Output Feedback Control via Linear Extended State Observer for an Uncertain Manipulator with Output Constraints and Input Dead-Zone

Duc Thien Tran 1,2 , Hoang Vu Dao 2, Truong Quang Dinh 3 and Kyoung Kwan Ahn 2,* 1 Automatic Control Department, Ho Chi Minh City University of Technology and Education, Ho Chi Minh City 700000, Vietnam; thientd@hcmute.edu.vn 2 School of Mechanical Engineering, University of Ulsan, Ulsan 44610, Korea; hoangvudaocsp@gmail.com 3 Warwick Manufacturing Group (WMG), University of Warwick, Coventry CV4 7AL, UK; T.Dinh@warwick.ac.uk * Correspondence: kkahn@ulsan.ac.kr; Tel.: +82-52-259-2282


Introduction
In recent years, robots have attracted the interest of many researchers in institutes, universities, and technology companies around the world [1]. Challenges such as highly nonlinear dynamics, modeling error, and external disturbances can degrade the control performance of the robotic manipulator. In order to improve the accuracy and reliability of the robotic manipulator, researchers have developed controller approaches to handle these problems. Some well-known robotic controllers such as computed torque control [2], backstepping control [3][4][5], and sliding mode control [3, [5][6][7][8], etc. have been widely applied in robotic applications.
Backstepping control is one of the most useful techniques for controlling nonlinear systems [9], regardless of the mismatched and matched uncertainties. In order to improve the effectiveness of the backstepping control, some advanced approximators, such as fuzzy logic systems (FLSs) [10][11][12], neural networks [13][14][15][16], and extended state observers [17,18], were applied to the backstepping control to compensate for the uncertainties. In Reference [10], an FLS was used in an advanced backstepping control to approximate the unknown nonlinearities of a manipulator. In Reference [19], Wang et al. designed an adaptive fuzzy backstepping control for an underwater vehicle manipulator system; the FLS was used to estimate the system parameters. The results in these papers proved the

Robotic Manipulator Dynamics
In this paper, we consider an n-DOF manipulator under the presence of unknown friction, input dead-zone, and external disturbance. Its dynamics in the joint space are expressed by [2] M(q) .. q + C(q, . q) where q, . q, .. q ∈ R n×1 present angular position, velocity, and acceleration vectors in the joint space of the manipulator, respectively; M(q) ∈ R n×n presents the inertia matrix; C(q, . q) ∈ R n×n expresses the Coriolis and centrifugal term matrix; G(q) ∈ R n×1 derives the gravity vector; J(q) is a nonsingular Jacobian matrix; f ext presents the external disturbance vector; τ f ric is the unknown friction vector; and H(τ) is the torque vector acting on joints within the dead-zone.
The friction model, τ f ric , is exhibited as where b ∈ R n×n and c ∈ R n×n are positive diagonal matrices; ψ is a positive constant; Assumption 1 [13]. The input dead-zone nonlinearity presented in Figure 1 can be derived as follows: where τ r and τ l are unknown constants for which: τ r > 0, τ l < 0, and H(τ) = [H(τ 1 ), . . . , H(τ n )] T .

Robotic Manipulator Dynamics
In this paper, we consider an n-DOF manipulator under the presence of unknown friction, input dead-zone, and external disturbance. Its dynamics in the joint space are expressed by [2] ( ) ( ) ( ) H τ is the torque vector acting on joints within the dead-zone.
The friction model, fric τ , is exhibited as where n n R × ∈ b and n n R × ∈ c are positive diagonal matrices; ψ is a positive constant; and Property 1 [34].
 is a skew-symmetric matrix, that is provided as Assumption 1 [13]. The input dead-zone nonlinearity presented in Figure 1 can be derived as follows: Electronics 2020, 9,1355 4 of 20 Based on Equations (3) and (4), the dead-zone functions can be represented as follows: where ξ(t) is an unknown vector whose elements are smooth functions. Additionally, their derivatives are bounded. Let x 1 = q ∈ R n and x 2 ∈ . q ∈ R n ; then the robotic dynamics (1) can be rewritten as follows: where including the unknown friction, external disturbance, and an unknown vector of the input dead-zone, and u depicts the torque control signal.
In this paper, we design an advanced control to guarantee that the output responses track a reference T are time-varying functions.
Assumption 3. In this study, we suppose that the manipulator operates in a bounded workspace. It means that the reference signals, x di , i = 1, 2, . . . , n, are bounded and known, |x di | ≤ X d0 , and X d0 is a positive constant.

Control Design
Before we design the LESO to estimate the lumped disturbance and the unmeasured states in the manipulator, an extra state x 3 ∈ R n×1 is added in the manipulator dynamics (6) to present the lumped disturbance, M −1 (x 1 )∆. The system state is presented as Note that the extra state x 3 is continuously differentiable and bounded as in [35,36].

Assumption 4.
The difference of the state x 3 (t) is assumed to be bounded, i.e., δ(t) ∞ ≤ δ, where δ is a positive constant.
The manipulator dynamics (6) is represented as follows: where δ(t) presents the derivative of the state x 3 (t);

Assumption 5.
We suppose that the functions F(x 1 , x 2 ) are locally Lipschitz for x 2 in its practical range.

Linear Extended State Observer Design
The LESO is designed to not only approximate the lumped disturbance, x 3 , but also estimate the unmeasured system state vector, x 2 . Letx express the estimated system state of x and let x present the estimation error, ( x = x −x). Now, we represent the robotic dynamics (7) as follows: Electronics 2020, 9,1355 5 of 20 The extended state observer is presented as follows: where φ(x) = [ 0 n×n F(x 1 ,x 2 ) 0 n×n ] T , κ = [ 3κ 0 I n×n 3κ 2 0 I n×n κ 3 0 I n×n ] T ∈ R 3n×n presents the observer gain matrix; and κ 0 > 0 is adjusted to enhance the observer performance. From Equations (8) and (9), the estimation error dynamics is computed as where , and then the LESO (10) is represented as From Assumption 5, the below inequality can be obtained: Theorem 1. When the LESO (9) is used to estimate the lumped disturbance of the unmeasured states, and the inequality (12) is guaranteed, then the estimation errors are bounded with the appropriate constant.
Proof of Theorem 1. A Lyapunov function is taken into account as follows: where P derives a positive definite matrix. It is chosen as a solution of the following Lyapunov equation: From (11), the differential Lyapunov function is presented as Electronics 2020, 9, 1355 6 of 20 The differential Lyapunov Function (15) λ max (P). The estimation errors in LESO are reducing and the stability of the ESO is guaranteed [37] when the bandwidth, κ 0 , increases. Figure 2 presents the structure of the proposed control with an n-DOF manipulator under the presence of unknown external disturbance, friction, input dead-zone, and output constraints. The proposed control consists of a full state feedback control based on the BLF, and a linear extended state observer. The backstepping control is designed with the barrier Lyapunov function to avoid the violation of the output constraint. Because all states, such as position and velocity, of the manipulator are used to design the backstepping control, this control is named as a full state feedback control-based barrier Lyapunov function. The extended state observer was employed to approximate the lumped disturbance in the manipulator dynamics and unmeasured variable,x 2 .

Proposed Control Design
The differential Lyapunov Function (15) is negative when The estimation errors in LESO are reducing and the stability of the ESO is guaranteed [37] when the bandwidth, 0 κ , increases. □ Figure 2 presents the structure of the proposed control with an n-DOF manipulator under the presence of unknown external disturbance, friction, input dead-zone, and output constraints. The proposed control consists of a full state feedback control based on the BLF, and a linear extended state observer. The backstepping control is designed with the barrier Lyapunov function to avoid the violation of the output constraint. Because all states, such as position and velocity, of the manipulator are used to design the backstepping control, this control is named as a full state feedback controlbased barrier Lyapunov function. The extended state observer was employed to approximate the lumped disturbance in the manipulator dynamics and unmeasured variable, 2 x .   The tracking errors, e i , (i = 1, 2), are defined as

Proposed Control Design
where x d ∈ R n×1 is the reference signal. The time-varying upper and lower boundary errors of e 1 are computed as: where k ci (t) ≤ x 1i (t) ≤ k ci (t), k ai is the time-varying upper boundary error, and k bi is the time-varying lower boundary error. The virtual control, α 1 is computed as where λ 1 ∈ R n×n is a positive diagonal matrix; λ 1 = diag([λ 11 , . . . , λ 1n ]) ∈ R n×n expresses a positive diagonal matrix. The elements in the matrix λ 1 are stated as Electronics 2020, 9, 1355 7 of 20 The control input is calculated as follows: where and K 2 ∈ R n×n is a positive diagonal matrix.

Stability Analysis
Lemma 1 [22]. The following inequality holds for any positive constant k ∈ R and x ∈ R so that |x| < k: Theorem 2. The control law in (18) and (20), which utilizes the estimation value of the unmeasurable state, and the lumped disturbance from the ESO in (9) guarantee the ultimately uniformly bounded tracking performance and satisfaction of the output constraint in (17) of the manipulator described by (1), under unknown friction, external disturbance, and an unknown vector of the input dead-zone.
Proof of Theorem 2.
Step 1. We take the time derivative of the position error e 1 shown as From (18), substituting the virtual control signal, α 1 , into (23), the result is represented as Based on Yu et al. [38], to guarantee the constrained performance of the joint angles, we can select a barrier Lyapunov function as follows: In order to simplify the BLF (25), we state variables by As a result, the Lyapunov Function (25) is rewritten as follows: The differential barrier Lyapunov function is calculated by Electronics 2020, 9,1355 8 of 20 Substituting (24) into (28), the result presents as Step 2. Based on (6), the time derivative of error, e 2 is expressed as follows From (30) and (20), the time derivative of error, . e 2 is represented as To consider the stability of the dynamics system (1), including not only the position tracking performance at Step 1, but also the speed tracking performance (31), the following Lyapunov function is investigated as Next, the time derivative of the Lyapunov function (32) is calculated as Replacing (29) and (31) into (33) with property 2, the result is expressed by In order to demonstrate the stability of the entire closed-loop system, including the estimation performance of the ESO, we choose a Lyapunov function via (15) and (32) as follows: The time derivative of the Lyapunov Function (35) is computed as as the disturbance estimation error. These inequalities hold: −e T 2 ε ≤ 1 2 e T 2 e 2 + 1 2 ε T ε and e T 2 K 2 x 2 ≤ 1 2 e T 2 K 2 e 2 + 1 2 x T 2 K 2 x 2 . Thus, when Lemma 1 is investigated, Equation (36) is rewritten as below: From Reference [39], we can state that when the system is controlled by the proposed control, it is ultimately uniformly bounded under the presence of unknown frictions, external disturbances, and input dead-zone. From (37), we can find the inequation as follows: where µ = D 2 c 0 . From (25), (26), (35) and (38), we achieve the results as After we take the exponentials on both sides of (40), the results are given as The following inequality can then be achieved: As a result, we can conclude that the output constraints are guaranteed. The proof is complete.

Simulation Descriptions
Some simulations were conducted on MATLAB Simulink with a 3-DOF planar manipulator to illustrate the superiorities of the proposed control. The MATLAB Simulink was configured with a sampling time of 0.001 s; the solver type was ODE3. Additionally, the simulation time was 30 s.
The 3-DOF planar manipulator presented in Figure 3 is a planar robot with 3 rotary actuators. The parameters of the manipulator are presented in Table 1. Additionally, all mass exists as a point mass at the distal end of each link, and the center of mass in each link is presented by i P C = l i X i , (i = 1,2,3) . By using the Newton iteration method in [2], the dynamics of the manipulator are presented in the Appendix A. The 3-DOF planar manipulator presented in Figure 3 is a planar robot with 3 rotary actuators. The parameters of the manipulator are presented in Table 1. Additionally, all mass exists as a point mass at the distal end of each link, and the center of mass in each link is presented by By using the Newton iteration method in [2], the dynamics of the manipulator are presented in the Appendix A.  Gravity constant -- The friction model vector includes the viscous and coulomb frictions, which is presented as follows: The friction model vector includes the viscous and coulomb frictions, which is presented as follows:

Simulation Results
The merits of the proposed controller are illustrated through comparisons with two other controllers: • The backstepping control (BC): • The linear extended state observer via backstepping control (LESOBC): wherex 3 is the estimated lumped disturbance. This estimated lumped disturbance is approximated by the LESO in (9).
The parameters of these controllers are exhibited in Table 2. In order to ensure equality in comparisons between the controllers, parameters of the backstepping are firstly selected. Next, some parameters of the LESOBC are inherited from the BC, and others, the observer gains, are adjusted. Finally, the proposed control copies the parameters in the LESOBC and uses the upper and lower boundaries.
The simulations are divided into two cases. In the first case, we conduct simulations with the low-frequency references which are set to 0.1 Hz. In the second case, the frequency of the reference is increased to 0.5 Hz.

Controllers Parameters
Backstepping control

The First Simulation Case
Trajectories in the Cartesian coordinate with frequency at 0.1 Hz are generated for the 3-DOF planar manipulator. These references in the joint space are computed through the inverse-kinematic equations of the manipulator, which are presented in Figure 4. The references of joint 1, joint 2, and joint 3 are exhibited by a dashed black line, a dashed red line, and a dashed blue line, respectively. Figure 5 presents the output responses of the planar robot at the joint spaces. The results of the reference, backstepping control, LESOBC, and the proposed control are exhibited by dashed black lines, black lines, dashed dot black lines, and red lines, respectively. The boundaries of the output responses are presented by dashed dot black lines. From these results, we easily realize that the output responses of the backstepping control transgressed the output constraints. In order to exhibit the effectiveness of the proposed method, the differences between the references and output responses are presented in Figure 6. These results prove that the LESOBC and proposed control guarantee the satisfaction of the output responses with the output constraints under the presence of an unknown friction and external disturbance which arises at the 15th second.
The responses of the lumped uncertainties, x 3 , at three joints are illustrated in Figure 7a with a dashed blue line, dashed dot black line, and dashed red line for joint 1, joint 2, and joint 3, respectively. In the first 15 s, the lumped uncertainties are unknown frictions at the joint space. In the last 15 s, an external force is applied at the end-effector along the x-axis. As a result, the uncertainties in each joint include not only the unknown friction but also the external disturbance. Therefore, they increased significantly and reduced the accuracy of the backstepping control. Figure 7b presents the estimated lumped disturbance,x 3 , which is the result of the LESO. Comparing to the lumped disturbance in Figure 7a, the results of the estimated lumped disturbance in Figure 7b proved that the LESO has a limited bandwidth, which means it cannot approximate the disturbance at high frequency. The accuracy of the LESOBC and proposed control is improved significantly by using LESO to estimate the lumped disturbance. The estimated results of the LESO are presented in Figure 8. The estimated errors in position, velocity, and lumped disturbance are bounded. lines, black lines, dashed dot black lines, and red lines, respectively. The boundaries of the output responses are presented by dashed dot black lines. From these results, we easily realize that the output responses of the backstepping control transgressed the output constraints. In order to exhibit the effectiveness of the proposed method, the differences between the references and output responses are presented in Figure 6. These results prove that the LESOBC and proposed control guarantee the satisfaction of the output responses with the output constraints under the presence of an unknown friction and external disturbance which arises at the 15th second.   The responses of the lumped uncertainties, 3 x , at three joints are illustrated in Figure 7a with a dashed blue line, dashed dot black line, and dashed red line for joint 1, joint 2, and joint 3, respectively. In the first 15 s, the lumped uncertainties are unknown frictions at the joint space. In the last 15 s, an external force is applied at the end-effector along the x-axis. As a result, the uncertainties       Figure 9a when the input dead-zones are not compensated in the BC. In the LESOBC and proposed control, the torque responses are different from the BC because the input dead-zones in two controllers are compensated by the LESO. Finally, in order to evaluate the effectiveness of the proposed control in detail, the root mean square error (RMSE) is used for evaluating the responses of the three controllers. The results are shown in Table 3.
Electronics 2020, 9, x FOR PEER REVIEW 15 of 22 not compensated in the BC. In the LESOBC and proposed control, the torque responses are different from the BC because the input dead-zones in two controllers are compensated by the LESO. Finally, in order to evaluate the effectiveness of the proposed control in detail, the root mean square error (RMSE) is used for evaluating the responses of the three controllers. The results are shown in Table 3.

The Second Simulation Case
The trajectories in Cartesian coordinate with frequency at 0.5 Hz are applied for the 3-DOF planar manipulator. Similarly, compared to the previous simulation, the trajectories in the joint space of the manipulator are calculated by using the inverse-kinematic equations. Figure 10 presents the output error responses of three controllers at three joints. The output error responses of the backstepping control, LESOBC, and proposed control are respectively presented by black lines, dashed dot blue lines, and red lines. Additionally, the upper and lower error boundaries are plotted by the dashed dot black lines. The results in this figure show that the backstepping control still

The Second Simulation Case
The trajectories in Cartesian coordinate with frequency at 0.5 Hz are applied for the 3-DOF planar manipulator. Similarly, compared to the previous simulation, the trajectories in the joint space of the manipulator are calculated by using the inverse-kinematic equations. Figure 10 presents the output error responses of three controllers at three joints. The output error responses of the backstepping control, LESOBC, and proposed control are respectively presented by black lines, dashed dot blue lines, and red lines. Additionally, the upper and lower error boundaries are plotted by the dashed dot black lines. The results in this figure show that the backstepping control still transgresses the constraints and the LESOBC begins breaking the output constraints because the accuracy of the LESOBC is improved by the LESO and it does not depend on the output boundaries. The proposed control is developed by integrating the output constraints into the control design. Therefore, its output responses are still guaranteed, although the working frequency increases. With this simulation, we see the effectiveness of the BLF in tackling the output constraints. The lumped uncertainties, in this case, are the unknown frictions in the first 15 s and a combination of the unknown frictions and external disturbances in the last 15 s. Electronics 2020, 9, x FOR PEER REVIEW 16 of 22 transgresses the constraints and the LESOBC begins breaking the output constraints because the accuracy of the LESOBC is improved by the LESO and it does not depend on the output boundaries. The proposed control is developed by integrating the output constraints into the control design. Therefore, its output responses are still guaranteed, although the working frequency increases. With this simulation, we see the effectiveness of the BLF in tackling the output constraints. The lumped uncertainties, in this case, are the unknown frictions in the first 15 s and a combination of the unknown frictions and external disturbances in the last 15 s. The lumped disturbance responses, 3 x , are presented in Figure 11a. Because the frequency of the trajectory is increased from 0.1 Hz to 0.5 Hz and the lumped uncertainties concern the velocity, the amplitude of the lumped disturbances increased if we compare it with the previous simulation. Figure 11b derives the estimated lumped disturbance, 3 x , of the LESO. The lumped disturbance responses, x 3 , are presented in Figure 11a. Because the frequency of the trajectory is increased from 0.1 Hz to 0.5 Hz and the lumped uncertainties concern the velocity, the amplitude of the lumped disturbances increased if we compare it with the previous simulation. Figure 11b derives the estimated lumped disturbance,x 3 , of the LESO. The effectiveness of the LESO is presented in Figure 12, with the estimated error responses of the position, the velocity, and the lumped disturbance. The torque signals of the BC, LESOBC, and proposed control are exhibited in Figure 13. In the LESOBC and the proposed control, the control responses are also different from the BC because the input dead-zones are overcome by the LESO. The effectiveness of the LESO is presented in Figure 12, with the estimated error responses of the position, the velocity, and the lumped disturbance. The torque signals of the BC, LESOBC, and proposed control are exhibited in Figure 13. In the LESOBC and the proposed control, the control responses are also different from the BC because the input dead-zones are overcome by the LESO. Finally, in order to evaluate the effectiveness of the proposed control in detail, the root mean square error (RMSE) is used for evaluating the responses of the three controllers. The results are shown in Table 4.
In summary, the proposed controller that integrates the BLF-backstepping control and ESO is effective to control the manipulator at different motion frequencies (0.1 Hz and 0.5 Hz) under a bunch of problems as unknown friction, external disturbance, and unknown input dead-zone. The proposed controller outperforms the ESOBC and BC in terms of the RMSE as mentioned in Tables 3 and 4, and especially entirely dominates the ESOBC at low frequency (0.1 Hz) and the BC at both low and high frequency, with respect to the prescribed-constraint performance as shown in Figures 6 and 10. The effectiveness of the LESO is presented in Figure 12, with the estimated error responses of the position, the velocity, and the lumped disturbance. The torque signals of the BC, LESOBC, and proposed control are exhibited in Figure 13. In the LESOBC and the proposed control, the control responses are also different from the BC because the input dead-zones are overcome by the LESO. Finally, in order to evaluate the effectiveness of the proposed control in detail, the root mean square error (RMSE) is used for evaluating the responses of the three controllers. The results are shown in Table 4.

Conclusions
This paper proposed an output feedback control via an extended state observer for an n-DOF robotic manipulator under the presence of unknown friction, external disturbances, input dead-zone, and the time-varying output constraints. These challenges are presented with n-DOF manipulator dynamics in mathematical equations. The proposed controller was developed from the LESO and the barrier Lyapunov function with the backstepping framework. The LESO estimated both the lumped disturbance and the unmeasured states in the robotic manipulator. Additionally, the BLF guaranteed that the output responses avoid violation of the constraints. Next, the Lyapunov approach was theoretically conducted to analyze the stability and robustness of the proposed control of the manipulator. Some simulations were conducted on the 3-DOF planar manipulator. The comparative results between the proposed control and the other controllers, such as backstepping control and the LESOBC, prove the superiority of the proposed control in improving accuracy against the lumped disturbances.
In future works, some advanced control can be developed from this algorithm to deal with other challenges such as finite-time convergence, the chattering effect, and input constraints, besides the output constraints. Some adaptive approximations can be investigated in this algorithm when the LESO is used as a fault detector or a force estimator for the manipulator.

Conflicts of Interest:
The authors declare no conflict of interest.