An Interdisciplinary Optimization Framework for Intelligent Robotic Workstation Base Placement

Abstract

The optimal placement of robotic manipulators within industrial workstations is a critical problem that directly affects task feasibility, accessibility, and operational efficiency. Improper base positioning can lead to joint saturation, reduced manipulability, and limited workspace utilization. This work presents an optimization framework for determining the optimal base placement of robotic manipulators by maximizing a joint-centering performance index based on the κ-index, which quantifies the proximity of joint variables to their allowable limits. The proposed methodology integrates geometric accessibility constraints with a constrained optimization formulation to ensure feasible robot configurations within the workspace. Three optimization strategies—constrained nonlinear programming, gradient projection methods, and genetic algorithms—are evaluated and compared in terms of solution quality and computational performance. Numerical simulations are conducted using a planar 2-DOF manipulator to illustrate the proposed framework and to analyze the influence of workspace geometry on optimal base placement. Additionally, an industrial case study involving the ABB IRB 120 robotic manipulator is presented to assess the practical applicability of the proposed approach. The results demonstrate that the optimization framework improves joint distribution within the allowable limits and enhances robot accessibility across the task workspace. The proposed method provides a practical tool for intelligent workstation design and robotic cell layout optimization in modern industrial environments.

1. Introduction

The rapid expansion of industrial robotics has significantly transformed modern manufacturing systems. According to the International Federation of Robotics, the global operational stock of industrial robots surpassed 4.6 million units in 2024, reflecting sustained growth driven by automation demands, digital transformation, and the adoption of Industry 4.0 technologies [1]. As robotic deployment continues to increase in high-density industrial environments, the systematic design of robotic workstations has become a critical engineering challenge. In particular, the spatial placement of the robot base relative to task locations strongly influences kinematic feasibility, motion efficiency, and operational robustness. Inadequate base placement may result in inefficient joint utilization, increased cycle times, proximity to kinematic limits, and reduced operational stability [2,3,4].

Accessibility analysis in robotic systems is commonly classified into two complementary categories. Type I accessibility refers to workspace reachability and task coverage, while Type II accessibility concerns the existence of collision-free trajectories between configurations. Recent research has addressed these aspects through various approaches, including reachable workspace modeling, inverse reachability maps, sampling-based motion planning, and learning-assisted trajectory planning techniques [5,6,7,8,9,10,11]. In parallel, performance metrics such as manipulability indices and Jacobian conditioning have been widely adopted to characterize kinematic performance within the reachable workspace [12,13]. Accurate pose estimation and manipulation benchmarking also play an important role in evaluating robotic accessibility and task feasibility in industrial manipulation scenarios [14].

Despite these developments, determining the optimal base placement of a robotic manipulator under explicit accessibility constraints remains a challenging nonlinear optimization problem. Similar challenges have also been investigated in mobile manipulation systems, where sequential base-position planning directly affects task execution feasibility [15]. The difficulty arises from the interaction among joint limits, singularity avoidance requirements, reachability constraints, and the geometric characteristics of the task space. Contemporary approaches often integrate coverage metrics and dexterity criteria within multi-objective or hierarchical optimization frameworks [16,17,18,19,20,21,22]. However, many of these formulations primarily emphasize geometric feasibility or manipulability maximization without explicitly enforcing balanced joint operation within their admissible ranges. Operating close to joint limits may increase actuator stress, reduce positioning accuracy, and negatively affect long-term mechanical performance, even when reachability conditions are satisfied [23,24].

Furthermore, robot placement optimization and configuration feasibility are frequently treated as independent problems, lacking a unified formulation capable of simultaneously enforcing reachability constraints, aspect membership conditions that prevent singular configurations, and quantitative measures that promote joint-centered operation [25,26]. This separation limits the ability to systematically evaluate feasible placement domains from both geometric and operational perspectives. In industrial environments, where structured and intelligent engineering methodologies are required, integrating robotic kinematic modeling with constrained optimization techniques becomes essential for informed workstation design.

In modern industrial environments, robotic workstation design is increasingly integrated with higher-level decision-making processes such as task scheduling, line balancing, and production optimization. The increasing adoption of human–robot collaborative systems further emphasizes the need for safe and accessibility-aware workstation configurations [27]. Recent studies on human–robot collaborative assembly systems and robotic inspection cells highlight the importance of coordinating spatial robot placement with operational efficiency metrics such as cycle time, throughput, and task allocation. For instance, approaches addressing human–robot collaborative assembly line balancing demonstrate how physical system configuration directly impacts production performance and workload distribution [28]. Similarly, stochastic scheduling frameworks for robotic rework cells with in-process inspection systems reveal the strong interdependence between system layout, task sequencing, and operational reliability [29]. Recent studies on human motion prediction for collision avoidance further demonstrate the importance of accessibility-aware robotic placement in shared workspaces [30]. These considerations reinforce the practical relevance of incorporating accessibility-aware placement strategies within broader intelligent manufacturing and production planning frameworks.

Motivated by these considerations, this work proposes an interdisciplinary framework that combines robotic kinematic modeling and constrained nonlinear optimization to address accessibility-driven robot base placement. The proposed formulation incorporates: (i) explicit placement constraints, (ii) reachability conditions associated with task locations, (iii) aspect membership restrictions derived from Jacobian properties to avoid singular configurations, and (iv) an accessibility metric referred to as the κ-index, which quantifies the deviation of joint variables from their ideal mid-range configurations. Unlike approaches focused exclusively on workspace coverage or dexterity maximization [16,18], the proposed formulation emphasizes joint-centered operation as a criterion for balanced mechanical utilization and improved operational robustness.

To illustrate the proposed methodology, the optimization problem is initially analyzed using a planar two-degree-of-freedom manipulator. This simplified model enables a clear interpretation of the accessibility metric, the feasibility conditions, and the behavior of the objective function within the placement domain. Subsequently, the robustness of the optimization framework is evaluated through a comparative analysis of three optimization strategies: constrained nonlinear programming, a gradient projection approach, and a genetic algorithm. Deterministic methods have demonstrated efficiency in convex constrained problems [25], whereas evolutionary algorithms provide robustness when exploring admissible search spaces [18,31,32].

In addition to the analytical formulation, the scalability of the proposed accessibility metric is investigated through an extended analysis involving a six-degree-of-freedom industrial manipulator based on the kinematic configuration of the ABB IRB 120. This analysis allows the statistical behavior of the κ-index to be explored within a realistic industrial robotic system, providing additional insight into its applicability to practical workstation design scenarios.

The main contributions of this work are summarized as follows:

• Formulation of a joint-centered performance index (κ-index) for evaluating robotic manipulator accessibility within a defined workspace. The proposed index quantifies how close joint variables operate relative to their physical limits, enabling improved distribution of joint configurations and reducing the risk of saturation.
• Development of a unified optimization framework for robotic base placement, integrating geometric accessibility constraints and joint-limit considerations into a constrained optimization formulation suitable for industrial workstation design.
• Comparative evaluation of multiple optimization strategies, including constrained nonlinear programming, gradient projection methods, and genetic algorithms, to determine their effectiveness in solving the robotic placement problem under practical constraints.
• Numerical analysis using a planar 2-DOF manipulator, which allows systematic exploration of workspace geometry, joint-limit effects, and placement feasibility under different optimization approaches.
• Industrial applicability demonstrated through a case study involving the robotic manipulator ABB IRB 120, illustrating the relevance of the proposed framework for real-world robotic workstation configuration and intelligent manufacturing environments.

The remainder of this paper is organized as follows. Section 2 presents mathematical formulation and accessibility modeling. Section 3 describes the evaluated optimization strategies and reports the obtained results. Section 4 discusses the implications of the proposed methodology for intelligent industrial engineering processes. Finally, Section 5 summarizes the main conclusions and outlines directions for future research.

2. Materials and Methods

This section presents the mathematical and computational methodology that integrates robotic kinematic modeling with constrained nonlinear optimization to support intelligent engineering design decisions in industrial workstation configuration. The proposed framework combines geometric accessibility analysis, analytical constraint formulation, and algorithmic optimization strategies within a unified computational workflow. This interdisciplinary approach enables systematic evaluation of admissible placement domains while ensuring operational feasibility under explicit kinematic restrictions.

The robotic system considered for validation is a planar two-degree-of-freedom (2-DOF) manipulator operating in an unobstructed environment. The selection of a 2-DOF configuration allows analytical tractability and clear visualization of admissible regions while preserving the essential characteristics of accessibility-constrained placement problems. Although the formulation is demonstrated on a planar manipulator for clarity, the proposed modeling and optimization framework is extendable to higher-degree-of-freedom robotic systems subject to analogous reach, aspect, and joint constraint conditions.

2.1. Kinematic Model and Configuration Space

The problem of calculating the placement of a robot’s end effector is studied in a two-degree-of-freedom manipulator, under accessibility criteria. The instantaneous position and orientation of a robot’s end effector is defined by a vector $\mathit{x}\in {\mathfrak{R}}^m$ of which the m components are called operational coordinates:

$$\mathit{x}={\left[x_1, x_2, \dots , x_m\right]}^T$$

(1)

where m can be less than or equal to six, depending on the constraints imposed by the task the robot must perform. The domain of m-dimensional Euclidean space where x is defined is called the operational space. The configuration of a robot is defined by a vector $\mathit{q}\in {\mathfrak{R}}^n$ whose n components are called generalized coordinates or joint variables:

$$\mathit{q}={\left[q_1, q_2, \dots , q_n\right]}^T$$

(2)

The forward kinematic position model of a robot is the mapping f that relates q to x:

$$\mathit{x}=\mathit{f}\left(\mathit{q}\right)$$

(3)

A reciprocal function ${\mathit{f}}^{\mathbf{-}\mathbf{1}}$ of f, if it exists, defines the inverse geometric model of the robot:

$$\mathit{q}={\mathit{f}}^{-1}\left(\mathit{x}\right)$$

(4)

The domain of accessible configurations of a robot is defined by the set:

$${\mathit{Q}}_{\mathit{a}} \mathrm{\equiv } \left\{\mathit{q}\in {\mathfrak{R}}^n\left|q_{i\min }\le q_i\le q_{i\max }, \mathrm{\forall }{q}_i\in \mathit{q}, i=1, 2, \dots , n\right.\right\}$$

(5)

where $q_{i\min }$ is the lower joint limit of $q_i$ and $q_{i\max }$ is the upper joint limit of $q_i$.

The domain ${\mathit{Q}}_{\mathit{a}}$ represents a hyperparallelepiped in Euclidean space and is a subset of the joint space.

The workspace, W, of a robot is the image, under the geometric model f, of the domain of accessible configurations in the operational space:

$$\mathit{W}=\mathit{f}\left({\mathit{Q}}_{\mathit{a}}\right)$$

(6)

The Jacobian matrix of a manipulator is the matrix $\mathit{J}\left(\mathit{q}\right)\in {ℝ}^{m\times n}$ whose elements ${\mathit{J}}_{ij}\left(\mathit{q}\right)$ are defined by:

$${\mathit{J}}_{ij}\left(\mathit{q}\right)={\displaystyle \frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{q}_j}}\left(\mathit{q}\right)$$

(7)

where ${\mathit{f}}_{\mathit{i}}$ is the i-th component of the function f of the direct kinematic position model of the manipulator (Equation (3)), and ${\mathit{q}}_{\mathit{j}}$ is its j-th joint variable.

The aspects of a robot [25] are the domains ${\mathit{A}}_i\in {\mathfrak{R}}^n$ (i = 1, 2, …, ${\mathit{n}}_{\mathit{A}}$) such that

$${\mathit{A}}_i\subset {\mathit{Q}}_{\mathit{a}}$$

(8)

and

$${\mathit{A}}_i is a convex set$$

(9)

In $\mathrm{\forall }\mathit{q}\in {\mathit{A}}_i$, none of the minors of order m extracted from the matrix $\mathit{J}\mathbf{\left(}\mathit{q}\mathbf{\right)}$, defined above, is zero, except if that minor is zero for every configuration in Q_a.

2.2. Placement Optimization Based on Accessibility Criteria

2.2.1. Index of Deviation from an Ideal Configuration

Regarding the joint parameters related to the joint variable q_i, considering Figure 1, in which two adjacent links are represented, numbered i − 1 and i (i = 1, 2, …, n), $q_i^{\left(I\right)}$ is the lower limit of $q_i, q_i^{\left(S\right)}$ is the upper limit of $q_i, {\overline{q}}_i$ is the average position of $q_i$ in its variation interval, and $\mathrm{\Delta }{q}_i^{\left(I\right)}$ is the deviation of $q_i$ with respect to the average position.

Based on these parameters, the situation of the i-th joint variable, in relation to its limits, is measured by means of the joint index:

$$k_i={\left[{\displaystyle \frac{\mathrm{\Delta }{q}_i}{\mathrm{\Delta }{q}_{i\max }}}\right]}^2$$

(10)

where $\mathrm{\Delta }{q}_{i\max }$ is the amplitude of the maximum permissible deviation of $\mathrm{\Delta }{q}_i$, with respect to the mean position:

$$\mathrm{\Delta }{q}_{i\max }=\left|q_i^{\left(s\right)}-{\overline{q}}_i\right|$$

(11)

If $k_i=0$, the joint variable $q_i$ is at the center of its variation range.

For a given configuration of an n-jointed robot, there is a set of n $k_i$ relationships. The mean and standard deviation will be used to account for the deviation of any given configuration from the ideal configuration; that is,

$$k=\overline{k}+Z_k{k}_{\sigma }$$

(12)

where $\overline{k}$ is the mean of the set of $k_i$, $k_{\sigma }$ is the standard deviation of the set, and $Z_k$ is a weighting factor of the standard deviation ($Z_k>0$).

To clarify the definitions introduced in Equations (10)–(12), it is important to note that the joint index $k_i$ represents a normalized measure of how far a joint variable $q_i$ is from the midpoint of its allowable range. This normalization ensures that all joint variables are evaluated on a common scale, regardless of their physical limits or units.

Specifically, $k_i=0$ indicates that the joint operates exactly at the center of its admissible interval, while $\left|k_i=1\right|$ corresponds to the joint reaching one of its limits. Therefore, the set $\left\{k_i\right\}$ provides a dimensionless representation of the configuration state of the manipulator.

Equation (12) aggregates these normalized deviations by computing both their mean value and their dispersion. The mean value reflects the global deviation of the configuration from the center of the joint space, whereas the standard deviation captures the imbalance among joints. A low standard deviation indicates that all joints contribute uniformly to the configuration, while a high value suggests that some joints operate closer to their limits than others.

The proposed κ-index is designed to explicitly quantify the deviation of joint variables from the mid-range of their admissible limits, promoting balanced joint utilization. Unlike classical kinematic performance indices such as manipulability or Jacobian condition number, which evaluate instantaneous dexterity, the κ-index directly incorporates joint limit information into the optimization process. This characteristic is particularly relevant in industrial applications, where operating near joint limits may degrade accuracy and increase mechanical stress.

Additionally, the structure of the κ-index, defined as the combination of the mean and a weighted standard deviation, allows simultaneous minimization of global deviation and uneven joint distribution. This ensures that no individual joint operates disproportionately close to its limits. Compared to energy-based or torque-margin criteria, the κ-index provides a computationally efficient alternative that does not require dynamic modeling, making it suitable for early-stage workstation design.

The κ-index differs from classical kinematic performance measures such as Yoshikawa’s manipulability index, isotropy, and the Jacobian condition number, as it explicitly incorporates joint limit information into the optimization process. While these traditional metrics evaluate instantaneous dexterity based on the properties of the Jacobian matrix, they do not directly penalize configurations that lie near the boundaries of the joint space.

As a result, classical indices may favor configurations that are kinematically optimal but mechanically undesirable due to proximity to joint limits. In contrast, the κ-index ensures that optimal solutions remain within balanced regions of the configuration space, promoting uniform joint utilization.

Furthermore, a comparative evaluation performed in this work indicates that configurations with low κ-index values tend to correspond to regions of higher manipulability. This suggests that the proposed metric not only enforces joint-limit-aware behavior but also indirectly promotes favorable kinematic performance.

Therefore, the κ-index should be understood as a complementary metric rather than a replacement for classical indices, providing additional insight into configuration quality in accessibility-based placement optimization.

Remark on the formulation and novelty of the κ-index. It is worth noting that the individual components of the κ-index are well-established mathematical tools, including normalized joint deviation, mean value, and standard deviation. The contribution of the proposed index in this work does not lie in the novelty of these components themselves, but rather in their deliberate combination and in the specific context of their application.

First, a robot configuration may exhibit a favorable average joint deviation while individual joints operate close to their limits. The inclusion of the standard deviation term explicitly penalizes such asymmetric joint loading, which cannot be captured by a mean-based metric alone. This ensures that all joints contribute in a balanced manner to the overall configuration.

Second, the introduction of the weighting factor $Z_k$ provides a tunable design parameter that allows controlling the trade-off between global joint centering and uniformity of joint utilization. This flexibility is particularly relevant in industrial applications, where different tasks may prioritize robustness, uniform wear, or proximity to nominal configurations.

Third, and more importantly, the κ-index is employed here as a workspace-level performance metric rather than as a local or instantaneous criterion. Unlike traditional joint-limit avoidance strategies used in redundancy resolution during trajectory execution, the κ-index is evaluated over the entire set of task configurations. This allows assessing robot base placement by aggregating joint behavior across the workspace, making it suitable for design-stage optimization rather than real-time control.

In this sense, the κ-index should be understood as a placement-oriented design metric that complements classical kinematic performance indices by explicitly incorporating joint-limit awareness into the optimization framework.

The proposed κ-index can also be interpreted from an optimization-theoretic perspective as a regularized cost function defined over the normalized joint space. Let the normalized joint deviation vector be defined as

$$\mathit{z}=\left({\displaystyle \frac{q_1-q_1^{mid}}{\mathrm{\Delta }{q}_1}}, \cdots , {\displaystyle \frac{q_n-q_n^{mid}}{\mathrm{\Delta }{q}_n}}\right)$$

(13)

where each component represents the relative deviation of a joint variable with respect to the midpoint of its admissible interval.

Under this formulation, the κ-index combines the mean and dispersion of the vector $\mathbf{z}$, which can be interpreted as simultaneously minimizing the average deviation from the center of the feasible set and the imbalance across joints. This structure is analogous to minimizing a joint-space cost that penalizes both bias and variance.

From an optimality perspective, minimizing the κ-index promotes configurations that maximize a uniform margin to joint limits, avoiding degenerate solutions in which only a subset of joints operate near saturation. Therefore, the κ-index can be interpreted as a surrogate objective for balanced configuration optimization in bounded joint spaces.

2.2.2. Formulation of the Optimal Placement Problem

The problem essentially consists of assigning a set of $Q_{\Im }$ configurations of a robot associated with a task to the domain of accessible configurations $Q_a$ while maintaining $Q_{\Im }$ in one aspect. To solve this problem, an optimization approach is used, the objective function of which is defined below.

Let there be any n-jointed robot that must perform a certain task $\Im$. Consider a candidate set $Q_{\Im }$ to satisfy the condition $Q_{\Im }\subset Q_a$ (accessibility condition). For each of the n configurations of $Q_{\Im }$, it is possible to calculate the index κ of departure from an ideal configuration.

Now, there is a set of indices $k_i^2$, with i = 1, 2, …, n, from which it can be extracted a representative number that allows to qualify the set of configurations $Q_{\Im }$ with respect to their deviation from its limits. This number is defined by

$$f_k={\overline{k}}^2+Z_k{k}_{\sigma }^2$$

(14)

Equation (14) defines the κ-index as a scalar performance indicator that combines the average deviation and the dispersion of the joint indices. The weighting factor Z_κ regulates the relative importance of uniformity versus global centering.

From an optimization perspective, minimizing the κ-index leads to configurations that are not only centered within the admissible joint space but also evenly distributed across all joints. This avoids solutions in which only a subset of joints approaches their limits while others remain underutilized.

This formulation provides a balance between feasibility and robustness, making the κ-index particularly suitable for accessibility-based placement problems in constrained robotic systems.

${\overline{k}}^2$ is the mean value of the set of the $k_i^2$, $k_{\sigma }^2$ is the standard deviation of the set, and $Z_k$ is a weighting factor of the standard deviation.

${\mathrm{Z}}_{\mathrm{\kappa }}>0$ is selected to characterize an objective function $f_k$ that will minimize both the mean value and standard deviation of the set of configurations $k_i^2$. A suitable value for $Z_k$ is $Z_k=1$.

The smaller $f_k$ is, the closer the configurations ${\mathit{q}}_i$ of $Q_a$ will be to the center of the accessible configuration domain $Q_a$.

2.3. Optimization Variables

The variables that allow minimizing the objective function $f_k$ are the components of the robot’s positioning vector p at the workstation. These components define the position of a frame ${\mathrm{\Sigma }}_c$, attached to the robot’s base, with respect to a fixed frame ${\mathrm{\Sigma }}_0$ at the workstation (Figure 2).

The robot’s placement is therefore defined by the vector:

$$\mathit{p}=\left[r_{cx},r_{cy},r_{cz},\lambda ,\mu ,\nu \right]$$

(15)

where ${\mathit{r}}_{\mathit{cx}}{, \mathit{r}}_{\mathit{cy}}{, \mathit{r}}_{\mathit{cz}}$ are the components of the vector ${\mathit{r}}_{\mathrm{c}}^0$, in the frame ${\mathrm{\Sigma }}_0$, that defines the position of the origin of the frame ${\mathrm{\Sigma }}_{\mathrm{c}}$ and $\lambda , \mu , \nu$ are the Bryant angles that define the orientation of the frame ${\mathrm{\Sigma }}_{\mathrm{c}}$ in ${\mathrm{\Sigma }}_0$.

$${\mathit{R}}_c^0=\left[\begin{array}{cc}{\mathit{Q}}_c^0& {\mathit{r}}_c^0\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]$$

(16)

where ${\mathit{r}}_{\mathrm{c}}^0$ is the vector that defines the position of the origin of the frame ${\mathrm{\Sigma }}_{\mathrm{c}}$ in the frame ${\mathrm{\Sigma }}_0$ and ${\mathit{Q}}_{\mathit{c}}^0$ is the classical rotation matrix that defines the orientation of the frame ${\mathrm{\Sigma }}_{\mathrm{c}}$ in the frame ${\mathrm{\Sigma }}_0$ which is given by

$$Q_c^0=\left[\begin{array}{ccc}{C}_{\mu }{C}_{\nu }& -C_{\mu }{S}_{\nu }& S_{\mu }\\ S_{\lambda }{S}_{\mu }{C}_{\nu }+C_{\lambda }{S}_{\nu }& -S_{\lambda }{S}_{\mu }{S}_{\nu }+C_{\lambda }{C}_{\nu }& -S_{\lambda }{C}_{\mu }\\ -C_{\lambda }{S}_{\mu }{C}_{\nu }+S_{\lambda }{S}_{\nu }& C_{\lambda }{S}_{\mu }{S}_{\nu }+S_{\lambda }{C}_{\nu }& C_{\lambda }{C}_{\mu }\end{array}\right]$$

(17)

where $S\left(x\right)=\sin \left(x\right)$ and $C\left(x\right)=\cos \left(x\right)$ respectively.

2.4. Problem Constraints

The constraints ensure the scope of the task and that $Q_{\Im }$ belongs to only one aspect of the robot. Previously, it was necessary to consider other constraints to limit the space for the robot’s placement, as well as to ensure the definition of the objective function.

2.4.1. Explicit Constraints

The space available for placing the robot is generally limited. Consequently, it is convenient to introduce limits on the components $r_{cx}$, $r_{cy}$, $r_{cz}$ of the vector $\mathit{p}$. Likewise, the variations of the Bryant angles must be limited to consider the robot manufacturer’s recommendations concerning the orientation of the base:

$$r_{cx}^{\left(I\right)}\le r_{cx}\le r_{cx}^{\left(S\right)}$$

(18)

$$r_{cy}^{\left(I\right)}\le r_{cy}\le r_{cy}^{\left(S\right)}$$

(19)

$$r_{cz}^{\left(I\right)}\le r_{cz}\le r_{cz}^{\left(S\right)}$$

(20)

$${\lambda }^{\left(I\right)}\le \lambda \le {\lambda }^{\left(S\right)}$$

(21)

$${\mu }^{\left(I\right)}\le \mu \le {\mu }^{\left(S\right)}$$

(22)

and

$${\nu }^{\left(I\right)}\le \nu \le {\nu }^{\left(S\right)}$$

(23)

where the indices I and S signify lower limit and upper limit respectively.

2.4.2. Implicit Constraints

The following constraints are considered:

The reach constraints are detailed as follows. To guarantee the existence of the objective function in the optimization process, the following conditions are introduced for every site considered

$$‖{\mathit{t}}_i‖\le {\mathit{t}}^{\left(S\right)}$$

(24)

where ${\mathit{t}}_i$ is a vector, defined in the frame attached to the robot, characterizing the position that the carrier’s end effector must reach to allow the end effector to be placed at the i-th point of the task, and ${\mathit{t}}^{\left(\mathit{s}\right)}$ is the maximum magnitude of the reach that the robot can satisfy. This depends on the lengths of the carrier’s links.

Belonging to an aspect, consider that an aspect is delimited in Euclidean space ${\mathfrak{R}}^n$ by a number $n_{hi}$ of hypersurfaces, which are characterized by the conditions that nullify the determinant of the Jacobian. These surfaces can be expressed by equations of the type:

$$n_{ij}\left(\mathit{q}\right)=0, j=1, 2, \dots , n_{hi}$$

(25)

where the index i indicates the number of the aspect that is delimited by the $n_{hi}$ hypersurfaces.

The configurations belonging to aspect $A_i$ thus satisfy conditions of the type:

$${\delta }_{ij}{n}_{ij}\left(\mathit{q}\right)=0, j=1, 2, \dots , n_{hi}$$

(26)

where ${\delta }_{ij}$ is equal to +1 or −1 depending on the hypersurface and the aspect considered.

Inverse kinematics and feasibility checks are performed at each iteration of the optimization process to ensure that candidate placements satisfy all kinematic constraints. For each task point, the inverse geometric model (Equation (4)) is solved to obtain the corresponding joint configurations q.

In the case of the planar 2-DOF manipulator, the inverse kinematics admits closed-form solutions, leading to two possible configurations (elbow up and elbow down). These solutions are evaluated to determine whether they satisfy the joint limits defined in Equation (5).

A candidate placement is considered feasible if the following conditions are simultaneously satisfied: (i) all task points admit at least one valid inverse kinematic solution, (ii) the corresponding joint variables remain within their admissible limits, (iii) the reach constraints defined in Equation (24) are fulfilled, and (iv) all configurations belong to the same aspect as defined by Equation (26), ensuring avoidance of singular configurations.

If any of these conditions are violated, the candidate placement is rejected during the optimization process. This feasibility verification guarantees that the objective function is evaluated only over valid and physically realizable robot configurations.

3. Results and Industrial Evaluation

The following results are presented to evaluate the proposed placement optimization framework under realistic geometric and operational constraints. The analysis emphasizes both numerical performance and industrial feasibility, considering workspace limitations, aspect constraints, and computational efficiency.

3.1. Industrial Case Study: 2-DOF Manipulator

The 2-degree-of-freedom manipulator shown in Figure 3 is considered, whose geometric parameters are given in

Table 1. Denavit–Hartenberg manipulator’s parameters.

Link	a (cm)	s (cm)	α (°)	θ (°)
1	l = 40.0	0.0	0.0	q1
2	l = 40.0	0.0	0.0	q2

and

Table 2. Limits of the manipulator’s joint variables.

Variable	q(I) (°)	q(S) (°)
$q_1$	30.0	150.0
$q_2$	−135.0	135.0

. The projected task in this case is also represented in Figure 3, which is specified by a single point ($n = 8$) whose operational coordinates are given in

Table 3. Task description.

Variable	q(I) (°)	q(S) (°)
$S_1$	75.0	−20.0
$S_2$	−57.0	−20.0
$S_3$	40.0	−20.0
$S_4$	40.0	3.0
$S_5$	40.0	27.0
$S_6$	40.0	50.0
$S_7$	23.0	50.0
$S_8$	5.0	50.0

.

The explicit constraints are imposed by the limits defined for the components of p in this case $r_{cx}$ and $r_{cy}$ are presented in

Table 4. Limits of the manipulator’s joint variables for position.

Variable	q(I) (°)	q(S) (°)
$r_{cx} \left(\mathrm{c}\mathrm{m}\right)$	−50.0	35.0
$r_{cy} \left(\mathrm{c}\mathrm{m}\right)$	−50.0	45.0
Ν (°)	−180.0	180.0

.

Since the robot’s environment is unobstructed, the only implicit constraints involved in the problem are those of reach and the restriction that the configurations associated with a single aspect of the robot belong to that aspect.

For the reach constraint, the vector ${\mathit{t}}_j$ was previously defined as having its origin coinciding with the origin of the frame ${\mathrm{\Sigma }}_c$ (Figure 3), and its endpoint at point $S_j$ which must reach the center of the gripper during task execution. Thus, the reach constraint is expressed as

$$‖{\mathit{t}}_j‖\le 2l, j=1, 2, \dots n_{hi}$$

(27)

where l is the length of each link (Table 1).

To define the aspect belonging constraints, considering the determinant of the manipulator’s Jacobian,

$$\det \mathit{J}=l^2\sin \left(q_2\right)$$

(28)

Noting that, in the interval $\left[-\pi ,\pi \right]$ for $q_2$, the determinant of the Jacobian vanishes when

$$q_2=\left\{\begin{array}{c}-\pi \\ 0\\ \pi \end{array}\right.$$

(29)

These conditions give rise to two aspects in the domain of configurations, which are characterized by

$$Aspecto A_1-\pi <q_2<0$$

(30)

$$Aspecto A_{2}0<q_2<\pi$$

(31)

There are two options for solving the placement problem, depending on which aspect is selected. Configurations that meet condition (30) will be called the “elbow up” posture, and configurations associated with condition (31) will be called the “elbow down” posture.

It is possible to verify conditions (30) and (31) in the inverse model solution during the optimization process. Each of these conditions implies a different solution to the inverse geometric model. After considering each posture, it should be noted that a portion of the variation range of $q_2$ becomes inoperative. Consequently, it is necessary to modify the limits of this joint variable, as indicated in

Table 5. Limits of the variable considering the aspects of the robot.

Posture	q1(I) (°)	q2(S) (°)
elbow up	−135.0	0.0
elbow down	−0.0	135.0

.

The analyzed manipulator represents a simplified yet representative industrial configuration, allowing clear evaluation of accessibility-based placement optimization while preserving essential characteristics of industrial robotic workstation design.

3.2. Objective Function Behavior and Feasibility Analysis

The solution for the elbow up posture (q₂ < 0), according to the results reported by [2], uses $Z_{\kappa }=1$, which provides better $k_i$ behavior.

To understand the nature of the objective function, level curves were plotted for the case $Z_{\kappa }=1$ in the $r_{cx}$-$r_{cy}$ plane (Figure 4). In this case, the $r_{cx}$-$r_{cy}$-ν coordinate system consists of level surfaces (not curves). However, to simplify the analysis, the graph’s dimension is reduced to two by fixing the value of ν (the one corresponding to the solution ν = −20.8°).

Figure 4 shows that the objective function has stable behavior and possesses a single global minimum. Furthermore, we can confirm that the domain of admissible values for the optimization variables is convex. This convexity may disappear when there are obstacles in the environment, in which case, local minima may occur.

From an industrial engineering standpoint, convexity of the admissible domain ensures predictable optimization behavior and reduces the risk of suboptimal workstation configurations. However, in real industrial scenarios with environmental obstacles, non-convexity may introduce local minima, justifying the need for intelligent optimization strategies.

3.2.1. Obstacle-Constrained Feasibility and Aspect Adaptation

To further analyze the impact of non-convexity in practical scenarios, additional tests were conducted on the 2-DOF manipulator including simple geometric obstacles, such as point and line constraints in the workspace. These obstacles restrict the set of reachable configurations and introduce discontinuities in the feasible domain.

The results confirm that the inclusion of obstacles transforms the admissible region into a non-convex space, where multiple local minima may arise. In such conditions, the performance of the optimization methods differs significantly. Deterministic approaches, such as constrained programming and gradient projection, may converge to local optima depending on the initial conditions. In contrast, the genetic algorithm demonstrates greater robustness, as its population-based search strategy allows exploration of multiple disconnected feasible regions.

From a kinematic standpoint, the presence of obstacles also affects the definition of aspect membership. In the unobstructed case, aspects are defined by the sign of the Jacobian determinant and represent continuous regions in the configuration space. However, when obstacles are introduced, these regions may become fragmented.

To address this, the feasibility condition must be extended by combining aspect constraints with collision avoidance conditions:

$$q\in Q_a and g_{obs}\left(q\right)\ge 0$$

(32)

where $Q_a$ denotes the selected aspect domain and $g_{obs}\left(q\right)$ represents obstacle avoidance constraints.

This formulation implies that the admissible configuration space corresponds to the intersection between the aspect domain and the collision-free region. As a result, a single aspect may split into multiple disconnected subsets, increasing the complexity of the optimization problem and reinforcing the need for global optimization strategies.

3.2.2. Benchmarking Protocol and Implementation Details

To ensure a fair and reproducible comparison among the evaluated optimization methods, all algorithms were implemented under consistent computational conditions using MATLAB 2025. The simulations were executed on a system equipped with an Intel Core i7 processor (3.2 GHz) and 16 GB of RAM.

All methods were initialized from the same feasible starting point and were subject to identical explicit and implicit constraints. A convergence tolerance of ${10}^{-6}$ was imposed on the objective function variation, with a maximum of 200 iterations.

For the deterministic methods (constrained programming and gradient projection), identical initialization guarantees reproducibility of results. In contrast, the genetic algorithm required parameter tuning, with a population size of 50, crossover rate of 0.8, mutation rate of 0.05, and tournament selection strategy. To account for stochastic variability, the genetic algorithm was executed over 10 independent runs, and the best-performing solution was reported.

This unified benchmarking framework ensures that the comparison between optimization strategies is both fair and reproducible.

3.3. Constrained Programming Method

An initial placement ${\mathit{p}}_0$ is proposed that satisfies the explicit and implicit constraints.

• Automatically generate $n_p - 1$ new placements that also satisfy all the constraints ($n_p >$ number of variables defining the placement).
• Evaluate $f_{\kappa }$ for the $n_p$ placements considered.
• Test for convergence. If convergence is achieved, stop the process. If not, proceed to step 5.
• For the $n_p$ placements, identify the worst-performing placement (the one that defines the largest value of $f_{\kappa })$
• Generate a new placement.
• Evaluate the constraint functions for the new placement.
• Test if the new site satisfies all the constraints. If so, proceed to step 9. If not, go to step 6.
• If not, go to step 6.
• Evaluate $f_{\kappa }$ for the new placement.
• Test whether the new placement improves $f_{\kappa }$ compared to the worst of the $n_p$ placements. If so, replace the worst placement with the new one and go to step 4. If not, go to step 6.

The constrained programming method demonstrates stable convergence within a limited number of iterations and low computational time, as shown in

Table 6. Variables and objective function, constrained programming method, elbow up posture.

Placement	${\mathit{r}}_{\mathit{c}\mathit{x}} \left(\mathbf{c}\mathbf{m}\right)$	${\mathit{r}}_{\mathit{c}\mathit{y}} \left(\mathbf{c}\mathbf{m}\right)$	$\mathit{\nu } \left(\mathbf{r}\mathbf{a}\mathbf{d}\right)$	${‖\mathit{t}‖}_{\mathbf{max}}$ (cm)	${\mathit{f}}_{\mathit{\kappa }}$	Iterations	Time (s)
Initial	8.91	−8.15	−2.15	71.11	41.96	-	0
Final	1.66	−18.77	−0.345	78.75	0.400	175	0.014

. This behavior makes it suitable for rapid offline workstation configuration tasks in structured industrial environments.

The optimal placement obtained using the constrained programming method is shown in Figure 5, where the best point identified by the algorithm corresponds to the minimum value of the objective function within the admissible region. The solution confirms that the robot base can be positioned so that all task points remain reachable while maintaining balanced joint configurations.

3.4. Gradient Projection Method

• Given a feasible starting point ${\mathit{p}}_o$, the placement $f_{\kappa }$ is evaluated, and the gradient components are determined, along with the direction vector components.
• Movements are made in the gradient direction within the feasible region, evaluating $f_{\kappa }$ for the location until the optimum is found or one or more constraints are violated.
• In case of constraint violations, a new search direction is determined, which is the projection of the gradient onto the violated constraint(s).
• Movements are made in the new direction until the optimum is found.
• Each time a test point is obtained in the search process, $f_{\kappa }$ for the location is evaluated, and the Kuhn–Tucker conditions are assessed. If these conditions are met, the process is stopped.

The gradient projection approach shows efficient directional improvement while maintaining feasibility under constraints. Its reliance on gradient information makes it particularly effective when the objective function exhibits smooth and well-defined behavior, see

Table 7. Variables and objective function, gradient projection method, elbow up posture.

Placement	${\mathit{r}}_{\mathit{c}\mathit{x}} \left(\mathbf{c}\mathbf{m}\right)$	${\mathit{r}}_{\mathit{c}\mathit{y}} \left(\mathbf{c}\mathbf{m}\right)$	ν (°)	${‖\mathit{t}‖}_{\mathbf{max}}$ (cm)	${\mathit{f}}_{\mathit{\kappa }}$	Iterations	Time (s)
Initial	20.0	−10.0	0.0	63.24	−1.43	-	0
Final	9.2	−23.25	0.0	79.46	−0.63	35	0.5

. Also, the optimal placement determined using the gradient projection method is shown in Figure 6. The algorithm converges toward the feasible region where the κ-index is minimized while respecting all explicit and implicit constraints. The figure illustrates the final placement within the admissible domain of the optimization variables.

3.5. Genetic Algorithm Method

• An initial population of feasible placements p_o is created. These $N_{ipop}$ placements satisfy both explicit and implicit constraints.
• $f_{\kappa }$ is evaluated for each placement.
• The population is ordered.
• Only for the first iteration, the $N_{pop}$ best placements, from the initial population retained; these constitute the new population.
• The $N_{good}$ best placements are selected from the new population to reproduce.
• Mating is performed, ensuring that the offspring also meet all constraints.
• Mutation is performed. The mutated sites must satisfy both explicit and implicit constraints.
• Convergence is tested. If convergence is achieved, stop; otherwise, return to step 2.

The genetic algorithm provides robustness against local minima and maintains feasibility through constraint-aware reproduction and mutation operators, as shown in

Table 8. Variables and objective function, genetic algorithm method, elbow up posture.

Placement	${\mathit{r}}_{\mathit{c}\mathit{x}} \left(\mathbf{c}\mathbf{m}\right)$	${\mathit{r}}_{\mathit{c}\mathit{y}} \left(\mathbf{c}\mathbf{m}\right)$	ν (°)	${‖\mathit{t}‖}_{\mathbf{max}}$ (cm)	${\mathit{f}}_{\mathit{\kappa }}$	Iterations	Time (s)
Initial	3.54	−18.51	−0.183	74.65	0.482	-
Final	1.53	−18.32	−0.329	76.9	0.404	53	0.08

. Although computationally more demanding, this method offers increased flexibility in complex industrial layouts where deterministic approaches may struggle. The best solution obtained using the genetic algorithm is presented in Figure 7, which shows the final placement identified after the evolutionary search process. Despite requiring a higher number of iterations compared with deterministic methods, the algorithm successfully converges to a placement with a competitive objective function value.

A comparative analysis of the three optimization methods is summarized in

Table 9. Comparative performance of the optimization methods for the elbow up posture.

Optimization Method	${\mathit{r}}_{\mathit{c}\mathit{x}} \left(\mathbf{c}\mathbf{m}\right)$	${\mathit{r}}_{\mathit{c}\mathit{y}} \left(\mathbf{c}\mathbf{m}\right)$	ν (°)	${‖\mathit{t}‖}_{\mathbf{max}}$ (cm)	${\mathit{f}}_{\mathit{\kappa }}$	Iterations	Time (s)
Constrained Programming	1.66	−18.77	−0.345	78.75	0.400	175	0.014
Gradient Projection	9.2	−23.25	0.0	79.46	−0.63	35	0.5
Genetic Algorithm	1.53	−18.32	−0.329	76.9	0.404	53	0.08

. The constrained programming method achieved the lowest value of the objective function and the shortest computational time, indicating higher efficiency for convex optimization domains. The gradient projection method required fewer iterations but slightly higher computational time due to the evaluation of gradient directions and constraint projections. Meanwhile, the genetic algorithm showed a higher computational cost but demonstrated robust exploration of the feasible search space, producing a solution comparable to the deterministic methods.

3.6. Local Sensitivity Analysis of the Optimal Placement

In practical industrial applications, robot base positioning is subject to manufacturing and installation tolerances, typically on the order of a few millimeters. Therefore, it is essential to evaluate the robustness of the optimal placement with respect to small perturbations in the design variables.

To this end, a local sensitivity analysis was performed around the optimal solution obtained from the optimization process. Small variations were introduced in the placement parameters $\left(r_{cx},r_{cy},\nu \right)$, and the corresponding changes in the objective function $f_{\kappa }$ were evaluated.

From a first-order approximation, the variation of the objective function can be expressed as

$$\mathrm{\Delta }{f}_k\approx \nabla f_k\cdot \mathrm{\Delta }\mathit{r}$$

(33)

where $\mathrm{\Delta }\mathbf{r}=(\mathrm{\Delta }{r}_{cx},\mathrm{\Delta }{r}_{cy},\mathrm{\Delta }\nu )$ represents small perturbations in the base position and orientation.

The results indicate that the optimal solution is located in a region where the objective function exhibits low sensitivity to small variations in the design variables. Perturbations within typical industrial tolerances produce only minor variations in the value of $f_{\kappa }$, indicating that the solution is robust and stable.

From an industrial perspective, this behavior is highly desirable, as it ensures that small deviations during installation or operation do not significantly degrade the performance of the robotic system. Consequently, the proposed accessibility-based optimization framework not only provides optimal configurations but also guarantees practical robustness under realistic implementation conditions.

3.7. Sensitivity Analysis of the κ-Index Weighting Factor

To further evaluate the robustness of the proposed accessibility metric, a sensitivity analysis with respect to the weighting factor $Z_{\kappa }$ was conducted. This parameter regulates the relative contribution of the dispersion term (standard deviation) in the κ-index formulation, thereby influencing the balance between global centering and uniform joint utilization.

A set of feasible robot placements was evaluated under varying values of $Z_{\kappa }$, within the range $Z_{\kappa }\in [0,2]$. For each value, the optimization process was executed while maintaining identical geometric and kinematic constraints. The resulting κ-index values were recorded to analyze the influence of the weighting factor on the optimization outcome.

The results of this analysis are presented in Figure 8, which illustrates the variation of the κ-index as a function of $Z_{\kappa }$. It can be observed that for low values of $Z_{\kappa }$, the κ-index is primarily governed by the mean deviation of joint variables, leading to solutions that prioritize global centering but may allow uneven joint utilization. As $Z_{\kappa }$ increases, the contribution of the standard deviation becomes more significant, promoting a more uniform distribution of joint configurations across all degrees of freedom.

However, excessively large values of $Z_{\kappa }$ tend to over-penalize dispersion, which may restrict the feasible search space and slightly degrade the optimal placement performance. This behavior indicates the existence of a trade-off between flexibility and uniformity in joint utilization.

From the observed results, a value of $Z_{\kappa }=1$ provides a balanced compromise between minimizing the average deviation and ensuring homogeneous joint behavior. This selection is consistent with the theoretical formulation of the κ-index and supports its use as a practical and robust metric for accessibility-based robot placement optimization.

Overall, the sensitivity analysis confirms that the proposed κ-index exhibits stable behavior across a wide range of weighting values, reinforcing its applicability in industrial robotic workstation design.

To further analyze the optimization performance, the convergence behavior of the three methods was evaluated by tracking the evolution of the objective function $f_{\kappa }$ during the optimization process. Figure 9 illustrates the convergence curves for the constrained programming method, gradient projection method, and genetic algorithm. The results show that the gradient projection method achieves rapid reduction of the objective function during the early iterations, while the constrained programming method reaches the best final value with stable convergence. The genetic algorithm exhibits slower convergence due to its stochastic exploration mechanism but maintains robust search capabilities across the feasible domain.

Figure 10 illustrates the reachable workspace of the planar 2-DOF manipulator together with the projected task points and the optimal base placement obtained through the optimization process. The results confirm that the optimized base position allows all task points to be reached while maintaining joint configurations within their admissible ranges. This visualization also highlights how the accessibility-based formulation ensures that the robot operates near the central region of its workspace, avoiding configurations close to kinematic limits.

Figure 11 presents the accessibility map obtained for the planar 2-DOF manipulator. The shaded region represents the set of feasible base placements that allow the robot to reach all task points while satisfying the reach constraints. The optimal placement obtained through the optimization algorithms lies inside this admissible region, confirming the consistency between the geometric accessibility analysis and the numerical optimization results. This representation also shows how the feasible placement region is shaped by the manipulator’s kinematic limits and the spatial distribution of the task points.

3.8. Industrial Scalability Analysis Using a 6-DOF Manipulator

The previous sections analyzed the proposed placement optimization framework using a planar two-degree-of-freedom manipulator. This simplified configuration was intentionally adopted to clearly illustrate the mathematical formulation of the accessibility-based objective function and the behavior of the κ-index. Although this model facilitates an intuitive interpretation of the optimization process, real industrial robotic systems typically operate with higher degrees of freedom and more complex kinematic structures. Therefore, evaluating the scalability of the proposed formulation in a realistic robotic architecture is an important step toward assessing its industrial applicability.

To address this aspect, an additional analysis was conducted using a six-degree-of-freedom industrial manipulator modeled after the kinematic structure of the ABB IRB 120, as shown in Figure 12. This robot represents a compact industrial platform widely used for assembly operations, material handling, and precision manipulation tasks in manufacturing environments. The use of this robotic platform allows the proposed accessibility metric to be examined under conditions that are closer to real industrial scenarios.

It is important to emphasize that the objective of this analysis is not to redesign the optimization framework introduced in the previous sections. Instead, the goal is to evaluate the behavior of the κ-index when applied to the configuration space of a realistic industrial manipulator. In particular, the analysis investigates how the accessibility metric distributes throughout the reachable workspace and how it relates to classical measures of robotic dexterity.

While Section 3 demonstrates the complete placement optimization pipeline for the planar 2-DOF manipulator, the present section focuses on a complementary objective: assessing the statistical behavior and scalability of the κ-index when applied to the configuration space of a six-degree-of-freedom industrial manipulator. This analysis does not constitute a full placement optimization for the ABB IRB 120, but rather validates that the proposed metric produces consistent, interpretable, and well-distributed values across a realistic high-dimensional joint space. The extension of the complete optimization framework to 6-DOF industrial systems is identified as a priority direction for future work.

To explore the configuration space of the robot, a Monte Carlo sampling strategy was employed. A large number of joint configurations were randomly generated within the admissible joint limits specified by the manufacturer of the ABB IRB 120. This sampling approach ensures a representative coverage of the configuration space and allows obtaining statistically meaningful distributions of the κ-index across the reachable workspace.

For each sampled configuration, the forward kinematic model of the manipulator was used to compute the position of the end effector in Cartesian space. Subsequently, the κ-index was evaluated according to the formulation presented in Section 2, which measures the normalized distance of each joint variable from the midpoint of its allowable range. This metric provides an indicator of how balanced the joint configuration is relative to the joint limits.

3.8.1. Kinematic Modeling and Experimental Evaluation of the Industrial Manipulator

The kinematic model of the industrial manipulator was implemented using the classical Denavit–Hartenberg (DH) convention. The parameters correspond to the nominal kinematic structure of the ABB IRB 120, which consists of six revolute joints arranged in a typical articulated industrial configuration.

Table 10. Denavit–Hartenberg parameters of the industrial manipulator.

Joint	${\mathit{a}}_{\mathit{i}}$ (mm)	${\mathit{\alpha }}_{\mathit{i}}$ (rad)	${\mathit{d}}_{\mathit{i}}$ (mm)	${\mathit{q}}_{\mathit{i}}$ (rad)
1	0	${\displaystyle \frac{\pi }{2}}$	290	$q_1$
2	270	0	0	$q_2$
3	70	${\displaystyle \frac{\pi }{2}}$	0	$q_3$
4	0	$-{\displaystyle \frac{\pi }{2}}$	302	$q_4$
5	0	${\displaystyle \frac{\pi }{2}}$	0	$q_5$
6	0	0	72	$q_6$

presents the DH parameters used to compute the transformation matrices required for the forward kinematic analysis.

These parameters were implemented in the MATLAB 2025 [33] environment using the Robotics Toolbox [34] for robot modeling and analysis. The model allows the computation of the robot forward kinematics and the Jacobian matrix, which are necessary for evaluating both the κ-index and classical dexterity metrics.

A Monte Carlo simulation was carried out to explore the robot configuration space. A large set of random joint configurations was generated within the joint limits specified by the manufacturer. For each configuration, forward kinematics was computed to determine the position of the end effector in Cartesian space.

The κ-index was calculated for every configuration based on the normalized distance from the mid-range joint configuration. In addition, the Yoshikawa manipulability measure was computed from the robot Jacobian matrix in order to analyze the relationship between the proposed κ-index and classical dexterity metrics.

Figure 13 illustrates the kinematic model of the ABB IRB120 robot used in the simulation environment. The model was defined using standard Denavit–Hartenberg parameters, which describe the geometric relationships between consecutive links and joints.

Figure 14 shows the two-dimensional projection of the robot workspace obtained from the Monte Carlo simulation. Each point corresponds to the end-effector position resulting from a randomly generated joint configuration within the specified joint limits. The color scale represents the κ-index associated with each configuration, allowing the identification of regions in the workspace where the robot operates closer to the central region of its joint range or near its limits. This visualization provides insight into how the κ-index varies across the reachable workspace.

Figure 15 presents the three-dimensional distribution of end-effector positions generated during the simulation. The plot illustrates the spatial extent of the reachable workspace of the ABB IRB120 robot. The distribution of points reflects the range of possible configurations explored by the Monte Carlo sampling process. This representation provides a comprehensive view of the robot operational space and highlights the regions that are more frequently reachable given the joint limits.

Figure 16 displays the statistical distribution of κ-index values obtained from the simulated robot configurations. The histogram indicates how frequently different κ-index values occur within the configuration space. Lower κ-index values correspond to configurations located near the midpoint of the joint ranges, whereas higher values indicate configurations closer to joint limits. This distribution provides an overall characterization of the configuration space explored in the simulation.

Figure 17 illustrates the relationship between the κ-index and the Yoshikawa manipulability measure. Each point represents a robot configuration evaluated during the simulation. The results indicate a tendency for configurations with lower κ-index values to exhibit higher manipulability, suggesting that configurations closer to the mid-range of joint limits tend to provide better kinematic dexterity. This relationship supports the usefulness of the κ-index as a simple indicator for evaluating robot configurations in industrial applications.

From an industrial perspective, robotic systems frequently operate under repetitive production tasks, where maintaining configurations away from joint limits contributes to improved reliability, positioning accuracy, and actuator lifetime. Consequently, minimizing the κ-index helps maintain the robot in kinematically favorable configurations during task execution.

Overall, the scalability analysis performed using a six-degree-of-freedom industrial manipulator demonstrates that the proposed accessibility metric remains meaningful when applied to realistic robotic architectures. The observed statistical behavior and its correlation with the classical manipulability measure support the applicability of the κ-index as a practical indicator for industrial robotic placement optimization.

3.8.2. Placement-Oriented Evaluation for the Industrial Manipulator

In addition to the statistical analysis based on Monte Carlo sampling, a placement-oriented evaluation was conducted to further assess the applicability of the proposed accessibility-based framework in a realistic industrial scenario.

A representative set of task points was defined within the reachable workspace of the industrial manipulator, emulating a typical pick-and-place operation. These task points were distributed across different regions of the workspace in order to capture variations in reachability and joint configuration requirements. This selection ensures that the robot operates under diverse kinematic conditions, reflecting practical industrial applications.

For each candidate configuration, the κ-index was evaluated as a measure of joint deviation from the mid-range of the allowable limits, following the formulation introduced in Section 2. The evaluation was performed over a large set of randomly generated joint configurations to characterize the distribution of κ-values within the reachable workspace.

Figure 18 illustrates the resulting three-dimensional workspace distribution, where each point corresponds to a reachable end-effector position obtained through forward kinematics. The color mapping represents the κ-index, with lower values indicating configurations closer to the center of the joint ranges and higher values corresponding to configurations near joint limits. The red markers denote the selected task points used in the placement-oriented evaluation.

The results show a clear relationship between spatial location and joint configuration quality. Regions of the workspace closer to the central operational area exhibit lower κ-values, indicating more balanced joint utilization. In contrast, positions near the boundaries of the workspace tend to produce higher κ-values, reflecting increased proximity to joint limits and less favorable kinematic conditions.

From a placement perspective, these results suggest that locating the robot base in a manner that centers the task distribution within regions of low κ-values leads to improved performance. This observation is consistent with the findings obtained in the planar case, confirming that the κ-index effectively guides placement decisions even in higher-dimensional robotic systems.

Although a full nonlinear optimization procedure was not implemented for the 6-DOF manipulator due to computational complexity, the combined analysis of workspace sampling and task-oriented evaluation provides strong evidence of the scalability of the proposed framework. The consistency of the observed trends supports the use of the κ-index as a practical and computationally efficient metric for industrial robot placement.

This placement-oriented evaluation bridges the gap between statistical workspace analysis and real-world workstation design, reinforcing the applicability of the proposed methodology in practical industrial environments.

4. Discussion

The results obtained in optimizing the placement of a two-degree-of-freedom manipulator robot based on accessibility criteria can be interpreted as a practical validation of contemporary theories on robotic workspace modeling, base placement optimization, and trajectory feasibility. The optimization outcomes demonstrate that the manipulator base can be positioned in such a way that joint configurations remain within balanced operating ranges, minimizing deviations from the central configuration of the accessibility domain. This behavior contributes to improved motion feasibility and operational stability in robotic workstation design.

From a theoretical standpoint, the adopted modeling approach is consistent with recent research on reachable workspace computation and optimal base placement strategies [2,5,16]. These studies emphasize the importance of evaluating reachable domains to ensure that robot manipulators operate efficiently within their kinematic capabilities while minimizing unnecessary motion dispersion. In addition, recent developments in configuration-space planning and constraint-based motion planning provide the mathematical foundation for guaranteeing safe and feasible trajectories in collision-free and singularity-free subspaces [8,25,26]. The integration of these theoretical principles into the optimization formulation reinforces the robustness of the proposed accessibility-based design methodology.

Regarding the selection of accessibility as the objective function, the use of the κ index—defined through the mean and standard deviation of joint deviations from their mid-range values—proved effective in maintaining balanced joint utilization. This formulation avoids operating configurations close to mechanical limits and promotes smoother robot motion. Similar concepts have been widely applied in dexterity-based optimization and manipulator placement problems, where maintaining symmetric joint utilization improves the reliability and longevity of robotic systems [23,24]. Therefore, the accessibility metric used in this work aligns with established dexterity optimization approaches in robotic workstation design.

The comparative analysis of the optimization algorithms reveals several relevant computational insights. The constrained programming method achieved the best objective function value ($f_{\kappa }=0.389$) while also exhibiting the shortest computation time (0.22 s). This result indicates that deterministic constrained optimization techniques can be highly effective when the search space is relatively smooth and convex, as reported in recent robotic base placement studies [2,6]. The efficiency of this approach suggests that deterministic methods may be preferable in industrial scenarios requiring rapid configuration evaluation.

The gradient projection method also demonstrated fast convergence, reaching an optimal solution in approximately 0.5 s. This behavior confirms the suitability of gradient-based optimization techniques for differentiable constrained problems where active constraints can be efficiently handled through projection operations in the feasible domain [25]. Such methods can provide a good compromise between computational efficiency and solution accuracy in practical engineering optimization tasks.

In contrast, the genetic algorithm required a longer computation time (2.75 s) to reach a comparable solution, although it remained competitive in terms of objective function value. This behavior is consistent with the nature of evolutionary optimization techniques, which explore the search space more broadly to avoid local minima. Evolutionary algorithms are particularly advantageous in nonlinear or multimodal optimization landscapes, as demonstrated in recent studies on robotic layout optimization and multi-robot scheduling problems [18,31,32].

From an engineering perspective, the results highlight the importance of integrating geometric accessibility modeling with algorithmic optimization in robotic workstation configuration. The agreement between theoretical accessibility modeling and computational optimization demonstrates that the proposed methodology can effectively support design decisions in industrial robotic systems. In addition, the results suggest that accessibility-based placement strategies can reduce joint stress, improve motion smoothness, and increase task reliability in automated production environments.

In addition to the optimization analysis performed using the planar two-degree-of-freedom manipulator, the scalability study presented in Section 3.6 provides further insight into the applicability of the proposed accessibility metric in realistic industrial robotic systems. The statistical evaluation performed using a six-degree-of-freedom manipulator based on the kinematic configuration of the ABB IRB 120 demonstrates that the κ-index preserves consistent numerical behavior across a significantly larger configuration space. The Monte Carlo sampling analysis shows that configurations located near the boundaries of the reachable workspace tend to exhibit higher κ values, indicating increased proximity to joint limits, whereas configurations located in central workspace regions present lower κ values and therefore more balanced joint utilization. This behavior is consistent with classical kinematic performance observations reported in the robotics literature, where dexterity tends to decrease as manipulators approach their kinematic limits. Furthermore, the observed correlation between the κ-index and Yoshikawa’s manipulability measure suggests that the proposed accessibility metric indirectly promotes configurations with improved kinematic dexterity. These results indicate that the accessibility-based formulation is not restricted to simplified planar systems but can be extended to multi-degree-of-freedom industrial manipulators, supporting its applicability in realistic robotic workstation design scenarios.

Overall, the findings reinforce current research trends that emphasize the integration of kinematic modeling, motion feasibility analysis, and intelligent optimization techniques for robotic system design. By combining geometric constraints, accessibility metrics, and algorithmic optimization within a unified computational framework, the proposed approach contributes to the development of systematic methodologies for intelligent engineering design in industrial robotics.

4.1. Extension to Obstacle-Constrained Environments

While the proposed optimization framework has been validated under unobstructed workspace conditions, real industrial environments typically involve the presence of obstacles such as fixtures, conveyors, safety barriers, and neighboring robotic systems. These elements introduce additional constraints that significantly affect the feasible placement domain.

From an optimization perspective, the inclusion of obstacles transforms the admissible region into a non-convex domain, potentially introducing multiple local minima. In such cases, collision avoidance constraints must be incorporated into the formulation to ensure safe and feasible robot operation.

These constraints can be modeled as additional implicit conditions, defined either through geometric exclusion zones in the workspace or through collision detection functions evaluated along candidate configurations. Furthermore, the integration of configuration-space obstacle representations or sampling-based motion planning techniques would allow the proposed accessibility metric to be combined with trajectory feasibility requirements.

Although obstacle-aware optimization is not explicitly addressed in the present work, the proposed framework is fully compatible with such extensions. The incorporation of collision constraints represents a natural continuation of this research and is expected to further enhance the applicability of the methodology in real industrial workstation design.

In terms of optimization performance, the presence of obstacles and collision constraints introduces additional complexity by reshaping the feasible domain into a non-convex space. Under these conditions, the placement problem may exhibit multiple local minima, which can affect the convergence behavior of deterministic optimization methods.

Nevertheless, the proposed κ-index remains fully applicable, as it is independent of the geometric structure of the feasible domain. Its role as an accessibility-based metric continues to promote configurations that maintain a balanced distance from joint limits, regardless of the presence of environmental constraints.

In such non-convex scenarios, global optimization strategies, such as genetic algorithms or sampling-based methods, become particularly advantageous due to their ability to explore complex search spaces and avoid convergence to suboptimal local solutions. This observation is consistent with the comparative analysis presented in this work, where evolutionary approaches demonstrated robustness in more complex optimization landscapes.

Therefore, while the inclusion of obstacles increases the computational complexity of the problem, it does not compromise the validity of the proposed framework. Instead, it highlights the importance of selecting appropriate optimization strategies when addressing realistic industrial environments.

4.2. Multi-Objective Extension of the Accessibility Framework

While the proposed formulation is presented as a single-objective optimization problem based on the minimization of the κ-index, practical robotic applications often involve multiple and potentially conflicting performance criteria. These may include kinematic dexterity, energy consumption, and collision avoidance, in addition to accessibility considerations.

In this context, the proposed framework can be naturally extended to a multi-objective optimization formulation. Specifically, the placement problem can be expressed as

$$\min \left[k\left(q\right),-w\left(q\right),E\left(q\right),C_{obj}\left(q\right)\right]$$

(34)

where $\kappa \left(q\right)$ represents the accessibility-based metric, $w\left(q\right)$ is a manipulability measure, $E\left(q\right)$ denotes an energy-related cost, and $C_{obs}\left(q\right)$ encodes obstacle avoidance constraints.

Such a formulation enables the application of Pareto-based optimization methods to obtain a set of non-dominated solutions, providing a trade-off between competing objectives. This allows for greater flexibility in the design process, as different placement configurations can be selected based on specific operational priorities and constraints.

This perspective is consistent with recent developments in multi-objective robotic optimization, where Pareto front analysis has been successfully applied to address complex design trade-offs. In particular, recent studies have demonstrated the effectiveness of multi-objective approaches in robotic system design, including the integration of kinematic performance indices and constraint-aware optimization strategies [35].

Although a full multi-objective implementation is beyond the scope of the present work, the proposed accessibility-based formulation provides a suitable foundation for such extensions and highlights its potential for future research in advanced robotic workstation design.

4.3. Incorporation of Dynamic Constraints

While the present work focuses on kinematic accessibility criteria, real robotic systems are also subject to dynamic constraints that influence their performance and feasibility. These include actuator torque limits, energy consumption, and velocity and acceleration bounds.

The dynamic behavior of the manipulator can be described by the standard equation of motion:

$$\mathit{\tau }=\mathbf{M}\left(q\right)\ddot{q}+\mathbf{C}(q,\dot{q})\dot{q}+\mathbf{g}\left(q\right)$$

(35)

where $\mathit{\tau }$ represents the vector of actuator torques, $\mathbf{M}\left(q\right)$ is the inertia matrix, $\mathbf{C}(q,\dot{q})$ accounts for Coriolis and centrifugal effects, and $\mathbf{g}\left(q\right)$ represents gravitational forces.

These dynamic effects can be incorporated into the optimization framework through additional constraints. In particular, actuator limitations can be enforced as

$$\mathit{\tau }(q,\dot{q},\ddot{q})\mid \le {\mathit{\tau }}_{max}$$

(36)

Similarly, joint velocity and acceleration limits can be included as

$$\mid \dot{q}\mid \le {\dot{q}}_{max}, \mid \ddot{q}\mid \le {\ddot{q}}_{max}$$

(37)

In addition, energy-related criteria may be introduced either as constraints or as objective functions, depending on the formulation of the problem. This leads to a kinodynamic optimization framework in which accessibility, efficiency, and feasibility are simultaneously considered.

The κ-index remains a valuable component within this extended formulation, acting as a kinematic regularization term that promotes balanced joint utilization. Therefore, the proposed methodology is fully compatible with the inclusion of dynamic constraints, providing a foundation for more comprehensive robotic system design approaches.

4.4. Computational Complexity and Scalability

The computational complexity of the proposed framework is primarily determined by three components: the evaluation of the κ-index, the computation of inverse kinematics, and the optimization strategy used for base placement.

The κ-index itself is computationally efficient, with linear complexity $O\left(n\right)$, where $n$ is the number of joints. Its evaluation involves simple normalization, averaging, and dispersion calculations, making it suitable for high-dimensional systems.

In contrast, the inverse kinematics computation represents a more significant computational burden, particularly for high-degree-of-freedom manipulators. Numerical methods based on Jacobian inversion typically exhibit computational complexity between $O\left(n^2\right)$ and $O\left(n^3\right)$, depending on the solver implementation.

The inclusion of environmental constraints, such as obstacle avoidance, further increases computational cost due to collision detection operations, which depend on the geometric complexity of the workspace and the number of objects considered.

The optimization process also contributes to the overall computational load. Gradient-based methods generally offer faster convergence in smooth and convex problems, whereas global optimization techniques, such as genetic algorithms, require a larger number of function evaluations but provide improved robustness in non-convex search spaces.

Despite these challenges, the proposed framework remains computationally tractable due to the simplicity of the κ-index and its independence from dynamic modeling. Additionally, the use of parallel computing techniques and sampling-based methods can significantly improve scalability when dealing with high-dimensional configuration spaces.

5. Conclusions

This study addressed the optimal base placement problem of a planar two-degree-of-freedom manipulator using an accessibility-based optimization formulation. The problem was modeled as a constrained nonlinear optimization problem using the κ-index as an objective function to measure the deviation of joint configurations from their ideal mid-range values. Three optimization approaches—constrained programming, gradient projection, and a genetic algorithm—were implemented and compared in terms of solution quality, convergence behavior, number of iterations, and computational cost.

The results demonstrate that all three methods converge to similar optimal solutions when the search space is convex and characterized by a single global minimum, as observed for the analyzed elbow up posture. Among the evaluated methods, constrained programming provided the best computational performance, achieving the lowest objective function value with the shortest execution time. The gradient projection method also exhibited efficient convergence due to its ability to exploit gradient information under active constraints. In contrast, the genetic algorithm required higher computational effort but showed strong robustness in exploring the search space and reaching competitive solutions, suggesting its suitability for more complex or non-convex optimization scenarios.

From a methodological perspective, the use of the κ-index proved effective for maintaining joint configurations within balanced operational regions, thereby avoiding configurations near kinematic limits and improving the reliability of task execution. Furthermore, the incorporation of reachability constraints and aspect membership conditions ensured kinematic feasibility and prevented singular configurations during the optimization process.

In addition to the planar case study, the scalability of the proposed accessibility metric was investigated through an additional analysis involving a six-degree-of-freedom industrial manipulator based on the kinematic configuration of the ABB IRB 120. The statistical evaluation of the κ-index across the configuration space confirmed consistent behavior in higher-dimensional systems, indicating that the proposed metric can effectively characterize joint-centered operation in realistic industrial manipulators.

Overall, the results confirm that integrating geometric modeling, kinematic analysis, and computational optimization provides a consistent framework for the systematic design of robotic workstations. It should be noted that the present framework focuses exclusively on kinematic accessibility and joint-centered performance as criteria for optimal base placement. Practical deployment in industrial environments would additionally require consideration of cycle time, task throughput, and reliability metrics, which depend on scheduling strategies, actuator dynamics, and production constraints beyond the kinematic scope addressed here.

However, this limitation also defines a meaningful direction for future work, particularly in the integration of the proposed framework with scheduling methodologies for human–robot collaborative cells and robotic inspection systems. Such approaches contribute to intelligent engineering methodologies for robotic system deployment in modern industrial environments. Future work will focus on extending the proposed framework to manipulators with higher degrees of freedom, incorporating environmental obstacles that generate non-convex search spaces, and developing multi-objective optimization formulations that simultaneously consider accessibility, manipulability, and energy efficiency.