Abstract
The precise tracking control problem for the robotic manipulator with flexible joints, subjected to system uncertainties and external disturbances, is addressed. A novel control scheme is presented that does not use link velocity measurements and high-order derivatives of the link states. The control scheme employs neural networks-based observers to estimate both motor velocity and link velocity. By using the virtually applied torque, the link controller is designed based on rigid link dynamics, and the motor controller is designed using the dynamic surface control technique. The proposed control scheme can guarantee that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded, and the tracking error eventually converges to a small neighborhood around zero. The simulation results confirm our theoretical analysis, and a comparison study demonstrates the advantages of the proposed control scheme compared to the standard DSC method.
1. Introduction
In recent years, robots have become standard collaborators not only in factories, hospitals, and offices but also in people’s homes, where they can play an important role in situations where a human cannot complete a task alone or needs the help of another person [1]. Flexible joints are extensively used in collaborative robots because, when a robotic manipulator with flexible joints (RMFJ) encounters obstacles during operation, the contact force between the RMFJ and obstacles may be relatively slight, allowing the RMFJ to stop immediately [2]. However, the introduction of joint flexibility in the robot model considerably complicates the equations of motion. The order of the related dynamics becomes twice that of rigid robots, and the number of degrees of freedom is larger than the number of control inputs. Therefore, achieving precise trajectory tracking control for RMFJ is difficult. Moreover, severe nonlinearities, coupling stemming from joint flexibility, structured and unstructured dynamical uncertainties, physical limitations, etc., are typical challenges that need to be addressed [3, 4].
To address the above problems, researchers have proposed a variety of control strategies for the tracking control of RMFJ including PD control [5], singular perturbation control [6, 7], backstepping control [8–10], adaptive control [11, 12], variable structure control [13, 14], intelligent control [15–18], and observer-based control [19, 20]. In particular, the backstepping technique is known as one of the popular techniques for controlling RMFJ. However, the backstepping technique has a drawback called the “explosion of complexity,” which is caused by the repeated differentiations of virtual controllers. Swaroop et al. [21] proposed a dynamic surface control (DSC) technique to solve this problem by introducing a first-order filter at each step of the traditional backstepping design. Many other useful research results have been reported for the control problems of various nonlinear real systems [22–24]. However, these proposed DSC schemes do not consider the errors caused by the introduction of filters, which may limit the performance of the system. As an alternative to DSC, command filtered-based control, proposed by Farrell et al. [25], can avoid the problem of the “explosion of complexity.” An error compensation mechanism can be constructed to compensate for filter errors and achieve better system tracking performance. The command-filtered technique has been successfully applied to RMFJ, and the desired control goals have been achieved by Ling et al. [26].
We all know that RMFJ inevitably suffers from nonlinear uncertainties due to unknown loads, inevitable friction, and electric motor aging. When these uncertainties are not addressed, the performance of the system will be seriously affected. With the development of intelligent control, approximating uncertain terms through neural networks (NNs) [16] and fuzzy logic systems (FLSs) [27] has become an effective control method for RMFJ. Shi et al. [18] propose a pattern-based control scheme for constrained flexible joint manipulators using the approximation and learning capabilities of NNs. The output constraint problem is handled through system transformation. Ling et al. [28] propose an adaptive fuzzy DSC scheme for single-link flexible-joint robotic systems with input saturation. The problem of input saturation is addressed by choosing a smooth function for approximation, and FLSs are used to approximate unknown continuous functions.
In addition to tracking issues, addressing the practical limitations of nonlinear systems during operation is very significant in applications. From a practical viewpoint, as pointed out in [29], in practical robotic systems, all the generalized coordinates can be precisely measured by the encoder for each joint. However, velocity measurements obtained through tachometers are easily perturbed by noise. Moreover, in today’s robotic applications, velocity sensors are frequently omitted due to considerable reductions in production costs, size, and weight of servo drives. Therefore, in order to align with economic and/or practical constraints, designing control strategies for robot dynamics based on nonlinear observers is of interest.
It is worth noting that, in order to achieve good control performance when designing the controller for RMFJ, it is necessary to consider two aspects. On one hand, the characteristics of the dynamic model need to be taken into account, such as high-order dynamics, uncertainty, nonlinearity, etc. On the other hand, practical limitations of the actual control system for RMFJ must also be considered. These limitations include immeasurable states, signal delays, unmodeled dynamics, periodic disturbances, and various constraints arising from physical limitations or security issues, especially during repetitive movements [9].
In this paper, we address the precise tracking control problem for RMFJ subjected to system uncertainties, external disturbances, and without link velocity. It is worth noting that there have been many excellent research results on this issue. Due to the introduction of flexible joints, the order of the RMFJ dynamics becomes twice that of rigid robots, and the RMFJ dynamics can be divided into robot dynamics and actuator dynamics. To solve the tracking control of flexible-joint manipulators, the concept of “virtually applied torque” is introduced [29]. The elastic force in robot dynamics is treated as a virtually applied torque, allowing the robot dynamics to be viewed as a conventional rigid robotic system. Inspired by this work, in this paper, we present a novel control scheme that does not require link velocity measurements and high-order derivatives of the link states. The proposed control scheme employs nonlinear observers to estimate both the motor velocity and link velocity. By using the concept of “virtually applied torque,” the link controller is designed based on rigid link dynamics, and the motor controller is designed using the DSC technique. NNs are used to approximate the uncertainties and disturbances of robot dynamics and motor dynamics. In comparison with the standard DSC method, the main contributions of this paper can be summarized in two parts as follows:(1)A novel NNs-based tracking control scheme for RMFJ, subjected to system uncertainties and external disturbances, is proposed. The control scheme consists of a rigid link controller and a motor controller. DSC only acts on motor dynamics, reducing the error accumulation caused by introducing first-order filters.(2)The proposed control scheme does not require link velocity measurements and high-order derivatives of the link states, such as acceleration and jerk. This makes it more suitable for practical applications.
It is shown that the proposed control scheme can guarantee the semiglobal uniform ultimate boundedness of all signals in the closed-loop system. A comparison is conducted to further demonstrate the main contribution of our proposed control scheme.
The rest of this paper is organized as follows: Section 2 presents the problem formulation and preliminaries, Section 3 states the control problems and the proposed solutions. Additionally, simulation results are shown in Section 4 to verify the effectiveness and potential of the proposed control scheme. Finally, Section 5 concludes with a summary of the obtained results.
2. Model Description and Problem Formulation
In general, the nominal dynamic models of an n-link RMFJ consist of robot dynamics and motor dynamics, which can be described using the following forms [26].where denotes the link position, is the inertia matrix. denotes the coriolis and centripetal forces, is the gravity vector, denotes the motor position, represents the joint flexibility, and is the motor inertia. The control vector is used as the torque input at each motor.
Property 1. The link inertia matrix is symmetric, positive definite. Both and are uniformly bounded as follows: and . Since is a constant matrix, , where , and are positive constants.
The nominal values , , , , and may be different from the actual values , , , and , respectively, due to model uncertainties and external disturbances. If the robot and the motor are further perturbed by external disturbances, the actual dynamics of the nominal RMFJ (1) and (2) can be expressed as follows:
Assumption 1. Suppose that, for a given RMFJ, only the nominal values , , , , and are known, but the actual values , , , and , including the model uncertainties and external disturbances, are unknown.
Assumption 2. The system states and are both available for feedback.
Property 2. is bounded as follows: , where is a positive constant.
Property 3. The matrix is skew-symmetric.
The actual dynamics of RMFJ (3) and (4) can be rewritten in the following formulation using the nominal model:where and denote the uncertainties of robot dynamics and motor dynamics of RMFJ, respectively, and are the external disturbances.
As a preliminary to the control design, if we define the state space variables as , , and , the RMFJ with uncertainty terms (5) and (6) is described as follows:
For Assumption 1, the uncertainty terms and cannot be directly evaluated. In this paper, NNs will be employed to observe the uncertainty terms. It has been previously recognized that any continuous function can be uniformly approximated by a linear combination of Gaussians. Hence, the following expressions exist as follows:where and are the input vectors, and denote the optimal weights in the approximate to ensure that the approximation error, and and are as small as possible. and are bounded approximation errors.
Remark 1. In recent years, NNs have undergone rapid development, and many advanced NN models have been proposed. This paper focuses on designing a new control scheme using the concept of “virtually applied torque.” It mainly demonstrates the development of controllers and NN tuning laws within a framework. Therefore, a simple RBF (radial basis function) NN model is chosen to approximate the unknown nonlinear system functions. In future work, more advanced NN models will be explored to investigate new control algorithms.
Definition 1. (SGUUB) [30]: The solution of the system is semiglobally uniformly ultimately bound (SGUUB) if for any compact set and all , there exists an and such that for all .
In this paper, the trajectory tracking control problem for RMFJ is investigated. The main objective of this research is to ensure that the RMFJ tracking errors are semiglobally uniformly ultimately bounded (SGUUB), even in the presence of system uncertainties, external disturbances, and the absence of link velocity signals.
3. Control Design
3.1. Control Scheme
In this section, we first focus on the rigid part (8) of RMFJ. By considering the elastic force as a virtually applied torque, this part can be treated as a conventional rigid robotic system. It is important to note that, compared to flexible robotic systems, there is a wealth of research available on the design of controllers for rigid robotic systems. Under Assumption 2, where the system state is available for feedback, the reference trajectory of the motor can be determined using the rigid control law for RMFJ tracking. Subsequently, the motor controller can be designed to track this trajectory.
Therefore, in this paper, a NNs-based tracking control scheme is developed for RMFJ, considering system uncertainties, external disturbances, and the absence of link velocity signals, by utilizing the concept of “virtually applied torque.” The schematic diagram of the proposed control scheme is shown in Figure 1. The control scheme employs a nonlinear observer to estimate both the motor velocity and link velocity. The link controller is designed based on rigid link dynamics, while the motor controller is designed using the DSC technique. Additionally, the NNs are utilized to approximate uncertainties and disturbances of robot dynamics and motor dynamics.
Remark 2. A similar control scheme is used in literature [29], but it adopts the backstepping method. To avoid the drawback of “explosion of complexity,” we utilize the DSC technique to design the motor controller.
Remark 3. A practical and effective controller for RMFJ is designed in literature [2] based on motor state feedback and the DSC technique. However, the use of too many first-order filters can lead to accumulated errors in the system’s boundary layer and reduce control accuracy. In this paper, by employing the concept of “virtually applied torque,” we only utilize the DSC technique to design the controller for motor dynamics. This approach helps to reduce the error accumulation caused by introducing first-order filters.
Since the link velocity is not available, nonlinear observers are used to estimate both the motor velocity and link velocity. In the development of the observers and the controllers, all quantities with “^” represent estimated quantities. In addition, quantities with “∼” represent estimation error vectors. The position estimation error is given by .
The estimates and of the system state are defined as follows:where is the link position estimation error, and is a positive constant.
The designed observer takes the form ofwhere , and are observer gain matrices. and represent the current values of the NNs weights as provided by the adaptive law.
Therefore, the link observer can be rewritten in terms of and as follows:
Considering , the link observer can be rewritten as follows:
The error dynamics of the link observer is obtained from (16) and (8) as follows:where , .
Similar to the design of the link observer, the estimates and are defined as follows:where is the motor position estimation error, and is a positive constant.
The designed motor observer takes the form of:where , and are observer gain matrix. represents the current values of the NNs weights as provided by the adaptive law.
Therefore, the motor observer can be rewritten as follows:
Considering , the motor observer can be rewritten as follows:
The error dynamics of the motor observer is obtained from (21) and (10).where , .
The link dynamics is the rigid dynamic model, and we take as the control input of the rigid dynamic model:
Considerable research has been conducted on controller design for rigid robots. In this study, we opted for the widely used computed torque method with NN compensation [31] to design a controller for a rigid link. Since the link velocity is not directly measurable, we utilized the estimated velocity value instead of the actual velocity as follows:where is the position gain matrix and is the velocity gain matrix.
We choose the adaptive law of as follows:where , is a positive constant matrix, is a positive constant.
Based on the structure of (3), we design the desired motor position vector to be:
Considering the structural complexity of , we design the controller based DSC method for motor dynamics.
Let pass through a first-order filter to obtain a new variable :where the time constant is the design constant.
Step 1: define dynamic surface error , whose time derivative along (27) is given by
To stabilize , a virtual control law is proposed as follows:where is a positive constant.
Let pass through a first-order filter to obtain a new variable :where the time constant is the design constant.
Step 2: define dynamic surface error , whose time derivative along (30) and (21) is given by:
A practical control law is proposed as follows:where is a positive constant.
We choose the adaptive law of as follows:where , is a positive constant matrix, is a positive constant.
3.2. System Stability Analysis
Letas the Lyapunov function candidate, where and are, respectively, Lyapunov functions of link observer and motor observer, and are, respectively, Lyapunov functions of link controller and actuator controller.
The purpose of link controller is to make converge to zero. Because cannot be measured directly, the link observer is designed to estimate . If the link observer can be guaranteed to converge, and the estimated link velocity can track the expected velocity , can track the expected position , it can be proved that the designed controller can guarantee the desired trajectory tracking.
Define new error variables as: and , that is, as long as converges, accurate tracking can be achieved as follows:
Let , and substituting (24) into (16), can be expressd as follows:where , .
Here we choose the appropriate matrix to ensure that the following equation holds:where is a positive definite symmetric matrix.
Hence, we can define the following Lyapunov function of link controller:
Notice that there is an error can be expressed as follows by substituting (24) into (26):
Notice that there is an error can be expressed as follows by substituting (39) into (29):
Substituting (40) into (28), the derivative of can be expressed as follows:
Substituting (32) into (31), the derivative of can be expressed as follows:
The derivative of can be expressed as follows:where
The derivative of can be expressed as follows:where , .
Hence, we can define the following Lyapunov function of motor controller:
Taking the time derivative of and using (17) produce:
Using Property 3 and (25), and considering , (46) can be simplified as follows:
Taking the time derivative of and using (22) produce:
Using (33), and considering , (48) can be simplified as follows:
Taking the time derivative of and using (36) produce:
Taking the time derivative of and using (41)–(44) produce:
Let , .
Taking the time derivative of and using (47)–(50) produce:
Then we can obtain by using Young’s inequality:where represents the minimum eigenvalue of matrix, represents the maximum eigenvalue of matrix, and and are maximum eigenvalue of the weight matrix.
Using Property 1, the following inequality holds:
Let , and they are all positive, and it follows that:where and .
By selecting the appropriate parameters, we can make , when , , is an invariant set. Therefore,
The inequality (57) illustrates the results that the system tracking errors, velocity observation errors, and NNs weights estimation errors will converge to a ball with a small radius determined by and , and then all the system states are SGUUB.
4. Simulation Results
This section shows a comparison study between our proposed scheme and the standard DSC method, to demonstrate the main contribution of our proposed control scheme.
To evaluate the dynamic behavior of the proposed control scheme, consider the single-link robotic manipulator with flexible joint [32]. The dynamic model is given by:
The nominal parameters are , , and . The initial states are set as . The parameters of link observer are set as . The parameters of motor observer are set as . The parameters of DSC controller are set as . The parameters of adaptive law (25) are set as The parameters of adaptive law (33) are set as
First, suppose that the dynamic model of RMFJ adopts the nominal parameters and the link velocity is not available, the proposed control scheme is used for the tracking control of the RMFJ and the desired trajectory is given by . Besides the proposed control scheme, the comparative studies are performed by using the standard DSC controller, the parameters of the standard DSC controller are set as , and the link velocity is available.
It is noting that if the input torque is directly used, it will lead to algebraic loop problems. The method we adopt is to pass through a low-pass filter to obtain . Because the actuator has low-frequency characteristics, is approximately equal to in the low-frequency range. Therefore, in the simulation, we use instead of .
The simulation results are shown in Figures 2–4. Figure 2 displays the system output trajectories () of the proposed and standard DSC method, and the given reference signal . We can see that the proposed control scheme and the DSC controller both achieve satisfactory control performance from Figure 2. Figure 3 shows the curves of tracking errors of both methods. Figure 4 shows that the error of the link observer. These results show that all the variables in the closed-loop system are bounded.
Figure 3 is a good illustration of the effectiveness of our proposed control scheme, where we choose the performance measure of maximum tracking error to compare the tracking performance of the system. The maximum tracking errors of these two methods in steady state are 0.0018 and 0.0191, respectively. It is shown that the tracking errors converge to a small neighborhood around the origin, and the tracking error of the proposed control scheme is smaller than that of the DSC controller. To further compare the tracking performance of the two control schemes, the ITAE index is calculated. The ITAE index of the proposed control scheme is 0.0093, and the ITAE index of the standard DSC method is 0.4513. It follows that the tracking performance of the proposed control scheme is better.
Remark 4. It is worth noting that the DSC method does not consider the errors caused by the introducing of first-order filters. In order to achieve precise control performance, in our proposed control scheme, the DSC acts only on motor dynamics, reducing the error accumulation caused by first-order filters. The simulation results, especially those in Figure 3, demonstrate the main contribution of our proposed control scheme.
Next, in order to illustrate the robustness of the control method, we made a change of 10% in the model nominal parameters. The actual model parameters and the external disturbances are set as , , , , , , and . The simulation results are shown in Figures 5–8. Figure 5 displays the system output trajectories () of the proposed and the standard DSC method, and the given reference signal . In Figure 5, the link position of the DSC controller shows an obvious vibration under the influence of parameter uncertainty and external disturbances. It can be observed from Figures 5–7 that even if the estimated parameters have large deviations and RMFJ is subjected to large external disturbance, the state observers can still accurately estimate the velocities of the link and the motor. The proposed control scheme in this paper can effectively achieve accurate trajectory tracking of link and suppress vibration of the flexible joints. The simulation results show that the proposed control scheme is robustness to external disturbance and the parameter uncertainty.
5. Conclusions
This paper proposed a novel control scheme for RMFJ. The issue associated with the tracking control of the RMFJ in the presence of system uncertainties and external disturbances is solved. Meantime, the proposed control scheme does not require link velocity measurements and high-order derivatives of the link states. The control scheme employs the NNs-based observers to estimate both motor velocity and link velocity. By using the virtually applied torque, the link controller is designed based on rigid link dynamics and the motor controller is designed by using DSC technique. The semiglobally uniformly ultimate boundedness of all signals within the closed-loop system is guaranteed by using the Lyapunov method. Numerical simulation results show that system output tracking errors converge to a small neighborhood around the origin. The standard DSC method is compared with the proposed control scheme, showing that the proposed control scheme can get better performance.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that there is no conflicts of interest.