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Article

A Stylus-Based Calibration Method for Robotic Belt Grinding Tools

Precision Measurement Laboratory, Zhejiang Sci-Tech University, Hangzhou 310018, China
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Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(19), 10846; https://doi.org/10.3390/app151910846
Submission received: 11 September 2025 / Revised: 6 October 2025 / Accepted: 7 October 2025 / Published: 9 October 2025

Abstract

To address the tool calibration challenge in robotic systems equipped with grinding tools, this paper proposes a high-precision method utilizing a stylus assembly and the Particle Swarm Optimization (PSO) algorithm. A global optimization strategy is implemented, which simultaneously identifies and compensates for coupled error sources, including the robot’s kinematic (DH) parameters, the tool coordinate frame (TCF), and the stylus tip’s spatial position. This approach transforms the complex calibration task into a constrained, high-dimensional optimization problem. The experimental results demonstrate the method’s effectiveness, reducing the final calibration Root Mean Square Error (RMSE) to below 0.1 mm. Validation through a practical grinding experiment confirmed a significant improvement in machining accuracy, with the workpiece’s axis deviation from the ideal model decreasing from 1.477° to 0.326°, and the maximum contour error being reduced from 1.4 mm to under 0.3 mm. This study provides a robust, low-cost technical solution for tool calibration in complex industrial applications.

1. Introduction

The growing deployment of industrial robots in high-precision machining and automated assembly has significantly increased the demand for high spatial accuracy at the tool end. Consequently, the accurate calibration of the Tool Coordinate Frame (TCF) and the Tool Center Point (TCP) has become a key technology for ensuring operational precision. The accuracy of TCF calibration directly affects the robot’s positioning error in tasks like gripping and vision measurement. Meanwhile, TCP calibration is critical for motion path and trajectory control, which is paramount in high-precision processes such as welding and grinding.
Accurate TCF calibration is fundamental to high-precision robot operations, as its accuracy directly affects the spatial localization and pose control of the end-effector. Existing research on TCF calibration encompasses various methods. For instance, vision-based approaches utilize sensors to capture tool features [1], but these methods are susceptible to lighting interference, and inherent errors from the vision system can propagate to the TCF results, compromising the final accuracy. Other studies focus on theoretical analysis, such as Hou et al. [2], who analyzed the effect of TCF calibration error on attitude servo performance and established a maximum allowable error model. However, this approach does not offer a direct error compensation scheme, limiting its practical application value.
In addition, some studies focus on TCF calibration in specific application scenarios. In arc welding, for instance, trajectories accuracy has been improved by optimizing the precise geometric relationship between the tool and the workpiece [3]. However, task-specific methods often exhibit limited generalizability and are difficult to extend to other machining or operational tasks. Moreover, calibration methods that use physical calibration tools [4], while simplifying the process and reducing costs, impose stringent manufacturing accuracy requirements on the calibrator and are less suitable for tools with complex geometries.
Research on TCP calibration can be divided into two categories. The first aims to improve calibration accuracy and computational efficiency by refining classical tool calibration methods. For example, optimization algorithms based on the four-point method [5], the five-point method [6,7], the seven-point method [8], and least-squares methods [9] are computationally efficient and do not require auxiliary measurement tools. However, because these methods deduce the tool’s pose solely from the robot motion, they are susceptible to error accumulation, demand precise trajectory control, and can exhibit instability when measurement errors are high [10].
The second category relies on external calibration devices—such as the use of vision sensors [11], laser measurement devices [12] and regular objects [13,14]—to determine the spatial position of the tool. These methods usually solve the transformation relationship between the tool coordinate system and the robot base coordinate system by building a geometric model [15] or introducing an optimization algorithm [16,17]. While such methods have achieved notable progress in positioning accuracy and system robustness, they also present clear limitations. High-precision measurement systems are costly and highly sensitive to environmental disturbances [18]. This issue is particularly pronounced in abrasive belt grinding, where harsh conditions such as dust, vibration, and temperature fluctuations are unavoidable. In addition, long error propagation chains in external measurement systems and the computational burden of complex image processing and geometric modeling further undermine their industrial feasibility [19]. Consequently, existing approaches are often unsuitable for robotic grinding applications. Therefore, there is an urgent need to explore a new method of tool calibration with higher versatility, accuracy and engineering feasibility to adapt to the dual demands for accuracy and efficiency in complex machining scenarios.
To address these challenges, this paper proposes a novel calibration framework for abrasive belt grinding tools based on a stylus assembly and the Particle Swarm Optimization (PSO) algorithm. In this framework, the robot guides the grinding tool into contact with the stylus, collects pose data, and employs the PSO algorithm to globally optimize the tool calibration and related kinematic parameters, thereby improving the accuracy and robustness of the tool calibration. A comparison between common calibration methods and the proposed method is shown in Table 1, which clearly shows the advantages of this work.
This work proposed a strategy that simultaneously identifies and optimizes multiple coupled error sources, including the robot’s kinematic (DH) parameters, the tool coordinate system, and the position of the external reference point, within a unified global optimization framework. The following sections detail the method’s principles, describe its implementation process, and validate its performance through simulations and experiments.

2. Calibration Principle

As mentioned previously, to address the tool calibration problem in robotic systems equipped with grinding tools, a calibration method based on a stylus assembly and the Particle Swarm Optimization (PSO) algorithm is proposed. The overall workflow of the proposed calibration method is summarized in Figure 1. In this process, stylus contact measurements are first collected and used for coarse calibration, followed by a PSO-based optimization to refine the tool and robot parameters. The calibrated TCP/TCF parameters are then validated through experiments to verify the accuracy and reliability of the method.
The experimental setup for calibration, shown in Figure 2, primarily consists of an industrial robot for precise motion, the grinding tool as the calibration target, and a stylus assembly serving as a fixed reference datum. The coordinate frames defined within the robotic grinding system include: robot base coordinate system { B } , robot end-flange coordinate system { E } , and grinding tool coordinate system { T } .
The central principle of the calibration process involves using the robot to guide the grinding tool into contact with a spatially fixed stylus across a range of poses. Variations in the robot’s pose induce corresponding changes in the orientation and position of the tool coordinate system relative to the robot base coordinate system. Although the stylus remains fixed within the base coordinate system, the contact point on the tool surface, as represented in the tool coordinate system, changes with each adjustment of the robot’s pose.
The positions of these contact points, obtained at varying robot poses P B , all adhere to a single coordinate transformation relationship within the tool coordinate system, as described by the following equation:
P B = T E B · T T E · P T
where P B is the position of the stylus tip in the robot base coordinate system, P T is the position of the stylus tip in the grinding tool coordinate system, T E B is the transformation matrix from the robot end-flange coordinate system to the base coordinate system, and T T E is the transformation matrix from the tool’s coordinate system to the end flange coordinate system.
Once these fundamental coordinate transformation relationships and their associated variables have been established, the calibration process can be executed by solving for the initial values of the unknown tool parameters. The known quantity is the transformation matrix from the base coordinate system to the robot end-flange coordinate system T E B . The primary unknown parameters to be determined include: the transformation matrix from the end-flange coordinate system to the tool coordinate system T T E , the precise position of the stylus tip in the base coordinate system P B , and potential errors in the robot’s kinematic model (e.g., DH parameters).

2.1. Definition of the Grinding Tool Coordinate System

The first step of the calibration procedure involves performing a contact experiment between the grinding tool and the stylus. Prior to conducting this experiment, it is essential to mathematically define the geometry of the grinding tool. To this end, this study establishes a tool coordinate system { T } using a conventional cylindrical grinding wheel as an example. The contact between the grinding wheel and the fixed stylus is shown in Figure 3.
During the calibration process, the robot guides the grinding wheel to bring its cylindrical surface into contact with the stylus tip at a series of distinct poses. For each instance of the contact between the grinding tool surface and the stylus, the corresponding pose of the robot’s end-effector is recorded. This pose data, combined with the known stylus position, constitutes the raw measurement data required to solve for the tool coordinate system’s transformation.
The specific definition of the grinding tool coordinate system { T } is shown in Figure 4. This coordinate system is established based on the principle of fully utilizing the tool’s geometric characteristics to facilitate the calibration process. The definition is as follows: the origin of the tool coordinate system is defined at an arbitrary point on a generatrix of the grinding wheel’s cylindrical surface (which also serves as the contact point). The X-axis is aligned with the cylinder axis, the Z-axis is oriented in the radial direction of the wheel and orthogonal to the X-axis, and the Y-axis is determined according to the right-hand rule to form a right-handed orthogonal coordinate system.
This definition ensures unambiguous interpretation and enables precise coordinate computations and accurate tool positioning during the grinding process. Moreover, since the coordinate system is aligned with the geometric features of the tool, the contact measurements data have yield spatially meaningful values, thereby simplifying the computations required by the calibration algorithm.

2.2. Coarse Calibration of the Tool Matrix

Based on the precise definition of the grinding tool coordinate system, stylus contact measurements can be conducted, enabling coarse calibration using the acquired data. The coarse calibration of the tool transformation matrix represents a critical step in the robotic grinding tool calibration procedure, as it provides initial estimates for the subsequent precise calibration, based on PSO. This method exploits the geometric properties of the cylindrical tool by performing the calibration through contact between a stylus and the tool’s cylindrical surface. By guiding the robot to make contact at multiple poses and recording the corresponding positions, a preliminary relationship between the tool coordinate system and the robot end-flange coordinate system is established.
The transformation for the tool coordinate system is governed by the following equation:
T 0 T B = T E B T 0   T E
This can be expanded into its rotational and translational components:
t 0 T B = r E B t 0 T E + t E B
r 0 T B = r E B r 0 T E
where T 0 T B is the initial value of the transformation matrix from the tool coordinate system to the base coordinate system, and t E B and r E B are the position and attitude of the robot’s flange end coordinate system relative to the base coordinate system, respectively. Once the transformation from the base to the tool coordinate system T E B is determined, the initial values for the tool’s position t 0 T E and orientation relative to the end-flange coordinate system r 0 T E can be computed using Equations (3) and (4). The coarse calibration procedure consists of two steps: coarse calibration of the TCP and coarse calibration of the TCF. Both steps are described as follows.

2.2.1. Coarse Calibration of the TCP

The purpose of TCP calibration is to determine the position of the tool center point in the robot end flange coordinate system. The fundamental principle is to maneuver the TCP, as defined in the tool coordinate system, to contact the same fixed reference point in space from various robot poses.
As shown in Figure 5, the robot is driven to adjust its pose, enabling the grinding wheel to contact the stylus tip from various directions and angles. At each contact, the position t n E B and attitude r n E B of the robot end flange are recorded relative to the base coordinate system. Since the TCP’s position is fixed relative to the tool and the stylus tip’s position is fixed in space, the position of the tool coordinate system relative to the base coordinate system t T B can be determined t T B . Therefore, based on the relationship in Equation (3), a system of linear equations (shown as Equation (5)) can be constructed using data from at least four different contact poses to solve for the TCP, and the position of the tool coordinate system can be calculated by using the least-squares method.
t T B = r 1 E B t 0 T E + t 1 E B t T B = r 2 E B t 0 T E + t 2 E B t T B = r n E B t 0 T E + t n E B
By subtracting the equation for one pose from another, it can be simplified as follows:
r 2 E B r 1 E B r 3 E B r 2 E B r n 1 E B r n E B t 0 T E = t 1 E B t 2 E B t 1 E B t 2 E B t n 1 E B t n 2 E B
This yields a system of linear equations of the form A x = B , which can be solved for x using the least-squares method:
x = A T A 1 A T b
where A is a matrix constructed from the rotation matrices of different poses, b is a vector representing the position differences of the end-flange, and x is the unknown TCP position vector relative to the end-flange coordinate system t 0 T E .

2.2.2. Coarse Calibration of the TCF

Following TCP calibration, this section presents the coarse calibration procedure of the TCF, which is based on controlled axial movements. The procedure entails guiding the robot to move the TCP along designated axes of the tool coordinate system. By recording the corresponding changes in the end-flange’s pose, the axis vectors of the tool coordinate system can be determined through inverse calculation. The specific steps are shown in Figure 6.
First, an initial pose is selected and designated as calibration point ‘a’. At this point, the robot end-flange’s pose T a E B = r a E B   ,   t a E B is recorded. Then, the TCP’s position in the base coordinate system P a B is then derived from its t 0 T E as follows:
P a B = r a E B t T E + t a E B
Starting from point ‘a’, the robot moves the TCP a certain distance along the tool’s X-axis to reach point ‘b’. The end-flange’s pose at this new point T b E B = r b B E   , t b B E is recorded, and the new TCP position in the base coordinate system P b B is calculated. Since the tool’s attitude remains constant during this translation, the resulting displacement vector of the TCP in the base coordinate system represents the direction of the tool’s X-axis. The axial vector of X-axis of tool coordinate system X T B can be expressed as:
X T B = P b B P a B
Following the same procedure, the robot moves the TCP from point ‘a’ along the tool’s Z-axis to a new point ‘c’. After recording the end-flange’s pose T c E B = r c E B   , t c E B and calculating the new TCP position in the base coordinate system P c B , the direction vector for the Z-axis Z T B can be determined.
Z T B = P c B P a B
After obtaining the direction vectors of the X-axis and Z-axis of the tool coordinate system in the base coordinate system, the axial vector of the Y-axis can be calculated according to the right-hand rule Y T B .
Y T B = X T B Z T B
The three calculated axis vectors are then normalized to obtain the unit vectors x T B , y T B and z T B . These unit vectors form the columns of the rotation matrix from the tool coordinate system to the base coordinate system r 0 T B :
r 0 T B = x T B   y T B   z T B
The rotation matrix of the tool coordinate system with respect to the end flange coordinate system is r 0 T E :
r 0 T E = r a E B 1 T B r 0
This completes the coarse calibration of the tool coordinate system. The resulting parameters (the TCP vector from Equation (7) and the rotation matrix from Equation (13)) together form the initial estimate for the complete transformation matrix T 0 T B :
T 0 T B = r 0 T E t 0 T E   0 1

2.3. Precise Calibration of Tool Matrix

2.3.1. Kinematic Modeling Based on the DH Method

Following coarse calibration, precise calibration based on a Particle Swarm Optimization (PSO) algorithm can be conducted. The procedure encompasses robot kinematic modeling and application of the PSO algorithm.
In robotic grinding, accurate tool positioning is critical to the final machining quality. The robot’s positioning error primarily manifests as a deviation between the actual and theoretical poses of the end-flange. Such deviations may arise from various factors, including thermal expansion due to prolonged operation, mechanical wear, and assembly inaccuracies. These factors can cause the robot’s end-effector (i.e., the grinding tool) to deviate from its ideal trajectory, thereby compromising grinding accuracy and the final surface quality.
To quantify and compensate for these complex geometric errors, the kinematic parameters of the robot must be accurately identified and globally optimized. This requires an effective kinematic model to present the errors and a robust optimization algorithm to solve for these unknown parameter deviations. In this study, the Denavit-Hartenberg (DH) modeling method is adopted for robot kinematic modeling, while a Particle Swarm Optimization (PSO) algorithm is employed to determine the parameter values.
The DH modeling method, originally proposed by Denavit and Hartenberg [20], is a widely used approach in robot kinematics, as shown in Equations (15) and (16).
T i i 1 = Rot ( x , α i 1 ) Trans ( a i 1 , 0 , 0 ) Rot ( z , θ i ) Trans ( 0 , 0 , d i )
i 1 T = c θ i s θ i 0 a i 1 s θ i c α i 1 c θ i c α i 1 s α i 1 d i s α i 1 s θ i s α i 1 c θ i s α i 1 c α i 1 d i c α i 1 0 0 0 1
where Trans ( ) is the translation transformation; Rot ( ) is the rotation transformation; c is the cos shorthand; s is the sin shorthand;
a i = Along the x i axis, the distance moved from z i to z i + 1 ;
α i = The angle along the x i axis, moving from z i to z i + 1 ;
d i = along the z i axis, the distance from x i 1 to x i ;
θ i = The angle along the z i axis, moving from x i 1 to x i .
The system’s error model can be expressed as:
T T B + Δ T = ( T E B + Δ T E B ( T T E + Δ T T E )
In this model, T E B is the transformation matrix from the robot base frame {B} to the end-effector frame {E}, and Δ E B T is its corresponding error matrix. Similarly, T T E represents the transformation from the end-effector frame to the grinding tool frame, with Δ T T E as its error matrix. The overall transformation from the base frame to the tool frame is denoted by T T B , and its cumulative error is Δ T . After neglecting higher-order error terms, the model simplifies to:
Δ T = ( T E B Δ T T E + T T E Δ T E B )

2.3.2. Stylus Tip Localization and Transformation Relationship

Prior to the global optimization, a preliminary estimate of the stylus tip position within the base coordinate system must be obtained. This estimate serves as the initial value, or defines the search range, for the stylus position parameter within the Particle Swarm Optimization (PSO) algorithm. Once the initial tool matrix is known, the initial position of the stylus tip P B in the base coordinate system can be calculated.
During the TCP calibration process, when the stylus tip makes contact with the fixed point (i.e., the origin of the tool coordinate system), its position within the tool coordinate system P T is defined as 0 ,   0 ,   0 .
To ensure measurement accuracy, the robot poses recorded during the multiple contacts, along with the TCP position in the tool frame P T are substituted into the following equation:
P Bi = T i E B T 0 T E P T
This yields multiple measurement results for the stylus tip’s position P B i . These results are then averaged to mitigate deviations caused by measurement errors or environmental interference, providing a robust initial estimate of the stylus tip’s position in the base coordinate system P B :
P B = 1 N i = 1 N P B i
The initial position of the stylus tip in the base coordinate system will be used as the initial value of one of the parameters to be optimized in the subsequent particle swarm optimization.
Since the transformation matrices T E B and T 0 T E are themselves optimization targets, the stylus tip’s position P B is dynamically recalculated as these parameters are adjusted. Therefore, the actual relationship is:
P B ( E B T + Δ E B T ) ( T E T + Δ T E T ) P T

2.3.3. Error Modeling and Identification for PSO

To simultaneously optimize the errors in the robot’s DH parameters, the tool transformation matrix, and the stylus tip’s position, this paper employs the Particle Swarm Optimization (PSO) algorithm. PSO proposed by Kennedy and Eberhart in 1995, is a swarm intelligence-based optimization algorithm [21]. Inspired by the foraging behavior of bird flocks, PSO searches for the optimal solution within a multi-dimensional space by facilitating information sharing among a population of candidate solutions (known as particles) The core concept is that each particle represents a potential solution and iteratively adjusts its position and velocity to converge towards the global optimum. During the algorithm’s initialization phase, a population of particles is randomly generated, each with an initial position and velocity. The fitness value of each particle is then calculated to update its personal best position ( p b e s t ) and the swarm’s global best position ( g b e s t ). Subsequently, each particle’s velocity is updated according to the following equation:
V i ( t + 1 ) = ω × V i ( t ) + c 1 × r a n d ( ) × ( p b e s t i ( t ) X i ( t ) ) + c 2 × r a n d ( ) × ( g b e s t i ( t ) X i ( t ) )
The position is updated with the formula.
X i ( t + 1 ) = X i ( t ) + V i ( t + 1 )
where X i ( t ) denotes the position of particle i at the t moment, X i ( t + 1 ) denotes the updated position, V i ( t + 1 ) denotes the updated velocity, ω is the inertia factor, c 1 and c 2 are the learning factors, and r a n d ( ) is a random number distributed between ( 0   ,   1 ) . Subsequently, the positions of the particles are iterated through the above process until the stopping condition (e.g., the maximum number of iterations or the error threshold is reached) is satisfied.
The objective function of the Particle Swarm Optimization is constructed from the geometric properties of the grinding tool. Since the grinding wheel is cylindrical, when the stylus tip precisely contacts the cylinder surface the perpendicular distance from that contact point (expressed in the tool coordinate system) to the cylinder’s central axis should theoretically equal the cylinder radius r.
This geometric relationship provides the key constraint for the optimization process. By incorporating this constraint into the optimization model, the PSO algorithm can identify the optimal set of parameters that best fits all the measurement points, thereby achieving a precise tool calibration. This approach enhances calibration accuracy while ensuring full consistency between the tool and the robot system.
The Equation of the axis of the cylinder is:
a x + b z = 0
Then the distance from the point to the axis is calculated as:
d = | a x + b z | a 2 + b 2
Each time the stylus tip makes contact with the cylindrical surface, the position of the contact point P b is recorded in the base coordinate system { B } . This position is equivalent to the stylus tip’s location in the base coordinate system and satisfies the following equation:
P t i = ( T + Δ T ) 1 P b
where P t i ( x t i , y t i , z t i ) is when the stylus tip and cylindrical surface contact, representing the contact point in the grinding tool coordinate system { T } under the position, and i is the number of contacts. In the acquisition of enough data, the contact point in the X t can be substituted into the Formula (12) to get:
d i = | a x t i + b z t i | a 2 + b 2
where d i is the position calculated from the contact surface to the axis of the cylinder every time the tip of the stylus comes into contact with the surface of the cylinder; at this point, the objective function can be defined as
f = min 1 n i = 1 n ( d i r ) 2
where n is the number of contact times and f is the root mean square error function to be calculated.
For each contact point i , the robot’s pose is used to transform the stylus tip’s position from the base coordinate system { B } to the tool coordinate system { T } . The distance from this transformed contact point to the cylinder’s central axis is then calculated. The objective of the PSO algorithm is to minimize the difference between this calculated distance and the known tool radius r .
To ensure efficient convergence, the search space for each parameter to be optimized was defined as a bounded range centered around the initial values derived from the coarse calibration, as detailed in Table 2.

2.3.4. PSO-Based Parameter Optimization

Traditional step-by-step calibration methods typically determine the kinematic parameters first and then calibrate the tool coordinate system. However, this sequential process is prone to error accumulation and propagation, making it difficult to achieve a globally optimal solution. To enhance the absolute positioning accuracy of robotic systems, it is therefore essential to adopt a global calibration strategy rather than addressing individual error sources in isolation. In this work, multiple coupled error sources—including the robot’s kinematic (DH) parameters, the tool coordinate system parameters, and the position of the external reference point—are simultaneously identified and optimized. This comprehensive approach transforms the calibration task into a high-dimensional nonlinear optimization problem, characterized by a complex solution space with numerous local optima. The specific parameters used to configure the PSO algorithm for this study are shown in Table 3.
The specific calibration process aims to systematically search for the optimal parameter combinations, and the detailed steps are presented in Algorithm 1:
Algorithm 1: Calibration Process using Particle Swarm Optimization (PSO)
Require:
1: { T i } i = 1 N : A set of N measured end-effector poses.
2: M : Swarm size (number of particles).
3: T max : Maximum number of iterations.
4: ϵ : Error threshold for convergence.
5: w , c 1 , c 2 : Inertia weight and learning factors.
6: Parameter Range: Search space for DH parameters, TCP offset, and stylus tip position.
Ensure: 7: BestParams: The optimal set of DH parameters, TCP offset, and stylus tip position.
8: Initialize a swarm of M particles with random positions ( p . x ) and velocities ( p . v ); set each particle’s best-known position p . p b e s t p . x .
9: Calculate the fitness value f( p . x ) for each particle (e.g., RMSE, see Equation (28)) and determine the global best position g b e s t .
10: for iteration = 1 to T max do
11:        for each particle p do // Update personal and global best
12:                if f( p . x )<f( p . p b e s t ) then
13:                           p . p b e s t p . x
14:                  end if
15:                if f( p . p b e s t ) < f( g b e s t ) then
16:                           g b e s t p . p b e s t
17:                  end if
18:          end for
19:        for each particle p do //Update velocity and position
20:                  Generate random numbers r 1 , r 2 ∼U(0,1)
21:                   p . v ←w⋅ p . v + c 1 r 1 ( p . p b e s t p . x ) + c 2 r 2 ( g b e s t p . x )
22:                   p . x p . x + p . v
23:                  Constrain p . x to be within the defined Parameter Range.
24:        end for
25:          if f( g b e s t ) ≤ ϵ then //Check for convergence
26:                break
27:        end if
28: end for
29: return  g b e s t
The calibration process systematically searches for the optimal parameter combination. Once the iterative process satisfies the termination condition, the algorithm’s output for the global best position ( g b e s t ) represents the identified optimal parameter set. This procedure incorporates the cylindrical geometry of the grinding tool as a hard constraint within the optimization process, ensuring that the final calibration result is not only mathematically optimal but also physically meaningful. The high-precision parameters obtained through this method provide a reliable data foundation for subsequent high-accuracy grinding operations and for the effective compensation of the system’s inherent errors.

3. Numerical Simulation

To validate the proposed mathematical model for the coarse calibration–PSO calibration procedure, a numerical simulation was performed. The simulation employed the classic six-degree-of-freedom PUMA560 robot model to access the feasibility of determining the tool coordinate system from multiple sets of contact point data. The PUMA560 model, widely used as a benchmark in MATLAB 2022b, was selected to facilitate straightforward algorithm validation [22]. The calibration principles are general and independent of any specific robot, thereby ensuring the relevance of the simulation to the experiments.
By introducing multi-dimensional simulated errors—such as deviations in link lengths and joint angles—into the model, the parameter identification accuracy of the system was evaluated under the presence of both random and systematic errors. The results demonstrated that the PSO algorithm effectively captured and compensated for these introduced errors, leading to a substantial reduction in the overall calibration error and thus validating the reliability of the proposed method.
First, the stylus tip position X point = [ x 0 , y 0 , z 0 ] T and the ideal tool matrix T i d e a l T E are set:
T i d e a l T E = 1 0 0 125 0 1 0 7 0 0 1 41 0 0 0 1
According to the inverse solution of Equation (2), the transformation matrix from the robot end to the base coordinate system T 0 E B is obtained:
T 0 E B = T 0 T B ·   T i d e a l 1 T E
When the tool’s center point touches the stylus tip, its spatial position remains constant. Therefore, the translation vector from the robot’s base coordinate system to the grinding tool’s coordinate system can be determined. However, since the grinding tool’s attitude is indeterminate, the rotation vector of the total transformation matrix cannot be determined directly.
To simulate the variability of real-world tool poses, 25 distinct rotation vectors were generated by applying random perturbations (within a range of ±20° for each axis) to an initial attitude in Euler angle space. Based on these assumed vectors, 25 corresponding transformation matrices for the robot’s end-flange were calculated, and the position of the stylus tip within the tool coordinate system was determined for each pose. Finally, to verify the correctness of the inverse kinematics, the calculated stylus tip positions were substituted back into the model to ensure self-consistency.
The inverse solution obtained by inverse kinematics is shown in Table 4:
Table 1 shows some of the data for each of the robot’s joint angles obtained by inverse kinematics solving for the above 25 sets of tool poses and stylus assembly positions. These data form the ideal input for the calibration algorithm in subsequent simulations.
To simulate the error introduced during a coarse calibration process, an initial tool matrix after coarse calibration,  T before T E was generated. This was achieved by left-multiplying the ideal tool matrix T i d e a l T E , with a predefined error matrix Δ T e r r . The error matrix consisted of a rotational error of 0.2° around each of the X, Y, and Z axes, and a translational error vector of   0.1   mm , 0.1   mm , 0.1   mm T .
T b e f o r e T E = 0.9605 0.1947 0.1987 125.0964 0.2334 0.9527 0.1947 6.9009 0.1514 0.2334 0.9605 41.1043 0 0 0 1
This coarse tool matrix T before T E was then optimized using the proposed method, yielding the final fine-calibrated tool matrix T after T E .
T after T E = 1.0000 0.0005 0.0084 124.995 0.0005 1.0000 0.0000 7.0000 0.0084 0.0000 1.0000 41.0569 0 0 0 1
As shown in Table 5, when the transformation matrix is converted into Euler angles and a position vector, the residual errors across the six pose parameters become clearly visible. This result demonstrates that the proposed PSO-based fine calibration method effectively compensates for initial coarse calibration errors and significantly improves the tool coordinate system’s calibration accuracy, ensuring reliable positioning for the subsequent high-precision operation of the robot.
To validate the robustness of the proposed method and to simulate geometric parameter deviations resulting from real-world factors such as manufacturing tolerances, assembly errors, or operational wear, the simulation was repeated with known errors introduced to the ideal DH parameters. Considering that parameter deviations in the intermediate links of the robot kinematic chain typically exert a greater influence on end-effector positioning accuracy, errors were specifically introduced. to the DH parameters of these links. These error parameters were incorporated into the robot kinematic model, as shown in Table 6.
As shown in Table 7, the simulation results indicate that even with these induced parameter errors, the algorithm was still able to accurately identify them. This demonstrates that the optimization algorithm can not only identify inherent deviations in the model but also maintain high parameter identification accuracy after compensating for these errors. These results confirm the algorithm’s strong robustness and its capability to accurately identify structural parameter errors within the robot’s kinematic model.
To assess the robustness of the proposed method, a sensitivity analysis was performed by adding Gaussian noise of varying magnitudes and introducing different proportions of outliers into the contact data. Outliers were defined as contact points deviating more than 3 mm from the ideal cylindrical surface. As presented in Table 8, the method showed strong robustness to moderate noise levels, while outliers had a more significant effect. In practical experiments, such outliers are identified and removed during preprocessing.

4. Experiments and Results

4.1. Calibration Experiment and Result Analysis

Following the validation through numerical simulation, the calibration procedure was subsequently validated experimentally. The experimental platform, as shown in Figure 7, consisted of an ARMYOUNG Z3-R650 industrial robot, a custom abrasive belt grinding tool, and a fixed-mounted stylus assembly. During the experiment, the industrial robot provided precise and repeatable motion control to guide the grinding tool through a series of predefined contact tasks. The grinding tool, serving as the primary object of calibration, featured a cylindrical grinding wheel at its front end, constituting the actual contact surface. The stylus assembly was securely mounted on an optical breadboard, with its tip serving as a high-precision, stationary reference point in space. To ensure robustness, all tests were conducted in a temperature-controlled laboratory, with the setup mounted on a rigid optical platform, and the stylus tip periodically inspected to minimize wear.
In the experiment, the robot was controlled to guide the grinding tool’s cylindrical surface into precise contact with the stylus tip at 200 distinct poses. To ensure data diversity and representativeness, the contact points were distributed as evenly as possible across three distinct axial segments of the grinding wheel’s cylindrical surface. The tool poses covered a range of ±20° in pitch, yaw, and roll angles. The high-dimensional optimization problem involves 29 parameters—including 20 DH error parameters, 6 tool transformation parameters, and 3 stylus position parameters—rendering the selection of a sufficiently diverse set of poses essential. The 200 poses provide the necessary overdetermination to effectively average random measurement errors and ensure solution stability, while the high diversity of poses ensures the problem is well-conditioned, allowing the algorithm to uniquely distinguish between different error sources, thus avoiding local minima and converging to a stable and physically meaningful global optimum. After undergoing preliminary processing, which included filtering outliers whose radial distance from the cylindrical fit exceeded ±3 mm and reformatting, the dataset was used as the input for the Particle Swarm Optimization (PSO) algorithm. The algorithm then optimized the system’s key parameters, identifying the compensation values for the DH parameters, the TCP, and the stylus tip’s position, with the results presented in the subsequent tables and matrices. The identified DH parameter compensation results are summarized in Table 9.
TCP parameter compensation is:
Δ T = 0.0001 0.0167 0.0023 0.1730 0.0167   0.0001 0 0.3095 0.0023   0.0001 0.0001   5.7590 0 0 0 1
The stylus tip position was compensated for:
Δ P = 0.476 7 0.034 2 1.529 0 0
To validate the method’s accuracy, calibration experiments were conducted on three distinct cylindrical segments. The robot maneuvered the grinding tool at various poses, bringing the stylus tip into contact with the grinding wheel’s surface. For each contact, the corresponding robot pose data were recorded for subsequent calculations. To ensure data reliability, multiple repeated contact measurements were conducted on each cylindrical segment, yielding a comprehensive set of contact point data.
Using the pre- and post-optimization parameters, the contact points were transformed into the tool coordinate system to generate three-dimensional point clouds, which were then fitted to a theoretical cylindrical surface. As shown in Figure 8, the pre-optimization point cloud is visibly more dispersed and exhibits poor conformity with the theoretical surface, with pronounced deviations particularly at the edges. In contrast, the post-optimization point cloud is significantly more concentrated and uniformly distributed around the theoretical surface. These results indicate that the optimization procedure effectively compensated for system errors and improved the accuracy of the tool model.
Figure 9 shows the results of cylindrical cross-section fitting before and after optimization. The distribution of the cylindrical contact points before optimization is more scattered, while the distribution of the cylindrical contact points after optimization is more concentrated, and the adherence to the fitted circle is significantly improved, indicating that the calibration results are more accurate after optimization.
In addition, the convergence behavior of the PSO is illustrated in Figure 10. The fitness value (RMSE) decreases rapidly during the first few iterations and stabilizes afterwards, confirming that the algorithm converged reliably and was not trapped in local minima.
To complement these visual results, Table 10 provides quantitative metrics, which clearly confirm the reduction in fitting errors and further validate the effectiveness of the proposed method. This not only improves the calibration accuracy but also enhances the robot’s reliability in industrial scenarios. The experimental results show that the method can effectively improve the accuracy of the grinding tool coordinate system.
Figure 11 compares the DH parameter identification performance before and after optimization with the PSO algorithm. Prior to optimization, the system’s standard deviation ranged from 0.20 to 0.30. While the overall fluctuation was small, the relatively high standard deviation indicated significant errors in the parameter identification process, leaving room for improvement in both accuracy and stability. The initial Root Mean Square Error (RMSE) was maintained between 0.10 and 0.15 mm, exhibiting some fluctuations. Following optimization, the RMSE was further reduced to approximately 0.10 mm, and its fluctuation was significantly dampened. This demonstrates that the optimization process led to higher stability and greater accuracy in parameter identification. A direct comparison with the classical four-point TCP calibration is shown in Table 11, and the results confirm that the proposed method achieves markedly higher accuracy than the four-point baseline.
For the dataset of 200 contact poses, the entire optimization process typically converged within approximately 5–7 min. This computation time is highly acceptable for offline calibration tasks, but the iterative nature of the global search makes the current implementation unsuitable for online or real-time adjustments. To verify repeatability, calibration experiments were conducted on different days, and the identified parameters showed negligible variation, confirming the method’s stability and repeatability.
To further evaluate repeatability and parameter uncertainty, five independent calibration runs were conducted under identical conditions, each using 50 contact points. The results are summarized in Table 12. The estimated TCP/TCF parameters across different runs show high consistency, with translational deviations below 0.02 mm and rotational deviations below 0.1°. In addition, the results are expressed as mean values and standard deviations (Mean ± STD) across runs, which provides a statistical measure of uncertainty. These findings demonstrate that the proposed method achieves both high repeatability and robust parameter estimation.
After verifying repeatability and parameter stability, an ablation study was carried out to analyze the contribution of different parameter groups to the final calibration accuracy. To this end, an ablation study was conducted with four separate optimization settings: (i) optimizing only DH parameters, (ii) optimizing only TCP parameters, (iii) optimizing only stylus localization, and (iv) full optimization of all parameters (DH + TCP + Stylus).
As summarized in Table 13, optimizing TCP alone achieves lower RMSE than optimizing DH or stylus parameters alone, which highlights the importance of precise tool offset estimation. Nevertheless, the best accuracy is obtained when all parameter groups are jointly optimized, with the RMSE reduced to 0.095 ± 0.009 mm. This confirms that the proposed integrated framework effectively leverages the complementary effects of different parameter groups to achieve the highest calibration performance.

4.2. Grinding Experimental Verification and Accuracy Assessment

To further validate the effectiveness of the proposed calibration method in a practical robotic grinding application, a series of grinding experiments were designed and conducted. The data acquisition setup is shown in Figure 12.
Using both the pre- and post-optimization tool matrices, the robot ground a workpiece following an identical path and using the same process parameters. Following grinding, the resulting surfaces from both tests were scanned with a line laser to acquire three-dimensional point cloud data. These point clouds were subsequently fitted to a cylindrical surface using the least-squares method to extract the axis parameters of each ground cylinder, which were then compared against an ideal model defined by the grinding task.
To geometrically evaluate the accuracy gain, Figure 13 compares the ideal model with the actual ground surfaces generated using both pre- and post-optimization parameters. An analysis of the XY cross-sectional projections of the cylinder axes reveals that the unoptimized parameters resulted in a spatial deviation of 1.477° from the ideal axis. In contrast, the path guided by the optimized parameters aligns closely with the ideal path. The quantitative error analysis in Table 14 further corroborates this visual observation.
For a detailed quantitative analysis of contour accuracy, the three-dimensional point cloud data was projected onto a two-dimensional plane. This process comprised three steps: first, the scanned point cloud of the as-ground surface was aligned with the ideal cylindrical model; second, the point cloud was sliced at specific intervals along the Z and X axes to generate two-dimensional cross-sectional contour point sets; and third, for each cross-section, the normal distance from each point on the actual contour to the ideal two-dimensional curve was calculated, with this distance defined as the contour error. As shown in Figure 14, when using the unoptimized parameters, the maximum contour deviation was 1.4 mm. After optimization, this error was effectively suppressed, numerically demonstrating the strategy’s effectiveness in improving final contour accuracy.

5. Discussion

The experimental results confirm the effectiveness of the proposed calibration method. However, there are some limitations of this work. The current model is predicated on a rigid-body kinematic assumption and does not explicitly incorporate nonlinear dynamic phenomena, including joint friction (Coulomb, viscous, and Stribeck) [23,24], gearbox backlash, and structural compliance. Such unmodeled factors may introduce pose-dependent residual errors, for example, friction effects becoming more pronounced during low-speed contact, backlash causing discontinuities at reversal points, and compliance leading to deflections under load, that cannot be fully compensated by parameter identification alone. In addition to these nonlinear effects, the accuracy of calibration may be affected by small errors in stylus tip localization. Although the numerical sensitivity analysis incorporating Gaussian noise of up to 0.5 mm implicitly accounts for such uncertainties, meticulous experimental procedures ensured that residual stylus errors remained below ±0.05 mm through periodic inspection and filtering. This level of error is negligible compared with other unmodeled factors and does not significantly affect the calibrated TCP/TCF parameters.
Another limitation concerns the identifiability of the parameters. In particular, correlations may exist between certain DH parameters and tool offset values, which can reduce the uniqueness of the identified solution [25]. For example, a minor variation in the tool offset can be partially compensated for by an adjustment in link length, resulting in parameter coupling. Although the optimization framework mitigates this issue by employing a large and diverse set of contact poses to constrain the solution space, complete independence of all parameters cannot be guaranteed.
Beyond the rigid-body assumption, the broader applicability of this method merits further discussion. The framework’s principles are inherently scalable to larger robots or multi-tool systems. However, such extensions would require strategies like hierarchical calibration to manage the increased parameter dimensionality. At present, the generality of this method is confined to cylindrical tools, as the cost function is geometry-dependent. Extending the formulation to other primitive shapes (e.g., spheres or planes) would require redefining the cost function, whereas adaptation to irregular tools would represent a greater challenge, likely necessitating CAD-model integration [26].
Tool wear can gradually alter the geometry of the grinding tool, leading to small but cumulative deviations in the tool coordinate frame and, consequently, minor accuracy drift over long-term use. To mitigate this effect, periodic or adaptive recalibration is recommended [27]. Regarding practical implementation, the computational cost of the global PSO search makes the current method an offline process, ideal for automated periodic recalibration. This approach directly addresses industrial needs like compensating for tool wear over time, maintaining long-term accuracy without requiring continuous in-process updates. The reliability of this process also depends on mitigating physical error sources; for instance, while the stylus tip was treated as an ideal point, its actual geometry and wear were managed during experiments through periodic inspection and data filtering to minimize systematic bias.
From a cost–benefit perspective, commercial metrology systems—such as laser trackers or coordinate measuring machines (CMMs)—can achieve sub-0.05 mm accuracy, but at very high cost and with specialized operation requirements. In contrast, the proposed method achieves an accuracy better than 0.1 mm (RMSE = 0.0952 mm) using only low-cost fixtures and readily available sensors, thereby offering a practical and economically viable solution for routine calibration in robotic grinding applications. In terms of deployment, although the current offline implementation may cause short production downtime, it remains well-suited for scheduled periodic recalibration to compensate for tool wear. Integration into automated lines would mainly require appropriate scheduling strategies rather than additional hardware. Future work could extend the framework by incorporating advanced error models to capture nonlinear effects, introducing lightweight dynamic compensation schemes to reduce residual deviations, and leveraging learning-based prediction methods [28,29] to further enhance calibration accuracy and robustness.

6. Conclusions

This paper presents a high-precision calibration method for industrial robotic grinding tools that employs a stylus assembly and the Particle Swarm Optimization (PSO) algorithm. By analyzing the tool geometry and the calibration process, a multi-stage calibration scheme is developed, utilizing the cylindrical tool axis as a key constraint. The effectiveness and feasibility are validated through simulations and physical experiments.
The experimental results closely align with the numerical simulations, confirming that the proposed method effectively enhances the calibration accuracy of the robot’s tool coordinate system. The slightly higher final RMSE observed in the physical experiments may be attributed to unmodeled systemic factors, including minor backlash and flexibility in the robot’s joints, subtle variations in the stylus’s trigger force across different contact directions, geometric imperfections of the grinding wheel surface, and the potential influence of ambient temperature fluctuations on the robot’s structural parameters. Despite these factors, the proposed method achieved a calibration error of less than 0.1 mm without relying on costly external measurement equipment such as laser trackers, thereby fulfilling the accuracy requirements for most industrial grinding applications.
Compared to conventional methods that rely solely on manufacturer-provided parameters or basic teach-pendant programming, the proposed method significantly enhances both accuracy and robustness through a global optimization strategy. Future work may focus on incorporating more refined error models or implementing online compensation techniques to further improve calibration performance.

Author Contributions

Writing—review and editing, supervision, D.C.; data curation, writing—original draft preparation, Y.W.; software, writing—original draft preparation, Y.C. Conceptualization, review and editing, L.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Overall workflow of the proposed calibration method.
Figure 1. Overall workflow of the proposed calibration method.
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Figure 2. Robotic grinding tool system.
Figure 2. Robotic grinding tool system.
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Figure 3. Schematic of contact between the grinding wheel and the stylus tip.
Figure 3. Schematic of contact between the grinding wheel and the stylus tip.
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Figure 4. Grinding tool coordinate system.
Figure 4. Grinding tool coordinate system.
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Figure 5. Grinding wheel contacting fixed stylus in different poses.
Figure 5. Grinding wheel contacting fixed stylus in different poses.
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Figure 6. Grinding wheel contacting fixed stylus in different attitudes.
Figure 6. Grinding wheel contacting fixed stylus in different attitudes.
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Figure 7. Experimental platform.
Figure 7. Experimental platform.
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Figure 8. Cylinder surface fitting before and after optimization.
Figure 8. Cylinder surface fitting before and after optimization.
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Figure 9. Cylinder cross-section fitting before and after optimization.
Figure 9. Cylinder cross-section fitting before and after optimization.
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Figure 10. PSO convergence curve.
Figure 10. PSO convergence curve.
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Figure 11. Relationship between the number of experimental groups and standard deviation.
Figure 11. Relationship between the number of experimental groups and standard deviation.
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Figure 12. Workpiece surface data acquisition using a line laser scanner.
Figure 12. Workpiece surface data acquisition using a line laser scanner.
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Figure 13. Comparison of the ideal axis, the axis before optimization, and the axis after optimization.
Figure 13. Comparison of the ideal axis, the axis before optimization, and the axis after optimization.
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Figure 14. Comparison of profile errors: (a) errors at different Z-section heights; (b) errors at different X-section heights.
Figure 14. Comparison of profile errors: (a) errors at different Z-section heights; (b) errors at different X-section heights.
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Table 1. Comparison of different tool calibration methods.
Table 1. Comparison of different tool calibration methods.
MethodPrincipleCostAccuracyEnvironmental Robustness
Classical Multi-PointManual teaching of a fixed pointLowMedium/Low, operator-dependentPoor
Vision-BasedCamera identifies tool featuresHighHigh, but sensitive to lightingPoor, unsuitable for dusty environments
Template-BasedContact with a calibration block of known geometryMediumHigh, dependent on template accuracyMedium
Proposed MethodStylus contact + Global optimizationLowHighHigh, unaffected by lighting/dust
Table 2. Search space for PSO parameters.
Table 2. Search space for PSO parameters.
Parameter CategoryParameters to Be OptimizedSearch Range
Robot Kinematic ErrorsDH Parameter Errors
( Δ a , Δ d , Δ α , Δ θ )
Lengths: ±5 mm,
Angles: ±2°
Tool Transformation ErrorsPositional Errors
( Δ x , Δ y , Δ z )
±10 mm
Orientational Errors
( Δ rx , Δ ry , Δ rz )
±2°
Stylus Position ErrorsPositional Errors
( Δ Px , Δ Py , Δ Pz )
±5 mm
Table 3. PSO algorithm parameters.
Table 3. PSO algorithm parameters.
ParameterValue
Swarm Size ( M )200
Inertia Weight ( w )0.8
Cognitive Factor ( c 1 )1.49
Social Factor ( c 2 )1.49
Maximum Iterations ( T max )500
Error Threshold ( ϵ )1 × 10−6
Table 4. Joint angle data for PSO calibration input.
Table 4. Joint angle data for PSO calibration input.
Joint Angle θ 1 / rad θ 2 / rad θ 3 / rad θ 4 / rad θ 5 / rad θ 6 / rad
1−0.068 8−1.664 1−1.741 70.000 0−2.874 40.313 8
2−0.226 4−1.670 4−1.616 00.000 0−2.997 81.012 8
3−0.258 6−1.683 0−1.533 20.000 1−3.062 71.242 3
4−0.277 7−1.745 0−1.271 90.000 03.018 11.737 9
50.022 4−1.772 4−1.525 50.013 8−2.737 0−0.009 0
60.019 4−1.912 7−1.183 00.020 9−2.599 50.001 7
24−0.018 0−1.666 6−1.749 10.000 0−2.867 50.128 6
250.024 6−1.666 1−1.748 40.000 0−2.867 7−0.024 6
Table 5. Pose deviation after fine calibration.
Table 5. Pose deviation after fine calibration.
X-Axis Angle
Difference/(°)
Y-Axis Angle
Difference/(°)
Z-Axis Angle
Difference/(°)
X-Axis Position Difference/(mm)Y-Axis Position Difference/(mm)Z-Axis Position Difference/(mm)
0.0000−0.00840.00050.00500.00000.0569
Table 6. Added error parameters.
Table 6. Added error parameters.
Link Δ θ i / rad Δ d i / mm Δ a i 1 / mm Δ α i 1 / rad
10000
20.10.10.10.1
30.10.10.10.1
40.10.10.10.1
50.10.10.10.1
60.10.10.10.1
Table 7. Identified DH parameter compensation in simulation.
Table 7. Identified DH parameter compensation in simulation.
Link Δ θ i / rad Δ d i / mm Δ a i 1 / mm Δ α i 1 / rad
10000
2−0.0998−0.0980−0.1200−0.0999
3−0.0982−0.0911−0.0930−0.1007
4−0.1004−0.0900−0.0900−0.0999
5−0.0967−0.0984−0.1200−0.0987
6−0.1048−0.0999−0.0900−0.1135
Table 8. Sensitivity analysis of calibration accuracy under noise and outliers (RMSE in mm).
Table 8. Sensitivity analysis of calibration accuracy under noise and outliers (RMSE in mm).
Outlier 0%Outlier 2%Outlier 5%Outlier 10%
Noise 0 mm3.2 × 10−81.44541.45382.6917
Noise 0.1 mm0.10040.92222.03522.3635
Noise 0.2 mm0.19651.50881.71983.0938
Noise 0.5 mm0.49491.02052.66223.0501
Table 9. Identified DH parameter compensation in experiment.
Table 9. Identified DH parameter compensation in experiment.
Link Δ θ i / rad Δ d i / mm Δ a i 1 / mm Δ α i 1 / rad
10000
20.0018−0.0162−0.0993−0.0041
3−0.0014−0.0292−0.02910.0003
40.00220.0736−0.01860.0038
5−0.00340.0993−0.09570.0013
6−0.0020−0.0332−0.0991−0.0046
Table 10. Quantitative comparison of fitting errors before and after optimization.
Table 10. Quantitative comparison of fitting errors before and after optimization.
RMSE/mmMAE/mmMax Deviation/mm
Before Optimization0.29320.21981.0071
After Optimization0.09520.07410.5157
Table 11. Comparison between proposed method and four-point baseline.
Table 11. Comparison between proposed method and four-point baseline.
RMSE/mmMAE/mmMax Deviation/mm
Proposed method0.09520.07410.5157
Four-point method0.42130.36461.1281
Table 12. Repeatability and statistical uncertainty of calibration parameters.
Table 12. Repeatability and statistical uncertainty of calibration parameters.
Group 1Group 2Group 3Group 4Group 5Mean ± STD
TCP-x (mm)128.135128.102128.128128.117128.124128.121 ± 0.013
TCP-y (mm)−6.352−6.379−6.365−6.370−6.360−6.365 ± 0.010
TCP-z (mm)35.30535.28135.29835.28735.29635.293 ± 0.009
TCF-roll (°)0.1820.1190.1440.1640.1350.149 ± 0.025
TCF-pitch (°)−0.210−0.142−0.171−0.189−0.165−0.175 ± 0.026
TCF-yaw (°)0.9920.9310.9590.9720.9480.960 ± 0.023
Needle-x (mm)449.820449.789449.815449.802449.809449.807 ± 0.012
Needle-y (mm)1.7521.7391.7471.7411.7441.745 ± 0.005
Needle-z (mm)44.53844.50844.52344.53244.51944.524 ± 0.011
Table 13. Ablation study on the contribution of different parameter groups.
Table 13. Ablation study on the contribution of different parameter groups.
SettingRMSE (mm)
DH-only0.210 ± 0.012
TCP-only0.120 ± 0.010
Stylus-only0.180 ± 0.015
DH + TCP+ Stylus0.095 ± 0.009
Table 14. Position and angle difference of the fitted cylinder axis.
Table 14. Position and angle difference of the fitted cylinder axis.
Ideal Grinding Circle Before Optimization After Optimization
Position difference/mm00.8150.096
Angle difference/°01.4770.326
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Chang, D.; Wang, Y.; Chen, Y.; Zhang, L. A Stylus-Based Calibration Method for Robotic Belt Grinding Tools. Appl. Sci. 2025, 15, 10846. https://doi.org/10.3390/app151910846

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Chang D, Wang Y, Chen Y, Zhang L. A Stylus-Based Calibration Method for Robotic Belt Grinding Tools. Applied Sciences. 2025; 15(19):10846. https://doi.org/10.3390/app151910846

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Chang, Di, Yichao Wang, Yi Chen, and Lieshan Zhang. 2025. "A Stylus-Based Calibration Method for Robotic Belt Grinding Tools" Applied Sciences 15, no. 19: 10846. https://doi.org/10.3390/app151910846

APA Style

Chang, D., Wang, Y., Chen, Y., & Zhang, L. (2025). A Stylus-Based Calibration Method for Robotic Belt Grinding Tools. Applied Sciences, 15(19), 10846. https://doi.org/10.3390/app151910846

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