A modified firefly algorithm for the inverse kinematics solutions of robotic manipulators

Abstract

The inverse kinematics of robotic manipulators consists of finding a joint configuration to reach a desired end-effector pose. Since inverse kinematics is a complex non-linear problem with redundant solutions, sophisticated optimization techniques are often required to solve this problem; a possible solution can be found in metaheuristic algorithms. In this work, a modified version of the firefly algorithm for multimodal optimization is proposed to solve the inverse kinematics. This modified version can provide multiple joint configurations leading to the same end-effector pose, improving the classic firefly algorithm performance. Moreover, the proposed approach avoids singularities because it does not require any Jacobian matrix inversion, which is the main problem of conventional approaches. The proposed approach can be implemented in robotic manipulators composed of revolute or prismatic joints of $n$ degrees of freedom considering joint limits constrains. Simulations with different robotic manipulators show the accuracy and robustness of the proposed approach. Additionally, non-parametric statistical tests are included to show that the proposed method has a statistically significant improvement over other multimodal optimization algorithms. Finally, real-time experiments on five degrees of freedom robotic manipulator illustrate the applicability of this approach.

Keywords

Denavit-Hartenberg Inverse kinematics Metaheuristics algorithms Multimodal optimization

1. Introduction

Many robotics applications such as visual guided tasks, object grasping, trajectory tracking, and control problems require the solution of the inverse kinematics of robot manipulators. Robot manipulators can have redundant joint configurations leading to the same end-effector pose [1]. Moreover, the inverse kinematics is a nonlinear problem, and due to its redundancy solution, this problem is a difficult task to solve [2]. Multiple joint configurations can be used to overcome issues such as robotic manipulators working in environments with the presence of obstacles, joint solutions of impossible execution due to physical restrictions, or joint configurations with abrupt changes during a trajectory planning task. For these reasons, it is crucial to find a set of joint configurations that reach the same end-effector pose to choose a suitable solution for the inverse kinematics task.

Conventional closed-form approaches such as the geometric and algebraic methods can be used to solve the inverse kinematics [3]. These approaches provide exact solutions. Moreover, some kinematics equations can provide multiple joint solutions for the same end-effector pose. However, as the total number of degrees of freedom (DOF) of the manipulator increase, the closed-form method becomes difficult to implement. Consequently, closed-form approaches are difficult to generalize. Other conventional approaches based on differential kinematics use a Jacobian matrix to solve the inverse kinematics [4]. These methods are commonly used because they provide remarkable approximations, and they can be generalized to different manipulator structures. The main inconvenience with these approaches is the singularities. These methods require the inversion of the Jacobian matrix, which depends on the actual joint configuration. However, some joint configurations provoke the Jacobian matrix to become rank deficient. This inconvenience produces a high demand for joint velocities of inadmissible execution. Additionally, these approaches only provide a single solution for the inverse kinematics. Due to the complexity and nonlinearities [5], metaheuristics algorithms should be considered to solve this problem [6].

Metaheuristics algorithms have been widely used to solve optimization problems in different researches areas [7]. In the field of robotics, researches implement these approaches to solve other robotic applications such as mobile robot control [8, 9], mobile robot navigation [10, 11], control of parallel robot platform [12], online optimization of manipulator trajectories [13], online tuning of a PID for mobile robots [14], training of a neural network to obtain the kinematic modeling of robot manipulators [15], and others [16]. These metaheuristics are often required to deal with problems that are generally difficult to solve.

The use of metaheuristics algorithms overcomes the inconvenience of singularities. Several authors have commonly implemented particle swarm optimization (PSO) to solve the inverse kinematics for different manipulator structures [2, 17, 18, 19]. The Genetic Algorithms (GA) and their modifications have also been implemented [20, 21, 22, 23]. Moreover, comparative studies have been realized to compare the performance of different metaheuristics such as Differential Evolution (DE), Gravitational Search Algorithm (GSA), Artificial Bee Colony (ABC), Firefly Algorithm (FA), Cuckoo Search (CS), and others [24, 25, 26]. The FA algorithm has been applied to solve the inverse kinematics for redundant manipulators [27, 28]. Other metaheuristic approaches have proved to be superior to other compared metaheuristics [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. However, these approaches are used for global optimization.

In multimodal optimization, we can simultaneously find many global and local optima for a given optimization problem [40]. There are several metaheuristics algorithms to search for multiple optima. Adaptations of DE and GA have been implemented based on the crowding methods [41, 42]. A modification to DE has been developed for multimodal optimization based on the fitness sharing scheme [41, 43]. There exists a version of PSO adapted to multimodal optimization based on speciation strategy [44]. Another strategy has been implemented in Flower Pollination Algorithm (FPA) and CS to provide multimodality. This strategy consists of three fundamental parts: a memory mechanism, an acceleration process for detecting potential local minima, and a depuration procedure to eliminate similar solutions. The modified algorithm is called MFPA [45] and MCS [46]. They have been compared against crowding, fitness sharing and speciation methods, where they proved to be superior regarding more consistent optima solutions. Other approaches based on niching methods such as real-coded GA [47, 5] have been implemented to solve inverse kinematics tasks. However, these approaches solved the required position of the end-effector without considering its orientation. The applicability of these methods is limited to manipulator structures with revolute joints and simulations without considering real joint limits.

Previously, we used metaheuristics algorithms to solve the inverse kinematics for robot manipulators [48], mobile manipulators [49], and dual-arm robotic systems [50]. These approaches provide single joint solutions since they solve the inverse kinematics as a global optimization problem. In this work, we propose to use a modified version of FA to provide multiples solutions simultaneously. The proposed modified version improves the FA multimodal capability without losing the balance between exploration and exploitation, where good communication among fireflies is maintained. The FA algorithm has been designed with multimodal capacities [51], an advantage over the revised multimodal algorithms. FA has an update mechanism that already deals with multimodality, while the revised methods were adapted to deal with multimodal optimization. Moreover, FA has been used to solve the inverse kinematics [26, 52]. However, these approaches provide only an optimal global solution. Similarly, there exist several modifications to FA [53, 54, 55], but these methods focus on improving its single global solution instead of improving its multimodality capability.

The contributions of this paper are given below. The proposed approach solves the inverse kinematics as a multimodal constrained optimization problem. Thanks to the modified FA version, the proposed method can provide multiple joints configurations under joint limits constrains, which reach the same end-effector pose. Moreover, the proposal is based on the Denavit-Hartenberg (DH) convention because the forward kinematics always have a solution. The proposed method does not require any inversion of a Jacobian matrix; for this reason, it does not suffer from singular configurations. Additionally, this method can be implemented under robotic manipulators with $n$ DOF composed of revolute and prismatic joints.

This paper is organized as follows: The next Section will introduce the basic concepts about the forward and inverse kinematics for robotic manipulators. In Section 3, the description of FA and its modification are given. Then, we give a detailed description of the proposed approach in Section 4. In Section 5, simulations, comparisons, non-parametric statistical tests, and real experiments are presented. Finally, we give conclusions in Section 6.

2. Robotic manipulator kinematics

Robotic manipulators are composed of a series of links connected by joints, which form an open kinematic chain [1]. The last element in the kinematic chain is called end-effector. Each joint is associated with an articulation, and they define a joint configuration $\vec{q}=[q_{1}\ q_{2}\ q_{3}\ \ldots\ q_{n}]^{T}$ , where $n$ defines the total degrees of freedom (DOF) of the manipulator, see Fig. 1.

Figure 1.

Kinematic chain of a serial robotic manipulator. ${}^{j-1}\mathbf{T}_{j}\left(q_{j}\right)$ is a homogeneous matrix that represents the coordinate frame of the link $j$ .

There are two types of articulations, the revolute and prismatic joints. A joint variable $q_{j}$ can be defined as

$\displaystyle q_{j}=\left\{\begin{array}[]{cl}\theta_{j}&\textbf{if}\ \textit{% joint j is a revolute}\\ d_{j}&\textbf{if}\ \textit{ joint j is a prismatic}\end{array}\right.$ (1)

where $\theta_{j}$ is a rotation angle and $d_{j}$ is a displacement.

Forward kinematics consists of computing the end-effector pose given a joint configuration. Forward kinematics is a straight forward problem that always has a solution. Forward kinematics can be expressed as follows

$\displaystyle{}^{0}\mathbf{T}_{n}(\vec{q})={}^{0}\mathbf{T}_{1}(q_{1})\>^{1}% \mathbf{T}_{2}(q_{2})\>\>\ldots\>\>^{n-1}\mathbf{T}_{n}(q_{n})$ (2) $\displaystyle{}^{0}\mathbf{T}_{n}(\vec{q})=\prod_{j=1}^{n}\;{}^{j-1}\mathbf{T}% _{j}(q_{j})$

where each link $j$ represents a homogeneous matrix ${}^{j-1}\mathbf{T}_{j}\left(q_{j}\right)$ , which transforms the coordinate frame attached to the link $j-1$ into the coordinate frame $j$ . The homogeneous matrix ${}^{0}\mathbf{T}_{n}(\vec{q})$ represents the pose of the end-effector.

Based on the Denavit-Hartenberg (DH) model, we can express the homogeneous matrix ${}^{j-1}\mathbf{T}_{j}$ as

${}^{j-1}\mathbf{T}_{j}=\left[\begin{array}[]{cccc}c\theta_{j}&-s\theta_{j}c% \alpha_{j}&s\theta_{j}s\alpha_{j}&a_{j}c\theta_{j}\\ s\theta_{j}&c\theta_{j}c\alpha_{j}&-c\theta_{j}s\alpha_{j}&a_{j}s\theta_{j}\\ 0&s\alpha_{j}&c\alpha_{j}&d_{j}\\ 0&0&0&1\end{array}\right]$ (3)

where the $\sin$ and $\cos$ operations are abbreviated as $s$ and $c$ , respectively. The parameter $\theta_{j}$ represents a joint angle, $a_{j}$ is a link length, $d_{j}$ is a link offset, and $\alpha_{j}$ is a link twist. For a revolute joint, the parameter $\theta_{j}$ becomes a joint variable. On the other hand, $d_{j}$ becomes a joint variable for prismatic joints.

The homogeneous matrix ${}^{j-1}\mathbf{T}_{j}$ can also be expressed as

${}^{j-1}\mathbf{T}_{j}=\left[\begin{array}[]{cccc}n_{x}&o_{x}&a_{x}&p_{x}\\ n_{y}&o_{y}&a_{y}&p_{y}\\ n_{z}&o_{z}&a_{z}&p_{z}\\ 0&0&0&1\end{array}\right]=\left[\begin{array}[]{cc}\mathbf{R}&\vec{p}\\ \vec{0}&1\end{array}\right]$ (4)

where $\vec{R}$ represents a rotation matrix of the frame $j$ , and the vector $\vec{p}$ represents its cartesian position.

Inverse kinematics consists of computing the required joint configuration to reach the given end-effector pose. Inverse kinematics is not a straight forward problem, and it may have multiple solutions. The geometric and algebraic approaches produce the exact solution for the inverse kinematics problem, and they can provide multiple joint solutions. However, these methods become complicated to implement in complex kinematics structures with high DOF, and there is not always a closed-form solution. Numerical methods produce significant approximations for the inverse kinematics. The main disadvantage in these approaches is the singularities. They require the inversion of a Jacobian matrix that can become rank deficient. Additionally, these methods provide a single joint solution.

The inverse kinematics can be solved by minimizing an error between the desired and the actual end-effector positions. Multiple joint configurations may lead to the same end-effector position since the inverse kinematics is a nonlinear problem. Figure 2 illustrates the multimodal function given by an error function of a 2 DOF planar manipulator. This manipulator is composed of two links with $0.5$ meters, and the desired end-effector position is $x_{d}=0.5$ and $y_{d}=0.5$ meters. The error function is $\sqrt{\left(x_{d}-x\right)^{2}+\left(y_{d}-y\right)^{2}}$ , where $x$ and $y$ are the actual positions at joint configuration $\left(\theta_{1},\theta_{2}\right)$ . As can be seen, two global minima represent different joint configurations to reach the same desired position. A solution is given by $\theta_{1}=0$ and $\theta_{2}=1.5708$ radians. The other solution is $\theta_{1}=1.5708$ and $\theta_{2}=-1.5708$ radians. Multimodal algorithms provide these two solutions, simultaneously. In contrast, global optimization algorithms only provide one of these solutions.

Figure 2.

The multimodal function of the inverse kinematics problem of a 2 DOF planar manipulator.

In this work, a modified version of FA is proposed to deal with multiples joint configuration. Moreover, the proposed approach solves the inconveniences of conventional methods.

A detailed description of the forward and inverse kinematics can be found in [1, 3, 4].

3. Firefly algorithm

The Firefly Algorithm (FA) is a metaheuristic method inspired by the attraction behavior among fireflies [51]. FA is conformed by a population of fireflies, where every firefly is associated with a brightness intensity. The landscape of an objective function produces this bright intensity. If a firefly is located near a local or global optimum, its bright intensity becomes higher. Those fireflies with high brightness intensity are more attractive. In other words, a firefly is attracted to another that is brighter than itself. Moreover, the attraction behavior exponentially decreases by the distance among fireflies.

In FA, a firefly $i$ represents a candidate solution $\vec{x}_{i}\in\mathbb{R}^{D}$ for a given problem with dimensionality $D$ , where $i=1,2,3,\ldots,N$ and $N$ is the total number of population members. If a firefly $i$ is attracted by another firefly $j$ more attractive, then the position $\vec{x}_{i}$ can be adjusted toward the position $\vec{x}_{j}$ as follows

$\displaystyle\vec{x}_{i}:=\vec{x}_{i}+\beta_{0}e^{\gamma_{i}r_{ij}^{2}}\left(% \vec{x}_{j}-\vec{x}_{i}\right)+\alpha\left(\vec{r}-{\displaystyle\frac{1}{2}}\right)$ (5)

where $\beta_{0}$ is an initial attraction factor, $\gamma$ is an absorption coefficient, and $\alpha$ is a randomness parameter. These elements are the particular parameter setting of FA, and the user must provide them. The vector $\vec{r}\in\mathbb{R}^{D}$ is a random component with values between $[0,1]$ , which are pondered by $\alpha$ to control the exploration. The term $\beta_{0}e^{\gamma_{i}r_{ij}^{2}}$ defines the attraction behavior, which exponentially decreases by the distance $r_{ij}$ and $\beta_{0}$ controls the exploitation.

When $\gamma_{i}r_{ij}^{2}\rightarrow 0$ fireflies are more attracted. On the other hand, as $\gamma_{i}r_{ij}^{2}\rightarrow\infty$ fireflies are not attracted at all. Based on this behavior, a firefly is more attracted by those fireflies that are closer to its current position. For this reason, the fireflies that are closer converge to the closest local or global optima. This behavior is important because it provides the FA the capability of multimodality.

The distance among two fireflies $i$ and $j$ is defined as

$\displaystyle r_{ij}=\sqrt{\sum_{k=1}^{D}\left(x_{ik}-x_{jk}\right)^{2}}$ (6)

Another important characteristic in FA is the reduction of the parameter $\alpha$ . This parameter setting is reduced in every generation as follows

$\displaystyle\alpha:=\delta\alpha$ (7)

where $\delta$ is a reduction parameter that must be defined by the user as well. The reduction parameter provides the FA a higher exploration at earlier generations.

For further information about FA, please go to references [51, 56].

3.1 Modified firefly algorithm

Classical FA has a good balance between exploration and exploitation controlled by the parameters $\alpha$ and $\beta_{0}$ , respectively. FA also has great behavior concerning communication among fireflies. Those fireflies located into some optima produce a higher bright intensity. Then, they provide the information on these locations to the other fireflies. However, the feedback characteristic in FA has poor performance. A firefly located in a global optimum attracts other fireflies, even if they are located in local optima. In consequence, the location of local optima may be lost after some iterations. We proposed a mechanism to avoid loosening the location of local or global optima, which is a weakness of FA. This mechanism improves the feedback characteristic of FA.

There are different strategies to provide a good performance about feedback behavior in metaheuristic. In PSO, every particle knows its current position but also knows its best position acquired so far. PSO uses this information to avoid losing the quality of the position in the swarm population. Other evolutionary computing algorithms, such as DE and Evolution Strategies (ES) [57], performed a selection operation to provide feedback characteristics. This selection operation consists of comparing offspring solutions against parent solutions. Offspring replace parent solutions only if they yield better solutions. Teaching-Learning Based Optimization (TLBO) [57] and ABC also performed a similar operation. These algorithms produce new candidate solutions, but they are abandoned if they perform poorly. This selection operation is simple, and it provides excellent results.

In this work, we proposed a simple modification to the classical FA to provide a better feedback characteristic inspired by the selection operation. By modifying the feedback behavior, FA will improve its multimodal capability without losing its performance in exploration, exploitation, and communication behaviors.

The modification to the classical FA is simple, and it is conducted as follows. We proposed to update the position of a firefly $i$ attracted by another firefly $j$ , based on the following

$\displaystyle\vec{v}_{i}:=\vec{x}_{i}+\beta_{0}e^{\gamma_{i}r_{ij}^{2}}\left(% \vec{x}_{j}-\vec{x}_{i}\right)+\alpha\left(\vec{r}-{\displaystyle\frac{1}{2}}\right)$ (8)

where $\vec{v}_{i}$ is a candidate position for firefly $i$ . Then, the solution $\vec{x}_{i}$ is compared against to $\vec{v}_{i}$ using the next scheme

$\displaystyle\vec{x}_{i}:=\left\{\begin{array}[]{ll}\vec{v}_{i}&\;\textbf{if }% f\left(\vec{v}_{i}\right)<f\left(\vec{x}_{i}\right)\\ \vec{x}_{i}&\;\textbf{otherwise}\end{array}\right.$ (9)

where firefly $i$ updates its position as $\vec{v}_{i}$ only if it yields a better solution. Otherwise, firefly $i$ retains its current position to avoid loosening its current bright intensity achieve so far. This simple modification provides a better feedback characteristic to FA.

In Eq. (8), a firefly $j$ located in a global optimum produces a candidate solution $\vec{v}_{i}$ toward its position, even if firefly $i$ is located in a local optimum. However, the solution $\vec{v}_{i}$ is accepted if it improves the current position $\vec{x}_{i}$ using the scheme in Eq. (9). Since closer fireflies produce a better solution $\vec{v}_{i}$ than farther fireflies, closer fireflies converge to the nearest local or global optima. This mechanism improves the multimodal capacities of FA. In this paper, we call this modification mFA. Several simulations are performed to compare the performance of the proposed mFA against the classical FA.

4. The proposed approach

The definition of an end-effector pose is needed to solve the inverse kinematics. The end-effector pose is represented as

$\displaystyle\mathbf{T}_{d}=\left[\begin{array}[]{cc}\mathbf{R}_{d}&\vec{p}_{d% }\\ \vec{0}&1\end{array}\right]$ (10)

where $\mathbf{T}_{d}$ is a homogeneous matrix, $\mathbf{R}_{d}$ defines the orientation and $\mathbf{p}_{d}$ the position of the end-effector. In FA, every firefly $i$ represents a candidate joint configuration $\vec{q}_{i}\in\mathbb{R}^{n}$ where $n$ defines the total DOF of the robotic manipulator, which is also the problem dimensionality. Based on the DH model of the manipulator, we can use its forward kinematics equations to compute the end-effector pose for the joint configuration $\vec{q}_{i}$ . The candidate end-effector pose for each firefly $i$ is represented as

$\displaystyle\mathbf{T}_{i}\left(\vec{q}_{i}\right)=\left[\begin{array}[]{cc}% \mathbf{R}_{i}&\vec{p}_{i}\\ \vec{0}&1\end{array}\right]$ (11)

An objective function is defined as an error between the desired $\mathbf{T}_{d}$ and the candidate end-effector pose $\mathbf{T}_{i}$ . This error can be computed as

$\displaystyle f\left(\vec{q}_{i}\right)=||{\vec{T}_{d}-\vec{T}_{i}\left(\vec{q% }_{i}\right)}||$ (12)

where $||{\cdot}||$ is the Frobenius norm. This function is used to solve the inverse kinematics for the desired position and orientation of the end-effector. If it is required to solve only the position of the end-effector, another objective function can be formulated as

$\displaystyle f\left(\vec{q}_{i}\right)=||{\vec{p}_{d}-\vec{p}_{i}\left(\vec{q% }_{i}\right)}||$ (13)

where it is considered the error between the desired $\vec{p}_{d}$ position and the candidate position $\vec{p}_{i}$ .

Robotic manipulators with at least 6 DOF can reach any desired end-effector pose inside of its workspace. For this case, the objective function Eq. (12) is highly recommended because the manipulator can solve the required end-effector position and orientation. However, several robotic manipulators have less than 6 DOF, where it is not possible to reach some specific end-effector poses. For this reason, most of these manipulators only solve the desired position. In this case, the objective function Eq. (13) is adequate to solve the inverse kinematics.

In this work, we proposed to solve the inverse kinematics as a constrained optimization problem defined as

$\displaystyle\min_{\vec{q}}f\left(\vec{q}_{i}\right),\>\>\text{ subject to \;}% \vec{q}_{l}<\vec{q}_{i}<\vec{q}_{u}$ (14)

where $\vec{q}_{l}$ and $\vec{q}_{u}$ are the lower and upper constraints. Indeed, the lower $\vec{q}_{l}$ and upper $\vec{q}_{u}$ joint limits of the robotic manipulator represent the constraints.

FA was designed to work in continuous spaces without considering constraints. To include the constrains in mFA optimization process, we propose to follow the next scheme

$\displaystyle q_{ij}:=\left\{\begin{array}[]{ll}q_{ij}&\;\textbf{if }q_{lj}% \leqslant q_{ij}\leqslant q_{uj}\\ q_{lj}+\left(q_{uj}-q_{lj}\right)r&\;\textbf{otherwise}\end{array}\right.$ (15)

where $r=[0,1]$ is a random value, and we have that

$\displaystyle\vec{q}_{i}=\left[\begin{array}[]{c}q_{i1}\\ q_{i2}\\ q_{i3}\\ \vdots\\ q_{in}\end{array}\right],\vec{q}_{l}=\left[\begin{array}[]{c}q_{l1}\\ q_{l2}\\ q_{l3}\\ \vdots\\ q_{ln}\end{array}\right],\vec{q}_{u}=\left[\begin{array}[]{c}q_{u1}\\ q_{u2}\\ q_{u3}\\ \vdots\\ q_{un}\end{array}\right]$ (16)

where $j=1,2,3,\ldots,n$ . The scheme in Eq. (15) determines if a candidate joint value $q_{ij}$ is a feasible or an unfeasible joint solution. For feasible solutions, the value of $q_{ij}$ remains without any modification. In contrast, for unfeasible solutions, the value $q_{ij}$ is outside of the allowed joint limits. In this case, these values are recalculated inside of the joint limits.

The proposed algorithm to solve the inverse kinematics of robot manipulator based on mFA is shown in Algorithm 4.

[htb] [1] initialize $\vec{q}_{i}$ where $i=1,2,3,\dots,N$ define $\beta_{0}$ , $\alpha$ , $\gamma$ , $\delta$ compute brightness intensity $I_{i}=f\left(\vec{q}_{i}\right)$ repeat for $i=1:N$ $\;$ for $j=1:N$ $\;$ $\;$ if $I_{i}<I_{j}$ $\;$ $\;$ $\;$ $r_{ij}=\sqrt{\sum_{k=1}^{D}\left(x_{ik}-x_{jk}\right)^{2}}$ $\;$ $\;$ $\;$ $\vec{v}_{i}:=\vec{x}_{i}+\beta_{0}e^{\gamma_{i}r_{ij}^{2}}\left(\vec{x}_{j}-% \vec{x}_{i}\right)+\alpha\left(\vec{r}-{\displaystyle\frac{1}{2}}\right)$ $\;$ $\;$ $\;$ if $f\left(\vec{v}_{i}\right)<f\left(\vec{x}_{i}\right)$ $\;$ $\;$ $\;$ $\;$ $\vec{q}_{i}=\vec{v}_{i}$ $\;$ $\;$ $\;$ end if $\;$ $\;$ end if $\;$ end for $j$ end for $i$ penalize unfeasible solutions using Eq. (15) update brightness intensity $I_{i}=f\left(\vec{q}_{i}\right)$ reduce $\alpha:=\delta\alpha$ until stop criteria met report results mFA algorithm for solving the inverse kinematics of robotic manipulators. $f$ is the objective function.

5. Simulation and experimental results

Simulation and real experiments were carried out to test the performance of the proposed approach for solving the inverse kinematics of robotic manipulators. We are interested in testing the capability for finding multiple joint solutions for the same desired end-effector pose. These joint solutions must provide accurate results concerning the desired end-effector pose. We also consider testing the robustness to solve the inverse kinematics of different robotic manipulator structures.

5.1 Simulation results

Simulations aim to test the performance of the proposed mFA against the classical FA for solving the inverse kinematics problems. We also consider comparing the mFA against MFPA and MCS multimodal algorithms. There is a constrained version of FA [58], where a random vector is computed considering each position of fireflies and the value of the search space boundaries. This random vector indicates how the firefly needs to move to handle the given constraints. Since the proposed approach also handles constraints, the constrained FA is also considered for comparisons. We called this method CFA.

The common parameter settings of the considered algorithms are the total numbers of iterations and population members. We defined a total of $1000$ iterations and $50$ members. With respect to the particular settings of these algorithms, the configuration was selected as follows: For the case of MPFA, the probability switch between global and local pollination was set to $sp=0.25$ . In the case of MCS, the probability factor was selected as $p_{a}=0.25$ . The parameter setting for FA and mFA are the same. In these cases, we fixed the attraction parameter $\beta_{0}=1.0$ . The absorption coefficient was selected as $\gamma=0.9$ . The randomness and reduction parameters were set to $\alpha=1.0$ and $\delta=0.97$ , respectively. In the case of CFA, we set the following setting $\beta_{0}=1.0$ , $\alpha=0.01$ and $\gamma=0.97$ . The CFA method does not consider the use of parameter $\delta$ . The parameter settings for MPFA, MCS, and CFA were selected following the recommendations proposed by the authors. However, the particular parameters of mFA, the total number of iterations, and population members have been selected by experimentation.

Simulations are conducted as follows: In the first part, the inverse kinematics tasks require solving the end-effector desired position. On the other hand, the second part requires both the desired position and orientation of the end-effector for the given inverse kinematics tasks.

To present the obtained results of simulations to show accuracy, we proposed to measure the following errors

$\displaystyle p_{\textit{error}}=||{\vec{p}_{d}-\vec{p}\left(\vec{q}\right)}||$ (17) $\displaystyle R_{\textit{error}}=||{\vec{R}_{d}-\vec{R}\left(\vec{q}\right)}||$

where $\vec{p}_{d}$ and $\vec{R}_{d}$ are the desired position and orientation for the end-effector pose, respectively. The position $\vec{p}$ and the orientation $\vec{R}$ are the produced results given by the joint configuration $\vec{q}$ found by the considered multimodal algorithms. The error $p_{\textit{error}}$ is a Euclidean distance and the error $R_{\textit{error}}$ is a Frobenius norm.

To estimate the number of joint solutions found by the multimodal algorithms, we proposed to use two well-known algorithms for clustering, which are Subtractive Clustering (SC) [59] and Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [60]. The input data are the final population members, and the center of each cluster represents the final joint solution. These clustering algorithms are necessary because no priority knowledge about the total number of joint solutions is provided. In the case of SC, we set a range of influence to 0.9. For DBSCAN, we set a search radius to 0.9 as well. It is important to mention that the joint solutions with position errors above 10 millimeters are not considered for clustering.

Table 1

DH table for a Planar manipulator of 2 DOF

Joint	$\theta\>$ (rad)	$d$ (m)	$a$ (m)	$\alpha\>$ (rad)
1	$\theta_{1}$	0.0	0.35	0.0
2	$\theta_{2}$	0.0	0.30	0.0

Table 2

DH table for a Planar manipulator of 3 DOF

Joint	$\theta\>$ (rad)	$a$ (m)
1	$\theta_{1}$	0.35
2	$\theta_{2}$	0.30
3	$\theta_{3}$	0.25

Table 3

DH table for an Anthropomorphic manipulator of 3 DOF

Joint	$\theta\>$ (rad)	$d$ (m)	$a$ (m)	$\alpha\>$ (rad)
1	$\theta_{1}$	0.35	0.0	$\pi/2$
2	$\theta_{2}$	0.0	0.30	0.0
3	$\theta_{3}$	0.0	0.25	0.0

The robotic manipulators considered for these simulations are two Planar manipulators of 2 DOF and 3 DOF, an Anthropomorphic manipulator of 3 DOF, a Scara manipulator of 4 DOF, and a Puma 560 manipulator of 6 DOF. The DH tables of these manipulators are presented in Tables 1–5, respectively. These robots are often used as educational examples in robot kinematics [1, 3].

Table 4

DH table for Scara manipulator of 4 DOF

Joint	$\theta\>$ (rad)	$d$ (m)	$a$ (m)	$\alpha\>$ (rad)
1	$\theta_{1}$	0.8	0.15	0.0
2	$\theta_{2}$	0.0	0.40	$\pi$
3	0.0	$d_{3}$	0.0	0.0
4	$\theta_{4}$	0.2	0.0	0.0

Table 5

DH table for Puma 560 manipulator of 6 DOF

Joint	$\theta\>$ (rad)	$d$ (m)	$a$ (m)	$\alpha\>$ (rad)
1	$\theta_{1}$	0.0	0.0	$\pi/2$
2	$\theta_{2}$	0.0	0.4310	0.0
3	$\theta_{3}$	0.15	0.0203	$-\pi/2$
4	$\theta_{4}$	0.4318	0.0	$\pi/2$
5	$\theta_{5}$	0.0	0.0	$-\pi/2$
6	$\theta_{6}$	0.0	0.0	0.0

Another important aspect to mention is the restrictions. For these simulations, the joint limits were selected as

$\displaystyle q_{lj}=\left\{\begin{array}[]{cl}-\pi&\textbf{if }\textit{ joint% j is a revolute}\\ 0&\textbf{if }\textit{ joint j is a prismatic}\end{array}\right.$ (18) $\displaystyle q_{uj}=\left\{\begin{array}[]{cc}\pi&\textbf{if }\textit{ joint % j is a revolute}\\ 1.0&\textbf{if }\textit{ joint j is a prismatic}\end{array}\right.$

where $q_{lj}$ and $q_{uj}$ are elements of the lower $\vec{q}_{l}$ and upper $\vec{q}_{u}$ joint limits, respectively.

The simulations were implemented in Matlab™. Every multimodal algorithm ran 25 times, and their results were illustrated using box plots. We used box plots to display graphically statistical variation for the considered algorithm results. In the simulations, the number of solutions found, the position and orientation errors are presented into box plots. We are interested in finding the algorithm with the smaller data distribution, the lower value results, and fewer outliers.

For the first part of the simulations, we considered the following robotic manipulators and their respective desired position of the end-effector

•

Planar manipulator of 2 DOF with the desired position $\vec{p}_{d}=[0.35\ 0.35\ 0.0]^{T}$

•

Anthropomorphic manipulator of 3 DOF with the desired position $\vec{p}_{d}=[0.30\ -0.20\ 0.35]^{T}$

•

Puma 560 manipulator of 6 DOF with the desired position $\vec{p}_{d}=[0.4\ -0.2\ 0.5]^{T}$

The simulation results for the Planar manipulator of 2 DOF are presented in Fig. 3. For both SC and DBSCAN algorithms, the MFPA, MCS, and mFA methods reported two joint configurations leading the same end-effector pose, while FA and CFA reported only one solution. With respect to the position error provided by SC algorithm in Fig. 3(c), the results reported a high value for MCS and CFA compared against MFPA, FA and mFA; however, MFPA presented outliers. Indeed, FA and mFA provided error values under 0.5 millimeters. The results of CFA, MFPA, and MCS also reported a large data distribution. In Fig. 3(d), CFA showed larger data distribution in the position error provided by DBSCAN. However, the other algorithms reported results below 0.1 millimeters of error.

Figure 3.

Inverse kinematics results for the Planar manipulator of 2 DOF, where only a desired position of the end-effector was required. The number of solutions clustered by SC is presented in (a) and by DBSCAN in (b). The position error for the center of the clusters provided by SC and DBSCAN is presented in (c) and (d), respectively.

In Fig. 4, the simulation results for the Anthropomorphic manipulator of 3 DOF are reported. MFPA, MCS, and mFA founded four joint solutions. In contrast, FA algorithms reported large data distribution with a median value of two solutions, see Fig. 4(a). CFA also reported a large data distribution. On the other hand, four joint solutions are also reported by DBSCAN for MFPA, MCS and mFA approaches. Only one solution is reported by FA, see Fig. 4(b). In Fig. 4(c), MCS reported the larger data distribution with 4 millimeters of error. FA and mFA showed similar results with an error below to 0.01 millimeters. Similar results for FA and mFA are presented in Fig. 4(d). In this case, these two approaches performed better than the others. MFPA and MCS illustrated a large data distribution, but the error results of both approaches are presented under an error of 0.8 millimeters. The performance of CFA for position error is poorly compared against the others.

Figure 4.

Inverse kinematics results for the Anthropomorphic manipulator of 3 DOF, where only a desired position of the end-effector was required. The number of solutions clustered by SC is presented in (a) and by DBSCAN in (b). The position error for the center of the clusters provided by SC and DBSCAN is presented in (c) and (d), respectively.

Figure 5 shows the simulation results for the Puma 560 manipulator of 6 DOF. MFPA and CFA did not report joint solutions in this test. The boxplots for both clustering algorithms showed outlier data for MFPA. Moreover, MFPA found a solution that is not considered as a result. Similarly, there are cases when CFA did not report solutions. For these reasons, Fig. 5(c) and (d) did not show any results for MFPA and CFA. MCS reported in Fig. 5(a) a large data distribution in the number of joint solutions found by SC. In this case, the minimum number of the solution was two, and the maximum was seven. However, Fig. 5(b) shows a small data distribution with a median value of 4 solutions. The number of clusters reported by SC might be more than expected. In Fig. 5(c) and Fig. 5(d), the FA and the mFA performed better than MCS with error values below to 1 millimeter. However, FA reported one solution clustered by DBSCAN and a median value of two solutions by SC. As we can see, MFPA performed poorly in this test. In contrast, the proposed mFA outperformed the other compared approaches.

Figure 5.

Inverse kinematics results for the Puma 560 manipulator of 6 DOF, where only a desired position of the end-effector was required. The number of solutions clustered by SC is presented in (a) and by DBSCAN in (b). The position error for the center of the clusters provided by SC and DBSCAN is presented in (c) and (d), respectively.

For the second part of simulations, we considered the following robotic manipulators and their respective desired position and orientation of the end-effector

•

Planar manipulator of 3 DOF with the desired pose $\vec{p}_{d}=[0.5\ -0.5\ 0.0]^{T}$ and

$\displaystyle\vec{R}_{d}=\left[\begin{array}[]{ccc}\cos\left(-\pi/4\right)&-% \sin\left(-\pi/4\right)&0\\ \sin\left(-\pi/4\right)&\cos\left(-\pi/4\right)&0\\ 0&0&1\end{array}\right]$

•

Scara manipulator of 4 DOF with the desired pose $\vec{p}_{d}=[0.4\ 0.2\ 0.1]^{T}$ and

$\displaystyle\vec{R}_{d}=\left[\begin{array}[]{ccc}\cos\left(\pi/4\right)&\sin% \left(\pi/4\right)&0\\ \sin\left(\pi/4\right)&-\cos\left(\pi/4\right)&0\\ 0&0&-1\end{array}\right]$

•

Puma 560 manipulator of 6 DOF with the desired pose $\vec{p}_{d}=[0.4\ -0.2\ 0.5]^{T}$ and

$\displaystyle\vec{R}_{d}=\left[\begin{array}[]{ccc}0&0&1\\ 0&1&0\\ -1&0&0\end{array}\right]$

The simulation results for the Planar manipulator of 3 DOF are given in Fig. 6. FA reported a larger data distribution for the joint solution clustered by SC, with a minimum value of one solution and a maximum of three solutions, see Fig. 6(a). However, DBSCAN reported only one joint solution, see Fig. 6(b). Moreover, MFPA, MCS, and mFA reported similar results where SC and DBSCAN provided two joint solutions. CFA presents the largest data distribution. The position and orientation errors for both SC and DBSCAN clusters are very similar, see Fig. 6(c)–(f). In all cases, FA and mFA outperformed the other approaches with position error values below 0.1 millimeters of error. CFA reported the larger data distribution in both the position and orientation errors. MFPA also presented a large data distribution.

Figure 6.

Inverse kinematics results for the Planar manipulator of 3 DOF, where a desired position and orientation of the end-effector were required. The number of solutions clustered by SC is presented in (a) and by DBSCAN in (b). The position error for the center of the clusters provided by SC and DBSCAN is presented in (c) and (d), respectively. Orientation errors are also presented in (e) and (f), respectively.

Figure 7 reports the simulation results for the Scara manipulator of 4 DOF. The behavior of the compared approaches is similar to the previous test. In Fig. 7(a) and 7(b), two joint solutions are reported for both SC and DBSCAN. Nevertheless, SC reported many joint solutions for FA, while DBSCAN reports one solution. SC poorly predicts the number of clusters for FA. We notice that CFA did not solve the inverse kinematics for this test. For this reason, the position and orientation results are not reported by CFA. The position error results are shown in Figs 7(c) and 7(d), where FA and mFA performed better than the other approaches with values below 0.1 millimeters. In contrast, MFPA and MSC presented bigger error results. All the algorithms reported small orientation errors, see in Fig. 7(e) and 7(f). However, FA and mFA showed the smallest data distribution. For both the position and orientation error, the MFPA presented the larger data distribution.

Figure 7.

Inverse kinematics results for the Scara manipulator of 4 DOF, where a desired position and orientation of the end-effector were required. The number of solutions clustered by SC is presented in (a) and by DBSCAN in (b). The position error for the center of the clusters provided by SC and DBSCAN is presented in (c) and (d), respectively. Orientation errors are also presented in e) and (f), respectively.

Finally, the simulation results for the Puma 560 manipulator of 6 DOF are presented in Fig. 8. FA reported the larger data distribution in the number of joint solutions clustered by CS, see Fig. 8(a). On the other hand, DBSCAN found just one solution, see Fig. 8(b). mFA outperformed the compared approaches by finding a median value of eight joint solutions reported by SC and DBSCAN. The MCS reported a maximum number of four solutions for both clustering algorithms with a large data distribution. Boxplots reported that there are some cases where MCS did not present any solution. Moreover, MFPA and CFA performed poorly. The results for both SC and DBSCAN reported that a few joint solutions were found, but they are considering noise data. In Fig. 8(c) and 8(d), FA and mFA reported error results with values below to 1 millimeter. However, FA reported the smallest data distribution. MCS had the largest data distribution, with error values of up to 7 millimeters. In the case of the orientation error in Fig. 8(e) and 8(f), the results reported small errors, which indicate that orientation problems were solved successfully for all compared approaches.

Figure 8.

Inverse kinematics results for the Puma 560 manipulator of 6 DOF, where a desired position and orientation of the end-effector were required. The number of solutions clustered by SC is presented in (a) and by DBSCAN in (b). The position error for the center of the clusters provided by SC and DBSCAN is presented in (c) and (d), respectively. Orientation errors are also presented in (e) and (f), respectively.

Based on simulations, FA and mFA performed similar respect to position and orientation error results. In contrast, mFA proved to be superior to FA for finding more joint solutions for the given problems. These results demonstrate that the proposed mFA highly improves the multimodal capability of classical FA without losing its exploration and exploitation capacities. Moreover, the statistical variations of mFA are better than the compared approaches. MFPA and CFA were not able to solve the given problems for the Puma 560 manipulator. The MCS also performed poorly for this manipulator.

Table 6

Comparison results between the proposed mFA algorithm and the geometric approach. $\vec{q}_{i}^{g}$ represents the solution of the geometric approach and $\vec{q}_{i}^{mFA}$ are the solution of mFA for the desired position $i$ . The operation $||\cdot||$ is the Euclidean norm

Test	Method	$\theta_{1}$ (rad)	$\theta_{2}$ (rad)	$\theta_{3}$ (rad)	$\|\|\vec{q}_{i}^{g}-\vec{q}_{i}^{\textit{mFA}}\|\|$	$p_{\textit{error}}$ (mm)
1	$\vec{q}_{1}^{g}$	0	$-$ 1.0748	3.1201	6.24 $\times 10^{-5}$	0.50770693
	$\vec{q}_{1}^{mFA}$	$-$ 4.18 $\times 10^{-6}$	$-$ 1.07484504	3.12005698
	$\vec{q}_{2}^{g}$	0	0.4313	0.1153	5.83 $\times 10^{-5}$	0.67553935
	$\vec{q}_{2}^{\textit{mFA}}$	5.82 $\times 10^{-6}$	0.43132101	0.11524593
	$\vec{q}_{3}^{g}$	$-$ 2.6545	$-$ 2.0668	0.1153	6.04 $\times 10^{-5}$	0.53647619
	$\vec{q}_{3}^{\textit{mFA}}$	$-$ 2.65449651	$-$ 2.0667681	0.11524885
	$\vec{q}_{4}^{g}$	$-$ 2.6545	2.7103	3.1201	7.44 $\times 10^{-5}$	0.60725650
	$\vec{q}_{4}^{\textit{mFA}}$	$-$ 2.65450124	2.71025835	3.12003838
2	$\vec{q}_{1}^{g}$	0.1754	$-$ 1.2425	$-$ 2.998	8.92 $\times 10^{-5}$	0.30338596
	$\vec{q}_{1}^{\textit{mFA}}$	0.17536148	$-$ 1.24251973	$-$ 2.998078
	$\vec{q}_{2}^{g}$	0.1754	0.4294	$-$ 0.0498	6.56 $\times 10^{-5}$	0.48010210
	$\vec{q}_{2}^{\textit{mFA}}$	0.17534783	0.4293959	$-$ 0.04983956
	$\vec{q}_{3}^{g}$	$-$ 2.4219	$-$ 1.8991	$-$ 0.0498	5.21 $\times 10^{-5}$	0.41886027
	$\vec{q}_{3}^{\textit{mFA}}$	$-$ 2.42191572	$-$ 1.8990721	$-$ 0.04984109
	$\vec{q}_{4}^{g}$	$-$ 2.4219	2.7122	$-$ 2.998	8.59 $\times 10^{-5}$	0.58748350
	$\vec{q}_{4}^{mFA}$	$-$ 2.42191949	2.71219639	$-$ 2.99808362
3	$\vec{q}_{1}^{g}$	0.1106	$-$ 1.2132	$-$ 3.1125	7.81 $\times 10^{-5}$	0.48755146
	$\vec{q}_{1}^{\textit{mFA}}$	0.11064843	$-$ 1.21322886	$-$ 3.11255402
	$\vec{q}_{2}^{g}$	0.1106	0.3437	0.0647	0.00010637	0.63621079
	$\vec{q}_{2}^{\textit{mFA}}$	0.11063925	0.34374072	0.06460991
	$\vec{q}_{3}^{g}$	$-$ 2.5105	$-$ 1.9284	0.0647	7.78 $\times 10^{-5}$	0.56034756
	$\vec{q}_{3}^{\textit{mFA}}$	$-$ 2.51045943	$-$ 1.92837093	0.06464031
	$\vec{q}_{4}^{g}$	$-$ 2.5105	2.7979	$-$ 3.1125	7.20 $\times 10^{-5}$	0.40562991
	$\vec{q}_{4}^{\textit{mFA}}$	$-$ 2.51045947	2.79786068	$-$ 3.11254467
4	$\vec{q}_{1}^{g}$	$-$ 1.4947	$-$ 1.6169	$-$ 2.9763	4.96 $\times 10^{-5}$	0.47516573
	$\vec{q}_{1}^{\textit{mFA}}$	$-$ 1.49474706	$-$ 1.61688435	$-$ 2.97630072
	$\vec{q}_{2}^{g}$	$-$ 1.4947	0.0769	$-$ 0.0716	5.37 $\times 10^{-5}$	0.50298926
	$\vec{q}_{2}^{\textit{mFA}}$	$-$ 1.49475341	0.0769036	$-$ 0.07159644
	$\vec{q}_{3}^{g}$	2.3405	$-$ 1.5247	$-$ 0.0716	5.74 $\times 10^{-5}$	0.54457964
	$\vec{q}_{3}^{\textit{mFA}}$	2.34044692	$-$ 1.52471644	$-$ 0.07158554
	$\vec{q}_{4}^{g}$	2.3405	3.0647	$-$ 2.9763	5.34 $\times 10^{-5}$	0.56073622
	$\vec{q}_{4}^{\textit{mFA}}$	2.34045477	3.06470836	$-$ 2.97632722
5	$\vec{q}_{1}^{g}$	1.6468	$-$ 0.5643	$-$ 2.6406	7.89 $\times 10^{-5}$	0.42982965
	$\vec{q}_{1}^{mFA}$	1.6468527	$-$ 0.56425351	$-$ 2.64063593
	$\vec{q}_{2}^{g}$	1.6468	1.4671	$-$ 0.4072	9.34 $\times 10^{-5}$	0.54973299
	$\vec{q}_{2}^{\textit{mFA}}$	1.64685594	1.46711375	$-$ 0.40727351
	$\vec{q}_{3}^{g}$	$-$ 0.8011	$-$ 2.5773	$-$ 0.4072	9.02 $\times 10^{-5}$	0.33093465
	$\vec{q}_{3}^{\textit{mFA}}$	$-$ 0.80113883	$-$ 2.57735002	$-$ 0.40726421
	$\vec{q}_{4}^{g}$	$-$ 0.8011	1.6745	$-$ 2.6406	5.34 $\times 10^{-5}$	0.36659398
	$\vec{q}_{4}^{\textit{mFA}}$	$-$ 0.80112816	1.67446782	$-$ 2.64063203

On the other hand, MFPA and MCS provided good results concerning position error, orientation error, and the number of joint solutions reported for the manipulators with 2 DOF and 3 DOF. These results indicate that the inverse kinematics for manipulators with 4 DOF and 6 DOF is more challenging to solve. We notice that CFA performed poorly in all tests, respecting the number of solutions founded and the position error results. For the clustering methods used to report the total number of joint solutions, SC performed poorly compared with DBCAN. DBSCAN reports more consistent, clustering results.

The computational complexity of FA algorithm is $O\left(m^{2}\right)$ [58]. This complexity is also the same for mFA. For larger population sizes, the computational time increases. The proposed approach can be used to solve the inverse kinematics of robot manipulators in offline applications. However, parallel computing has been used to reduce the computational time of FA [61]. By implementing this approach, the computational cost of the proposed approach can be improved.

5.2 The comparison against the geometric approach

The geometric approach provides exact solutions for the inverse kinematics. This method also provides multiples solutions. However, complicated analysis of kinematics structures is required for high DOF. In this work, the proposed mFA is compared to the geometric approach. This comparison demonstrates the accuracy of the proposal compared to the exact solutions.

The robot manipulator Puma 560 is considered in this test. Give a desired position of the end-effector, the geometric approach may compute up to four joint configurations for this manipulator. In [47], the author reported five desired positions where four joint solutions can be found. The desired positions are listed below

1.
$\vec{p}_{d}=[0.6\ 0.4\ 0.13]^{T}$
2.
$\vec{p}_{d}=[0.4\ -0.6\ 0.13]^{T}$
3.
$\vec{p}_{d}=[0.35\ 0.35\ 0.13]^{T}$
4.
$\vec{p}_{d}=[-0.1\ 0.7\ 0.13]^{T}$
5.
$\vec{p}_{d}=[0.65\ -0.45\ 0.13]^{T}$

Since only the desired position is required, the first 3 DOF of Puma 560 are needed to solve the inverse kinematics. The parameter settings for mFA are the same from simulations. This test was perform using Matlab™, where the mFA only ran once for each desired position. More details about the geometric approach for Puma 560 can be found in [47].

The results of the comparison are presented in Table 6. Every test corresponds to a desired position of the end-effector. As can be seen, the Euclidean norm results indicate that mFA provides joint configuration results with high accuracy compared with the exact solution computed by the geometric approach. Moreover, the position error results $p_{\textit{error}}$ of mFA have an error below to 1 millimeter. The proposed mFA also demonstrated that can find the multiples joint solution similar to the geometric approach in all the given problems.
5.3 Non-parametric statistical test

Non-parametric statistical tests are usually employed to decide when an algorithm shows a significant difference compared to other algorithms [62]. In the non-parametric procedure, the null hypothesis and the alternative hypothesis are defined. The null hypothesis indicates that the algorithms are statistically equivalent, while the alternative hypothesis indicates that they are statistically different. Moreover, a p-value is computed to provide information about the significance of the statistical hypothesis. Smaller $p$ -values indicate strong evidence against the null hypothesis. A level of significance $\alpha$ is defined to determine at which level the null hypothesis may be rejected. Base on the results obtained from simulations, we consider showing that the mFA algorithm is statistically different than the FA and MCS algorithms.

Table 7
Pairwise comparison of mFA against FA and MCS algorithms under the Sign test. The position error results of Planar of 2 DOF, Anthropo-morphic of 3 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Simulation

mFA

MCS

Planar of 2 DOF

Wins (+)

Loses (-)

p

-vale

1.9431

\times 10^{-5}

5.9605

\times 10^{-8}

Antropomorphic

of 3 DOF

Wins (+)

Loses (-)

p

-vale

5.9605

\times 10^{-8}

5.9605

\times 10^{-8}

Puma 560

of 6 DOF

Wins (+)

Loses (-)

p

-vale

1.9431

\times 10^{-5}

5.9605

\times 10^{-8}

Table 8

Pairwise comparison of mFA against FA and MCS algorithms under the Wilcoxon signed ranks test. The position error results of Planar of 2 DOF, Anthropomorphic of 3 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Simulation

mFA

MCS

Planar of 2 DOF

{}_{+}

321

325

{}_{-}

p

-vale

4.1723

\times 10^{-7}

5.9605

\times 10^{-8}

Antropomorphic

of 3 DOF

{}_{+}

325

{}_{-}

p

-vale

5.9605

\times 10^{-8}

5.9605

\times 10^{-8}

Puma 560

of 6 DOF

{}_{+}

316

325

{}_{-}

p

-vale

1.967

\times 10^{-6}

5.9605

\times 10^{-8}

Table 9

Pairwise comparison of mFA against FA and MCS algorithms under the Sign test. The position error results of Planar of 3 DOF, Scara of 4 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Simulation

mFA

MCS

Planar of 2 DOF

Wins (+)

Loses (-)

p

-vale

1.5497

\times 10^{-6}

5.9605

\times 10^{-8}

Antropomorphic

of 3 DOF

Wins (+)

Loses (-)

p

-vale

1.9431

\times 10^{-5}

5.9605

\times 10^{-8}

Puma 560

of 6 DOF

Wins (+)

Loses (-)

p

-vale

0.00091052

5.9605E-08

In this paper, we proposed to use the Sign test and the Wilcoxon signed ranks test for pairwise comparisons [63]. Additionally, a level of significance $\alpha=0.05$ was selected. The Sign and the Wilcoxon tests were performed using Matlab™.

The statistical results provided by the DBSCAN algorithm from simulations were used in these tests. The position error results of the Planar manipulator of 2 DOF, the Anthropomorphic manipulator of 3 DOF, and the Puma 560 manipulator of 6 DOF were considered for pairwise comparisons. Additionally, the position and the orientation error results of the Planar manipulator of 3 DOF, Scara manipulator of 4 DOF, and the PUMA 560 manipulator of 6 DOF were also considered. The results of the Sign test and the Wilcoxon signed ranks test are reported in Tables 7–12.

The Sign tests and the Wilcoxon signed ranks tests for the position error results of the Planar, Anthropomorphic and Puma 560 manipulators are provided in Tables 7 and 8, respectively. In these cases, the small $p$ -values reported show strong evidence that mFA is statistically different than the compared algorithms for all simulations.

Table 10

Pairwise comparison of mFA against FA and MCS algorithms under the Wilcoxon signed ranks test. The position error results of Planar of 3 DOF, Scara of 4 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Simulation

mFA

MCS

Planar of 2 DOF

{}_{+}

323

325

{}_{-}

p

-vale

1.7881

\times 10^{-7}

5.9605

\times 10^{-8}

Antropomorphic

of 3 DOF

{}_{+}

316

325

{}_{-}

p

-vale

1.967

\times 10^{-6}

5.9605

\times 10^{-8}

Puma 560

of 6 DOF

{}_{+}

325

{}_{-}

296

p

-vale

0.000103

5.9605

\times 10^{-8}

Table 11

Pairwise comparison of mFA against FA and MCS algorithms under the Sign test. The orientation error results of Planar of 3 DOF, Scara of 4 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Simulation

mFA

MCS

Planar of 2 DOF

Wins (+)

Loses (-)

p

-vale

1.5497

\times 10^{-6}

5.9605

\times 10^{-8}

Antropomorphic

of 3 DOF

Wins (+)

Loses (-)

p

-vale

0.00015652

5.9605

\times 10^{-8}

Puma 560

of 6 DOF

Wins (+)

Loses (-)

p

-vale

0.22952294

1.9431

\times 10^{-5}

Table 12

Pairwise comparison of mFA against FA and MCS algorithms under the Wilcoxon signed ranks test. The orientation error results of Planar of 3 DOF, Scara of 4 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Simulation

mFA

MCS

Planar of 2 DOF

{}_{+}

323

325

{}_{-}

p

-vale

1.7881

\times 10^{-7}

5.9605

\times 10^{-8}

Antropomorphic

of 3 DOF

{}_{+}

316

325

{}_{-}

p

-vale

1.967

\times 10^{-6}

5.9605

\times 10^{-8}

Puma 560

of 6 DOF

{}_{+}

322

{}_{-}

253

p

-vale

0.01355469

2.9802

\times 10^{-7}

Figure 9.

KUKA Youbot™ manipulator of 5 DOF.

The position error results for the Planar, Scara, and Puma 560 manipulators, the Signed tests, and the Wilcoxon signed ranks tests are presented in Tables 9 and 10, respectively. The provided $p$ -values indicate that mFA algorithm shows a significant statistical difference over the compared algorithms. We can also notice that FA shows a significant improvement over mFA with respect to the Puma 560 simulation, for both the Sign and Wilcoxon tests.

Finally, the Sign tests and the Wilcoxon signed ranks tests for the orientation error results of the Planar, Scara, and Puma 560 manipulators are provided in Tables 11 and Table 12, respectively. In the Puma 560 simulation of the Sign test, the obtained $p$ -value $=0.22952294$ indicates that mFA and FA are considered statistically equivalent due to the level of significance $alpha=0.05$ . On the other hand, the Wilcoxon test reports that mFA and FA are statistically different, with a $p$ -value $=0.01355469$ . For the other cases, the mFA is considered statistically different than the compared algorithms.

Based on the results obtained by the non-parametric statistical tests, we can conclude that mFA shows a significant difference over FA and MCS. Additionally, mFA also shows an improvement over the compared algorithms in general.

5.4 Experiment results

The main purpose of the experiments is to show the applicability of the proposed approach to solve inverse kinematics tasks under real joint restrictions. A comparison study among FA, MCS, and mFA are also considered for these experiments. The particular parameters setting for these methods are the same from simulations. However, it was considered a total of 300 iterations and 50 population members. DBSCAN algorithm was also selected to cluster to solutions provided by the considered algorithms. Since statistical results are not considered in these tests, every algorithm only ran once.

The robotic manipulator for these experiments is the KUKA Youbot™ illustrated in Fig. 9. This manipulator is composed of five revolute joints in an open kinematic chain. The DH model of this manipulator is presented in Table 13. Since the considered manipulator is composed of 5 DOF, complex geometrical analysis is required to solve the inverse kinematics by conventional approaches. However, that analysis is not required by the proposed approach. Indeed, the proposed method avoids the algebraic analysis of manipulator kinematics. In these experiments, conventional approaches are not included.

Table 13
DH table for KUKA Youbot™ manipulator of 5 DOF

Joint	$\theta\>$ (rad)	$d$ (m)	$a$ (m)	$\alpha\>$ (rad)
1	$\theta_{1}$	0.147	0.033	$\pi/2$
2	$\theta_{2}$	0.0	0.155	0.0
3	$\theta_{3}$	0.0	0.135	0.0
4	$\theta_{4}$	0.0	0.0	$\pi/2$
5	$\theta_{5}$	0.2174	0.0	0.0

The joint limits for the KUKA Youbot™ manipulator are defined as follows

$\displaystyle\vec{q}_{l}=\left[\begin{array}[]{c}-2.9496\\ 0\\ -2.5482\\ -0.2007\\ -2.9234\end{array}\right],\vec{q}_{u}=\left[\begin{array}[]{c}2.9496\\ 2.7053\\ 2.6354\\ 3.3598\\ 2.9234\end{array}\right]$ (19)

The experiments were performed using C++ and the Robot Operation System (ROS). There is an existing ROS component to access the KUKA Youbot™ hardware. ROS also provides a generic PID controller to control the desired joint positions. It is needed to add an offset $\vec{q}^{\textit{offset}}$ to use this PID controller. The offset is defined as

$\displaystyle\vec{q}^{\textit{PID}}=-\vec{q}^{\textit{mFA}}+\vec{q}^{\textit{% offset}}$ (20)

where $\vec{q}^{\textit{mFA}}$ are the joint values obtained by the proposed approach.

Figure 10.

Joint configuration solutions of the proposed approach for Experiment 1. Two joint solutions are reported in this experiment.

Figure 11.

Joint configuration solutions of the proposed approach for Experiment 2. A joint solution is reported in this experiment.

Figure 12.

Joint configuration solutions of the proposed approach for Experiment 3. Four joint solutions are reported in this experiment.

The experiments are conducted as follows

•

Experiment 1

It is required to solve both the position and orientation of the end-effector. The first 4 DOF of the Youbot™ manipulator are required for this experiment. The position $\vec{p}_{d}$ and orientation $\vec{R}_{d}$ are defined as

$\displaystyle\vec{p}_{d}=\left[\begin{array}[]{c}0.35\\ 0.15\\ 0.35\end{array}\right],\;\vec{R}_{d}=\left[\begin{array}[]{ccc}0&p_{y}/\rho&p_% {x}/\rho\\ 0&-p_{x}/\rho&p_{y}/\rho\\ 1&0&0\end{array}\right]$

where $\rho=\sqrt{p_{x}^{2}+p_{y}^{2}}$ .

•

Experiment 2

Table 14

Inverse kinematics results for Experiment 1. The joint configurations $\vec{q}_{i}^{\textit{FA}}$ , $\vec{q}_{i}^{\textit{MCS}}$ and $\vec{q}_{i}^{\textit{mFA}}$ represent the solutions provided by FA, MCS, and mFA algorithms, respectively. The value $i$ indicate the solution number. Bold results indicate the best results

Method	$\theta_{1}$ (rad)	$\theta_{2}$ (rad)	$\theta_{3}$ (rad)	$\theta_{4}$ (rad)	$\theta_{5}$ (rad)	$p_{\textit{error}}$ (mm)	$R_{\textit{error}}$
$\vec{q}_{i}^{\textit{FA}}$	0.404876	1.54355	$-$ 1.17956	1.20682	$-$ 2.4582 $\times 10^{-6}$	0.01373984	2.9739 $\times 10^{-5}$
$\vec{q}_{i}^{\textit{MCS}}$	0.402037	1.54246	$-$ 1.16916	1.19666	$-$ 2.4582 $\times 10^{-6}$	1.49869083	0.00420696
$\vec{q}_{i}^{\textit{mFA}}$	0.404886	1.54352	$-$ 1.17955	1.20686	$-$ 2.4582 $\times 10^{-6}$	0.0096473	4.8444 $\times 10^{-5}$
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	–	–
$\vec{q}_{i}^{\textit{MCS}}$	0.404947	0.451983	1.17409	$-$ 0.0548869	-2.4582 $\times 10^{-6}$	1.64417305	0.00055674
$\vec{q}_{i}^{\textit{mFA}}$	0.404886	0.456263	1.17941	$-$ 0.0648869	-2.4582 $\times 10^{-6}$	0.00744975	1.6977 $\times 10^{-5}$

Table 15

Inverse kinematics results for Experiment 2. The joint configurations $\vec{q}_{i}^{\textit{FA}}$ , $\vec{q}_{i}^{\textit{MCS}}$ and $\vec{q}_{i}^{\textit{mFA}}$ represent the solutions provided by FA, MCS, and mFA algorithms, respectively. The value $i$ indicate the solution number. Bold results indicate the best resultss

Method	$\theta_{1}$ (rad)	$\theta_{2}$ (rad)	$\theta_{3}$ (rad)	$\theta_{4}$ (rad)	$\theta_{5}$ (rad)	$p_{\textit{error}}$ (mm)	$R_{\textit{error}}$
$\vec{q}_{i}^{\textit{FA}}$	0.540408	0.630661	$-$ 0.786753	0.156109	0.540396	0.01373018	2.9428 $\times 10^{-5}$
$\vec{q}_{i}^{\textit{MCS}}$	0.540982	0.634629	$-$ 0.811948	0.17748	0.541944	2.45931071	0.00137939
$\vec{q}_{i}^{\textit{mFA}}$	0.540408	0.630641	$-$ 0.786753	0.156131	0.540418	0.00876416	3.0364 $\times 10^{-5}$

Table 16

Inverse kinematics results for Experiment 3. The joint configurations $\vec{q}_{i}^{\textit{FA}}$ , $\vec{q}_{i}^{\textit{MCS}}$ and $\vec{q}_{i}^{\textit{mFA}}$ represent the solutions provided by FA, MCS, and mFA algorithms, respectively. The value $i$ indicate the solution number. Bold results indicate the best results

Method	$\theta_{1}$ (rad)	$\theta_{2}$ (rad)	$\theta_{3}$ (rad)	$\theta_{4}$ (rad)	$\theta_{5}$ (rad)	$p_{\textit{error}}$ (mm)
$\vec{q}_{i}^{\textit{FA}}$	0.404896	1.09609	$-$ 1.69398	2.8752	$-$ 2.4582 $\times 10^{-6}$	0.00317329
$\vec{q}_{i}^{\textit{MCS}}$	0.405802	1.54525	$-$ 1.18914	1.21945	$-$ 2.4582 $\times 10^{-6}$	0.36377878
$\vec{q}_{i}^{\textit{mFA}}$	0.404886	1.54141	$-$ 1.17091	1.19653	$-$ 2.4582 $\times 10^{-6}$	0.00643659
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	–
$\vec{q}_{i}^{\textit{MCS}}$	$-$ 2.73701	2.01441	0.894172	1.65757	$-$ 2.4582 $\times 10^{-6}$	0.85232152
$\vec{q}_{i}^{\textit{mFA}}$	$-$ 2.7367	2.01868	0.933851	1.58447	$-$ 2.4582 $\times 10^{-6}$	0.01815268
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	-
$\vec{q}_{i}^{\textit{MCS}}$	0.405762	0.477408	$-$ 0.980589	3.20821	$-$ 2.4582 $\times 10^{-6}$	0.362999
$\vec{q}_{i}^{\textit{mFA}}$	0.404896	0.831522	$-$ 1.47883	3.11498	$-$ 2.4582 $\times 10^{-6}$	0.00353298
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	-
$\vec{q}_{i}^{\textit{MCS}}$	0.407615	0.677705	0.807464	$-$ 0.0459931	$-$ 2.4582 $\times 10^{-6}$	1.04949472
$\vec{q}_{i}^{\textit{mFA}}$	0.404896	0.610603	0.927646	$-$ 0.0632719	$-$ 2.4582 $\times 10^{-6}$	0.00187171

Table 17

Inverse kinematics results for Experiment 4. The joint configurations $\vec{q}_{i}^{\textit{FA}}$ , $\vec{q}_{i}^{\textit{MCS}}$ and $\vec{q}_{i}^{\textit{mFA}}$ represent the solutions provided by FA, MCS, and mFA algorithms, respectively. The value $i$ indicate the solution number. Bold results indicate the best results

Method	$\theta_{1}$ (rad)	$\theta_{2}$ (rad)	$\theta_{3}$ (rad)	$\theta_{4}$ (rad)	$\theta_{5}$ (rad)	$p_{\textit{error}}$ (mm)
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	–
$\vec{q}_{i}^{\textit{MCS}}$	$-$ 0.40492	1.56592	$-$ 1.56301	1.78834	$-$ 0.637038	0.12657858
$\vec{q}_{i}^{\textit{mFA}}$	$-$ 0.404879	1.54083	$-$ 1.16838	1.19352	$-$ 0.0621485	0.00937878
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	–
$\vec{q}_{i}^{\textit{MCS}}$	2.7368	2.03592	1.01063	1.42416	0.424644	1.26441184
$\vec{q}_{i}^{\textit{mFA}}$	2.73669	2.02418	1.01487	1.44202	0.0480949	0.00533076
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	–
$\vec{q}_{i}^{\textit{MCS}}$	$-$ 0.409572	0.83848	0.500876	0.0302901	$-$ 1.06551	1.80454871
$\vec{q}_{i}^{\textit{mFA}}$	$-$ 0.4049	0.720297	0.72839	$-$ 0.0313692	$-$ 0.341064	0.00350861
$\vec{q}_{i}^{\textit{FA}}$	–	–	–	–	–	–
$\vec{q}_{i}^{\textit{MCS}}$	$-$ 0.40356	0.325635	$-$ 0.706501	3.16173	1.26856	1.24839478
$\vec{q}_{i}^{\textit{mFA}}$	$-$ 0.4049	0.316452	$-$ 0.690683	3.16445	1.85028	0.00445909
$\vec{q}_{i}^{\textit{FA}}$	$-$ 0.4049	1.00687	$-$ 1.63702	2.97233	1.35307	0.00699317
$\vec{q}_{i}^{\textit{MCS}}$	–	–	–	–	–	–
$\vec{q}_{i}^{\textit{mFA}}$	$-$ 0.4049	1.22712	$-$ 1.74753	2.69925	$-$ 0.934339	0.00386831

In this case, it is required to solve both the position and orientation of the end-effector. The 5 DOF of the Youbot™ manipulator are required for this experiment. The position $\vec{p}_{d}$ and orientation $\vec{R}_{d}$ are defined as

$\displaystyle\vec{p}_{d}=\left[\begin{array}[]{c}0.25\\ 0.15\\ 0.0\end{array}\right],\;\vec{R}_{d}=\left[\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&-1\end{array}\right]$

•

Experiment 3

For this experiment, it is required to solve only the position of the end-effector. The first 4 DOF of the Youbot™ manipulator are also required. The position $\vec{p}_{d}$ is defined as $\vec{p}_{d}=[0.35\ 0.15\ 0.35]^{T}$ .

•

Experiment 4

Finally, it is required to solve only the position of the end-effector for this experiment. In this case, 5 DOF of the Youbot™ manipulator are required. The position $\vec{p}_{d}$ is defined as $\vec{p}_{d}=[0.35\ -0.15\ 0.35]^{T}$ .

The reason for using 4 DOF or 5 DOF in the experiments is to demonstrate the robustness of the proposed approach.

In the results of the experiments, every considered algorithm ran under the Youbot™ hardware. Then, the computed joint configurations were measured by the encoders. These measured values are reports as results. The inverse kinematics results are given in Tables 14–17 for Experiments 1–4, respectively.

Figure 13.

Joint configuration solutions of the proposed approach for Experiment 4. Five joint solutions are reported in this experiment.

In Table 14, the FA reports one solution while MCS and mFA report two solutions. The FA has the best orientation results for solution $i=1$ . In contrast, the mFA has the best position error for solution $i=1$ . Moreover, mFA has the best results for solution $i=2$ .

In the case of Experiment 2, only one solution is reported by the algorithms, see Table 15. In this case, mFA performed better than the others.

In Table 16, mFA and MCS were able to find four joint solutions while FA found one solution. FA reported the best position error for solution $i=1$ . On the other hand, mFA provided the best position results for solutions $i=2,3,4$ .

In experiment 4, five solutions are reported by mFA, four solutions by MCS, and one solution by FA. The proposed mFA outperformed the other algorithms with better position error results in all cases, see Table 17.

In general, the position error results of FA and mFA are similar, with values below 0.1 millimeters of errors. However, the mFA outperformed the FA for the multimodal capacities since mFA reported a greater number of joint solutions for the given problems. When the position and orientation of the end-effector are required, then the number of solutions is reduced. On the other hand, when only the position of the end-effector is required, the numbers of the solution to yield the same end-effector pose increase.

As a visual demonstration, the results of the joint configuration for Experiment 1–4 are illustrated in Figs 10–13, respectively.

6. Conclusions

This paper introduced the mFA algorithm to solve the inverse kinematics of robotic manipulators as a multimodal problem. The mFA is a modified version of FA, which improves multimodal capacities keeping the exploration and exploitation characteristic of FA. The mFA provides multiples joint solutions leading the same end-effector pose with accuracy and robustness. Simulation and experiments were carried out to show the effectiveness of the proposed approach. Manipulator structures composed of 2 DOF, 3 DOF, 4 DOF and 6 DOF were considered to demonstrate robustness. The position and orientation error results of the inverse kinematics to determine accuracy.

In simulations, we compared the performance of mFA against other metaheuristics such as FA, CFA, MFPA, and MCS. The mFA algorithm performed better than CFA, MFPA, and MCS with the most consistent number of joint solutions and best results in position and orientation errors. The error results of FA and mFA were similar. However, the number of joint solutions proved that mFA outperformed the multimodal capacities of FA.

The performance of mFA was also compared to the geometric approach. This comparison demonstrated that mFA finds multiple joint configurations with high accuracy, similarly to the geometric approach exact solution.

The Sign test and the Wilcoxon signed ranks tests were included for pairwise statistical comparison. The mFA algorithm showed a significant statistical difference against FA, and MCS concerning the position and orientation errors results from simulations.

The proposed approach applicability was also tested in real experiments. The 5 DOF KUKA Youbot™ arm manipulator with real joint limits constrains was considered in these tests. The mFA algorithm shows a significant improvement over the FA and MCS methods with more joint solutions and better error results.

In future works, the proposed approach can be extended to solve the inverse kinematics of mobile manipulators [49] and dual-arm systems [50]. Moreover, mobile robot applications [64, 65, 66] may be solved as multimodal optimization problems. These algorithms have proven to solve many applications in the field of robotics, so it is interesting to propose the use of these algorithms to solve problems of recent applications such as, cognitive developmental robotics [67], robotic motion planning [68], human-robot interaction [69], coordination controller design [70], and visual recognition for social robot [71].

Footnotes

Acknowledgments

Authors would like to thank the University of Guadalajara, and the support of CONACYT Mexico, through the projects CB256769, CB258068, and FOINS 241246.

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A modified firefly algorithm for the inverse kinematics solutions of robotic manipulators

Abstract

Keywords

1. Introduction

2. Robotic manipulator kinematics

5.1 Simulation results

Table 7 Pairwise comparison of mFA against FA and MCS algorithms under the Sign test. The position error results of Planar of 2 DOF, Anthropo-morphic of 3 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Table 13 DH table for KUKA Youbot™ manipulator of 5 DOF

Footnotes

Acknowledgments

References

Table 7
Pairwise comparison of mFA against FA and MCS algorithms under the Sign test. The position error results of Planar of 2 DOF, Anthropo-morphic of 3 DOF, and Puma 560 of 6 DOF manipulators were used in this test

Table 13
DH table for KUKA Youbot™ manipulator of 5 DOF