The Method of Mass Estimation Considering System Error in Vehicle Longitudinal Dynamics

energies

Article The Method of Mass Estimation Considering System Error in Vehicle Longitudinal Dynamics

Nan Lin 1,2 , Changfu Zong 1 and Shuming Shi 2,*

1 State Key Laboratory of Automotive Simulation and Control, Jilin University, Jilin 130022, China; [email protected] (N.L.); [email protected] (C.Z.) 2 Transportation College, Jilin University, Jilin 130022, China * Correspondence: [email protected]; Tel.: +86-15948022008

Received: 27 November 2018; Accepted: 21 December 2018; Published: 25 December 2018

Abstract: Vehicle mass is a critical parameter for economic cruise control. With the development of active control, vehicle mass estimation in real-time situations is becoming notably important. Normal state estimators regard system error as white noise, but many sources of error, such as the accuracy of measured parameters, environment and vehicle motion state, cause system error to become colored noise. This paper presents a mass estimation method that considers system error as colored noise. The system error is considered an unknown parameter that must be estimated. The recursive least squares algorithm with two unknown parameters is used to estimate both vehicle mass and system error. The system error of longitudinal dynamics is analyzed in both qualitative and quantitative aspects. The road tests indicate that the percentage of mass error is 16%, and, if the system error is considered, the percentage of mass error is 7.2%. The precision of mass estimation improves by 8.8%. The accuracy and stability of mass estimation obviously improves with the consideration of system error. The proposed model can offer online mass estimation for intelligent vehicle, especially for heavy-duty vehicle (HDV).

Keywords: mass estimation; system error; colored noise; recursive least squares; heavy-duty vehicle

1. Introduction Vehicle mass significantly affects the power and efficiency performance. For HDVs, advanced control systems such as automated gear shift strategies [1] and economic cruise control [2] need the information of vehicle mass. For vehicle longitudinal dynamics, acceleration resistance, rolling resistance and grade resistance are all influenced by vehicle mass. Vehicle mass is the most important parameter and potentially has notably large changes [3]. For a HDV, the vehicle total mass can be 10–50 t, with or without a trailer. A variation up to 400% in operation is normal. Hence, it is important to design an online mass estimation method for HDVs. Generally, mass estimation methods are based on longitudinal dynamics. State estimation algorithms such as the recursive least squares algorithm (RLS) [4] and Kalman filter (KF) [5] are commonly used technologies. Fathy [6] is inspired by perturbation theory and simplifies the mass estimation model with the differential equation of longitudinal dynamics. Similarly, Chu et al. [7] established an electric vehicle (EV) mass estimation model with a high-frequency information extraction method. EVs have advantages in measuring the high-quality tractive force, which guarantees the estimation accuracy. Altmannshofer and Endisch [8] presented a robust parameter algorithm for the vehicle mass and driving resistance estimation. The algorithm overcomes the drawbacks of outliers and insufficient excitation. The error estimation [9] and torque observer [10] are also applied to further improve the accuracy and robustness of the mass estimation.

Energies 2019, 12, 52; doi:10.3390/en12010052 www.mdpi.com/journal/energies Energies 2019, 12, 52 2 of 15

Energies 2018, 11, x FOR PEER REVIEW 2 of 16 Many scholars simultaneously estimate the mass and road grade [11–15]. Vahidi [16,17] designed different parameters, thus enabling the simultaneous estimation of the time-varying grade and the RLS with multiple forgetting factors to account for different rates of change of different parameters, piece-wise constant mass. Lei et al. [18] used an extended KF to estimate both mass and road grade. thus enabling the simultaneous estimation of the time-varying grade and piece-wise constant mass. Although it is notably efficient and convenient to obtain both mass and grade, the disadvantage is Lei et al. [18] used an extended KF to estimate both mass and road grade. Although it is notably also obvious. Since both mass and grade are calculated from longitudinal dynamics, it is difficult to efficient and convenient to obtain both mass and grade, the disadvantage is also obvious. Since both guarantee accuracy when so many variables are left unknown. For other area of moving objects, mass and grade are calculated from longitudinal dynamics, it is difficult to guarantee accuracy when mass and other structure parameters may be estimated simultaneously. Such as robots [19] and so many variables are left unknown. For other area of moving objects, mass and other structure excavator arm [20]. parameters may be estimated simultaneously. Such as robots [19] and excavator arm [20]. Although the literature has provided many reasonable solutions, good results always rely on Although the literature has provided many reasonable solutions, good results always rely on precise inputs. Without using laboratory sensors, how to maintain the accuracy and stability precise inputs. Without using laboratory sensors, how to maintain the accuracy and stability remains remains a challenge. a challenge. Too many parameters make it difficult to maintain the precision. The longitudinal dynamics Too many parameters make it difficult to maintain the precision. The longitudinal dynamics contains approximately 10 constants and 5 variables. Constants such as the roll resistance coefficient are contains approximately 10 constants and 5 variables. Constants such as the roll resistance coefficient are measured in standard test situations. These constants cannot self-adapt to different driving conditions. measured in standard test situations. These constants cannot self-adapt to different driving conditions. The most important variable, which is the driving torque, is also calculated by a complicated state The most important variable, which is the driving torque, is also calculated by a complicated state estimator [10,21], which may contain many errors. Improving the precision of one or two parameters is estimator [10,21], which may contain many errors. Improving the precision of one or two parameters not sufficient, and the system error of longitudinal dynamics should be considered. is not sufficient, and the system error of longitudinal dynamics should be considered. The system error is always considered a Gaussian distribution in RLS and KF. However, there The system error is always considered a Gaussian distribution in RLS and KF. However, there is a is a fixed deviation in different driving conditions in longitudinal dynamics. Different fixed fixed deviation in different driving conditions in longitudinal dynamics. Different fixed deviations deviations can be described as colored noise. Since this fixed deviation represents a tractive or can be described as colored noise. Since this fixed deviation represents a tractive or braking force in braking force in longitudinal dynamics, we call the system error Fse. If the system error is mixed longitudinal dynamics, we call the system error Fse. If the system error is mixed with another force in with another force in longitudinal dynamics, the accuracy of the mass estimation may decrease. longitudinal dynamics, the accuracy of the mass estimation may decrease. This study considers the system error as an unknown parameter to identify. The RLS is used to This study considers the system error as an unknown parameter to identify. The RLS is used to simultaneously identify the vehicle mass and system error. The system error of longitudinal simultaneously identify the vehicle mass and system error. The system error of longitudinal dynamics dynamics is qualitatively and quantitatively analyzed. The road test results indicate that the error is qualitatively and quantitatively analyzed. The road test results indicate that the error percentage percentage can decrease by 8.8% by considering the system error. can decrease by 8.8% by considering the system error. This paper is organized as follows. In section 2, the mass estimation method that considers the This paper is organized as follows. In Section2, the mass estimation method that considers system error in vehicle longitudinal dynamics is built. Then, the model is verified by field tests in the system error in vehicle longitudinal dynamics is built. Then, the model is verified by field tests Sections 2 and 3. In Section 4, the system error of longitudinal dynamics is qualitatively and in Sections2 and3. In Section4, the system error of longitudinal dynamics is qualitatively and quantitatively analyzed. The off-line fitting data explain how Fse causes the lower-precision problem. quantitatively analyzed. The off-line fitting data explain how Fse causes the lower-precision problem. The road test results indicate that the precision of the mass estimation obviously improves when The road test results indicate that the precision of the mass estimation obviously improves when the the system error is considered. system error is considered.

2.2. MassMass EstimationEstimation ThatThat ConsidersConsiders SystemSystem ErrorError inin VehicleVehicle LongitudinalLongitudinal DynamicsDynamics

2.1.2.1. LongitudinalLongitudinal DynamicsDynamics withwith SystemSystem ErrorError MassMass estimationestimation is isbased based on vehicle on vehicle longitudinal longitudinal dynamics, dynamics, which whichrepresents represents the balance the between balance thebetween tractive the force tractive and force four typesand four of driving types of resistance. driving resistance. Shown as Shown Figure 1as. Figure 1.

Fw Ft F Fj+Ff i

i mg

Figure 1. Vehicle longitudinal dynamics. Figure 1. Vehicle longitudinal dynamics.

In addition, we express the system error of longitudinal dynamics as follows,

Ft  Ff  Fw  Fi  Fj  Fse (1)

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In addition, we express the system error of longitudinal dynamics as follows,

Ft = Ff + Fw + Fi + Fj + Fse (1) where Fse is the system error of longitudinal dynamics. System error represents accumulative errors caused by sensor drift or uncertain environment disturbance in vehicle longitudinal dynamics. It contains the error of both variables and parameters in the tractive force and driving resistance. Different with normal measured noise, such as white noise, system error may change with different driving condition. It is a kind of color noise. In one independent test, its value is considered a constant. Other tractive forces or resistances are: tractive force Ft; roll resistance Ff ; air drag Fw; grade resistance Fi; inertia force Fj. For roll resistance, Ff = mg f (2) here m is vehicle mass. f is roll resistance coefﬁcient. g is acceleration of gravity. For air drag,

1 F = C Aρv2 (3) w 2 D here CD is air drag coefﬁcient, A is front area and ρ is air density. For tractive force,

Ttqi0igη F = (4) t r here Ttq is engine torque. i0 is the ﬁnal drive. ig is the transmission. η is the efﬁciency of power train. r is radius of the tire. With the engine speed, vehicle speed and tire radius, the gear ratio of the entire powertrain is, rn i i = 0.377 (5) 0 g v here n is the engine speed, which can be obtained from the Controller Area Network (CAN) bus. Bringing (5) into the expression of the tractive force (4), we have a new expression of the tractive force as (6). The tractive force can be calculated without the transmission ratio.

Ttqηn F = 0.1047 (6) t v For grade resistance, Fi = mgi (7) here i is the sine of road slope angle. For inertia force,

Fj = δmav (8) av is the acceleration of vehicle. δ is the correction coefﬁcient of the rotating mass, which is illustrated in (9). The correction coefﬁcient of the rotating mass can be expressed as [22],

2 2 ! 1 I 1 If i igη δ = 1 + w + 0 (9) m r2 m r2 where If is the rotational inertia of the ﬂywheel, Iw is the rotational inertia of all tires, and the inertial force can be separated into three parts and represented by:

Fja = mav (10)

I F = w a (11) jw r2 v Energies 2019, 12, 52 4 of 15

2 2 2 If i igη 0.1047n F = 0 a = a I η (12) j f r2 v v f v Then, Equation (1) becomes

Ft = Ff + Fw + Fi + Fja + Fjw + Fj f + Fse (13)

According to whether each tractive/resistance is related to the vehicle mass, we arrange them on each side of the equation. On the left side,

Fet = Ft − Fw − Fjw − Fj f (14)

Fet is the equivalent tractive force. This variable represents the tractive force without resistances, which is not related to vehicle mass. On the right side, we have

Ff + Fi + Fja + Fse = m(g f + gi + av) + Fse (15)

To measure the road grade with high accuracy, a longitudinal acceleration sensor is used [23,24].

asen = gi + av (16) asen is the acceleration measured by the sensor. Since this signal can be measured with high quality and reliability, the system error in this model will not be considered. Its measurement noise is white noise, which can be ﬁltered by the RLS algorithm. Substituting (16) into (15), we can obtain the ﬁnal form of the mass estimation model,

Fet = m(g f + asen) + Fse (17)

Equation (17) can be used to estimate the vehicle mass and system error.

2.2. Recursive Least Squares Model The RLS model is used to simultaneously identify the vehicle mass and system error. The classic form of RLS with the forgetting factor is used to estimate the vehicle mass, which can be expressed as,

Z = ϕθ + em (18)

T where ϕ = (gf + asen 1) and Z = (Fet) are the system input and output, respectively, and θ = (m Fse) is the parameter to be estimated. em is the measurement noise. At each time step from t to t + 1, the estimation process in discrete form can be expressed as [17],

−1 γ(t + 1) = L(t)ϕ(t + 1)ϕT(t + 1)L(t)ϕ(t + 1) + λ 1 T (19) L(t + 1) = λ 1 − γ(t + 1)ϕ (t + 1) L(t) θˆ(t + 1) = θˆ(t) + γ(t + 1)z(t + 1) − ϕT(t + 1)θˆ(t) where λ is the forgetting factor, and L is the covariance matrix. The initial value of the covariance matrix is L0 = 100. L0 is selected based on the test data. The value of λ can be constant or vary with time. A vary value of λ can solve the problem of data saturation. In our research, based on the principle of valid data judgment, much of the data will be ﬁltered. Data saturation is not obvious; thus, we use λ = 1. Not all the measured data are qualiﬁed for mass estimation. Since the test vehicle cannot offer a braking force, only the acceleration section can be used. The measured data at low speed are generally inaccurate because the gear frequently shifts when the vehicle begins to run. The system input and Energies 2019, 12, 52 5 of 15 outputEnergies 2018 also, 11 have, x FOR boundaries. PEER REVIEW Based on these reasons, the valid data principle is formed in Table5 of 116. The value of each principle is selected based on the test data. Table 1. Valid data principle. Table 1. Valid data principle. Principle Value Reason 1 PrincipleMinimum speed >5 m/s Value stable running Reason 12 Clutch Minimum engaged speed <1 >5 m/stractive stableforce output running 23 Braking Clutch pedal engaged released <1 <1no tractivebraking forceforce output 3 Braking pedal released0.05 m/s2< X <1 < 0.8 no braking force 44 System System input input saturation( saturationX (X)) 0.05 m/s2 < X < 0.8 m/seliminate2 eliminate abnormal abnormal data data m/s2 5 System output saturation (Y) >500 N eliminate abnormal data 5 System output saturation(Y) >500N eliminate abnormal data

DataData selectionselection will will make make the the RLS RLS method method discontinuous. discontinuous. At valid At valid sample sample times, times, (17) can(17) be can used, be whereas,used, whereas, at invalid at invalid sample sample times, the times, result the is resultonly maintained is only maintained until the next until sample the next time. sample When time. the validWhen data the reach valid 100 data s, the reach mass 100 estimation s, the mass stops, estimation which may stops, take which 200–500 may s from take the 200 beginning.–500 s from When the thebeginning. estimation When time the exceeds estimation 600 s, time the calculationexceeds 600 is s also, the stopped.calculation The is framealso stopped. of the mass The estimationframe of the is shownmass estimation in Figure2 is. shown in Figure 2.

FigureFigure 2.2. Frame of mass estimation.estimation.

3.3. Road Test ToTo verifyverify thethe constructedconstructed model, model, road road tests tests were were performed. performed. The The testing testing ground ground was was a sectiona section of anof expressway an expressway to avoid to avoid the controlled the controlled environment environment of a field of test. a field The test. driver The finished driver his finished transport his tasktransport in real task traffic in real conditions. traffic conditions. TheThe test test vehicle vehicle is is a heavy-duty a heavy-duty vehicle vehicle with with a diesel a diesel engine engine and 12-gear and 12 transmission.-gear transmission. All vehicle All parametersvehicle parameters in Section in2 areSection provided 2 are by provided the manufacturer by the manufacturer (Table2). One (Table 1.6-g 2 chassis). One acceleration 1.6-g chassis sensoracceleration was used. sensor It was is a kindused. of It Micro is a kind Electro of Micro Mechanical Electro SystemsMechanical (MEMS) Systems acceleration (MEMS) acceleration sensor. The sensorsensor. was The installed sensor was behind installed the cab behind as shown the cab in Figure as shown3. This in positionFigure 3 is. T convenienthis position to is install convenient and less to affectedinstall and by the less pitching affected motion. by the The pitching type of motion. sensor is T suitablehe type for of production sensor is suitable without for high production cost. without high cost.

Table 2. Parameters in road test.

Constants Symbol Meaning Unit Value η efficiency of power train - 0.93 r radius of the tire m 0.52 f roll resistance coefficient - 0.0046

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Table 2. Parameters in road test.

Constants Symbol Meaning Unit Value η efficiency of power train - 0.93 Energies 2018, 11, x FOR PEER REVIEW 6 of 16 r radius of the tire m 0.52 f roll resistance coefficient - 0.0046 g g accelerationacceleration of of gravity gravity m/sm/s2 2 9.89.8 CDCADρAρ airair drag drag coefficient coefficient (combined) (combined) -- 10.65 10.65 I · 2 f If rotationalrotational inertia inertia of of the the flywheel flywheel kgkg·mm 2 1.71.7 2 Iw rotational inertia of all tires kg·m 398.3 Iw rotational inertia of all tires kg·m2 398.3

FigureFigure 3.3. Test vehiclevehicle andand thethe installationinstallation positionposition ofof thethe accelerationacceleration sensor.sensor.

OneOne setset ofof Dewe-43Dewe-43 fromfrom DewetronDewetron companycompany isis usedused toto simultaneouslysimultaneously recordrecord thethe CANCAN busbus signalsignal and and sensor sensor signal. signal. The The speciﬁc specific channel channel of measured of measured variables variables is shown is shown in Table in3 . Table The sample 3. The frequencysample frequency is 20 Hz. is 20 Hz.

TableTable 3.3. ChannelChannel ofof measuremeasure variables.variables.

NONO Variables Variables UnitUnit SourceSource 1 1S Sampleample timetime s s 2 2Engine Engine speed rpmrpm 3 Engine torque Nm 3 Engine torque Nm CAN bus signal 4 Clutch signal 0/1 CAN bus signal 4 Clutch signal 0/1 5 Braking pedal signal 0/1 5 6Braking Vehicle pedal speed signal km/h0/1 6 VeLongitudinalhicle speed km/h 7 m/s2 Acceleration sensor 7 Longitudinalacceleration acceleration m/s2 Acceleration sensor

EachEach independentindependent testtest comprisescomprises a completecomplete driving task, which includes starting the vehicle,vehicle, acceleration,acceleration, cruisingcruising andand stopping.stopping. The minimum time for each test isis 3030 min.min. ThreeThree differentdifferent loadload teststests areare performed.performed. TheThe listlist ofof roadroad teststests isis shownshown inin TableTable4 .4.

TableTable 4.4. RoadRoad testtest databasedatabase withwith differentdifferent loads.loads.

DatabaseDatabase MassMass (t) (t) Number Number of of tests Tests Notes Notes Tractor with full load trailer 48.0 16 - Tractor with full load trailer 48.0 16 - TractorTractor with with empty empty trailer trailer 23.223.2 1717 Winter Winter test test TractorTractor without without trailer trailer 9.59.5 1515 - -

4. Mass Estimation Result

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4. Mass Estimation Result

4.1. Single Test The time series of road tests are analyzed in this section. Three different loads are illustrated. Each set contains vehicle speed (v) system input (gf + asen), output (Fet), estimated vehicle mass (m) and system error (Fse). The yellow backgrounds indicate those valid sample times. Figure4 is the result of a tractor with a full-load trailer. The whole mass estimation algorithm starts when the engine is ignited. The oscillation of measured data at the beginning is obvious, which may be caused by gear shifts. When the speed reaches approximately 20 km/h, the RLS starts to run. Energies 2018, 11, x FOR PEER REVIEW 7 of 16 The response speed of both m and Fse is notably fast. m gradually approaches the real mass, while Fse oscillates one or two times. m and Fse become stable simultaneously. The algorithm runs for 240 s. Figure 5 represents one set result of the tractor with an empty trailer. Fet significantly decreases. The The estimated vehicle mass is 48.9 t, and the error is 0.9 t. The estimated system error is −1115 N. vehicle mass has significant effects on the engine demand torque. The outliers of Fet at the beginning are Figure5 represents one set result of the tractor with an empty trailer. F signiﬁcantly decreases. obvious. The RLS starts until the vehicle speed is 60 km/h. The algorithm runset for 433 s. The estimated The vehicle mass has signiﬁcant effects on the engine demand torque. The outliers of F at the vehicle mass is 23.9 t, and the error is 0.7 t. The estimated system error is 1045 N. et beginning are obvious. The RLS starts until the vehicle speed is 60 km/h. The algorithm runs for 433 s. The estimated vehicle mass is 23.9 t, and the error is 0.7 t. The estimated system error is 1045 N.

Figure 4. One set of the full-load test. Figure 4. One set of the full-load test.

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Figure 5. One set of the empty trailer test. Figure 5. One set of the empty trailer test. Figure6 is one set of results of the tractor without a trailer. This test contains many deceleration sections. TheFigure valid 6 data is one principle set of excludesresults of thethe inﬂuencetractor without of the a deceleration. trailer. This test The contains discontinuity many caused deceleration sections. The valid data principle excludes the influence of the deceleration. The by dataEnergiesdiscontinuity selection 2018, 11, doesx causedFOR PEER not by REVIEW cause data selection any stability does not problem cause any of thestability RLS. problem The algorithm of the RLS. runs9 ofT 16he for 241 s. The estimatedalgorithm vehicle runs for mass 241 s is. T 9.1he estimated t, and the vehicle error mass is − 0.4is 9.1 t. Thet, and estimated the error is system −0.4 t. The error estimated is −641 N. system error is −641 N.

Figure 6. One set of the no-trailer test. Figure 6. One set of the no-trailer test.

4.2. Statistical Analysis of Road Tests All estimation results of the three loads are shown in Figure 7. Each independent test contains the estimated mass, mass error, system error and duration time from the algorithm set up until the termination condition is satisfied. The error between the estimated and real masses is used to analyze the accuracy and stability of the proposed model. The statistical analysis of the mass error is shown in Table 5. The absolute error is used to avoid positive and negative errors canceling each other out.

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4.2. Statistical Analysis of Road Tests All estimation results of the three loads are shown in Figure7. Each independent test contains the estimated mass, mass error, system error and duration time from the algorithm set up until the termination condition is satisﬁed. The error between the estimated and real masses is used to analyze the accuracyEnergies 2018 and, 11 stability, x FOR PEER of theREVIEW proposed model. The statistical analysis of the mass error is shown10 of in16 Table5. The absolute error is used to avoid positive and negative errors canceling each other out.

Full load trailer

Empty trailer Without trailer

(Number)

Figure 7. Results of the online estimation. Figure 7. Results of the online estimation. For the estimated vehicle mass, in terms of stability, 98% of the estimated mass error can be For the estimated vehicle mass, in terms of stability, 98% of the estimated mass error can be controlled within 5 t. The maximum absolute value is 4.6 t. In terms of accuracy, the average absolute controlled within 5 t. The maximum absolute value is 4.6 t. In terms of accuracy, the average value is 1.6 t. The system error has a wide range, from approximately −2000 N to 2000 N, and its value absolute value is 1.6 t. The system error has a wide range, from approximately −2000 N to 2000 N, at any specific test is random. Most tests can be finished within 500 s. Full-load tests can finish the and its value at any specific test is random. Most tests can be finished within 500 s. Full-load tests estimation in a shorter time because the heavy load reduces the vehicle power performance. The mild can finish the estimation in a shorter time because the heavy load reduces the vehicle power accelerationperformance. improves The m theild percentage acceleration of improv the valides the sample percentage time. of the valid sample time. For theFor full the load,full load, its average its average absolute absolute value value is the is bestthe best (1.9 (1.9 t), which t), which is only is only 4% 4% ofits of realits real load. load. Its stabilityIts stability is good, is with good, all with estimated all estimated errors errors less than less5 than t. 5 t. The resultsThe results of the of emptythe empty trailer trailer are are not not notably notably satisfactory. satisfactory. Its Its maximum maximum absolute absolute errorerror is 7.3 t t,, whichwhich is greater is greater than than that ofthat the of other the other two loads.two loads. There There is an is approximately an approximately 94.1% 94.1% possibility possibility that that the errorthe can error be constrained can be constrained to 5 t. The to 5 average t. The average absolute absolute error iserror 2.1 t,is which2.1 t, which is 9.1% is 9.1% of the of real the mass.real mass.

TableTable 5. Statistical 5. Statistical analysis analysis of theof the mass mass error. error.

Average Average absolute value MaximumMaximumPercentage PercentagePercentage Percentage Average Average Absolute Value Tests condition Absolute within 3 t within 5 t Value Percentage in Percentage in Tests Condition Absolute within 3 t within 5 t Percentage in Value (t) Percentage in Value (N) (%) (%) Value (t)(t) Real Load (%) Value (t) Real Load (%) Value (N) (%) (%) Real Load (%) Real Load (%) Full-load trailer 4.6 81.3 100.0 0.1 0.2 1.9 4.0 Full-loadempty trailer trailer 4.67.3 81.382.4 100.094.1 0.11.5 0.26.5 1.92.1 4.09.1 emptywithout trailer trailer 7.31.9 82.4100.0 94.1100.0 1.50.3 6.53.2 2.10.8 9.18.5 without trailerTotal 1.94.6 100.087.9 100.098.0 0.30.6 3.23.3 0.81.6 8.57.2 Total 4.6 87.9 98.0 0.6 3.3 1.6 7.2 The tractor without a trailer has the best stability. The error in all tests is limited to 3 t, with its Theaverage tractor absolute without error a trailerof only has 0.8 thet. Since best stability.its real load The is error only in9.8 all t, teststhe percentage is limited of to 3the t, withaverage its averageabsolute absolute error error remains of only higher 0.8 thant. Since that its of real the loadfull load. is only The 9.8 percentage t, the percentage of the average of the absoluteaverage absoluteerror error of the remains overall highertest is approximately than that of the 7.2%. full load. The percentage of the average absolute error of the overall test is approximately 7.2%. 5. Analysis of System Error in Vehicle Longitudinal Dynamics The online estimation results show that the system error considerably varies. Its value is random in each independent test and cannot be ignored. In this section, the formation of the system error will be illustrated in both qualitative and quantitative aspects. Offline fitting and experiments without considering the system error are used for demonstration.

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5. Analysis of System Error in Vehicle Longitudinal Dynamics The online estimation results show that the system error considerably varies. Its value is random in each independent test and cannot be ignored. In this section, the formation of the system error will be illustrated in both qualitative and quantitative aspects. Ofﬂine ﬁtting and experiments without considering the system error are used for demonstration.

5.1. Source of System Error The foundation of longitudinal dynamics is steady-state force equilibrium. This approach is always used for ofﬂine research or analyses such as vehicle performance optimization and vehicle design. Under online driving conditions, the numerical precision of this theoretical force equilibrium is reduced by many situations such as the measured data accuracy, road and weather environment, and vehicle moving status. The meaning of the system error is to self-adapt these disturbances with one parameter. The main source of system error is the following aspect.

5.1.1. Accuracy Limitation of Engine Torque from CAN Bus Signal For traditional internal combustion engines, it is difﬁcult to acquire their output engine torque in online situations. Torque sensors can be used in the laboratory but are not suitable for vehicles on the market. Online torque can be estimated by models such as mean value engine models [21,25]. The characteristic of an engine may change over time; thus, the accuracy of engine torque is difﬁcult to estimate. During online road tests, it is impractical to measure the engine torque of the test vehicle with high accuracy. This problem can be solved for EVs. The tractive force of EVs can be more easily and accurately measured so that the mass estimation with higher accuracy and stability can be obtained for EVs [7].

5.1.2. Constant Parameters Driving conditions vary considerably, but the vehicle parameters are measured under certain circumstances. It is impossible to make all parameters self-adapt. The mass estimation with one set of vehicle parameters inevitably contains system error. Most vehicle parameters vary with environmental conditions. The tire radius changes with tire pressure. The roll resistance coefficients change with vehicle speed and road surfaces. For the HDV, the trailer size and the dimensions of transported goods will change the front area and drag coefficient. The total wheel inertia and drag coefficient varies by over 200%.

5.1.3. Lateral Motion and Tire Slip Longitudinal dynamics is based on longitudinal movement. Lateral motion and tire slip are ignored. Since the test vehicle cannot offer the steering wheel angle and lateral acceleration through the CAN bus signal, it is difﬁcult to precisely identify lateral motion. The longitudinal force equilibrium is disturbed while turning. Figure8 summarizes the sources of system error. There are too many aspects that may generate system error. To improve the accuracy of mass estimation, the system error without a linear relationship with vehicle mass should be separated from the tractive force. The estimation of system error is a system error self-adapting mechanism, which improves the accuracy and stability of mass estimation. In addition, for the test HDV, the main source of system error is the engine torque. The percentage of roll resistance is much smaller than other resistances. The inertia from the road slope and acceleration can be measured from sensors, which is highly accurate. The air drag becomes serious when cruising at high speed. The estimation is mostly performed at the beginning of the acceleration; thus, the average speed is not notably high. Based on the results of the test vehicle, the system error of longitudinal dynamics mostly originates from the engine torque. Energies 2019, 12, 52 11 of 15 Energies 2018, 11, x FOR PEER REVIEW 12 of 16 Energies 2018, 11, x FOR PEER REVIEW 12 of 16

Figure 8. Source of system error. FFigureigure 8 8.. SSourceource of of system system error error..

5.2.5.2. Data Data Fitting Fitting in in Offline Offline Situation Situation 5.2. Data Fitting in Offline Situation InIn addition, addition, for for the the test test HDV, HDV, the main the source main source of system of error system is the error engine is the torque. engine The torque. percentage The In addition, for the test HDV, the main source of system error is the engine torque. The ofpercentage roll resistance of roll is much resistance smaller is thanmuch other smaller resistances. than other The resistances. inertia from The the roadinertia slope from and the acceleration road slope percentage of roll resistance is much smaller than other resistances. The inertia from the road slope canand be acceleration measured from can sensors, be measu whichred isfrom highly sensors, accurate. which The is air highly drag becomes accurate. serious The air when drag cruising becomes at and acceleration can be measured from sensors, which is highly accurate. The air drag becomes highserious speed. when The cruising estimation at high is mostly speed. performed The estimation at the beginning is mostly of performed the acceleration; at the thus,beginning the average of the serious when cruising at high speed. The estimation is mostly performed at the beginning of the speedacceleration; is not notably thus, the high. average Based speed on the is not results notably of the high. test Based vehicle, on thethe systemresults of error the oftest longitudinal vehicle, the acceleration; thus, the average speed is not notably high. Based on the results of the test vehicle, the dynamicssystem error mostly of longitudinal originates from dynamics the engine mostly torque. originates from the engine torque. system error of longitudinal dynamics mostly originates from the engine torque. TheThe online online estimation estimation can can be be achieved achieved using using the the recursive recursive least least squares squares algorithm, algorithm, and and it it is is The online estimation can be achieved using the recursive least squares algorithm, and it is inconvenientinconvenient to to observe observe the the distribution distribution of of all all measured measured data. data. Only Only the the timetime series series of of estimated estimated inconvenient to observe the distribution of all measured data. Only the time series of estimated parametersparameters can can be be obtained. obtained. Offline Offline data data fitting fitting is used is used in this in section, this section, which illustrateswhich illustrates how the how system the parameters can be obtained. Offline data fitting is used in this section, which illustrates how the errorsystem reduces error thereduces accuracy the accuracy of the mass of the estimation. mass estimation. system error reduces the accuracy of the mass estimation. IfIf we we ignore ignore the the system system error, error, (17) (17 can) can be be rewritten rewritten as, as, If we ignore the system error, (17) can be rewritten as,

Fet  mgf  asen  (20) F  mFetgf=ma(g f+ asen ) (20) (20) et sen Consider this equation as the contrast model of mass estimation, which is the common form ConsiderConsider this this equation equation as asthe the contrast contrast model model of of mass estimation, whichwhich isis the the common common form form used used by other scholars; the offline least squares algorithm is used, where φ’ = (gf+asen) and Z’ = (Fet) usedby by other other scholars; scholars; the the ofﬂine offline least least squares squares algorithm algorithm is used, is used, where whereϕ’ = ( gfφ’+ =a (gf+a) andsen) andZ’ = Z’(F =) ( areFet) the are the system input and output, respectively, and θ’ = (m)T is the parametersen to identify. et aresystem the system input input and and output, output respectively,, respectively, and θand’ = (θ’m) =T (ism the)T is parameter the parameter to identify. to identify. Based on the principle of valid data, all valid data of each independent test were selected for the BasedBased on onthe theprinciple principle of valid of valid data, data, all allvalid valid data data of each of each independent independent test test were were selected selected for forthe the offline estimation. Two kinds of estimation models, (17) and (20), were used. Two typical cases of offlineofﬂine estimation. estimation. Two Two kinds kinds of estimation of estimation models, models, (17)(17) and and (20), (20), were were used. used. Two Two typical typical cases cases of of full-load situations are shown in Figures 9 and 10. The blue dots are valid data. Both real mass and two fullfull-load-load situations situations are shown are shown in Figures in Figures 9 and9 10and. The 10. blue The dots blue are dots valid are validdata. Both data. real Both mass real and mass two and estimated masses are drawn as lines. The slope of each line represents different mass values. estimatedtwo estimated masses are masses drawn are as drawn lines. The as lines. slope Theof each slope line of represents each line representsdifferent mass different values. mass values.

FigureFigure 9. 9.Total Total valid valid input input and and output output of of the the ofﬂine offline estimation estimation in in one one independent independent test test (full (full load load 1). 1). Figure 9. Total valid input and output of the offline estimation in one independent test (full load 1).

EnergiesEnergies 20192018,, 1211,, 52x FOR PEER REVIEW 1213 ofof 1516

FigureFigure 10. 10Total. Total valid valid input input and and output output of of thethe ofﬂineoffline estimation in in one one independent independent test test (full (full load load 2). 2).

FigureFigure9 9 shows shows that that there there is is only only a a slight slight difference difference between between the the two two models models when when the the system system errorerror is small. Its Its estimated estimated system system error error is θ is2 =θ 2428= 428N, and N, andthe estimated the estimated mass massis θ1 = is θ’=θ1 46= θt.’= Figure46 t. Figure10 shows 10 shows that if that the if system the system error error cannot cannot be be ignored, ignored, the the accuracy accuracy is is reduced reduced with with only one one estimatedestimated parameter. parameter. The The value value of ofthe the system system error error is − is1462 −1462 N; thus, N; thus, the absolute the absolute value valueis large. is When large. consideringWhen considering the system the error, system the error, estimated the massestimated is 49 t. mass The mass is 49 valuet. The is mass 43 t without value isconsidering 43 t without the systemconsidering error, the which system is far error, from which its real is mass.far from its real mass. RethinkingRethinking this this question question by by data data fitting, fitting, there there are twoare fittingtwo fitting parameters parameters if we considerif we consider the system the error:system one error: linear one coefficient linear coefficient and one constantand one value.constant The value. position The of position the system of the error system in the error coordinate in the spacecoordinate is unconstrained. space is unconstrained. With only one With fitting only parameter, one fitting the result parameter, must cross the the result original must point. cross The the residualoriginal errorpoint. of The the residual two parameters error of isthe clearly two smallerparameters than is that clearly with smaller only one than parameter. that with only one parameter.Since the system error can be estimated from the model, it is helpful to quantitatively analyze the systemSince the error system of longitudinal error can dynamics.be estimated The from statistical the model, analysis it is of helpful the system to quantitatively error is illustrated analyze in Tablethe system6. The error percentage of longitudinal of system dynamics. error in each The independent statistical analysis test is calculated of the system as (21). error is illustrated in Table 6. The percentage of system error in each independent test is calculated as (21). Fse εi = × 100% (21) Fse Fet εi  100% (21)

Fet where εi is the percentage of system error in test i. Because Fet of each test is a variable, we calculate its mean value. Fet is the mean equivalent tractive force of test; ε is the percentage of system error of where εi is the percentage of system error in test i. Because Fet ofi each test is a variable, we calculate each test. its mean value. Fet is the mean equivalent tractive force of test; εi is the percentage of system error of each test. Table 6. Statistical analysis of the system error.

Tests Standard Average of Average Average of Absolute Average (N) Condition Table 6Deviation. Statistical analysisAbsolute of the (N) systemPercentage error. (%) Percentage (%) full loadTests trailer Average−798.8 Standard 869.7 Average of 919.8 Average− 9.9Average of Absolute 10.5 condition (N) Deviation Absolute (N) Percentage (%) Percentage (%) empty trailer 240.3 1158.5 872.5 8 16.8 full load −798.8 869.7 919.8 −9.9 10.5 trailer Theempty value trailer of system240.3 error is1158.5 random; its (absolute)872.5 average8 and standard deviation16.8 are large. For the mean system error, the full load is negative, whereas the empty trailer is positive. The absolute The value of system error is random; its (absolute) average and standard deviation are large. average percentages of system error are 10.5% and 16.8% for two loads, which indicates that the For the mean system error, the full load is negative, whereas the empty trailer is positive. The measured variables and parameters have unneglectable error. The system error has no linear absolute average percentages of system error are 10.5% and 16.8% for two loads, which indicates relationship system input or output. The system error will inevitably introduce error when this that the measured variables and parameters have unneglectable error. The system error has no constant value is ﬁtted into the linear coefﬁcient, which represents the vehicle mass. linear relationship system input or output. The system error will inevitably introduce error when this constant value is fitted into the linear coefficient, which represents the vehicle mass.

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5.3. Comparison of the Consideration of System Error Although the measured data are not sufﬁciently qualiﬁed, it is convenient to achieve an acceptable mass result by modifying the model structure. (17) and (20) were used to estimated vehicle mass. The results are shown in Table7.

Table 7. Statistical analyses of the estimated mass of the two models

Average Maximum Average Absolute Error Estimated Tests Condition Error (t) Absolute Error Absolute Error Percentage in Real Load Mass (t) (t) (t) (%) Full-load trailer θ’ = [m]T 44.8 −3.2 4.5 14.6 9.4 T empty trailer θ = [m Fe] 48.1 0.1 1.9 4.6 4 without trailer θ’ = [m]T 26.9 3.7 5.3 19.9 22.8 T full-load trailer θ = [m Fe] 24.7 1.5 2.1 7.3 9.1 θ’ = [m]T 8 -1.5 1.5 3.5 15.8 empty trailer T θ = [m Fe] 9.8 0.3 0.8 1.9 8.4 θ’ = [m]T - -0.3 3.8 12.7 16 total T θ= [m Fe] - 0.6 1.6 4.6 7.2

Without system error, only 57.1% of the estimated mass error can be constrained in 3 t. This number can be improved to 87.9% with system error. Without system error, the total average absolute error percentage in real load is 16%. When the system error is considered, this number can be reduced to 8.8%. Above all, the mass estimation with system error consideration in vehicle longitudinal dynamics performs much better than that without system error.

6. Conclusions Vehicle longitudinal dynamics contains too many constants and variables, and it is difﬁcult to adapt various driving situations. Normal state estimators consider system error as white noise, but many sources of error, such as the accuracy of measured parameters, environment and vehicle motion state, make the system error become colored noise. This paper considers system error as a parameter to be estimated. The RLS with two unknown parameters was used to identify both vehicle mass and system error. The source of system error was analyzed in both qualitative and quantitative aspects. The road tests indicate that the percentage of mass error is 16%, and, if the system error is considered, the percentage of mass error is 7.2%. When the system error is considered, the precision of mass estimation improves by 8.8%. The proposed model can offer online mass estimation for intelligent vehicle, especially for heavy-duty vehicle (HDV). Vehicle mass is an important parameter for advanced control systems such as automated gear shift strategies and economic cruise control. Future research may focus on modeling system error with other reasonable methods. Here, we consider the system error as a constant value that is a balance between accuracy and efﬁciency. If the mechanism of system error can be expressed by a more reasonable form without increasing calculations, progress can be made in mass estimation.

Author Contributions: N.L. and S.S. conceived the main concept of the control structure and developed the entire control system. N.L. carried out the research and analyzed the numerical data with guidance from C.Z. N.L. and S.S. collaborated to prepare the manuscript. Funding: This research was funded by Natural Science Foundation of China (51805202). Conﬂicts of Interest: The authors declare no conﬂict of interest. Energies 2019, 12, 52 14 of 15

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