Dual-Manipulator Optimal Design for Apple Robotic Harvesting

Abstract

In order to ensure canopy area coverage with the most compact mechanical configuration possible, this paper proposes a configuration optimization design method of dual-manipulator to meet the research and development needs of an apple-efficient harvesting robot using the typical tree shape of a “high spindle” in China as the object. A Cartesian coordinate dual-manipulator with two groups of vertically synchronous operations and a three-degree range of motion based on the features of the spatial distribution of fruits under a typical canopy of dwarf and close planting was designed. Two-stage telescoping components that can be driven by both gas and electricity are employed to ensure the picking robotic arm’s quick response and accessibility to the tree crown. Based on the quantitative description of the working space and configuration parameters of the dual-manipulator, a multi-objective optimization model of the major configuration parameters is constructed. A comprehensive evaluation method of the dual-manipulator configuration based on the CRITIC–TOPSIS combined method is proposed. The optimal solutions of the lengths and elevations of upper and lower telescopic parts of the dual-manipulator and the distance from the mounting base of the outer frame of the dual-manipulator to the center of the tree trunk are determined, which are 1119.3 mm and 39.4°, 898.7 mm and 26°, 755.3 mm, respectively. The interaction between the configuration parameters of the dual-manipulator and its working area is then simulated and examined in order to verify the rationality of the optimum configuration settings. The results show that the optimal configuration of the dual-manipulator can fully cover the target working space, and the redundancy rate is 16.62%. The results of this study can be utilized to advance robotic fruit-picking research and development.

1. Introduction

China leads the world in apple planting area and production, generating 55% of all apples [1], which is a crucial assurance for the secure supply of fruits for humans and a rise in farmers’ incomes. However, in recent years, with the decrease in the agricultural labor force, the labor cost of apple planting and production has been rising, reaching about 66% of the total production cost [2]. In orchard production, the mechanization rate of fruit harvesting is the lowest, less than 3%, especially in the harvest season of short-term explosive employment demand, and the problem of “difficult and expensive employment” is particularly prominent. Because fresh apples need to have good appearance quality, developing robots with selective and accurate harvesting ability is expected to solve the problems faced by apple production at present and achieve the goal of alleviating labor intensity and improving harvesting efficiency. This research direction has been a wide concern in the world in recent years [3,4].

A picking manipulator is the key component of a picking robot to locate the position and posture of the picking gripper according to the growth posture of fruit [5], which directly determines the working space and efficiency of the picking robot. To guarantee proper functioning of the robot, it is crucial to combine the planting mode and production efficiency needs of a particular picking object, choose the configuration of the picking robotic manipulator and optimize the design of its structural parameters.

According to the number of manipulators that can be driven, there are currently two types of fresh-fruit-picking robots: single-manipulator configuration and multi-manipulator configuration. When choosing goods with a relatively narrow distribution area in greenhouses, such as strawberries, tomatoes, sweet peppers, etc., the single-manipulator picking robot’s configuration is frequently utilized. The single-manipulator configuration is mainly articulated to adapt to the narrow operating space in the greenhouse and meet the harvesting requirements of mature fruits in specific areas. The “RUBION” automated damage-free strawberry-picking robot was created by Preter et al. [6] and has a customized five-DoF (degree of freedom) articulated mechanical arm. Its prototype can pick a strawberry in 4 s and is mostly employed in tiny working spaces, such as greenhouse scenes. Arad et al. [7] developed a sweet pepper harvesting robot named “SWEEPER”, and its prototype includes a six-DoF articulated industrial manipulator model Fanuc LR Mate 200iD. The robotic manipulator of “SWEEPER” is installed on a movable trolley with a scissor lift mechanism and harvests sweet peppers within a 200–290 mm reachable three-dimensional area in front of the robot body. The average cycle time for harvesting a single fruit (including fruit localization, obstacle localization, visual serving, detaching fruit) is 24 s. The cherry tomato picking robot developed by Feng et al. [8] used a DENSO VS-6556 six-DoF articulated robotic manipulator to pick tomatoes in the 600–1200 mm area with a distance of 550 mm and put the fruits into the basket after picking. Its picking efficiency is 8 s in each cycle. The apple harvesting robot by Abundant Robotics [9] adopts a parallel mechanical arm and negative pressure air suction picking end, and the average picking efficiency is one apple per second, which greatly reduces manpower work.

For the fruits distributed in a tall tree canopy, the picking robot needs to have a larger working area and picking efficiency. The standard mechanical manipulator configuration used by the greenhouse fruit and vegetable picking robot is difficult to meet the needs, and the multi-manipulator configuration with parallel operation capability [10,11,12,13,14] has been continuously applied to picking robots [15].

Xiong et al. [16] developed an automatic strawberry-picking robot. The overall picking manipulator is composed of two three-DoF Cartesian coordinate picking robotic arms running on a single track and a self-developed and designed end. The machine vision system is used to detect the area of 1200 mm × 500 mm within 500 mm of the robot body for positioning and picking ripe strawberries. Working with both arms at the same time can improve the work efficiency by half, and the average picking time of a strawberry is 4.6 s. Williams et al. [17] developed a kiwifruit selective intelligent harvesting robot. The robot has four separate platforms, and each platform is installed with a three-DoF articulated harvesting robotic arm. Each robot arm independently picks kiwifruit cultivated in the pergola-style frame within a range of about 1.7–2.0 m from the ground and about 3 m wide. Compared with the single-arm picking cycle of 3.36 s per piece, the picking robotic arms on the four platforms work at the same time, and the average harvest time of kiwifruit is only 0.84 s per piece. Israeli company FFRobotics (Gesher HaEts, Israel) has developed an apple harvesting robot for apple orchards that adopts fence cultivation mode [18]. Three parallel tracks are arranged on each side of the prototype, and two picking robotic arms are installed on each track, a total of 12 three-DoF robots. The Cartesian coordinate picking robotic manipulator can simultaneously pick multiple apple tree crowns in the fruit rows on both sides of the fruit road. FFRobotics company claims that the picking efficiency of high-quality apples is 10 times higher than that of manual work. AGROBOT Robotics of Spain (Huelva, Spain) has developed a picking robot for strawberry high-ridge cultivation and frame cultivation [19]. Six independent parallel tracks are set on the main body, and each track runs four three-DoF Cartesian coordinate picking robotic arms (a total of 24 independent arms) working together on four rows of strawberries at the same time, which greatly improves the efficiency of strawberry picking. In view of this, a brief summary of the above-mentioned manipulator configuration is carried out, as shown in

Table 1. A table with shortcomings in representative manipulators and products.

Classification of Existing Manipulators	Picking Object	Large-Scale Operation Area	Compact Structure	Representative Product	Shortcoming
Articulated manipulator	Greenhouse strawberry, sweet pepper	No	Yes	RUBION [7], SWEEPER [8]	Used for greenhouse fruits, which cannot cover a large working area
Parallel manipulator	Apple	Yes	Yes	Abundant Robotics [10]	The manipulator has complex configuration and difficult to design multiple parallel manipulators
Multi-Cartesian manipulator	Apple	Yes	Yes	FFRobotics [18]	Mainly aimed at the American vertical trellis cultivation mode

. By selecting representative manipulator configurations and products and analyzing their shortcomings, some ideas are provided for the configuration of this paper.

The optimization and determination of its structural characteristics are also a prerequisite for the design of the manipulator in terms of workspace coverage, dynamic control precision and cloud route planning depending on the requirements of various jobs. According to the research and development needs of the greenhouse cucumber picking robot, Feng Qingchun [20] et al. established the quantitative relationship of each arm length, installation position and picking space of a five-DoF articulated picking arm and selected the optimal arm length parameter, which effectively covered over 90.5% of the fruit distribution area. Using finite element virtual simulation technology, Sun Feng et al. [21] constructed a multi-objective optimization model according to the dynamic characteristics of six-DoF articulated manipulator joints and solved the optimal combination of manipulator structural parameters, which reduced the average angular velocity of each manipulator’s joint to varying degrees. The maximum descending range can reach 9.3654% compared with that before optimization, which effectively improves the stability of the end position and posture of the manipulator. Zhao Jiangbo [22] et al. established a multi-objective optimization model with a linear weighting of three factors in order to comprehensively optimize the cooperative workspace, load capacity and end motion accuracy of the manipulator with two six-DoF joints and obtained the optimal structural parameters of the manipulator by using the particle swarm optimization algorithm. The cooperative workspace of the two arms increased by 9.5%, the end motion speed increased by 7.8%, and the load capacity of the mechanical arm increased by 6.1%.

For the apple-picking robot, the effective coverage of the working area of the tall tree canopy by the robot working space is a necessary condition for the design of the picking robotic manipulator. In this paper, the high-spindle-shaped apple tree widely planted in China is taken as the object, and the vertical parallel operation of the dual-manipulator picking mechanism is studied according to the spatial shape of the tree canopy. According to the relationship between the structural parameters of the manipulator and its working space, a multi-objective optimization model of the parameters of its key components is established. By solving the optimal configuration parameter combination, the purpose of covering the fruit picking area with a compact machine is achieved. On this basis, the optimal configuration is analyzed and verified by simulation experiments. This research can provide a design basis for research and development of apple-picking robots.

2. Design Specifications for Apple Harvesting Robot

2.1. Standard Spindle-Shaped Tree

In recent years, application of dwarf and close planting cultivation technology in standard apple orchards has become a hot trend in different apple-producing areas in China [23], which also laid a good foundation for intelligent and mechanized operation of orchard production. To improve the efficiency of orchard production management and facilitate mechanized operations, the spindle-shaped standard fruit tree shape is widely used in China [24], which has the characteristics of early fruiting, high land utilization and large yield. As shown in Figure 1a, the fruit branches are pruned in a standardized manner, and there are support rods behind the fruit trees, which are fixed and constrained by vertical equal-spaced steel wire ropes installed along the tree row so that about 25 branches of each fruit tree grow along the steel wire ropes. The crown is a high spindle shape. The row spacing of fruit trees is 3500 mm, the plant spacing is 1000 mm and the tree height is 3000–3500 mm. Fruit trees have a crown width between 1000 and 1500 mm during the fruiting season and a side thickness between 400 and 500 mm. A vertical fruit wall is created when the crowns of fruit trees next to one another touch and cross over one another. To collect fruit from either side of the crown, the robot alternately goes back and forth between the rows of fruit trees. (For an illustration, see Figure 1a,b).

2.2. Definition of Harvesting Workspace

Constrained by the shape of spindle-shaped branches, the spatial distribution densities of fruits in the canopy vary (as shown in Figure 2a,b). The canopy of 3-year-old Fuji apple trees in northern China was divided into sampling blocks with length, width and height of 50 cm × 20 cm × 50 cm in different layers and orientations, and the sampling blocks with height less than or equal to 1000 mm from the ground belonged to the bottom of the canopy, 1000–2000 mm belonged to the middle of the canopy and more than or equal to 2000 mm belonged to the top of the canopy. Each layer is divided into the inner chamber (<200 mm away from the trunk) and outer chamber (>200 mm away from the trunk), with the trunk as the central axis. The number of fruits in each sampling block and canopy level was counted separately. It is obvious that the fruit density of different canopy layers of the apple high-spindle canopy gradually increases from top to bottom. The fruit density of the upper layer was lower, 5 fruits/m², and the density of the middle and lower layers was higher, 12 fruits/m² and 13 fruits/m². In general, the fruits in the canopy are mainly distributed at 1000 mm–2500 mm from the ground, accounting for about 95% of the total number of fruits.

Because of the low density of fruit distribution at the edge of the canopy, combined with the compact design requirements of the robot picking manipulator, it is considered that the middle and lower areas of the canopy with high fruit density are the picking operation space, so the target space of picking operation is determined as follows: the trunk of the fruit tree as the center, a space with a length of 1500 mm, a width of 400 mm and a height of 1000–2500 mm from the ground (as shown in Figure 2c,d).

3. Dual-Manipulator Prototypes for Apple Harvesting

3.1. Mechanical Configuration Design

The design requirement of the picking robotic arm is to drive the picking claws to move between the tree crown and the robot body according to the spatial position of the fruit in the robot’s field of view to separate the fruit from the branches and put it into the fruit collection basket. For the standard apple tree shape as described above, the apple fruits are concentrated in the area of a tall and large cuboid. The Cartesian coordinate manipulator combined with the linear motion mechanism has a good match between the working space and the fruit growing area, so it has good applicability. In addition, unlike the manipulator with a tandem joint coordinate configuration, the Cartesian coordinate manipulator has no singularity problem. The current picking scene does not require the rotation of the manipulator, and the Cartesian coordinate manipulator can have a larger picking work area. To improve the working space and efficiency of the traditional Cartesian coordinate manipulator, this paper designs a dual-manipulator configuration arranged vertically in parallel to meet the needs of picking robots for fruit picking in tall tree canopies.

The dual-manipulator prototype (as shown in Figure 3a) primarily consists of two sets of picking mechanical arms aligned longitudinally and their respective independent driving components. Three degrees of freedom—horizontal, vertical and telescopic—are available for each picking arm. The mounting base of the outer frame is set as the origin $O$, the horizontal movement is parallel to the $X$-axis direction and the vertical movement is parallel to the $Z$-axis direction. Two sets of horizontal driving components travel up and down along the vertical driving components in turn, and its vertical motion driving mechanism is fixed to the picking robot body. The translating picking arms move along the corresponding horizontal drive members. The picking arm’s telescoping movement is a two-stage driving system that consists of a pneumatic telescopic component and an electric drive servo component. The pneumatic telescopic part is driven by a cylinder, which is used to drive the picking gripper to move back and forth quickly in the non-picking area between the robot and the crown, improving the positioning efficiency of the robot to the picking end gripper. The electric drive servo component is driven by a servo motor, and the motion displacement and speed are precisely controllable (as shown in Figure 3b).

The picking robot controller receives the three-dimensional position data of the fruit in the canopy obtained by the vision sensor and divides the picking area of the dual-manipulator in real time. According to the close-up cameras installed on the telescopic parts, the two sets of robotic arms pick and collect the fruits in the picking area in turn. The picking gripper that performs the picking task has two degrees of freedom: rotation and opening and closing, which can “screw” the target fruit. The main material of the claws is made of aluminum alloy, which can ensure rigidity and light weight. The contact part of the fruit is made of flexible material, and it is coated on the aluminum alloy frame of the claw, which can well protect the surface of the picked fruit. Conveying devices for fruit collection tasks are divided into horizontal and vertical. Horizontal conveying comprises upper and lower transverse conveying devices, wherein the device is provided with a conveying channel, flexible baffles are arranged on both sides of the conveying direction and transposition elbows are installed at the conveying endpoint, which, respectively, follows the upper and lower horizontal track for synchronous lifting, collect fruits harvested by picking grippers and transport them to the vertical conveying device through the transposition elbows. The vertical conveying device is designed with two conveying passages, which, respectively, convey the fruits delivered by the upper and lower horizontal transposed elbows. The fruits flow out from the back outlet of the vertical conveying device and are collected by the universal fruit transfer frame.

3.2. Definition of Dual-Manipulator’s Workspace

For the picking operation of spherical fruits, the picking robotic manipulator drives and positions the picking grippers with the three-dimensional coordinates of the center of the fruit as the target position. The working space of the picking manipulator refers to the set of spatial points that its end can reach due to its configuration and structure [25]. Obtaining the maximum coverage of the target working space with a compact configuration size is an important index to evaluate the configuration design of the manipulator. Therefore, establishing the relationship between the configuration parameters of the picking manipulator and its working space is the premise of the configuration optimization design. Especially, for the approximately rectangular canopy area, the accessibility of the picking robotic manipulator in the canopy height and depth directions is a quantitative representation of the picking robot’s working space.

The picking manipulator and the canopy target working space are parameterized (as shown in Figure 4). The height of the mounting base of the picking robot arm from the ground (denoted by $B$) is 300 mm; taking the mounting base of the robotic arm as the origin $O$, the picking direction of the picking arm is the $X$-axis, the height direction of the robot body frame is the $Z$-axis and, on the $XOZ$ plane, create a coordinate system. Assume that the depth of the target picking canopy is represented by $W$, the height of the upper limit position of the target picking canopy from the ground is represented by $T$ and the height of the lower limit position of the target picking canopy from the mobile platform is represented by $K$.

The quadrilateral region produced by extending the edge of the canopy to the center of the canopy at each elevation angle is designated as $A$ on the $XOZ$ plane when the double picking robotic arms are at the upper and lower height limit locations. On the $XOZ$ plane, the projected area of the chosen target working space is $A_0$. The double-arm invalid picking space $A_1$ is the remaining space, and the predicted invalid picking areas for the robotic upper and lower picking arms are $A_u$ and $A_d$, respectively. The center distance between the outer frame installation base and the trunk of the tree is $G$, and the maximum reachable height of the upper arm is $H$. Set the length of the upper picking manipulator arm as $L_u$ and the inclination angle with the horizontal as ${\theta }_u$. The length of the lower picking manipulator arm is $L_d$, and the inclination angle to the horizontal is ${\theta }_d$. The total structure length $L_m$ is the sum of $H$, $L_u$, $L_d$. When the lower picking arm is at the lower limit position, the height from the bottom of the mobile platform is $L_h$. The length $\beta$ of the telescopic module in the rear section of each picking arm is set to 3/4 of the length of its associated picking arm (when the picking task is completed, the picking arm can be retracted into the frame as much as possible to improve the utilization rate of the internal space of the frame).

Taking the trunk of the tree as the center, the fruits on one side of the tree are distributed in the blue rectangular area. Four boundary points $P_i ( i$ = 1, 2, 3, 4) are the initial conditions, and the reachable space at the end of the picking robot arm is the red area $R_j (j$ = 1, 2, 3, 4), the upper picking limit point at the end of the upper picking robot arm is $R_{umax}$($R_2$) and the lower picking limit point at the end of the lower picking robot arm is $R_{dmax}$($R_3$).

4. Optimization and Evaluation of Key Configuration Parameters

4.1. Two-Objective Optimization Model

Considering that the size of the component configuration directly determines the overall volume, load and stiffness of the manipulator, it is necessary to use the compact picking manipulator structure to cover the picking target area. In view of the fact that the working space of the picking manipulator is mainly determined by the length of the upper and lower picking arms and their elevation angle and is directly related to the overall size of the picking manipulator, this paper takes the configuration parameters as the parameters that need to be optimized.

As shown in Figure 4, when the upper and lower picking arms are at their respective extreme positions, this makes the $R_{umax}$ point on the extension line of the $P_2{P}_3$ line $R_{umax}$ coincide with point $P_3$ or be below point $P_3$, and the lower picking arm is at the bottom. When the limit position is retracted, the lower endpoint $P_m$ of the rear section of the manipulator may interfere with the moving platform. In order to minimize the redundancy ratio $\gamma$ of the workspace (i.e., the invalid workspace $A_1$) of the picking robot, an objective optimization function of “avoiding invalid workspace” should be established, which accords with the structural parameters of each picking arm and the conditions of three limit feature points: $R_{umax}$, $R_{dmax}$ and $P_m$. Specifically:

$$\{\begin{array}{c}\delta \left(L_u,{\theta }_u,L_d,{\theta }_u,G\right)=A_u+A_d\hfill \\ A_u=\frac{1}{2}W\times \left[G\times \mathit{tan}{\theta }_u+H-T\right]\hfill \\ A_d=\frac{1}{2}W\times \left[2K+\left(W-2G\right)\times \mathit{tan}{\theta }_d-2L_h\right]\hfill \end{array}$$

where $H=T-\left(G-W\right)\times \mathit{tan}{\theta }_u$, $L_h=\beta \times L_d\times \mathit{sin}{\theta }_d$, $\gamma =\frac{A_1}{A}$;

At the same time, to make the picking robot move more flexibly when performing the picking task, the configuration should be compact on the premise of meeting the requirements of the target picking space task of the dual-arm picking robot. Therefore, an objective optimization function is established to make the total structure length $L_m$ small and achieve a “compact structure”.

The three characteristic limit points $R_{umax}$, $R_{dmax}$, $P_m$ and the proposed objective optimization function are used to limit the optimization conditions and set the parameter change interval of the distance between the outer frame mounting base and the center of the tree trunk as $G$ as $\left[G_{io},G_{ic}\right]$; the parameter variation interval of the upper picking arm length $L_u$ and the lower picking arm length $L_d$ is $\left[L_{io},L_{ic}\right]$; the parameter variation intervals of the inclination angle ${\theta }_u$ between the upper arm and the horizontal and the inclination angle ${\theta }_d$ between the lower arm and the horizontal are $\left[{\theta }_{io},{\theta }_{ic}\right]$. Reasonably limit the range of these five design variables so that the two objectives of “avoiding invalid workspace” $\delta \left(L_u,{\theta }_u,L_d,{\theta }_u,G\right)$ and “compact structure” $\phi L_u,{\theta }_u,L_d,{\theta }_d,G$ can be minimized when meeting the picking requirements, namely:

$$\mathit{min}\{\begin{array}{c}\delta \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)\\ \phi \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)\end{array}$$

$$s.t.\left\{\begin{array}{c}{L}_{io}\le L_u,L_d\le L_{ic};{\theta }_{io}\le {\theta }_u,{\theta }_d\le {\theta }_{ic};\\ G_{io}\le G\le G_{ic};L_u,L_d,{\theta }_u,{\theta }_d,G\in 0.1\times N;\end{array}\right\}$$

The multi-objective optimization equation in the above formula is solved by the non-dominated sorting genetic algorithms (NSGA-II) [26,27]. NSGA-II is a genetic algorithm method that uses fast non-dominated sorting and elite mechanism to solve multi-objective optimization problems. The implementation steps are as follows:

(1) Taking $a_w=\left\{1, 1, 1, 1\right\}$ as the corresponding fusion weighted initial values of images in different wavebands, adding random numbers $r\in \left(-1,1\right)$ to each coefficient to generate an initial population of fusion coefficients and performing fast non-dominant sorting on the initial population; (2) carrying out the binary selection, crossover and variation on the initial population to obtain a new population; (3) merging the new population with the initial population into a new population, performing non-dominant ranking on all individuals and calculating the crowding degree in the non-dominant set; (4) when generating a proper number of first-generation populations, the individuals with low non-dominant level are preferred, and, when they are at the same level, the individuals with high crowding distance are preferred to ensure individual diversity; (5) if the evolutionary algebra reaches 1000, the algorithm is stopped and the contemporary population is taken as the optimal Pareto solution set. Otherwise, go to step (2) to continue the loop. In order to better understand the NSGA-II algorithm, its flowchart is as follows (Figure 5).

4.2. Optimal Solution Selection Based on CRITIC–TOPSIS

Because multi-objective optimization problems often obtain a solution set consisting of many non-inferior solutions (effective solutions), it is necessary to combine the actual needs to determine the optimal objective solution. The TOPSIS method [28] (technique for order preference by similarity to an ideal solution method) is an effective method for multi-objective decision analysis of finite programs. It obtains a comprehensive evaluation of the individuals in the target solution set by calculating the distance between the evaluation-specific solution and the corresponding positive and negative ideal targets. In particular, considering the difference in volatility and correlation of $\delta \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)$, $\phi \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)$ solution sets (hereinafter referred to as $A_1$, $L_m$), their contributions to the optimal results are different. This paper introduces the CRITIC weight method [29], which normalizes the solution sets of the two optimization objectives and then weighs and evaluates them. The calculation steps are:

(1) Pareto solution sets have different schemes. The solution sets of $A_1$, $L_m$ are composed of decision matrix $S$:

$$S={\left(S_{ij}\right)}_{n\times 2} \left(i=1, 2,\cdot \cdot \cdot ,n;j=1, 2\right)$$

(2) To eliminate the influence of dimensions on the evaluation results, each element in the decision matrix is standardized and both objectives are reverse indicators:

$${\beta }_{ij}^{*}=\frac{S_{jmax}-S_{ij}}{S_{jmax}-S_{jmin}}$$

where $S_{jmax}$, $S_{jmin}$ are the largest and smallest elements in column $j$.

(3) Calculate the standard deviation of each optimization objective in the decision matrix as follows:

$$D_j=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{(S_{ij}-{\overline{X}}_j)}^2}{n-1}}$$

where ${\overline{X}}_j$ is the average value of the elements in column $j$, and $D_j$ is the standard deviation of the indicators in column $j$, which is used to express the difference and fluctuation of the values within each indicator. If the standard deviation of $A_1$ solution set is larger, it reflects that the numerical difference of $A_1$ solution set is larger, more information can be reflected and the evaluation intensity of index $A_1$ is stronger. When assigning weights to $A_1$, more weights will be distributed, which also means that this goal has a greater influence on the selection of the whole optimal goal solution.

(4) Calculate the correlation and conflict quantitative index values among optimization objectives:

$$r_{ij}=\frac{{{\displaystyle \sum }}_i^n\left(S_{ij}-{\overline{X}}_j\right)\left(S_{ik}-{\overline{X}}_k\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{\left(S_{ij}-{\overline{X}}_j\right)}^2{{\displaystyle \sum }}_{i=1}^n{\left(S_{ik}-{\overline{X}}_k\right)}^2}}$$

$$R_j={\displaystyle \sum }_{j=1}^2\left(1-r_{ij}\right) \left(i=1,2,\cdot \cdot \cdot ,n;j=1;k=2\right)$$

where $r_{ij}$ is the correlation coefficient between the $i$-th index and the $j$-th index, and $j$ is the conflict between the $j$-th decision index and other indexes. The smaller the value of $R_j$ is, the smaller the conflict is, which means that the information of the optimization goal is repeated too much in the solution set and the smaller the contribution to determining the optimal solution.

(5) The weight of each optimization target is quantified after comprehensively measuring the contrast intensity and conflict among indexes, and its comprehensive information amount $C_j$ is calculated as follows:

$$C_j=D_j{\displaystyle \sum }_{j=1}^2\left(1-r_{ij}\right)=D_j\times R_j$$

where $C_j$ is the amount of information contained in the solution set of the $j$-th optimization objective. The greater the value of $C_j$, the greater the contribution of the $j$-th objective to the determination of the optimal solution and the greater its corresponding weight.

(6) The weights ${\omega }_j^{*}$ of the two optimization objectives are calculated by using the obtained information $C_j$:

$${\omega }_j^{*}=\frac{C_j}{{{\displaystyle \sum }}_{j=1}^2{C}_j}$$

(7) Each element in the decision matrix is weighted to obtain a weighted decision matrix $T$:

$$T={(t_{ij})}_{n\times 2} \left(i=1, 2, \dots ,n;j=1, 2\right)$$

where $t_{ij}={\beta }_{ij}^{*}\times {\omega }_j^{*}$.

(8) The ideal optimal solution $V_j^{+}=\underset{1\le i\le n}{max}{T}_{ij }$ and the worst solution $V_j^{-}=\underset{1\le i\le n}{min}{T}_{ij }$ are obtained according to the weighting matrix $T$, and the distances $P_i^{+}$, $P_i^{-}$ of each element $t_{ij}$ of the weighting matrix T to $V_j^{+}$, $V_j^{-}$ are calculated:

$$P_i^{+}=\sqrt{{\displaystyle \sum }_{j=1}^2{(t_{ij}-V_j^{+})}^2} P_i^{-}=\sqrt{{\displaystyle \sum }_{j=1}^2{(t_{ij}-V_j^{-})}^2} \left(i=1,2,\cdot \cdot \cdot ,n\right)$$

The approximation degree between the $i$-th solution and the ideal optimal solution is expressed as $K_i$, and the value of $K_i$ is in the interval of (0, 1). The more $K_i$ tends to 1, the closer the $i$-th scheme is to the optimal scheme:

$$K_i=\frac{P_i^{+}}{P_i^{+}+P_i^{-}}$$

5. Test and Results

5.1. Test

According to the final solution of the configuration parameters of the dwarf and close planting apple orchard picking robot and the structural parameters of the manipulator, combined with the picking target workspace, the simulation verification of the picking working space of the robot is carried out.

The parameters of the picking operation area in the canopy of a typical fruit tree in Section 2.2 are substituted into the multi-objective optimization model. Input the standard orchard environmental working parameters and the actual task requirements of the dual-manipulator picking robot and set the numerical variation range of the relevant variables. Considering the movement limit range of the manipulator extending and retracting into the frame, and referring to the average thickness of the fruit tree crown on one side of 400–500 mm, the parameter interval of the length $L_u$ of the upper arm and the length $L_d$ of the lower arm is set as 700–1500 mm. In particular, since the average height of the canopy from the ground is 1000–1200 mm, the lower arm must meet the premise of accommodating the target workspace. The parameter variation range of the inclination angle ${\theta }_d$ between the lower arm and the horizontal is 20–40°. The difference between the horizontal inclination angle of the upper arm and the horizontal inclination angle of the lower arm should not be too large, which will affect the lifting stroke of the horizontal linear tracks (components 1 and 5 in Figure 3a) and the picking of fruits in the middle area of the lifting stroke. Therefore, the parameter variation range of the inclination angle ${\theta }_u$ between the picking manipulator and the horizontal is set to be 20–50°. The row spacing of the orchard is about 3500 mm. Due to the limitation of the width of the bottom of the car body and the position of the picking robot between the fruit rows when working, the parameter variation range of the distance $G$ between the outer frame mounting base and the center of the tree trunk is 450–1000 mm.

$$s.t.\left\{\begin{array}{c}700\le L_u{,\mathrm{L}}_d\le 1500;20\le {\theta }_u\le 50;20\le {\theta }_d\le 40;\\ 450\le G\le 1000;L_u,L_d,{\theta }_u,{\theta }_d,G\in 0.1\times N;\end{array}\right\}$$

In the parameter setting of the NSGA-II genetic algorithm, the choice of crossover probability and mutation probability affects the behavior and solution performance of the whole algorithm and even directly affects the convergence of the algorithm. The crossover probability is generally selected in the range of 0.9–0.97, The greater the crossover probability, the faster the new individuals will be produced. However, when the crossover probability is too high, then there is a greater possibility that the genetic model will be destroyed, which will make the individual structure with high fitness quickly destroyed. However, if the crossover probability is too small, the whole algorithm will slow down or even stagnate. As far as the variation probability is concerned, the general value range is 0.01–0.1, and, if the value is too small, it is difficult to produce a new individual structure. If the value is too large, the genetic algorithm loses its constraint and becomes a random search algorithm.

Therefore, in this solution, the initial population size of the NSGA-II genetic algorithm is set as 50, the crossover probability is 0.9, the crossover distribution index is 20, the mutation probability is 0.04 and the maximum iterative generation is 1000 generations (it is recommended to set it to 20 times the number of design variables, and the number of variables in this paper is five). The objective optimization function and constraints are compiled and the Pareto solution set and the changes in various parameters are obtained as shown in Figure 6.

The weight of total structure length $L_m$ is 23.4%, and the weight of dual-manipulator invalid picking space $A_1$ is 76.6% by objective weight calculation with the CRITIC method. The top 10 solutions in TOPSIS’s sorting of Pareto solution sets are shown in the following

Table 2. Top 10 Pareto optimal solutions after TOPSIS ranking.

Sub-Order Number	Design Variables					Design Objective		Combined Score Index (Ki)
	Lu (mm)	θu (°)	Ld (mm)	θd (°)	G (mm)	A1 (dm2)	Lm (cm)
1	1119.3	39.4	898.7	26.0	755.3	11.96	422.6	0.784
2	1119.1	42.2	881.7	26.1	753.1	12.78	418.1	0.783
3	1132.3	40.5	887.1	26.1	753.6	12.28	421.7	0.780
4	1115.7	42.5	858.9	26.0	750.3	13.26	415.4	0.775
5	1137.2	38.3	902.4	26.1	754.9	11.49	426.0	0.773
6	1129.1	44.4	838.5	26.1	751.6	13.97	412.3	0.751
7	1137.2	37.2	942.4	25.6	755.7	11.20	431.0	0.747
8	1116.1	45.6	831.9	26.1	741.9	14.59	409.9	0.722
9	1111.4	48.2	832.1	26.3	740.3	15.23	406.2	0.693
10	1155.9	36.5	1040.7	24.4	760.6	10.84	442.9	0.688

. Finally, the first structural parameter is determined as the final optimal structural parameters, which are bold in Table 2.

The solution with the highest comprehensive score index (Figure 7a) is the optimal configuration parameter for this optimization. The position of this solution in the Pareto optimal solution set is shown in Figure 7b.

By investigating the actual orchard working parameters, the orchard 3D simulation environment was drawn and the working space and target picking space of the picking robot were analyzed (as shown in Figure 8). The main section of the accessible picking working area formed by the upper limit position and the lower limit position of the picking robot arm can completely contain the target picking area, which can not only meet the requirements of the best recovery ratio of the picking robot but also reach a larger accessible area with a smaller structural length of the robot arm. The overall configuration is compact, which is suitable for working in an unstructured orchard environment.

5.2. Analysis of the Validity of the Optimal Parameters

In order to verify the unique validity of the optimal parameter solution, it is necessary to clarify the relationship between the optimization objective and each parameter. On the premise that the two-arm working space covers the target space, the optimal results of $\phi \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)$ (i.e., $L_m$) and $\delta \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)$ (i.e., $A_1$) are set as fixed constant values $C_{L_m}$ and $C_{A_1}$, respectively, to verify whether another optimization objective has a unique optimal solution within the numerical range of each configuration parameter, that is, to transform a multi-objective optimization into a single-objective optimization problem for comparison and verification. The curve of the result value of the optimization objective function $\min \delta \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)$ varying with each structural parameter is shown in Figure 9a–e.

$$\{\begin{array}{c}\mathit{min}\delta \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)\hfill \\ \phi \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)=C_{L_m}\hfill \end{array} \{\begin{array}{c}\delta \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)=C_{A_1}\hfill \\ \mathit{min}\phi \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)\hfill \end{array}$$

$$s.t.\left\{\begin{array}{c}700\le L_u{,L}_d\le 1500;20\le {\theta }_u\le 50;20\le {\theta }_d\le 40;\\ 450\le G\le 1000;L_u,L_d,{\theta }_u,{\theta }_d,G\in 0.1*N;\end{array}\right\}$$

It can be seen in Figure 9a that the projected area $A_1$ of the robot’s invalid picking space increases with the length of the upper arm $L_u$, which presents an approximate “U”-shaped curve state. $L_u$ is in the range of 1000–1200 mm, and there is a low point of the downward trend of $A_1$, which means that the optimal $L_u$ is generated in the range of 1000–1200 mm. Different from Figure 9a, it can be seen in Figure 9b that, when the inclination angle ${\theta }_u$ between the upper arm and the horizontal increases, the projected area $A_1$ of the invalid picking space can be changed to about 40° at ${\theta }_u$ when there is a trend change inflection point that first descends and then rises, and the overall curve is similar to the letter “V” shape, which means that the optimal ${\theta }_u$ is generated within a small area of about 40°. Compared with the curve change in Figure 9a, the curve of Figure 9c oscillates less, and the projected area $A_1$ of the invalid picking space has several lower values of $A_1$ in the range of 900–1100 mm of $L_d$, and the range of variation interval is large, indicating that the optimal $L_d$ has little influence on the result value of $A_1$. It can be seen in Figure 9d that the corresponding change curve of the inclination angle ${\theta }_d$ between the lower arm and the horizontal and the projected area $A_1$ of the invalid picking space is similar to the letter “W” as a whole, at 25° and 35°. There is an inflection point of the changing trend of $A_1$ value on the left and right of the angle, and, after 35°, the upward amplitude of $A_1$ increases sharply, which means that, when the inclination angle ${\theta }_d$ between the lower arm and the horizontal is greater than 35°, it is difficult to obtain the optimal ${\theta }_d$ value. It can be seen from Figure 9e that, when the center distance $G$ is in the range of 650–800 mm, the change range of the ineffective picking space projected area $A_1$ is small and the low value of $A_1$ appears intensively; especially in the range of 750–800 mm, the trend is more obvious.

Similarly, the optimization objective function $\min \phi \left(L_u,{\theta }_u,L_d,{\theta }_d,G\right)$ shows the curve of the result value changing with each structural parameter, as shown in Figure 9f–j.

It can be seen in Figure 9f that the optimal $L_u$ value is located in the region of 1050–1200 mm in the middle of the variation scheme. Although the total structure length $L_m$ corresponding to this range shows an upward trend with the increase in $L_u$, the change range is small. It can be seen in Figure 9g that, when the inclination angle ${\theta }_u$ between the upper picking manipulator and the horizontal increases, the total structure length $L_m$ will first decrease and then increase when θ_u changes to about 40°. The trend change inflection point, the optimal ${\theta }_u$ is at this angle. The unique solution produced around 40° is characteristically distinct. It can be seen in Figure 9h that the total structure length $L_m$ increases correspondingly with the increase in $L_d$, but the increased range is different. The optimal $L_d$ is in the range of 800–900 mm, and the value of the total structure length $L_m$ remains in this range, at lower levels, and the increase is small. It can be seen in Figure 9i that, when the inclination angle ${\theta }_d$ between the lower arm and the level changes from 20° to 24°, the total structure length $L_m$ is greatly disturbed, and the obtained $L_m$ value is at a high level. In contrast, in the 24–28° region, the variation in $L_m$ is small and there are many low points of $L_m$ in this region, and the optimal ${\theta }_d$ is more likely to be generated in this region. It is not difficult to see from Figure 9j that, when the center distance $G$ varies from 700 mm to 800 mm, the total structure length $L_m$ presents a “U”-shaped curve, the low value of $L_m$ frequently appears around 750 mm and the optimal $G$ has obvious characteristics around 750 mm.

To sum up, the results of the two optimization objectives are fixed by constants, and whether the other optimization objective has a unique optimal solution within the numerical range of each configuration parameter is explored. By comparing the result intervals of each optimal configuration parameter after analysis, it is found that the intersection part of the intervals is the result of the final selected optimization parameter scheme, which can verify the effectiveness of the bi-objective optimization. When $L_u$ is 1119.3 mm, ${\theta }_u$ is 39.4, $L_d$ is 898.7 mm, ${\theta }_d$ is 26 and $G$ is 755.3 mm, the redundant working space of the upper and lower picking manipulators can be minimized, and the structure length of the dual-arm picking robot is small and the overall configuration is compact.

6. Conclusions

This paper investigates the picking conditions of the dwarf and close planting apple orchard in the Beijing area and obtains the distribution law of mature fruit in the apple canopy by using the crown three-dimensional zoning method. The main configuration of the apple picking robot was determined through analysis of the orchard target workspace, and the structure of the picking robot arm driven by electric and pneumatic telescope was designed by using three-dimensional mechanical design software. Parametric analysis of the relationship between the configuration of the apple-picking robot, the structure parameters of the dual-manipulator and the target workspace is established, and a multi-objective optimization function is established with “avoiding invalid workspace” and “compact structure” as performance indexes.

The CRITIC–TOPSIS combined method is used to evaluate and screen the multi-objective optimal solution. The final optimization result is that the length of the upper picking arm is 1119.3 mm, the horizontal elevation angle is 39.4°, the length of the lower picking arm is 898.7 mm, the horizontal elevation angle is 26° and the outer frame installation base is 755.3 mm from the center of the tree trunk. The working space of the dual-manipulator completely covers the target working space, and the redundancy rate of the working space is 16.62%.

The experimental results show that the sensitivity of the two optimization objectives to the change in configuration parameters is different, and the optimal solution has unique validity within the numerical range of composition parameters, thus achieving the design goal of minimizing the invalid working space and compacting the structure of the dual-manipulator picking robot. The optimization method in this paper can be used for a standardized scene, similar to the standardized dwarf densely planted orchard. At the same time of research and development of picking robots, agronomic requirements need to be completed simultaneously, and the research and development of robots and agronomy complement each other. The dual-manipulator apple picking robot proposed in this paper can also be iterated into a four-manipulator apple picking robot; that is, two groups of arms are combined into two groups of four-arms, and the upper and lower group arms have the same structural parameters, so this iteration has certain practical application significance.