Individual Drive-Wheel Energy Management for Rear-Traction Electric Vehicles with In-Wheel Motors

applied sciences

Article Individual Drive-Wheel Energy Management for Rear-Traction Electric Vehicles with In-Wheel Motors

Jose del C. Julio-Rodríguez * , Alfredo Santana-Díaz. * and Ricardo A. Ramirez-Mendoza *

School of Engineering and Sciences, Tecnologico de Monterrey, Toluca 50110, Mexico * Correspondence: [email protected] (J.d.C.J.-R.); [email protected] (A.S.-D.); [email protected] (R.A.R.-M.)

Abstract: In-wheel motor technology has reduced the number of components required in a vehicle’s power train system, but it has also led to several additional technological challenges. According to kinematic laws, during the turning maneuvers of a vehicle, the tires must turn at adequate rotational speeds to provide an instantaneous center of rotation. An Electronic Differential System (EDS) controlling these speeds is necessary to ensure speeds on the rear axle wheels, always guaranteeing a tractive effort to move the vehicle with the least possible energy. In this work, we present an EDS developed, implemented, and tested in a virtual environment using MATLAB™, with the proposed developments then implemented in a test car. Exhaustive experimental testing demonstrated that the proposed EDS design signiﬁcantly improves the test vehicle’s longitudinal dynamics and energy consumption. This paper’s main contribution consists of designing an EDS for an in-wheel motor electric vehicle (IWMEV), with motors directly connected to the rear axle. The design demonstrated effective energy management, with savings of up to 21.4% over a vehicle without EDS, while at the same time improving longitudinal dynamic performance.

Citation: Julio-Rodríguez, J.d.C.; Keywords: electric vehicles; electromobility; in-wheel motors; electronic differential; wheel-speed Santana-Díaz., A.; Ramirez-Mendoza, control; powertrain; energy consumption; automotive control; vehicle dynamics control R.A. Individual Drive-Wheel Energy Management for Rear-Traction Electric Vehicles with In-Wheel Motors. Appl. Sci. 2021, 11, 4679. 1. Introduction https://doi.org/10.3390/app11104679 As transport technology evolves, new opportunities arise. Formerly, the efficiency of energy usage in street vehicles was close to being maxed out, up to a point limited Academic Editor: Jordi Cusido by the internal combustion engine’s physical principles. With the renewed interest in electric vehicles (EVs) and the mass production of them, a vast new field for improvement Received: 13 April 2021 has opened up for research engineers, physicists, and designers in many areas, including Accepted: 12 May 2021 Published: 20 May 2021 battery technology, charging stations, power train mechanics, and of course vehicle energy management [1]. This era of in-vehicle technology has seen the advent of two significant

Publisher’s Note: MDPI stays neutral paradigm changes—the electrification of street vehicles and the implementation of higher with regard to jurisdictional claims in levels of autonomous driving—thanks to the increased availability of high computational published maps and institutional affil- power onboard the vehicles and cloud computing connectivity [2]. iations. The development of intelligent and connected vehicles and their integration with energy management technology in their power train can further increase energy efficiency [3,4]. The integration of all the methods mentioned earlier comprises a synergy of great interest for studying, evaluating, and implementing, taking energy usage optimization to new high possibilities [5]. The transformation of ground transport to electromobility Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. is an undeniable reality [6]. There is interest in increasing vehicle energy efficiency for This article is an open access article environmental, technical, and economic reasons. distributed under the terms and In-wheel motor technology was first introduced to commercial vehicles by Ferdinand conditions of the Creative Commons Porsche around the year 1900. With this technology, it is possible to eliminate extra Attribution (CC BY) license (https:// components and simplify the power train system on hybrid and fully electric vehicles [7,8]. creativecommons.org/licenses/by/ One of the main characteristics of an IWMEV is the absence of a transmission combined 4.0/). with the wheel’s differential, which has been essential in every four-wheeled vehicle since

Appl. Sci. 2021, 11, 4679. https://doi.org/10.3390/app11104679 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 4679 2 of 16

they were first developed more than one hundred years ago. The mechanical transmission has been improved over the years, aiming to maximize engine performance through its speed range. It is still being developed, and the possibility of its pertinency in electric vehicles is being researched [9]. On the other hand, the differential has been left alone as part of a specific and straight- forward function. It is to maintain the freedom of spin of the rear axle wheels, allowing them to rotate at different speeds, one relatively against the other according to the car-specific requirements at any given time while still being capable of delivering a tractive effort to the ground to move the vehicle. The in-wheel motor technology does not allow a mechanical differential to be used in the vehicle [10]. However, the car still needs to control the wheel spin velocity during turning maneuvers. The development of an EDS presents the opportunity to solve this problem while also improving the vehicle’s energy efficiency by addressing the energy management requirements of the IWMEV. The re-emerging in-wheel motor technology promises to expand the powertrain architecture possibilities for electric vehicles, addressing particularities like moving the motor’s weight out of the car’s suspended mass. At the same time, it increases the inertia of the wheels, which has undesired consequences that are out of the scope of this work. These dif- ferences bring new challenges like the one addressed in this work: the differentiation and power distribution in the individual motors of the IWMEV. This document explains the development, setup, and validation of the electric vehicle’s onboard EDS with rear traction. The modeling was started using MATLAB™, and following this, tests were carried out on a chassis dynamometer and in real circuit driving experiments. The scope of this paper is to evaluate the performance of this EDS developed around vehicle geometry theory and dynamics equations to improve the handling behavior and energy control of an IWMEV. This process starts with the programming and simulation of the EDS in a numerical computing environment, followed by designing and constructing the required hardware for the system to be implemented in the real electric vehicle for further testing and evaluation. The approach followed was to take the vehicle’s geometric and dynamic equations as the core of the EDS because it was a simple model to put on board and observe the power train behavior with and without differential [11]. This evaluation was carried out by measuring the directly implied variables like the wheels’ angular speed, the throttle input signal, the steering angle, and the longitudinal speed. This paper is organized as follows: Section2 describes the state of the art in EDS. Section3 presents the methodology employed in this work. Section4 presents the EV prototype used for testing the EDS. Section5 introduces the vehicle dynamics fundamentals considered for the EDS design. Section6 describes the EDS development process. Section7 details the experimental procedures performed on the vehicle. Section8 presents the EDS test results. Finally, the findings are discussed in the last section.

2. State of the Art in EDS Many studies have been conducted about EV efficiency [12–14], but few focus on the particular case of IWMEVs [15]. The Ackermann steering geometry is the starting point of most proposed models for wheel speed differentiation [16]. It specifies the geometric relationships that should be complied with to achieve an optimum vehicle performance during turning maneuvers, as stated in the vehicle dynamics literature [17]. The essential inputs for the EDSs proposed in other works are the steering angle and the throttle position [18]. For the EDS presented in this work, the inputs mentioned earlier are also used, complete with the vehicle’s longitudinal speed information. The modeling of the IWMEV results in a complex nonlinear system due to the tire characteristics during a turning or direction-changing process [19]; rapid steering or a low-adhesion-coefficient road can cause vehicle instability like tire slipping causing the driver to lose control. The testing of the EDS proposed in this work was performed at low speeds and maintaining low mechanical power demand on the testing vehicle; this guaranteed vehicle stability and eliminated the Appl. Sci. 2021, 11, 4679 3 of 16

possibility of slipping during testing [20], simplifying the vehicle dynamics modeling and allowing for energy interactions to be better observed. The dynamics stability control approach is used by Hua et al. in their work [21]. Their proposed EDS follows the yaw rate’s value closely and distributes the torque on both sides to ensure steering performance and driving stability. Using an input signal, the steering angle values, and the vehicle speed, their proposed fuzzy logic controller calculates the additional yaw moment necessary to maintain the vehicle’s stability. An- other proposed approach is the one presented by Daya et al. [22]. They introduce an EDS based on wavelength control, transforming the steering angle and the throttle input into frequency components to calculate the respective gains fed into the system as input to the motors. Daya et al. also compare the wavelength controller with their own PID controller. It is noticeable that the wavelength controller provides a smoother response with less overshot and a faster settling time than the PID controller. Zhou et al. [23] acknowledge the nonlinearity of the vehicle dynamics in their proposed EDS, including the combined effects and interactions of the longitudinal, lateral, and yaw dynamics together with tire dynamics. The EDS calculations are made by a Model Predictive Control (MPC) controller that pro- cesses the vehicle dynamics variables, achieving suitable stability parameters. MPC theory is a recurrent method in vehicle dynamics control [24]. Hu et al. [25] used MPC for torque distribution in a IWMEV that included the motor’s efficiency map to reduce energy consumption, achieving savings of 4.5% and 0.8% for lane changing and straight acceleration maneuvers, respectively. Xu et al. [26] also used MPC for torque control in their IWMEV simulation model to optimize their in-wheel motors’ regenerative braking function. EDS works focus on vehicle dynamics; however, there have been efforts in combining dynamics control with energy optimization since both tasks imply powertrain control and actuation. Ma et al. [27] use torque allocation to reduce side slipping by implementing a penalty function within its controller; reducing side slip angle has positive effects on the vehicle dynamics and reduces energy losses in the tire due to tire wear. Zhao et al. [28] proposed a system with a dynamics stability approach, using a holistic corner controller for a vehicle with four in-wheel motors to reduce tire slipping and thus tire wear, controlling the yaw rate of the car and finally achieving a reduction of tire slip energy of 16.62%. This paper’s main contribution consists of an EDS focused on energy management and implementation in an IWMEV test car. The EDS manages energy by calculating the optimal speed for each drive wheel, using proportional control and a deterministic vehicle model. The EDS then allocates the required power to each in-wheel motor through the motor controllers, while these handle the torque applied at the contact patch of the wheels with the ground. Exhaustive experimental testing has demonstrated that the proposed EDS design significantly improves vehicle longitudinal dynamics and powertrain energy. In particular, the design has been shown savings of up to 21.4% over a vehicle without EDS. The tests and analyses that lead to these results are widely described in the following sections of the document.

3. Methodology The methodology for EDS development is inspired by the classical design methodology [29]. This methodology suits the project’s scope because it includes a preparatory phase, conceptualizing an idea based on a given problem, conceptual design, embodiment design, system development, physical model implementation, and testing (see Figure1). Appl.Appl.Appl. Sci. Sci.Sci.2021 20212021, ,11, 1111,, 4679, xx FORFOR PEERPEER REVIEWREVIEW 44 ofofof 161717 Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 17

FigureFigureFigure 1. 1.1. Design DesignDesign methodology methodologymethodology of ofof the thethe EDS. EDS.EDS.

4.4.4. Experimental ExperimentalExperimental Test TestTest VehicleVehicle VehicleVehicle ThisThisThis work’swork’swork’s experimental experimentalexperimentalexperimental platform platformplatform is is isis a a aa test test testtest vehicle vehicle vehiclevehicle available available availableavailable at at atat the the thethe Research Research ResearchResearch Center Center CenterCenter forforfor Automotive AutomotiveAutomotive MechatronicsMechatronics MechatronicsMechatronics (CIMA) (CIMA)(CIMA) (Figure (Figure(Figure 22). ).2).2). This ThisThis test testtest vehicle vehiclevehicle is isis a aarecreational recreational recreationalrecreational vehicle vehicle vehiclevehicle thatthatthat features featuresfeatures a aaa lightweight lightweightlightweightlightweight aluminumaluminum aluminumaluminum chassis. chassis.chassis. The The TheThe car car carcar was was waswas specially specially speciallyspecially designeddesigned designeddesigned and and andand built built builtbuilt to to to to researchresearchresearch electric electricelectric cars, cars,cars,cars, in ininin particular particular particularparticular IWMEVs. IWMEVs IWMEVsIWMEVs Vehicle. ..Vehicle VehicleVehicle components components componentscomponents have have havehave been been beenbeen adapted adapted adaptedadapted to theto to to chassisthethe chassischassis and are,andandand for are,are,are, the forforfor most the thethe most part,mostmost commercialpart, part,part, comme commecomme components.rcialrcialrcial components. components.components. The purpose The TheThe purpose purposepurpose of this of test ofof this this vehiclethis test testtest is primarily to experience realistic vehicle dynamics conditions. vehiclevehicle isis primarilyprimarily to toto experience experienceexperience re rerealisticalisticalistic vehicle vehiclevehicle dynamics dynamicsdynamics conditions. conditions.conditions.

Figure 2. Test car for IWMEV research. FigureFigureFigure 2. 2.2. Test TestTest car carcar for forfor IWMEV IWMEVIWMEV research. research.research. The powertrain consists of two electric motors with a nominal power of 3 kW each. TheTheThe powertrain powertrainpowertrain consists consistsconsists ofofof two twotwo electric electricelectric motorsmomotorstors withwithwith aaa nominalnominalnominal powerpowerpower ofofof 333 kWkWkW each.each.each. Electric motors are commercial [ref: www.qsmotor.com, accessed on 12 June 2020] and are ElectricElectricElectric motorsmotors are are commercialcommercial [ref:[ref: [ref: www.qsmo www.qsmowww.qsmotor.comtor.com,tor.com, accessed,accessed accessed onon on 1212 12 JuneJune June 2020]2020] 2020] andand and areare direct-current brushless motors. The motors are installed in each of the vehicle’s rear aredirect-currentdirect-current direct-current brushlessbrushless brushless motors.motors. motors. TheThe The motorsmotors motors arar areee installedinstalled installed inin in eacheach each ofof thethe vehicle’svehicle’svehicle’s rearrearrear wheels. The operating region of the electric motors is in the voltage range of 48 to 72 volts. wheels.wheels.wheels. The TheThe operating operatingoperating region regionregion of ofof the thethe electric electricelectric motors motorsmotors is isis in inin the thethe voltage voltagevoltage range rangerange of ofof 48 4848 to toto 72 7272 volts.volts.volts. These motors’ behavior curves are shown in Figure 3, where the torque behavior versus TheseTheseThese motors’ motors’motors’ behavior behaviorbehavior curves curvescurves are areare shown shownshown in inin Figure FiFiguregure3 ,3,3, where wherewhere the thethe torque torquetorque behavior behaviorbehavior versus versusversus speed and efficiency are described in the corresponding figure. speedspeedspeed and andand efﬁciency efficiencyefficiency are areare described describeddescribed in inin the thethe corresponding correspondingcorresponding ﬁgure. figure.figure.

Figure 3. In-wheel motor characteristic curve (www.qsmotor.com, accessed on 12 June 2020). FigureFigureFigure 3. 3.3. In-wheel In-wheelIn-wheel motor motormotor characteristic characteristiccharacteristic curve curvecurve ( (www.qswww.qsmotor.com(www.qsmotor.com,motor.com,, accessed accessedaccessed on onon 12 1212 June JuneJune 2020). 2020).2020). Appl.Appl. Sci.Sci.2021 2021,,11 11,, 4679 x FOR PEER REVIEW 55 of 1617

FigureFigure4 4shows shows the the electronic electronic architecture architecture to implement to implement the EDS,the EDS, with with the components the compo- labelednents labeled with numbers with numbers from onefrom to one six. to In six. particular, In particular, one encoder one encoder sensor sensor (1) measures (1) measures the steeringthe steering wheel wheel angle. angle. The The corresponding corresponding angular angular speeds speeds are are measured measured using using the the threethree hallhall effecteffect sensorssensors installedinstalled onon eacheach rearrear wheelwheel (2).(2). InIn addition,addition, aa potentiometerpotentiometer (3)(3) hashas beenbeen usedused toto measuremeasure thethe throttlethrottle (accelerator(accelerator pedal).pedal). AllAll sensorssensors connectconnect toto twotwo systems,systems, ArduinoArduino™™ UNO UNO boards.boards. TheseThese provideprovide connectivityconnectivity toto thethe onboardonboard computercomputer(4) (4) withwith anan applicationapplication inin MATLAB’s MATLAB’s App App Designer Designer™™ with with the the algorithm algorithm programming programming code. code. Finally, Fi- twonally, motor two motor controllers controllers (5) and (5) the and battery the batte packry (6) pack are also(6) are shown also shown in the ﬁgure.in the figure.

FigureFigure 4.4. EDSEDS hardwarehardware schematic.schematic.

TheThe operation of the the entire entire system system is is desc describedribed as as follows. follows. The The EDS EDS reads reads inputs inputs and andcalculates calculates angular angular speed speedss required required at the at thedrive drive wheels. wheels. The The motor motor controllers controllers receive receive in- instructionsstructions from from the the EDS EDS and and manage manage the the energy energy delivered delivered to the motors.motors. AA batterybattery packpack storesstores andand provides provides the the energy energy required required by by the th electrice electric motors. motors. The The electric electric in-wheel in-wheel motors mo- havetors have the mechanical the mechanical power power as instructed as instructed by the by controllers the controllers and report and report their angular their angular speed tospeed the EDS.to the The EDS. throttle The throttle pedal pedal reads reads the acceleration the acceleration desired desired by the by user the user and sendsand sends the voltagethe voltage signal signal to the to EDS.the EDS. The rotaryThe rotary encoder encoder reads reads the position the position of the of steering the steering wheel wheel and sends the information to the EDS. Figure4 shows signal communication and electric power and sends the information to the EDS. Figure 4 shows signal communication and electric delivery in the direction of the arrows. power delivery in the direction of the arrows. Table1 summarizes the vehicle’s characteristics and the results obtained during the per- Table 1 summarizes the vehicle’s characteristics and the results obtained during the formance tests applied using CIMA’s laboratory equipment, such as the chassis dynamometer. performance tests applied using CIMA’s laboratory equipment, such as the chassis dyna- Tablemometer. 1. Physical parameters of the car test.

Table 1. Physical parameters of the car Vehicletest. Information Power train Rear-drive, in-wheel motors WeightVehicle 400 Information kg (44% front axle–56% rear axle) DimensionsPower train 2.96 m long Rear-drive,× 2.14 m in-wheel wide × 1.53 motors m high PowerWeight source 59 400 volts—16 kg (44% Li-ion front Cells axle–56% Battery rear pack axle) Frontal area 1.82 m2 C : 0.428 Dimensions 2.96 m long × 2.14d m wide × 1.53 m high Tire effective radius 0.25 m Rolling Resistance coefﬁcient: 0.027 Tire corneringPower coefﬁcientsource 59 volts—16 0.19 (1/deg) Li-ion (normalized) Cells Battery pack WheelbaseFrontal area 1.72 1.82 m m2 Inter-axle distance:Cd: 1.890.428 m * CM toTire rear effective wheels distance radius 0.83 0.25 m m Rolling *Resistance CM heigh: coefficient: 0.3 0.027 m Max torque 350 N-m TireMax cornering acceleration coefficient 1.54 m/s2 0.19 [1/deg]Max speed: (normalized) 64 km/h * CM refers toWheelbase the center of mass of the vehicle. 1,72 m Inter-axle distance: 1.89 m *CM to rear wheels distance 0.83 m * CM heigh: 0.3 m 5. Vehicle DynamicsMax torque Modeling 350 N-m ThisMax section acceleration describes the vehicle 1.54 dynamicsm/s2 characteristicsMax speed: and the testing performed64 km/h to* CM gather refers this to the information. center of mass of the vehicle. Appl. Sci. 2021, 11, 4679 6 of 16

Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 17

5.1. Steady-State Handling Characteristics The steady-state handling characteristics of a vehicle provide information on how the5. Vehicle vehicle Dynamics handles the Modeling steering and turning maneuvers and provides a classification into threeThis categories: section understeer,describes the neutral vehicle steer, dynamics and oversteer. characteristics The classificationand the testing depends performed on theto gather vehicle’s this understeer information. coefficient, Kus, a factor calculated as a function of the weight distribution and the tire cornering stiffness, Cα. The normalized cornering coefficient (defined5.1. Steady-State as the corneringHandling Characteristics stiffness per unit normal load) of all the test car wheels was estimated to have a value of 0.19 (1/deg). The steady-state handling characteristics of a vehicle provide information on how the The steering angle required for the vehicle to perform a given curve depends on vehicle handles the steering and turning maneuvers and provides a classification into the wheelbase, turning radius, forward speed, and the mentioned understeer coefficient. three categories: understeer, neutral steer, and oversteer. The classification depends on The test vehicle’s weight distribution has been measured as 44% and 56% for the front the vehicle’s understeer coefficient, Kus, a factor calculated as a function of the weight dis- and rear axles, respectively, making its understeer coefficient have a value of −0.632 [deg], tribution and the tire cornering stiffness, Cα. The normalized cornering coefficient (de- which indicates that the vehicle is classified as oversteer. If an oversteer vehicle is acceler- fined as the cornering stiffness per unit normal load) of all the test car wheels was esti- ated in a constant radius, the driver must decrease the steering angle [17]. This characteristic ismated considered to have in a the value EDS of operation. 0.19 (1/deg). The EDS will respond in real-time to any changes in the user’sThe steering steering angle angle to maintainrequired thefor vehiclethe vehicle in the to desiredperform path a given and curve at the depends desired speed. on the wheelbase, turning radius, forward speed, and the mentioned understeer coefficient. The 5.2.test Preliminaryvehicle’s weight Dynamics distribution Testing of has VR1 been measured as 44% and 56% for the front and rear Theaxles, vehicle’s respectively, preliminary making testing its un providesdersteer critical coefficient information have latera value used of in − developing0.632 [deg], thewhich EDS, indicates like the that power the train vehicle capabilities is classified and as dynamics oversteer. response If an oversteer to handling vehicle maneuvers. is acceler- Theated following in a constant figures radius, present the driver the testing must resultsdecrease of the the steering test car angle using [17]. the dynamometer.This character- Figureistic is5 considered shows the acceleration in the EDS operation. and speed The curves, EDS and will Figure respond6 contains in real-time the torque to any response changes andin the mechanic user’s steering power developmentangle to maintain of the the powertrain. vehicle in the desired path and at the desired speed. 5.3. Kinematics 5.2. PreliminaryThe basis for Dynamics the EDS isTesting the geometry of VR1 implicit in the Ackermann equations. These equa- tionsThe describe vehicle’s the geometricpreliminary and testing dynamic provides relationships critical information that need to later be achieved used in develop- to guaranteeing the that EDS, a no-slipping like the power condition train capabiliti occurs ines the and drive dynamics wheels response during a to turning handling maneuver. maneu- Thevers. base The equationsfollowing figures were taken present from the Jazartesting [30 results], and of their the test development car using the to obtaindynamom- the driveeter. Figure wheels’ 5 angularshows the speed acceleration is shown and below. speed The curves, symbols and usedFigure in 6 the contains equations the torque are in Appendixresponse Aand. mechanic power development of the powertrain.

FigureFigure 5.5. AccelerationAcceleration andand aa toptop speedspeed ofof thethe testtest carcar inin chassischassis dynamometer.dynamometer.

Figure7 shows the existing relationship between the parameters of a car making a turn with a radius R1. The angles δi and δo must be such that the line coming from the turning center-point O is perpendicular to the left and right wheel’s lateral plane, which guarantees that both turning wheels are in full rolling behavior, not slipping. The rear axle is of interest to establish the correspondent geometric and dynamics equations for the drive wheels. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 17

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Figure 6. Torque and power characteristic curves of the test car power train.

5.3. Kinematics The basis for the EDS is the geometry implicit in the Ackermann equations. These equations describe the geometric and dynamic relationships that need to be achieved to guarantee that a no-slipping condition occurs in the drive wheels during a turning maneuver. The base equations were taken from Jazar [30], and their development to obtain the drive wheels’ angular speed is shown below. The symbols used in the equations are in Appendix A. Figure 7 shows the existing relationship between the parameters of a car making a turn with a radius R1. The angles 𝛿i and 𝛿o must be such that the line coming from the turning center-point O is perpendicular to the left and right wheel’s lateral plane, which guarantees that both turning wheels are in full rolling behavior, not slipping. The rear axle is of interest to establish the correspondent geometric and dynamics equations for the Figure 6. Torque and power characteristic curves of the test car power train. driveFigure wheels. 6. Torque and power characteristic curves of the test car power train. 5.3. Kinematics The basis for the EDS is the geometry implicit in the Ackermann equations. These equations describe the geometric and dynamic relationships that need to be achieved to guarantee that a no-slipping condition occurs in the drive wheels during a turning maneuver. The base equations were taken from Jazar [30], and their development to obtain the drive wheels’ angular speed is shown below. The symbols used in the equations are in Appendix A. Figure 7 shows the existing relationship between the parameters of a car making a turn with a radius R1. The angles 𝛿i and 𝛿o must be such that the line coming from the turning center-point O is perpendicular to the left and right wheel’s lateral plane, which guarantees that both turning wheels are in full rolling behavior, not slipping. The rear axle is of interest to establish the correspondent geometric and dynamics equations for the drive wheels. Figure 7. AA front-wheel-steering vehicle and the Ackermann condition (Jazar, 2008).

Figure 88 showsshows thethe samesame rear-driverear-drive carcar withwith moremore parameters,parameters, likelike thethe distancedistance aa2 between thethe centercenter ofof mass massC Cand and the the rear rear axle, axle, the the distance distanceR from R from the turningthe turning center-point center- pointO to the O centerto the ofcenter mass, of the mass, angular the speed angular of each speed rear of wheel each ωreari and wheelωo, and ωi and the longitudinalωo, and the longitudinalspeed vector speed of the vector inner andof the outer inner rear and wheel outerV irearand wheelVo, respectively. Vi and Vo, respectively. The relationship The relationshipbetween these between parameters these allowsparameters the construction allows the construction of the following of the equations: following equations: q 2 2 2 R𝑅== a𝑎2+ +l 𝑙cot 𝑐𝑜𝑡δ 𝛿 (1)

Equation (1) shows the the vehicle’s vehicle’s tu turningrning radius radius as as a a function function of of δδ,, where where δδ isis the the cot- cot- average of the inner and and outer outer steer steer angles angles and and l l distance distance between between the the front front and and rear rear axles. axles. Equation (2) shows the equivalent angle δ when the front wheels have different values.values.

cot(δ ) + cot(δ ) Figure 7. A front-wheel-steering vehiclecot(δ) and= the Ackermanno icondition (Jazar, 2008). (2) 2 a2 TheFigure test 8 car’s shows steering the same mechanism rear-drive characteristics car with more classify parameters, it as a parallel like steerthe distance (Figure9 ). Thisbetween means the that center the steering of mass angle C and of the both rear front axle, wheels the isdistance equal (Equation R from the (3)). turning The effects center- of havingpoint O a steeringto the center mechanism of mass, that the does angular not adhere speed to of the each Ackerman rear wheel geometry ωi and are ω increasedo, and the tirelongitudinal scrub during speed cornering, vector of tire the wear, inner and and increased outer rear steering wheel effort. Vi and The Vo, magnitude respectively. of notThe havingrelationship Ackerman between steering these is parameters variable and allows depends the construction on the steering of the angle following or the radii equations: of the turning maneuver being performed. For all the tests considered in this work, the steering angles were at or below 16 (deg), so𝑅= the deviation𝑎 + 𝑙 𝑐𝑜𝑡 from 𝛿 the ideal curve was minimal,(1) and the effects can be neglected. Equation (1) shows the vehicle’s turning radius as a function of δ, where δ is the cot- δi = δo = δ (3) average of the inner and outer steer angles and l distance between the front and rear axles. Equation (2) shows the equivalent angle δ when the front wheels have different values. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 17

Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 17

𝑐𝑜𝑡(𝛿) + 𝑐𝑜𝑡(𝛿) 𝑐𝑜𝑡(𝛿) = (2) 𝑐𝑜𝑡(𝛿)2+ 𝑐𝑜𝑡(𝛿) 𝑐𝑜𝑡(𝛿) = (2) 2 The test car’s steering mechanism characteristics classify it as a parallel steer (Figure 9). ThisThe means test car’s that steering the steering mechanism angle ofcharacteristic both front swheels classify is it equal as a parallel (Equation steer (3)). (Figure The effects9). This of means having that a steering the steering mechanism angle ofthat both does front not wheelsadhere isto equalthe Ackerman (Equation geometry (3)). The areeffects increased of having tire ascrub steering during mechanism cornering, that tire does wear, not andadhere increased to the Ackermansteering effort. geometry The magnitudeare increased of nottire having scrub Ackermanduring cornering, steering tireis variable wear, andand increaseddepends on steering the steering effort. angle The ormagnitude the radii of notthe havingturning Ackerman maneuver steering being performed. is variable Forand all depends the tests on considered the steering in angle this Appl. Sci. 2021, 11, 4679 work,or the theradii steering of the turningangles were maneuver at or below being 16 performed. (deg), so theFor deviation all the tests from considered the ideal incurve8 this of 16 waswork, minimal, the steering and theangles effects were can at be or neglected.below 16 (deg), so the deviation from the ideal curve was minimal, and the effects can be neglected.

Figure 8. A front-wheel-steering vehicle and steer angles of the inner and outer wheels (Jazar, 2008).FigureFigure 8. 8. AA front-wheel-steering front-wheel-steering vehicle vehicle and and steer steer an anglesgles of the inner and outer wheels (Jazar,(Jazar, 2008). 2008).

FigureFigure 9. 9. ComparisonComparison of of Ackerman Ackerman steering steering and and parallel parallel steering steering (Jazar, (Jazar, 2008). 2008). Figure 9. Comparison of Ackerman steering and parallel steering (Jazar, 2008). From the geometric relations in Figure8, we obtain Equation (4). 𝛿 =𝛿 =𝛿 (3) 𝛿 =𝛿 =𝛿R1 (3) Cot δ = (4) From the geometric relations in Figure 8, wel obtain Equation (4). From the geometric relations in Figure 8, we obtain Equation (4). 𝑅 The dynamics equations for each rear𝐶𝑜𝑡 wheel’s 𝛿= longitudinal speed (Vi and Vo) and(4) the 𝑅𝑙 geometric proportions of a vehicle turning𝐶𝑜𝑡 left can 𝛿= be written as shown in Equations (5) and(4) (6) 𝑙 for theThe innerdynamics wheel equations and the for outer each wheel, rear wheel’s respectively. longitudinal The wheel’s speed (V effectivei and Vo) radiusand the is i o geometricdenotedThe dynamics by proportionsRw and equations the of Yaw-rate a vehi forcle each of turning the rear vehicle wheel’s left bycanY longitudinal Rbe. written as speed shown (V inand Equation V ) and the(5) andgeometric Equation proportions (6) for the of inner a vehi wheelcle turning and the leftouter can wheel, be written respectively. as shown The in wheel’s Equation effec- (5) and Equation (6) for the inner wheel and the outerw wheel, respectively. The wheel’s effective radius is denoted by Rw andV thei = Yaw-rateYR R1 − of the= vehicleωiRw by YR. (5) tive radius is denoted by Rw and the Yaw-rate of2 the vehicle by YR. w V = Y R + = ω R (6) o R 1 2 o w Replacing Equation (4) with Equations (5) and (6) and solving the angular speed of the inner (ωi) and outer (ωo) wheels of the rear axle of a rear-drive car, we obtain Equations (7) and (8). w YR l ∗ Cot(δ) − 2 ωi = (7) Rw w YR l ∗ Cot(δ) + 2 ωo = (8) Rw The former equations are the base for the EDS and are programmed in a series of successive code lines that calculate the required speed for each of the rear wheels as a function of the car’s instantaneous turning angle and velocity. The motor controllers set Appl. Sci. 2021, 11, 4679 9 of 16

the voltage on the motors to achieve the individual speed calculated for each rear wheel, ωi, and ωo. This information is then sent to the EDS to recalculate the power parameters for the wheels in the next iteration.

6. EDS Development After obtaining the critical characteristics of the vehicle dynamics, the EDS could be developed.

EDS Development Using MATLAB The program’s main purpose is to maintain the required angular velocity differential between the two rear wheels. However, it also needs to control some other aspects, like ac- knowledging the motors’ maximum speed capabilities and making smooth transitions Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 17 between speeds to reduce excessive vehicle accelerations. The main functions of the EDS algorithm are shown in Figure 10.

Figure 10. EDS general diagram. Figure 10. EDS general diagram. 7. Experimental Setup Specifications The EDS must acknowledge the limited capabilities of the hardware. In this case, the This section provides information on the testing performed to evaluate the EDS. It is electric motors have a maximum angular speed of 65 (rad/s). The motor’s manufacturer of particular interest to evaluate its functionality in terms of longitudinal speed control specified thisand value, energy and management it was confirmed capabilities. byFor testingthe testing the design, electric it was vehicle necessary in ato chassishave the dynamometer.vehicle In principle, perform turning the EDS maneuver must guarantees to notice the that EDS the effects. two wheels’ This section speed presents sustains the a calculateddifferent difference. tests If and a turning the specific maneuver conditions at of a giventhe experiments. speed requires It includes that graphic the external profiles wheel goes atof andifferent angular variables speed and higher detailed than inst theructions maximum of the allowedtesting procedures. by the in-wheel motor, the total longitudinalTwo different speed of tests the were car designed must be and reduced were applied so that to the the system testing vehicle. keeps workingThe speed within its limits.control and acceleration testing provided information on the longitudinal performance of Consideringthe test the vehicle, sensors’ providing specifications, data for both the speed sampling and acceleration period foranalysis. the analog The energy signals management test provided data for the energy allocation performance analysis. measured by the EDS was defined so that the aliasing was avoided [31]. The variables in 7.1. EDS Speed Control and Acceleration Testing The longitudinal dynamics tests were performed on even terrain by accelerating the vehicle to a moderated speed and performing turning maneuvers. A zig-zag steering angle pattern was applied, starting at zero degrees and then turning left and right seven degrees in both directions with the constant throttle. The complete specifications of the performed experiments are described in Figure 11. Appl. Sci. 2021, 11, 4679 10 of 16

the control loop and the values for the motor’s speed are calculated for each computational cycle (490 Hz). Still, the actuation introduces a delay so that the motor’s speeds are set 10 times per second.

7. Experimental Setup Specifications This section provides information on the testing performed to evaluate the EDS. It is of particular interest to evaluate its functionality in terms of longitudinal speed control and energy management capabilities. For the testing design, it was necessary to have the vehicle perform turning maneuvers to notice the EDS effects. This section presents the different tests and the specific conditions of the experiments. It includes graphic profiles of different variables and detailed instructions of the testing procedures. Two different tests were designed and were applied to the testing vehicle. The speed control and acceleration testing provided information on the longitudinal performance of the test vehicle, providing data for both speed and acceleration analysis. The energy management test provided data for the energy allocation performance analysis.

7.1. EDS Speed Control and Acceleration Testing The longitudinal dynamics tests were performed on even terrain by accelerating the vehicle to a moderated speed and performing turning maneuvers. A zig-zag steering angle pattern was applied, starting at zero degrees and then turning left and right seven degrees Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 17 in both directions with the constant throttle. The complete speciﬁcations of the performed experiments are described in Figure 11.

FigureFigure 11. 11.Speed Speed and and acceleration acceleration testing. testing.

7.2.7.2. Energy Energy Management Management Testing Testing AnAn onboardonboard currentcurrent and and voltage voltage measurement measurement device device (Dewesoft’s (Dewesoft’s SIRIUS SIRIUS™)™) was was usedused to to register register the the energy energy consumed consumed by by each each motor motor and and the the instantaneous instantaneous electric electric power power demanded.demanded. ThisThis setset ofof teststests consistedconsisted ofof drivingdriving inin circlescircles with with a a constant constant radius radius while while recordingrecording dynamicsdynamics and energy energy data. data. For For the the energy energy consumption consumption tests, tests, the throttle the throttle posi- positiontion is kept is kept constant constant for the for duration the duration of the of tests, the tests,and the and maneuver the maneuver starts with starts the with vehicle the vehiclespeed at speed 11 (km/h). at 11 (km/h). In addition, In addition, no braking no braking is applied, is applied, and the and turning the turning maneuver maneuver is per- isformed performed with with a constant a constant steering steering angle angle of of16 16 (deg) (deg) on on even even ground. ground. Again,Again, thethe turningturning maneuvermaneuver tests tests were were performed performed with with the the EDS EDS activated activated and and deactivated. deactivated. The The details details of the of experiments are presented in Figure 12. the experiments are presented in Figure 12.

Figure 12. Energy management testing. Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 17

Figure 11. Speed and acceleration testing.

7.2. Energy Management Testing An onboard current and voltage measurement device (Dewesoft’s SIRIUS™) was used to register the energy consumed by each motor and the instantaneous electric power demanded. This set of tests consisted of driving in circles with a constant radius while recording dynamics and energy data. For the energy consumption tests, the throttle position is kept constant for the duration of the tests, and the maneuver starts with the vehicle speed at 11 (km/h). In addition, no braking is applied, and the turning maneuver is per- Appl. Sci. 2021, 11, 4679 formed with a constant steering angle of 16 (deg) on even ground. Again, the turning11 of 16 maneuver tests were performed with the EDS activated and deactivated. The details of the experiments are presented in Figure 12.

FigureFigure 12.12.Energy Energy managementmanagement testing.testing.

8. Results and Discussion This section presents a further analysis of the EDS, looking at its impact on the vehicle’s dynamic and energetic performance. The following results were obtained by employing the test procedures described in Section7.

8.1. EDS Results on Speed Control during the Turning Maneuvers Tests were performed on the vehicle with the EDS disabled to be used as a baseline; results are shown in Figure 13. As we could observe, the vehicle’s longitudinal velocity presented a variation of 8.55% from its original velocity during the turning maneuvers. This is an undesired effect. It is the EDS’s objective to compensate the relative angular velocities of both wheels to maintain the vehicle’s net longitudinal speed constant unless the user changes the throttle position. The same tests were carried out again with the EDS activated (Figure 14). In this case, the EDS successfully reduced longitudinal velocity variation during the turning maneuvers to 4.85%.

8.2. Acceleration Analysis Acceleration is a critical factor to consider while evaluating the EDS performance. In this case, the vehicle should not present any signiﬁcant acceleration spikes that the passengers perceive. The accelerations to observe are the longitudinal acceleration, the individual angular accelerations, and the lateral acceleration. Results for longitudinal acceleration are shown in Table2, with and without the EDS. The highest values for longitudinal acceleration were around 1.5 (m/s2), both positive and negative. During most of the acceleration tests, the longitudinal acceleration stayed between −1.5 and 0.5 [m/s2], a comfortable value for the passengers. During data analysis, it was also observed that the magnitudes of the instant acceleration values were similar for activated EDS and deactivated EDS; the difference is the allocation of this acceleration event performed by the EDS when it is active, controlling the longitudinal speed and reducing its undesired variation. The acceleration results are presented in Figure 15. The turning maneuver tests are described in Section7. Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 17 Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 17

8. Results and Discussion 8. Results and Discussion This section presents a further analysis of the EDS, looking at its impact on the vehi- This section presents a further analysis of the EDS, looking at its impact on the vehicle’s dynamic and energetic performance. The following results were obtained by employ- cle’s dynamic and energetic performance. The following results were obtained by employing the test procedures described in Section 7. ing the test procedures described in Section 7. 8.1. EDS Results on Speed Control During the Turning Maneuvers 8.1. EDS Results on Speed Control During the Turning Maneuvers Tests were performed on the vehicle with the EDS disabled to be used as a baseline; Tests were performed on the vehicle with the EDS disabled to be used as a baseline; results are shown in Figure 13. As we could observe, the vehicle’s longitudinal velocity results are shown in Figure 13. As we could observe, the vehicle’s longitudinal velocity presented a variation of 8.55% from its original velocity during the turning maneuvers. presented a variation of 8.55% from its original velocity during the turning maneuvers. This is an undesired effect. It is the EDS’s objective to compensate the relative angular This is an undesired effect. It is the EDS’s objective to compensate the relative angular velocities of both wheels to maintain the vehicle’s net longitudinal speed constant unless velocities of both wheels to maintain the vehicle’s net longitudinal speed constant unless the user changes the throttle position. The same tests were carried out again with the EDS Appl. Sci. 2021, 11, 4679 the user changes the throttle position. The same tests were carried out again with the12 ofEDS 16 activated (Figure 14). In this case, the EDS successfully reduced longitudinal velocity var- activated (Figure 14). In this case, the EDS successfully reduced longitudinal velocity variation during the turning maneuvers to 4.85%. iation during the turning maneuvers to 4.85%.

Figure 13. Disabled electronic differential test drive results. FigureFigure 13. 13.Disabled Disabled electronic electronic differential differential test test drive drive results. results.

Figure 14. Reduced variation of the speed with enabled EDS. FigureFigure 14. 14.Reduced Reduced variation variation of of the the speed speed with with enabled enabled EDS. EDS. 8.2. Acceleration Analysis Table8.2. Acceleration 2. Acceleration Analysis values for zig-zag maneuvers with EDS activated. Acceleration is a critical factor to consider while evaluating the EDS performance. In Acceleration is a critical factor to consider while evaluating the EDS performance. In this case, the vehicle should not present anyLongitudinal significant Accelerationacceleration (m/s spikes2) that the pas- this case, the vehicle should not present any significant acceleration spikes that the passengers perceive. The accelerations to observe are the longitudinal acceleration, the indi- sengers perceive. The accelerations Deactivatedto observe are EDS the longitudinal Activated acceleration, EDS the individual angular accelerations, and the lateral acceleration. vidual angularMax accelerations, and the lateral 0.5491 acceleration. 0.5374 Min −1.5168 −1.2662 mean + 0.1925 0.1495 mean − −0.5198 −0.1420

8.3. Energy Management Results The EDS directly affects the power train system and the energy distribution. The measured energy data of a sustained right-turning maneuver for both cases (activated and deactivated EDS) are presented in Table3. Here we can observe that the power train system’s total consumed energy was 21.4% less with the EDS activated. The power distribution noticeably changed between the two in-wheel electric motors (left and right). For a right turning maneuver, the right wheel required less energy than the left wheel. The electric power delivered to each motor and the totals are presented in Figure 16A,B for deactivated and activated EDS. The EDS guarantees that the power is delivered according to each motor’s speciﬁc energy requirements. The energy management functionality is observed in the orange and gray lines in the power distribution graphs; when the EDS was Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 17

Results for longitudinal acceleration are shown in Table 2, with and without the EDS. The highest values for longitudinal acceleration were around 1.5 (m/s2), both positive and negative. During most of the acceleration tests, the longitudinal acceleration stayed between −1.5 and 0.5 [m/s2], a comfortable value for the passengers.

Table 2. Acceleration values for zig-zag maneuvers with EDS activated.

Longitudinal Acceleration (m/s^2) Deactivated EDS Activated EDS Max 0.5491 0.5374 Min −1.5168 −1.2662 Appl. Sci. 2021, 11, 4679 mean + 0.1925 13 0.1495 of 16 mean - −0.5198 −0.1420 During data analysis, it was also observed that the magnitudes of the instant acceleration values were similar for activated EDS and deactivated EDS; the difference is the deactivated, theseallocation lines repeatedly of this acceleration crossed during event testing,performed which by the means EDS that when both it is motors active, controlling were receiving nearlythe longitudinal the same power.speed and Meanwhile, reducing withits undesi the EDSred enabled,variation. the The orange acceleration and results are gray lines stay noticeablypresented in separated. Figure 15. The turning maneuver tests are described in Section 7.

Figure 15.FigureLongitudinal 15. Longitudinal acceleration acceleration results results without with EDSout (left) EDS and (left) with and EDS with (right). EDS (right).

8.3. Energy Management Results Table 3. Energy and power result in a right turning maneuver. The EDS directly affects the power train system and the energy distribution. The measured energyRight data Turn of a withsustained EDS Activated right-turning maneuver for both cases (activated and deactivated EDS)Left are Motor presented in Table Right 3. Motor Here we can observe Total that the power train system’s total consumed energy was 21.4% less with the EDS activated. The power distribu- Energy (Wh) 2.12 0.89 3.01 tion noticeably changed between the two in-wheel electric motors (left and right). For a Average Powerright (W) turning maneuver,621.69 the right wheel 262.99 required less energy than 884.68 the left wheel. Right Turn with EDS Deactivated Table 3. EnergyLeft and Motor power result in a right Right turning Motor maneuver. Total Energy (Wh) 1.83Right Turn 2.00with EDS Activated 3.83 Average Power (W) 657.24Left Motor 721.41 Right Motor 1378.65 Total Energy (Wh) 2.12 0.89 3.01 Average Power (W) 621.69 262.99 884.68 Another interesting remark is that, in both cases, wheel speed differentiation was observed, but the causes were different. In the ﬁrst case, with the EDS deactivated, cinematic and mechanical restrictions caused the differentiation.Right Turn with Thus, EDS the Deactivated excess power delivered to the inner wheel was lost as wheel slipping or motor overcurrent heat. On the other hand, when the EDS was activated, the energy was correctly allocated and effectively used to accelerate the vehicle. Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 17

Left Motor Right Motor Total Energy (Wh) 1.83 2.00 3.83 Average Power (W) 657.24 721.41 1378.65

The electric power delivered to each motor and the totals are presented in Figure 16A,B for deactivated and activated EDS. The EDS guarantees that the power is delivered according to each motor’s specific energy requirements. The energy management functionality is observed in the orange and gray lines in the power distribution graphs; when the EDS was deactivated, these lines repeatedly crossed during testing, which means that both motors were receiving nearly the same power. Meanwhile, with the EDS enabled, the orange and gray lines stay noticeably separated. Another interesting remark is that, in both cases, wheel speed differentiation was observed, but the causes were different. In the first case, with the EDS deactivated, cinematic and mechanical restrictions caused the differentiation. Thus, the excess power de- Appl. Sci. 2021, 11, 4679 livered to the inner wheel was lost as wheel slipping or motor overcurrent heat. On14 ofthe 16 other hand, when the EDS was activated, the energy was correctly allocated and effectively used to accelerate the vehicle.

(A) (B)

Figure 16. ((A)A) Power Power distribution distribution and and individual individual wheel-speed wheel-speed profil proﬁlee for for a a right right turning turning maneuver maneuver with with EDS EDS deactivated. deactivated. (B) Power distribution andand individualindividual wheel-speedwheel-speed proﬁleprofile forfor aa rightright turningturning maneuvermaneuver withwith EDSEDS activated.activated.

9. Conclusions Conclusions By closely closely controlling controlling the the relative relative angular angular velocities velocities of both of both rear rear wheels, wheels, the EDS the EDSreg- ulatedregulated the thevehicle’s vehicle’s longitudinal longitudinal acceleration acceleration and and had had a anoticeable noticeable effect effect on on the the vehicle’s longitudinallongitudinal speed speed during during turning turning maneuvers, maneuvers, achieving a decrease in longitudinal speed variation from 8.55% to an average of 4.85% for the same driving maneuvers. The EDS also improved the power train’s energy performance by avoiding sending excess electric powerpower toto the the physically physically restricted restricted inner inner motor, motor, noticeably noticeably minimizing minimizing energy en- ergylosses losses due due to overcurrent to overcurrent heat, heat, wheel wheel slip, slip, and and friction. friction. In In addition, addition, thethe outerouter motor performed better as it now received the correct amount of energy. Depending on the turning radius, the electric power misallocation can account for up to 21.4% of the turning maneuver’s energy. With the EDS’s help, this lost energy was minimized and saved, dramatically improving the vehicle’s energy efﬁciency.

10. Future Work There is still plenty of research to be conducted on IWMEV technology; the following are ideas that can further develop electronic differential capabilities: • Evaluate advanced control strategies for the EDS, like machine learning models to predict optimal power strategies. The optimization parameters can be measurements provided by onboard sensors, while the model aims to minimize energy in real- time driving. • Study the effects of integrating regenerative braking functionality into the EDS. • Increase the number of onboard sensors and include IMUs to enhance vehicle dynamics together with energy management. • Evaluate EDS capabilities in new terrains and driving scenarios, like uneven ground and steep slopes, to ﬁnd new ﬁelds for potential improvement. Appl. Sci. 2021, 11, 4679 15 of 16

Author Contributions: Conceptualization, A.S.-D. and J.d.C.J.-R.; methodology, A.S.-D. and J.d.C.J.-R.; software, J.d.C.J.-R.; validation, J.d.C.J.-R., R.A.R.-M. and A.S.-D.; formal analysis, A.S.-D. and R.A.R.-M.; investigation, J.d.C.J.-R.; resources, A.S.-D.; data curation, J.d.C.J.-R.; writing—original draft preparation, J.d.C.J.-R.; writing—review and editing, J.d.C.J.-R., R.A.R.-M. and A.S.-D.; supervi- sion, A.S.-D., and R.A.R.-M.; project administration, A.S.-D.; funding acquisition, A.S.-D. All authors have read and agreed to the published version of the manuscript. Funding: Funding was provided by Tecnológico de Monterrey (Grant No. a01368435) and Consejo Nacional de Ciencia y Tecnologia (CONACYT) by the scholarship 867164. Acknowledgments: We appreciate Martín Rogelio Bustamante Bello’s support and collaboration during the final stages of the project, and we thank Pedro Camelo Romero for proofreading this document. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A

Table A1. List of acronyms.

Acronym Deﬁnition EV Electric Vehicle IWMEV In-Wheel Motor Electric Vehicle EDS Electronic Differential System

Table A2. Nomenclature.

Variable Description

Cd aerodynamic drag coefﬁcient CM center of mass Cα cornering stiffness Kus understeer coefﬁcient l length between the front and rear axle R turning radius Rw tire effective radius V longitudinal speed Vi speed of the inner rear wheel Vo speed of the outer rear wheel w wheelbase or track of the vehicle YR yaw rate of the car rad/s ωi the angular speed of the inside rear tire ωo the angular speed of the outside rear tire δi steer angle of the inside front tire δo steer angle of the outside front tire δ the equivalent turning angle of the front wheels

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