Dwarf and high-density planting is a core cultivation mode for the intensive development of the modern apple industry^[23], offering significant advantages such as high yield, superior fruit quality, and strong adaptability to mechanized operations. To quantitatively characterize the three-dimensional spatial distribution of apples in such orchards and to facilitate subsequent workspace analysis and mechanism parameter design, A tree-based coordinate system is defined by taking the intersection of the trunk and the ground as the origin [Figure 1]. In this coordinate system, the X_t-axis is aligned with the inter-row direction to describe the distribution of fruits across row spacing; the Y_t-axis is aligned with the in-row (plant spacing) direction to represent the spatial position of fruits along the tree row; and the Z_t-axis is defined as vertically upward from the ground to characterize the vertical distribution of fruits. For the orchard conditions considered in this study, mature tree height is approximately 2.8-3.1 m. In the orchard used for field harvesting experiments, the average inter-row spacing and intra-row tree spacing were approximately 3.0 and 2.0 m, respectively. Fruits are primarily distributed at heights of 700-2,600 mm above the ground and within a horizontal distance of 0-800 mm from the trunk. Accordingly, the picking workspace of a single tree can be approximated as a cuboid with dimensions of 1,600 mm × 300 mm × 1,900 mm.

Figure 2 illustrates the overall configuration of the two-stage partitioned apple harvesting robot developed for dwarf and high-density orchards. The system includes a chassis, a support frame, a positioning module, and a picking module. The positioning module is composed of a support arm, a lifting arm, and a rotating main arm. The picking mechanism consists of a rotating secondary arm, a translational arm, a picking main arm, and a picking secondary arm.

The positioning mechanism adopts a 3R serial manipulator configuration, which is primarily used to achieve rapid positioning and posture adjustment of the picking base point within a single workspace. The picking mechanism employs an R–P–R–R joint configuration, enabling precise grasping and detachment of fruits within the picking region after base-point positioning is completed. The control system consists of a coordinated upper-level and lower-level architecture. The upper-level system, developed based on MATLAB and Python, is responsible for motion planning and task scheduling, while the lower-level system, centered on a microcontroller, performs motion control and execution feedback. This configuration enables the integration of autonomous harvesting and human–machine collaborative operations.

During operation, the center point at the end of the rotating main arm is defined as the positioning reference point of the device, whereas the grasping center of the end-effector is defined as the actual picking point. The main technical parameters of the robot are listed in Table 2.

The robot adopts a hierarchical motion-execution architecture. Target fruit coordinates are treated as task-level inputs to the motion system. The upper-level program, developed using MATLAB and Python, performs coordinate transformation, inverse-kinematic calculation, joint-command generation, and sequential task scheduling. The lower-level microcontroller executes the prescribed joint commands and returns motion-execution status to the upper level. The present study focuses on model-based motion generation and coordinated execution after a target coordinate has been provided. Fruit detection, multimodal sensor fusion, online obstacle avoidance, adaptive control, and autonomous target-selection strategies are outside the scope of the current work and are not evaluated experimentally. This scope distinction is important because the reported harvesting performance reflects the combined influence of the mechanical system and the prescribed motion-execution strategy rather than the performance of a complete autonomous perception-and-decision system.

Considering the spatial distribution of fruits and the canopy thickness in dwarf and high-density apple orchards, the picking workspace for a single tree can be simplified.To ensure full coverage of the fruiting area, the workspace is approximated as a cuboid measuring 1,600 mm × 300 mm × 1,900 mm. To reduce redundant motions and posture adjustments caused by cross-region operations of the manipulator, the cuboid workspace is symmetrically divided along the trunk direction into two independent target picking regions, denoted as U₁ and U₂, with the fruit partition plane serving as the symmetry reference, as shown in Figure 3A.

To minimize time loss caused by frequent repositioning and repeated adjustments during harvesting, and to improve the overall efficiency within a single picking region, a “single positioning–multiple picking” operation mode is adopted for each target region. Using the left picking region U₁ as an example, the geometric center of its front surface is taken as point A₀, representing the spatial center of the region. Taking into account the safety clearance between the end-effector and the fruit, along with structural installation constraints, point A₀ is shifted 300 mm along the positive X_t-axis to obtain point A₁. Point A₁ serves as the positioning reference for single-region harvesting, and its coordinates are expressed as (X_A, Y_A, Z_A) [Figure 3B].

To accurately describe the spatial configuration of the harvesting robot, a base coordinate system (denoted as the X−Z−Y coordinate system) is established with the center of the support base as the origin. During operation, the chassis is positioned between two rows of trees, aligning the vertical column with one-quarter of the canopy width, as shown in Figure 4. During harvesting, the target joint angles of the lifting arm and the rotating main arm are determined via inverse kinematics based on the coordinates of the picking base point. The control system commands the motors to move the positioning reference point - corresponding to the end of the rotating main arm - to the specified base location. Target fruit coordinates expressed in the X_tY_tZ_t frame are then converted into the base coordinate system using MATLAB. Joint variables are subsequently obtained from the inverse kinematics of the picking mechanism and sent to the microcontroller through a Python interface, enabling the mechanism to complete automated harvesting.

Compared with the conventional Denavit-Hartenberg formulation, the POE representation describes each joint directly through its screw axis in a common reference frame, without requiring the successive construction of link-attached coordinate frames or the selection of different DH conventions. This feature is particularly suitable for the present robot because it consists of two structurally different serial subsystems, namely a 3R positioning mechanism and an R-P-R-R picking mechanism. The screw-axis representation allows both mechanisms to be modeled within a consistent mathematical framework and enables the corresponding spatial Jacobians and singularity conditions to be derived directly from the same set of screw parameters. Therefore, the advantage considered in this study is primarily modeling consistency, geometric interpretability, and convenient singularity analysis, rather than experimentally demonstrated computational-speed improvement.

Based on the established geometric configuration and dimensions of the robot^[22] (with key parameters listed in Table 3), the positioning and picking mechanisms are modeled separately. Their forward kinematics are established based on the segmented structural design.

The instantaneous motion of a rigid body is characterized by its linear and angular velocity components. For a clearer representation within screw theory, the twist ξ is normalized and formulated using the PoE^[24] approach. Accordingly, the rotational motion of the rigid body can be expressed as:

(1) $$ \exp \left(S_{i} \theta_{i}\right)=\left[\begin{array}{cc} e^{\left[\omega_{i}\right] \theta_{i}} & \left(I \theta_{i}+\left(1+\cos \theta_{i}\right)\left[\omega_{i}\right]+\left(\theta_{i}-\sin \theta_{i}\right)\left[\omega_{i}\right]^{2}\right) v_{i} \\ 0 & 1 \end{array}\right] $$

Note: All arguments of the matrix exponential and the trigonometric functions in Equations (1)-(14) are dimensionless, and all revolute-joint variables are therefore evaluated in radians. where S_i denotes the normalized motion twist in the initial configuration; I is the 3 × 3 identity matrix; θ_i represents the joint variable; ω_i is the angular velocity about the screw axis of the i-th joint; v_i is the linear velocity along the screw axis of the i-th joint; [ω_i] is defined as the 3 × 3 skew-symmetric matrix derived from the angular velocity ω, $$ [\omega_i]=\begin{bmatrix} 0 & -\omega_z & \omega_y\\ \omega_z & 0 & -\omega_x\\ -\omega_y & \omega_x & 0 \end{bmatrix} $$; For a revolute joint, θ_i is expressed in radians in all product-of-exponentials, trigonometric, Jacobian, and dynamic calculations. When a joint command or measurement is reported in degrees, it is converted before numerical evaluation according to

$$ \theta _i(rad)=\theta _i(^\circ)\pi /180 $$

Angular velocity and angular acceleration are expressed in rad·s^-1 and rad·s^-2, respectively. For a prismatic joint, θ_i denotes translational displacement; millimeters are used for engineering reporting, while meters are used in SI-based dynamic calculations.

At the initial configuration, the PoE-based forward kinematic description is fully determined by specifying the base frame {s} and the end-effector frame {b}. Under this condition, the forward kinematics takes the following POE form:

(2) $$ T=M \prod_{i=1}^{n} \exp \left(S_{i} \theta_{i}\right) $$

Here, T corresponds to the end-effector pose relative to the base frame, while M specifies its initial configuration. The symbol n indicates the total number of joints in the mechanism, and this notation is used consistently throughout the remainder of the paper.

Let the initial pose matrices of the positioning mechanism and the picking mechanism be denoted as M₁ and M₂, respectively, where the reference point for each is defined as the origin of its end-effector frame. Their corresponding coordinates are given by:

$$ M_{1}=\left[\begin{array}{cccc} 1 & 0 & 0 & l_{2}+l_{3} \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & d_{0}+l_{1} \\ 0 & 0 & 0 & 0 \end{array}\right] ; M_{2}=\left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & l_{4}+l_{5}+l_{6}+l_{7} \\ 0 & 0 & 0 & 1 \end{array}\right] . $$

Figure 5 presents the coordinate frames at the initial configuration of each mechanism, while the associated screw axis parameters S are provided in Tables 4 and 5.

Using Equation (2) together with the parameters in Tables 4 and 5, the forward kinematics of the positioning and picking mechanisms are derived as follows:

(3) $$ \begin{aligned} T_{0}^{3} &= M_{1} \prod_{i=1}^{3} \operatorname{esp}\left(S_{i}\theta_{i}\right) \\ &= \left[ \begin{array}{cccc} c_{1}c_{23} & -c_{1}s_{23} & s_{1} & c_{1}s_{2}\left(d_{0}+l_{1}\right) +\left(l_{2}+l_{3}\right)c_{1}c_{23} -\left(d_{0}+l_{1}\right)c_{1}s_{23} -c_{1}c_{2}\sigma_{1} +c_{1}s_{2}\sigma_{2} \\ s_{1}c_{23} & -s_{1}s_{23} & -c_{1} & s_{1}s_{2}\sigma_{2} -\left(d_{0}+l_{1}\right)s_{1}s_{23} +\left(l_{2}+l_{3}\right)s_{1}c_{23} +s_{1}s_{2}\left(d_{0}+l_{1}\right) -c_{2}s_{1}\sigma_{1} \\ c_{23} & s_{23} & 0 & \left(d_{0}+l_{1}\right)c_{23} -s_{2}\sigma_{1} -c_{2}\sigma_{2} +\left(l_{2}+l_{3}\right)s_{23} -\left(d_{0}+l_{1}\right)\left(c_{2}-1\right) \\ 0 & 0 & 0 & 1 \end{array} \right] \end{aligned} $$

where σ₁ = l₂(c₃ - 1) - s₃(d₀ + l₁); σ₂ = l₂s₃ + (d₀ + l₁)(c₃ - 1); s₂₃ = sin(θ₂ + θ₃); c₂₃ = cos(θ₂ + θ₃), and the same notation applies hereafter.

(4) $$ \begin{aligned} T_{0^{\prime}}^{7} &=M_{2} \prod_{i=4}^{5} \operatorname{esp}\left(S_{i} \theta_{i}\right) \\ &=\left[\begin{array}{cccc} s_{4} s_{67} & s_{4} c_{67} & c_{4} & s_{4} s_{67}l_{7}-\theta _{5}s_{4}-s_{4}s_{6}l_{6}\\ -c_{4} s_{67} & -c_{4} c_{67} & s_{4} & \theta _{5}c_{4}-c_{4}s_{67}l_{7}-c_{4}s_{6}l_{6}\\ \sigma_{2} & -s_{67} & 0 & l_{4}+l_{5}+c_{6}l_{7}-c_{67}(l_{4}+l_{5}+l_{6})\\ 0 & 0 & 0 & 1 \end{array}\right] \end{aligned} $$

where θ₅ denotes the displacement of the prismatic joint (mm), s₆₇ = sin(θ₆ + θ₇), c₆₇ = cos(θ₆ + θ₇), and the same notation applies hereafter.

The accuracy of the screw-theory-based forward kinematic models is examined by simplifying the fourth columns of Equations (3) and (4) and comparing them with the corresponding entries in the pose matrices reported in Ref.^[22]. Full agreement is observed, which validates the correctness of the models.

According to the structural characteristics of the robot, the inverse kinematics are solved using the variable separation method^[25] and the geometric–random point method (see Ref.^[22] for details), respectively. Based on the variable separation approach, let the spatial position of the positioning reference point [i.e., the origin of the (X₃Y₃Z₃) coordinate frame] in the (X₀Y₀Z₀) coordinate frame be (x, y, z). According to Equation (2), the corresponding transformation matrices can be derived as follows:

$$ T_{0}^{1}=\left[\begin{array}{cccc} c_{1} & 0 & s_{1} & 0 \\ s_{1} & 0 & -c_{1} & 0 \\ 0 & 1 & 0 & l_{1}+d_{0} \\ 0 & 0 & 0 & 1 \end{array}\right] ; T_{1}^{2}=\left[\begin{array}{cccc} c_{2} & -s_{2} & 0 & l_{2} c_{2} \\ s_{2} & c_{2} & 0 & l_{2} s_{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] ; T_{2}^{3}=\left[\begin{array}{cccc} c_{3} & -s_{3} & 0 & l_{3} c_{3} \\ s_{3} & c_{3} & 0 & l_{3} s_{3} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] $$

To ensure dimensional consistency in the matrix operations, the position of the positioning base point is reformulated in homogeneous coordinates as (x, y, z, 1)^T. Accordingly, the elements of the known end-effector pose matrix are equated with the corresponding elements of the positioning base point vector.

(5) $$ {\left[\begin{array}{l} x \\ y \\ z \\ 1 \end{array}\right]=T_{0}^{1} T_{1}^{2} T_{2}^{3}\left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}\right]=T_{0}^{1} T_{1}^{2}\left[\begin{array}{c} f_{1}\left(\theta_{3}\right) \\ f_{2}\left(\theta_{3}\right) \\ f_{3}\left(\theta_{3}\right) \\ 1 \end{array}\right]=T_{0}^{1}\left[\begin{array}{c} g_{1}\left(\theta_{2}, \theta_{3}\right) \\ g_{2}\left(\theta_{2}, \theta_{3}\right) \\ g_{3}\left(\theta_{2}, \theta_{3}\right) \\ 1 \end{array}\right]} $$

where $$ {\left[\begin{array}{c} f_{1}\left(\theta_{3}\right) \\ f_{2}\left(\theta_{3}\right) \\ f_{3}\left(\theta_{3}\right) \\ 1 \end{array}\right]=\left[\begin{array}{c} l_{3} \cos \theta_{3}+l_{2} \\ l_{3} \sin \theta_{3} \\ 0\\ 1 \end{array}\right] ;\left[\begin{array}{cc} g_{1}\left(\theta_{2}, \theta_{3}\right) \\ g_{2}\left(\theta_{2}, \theta_{3}\right) \\ g_{3}\left(\theta_{2}, \theta_{3}\right) \\ 1 & \end{array}\right]=\left[\begin{array}{c} f_{1}\left(\theta_{3}\right) \cos \theta_{3}-f_{2}\left(\theta_{3}\right) \sin \theta_{2} \\ -f_{3}\left(\theta_{3}\right) \\ f_{1}\left(\theta_{3}\right) \sin \theta_{3}+f_{2}\left(\theta_{3}\right) \cos \theta_{2} \\ 1 \end{array}\right] .} $$

Let r = x² + y² + z² be defined accordingly and substitute it into Equation (5), yielding r = l₃² + l₂² + 2l₂l₃cosθ₃. Therefore, the joint angle θ₃ can be expressed as: $$ \theta_{3}=\arccos \left(\frac{r-l_{3}^{2}-l_{2}^{2}}{2 l_{2}^{2} l_{3}^{2}}\right). $$

(6) $$ z=g_{3}\left(\theta_{2}, \theta_{3}\right)=f_{1}\left(\theta_{3}\right) \sin \theta_{2}+f_{2}\left(\theta_{3}\right) \cos \theta_{2}=R_{1} \sin \left(\theta_{2}+\phi\right) $$

From Equation (6), let R₁ = $$ \sqrt{f_1^2(\theta _3)+f_2^2(\theta _3)} $$ and ϕ = arctan$$ \left ( \frac{f_2(\theta _3)}{f_1(\theta _3)} \right ) $$ be defined accordingly. The joint angle θ₂ can be obtained; when $$ \left | \frac{z}{R_1} \right | $$ > 1, no solution exists for θ₂. When $$ \left | \frac{z}{R_1} \right | $$ ≤ 1, θ₂ has two possible solutions, i.e., θ₂ = arcsin$$ \left ( \frac{z}{R_1} \right ) $$ - ϕ or θ₂ = π - arcsin$$ \left ( \frac{z}{R_1} \right ) $$ - ϕ.

(7) $$ x=g_{1}\left(\theta_{2}, \theta_{3}\right) \cos \theta_{1}-g_{2}\left(\theta_{2}, \theta_{3}\right) \sin \theta_{1}=R_{2} \cos \left(\theta_{1}+\varphi\right) $$

From Equation (7), let R₂ = $$ \sqrt{g_1^2(\theta _2,\theta _3)+g_2^2(\theta _2,\theta _3)} $$ and φ = arctan$$ \left ( \frac{g_2(\theta _2,\theta _3)}{g_1(\theta _2,\theta _3)} \right ) $$ be defined accordingly. Based on the X-coordinate of the positioning reference point, the joint angle θ₁ can be obtained; when $$ \left | \frac{z}{R_2} \right | $$ > 1, no solution exists for θ₁. When $$ \left | \frac{z}{R_2} \right | $$ ≤ 1, θ₁ = ±arccos$$ \left ( \frac{x}{R_2} \right ) $$ - φ.

Kinematic singularities cause a robot’s end-effector to lose mobility in certain translational or rotational directions. To quantitatively evaluate this phenomenon and enable visualization, the Jacobian matrix^[26] is used to establish the mapping between joint velocities and end-effector velocity. A manipulability index then characterizes the robot’s proximity to singular configurations at a given pose. The spatial velocity V_s can be expressed as:

(8) $$ V_s=J_s(\theta)\dot{\theta} $$

Here, J_s(θ) represents the space Jacobian defined in the fixed (space) frame, while $$ \dot{\theta} $$ corresponds to the joint velocity vector.

The i-th column J_si(θ) of the Jacobian corresponds to the twist associated with the i-th joint axis in the fixed (space) frame. Its expression is given by

(9) $$ J_{s i}(\theta)=A d_{T_{i-1}}\left(S_{i}\right) $$

where the first column J_s1 = S₁; T_i-1 = $$ \prod_{i=1}^{i-1} $$esp(S_iθ_i); and T = $$ \begin{pmatrix} R & P\\ 0 & 1 \end{pmatrix} $$ is a 4 × 4 matrix.

In this formulation, the twist coordinates of the i-th joint after the rigid-body transformation T_i-1 are denoted by $$ Ad_{T_{i-1}} $$(S_i). The matrix T_i-1 describes the corresponding pose transformation, with R representing the 3 × 3 rotation matrix and P the 3 × 1 position vector. This notation is used consistently in the following sections.

Based on Equations (8) and (9), the Jacobian matrices of the positioning mechanism (denoted as J_c) and the picking mechanism (denoted as J_z) can be expressed as follows:

(10) $$ J_{C}=\left[\begin{array}{ccc} 0 & \mathrm{~s} \theta_{1} & \mathrm{~s} \theta_{1} \\ 0 & -c \theta_{2} & -c \theta_{2} \\ 1 & 0 & 0 \\ 0 & \left(d_{0}+l_{1}\right) c \theta_{1} & \left(d_{0}+l_{1}\right) c \theta_{1}+l_{2} c \theta_{1} s \theta_{2} \\ 0 & \left(d_{0}+l_{1}\right) s \theta_{1} & \left(d_{0}+l_{1}\right) s \theta_{1}+l_{2} s \theta_{1} s \theta_{2} \\ 0 & 0 & -l_{2} c \theta_{2} \end{array}\right] $$

(11) $$ J_{Z}=\left[\begin{array}{cccc} 0 & 0 & c \theta_{4} & c \theta_{4} \\ 0 & 0 & s \theta_{4} & s \theta_{4}\\ 1 & 0 & 0 & 0\\ 0 & -s \theta_{4} & -\left(l_{4}+l_{5}\right) s \theta_{4} & -\left(l_{4}+l_{5}\right) s \theta_{4}-l_{6}s \theta_{4}c \theta_{6}\\ 0 & c \theta_{4} & \left(l_{4}+l_{5}\right) c \theta_{4} & \left(l_{4}+l_{5}\right) c \theta_{4}+l_{6}c\theta_{4}c \theta_{6}\\ 0 & 0 & -e x_{5} & \left(l_{4}+l_{5}\right) s \theta_{6}(s \theta_{4}+c \theta_{4})+\left(l_{4}+l_{5}+l_{6}\right)s \theta_{6}-ex_5 \end{array}\right] $$

To evaluate the motion performance of the mechanism and avoid the occurrence of singular configurations, singularity analysis is conducted based on the rank properties of the Jacobian matrix and the manipulability index. The Jacobian matrix of the positioning mechanism, J_c, is a 6 × 3 matrix. When the second and third column vectors become linearly dependent, the joint angle θ₂ takes a value of 90°. In addition, the fourth and fifth elements of the corresponding column vectors must be equal, satisfying the following condition:

(12) $$ \left\{\begin{array}{l} \left(d_{0}+l_{1}\right) c \theta_{1}=\left(d_{0}+l_{1}\right) c \theta_{1}+l_{2} c \theta_{1} s \theta_{2} \\ \left(d_{0}+l_{1}\right) s \theta_{1}=\left(d_{0}+l_{1}\right) s \theta_{1}+l_{2} s \theta_{1} s \theta_{2} \\ \end{array}\right. $$

The solution shows that no valid value exists for θ₁, indicating that the second and third column vectors are linearly independent. Furthermore, the third column vector cannot be expressed as a linear combination of the other two columns; therefore, the rank of J_c is 3 (full rank), and the positioning mechanism does not exhibit kinematic singularities.

Similarly, for the Jacobian matrix J_z, if the third and fourth column vectors are linearly dependent, the following condition must be satisfied:

(13) $$ \begin{aligned} \left\{\begin{array}{l} -\left(l_{4}+l_{5}\right) s \theta_{4}=-\left(l_{4}+l_{5}\right) s \theta_{4}-l_{6} s \theta_{4} c \theta_{6} \\ \left(l_{4}+l_{5}\right) c \theta_{4}=\left(l_{4}+l_{5}\right) c \theta_{4}+l_{6} c \theta_{4} c \theta_{6}\\ -ex_5=(l_4+l_5)s\theta_{6}(s\theta_{4}+c\theta_{4})+(l_4+l_5+l_6)s\theta_{6}-ex_5 \end{array}\right. \end{aligned} $$

Solving the above condition yields θ₆ = π/2 and θ₄ = 0.81π. Further examination of the third row elements indicates that the first three column vectors are linearly independent. Therefore, the rank of J_z is 3 (less than its column number of 4), implying that the mechanism is rank-deficient and a kinematic singularity exists at this configuration.

Manipulability is an important metric for quantifying the motion capability of a robot end-effector in task space. Based on this index, the motion capabilities of the picking mechanism along the X-, Y-, and Z-axes are quantitatively evaluated. The manipulability ratio μ₁ is adopted as the evaluation parameter, which characterizes the relative ease of motion in different directions at a given pose (μ₁ ≥ 1). The specific formulation is given as:

(14) $$ \mu_{1}=\frac{\sqrt{\lambda_{\max }\left(J J^{T}\right)}}{\sqrt{\lambda_{\min }\left(J J^{T}\right)}} $$

Here, A is a 6 × 6 matrix, and J represents the Jacobian. The quantities λ_max and λ_min correspond to the largest and smallest eigenvalues of JJ^T, respectively.

Based on Equation (14), the analysis is conducted in conjunction with the picking mechanism’s initial configuration. The initial configuration parameters are summarized in Table 6.

The geometric characteristics of the manipulability ellipsoid indicate that when μ₁ approaches 1, the ellipsoid tends toward a spherical shape, exhibiting strong isotropy, and the mechanism’s manipulability becomes nearly uniform in all directions. As μ₁ increases, the ellipsoid becomes increasingly elongated, indicating that motion along its principal axis is significantly better than in other directions. When μ₁ approaches infinity, the mechanism tends toward a kinematic singular configuration.

Based on Equation (11) and Table 6, the Jacobian matrix of the picking mechanism at the initial configuration (with arm length parameters expressed in SI units, i.e., meters) yields eigenvalues of (0, 0, 0, 1, 3.15) and (0.75). Substituting these into Equation (11) shows that μ₁ approaches infinity at the initial configuration, indicating the presence of a kinematic singularity.

To further investigate the motion characteristics of the positioning mechanism, a dynamic analysis is conducted^[27,28]. Considering its operating requirements and structural features, a dynamic model is established using the Lagrangian formulation^[29]. The independent generalized coordinates are chosen as q₁ and q₂, corresponding to the rotation angles of the lifting arm and the rotating main arm, respectively. The Lagrangian function L(q,$$ \dot{q} $$) is expressed as:

(15) $$ L(q, \dot{q})=K(q, \dot{q})-P(q) $$

where K is the total kinetic energy of the mechanism, and P is the total potential energy.

Based on the fundamental form of the Lagrangian function, the dynamic equations can be expressed as:

(16) $$ \frac{d}{d t} \frac{\partial K}{\partial \dot{q}}-\frac{\partial K}{\partial q}+\frac{\partial P}{\partial q}=\tau $$

where τ denotes the non-conservative generalized force acting on the mechanism.

Due to its limited rotational range and negligible impact on overall system dynamics, the support arm is not considered in the analysis; the dynamic modeling focuses on the lifting arm and rotating main arm (hereinafter referred to as the “key links”). The support arm remains stationary during the harvesting operation and serves only as a structural base for the positioning mechanism. It is retained in the geometric description to define the installation height and base coordinate frame, but it is not assigned an independent generalized coordinate and does not contribute a kinetic-energy term to the dynamic equations. Therefore, the dynamic analysis focuses on the lifting arm and the rotating main arm.

The corresponding model is shown in Figure 6. The positioning mechanism is represented by three physical links: the support arm, lifting arm, and rotating main arm, with masses m₁, m₂, and m₃, respectively. The two moving links are the lifting arm and rotating main arm, whose generalized coordinates are q₁ and q₂, respectively. The center-of-mass distances of the three physical links are denoted by lc₁, lc₂, and lc₃.

The bottom surface of the support arm is selected as the zero-potential-energy reference plane. With this choice of reference plane, the gravitational potential energy of m₁ must be included. Therefore, the total potential energy of the system is expressed as:

(17) $$ P=-m_{1} g\left(l_{1}-l_{c 1}\right)+\left(m_{2}+m_{3}\right) g l_{1}+m_{2} g l_{c 2} \sin q_{1}+m_{3} g\left(l_{2} \sin q_{1}+l_{c 3} \sin \left(q_{1}+q_{2}\right)\right) $$

With the X₁O₁Y₁ frame [Figure 6] defined as the reference frame, the rotation matrix and position vector of the centroid of the i-th link relative to this frame are obtained as follows, where the rotation matrix is denoted as R_i and the centroid displacement vector is denoted as O_ci⁰, respectively:

$$ R_{1}=\left[\begin{array}{ccc} c_{1} & -s_{1} & 0 \\ s_{1} & c_{1} & 0 \\ 0 & 0 & 1 \end{array}\right] ; R_{2}=\left[\begin{array}{ccc} c_{12} & -s_{12} & 0 \\ s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{array}\right] ; O_{c 2}^{0}=\left[\begin{array}{c} l_{c 2} c_{1} \\ l_{c 2} s_{1} \\ 0 \end{array}\right] ; O_{c 3}^{0}=\left[\begin{array}{c} l_{2} c_{1}+l_{c 3} c_{12} \\ l_{2} s_{1}+l_{c 3} s_{12} \\ 0 \end{array}\right] $$

Using Equation (9), the centroid Jacobian J_ci is formulated to relate joint velocities to the centroid velocities of each link. On this basis, the total kinetic energy of the system is given by:

(18) $$ K=\frac{1}{2}\dot{q}^T\left [ \sum_{i=1}^n\{m_iJ_{vi}^TJ_{vi}+J_{\omega i}^TR_iI_iR_i^TJ_{\omega i}\} \right ] \dot{q}=\frac{1}{2}\dot{q}^TM\dot{q} $$

In this formulation, M refers to the 2 × 2 inertia matrix, while $$ \dot{q} $$ indicates the vector of joint variables, I_i = $$ \begin{bmatrix} I_{xxi} & I_{xyi} & I_{xzi}\\ I_{yxi} & I_{yyi} & I_{yzi}\\ I_{zxi} & I_{zyi} & I_{zzi} \end{bmatrix} $$

Each link is treated as a rigid body, and joint friction is neglected. Under these conditions, the dynamic equation of the mechanism is given by

(19) $$ M(q) \ddot{q}+C(q, \dot{q}) \dot{q}+G(q)=\tau $$

Here, M denotes a symmetric positive-definite matrix reflecting the effect of joint accelerations on the system dynamics. Based on Equation (18), it can be obtained that:

$$ M=\begin{bmatrix} m_2l_{c2}^2+m_3(l_2^2+l_{c3}^2+2l_2l_{c3}c_2) +I_{zz2}+I_{zz3} & m_3(l_{c3}^2+l_1l_{c3}c_2)+I_{zz3}\\ m_3(l_{c3}^2+l_2l_{c3}c_2) +I_{zz3} & m_3l_{c3}^2+I_{zz3} \end{bmatrix} $$

From Equation (19), it can be seen that C(q,$$ \dot{q} $$) is a 2 × 2 matrix, and its (k,j)-th element is defined as:

(20) $$ c_{kj} = \sum_{i=1}^{n} c_{ijk}(q)\dot{q}_{i} = \sum_{i=1}^{n} \frac{1}{2} \left\{ \frac{\partial m_{kj}}{\partial q_{i}} + \frac{\partial m_{ki}}{\partial q_{j}} - \frac{\partial m_{ij}}{\partial q_{k}} \right\} \dot{q}_{i}$$

where c_ijk denotes the Christoffel symbols of the first kind. Based on the above equation, it can be obtained that C = $$ \begin{bmatrix} -m_3l_2l_{c3}s_2\dot{q}_2 & -m_3l_2l_{c3}s_2(\dot{q}_1+\dot{q}_2)\\ m_3l_2l_{c3}s_2\dot{q}_1 & 0 \end{bmatrix}. $$

Based on Equations (19) and (20), it can be obtained that G = $$ \begin{bmatrix} m_2l_{c2}\cos q_1+m_3g(l_2\cos q_1+l_{c3}\cos(q_1+q_2))\\ m_3gl_{c3}\cos(q_1+q_2) \end{bmatrix}. $$

The dynamic model of the positioning mechanism establishes the mapping relationship between the driving torques and the motion states based on the matrices M, C, and G, providing a fundamental theoretical basis for dynamic performance analysis and parameter identification.

The picking mechanism exhibits a multi-degree-of-freedom serial configuration, and its dynamics are modeled using the Newton–Euler recursive approach^[30]. Geometric parameters of the links are summarized in Table 3, while their corresponding masses are provided in Table 7. A schematic representation of the dynamic model is presented in Figure 7.

Each link is assumed to be a symmetric rigid homogeneous body, with its mass concentrated at the geometric center. The distances between each center of mass and its corresponding joint are defined as l_c4, l_c5, l_c6, and l_c7.

The Newton–Euler dynamic formulation, derived from rigid-body kinematics and dynamics, is carried out in two stages: first, a forward recursion is performed to compute the velocities and accelerations of each link; then, a backward recursion is carried out to determine the joint driving forces (torques). The transformation matrix T enables the propagation of kinematic variables between adjacent links. The pose transformation between neighboring joints can be obtained from Equation (2).

Considering the actual picking operation state and the structural characteristics of the robot, the initial conditions and joint motion types are defined. When the end-effector grasps an apple, it is subjected to the gravitational vector g acting vertically downward. At the initial operation stage, the mechanism is in a static state, and the initial kinematic parameters are given as follows: All θ₁, $$ \dot{\theta} $$₁, d₁ and $$ \dot{d} $$₁ are [0 0 0]^T.

The forward recursion starts from the rotating secondary arm and proceeds sequentially along each link to the end-effector (picking point), yielding the angular velocity, angular acceleration, linear velocity, and linear acceleration of each link. The corresponding kinematic equations are given as follows:

(21) $$ {}^{i+1}\omega_{i+1} = R_{i}^{i+1}{}^{i}\omega_{i} + \dot{\theta}_{i+1}\,{}^{i+1}\hat{Z}_{i+1} $$

(22) $$ {}^{i+1}\dot{\omega}_{i+1} = R_{i}^{i+1}{}^{i}\dot{\omega}_{i} + R_{i}^{i+1}{}^{i}\omega_{i} \times \dot{\theta}_{i+1}\,{}^{i+1}\hat{Z}_{i+1} + \ddot{\theta}_{i+1}\,{}^{i+1}\hat{Z}_{i+1} $$

(23) $$ {}^{i+1}v_{i+1} = R_{i+1}^{i} \left( {}^{i}v_{i} + {}^{i}\omega_{i} \times {}^{i}P_{i+1} \right) + \dot{d}_{i+1}\,{}^{i+1}\hat{Z}_{i+1} $$

(24) $$ {}^{i+1}\dot{v}_{i+1} = R_{i}^{i+1} \left( {}^{i}\dot{\omega}_{i} \times P_{i+1}^{i} + {}^{i}\omega_{i} \times \left( {}^{i}\omega_{i} \times P_{i+1}^{i} \right) + {}^{i}\dot{v}_{i} \right) $$

Here, i represents the joint index (i = 3, 4, 5, 6, 7). The unit vector along the Z-axis of joint i+1 is given by ⁱ⁺¹$$ \hat{Z} $$_i+1, and the rotation matrix R_iⁱ⁺¹ maps the coordinate frame of joint i to that of joint i+1. The vector P_i+1ⁱ defines the position from the i-th to the i+1-th joint frame. The initial linear and angular velocities of link i+1 in the base frame are denoted by $$ \dot{d} $$_i+1 (m/s) and $$ \dot{\theta} $$_i+1 (rad/s), respectively; the former vanishes for revolute joints, while the latter is zero for prismatic joints. The initial angular acceleration is given by $$ \ddot{\theta} $$_i+1, (rad/s²). Within the i+1-th joint frame, ⁱ⁺¹ω_i+1, and ⁱ⁺¹$$ \dot{\omega} $$_i+1, denote the angular velocity and angular acceleration of link i+1, respectively. Similarly, ⁱ⁺¹v_i+1, and ⁱ⁺¹$$ \dot{v} $$_i+1, represent its linear velocity and linear acceleration, with units of m/s and m/s².

Based on Equations (21)-(24), the kinematic equations of the centroid acceleration are given as follows:

(25) $$ {}^{i+1}\dot{v}_{C_{i+1}} = {}^{i+1}\dot{\omega}_{i+1} \times {}^{i+1}P_{C_{i+1}}^{i+1} + {}^{i+1}\omega_{i+1} \times \left( {}^{i+1}\omega_{i+1} \times P_{C_{i+1}}^{i+1} \right) + {}^{i+1}\dot{v}_{i+1} $$

The centroid linear acceleration of link i+1, expressed in the i+1-th joint frame, is given by ⁱ⁺¹$$ \dot{v}_{C_{i+1}} $$ (m/s²). In the same coordinate system, $$ P_{C_{i+1}}^{i+1} $$ represents the position vector of the centroid, defined in an analogous manner.

Based on Equation (25), the force and moment equations at the link centroid are given as follows:

(26) $$ {}^{i+1}F_{i+1} = m_{i+1}\,{}^{i+1}\dot{v}_{C_{i+1}} $$

(27) $$ {}^{i+1}N_{i+1} = {}^{C_{i+1}}I_{i+1}\,{}^{i+1}\dot{\omega}_{i+1} + {}^{i+1}\omega_{i+1} \times {}^{C_{i+1}}I_{i+1}\,{}^{i+1}\omega_{i+1} $$

The mass of link i+1 is given by m_i+1 (kg), and its centroidal inertia is described by the matrix $$ {}^{C_{i+1}}I_{i+1} $$. The interaction force exerted by link iii on the centroid of link i+1, expressed in the i+1-th joint frame, is represented by ⁱ⁺¹F_i+1 (N). The corresponding moment in the same frame is denoted by ⁱ⁺¹N_i+1 (N·m). Here, quantities are defined analogously for the subsequent terms.

Based on Equations (26) and (27), the forces and moments acting on each link are first computed. Then, the corresponding joint driving forces and torques under the given loading conditions are obtained using the backward recursion formulation.

(28) $$ {}^{i}f_{i} = R_{i+1}^{i}\,{}^{i+1}f_{i+1} + {}^{i}F_{i} $$

(29) $$ {}^{i}n_{i} = {}^{i}N_{i} + R_{i+1}^{i}\,{}^{i+1}n_{i+1} + P_{C_i}^{i} \times {}^{i}F_{i} + P_{i+1}^{i} \times R_{i+1}^{i}\,{}^{i+1}f_{i+1} $$

The driving force acting on link i, expressed in the i-th joint coordinate frame, is represented by ⁱf_i (N), with analogous definitions applied to subsequent terms. The corresponding driving torque in the same frame is denoted by ⁱ⁺¹f_i+1 (N·m).

Using Equations (28) and (29), the joint forces and torques are derived by projecting the forces (and moments) between adjacent links onto the Z-axis, as expressed below:

(30) $$ {}^{q}\tau_{i} = {}^{i}n_{i}^{T}\,{}^{i}\hat{Z}_{i} $$

(31) $$ {}^{q}f_{i} = {}^{i}f_{i}^{T}\,{}^{i}\hat{Z}_{i} $$

Here, ^qf_i represents the driving force of the i-th joint, while ^qτ_i corresponds to the driving torque of the same joint.

From the above dynamic equations, the dynamic model of the picking mechanism is obtained. Given that the harvesting task primarily depends on the main functional components, this work derives the joint torque expressions for the picking main arm and secondary arm. For brevity, the dynamic equations of the remaining joints are not presented.

(32) $$ \tau_{6} = c_{4}\sigma_{4} - s_{4}s_{6}\sigma_{2} - c_{6}s_{7}\sigma_{3} $$

where τ₆ denotes the joint torque of the picking main arm;σ₁ = 118.0451c₆s₇ + 92.563438s₆₇c₆₇ + 0.6651c₆σ₈ + 203.4798c₇s₆₇c₆₇+ 1.02s₇s₆₇s₆₇ + 1.02σ₄ + 1500c₇ - 0.3462s₇σ₅ + 0.3462c₇σ₄ + 880.88s₇ + 1000;σ₂ = 1652.5119c₆ + 16739.2266s₄ + 830.8915c₆₇ + c₇σ₆ + s₇σ₇ + 3.4127;σ₃ = 166.8981s₆ - s₇σ₆ + c₇σ₇;σ₄ = 600c₆s₆ + c₆σ₈;σ₅ = 600s₆s₆ + s₆σ₈ + 600;σ₆ = 415.4458c₆ + 3396.2691s₄ + 337.6656c₆₇ + 0.6924;σ₇ = 7.1967s₆₇;σ₈ = θ₅ + 9810c₄.

(33) $$ \tau_{7} = -s_{4}c_{67}\sigma_{1} - c_{4}\sigma_{2} - 7.1967s_{4}s_{67} $$

where τ₇ denotes the joint torque of the picking secondary arm;σ₁ = 415.4458c₇ + 3396.2691s₄ + 377.6656c₆₇ + 0.6924;σ₂ = 92.564338s₆₇c₆₇ + 207.72s₆c₆₇ + 0.3462c₆₇σ₆ - 207.72s₇;σ₃ = θ₅ + 9810c₄.

To evaluate the numerical consistency of the analytical Newton–Euler implementation, a corresponding rigid-body multibody model was established in ADAMS. The analytical and ADAMS models use the same link geometry, mass properties, gravitational acceleration, prescribed joint trajectories, ideal-joint definitions, and external payload. Therefore, the comparison presented in this section is intended as numerical cross-verification between two independently implemented formulations under matched assumptions, rather than as independent physical validation of the prototype.

Simulation studies of joint dynamic parameters are conducted under representative operating conditions to assess the validity of the dynamic equations of the picking mechanism. A simplified three-dimensional model is first established in SolidWorks and then imported into ADAMS. Material properties and spatial pose parameters are assigned, and appropriate kinematic joints are defined according to the joint types to construct a multibody dynamic model, as shown in Figure 8.

Based on the simplified model, key simulation parameters are specified. The gravitational acceleration is set to g = 9.81 m/s², while the link lengths and masses are given in Tables 3 and 7, respectively. Considering the actual harvesting conditions, this section focuses on the key links directly involved in fruit grasping and picking, namely the translational arm, the picking main arm, and the picking secondary arm, and analyzes their dynamic characteristics. For engineering readability, the commanded revolute-joint velocity is reported as 30°·s^-1, corresponding to 0.5236 rad·s^-1 in the analytical calculation. The prismatic-joint velocity is 0.05 m·s^-1. Angular data exported from ADAMS are converted into radians, rad·s^-1, and rad·s^-2 before comparison with the MATLAB results. Taking the weight of a single fruit as the reference, a force of 2.5 N is applied at the end of the picking secondary arm (in the direction of gravity and moving synchronously with the link). Joint dynamic data are obtained from ADAMS and compared with the results calculated using the Newton–Euler recursive formulation.

To quantitatively evaluate model accuracy and eliminate the influence of magnitude differences, 240 sampling points are selected at intervals of 0.1 s. The relative error η_i at each time instant, as well as the mean relative error η_a, between the simulation data and the theoretical results are then calculated.

(34) $$ \eta_{i} = \left| \frac{D_{Mi}-D_{Ai}}{D_{Ai}} \right| \times 100\% $$

(35) $$ \eta_{a} = \frac{1}{n} \sum_{i=1}^{n}\eta_{i} $$

Here, D_Mi represents the value obtained from the MATLAB-based dynamic equations at the i-th sampling instant, while D_Ai corresponds to the simulation results generated by ADAMS at the same time step. The same notation is used consistently in the following sections. η_i corresponds to the relative error at the i-th sampling instant, and n indicates the total number of sampling points. This notation is maintained throughout the remainder of the paper.

Based on the simulation data and Equations (34) and (35), the mean relative error is calculated, and comparative plots of the theoretical and simulated joint forces and torques of the picking mechanism are generated [Figure 9].

As shown in Figure 9C, F, and I, the mean relative errors of each link are 0.00152%, 0.598%, and 0.0856%, respectively. These errors mainly arise from the calculation of centroid inertia and the rounding errors introduced by truncation to three decimal places in the error analysis. The largest error occurs at the joint of the picking main arm, but it remains below 0.6%, which satisfies the accuracy requirements. This confirms the validity of the MATLAB-based dynamic formulations.

To quantitatively evaluate the linear correlation between the two dynamic curves corresponding to the same joint, the linear correlation coefficient r is adopted as the evaluation metric.

(36) $$ r = \frac{ \displaystyle \sum_{i=1}^{n} \left[ \left(D_{Mi}-\overline{D}_{M}\right) \left(D_{Ai}-\overline{D}_{A}\right) \right] }{ \displaystyle \sqrt{\sum_{i=1}^{n}\left(D_{Mi}-\overline{D}_{M}\right)^2} \sqrt{\sum_{i=1}^{n}\left(D_{Ai}-\overline{D}_{A}\right)^2} } $$

where $$ \bar{D}_M $$ denotes the mean value of the data computed from the MATLAB-based dynamic equations over all sampling instants, and $$ \bar{D}_A $$ denotes the mean value of the ADAMS simulation data over all sampling instants.

Based on Equation (36), the linear correlation coefficients of the joint curves are calculated as 0.999375, 0.999998, and 0.999999, respectively. These results indicate that all r values are close to 1, demonstrating a strong linear correlation between the corresponding curves.

To verify the accuracy of the model, we introduce the following calculation indicators.

Let the residual at the iii-th sampling instant be defined as

$$ e_i=D_{Mi}-D_{Ai} $$

where D_Mi and D_Ai denote the analytical and ADAMS results, respectively. The following metrics are used:

$$ MAE=\frac{1}{n}\sum_{i=1}^n|e_i| $$

$$ RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^ne_i^2} $$

$$ MaxAE=\max|e_i| $$

Pearson’s correlation coefficient is calculated from the unrounded data. The phase lag is determined from the time shift corresponding to the maximum cross-correlation between the two curves. The calculation data are shown in Table 8 below.

The force/torque errors of the translational arm, picking main arm, and picking secondary arm were evaluated using the mean absolute error (MAE), root mean square error (RMSE), and maximum absolute error (MaxAE). The results show that the translational arm has the lowest values for all three error metrics, indicating a high degree of agreement between the analytical and simulation results. The error level of the picking secondary arm lies between those of the translational arm and the picking main arm, and its overall agreement is better than that of the picking main arm. In comparison, the picking main arm exhibits the highest MAE, RMSE, and MaxAE among the three mechanisms, together with relatively pronounced instantaneous error peaks during motion, indicating comparatively large local deviations in its dynamic response.

Considering the three error metrics collectively, the errors of the translational arm, picking main arm, and picking secondary arm are all below the allowable error thresholds specified for the system. This result demonstrates good overall agreement between the dynamic-model predictions and the simulation results. Therefore, the accuracy and reliability of the established dynamic models are validated under the prescribed motion trajectory and operating conditions examined in this study.

The present model does not explicitly account for joint friction, reducer and transmission efficiency, mechanical backlash, structural compliance, motor and reducer inertia, closed-loop controller dynamics, or fruit–branch contact. These factors may cause differences between the predicted rigid-body actuator loads and the actual prototype responses. Calibrated joint-load or motor-current measurements are therefore required before the model can be considered physically validated.

The picking mechanism serves as the primary execution unit of the harvesting system, and precise control of its joint motions is essential for reliable operation. To compare candidate joint-velocity parameter configurations, numerical simulations were performed based on the established dynamic model and the prescribed boundary positions of the picking trajectory. The one-way reaching duration, defined as the prescribed time for the picking mechanism to move from its initial configuration to a boundary reference point, was fixed at T_r =7 s as an engineering constraint for the joint-time allocation analysis. This prescribed reaching duration should be distinguished from both the out-and-return simulation cycle used in Sections 5.2-5.3 and the experimentally measured single-fruit picking time reported in the orchard test. This value was selected with reference to the picking cycle time reported in Ref.^[22], while also considering the motion range and actuator capability of the proposed mechanism. Throughout this manuscript, the terms “one-way reaching duration”, “out-and-return simulation cycle”, and “field single-fruit picking time” refer to different quantities and are used accordingly.

The joint-motion timing directly affects actuator loads and motion smoothness. Rather than performing continuous numerical optimization, we compare three predefined engineering allocation rules under the same fixed one-way reaching-duration constraint. These rules are equal-time allocation, link-length-proportional allocation, and mass-length-combined allocation. For each scheme, the motion time assigned to joint i is denoted by t_i, satisfying

$$ \sum_{i=4}^7t_i=T $$

and the corresponding commanded joint velocity is calculated from

$$ v_i=\frac{\Delta q_i}{t_i} $$

where Δq_i is the prescribed motion range of joint i. The three schemes are evaluated under identical motion ranges, payload, and simulation conditions by comparing the amplitude, abruptness, and duration of the resulting joint-force and joint-torque fluctuations. Accordingly, the procedure is an engineering comparison of parameter-allocation rules rather than a mathematical optimization involving an iterative search algorithm. Table 9 summarizes the motion ranges of each joint at the boundary picking point of the mechanism.

Based on the one-way reaching duration and the parameters listed in Table 9, three predefined allocation strategies - namely, equal-time allocation, link-length proportional allocation, and mass–length combined allocation - are adopted to determine the joint velocity parameters, and simulations are conducted to compare the resulting joint-motion parameter configurations.

In the equal-time allocation strategy, the prescribed 7 s one-way reaching duration is equally allocated among the four joint motions. The parameters for the link-length proportional scheme and the mass–length combined scheme are defined as follows:

(37) $$ d_{Li} = \frac{l_i}{\displaystyle\sum_{i=4}^{7}l_i} \times 100\% $$

(38) $$ d_{Mi} = \frac{m_i l_i}{\displaystyle\sum_{i=4}^{7}m_i l_i} \times 100\% $$

where l_i denotes the link length of the i-th arm of the picking mechanism (i = 4, 5, 6, 7); d_Li is the allocation ratio based on the link-length proportion; and d_Mi is the optimization parameter based on the mass–length combined criterion.

According to the above allocation strategies, the joint-motion time allocation within the prescribed 7 s one-way reaching phase is obtained. The corresponding results are summarized in Table 10.

Based on the ADAMS dynamic model established above and the parameters listed in Table 10, dynamic simulation experiments are carried out under different optimization schemes. ADAMS simulation is conducted for the end-effector motion as it reaches the boundary picking point. The joint forces and torques of each link are then obtained from the simulation results. A comparative analysis [Figure 10] is then conducted to investigate the distribution patterns of joint forces/torques under different parameter allocation schemes.

Using the amplitude and duration of the force/torque fluctuations as evaluation indices, the responses shown in Figure 10A were compared for the three predefined joint-time allocation strategies. During the first half of the motion cycle (0-3.5 s), the mass–length-proportional allocation strategy exhibits the largest peak-to-peak variation in force/torque. In particular, a pronounced abrupt change occurs between 0.5 and 1.0 s, indicating stronger transient fluctuations than those produced by the other two strategies. During the second half of the motion cycle (3.5-7.0 s), the equal-time allocation strategy shows the largest peak-to-peak variation, characterized by a sharp spike at approximately 3.75 s followed by a rapid decline. By comparison, the link-length-proportional allocation strategy exhibits relatively moderate force/torque fluctuation amplitudes and avoids the pronounced transient variations observed under the mass–length-proportional and equal-time strategies. At 7.0 s, the torque responses of all three strategies converge to the same steady-state value of approximately 22,000 N·mm. Similar response trends are observed for the remaining joints.

The differences among the three strategies can be attributed to their respective joint-time distributions. Under the mass–length-proportional allocation strategy, relatively short motion durations are assigned to the secondary rotational joint and the translational joint, resulting in comparatively large initial force/torque peaks. The equal-time allocation strategy produces a pronounced abrupt variation during the later stage of the motion cycle. In contrast, the link-length-proportional allocation strategy avoids the largest transient peaks observed in the other two strategies and provides a comparatively smooth actuator-load response under the prescribed boundary-point trajectory. Accordingly, it was selected for the subsequent payload-dependent analysis, motion-cycle analysis, simulations, and physical experiments. This selection represents an engineering choice among the three predefined allocation strategies under the tested trajectory and operating conditions. Because the allocation ratios were specified in advance and no numerical search or formal optimization procedure was employed, the comparison does not demonstrate local or global mathematical optimality.

The influence of varying loads on the dynamic characteristics of the picking mechanism is systematically investigated. Based on a link-length proportional optimization strategy, the joint motion time allocation is obtained, and simulations under different load cases are performed. The simulation process is divided into an outward reaching phase (0-7 s) and a return phase (7-14 s), with a total duration of 14 s. The payload-dependent simulation consists of a 7 s outward-reaching phase and a 7 s return phase, for a complete out-and-return simulation cycle of 14 s. The 14 s duration is a reciprocating simulation interval and does not represent the field single-fruit picking time.

In accordance with the actual single-fruit picking procedure, the joint motion ranges and time allocation are specified as listed in Table 11.

The picking mechanism is a serial structure, and the joints complete the entire single-fruit harvesting process sequentially. To simulate the end-effector load after fruit picking and ensure consistency with field operating conditions, equivalent additional loads of 0 kg (no load), 0.25 kg (light load), and 0.5 kg (full load) are applied at the gripper at t = 7 s. The load acts vertically downward and moves synchronously with the links.

To study how varying loads affect the driving joints, a 14 s complete working cycle is selected for analysis. The joint responses are examined over the entire motion period (0-14 s) under an identical trajectory, and the maximum forces and torques for different load conditions are summarized in Table 12.

As shown in Figure 11, the picking mechanism operates under no-load conditions during 0-7 s, during which the joint force (torque) curves corresponding to the three preset loads completely overlap. This indicates that, in the absence of load, the joint dynamic characteristics are governed solely by kinematic constraints and are independent of the preset load parameters.

At t = 7 s, a sudden application of load causes a transient fluctuation in all curves. If this instantaneous disturbance is neglected, the variation patterns of the curves during 7-14 s remain consistent and symmetric with those observed during 0-7 s. This behavior is closely related to the symmetry of the motion trajectory and the sequential symmetry of joint motions in the serial mechanism.

As shown in Table 12, when the load increases from 0 to 0.5 kg, the differences in the maximum joint forces (torques) are 5,120.34 N·mm, 3.4648 N, 4,153.41 N·mm, and 1,830.84 N·mm, respectively. To analyze the influence of the maximum force (or torque) on the joint mechanical characteristics and eliminate the effect of dimensionality, a normalized secant load-response index (χ) is introduced to compare the finite change in the maximum joint response over the tested payload interval.

(39) $$ \chi = \frac{D_{0.5}-D_0}{m_{0.5}-m_0} \times \frac{m_{0.25}}{D_{0.25}} \times 100\% $$

where m₀, m_0.25, and m_0.5 denote the load masses, with values of 0, 0.25, and 0.5, respectively. In the equation, D₀ represents the maximum joint force (torque) under a 0 kg load; D_0.25 represents the maximum joint force (torque) under a 0.25 kg load; and D_0.5 represents the maximum joint force (torque) under a 0.5 kg load. The index χ represents a normalized finite-difference response between the tested payload levels. It is not a local derivative or elasticity and should not be interpreted as the percentage response caused by a 1% infinitesimal change in payload.

According to Equation (39), the normalized secant load-response indices over the tested payload interval are 6.83% for the rotating secondary arm, 48.43% for the translational arm, 9.17% for the picking main arm, and 20.83% for the picking secondary arm. Among the four joints, the translational joint exhibits the largest normalized finite change between the three tested payload conditions.

These values are descriptive indices based on payloads of 0, 0.25, and 0.5 kg. They do not represent local elasticities, and 48.43% must not be interpreted as indicating that a 1% payload change causes a 48.43% change in force.

Because the payload is applied as a step input at t = 7 s, the reported global maxima may include both the transient impact of load application and the subsequent loaded motion response. A more detailed separation of static and transient load effects will require additional payload levels and dedicated regression experiments.

Over the three tested payload levels, the translational joint exhibited the largest normalized secant change in maximum actuator load. This result provides a qualitative indication that the translational drive deserves particular attention in actuator selection. The reported index is not interpreted as a local payload elasticity.

Three representative simulation durations of 14 s, 18 s, and 22 s are adopted to examine the effect of different motion cycles on the kinematic behavior of the picking mechanism. In this section, “simulation cycle” refers specifically to the complete out-and-return reciprocating simulation duration. Three complete simulation cycles of 14, 18, and 22 s are evaluated. For each test group, the motion time of each step is reduced by 0.5 s relative to the previous group, and comparative experiments are conducted under identical loading conditions. To ensure comparability across tests, a constant working load of 0.25 kg was applied in all cases, so that differences in the motion response of the mechanism could be attributed to the tested motion parameters rather than variations in load.

In contrast to the boundary-reaching trajectories used in Sections 5.1 and 5.2, Section 5.3 employs a fixed reciprocating trajectory to investigate the sensitivity of the mechanism response to motion-cycle duration. The prescribed motion ranges are 30° for the rotating secondary arm, 24 mm for the translational arm, 22.5° for the picking main arm, and 45° for the picking secondary arm. These ranges remain unchanged across the three simulated cycle-duration conditions, with only the corresponding motion durations being varied. This controlled setup isolates the effect of motion-cycle duration from that of joint displacement, thereby enabling the influence of cycle duration on the motion and dynamic responses of the mechanism to be systematically evaluated. Accordingly, the analysis is intended to reveal the response characteristics associated with different motion-cycle durations under a prescribed trajectory rather than to reproduce the full boundary-reaching trajectories reported in Tables 9 and 11. The specific simulation parameters are summarized in Table 13.

Based on its motion characteristics, the dynamic parameters of the translational arm, the picking main arm, and the picking secondary arm are analyzed [Figure 12].

As shown in Figure 12, the motion parameter curves of each joint exhibit an approximately symmetric distribution about the midpoint of the motion cycle, which is highly consistent with the temporal characteristics of the “out-and-return” reciprocating motion in the picking operation. When the centroid velocity reaches its maximum value, the momentum simultaneously attains its peak. The maximum values occur at the picking secondary arm during the 14 s motion cycle, as shown in Figure 12G-I, with values of 235.47 mm/s, 470.94 mm/s², and 0.41 N·s, respectively. From Figure 12A, B, D, E, G, and H, it can be further observed that as the complete out-and-return simulation cycle is shortened from 22 to 14 s, the peak centroid velocity and acceleration increase correspondingly.

These results reveal the intrinsic coupling relationship among motion cycle, dynamic parameters (velocity, acceleration, and momentum), and system vibration. This provides a theoretical basis and engineering guidance for subsequent optimization of structural stability and vibration suppression of the picking mechanism.

The kinematic and dynamic calculations were performed using MATLAB R2022b (MathWorks, Natick, MA, USA). The multibody dynamic model was established and solved using MSC Adams 2021 (MSC Software Corporation, USA), and the three-dimensional mechanical model was prepared using SolidWorks 2020 (Dassault Systèmes, France). The robot-control and data-transfer programs were implemented using Python 3.10.9. The recorded high-speed images were processed using TEMA Motion 2022 (Image Systems AB, Sweden). All analyses were performed on a Lenovo ThinkBook computer equipped with an Intel Core i7-12700H processor, 32 GB of RAM, and the 64-bit Windows 10 operating system.

The boundary-point tracking experiment was conducted on an open hard-ground test site at the Agricultural Experimental Station of Shandong Agricultural University in June 2025. The open site was selected to minimize interference from uneven orchard terrain, branches, foliage, wind-induced fruit motion, and variable canopy occlusion, thereby facilitating stable observation of the prescribed reference-point trajectories. The experiment was designed to evaluate the motion-execution accuracy of the picking mechanism under controlled conditions while avoiding interference from orchard terrain and environmental variability. The planar tracking errors were calculated from the deviations between the measured and theoretical coordinates in the calibrated Y–Z measurement plane. Because the present single-view arrangement did not provide independently calibrated depth information along the X-axis, a complete three-dimensional Euclidean error was not calculated. The high-speed camera was positioned approximately perpendicular to the Y–Z measurement plane and was used to track the marker corresponding to the gripper reference point. The inclinometer and laser displacement sensor measured the revolute-joint angles and prismatic-joint displacement, respectively. The purpose of this experiment was to evaluate boundary-reference-point reaching accuracy, joint-motion accuracy, and arrival-time accuracy under controlled measurement conditions. No fruit-contact or fruit-detachment criterion was involved in this experiment. The experiment was conducted using the developed harvesting robot, which was equipped with a Lenovo ThinkBook laptop, a VEO410L high‑speed camera (York Technology Co., Beijing; 1280 × 800 resolution, 5200 fps, 190 μs exposure), a DL1903 inclinometer (0.05° accuracy), an HG‑C1100 laser displacement sensor (0.01 mm accuracy; Panasonic, Japan), a handheld temperature and humidity meter (0.1 °C accuracy), a vernier caliper, a measuring tape, a stopwatch, and related auxiliary equipment.

The orchard harvesting experiment was conducted in October 2025 at the orchard experimental base of the Horticulture Experimental Station of Shandong Agricultural University to evaluate the harvesting performance of the robot under actual orchard conditions. The experimental orchard was planted with ‘Golden Delicious’ apple trees. The average intra-row spacing and inter-row spacing were approximately 2.0 and 3.0 m, respectively. The measured inter-row spacing of approximately 3.0 m falls within the robot’s specified applicable inter-row operating range of 2.5-3.5 m. The intra-row spacing of approximately 2.0 m represents the actual spacing between adjacent trees in the experimental orchard and is not a machine operating-range parameter. The mean tree height was approximately 3.1 m, and the mean canopy diameter was approximately 1.6 m. The orchard terrain was relatively flat, providing suitable operating conditions for the mobile harvesting platform.

Before the experiment, the robot was positioned between adjacent tree rows and operated using the joint-motion parameters selected from the preceding motion-parameter allocation analysis. Apples located within the effective picking regions of the robot were selected as harvesting targets. For each target, the positioning mechanism first moved the picking mechanism to the corresponding operating region, after which the picking mechanism executed the prescribed reaching, grasping, and detachment motions. A harvesting attempt was defined as successful when the target apple was detached from the tree and securely retained by the end-effector after completion of the picking operation.

The harvesting performance was evaluated using two indicators: picking success rate and single-fruit picking time. The picking success rate was calculated as the ratio of the number of successfully harvested apples to the total number of attempted targets. The single-fruit picking time was defined as the elapsed time required to complete one fruit-picking operation and was recorded for each tested picking region. The orchard experiment was intended to evaluate the practical harvesting performance of the robot under the tested orchard conditions and was not used to quantify geometric tracking errors.

Descriptive variability among the ten tested orchard domains was quantified using the population standard deviation,

$$ \sigma =\sqrt{\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2} $$

where n = 10 is the total number of experimental domains included in the field campaign. The coefficient of variation was calculated as

$$ CV=\frac{\sigma}{\bar{x}}\times 100\% $$

Population rather than sample standard deviation was used because these statistics were intended to descriptively characterize the complete set of domains tested in the present experimental campaign.