2. Problem statement

Consider a robotic manipulator that is requested to operate in a 2D (or 3D) environment confined by static obstacles characterized by known arbitrary shape, size, and location, as illustrated in Figure 1. The primary objective for the robot’s end-effector is to traverse a specified set of M task-points within this environment, ensuring each task-point is visited exactly once. In this study, our emphasis lies exclusively on the kinematic aspects of the robotic manipulator, disregarding dynamic considerations. This strategic decision aims to simplify the optimization process and alleviate computational complexity, ensuring practical feasibility for real-time implementation in industrial environments.

Figure 1. A robotic manipulator with $n$ joints in a 2D environment cluttered with obstacles assigned to reach M task-points.

Without accounting for the dynamic properties, the robot’s kinematic model determines its motion, while being restricted by both the obstacles within the work cell and the torque limits of the actuators. Additionally, the optimization of the total cycle time does not take into consideration the time spent by the robot at each task-point for performing operations, since this duration depends on the task process and can be regarded as fixed and known.

Under the above analysis and assumptions, the aim is threefold:

1. A. To determine the end-effector’s feasible tour that minimizes the cycle time.

2. B. To determine an end-effector’s path to ensure safe movement of the manipulator.

3. C. To achieve mechanical advantage at the manipulator’s end-effector during the traversal of the task-points.

The first optimization criterion centers on addressing the TSP within robotic applications, aiming to establish the optimal sequence among task-points by minimizing time rather than distance. To adapt TSP to Robotics, consider a robotic manipulator with n rotational joints in a 2D (or 3D) Cartesian space designated to visit M task-points, each defined by known coordinates. The determination of the optimal cycle time involves accounting for the multiplicity of robot configurations corresponding to each task-point, along with the presence of obstacles within the environment. Consequently, selecting the robot’s configuration depends on both the criterion for optimal cycle time and the characteristics of obstacles, including shape and location, which limit the available configuration choices.

The total cycle time $t_{c}$ required for the manipulator to travel between M designated task-points and return to the initial task-point is expressed as:

(1) \begin{align} t_{c}=\sum _{i=2}^{M\left(1+\gamma \right)}t_{i}+ {\max}_{j}\left(\frac{\left| \boldsymbol{\theta }_{j1}-\boldsymbol{\theta }_{j\left[M\left(1+\gamma \right)\right]}\right| }{\dot{\theta }_{j}}\right) \end{align}

in which the first term $(\sum _{i=2}^{M(1+\gamma )}t_{i})$ expresses the time required by the manipulator for its transition between the successive configurations, and the second term $\left(\frac{\left| \boldsymbol{\theta }_{j1}-\boldsymbol{\theta }_{j\left[M\left(1+\gamma \right)\right]}\right| }{\dot{\theta }_{j}}\right)$ expresses the time required for its transition from the last configuration back to the initial one, where,

• • γ is the number of intermediate configurations along a specific end-effector’s path between two successive task-points [Reference Saha, Roughgarden, Latombe and Sánchez-Ante15] and

• • $t_{i}$ is the time needed by the manipulator to travel from the point $(i-1)$ to the point $i$ is given by,

(2) \begin{align} t_{i}= {\max}_{j}\left(\frac{\left| \boldsymbol{\theta }_{ji}-\boldsymbol{\theta }_{j\left(i-1\right)}\right| }{\dot{\theta }_{j}}\right) \end{align}

where

• • $\boldsymbol{\theta }_{ji}$ is the $j^{th}$ -joint ( $j=1,2,\ldots, n$ ) displacement for the $i^{th}$ ( $i=1,2,\ldots, M$ ) end-effector location and

• • $\dot{\theta }_{j}$ is the average velocity of the $j^{th}$ joint considered as constant.

The second optimization criterion involves identifying a safe path for the robotic manipulator during its transition from one task-point to another. To account for the manipulator’s shape and ensure an obstacle-free trajectory for the end-effector, we designate a set of random points $l_{v}^{j}, v=1,\ldots, N$ and $j=1,\ldots .,n$ on the surface of each manipulator’s link, where $N$ represents the total number of points on each link and $n$ is the total number of the manipulator’s links.

Following the methodology presented in ref. [Reference Xidias25], each point $l_{v}^{j},$ traces a path in robot’s workspace $\mathcal{W}\equiv \mathbb{R}^{2}$ or $\mathbb{R}^{3}$ . A collision-free path $l_{v}^{j},(s)$ should be searched in the “flat” areas of the Bump-Surface. The image $S(l_{v}^{j} (s))$ that lies on the bumps of $S$ yields a path $l_{v}^{j}(s)$ that avoids penetrating the obstacles in $\mathcal{W}$ . Thus, the goal for collision avoidance entails a path $l_{v}^{j}(s)$ on $S$ with minimal flatness H, which is described by,

(3) \begin{align} H=\max _{\boldsymbol{\theta }_{\textit{total}}} \left(S_{w}\left(l_{1}^{1}\left(s\right)\right),\ldots, S_{w}\left(l_{N}^{1}\left(s\right)\right),\ldots, S_{w}\left(l_{N}^{n}\left(s\right)\right)\right) \end{align}

with respect to the joint variables $\boldsymbol{\theta }_{\textit{total}}$ , where,

• • if $\mathcal{W}\equiv \mathbb{R}^{2}$ , then $\boldsymbol{\theta }_{\textit{total}}=(\boldsymbol{\theta }_{0},\ldots, \boldsymbol{\theta }_{m},\ldots, \boldsymbol{\theta }_{M+(M-1)\gamma })$ and $\boldsymbol{\theta }_{m}\boldsymbol{\epsilon }\mathbb{R}^{n}$ for a planar manipulator with $n$ links and,

• • if $\mathcal{W}\equiv \mathbb{R}^{3}$ , then $\boldsymbol{\theta }_{\textit{total}}=(\theta _{0},\theta _{1},\ldots, \theta _{m},\ldots, \theta _{M+(M-1)\gamma })$ and $\theta _{m}\epsilon \mathbb{R}^{6}$ for a Programmable Universal Manipulation Arm (PUMA) manipulator,

The third optimization criterion relates to MMA. MMA encompasses information related to the robot’s ability to exert force at the end-effector in a specific direction, providing a quantification of the robot’s force efficiency along that particular vector. MMA relies on both the configuration and the end-effector force vector. A higher MMA value indicates a relatively high end-effector force in comparison to the joint torques. Conversely, a lower MMA value corresponds to a requirement for relatively high joint torques to achieve high end-effector force. Thus, the minimum MMA represents the most unfavorable force/torque ratio along the path. Consequently, by maximizing the worst force/torque ratio, MMA is consistently maintained at a high level along the path.

The expression for the minimum MMA value can be articulated as follows:

(4) \begin{align} {\textit{min}}(MMA)={\textit{min}} \{(r_{m})_{i}\},\ i=1,\ldots, M \end{align}

and

(5) \begin{align} r_{m}=\sqrt{\frac{ {\boldsymbol{f}} _{m}^{T} {\boldsymbol{f}}_{m}}{\boldsymbol{\tau }_{m}^{T}\boldsymbol{\tau }_{m}}} \end{align}

where,

• • ${\boldsymbol{f}}_{m}$ represents the weighted vectors of end-effector force and

• • $\boldsymbol{\tau }_{m}$ represents joint torque.

The objective of this work is threefold: (a) the minimization of the overall cycle time $t_{c}$ (Eq.(2)), (b) the minimization the flatness $H$ (Eq.(3)), and (c) the maximization of the $min (MMA)$ (Eq.(4)). Given the conflicting nature of these three objectives, it is imperative to appropriately combine the objective functions to address the overall problem.

For non-redundant robots, where the possible configurations at specific points can be determined by the inverse kinematics, the objective function is expressed as a ratio of these three objectives. For redundant robots, where direct kinematics is applied to determine potential (infinite) configurations at the points, an additional parameter to account for the position error $(D)$ at the end-effector should be considered. Therefore, the objective function is formulated as:

(6) \begin{align} \min _{\theta } \left(E \right)=\begin{cases} \min _{\theta }\left(\frac{t_{c}\left(\boldsymbol{\theta }_{\textit{total}}\right)e^{H\left(\boldsymbol{\theta }_{\textit{total}}\right)}}{MMA}\right)\!,\ for\ non-\textit{redundant}\ \textit{manipulators}\\ \min _{\theta }\left(\frac{t_{c}\left(\boldsymbol{\theta }_{\textit{total}}\right)e^{H\left(\boldsymbol{\theta }_{\textit{total}}\right)}e^{D\left({\boldsymbol{\theta }_{\textit{total}}}\right)}}{MMA}\right)\!,\ for\ \textit{redundant}\ \textit{manipulators} \end{cases} \end{align}

where $D=\| {\boldsymbol{p}}_{c}-{\boldsymbol{p}}_{d}\|$ is characterized as the position error between the current position ${\boldsymbol{p}}_{c}$ of the manipulator’s end effector and the desired position ${\boldsymbol{p}}_{d}$ ( ${\boldsymbol{p}}_{c}, {\boldsymbol{p}}_{d}\in [0,1]^{2}$ ). Additionally, $D(\boldsymbol{\theta }_{\textit{total}})$ represents the minimum distance between the manipulator’s links and the obstacles.

3. Preliminaries

3.1. Theoretical background of manipulator mechanical advantage

Dubey and Luh [Reference Dubey and Luh6] introduced the MMA as a performance measure, representing the robot’s capability to exert force and moment at the end-effector. MMA is visualized through an ellipsoid termed as MMA-ellipsoid (MMAE), serving as an indicator of the robot’s capability to exert forces at various points within its workspace. A higher mechanical advantage, indicative of enhanced performance in applying forces and moments, is achieved in the direction of force and moment. Enhancing MMA in the direction of force and moment exerted by the robot at the end-effector leads to reduced joint force requirements. Enhancing the mechanical advantage is particularly recommended for applications like grinding and deburring, where increased capability for force exertion at the end-effector is crucial.

MMA delineates the relationship between the end-effector’s force and the joint torques contingent upon the robot’s configuration and the direction of the end-effector force vector. A high MMA value implies a relatively high end-effector’s force compared to the joint torques, while a low MMA value necessitates relatively high joint torques to succeed a high end-effector’s force. In other words, the minimum MMA signifies the worst ratio and by maximizing the worst ratio, the MMA is consistently maintained at a high level across all the visited task-points.

According to Dubey and Luh [Reference Dubey and Luh6], MMA is defined as the ratio of the norm of the end-effector force vector to the norm of joint torque vector, and it is denoted as $r_{m}$ . Thus,

(7) \begin{align} r_{m}=\sqrt{\frac{f_{m}^{T}f_{m}}{\tau _{m}^{T}\tau _{m}}} \end{align}

where $f_{m}$ and $\tau _{m}$ are the weighted vectors of end-effector force and joint torque, respectively.

Let $\tau$ , $f$ , and $\mathrm{J}$ be the end-effector force vector, the joint torque vector and the robot’s Jacobian matrix, respectively, the transformations $J_{m}=W_{f}^{-1/2}JW_{\tau }^{1/2}$ , $f_{m}=W_{f}^{1/2}f$ , and $\tau _{m}=W_{\tau }^{1/2}\tau$ are used, where $W_{f}\in R^{m\times m}$ and $W_{\tau }\in R^{n\times n}$ are symmetric and positive definite weighting matrices. In most instances, these will be taken as diagonal matrices for simplicity.

The joint torque vector is given by

(8) \begin{align} \tau _{m}=J_{m}^{T}f_{m} \end{align}

Substituting (2) into (1), we get

(9) \begin{align} r_{m}=\sqrt{\frac{f_{m}^{T}f_{m}}{\left(f_{m}^{T}J_{m}J_{m}^{T}f_{m}\right)}} \end{align}

If the transformed vector $f_{m}$ belongs to the null space of $J_{m}^{T}$ , the torques generated at the joints reduce to zero and r_m tends to infinity. If $\boldsymbol{u}_{\boldsymbol{m}}$ is defined as a unit vector oriented in the direction of $\boldsymbol{f}_{\boldsymbol{m}}$ , then $\boldsymbol{r}_{\boldsymbol{m}}$ can be reformulated as:

(10) \begin{align} \boldsymbol{r}_{\boldsymbol{m}}=\mathbf{1}/\boldsymbol{u}_{\boldsymbol{m}}^{\boldsymbol{T}} \boldsymbol{J}_{\boldsymbol{m} }\boldsymbol{J}_{\boldsymbol{m}}^{\boldsymbol{T}} \boldsymbol{u}_{\boldsymbol{m}} \end{align}

where

(11) \begin{align} \boldsymbol{u}_{\boldsymbol{m}}=\boldsymbol{f}_{\boldsymbol{m}}/\boldsymbol{f}_{\boldsymbol{m}}^{\boldsymbol{T}} \boldsymbol{f}_{\boldsymbol{m}} \end{align}

The singular value decomposition of $J_{m}$ can be written as:

(12) \begin{align} J_{m}=\boldsymbol{S}\boldsymbol{\Sigma }_{\boldsymbol{m}}\boldsymbol{T}^{\mathrm{T}} \end{align}

where $\boldsymbol{S}\in \mathbb{R}^{\boldsymbol{mxm}}$ and $\boldsymbol{T}\in \mathbb{R}^{\boldsymbol{nxn}}$ are orthogonal matrices, and $\boldsymbol{\Sigma }_{\boldsymbol{m}}\in \mathbb{R}^{\boldsymbol{mxn}}$ , $m\lt n$ , is a diagonal matrix of the following form:

(13) \begin{align} {\unicode[Arial]{x03A3}} _{m}=\left[\begin{array}{lllllll} \xi _{1} & & & & & | & \\ & \xi _{2} & & 0 & & | & \\ & & . & & & | & 0\\ & 0 & & . & & | & \\ & & & & \xi _{m} & | & \end{array}\right] \end{align}

where ξ ₁ ≥ ξ ₂ ≥ ^… ≥ ξ _m ≥ 0. The non-zero elements of Σ _m represent the singular values of J _m and their number is equal to the rank of J _m . The manipulator mechanical advantage r_m can be demonstrated to fall within the following range [Reference Nakamura and Hanafusa26]:

(14) \begin{align} \frac{1}{\xi _{1}}\leq r_{m}\leq \frac{1}{\xi _{m}} \end{align}

The actual value of r _m is influenced not only by the robot configuration and the weighting matrices W _τ και W _f but also by the direction of the force applied by the robot end-effector. The end-points of the vectors ${\boldsymbol{r}}_{\boldsymbol{m}} {\boldsymbol{u}}_{\boldsymbol{m}}$ for a given robot configuration can be depicted as a surface (or curve for m = 2). This surface can be accurately represented by an ellipsoid with principal axes (1/ξ ₁ ) $\boldsymbol{s}_{\mathbf{1}}$ , (1/ξ ₂ ) $\boldsymbol{s}_{\mathbf{2}}$ , _… , (1/ξ _m ) $\boldsymbol{s}_{\boldsymbol{m}}$ , where $\boldsymbol{s}_{\boldsymbol{i}}$ ∈R ^m denotes the i ^th column vector of S in (12). This ellipsoid is denoted as the MMAE and both its shape and volume are contingent upon the robot configuration. In the case that one or more ξ_i are equal to zero, the manipulator mechanical advantage in the direction(s) of $\boldsymbol{s}_{\boldsymbol{i}}$ vector(s) becomes infinite. If the external force vector applied at the end-effector aligns with the direction(s) of vector(s) $\boldsymbol{s}_{\boldsymbol{i}}$ corresponding to zeros ξ_i, then the resulting torques at the joints will be null.

A redundant robot provides the capability to derive diverse MMAEs for the same end-effector position through self-motion. The overall ability of a redundant robot to exert force at the end-effector in a desired direction can be enhanced by reshaping and reorienting MMAE.

As a general remark, the mechanical advantage becomes essential for performance, especially in cluttered environments, where obstacles may impede the motion of robotic manipulators. This performance measure determines the manipulator’s ability to exert force effectively. A higher mechanical advantage enables the manipulator to move in cluttered environments by optimizing motion to avoid obstacles and achieve task objectives. The consideration of the manipulability index is essential for assessing the manipulator’s performance in cluttered environments and optimizing its capabilities to perform tasks efficiently.

3.2. An introduction to the bump-surface concept

The Bump-Surface concept proves to be a powerful tool for representing the robot’s workspace utilizing a unified mathematical entity [Reference Azariadis and Aspragathos27]. As a global method, this concept characterizes the robot’s workspace by employing a B-Spline surface incorporated within a higher dimensional space. The Bump-Surface concept operates under the assumption that the environment of robot is confined within a region denoted as $[0,1]^{K}, K=2,3$ , which represents the normalized space. This space is discretized into evenly spaced subintervals along its $n$ orthogonal directions, thereby creating a grid of points ${\boldsymbol{p}}\epsilon [0,1]^{K+1}, K=2,3$ . The $K+1$ coordinate of each grid point ${\boldsymbol{p}}\epsilon (0,1],$ if ${\boldsymbol{p}}$ lies within an obstacle, and it is equal to 0, otherwise. For example, in a 2D environment $(K=2)$ , a 2D B-Spline surface is depicted within 3D Euclidean space, as illustrated in Figure 2.

Figure 2. (a) A 2D environment with two rectangular obstacles. (b) The corresponding Bump-Surface.

A collision-free path needs to be identified within the “flat” regions. Choosing a path that “ascends” the bumps of the surface leads to a path that traverses the obstacles in the initial robot’s workspace (Figure 3).

Figure 3. (a) An example two paths on the Bump-Surface, the path with red color “climbs” the bump while the curve with yellow color is collision free. (b) Another point of view.

Figure 4. A feasible chromosome for a non-redundant manipulator indicating that the robot first visits the task-point three with a configuration denoted by the alleles (1 0 1), then visits the task-point one with a configuration denoted by the alleles (111) etc. The second segment of the chromosome delineates the manipulator’s configuration at the task-points, where each three-digit allele of (000, 001, 010, …, 111) designates one out of eight configurations of the manipulator.

Figure 5. A feasible chromosome for a redundant manipulator.

Figure 6. (a) Robot’s configurations at the task-points. (b) The manipulator’s end-effector trace while moving in 2D environment.

Figure 7. (a) Minimal distance between the robot’s end-effector and the nearest obstacle during its motion from task-point to task-point. (b) Normalized manipulability index during task.

Figure 8. (a) A representative example of manipulator’s workspace. (b) Another point of view.

Figure 9. (a) The proposed solution. (b)-(f) Some points of view.

Figure 10. (a) Minimal distance between the robot’s end-effector and the nearest obstacle during its motion. (b) Normalized manipulability index during task.

Figure 11. The CPU time versus the number of task-points for the 1st scenario.

Figure 12. The CPU time versus the number of task-points for the 2nd scenario.

4. The optimization search algorithm

Multi-goal motion planning poses a challenge as an NP-complete optimization problem. GAs [Reference Holland28, Reference Michalewitz29] emerge as a robust optimization tool for addressing problems in high-dimensional search spaces when conventional techniques prove too intricate or time-consuming. The essential components of GAs are elucidated in the subsequent sections.

The representation mechanism: The choice of an encoding mechanism is crucial, with integer encoding proving especially advantageous in cases where binary encoding would require specialized repair algorithms. In this regard, integer encoding aligns well with the TSP. Conversely, binary or floating encoding is more suitable for handling the multiplicity of robot configurations at task-points depending on whether the robot exhibits redundancy or not. As a result, a mixed encoding scheme is preferable. This proposed representation scheme reduces the required number of genes to encompass robot configurations at the task-points. As a result, the proposed chromosome is comprised of three segments:

Segment # 1: utilizes integer encoding to represent the sequence of the robot’s movement between the task-points (integer encoding).

Segment # 2: represents the configurations corresponding to the task-points specified in Segment#1 (binary or floating encoding for non-redundant and redundant robots, respectively).

Segment # 3: expresses the intermediate configurations between successive task-points, as outlined in Segment#1 (floating encoding).

When considering the two distinct cases separately (non-redundant and redundant robots), it becomes apparent that a distinct representation scheme is required to accommodate the different characteristics of each robot type. This distinction is necessary since the possible configurations at each task-point for non-redundant robots are known and finite, whereas for redundant robots, the possible configurations at each task-point are infinite. For the case of non-redundant robots, the binary encoding is employed to express the finite configurations corresponding to the points, while for the case of redundant robots, the floating encoding is used to express the joint angles constituting the configurations at the points. It should also be noted that the manipulator transitions between successive points on a path are determined by linear interpolation between two successive configurations.

For example, let a three-DOF non-redundant manipulator operating in the 3D space that is assigned to visit four task-points through a path defined by $\gamma =2$ intermediate points between successive task-points. Assuming that the manipulator can reach each task-point with eight different configurations, a feasible chromosome can be formulated as follows (Figure 4):

Now, let a redundant manipulator with three rotational joints, that is requested to reach $M=4$ task-points in a 2D workspace through a path defined by γ $=2$ intermediate points. A plausible chromosome can be formulated accordingly (Figure 5):

where TP1, TP2, TP3, and TP4 are the task-points 1,2,3, and 4, respectively, and IC is the intermediate configuration corresponding to the successive task-points.

The evaluation mechanism assesses the quality (i.e. the fitness) of each chromosome within the population and is given by:

(15) \begin{align} \textit{fitness}=\frac{1}{E} \end{align}

Genetic operators play a crucial role in the proposed GA. The reproduction process employs the roulette wheel scheme, where chromosomes are chosen for replication with probabilities proportional to their fitness. Following reproduction, crossover serves as a recombination operator. In this work, the order crossover is employed for the integer encoding, while a one-point crossover is applied to both the binary and floating encoding. Mutation is introduced to infuse new genetic material into the population, preventing premature convergence to local minima. For the integer encoding, the inversion mutation operator is utilized. For the binary encoding, the mutation operator involves altering a randomly selected gene with a digital value of “0” to “1” and vice versa. For the floating encoding, the mutation operator is implemented to change a random gene to another lying within the search space.

5. Simulation results

To evaluate the performance of the proposed approach, two scenarios have been carried out. The first scenario, which considers a redundant robot operating in a 2D environment, and the second scenario, which considers a non-redundant robot operating in a 3D environment, are indicatively presented.

In the first scenario , we consider that a three-dof manipulator is requested to visit a set of task-points in an environment that is cluttered with convex and non-convex polygonal obstacles. In this scenario, a three-dof manipulator with rotational joints is requested to approach 10 task-points. The robot’s environment contains four polygonal obstacles and the task-points are located on vertices and edges of the obstacles. Figure 6(a) presents the robot’s configurations at the task-points and the optimum sequence of the task-points provided from the proposed approach is as follows: 7 – 6 – 5 – 1 – 3 – 4 – 9 – 2 – 8 – 10. Figure 6(b) presents the end-effector’s trace while the robot is moving in the 2D environment between the task-points (the black triangle shows the robot’s base). The minimum distance between the robot’s end-effector and the obstacles while moving in the 2D environment is presented in Figure 7(a). Figure 7(b) presents the performance of the manipulability index, while the manipulator’s end-effector follows the proposed trajectory. It is noticed that the manipulability index takes small values while the manipulator is moving close to the obstacles.

In a second scenario , we assume that a PUMA manipulator is requested to accomplish a similar task. In both simulations, the proposed approach provides a trajectory for the manipulator’s end-effector while simultaneously the manipulability index is high even avoiding an obstacle. In this experiment, the robotic arm (Puma 762) is required to perform its task in a 3D workspace. The manipulator is requested to visit seven task-points in an environment that is cluttered with four convex and non-convex polyhedral obstacles. An illustrative figure of the manipulator’s workspace is shown in Figure 8. Here, the obstacles are represented by red and dark green, and the gray area shows the operation space. The task-points are represented by orange points and are placed on some of the vertices of the obstacles. The proposed solution is presented in Figure 9(a)-(f) shown from different points of view, where the manipulator’s end-effector is requested to visit the task-points in the following order: 1 – 2 – 5 – 3 – 4 – 7 – 6. In Figure 9, the blue dots represent the end-effector’s trace.

To demonstrate the efficacy of the proposed methodology, the calculation of the minimum distance between the robot’s end-effector and obstacles during movement in the 3D environment is conducted and presented in Figure (a). It is noticed from Figure 10(a) that the distance takes the value 0 only on the task-points and takes values other than zero when the robot is moving between the task-points. Figure 10(b) presents the performance of the manipulability index, while the manipulator’s end-effector follows the proposed trajectory. It is observed that the manipulability index exhibits small values when the manipulator is in close proximity to obstacles.

A noteworthy insight into problem complexity arises from the variability in calculation time concerning the number of task-points. Numerous experiments were executed, utilizing a three-DOF manipulator navigating a 2D environment with four obstacles. The task assigned was to reach a specified number of task-points while ensuring the minimum cycle time and avoiding obstacles. To evaluate how the increasing number of task-points affects computational time, a series of experiments were conducted involving a Puma manipulator working in a 3D environment with four obstacles and assigned to reach a specific number of task-points. Figure 11 and Figure 12 illustrate the relationship between computational time and the number of task-points, suggesting that central processing unit (CPU) time experiences an almost linear increase with the growing number of task-points.