A Computational Tool for Three-Point Hitch Geometry Optimisation Based on Weight-Transfer Minimisation

Abstract

The weight-transfer effect, consisting of the change in dynamic load distribution between the front and the rear tractor axles, is one of the most impairing phenomena for the performance, comfort, and safety of agricultural operations. Excessive weight transfer from the front to the rear tractor axle can occur during operation or manoeuvring of implements connected to the tractor through the three-point hitch (TPH). In this respect, an optimal design of the TPH can ensure better dynamic load distribution and ultimately improve operational performance, comfort, and safety. In this study, a computational tool (the Optimiser) for the determination of a TPH geometry which minimises the weight-transfer effect is developed. The Optimiser is based on a constrained minimisation algorithm. The objective function to be minimized is related to the tractor front-to-rear axle load transfer during a simulated reference manoeuvre performed with a reference implement on a reference soil. Simulations are based on a dynamic model of the tractor-TPH-implement aggregate. The geometry determined by the Optimiser complies with the ISO-730 Standard functional requirements and other design requirements. The interaction between the soil and the implement during the simulated reference manoeuvre was successfully validated against experimental data. The simulation results show that the adopted reference manoeuvre is effective in triggering the weight-transfer effect, with the front axle load exhibiting a peak-to-peak value of $27.1 \mathrm{kN}$ during the manoeuvre. A benchmark test was conducted starting from the geometry of a commercially available TPH; the test showed that the Optimiser, after 36 iterations, was able to find an optimised TPH geometry which allows to reduce the weight-transfer effect by 14.9%.

1. Introduction

Nowadays, the topics of operational performance, efficiency, comfort, and safety have become of utmost importance for the designers of agricultural machinery and for the market as well [1,2]. One of the most impairing phenomena in terms of comfort, safety, and performance during agricultural operations is the weight-transfer effect [3], consisting of the dynamic change in load distribution between the front and rear wheels of the tractor, occurring during tillage operations or implement manoeuvring.

The loads resulting from the interaction between the implement and the soil are transmitted to the tractor through a hitching system. Especially in the case of implements connected to the tractor through the three-point hitch (TPH), an optimal design can ensure better dynamic load distribution and, consequently, can offer advantages in terms of operational performance, comfort, and safety; improve tractor lateral stability and levelling of the implement; and help prevent fatigue damage [4,5]. Indeed, implement extraction from the soil during headland turns is the primary cause of fatigue damage during field operation since it induces a significant load change in tractor axles [6].

Considering the range of variation of the TPH geometrical dimensions and the requirements prescribed by the Standards [7,8], designers are faced with the task of defining the TPH geometry in order to meet several kinematic, functional, structural, working, and economical requirements [9,10]. With the purpose of determining the optimal TPH geometry, Ambike and Schmiedeler [11] proposed the application of Geometric Constraint Programming for the determination of kinematic TPH configurations that satisfy the constraints imposed by the ASAE Standard S217.12 [7]. The design tool developed in their study also allows the user to graphically visualize the effects of the different parameters on the resulting design geometry through a CAD package software. Kumar et al. [12] developed a computer program that locates the virtual hitch point of the tractor with respect to the depth of operation. The program ultimately optimizes the TPH geometry by making the virtual hitch point lie on the line of pull. Prasanna Kumar [13] developed a Newton–Raphson-based algorithm capable of generating the trajectory of the lower, upper, and virtual hitch points. The algorithm also determines the geometric performance parameters of the TPH and its mechanical advantage, expressed in terms of the ratio of the output force to the input force of the mechanism. The algorithm was run on a group of 165 different TPHs collected from Nebraska tractor test reports and, albeit no objective functions were formally defined, optimal designs were selected based on the kinematic performance parameters and the maximum mechanical advantage of the TPH. In a similar fashion, Dhruw et al. [14] developed a design tool in which the mechanical advantage of the mechanism acted as the performance parameter of the TPH. The mechanical advantage of the TPH was not the only parameter of mechanical performance assumed for the optimization of the TPH geometry. Molari et al. [15] proposed a design methodology based on a constrained optimization technique where the objective function to be maximised was related to the TPH lifting performance, as defined by the OECD Code 2 [16]. Constraints to the optimisation were provided by the ISO-730 [8] Standard requirements.

Albeit having provided valuable results in terms of TPH geometry optimization, the previous studies have not accounted for the influence of the TPH geometry on the transmission of the implement loads to the tractor. Notably, Bauer at al. [17] developed a model to investigate the effect of the TPH setup on the load distribution between the tractor rear wheels during in-furrow ploughing; from their study, it was possible to conclude that the load difference between the in-furrow and the on-land wheels can be significantly reduced by adjusting the length of the TPH upper link. Their work highlighted the central role of the TPH in transmitting the loads acting on the implement to the tractor; however, their focus was on the TPH setup and not on optimising the TPH design in order to improve the overall dynamic behaviour of the tractor.

The aim of the present work is to fill this gap by developing a design tool for the determination of an optimised TPH geometry which minimises the weight-transfer effect, thus improving the dynamic performance of the tractor-implement system. The geometry optimisation tool, referred to as the Optimiser, is based on a constrained mimisation algorithm. The objective function to be minimized by the algorithm is related to the tractor front-to-rear axle load transfer during a simulated reference manoeuvre performed with a reference implement on a reference soil. Simulations are based on a dynamic model of the aggregate system constituted by the tractor, the TPH, and the implement. Constraints to the minimisation problem arise from the ISO-730 Standard functional requirements and other design requirements.

2. Materials and Methods

A model of an agricultural tractor equipped with a front axle suspension and bearing an implement mounted on the rear TPH (Figure 1) was developed to evaluate the influence of the TPH geometry on the weight-transfer effect. The model was kept as simple as possible in order to reduce computation time; nevertheless, all the features responsible for describing the weight transfer effect were included. The external loads due to the interaction between the implement and the soil were also modelled. The following assumptions were made:

• The tractor is modelled as a system constituted by the tractor chassis and the front axle;
• The tractor chassis, the rear axle, each TPH link, and the implement behave as rigid bodies;
• Friction at the TPH joints can be neglected;
• Front and rear wheels are massless, each modelled as a linear spring and a viscous damper set in parallel between the vehicle and the ground;
• The front axle suspension system is modelled as a linear spring in parallel with a viscous damper;
• The ground surface is assumed to be horizontal;
• The tractor performs the reference manoeuvre while moving forward on a straight line at constant speed.

In order to reduce load transfer from the front to the rear tractor axle during implement operation or manoeuvring, a computational tool (the Optimiser) was developed, which optimises the TPH geometry while satisfying all the prescriptions of the ISO-730 Standard for TPH design. Scheme 1 depicts the structure of the Optimiser, whose kernel is constituted by a constrained minimisation algorithm. The inputs of the Optimiser are the parameters of the tractor equipped with the TPH subject to optimisation, the TPH category, and its initial geometry. The optimisation process is iterative, with the algorithm evaluating a trial TPH geometry (referred to as the current TPH geometry) at each step of the iteration. The evaluation is carried out through the tractor–TPH–implement model and consists of simulating a prescribed manoeuvre (referred to as the reference manoeuvre) performed with a reference implement while the tractor is running over a reference soil. The simulation allows to determine, for the current TPH geometry, the load on the tractor front axle as a function of time and, ultimately, to extract a measure of the weight transfer during the reference manoeuvre, which acts as the objective function associated with the current TPH geometry. The iterative optimisation process ends when a TPH geometry that minimises the objective function is found. Such optimised TPH geometry is the output of the Optimiser. The determination of the objective function follows the algorithm described in detail in Section 2.1, Section 2.2, Section 2.3, Section 2.4 and Section 2.5 and summarised in Scheme 2.

2.1. TPH Kinematic Analysis

Performing the kinematic analysis of the TPH is essential for simulating the reference manoeuvre and locating the position of the links where forces are exchanged between the implement and the TPH, and between the TPH and the tractor.

The kinematic analysis is performed in the median vertical–longitudinal plane (X–Y plane, Figure 2); the center of the rear axle is assumed as the origin of the reference frame. The mechanism is considered symmetrical about the X–Y plane and is divided into three subsystems: the triangle whose vertices are the points $P_{hcp},P_{hcd},P_{cp}$; the quadrilateral whose vertices are the points $P_{lap},P_{lam},P_{cd},P_{cp}$; and the quadrilateral whose vertices are the points $P_{lap},P_{lad},P_{uad},P_{uap}$. As proposed by Molari et al. [15], the analysis is based on the solution of a system of six nonlinear equations representing the condition of closure of the polygons which compose the three subsystems:

$$\{\begin{array}{l}{X}_{P_{hcp}}+l_{hc}\cos {\phi }_{hc}-l_{chc}\cos \left({\phi }_c-{\phi }_{chc}\right)-X_{P_{cp}}=0\\ Y_{P_{hcp}}+l_{hc}\sin {\phi }_{hc}-l_{chc}\sin \left({\phi }_c-{\phi }_{chc}\right)-Y_{P_{cp}}=0\\ X_{P_{cp}}+l_c\cos {\phi }_c-l_{lr}\cos {\phi }_{lr}-l_{lalr}\cos {\phi }_{la}-X_{P_{lap}}=0\\ Y_{P_{cp}}+l_c\sin {\phi }_c-l_{lr}\sin {\phi }_{lr}-l_{lalr}\sin {\phi }_{la}-Y_{P_{lap}}=0\\ X_{P_{uap}}+l_{ua}\cos {\phi }_{ua}-m_h\cos {\phi }_i-l_{la}\cos {\phi }_{la}-X_{P_{lap}}=0\\ Y_{P_{uap}}+l_{ua}\sin {\phi }_{ua}-m_h\sin {\phi }_i-l_{la}\sin {\phi }_{la}-Y_{P_{lap}}=0\end{array}$$

(1)

In order for the position analysis of the TPH to be correctly performed, the system of Equation (1) must have six unknowns. Depending on the specific analysis requested at the different stages of the algorithms in Scheme 1 and Scheme 2 (e.g., simulating the reference manoeuvre, or enforcing one of the Optimizer nonlinear constraints), the six dimensional parameters playing the role of unknowns may vary.

2.2. Definition of the Reference Manoeuvre

The reference manoeuvre simulates the extraction of a heavy-duty implement (such as a plough or a subsoiler) from the soil and was chosen to trigger weight transfer from the front to the rear axle of the tractor. It was defined in a standardised manner, according to the dimensional requirement of the ISO-730 Standard; in this way, the manoeuvre is adaptable to all the TPH categories. The criteria upon which the manoeuvre is based are the following:

$$\{\begin{array}{l}{H}_{lad,min}=1.5 L_{14}\\ H_{lad.max}=H_{lad,min}+0.2 L_{18}\\ l_{lr}=l_{lr,max}\\ l_{ua}\mathrm{such} \mathrm{that} {\phi }_i=\frac{\pi }{2} at H_{lad,min}\end{array}$$

(2)

where $H_{lad,min}$ is the minimum height of the point $P_{lad}$ above the ground (Figure 3a), $H_{lad,max}$ is the maximum height of the point $P_{lad}$ above the ground (Figure 3b), the dimensions $l_{lr}$, $l_{ua}$, ${\phi }_i$ are depicted in Figure 2, and the dimensions $L_{14}$ and $L_{18}$ are, respectively, the lower hitch point height and the movement range, as defined by the ISO-730 Standard. Based on criteria (2), the simulated manoeuvre is performed setting the lift rods at their maximum extension and the upper arm in such a way that the implement mast is vertical when the implement is at its minimum height (Figure 3a). This is not a common TPH setup for real applications; however, it was defined in this way for the sake of robustness of the Optimizer: the manoeuvre defined in (2) can be successfully performed with any TPH trial geometries the optimiser might consider during the automated optimisation process.

Through the kinematic analysis of the TPH (system of Equation (1)), the values of the hydraulic lift cylinders length when the TPH is at its lower and higher height, namely $l_{hc,min}$ and $l_{hc,max}$, are determined. Then, the speed at which the reference manoeuvre is performed is set by prescribing the extension law of the hydraulic lift cylinders (Figure 3c) as follows:

$$l_{hc}\left(t\right)=\frac{l_{hc, max}-l_{hc, min}}{2} ·\mathrm{erf}\left(\frac{t-t_0}{T}\right)+\frac{l_{hc, max}+l_{hc,min}}{2}$$

(3)

where $\mathrm{erf}$ is the Gauss error function, $T$ is a characteristic time to be set based on the flow rate of the hydraulic circuit actuating the hydraulic lift cylinders, $t_0$ is an offset time for setting the manoeuvre onset, and $t$ is the simulation elapsed time. The choice of the $\mathrm{erf}$ function to model the cylinders extension is based on experimental observations (Section 3).

Once the hydraulic lift cylinders extension law is set, a second kinematic analysis is performed to determine the following quantities (Scheme 2):

$$\left\{{\phi }_c\left(t\right), {\phi }_{hc}\left(t\right), {\phi }_{ua}\left(t\right), {\phi }_{lr}\left(t\right), {\phi }_{la}\left(t\right), {\phi }_i\left(t\right)\right\}.$$

(4)

In this way, the position of each link of the TPH during the entire reference manoeuvre is completely known.

2.3. Kinematic Analysis of the Implement

Knowing, from (4), the values of the angles ${\phi }_{la}$ and ${\phi }_i$, the position of the implement COG,$G_i$, can be determined:

$$\{\begin{array}{l}{X}_{G_i}=X_{P_{lad}}+ R_i\sin \left({\gamma }_i +{\phi }_i\right)=X_{P_{lap}}+l_{la}\cos \left({\phi }_{la}\right) + R_i\sin \left({\gamma }_i +{\phi }_i\right)\\ Y_{G_i}=Y_{P_{lad}}- R_i\cos \left({\gamma }_i +{\phi }_i\right)=Y_{P_{lap}}+l_{la}\sin \left({\phi }_{la}\right)- R_i\cos \left({\gamma }_i +{\phi }_i\right)\end{array}$$

(5)

$R_i$ and ${\gamma }_i$ being the polar coordinates of the implement COG with respect to point $P_{lad}$ (Figure 4) and having computed the coordinates of $P_{lad}$ according to the TPH geometry in Figure 2.

To complete the kinematic analysis of the implement (Scheme 2), the first-order and second-order derivatives of $X_{G_i}$, $Y_{G_i}$, and ${\phi }_i$ are computed using the central finite difference scheme.

2.4. Soil–Implement Interaction

Since the implement is assumed to behave as a rigid body, the forces exerted by the soil can be represented (Figure 4) as an equivalent system of forces composed by a horizontal force ($F_{wx}$) and a vertical force ($F_{wy}$) applied to a reference point of the system, plus a moment ($M_w$). The point $P_{lad}$ was chosen as the reference point.

The forces exchanged between the soil and the implement depend on constitutive parameters like the geometry of the tillage tools and the soil composition and condition, as well as on operational parameters like the working depth and tractor speed [18,19,20,21,22,23]. Since the Optimizer simulates a reference manoeuvre performed with a reference implement on a reference soil, and with the tractor moving at constant speed, the forces exerted by the soil on the implement may be assumed to vary only as functions of the working depth and of the implement vertical speed, while all the other parameters remain constant. Hence, the following relations are assumed:

$$\{\begin{array}{l}{F}_{wx}=F_{0x}+K_{sx}·Y_{P_{lad}}\\ F_{wy}=F_{0y}+K_{sy}·Y_{P_{lad}}+C_{sy}·{\dot{Y}}_{G_i}\\ M_w=M_0+K_{sm}·Y_{P_{lad}}\end{array}$$

(6)

where $F_{0x}$, $F_{0y}$, and $M_0$ are offset values accounting for the fact that $Y_{P_{lad}}$ is not zero when the implement tools approach the soil, while $K_{sx}$, $K_{sy}$, and $K_{sm}$ are proportionality coefficients and $C_{sy}$ is a viscous coefficient defined as:

$$C_{sy}=\{\begin{array}{l}{C}_{sy}^{-}, \mathrm{if} {\dot{Y}}_{G_i}\le 0\\ C_{sy}^{+}, \mathrm{if} {\dot{Y}}_{G_i}>0\end{array}$$

(7)

in order to account for the fact that soil drag between implement penetration and extraction is different. The exact values of the coefficients appearing in Equations (6) and (7) were determined at the model validation stage (Section 3).

Many studies report a nonlinear dependence of the soil loads on the working depth [23,24,25,26]; however, for the sake of simplicity and without loss of generality, a linear dependence is chosen here.

2.5. Tractor–TPH–Implement Model

The equations of motion for the implement read (Figure 4):

$$\{\begin{array}{c}{F}_{ladx}-F_{ua}\cos \left({\phi }_{ua}\right)+F_{wx}-M_i{\ddot{X}}_{G_i}=0\hfill \\ F_{lady}-F_{ua} \sin \left({\phi }_{ua}\right)-F_{wy}-M_i g-M_i {\ddot{Y}}_{G_i}=0\hfill \\ -F_{ladx} R_i \cos \left({\phi }_i+{\gamma }_i\right)-F_{lady} R_i \sin \left({\phi }_i+{\gamma }_i\right)\hfill \\ +F_{ua}\left[m_h\text{ sin}\left({\phi }_i-{\phi }_{ua}\right)+R_i \cos \left({\phi }_i-{\phi }_{ua}+{\gamma }_i\right)\right]+M_w\\ -F_{wx} R_i\cos \left({\phi }_i+{\gamma }_i\right)+F_{wy} R_i \sin \left({\phi }_i+{\gamma }_i\right)-I_i^{Gz} {\ddot{\alpha }}_i=0\end{array}$$

(8)

where $F_{ua}$ is the force exerted by the TPH upper arm on the implement, $F_{ladx}$ and $F_{lady}$ are, respectively, the horizontal and vertical components of the forces exchanged by the implement and the TPH at the two lower hitch points, $M_i$ is the implement mass, and $I_i^{Gz}$ its moment of inertia with respect to an axis parallel to $Z$ and passing through $G_i$ (Figure 4). $g$ is the gravitational acceleration. Note that the force $F_{ua}$ lies in the same direction as the upper arm.

Once the kinematic analysis of the implement has been performed and the soil–implement loads have been computed, the values of $F_{ladx}$, $F_{ladx}$ and $F_{ua}$ as functions of time during the entire reference manoeuvre are determined from the system of Equation (8) (Scheme 2).

As regards the TPH model, it is sufficient to write equilibrium equations for the lower arms (Figure 5a) and for the lift arms (Figure 5b), as inertial effects of the TPH links have been neglected:

$$\{\begin{array}{l}{F}_{lapx}+F_{lr}\cos \left({\phi }_{lr}\right)-F_{ladx}=0\\ F_{lapy}+F_{lr}\sin \left({\phi }_{lr}\right)-F_{lady}=0\\ F_{lr} l_{lalr}\sin \left({\phi }_{lr}-{\phi }_{la}\right)-F_{lady} l_{la}\cos \left({\phi }_{la}\right)+F_{ladx} l_{la} sin\left({\phi }_{la}\right)=0\end{array}$$

(9)

where $F_{lapx}$ and $F_{lapy}$ are, respectively, the horizontal and vertical components of the force that the tractor exerts on the TPH through the two lower link points, and $F_{lr}$ is the force acting on the two lower arms due to the lift rods. The force $F_{lr}$ lies in the same direction as the lift rods, and the inclination of the lift rods in the vertical–transversal plane (Y-Z plane, Figure 2) has been neglected for simplicity.

The equilibrium equations of the lift arms read:

$$\{\begin{array}{l}{F}_{cpx}+F_{hc}\cos \left({\phi }_{hc}\right)-F_{lr}\cos \left({\phi }_{lr}\right)=0\\ F_{cpy}+F_{hc}\sin \left({\phi }_{hc}\right)-F_{lr}\sin \left({\phi }_{lr}\right)=0\\ F_{hc} l_{chc}\sin \left({\phi }_{hc}-{\phi }_c+{\phi }_{chc}\right)-F_{lr} l_c\sin \left({\phi }_{lr}-{\phi }_c\right)=0\end{array}$$

(10)

where $F_{cpx}$ and $F_{cpy}$ are, respectively, the horizontal and vertical components of the force that the tractor exerts on the TPH through the two lift arm link points, and $F_{hc}$ is the force exerted by the two hydraulic lift cylinders on the TPH, lying in the direction of the cylinders. From the systems of Equations (9) and (10), the forces $F_{lapx}$, $F_{lapy}$, $F_{lr}$, $F_{cpx}$, $F_{cpy}$, and $F_{hc}$ as functions of time during the entire reference manoeuvre can be calculated (Scheme 2).

As regards the tractor, a 3-degrees-of-freedom model was developed (Figure 6a), taking the vertical displacement of the tractor COG $Y_{G_t}$, the pitch angle ${\theta }_t$, and vertical displacement of the front axle unsuspended mass $Y_a$ as the degrees of freedom.

Naming $P_{tf}$ the link points of the front axle suspension on the tractor chassis, $P_{tr}$ the rear wheels hub, and in accordance with the hypotheses on which the model lays (Section 2), the front tyres, the rear tyres, and the front axle suspension transmit the following visco–elastic forces, respectively:

$$F_{fw}=k_f{Y}_a+c_f {\dot{Y}}_a$$

(11)

$$F_{rw}=k_r{Y}_{tr}+c_r {\dot{Y}}_{tr}$$

(12)

$$F_s=k_s\left(Y_{tf}-Y_a\right)+c_s \left({\dot{Y}}_{tf}-{\dot{Y}}_a\right)$$

(13)

where $k_f$, $k_r$, and $k_s$ are spring constants; $c_f$, $c_r$, and $c_s$ are damping coefficients; and $Y_{tf}$, $Y_{tr}$ are the vertical displacements of the points $P_{tf}$ and $P_{tr}$, which can be calculated as the sum of two contributions: the displacement induced by the tractor COG vertical motion and the vertical displacement induced by the pitch motion of the tractor (Figure 6b):

$$Y_{tf}=Y_{G_t}+h_f\left(1-\cos {\theta }_t\right)-l_f\sin {\theta }_t$$

(14)

$$Y_{tr}=Y_{G_t}+h_r\left(1-\cos {\theta }_t\right)+l_r\sin {\theta }_t$$

(15)

where $h_f$, $h_r$, $l_f$, and $l_r$ are the dimensions depicted in Figure 6.

By taking the derivatives of Equations (14) and (15), the expressions for the vertical velocity of $P_{tf}$ and $P_{tr}$ are obtained:

$${\dot{Y}}_{tf}={\dot{Y}}_{G_t}+{\dot{\theta }}_t \left(h_f\sin {\theta }_t-l_f\cos {\theta }_t\right)$$

(16)

$${\dot{Y}}_{tr}={\dot{Y}}_{G_t}+{\dot{\theta }}_t \left(h_r\sin {\theta }_t+l_r\cos {\theta }_t\right) .$$

(17)

The resulting equations of motion for the tractor model are:

$$\{\begin{array}{c}{M}_a {\ddot{Y}}_a=F_s-F_{fw}-M_a g\hfill \\ M_t {\ddot{Y}}_{G_t}=-F_s-F_{rw}+F_{ua}\sin {\phi }_{ua}-F_{lapy}-F_{cpy}-F_{hc}\sin {\phi }_{hc}-M_t g\hfill \\ I_t^{Gz} {\ddot{\theta }}_t=F_s l_f-F_{rw} l_r+ F_{ua}\left[\left(l_r+X_{P_{uap}}\right)\sin {\phi }_{ua}+\left(h_r-Y_{P_{uap}}\right)\cos {\phi }_{ua}\right]-F_{cpy} \left(l_r+X_{P_{cp}}\right)+\hfill \\ -F_{cpx} \left(h_r-Y_{P_{cp}}\right)-F_{hc}\left[\left(l_r+X_{P_{hcp}}\right)\sin {\phi }_{hc}+\left(h_r-Y_{P_{fhc}}\right)\cos \left({\phi }_{hc}\right)\right]+\\ -F_{lapy} \left(l_r+X_{P_{lap}}\right)-F_{lapx} \left(h_r-Y_{P_{lap}}\right)-H_T \left(h_r+r_{rw}+Y_{tr}\right)\end{array}$$

(18)

where $M_a$ is the front axle unsuspended mass, $M_t$ is the tractor chassis mass, $I_t^{Gz}$ its moment of inertia with respect to an axis parallel to $Z$ and passing through $G_t$ (Figure 6a), $H_T$ is the total traction force developed at the interface between the soil and the tractor wheels, and $r_{rw}$ is the static loaded radius of the rear wheels. As it concerns the third equation in the system (18), considering the total traction force $H_T$ is equivalent to considering the traction forces and the driving torques at the wheel hubs. However, using $H_T$ in the calculation is easier, as there is no need to determine how traction forces and driving torques are distributed between the front and rear wheels.

The total traction force can be determined through the balance of linear momentum of the tractor along the horizontal direction. Since the tractor is assumed to move forward on a straight line at constant speed, the acceleration of the tractor COG is null along the horizontal direction, and the balance of linear momentum reduces to an equilibrium of the horizontal components of the forces acting on the system, from which $H_T$ can be obtained (Figure 6a):

$$H_T=F_{ua}\cos {\phi }_{ua}-F_{hc}\cos {\phi }_{hc}-F_{lapx}-F_{cpx} .$$

(19)

Upon substituting Equations (11)–(13) and (19) into (18), and accounting for Equations (14)–(17), a system of three-second order ODEs is obtained, which constitutes the tractor model in the algorithm depicted in Scheme 2. The values of the loads exerted by the TPH on the tractor during the entire reference manoeuvre are known from the previous steps of the algorithm, and solving the system of Equation (18) allows to determine the quantities:

$$\left\{Y_a\left(t\right), Y_{G_t}\left(t\right), {\theta }_t\left(t\right)\right\} .$$

(20)

The system of Equation (18) is solved using an explicit Runge–Kutta method through the MATLAB built-in function ode45 (MATLAB^®, Mathworks, Inc., Natick, MA, USA).

Once the values of the quantities (20) have been determined, the load on the tractor front axle as a function of time during the entire reference manoeuvre can be reconstructed (Scheme 2). Accounting for Equations (13), (14), and (16), the front axle load takes the form:

$$F_s=k_s\left(Y_{G_t}+h_f\left(1-\cos {\theta }_t\right)-l_f\sin {\theta }_t-Y_a\right)+c_s \left({\dot{Y}}_{G_t}+{\dot{\theta }}_t \left(h_f\sin {\theta }_t-l_f\cos {\theta }_t\right)-{\dot{Y}}_a\right)$$

(21)

2.6. Optimiser

The Optimiser solves the following mathematical problem (Scheme 1):

$$\{\begin{array}{l}\underset{d}{\min }\mathsf{\Phi }\left(d\right)\\ C_k\left(d\right)\le 0, k=1,\cdots ,N_C\\ L_j\le d_j\le U_j, j=1,\cdots , N_D\end{array}$$

(22)

where $d$ is a vector containing all the TPH dimensions subject to optimisation, $\mathsf{\Phi }$ is the objective function to be minimised, $C_k$ are the constraints that the TPH has to satisfy, $N_C$ is the number of constraints, $L_j$ and $U_j$ are, respectively, the lower and upper bounds on the dimension $d_j$, and $N_D$ is the number of TPH dimensions subject to optimisation. The dimensions vector is composed by $N_D=19$ TPH dimensions, namely:

$$d=\left[X_{P_{cp}};Y_{P_{cp}};X_{P_{hcp}};Y_{P_{hcp}};X_{P_{lap}};Y_{P_{lap}};X_{P_{uap}};Y_{P_{uap}};l_c;l_{chc};l_{la};l_{lalr};l_{l{r}_{max}};l_{l{r}_{min}};l_{u{a}_{max}};l_{u{a}_{min}};{\phi }_{chc};{\phi }_{c_{max}};{\phi }_{c_{min}}\right]$$

The objective function is the peak-to-peak (P2P) value of the front axle load (Equation (21)) during the reference manoeuvre and is determined through the algorithm depicted in Scheme 2:

$$\mathsf{\Phi }=\mathrm{P}2\mathrm{P}\left(F_s\right) .$$

(23)

Ensuring that the optimiser minimises $\mathsf{\Phi }\left(d\right)$ will result in finding the TPH optimal geometry, which minimises the weight-transfer effect during the reference manoeuvre. Problem (22) is solved using an active-set sequential quadratic programming method through the MATLAB built-in function fmincon (MATLAB^®, Mathworks, Inc., Natick, MA, USA).

2.6.1. Optimiser Constraints

The Optimiser accounts for $N_C=36$ constraints (

Table 1. Optimiser constraints.

Nr.	Equation	Significance
C1	${\phi }_{c,max}\ge {\phi }_{c,min}$	Logical constraints
C2	$l_{lr,max}\ge l_{lr,min}$
C3	$l_{ua,max}\ge l_{ua,min}$
C4	$l_c+l_{lr,min}+l_{lalr}\ge \left|P_{lap}-P_{cp}\right|$	Robustness constraints
C5	$l_{lr,min}+l_{lalr}+\left|P_{lap}-P_{cp}\right|\ge l_c$
C6	$l_c+l_{lr,min}+\left|P_{lap}-P_{cp}\right|\ge l_{lalr}$
C7	$l_c+\left|P_{lap}-P_{cp}\right|+l_{lalr}\ge l_{lr,max}$
C8	$m_h+l_{la}+l_{ua, min}\ge \left|P_{lap}-P_{uap}\right|$
C9	$m_h+\left|P_{lap}-P_{uap}\right|+l_{ua, min}\ge l_{la}$
C10	$m_h+l_{la}+\left|P_{lap}-P_{uap}\right|\ge l_{ua, max}$
C11	$l_c-l_{chc}\ge 50 \mathrm{mm}$	Functional constraints
C12	$l_{la}-l_{lalr}\ge 200 \mathrm{mm}$
C13	$Y_{P_{cp}}-Y_{P_{lap}}\ge 50 \mathrm{mm}$
C14	$Y_{P_{uap}}-Y_{P_{lap}}\ge 300 \mathrm{mm}$
C15	$Y_{P_{hcp}}-Y_{P_{lap}}\ge 0 \mathrm{mm}$
C16	${\phi }_{hc, max}\le 85°$
C17	$\frac{l_{hc, min}}{l_{hc,max}}\ge 0.6$	Proportioning constraints
C18	$\frac{l_{lr, min}}{l_{lr,max}}\ge 0.7$
C19	$\frac{l_{ua, min}}{l_{ua,max}}\ge 0.6$
C20	$X_{P_{lap}}+l_{la}-X_{PTO}\ge L_{MIN}$	Tractor PTO distance from lower hitch points (ISO-730, Figure 2 and Table 2)
C21	$X_{P_{lap}}+l_{la}-X_{PTO}\le L_{MAX}$
C22	$H_{lad,min}\le L_{14}$	$\mathrm{Lower}\mathrm{hitch}\mathrm{points}\mathrm{height},\mathrm{TPH}\mathrm{in}\mathrm{configuration}\mathrm{A}(\mathrm{dimension}{L}_{14}$ as per ISO-730)
C23	$H_{lad,min}\ge 50 \mathrm{mm}$
C24	${\phi }_{lr}\ge 95°$	Functional requirements for TPH in configuration A
C25	${\phi }_i\le 90°$
C26	$H_{lad,max}\ge L_{19}$	$\mathrm{Transport}\mathrm{height},\mathrm{TPH}\mathrm{in}\mathrm{configuration}\mathrm{B}(\mathrm{dimension}{L}_{19}$ as per ISO-730)
C27	${\mathsf{\Delta }}_{min}\ge L_{20}$	$\mathrm{Lower}\mathrm{hitch}\mathrm{points}\mathrm{clearance},\mathrm{TPH}\mathrm{in}\mathrm{configuration}\mathrm{B}(\mathrm{dimension}{L}_{20}$ as per ISO-730)
C28	$l_{ua}\ge l_{ua, min}$	Functional constraints for configuration C
C29	$l_{ua}\le l_{ua, max}$
C30	$\frac{l_{lr, max}+l_{lr, min}}{2}\ge l_{lr, lim}$
C31	$L_{CV}\ge 0.9 \left(l_f+l_r\right)$	Constraints on vertical convergence distance
C32	$L_{CV}\le 3 \left(l_f+l_r\right)$
C33	$\frac{H_{lad, max}-H_{lad,min}}{2}\ge L_{15}$	$\mathrm{Levelling}\mathrm{adjustment},\mathrm{TPH}\mathrm{in}\mathrm{configuration}\mathrm{C}(\mathrm{dimension}{L}_{15}$ as per ISO-730)
C34	$H_{lad, max}-H_{lad,min}\ge L_{18}$	$\mathrm{Movement}\mathrm{range},\mathrm{TPH}\mathrm{from}\mathrm{configuration}{\mathrm{D}}_1\mathrm{to}{\mathrm{D}}_2(\mathrm{dimension}{L}_{18}$ as per ISO-730)
C35	${\phi }_{i, max}\ge 95° \left(90°\mathrm{for}\mathrm{category}1\mathrm{N}\mathrm{TPH}\right)$	Mast adjustment, TPH in configurations E1 and E2 (ISO-730, comma 3.2.22)
C36	${\phi }_{i, min}\le 85° \left(80°\mathrm{for}\mathrm{category}1\mathrm{N}\mathrm{TPH}\right)$

), implemented in the non-dimensional form $C_k\left(x\right)\le 0$. For the sake of readability, constraints will not be presented in this form in Table 1, but in the form they were naturally derived.

Constraints C1–C3 are logical constraints on some of the elements of $d$: for obvious reasons, the maximum extension of the lift rods and of the upper arm cannot be less than their minimum extension; similarly, the maximum value of the lift arms angle cannot be less than its minimum value. Constraints C4–C10 are robustness constraints: they prescribe conditions for the existence of the closed polygons, which constitute the TPH kinematic subsystems described in Section 2.1, thus impeding the Optimiser from choosing trial geometries that would result in unfeasible TPH mechanisms. Constraints C11–C16 are functional constraints: for manufacturing and accessibility reasons, there needs to be a minimum ensured distance between some link points; moreover, the hydraulic lift cylinders must not reach a vertical position when fully extended. Constraints C17–C19 are proportioning constraints on the ratio of minimum to maximum length of the extensible links, set to avoid disproportioning.

The other constraints are derived from the requirements contained in the ISO-730 Standard: constraints C20–C21 account for the tractor power take-off (PTO) location with respect to the TPH, while constraints C22–C36 concern the functional performance of the TPH and are enforced by evaluating, through the kinematic analysis described in Section 2.1, the TPH geometry in the different configurations described in Appendix A.

3. Model Validation

To validate the soil–implement interaction model and to determine the values of the coefficients in Equations (6) and (7), a series of tests was conducted (Figure 7) in which a tractor with a sensorised TPH performed a manoeuvre similar to the reference manoeuvre described in Section 2.2. Tests were performed on a clay–loam untilled soil, in a farm located in northern Italy. The tractor used for the tests was a 96 kW MFWD tractor with a rear-mounted category 3 TPH. A 7-shank subsoiler (

Table 2. Implement used in the validation tests.

Implement Type Number of Tools	Subsoiler 7 Shanks
$M_i$	$768 \mathrm{kg}$
$I_i^{Gz}$	$314 {\mathrm{kg}\mathrm{m}}^2$
$R_i$	$795 \mathrm{mm}$
${\gamma }_i$	$5.19°$
$m_h$	$690 \mathrm{mm}$

) was attached to the tractor and the loads at different locations of the TPH were measured during implement manoeuvring. The tractor was equipped with the following sensors: a pressure sensor mounted on the hydraulic lift cylinders; a load cell placed on the TPH upper arm; a load cell that measured the horizontal component of the force at the lower link points; a sensor for measuring the lift arms angle with respect to the horizontal direction, which allowed to reconstruct the TPH configuration; and a pressure sensor on the front-axle hydropneumatic suspension.

The manoeuvre consisted of raising and lowering the subsoiler while the tractor was running at a constant speed of 2 km/h. Several consecutive repetitions of the manoeuvre were performed. For the whole extent of the manoeuvre, the subsoiler shanks were sunk into the ground.

Figure 8 shows a comparison between the measured and simulated TPH configuration in terms of the hydraulic lift cylinders’ extension for three test repetitions; the simulated values were determined using Equation (3). It can be observed that the model is able to reproduce the experimental manoeuvre and, in particular, the erf function appears to be an effective choice for approximating the cylinder kinematics observed in the field tests.

Figure 9 shows a comparison between the force measured at different locations of the TPH and the force predicted through the soil–implement interaction model in Equations (6) and (7), the implement model in Equation (8), and the TPH model in Equations (9) and (10). From the comparison, it can be concluded that the soil–implement interaction model is suitable for describing the loads that the soil exerts on the subsoiler shanks during a raising/lowering manoeuvre. In particular, the peak in the hydraulic lift cylinders force during implement extraction from the soil is reproduced. Further analysis shows that such peak is aligned with the peak in the implement vertical speed (Figure 10), suggesting that the loads exerted by the soil on the implement should depend not only on the working depth, but also on the speed of penetration/extraction of the implement into/from the soil. This is the reason that led to the introduction of the viscous term in the expression of $F_{wy}$ in Equation (6).

From the field tests, the parameters for the reference manoeuvre and the reference soil were also determined by taking the values that allowed to best match the experimental measurements (

Table 3. Parameters used in the simulations.

Reference manoeuvre	$T$	$0.444 \mathrm{s}$
	$t_0$	$5 \mathrm{s}$
	$Y_{P_{lad},\min }$	$-451 \mathrm{mm}$ (TPH down)
	$Y_{P_{lad}, \max }$	$-304 \mathrm{mm}$ (TPH up)
Reference implement	Table 2
Reference soil	$F_{0x}$	$-125.9 \mathrm{kN}$
	$F_{0y}$	$14.23 \mathrm{kN}$
	$M_0$	$-21.8\times {10}^3 \mathrm{N}\mathrm{m}$
	$K_{sx}$	$0.445 {\mathrm{kN}\mathrm{mm}}^{-1}$
	$K_{sy}$	$26.4\times {10}^{-3} {\mathrm{kN}\mathrm{mm}}^{-1}$
	$K_{sm}$	$68.6 \mathrm{kN}$
	$C_{sy}^{-}$	$65 {\mathrm{N}\mathrm{s}\mathrm{mm}}^{-1}$
	$C_{sy}^{+}$	$140 {\mathrm{N}\mathrm{s}\mathrm{mm}}^{-1}$

). The subsoiler used in the validation tests was taken as the reference implement for the model.

4. Results and Discussion

Simulations were performed to show the capabilities of the Optimiser and of the underlying tractor–TPH–implement model. The tractor used for the validation tests was taken as the benchmark for the simulations. As described in Section 2, the Optimiser simulates the reference manoeuvre performed on the reference soil with the reference implement; the values of the manoeuvre, soil, and implement parameters are listed in Table 3. The values of the tractor parameters used in the simulations are listed in

Table 4. Tractor parameters used in the tractor dynamic model.

$M_t$	$5975 \mathrm{kg}$
$I_t^{Gz}$	$5220 {\mathrm{kg}\mathrm{m}}^2$
$M_a$	$298 \mathrm{kg}$
$l_f$	$1069 \mathrm{mm}$
$l_r$	$1491 \mathrm{mm}$
$h_f$	$195 \mathrm{mm}$
$h_r$	$20 \mathrm{mm}$
$r_{rw}$	$877 \mathrm{mm}$
$k_s$	$1.10\times {10}^5{\mathrm{N}\mathrm{m}}^{-1}$
$k_f$	$1.00\times {10}^6{\mathrm{N}\mathrm{m}}^{-1}$
$k_r$	$2.10\times {10}^6{\mathrm{N}\mathrm{m}}^{-1}$
$c_s$	$4.60\times {10}^3 {\mathrm{N}\mathrm{s}\mathrm{m}}^{-1}$
$c_f$	$9.10\times {10}^3 {\mathrm{N}\mathrm{s}\mathrm{m}}^{-1}$
$c_r$	$1.00\times {10}^4{\mathrm{N}\mathrm{s}\mathrm{m}}^{-1}$

. Spring constants, damping coefficients, and tractor moment of inertia were set through a parameter identification procedure that involved measuring the front axle load during an in-filed manoeuvre similar to the reference manoeuvre, performed with the subsoiler used for the validation tests (Table 2). The parameter identification procedure consisted of choosing the parameters values that allowed to best match the higher and lower peaks in the simulated front axle response with those of the experimental signal. The starting values for the parameter identification procedure were taken from data available in the literature [27,28,29,30].

4.1. Behaviour of the Tractor–TPH–Implement Aggregate during the Reference Manoeuvre

The predicted loads exchanged between the soil and the implement during the reference manoeuvre are depicted in Figure 11. It can be observed that the horizontal component $F_{wx}$, Figure 11a, points rearward, exerting a drag force that reaches its maximum when the working depth of the implement is maximum and decreases as the implement is lifted. As regards the vertical component $F_{wy}$, Figure 11b, a peak is observed during the reference manoeuvre, which is induced by the behaviour of the implement vertical speed (Figure 10) and by the fact that the soil–implement loads depend on the speed of extraction of the implement from the soil (Equation (6)). The fact that $F_{wy}$ points downward is consistent with the literature [31] and is due to the value of the rake angle of the reference implement shanks, which is smaller than the cross-over value discussed by Goodwin [22,24]. Furthermore, the sign of the moment of the resultant about the point $P_{lad}$, Figure 11c, indicates that the line of pull lies below $P_{lad}$ when the implement working depth is maximum, whereas when the working depth is minimum, the horizontal component of the resultant decreases significantly, causing the line of pull to move rearward with respect to $P_{lad}$ and, ultimately, the moment $M_w$ to be negative.

Figure 12 shows the behaviour of the load on the tractor front axle during the reference manoeuvre. It can be observed that lifting the implement causes the load on the front axle to decrease drastically. This is ultimately due to the peak in the vertical force $F_{wy}$ (Figure 11b). At the end of the manoeuvre, the front axle load exhibits an overshoot, and then a damped oscillation occurs, due to the viscoelastic nature of tyres and front axle suspension. After oscillations are dampened out, the front axle load reaches a higher value compared to the beginning of the manoeuvre, as the implement drag is lower at the end of the manoeuvre than at the beginning (Figure 11). The reference manoeuvre thus proves to be effective in inducing weight transfer from the front to the rear axle and the P2P value of the force on the front axle appears to be a suitable measure of this phenomenon.

4.2. TPH Optimisation

To test the capabilities of the Optimiser, the geometry of a commercially available category 3 TPH was chosen as the initial TPH geometry (Scheme 1). Design considerations led to establish lower and upper bounds on the TPH dimensions subject to optimisation (

Table 5. Lower and upper bounds on the TPH dimensions subject to optimisation.

TPH Dimension	Lower Bound	Upper Bound	Unit
$X_{P_{cp}}$	$140$	$200$	mm
$Y_{P_{cp}}$	$325$	$475$	mm
$X_{P_{hcp}}$	$235$	$300$	mm
$Y_{P_{hcp}}$	$-150$	$-90$	mm
$X_{P_{lap}}$	$50$	$280$	mm
$Y_{P_{lap}}$	$-270$	$-180$	mm
$X_{P_{uap}}$	$275$	$475$	mm
$Y_{P_{uap}}$	$200$	$550$	mm
$l_c$	$300$	$500$	mm
$l_{chc}$	$150$	$350$	mm
$l_{la}$	$500$	$1500$	mm
$l_{lalr}$	$250$	$1000$	mm
$l_{l{r}_{min}}$	$400$	$1200$	mm
$l_{l{r}_{max}}$	$400$	$1200$	mm
$l_{u{a}_{min}}$	$400$	$1200$	mm
$l_{u{a}_{max}}$	$400$	$1200$	mm
${\phi }_{chc}$	$10$	$45$	°
${\phi }_{c_{max}}$	$30$	$80$	°
${\phi }_{c_{min}}$	$-20$	$50$	°

), which were imposed on the Optimiser (Problem 22). The constraints listed in Table 1 were enforced, setting the limit values prescribed by the ISO-730 Standard for a category 3 TPH (

Table 6. ISO-730 functional prescriptions for a category 3 TPH (Section 2.6.1 and Table 1).

$L_{min}$	$560$	mm
$L_{max}$	$775$	mm
$L_{14}$	$230$	mm
$L_{15}$	$125$	mm
$L_{18}$	$735$	mm
$L_{19}$	$1065$	mm
$L_{20}$	$100$	mm
$H_{ma, min}$	$230$	mm
$H_{ma, max}$	$660$	mm

).

Table 7. Optimiser performance.

${\mathsf{\Phi }}_{initial}$	$27.07$	$\mathrm{kN}$
${\mathsf{\Phi }}_{optimised}$	$23.05$	$\mathrm{kN}$
$\mathrm{reduction}$	$14.9$	$\%$
no. of iterations	$36$

summarises the performance of the Optimizer: after 36 iterations, an optimised TPH geometry was found that allowed to reduce the P2P value of the front axle load by 14.9%. A comparison between the initial and optimised TPH geometry is shown in Figure 13. In the optimised geometry, the point $P_{lap}$ is shifted towards the rear axle centre as much as the bounds on the dimensions $X_{P_{lap}}$ and $Y_{P_{lap}}$ allowed. Furthermore, the point $P_{cp}$ is shifted upward and rearward, while the angle ${\phi }_{chc}$ is considerably smaller than it was in the starting TPH geometry; as a consequence, the value of the dimension ${\phi }_{c_{min}}$ decreased to its lower admissible value. Figure 13 also shows that the point $P_{uap}$ is considerably shifted upward and rearward, the value of the dimension $X_{P_{uap}}$ being equal to its upper admissible value.

5. Conclusions

The paper presents a computational tool (the Optimiser) developed to optimise the geometry of a tractor rear-mounted TPH in order to minimise the weight transfer from the front to the rear tractor axle during implement operation or manoeuvring. To this end, the Optimiser simulates a reference manoeuvre performed with a reference implement on a reference soil. Simulations are based on a dynamic model of the tractor–TPH–implement aggregate. The following conclusions can be drawn from the study:

• The proposed reference manoeuvre, defined in a general manner based on ISO-730 functional requirements, is suitable for all TPH categories and can accurately reproduce real infield implement manoeuvring;
• The soil–implement interaction model developed in the study and implemented in the Optimiser was successfully validated against infield test data;
• The reference manoeuvre is effective in triggering the weight-transfer effect and the P2P value of the front axle load during the reference manoeuvre appears to be a suitable measure of this phenomenon;
• The Optimiser was subjected to a benchmark test starting from the geometry of a commercially available TPH; the test showed that the Optimiser was able to find an optimised TPH geometry which allows to reduce the weight-transfer effect by 14.9%.