A method of optimal traction control for farm tractors with feedback of drive torque

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

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Research Paper

A method of optimal traction control for farm tractors with feedback of drive torque Pavel V. Osinenko*,1, Mike Geissler 2, Thomas Herlitzius 3 Chair of Agricultural Systems and Technology (AST), Institute of Processing Machines and Mobile Machinery, € t Dresden (TU Dresden), Dresden, Germany P.O. Box: 01069, Technische Universita

article info

Traction efficiency of farm tractors barely reaches 50% in field operations (Renius et al.,

Article history:

1985). On the other hand, modern trends in agriculture show growth of the global tractor

Received 20 January 2014

markets and at the same time increased demands for greenhouse gas emission reduction

Received in revised form

as well as energy efficiency due to increasing fuel costs. Engine power of farm tractors is

3 September 2014

growing at 1.8 kW per year reaching today about 500 kW for the highest traction class

Accepted 17 September 2014

machines. The problem of effective use of energy has become crucial. Existing slip control

Published online

approaches for farm tractors do not fulfil this requirement due to fixed reference set-point. This paper suggests an optimal control scheme which extends a conventional slip

Keywords:

controller with set-point optimisation based on assessment of soil conditions, namely,

Slip control

wheel-ground parameter estimation. The optimisation considers the traction efficiency

Optimal control

and net traction ratio and adaptively adjusts the set-point under changing soil conditions.

Infinitely variable transmissions

The proposed methodology can be mainly implemented in farm tractors equipped with

Traction efficiency

hydraulic or electrical infinitely variable transmissions (IVT) with use of the drive torque

Traction parameters

feedback. © 2014 IAgrE. Published by Elsevier Ltd. All rights reserved.

1.

Introduction

1.1.

Brief description of traction dynamics

In this section, the main factors contributing to traction efficiency are discussed. First, the wheel dynamics are briefly described. The corresponding force diagram is given in Fig. 1. The soil reaction force Fz acts against the axle load Fz,axle and

the wheel weight. The horizontal soil reaction Fh (or horizontal force) is exerted by the driving torque Md. An opposite force on the wheel, namely, reaction of the vehicle body, is denoted by Fx,axle. The point of application of the soil reaction is shifted by Dlz in direction of motion due to tyre deformation which characterises the internal rolling resistance. Another part of the rolling resistance Frr,e is external, due to soil deformation, and should not be confused with the internal resistance (Schreiber & Kutzbach, 2007).

* Corresponding author. E-mail addresses: [email protected], [email protected] (P.V. Osinenko), [email protected] (M. Geissler), [email protected] (T. Herlitzius). 1 Graduate student. 2 Scientific staff member. 3 Chairman. http://dx.doi.org/10.1016/j.biosystemseng.2014.09.009 1537-5110/© 2014 IAgrE. Published by Elsevier Ltd. All rights reserved.

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

Nomenclature ht k m uw r az bt Fh Fz Jw m Md mw rd s v vw

Traction efficiency Net traction ratio Horizontal force coefficient Wheel revolution speed, rad s1 Rolling resistance coefficient Wheel vertical acceleration, m s2 Tyre section width, m Horizontal force, N Normal force, N Wheel moment of inertia around lateral axis, kg m2 Vehicle mass, kg Drive torque, Nm Wheel mass, kg Tyre dynamic rolling radius, m Slip Vehicle travelling velocity, m s1 Wheel travelling velocity, m s1

m¼

Fh ; Fz

(2)

ri ¼

Frr;i Frr;e ;r ¼ Fz e Fz

(3)

k ¼ m  re ;

(4)

The rolling resistance coefficient is computed as sum of re and ri in (3): r ¼ re þ ri. The wheel slip is defined as follows: s ¼1

jvj ; rd juw j

s ¼ 1 þ

rd juw j ; jvj





if
v
 rd
uw
;



if
v
> rd
uw
:

(5)

It ranges from 1 (locked wheel) to 1 (spinning on the spot). The traction efficiency is defined as follows: ht ¼

The equations of motion are written as follows: mw v_w ¼ Fh  Frr;e  Fx;axle ; Jw u_ w ¼ Md  rd Fh  Dlz Fz ; mw az ¼ Fz  mw g  Fz;axle :

21

(1)

The term DlzFz is substituted by rdFrr,i where Frr,i denotes the internal rolling resistance (due to tyre deformation). Longitudinal dynamics are characterised by several parameters: the horizontal force coefficient m, the internal and external rolling resistance coefficients ri,re respectively and the net traction ratio k. They are computed with the following formulas:

Fig. 1 e Forces and torques acting on a wheel in longitudinal motion. ! v w is the wheel travelling velocity, uw is the wheel revolution speed, mw is the wheel mass, Jw is the wheel moment of inertia around the lateral axis, rd is the dynamic rolling radius which is the distance between the wheel's centre and bottom points, az is the wheel vertical acceleration.

k ð1  sÞ: kþr

(6)

Usually, the traction parameters k,r and the traction efficiency ht are considered as functions of slip. Some characteristic curves for different soil types are illustrated in Fig. 2. The curves of the net traction ratio are shown without bias at zero for simplicity. Details of zero-slip conditions have been described by Schreiber and Kutzbach (2007). It can be seen that, in general, maxima of ht(s) as well as maximum achievable traction effort, characterised by k, are different for different soil types.

1.2.

Improvement of traction

The main factors, which affect the traction efficiency of farm tractors, include the tyre pressure, properties of tyres or tracks, the vertical load and the drive train slip. In most cases, only the drive train slip is adjusted during the field operation, i.e. online. The main possibilities of balancing traction efficiency and productivity include drive train slip control, dynamic vertical load adjustment, automatic tyre pressure

Fig. 2 e Modelled traction characteristics for different soil types (Wu ¨ nsche, 2005). Solid lines e stubble, dashed lines e wet loam, dotted lines e muddy soil. r is the rolling resistance coefficient, ht is the traction efficiency and k is the net traction ratio.

22

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control, ballasting and traction prediction. Dynamic axle load adjustment as well as automatic tyre pressure control remain technically difficult and are not considered in the framework of the present paper. Traction prediction is a technique which is used for optimising the machine configuration including ballasting and wheel parameters based on empirical models relating tyres and soil properties. Considerable research on tyre empirical models and traction prediction has been conducted at the US Army Engineer Waterways Experiment Station. In characterising the tyre flexibility, a dimensionless number, which is equal to the ratio of the tyre deflection to the section height, was introduced. This ratio and its square, a parameter establishing the relation of wheel load, tyre section width and diameter, and the cone-index (CI to characterise the soil strength) were introduced by Freitag (1965). Wismer and Luth (1973) suggested equations with which the tyre section width and diameter and wheel load can be chosen from a set of parameters for high traction efficiency. Among these parameters, CI plays the most important role. It is obtained with a cone penetrometer in a field test. The relation between CI, tyre parameters and wheel load is summarised in a so-called wheel numeric. Based on this parameter, the horizontal force coefficient as a function of slip can be predicted. Brixius (1987) developed a more advanced approach to traction prediction for bias-ply pneumatic tyres using curve fitting to field test measurements. This approach is based on a combination of the wheel numeric with tyre geometric parameters e deflection to section height ratio and width to diameter ratio. The resulting dimensionless numeric was called a tyre mobility number. The horizontal force coefficient is estimated as a function of slip and mobility number. The advantages of traction prediction have also been utilised by some researchers in the form of computer programs. Al-Hamed and Al-Janobi (2001) developed a tractor performance program in Visual Cþþ with which the user can choose a suitable configuration of a tractor by prediction of performance parameters given the machine and tyre dimensions, static wheel loads, transmission energy efficiency and some other parameters as well as CI. There have also been several modifications of the wheel numeric and mobility numbers (see, for example, Maclaurin, 1990; Rowland & Peel, 1975). One of the recent advances in the development of tyre mobility models was made by Hegazy and Sandu (2013). A new mobility number was proposed based on analysis of existing formulas as well as on experimental data. This parameter is defined via the wheel numeric and the square root of the difference between tyre section height and tyre deflection divided by tyre diameter. Multiple tests have shown great improvement of prediction of the net traction ratio characteristic curve compared to existing approaches including Freitag (1965); Rowland and Peel (1975); Brixius (1987). Schreiber and Kutzbach (2008) suggested an empirical model of the net traction ratio and rolling resistance coefficient as functions of slip with parameters computed from a set of six factors taken as inputs e one for the tyre and five for the soil. These factors can be easily obtained by measurement or estimation for basic soil types. The advantage of this model is that the parameters in mathematical equations for the net traction ratio and rolling resistance coefficient, which are

abstract, are related to certain factors which have physical meaning. The corresponding relationships were established by analysing the characteristic curves obtained in experiments. As was mentioned above, in most cases only the drive train slip is the subject of control and this can be performed online. Renius (1985) made a recommendation for slip to be observed and kept at about 10% for 4 wheel drive and 15% for two wheel drive vehicles. Slip control can be implemented as an additional function of the three point hitch control or by means of a traction control system (TCS) (for some recent technical solutions and methods, refer to Boe, Bergene, & Livdahl, 2001; Hrazdera, 2003; Ishikawa, Nishi, Okabe, & Yagi, 2012; Pranav, Tewari, Pandey, & Jha, 2012). The problem of optimal slip control has recently been a field of interest for some research. Pichlmaier (2012) addresses methods of determining drive torque in a Fendt power-split transmission and suggests calculating the actual net traction ratio and rolling resistance coefficient from these data together with draft force and wheel load measurements. This information is used to make recommendations on optimal ballasting of the tractor. Due to changes in soil conditions, all the approaches with a fixed set-point are suboptimal and might lead to unreasonably high fuel consumption or, otherwise, low productivity. The major objective of this paper is, therefore, to develop an algorithm to find optimal slip set-points under changing soil conditions during field operation. Such an approach should overcome some disadvantages of the traction prediction methods related to the lack of adaptation to the environment. It may be used in combination with the existing slip control algorithms. The paper is organised as follows: Section 2 discusses methods and techniques of obtaining the information on the current soil conditions via the traction parameters k and r. Section 3.1 describes the newly suggested strategy of optimal traction control. Sections 3.2 and 3.3 introduce details of the suggested algorithms including the net traction ration characteristic curve estimation and the optimisation procedure. The simulation results and general discussion on algorithm tuning are presented in Section 3.4. Possible future improvements of the suggested methodology are mentioned. Section 3.5 discusses the possibilities of experimental verification.

2.

Materials and methods

For a traction control algorithm, which is able to adapt to changing soil conditions, the estimation of the traction parameters k,r play a crucial role. The most important information used in this estimation process is the drive torque feedback which can be obtained for hydraulic or electrical drive trains without installation of expensive torque sensors. Low-cost solutions for torque measurement and calculation in conventional mechanical drives are being developed. For example, Li, Hebbale, Lee, Samie, and Kao (2011) suggest usage of existing speed sensors for estimation of torque variations on the transmission output shaft in the set-up called “virtual torque sensor” (VTS). Wellenkotter and Li (2013) used a set of speed sensors for estimation of the wheel torque from the

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

variables: the dynamic rolling radii rd,1,…rd,4, the wheel loads Fz,1,…Fz,4 and the internal rolling resistance coefficients ri,1,… ri,4. These are arranged into an auxiliary vector ðrd;1 ; …rd;4 ; Fz;1 ; …Fz;4 ; ri;1 ; …ri;4 ÞT ¼ w. The parameter vector is defined as Q ¼ ðt; wÞT . The estimation problem can be considered in terms of an extended state vector c ¼ ðx; QÞT 2ℝ5þ5þ12 :

relative position of the driven and undriven wheels. These approaches give only relative values of the torque, while for traction parameter estimation, absolute values are necessary. For this purpose, an improved VTS was suggested by Li, Samie, Hebbale, Lee, and Kao (2012). However, it requires not only software modifications, but also an additional speed sensor and a gear on the transmission propeller shaft before the differential. In hydraulic drive trains, torque estimation can be provided by oil pressure sensors. For some details and corresponding aspects of traction parameter estimation, refer to Pichlmaier (2012). The methods of drive torque estimation in mechanical or hydraulic drive trains usually refer to calculation/measurement of the torque at the transmission output shaft. This is appropriate if the tractor operates with a passive implement or if the power take-off is independent of the wheel drive. Electrified wheel drive (Barucki, 2001) is a promising candidate to substitute conventional mechanical drives with more controllable ones. One of its configurations, electrical single wheel drive (Wu¨nsche, 2005), was implemented in RigiTrac EWD 120 with 80 kW drive train power developed by the AST of TU Dresden together with EAAT GmbH Chemnitz (Geißler, Aumer, Lindner, & Herlitzius, 2010). It provides options to optimise construction of the vehicle by installing drives directly into wheel rims. Electrical drives are also used in construction machinery, in particular in some bulldozers where optimal slip control problems are somewhat similar to those of farm tractors. Drive torque feedback is obtained from the motor electrical current and position (refer, for example, to Meyer, Grote, & Bocker, 2007 for details). For a four-wheel tractor, the equations of the vehicle dynamics in longitudinal motion in terms of traction parameters can be written as follows:

u_ w;j

  1  Md;j  rd;j Fz;j mj þ ri;j ; ¼ Jw;j

j ¼ 1…4v_ ¼

1 m

n X

mk Fz;k 

x_ ¼ f ðx; u; QÞ; _ ¼ 0171 ; Q y ¼ x;

where indices j ¼ 1,2,3,4 correspond to the real left, rear right, front left, front right wheel, Jw,1 ¼ Jw,2 and Jw,3 ¼ Jw,4 denote the rear and front wheel inertia moments around the lateral axis respectively, m is the tractor mass, Fd is the hitch draft force. The values of Jw,j for j ¼ 1…4 and m are supposed to be known and the drive torques u ¼ (Md,1,…Md,4)T are obtained via the drive torque feedback and can be considered as exogenous input. The wheel revolution speed and the vehicle travelling velocity are measured which means that the output vector and the state vector are equal: x ¼ y ¼ (uw,1,…uw,4,v)T. In general, every single wheel has its own soil conditions and, therefore, its own re. However, for the purposes of this paper, it suffices to identify average re for the whole vehicle. Using this assumption, the equation of longitudinal dynamics of the tractor can be written as follows: n X mk Fz;k  Fd  re mg: (8) mv_ ¼ k¼1

The unknown traction parameter vector is, therefore, (m1,… m4,re)T ¼ t. Besides this, there are twelve extra unknown

(9)

where f ðx; u; QÞ consists of the right-hand side of the first four equations of (7) and equation (8), 0l1 denotes an l-length zero vector. It is straightforward to see that (9) is not observable, i.e. it is impossible to reconstruct x,t and w. Indeed, in order to be observable, (9) must have an observability matrix of rank 22 (Del Vecchio & Murray, 2003). Since the original system (7) is observable in terms of x, it is easily seen that an extended system of type (9) is observable if the number of parameters Q equals the number of states x which is 5. This amounts to, for example, finding a means of eliminating w from the list of unknowns by computing/measuring them outside of estimation problem (9). There are methods of estimating the rolling radii rd,1,…rd,4 and internal rolling resistance coefficients ri,1,… ri,4, while the front wheel load is typically measured in the suspension. The rear wheel load can be, thus, calculated using the vehicle parameters. The details are discussed further in this section. To summarise, the measurement signals required in the estimation process are vertical load on vehicle corners with suspension, draft force, wheel revolution speed and vehicle traveling velocity. All these are obtainable with conventional and/or easily installed inexpensive sensors. The draft force measurement is

n X

k¼1

23

! re;k Fz;k  Fd ;

(7)

k¼1

typically used in the three point hitch control and is performed by, for example, magnetoelastic sensors or strain gauges installed in load pins. Usually, farm tractors have front suspension and some have rear suspension as well which allows wheel load to be measured using pressure sensors and, possibly, induction sensors to measure the stroke displacement. Gyroscopes, yaw rate sensors, accelerometers or other relatively cheap measurement devices may be additionally used to improve estimation. If measurement of the rear wheel load is not available, it can be computed using the force diagram in Fig. 3. Using D'Alembert's principle for the sum of torques around D0 in Fig. 3 yields: Fz;r ¼

    1  Fg þ maz l þ lr  Fz;f ld þ l  Fd;x hd þ max hCG ld     2 €y : þ Jyy þ m ðld þ lr Þ þ h2CG 4

(10)

The moment of inertia around D0 is computed using the € y ; ax ; az parallel axis theorem. The dynamical components 4

24

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Fig. 3 e Force diagram of a tractor where Jyy is the moment of inertia around the lateral axis, Fx ¼ Fh ¡ Frr,e is the driving force, 4y is the pitch angle. can be taken into account if corresponding sensors are available, e.g. gyroscope and/or accelerometer. Otherwise, they may be ignored. For some technical solutions of piston position measurement, refer to Albright, Bares, Shelbourn, and Mason (2005) and Brown and Richter (2003). On the other hand, there are many approaches to estimate wheel vertical loads more exactly d using model identification Doumiati, Victorino, Charara, and Lechner (2008) developed an identification approach for vehicle vertical dynamics using only standard sensors: accelerometers and relative suspension sensors. Here, lateral load transfer is considered and all four wheel loads are estimated. The approach is based on Kalman filter. Moshchuk, Nardi, Ryu, and O'dea (2008) used suspension displacement sensors, which are cheap and easy-to-install, to estimate wheel load together with vehicle vertical acceleration using simple formulas and a differentiator filter. Ray (1995) suggested an extended Kalman filter for the same purposes. As in the previous case, suspension displacement sensors are used. The tyre dynamic rolling radius is defined as follows: rd ¼ r0  Df  ;

(11) *

where r0 is the tyre unloaded radius and Df is the tyre deflection on a loose soil which can be estimated using some geometric tyre-ground contact model (for example, cylindrical). It is usual to approximate Df* from that on a rigid surface. The latter is an open subject of investigations which include both empirical models and sensor design. Generally, Df depends nonlinearly on the vertical load and the nonlinearity is due to the tyre material and construction. Schmid (1995) developed iterative numerical algorithms to derive the tyre deflection Df* on a loose soil and contact surface length from Df and tyre spring constant using a cylindrical model. In this paper, the tyre dynamic rolling radius (11) is approximated using Df instead of Df*. Guskov et al. (1988, p. 40) uses a linear empirical formula for the tyre deflection on a rigid surface Df as follows: Df ¼

Fz pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; bt =2r0

2p$105 $pt

(12)

where pt is the tyre inflation pressure in bar and bt is the tyre section width. Example of application of this formula is shown in Fig. 4.

Fig. 4 e An example of application of the empirical formula. Tyre radial deformation of Michelin AGRIBIB 18.4 R30. Solid lines show measurements, dashed lines show approximations.

It is seen that at vertical loads recommended for certain inflation pressures, the empirical formula provides estimates which may be appropriate in some applications. However, for some tyres, the accuracy might be poor and vertical deflection measurement followed by regression analysis might be necessary. For some further estimation approaches, refer to Rashidi, Azadeh, Jaberinasab, Akhtarkavian, and Nazari (2013), Rashidi, Sheikhi, & Abdolalizadeh (2013), Lyasko (1994). Lyasko (1994) also provides methods of estimating the tyre contact area width and length. The internal rolling resistance coefficient ri does not change significantly and mainly depends on the tyre inflation pressure. On a loose soil, it can be estimated from that on a rigid surface (see, for example, Schreiber & Kutzbach, 2007; Schreiber & Kutzbach, 2008). In this paper, ri is assumed as a known parameter. For it, the estimation of traction parameters t the wheel and wheel inertia indices can be omitted. The wheel rotational dynamical component Jw u_ w can be estimated from the wheel speed measurement using a differentiator filter. Otherwise, model identification approaches can be used (for some of them, refer to Ono et al., 2003; Dakhlallah, Glaser, Mammar, & Sebsadji, 2008; Osinenko, 2013; Canudas-de Wit, Petersen, & Shiriaev, 2003). Finally, m is computed by: m¼

Md  Jw u_ w  ri : rd Fz

(13)

Supposing that the horizontal force coefficients mk are estimated, re can be computed similarly to (13) as follows: 1 re ¼ mg

n X

! mk Fz;k  Fd

k¼1

v_  : g

(14)

The net traction ratio k in terms of the whole vehicle can be computed by: k¼

n 1 X m Fz;k  re : mg k¼1 k

(15)

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

3.

Results and discussion

3.1.

Optimal traction control strategy

The suggested methodology of this paper extends a slip control algorithm, realised either by the three point hitch, TCS or some other method, by incorporating an optimality condition depending on two factors: the traction efficiency and performance. The modification is made in the form of a supervisor which estimates the traction parameters online using the drive torque feedback and measurement signals from sensors which are often available. The estimated values are utilised in estimation of the net traction ratio characteristic curve. The optimality functional is formulated in terms of this curve, the corresponding traction efficiency curve and one parameter to balance these two factors. Uniqueness of a maximum of the functional is shown. The computed optimal drive train slip set-point is transmitted to the slip control method. The latter together with the supervisor constitute the suggested optimal traction control. This strategy is not used to predict the optimal operating conditions or to define the machine and/or tyre dimensioning as it is performed in traction prediction. The goal of the approach is to change the set-point adaptively during the field operation. To summarise, the suggested optimal traction control includes the following steps: 1. obtain machine parameters (wheel radius, dimensioning etc.) and operation strategy (efficiency or productivity), 2. perform measurements, 3. estimate the traction parameters, 4. estimate the net traction ratio characteristic curve, 5. compute the optimum of slip, 6. perform slip control with the computed optimal set-point 7. check soil condition change. The optimisation in step 5 is one-dimensional and has polynomial time complexity which indicates that the algorithm is efficient (Cobham, 1965). Supposing the optimum is located within the unit interval and given the tolerance of 1/n for some natural number n (that is, the outcome of the algorithm and the actual optimum will differ at most by 1/n), the worst-case time complexity is O(n). The estimation process in step 4 cannot be unambiguously performed with classical model identification approaches from the current operating point and generally requires some curve fitting algorithm from a set of estimated points. Such a procedure may comprise multidimensional optimisation which might be computationally expensive. On the other hand, some parts of the estimation can be carried out offline and the obtained parameters can be used further online without considerable hardware requirements. A variant of a such method is currently used in the suggested optimal traction control and comprises a set of 15 parameters obtained offline from typical net traction ratio characteristic curves. The proposed algorithm is able to estimate the curve given one slip-k tuple. The set of 15 parameters is built-in and not used as an input. This is different from several traction prediction algorithms where the user defines some empiric or measured wheel and soil parameters with which the characteristic curve can be

25

obtained. Instead, the user only defines the strategy via one parameter ranging from zero to one which corresponds to emphasising traction efficiency or performance. The only purpose of the parameter set used in step 4 is to reduce computational load and to make the algorithm appropriate for conventional microcontrollers. The same goal may be achieved by tuning the tolerance of the method in step 7 where the soil condition changes are detected. Increasing a threshold, beyond which changes in soil conditions are indicated, allows for more sparse optimum calculations and less computational load. The details of the soil condition change checking are described in Section 3.3.

3.2. curve

Estimation of the net traction ratio characteristic

The traction parameter estimation discussed in Section 2 provides information only about current operation conditions, i.e. tuples of type (s,k) and (s,ht) where s denotes current slip. On the other hand, in order to define the optimal traction control set-point, it is reasonable to obtain information of the characteristic curves k(s),ht(s) over a wide range of slip. This can be done purely online by gathering a set of estimated r e denotes tuples including the zero-slip tuple ð0; b r e Þ where b the estimated external rolling resistance coefficient. The set of tuples can be approximated using some suitable mathematical model. Such an algorithm can be roughly classified as “expensive” when considering computational complexity compared to a “cheap” algorithm, which is now discussed. Several models for k as a function of slip that can be found in the literature consist of a constant, a linear and an exponential term. For example, Schreiber and Kutzbach (2007) used the following equation: kðsÞ ¼ a þ ds  b expðcsÞ;

(16)

where a, b, c, d are the unknown parameters. In general, characteristic curves have a bias at zero depending on the external rolling resistance coefficient. Therefore, Schreiber and Kutzbach (2007) substituted b  a with re. A similar function was used by Burckhardt and Reimpell (1993) for the horizontal force coefficient. In this paper, the bias is considered separately by introducing the estimated re. The formula (16) is modified by excluding the bias and introducing the second exponential term instead of the linear term in the following way: k0 ðsÞ ¼ a0  c0 expðb0 sÞ  c1 expðb1 sÞ;

(17)

where a0, c0, b0, c1, b1 are the unknown parameters. With such a formula, appropriate accuracy of approximation can still be achieved and different behaviour in the low- and in the highslip range can be captured. On the other hand, it can provide the necessary convexity property for the optimisation problem to guarantee uniqueness of solution. Details will be discussed in the next section. The resulting characteristic curve k(s) is equal to k0 (s)re. The idea of the “cheap” algorithm is to provide parameters q ¼ (a0,c0,b0,c1,b1)T of a k0 -curve given one user-defined point (s,k0 ), i.e. to find q ¼ q(s,k0 ). For this purpose, a set of simplified characteristic curves which roughly classify soil conditions

26

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from “bad” to “good” was assumed. Such classification has been used by several authors (see, for example, Kutzbach, 1982; Renius, 1985). In the current set-up, seven typical curves out of a range from stubble to muddy soil given by € hne (1964, p. 45) were assumed. So Such a set of curves approximately describes the behaviour of the net traction ratio in a wide range of soil conditions. First, the model (17) is fitted to the given curves. Second, additional curves are constructed between the original set. When the input tuple (s,k0 ) is received, two neighbouring curves are found. The estimated curve is obtained via interpolation. This procedure is somewhat analogous to forming a lookup table of curves and serves for computational load reduction. The initial curves are shown in Fig. 5. The bias at zero is removed at this stage and introduced after the approximation process. The curves are given for the range of slip between zero and 50% which is supposed to be enough for practical use of traction control. At the first step, parameters q of model eq:kappa-model were fitted to given k0 -curves numerically using LevenbergeMarquardt algorithm (Marquardt, 1963): 0

2 c1 expðb1 sÞÞk2 ;

minimise kk  ða0  c0 expðb0 sÞ  (18) subject to a0 ; c0 ; b0 ; c1 ; b1  0;



where



2 denotes Euclidean norm. For a discrete set of N points of a given k0 -curve, the objective amounts to: N  X     2 k0j  a0  c0 exp b0 sj  c1 exp b1 sj :

(19)

NRMSE ¼

1     maxj k0j  minj k0j

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 uP  u N t j¼1 k0j  k0 ða0 ;c0 ;b0 ;c1 ;b1 ;sÞ N (20) 0

was below 0.1%. Number of data points of each k -curve was b0j ; b b1j Þ; i.e. b0j ; b c 0j ; b c 1j ; b N ¼ 50. The solutions are denoted by b qj ¼ ða b b b b b b b b b b q 1j ¼ a 0j ; q 2j ¼ c 0j ; q 3j ¼ b 0j ; q 4j ¼ c 1j ; q 5j ¼ b 1j ; j ¼ 1…7 and first index denotes the number of the parameter, second denotes the number of the curve. Further, each parameter was fitted as a function of the net traction ratio k0 at s1 ¼ 50%. These values are indexed for each of seven curves in the following manner: k01 ;…k07 . It was observed that appropriate accuracy could be achieved using quadratic polynomial model: 

 ai;0 þ ai;1 k0 þ ai;2 k02 ;

i ¼ 1:::5;

(21)

where ai,0, ai,1, ai,2 are the subparameters. In this case, fitting was done using polynomial approximation by means of Vandermonde matrix for each parameter: 0

1 k01 B 0 B Vi ¼ B 1 k 2 @« « 1 k07

 0 2 1 k  10 2 C k2 C C; « A  0 2 k7

i ¼ 1…5:

(22)

Further, the following matrix equations are solved: T  q i7 ; Vi pi ¼ b q i1 …b

i ¼ 1…5;

(23)

j¼1

This problem is non-convex since (a0  c0 exp(b0s)  c1 exp(b1s)) is a non-convex function for arbitrary (a0, c0, b0, c1, b1), therefore, only a local solution is possible. Nevertheless, a global solution is not crucial at this stage. Satisfactory accuracy can be achieved by changing initial conditions and running the optimisation algorithm repeatedly. For all given curves, the normalised root-meansquare error (NRMSE) of fitting.

where pi ¼ (ai,0,ai,1,ai,2)T is the polynomial coefficient vector. The results for parameters q depending on k0 at 50% slip for seven curves are shown in Fig. 6. Approximation NRMSE of parameters a0,c0,b0,c1,b1 was 0.04, 0.12, 0.94, 1.5 and 0.07% respectively. Using these approximations, n ¼ 25 curves were built (see Fig. 5). The parameters q are now approximated as functions of curve index k:k0 k(s), k ¼ 1…n. As in (21), usage of quadratic polynomial models. qi ðkÞ ¼ bi;0 þ bi;1 k þ bi;2 k2 ;

i ¼ 1…5;

(24)

provided appropriate accuracy. Here, bi,0, bi,1, bi,2 are the subparameters. Approximation NRMSE of parameters a0, c0, b0, c1,

Fig. 5 e Initial set of k′-curves.

Fig. 6 e Parameters q depending on k′ at s ¼ 50%. Dotted lines show quadratic approximation. Parameter values are indicated by circles.

27

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

b1 were 0.07, 0.15, 1.14, 0.05 and 0.18% respectively. The last step is to determine curve index k for the given point (s,k0 ). This is performed using the algorithm in Fig. 7. If a relevant curve index is not found, the algorithm returns flag UPD_CON ¼ 0 and UPD_CON ¼ 1 otherwise. In the latter case, the set-point will not be updated. Further explanation is given in the next section. All numerical procedures were performed in MATLAB©R2010a on a platform with AMD Athlon™Processor/2148 Mhz and 1 Gb RAM. The worst-case time complexity of the algorithm in Fig. 7 is O(n). The k0 -curves estimated with use of this methodology are without bias at zero which, in fact, corresponds to the external rolling resistance coefficient re (see Schreiber & Kutzbach, 2007 for detail). Therefore, an offset should be performed before estimation, namely, the current operating point (s,k) should be set to (s,k þ re) as input to the algorithm in Fig. 7.

3.3.

The suggested methodology of this paper implies a slip control algorithm with optimal set-point computation. Slip control itself can be performed by means of the three point hitch or drive trains. Consider the following optimisation problem: ht ðs; q; re ; ri Þ; kðs; q; re Þ; 0  s  s1 ;

kðs; q; re Þ ¼ a0  c0 expðb0 sÞ  c1 expðb1 sÞ  re ; ht ðs; q; re ; ri Þ ¼

kðs; q; re Þ ð1  sÞ: kðs; q; re Þ þ re þ ri

(26)

(27)

The optimisation problem can be scalarised and reformulated in the following way: maximise ðover sÞ skðs; q; re Þ þ ð1  sÞht ðs; q; re ; ri Þ; subject to 0  s  s1 ;

(28)

where s ¼ 0…1 is a user-defined parameter which characterises the operation strategy ranging from maximal traction efficiency to maximal productivity. Theorem 1. Consider optimisation problem (28) together with (26), (27). Let the following conditions hold:

Optimal traction control algorithm

maximise ðover sÞ maximise ðover sÞ subject to

where s1 ¼ 50 % as in the previous section. The objectives are defined as follows:

(25)

1. a0, c0, b0, c1, b1  0, 2. at least one of tuplets (c0,b0) and (c1,b1) is not equal to (0,0), 3. k(s1,q,re) þ re þ ri > 0, then the objective function of (28) has a unique maximum on ðs~; s1  for ~s defined by a0  c0 expðb0 s~Þ  c1 expðb1 ~sÞ þ ri ¼ 0: Proof. Consider function kðsÞ ¼ a0  c0 expðb0 sÞ  c1 expðb1 sÞ  re : Its derivative ðkðsÞÞ0 ¼ b0 c0 expðb0 sÞ þ b1 c1 expðb1 sÞ

(29)

is strictly positive since b0c0, b1c1  0 and at least one of the exponential terms is strictly positive. The second derivative is: 00

ðkðsÞÞ ¼ b20 c0 expðb0 sÞ  b21 c1 expðb1 sÞ < 0

(30)

for any s. Therefore, (26) is strictly concave and increasing. Consider now function: hðsÞ ¼

kðsÞ ; kðsÞ þ r

(31)

where r ¼ re þ ri. Function k(s) þ r ¼ a0  c0 exp(b0s)  c1 exp(b1s) þ ri is strictly increasing and since lim ðkðsÞ þ rÞ ¼ ∞ and kðs1 Þ þ r > 0

(32)

s/∞

has a unique zero s~. The derivative of h(s) is computed as follows:

0

ðhðsÞÞ ¼

ðkðsÞÞ0 ðkðsÞ þ rÞ  k0 ðsÞkðsÞ 2

ðkðsÞ þ rÞ

¼

rðkðsÞÞ0 ðkðsÞ þ rÞ2

:

(33)

It can be seen that ðhðsÞÞ0 > 0 for any sss~ since ðkðsÞÞ0 > 0. The second derivative is:  2 ! ðkðsÞÞ0  2 : ðkðsÞ þ rÞ2 ðkðsÞ þ rÞ3 00

00

ðhðsÞÞ ¼ r

ðkðsÞÞ

(34) 00

Fig. 7 e Flowchart of the algorithm for finding a relative curve index and parameters k′-curve parameters.

The first term in parentheses ðkðsÞÞ =ðkðsÞ þ rÞ2 is strictly negative for any s according to (30). Term 2ðððkðsÞÞ0 Þ2 = ðkðsÞ þ rÞ3 Þ is strictly negative for s > ~s. Therefore, h(s) is strictly concave for s > ~s. Consider function:

28

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

gðsÞ ¼ hðsÞð1  sÞ;

(35)

Its second derivative reads as: 00

00

0

ðgðsÞÞ ¼ ðhðsÞÞ ð1  sÞ  2ðhðsÞÞ :

(36)

It can be noticed that (1  s) > 0 for s⩽s1 ¼ 0:5. Therefore, 00 ðhðsÞÞ ð1  sÞ < 0 and since ðhðsÞÞ0 > 0 for s~ < s⩽s1 ; function ht(s,q,re,ri) ¼ g(s) is strictly concave for s~ < s⩽s1 . For any s ¼ 0…1, the objective of (28) is either equal to k(s) or g(s) or their positive weighted sum. On interval ð~s; s1 , it has a unique maximum. ∎ Remark 2. Conditions 1., 2. and 3. of the theorem imply that the k-curve is not a constant and k(s,q,re) þ re þ ri has a zero ~ s < s1 . Normally, according to Schreiber and Kutzbach (2007), the net traction ratio is equal to the external rolling resistance coefficient at zero slip: k(0,q,re) ¼ re. In this case, s~ < 0 and optimisation problem (28) together with its constraint are well-defined, i.e. 0  s  s1 is within the domain of (27) and there is a unique solution. However, due to inaccuracy of kcurve approximation (see Section 3.2), equality k(0,q,re) ¼ re might not hold. Therefore, the lower bound of constraint

0  s  s1 might need to be tightened to some s0. This can be done with the following algorithm: 1. set s0: ¼ 0 %, 2. if k(s0,q,re) > (re þ ri), then finish, else s0: ¼ s0 þ Ds, repeat, where Ds is a tuning parameter which can be set to 0.5% for instance. A solution to (28) can be found by some algorithm which would not “fall off” the constraint, e.g. by Golden Section method (Kiefer, 1953). If s ¼ 0, the optimisation problem amounts to finding the maximum of ht(s)-curve. If s ¼ 1, the solution is the maximum of k(s)-curve. Further, several strategies of optimal traction control are possible. Optimisation can be performed continuously during the operation which might require considerable computational resources. On the other hand, it is reasonable to compute set-points for a slip control system at discrete moments of time when changes in soil conditions are noticeable. The suggested methodology is summarised in the flowchart in Fig. 8 which is a modified variant from Osinenko (2013). The algorithm starts by acquiring vehicle and tyre parameters. Some of the values, like tyre unloaded radius,

Fig. 8 e Flowchart of the optimal slip control algorithm.

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

29

Fig. 9 e Traction efficiency, net traction ratio, drive train (thick solid lines), traction power (thin solid lines) and power losses (dashed lines) as functions of slip for three soil conditions.

section width etc., can be programmed into ROM of a microcontroller since they are changed rarely, e.g. when the vehicle is equipped with other wheels. Information about the tyre air pressure must be provided by the operator, i.e. driver, or by means of sensors. In the case where a relevant characteristic curve is not found (UPD_CON ¼ 0), the algorithm does not update the setpoint and slip control is performed with the previously computed reference. This step is needed to process a failure in Fig. 7 and lasts for STD_T seconds after which the algorithm tries to find a curve again. The parameter STD_T can be adjusted. Tuning parameter Dk is used to detect noticeable changes in soil conditions. It can be adjusted by the user. Lower values would mean more frequent computation of setpoints and make the control system more sensitive to changes in soil conditions and vice versa.

3.4.

Simulation results

RigiTrac EWD 120 was used as an example tractor for testing the suggested control scheme. Three soil conditions roughly

ranging from “bad” to “good” were simulated. They are denoted as Soil I, II and III and for each, simulation was performed to obtain curves (s,k),(s,ht) as well as traction power (s,Ptr), drive train power (s,Pdrive) and power losses (s,Ploss) ¼ (s,Pdrive  Ptr). Results are shown in Fig. 9. First, the characteristic k-curves were approximated offline using (18) to investigate the influence of the user-defined strategy s on the traction efficiency and performance. The results are shown in Fig. 10. The values at s ¼ 0 have a clear meaning, they correspond to the maxima of ht. In most applications, suitable operating points, which provide a reasonable trade-off between the traction efficiency and performance, lie slightly beyond these values (Wismer & Luth, 1973). Therefore, s should be set slightly above zero. It is seen that values of 0.2e0.3 roughly correspond to the slip at which the growth of the power losses is moderate for all three soils. Beyond these values, the growth of Ploss increases as ht plays a less dominant role. Therefore, s z 0.25 should satisfy a wide range of applications. In the online phase, traction parameters were estimated as described in Section 2. Dynamical processes Jw u_ w and mv_ were

Fig. 10 e Computed optimal slip (dashed lines), traction power (solid black lines) and power losses (solid grey lines) as functions of the user-defined strategy s.

30

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

estimated by filtering the velocity and wheel speed measurements using a 4th-order Butterworth low-pass filter with cutoff frequency equal to 5 Hz and taking discrete derivatives using the following discrete transfer function: WðzÞ ¼

z1 ; Ts z

where z denotes unit delay and Ts is the simulation step. Change of soil conditions was simulated by step functions. Parameter Dk was set to 0.075, the working depth and working width were fixed at 75 mm and 5 m respectively. Slip control was performed by means of a TCS using the algorithm from Sunwoo (2004). The vehicle starts on soil I with a conventional set-point of 10%. For 5 s, the supervisor is switched off for initial collecting of information. The following phases are of interest (see Fig. 11):  phase 1 (0e5 s): the supervisor is switched off, the slip controller works with 10% reference which corresponds to the conventional control, the soil conditions correspond to Soil I;  phase 2 (5e16 s): supervisor computes and sets the reference for the slip control system;  phase 3 (16e29 s): at 16 s, soil conditions change from I to II, a new set-point is computed in about 1 s and then stays fixed;  phase 4 (29e40 s): at 29 s, soil conditions change from II to III, a new set-point is computed in about 1.5 s and then stays fixed; Due to the vertical load transfer, the drive torques on rear and front wheels are different in order to keep the desired slip. Estimation of traction parameters is shown in Fig. 12. It is seen that the transient phases in computation of new set-points in

Fig. 11 e Vehicle dynamics under changing soil conditions and optimal traction control. Rear drive torques are shown as black solid lines, front drive torques are shown as grey solid lines.

Fig. 12 e Estimated external rolling resistance coefficient and net traction ratio (dashed grey lines). True values are shown as solid black lines.

Fig. 11 roughly correspond to the transient phases in estimated re and k. After the soil conditions stabilse, the updating of the set-points stops. The results of k-curve approximation for soils I, II and III are shown in Fig. 13. NRMSE for all three cases is below 1.3%. Optimal traction control was performed with s ¼ 0.25. Results for soil I are shown in Table 1. It is seen that with optimal traction control, the traction efficiency is almost the same as for conventional traction control, while the net traction ratio and traction power are 23% and 43% higher respectively. The productivity is 8% higher. The growth of the power losses is about 44%, which is about as high as growth of the traction power. With no control, operating at full drive train power, the traction power grew 19%, while the power losses were 33% higher. For some practical purposes, such excessive growth of power losses might be unreasonable since the increase of traction power is only one half the increase in loss. Therefore, the value of slip at about 13e14 % can be recognised as optimal for soil I. Table 2 contains results for soil II. In this case, even the maximum of the traction efficiency is not achieved with conventional traction control. Optimal control showed 2.4 times the traction power than with 10% slip. The growth of power losses is less, i.e. twice as high as with conventional traction control. The productivity is 56% higher. Working at 10% slip is unreasonable and the tractor simply does not achieve effective drive train power. However, further increase of Pdrive becomes unprofitable since the traction power grows 42% higher, while the power losses are 61% higher.

Fig. 13 e Online estimation of k-curves (dashed grey lines). Results of the simulation are shown as solid black lines.

31

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

Table 1 e Comparative table of conventional and optimal traction control. Soil I. Type of control Convention. s ¼ 10% Opt. s ¼ 13:9% Full Pdrive , s ¼ 16:5%

k

ht , %

0.3

72.5

0.37 0.4

Ptr , kW

Ploss , kW

Productivity, ha h1

33

12.5

3.38

72.4

47.1

18

3.65

70

56

24

3.81

Results of simulation for soil III are summarised in Table 3. Conventional traction control is unreasonable since the traction characteristic curve is “bad” and more slip is needed to achieve normal working conditions. On the other hand, since this “bad” curve lies below that for soils I and II and is, thus, relatively “flat”, the tractor can travel with high slip at full drive train power. Increasing the operating point above 20e23 % slip causes excessive growth of power losses, at rates which exceed that for traction power. According to Table 3, Ploss is 3.2 times that with optimal slip control, while the growth of Ptr is only twice. To summarise, the algorithm for optimal traction control provides a reasonable trade-off between the traction efficiency and performance for all three soil types. It is assumed to investigate and introduce more models of k-curves with different shapes corresponding to different propelling units (for example, tracks). The traction parameter estimation procedure should be improved with use of methods which process dynamical components more accurately, than combinations of low-pass filters and differentiators, and address measurement failures and disturbances (for example, adaptive Kalman filters). Better traction parameter estimations should offer more possibilities for implementation and optimisation of pure online algorithms for approximation of characteristic curves. Furthermore, estimation of the auxiliary parameters e the tyre deflection, contact area, internal rolling resistance etc. e with use of tyre contact geometric models and spring parameters should be incorporated. The sensitivity of the proposed algorithm, which is related to the threshold Dk, should be also investigated in more detail. For this purpose, more sophisticated imitation of changes in k-curve parameters, as well as in re, than step-wise functions used in the current paper should be incorporated.

Table 2 e Comparative table of conventional and optimal traction control. Soil II. Type of control Convention. s ¼ 10% Opt. s ¼ 18:6% Full Pdrive , s ¼ 27%

k

ht , %

0.155

50

12

12

2.07

0.27

56

28.6

24.3

3.24

0.33

51

40.8

39.2

3.67

Ptr , kW

Ploss , kW

Productivity, ha h1

Table 3 e Comparative table of conventional and optimal traction control. Soil III. Type of control Convention. s ¼ 10% Opt. s ¼ 20:53% Full Pdrive , s ¼ 47%

3.5.

k

ht , %

Ptr , kW

Ploss , kW

Productivity, ha h1

6

7

1.71

0.1

46

0.195

53.5

17

14.8

2.54

0.23

41

32.8

47.2

3.11

Possibilities of experimental verification

Future experiments are possible in which the abilities of optimal slip control can be tested. Following options of experimental verification might be suggested for the parameter estimation process:  performing conventional traction tests with a tractor and an attached braking machine. The estimated traction parameters can be compared to the measurement results including the net traction ratio characteristic curve. The latter is typically obtained after repeated traction tests by averaging the results. Therefore, this method could verify the parameter estimation indirectly as it does not take into account changing soil conditions. Using homogeneous areas of the field would improve the accuracy. For the supervisor, it can be indicated by not violating the Dkthreshold (see Fig. 8);  using force and torque sensors mounted on the wheel rim. This option allows for a direct verification of the traction parameter estimation. The measurement results can be used to test the supervisor offline as well. That is, the simulation model can be fed with the measured/computed drive torque, travelling velocity, wheel speed etc. The estimated traction parameters may be compared to those obtained from the wheel force measurement. Some preliminary results on the identification of the wheel load torque, which plays a crucial role in the traction parameter estimation, were obtained in an electrical single wheel drive test stand (Osinenko, 2013). Further possible stages of experiments are related to the optimal slip control. They would verify the estimation process indirectly, but the performance of the reference slip computation may be tested explicitly:  at the first stage, the feedback of the supervisor should be switched off. The goal is only to validate the computed reference set-points if they are plausible. This can be performed along with torque and force measurements as described above. For example for s ¼ 0 and if the threshold Dk was not violated, the computed set-point should approximately match the maximum of the averaged traction efficiency curve obtained from the force measurement;  the ability of the supervisor to detect a soil condition change can be checked by the Dk-indicator. The simplest

32

b i o s y s t e m s e n g i n e e r i n g 1 2 9 ( 2 0 1 5 ) 2 0 e3 3

variant of such tests would be to drive on tilled/untilled areas of the field;  different optimal set-points provided by the supervisor may be checked for general types of soil, e.g. wet/dry, sandy/clay etc. The values for the “worse” soils should be greater, than for the “better” ones;  if the values computed by the supervisor are plausible in all previous experiments, the optimal slip control can be tested fully online. In this case, such parameters as the fuel consumption per unit area or productivity may be compared. It is expected that even under uncertainties of the tyre contact parameters, internal rolling resistance etc., the proposed control scheme should be able to produce reasonable reference set-points. All suggested experiments may be similarly carried out with a construction machine like a bulldozer. Another important possibility is to verify the optimal slip control in a laboratory for testing tyres on a loose soil. Such a laboratory would contain, for example, a specially built gutter filled with a soil and equipment able to drive the tyre in the gutter.

4.

Conclusions

In this paper, a new strategy for optimal traction control is suggested. The approach is based on traction parameter estimation via drive torque feedback. It is able to estimate whole characteristic curves of the net traction ratio against slip using a set of model parameters. Traction control is based on optimisation which is performed in periods of noticeable change of soil conditions. The suggested methodology can be used for off-road vehicles where the problems of traction efficiency are crucial. Simulation results showed better performance of optimal traction control than existing methods.

Acknowledgement The authors would like to thank A.Gu¨nther, the AST, and K. € benack, the Chair for Control Engineering of TU Dresden, Ro for their valuable suggestions and comments. Special thanks € ll for the major ideas in developing the algorithm of go to H. Do the net traction ratio characteristic curve approximation. The research was conducted in the framework of the agriculture electrification project at the AST of TU Dresden.

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