Patent Publication Number: US-11046323-B2

Title: Architecture and reconfigurable tire force estimation for various powertrain configurations

Description:
INTRODUCTION 
     The present disclosure relates methods and systems for vehicle tire force estimation and, more particularly to an architecture and reconfigurable tire force estimation for various powertrain configurations. 
     Exiting tire force estimation methods are tire-based or torque-based. The estimations method may provide inaccurate results due to uncertainties in tire model parameters and variations in road friction. Therefore, there is a need for a robust tire force estimation method. 
     SUMMARY 
     Tire force estimation is useful for most of the controls and diagnostics methodologies. One of the challenges for the existing algorithms is the sensitivity to the road surface condition. The other difficulty is how to handle different driveline configurations with totally different actuation systems. This presently disclosed method addresses the issues mentioned above in a robust manner. 
     The present disclosure describes a reconfigurable algorithm for estimation of tire longitudinal and lateral tire forces at each corner of a vehicle that is proposed and tested on different roads and under various driving conditions. The presently disclosed algorithm is independent from the road condition and may be used for different configurations of AWD, RWD, or FWD. The estimation method may be used for various actuation/transmission types, i.e., Electronic Limited Slip Differential (eLSD), open differentials, electric motors, gasoline engines, etc. The presently disclosed method estimates longitudinal and lateral tire force without requiring extra or uncommon vehicle sensors for different AWD/FWD/RWD driveline configurations. The road friction condition is not required for the algorithm. The presently disclosed method is also reconfigurable for different actuation/transmission types, i.e., eLSD, open differential, electric motor, gasoline engines, etc. The presently disclosed method provides good estimation outcomes in drift maneuvers as well as lane-change, harsh steering, and more specifically combined-slip maneuvers. The presently disclosed method considers driving conditions for more accurate and reliable estimation at each corner. Accelerators in vehicles may generate noises, and the signals generated by accelerometers may fluctuate. The presently disclosed method is robust to the accelerometers&#39; noises and fluctuations. The presently disclosed method is robust to the tire parameter changes (changes due to wear, aging and temperature changes has no effect on the longitudinal and lateral tire force estimation). The presently disclosed method employs slip/excitation monitoring time windows to remove outliers in different driving conditions. The presently disclosed method lowers computational complexity compared to existing approaches. By employing this method, the tire fore estimate may be used to significantly improve the performance of the vehicle motion. 
     An integrated method for estimation of vehicle (longitudinal and lateral) tire forces includes receiving, by a controller of a vehicle, a measured vehicle acceleration of the vehicle; receiving, by a controller, a measured yaw rate of the vehicle; receiving, by the controller, a measured wheel speed of the vehicle; forming, by the controller, inertia matrices based on an inertia of rotating components of the vehicle based on the measured wheel speed and the measured vehicle acceleration; calculating torques at corners of the vehicle using the inertia matrices and the torque generated by an engine (or electric motor) by various differential configurations, including open differential and eLSD; estimating tire forces of the vehicle based on the measured vehicle acceleration, the measured wheel speed, and the inertia matrices; and controlling, by the controller, the vehicle, based on the plurality of estimated tire forces. 
     The measured acceleration may be referred to as the measured longitudinal acceleration of the vehicle. The method may further include receiving, by the controller, a measured lateral acceleration of the vehicle. The method may further include receiving, by the controller, a road bank angle and a road grade angle. The method may further include correcting the measured longitudinal acceleration and the measured lateral acceleration using the road grade angle and the road bank angle to determine a corrected longitudinal acceleration of the vehicle and a corrected lateral acceleration of the vehicle. The rotating components may include a plurality of wheels of the vehicle. The plurality of wheels may include a first wheel, a second wheel, a third wheel, and a fourth wheel. The vehicle may include a central drive shaft coupled to the transmission system and an Electronic Limited Slip Differential (eLSD) coupled to the central drive shaft. The eLSD may include a differential case and a clutch coupled to the differential case. 
     The vehicle may include a first shaft interconnecting the eLSD and the fourth wheel. The vehicle may include a second shaft interconnecting the eLSD and the third wheel. The controller may form the inertia matrices by determining moments of inertia of the differential case and the clutch; and determining moments of inertia of the third wheel and the fourth wheel as a function of the moments of inertia of the differential case and the clutch. The plurality of tires may include a first tire, a second tire, a third tire, and a fourth tire. The torques at the corners of the vehicle may include a first corner torque at the first tire, a second corner torque at the second tire, a third corner torque at the third tire, and a fourth corner torque at the fourth tire. The first corner torque may be calculated as a function of a first brake torque at the first tire. The second corner torque may be calculated as a function of a second brake torque at the second tire. The third corner torque may be calculated as a function of the inertia matrices. The fourth corner torque may be calculated as a function of the inertia matrices. 
     The method may further include estimating the virtual wheel speed at the corners of the vehicle as a function of the torques at the corners of the vehicle. The method may further include determining longitudinal and lateral tire forces at the corners of the vehicle as a function of the virtual wheel speed, corrected longitudinal/lateral accelerations (by road angles), and measured yaw rate. The method may further include forming augmented state matrices using the virtual wheel speed at the corners of the vehicle and the longitudinal and lateral tire forces at the corners of the vehicle. The method may further include filtering the longitudinal and lateral tire forces at the corners of the vehicle. 
     The present disclosure further describes a vehicle. The vehicle includes a plurality of rotating components. The plurality of rotating components includes a plurality of tires. The vehicle further includes a transmission system coupled to the plurality of tires and an actuator coupled to the plurality of tires. The vehicle further includes a plurality of sensors and a controller in communication with the plurality of sensors. The controller is programmed to execute the method described above. 
     The above features and advantages, and other features and advantages, of the present teachings are readily apparent from the following detailed description of some of the best modes and other embodiments for carrying out the present teachings, as defined in the appended claims, when taken in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic block diagram of a vehicle. 
         FIG. 2  is a schematic diagram of the vehicle of  FIG. 1 . 
         FIG. 3  is a flowchart of a method for tire force estimation. 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description is merely exemplary in nature and is not intended to limit the application and uses. Furthermore, there is no intention to be bound by expressed or implied theory presented in the preceding technical field, background, brief summary or the following detailed description. As used herein, the term “module” refers to hardware, software, firmware, electronic control component, processing logic, and/or processor device, individually or in a combination thereof, including without limitation: application specific integrated circuit (ASIC), an electronic circuit, a processor (shared, dedicated, or group) and memory that executes one or more software or firmware programs, a combinational logic circuit, and/or other suitable components that provide the described functionality. 
     Embodiments of the present disclosure may be described herein in terms of functional and/or logical block components and various processing steps. It should be appreciated that such block components may be realized by a number of hardware, software, and/or firmware components configured to perform the specified functions. For example, an embodiment of the present disclosure may employ various integrated circuit components, e.g., memory elements, digital signal processing elements, logic elements, look-up tables, or the like, which may carry out a variety of functions under the control of one or more microprocessors or other control devices. In addition, those skilled in the art will appreciate that embodiments of the present disclosure may be practiced in conjunction with a number of systems, and that the systems described herein are merely exemplary embodiments of the present disclosure. 
     For the sake of brevity, techniques related to signal processing, data fusion, signaling, control, and other functional aspects of the systems (and the individual operating components of the systems) may not be described in detail herein. Furthermore, the connecting lines shown in the various figures contained herein are intended to represent example functional relationships and/or physical couplings between the various elements. It should be noted that alternative or additional functional relationships or physical connections may be present in an embodiment of the present disclosure. 
     As depicted in  FIG. 1 , the vehicle  10  generally includes a chassis, a body  14 , and front and rear wheels  17 . The plurality of wheels includes a first (front-left) wheel  17   a , a second (front-right) wheel  17   b , a third (rear-left) wheel  17   c , and a fourth (rear-right) wheel  17   d . The body  14  is arranged on the chassis and substantially encloses components of the vehicle  10 . The body  14  and the chassis may jointly form a frame. The wheels  17  are each rotationally coupled to the chassis near a respective corner of the body  14 . The vehicle  10  also includes a plurality of rotating components, such as the wheels  17 , a central drive shaft  12 , a differential case  52  of a differential  50 , a first (or right) shaft  54  interconnecting the differential  50  to the fourth wheel  17   d , and a second (or left) shaft  56  interconnecting the differential  50  to the third wheels  17   c . The central drive shaft  12  transmits torque from the transmission system  22  to the differential  50 . The differential  50  may be an open differential or an Electronic Limited Slip Differential (eLSD) and may include the differential case  52  and a clutch  58  coupled to the differential case  52 . The differential case  52  may include a rotating rear gear and therefore rotates. During rotation, the differential case transfers torque from the differential  50  to the first shaft  54  and the second shaft  56 . 
     In various embodiments, the vehicle  10  may be an autonomous vehicle and a control system  98  is incorporated into the vehicle  10 . The control system  98  may be simply referred to as the system. The vehicle  10  is, for example, a vehicle that is automatically controlled to carry passengers from one location to another. The vehicle  10  is depicted in the illustrated embodiment as a passenger car, but it should be appreciated that another vehicle including motorcycles, trucks, sport utility vehicles (SUVs), recreational vehicles (RVs), marine vessels, aircraft, etc., can also be used. In an exemplary embodiment, the vehicle  10  may be a partially or fully automated vehicle. A Level Four system indicates “high automation”, referring to the driving mode-specific performance by an automated driving system of aspects of the dynamic driving task, even if a human driver does not respond appropriately to a request to intervene. A Level Five system indicates “full automation”, referring to the full-time performance by an automated driving system of aspects of the dynamic driving task under a number of roadway and environmental conditions that can be managed by a human driver. 
     As shown, the vehicle  10  generally includes a propulsion system  20 , a transmission system  22 , a steering system  24 , a brake system  26 , a sensor system  28 , an actuator system  30 , at least one data storage device  32 , at least one controller  34 , and a communication system  36 . The propulsion system  20  may, in various embodiments, include an electric machine such as a traction motor and/or a fuel cell propulsion system. The vehicle  10  further includes a battery (or battery pack)  21  electrically connected to the propulsion system  20 . Accordingly, the battery  21  is configured to store electrical energy and to provide electrical energy to the propulsion system  20 . Additionally, the propulsion system  20  may include an internal combustion engine. The transmission system  22  is configured to transmit power from the propulsion system  20  to the vehicle wheels  17  according to selectable speed ratios. According to various embodiments, the transmission system  22  may include a step-ratio automatic transmission, a continuously-variable transmission, or other appropriate transmission. The brake system  26  is configured to provide braking torque to the vehicle wheels  17 . The brake system  26  may, in various embodiments, include friction brakes, brake by wire, a regenerative braking system such as an electric machine, and/or other appropriate braking systems. The steering system  24  influences a position of the vehicle wheels  17 . 
     The sensor system  28  includes one or more sensors  40  (i.e., sensing devices) that sense observable conditions of the exterior environment and/or the interior environment of the vehicle  10 . The sensors  40  are in communication with the controller  34  and may include, but are not limited to, one or more radars, one or more light detection and ranging (lidar) sensors, one or more global positioning systems (GPS) devices, one or more cameras (e.g., optical cameras and/or thermal cameras), ultrasonic sensors, yaw rate sensors, gyroscopes, one or more inertial measurement units (IMUs), one or more steering angle sensor (SAS) for measuring the steering wheel position angle and rate of turn and/or other sensors. The actuator system  30  includes one or more actuator devices  42  that control one or more vehicle features such as, but not limited to, the propulsion system  20 , the transmission system  22 , the steering system  24  (which includes a steering wheel  25 ), active aerodynamic devices  60  ( FIG. 2 ), and the brake system  26 . In various embodiments, the vehicle features can further include interior and/or exterior vehicle features such as, but are not limited to, doors, a trunk, and cabin features such as air, music, lighting, etc. (not numbered). The sensor system  28  includes one or more Global Positioning System (GPS) devices  40   g  configured to detect and monitor the route data (i.e., route information). The GPS device  40   g  is configured to communicate with a GPS to locate the position of the vehicle  10  in the globe. The GPS device  40   g  is in electronic communication with the controller  34 . The GPS device  40   g  includes a GPS-transmitter to receive data (such as topographical data) from a GPS. The topographical data includes the road grade and road bank angle of the road that the vehicle  10  is on. 
     The data storage device  32  stores data for use in automatically controlling the vehicle  10 . In various embodiments, the data storage device  32  stores defined maps of the navigable environment. In various embodiments, the defined maps may be predefined by and obtained from a remote system. For example, the defined maps may be assembled by the remote system and communicated to the vehicle  10  (wirelessly and/or in a wired manner) and stored in the data storage device  32 . As can be appreciated, the data storage device  32  may be part of the controller  34 , separate from the controller  34 , or part of the controller  34  and part of a separate system. 
     The controller  34  includes at least one processor  44  and a non-transitory computer readable storage device or media  46 . The processor  44  can be a custom made or commercially available processor, a central processing unit (CPU), a graphics processing unit (GPU), an auxiliary processor among several processors associated with the controller  34 , a semiconductor-based microprocessor (in the form of a microchip or chip set), a macroprocessor, a combination thereof, or generally a device for executing instructions. The computer readable storage device or media  46  may include volatile and nonvolatile storage in read-only memory (ROM), random-access memory (RAM), and keep-alive memory (KAM), for example. KAM is a persistent or non-volatile memory that may be used to store various operating variables while the processor  44  is powered down. The computer-readable storage device or media  46  may be implemented using a number of memory devices such as PROMs (programmable read-only memory), EPROMs (electrically PROM), EEPROMs (electrically erasable PROM), flash memory, or another electric, magnetic, optical, or combination memory devices capable of storing data, some of which represent executable instructions, used by the controller  34  in controlling the vehicle  10 . 
     The instructions may include one or more separate programs, each of which comprises an ordered listing of executable instructions for implementing logical functions. The instructions, when executed by the processor  44 , receive and process signals from the sensor system  28 , perform logic, calculations, methods and/or algorithms for automatically controlling the components of the vehicle  10 , and generate control signals to the actuator system  30  to automatically control the components of the vehicle  10  based on the logic, calculations, methods, and/or algorithms. Although a single controller  34  is shown in  FIG. 1 , embodiments of the vehicle  10  may include a number of controllers  34  that communicate over a suitable communication medium or a combination of communication mediums and that cooperate to process the sensor signals, perform logic, calculations, methods, and/or algorithms, and generate control signals to automatically control features of the vehicle  10 . 
     In various embodiments, one or more instructions of the controller  34  are embodied in the control system  98 . The vehicle  10  includes a user interface  23 , which may be a touchscreen in the dashboard. The user interface  23  is in electronic communication with the controller  34  and is configured to receive inputs by a user (e.g., vehicle operator). Accordingly, the controller  34  is configured to receive inputs from the user via the user interface  23 . The user interface  23  includes a display configured to display information to the user (e.g., vehicle operator or passenger). 
     The communication system  36  is in communication with the controller  34  and is configured to wirelessly communicate information to and from other entities  48 , such as but not limited to, other vehicles (“V2V” communication), infrastructure (“V2I” communication), remote systems, and/or personal devices. In an exemplary embodiment, the communication system  36  is a wireless communication system configured to communicate via a wireless local area network (WLAN) using IEEE 802.11 standards or by using cellular data communication. However, additional or alternate communication methods, such as a dedicated short-range communications (DSRC) channel, are also considered within the scope of the present disclosure. DSRC channels refer to one-way or two-way short-range to medium-range wireless communication channels specifically designed for automotive use and a corresponding set of protocols and standards. Accordingly, the communication system  36  may include one or more antennas and/or transceivers for receiving and/or transmitting signals, such as cooperative sensing messages (CSMs). The communication system is configured to wirelessly communicate information I between the vehicle  10  and a second vehicle. 
       FIG. 1  is a schematic block diagram of the control system  98 , which is configured to control the vehicle  10 . The controller  34  of the control system  98  is in electronic communication with the braking system  26 , the propulsion system  20 , and the sensor system  28 . The braking system  26  includes one or more brake actuators (e.g., brake calipers) coupled to one or more wheels  17 . Upon actuation, the brake actuators apply braking pressure on one or more wheels  17  to decelerate the vehicle  10 . The propulsion system  20  includes one or more propulsion actuators for controlling the propulsion of the vehicle  10 . For example, as discussed above, the propulsion system  20  may include an internal combustion engine and, in that case, the propulsion actuator may be a throttle specially configured to control the airflow in the internal combustion engine. The sensor system  28  is in electronic communication with the controller  34  and may include one or more accelerometers (or one or more gyroscopes) coupled to one or more wheels  17 . The accelerometer is in electronic communication with the controller  34  and is configured to measure and monitor the longitudinal and lateral accelerations of the vehicle  10 . The sensor system  28  may include one or more speed sensors configured to measure the speed (or velocity) of one or more wheels  17 . The speed sensor (i.e., one of the sensors  40 ) is coupled to the controller  34  and is in electronic communication with one or more wheels  17 . 
     With reference to  FIG. 2 , the vehicle  10  may include one or more active aerodynamic devices  60 . The term “active aerodynamic device” means a physical, tangible structure (such as a spoiler, a wing, air dams, underbody panels, guide vanes, winglets) that is specifically configured to disrupt existing airflow patterns around the vehicle  10  and that is movable relative to the body  14 . Each wheel  17  ( FIG. 1 ) is coupled to a tire, such as a first tire  19   a  (i.e., the front left tire), a second tire  19   b  (i.e., the front right tire), a third tire  19   c  (i.e., the rear left tire), and a fourth tire  19   d  (i.e., the rear right tire). The tires are located at the corners of the vehicle  10 .  FIG. 2  also illustrates the tire forces (i.e., a first longitudinal tire force F x1 , a first lateral tire force F y1 , a second longitudinal tire force F x2 , a second lateral tire force F y2 , a third longitudinal tire force F x3 , a third lateral tire force F y3 , a fourth longitudinal tire force F x4 , and a fourth lateral tire force F y4 , which are also referred to as the forces at corners of the vehicle  10 ).  FIG. 2  also illustrates a yaw direction ψ of the vehicle  10 . 
       FIG. 3  is a flowchart of a method  100  for tire force estimation. The method  100  may be executed by the controller  34  and begins at a start block  102 . Then, the method  100  proceeds to block  104 . Block  104  entails parameters definitions and initialization. The parameters definitions may be stored on non-transitory computer readable storage device or media  46  and may include vehicle mass and geometry parameters p v , inertia and effective radius matrices (including tire effective rolling radius R e     i   ), initial augmented states (tire forces) {circumflex over (x)} 0  in the state observer, longitudinal force observer&#39;s internal constant η_i, iϵ{1,2,3,4}, observer gain matrices (such as longitudinal force observer gains at each corner (related to wheel speed) k p     i   , longitudinal force observer gain matrix at each corner (related to acceleration and yaw rate correction)    i   x , lateral force observer gain matrix at each corner (related to acceleration and yaw rate correction)    q   y  q ϵ {5,6}. After block  104 , the method  100  proceeds to block  106 . 
     At block  106 , the controller  34  receives sensor measurements and estimates. For example, at block  106 , the controller  34  receives a measured vehicle acceleration of the vehicle  10  as measured by one or more sensors  40  (e.g., accelerometers and/or IMU). Specifically, the controller  34  receives a measured longitudinal acceleration a x  and a measured lateral acceleration a y  of the vehicle  10 . The controller  34  also receives a measured steering wheel angle δ of the steering wheel  25  as measured by one of the sensors  40  (e.g., steering angle sensor). The controller  34  also receives a measured yaw rate r of the vehicle  10  as measured by one or more sensors  40  (e.g., yaw rate sensors, gyroscopes, IMU). The controller  34  also receives a measured wheel speed (ω i , i ϵ {1,2,3,4}) of one or more wheels  17  as measured by one or more of the sensors  40  (e.g., a speed sensor). The controller  34  receives a road grade θ r  and a bank angle ϕ r  of the road that the vehicle  10  is on if available to correct measured longitudinal acceleration a x  and the measured lateral acceleration a y . The controller  34  receives the road grade θ r  and the bank angle ϕ r  from one of the sensors  40  (e.g., GPS device). At block  106 , the controller  34  also estimates several torques, including the brake torque at each corner/tire T bi , engine torque, electric motor torque, differential torques (such as the central drive shaft torque T g  to the differential  50 , and clutch torque T c  (e.g., eLSD control clutch torque)). After block  106 , the method  100  proceeds to block  108 . 
     At block  108 , the controller  34  corrects the measured longitudinal acceleration a x  and the measured lateral acceleration a y  using the road grade angle and the road bank angle to determine a corrected longitudinal acceleration and a corrected lateral acceleration of the vehicle  10 . A time window is used to refine the measured longitudinal acceleration a x  and the measured lateral acceleration a y  for the transient regions and maneuvers on gravel surfaces with high fluctuations in the acceleration measurement. Then, the method  100  proceeds to block  110 . 
     At block  110 , the controller  34  determines whether the stationary condition on the corrected longitudinal acceleration, the corrected lateral acceleration, and the measured wheel speed ω ij  at each corner/tire of the vehicle  10  is satisfied. In other words, at block  110 , if at least one of the corrected longitudinal acceleration and the corrected lateral acceleration, and the measured wheel speed ω i  of the vehicle  10  is equal to or less than respective predetermined thresholds, then the method  100  returns to block  104 . On the other hand, if one or more of the corrected longitudinal acceleration, the corrected lateral acceleration, and the measured wheel speed ω ij  at each corner/tire of the vehicle  10  is greater than the respective predetermined thresholds, then the method  100  proceeds to block  112 . 
     At block  112 , the controller  34  forms inertia matrices based on an inertia of rotating components of the vehicle  10  (e.g., the wheels  17 , the central drive shaft  12 , the differential case  52  of the differential  50 , the first (or right) shaft  54  interconnecting the differential  50  to one of the wheels  17 , and the second (or left) shaft  56  interconnecting the differential  50  to one of the wheels  17 ) based on the measured wheel speed ω i  and the measured or corrected vehicle acceleration of the vehicle  10  (e.g., measured longitudinal acceleration a x  and a measured lateral acceleration a y  or the corrected longitudinal acceleration and the corrected lateral acceleration determined at block  108 ). If the differential  50  is an eLSD at the rear track of the vehicle  10 , the controller  34  may employ equations (1) and (2) to calculate the inertia components of the inertia matrices: 
     
       
         
           
             
               
                 
                   
                     
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             I sh     L   , I sh     R    are moments of inertia of the rear-left &amp; -right shafts from eLSD to the third and fourth wheels  17   c  and  17   d  (i.e., the first shaft  54  and the second shaft  56 ); and 
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     As described above, the inertia matrices are formed based on inertias of: wheels  17 , the differential case  52 , the central drive shaft  12 , and right/left shafts (i.e., the first shaft  54  and the second shaft  56 ) from the differential  50  (e.g., eLSD or open differential) to the third and fourth wheels  17   c  and  17   d , among other things. Gain matrices are formed and latest high-slip conditions are incorporated for any gain update. The algorithm detects saturation condition to allocate adaptive observer gains. The threshold for saturation detection changes according to the driving conditions. At block  112 , the controller  34  forms a discretized system, inertia, and input matrices (with augmented wheel speed and force states). The discretized system matrices and augmented states of the generalized force estimation structure are generated. The discretization is done by step-invariance for exact solution. After block  112 , the method  100  proceeds to block  114 . 
     At block  114 , the controller  34  calculates torques at corners of the vehicle  10  using the inertia matrices. To do so, the controller  34  employs equations (3), (4), and (5): 
     
       
         
           
             
               
                 
                   
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                         ⁢ 
                         sgn 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               ω 
                               3 
                             
                             - 
                             
                               ω 
                               4 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           I 
                           
                             t 
                             4 
                           
                         
                         ⁢ 
                         
                           
                             ω 
                             . 
                           
                           4 
                         
                       
                       - 
                       
                         
                           
                             n 
                             2 
                           
                           4 
                         
                         ⁢ 
                         
                           
                             I 
                             _ 
                           
                           
                             i 
                             ⁢ 
                             n 
                           
                         
                         ⁢ 
                         
                           
                             ω 
                             . 
                           
                           3 
                         
                       
                       - 
                       
                         T 
                         
                           b 
                           4 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     rear 
                     ⁢ 
                     
                       - 
                     
                     ⁢ 
                     right 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     tire 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
         
         
           
             where: 
             T g  is the estimated output torque from transmission system  22  to the central drive shaft  12 ; 
             T c  is estimated clutch torque of the differential clutch  58  (e.g., the eLSD clutch torque); 
             ω 3  is the measured wheel speed of the third wheel  17   c;    
             ω 4  is the measured wheel speed of the fourth wheel  17   d;    
             n is the eLSD gear ratio; 
             I t3 , I t4  are moments of inertia of the rear eLSD side shafts; 
             Ī in  is the moment of inertia of the input shaft (to eLSD) inertia; 
             T b     1,   T b     2,   T b     3,   T b     4,    are the braking torque at the first tire  19   a , second tire  19   b , third tire  19   c , and fourth tire  19   d , respectively; 
             T t     1,   T t     2,   T t     3,   T t     4,    are the corner torques at the first tire  19   a , second tire  19   b , third tire  19   c , and fourth tire  19   d , respectively. 
           
         
       
    
     As described above, the torques at the corners of the vehicle  10  include the first corner torque T t     1    at the first tire  19   a , the second corner torque T t     2    at the second tire  19   b , the third corner torque T t     3    at the third tire  19   c , and the fourth corner torque T t     4,    at the fourth tire  19   d . In view of equations (3), (4), and (5), the first corner torque T t     1    is calculated as a function of the first brake torque T b     1    at the first tire  19   a ; the second corner torque T t     2    is calculated as a function of the second brake torque T b     2    at the second tire  19   b ; the third corner torque T t     3    is calculated as a function of the inertia matrices (e.g., the moment of inertia of the input shaft (to eLSD) inertia Ī in ); and the fourth corner torque T t     4    is calculated as a function of the inertia matrices (e.g., the moment of inertia of the input shaft (to eLSD) inertia Ī in ). The corner torques are estimated by the torque generation module (block  114 ) based on the power transmission and actuator torques. The reconfigurable torque generation structure facilitates the application of the tire force estimator for various powertrain configurations (FWD/RWD/AWD) and actuation type (combustion engines, electric motors, open differentials, eLSD, etc.). Input matrices are formed using an approximated output torque from actuator and transmission (for example, from transmission system  22  to the central drive shaft  12  and clutch torque for the eLSD case). The developed observer with modular estimators at each corner tackles uncertainties in the estimated torques from engines, at central drive shaft  12 , open differential shafts, and at eLSD clutch. After block  114 , the method  100  proceeds to block  116 . 
     At block  116 , the controller  34  forms augmented state matrices by using a virtual wheel speed estimator and vehicle planar kinetics. To do so, the controller  34  may employ equations (6), (7), and (8): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ω 
                           ^ 
                         
                         . 
                       
                       i 
                     
                     = 
                     
                       
                         
                           
                             - 
                             
                               k 
                               
                                 p 
                                 i 
                               
                             
                           
                           
                             I 
                             
                               w 
                               i 
                             
                           
                         
                         ⁢ 
                         
                           
                             ω 
                             ^ 
                           
                           ij 
                         
                       
                       + 
                       
                         
                           1 
                           
                             I 
                             
                               w 
                               i 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               T 
                               
                                 t 
                                 i 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   e 
                                   i 
                                 
                               
                               ⁢ 
                               
                                 η 
                                 i 
                               
                             
                             + 
                             
                               
                                 k 
                                 
                                   p 
                                   i 
                                 
                               
                               ⁢ 
                               
                                 ω 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     virtual 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     wheel 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     speed 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     at 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     corners 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           F 
                           ^ 
                         
                         . 
                       
                       
                         x 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               k 
                               
                                 p 
                                 i 
                               
                             
                             2 
                           
                           
                             R 
                             
                               
                                 e 
                                 i 
                               
                               ⁢ 
                               
                                 w 
                                 i 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             ω 
                             ~ 
                           
                           i 
                         
                       
                       + 
                       
                         
                           
                             k 
                             
                               p 
                               i 
                             
                           
                           
                             I 
                             
                               w 
                               i 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   T 
                                   
                                     t 
                                     i 
                                   
                                 
                                 - 
                                 
                                   
                                     I 
                                     
                                       w 
                                       i 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ω 
                                       . 
                                     
                                     i 
                                   
                                 
                               
                               
                                 R 
                                 
                                   e 
                                   i 
                                 
                               
                             
                             - 
                             
                               η 
                               i 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           𝕃 
                           i 
                           x 
                         
                         ⁢ 
                         
                           y 
                           ˜ 
                         
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Longitudinal 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     forces 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     at 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     corners 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   F 
                                   ^ 
                                 
                                 . 
                               
                               
                                 y 
                                 f 
                               
                             
                             = 
                             
                               
                                 𝕃 
                                 5 
                                 y 
                               
                               ⁢ 
                               
                                 y 
                                 ˜ 
                               
                             
                           
                           , 
                           
                             
                               
                                 
                                   F 
                                   ^ 
                                 
                                 . 
                               
                               
                                 y 
                                 r 
                               
                             
                             = 
                             
                               
                                 𝕃 
                                 6 
                                 y 
                               
                               ⁢ 
                               
                                 y 
                                 ˜ 
                               
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                           
                             Lateral 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Forces 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             at 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             front 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             rear 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             tracks 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               where 
                               : 
                               
                                  
                               
                               ⁢ 
                               
                                 
                                   ω 
                                   ~ 
                                 
                                 i 
                               
                             
                           
                           = 
                           
                             
                               ω 
                               i 
                             
                             - 
                             
                               
                                 ω 
                                 ^ 
                               
                               i 
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                       ⁢ 
                       
                         
                           𝕃 
                           i 
                           x 
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   L 
                                   i 
                                   1 
                                 
                               
                               
                                 
                                   L 
                                   i 
                                   2 
                                 
                               
                               
                                 
                                   L 
                                   i 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                       
                       , 
                       
                         
                           𝕃 
                           q 
                           y 
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     L 
                                     q 
                                     1 
                                   
                                 
                                 
                                   
                                     L 
                                     q 
                                     2 
                                   
                                 
                                 
                                   
                                     L 
                                     q 
                                     3 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           for 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         q 
                         ∈ 
                         
                           { 
                           
                             5 
                             , 
                             6 
                           
                           } 
                         
                       
                       ; 
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           y 
                           ˜ 
                         
                         = 
                         
                             
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       a 
                                       x 
                                     
                                     - 
                                     
                                       
                                         Σ 
                                         ⁢ 
                                         
                                           F 
                                           ^ 
                                         
                                       
                                       m 
                                     
                                   
                                 
                                 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         a 
                                         y 
                                       
                                       - 
                                       
                                         
                                           Σ 
                                           ⁢ 
                                           
                                             F 
                                             ^ 
                                           
                                         
                                         m 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       r 
                                       . 
                                     
                                     - 
                                     
                                       
                                         Σ 
                                         ⁢ 
                                         
                                           
                                             M 
                                             ^ 
                                           
                                           Z 
                                         
                                       
                                       
                                         I 
                                         z 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           T 
                         
                       
                       ; 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
         
         
           
             i ϵ {1, 2, 3, 4}; 
             {circumflex over ({dot over (ω)})} i  is a virtual wheel speed at each corner; 
             ω i  is the measured wheel speed at each corner; 
             total torque at each corner, T t     i   , is obtained from (3)-(5); 
             I W     i    is the moment of inertia of the wheels  17  (at each corner); 
             R e     i    is a tire effective rolling radius; 
             η i  is longitudinal force observer&#39;s internal constant (one of the input observer gains); 
             k p     i    is a longitudinal force observer gains at each corner related to wheel speed (one of the input observer gains); 
             {circumflex over (ω)} ij  is the wheel speed at each corner; 
             m is the total vehicle mass; 
             {circumflex over (M)} z  is the total yaw moment at vehicle CG; 
                 i   x  is the longitudinal force observer gain matrix at each corner (related to acceleration and yaw rate correction); 
                 q   y  is the lateral force observer gain matrix at each corner (related to acceleration and yaw rate correction); 
             {circumflex over (F)} x     i    is estimated longitudinal tire forces at each corner/tire; and 
             {circumflex over (F)} y     i    is estimated lateral tire forces at each corner/tire. 
           
         
       
    
     As described in the equations above, the virtual wheel speed at the corners of the vehicle is calculated as a function of the torques at the corners of the vehicle  10 . At block  118 , the controller  34  determines the longitudinal and lateral tire forces {circumflex over (F)} x     i   , {circumflex over (F)} y     i    at the corners of the vehicle  10  as a function of the virtual wheel speed {circumflex over ({dot over (ω)})} i . The controller  34  estimates tire forces of the vehicle  10  based on the measured vehicle accelerations, the measured yaw rate, the measured wheel speed, and the inertia matrices at block  116 . 
     The augmented state vector with dynamic forces may be defined as:
 
{circumflex over ( x )}=[{circumflex over (ω)} 1 {circumflex over (ω)} 2 {circumflex over (ω)} 3 {circumflex over (ω)} 4 {circumflex over (F)} x     1   {circumflex over (F)} x     2   {circumflex over (F)} x     3   {circumflex over (F)} x     4   {circumflex over (F)} y     f   {circumflex over (F)} y     r   ] T   (9)
         where:   {circumflex over (ω)} 1 , {circumflex over (ω)} 2 , {circumflex over (ω)} 2 , {circumflex over (ω)} 4  are the virtual wheel speeds at each corner of the vehicle  10 ;   {circumflex over (F)} x     1    is the estimated longitudinal tire force at the first tire  19   a;      {circumflex over (F)} x     2    is the estimated longitudinal tire force at the second tire  19   b;      {circumflex over (F)} x     3    is the estimated longitudinal tire force at the third tire  19   c;      {circumflex over (F)} x     4    is the estimated longitudinal tire force at the fourth tire  19   d;      {circumflex over (F)} y     f    is the estimated lateral tire forces at each front tires (first and second tires  19   a ,  19   b ) of the vehicle  10 ;   {circumflex over (F)} y     r    is the estimated lateral tire forces at each rear tires (first and second tires  19   c ,  19   d ) of the vehicle  10 ; and   {circumflex over (x)} is the augmented state vector, which is estimated at each time step.       

     The direct estimator formulation with simultaneous correction/observation is defined as:
 
{circumflex over ({dot over ( x )})}= A   c {circumflex over ( x )}+ B   c   (10)
         where:   {circumflex over (x)} is the augmented state vector;   A c  is a continuous-time state matrix; and   B c  is a continuous-time input matrix.       

     At block  116 , the new augmented states and direct estimator formulation with simultaneous correction/observation are computationally efficient because of the form of the state matrix. The observation is done on the wheel speed, IMU data (acceleration and yaw rate) by using disturbance observers, wheel dynamics, and longitudinal/lateral dynamic models. The observer gain allocation defines the order of reliance on the model or on the measurements (wheel speed and acceleration). Gains changes are based on the high-slip status to avoid fluctuations (caused by nonlinearities/disturbances) during the transient regions of harsh maneuvers. After block  116 , the method  100  proceeds to block  118 . 
     At block  118 , the controller  34  employs input and linear observers. The vehicle planar kinetics, wheel dynamics, and input observers at each corner (tire) are combined to ensure reliable estimation in low and high-excitation regions. Specifically, the controller  34  forms the continuous-time state matrix A c  and the continuous-time state matrix B c  using inertia, gain, and model input matrices as defined by equations (11)-(16): 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         A 
                         C 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 A 
                                 _ 
                               
                             
                             
                               
                                 0 
                                 
                                   4 
                                   × 
                                   6 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   A 
                                   _ 
                                 
                                 2 
                               
                             
                             
                               
                                 
                                   A 
                                   _ 
                                 
                                 3 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       𝕃 
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 L 
                                 1 
                                 1 
                               
                             
                             
                               
                                 L 
                                 2 
                                 1 
                               
                             
                             
                               
                                 L 
                                 3 
                                 1 
                               
                             
                             
                               
                                 L 
                                 4 
                                 1 
                               
                             
                             
                               
                                 L 
                                 5 
                                 1 
                               
                             
                             
                               
                                 L 
                                 6 
                                 1 
                               
                             
                           
                           
                             
                               
                                 L 
                                 1 
                                 2 
                               
                             
                             
                               
                                 L 
                                 2 
                                 2 
                               
                             
                             
                               
                                 L 
                                 3 
                                 2 
                               
                             
                             
                               
                                 L 
                                 4 
                                 2 
                               
                             
                             
                               
                                 L 
                                 5 
                                 2 
                               
                             
                             
                               
                                 L 
                                 6 
                                 2 
                               
                             
                           
                           
                             
                               
                                 L 
                                 1 
                                 3 
                               
                             
                             
                               
                                 L 
                                 2 
                                 3 
                               
                             
                             
                               
                                 L 
                                 3 
                                 3 
                               
                             
                             
                               
                                 L 
                                 4 
                                 3 
                               
                             
                             
                               
                                 L 
                                 5 
                                 3 
                               
                             
                             
                               
                                 L 
                                 6 
                                 3 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         A 
                         ¯ 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     
                                       p 
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 
                                   I 
                                   w 
                                 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     
                                       p 
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 
                                   I 
                                   w 
                                 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     
                                       p 
                                       ⁢ 
                                       3 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 
                                   I 
                                   w 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     
                                       p 
                                       ⁢ 
                                       4 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 
                                   I 
                                   w 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                         
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           A 
                           ¯ 
                         
                         2 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     I 
                                     w 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     _ 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 0 
                                 
                                   2 
                                   × 
                                   4 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           A 
                           ¯ 
                         
                         
                           3 
                           
                             j 
                             ⁢ 
                             1 
                           
                         
                       
                       = 
                       
                         
                           
                             - 
                             
                               1 
                               m 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                                 ⁢ 
                                 
                                   L 
                                   j 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                                 ⁢ 
                                 
                                   L 
                                   j 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             
                               L 
                               j 
                               3 
                             
                             
                               I 
                               z 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     W 
                                     f 
                                   
                                   2 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                               
                               + 
                               
                                 
                                   l 
                                   f 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ) 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           A 
                           ¯ 
                         
                         
                           3 
                           
                             j 
                             ⁢ 
                             2 
                           
                         
                       
                       = 
                       
                         
                           
                             - 
                             
                               1 
                               m 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                                 ⁢ 
                                 
                                   L 
                                   j 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                                 ⁢ 
                                 
                                   L 
                                   j 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             
                               L 
                               j 
                               3 
                             
                             
                               I 
                               z 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     - 
                                     
                                       W 
                                       f 
                                     
                                   
                                   2 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                               
                               + 
                               
                                 
                                   l 
                                   f 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           A 
                           _ 
                         
                         
                           3 
                           j3 
                         
                       
                       = 
                       
                         
                           
                             - 
                             
                               1 
                               m 
                             
                           
                           ⁢ 
                           
                             L 
                             j 
                             1 
                           
                         
                         - 
                         
                           
                             
                               W 
                               r 
                             
                             
                               2 
                               ⁢ 
                               
                                 I 
                                 z 
                               
                             
                           
                           ⁢ 
                           
                             L 
                             j 
                             3 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           A 
                           ¯ 
                         
                         
                           3 
                           
                             1 
                             ⁢ 
                             4 
                           
                         
                       
                       = 
                       
                         
                           
                             - 
                             
                               1 
                               m 
                             
                           
                           ⁢ 
                           
                             L 
                             j 
                             1 
                           
                         
                         + 
                         
                           
                             
                               W 
                               r 
                             
                             
                               2 
                               ⁢ 
                               
                                 I 
                                 z 
                               
                             
                           
                           ⁢ 
                           
                             L 
                             j 
                             3 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           A 
                           ¯ 
                         
                         
                           3 
                           
                             j 
                             ⁢ 
                             5 
                           
                         
                       
                       = 
                       
                         
                           
                             1 
                             m 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                   δ 
                                   f 
                                 
                                 ⁢ 
                                 
                                   L 
                                   j 
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             where y m =[a x  a y  {dot over (r)}] T    
           
         
       
    
     Equations (11)-(16) are the system matrices in the transformed input/output form. At block  118 , the continuous-time state matrix A c  and the continuous-time state matrix B c  are discretized using equation (17):
 
{circumflex over ( x )} k+1   =A   d {circumflex over ( x )} k   +B   d   ,A   d   =e   A     c     (t)T     s     ,B   d =∫ 0   T     s     e   A     c     (t)τ   B   c ( t ) dτ   (17)
         where:   A c , A d  are continuous-time and discrete-time state matrices, respectively; and   B c , B d  are continuous-time and discrete-time input matrices, respectively.       

     At block  118 , the controller  34  then estimates the augmented states (longitudinal and lateral forces {circumflex over (F)} x     i   , {circumflex over (F)} y     i   ) and virtual wheel speeds {circumflex over (ω)} i  using the direct input observer. After block  118 , the method  100  proceed to block  120 . 
     At block  120 , the controller  34  determines whether the persistence criterion on a saturation condition is satisfied. To do so, the controller  34  inspects a persistence criterion (through a time window) to remove short-term outliers and avoid incorrect gain allocation. Also, the controller  34  may detect a saturation condition (obtained by wheel speed/acceleration) and level of excitation (steering &amp; acceleration measurements). In other words, the controller  34  determines whether the tire forces are less than or greater than maximum thresholds and minimum thresholds, respectively, on a saturation condition. If the persistence criterion is not satisfied, the controller  34  proceeds to block  122 . At block  122 , the controller  34  filters the tire forces at each corner. On the other hand, if the persistence criterion is satisfied, the method  100  proceeds to block  124 . At block  124 , the controller  34  passes the estimated virtual wheel speeds (Equation (6)) to the estimator (i.e., equations (7) and (8)). After block  124 , the method  100  proceeds to block  126  and block  130 . 
     At block  126 , each corner&#39;s augmented state and system matrices are re-initialized. At block  130 , the tire forces {circumflex over (F)} x     i   , {circumflex over (F)} y     f   , and {circumflex over (F)} y     r    are determined using the equations above. After block  126 , the method  100  proceeds to block  128 . Each corner&#39;s longitudinal force is estimated by the direct (discretized) estimator formulation and variable observer gains is refined to have reliable estimation during large steering maneuvers or low-excitation cases, and avoid oscillations produced by harsh brake/acceleration. 
     At block  128 , the controller  34  waits for the next sampled data (i.e., the data collected at block  106 ) and returns to block  106 . After block  130 , the method  100  proceeds to block  132 . At block  132 , the controller  34  controls the vehicle  10  using the longitudinal tire forces {circumflex over (F)} x     i    or lateral tire forces {circumflex over (F)} y     f   , {circumflex over (F)} y     r    (by active front/rear steering). For instance, the controller  34  may control one or more actuator devices  42  (e.g., active aerodynamic devices  60 , traction/stability control systems, the propulsion system  20 , the transmission system  22 , the steering system  24  and/or the brake system  26 ) based on the estimated tire forces {circumflex over (F)} x     i    at each corner and axle lateral forces {circumflex over (F)} y     f   , {circumflex over (F)} y     r    of the vehicle  10 . 
     The detailed description and the drawings or figures are a supportive description of the present teachings, but the scope of the present teachings is defined solely by the claims. While some of the best modes and other embodiments for carrying out the present teachings have been described in detail, various alternative designs and embodiments exist for practicing the present teachings defined in the appended claims.