1. Introduction
Industrial robots have been extensively implemented for several decades, with different applications of their mechanisms in operations like drilling, painting, screwing and welding across different industrial environments. It is important to mention that industrial robots are classified into two main categories: serial robots and parallel robots. Serial robots are the type of robots in which each link is connected to the other, successively, and parallel robots are those in which each link is connected in a parallel fashion. Among the most important serial robots implemented for industrial environments are the many types of six-degree-of-freedom mechanisms used for different tasks. Meanwhile, parallel robots, which are commonly implemented in different kinds of industrial applications, include the delta robot and the Stewart–Gough platform, which provide the necessary degrees of freedom (DOF) and flexibility in comparison with serial robots.
As we know, serial robots are implemented and studied due to the importance they have in the implementation of different mechanisms across different tasks, increasing productivity and reliability on assembly lines. Nevertheless, it is important to remark that many experimental robots are currently being developed, taking into consideration the necessity for new kinematic and dynamic analyses. Additionally, novel control strategies need to be designed to obtain optimal performance for trajectory tracking of the end effector, especially in diverse industrial applications. For these reasons, it is important to analyze the kinematic and dynamic properties of experimental or laboratory robotic manipulators in order to provide relevant theoretical, experimental, and practical conclusions about any type of implementation of robot manipulators.
Kinematics plays an important role in the mathematical modeling of robot manipulators. There are many kinematic model methodologies that define the direct and inverse kinematics of a robot manipulator. Among these strategies are the Denavit-–Hartenberg convention, Hamilton quaternions, and screw theory. It is crucial to mention the following research papers in which these kinematic model methodologies are found. So, for example, in [
1], a novel inverse kinematic model was utilized for a novel 6R manipulator by implementing quaternions. Additionally, in [
2], the kinematics of a parallel robot were evinced using lie group theory. Other interesting results are found in papers like [
3], in which the forward kinematics of robot-assisted surgery are presented. Moreover, in [
4], the kinematic reliability of an industrial robots is evinced. In [
5], a forward kinematics analysis of a general Stewart–Gough platform using a multibody formulation is performed; in [
6], a 7R six-degree-of-freedom robot with a non-spherical wrist kinematic calibration procedure is presented. Furthermore, another interesting piece of research is shown in [
7], in which the kinematic calibration of a 5-DOF machining robot is performed using the Kalman filter approach. All these results are important to the present study, taking into consideration that they provide the theoretical framework for the analyzed robotic mechanisms. It is important to consider that in the above-mentioned research studies, there are some strategies, such as quaternions, that are very important to the design of a kinematic model for an RV-3SB robot using screw theory.
One of the main issues on which the present paper is focused is the dynamic modeling of an RV-3SB robot. The approach used in this research study presents their fundamentals utilizing an Euler–Lagrange approach, but considering the kinematic model obtained by screw theory for the analyzed robots, it is important to consider that the dynamic model derivation is significantly reduced, which is a must when dynamic model derivation for a new robotic mechanism is obtained. The following references are important for the present study, taking into consideration the contribution to the dynamic model derivation of the RV-3SB robot. For example, in [
8], bi-stable dynamics in a two-module vibration-driven robot are presented. Additionally, in [
9], the dynamics equations of a Hexabot robot are obtained using the Lagrange equation of a second kind. Furthermore, in [
10], neural dynamics-driven control of a redundant robot is provided. In [
11], remote teaching of dynamics and control of robots is performed. Other studies that have provided significant results can be found in papers such as [
12,
13], according to the dynamic model mathematical derivation for different types of robots. The implementation of different dynamic model techniques in the above studies is an important contribution to the present paper, considering that dynamic models based on the Euler–Lagrange formulation are significant for the dynamic model derivation of the RV-3SB robot analyzed in this research.
Screw theory is fundamental to the derivation of feasible kinematic models for forward kinematics and inverse kinematics; for this reason, it is important to mention the following studies. For example, in [
14], a complete review of screw theory for the kinematic modeling of serial and parallel robots is presented. In this chapter, it is stressed that reciprocal screw theory facilitates obtaining compact and elegant kinematic model representations for the previously mentioned types of robots. Meanwhile, in [
15], it is shown how screw theory is implemented for a force model of a crane. In [
16], a configuration synthesis of deployable antennas based on screw theory is proposed. Another interesting paper that is worthy of citation is found in [
17], in which a higher-order representation of metamorphic mechanisms based on screw theory is presented. In [
18], screw theory is used to conduct a vibration analysis of a space parallel robot. Furthermore, in [
19], a mobility analysis of scissor-like elements is presented via the implementation of reciprocal screw theory. As we know, the screw consists of defining the velocities and moment or force vectors which represents the rotation about an specified axis. Screw theory provides a compact and simplified model that is used for the forward and inverse kinematics derivations of our studied and analyzed robot. The previous cited research results provides the mathematical fundamentals for this research study.
Nowadays, there is a vast amount of control strategies for different types of robots, which are independent if they are serial chain or parallel chain configurations. Therefore, it is important to mention several research studies related to this topic in order to provide an introductory theoretical framework for the passivity-based control strategy of the RV-3SB robot. Among the papers found in the scientific literature, [
20] presents the visual control of a robot using a Deep Reinforcement Learning technique. In [
21] the control of a laparoscopic robot is performed using a leap motion sensor. In [
22], the coordinated control of a space robot manipulator is evinced. Another interesting result is shown in [
23], in which the robust variable admittance control of a human–robot manipulator is shown. In [
24], a non-singular terminal sliding mode controller for a prosthetic leg robot is described. Ultimately, in [
25], the robust control of a planar snake robot using Takagi–Sugeno fuzzy control is presented.
Passivity-based control is the main control strategy implemented in this research study, so it is worth mentioning the following passivity-based control strategies. In [
26] the control of a piezoelectric actuator is achieved using a Krasovskii passivity-based approach. Then, in [
27], the instability problem caused by the constant power load in a Buck converter is addressed. Other important results are found in papers such as [
28], in which a self-balancing robot is controlled by a proportional–integral–derivative (PID) passivity-based control strategy. In [
29], an underactuated three-dimensional crane is controlled by a passivity-based controller. Then, in [
30], a three degrees of freedom 3-DOF crane is controlled using a passivity-based adaptive trajectory tracking controller. Finally in [
31], a spacecraft attitude simulator is controlled using a passivity-based sliding mode controller. Passivity-based control, as we know, consists of a control strategy based on the energy consideration of the nonlinear dynamic system to be controlled. It is important to mention that this is the main control strategy for this research paper. This strategy is novel and considers the energy properties of the nonlinear dynamics of the RV-3SB robot mechanism.
In this paper, we propose the passivity-based control of a Mitsubishi RV-3SB robot. First, the kinematic model of the robot with the implementation of screw theory is obtained. Then, the dynamic model of the robot manipulator is obtained via the implementation of the Euler–Lagrange formulation, yielding a decoupled non-linear dynamic model for the position and orientation of the Mitsubishi RV-3SB robot. The passivity-based controller of the robot is then obtained by selecting a suitable storage function. This choice ensures that the passivity-based condition is met by implementing the selected control law, thereby satisfying the closed-loop stability requirements. It is important to remark that the passivity controller consists of the passivity-based control law along with a output feedback controller, allowing it to achieve the desired trajectory tracking system performance. A numerical experiment is conducted in order to validate the theoretical results provided in this research study. Finally, a discussion and conclusion section is presented to discuss the results of this research study.
The advantages of the proposed controller, as demonstrated in this research study, are summarized in comparison to other control strategies found in the literature, as follows:
The passivity-based control law is relatively simple in comparison with other strategies found in the literature.
Considering that this control strategy is novel and has not been found in the literature, it offers a suitable approach to solving this kind of problem.
Naturally, dynamical systems established in the Lagrangian and Hamiltonian mathematical representations possess energy considerations. Therefore, the passivity-based controller proposed in this paper offers an appropriate control strategy, taking into account the dynamics of the RV-3SB robot.
As verified experimentally and theoretically, the passivity-based control strategy presented in this paper yields a better closed loop performance in comparison with other control strategies.
The passivity-based controller evinced in this research study yields a better low computational effort considering the implementation in real-time hardware such as hardware in the loop.
The parameters of this passivity-based controller are easy to tune with some evolutionary and/or optimization algorithm. This will be considered as a future direction of this research study.
Table 1 shows a qualitative comparative analysis of four control strategies for robotic manipulators. These are the most relevant references mentioned in this research study, so it is important to evince this comparative analysis that will be corroborated later in the numerical simulation section. The advantages and disadvantages of these control techniques are presented in order to demonstrate the weakness and strengths of these control approaches.
This paper is divided into the following sections: in
Section 2, the relevant related work is presented; meanwhile, in
Section 3, the theoretical derivations of the dynamic model of the robot and the control strategy are presented; then, in
Section 4, two numerical experiments are performed; finally, in
Section 5 and
Section 6, the discussion and conclusions of this research strategy are presented.
2. Related Work
This section provides a comprehensive review of closely related work in order to demonstrate the importance of the present research study. This section is focused on the following issues, which are important in order to evince a detailed overview of the most relevant research studies found in the scientific literature:
To begin this literature review, we start with some kinematics techniques for industrial robots, taking into consideration that in this present research study, the screw theory is considered for both the kinematic and dynamic modeling of the RS-3SB robot. For instance, in [
34], a differential kinematic modeling of a mobile robot is presented. Meanwhile, in [
35], a kinematic workspace model of 3R, 4R, 5R, or 6R robots is presented. Then, in [
36], the meaning of four co-reciprocal screws is presented in terms of its kinematic significance. In [
37], the identification of kinematics parameters of a multi-link manipulator is presented. Furthermore, several other research studies explore the kinematics of diverse types of robotic manipulators. For instance, in [
38], a new formulation of the inverse dynamic of a parallel delta robot is provided. Similarly, in [
39], a closed form for inverse dynamic model of the delta parallel robot is presented.
Other interesting strategies for kinematic modeling of serial and parallel robots are found in papers such as [
40], in which Clifford algebra is implemented for kinematic control of a serial robot. Then, in [
41], an automatic approach to identify the parameters of a serial link chains using reciprocal screws is presented. Meanwhile, in [
42], screw theory and dual quaternions are used for motion controllers. In [
43], conformal geometric algebra is implemented in order to obtain the inverse kinematic of serial robots. Furthermore, in [
7], an extended Kalman filter is implemented in order to obtain the kinematic calibration of a 5-DOF hybrid machining robot. Finally, in [
1], the inverse kinematic formula for a new class of 6R robot is presented.
Industrial and many kinds of novel robot manipulators are crucial to mention in this research paper, taking into consideration the results that are important to the present research study. For instance, in papers such as [
44], a bipedal walking robot is shown. Other interesting results are found in [
45], in which a model control for industrial robots is presented. In [
46], the adaptive output feedback tracking control of a non-holonomic mobile robot is presented. Meanwhile, in [
47], a control strategy for two link underactuated planar robots is presented. In [
48], a stable controller for a robot manipulator is evinced.
Dissipation properties of nonlinear dynamics systems are important, considering the characteristics of passivity-based controllers. For instance, in [
49], the dissipative characteristics of a negative imaginary system are exploited in order to design a control strategy. In [
50], the dissipative properties required for infinite dimensional continuous control are studied. Meanwhile, in [
51], the dissipative output feedback control of Markovian jump systems is presented. In [
52], the dissipative optimal infinite dimensional control is studied. Furthermore, interesting results are found in [
30], in which dissipative discrete time stochastic delayed systems are analyzed. Finally, a discrete-time neural network and its dissipative control are designed in [
53].
Passivity-based control is one of the control strategies developed for various types of physical systems, including electrical and mechanical systems. As mentioned earlier, this strategy involves selecting a storage function to ensure that the chosen control law satisfies the passivity requirements. In [
54] the passivity and power-based control of a robot is presented; in [
55], the adaptive passivity-based force controller for an uncertain system is studied; in [
56], the passivity-based control of hydraulic robots is evinced. Furthermore, a distributed passivity-based controller for constrained robotics networks is shown in [
57], and in [
58], new results in passivity-based control for robots are presented. Finally, other control strategies such as backstepping control, sliding mode control, output feedback robust control, among others, can be found in [
23,
59,
60,
61,
62,
63].
3. Dynamic Modeling of the RV-3SB Robot Manipulator and Controller Design
The dynamic modeling of the RV-3SB (
Figure 1) robot is presented in this section. In order to obtain a simple dynamic model, the Euler–Lagrange formulation is applied in this section. Considering that the last three axes of the robot intersect each other, the dynamic model of this robot is decoupled, which implies that the robot’s position and orientation can be analyzed independently. Furthermore, this dynamic model formulation is suitable for trajectory tracking controller design, as verified in the control design section. Consider the following kinetic and potential energy functions:
In which
I is the inertia matrix,
M is the mass matrix,
g is the gravity constant, and
is the total mass of the robot, as stated below:
with the following vectors:
In which
, for
, are the actuator angles, and
x,
y and
z are the end effector coordinates. Basically, the vector
q is related to the orientation and
X is related to the position of the designed robot. Consider the following Lagrangian:
Now consider the Euler–Lagrange formulation:
obtaining the following dynamic system:
In which
,
is a virtual force and
is the actuator torques. So, the previous equation can be transformed into a state space formulation as follows:
For
,
and
with the following matrix:
The parameters of the robot are given in the following paragraph, as shown in the paper [
64]
Six Degrees of Freedom.
Repeatability: ±0.02 mm.
Maximum speed: 5500 mm/s.
Range of Motion (degrees): J1 = 340, J2 = 225, J3 = 191, J4 = 320, J5 = 240, J6 = 720.
Maximum speed in each joint (deg/seg): J1 = 250, J2 = 187, J3 = 250, J4 = 412, J5 = 412, J6 = 660.
Weight: 37 kg.
It is important to remark that the kinematic model of this robot is obtained via screw theory; the following screws are defined in
Table 2 as shown in [
64]:
Passivity-Based Control of the RV-3SB Robot Manipulator
For the passivity-based control strategy of the RV-3SB robot manipulator, the required energy considerations are taken into account in order to ensure the closed loop stability. It is important to recall that passivity-based control is an important technique for robotic systems that has been implemented over decades for different types of mechanisms. To obtain the obtained passivity-based control law consider the following theorem:
Theorem 1. The dynamic model of the robot RV-3SB (7) represented in the state-space model, as appears in (8), is passive if the following control law is implemented: In which the error variable is given as follows: is the reference variable for trajectory tracking purposes, and . The matrices gains are defined as and . The control input is given by in which is a gain matrix.
Proof. Consider the following Lyapunov functional:
By obtaining the first derivative of the previous Lyapunov functional and making the appropriate substitutions, we yield:
Then, by substituting (
10) into (
13), we obtain:
Thus, the system is passive and stable and the proof is completed. □
Now, in order to obtain the virtual force input , the following property is necessary.
Property 1. Consider the following transformation matrices [64] in which the following matrices establish the necessary coordinate transformation to each axis: Then, considering that the control input vector is given by and the input vector is given by ; the virtual control input is found by solving the following system of equations: In which is the fourth column of the transformation matrices, meanwhile, is a zero row vector of three elements. The inverse of the matrix which appears in (16) is solved by the Moore–Penrose pseudo-inverse.