A Method of Optimal Traction Control for Farm... Osinenko 2015

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 3Available online at www.sciencedirect.com ScienceDirect journal homepage: www.elsevier.com/locate/issn/15375110 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. Improvement of traction The main factors. ht is the traction efficiency and k is the net traction ratio. The main possibilities of balancing traction efficiency and productivity include drive train slip control. The curves of the net traction ratio are shown without bias at zero for simplicity. online.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. i. m s2 Tyre section width. Some characteristic curves for different soil types are illustrated in Fig.axle : 21 (1) The term DlzFz is substituted by rdFrr. Longitudinal dynamics are characterised by several parameters: the horizontal force coefficient m.e. Solid lines e stubble. 1 e Forces and torques acting on a wheel in longitudinal motion. ! v w is the wheel travelling velocity.2.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. It can be seen that. rd is the dynamic rolling radius which is the distance between the wheel's centre and bottom points. kg Drive torque. dynamic vertical load adjustment. m s1 m¼ Fh . . N Wheel moment of inertia around lateral axis. 2005). dotted lines e muddy soil. m Horizontal force. The traction efficiency is defined as follows: ht ¼ The equations of motion are written as follows: mw v_w ¼ Fh  Frr. 1. The wheel slip is defined as follows: s ¼1 jvj .e . the vertical load and the drive train slip. m Slip Vehicle travelling velocity. mw az ¼ Fz  mw g  Fz.e  Fx. They are computed with the following formulas: Fig. which affect the traction efficiency of farm tractors. kg Tyre dynamic rolling radius. Details of zero-slip conditions have been described by Schreiber and Kutzbach (2007). rd juw j s ¼ 1 þ rd juw j . only the drive train slip is adjusted during the field operation. Jw is the wheel moment of inertia around the lateral axis. uw is the wheel revolution speed. az is the wheel vertical acceleration. In most cases.i Frr. maxima of ht(s) as well as maximum achievable traction effort. dashed lines e wet loam. the traction parameters k. properties of tyres or tracks. the internal and external rolling resistance coefficients ri.axle . mw is the wheel mass.re respectively and the net traction ratio k.i where Frr. 2. automatic tyre pressure Fig. Nm Wheel mass. N Normal force. 2 e Modelled traction characteristics for different soil types (Wu ¨ nsche. characterised by k. are different for different soil types. kg m2 Vehicle mass. Jw u_ w ¼ Md  rd Fh  Dlz Fz .i denotes the internal rolling resistance (due to tyre deformation). r is the rolling resistance coefficient.



if
v
> rd
uw
: (5) It ranges from 1 (locked wheel) to 1 (spinning on the spot). Fz (2) ri ¼ Frr. include the tyre pressure. m s1 Wheel travelling velocity. jvj



if
v
 rd
uw
. k ð1  sÞ: kþr (6) Usually. rad s1 Rolling resistance coefficient Wheel vertical acceleration.r and the traction efficiency ht are considered as functions of slip. in general. 3 introduce details of the suggested algorithms including the net traction ration characteristic curve estimation and the optimisation procedure. The relation between CI. Possible future improvements of the suggested methodology are mentioned. Sections 3. Considerable research on tyre empirical models and traction prediction has been conducted at the US Army Engineer Waterways Experiment Station. Okabe. Hrazdera. which is able to adapt to changing soil conditions. The horizontal force coefficient is estimated as a function of slip and mobility number. which are abstract. tyre parameters and wheel load is summarised in a so-called wheel numeric. & Yagi. which is equal to the ratio of the tyre deflection to the section height.r play a crucial role. & Livdahl. These factors can be easily obtained by measurement or estimation for basic soil types. and the cone-index (CI to characterise the soil strength) were introduced by Freitag (1965).2 and 3. 2012). 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. Brixius (1987). Li. Among these parameters. The advantage of this model is that the parameters in mathematical equations for the net traction ratio and rolling resistance coefficient. all the approaches with a fixed set-point are suboptimal and might lead to unreasonably high fuel consumption or. It is obtained with a cone penetrometer in a field test. This ratio and its square. The advantages of traction prediction have also been utilised by some researchers in the form of computer programs. 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. 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. therefore. 2001. are related to certain factors which have physical meaning. For example. & Jha. 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. Maclaurin. the horizontal force coefficient as a function of slip can be predicted. the estimation of the traction parameters k. Wellenkotter and Li (2013) used a set of speed sensors for estimation of the wheel torque from the . 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. One of the recent advances in the development of tyre mobility models was made by Hegazy and Sandu (2013). Based on this parameter. otherwise. a parameter establishing the relation of wheel load. to develop an algorithm to find optimal slip set-points under changing soil conditions during field operation. Materials and methods For a traction control algorithm. Tewari. Brixius (1987) developed a more advanced approach to traction prediction for bias-ply pneumatic tyres using curve fitting to field test measurements. 2003. Section 3. static wheel loads. Nishi. The corresponding relationships were established by analysing the characteristic curves obtained in experiments.5 discusses the possibilities of experimental verification. Pandey. Due to changes in soil conditions. tyre section width and diameter. transmission energy efficiency and some other parameters as well as CI. 1975). Pranav. 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. There have also been several modifications of the wheel numeric and mobility numbers (see. A new mobility number was proposed based on analysis of existing formulas as well as on experimental data.4.22 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 control. 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. 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. in most cases only the drive train slip is the subject of control and this can be performed online. The simulation results and general discussion on algorithm tuning are presented in Section 3. Lee. 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). a dimensionless number. 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. low productivity. 1990. Low-cost solutions for torque measurement and calculation in conventional mechanical drives are being developed. Samie. Section 3.1 describes the newly suggested strategy of optimal traction control. ballasting and traction prediction. Multiple tests have shown great improvement of prediction of the net traction ratio characteristic curve compared to existing approaches including Freitag (1965). The major objective of this paper is. Ishikawa. 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. 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. 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. Such an approach should overcome some disadvantages of the traction prediction methods related to the lack of adaptation to the environment. for example. 2. The problem of optimal slip control has recently been a field of interest for some research. CI plays the most important role. Hebbale. Rowland & Peel. The resulting dimensionless numeric was called a tyre mobility number. This information is used to make recommendations on optimal ballasting of the tractor. As was mentioned above. 2012. Rowland and Peel (1975). Bergene. It may be used in combination with the existing slip control algorithms. was introduced. In characterising the tyre flexibility. refer to Boe. 1.4)T are obtained via the drive torque feedback and can be considered as exogenous input.j Fz.j .j  rd. For a four-wheel tractor. u.4 denote the rear and front wheel inertia moments around the lateral axis respectively. _ ¼ 0171 . Lee. These are arranged into an auxiliary vector ðrd. Q y ¼ x. The estimation problem can be considered in terms of an extended state vector c ¼ ðx. The dynamical components 4 . Lindner. Besides this. For some details and corresponding aspects of traction parameter estimation. (9) must have an observability matrix of rank 22 (Del Vecchio & Murray. the equations of the vehicle dynamics in longitudinal motion in terms of traction parameters can be written as follows: u_ w.1 . There are methods of estimating the rolling radii rd.1. where indices j ¼ 1. while for traction parameter estimation. finding a means of eliminating w from the list of unknowns by computing/measuring them outside of estimation problem (9).…uw. for example.4. for example.1.j   1  Md.2.3 ¼ Jw. rear right. 3. Drive torque feedback is obtained from the motor electrical current and position (refer. in order to be observable. Aumer. Electrical drives are also used in construction machinery. Fz.1. accelerometers or other relatively cheap measurement devices may be additionally used to improve estimation.4. The details are discussed further in this section. Usually.k Fz. az parallel axis theorem. It is straightforward to see that (9) is not observable. but also an additional speed sensor and a gear on the transmission propeller shaft before the differential. it can be computed using the force diagram in Fig.4. and Kao (2012). Using this assumption. 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. The values of Jw. wheel revolution speed and vehicle traveling velocity. for the purposes of this paper.j for j ¼ 1…4 and m are supposed to be known and the drive torques u ¼ (Md. draft force. front left. it requires not only software modifications. calculated using the vehicle parameters.4 ÞT ¼ w. To summarise.4. there are twelve extra unknown (9) where f ðx. Hebbale. it suffices to identify average re for the whole vehicle. The draft force measurement is n X k¼1 23 ! re. to Meyer. …Fz.k  Fd . the wheel loads Fz. Indeed. front right wheel. an improved VTS was suggested by Li. (7) k¼1 typically used in the three point hitch control and is performed by. …rd.re)T ¼ t. The rear wheel load can be. for example. Using D'Alembert's principle for the sum of torques around D0 in Fig.1 . …ri.k  Fd  re mg: (8) mv_ ¼ k¼1 The unknown traction parameter vector is. therefore.4 correspond to the real left.…Md. For this purpose.r ¼     1  Fg þ maz l þ lr  Fz.4 .… ri. Fd is the hitch draft force. refer to Pichlmaier (2012). & Herlitzius.… ri. This is appropriate if the tractor operates with a passive implement or if the power take-off is independent of the wheel drive. ri.… m4. 2005). 2010). Since the original system (7) is observable in terms of x. wÞT . induction sensors to measure the stroke displacement. These approaches give only relative values of the torque. Gyroscopes. QÞ consists of the right-hand side of the first four equations of (7) and equation (8). If measurement of the rear wheel load is not available. QÞ. Samie. 3 yields: Fz.1. in particular in some bulldozers where optimal slip control problems are somewhat similar to those of farm tractors. This amounts to.1 . (m1.2 and Jw. torque estimation can be provided by oil pressure sensors. 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. ¼ Jw. while the front wheel load is typically measured in the suspension.1 ¼ Jw. i. farm tractors have front suspension and some have rear suspension as well which allows wheel load to be measured using pressure sensors and. Grote. the equation of longitudinal dynamics of the tractor can be written as follows: n X mk Fz.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. possibly. One of its configurations. All these are obtainable with conventional and/or easily installed inexpensive sensors. magnetoelastic sensors or strain gauges installed in load pins.4 .4 and the internal rolling resistance coefficients ri. 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. 2001) is a promising candidate to substitute conventional mechanical drives with more controllable ones. In general. it is impossible to reconstruct x. The parameter vector is defined as Q ¼ ðt. therefore. However.1.f ld þ l  Fd.j j ¼ 1…4v_ ¼ 1 m n X mk Fz.…rd.k  x_ ¼ f ðx.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 . Jw. However. Electrified wheel drive (Barucki.…Fz. In hydraulic drive trains. thus. & Bocker.…rd. its own re.t and w. It provides options to optimise construction of the vehicle by installing drives directly into wheel rims. QÞT 2ℝ5þ5þ12 : relative position of the driven and undriven wheels. 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 5.1. 2003).j mj þ ri. yaw rate sensors.4 and internal rolling resistance coefficients ri.e. m is the tractor mass. 0l1 denotes an l-length zero vector.3. electrical single wheel drive (Wu¨nsche. the measurement signals required in the estimation process are vertical load on vehicle corners with suspension. absolute values are necessary. ax . every single wheel has its own soil conditions and.v)T. 2007 for details). u. suspension displacement sensors are used. It is usual to approximate Df* from that on a rigid surface. Ryu. Schreiber & Kutzbach. Dakhlallah. (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. Osinenko. which are cheap and easy-to-install. 2008). Moshchuk. 4y is the pitch angle. Petersen. Lyasko (1994). 2013. 3 e Force diagram of a tractor where Jyy is the moment of inertia around the lateral axis.g. For some technical solutions of piston position measurement. and Lechner (2008) developed an identification approach for vehicle vertical dynamics using only standard sensors: accelerometers and relative suspension sensors. m is computed by: m¼ Md  Jw u_ w  ri : rd Fz (13) Supposing that the horizontal force coefficients mk are estimated. However. Rashidi. & Shiriaev. Canudas-de Wit.4 R30.e is the driving force. p. For some further estimation approaches. The wheel rotational dynamical component Jw u_ w can be estimated from the wheel speed measurement using a differentiator filter.24 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 Fig. can be taken into account if corresponding sensors are available. 2003. Ray (1995) suggested an extended Kalman filter for the same purposes. to estimate wheel load together with vehicle vertical acceleration using simple formulas and a differentiator filter. On a loose soil. Charara. Jaberinasab. Shelbourn. Otherwise. Akhtarkavian. and Nazari (2013). Finally. and Mason (2005) and Brown and Richter (2003). ri is assumed as a known parameter. Victorino. for some tyres. cylindrical). the estimation of traction parameters t the wheel and wheel inertia indices can be omitted. 2007.k  re : mg k¼1 k (15) . they may be ignored. Solid lines show measurements. Tyre radial deformation of Michelin AGRIBIB 18. Schreiber & Kutzbach. dashed lines show approximations. Example of application of this formula is shown in Fig. lateral load transfer is considered and all four wheel loads are estimated. & Abdolalizadeh (2013). Bares. On the other hand. The latter is an open subject of investigations which include both empirical models and sensor design. there are many approaches to estimate wheel vertical loads more exactly d using model identification Doumiati. Glaser. In this paper. 4 e An example of application of the empirical formula. Mammar. 4. bt =2r0 2p$105 $pt (12) where pt is the tyre inflation pressure in bar and bt is the tyre section width. As in the previous case. In this paper. re can be computed similarly to (13) as follows: 1 re ¼ mg n X ! mk Fz. Sheikhi. Fx ¼ Fh ¡ Frr. For it. & Sebsadji. refer to Rashidi. e. Fig. 2008. Here.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. refer to Albright. the empirical formula provides estimates which may be appropriate in some applications. Lyasko (1994) also provides methods of estimating the tyre contact area width and length. The tyre dynamic rolling radius is defined as follows: rd ¼ r0  Df  . The internal rolling resistance coefficient ri does not change significantly and mainly depends on the tyre inflation pressure. Otherwise. 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. Guskov et al. the tyre dynamic rolling radius (11) is approximated using Df instead of Df*. model identification approaches can be used (for some of them. for example. The approach is based on Kalman filter. the accuracy might be poor and vertical deflection measurement followed by regression analysis might be necessary. Df depends nonlinearly on the vertical load and the nonlinearity is due to the tyre material and construction. it can be estimated from that on a rigid surface (see. Nardi. gyroscope and/or accelerometer.. Azadeh. 40) uses a linear empirical formula for the tyre deflection on a rigid surface Df as follows: Df ¼ Fz pffiffiffiffiffiffiffiffiffiffiffiffiffiffi . refer to Ono et al. It is seen that at vertical loads recommended for certain inflation pressures. (1988. Generally. and O'dea (2008) used suspension displacement sensors. 2003). the user only defines the strategy via one parameter ranging from zero to one which corresponds to emphasising traction efficiency or performance. estimate the net traction ratio characteristic curve. the corresponding traction efficiency curve and one parameter to balance these two factors. c. The optimality functional is formulated in terms of this curve. Several models for k as a function of slip that can be found in the literature consist of a constant.b0. a set of simplified characteristic curves which roughly classify soil conditions . TCS or some other method. Supposing the optimum is located within the unit interval and given the tolerance of 1/n for some natural number n (that is.e. curve Estimation of the net traction ratio characteristic The traction parameter estimation discussed in Section 2 provides information only about current operation conditions. To summarise. Optimal traction control strategy The suggested methodology of this paper extends a slip control algorithm. 3. The same goal may be achieved by tuning the tolerance of the method in step 7 where the soil condition changes are detected.e. 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. On the other hand. This can be done purely online by gathering a set of estimated r e denotes tuples including the zero-slip tuple ð0. the outcome of the algorithm and the actual optimum will differ at most by 1/n). 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. Therefore.k0 ). The proposed algorithm is able to estimate the curve given one slip-k tuple.c1. 3. d are the unknown parameters. Instead. In this paper.1. some parts of the estimation can be carried out offline and the obtained parameters can be used further online without considerable hardware requirements. perform measurements. Uniqueness of a maximum of the functional is shown. Details will be discussed in the next section. b1 are the unknown parameters. The goal of the approach is to change the set-point adaptively during the field operation. 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. compute the optimum of slip. dimensioning etc. Schreiber and Kutzbach (2007) used the following equation: kðsÞ ¼ a þ ds  b expðcsÞ.3. In general. On the other hand. For this purpose. c1.ht(s) over a wide range of slip.c0. 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. b0. c0. The set of tuples can be approximated using some suitable mathematical model. Increasing a threshold. i. 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.b1)T of a k0 -curve given one user-defined point (s. (17) where a0. appropriate accuracy of approximation can still be achieved and different behaviour in the low. 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.and in the highslip range can be captured. tuples of type (s. 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. realised either by the three point hitch. The resulting characteristic curve k(s) is equal to k0 (s)re. 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. Such a procedure may comprise multidimensional optimisation which might be computationally expensive. The estimated values are utilised in estimation of the net traction ratio characteristic curve. 5. it can provide the necessary convexity property for the optimisation problem to guarantee uniqueness of solution. A similar function was used by Burckhardt and Reimpell (1993) for the horizontal force coefficient. b r e Þ where b the estimated external rolling resistance coefficient. to find q ¼ q(s. (16) where a. the bias is considered separately by introducing the estimated re. allows for more sparse optimum calculations and less computational load.ht) where s denotes current slip.2. beyond which changes in soil conditions are indicated. For example. by incorporating an optimality condition depending on two factors: the traction efficiency and performance. The details of the soil condition change checking are described in Section 3. the worst-case time complexity is O(n). 2. which is now discussed. a linear and an exponential term. b. the suggested optimal traction control includes the following steps: 1. perform slip control with the computed optimal set-point 7. estimate the traction parameters. Results and discussion 3. it is reasonable to obtain information of the characteristic curves k(s). 6. 4. The optimisation in step 5 is one-dimensional and has polynomial time complexity which indicates that the algorithm is efficient (Cobham. obtain machine parameters (wheel radius. The set of 15 parameters is built-in and not used as an input.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. in order to define the optimal traction control set-point. With such a formula. Such an algorithm can be roughly classified as “expensive” when considering computational complexity compared to a “cheap” algorithm. The idea of the “cheap” algorithm is to provide parameters q ¼ (a0.k0 ). i.) and operation strategy (efficiency or productivity). characteristic curves have a bias at zero depending on the external rolling resistance coefficient. 1965). Schreiber and Kutzbach (2007) substituted b  a with re.k) and (s. check soil condition change. On the other hand. c1. Renius. b b1j Þ. n ¼ 25 curves were built (see Fig.b0.



where



2 denotes Euclidean norm. q 2j ¼ c 0j . b0j . It was observed that appropriate accuracy could be achieved using quadratic polynomial model:   ai. Nevertheless.0. The solutions are denoted by b qj ¼ ða b b b b b b b b b b q 1j ¼ a 0j . b0.1. 5 e Initial set of k′-curves.b0 . ai.1 k0 þ ai. additional curves are constructed between the original set. 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.12. 0.07% respectively. 5). a global solution is not crucial at this stage. 0. only a local solution is possible. bi. Such classification has been used by several authors (see.94.04.0 þ bi. « A  0 2 k7 i ¼ 1…5: (22) Further.0.26 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 from “bad” to “good” was assumed. c0. the model (17) is fitted to the given curves. . therefore. seven typical curves out of a range from stubble to muddy soil given by € hne (1964. each parameter was fitted as a function of the net traction ratio k0 at s1 ¼ 50%.2 k2 . b1  0. These values are indexed for each of seven curves in the following manner: k01 . q 4j ¼ c 1j . The bias at zero is removed at this stage and introduced after the approximation process. q 5j ¼ b 1j .c1. Second. b c 0j . Kutzbach. 6 e Parameters q depending on k′ at s ¼ 50%. c1. b1).k0 ) is received.ai. k ¼ 1…n.b1 . Approximation NRMSE of parameters a0.ai.1. 1985). i ¼ 1:::5. In the current set-up. usage of quadratic polynomial models.c1 . At the first step.sÞ N (20) 0 was below 0. This procedure is somewhat analogous to forming a lookup table of curves and serves for computational load reduction. bi. The parameters q are now approximated as functions of curve index k:k0 k(s). Satisfactory accuracy can be achieved by changing initial conditions and running the optimisation algorithm repeatedly. In this case.c0.1%. j ¼ 1…7 and first index denotes the number of the parameter. So Such a set of curves approximately describes the behaviour of the net traction ratio in a wide range of soil conditions. q 3j ¼ b 0j .0 þ ai. The results for parameters q depending on k0 at 50% slip for seven curves are shown in Fig. ai. p.1 k þ bi.2 k02 . Here. Fig. b0. (21) where ai. c1 . the following matrix equations are solved: T  q i7 . First.0. Dotted lines show quadratic approximation. i ¼ 1…5. b N ¼ 50.b1 was 0.2)T is the polynomial coefficient vector. parameters q of model eq:kappa-model were fitted to given k0 -curves numerically using LevenbergeMarquardt algorithm (Marquardt. Vi pi ¼ b q i1 …b i ¼ 1…5. qi ðkÞ ¼ bi. Fig. 45) were assumed. c0. Using these approximations. where pi ¼ (ai. 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. b c 1j . c0 .1. 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 . Further. minimise kk  ða0  c0 expðb0 sÞ  (18) subject to a0 . 1982. As in (21).2 are the subparameters.c0 . b0 . 1. for example. Number of data points of each k -curve was b0j . (23) j¼1 This problem is non-convex since (a0  c0 exp(b0s)  c1 exp(b1s)) is a non-convex function for arbitrary (a0. For all given curves.…k07 . Approximation NRMSE of parameters a0. 6. bi.5 and 0. second denotes the number of the curve. Parameter values are indicated by circles. 5. 1963): 0 2 c1 expðb1 sÞÞk2 .e. The initial curves are shown in Fig. i.2 are the subparameters. two neighbouring curves are found. (24) provided appropriate accuracy. The estimated curve is obtained via interpolation. the normalised root-meansquare error (NRMSE) of fitting. For a discrete set of N points of a given k0 -curve. When the input tuple (s. b1) is not equal to (0. q. in fact. q. 2. c0. kðsÞ þ r (31) where r ¼ re þ ri. q. Therefore. Therefore.07.0). q. ri Þ. the current operating point (s. 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. ht ðs.q. q. The second derivative is:  2 ! ðkðsÞÞ0  2 : ðkðsÞ þ rÞ2 ðkðsÞ þ rÞ3 00 00 ðhðsÞÞ ¼ r ðkðsÞÞ (34) 00 Fig. 7.3. (28) where s ¼ 0…1 is a user-defined parameter which characterises the operation strategy ranging from maximal traction efficiency to maximal productivity. b1  0. kðs. corresponds to the external rolling resistance coefficient re (see Schreiber & Kutzbach.15. q. ri Þ ¼ kðs. 0  s  s1 . kðs. The second derivative is: 00 ðkðsÞÞ ¼ b20 c0 expðb0 sÞ  b21 c1 expðb1 sÞ < 0 (30) for any s. c1. Consider optimisation problem (28) together with (26).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. a0. re Þ. 7. 3. k(s1. the set-point will not be updated. namely.k) should be set to (s. 7 e Flowchart of the algorithm for finding a relative curve index and parameters k′-curve parameters. In the latter case. re Þ þ ð1  sÞht ðs. re Þ þ re þ ri (26) (27) The optimisation problem can be scalarised and reformulated in the following way: maximise ðover sÞ skðs. re Þ ¼ a0  c0 expðb0 sÞ  c1 expðb1 sÞ  re . The worst-case time complexity of the algorithm in Fig. at least one of tuplets (c0. q. (27). b1c1  0 and at least one of the exponential terms is strictly positive. 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~. Slip control itself can be performed by means of the three point hitch or drive trains. re Þ ð1  sÞ: kðs. re . Further explanation is given in the next section. Consider now function: hðsÞ ¼ kðsÞ . 0. ri Þ. The first term in parentheses ðkðsÞÞ =ðkðsÞ þ rÞ2 is strictly negative for any s according to (30). (26) is strictly concave and increasing.05 and 0. 1. then the objective function of (28) has a unique maximum on ðs~. an offset should be performed before estimation. Term 2ðððkðsÞÞ0 Þ2 = ðkðsÞ þ rÞ3 Þ is strictly negative for s > ~s. the algorithm returns flag UPD_CON ¼ 0 and UPD_CON ¼ 1 otherwise. The last step is to determine curve index k for the given point (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. 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.k þ re) as input to the algorithm in Fig.b0) and (c1. subject to 0  s  s1 . Consider function: . h(s) is strictly concave for s > ~s. 0. The k0 -curves estimated with use of this methodology are without bias at zero which. If a relevant curve index is not found. b0. The suggested methodology of this paper implies a slip control algorithm with optimal set-point computation.18% respectively. The objectives are defined as follows: (25) 1.k0 ). Therefore. s1  for ~s defined by a0  c0 expðb0 s~Þ  c1 expðb1 ~sÞ þ ri ¼ 0: Proof. 7 is O(n). 3.re) þ re þ ri > 0. This is performed using the algorithm in Fig. Consider the following optimisation problem: ht ðs. 2007 for detail). re . All numerical procedures were performed in MATLAB©R2010a on a platform with AMD Athlon™Processor/2148 Mhz and 1 Gb RAM. re . Theorem 1. q.14. repeat. then finish. (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. 00 ðhðsÞÞ ð1  sÞ < 0 and since ðhðsÞÞ0 > 0 for s~ < s⩽s1 . and 3. by Golden Section method (Kiefer. If s ¼ 1. 8 e Flowchart of the optimal slip control algorithm. Normally.re) ¼ re. the optimisation problem amounts to finding the maximum of ht(s)-curve. . where Ds is a tuning parameter which can be set to 0.re) ¼ re might not hold. it has a unique maximum.5% for instance.q. Optimisation can be performed continuously during the operation which might require considerable computational resources. 2.28 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. For any s ¼ 0…1. 2.q. function ht(s.ri) ¼ g(s) is strictly concave for s~ < s⩽s1 . it is reasonable to compute set-points for a slip control system at discrete moments of time when changes in soil conditions are noticeable.2). Fig.e. In this case.q. else s0: ¼ s0 þ Ds.q. equality k(0. according to Schreiber and Kutzbach (2007). s~ < 0 and optimisation problem (28) together with its constraint are well-defined. i. On interval ð~s. ∎ Remark 2. Conditions 1. Therefore. If s ¼ 0. 0  s  s1 is within the domain of (27) and there is a unique solution.re.. like tyre unloaded radius. The algorithm starts by acquiring vehicle and tyre parameters. due to inaccuracy of kcurve approximation (see Section 3. Some of the values. 1953). Therefore. the objective of (28) is either equal to k(s) or g(s) or their positive weighted sum. The suggested methodology is summarised in the flowchart in Fig. s1 .g. Further. e. several strategies of optimal traction control are possible.re) > (re þ ri).q. 8 which is a modified variant from Osinenko (2013). This can be done with the following algorithm: 1. However. of the theorem imply that the k-curve is not a constant and k(s. if k(s0.re) þ re þ ri has a zero ~ s < s1 . On the other hand. the lower bound of constraint 0  s  s1 might need to be tightened to some s0. set s0: ¼ 0 %. the net traction ratio is equal to the external rolling resistance coefficient at zero slip: k(0. the solution is the maximum of k(s)-curve. A solution to (28) can be found by some algorithm which would not “fall off” the constraint. 9. .4. Three soil conditions roughly ranging from “bad” to “good” were simulated. Results are shown in Fig. It is seen that values of 0. In most applications. 9 e Traction efficiency.Ptr).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. Lower values would mean more frequent computation of setpoints and make the control system more sensitive to changes in soil conditions and vice versa. section width etc. Beyond these values. Simulation results RigiTrac EWD 120 was used as an example tractor for testing the suggested control scheme.3 roughly correspond to the slip at which the growth of the power losses is moderate for all three soils. traction parameters were estimated as described in Section 2. The results are shown in Fig. or by means of sensors. traction power (thin solid lines) and power losses (dashed lines) as functions of slip for three soil conditions. 10. they correspond to the maxima of ht. lie slightly beyond these values (Wismer & Luth.k).e. can be programmed into ROM of a microcontroller since they are changed rarely. In the online phase. 3. e.g. suitable operating points. Tuning parameter Dk is used to detect noticeable changes in soil conditions. 10 e Computed optimal slip (dashed lines). Therefore.ht) as well as traction power (s.. This step is needed to process a failure in Fig.Pdrive) and power losses (s. 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. drive train (thick solid lines).Ploss) ¼ (s. In the case where a relevant characteristic curve is not found (UPD_CON ¼ 0).2e0. s z 0. Therefore. s should be set slightly above zero. They are denoted as Soil I. traction power (solid black lines) and power losses (solid grey lines) as functions of the user-defined strategy s. i. The parameter STD_T can be adjusted.Pdrive  Ptr). II and III and for each. Information about the tyre air pressure must be provided by the operator. net traction ratio. drive train power (s. Dynamical processes Jw u_ w and mv_ were Fig. the growth of Ploss increases as ht plays a less dominant role. the algorithm does not update the setpoint and slip control is performed with the previously computed reference. 1973). 7 and lasts for STD_T seconds after which the algorithm tries to find a curve again. driver.(s. simulation was performed to obtain curves (s. It can be adjusted by the user. The values at s ¼ 0 have a clear meaning. First. when the vehicle is equipped with other wheels. which provide a reasonable trade-off between the traction efficiency and performance.25 should satisfy a wide range of applications. The vehicle starts on soil I with a conventional set-point of 10%. the soil conditions correspond to Soil I. II and III are shown in Fig. NRMSE for all three cases is below 1. while the power losses are 61% higher.  phase 3 (16e29 s): at 16 s. the traction efficiency is almost the same as for conventional traction control. The results of k-curve approximation for soils I. Slip control was performed by means of a TCS using the algorithm from Sunwoo (2004). The growth of the power losses is about 44%.075. 13 e Online estimation of k-curves (dashed grey lines). soil conditions change from I to II. the traction power grew 19%. a new set-point is computed in about 1 s and then stays fixed. soil conditions change from II to III. Parameter Dk was set to 0. twice as high as with conventional traction control. Working at 10% slip is unreasonable and the tractor simply does not achieve effective drive train power. 13. Due to the vertical load transfer. a new set-point is computed in about 1. the updating of the set-points stops. while the net traction ratio and traction power are 23% and 43% higher respectively. Table 2 contains results for soil II. Optimal traction control was performed with s ¼ 0. the slip controller works with 10% reference which corresponds to the conventional control. front drive torques are shown as grey solid lines. In this case. True values are shown as solid black lines. After the soil conditions stabilse. such excessive growth of power losses might be unreasonable since the increase of traction power is only one half the increase in loss. the supervisor is switched off for initial collecting of information. 12 e Estimated external rolling resistance coefficient and net traction ratio (dashed grey lines). For 5 s. The productivity is 56% higher.4 times the traction power than with 10% slip. For some practical purposes. 11 roughly correspond to the transient phases in estimated re and k.3%. i. Therefore. The following phases are of interest (see Fig. Change of soil conditions was simulated by step functions. which is about as high as growth of the traction power. Results for soil I are shown in Table 1. Optimal control showed 2. 11):  phase 1 (0e5 s): the supervisor is switched off. Fig. It is seen that the transient phases in computation of new set-points in Fig. .  phase 2 (5e16 s): supervisor computes and sets the reference for the slip control system. the value of slip at about 13e14 % can be recognised as optimal for soil I. Ts z where z denotes unit delay and Ts is the simulation step. Estimation of traction parameters is shown in Fig. Fig. The productivity is 8% higher. operating at full drive train power.e. Results of the simulation are shown as solid black lines. Rear drive torques are shown as black solid lines. 12.30 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 .5 s and then stays fixed. With no control.  phase 4 (29e40 s): at 29 s. further increase of Pdrive becomes unprofitable since the traction power grows 42% higher. the working depth and working width were fixed at 75 mm and 5 m respectively. It is seen that with optimal traction control. even the maximum of the traction efficiency is not achieved with conventional traction control. while the power losses were 33% higher. The growth of power losses is less. However. Fig. 11 e Vehicle dynamics under changing soil conditions and optimal traction control.25. the drive torques on rear and front wheels are different in order to keep the desired slip. The traction parameter estimation procedure should be improved with use of methods which process dynamical components more accurately. Table 2 e Comparative table of conventional and optimal traction control.2 3. ha h1 Table 3 e Comparative table of conventional and optimal traction control. estimation of the auxiliary parameters e the tyre deflection.33 51 40. That is.4 47. Soil II. the simulation model can be fed with the measured/computed drive torque.71 0. kW Ploss . s ¼ 13:9% Full Pdrive . it can be indicated by not violating the Dkthreshold (see Fig. % 0. e with use of tyre contact geometric models and spring parameters should be incorporated. but the performance of the reference slip computation may be tested explicitly:  at the first stage.65 70 56 24 3. the feedback of the supervisor should be switched off.2 3.  the ability of the supervisor to detect a soil condition change can be checked by the Dk-indicator. tracks). Soil III. Furthermore. Some preliminary results on the identification of the wheel load torque. Increasing the operating point above 20e23 % slip causes excessive growth of power losses.5 0.4 Ptr . should be also investigated in more detail.23 41 32. The estimated traction parameters can be compared to the measurement results including the net traction ratio characteristic curve. which is related to the threshold Dk. The sensitivity of the proposed algorithm. kW Ploss . 8). travelling velocity. s ¼ 10% Opt. This can be performed along with torque and force measurements as described above.54 0.3 3. 2013). ha h1 33 12. For the supervisor.3 72. s ¼ 16:5% k ht . Type of control Convention. % Ptr . It is assumed to investigate and introduce more models of k-curves with different shapes corresponding to different propelling units (for example. while the growth of Ptr is only twice.38 72. kW Productivity. This option allows for a direct verification of the traction parameter estimation.67 Ptr . this method could verify the parameter estimation indirectly as it does not take into account changing soil conditions.27 56 28. The estimated traction parameters may be compared to those obtained from the wheel force measurement. They would verify the estimation process indirectly. the computed set-point should approximately match the maximum of the averaged traction efficiency curve obtained from the force measurement. internal rolling resistance etc. k ht . were obtained in an electrical single wheel drive test stand (Osinenko. relatively “flat”.07 0. Soil I. On the other hand. Type of control Convention. kW Productivity. To summarise. Type of control Convention. than combinations of low-pass filters and differentiators.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. s ¼ 10% Opt. Further possible stages of experiments are related to the optimal slip control.195 53. Therefore. s ¼ 18:6% Full Pdrive . % 0.24 0.2 times that with optimal slip control. 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.1 46 0. more sophisticated imitation of changes in k-curve parameters. wheel speed etc. The latter is typically obtained after repeated traction tests by averaging the results. s ¼ 27% k ht . which plays a crucial role in the traction parameter estimation. s ¼ 47% 3. Better traction parameter estimations should offer more possibilities for implementation and optimisation of pure online algorithms for approximation of characteristic curves. kW Ploss . Using homogeneous areas of the field would improve the accuracy. According to Table 3.81 Results of simulation for soil III are summarised in Table 3. at rates which exceed that for traction power.6 24. thus. kW Productivity.5 3. since this “bad” curve lies below that for soils I and II and is.5.8 47. the tractor can travel with high slip at full drive train power.11 Possibilities of experimental verification Future experiments are possible in which the abilities of optimal slip control can be tested. the algorithm for optimal traction control provides a reasonable trade-off between the traction efficiency and performance for all three soil types. as well as in re. ha h1 6 7 1. The measurement results can be used to test the supervisor offline as well. The simplest . s ¼ 10% Opt. The goal is only to validate the computed reference set-points if they are plausible. s ¼ 20:53% Full Pdrive .155 50 12 12 2. For example for s ¼ 0 and if the threshold Dk was not violated. Conventional traction control is unreasonable since the traction characteristic curve is “bad” and more slip is needed to achieve normal working conditions. and address measurement failures and disturbances (for example.8 39.37 0. adaptive Kalman filters).8 2.  using force and torque sensors mounted on the wheel rim. contact area. Ploss is 3.1 18 3. For this purpose.5 17 14. than step-wise functions used in the current paper should be incorporated. (in German). Computers and Electronics in Agriculture.005. Kutzbach.  different optimal set-points provided by the supervisor may be checked for general types of soil. Journal of Terramechanics. 4(3). M. Del Vecchio. Mammar. Cobham. & Richter. 137e149. N. references Al-Hamed. and K. (1953). Nishi.. 4. 50(5). operation.. M.. & Lechner.. measurement of wheel rotational dynamics. D. M. The values for the “worse” soils should be greater.. M.. 42nd IEEE Conference (Vol. & Murray. WO Patent 2.. North Holland.. (2001).. 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