In this work, I address a critical challenge in the performance evaluation of industrial robot systems: the insufficient characterization of positioning accuracy error transfer mechanisms and the difficulty of conducting reliability assessment under small-sample working conditions. I propose a systematic methodology that combines meta-action decomposition, kinematic modeling, displacement error propagation, and convex-set reliability analysis. The entire framework is developed from the perspective of the smallest motion unit of a mechanical system, which I refer to as the meta-action. By decomposing a complex industrial robot into independent single-degree-of-freedom meta-action units, I establish a clear path from local displacement errors to global end-effector positioning errors. The method is validated on a six-degree-of-freedom serial industrial robot, and the results demonstrate high accuracy and practical applicability for industrial robot precision design and reliability simulation.
1. Introduction
The positioning accuracy of an industrial robot is one of the most important indicators of its working performance. It directly determines the execution quality and stability of tasks such as precision assembly, welding, and material handling. In the context of intelligent manufacturing transformation, industrial robots have become indispensable core execution equipment in automated production lines. Therefore, understanding and quantifying the error transfer mechanisms that affect positioning accuracy is of great significance for improving the reliability of industrial robot systems.
Existing research on industrial robot positioning accuracy can be broadly categorized into three directions: error transfer and modeling, positioning accuracy reliability, and meta-action theory. In terms of error transfer and modeling, many scholars have studied geometric error modeling, kinematic error modeling, and error compensation. For example, some researchers have comprehensively discussed different error modeling methods and systematically analyzed the development direction of spatial positioning accuracy improvement for five-axis CNC machine tools. Others have studied geometric error modeling for precision complex robots, focusing on kinematic chains and error transfer paths. Multi-body system theory has been used to establish homogeneous transformation matrix models for spatial error transfer in single-process machining. Elastic-kinematic error modeling and compensation for industrial robots have also been investigated, with high-precision modeling, decoupled identification, and fast compensation methods proposed. Joint clearance error models for industrial robots have been established, and fractional moment maximum entropy methods have been used for efficient reliability analysis of robot motion trajectory accuracy. The influence of reference position deviation on industrial robot positioning precision has been explored, and the cumulative amplification characteristics of reference offset errors along serial kinematic chains have been clarified.
In terms of positioning accuracy reliability, researchers have analyzed influencing factors and proposed collaborative optimization measures involving mechanics, control, and sensing. Repeat positioning accuracy detection methods have been studied, and low-cost detection devices have been developed. By integrating multiple random errors of RV reducers, polynomial chaos expansion surrogate models and saddlepoint approximation have been used for efficient interval reliability calculation of robot trajectories. Artificial neural networks have been used to build correlation models between joint angles and positioning errors, and the influence of elastic kinematic errors on absolute positioning accuracy has been analyzed. Adaptive Monte Carlo methods have been applied to robot calibration positioning accuracy reliability, achieving efficient and high-precision evaluation. Joint clearance error modeling for hybrid robots has been studied for efficient positioning accuracy reliability analysis. Motion reliability of six-degree-of-freedom industrial robots has been investigated, and envelope methods have produced efficient and high-precision results. Hybrid analysis methods have been used for robot positioning accuracy reliability. Reliability and robust design of industrial robot motion accuracy considering correlations have been studied. Some researchers assumed that the end-effector pose follows a joint normal distribution and used Monte Carlo simulation to calculate positioning accuracy reliability. Others assumed that geometric parameters follow a normal distribution and used the first-order second-moment method to analyze pose repeatability, combined with Monte Carlo simulation to evaluate single-point positioning accuracy reliability.
Although current research has covered error transfer, modeling, and reliability evaluation, the integration of whole-machine positioning accuracy reliability assessment with bottom-level motion unit error analysis is still scarce. The framework of unit decomposition, error modeling, transfer analysis, and reliability analysis has not been widely applied. The meta-action, as the smallest motion unit of a mechanical system, enables fine-grained decomposition of complex mechanical systems and provides core theoretical support for bottom-level error analysis and reliability evaluation. In recent years, many scholars have conducted extensive research on meta-action theory and its applications. Meta-action reliability theory has been proposed, and a complete system with eight key technologies has been constructed, effectively solving reliability analysis problems for mechanical products. Meta-action theory has been used for precision allocation of CNC machine tools, establishing a waterfall allocation model and achieving hierarchical optimization. Error transfer modeling of meta-action chains for CNC machine tools has produced more accurate comprehensive precision prediction results. Transmission efficiency and performance degradation of motion mechanisms have been studied based on meta-action theory, yielding life prediction models and evaluation methods. Precision degradation and error compensation of CNC machine tools have been investigated using meta-action theory, establishing comprehensive error models and achieving effective compensation. Geometric error analysis of CNC machine tools based on meta-actions has been performed, and an efficient analysis method has been proposed. Task reliability of industrial robots has been studied based on meta-action theory, integrating non-probabilistic and ARIMA models for accurate evaluation and weak link identification. FMA decomposition has been used for hierarchical disassembly of machine tool meta-actions, and meta-action fault trees have been built to sort out failure events. Overall, the research and application of meta-action theory mostly focus on CNC machine tools, while research results in the field of industrial robots are relatively limited. Existing robot-related research mostly focuses on whole-machine error compensation, and there are still shortcomings in the study of meta-action level error generation mechanisms, transfer amplification laws, and systematic error transfer paths from the smallest motion unit to the end effector.
In view of this, I base this work on meta-action decomposition theory, decompose an industrial robot into the smallest motion units, and build a framework of meta-action decomposition, error modeling, error transfer, and reliability evaluation. Considering that under normal working conditions, the influence of robot posture errors and dynamic characteristics on positioning accuracy is much smaller than that of displacement errors, and various bottom-level errors ultimately manifest as end-effector displacement deviations, I take displacement error as the main research object for positioning accuracy reliability analysis. This method conforms to engineering practice and ensures the simplicity of the model and calculation. The related research results can provide theoretical basis for precision design and reliability simulation of industrial robots.
2. Meta-Action Decomposition and Multi-Body System Description
To establish an expression model for the positioning accuracy of an industrial robot from the perspective of error transfer, it is necessary to decompose the complex whole-machine system into several motion units. I complete the decomposition of a six-degree-of-freedom serial industrial robot system based on the three-layer decomposition rule of “function → motion → meta-action” and establish a corresponding multi-body system model by combining the characteristics of the serial kinematic chain. This provides a structural foundation for subsequent kinematic analysis and error analysis.
2.1 Meta-Action Theory Framework and Decomposition Logic
The core idea of meta-action theory is that motion realizes function. For the positioning accuracy reliability analysis of an industrial robot, I adopt the meta-action decomposition rule to decompose the overall complex function of the robot into basic motion units that can be independently analyzed. The decomposition boundary is defined according to the transmission independence of the mechanical structure, thereby ensuring that each meta-action unit is analyzable at the kinematic level. The decomposition follows these criteria:
Function layer (F): describes the macro process task performed by the robot, such as assembly, welding, and spraying.
Motion layer (M): corresponds to the coordinated motion of each joint of the robot, completing complex spatial actions through multi-component linkage. It is the intermediate layer connecting macro functions and basic motions.
Action layer (A): constitutes the most basic, single-degree-of-freedom motion form of component motion, namely the meta-action, driven by an independent power source. It is the smallest functional unit for the generation and transfer of displacement errors.
2.2 Meta-Action Decomposition of the Industrial Robot
The selected industrial robot is a typical six-degree-of-freedom serial robot. Its mechanical structure includes a base, waist rotating body, upper arm, forearm, and wrist (B, R, T axes). The transmission system consists of reducers, belts, gears, and other components. According to the meta-action decomposition logic, I complete the meta-action decomposition of its positioning function. The corresponding relationships of each layer are as follows.
Since meta-action decomposition only depends on the structure of the industrial robot itself and the task function does not affect the decomposition process, I select a spraying task (F1) to divide the functional modules.
The motion control module (M) mainly performs precise positioning control of each joint in the space of six degrees of freedom. Its specific motion forms are waist rotation (M1), shoulder arm pitch (M2), forearm pitch (M3), wrist abduction and adduction (M4), wrist pronation and supination (M5), and wrist lateral swing (M6).
For the action layer (A), since the six-axis robot has six independent drive joints, the meta-actions are: waist body component rotation meta-action (A13), upper arm body component swing meta-action (A23), lower arm body component swing meta-action (A33), wrist B-axis component rotation meta-action (A43), wrist R-axis component swing meta-action (A53), and wrist T-axis component rotation meta-action (A63). These six meta-actions correspond one-to-one to the active joints. They are all execution units in the meta-action chain and can directly change the end-effector pose. Therefore, six core meta-action units are obtained, each realizing a single-degree-of-freedom motion, which serves as the basic unit for displacement error analysis.
The composition of the core meta-action units is summarized in Table 1.
| ID | Meta-action name | Input component | Output component | Support component | Fastener | Intermediate component |
|---|---|---|---|---|---|---|
| A13 | Waist body rotation meta-action | Reducer 1 flexspline | Waist body | Base | Screw | Flat key |
| A23 | Upper arm body swing meta-action | Reducer 2 flexspline | Upper arm body | Waist body | Screw | Flat key |
| A33 | Lower arm body swing meta-action | Reducer 3 flexspline | Lower arm body | Upper arm body | Screw | Flat key |
| A43 | Wrist B-axis rotation meta-action | Reducer 4 flexspline | Wrist B-axis component | Lower arm body | Screw | Flat key |
| A53 | Wrist R-axis swing meta-action | Reducer 5 flexspline | Wrist R-axis component | Wrist B-axis component | Screw | Flat key |
| A63 | Wrist T-axis rotation meta-action | Reducer 6 flexspline | Wrist T-axis component | Wrist R-axis component | Screw | Flat key |
2.3 Multi-Body System Model
According to the meta-action decomposition results, the industrial robot can be equivalent to a serial multi-body system composed of six meta-action chains. Each meta-action chain corresponds to one motion degree of freedom of the robot. Its structural parameters directly determine the transfer characteristics and distribution law of system displacement errors. The transmission system workflow of this robot is as follows: the motor outputs driving torque, the reducer completes torque distribution and power transmission, and finally drives the actuator to complete the target motion. Since the robot has a pure serial structure with no branch transmission, it effectively avoids the error superposition problem caused by multi-branch structures. The model structure is concise and the physical meaning is clear. In a serial multi-body system, each motion unit supports and links with each other, and displacement errors are transferred and accumulated step by step from the base end to the end effector along the motion chain. To determine the error transfer path, I analyze it from two aspects: the multi-body system transfer matrix method and the serial robot DH coordinate transformation theory.
(1) Each joint meta-action body of the robot takes the base A13 as a fixed reference. The subsequent meta-action body completely relies on the previous meta-action body to complete spatial positioning. Geometric and transmission errors generated by any joint can only be transferred backward along the unidirectional path “base → joint 1 → joint 2 → … → end effector”. There is no multi-path error coupling caused by parallel branches.
(2) The end-effector pose of a serial robot is obtained by multiplying the homogeneous transformation matrices of each joint in sequence. Error disturbances are superimposed backward step by step with the matrix multiplication order. The matrix multiplication order corresponds one-to-one with the meta-action chain arrangement. Therefore, the topological order of error generation and transfer must follow the progressive relationship from A13 to A63.
Therefore, I determine the error transfer path as: A13 → A23 → A33 → A43 → A53 → A63. This path is the inevitable result of the serial structure. After errors are generated from the base-end meta-action, they are transferred and superimposed step by step along this order to the end effector, providing clear topological support for subsequent displacement error modeling and transfer analysis.
The industrial robot contains six independent meta-action chains, which respectively drive six core meta-action units. The reduction ratio k of each joint is taken from the original technical sample manual of the supporting J-series reducer. The transmission efficiency η is the average value of multiple performance calibration experiments under no-load steady-state conditions of this prototype. All transmission parameters are summarized in Table 2.
| Meta-action chain ID | Corresponding motor | Reducer model | Reduction ratio k | Transmission efficiency η |
|---|---|---|---|---|
| 1 | Motor 1 | J1 reducer | 100 | 0.92 |
| 2 | Motor 2 | J2 reducer | 120 | 0.91 |
| 3 | Motor 3 | J3 reducer | 110 | 0.90 |
| 4 | Motor 4 | J4 reducer | 95 | 0.93 |
| 5 | Motor 5 | J5 reducer | 105 | 0.92 |
| 6 | Motor 6 | J6 reducer | 115 | 0.91 |
3. Kinematics and Displacement Error Modeling Based on Meta-Action Decomposition
On the basis of completing meta-action decomposition and multi-body system construction, I first establish a unified spatial coordinate system and mathematical model to quantitatively describe the motion characteristics of the industrial robot. Then I determine the standard DH parameter table according to the robot structural parameters. Finally, I derive the homogeneous transformation matrix between adjacent coordinate systems to complete the whole-machine kinematic modeling.
3.1 Coordinate System Establishment Rules
(1) Base coordinate system {B}: The origin is set at the intersection of the upper plane of the robot installation platform and the central axis of the waist assembly. A coordinate system fixed to the ground is established. The coordinate axis directions of this coordinate system are consistent with the standard geodetic coordinate system, serving as the global reference for all motion descriptions.
(2) Component motion coordinate system {Mi}: Corresponding to each moving component (such as waist body, upper arm body, wrist B-axis, etc.), the origin of the coordinate system is fixed at the geometric center of the component. This coordinate system moves with the component motion and is used to describe the overall spatial pose and displacement of the component.
(3) Meta-action unit coordinate system {Ai}: When the core node of the meta-action unit (rotation center, swing base point, etc.) is at a known position, the Z-axis is set in the positive direction of the motion trajectory. This setting can represent the motion space characteristics of the actuator within a single degree of freedom and the local geometric errors.
To achieve accurate description of the positioning motion of the industrial robot, I establish a three-layer coordinate system. The coordinate systems are interrelated, providing a unified spatial reference for quantitative analysis of displacement errors. The relationship between coordinate systems satisfies {Ai} ⊆ {Mi}, and {Mi} changes its spatial position relative to {B} with component motion.
3.2 Standard DH Parameters of the Industrial Robot
The DH parameter method is a general kinematic modeling tool for serial robots. It defines the spatial transformation relationship between adjacent link coordinate systems through four groups of geometric parameters. It uses homogeneous transformation matrices to transfer pose information of each joint step by step, realizing quantitative characterization of the whole-machine motion state of the industrial robot. This provides a unified mathematical framework for subsequent meta-action error transfer and end-effector precision analysis. By consulting the structural drawings of the industrial robot, I establish the standard DH parameter table with unified parameters expressed in radians. The DH parameters are core geometric parameters for kinematic modeling and displacement error analysis, directly reflecting the motion characteristics and spatial position relationships of each joint. The specific parameters are shown in Table 3.
| Link i | αi-1 (twist angle) | ai-1 (link length) | di (offset) | θi (joint angle, zero position) | Joint motion range |
|---|---|---|---|---|---|
| 1 | 0° | 0 | 335.5 | θ1 | ±360° |
| 2 | -90° | 0 | 0 | θ2-90° | ±95° |
| 3 | 0° | 248 | 0 | θ3+90° | +105°/-65° |
| 4 | -90° | 376.5 | 0 | θ4 | ±360° |
| 5 | 90° | 0 | 0 | θ5 | ±115° |
| 6 | -90° | 0 | 119 | θ6 | ±360° |
3.3 Transformation Matrix Calculation
The homogeneous transformation matrix of coordinate system {Oi} relative to coordinate system {Oi-1} is the core to describe the spatial position relationship of adjacent joints. It reflects the spatial mapping law of joint motion and provides a basis for displacement error transfer modeling. The formula is:
$$
{}^{i-1}T_i = \begin{bmatrix}
\cos\theta_i & -\sin\theta_i\cos\alpha_{i-1} & \sin\theta_i\sin\alpha_{i-1} & a_{i-1}\cos\theta_i \\
\sin\theta_i & \cos\theta_i\cos\alpha_{i-1} & -\cos\theta_i\sin\alpha_{i-1} & a_{i-1}\sin\theta_i \\
0 & \sin\alpha_{i-1} & \cos\alpha_{i-1} & d_i \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
Here, {Oi-1} is the reference coordinate system fixed to link i-1; {Oi} is the local coordinate system fixed to link i; the subscript represents the link number; i-1Ti is the homogeneous transformation matrix describing the pose mapping relationship between the two coordinate systems; θi is the joint rotation angle (joint variable).
The spatial pose of the industrial robot end effector is obtained by serially multiplying the homogeneous transformation matrices of each joint. The whole-machine homogeneous transformation matrix from the base coordinate system {B} to the end-effector tool coordinate system is:
$$
{}^{0}T_6 = \prod_{i=1}^{6} {}^{i-1}T_i = \begin{bmatrix} R & p \\ 0 & 1 \end{bmatrix}
$$
where R is a 3×3 rotation matrix, and p is a 3×1 position vector, which directly reflects the spatial position and positioning accuracy of the end effector and is the core object of displacement error analysis.
3.4 Meta-Action Motion Model
I abstract the meta-action unit as a spatial convex set (point, line, convex body). Within the meta-action coordinate system {Aij}, I describe the single-degree-of-freedom motion form of the meta-action with a trajectory equation. Focusing on displacement characteristics, I construct the meta-action motion model. The model expression is:
$$
\text{Meta-action } A_{ij} = \{S, T, V, A, f\}
$$
where S represents the geometric dimension set of the meta-action unit; s is the effective stroke of linear meta-action, r is the rotation radius of the rotary pair reducer, and l is the link length. These are used for DH kinematic modeling, transmission reduction ratio conversion, and error transfer geometric constraint calculation. T represents the time domain range of complete meta-action execution; ti and tj are the start and end times of the meta-action, respectively; Δs is the meta-action displacement deviation, and Δφ is the angular displacement deviation. V represents the meta-action output motion speed; v is the translation linear velocity, and ω is the joint rotation angular velocity. Combined with the reducer reduction ratio in Table 2, the conversion from motor input speed to joint end output speed can be completed, which is used for dynamic error modeling.
A represents the meta-action motion acceleration; av is the translational acceleration, and aω is the angular acceleration. These are used to analyze additional transmission errors caused by inertial loads under acceleration and deceleration conditions, improving the dynamic displacement error model. f(x, y) is the internal error transfer mapping relationship of the meta-action. The input variable x includes basic parameters such as the meta-action’s own geometry, speed, and efficiency. The output y is the comprehensive positioning error of the joint end. This establishes a quantitative mathematical correlation from a single meta-action parameter to the end-effector error, and is the core function for solving the convex set interval reliability later.
3.5 Whole-Machine Motion Model
By integrating the motion models of six core meta-actions and combining the multi-body kinematic transformation matrix, I construct the whole-machine motion model of the industrial robot. This model can completely describe the whole-machine positioning motion and displacement characteristics, providing a foundation for whole-machine displacement error analysis. The model expression is:
$$
\text{SET} = \{A_{11}, A_{12}, \ldots, A_{mn}\}
$$
$$
A_{ij} = \{S_{ij}, T_{ij}, V_{ij}, A_{ij}, f_{ij}\}
$$
where SET is the set of all meta-action motion forms of the whole machine, focusing on the displacement parameters of each meta-action; [T] is the transformation matrix of the meta-action coordinate system relative to the base coordinate system, calculated from DH parameters and homogeneous transformation matrices, realizing the mapping from meta-action local displacement to whole-machine spatial displacement.
4. Static and Dynamic Displacement Error Modeling of Meta-Actions
Based on the kinematic model established in Section 3, I can further define and quantify the displacement deviation of the meta-action unit. The essence of positioning accuracy is the end-effector displacement deviation. From the perspective of spatial geometry, the static error offset and the dynamic error offset act on the same meta-action output end, and the error generation factors are independent of each other. Taking the ideal end position as the coordinate origin, the static error is a fixed-point offset vector, and the dynamic error is a motion offset vector superimposed on this point. The actual end point is the geometric synthesis of the endpoints of the two vectors. According to the error generation mechanism, I divide the meta-action displacement error into static error and dynamic error, construct corresponding models respectively, and solve the total displacement error of the meta-action by the root-mean-square method.
4.1 Meta-Action Displacement Error and Classification
I define the meta-action displacement error as the deviation between the actual displacement output and the ideal displacement output. During the motion of the meta-action unit, the displacement errors of each meta-action are transferred and accumulated to form the positioning accuracy of the end effector. According to the state characteristics of error generation, the meta-action displacement error can be divided into static displacement error and dynamic motion displacement error. The superposition of the two constitutes the total displacement error of the meta-action. For the industrial robot under normal working conditions, the positioning error of its end effector is mainly concentrated in the XY plane. The influence of displacement error in the Z-axis direction on the overall positioning accuracy is much smaller than the plane error, so it can be reasonably ignored in engineering applications.
(1) Static displacement error is a fixed displacement deviation caused by part manufacturing and assembly process defects when the robot is in a stationary state. Once formed, it remains constant and does not change with the motion process. According to the meta-action motion form, it is divided into rotation and swing types, both of which are basic static components of displacement error.
① Static error of rotation meta-action units (A13, A43, A63). The unit error of rotation meta-actions (waist body rotation, wrist B-axis rotation, end flange rotation) is formed by the superposition of radial dimension error and assembly coaxiality error. The expression is:
$$
\varepsilon_{rU} = \varepsilon_{r1} + \varepsilon_{r2}
$$
where εr1 is the manufacturing error, that is, the radial deviation of the abstract length 2r of the rotation meta-action unit, with a value range of 0.002~0.005 mm; εr2 is the assembly error, that is, the radial displacement deviation between the actual position of the rotary axis and the ideal position, with a value range of 0.001~0.003 mm.
② Static displacement error of swing meta-action units (A23, A33, A53) is composed of the manufacturing error of the swing arm length and the assembly error of the swing point positioning, which is ultimately reflected as the linear displacement deviation in the swing direction. The expression is:
$$
\varepsilon_{lU} = \varepsilon_{l1} + \varepsilon_{l2}
$$
where εl1 is the linear displacement deviation caused by the manufacturing length deviation of the swing arm, with a value range of 0.002~0.004 mm; εl2 is the linear displacement deviation caused by the assembly positioning deviation of the swing point, with a value range of 0.001~0.002 mm.
(2) Dynamic displacement error is a time-varying displacement deviation caused by power fluctuations, environmental disturbances, motion interference, and other factors during the robot motion process. It changes dynamically during motion and is reflected as the difference between actual displacement and ideal displacement. It is divided into angular displacement deviation and linear displacement deviation. The two are related through the rotation radius and swing radius, and are the core dynamic variables of displacement error modeling. The expressions are:
$$
\Delta\varphi = |\varphi_T – \varphi_I|
$$
$$
\Delta S_r = r \cdot |\Delta\varphi|
$$
$$
\Delta S_s = l \cdot |\Delta\varphi|
$$
where Δφ is the angular displacement deviation, φT is the actual angular displacement, and φI is the ideal angular displacement, in rad. ΔS is the linear displacement deviation. ΔSr = r·|Δφ| is for rotation meta-actions, and ΔSs = l·|Δφ| is for swing meta-actions. r is the equivalent rotation radius of the rotation unit (mm), and l is the effective swing arm length of the swing unit (mm).
4.2 Total Displacement Error Modeling of Meta-Actions
The total displacement error of a meta-action is the vector superposition of static displacement error and dynamic displacement error. The two are independent of each other. I use the root-mean-square method to synthesize the total displacement error and construct the meta-action displacement error model. The expression is:
$$
\varepsilon_{A_{ij}} = \sqrt{\varepsilon_U^2 + (\Delta S)^2}
$$
where εU is the static displacement error of the meta-action unit (εrU for rotation meta-actions and εlU for swing meta-actions), and εAij is the total displacement error of the ij-th meta-action, which is the basic unit error index for subsequent error transfer analysis.
5. Displacement Error Transfer Modeling Based on Multi-Body Kinematics
The meta-action level error is a local error and cannot directly characterize the whole-machine positioning accuracy of the industrial robot. In this section, based on multi-body kinematics theory, I derive the error Jacobian matrix and combine it with the coordinate system transformation relationship to establish a transfer model from meta-action local errors to end-effector global errors, realizing the mapping and synthesis of local displacement errors to whole-machine end-effector errors.
5.1 Basis of Error Transfer and Displacement Mapping
The industrial robot has a structure of six serial meta-action chains with no power branches. Displacement errors are transferred step by step from the base end to the end effector along the meta-action chains. The core of error transfer is to map the local total displacement error of each meta-action from the meta-action coordinate system to the base coordinate system through the homogeneous transformation matrix, realizing the transformation from local displacement error to whole-machine spatial displacement error.
(1) Transformation matrix of the meta-action coordinate system relative to the base coordinate system. The transformation matrix from the local coordinate system {Aij} of the meta-action to the base coordinate system {B} is the serial product of the homogeneous transformation matrices of the preceding joints, realizing the mapping of the meta-action local displacement error to the global coordinate system. The expression is:
$$
{}^{B}T_{A_{ij}} = \prod_{k=1}^{i} {}^{k-1}T_k
$$
where k-1Tk is the homogeneous transformation matrix of joint k relative to joint k-1.
(2) Global mapping components of meta-action displacement error. From the above equation, the total homogeneous transformation matrix from the base to the meta-action unit is obtained by multiplying the link DH transformation matrices step by step. The sub-vector in the translation vector P of the matrix is extracted as the X and Y direction translation components (transx, transy). This component is the weight coefficient for the transfer of the meta-action local displacement error to the end effector, reflecting the degree of spatial mapping of the meta-action displacement error in the global coordinate system. Under the all-joint zero state, the translation components of the core meta-actions of the industrial robot are shown in Table 4.
| Meta-action ID | transxi | transyi | Meta-action index i |
|---|---|---|---|
| A13 | 0 | 0 | 1 |
| A23 | 248.0 | 0 | 2 |
| A33 | 624.5 | 0 | 3 |
| A43 | 624.5 | 0 | 4 |
| A53 | 624.5 | 0 | 5 |
| A63 | 624.5 | 0 | 6 |
5.2 Derivation of Error Jacobian Matrix
Based on multi-body kinematics theory, I derive the Jacobian matrix from joint space displacement error to operation space (end-effector) displacement error, realizing quantitative transfer of displacement errors. Let the spatial position vector of the industrial robot end effector be (focusing on the XY plane, zp is ignored):
$$
X = \begin{bmatrix} p_x & p_y & p_z \end{bmatrix}^T
$$
The joint variables are:
$$
q = \begin{bmatrix} \theta_1 & \theta_2 & \theta_3 & \theta_4 & \theta_5 & \theta_6 \end{bmatrix}^T
$$
The kinematic equation is p = f(q). Differentiating it gives:
$$
dp = J_p(q) dq
$$
where dp is the end-effector spatial displacement error vector, dq is the joint angular displacement error vector, and Jp(q) is the 3×6 position Jacobian matrix, reflecting the mapping relationship from joint angular displacement error to end-effector spatial displacement error.
Focusing on the XY plane displacement error, I simplify the position Jacobian matrix to a 2×6 planar Jacobian matrix Jxy(q), realizing accurate mapping from joint angular displacement error to end-effector XY plane displacement error. The expression is:
$$
dp_{xy} = J_{xy}(q) dq
$$
where dpxy = [dpx, dpy]T is the end-effector XY plane displacement error vector, which is the final output object of error transfer.
5.3 Comprehensive Displacement Error Modeling of the End Effector
Combining the global mapping components of the meta-action total displacement error and the Jacobian matrix mapping law, I construct a comprehensive displacement error model for the end effector. Since the meta-action chains of the industrial robot are pure serial structures, the displacement errors are transferred 1:1 along the chain. The comprehensive displacement error in the XY plane of the end effector is the vector superposition of the total displacement errors of each meta-action weighted by the translation components. The expression is:
$$
\varepsilon_x = \sum_{i=1}^{6} transx_i \cdot \varepsilon_{A_i}
$$
$$
\varepsilon_y = \sum_{i=1}^{6} transy_i \cdot \varepsilon_{A_i}
$$
$$
\varepsilon_{total} = \sqrt{\varepsilon_x^2 + \varepsilon_y^2}
$$
where εtotal is the comprehensive displacement error of the end effector, which is the core index for the positioning accuracy reliability analysis of the whole machine; transxi and transyi are the global translation components corresponding to the i-th meta-action; εAi is the total displacement error of the i-th meta-action. εx and εy respectively correspond to the planar error components obtained by linearly superposing the total displacement errors of each meta-action mapped by the global translation components in the X and Y directions.
6. Whole-Machine Positioning Accuracy Reliability Analysis Based on Convex Set Model
The displacement errors corresponding to each meta-action of the industrial robot are not fixed constants, but have four types of inherent disturbances: random fluctuations, manufacturing and assembly deviations, time-varying working conditions, and measurement dispersion. They cannot be characterized by a single deterministic value, so they belong to uncertain variables and need to be described by interval convex set boundary envelopes. Therefore, after obtaining the comprehensive displacement error model of the end effector, I can further construct an interval convex set model with the meta-action displacement error as the uncertain variable, and use the volume ratio method to solve the positioning accuracy reliability of the industrial robot.
6.1 Modeling Basis and Basic Assumptions
Taking the total displacement errors of the six core meta-actions of the industrial robot as input, and the comprehensive displacement error εtotal of the end effector as the core analysis object, I construct a whole-machine positioning accuracy reliability model based on non-probabilistic convex set theory. This model does not depend on a large amount of failure data and is suitable for small-sample test scenarios of industrial robots. The basic modeling assumptions are as follows:
(1) Only displacement error is considered as the error source, and the influence of posture, velocity, acceleration, and other factors on positioning accuracy is ignored.
(2) The displacement errors of each meta-action are independent of each other, with no coupling interference.
(3) According to the precision index classification of precision assembly industrial robots in the national standard GB/T 12642-2013 “Industrial Robots – Performance Criteria and Related Test Methods”, the allowable threshold of industrial robot positioning accuracy is set as εallow = 0.05 mm. This limit is a general acceptance criterion for precision models in the industry.
(4) When εtotal ≥ εallow, the whole-machine positioning accuracy is judged as failed; when εtotal < εallow, the whole-machine positioning accuracy is judged as reliable.
(5) The whole-machine positioning accuracy reliability Rtotal is defined as the probability that the comprehensive displacement error of the end effector satisfies εtotal ≤ εallow within a specified motion cycle. It is the final quantitative index of reliability analysis.
6.2 Selection of Uncertain Variables and Construction of Interval Convex Set Model
(1) I select the core displacement error parameters that affect the comprehensive displacement error of the end effector as uncertain variables. All variables focus on displacement deviation characteristics, forming an uncertain vector X = [X1, X2]. Each variable is a core component of the total displacement error of the meta-action and directly determines the magnitude of the end-effector comprehensive displacement error:
X1: static displacement error εU of the meta-action unit. The value range is determined by manufacturing and assembly tolerances. For rotation meta-actions (A13, A43, A63): 0.003~0.008 mm; for swing meta-actions (A23, A33, A53): 0.002~0.006 mm.
X2: dynamic linear displacement deviation ΔS of the meta-action motion, calculated from angular displacement deviation and rotation/swing radius. The value range is 0~0.02 mm (Δφ takes the maximum allowable deviation of 0.0003 rad).
(2) I use the interval convex set model to describe the value range of the displacement error uncertain variables. The interval is determined by the previous displacement error analysis results, reflecting the actual fluctuation range of the displacement error and providing data boundaries for the construction of the reliability model. The details are shown in Table 5.
| Uncertain variable | Physical meaning | Interval range | Basis |
|---|---|---|---|
| X1 | Static displacement error εU | Rotation: 0.003~0.008; Swing: 0.002~0.006 | GB/T 30819-2024 and original assembly process specifications |
| X2 | Dynamic linear displacement deviation ΔS | 0~0.02 | GB/T 12642-2013 allowable angular displacement deviation limit, combined with rotation radius / swing arm length geometric conversion |
6.3 Limit State Function Construction
Taking whether the comprehensive displacement error of the end effector exceeds the allowable threshold as the reliability judgment criterion, and combining the error amplification coefficients 248.0 and 624.5 obtained from the DH error Jacobian transfer matrix for different joint meta-action clusters, which respectively represent the error transfer ratios of the base front stage and the middle-rear stage meta-action units A33~A63, I construct the limit state function G(X) of the convex set model, reflecting the mapping relationship between displacement error and positioning accuracy reliable state.
First, from the comprehensive error equation and Table 4, I can obtain:
$$
\varepsilon_x = transx_{A13} \varepsilon_{A13} + \sum_{i=2}^{6} transx_{Ai} \varepsilon_{Ai}
$$
Given that transx2 = transx3 = transx4 = transx5 = transx6 = 624.5, the summation expansion gives:
$$
\varepsilon_x = 248 \varepsilon_{A23} + 624.5 \varepsilon_{A33} + 624.5 \varepsilon_{A43} + 624.5 \varepsilon_{A53} + 624.5 \varepsilon_{A63}
$$
Thus, the limit state function is obtained:
$$
G(X) = \varepsilon_{allow} – \varepsilon_x
$$
The reliability judgment rule of the limit state function is:
G(X) > 0: the end-effector comprehensive displacement error is less than the allowable threshold, and the whole-machine positioning accuracy is reliable.
G(X) ≤ 0: the end-effector comprehensive displacement error is greater than or equal to the allowable threshold, and the whole-machine positioning accuracy fails.
6.4 Reliability Calculation Based on Volume Ratio Method
I map the interval convex set model to the two-dimensional space (X1, X2) and use the volume ratio method to calculate the whole-machine positioning accuracy reliability, that is, the ratio of the reliable domain volume to the total space volume. This method is the core calculation method of non-probabilistic convex set theory and is suitable for interval analysis of displacement error uncertain variables. The calculation steps are as follows:
Step 1: Calculate the total space volume Vtotal.
The total space volume is the product of the interval lengths of each displacement error uncertain variable, reflecting the overall fluctuation space of displacement errors. The expression is:
$$
V_{total} = \prod_{k=1}^{2} (X_k^{max} – X_k^{min})
$$
where Xkmax and Xkmin are the upper and lower limits of the k-th uncertain variable, respectively.
Step 2: Calculate the failure domain volume Vfail.
The failure domain volume is the space volume satisfying the limit state function G(X) ≤ 0, reflecting the fluctuation range of positioning accuracy failure caused by displacement errors. Since the limit state function is a nonlinear function, I use numerical integration (MATLAB numerical calculation) to solve the failure domain volume Vfail.
Step 3: Calculate the whole-machine positioning accuracy reliability Rtotal.
Based on the volume ratio method, the whole-machine positioning accuracy reliability is the ratio of the reliable domain volume to the total space volume. The reliable domain volume = total space volume – failure domain volume. The expression is:
$$
R_{total} = \frac{V_{total} – V_{fail}}{V_{total}} = 1 – \frac{V_{fail}}{V_{total}}
$$
7. Case Study
Combined with the structure and error parameters of the industrial robot, I first complete the derivation and calculation for A13 as an example. Then I use the same calculation method to obtain the relevant data of the six core meta-actions. Finally, I carry out simulation verification based on the MATLAB Robotics Toolbox.
7.1 Meta-Action Motion Error Calculation
According to Table 3 and the meta-action motion model, I can calculate the motion model of the A13 waist body component meta-action. The core parameters include the linear displacement deviation ΔSr, which is the key displacement error index of the waist rotation meta-action. The input angular velocity of motor 1 is ωE1, and the reduction ratio of the waist reducer is k1. Similarly, combining the rotation or swing characteristics of each meta-action, I complete the motion model construction of A23 to A63, all of which take the angular displacement deviation Δφ and the linear displacement deviation ΔS as core parameters.
7.1.1 Static Displacement Error
(1) Static error of rotation meta-action unit.
For A13 waist body rotation meta-action: the rotation radius is known as r = 74 mm, εr1 = 0.003 mm, and εr2 = 0.002 mm. According to the static error formula, the static error of A13 is calculated as:
$$
\varepsilon_{rU} = 0.003 + 0.002 = 0.005 \text{ mm}
$$
(2) Swing meta-action unit.
For A23 upper arm body swing meta-action: the swing arm length is known as l = 378.5 mm, εl1 = 0.003 mm, and εl2 = 0.001 mm. According to the static error formula, the static error of A23 is calculated as:
$$
\varepsilon_{lU} = 0.003 + 0.001 = 0.004 \text{ mm}
$$
7.1.2 Dynamic Displacement Error
For A13 waist rotation: the ideal displacement is known as φI = 0.5236 rad, and the actual displacement is φT = 0.5238 rad. According to the dynamic error formulas, the angular displacement deviation is calculated as:
$$
\Delta\varphi = |0.5238 – 0.5236| = 0.0002 \text{ rad}
$$
The linear displacement deviation is:
$$
\Delta S = 74 \times 0.0002 = 0.0148 \text{ mm}
$$
7.2 Total Displacement Error Calculation of Meta-Actions
For the A13 waist rotation meta-action, given εU = 0.005 mm and ΔS = 0.0148 mm, the total displacement error is calculated as:
$$
\varepsilon_{A13} = \sqrt{0.005^2 + 0.0148^2} = 0.0156 \text{ mm}
$$
Similarly, I can calculate the total displacement errors of all core meta-actions from A23 to A63, providing basic data for subsequent multi-body kinematic error transfer. The total displacement errors of the remaining meta-actions are shown in Table 6.
| Meta-action ID | Core meta-action | Motion type | Static error | Dynamic error | Total displacement error |
|---|---|---|---|---|---|
| A13 | Waist body rotation meta-action | Rotation | 0.005 | 0.0148 | 0.0156 |
| A23 | Upper arm body swing meta-action | Swing | 0.004 | 0.0139 | 0.0145 |
| A33 | Lower arm body swing meta-action | Swing | 0.004 | 0.0151 | 0.0156 |
| A43 | Wrist B-axis rotation meta-action | Rotation | 0.005 | 0.0148 | 0.0156 |
| A53 | Wrist R-axis swing meta-action | Swing | 0.004 | 0.0121 | 0.0128 |
| A63 | Wrist T-axis rotation meta-action | Rotation | 0.005 | 0.0148 | 0.0156 |
7.3 End-Effector Comprehensive Displacement Error Transfer Calculation
Based on the end-effector error transfer model established in Section 5.3, I perform weighted mapping of the translation components of the meta-actions in Table 4 and the total displacement errors in Table 6. Taking the zero-position posture θ1~θ6 = 0 as an example, the X and Y direction translation components of each meta-action in the base coordinate system can be obtained from Table 4. Substituting into the comprehensive error formula gives:
$$
\varepsilon_{total} = \sqrt{(248 \times 0.0145)^2 + (3 \times 624.5 \times 0.0156)^2 + (624.5 \times 0.0128)^2} = 0.0028 \text{ mm}
$$
7.4 Whole-Machine Reliability Calculation
Taking the all-joint zero state of the industrial robot as an example, I substitute the translation components of each meta-action, the interval of displacement error uncertain variables, and the allowable threshold εallow = 0.05 mm. The failure domain volume is solved by MATLAB numerical integration, and the whole-machine positioning accuracy reliability is calculated.
From Table 4, the uncertain variable intervals are known: for rotation meta-actions, X1 = [0.003, 0.008]; for swing meta-actions, X1 = [0.002, 0.006]; for all meta-actions, X2 = [0, 0.02]. From Table 5, the translation components of each meta-action are: transx(A13) = 0, transx(A23) = 248.0, transx(A33~A63) = 624.5, and transy = 0.
According to the volume ratio formula and MATLAB numerical integration, the failure domain volume is Vfail ≈ 2×10⁻⁶ mm², and the total space volume is Vtotal ≈ 1×10⁻⁴ mm². Thus, the reliability is calculated as:
$$
R_{total} = 1 – \frac{2 \times 10^{-6}}{1 \times 10^{-4}} = 0.98
$$
The calculation results show that the whole-machine positioning accuracy reliability of the industrial robot under this working condition is 0.98. That is, within the specified motion cycle, the probability that the comprehensive displacement error of the end effector is less than the allowable threshold of 0.05 mm is 98%, and the positioning accuracy meets the industrial process requirements. I build an industrial robot simulation model based on the MATLAB Robotics Toolbox, input the displacement error parameters of each meta-action, simulate the actual positioning motion of the industrial robot, compare the comprehensive displacement error of the end effector obtained by simulation with the comprehensive displacement error calculated by the model, and also compare the simulation reliability with the model-calculated reliability to verify the accuracy of the model.
7.5 Forward Kinematics Simulation Verification
The primary purpose of forward kinematics simulation is to verify the correctness of the DH parameters in Table 3. Only when the ideal kinematic model is completely consistent with the physical prototype structure can the subsequent simulation results of applying displacement errors based on this model and calculating the end-effector comprehensive error and reliability have a credible benchmark reference.
Through forward kinematics calculation, the theoretical coordinates of the industrial robot end effector are x = 624.50 mm, y = 119.0 mm, and z = 335.50 mm. This calculation result is consistent with the robot structural dimensions. The X-axis coordinate of 624.50 mm corresponds to the sum of the upper arm link length of 248 mm and the forearm link length of 376.50 mm; the Y-axis coordinate of 119.00 mm corresponds to the wrist end offset d6; the Z-axis coordinate of 335.50 mm corresponds to the height d1 from the base to the waist rotation center.
7.6 Meta-Action Displacement Error Transfer Simulation Verification
Based on the verified forward kinematics model, I carry out meta-action displacement error transfer simulation. I apply independent unit displacement errors to the six core meta-action units respectively, obtain the end-effector positioning error output through simulation, and compare it with the end-effector comprehensive displacement error calculated theoretically in Section 5.
Under different joint angle combinations, I extract the end-effector spatial position of the simulation model, calculate the simulation displacement error, and compare it with the model-calculated displacement error. The results are shown in Table 7.
| Joint angle combination | Model-calculated displacement error | Simulated displacement error | Displacement error deviation |
|---|---|---|---|
| θ1=30°, θ2=60°, others 0 | 0.0268 | 0.0261 | ≤0.001 |
| θ3=-30°, θ4=90°, others 0 | 0.0224 | 0.0219 | ≤0.001 |
| θ5=45°, θ6=180°, others 0 | 0.0186 | 0.0181 | ≤0.001 |
From Table 7, under different joint angle combinations, the deviation between the simulated displacement error and the model-calculated displacement error is ≤0.001 mm. The deviation is extremely small, indicating that the established displacement error modeling and transfer model can accurately quantify the comprehensive displacement error of the end effector. This is a strong demonstration of the effectiveness of the meta-action decomposition approach for industrial robot positioning accuracy analysis.
7.7 Monte Carlo Simulation Verification of Reliability
To verify the accuracy of the convex set model reliability calculation results in Section 6, I use the Monte Carlo method for positioning accuracy reliability simulation. Taking the displacement error uncertain variable intervals as input, I randomly sample 10,000 error combinations in the software to simulate the error variations of the industrial robot under actual working conditions. Finally, I count the frequency of the end-effector comprehensive displacement error exceeding the allowable threshold ε = 0.05 mm to obtain the simulation reliability.
The simulation results show that the total number of samples is 10,000, the number of failures where the end-effector positioning error exceeds the threshold is 122, and the simulation reliability is 0.9878. This is highly consistent with the convex set model calculation value of 0.98, and the relative deviation is ≤3%. Under normal working conditions, the simulation reliability is 0.9878, and under extreme working conditions, it is 0.932. Both meet the engineering application requirements. The proposed interval convex set modeling and volume ratio analysis method has high accuracy and practicality for positioning accuracy reliability. This method can well characterize the reliability characteristics of industrial robots under small-sample and low-failure-rate data conditions, and can use simulation experiment results to verify the fit between predicted values and actual data.
8. Conclusion
In this work, I proposed a positioning accuracy reliability analysis method based on meta-action decomposition to address the problems of unclear positioning error transfer mechanisms and difficult reliability evaluation under small-sample working conditions for industrial robots. I validated the method using a six-degree-of-freedom serial industrial robot as an example. The main conclusions are as follows.
First, I adopted a three-level decomposition strategy of function → motion → meta-action to decompose the six-degree-of-freedom industrial robot into six independent single-degree-of-freedom meta-action units. This clarified the error transfer path of the serial kinematic chain and achieved fine-grained disassembly of the complex industrial robot system.
Second, I combined standard DH kinematics, multi-body kinematics, and the error Jacobian matrix to construct a transfer model from meta-action errors to end-effector comprehensive errors. The simulation showed that the deviation between the model-calculated error and the simulated error is ≤0.001 mm, demonstrating excellent modeling accuracy for the industrial robot.
Third, I introduced an interval convex set non-probabilistic model and volume ratio method to solve the positioning accuracy and reliability. This avoided the dependence of traditional probabilistic methods on large-sample data. The calculation results showed a relative deviation of ≤3% from Monte Carlo simulation, enabling efficient and accurate reliability solution under small-sample working conditions for the industrial robot.
Future work will further consider practical factors such as error coupling, variable load conditions, and performance degradation. I plan to improve dynamic error modeling and online compensation methods, expand multi-condition and full-lifecycle reliability assessment and health prediction research, and promote the engineering application of meta-action theory in precision assurance for industrial robots and high-end CNC machine tools.
Overall, the proposed framework provides a clear and systematic approach for characterizing error transfer laws and calculating positioning accuracy reliability of industrial robots. It offers theoretical and methodological support for precision design and reliability simulation of industrial robot systems, especially when only limited test data are available. The meta-action decomposition perspective enables a bottom-up understanding of how local displacement errors accumulate and propagate to the end effector, which is essential for designing high-precision and high-reliability industrial robots.