Integrated Optimization of Structure and Control Parameters for the Height Control System of a Vertical Spindle Cotton Picker

Abstract

Vertical picking method is a predominate method used to harvest cotton crop. However, a vertical picking method may cause spindle bending of the cotton picker if spindles collide with stones on the cotton field. Thus, how to realize a precise height control of the cotton picker is a crucial issue to be solved. The objective of this study is to design a height control system to avoid the collision. To design it, the mathematical models are established first. Then a multi-objective optimization model represented by structure parameters and control parameters is proposed to take the pressure of chamber without piston, response time and displacement error of the height control system as the optimization objectives. An integrated optimization approach that combines optimization via simulation, particle swarm optimization and simulated annealing is proposed to solve the model. Simulation and experimental test results show that the proposed integrated optimization approach can not only reduce the pressure of chamber without piston, but also decrease the response time and displacement error of the height control system.

1 Introduction

Cotton is one of the most important crops [1]. According to the data from International Cotton Advisory Committee, the world cotton production was more than 24 million tons in 2020. Increasing cotton production creates an urgent need for high efficiency harvesting machines. Due to the high efficiency in cotton harvesting, in recent years, the mechanical cotton pickers have gained more and more application. Among these mechanical cotton pickers, the most representative one is the horizontal spindle cotton pickers produced by John Deere and CaseIH, which can work not only reliably and stably, but also harvest with low cotton loss and less cotton leaf [2].

However, during the cotton harvesting process, the traditional horizontal spindle cotton picker is parallel to the cotton crop, which may lead to a low fiber quality of the cotton. To solve this problem, vertical spindle cotton pickers are developed in recent years [3]. It adopts a telescopic cotton removal method to avoid the reverse pulling by the cotton removal tray, which can efficiently improve the cotton fiber quality. Nevertheless, when using the vertical spindle cotton pickers, the spindles of cotton pickers may touch the soil. This may cause a deformation of spindles and decrease the cotton picking efficiency. Therefore, it is important to develop a height control system for the vertical spindle cotton picker.

Usually, there are two kinds of height control system, i.e., mechanical height control system and electro-hydraulic height control system. In early agricultural practice, the mechanical height control system is widely used in the no-till seeding [4] and transplanting machines due to its simple structure, easy control logic and low cost [5,6,7,8]. Unfortunately, as the control accuracy and response time of the mechanical height control system are poor, it cannot be used in cases that need high control accuracy and rapid response. To solve this problem, electro-hydraulic height control systems are developed by researchers [9, 10].

However, although the electro-hydraulic height control system has higher control accuracy than the mechanical height control system, the control of its hydraulic cylinder is very complex. In the past, many researchers have been engaged in this area and a variety of control methods have been proposed, such as proportion integration differentiation (PID) control algorithm [11], sliding-mode fuzzy control method [12], etc. Among these methods, PID control algorithm is widely used because of its features of simplicity, robustness and reliability [13]. Many improved PID algorithms are proposed based on the basic PID to enhance the probability of the suitable selection of the controller gains [14]. For example, in the work presented by Peng et al., they combined traditional PID with fuzzy control algorithm for planning trajectory of an electro-hydraulic robot [15].

A difficulty in a fuzzy PID control method is the design of fuzzy rules because it highly depends on the experience of experts or large amounts of experimental data [16]. Hence, the combinations with various nature-inspired optimization algorithms (NIOAs) have a significant impact on the performance of PID control. Other researchers used the NIOAs to optimize PID parameters and fuzzy controller, which can improve the control efficiency of the system. Examples of the approaches can be found in the works reported in References [17,18,19,20], such as Grey Wolf Optimizer algorithm [17], Particle Swarm Optimization [18, 19] and Simulated Annealing algorithm [20].

Summarizing the findings of the above discussion, it can be found that PID parameters have been optimized for hydraulic actuators to obtain accurate and precise control performance. However, the height control system of the cotton picker is a mechanical-electric-hydraulic coupled system. Its design should consider interactions between mechanical and electrical parts of the complete system. Several works about integrated optimization approach have been carried out in the past decades, such as parallel robots [21, 22], mesh reflector deployable space antennas [23], pipeline leak detection [24] and multistage synchronous induction coilgun [25]. It is confirmed that there is a lack of work in integrated optimization approach for the height control system.

Generally, for a height control system, its performance is dependent on its mechanical structure design and controller design. In designing such a system, the mechanical structure cannot be optimized without considering its influence on control. Conversely, the control performance, which can be optimized by using control design methods, can further be improved by modifying the mechanical structure. Hence, optimization of mechanical structure parameters and control parameters in an integrated manner can further improve the performance of the height control system [26, 27]. Despite the success achieved by integrated optimization approach, little work has been devoted to height control system. Motivated by those findings, this paper aims to bridge the research gap and make the following contributions. Firstly, an integrated optimization model of mechanical structure parameters and control parameters is proposed to take the pressure of the chamber without piston, response time and displacement error of height control system as the optimization objectives. Secondly, this model is solved by Optimization via Simulation method (OvS), Particle Swarm Optimization (PSO) and Simulated Annealing (SA). To our best knowledge, it is the first time that both mechanical structure parameters and control parameters are simultaneously considered in the proposed integration problem for the height control system.

The rest of the paper is organized as follows. Section 2 establishes mathematical models of this system. Section 3 presents an integrated optimization method. Section 4 gives the case study, followed by the conclusion in Section 5.

2 Mathematical Models

2.1 Structure of Height Control System

Figure 1 shows the schematic of the proposed height control system, which consists of signal acquisition device, control system, hydraulic system and lifting mechanism. The signal of height is obtained via the angle sensor installed on the signal acquisition device. Then the signal of height is used to govern the electro-hydraulic servo valve. The hydraulic cylinder controlled by electro-hydraulic servo valve is used as the power sector of lifting mechanism. With cylinder controlled by valve to lift the spindles of the cotton picker, it can avoid spindles colliding with stones on the cotton field.

Figure 1

Schematic graph of the height control system

2.2 Modeling of the Lifting Mechanism

The diagram of lifting mechanism of the height control system is shown in Figure 2. Line AB represents the hydraulic cylinder. Point B and point C are articulated on the frame of the spindle cotton picker corresponding to the installation position. L₁ represents the total length of the hydraulic cylinder, L₂ is the length of articulated rod AC, L₃ is the installation position on the frame, and L₄ is the distance of picking head centroid from articulated point A. L₁ can be further expressed as follows:

$$L_{1} = L_{0} + L,$$

(1)

where L₀ and L represent cylinder length and piston displacement of the hydraulic cylinder, respectively.

Figure 2

Schematic diagram of the lifting mechanism

For convenience of analyzing the dynamics of the lifting mechanism, it is simplified to an equivalent diagram as shown in Figure 3. Then, the dynamic model of the equivalent lifting mechanism is formulated as follows:

$$A_{1} P_{1} - A_{2} P_{2} = M\frac{{{\text{d}}^{2} L}}{{{\text{d}}t^{2} }} + B\frac{{{\text{d}}L}}{{{\text{d}}t}} + F_{L} ,$$

(2)

where $A_{1}$ is the area of the end face of piston, $A_{2}$ is the effective area of the piston in cylinder chamber with piston, P₁ is the input oil pressure of the chamber without piston, P₂ is the pressure of chamber with piston, M is the equivalent mass of the lifting mechanism, t is the elapsed time of the lifting mechanism from starting to move to the end of lifting, B is viscous damping coefficient of the piston, $F_{L}$ is the external load of the lifting mechanism.

Figure 3

Equivalent model of the lifting mechanism

Equivalent mass M is calculated based on the theory that kinetic energy of the equivalent lifting mechanism is equal to the original lifting mechanism. Mass of hydraulic cylinder, piston and articulated rod AC is negligible compared to the mass of picking unit. Besides, velocity of hydraulic cylinder, piston and articulated rod AC is same order of magnitude with velocity of picking unit. So kinetic energy of hydraulic cylinder, piston and articulated rod AC is ignored. Equivalent mass M is calculated based on the following Eq. (3):

$$\frac{1}{2}M\dot{L}^{2} = \frac{1}{2}m\left( {\dot{x}^{2} + \dot{y}^{2} } \right),$$

(3)

where m is the mass of picking unit, $m = {G \mathord{\left/ {\vphantom {G g}} \right. \kern-\nulldelimiterspace} g}$, G is gravitational force of picking unit, g is gravitational acceleration, $\dot{x}$ and $\dot{y}$ are two components of barycenter velocity of picking unit. $x$ and $y$ are calculated as follows:

$$\left\{ \begin{aligned} x & = \left( {L_{2} + L_{4} } \right)\sin \alpha , \\ y & = L_{3} - \left( {L_{2} + L_{4} } \right)\cos \alpha , \\ \end{aligned} \right.$$

(4)

$$\alpha = \arccos \left( {\frac{{L_{2}^{2} + L_{3}^{2} - L_{1}^{2} }}{{2L_{2} L_{3} }}} \right).$$

(5)

External load $F_{L}$ is calculated as shown in Eq. (6):

$$F_{L} = G\frac{{{\text{d}}y}}{{{\text{d}}L}}.$$

(6)

Based on Eqs. (1)‒(6), the dynamic model of the equivalent lifting mechanism is expressed as follows:

$$\begin{aligned} \frac{{\uppi }}{4}D^{2} P_{1} - \frac{{\uppi }}{4}\left( {D^{2} - d^{2} } \right)P_{2} & { = }\frac{{4L_{1}^{2} \left( {L_{2} + L_{4} } \right)^{2} }}{{4L_{2}^{2} L_{3}^{2} - \left( {L_{2}^{2} + L_{3}^{2} - L_{1}^{2} } \right)^{2} }}\frac{G}{g}\frac{{{\text{d}}^{2} L}}{{{\text{d}}t^{2} }} \\ & \quad + B\frac{{{\text{d}}L}}{{{\text{d}}t}}{\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{G\left( {L_{2} + L_{4} } \right)\left( {L_{0} + L} \right)}}{{L_{2} L_{3} }}. \\ \end{aligned}$$

(7)

The lifting mechanism of the height control system realizes the lifting function of the picking unit by extending the piston of the hydraulic cylinder. Thus, it can prevent the spindle from being bended in the collision with stones and metals. In Figure 4, point A is the articulation of the hydraulic cylinder and the picking unit. When the articulation is in the state of point A, the picking unit works in the normal position. The piston of the hydraulic cylinder does not extend and its stroke is zero. When the piston of the hydraulic cylinder moves to point A₁, the picking unit is lifted. In this process, the profile height h can be calculated as:

$$h = \frac{{L^{2} + 2L_{0} L}}{{2L{}_{3}}}.$$

(8)

Figure 4

Kinematic analysis of lifting mechanism

When the surface of cotton field changes, the angle variation of profiling slider is detected by the angle sensor. The angle sensor transmits the angle variation into the controller. The controller will output signal to make the piston rod of the hydraulic cylinder to extend out. The profile height h of the lifting mechanism is determined by the angle change of the angle sensor, which can be modelled as shown in Eq. (9):

$$h = l\Delta \theta ,$$

(9)

where l is the distance between angle sensor and touchpoint of the profiling slide, Δθ is the angle change of angle sensor.

2.3 Modeling of the Hydraulic Control System

When the hydraulic fluid is pumped into the cylinder of the height control system, the hydraulic pressure will force the piston to move back and forth. The displacement and speed of the hydraulic cylinder is determined by the opening size of the electro-hydraulic servo valve. Therefore, the height and rising speed of the picking unit are adjusted by the electro-hydraulic servo valve. In order to achieve precise position tracking control, a PID controller, which is widely used in various industries due to its high reliability and robustness [28,29,30,31], is adopted to control the electro-hydraulic servo valve.

As shown in Figure 5, the input of the PID controller is ΔU, whose value is the difference between the actual displacement signal U₂ and the input displacement signal U₁. After a proportional, integral and differential processing of the PID controller, the new displacement signal U is the output. Then the displacement signal U is converted into control current I of the electro-hydraulic servo valve by the servo amplifier. The transfer function of the servo amplifier is expressed in Eq. (10):

$$\frac{I}{U} = \frac{{K_{A} \omega_{A} }}{{s + \omega_{A} }},$$

(10)

where K_A is the amplification factor of servo amplifier, ω_A is the natural frequency of the amplifier.

Figure 5

Control block diagram of hydraulic system

The control current I of the electro-hydraulic servo valve controls the displacement of the valve core, thus resulting in the change of valve opening, and finally realizing the control of the flow q_v of the electro-hydraulic servo valve. The transfer function of the electro-hydraulic servo valve is [32]:

$$\frac{{q_{v} }}{I} = \frac{{K_{sv} \omega_{sv}^{2} }}{{s^{2} + 2\xi_{sv} \omega_{sv} s + \omega_{sv}^{2} }},$$

(11)

where K_sv is the gain of the electro-hydraulic servo valve, ω_sv is the natural frequency of the amplifier, ξ_sv is the damping ratio of the electro-hydraulic servo valve.

When the output fluid of the electro-hydraulic servo valve flows into the hydraulic cylinder, the piston of the hydraulic cylinder will extend out of the cylinder block. So that the picking unit will be lifted to the position of the height required by the system. The transfer function of the hydraulic cylinder is:

$$\frac{L}{{q_{v} }} = \frac{{K_{h} \omega_{h}^{2} }}{{s\left( {s^{2} + 2\xi_{h} \omega_{h} s + \omega_{h}^{2} } \right)}},$$

(12)

where K_h is the gain of hydraulic cylinder, ω_h is the natural frequency of hydraulic cylinder, ξ_h is the damping ratio of hydraulic cylinder.

When the displacement L of the piston rod is detected by the sensor installed on the hydraulic cylinder, it will be converted into actual displacement signal. The difference between the actual displacement signal and the input displacement signal are sent into the PID controller to adjust the displacement of the piston of the hydraulic cylinder. The transfer function of the displacement sensor is:

$$\frac{{U_{2} }}{L} = \frac{{K_{se} \omega_{se} }}{{s + \omega_{se} }},$$

(13)

where K_se is the gain of displacement sensor, ω_se is the natural frequency of displacement sensor.

3 Multi-objective Optimization of Height Control System

The height control system is a mechanics-electronics-hydraulics coupled system that consisting of signal acquisition device, control system, hydraulic system and lifting mechanism. For such a system, pressure in the chamber without piston, response time and displacement error are the most important performance indicators. A reduction of the pressure in the chamber without piston will decrease the pressure required by the hydraulic system and hence improve the reliability of the hydraulic system. Moreover, minimization the response time and displacement error can improve response speed and precision of the height control system. As a result, the probability of the spindle cotton picker, whose off-ground height can be adjusted by the height control system, colliding to the ground will be reduced. Hence, in order to reduce pressure in the chamber without piston and minimize response time and displacement error of the hydraulic system, a multi-objective optimization is adopted in this paper. Moreover, for the height control system, the pressure of the chamber without piston is highly dependent on the mechanical structure parameters, i.e., inner diameter of hydraulic cylinder D, length of hydraulic cylinder barrel L₀ and the length of rod AC L₂. The response time and displacement error is related to the combination of the mechanical structure parameters (i.e., inner diameter of hydraulic cylinder D, length of hydraulic cylinder barrel L₀) and control parameters. (i.e., proportion coefficient K_P, integration coefficient K_I and differential coefficient K_D). Hence, during the optimization process, the mechanical structure parameters and control parameters are both taken as the optimization variables. With the obtained optimal mechanical structure parameters and control parameters, a better height control system can be designed to reduce the probability of the spindle cotton picker colliding to the ground. In this section, a multi-objective optimization model is proposed firstly. Then an algorithm combined OvS, PSO and SA is proposed to solve the model.

3.1 Optimization Objectives and Variables

3.1.1 Pressure of the Chamber without Piston

In a hydraulic system, the lower the pressure in the chamber without piston is, the lower the pressure required by the hydraulic system is. As a result, the reliability of the hydraulic system will be improved. Hence, in this paper, the pressure of the chamber without piston $P_{1}$ is set as an optimization objective.

Oil return circuit is directly connected to the oil tank from the hydraulic schematic diagram, so P₂ = 0. According to Eq. (7), $P_{1}$ is expressed as follows:

$$\begin{aligned} P_{1} & { = }\frac{4}{{{\uppi }D^{2} }}\frac{{4L_{1}^{2} \left( {L_{2} + L_{4} } \right)^{2} }}{{4L_{2}^{2} L_{3}^{2} - \left( {L_{2}^{2} + L_{3}^{2} - L_{1}^{2} } \right)^{2} }}\frac{G}{g}\frac{{{\text{d}}^{2} L}}{{{\text{d}}t^{2} }} \\ & \quad { + }\frac{4}{{{\uppi }D^{2} }}\left[ { + B\frac{{{\text{d}}L}}{{{\text{d}}t}} + \frac{{G\left( {L_{2} + L_{4} } \right)\left( {L_{0} + L} \right)}}{{L_{2} L_{3} }}} \right]. \\ \end{aligned}$$

(14)

3.1.2 Response Time and Displacement Error

For the height control system of the spindle cotton picker, response time t₀ and displacement error e are two vital performance indicators. With the aim to improve the profiling precision, reduce response time of hydraulic cylinder and decrease the probability of spindle colliding to the ground, the response time t₀ and displacement error e are chosen to be optimization objectives. For the sake of algorithmic convergence, the response time t₀ and displacement error e are merged into one normalized objective H, which can be seen in Eq. (15):

$$H = \omega_{1} \frac{{t_{0} }}{{t_{opt} }} + \omega_{2} \frac{e}{{e_{opt} }},$$

(15)

where t_opt and e_opt are the response time and displacement error required by the height control system, ω₁ and ω₂ are the corresponding weights.

The hydraulic system of the height control system is complex with nonlinear characteristics. Its response time t₀ and displacement error e are influenced by multiple factors, such as the mechanical structure parameters and control parameters. During the optimization process, it is difficult to establish a precise mathematical model of the hydraulic system to acquire the response time and displacement error. Hence, a simulation model based on AMESim is proposed to acquire the response time and displacement error of the hydraulic system.

The AMESim is a kind of software serving for advanced modeling and simulation of engineering systems based on intuitive graphical interface. It is specialized in multi-disciplinary collaborative simulation especially for mechanics, electronics and hydraulics. As the height control system is a typical mechanical-electro-hydraulic system, several libraries are involved to build the whole model, like hydraulic component design library for the hydraulic pump, planner mechanical library for mechanical attachments and control library for control operations. Meanwhile, the required parameters such as the inner diameter of hydraulic cylinder D and length of hydraulic cylinder barrel L₀ are input in the model. The simulation model of the height control system can be seen in Figure 6.

Figure 6

AMESim model of the height control system

3.1.3 Variables

The decision variables considered for the integrated optimization problem can be classified into two categories. One is related to the structure and the variables needed to be decided are the inner diameter of hydraulic cylinder D, length of hydraulic cylinder barrel L₀ and the length of rod AC L₂. Another is the PID control parameters, i.e., proportion coefficient K_P, integration coefficient K_I and differential coefficient K_D.

3.2 Constraints

• (1) The inner diameter of hydraulic cylinder D should be selected from the Standard GB/T 2348, which can be seen in Eq. (16):

  $$D = \left[ {63,\;80,\;90,\;100,\;110,\;125,\;140,\;160,\;180} \right].$$

  (16)

• (2) The overall length of hydraulic cylinder L₁, length of rod AC L₂ and installation dimension L₃ are restricted to the geometry constraints of a triangle:

  $$\left\{ {\begin{array}{*{20}c} {L_{0} + L + L_{2} > L_{3} ,} \\ {L_{0} + L + L_{3} > L_{2} ,} \\ {L_{2} + L_{3} > L_{0} + L,} \\ \end{array} } \right.$$

  (17)

  where the overall length of hydraulic cylinder L₁ is the sum of the length of hydraulic cylinder barrel L₀ and travel range of piston L, installation dimension L₃ is constant.

  The length of hydraulic cylinder barrel L₀ is:

  $$L_{0} = L_{max } + B + L_{A} + L_{M} + L_{C} ,$$

  (18)

  where B is width of rod, B = (0.6‒0.8)D, L_A is the guiding distance of hydraulic rod, L_A = (0.6‒1.5)D, L_M is sealing distance of hydraulic rod determined by sealing method, L_C is the rest length of hydraulic cylinder barrel.

• (3) The length of hydraulic cylinder barrel L₀ is within twenty times the length of its inner diameter:

  $$L_{0} < 20D.$$

  (19)

• (4) The lifting height should be more than 100 mm when the hydraulic cylinder reaches its maximum travel range:

  $$\frac{{L_{max }^{2} + 2L_{0} L_{max } }}{{2L{}_{3}}} > 100,$$

  (20)

  where L_max is the maximum travel range of hydraulic cylinder.

• (5) The PID parameters are restricted by the reasonable ones:

  $$\left\{ \begin{aligned} 0 & < K_{P} < K_{Pmax} , \\ 0 & < K_{I} < K_{Imax} , \\ 0 & < K_{D} < K_{Dmax} , \\ \end{aligned} \right.$$

  (21)

  where K_Pmax, K_Imax and K_Dmax are the maximum values of K_P, K_I and K_D. The widely used method for acquiring these values is simulation tests. During each test, one of the PID parameters is changed while another two parameters remain unchanged. If the response time and displacement error of the height control system are within a reasonable range, then the value of the changed parameter will be recorded. The tests will last for several times until all the maximum values of the three PID parameters are settled.

3.3 Optimization Model

Based on the analysis above, a multi-objective optimization model with the aim to minimize the pressure of chamber without piston, response time and displacement error of the height control system, is then formulated as shown in Eq. (22):

$$\begin{aligned} & \min \;[P_{1} (L_{0} ,\;D,\;L_{2} ),\;H(L_{0} ,\;D,\;K_{P} ,\;K_{I} ,\;K_{D} )], \\ & {\text{s}}.{\text{t}}.,\left\{ {\begin{array}{*{20}ll} {D = \left[ {63,\;80,\;90,\;100,\;110,\;125,\;140,\;160,\;180} \right],} \\ {L_{0} + L + L_{2} > L_{3} ,} \\ {L_{0} + L + L_{3} > L_{2} ,} \\ {L_{2} + L_{3} > L_{0} + L,} \\ {\frac{{L_{max }^{2} + 2L_{0} L_{max } }}{{2L_{3}}} > 100,} \\ {L_{0} < 20D,} \\ {0 < K_{P} < K_{Pmax} ,} \\ {0 < K_{I} < K_{Imax} ,} \\ {0 < K_{D} < K_{Dmax} .} \\ \end{array} } \right. \end{aligned}$$

(22)

3.4 Optimization Solution

In this paper, the optimization model is solved via a simulation based optimization method. The response time and displacement error of the hydraulic cylinder are firstly acquired by the AMESim platform. Then the simulation results are chosen to be fitness functions of the multi-objective optimization model. For the discrete structure parameters, i.e., inner diameter of hydraulic cylinder D, length of hydraulic cylinder barrel L₀ and length of rod AC L₂, they are firstly searched using SA and the optimal control parameters K_P, K_I and K_D for each set of structure parameters are then obtained through PSO algorithm. The reason for choosing SA and PSO is that the SA is capable of high searching efficiency and fast convergence speed for solving discrete optimization problem [33] and so does the PSO for continuous optimization problems [34, 35]. The flowchart of the algorithm is shown in Figure 7 and the detailed procedure is described as follows.

Figure 7

Simulation based optimization algorithm flow

• Step 1: Set the parameters of SA and PSO. Generate initial solution x_current(D, L₀, L₂, K_P, K_I, K_D) and calculate objective function object(P₁, H). Set x_SA_current as x_current(1:3).

• Step 2: Enter the outer loop of SA. Judge whether the temporary temperature temp is greater than the minimum temperature temp_min set in Step 1 or not. If so, go to Step 3. Otherwise, go to Step 10.

• Step 3: Judge whether the current iteration count g is less than the maximum iteration count g^M or not. If so, go to Step4. Otherwise, go to Step 9.

• Step 4: Randomly generate the adjacent solution x_SA_new of x_SA_current in the outer loop of SA and calculate the pressure object _SA_new(P₁) of the chamber without piston in the hydraulic cylinder.

• Step 5: Enter the inner loop of PSO and calculate the optimal PID parameters corresponding to the structural parameters x_SA_new generated in Step 4. The details are as follows.

  • Step 5.1: Initialize the PSO, generate the initial solution x_pos_current(K_Pi, K_Ii, K_Di). Invoke the model in AMESim and get the output results of response time and displacement error of hydraulic cylinder, namely the value of objective function object_pso_current(H).

  • Step 5.2: Update the velocity v_id and position x_id of every particle according to Eqs. (23) and (24) and generate the adjacent solution x_pos_new(K_Pi, K_Ii, K_Di) of PSO. Invoke the model in AMESim and obtain the output results of response time and displacement error of profiling hydraulic cylinder, namely the value of objective function object_pso_new(H).

    $$v_{id}^{T + 1} = \omega v_{id}^{T} + c_{1} r_{1} (p_{id} - x_{id}^{T} ) + c_{2} r_{2} (p_{gd} - x_{id}^{T} ),$$

    (23)
    $$x_{id}^{T + 1} = x_{id}^{T} + v_{id}^{T + 1} ,$$

    (24)

    where $v_{id}^{T}$ and $x_{id}^{T}$ are velocity and position of the dth (d = 1, 2 or 3) dimension of the ith particle in the Tth iteration, p_id is the best position of each particle during the iterative process, p_gd is the best position of all particles in the temporary iterative procedure, c₁ and c₂ are learning factors, r₁ and r₂ are two random numbers, ω is inertia weight.

  • Step 5.3: Update the iteration count of PSO, T = T + 1.

  • Step 5.4: Judge whether the current iteration count T is greater than the maximum iteration count T_max or not. If so, go back to Step 5.2, otherwise go to Step 5.5.

  • Step 5.5: Generate the optimal PID parameters x_pos_new that are related to the structure parameters from the outer loop of SA.

• Step 6: Set x_new(1:3) as x_SA_new, x_new(4:6) as x_pos_new and value of objective function object_new as (P₁, H).

• Step 7: Judge the dominating relationship of the solutions in SA. The details are as follows.

  • Step 7.1: Judge whether x_new is dominated by x_current or not. If not, set x_current as x_new and add x_new to Pareto Archive, otherwise, go to Step 7.2.

  • Step 7.2: Calculate the probability of acceptance Prob according to Eq. (25):

    $$Prob = \exp \left( {\frac{ - \Delta E}{{temp}}} \right),$$

    (25)

    where ΔE = E(f(x_new), λ) − E(f(x_current), λ), E(f(x), λ) = λ₁P₁ + λ₂H. λ₁ and λ₂ are set as 0.5.

  • Step 7.3: Generate a random number Rand and judge whether Rand is less than Prob or not. If so, set x_current as x_new and add x_new to the Pareto Archive. Otherwise, retain x_current.

• Step 8: Update the iteration count of SA, g = g + 1. Go back to Step 3.

• Step 9: A commonly-used exponent cooling strategy is adopted in the outer loop of SA:

  $$temp = a \times temp,$$

  (26)

  where a is the cooling factor of SA and faster cooling speed is achieved with less value of a. Go back to Step 2.

• Step 10: Output the Pareto Archive.

4 Case Study

To validate the proposed multi-objective optimization model and algorithm and gain a better understanding of the influence of structure parameters and control parameters on pressure of the chamber without piston, response time and displacement error, a series of experiments are carried out in this section.

4.1 Optimization Results

To verify the effectiveness of the proposed approach, three optimization cases are studied. Hydraulic parameters and structure parameters of the height control system for the cotton picker are shown in Table 1. Parameters of SA and PSO are given in Table 2. Optimization results are shown in Table 3.

Case 1: Height control system with original structure parameters and PID control parameters.

Case 2: Height control system with optimal PID control parameters but original structure parameters.

Case 3: Height control system with optimal structure parameters and PID control parameters obtained by the proposed integration optimization approach.

Table 1 Hydraulic parameters and structure parameters of height control system

Table 2 Algorithm parameters of height control system

Table 3 Optimization results

4.1.1 Simulation Results under an Identical Height

According to the structural parameters of the height control system and displacement of the hydraulic cylinder, the pressure of the chamber without piston P₁ for the three cases is calculated and plotted in Figure 8.

Figure 8

Pressure of hydraulic cylinder for different cases

From Figure 8, it can be seen that the maximum pressure of Case 1, Case 2 and Case 3 is 29.07 MPa, 29.07 MPa and 24.81 MPa, respectively. Compared with Case 1 and Case 2, the maximum pressure of Case 3 is reduced by 14.65%. Furthermore, it also can be found that the average pressure of Case 1, Case 2 and Case 3 is 11.82 MPa, 11.82 MPa and 12.31 MPa, respectively. The average pressure is slightly increased by 4.15% when compared with Case 1 and Case 2. Note that the maximum pressure and average pressure of Case 1 and Case 2 are the same. This is because the load of the hydraulic cylinder will not be changed if only control parameters are optimized but structural parameters remains unchanged.

To study the response characteristics of the three cases under the same height, an AMESim simulation is performed. During the simulation process, the output load of the hydraulic cylinder is set to be the maximum load. As the step signal is the most common and useful reference signal for judging the response characteristics of a controlled system, during the simulation, the step signal is used to study the response characteristics of the height control system. The lifting height of spindle cotton picker is set to be 100 mm. The displacement of the hydraulic cylinder for the three cases is calculated by Eq. (8) and the piston displacement of Case 1, Case 2 and Case 3 is 80 mm, 80 mm and 72 mm. Simulation is carried out with step signal as shown in Figure 9.

Figure 9

Displacement of the hydraulic cylinder before and after optimization

In Figure 10, the response curves of Case 1, Case 2 and Case 3 is given. The response time of the three cases are 0.98s, 0.75s and 0.48s respectively. Moreover, the displacement error of the three cases is 4.0%, 2.6% and 1.1%. The response time and displacement error of Case 3 are less than that of Case 1 or Case 2. This is because the displacement of the piston rod L for Case 3 is smaller than that of Case 1 and Case 2 under the same required height. Moreover, from Figure 10, it can be found that the control parameters has a great influence on the response time and displacement error of the height control system.

Figure 10

Comparison of response curves under the same height

4.1.2 Simulation Results under an Identical Displacement

In order to verify the performance of the height control system, comparative simulations are made on the three cases with the same displacement of 80 mm. From Figure 11, it can be seen that the maximum pressure of the chamber without piston for Case 1 and Case 2 is 29.07 MPa, while the maximum pressure of Case 3 is only 24.81 MPa. Compared to Case 1 and Case 2, the maximum pressure of Case 3 is reduced by 14.65%. From Figure 12, it can be found that the response time of Case 3 is 0.62 s, which is reduced by 18.42% compared with Case 1 and Case 2. Besides, the displacement error of Case 3 is reduced by 1.4%. It can be found that significant improvement is achieved with the proposed approach.

Figure 11

Pressure comparison of the three cases

Figure 12

Comparison of displacement response

From the analysis above, it can be found that the optimal control parameters will vary with different structure parameters, optimization of structure parameters and control parameters in an integrated manner can further reduce the pressure, shorten the response time, and improve the positioning accuracy of the height control system.

4.2 Experimental Validation

According to the optimal structure and PID control parameters, as shown in Figure 13, a height control system for a cotton picker is constructed. In order to verify its performance, a practical test is carried out. Obstacles with the height of 100 mm, 80 mm and 50 mm are put on a smooth road to simulate the fluctuation of cotton field ground. The cotton picker travels through obstacles at a working speed of 5.8 km/h. The response time of the height control system is recorded as shown in Table 4. From Table 4, it can be found that the response time is 0.55 s, 0.43 s and 0.28 s under the height of 100 mm, 80 mm and 50 mm. All the response time is less than 0.93 s, which is the maximum response time to lift the picking unit to avoid colliding the obstacles at a speed of 5.8 km/h. This proves that the optimal structure and control parameters of the height control system can meet the requirements.

Figure 13

Height control system of spindle cotton picker

Table 4 Response time of the height control system

5 Conclusions

In this paper, a vertical spindle cotton picker has been presented to solve existing problems in the traditional spindle cotton picker. A mechanical-electric-hydraulic coupled height control system is designed for the vertical cotton picker to avoid collision between spindles and bumps on the cotton field, and improve the efficiency of cotton picking. The conclusions are as follows.

• (1) Both structure and control parameters are optimized to improve terrain-tracking performance of the height control system. A multi-objective integrated optimization model is established, which takes the pressure, response time and displacement error as the optimization objectives.

• (2) The multi-objective integrated optimization model is solved by using the optimization method based on AMESim simulation. Experimental results show that the maximum pressure of the hydraulic cylinder, the response time and the displacement error can be effectively reduced by 14.65%, 18.42% and 1.4%, respectively.

• (3) Future work will be focused on the following aspects. Effects of the SA and PSO algorithm parameters on the optimization results of the height control system are not analyzed in the current study. It is expected that these works will be the future work. More improved optimization algorithms such as Artificial Bee Colony and Cuckoo Search algorithm [36,37,38] will be proposed to solve the proposed multi-objective optimization model.