Region-Based Approach for Machining Time Improvement in Robot Surface Finishing

Abstract

Traditionally, in robotic surface finishing, the entire workpiece is processed at a uniform speed, predetermined by the operator, which does not account for variations in the machinability across different regions of the workpiece. This conventional approach often leads to inefficiencies, especially given the diverse geometrical characteristics of workpieces that could potentially allow for different machining speeds. Our study introduces a region-based approach, which improves surface finishing machining time by allowing variable speeds and directions tailored to each region’s specific characteristics. This method leverages a task-oriented strategy integrating robot kinematics and workpiece surface geometry, subdivided by the clustering algorithm. Subsequently, methods for optimization algorithms were developed to calculate each region’s optimal machining speeds and directions. The efficacy of this approach was validated through numerical results on two distinct workpieces, demonstrating significant improvements in machining times. The region-based approach yielded up to a 37% reduction in machining time compared to traditional single-direction machining. Further enhancements were achieved by optimizing the workpiece positioning, which, in our case, added up to an additional 16% improvement from the initial position. Validation processes were conducted to ensure the collaborative robot’s joint velocities remained within safe operational limits while executing the region-based surface finishing strategy.

1. Introduction

In the rapidly evolving landscape of modern manufacturing, the transition towards custom-based production and low-volume, high-mix manufacturing has become a significant trend [1]. This shift requires adopting advanced and flexible automation solutions to meet the increasingly diverse and dynamic market needs. Integrating collaborative robots (cobots) into various manufacturing processes is one of the most promising avenues for achieving this flexibility. Unlike traditional industrial robots, cobots are designed to work safely alongside human operators, facilitating a harmonious blend of human ingenuity and robotic precision [2]. However, integrating cobots into machining processes presents several challenges. One of the primary obstacles is the limited motion capability of cobots, which demands careful task planning and optimization, to ensure feasible and efficient operations. In robotics machining applications [3], such as milling, polishing, or hammer peening, where the robot tool is constrained to follow a prescribed path on the workpiece surface, understanding the robot’s kinematics and dynamics is highly important to optimize the robot’s task performance [4].

Repeated action, especially in machine hammer peening (MHP), is necessary to achieve the desired surface treatment and demands continuous, precise movements from the robot, exerting considerable strain on its mechanical components. MHP is an advanced application utilized primarily in manufacturing and toolmaking to improve the metallic components’ surface characteristics. This process can involve using a robotic arm equipped with a pneumatic or electromagnetic hammering tool that impacts the surface of a workpiece frequently [5]. The primary goal of MHP is to enhance the surface’s mechanical properties related to the surface quality, fatigue strength of metal parts, and improved overall durability [6,7]. Existing MHP research primarily explores its impact on workpieces’ mechanical and structural properties [8]. These studies extensively report enhancements such as improved surface hardness and smoothness, increased compressive residual stresses, and beneficial microstructural changes, as shown in recent research by [9], which analyzed the effects on stainless steel and mild steel. The authors in [10] explored the surface integrity of bearing steel, and another study by [11] expanded on integrating laser cladding with MHP for repairing and strengthening surfaces, further highlighting the process’s capability to enhance wear and fatigue resistance. Additionally, research in [12] demonstrates that MHP introduces plastic deformation in the surface, effectively reducing residual tensile stress and enhancing grain refinements. However, these studies typically do not address the optimization of machining time, which is a critical factor given the inherently time-consuming nature of the MHP process due to the very dense machining toolpaths. Standard practice in robotic MHP often involves using classic zig-zag machining patterns [13], which are not specifically optimized for the unique geometrical characteristics of each workpiece and the robot kinematics involved.

Further complications arise when less capable robots are paired with complex workpieces. In such scenarios, the robot may not have the reach or dexterity to access all points on the workpiece to follow the machining path successfully. This is a common challenge in manufacturing, prompting numerous research efforts to optimize the workpiece positioning or the robotic system to enhance the machining process. A placement optimization of the robot—workpiece or vice versa—is performed for several reasons, such as for energy consumption [14], wear reduction [15], cycle time optimization [16], or stiffness reduction [17]. In [18], the optimal workpiece placement is searched by maximizing the achievable linear velocity along the predefined machining toolpath based on the Decomposed Twist Feasibility (DTF) method [19]. The authors in [20] used two multi-objective particle swarm optimization loops, focusing on finding the proper robot base placement and configuration. The first loop minimizes all the robot’s constraints, such as singularities, joint limitations, and collisions. Redundancy data were then fed into the second loop to find the optimal robot base placement, such that all trajectories can be followed by continuous motion. In [21], two genetic algorithms were employed to improve the robot tool’s initial position in the robotic milling. The first algorithm maximized kinematic and dynamic manipulability, and the second minimized total joint torques. Both optimizations were executed along the milling trajectory. A feasible workpiece placement through multiple constraint violation functions, which are applied and minimized incrementally, was researched in [22]. Much research on workpiece positioning was conducted in the milling section, such as in [23], where, based on milling cutting forces and an elastostatic model of the robot, the optimal workpiece position is searched with the help of genetic and interior-point algorithms. Deburring applications are also considered, such as in [24], where the authors used a genetic algorithm to optimize the workpiece’s placement based on the robot’s stiffness.

Due to the geometric complexity of freeform surface machining, tool orientations typically vary dynamically when using a 5-axis machine tool; because of this, clustering is recognized increasingly as a strategic method for optimizing the machining of large workpieces or those with varied surface curvatures. The authors in [25] aimed to improve CNC machining efficiency by optimizing tool orientations and feed directions. The surface was segmented into optimized sub-regions using K-means clustering, ensuring machinability through gouging and collision-free orientations. Similar work was performed in [26], where the workpiece was partitioned based on convex, concave, and saddle surface curvature regions and processed further by the Fuzzy-means clustering algorithm. The complexity of robot machining increases significantly when dealing with workpieces with intricate geometries. Traditional CAD/CAM programs often fall short in these scenarios due to the robots’ physical limitations and the tasks’ complexities. Thus, there is a need for novel approaches that can evaluate and optimize the robot’s kinematic capabilities accurately, ensuring that the machining tasks are performed efficiently. Because of the relatively low stiffness of industrial robots, numerous research efforts have focused on enhancing robot stiffness through optimizing robot posture [27,28]. Here, the authors propose a region-based toolpath generation method for the robotic milling of freeform surfaces to optimize robot stiffness. The method divides a surface into sub-regions with similar robot postures to enhance stiffness, and then generates toolpaths tailored to these regions. The article by [29] developed a robotic belt grinding approach that segments workpieces with complex shapes into regions to simplify the generation of gouge-avoidance toolpaths. It introduced algorithms for both wheel face grinding and wheel edge grinding to analyze and avoid potential gouges, enhancing the robotic grinding process’s efficiency and accuracy. Region-based path planning is also used in achieving proper welding layers on complex workpieces [30] and in [31], where partitioning with the help of clustering was used to machine giant complex sculptures.

None of the existing robotic region-based approaches incorporate robotic kinematic analysis directly with free-form surface geometry in a task-oriented way. A novel approach to robot kinematic analysis incorporating the constraints imposed by the workpiece surface geometry has been developed to address these challenges [32]. The proposed methodology extends the well-known concept of manipulability, which is used traditionally to assess robot motion capabilities. Introducing the workpiece surface constraint into the kinematic analysis provides a more accurate evaluation of the robot’s available velocity capabilities in a reduced-dimensional space. The concept of manipulability, first introduced by Yoshikawa [33], provides a qualitative method for assessing the ease with which a robot can move in its operational workspace. This concept is typically represented by the manipulability ellipsoid, derived from the robot’s velocity kinematics description using the singular value decomposition of the associated Jacobian matrix [34]. However, traditional manipulability indices often suffer from physical inconsistencies due to the mixing of different units of linear and angular velocity [35]. To overcome these limitations, the author’s work [32] introduces a task-oriented approach to robot kinematics, focusing on the surface constraints imposed by the workpiece geometry. The strategy involves the development of a task-specific augmented inverse Jacobian matrix, which incorporates motion constraints derived from the differential surface geometry of the workpiece. This matrix transforms the robot’s task space into a two-dimensional surface tangent plane, effectively avoiding the physical inconsistencies associated with mixed linear and angular velocity units. The resulting manipulability analysis focuses solely on linear velocities, providing a more consistent and meaningful evaluation of the robot’s motion capabilities.

Building on this theoretical framework, our study employs clustering techniques, specifically K-means clustering [36], to segment the workpiece surface into regions with similar kinematic characteristics. Clustering the surface into manageable sections allows for more precise and optimized surface finishing strategies. An optimization algorithm determines the optimal machining directions and maximum feasible speeds for each region, which enables optimal surface finishing path planning concerning the machining time. The practical implications of this approach are demonstrated through numerical experiments involving semi-complex workpiece geometries. These experiments validate the effectiveness of the proposed method in enhancing the robot’s machining time performance, while ensuring task feasibility within the constraints of the cobot’s motion capabilities. By evaluating the robot’s kinematic capabilities accurately and optimizing the machining paths, the proposed method improves the efficiency of the robot machining task significantly. In addition to improving the robot’s machining performance, this approach addresses the critical issue of workpiece positioning. The optimal positioning of the workpiece is crucial for maximizing the robot’s kinematic capabilities and ensuring that the machining tasks are performed efficiently. The proposed method uses an optimization algorithm to determine the best position for the workpiece, considering the robot’s motion constraints and the geometry of the workpiece. This ensures that the robot can perform the machining tasks with maximum efficiency, thus additionally reducing the overall machining time. Reducing machining time is a measure of efficiency and a critical factor in enhancing production throughput and reducing operational costs. Lower machining times translate directly into faster cycle times, allowing for increased output within the same operational hours. This is particularly significant in MHP, where the process is time-consuming and can extend over several hours.

Combining clustering techniques into the kinematic analysis represents a significant advancement in robot machining. The method allows for more detailed and precise kinematic evaluations by dividing the workpiece surface into regions. This segmentation enables the optimization algorithm to tailor the machining strategies to the specific characteristics of each region, ensuring that the robot operates at its maximum capability throughout the entire machining process.

In conclusion, this study presents a comprehensive and innovative approach to robot machining that addresses the key challenges of integrating cobots into complex machining tasks. By incorporating workpiece surface constraints into the kinematic analysis and leveraging clustering techniques and optimization algorithms, the proposed method provides a robust framework for enhancing the performance and efficiency of robot machining. The practical implications of this approach are demonstrated through numerical experiments in the presented case study, validating its effectiveness and highlighting its potential for improving the flexibility and adaptability of modern manufacturing processes. As the industry continues to evolve, such innovative approaches will play a crucial role in meeting the diverse and dynamic demands of the market, paving the way for more efficient and flexible manufacturing systems.

The rest of the paper is organized as follows: Section 2 provides the necessary background information on the surface differential geometry and the theoretical method. Section 3 presents the development of the proposed region-based robot machining method. Section 4 includes the numerical experiments and validation within the case study. Finally, Section 5 discusses the results and concludes the paper.

2. Preliminaries

A free-form surface S in a three-dimensional space can be represented parametrically by the following vector function:

$$r(u,v)=\left[x(u,v),y(u,v),z(u,v)\right]$$

(1)

where u and v are parameters spanning the surface [37]. The explicit surface is examined, where the z-coordinate is represented as a function of the x- and y-coordinates:

$$z=f(x,y)$$

(2)

where f is a real-valued function that is at least twice continuously differentiable. The parametric form is obtained by setting $x=u$ and $y=v$, where $\left[x,y\right]\in D$, $D$ being a connected domain in the xy-plane. The surface can be described by

$$r=\left[x,y,f(x,y)\right]$$

(3)

The unit normal vector at any point on the surface, necessary for many applications, including robot machining, is given by

$$\widehat{n}={\displaystyle \frac{r_x\times r_y}{‖r_x\times r_y‖}}$$

(4)

where vectors $r_x$ and $r_y$ span the tangent plane to the surface at a given point (x,y) and are used to compute the surface normal.

The first fundamental form, which encapsulates the surface’s metric properties, is defined by the following matrix [37]:

$$I=\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]$$

(5)

with the first fundamental form coefficients $E=r_x\circ r_x,F=r_x\circ r_y,G=r_y\circ r_y$.

The second fundamental form, providing information crucial for understanding the surface curvature, is given by

$$II=\left[\begin{array}{cc}L& M\\ M& N\end{array}\right]$$

(6)

The so-called second-form coefficients are derived from the directional derivatives of the normal vector $L=r_{xx}\circ \widehat{n},M=r_{xy}\circ \widehat{n},G=r_{yy}\circ \widehat{n}$.

The relationship between the joint velocities and the end-effector’s motion in the workspace is represented through the following velocity kinematics equation [34]:

$$v=J(\theta )\dot{\theta }$$

(7)

Here, $J\in {ℝ}^{6\times n}$ denotes the geometric Jacobian matrix, which transforms the joint velocity vectors $\dot{\theta }$ into operational velocity twist vector $v\in {ℝ}^6$, encapsulating both the linear $v\in {ℝ}^3$ and angular $\omega \in {ℝ}^3$ velocities:

$$v=\left[\begin{array}{c}v\\ \omega \end{array}\right]$$

(8)

In robotic machining, especially in configurations where the robot manipulator is an open chain with six degrees of freedom (DOF), i.e., n = 6, kinematic analysis becomes crucial for effective operation. The manipulator’s base is typically stationary, while the end-effector moves freely within the operational space to perform tasks accurately. The inverse kinematics problem must be solved to achieve the desired end-effector velocity. This involves determining the necessary joint velocities $\dot{\theta }$ from given twist vector $v$. The solution can be obtained by inverting the following full-rank, non-singular quadratic Jacobian matrix:

$$\dot{\theta }=J^{-1}v$$

(9)

The essence of manipulability in robotics is to understand how effectively a robot can change its position or orientation in space, which is quantified using the Jacobian matrix of the system. The manipulability measure is derived from the following condition:

$${\dot{\theta }}^T\dot{\theta }\le 1$$

(10)

which maps the unit sphere in the joint velocity space onto an ellipsoid in the task space, described by

$$v^T{(J{J}^T)}^{-1}v\le 1$$

(11)

which is the equation of points on an ellipsoid’s surface within the end-effector velocity space [38].

To analyze and simplify the manipulability condition, the Jacobian matrix J can be decomposed using the singular value decomposition (SVD) into

$$J=U\sum V^T$$

(12)

In this decomposition, U and V are unitary matrices representing the orthonormal bases of the Jacobian column and row space. $U=\left[u_1\dots u_m\right]$ is a unitary matrix with the column vectors that span the m-dimensional robot operational space and present the ellipsoid semi-axis directions, $\sum$ is a diagonal matrix with singular values ${\sigma }_1\ge {\sigma }_2\ge \dots \ge {\sigma }_n$ representing the semi-axis lengths, and $V=\left[v_1\dots ,v_n\right]$ contains the orthonormal basis vectors for the joint velocity space. The end-effector can move at high velocities along the major axis of the ellipsoid, whereas it achieves lower velocities along the minor axis.

In this paper, the inverse Jacobian matrix $J^{-1}$ is replaced and used in the SVD (12) by the so-called augmented inverse Jacobian matrix $J_C^{\#}$ of constrained kinematics [32], which incorporates differential characteristics of the workpiece surface:

$$J_C^{\#}={\tilde{J}}_T^{†}{S}_v{S}_v^{†}+{\tilde{J}}_R^{†}{S}_{\omega }{S}_v^{†}$$

(13)

where ${\tilde{J}}_T$ and ${\tilde{J}}_R$ are strong translational and strong rotational Jacobian matrices [39], and $S_v$, $S_{\omega }$ are defined by (14) and (15), respectively. Operator ${(.)}^{†}$ stands for the Moore–Penrose pseudoinverse of a matrix.

$$S_v=\left[\begin{array}{cc}{r}_x& r_y\end{array}\right]$$

(14)

$$S_{\omega }={\displaystyle \frac{\left[\begin{array}{cc}{r}_x& r_y\end{array}\right]}{\sqrt{EG-F^2}}}\cdot \left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\cdot II$$

(15)

Matrix $S_v$ maps a 2D linear velocity vector in the Cartesian plane to a 3D velocity vector in the Cartesian space, whereas $S_{\omega }$ maps a 2D linear velocity vector in the Cartesian plane to a 3D angular velocity in the Cartesian space. More detailed derivations and explanations can be found in the article [32].

Matrix $J_C^{\#}$(13) is homogeneous in units and rank deficient, i.e., $r=rank(J_C^{\#})=2$, meaning that the constrained manipulability ellipsoid is reduced to an ellipse in the tangent plane of the workpiece surface. It has two non-zero singular values, ${\sigma }_{\min }$ and ${\sigma }_{\max }$, along the corresponding minor and major ellipse semi-axis directions $u_{\min }$ and $u_{\max }$, respectively. In Figure 1, the ellipses with their major axes directions are drawn in pink.

3. Materials and Methods

3.1. Problem Formulation

In workpiece surface finishing, the robot path with its tool oriented perpendicular to the workpiece surface can be determined arbitrarily for relatively simple surfaces, but for workpieces featuring more curved surfaces, identifying the optimal machining direction becomes more complex. Figure 1 illustrates the directions of the major ellipse semi-axes $u_{\max }$ and ${-u}_{\max }$ for points on the workpiece with black arrows, utilizing a collaborative robot arm UR5e and sample workpiece (see Section 3.4). Note that the robot could, theoretically, move with the same capability in both directions. The color map on the surface represents the maximum possible machining speeds ${\sigma }_{\max }$ in the considered surface points, where the machining speed is low in the blue areas and is high in the red areas. Ideally, path planning would utilize these vectors directly, but since their direction can change considerably from one point to another, this requires the development of an advanced algorithm that carefully considers all the points and integrates them into a continuous path. Therefore, the challenge is creating optimal machining paths based on the ellipse characteristics (see Figure 1), incorporating robot kinematics to complete surface finishing tasks successfully.

3.2. Optimal Machining Path Direction

During the machining of the workpiece, the tooltip must follow a specific path embedded in the workpiece surface. Both position and orientation constrain this toolpath, assuming the tooltip must maintain a perpendicular orientation relative to the curved workpiece surface. In similar articles, the robot surface finishing trajectory is usually predetermined, like in [18], where only the workpiece position is optimized based on the existing toolpath. In our scenario, we take into account the robot-constrained velocity kinematics featured by (13), and simultaneously determine both the optimal robot direction and speed using the modified DTF method [19] to find the machining paths with the maximum feasible end-effector linear speed $V_{\max }$ (17) for the workpiece surface point under consideration.

First, for each point $P_i$, the searched two-dimensional Cartesian direction vector $u_2={\left[\cos (\phi ),\sin (\phi )\right]}^T$ is mapped to obtain a three-dimensional Cartesian direction vector for machining on a workpiece surface tangent plane:

$$u_T={\displaystyle \frac{S_v{u}_2}{‖S_v{u}_2‖}}$$

(16)

Then, $u_T$ is used to determine the maximum surface-constrained feasible speed at which the robot surface finishing can occur at each point. The concept of the DTF method [19] is utilized here, which yields

$$V_{\max }={{\displaystyle \frac{1}{‖J_C^{\#}\cdot u_T‖}}}_{\infty }$$

(17)

where $J_C^{\#}$ is the constrained inverse Jacobian matrix defined in (13), and $u_T$ is the direction vector (16).

To determine the optimal machining direction and speed, the following optimization problem is set:

$$V_{\max }^{opt}=\underset{u_2}{\max }\left(\underset{i=1,\dots ,N}{\min }(V_{\max }(P_i))\right)$$

(18)

where $V_{\max }^{opt}$ represents the optimal maximum feasible linear speed on the surface at the optimal direction $u_2$, and N is the number of the surface points under consideration. Additionally, Equation (18) can be described as

• Inner optimization procedure with the expression $\underset{i=1,\dots ,N}{\min }(V_{\max }(P_i))$: This step identifies the maximum feasible speed across all points $P_i,i=1,\dots ,N$ under consideration within a selected surface path. This speed represents the limiting factor for the machining process since the robot cannot exceed this speed in the specified direction.
• Outer maximization: After computing the maximum feasible speed for each direction, this step searches for the optimal direction $u_2$ that maximizes the machining speed on the surface patch. Maximizing this value ensures that the robot can achieve the highest feasible speed across the workpiece points in the optimal direction.

The optimal machining directions enable the forming of a velocity vector field (VVF) used for the path planning process.

3.3. Surface Subdivision and Machining Time Estimation

3.3.1. Clustering

As observed in Figure 1, the ellipse data already form distinct areas of similarity, but relying solely on intuition and visual inspection is insufficient. Hence, clustering algorithms such as K-means can be used to subdivide the workpiece into clusters. K-means clustering is a popular unsupervised machine learning algorithm used to partition a dataset into distinct clusters with similar data subsets, where each data point belongs to the cluster with the nearest mean (centroid). The algorithm assigns data points to clusters iteratively, and updates the centroids by calculating the mean of all points within each cluster. This process repeats until the centroids stabilize or a predefined number of iterations is reached [40].

The data are partitioned into clusters using the squared Euclidean distance metric, where K is the number of clusters. The algorithm allows up to an arbitrary number of iterations to converge. Empty clusters are handled by creating a new cluster with a single point, and the algorithm runs several times with different initial centroids to avoid the local minima. The centroids can be initialized using the K-means++ algorithm [41], which ensures better initial placement.

For each surface point $P_i$ on the workpiece, a multidimensional feature vector $F_i$ (19) is constructed by combining the position vector $P_i$, surface unit normal vector ${\widehat{n}}_i$, minimal possible speed ${\sigma }_{\min ,i}$ (corresponding to the minor ellipse semi-axis length), and maximal possible speed ${\sigma }_{\max ,i}$ (corresponding to the major ellipse semi-axis length).

$$F_i=\left[P_{ix}{P}_{iy}{P}_{iz}{\widehat{n}}_{ix}{\widehat{n}}_{iy}{\widehat{n}}_{iz}{\sigma }_{\min ,i}{\sigma }_{\max ,i}\right]$$

(19)

The Z-Score normalization [42] is used as

$${\tilde{F}}_i={\displaystyle \frac{F_i-\mu }{s}}$$

(20)

${\tilde{F}}_i$ represents the normalized clusterization vector $F_i$, while $\mu$ and $s$ denote the feature vectors’ mean value and Standard Deviation.

3.3.2. Surface Regionalization and Boundary Definition

In the previous chapter, the K-means algorithm was employed to partition the dataset into clusters, as seen in Figure 2a. These clusters are essentially groups of data points that share similar characteristics, but do not necessarily form coherent spatial boundaries. To interpret the results in a spatial context better, these clusters are processed further to create what will be referred to as regions. A critical step in this process involves defining clear boundaries for these regions to facilitate efficient region-based analysis or machining operations. To achieve this, the alpha shape algorithm [43] was applied, which refines the clusters by generating well-defined boundaries for each region. The algorithm works by constructing a shape, enveloping the set of points with the smallest possible predefined radius $\alpha$, effectively capturing the concavities of the region, and ignoring outliers or noise. Figure 2b showcases the refined regions after applying the alpha shape algorithm, with each distinct region outlined clearly and separated from others, thus creating separation machining regions, each representing a single continuous space with VVF.

3.3.3. Region Merging

Segmenting machining workpieces into regions can sometimes yield impractical small areas for effective machining. Such regions lead to the generation of numerous short, discontinuous toolpaths, which can compromise the efficiency of the machining process [29]. A systematic approach to merging smaller regions into adjacent, larger ones addresses this issue.

The first step in region merging involves defining and identifying small regions based on a predefined area threshold. Regions whose areas fall below this threshold are flagged for potential merging to enhance the compactness and continuity of the machining paths. Once small regions are identified, their neighboring regions are examined. This involves analyzing the spatial adjacency and connectivity of the regions based on their proximity and shared boundaries.

The small region is merged with each neighboring region, and the impact on the workpiece machining time is calculated. The adjacent region that leads to the most favorable reduction in machining time is chosen as the optimal candidate for merging. The results can be seen in Figure 2c, where the initial number of 13 regions from Figure 2b was lowered to 9.

3.3.4. Single Boundaries

Each region is enveloped individually by its boundaries in Figure 2c, creating double boundaries. While effective for our previous analytical purposes, this configuration can leave unattended gaps that hinder the machining process, leading to unmachined regions on the workpiece. To address this, a single boundary was created between the existing double boundaries. In this approach, the distances between all the boundary points of each pair of adjacent region boundaries are calculated using the Euclidean distance metric.

Let A and B represent matrices of boundary points from two different regions, where A and B are $m\times n$ and $p\times n$ matrices, respectively. Each row of A and B corresponds to the coordinates of a boundary point in an n-dimensional space. The Euclidian distance between any two points $a_i$ from A and $b_j$ from B is calculated as follows:

$$d_{ij}={\displaystyle \sum _{k=1}^n{(a_{ik}-b_{jk})}^2}$$

(21)

where $d_{ij}$ is the distance between the i-th point in A and the j-th point in B, and $a_{ik}$ and $b_{jk}$ are the k-th components of points $a_i$ and $b_j$, respectively. Pairs of boundary points whose distances fall below a specified threshold (adjacent regions) are identified as candidates for creating new boundary segments. Then, the midpoints between each pair of closely located boundary points are calculated and sorted based on their spatial coordinates to maintain a logical sequence:

$$m_{ij}={\displaystyle \frac{a_i+b_j}{2}}$$

(22)

where $m_{ij}$ represents the coordinate-wise average of the points $a_i$ and $b_j$. The midpoints $m_{ij}$ are then smoothed using the Savitzky–Golay filter [44] to create a smoother transition between regions, as depicted in Figure 2c.

3.4. Optimal Workpiece Placement

In robotic machining, the feasibility of the operation can be influenced significantly by the position and orientation of the workpiece relative to the robot base [18]. In the optimization setup, workpiece {W} has its local coordinate system in its center with relation to the robot’s base frame {B}, as illustrated in Figure 3. The pose of the workpiece on the worktable relative to the robot’s base frame {B} can be determined by three variables $〈x,y,\gamma 〉$, which are subject to optimization. Here, x and y represent the offsets of the workpiece in the Cartesian xy-directions from the robot base, while $\gamma$ denotes the rotation of the workpiece about its z-axis; the z-coordinate is constant, and indicates the fixed distance between the worktable and the robot base.

To evaluate the machining performance suitable for optimization, we introduce the machining time index (MTI) concept, which provides a reference metric that estimates the machining time based on the calculated $V_{\max }^{opt}$ (18) and the density of design points on the workpiece:

$$\tau ={\displaystyle \sum _{i=1}^N\frac{d_i}{V_{\max }^{opt}}}$$

(23)

where $\tau$ stands for MTI, $d_i$ is the so-called standard distance between points (standard measure), N is the number of surface points under consideration, and $V_{\max }^{opt}$ is the maximum feasible linear speed in the tangent plane attached to the point on the workpiece surface.

The MTI can also be calculated for each region individually by adjusting the surface points and speeds for that region. The total workpiece MTI is then the sum of the regional MTIs, where lower values are better and correlated to shorter machining times.

Again, an optimization algorithm is used to find the optimal workpiece pose based on minimizing the total workpiece MTI (23). The objective function is as follows:

$$\underset{〈x,y,\gamma 〉}{\min }F(\tau )$$

(24)

which is subject to the following constraints:

• Robot joint angles ${\theta }_i$ are within limits:

$${\theta }_{i,\min }\le {\theta }_i\le {\theta }_{i,\max ,}i=1,\dots n$$

(25)

• The workpiece must be located within a working area:

$$\begin{array}{l}{x}_{\min }\le x\le x_{\max },\\ y_{\min }\le y\le y_{\max },\\ {\gamma }_{\min }\le \gamma \le {\gamma }_{\max }.\end{array}$$

(26)

• All points on the workpiece must be reachable, i.e., the inverse kinematic (IK) solutions must exist for all points.
• The robot must not be in self-collision, collision with the workpiece, or collision with the movable platform.
• The workpiece must be on the working table with its full area.

4. Case Study

This section presents a case study to evaluate the effectiveness and applicability of our region-based surface finishing method compared to traditional single-direction finishing. Both methods use the zig-zag (bidirectional) machining strategy [45], which is a commonly used technique in machine hammer peening (MHP) surface finishing. Firstly, the initial velocity vector field (VVF) model was deployed to generate optimal machining paths with maximum feasible speeds. The comparison begins by analyzing machining times for two different workpieces in their initial positions, using both the region-based and traditional fixed-direction approaches. Additionally, we also explored workpiece optimization to highlight any further performance improvements. Lastly, a validation process compared the initial VVF model predictions against the actual machining speeds to validate the accuracy of our path planning method.

4.1. Experimental Setup

Our numerical experimental setup featured the Universal Robots UR5e collaborative robot, known for its adaptability and widespread use in industries requiring precision in low-volume/high-mix production settings. The UR5e has a reach radius of 850 mm and a payload capability of up to 5 kg. Mounted on the robot is an MHP tool with a length of 284.5 mm, used for simulating the robot hammer peening machining task. The full layout of our flexible workstation setup used in the experiments is shown in Figure 3.

The proposed method will be tested on two specific workpieces in the Autodesk Fusion 360 CAM (v.2.0.19994) and RoboDK (v5.6.8) software to compare region-based versus single-direction machining strategies’ machining times. The impact of varying stepover distances was also analyzed, offering insights that could benefit various machining operations. The main goal was to achieve complete machining of the workpiece in the shortest time possible without exceeding the robot’s maximum joint velocities.

The approach and retract speed parameters control the feed rate of the robotic tool during the initial approach to the surface before machining, and the retraction away from the surface after machining, respectively. The transition speed is the feed rate used when moving between different regions of the workpiece, typically occurring at a higher speed to reduce non-cutting time. The stepover speed refers to the rate at which the tool moves horizontally between adjacent passes (stepover), where the speed is usually lowered. For MHP, the recommended machining stepover distance typically ranges from 0.1 to 2 mm [46]. The detailed machining parameters are shown in

Table 1. Machining parameters.

Approach speed [mm/s]	500
Retract speed [mm/s]	500
Transition speed [mm/s]	500
Stepover speed [mm/s]	20
Stepover [mm]	0.1–2

.

Within this setup, machining paths were generated for two distinct scenarios to evaluate the effectiveness of our subdivision region-based approach on two workpieces, i.e., Workpiece No. 1 (WP1) and Workpiece No. 2 (WP2). Figure 4 shows four different machining toolpaths:

A.

Arbitrary Single-Directional Machining

Initially, machining paths were created to operate solely along the x- and y-axes, respectively, as seen in Figure 4a–d. This scenario represents a more traditional approach, where the MHP tool moves linearly across the workpiece surface without altering its machining direction based on the surface topology or the underlying robotic kinematics.

B.

Machining with Optimized Directional Methods

• Single optimal direction machining (Figure 4e,f): The machining direction was determined based on a calculated workpiece MTI, optimizing the toolpath for the most efficient single direction across the entire workpiece.
• Region-based optimal directional machining (Figure 4g,h): This approach involves creating distinct optimal machining toolpaths for different regions within the workpiece. Each region was machined according to a direction that maximized the region MTI of the corresponding segment of the workpiece and the overall workpiece MTI.

4.2. Region-Based Machining Validation

Maintaining a controlled experimental environment is crucial to assess the impact of machining subdivided surfaces compared to traditional single-direction machining. For this purpose, the position of the workpiece was the same across all the tests. This consistency ensures that any variations in machining time can be attributed directly to the machining method employed, rather than differences in workpiece positioning or orientation. Both workpieces were placed with their centers at [0, −570, −96] (mm) from the robot reference frame and rotated 0 degrees. The workpiece positions were found and fixed manually to provide a stable basis for comparison; here, it must be added that finding the suitable position with trial and error, where the whole workpiece is reachable and the robot is not in self-collision, is difficult and time-consuming.

Determining the optimal number of clusters for the K-means algorithm is an important step in our region-based machining strategy. After extensive testing, setting the number of clusters to 10 (K = 10) provided the best results for both workpieces based on the MTI.

Post-clustering, we identified and targeted smaller regions for potential merging to enhance the machining time index (MTI). Each small region was assessed for merging with adjacent regions, with the decision to merge based on achieving a more favorable MTI again.

After regionalization, MATLAB’s built-in global optimization function, particle swarm optimization (PSO) [33] was used to explore the potential directions comprehensively by maximizing the objective function $F(\phi )=V_{\max }^{opt}$, defined to capture the constraints and demands of the machining process. The optimal machining directions enable the forming of a velocity vector field (VVF) used for the path planning process.

Our experimental analysis demonstrates the efficiency of machining methods, particularly highlighting the advantages of a region-based approach over single-direction machining paths. The results for WP1 are in

Table 2. Machining results for WP1 in the initial position with different machining strategies. The improvement (%) is between regions and a single optimal machining path.

	X-Direction	Y-Direction	Single Optimal	Regions	Impr. %
MTI [s]	114.46	151.89	113.59	66.82
Max. feasible speed [mm/s]	87.36	65.83	88.03	R3: 340.3
	Stepover 2 mm
Machining time [min:s]	4:32	5:53	4:30	3:35	20.4
	Stepover 1 mm
Machining time [min:s]	8:39	11:20	8:25	6:06	27.5
	Stepover 0.5 mm
Machining time [min:s]	16:52	22:15	16:16	10:58	32.6
	Stepover 0.1 mm
Machining time [h:min:s]	1:22:38	1:49:31	1:18:59	49:18	37.6

, where we can see a decrease in machining time from approximately 20–37%, and the results for WP2 are in

Table 3. Machining results for WP2 in the initial position with different machining strategies. The improvement (%) is between regions and a single optimal machining path.

	X-Direction	Y-Direction	Single Optimal	Regions	Impr. %
MTI [s]	68.31	106.89	65.02	42
Max. feasible speed [mm/s]	146.37	93.55	153.8	R11: 568.8
	Stepover 2 mm
Machining time [min:s]	3:11	4:09	3:07	2:57	5.3
	Stepover 1 mm
Machining time [min:s]	5:12	7:51	4:59	4:32	9
	Stepover 0.5 mm
Machining time [min:s]	9:53	15:15	9:28	7:28	21.1
	Stepover 0.1 mm
Machining time [h:min:s]	47:40	1:14:24	45:31	30:12	33.7

, where the decrease ranges from 5 to 33%. Workpieces with regions are shown in Figure 4g,h and are more detailed in Figure 5. Figure 5a represents the 3D regions of WP1 with optimal machining directions; in Figure 5b, a 2D visualization is shown, with added maximum feasible speeds for each region; similar results can be seen for WP2 in Figure 5c,d.

The comprehensive data show that region-based machining outperformed other strategies in reducing machining time, especially as the stepover values decreased. However, the reduction in machining time with smaller stepovers comes with an increase in machining distance. This increase is attributable to the higher density of toolpaths, which requires highly optimized feed rates. By subdividing the workpiece into regions, it is possible to avoid the bottleneck of machining the entire workpiece at the minimal (maximal feasible) speed dictated by the single-direction approach. Instead, each region can be machined at an optimized speed most suitable for its specific characteristics. In robot hammer peening, the oscillating speed of the metal ball on the electromagnetic hammer peening tools can be adjusted [47], allowing for synchronization with the tool’s machining speed. This synchronization ensures that each region is treated appropriately, enhancing the overall quality and efficiency of the machining process.

4.3. Workpiece Position Optimization

As discussed in earlier sections, the positioning of the workpiece can affect robotic surface machining, leading to the use of an optimization algorithm to determine the optimal workpiece position. Also, given the numerous constraints, finding a workpiece position manually that satisfies all the constraints through trial and error proved challenging. The PSO algorithm was utilized to minimize the optimization problem described in (24), enhancing the robot’s operational efficiency under the optimization constraints shown in

Table 4. Optimization constraints.

Optimization Constraint	Value
UR5e robot joint limits	$360°\le {\theta }_i\le 360°,i=1,\dots ,6$
Working area	$\begin{array}{l}-350\mathrm{mm}\le x\le 350\mathrm{mm}\\ -300\mathrm{mm}\le y\le -700\mathrm{mm}\end{array}$
Workpiece rotation	$-180°\le \gamma \le 180°$

. Additional workpiece placements were also analyzed to validate the proposed objective function’s effectiveness, confirming our optimization approach’s robustness and practical applicability.

Figure 6 shows the different placements of the two workpieces in the working area. Figure 6a represents the placements for WP1, where green is the optimal position found by the optimization algorithm and orange is the initial middle position as in Section 4.2, whereas blue and violet are randomly picked poses; the same goes for WP2 in Figure 6b. Actual positions in the form $x\left[mm\right];y\left[mm\right];\gamma \left[°\right]$ are presented in

Table 5. Comparison of minimal (maximum feasible speeds) and machining times for different region-based workpiece positions $x\left[mm\right];y\left[mm\right];\gamma \left[°\right]$.

	Ran. Pos. 1	Ran. Pos. 2	Initial Pos.	Optimal Pos.	Impr. %
	Workpiece 1
Position [mm; mm; °]	−200; −525.2; 30	125.1; −540.5; 147.5	0; −570; 0	−17.4; −564.6; 223.1
MTI [s]	69.04	71.24	66.82	60.98
Machining time [min:s]	6:19	6:28	6:06	5:51	4.1
	Workpiece 2
Position [mm; mm; °]	55.4; −566.3; 153.5	−165.6; −566.3; 153.5	0; −570; 0	−104.9; −407.9, 301.8
MTI [s]	48.18	43.45	42.06	34.81
Machining time [min:s]	4:41	4:35	4:32	3:47	16.5

.

A machining time comparison was made for region-based machining paths, with a stepover of 1 mm for all positions; the results are shown in Table 5. As we can see, the best results were observed in the optimal position identified by the PSO, where the machining time improved by approximately 4% for WP1 and 16% for WP2 compared to the second-best initial positions, which correlates with the calculated workpiece MTIs. This modest improvement can be attributed to the relative proximity of all the tested positions; the random positions were near a global minimum. This suggests that the minor improvements are because no other random position, significantly further from the global minimum, could be found manually, confirming our challenges and limitations of manual positional adjustments. The final region-based subdivision resulted in the optimal positions, which are shown in Figure 7.

4.4. Path Speed Validation

4.4.1. Path Point—Wise Design Speed Validation

Our methodology for calculating the maximum feasible robot surface machining speed $V_{\max }$ (17) in the optimal direction was developed in the surface tangent plane. Note that, in this approach, the $V_{\max }$ was calculated for all surface points, essentially creating a velocity vector field (VVF), as seen in the left columns of Figure 8 and Figure 9. The surface machining paths seen in the right columns of Figure 8 and Figure 9 were created based on these directions. The maximum feasible speeds for those machining paths were calculated by the DTF method [19], which computes the robot’s maximum feasible linear speed between consecutive points along the machining path:

$$V_{\max }^{path}={{\displaystyle \frac{1}{‖{\tilde{J}}_T^{†}{\widehat{u}}_T^{path}+h^{-1}{\tilde{J}}_R^{†}{\widehat{u}}_R^{path}‖}}}_{\infty }$$

(27)

where ${\widehat{u}}_T^{path}$ and ${\widehat{u}}_R^{path}$ are linear and angular directions along the path, and $h=V/\mathsf{\Omega }$ is a task-dependent velocity ratio factor that reflects the synchronization between the translational and rotational motion, calculated as the ratio of the linear velocity to the angular velocity.

In this validation process, we wanted to test if the actual machining paths and their maximum feasible speeds corresponded to the initial design calculations. In Figure 8, there is a comparison for WP1 in the X-direction (Figure 8a,b), Y-direction (Figure 8c,d), optimal direction (Figure 8e,f), and region-based directions (Figure 8g,h). This applies similarly to WP2, which is depicted in Figure 9, where the X-direction is shown in Figure 9a,b, the Y-direction in Figure 9c,d, the optimal direction in Figure 9e,f, and region-based directions in Figure 9g,h.

The mapping from the model-based VFF to the actual machining path can result in deviations in the maximum feasible speeds. These deviations arise due to the disparity in the density of model points and generated path points and their different positions. However, the speed deviation should converge to zero with a higher point density. In Figure 8 and Figure 9, the dark gray crosses indicate maximum feasible speeds in mm/s. At the same time, the colormap represents the full range of speeds across the surface, where red means lower speed and blue means higher speed. Despite minor deviations, the validation results generally demonstrated a good alignment, confirming the method’s ability to predict the optimal directions and speeds accurately before the path creation phase. It should be noted that the figures have been modified for enhanced visual clarity. The actual tests were conducted on model workpiece surfaces with 2500 design points, while the surface finishing paths were generated with a 2 mm stepover, resulting in approximately 22,000–24,000 points, depending on the specific case. Both workpieces were positioned optimally within their respective regions.

All the results for feasible speeds are summarized in

Table 6. Validation results for maximum feasible speeds in mm/s between the initial design VVF model and machining toolpath for multiple machining cases.

Path Directions	VVF Model	Machining Toolpath	Difference (%)
Workpiece 1
X-direction	82.3	81.9	0.5
Y-direction	72.2	71.2	1.3
Optimal direction	93.3	90.2	3.4
Region-based directions	R1: 179.1, R2:213.4, R3: 120, R4: 292.2, R5: 203.6, R6: 91.6, R7: 104.5, R8: 228.4, R9: 155.7	R1: 172.5, R2: 211.1, R3: 117.1, R4: 281, R5: 202.4, R6: 89.6, R7: 101.9, R8: 224, R9: 150	R1: 3.8, R2: 1.1, R3: 2.4, R4: 3.9, R5: 0.6, R6: 2.2, R7: 2.5, R8: 1.9, R9: 3.7
Workpiece 2
X-direction	124.6	123	1.3
Y-direction	115.3	113.1	1.9
Optimal direction	155.1	151.4	2.4
Region-based directions	R1: 233.1, R2: 271.2, R3: 276.1, R4: 262.6, R5: 327.4, R6: 240.4, R7: 454, R8: 415.3 R9: 345, R10: 591.6, R11: 216	R1: 227.4, R2: 264.5, R3: 264.9, R4: 261.2, R5: 319.4, R6: 234.8, R7: 443.8, R8: 404.1, R9: 339.3, R10: 583.4, R11: 209.1	R1: 2.5, R2: 2.5, R3: 4.1, R4: 0.5, R5: 2.5, R6: 2.4, R7: 2.3, R8: 2.7, R9: 1.7, R10: 1.4, R11: 3.2

, where it is evident that the maximum deviation was approximately 4% or less. This confirms the robustness of our method further, as these deviations remained minimal and within acceptable limits, demonstrating that the method performed well, even in scenarios involving region-specific optimization.

4.4.2. Region-Based Design Speed Validation

To validate the practicality of our region-based design path speed method combined with optimal workpiece placement, we also analyzed the infinity norm of the joint velocities representing the maximum value of the joint velocity vector at each step of all the generated trajectories. The primary objective was to confirm that the robot’s movements adhered to the calculated machining speeds and that these velocities remained within the limits of the UR5e robot, specifically not exceeding its maximum joint speed limit ${\dot{\theta }}_{\max }$ of 3.14 rad/s across the four workpiece placements. Regions in the initial position for both workpieces can be seen in Figure 5, and regions in the optimal positions can be seen in Figure 7.

The analysis revealed that the actual joint speeds in certain regions exceeded the maximum allowable limit slightly, as seen in Figure 10, for all workpiece placements. Still, the machining time was the lowest in the optimum position, confirmed in Table 5, which aligns with our previous observation regarding maximum feasible machining speeds. This finding indicates that, while the robot operated close to its kinematic threshold to achieve the highest possible linear velocities per our optimization, it occasionally surpassed the safety margins. This is because the VVF is designed in the tangent planes attached to the surface design points, whereas the maximum speeds on the path are calculated on the path point-to-point principle. Given these discrepancies and the minor overages in joint speeds, we recommend setting the surface finishing speeds at, e.g., 10% lower than those calculated by our method for future implementations in practice. However, here, we should consider that the safety margin depends on the density of the surface points considered within the design phase and the density of the generated path points. This adjustment will ensure that machining operations remain safely within the robot’s mechanical capabilities, mitigating the risk of exceeding joint speed limits and enhancing the overall feasibility of the machining process.

5. Discussion and Conclusions

This study delves into robot kinematics’ capabilities and velocity performance in machining workpieces with semi-complex surface geometries. We initiated our analysis using an augmented robot Jacobian kinematics matrix incorporating workpiece surface constraints, facilitating a well-known manipulability analysis. Building on this fundamental analysis, a task-constrained method for calculating surface path directions represented by VVF was developed, which enables generating speed-optimal machining paths and implemented it into region-based machining, which enhanced the efficiency of the machining process.

While existing studies on robotic hammer peening have primarily focused on the mechanical and structural enhancements of workpieces’ surfaces, our research introduces a novel approach by emphasizing the optimization of machining time—a critical factor often overlooked in traditional methodologies. Our study demonstrates that region-based machining strategies enhanced efficiency over traditional single-direction machining significantly, reducing the machining time from 5% to 37%, depending on the stepover distance. Notably, the greatest improvements were observed as the stepover distance decreased, indicating that region-based strategies become increasingly effective with denser toolpaths. This suggests that the benefits of a region-based approach might not justify the implementation complexity for applications with larger stepovers. In particular, while machining times for workpiece WP2 showed only modest improvements of about 5% at the largest stepover distance, this can be attributed to the increased number of regions that required more frequent transitions between machining zones. However, as the stepover distance decreased, the machining times for WP2 tended to converge with those of WP1, highlighting the potential for significant efficiency gains in more finely detailed machining tasks.

Further enhancements were realized through optimal workpiece positioning, which improved machining times by an additional 4–16%. Although these gains may appear modest, the primary challenge with using lower-capability collaborative robots lies not just in optimizing the position for efficiency, but in finding any feasible position where successful machining can occur. In conclusion, our validation analysis discovered slight discrepancies in machining and robot joint speeds between the model-based design calculations and machining paths, which may have occurred due to low surface model point density, leading us to recommend reducing the surface finishing speeds at a certain margin to ensure safe operation within the robot’s kinematical limits.

Looking ahead, our research will focus on refining the region-based machining approach by integrating a compactness index into the clustering algorithm. This enhancement aims to create more compact machining regions, minimizing transitions between them, to reduce machining times further and improve operational efficiency. Optimizing the order in which these regions are machined could also streamline the process by reducing non-productive tool movements between regions. Beyond these improvements, the next step also involves applying these optimized strategies to robots in real-world manufacturing settings, allowing for a practical validation and further refinement of the proposed method.