of the present disclosure. For example, the motor vehicle  12  may have an ICE  22  acting only on the front axle  30 , while one or more electric motors  24  acts solely on the rear axle  28 . In further examples, the ICE  22  may act on both front and rear axles  30 ,  28 , and an electric motor may act on both front and rear axles  30 ,  28 . 
     In several aspects, the drivetrain  20  includes one or more in-plane actuators  32 . In-plane actuators  32  may include all-wheel drive (AWD) systems including electronically-controlled or electric AWD (eAWD)  34  systems as well as limited slip differentials (LSDs) including electronically-controlled or electric LSD (eLSD)  36  systems. In-plane actuators  32  including eAWD  34  and eLSD  36  systems can generate and/or modify force generation in X and/or Y directions at a tire  18  to road surface contact patch  38  within a certain predetermined capacity. An eAWD  34  system may transfer torque from front to rear of a motor vehicle  12  and/or from side-to-side of the motor vehicle  12 . Likewise, an eLSD  36  system may transfer torque from side-to-side of the motor vehicle  12 . In some examples, the eAWD  34  and/or eLSD  36  may directly alter or manage torque delivery from the ICE  22  and/or electric motors  24  and/or the eAWD  34  and eLSD  36  may act on a braking system  40  to adjust a quantity of torque delivered to each of the tires  18  of the motor vehicle  12 . 
     In further examples, the motor vehicle  12  may include a means of altering a normal force on each of the tires  18  of the motor vehicle  12  via one or more out-of-plane actuators  42  such as active aerodynamic actuators  44  and/or active suspension actuators  46 . The active aerodynamic actuators  44  may actively or passively alter an aerodynamic profile of the motor vehicle via one or more active aerodynamic elements  48  such as wings, spoilers, fans or other suction devices, actively-managed Venturi tunnels, and the like. The active suspension actuators  46  such as active dampers  50  or the like. In several aspects, the active dampers  50  may be magnetorheological dampers or other such electrically, hydraulically, or pneumatically-adjustable dampers without departing from the scope or intent of the present disclosure. For the sake of simplicity in the description that follows, ICEs  22 , electric motors  24 , eAWD  34 , eLSD  36 , the braking system  40 , aerodynamic control system including active aerodynamic elements  48 , active dampers  46 , and the like will be referred to more broadly as actuators  52 . 
     The terms “forward”, “rear”, “inner”, “inwardly”, “outer”, “outwardly”, “above”, and “below” are terms used relative to the orientation of the motor vehicle  12  as shown in the drawings of the present application. Thus, “forward” refers to a direction toward a front of a motor vehicle  12 , “rearward” refers to a direction toward a rear of a motor vehicle  12 . “Left” refers to a direction towards a left-hand side of the motor vehicle  12  relative to the front of the motor vehicle  12 . Similarly, “right” refers to a direction towards a right-hand side of the motor vehicle  12  relative to the front of the motor vehicle  12 . “Inner” and “inwardly” refers to a direction towards the interior of a motor vehicle  12 , and “outer” and “outwardly” refers to a direction towards the exterior of a motor vehicle  12 , “below” refers to a direction towards the bottom of the motor vehicle  12 , and “above” refers to a direction towards a top of the motor vehicle  12 . Further, the terms “top”, “overtop”, “bottom”, “side” and “above” are terms used relative to the orientation of the actuators  52 , and the motor vehicle  12  more broadly shown in the drawings of the present application. Thus, while the orientation of actuators  52 , or motor vehicle  12  may change with respect to a given use, these terms are intended to still apply relative to the orientation of the components of the system  10  and motor vehicle  12  components shown in the drawings. 
     The controllers  14  are non-generalized, electronic control devices having a preprogrammed digital computer or processor  54 , non-transitory computer readable medium or memory  56  used to store data such as control logic, software applications, instructions, computer code, data, lookup tables, etc., and input/output (I/O) ports  58 . Computer readable medium or memory  56  includes any type of medium capable of being accessed by a computer, such as read only memory (ROM), random access memory (RAM), a hard disk drive, a compact disc (CD), a digital video disc (DVD), or any other type of memory. A “non-transitory” computer readable memory  56  excludes wired, wireless, optical, or other communication links that transport transitory electrical or other signals. A non-transitory computer readable memory  56  includes media where data can be permanently stored and media where data can be stored and later overwritten, such as a rewritable optical disc or an erasable memory device. Computer code includes any type of program code, including source code, object code, and executable code. The processor  54  is configured to execute the code or instructions. The motor vehicle  12  may have controllers  14  including a dedicated Wi-Fi controller or an engine control module, a transmission control module, a body control module, an infotainment control module, etc. The I/O ports  58  may be configured to communicate via wired communications, wirelessly via Wi-Fi protocols under IEEE 802.11x, or the like without departing from the scope or intent of the present disclosure. 
     The controller  14  further includes one or more applications  60 . An application  60  is a software program configured to perform a specific function or set of functions. The application  60  may include one or more computer programs, software components, sets of instructions, procedures, functions, objects, classes, instances, related data, or a portion thereof adapted for implementation in a suitable computer readable program code. The applications  60  may be stored within the memory  56  or in additional or separate memory. Examples of the applications  60  include audio or video streaming services, games, browsers, social media, etc. In other examples, the applications  60  are used to manage body control system functions, suspension control system functions, aerodynamic control system functions, or the like in an exemplary motor vehicle  12 . 
     In several aspects, to manage tire  18  performance the system  10  makes use of one or more applications  60  for modeling the tires  18 . In an example, the applications  60  include an offline adaptation algorithm  61  and an online adaptation algorithm  63 . The offline adaptation algorithm  61  models tire  18  parameters and compensates based on tire  18  slip effects and actual tire  18  grip capacity. By contrast, the online adaptation algorithm  63  models tire  18  parameters and compensates based on tire  18  wear, tire  18  pressure, tire  18  temperature, road surface or contact patch  38  coefficients of friction, and the like. In several aspects, the applications  60  may be model predictive control (MPC) algorithms  62  or the like, or other known techniques used to model and predict motor vehicle  12  behavior over a short term horizon. In some examples, the short-term prediction horizon is 10-15 sample times, or approximately 100-150 milliseconds, however the precise number of sample times and/or time defining the prediction horizon may differ from the above-recited values without departing from the scope or intent of the present disclosure. In the MPC algorithm  62 , the system  10  operates primarily in a feedback control model to adjust a real-time constraint to optimize tire  18  capacity and actuator  52  capacity for maintaining motor vehicle  12  stability, yaw rate, lateral velocity, and the like. That is, the controller  14  combines the offline adaptation algorithm  61 , online adaptation algorithm  63  and the motor vehicle state information in a hybridized tire force calculation  65  that can be used to generate a real time optimization  67  that that maximizes tire  18  capacity and actuator  52  capacity for maintaining motor vehicle  12  performance in a wide variety of driving conditions. 
     The controller  14  receives data from a variety of sensors  64  equipped to the motor vehicle  12  and obtaining vehicle state information. The sensors  64  may measure and record a wide variety of motor vehicle  12  data. In several examples, the sensors  64  may include inertial measurement units (IMUs)  66 , suspension control units such as Semi Active Damping Suspension (SADS)  68 , global positioning system (GPS) sensors  70 , wheel speed sensors  72 , throttle position sensors  74 , accelerator pedal position sensors  76 , brake pedal position sensors  78 , steering position sensors  80 , tire pressure monitoring sensors  82 , aerodynamic element position sensors  84 , 
     A piecewise affine tire model  86  defines another of the applications  60  stored in memory  56 . The piecewise affine tire model  86  obtains longitudinal and lateral forces at each tire  18  at each time step of the prediction horizon. The piecewise affine tire model  86  is adaptive and takes into account the impact of variation of normal forces and tire deformation in real-time. In an example, as the motor vehicle  12  traverses a road or off-road surface, the tires  18  may encounter a variety of surface perturbations present in the road or off-road surface. The surface perturbations may be potholes, pavement discontinuities, road camber and/or crown, rocks, dirt, standing water, oil, or any of a wide variety of other changes in surface coefficient of friction and/or shape. As the tires  18  encounter the perturbations, and due to the elasticity built into each tire  18 , and the suspension system components to which the wheels and tires  18  are attached, the tires  18  may deflect from a given path of travel on the surface in one or more of the X, Y, and Z directions. Likewise, the elasticity of tires  18  may cause deformations from a round shape when a surface perturbation is encountered. Actual forces on each of the tires  18  are obtained through online estimations at the beginning of each of the prediction horizons. Predictive tire  18  forces are then calculated using an adaptive tire model that relies on nominal known tire parameters and a potential compensation to match force with real-time estimations in different driving conditions. 
     In several aspects, the piecewise affine tire model  86  is a linear approximation of a so-called “magic formula” (MF) based tire model. The MF tire model fits a wide variety of tire  18  types, constructions, and operating conditions. In the MF tire model, each tire  18  is characterized by a plurality of coefficients for each force relevant to tire  18  performance. In some examples, the plurality of coefficients relate to contact patch, lateral and longitudinal forces, self-aligning torque, and the like. These coefficients are used as a best-fit between experimentally-determined tire  18  performance data and the MF model. The coefficients then may be used to generate equations showing how much force is generated for a particular vertical load on the tire  18 , as well as camber angle, slip angle α i,j , and the like. An original formula for longitudinal forces in the MF tire model may be presented as follows: 
     
       
         
           
             
               
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     where for combined slip in both longitudinal and lateral directions, the parameter G xa  can be calculated as: 
     
       
         
           
             
               
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                     Hx 
                     ⁢ 
                     α 
                   
                 
               
             
             ⁢ 
             
 
             
               
                 B 
                 
                   x 
                   ⁢ 
                   α 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       r 
                       
                         Bx 
                         ⁢ 
                         1 
                       
                     
                     + 
                     
                       
                         r 
                         
                           Bx 
                           ⁢ 
                           3 
                         
                       
                       ⁢ 
                       
                         γ 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 cos 
                 ⁢ 
                 
                   { 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     [ 
                     
                       
                         r 
                         
                           Bx 
                           ⁢ 
                           2 
                         
                       
                       ⁢ 
                       κ 
                     
                     ] 
                   
                   } 
                 
                 ⁢ 
                 
                   λ 
                   
                     x 
                     a 
                   
                 
               
             
             ⁢ 
             
 
             
               
                 C 
                 xa 
               
               = 
               
                 r 
                 
                   Cx 
                   ⁢ 
                   1 
                 
               
             
             ⁢ 
             
 
             
               
                 E 
                 xa 
               
               = 
               
                 
                   r 
                   
                     Ex 
                     ⁢ 
                     1 
                   
                 
                 + 
                 
                   
                     r 
                     
                       Ex 
                       ⁢ 
                       2 
                     
                   
                   ⁢ 
                   
                     df 
                     z 
                   
                 
               
             
             ⁢ 
             
 
             
               
                 S 
                 
                   Hx 
                   ⁢ 
                   α 
                 
               
               = 
               
                 r 
                 
                   Hx 
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     However, when combined slip in longitudinal and lateral directions is not used, G xa =1. Likewise, when turning-slip is not used in the formulae above, ζ 1 =1. 
     In order to calculate derivatives of the longitudinal forces on each tire  18  in an analytical fashion, and to reduce computational time, effort, and resource utilization, the piecewise affine linear approximation is utilized as: 
     
       
         
           
             
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               x 
             
             = 
             
               
                 κ 
                 
                   
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                   κ 
                   
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                         2 
                       
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     where, the tire parameters are found such that linear approximation provides an accurate tire force behavior both in linear and nonlinear regions. 
     Tire  18  deformation can be quantified in terms of longitudinal and lateral slip  87 . To calculate longitudinal and lateral slip  87 , first longitudinal and lateral velocities coordinates relating to a body  88  of the motor vehicle  12  are calculated according to the equations below. 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       x 
                       FL 
                     
                   
                   = 
                   
                     
                       V 
                       x 
                     
                     - 
                     rLwf 
                   
                 
               
               
                 
                   
                     V 
                     
                       y 
                       FL 
                     
                   
                   = 
                   
                     
                       V 
                       y 
                     
                     + 
                     rLf 
                   
                 
               
             
             
               
                 
                   
                     V 
                     
                       x 
                       FR 
                     
                   
                   = 
                   
                     
                       V 
                       x 
                     
                     + 
                     rLwf 
                   
                 
               
               
                 
                   
                     V 
                     
                       y 
                       FR 
                     
                   
                   = 
                   
                     
                       V 
                       y 
                     
                     + 
                     rLf 
                   
                 
               
             
             
               
                 
                   
                     V 
                     
                       x 
                       RL 
                     
                   
                   = 
                   
                     
                       V 
                       x 
                     
                     - 
                     rLwr 
                   
                 
               
               
                 
                   
                     V 
                     
                       y 
                       RL 
                     
                   
                   = 
                   
                     
                       V 
                       y 
                     
                     - 
                     rLr 
                   
                 
               
             
             
               
                 
                   
                     V 
                     
                       x 
                       RR 
                     
                   
                   = 
                   
                     
                       V 
                       x 
                     
                     + 
                     rLwr 
                   
                 
               
               
                 
                   
                     V 
                     
                       y 
                       RR 
                     
                   
                   = 
                   
                     
                       V 
                       y 
                     
                     - 
                     rLr 
                   
                 
               
             
           
         
       
     
     Where V xFL  and V yFL  define a velocity of the tire  18  at a front left of the motor vehicle  12  in X and Y directions, respectively. Likewise, V XFR  and V yFR  define a velocity of the tire  18  at a front right of the motor vehicle  12  in X and Y directions, respectively. Similarly, V xRL  and V yRL  and V xRR  and V yRR  define velocities of the tires  18  at the left rear and right rear of the motor vehicle  12  in X and Y directions, respectively. A rotation matrix may be used to transfer velocities into wheel coordinates: 
     
       
         
           
             
               
                 ϕ 
                 ij 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         cos 
                         ⁡ 
                         ( 
                         
                           θ 
                           
                             f 
                             , 
                             r 
                           
                         
                         ) 
                       
                     
                     
                       
                         sin 
                         ⁡ 
                         ( 
                         
                           θ 
                           
                             f 
                             , 
                             r 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         - 
                         
                           sin 
                           ⁡ 
                           ( 
                           
                             θ 
                             
                               f 
                               , 
                               r 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         cos 
                         ⁡ 
                         ( 
                         
                           θ 
                           
                             f 
                             , 
                             r 
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 v 
                 
                   x 
                   , 
                   
                     y 
                     
                       i 
                       , 
                       j 
                     
                   
                 
               
               = 
               
                 
                   
                     ϕ 
                     ij 
                   
                   [ 
                   
                     
                       
                         
                           V 
                           
                             x 
                             
                               i 
                               , 
                               j 
                             
                           
                         
                       
                     
                     
                       
                         
                           V 
                           
                             y 
                             
                               i 
                               , 
                               j 
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
     Then, slip angle and slip ratio can be calculated as: 
     
       
         
           
             
               
                 α 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 
                   tan 
                   
                     - 
                     1 
                   
                 
                 ( 
                 
                   
                     v 
                     
                       y 
                       ij 
                     
                   
                   
                     v 
                     
                       x 
                       ij 
                     
                   
                 
                 ) 
               
             
             , 
             
               
                 κ 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         
                           R 
                           e 
                         
                         ⁢ 
                         
                           ω 
                           
                             i 
                             , 
                             j 
                           
                         
                       
                       - 
                       
                         v 
                         
                           x 
                           ij 
                         
                       
                     
                     ) 
                   
                   
                     v 
                     
                       x 
                       ij 
                     
                   
                 
                 . 
               
             
           
         
       
     
     As side slip angle α i,j  is only a function of motor vehicle  12  lateral and longitudinal speeds at each tire  18  and only dependent on body  88  states, rather than states of the wheels and tires  18  themselves, during a short-duration prediction horizon, the side slip angle α i,j  is unlikely to change dramatically. Therefore, the side slip angle α i,j  may be, for the purposes of certain calculations, assumed to be constant for the length of a short-duration prediction horizon. 
     The piecewise affine tire model  86  further includes a tire curve fitting process  90 . The tire curve fitting process  90  is a computer executable program code portion or algorithm that matches forces at each tire  18  and calculated from the MF tire model to the simplified piecewise affine tire model  86  for different slip angles α i,j  and normal forces. The result of the above analysis is the plurality of coefficients describing the various forces on each tire  18 . More specifically, the coefficients obtained are based on the actual forces on each tire  18  in different slip angles α i,j  and normal forces using a nonlinear least squares data fitting method mathematically defined by the formula below: 
     
       
         
           
             
               min 
               
                 
                   c 
                   1 
                 
                 , 
                 
                   c 
                   2 
                 
                 , 
                 
                   c 
                   3 
                 
                 , 
                 
                   c 
                   4 
                 
                 , 
                 
                   c 
                   5 
                 
               
             
             
               
                 ∑ 
                 i 
                 n 
               
                 
               
                 
                   ( 
                   
                     
                       
                         F 
                         z 
                       
                       ⁢ 
                       
                         μ 
                         
                           x 
                           i 
                         
                       
                     
                     - 
                     
                       y 
                       i 
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
     
     where y i  represents the MF force calculation. Once the coefficients c 1 , c 2 , c 3 , c 4 , c 5  are obtained for different slip angles and normal forces, a lookup table  92  is designed which provides the coefficients c 1 , c 2 , c 3 , c 4 , c 5  for a certain range of slip angles α i,j  and normal forces to be used in the piecewise affine tire model  86 . 
     To calculate nominal forces on each tire  18 , the system  10  estimates and/or predicts the state variables of the motor vehicle  12 . Then, longitudinal and lateral slip  87  information of the tires  18  of the motor vehicle  12  are calculated. More specifically, slip ratios and slip angles α i,j  are calculated based on available estimates at the beginning of the prediction horizon. Then, the lookup table that contains tire  18  curve information is utilized to calculate forces on each tire  18  and to predict body  88  and wheel  27  state variables within the piecewise affine prediction model  86 . Predicted state variables are used in the context of sampling time to calculate updated predicted slip ratios and slip angles α i,j  and then, to calculate new forces on each tire  18 . The system  10  continuously and recursively updates the estimates and/or predictions of the state variables throughout while the motor vehicle  12  is being driven. Accordingly, the system  10  generates real-time computations of longitudinal and lateral forces on the tires  18  of the motor vehicle  12 . 
     Turning now to  FIG.  3   , and with continuing reference to  FIGS.  1 - 2     b , to provide a robust tire  18  force prediction and achieve optimal control performance in different road and environmental conditions, both controlled and uncontrolled, a compensator algorithm  100  captures discrepancies between real-time force estimation and nominal tire  18  force calculations. In broad terms, a formulation of the compensator algorithm  100  may be presented as follows: 
     
       
      
       F 
       x,tire model 
       ={circumflex over (F)} 
       x,estimated  
      
     
     
       
      
       F 
       y,tire model 
       ={circumflex over (F)} 
       y,estimated  
      
     
     where, 
         F   x,tire model   =F   z μ x μ x *
 
         F   y,tire model   =F   z μ y μ y *
 
     In fact, μ x *and μ y *are compensation parameters that match real-time force estimation with tire  18  model calculations to be robust against road surface, tire  18  temperature and pressure changes, and the like. Real time estimations are provided to the compensator algorithm  100  through the controller  14 . More specifically, in some aspects, the controller  14  is a vehicle dynamic controller (VDC)  102  or control module. The VDC  102  provides the compensator algorithm  100  with estimations of lateral and longitudinal forces on each tire  18 . The compensator algorithm  100  also receives state estimations from an extended Kalman filter (EKF)  104 . In several aspects, the EKF  104  is an optimal estimation algorithm that estimates states of the system  10  from indirect and/or uncertain measurements. A compensation calculator  106  determines compensation parameters μ x *and μ y *from the outputs from EKF  104  and the VDC  102 . The compensation parameters μ x *and μ y *are then fed into a linear time-varying (LTV) MPC  108  to match real-time force estimation with tire  18  model calculations during the prediction horizon. 
     Forces on each tire  18  may then be predicted based on predicted state variables and the tire models as shown in the modified MF equations below: 
     
       
         
           
             
               F 
               
                 x 
                 
                   i 
                   , 
                   j 
                 
               
             
             = 
             
               
                 F 
                 
                   z 
                   
                     i 
                     , 
                     j 
                   
                 
               
               ⁢ 
               
                 μ 
                 
                   x 
                   
                     i 
                     , 
                     j 
                   
                 
                 * 
               
               ⁢ 
               
                 
                   κ 
                   
                     i 
                     , 
                     j 
                   
                 
                 
                   
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                     κ 
                     
                       i 
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               ⁢ 
               
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                 ⁡ 
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                       3 
                     
                   
                 
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               F 
               
                 y 
                 
                   i 
                   , 
                   j 
                 
               
             
             = 
             
               
                 F 
                 
                   z 
                   
                     i 
                     , 
                     j 
                   
                 
               
               ⁢ 
               
                 μ 
                 
                   y 
                   
                     i 
                     , 
                     j 
                   
                 
                 * 
               
               ⁢ 
               
                 
                   α 
                   
                     i 
                     , 
                     j 
                   
                 
                 
                   
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                     α 
                     
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     Likewise, tire  18  forces can be utilized to calculate state variables such as wheel angular velocities: 
         J   w {dot over (ω)} i,j   =T   i,j   −R   e - F   xi,j  
 
     Moreover, body  88  state variables can be calculated with a similar approach. 
     The compensation parameters μ x *and μ y *may be estimated using several different methods. In one example, a Least Square Method (LSM) is used for real-time estimation of the compensation parameters to capture discrepancies between real-time force estimation and nominal tire  18  force calculations. However, it should be appreciated that the LSM may be replaced with any optimization method capable of finding the compensation parameters with minimum possible error without departing from the scope or intent of the present disclosure. In an example, given data {(x 1 , y 1 ) . . . (x n , y n )}, the error associated with y=ax+b is defined as E(a, b)=Σ n=1   n (y n −(ax n +b)) 2 . The goal then, is to find values of a and b such that they minimize the error E(a,b). In order to accomplish this goal, a and b are calculated such that 
     
       
         
           
             
               
                 
                   ∂ 
                   E 
                 
                 
                   ∂ 
                   a 
                 
               
               = 
               0 
             
             , 
             
               
                 and 
                 ⁢ 
                 
                   
                     ∂ 
                     E 
                   
                   
                     ∂ 
                     b 
                   
                 
               
               = 
               0. 
             
           
         
       
     
     The analytical solution for the minimum possible values of a and b via the LSM estimate is, then: 
     
       
         
           
             
               ( 
               
                 
                   
                     a 
                   
                 
                 
                   
                     b 
                   
                 
               
               ) 
             
             = 
             
               
                 
                   ( 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             N 
                           
                             
                           
                             x 
                             n 
                             2 
                           
                         
                       
                       
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             N 
                           
                             
                           
                             x 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             N 
                           
                             
                           
                             x 
                             n 
                           
                         
                       
                       
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             N 
                           
                             
                           1 
                         
                       
                     
                   
                   ) 
                 
                 
                   - 
                   1 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           N 
                         
                           
                         
                           
                             x 
                             n 
                           
                           ⁢ 
                           
                             y 
                             n 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           N 
                         
                           
                         
                           y 
                           n 
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Referring now to  FIG.  4   , and with continuing reference to  FIGS.  1 - 3   , a method  200  of adaptive tire force prediction for integrated vehicle motion control is shown in flowchart form. The method  200  begins at block  202  where the one or more sensors  64  equipped to the motor vehicle  12  continuously in real-time measure vehicle state information including static and dynamic motor vehicle  12  state data. At block  204 , the controller  14  executes a first program code portion to receives the vehicle state information from the sensors. 
     At block  206 , the controller  14  executes a second program code portion that uses a piecewise affine model  86  to determine forces at each tire  18  of the motor vehicle  12  at one or more incremental time steps. More specifically, the piecewise affine tire model  86  is a linear approximation of a so-called “magic formula” (MF) based tire model. In the MF tire model, each tire  18  is characterized by a plurality of coefficients for each force relevant to tire  18  performance. In some examples, the plurality of coefficients relate to contact patch, lateral and longitudinal forces, self-aligning torque, and the like. These coefficients are used as a best-fit between experimentally-determined tire  18  performance data and the MF model. The coefficients then may be used to generate equations showing how much force is generated for a particular vertical load on the tire  18 , as well as camber angle, slip angle α i,j , and the like. 
     In several aspects, the piecewise affine linear approximation is utilized as: 
     
       
         
           
             
               μ 
               x 
             
             = 
             
               
                 κ 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   κ 
                   
                     ❘ 
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               ⁢ 
               
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                       c 
                       1 
                     
                     ⁢ 
                     
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                       ⁡ 
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                               c 
                               4 
                             
                             ⁢ 
                             
                               
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                               κ 
                               
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                         c 
                         2 
                       
                       ⁢ 
                       
                         min 
                         ⁡ 
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                                 4 
                               
                               ⁢ 
                               
                                 
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                       3 
                     
                   
                 
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     where, the tire parameters are found such that linear approximation provides an accurate tire force behavior both in linear and nonlinear regions 
     At block  208 , the controller executes a third program code portion that estimates actual forces on each tire  18  of the motor vehicle  12  at each of the one or more incremental time steps. More specifically the system  10  estimates and/or predicts the state variables of the motor vehicle  12 . Then, longitudinal and lateral slip  87  information of the tires  18  of the motor vehicle  12  are calculated. Slip ratios and slip angles α i,j  are calculated based on available estimates at the beginning of the prediction horizon. Then, the lookup table that contains tire  18  curve information is utilized to calculate forces on each tire  18  and to predict body  88  and wheel  27  state variables within the piecewise affine prediction model  86 . Predicted state variables are used in the context of sampling time to calculate updated predicted slip ratios and slip angles α i,j  and then, to calculate new forces on each tire  18 . The system  10  continuously and recursively updates the estimates and/or predictions of the state variables throughout while the motor vehicle  12  is being driven. Accordingly, the system  10  generates real-time computations of longitudinal and lateral forces on the tires  18  of the motor vehicle  12 . 
     At block  210 , the controller  14  executes a fourth program code portion that adaptively predicts forces at each tire  18  of the motor vehicle  12  at each of the one or more incremental time steps. More specifically, the fourth program code portion adapts to tire  18  parameters that may change over the lifespan of a given tire  18 . For example, the fourth program code portion compensates for the effects of tire  18  wear, tire  18  temperature, tire  18  inflation pressure, a coefficient of friction of a surface in contact with the tire  18 , and so forth. 
     At block  212 , the controller  14  executes a fifth program code portion generates one or more control commands for the one or more actuators  52  of the motor vehicle  12 . The control commands may include a torque request, a steering input, or the like. At block  214 , the controller  14  executes a sixth program code portion that captures discrepancies between real-time force estimations and nominal force calculations at each tire  18  of the motor vehicle  12 . The sixth program code portion also applies compensation parameters to reduce tracking errors in the one or more control commands to the one or more actuators  52  of the motor vehicle  12  so that adhesion between the tire  18  and road surface at the contact patch  38  is maximized for the motor vehicle  12  in any given dynamic or complex driving situation. 
     At block  216 , the method ends and returns to block  202  where the method  200  runs again iteratively, continuously, and/or recursively while the motor vehicle  12  is being driven. 
     The system  10  and method  200  of adaptive tire force prediction for integrated vehicle motion control of the present disclosure offer several advantages. These include providing a motor vehicle  12  driver or operator maximum feasible performance, stability, handling, maneuverability, steerability of the motor vehicle  12  in a wide variety of conditions, including in inclement weather, instances of tire  18  deformation, tire  18  wear, tire  18  temperature variations, tire  18  inflation levels, and the like. Additionally, the system  10  and method  200  may operate on a motor vehicle  12  in complex driving scenarios, including performance driving situations in which the driver may attempt a powerslide, drift, or the like, and the system  10  and method  200  will operate to provide an appropriate quantity of force generation at the tire  18 /road interface or contact patch  38 , while also providing maximum tire  18 /road interface or contact patch  38  adhesion in driving scenarios in which maximum grip is desired. These benefits may all be obtained using the system  10  and method  200  described herein while maintaining or reducing cost and complexity, reducing calibration efforts, and improving simplicity, and while also providing increased redundancy and robustness. 
     The description of the present disclosure is merely exemplary in nature and variations that do not depart from the gist of the present disclosure are intended to be within the scope of the present disclosure. Such variations are not to be regarded as a departure from the spirit and scope of the present disclosure.