Computational Efficiency-Based Adaptive Tracking Control for Robotic Manipulators with Unknown Input Bouc–Wen Hysteresis

In order to maintain robotic manipulators at a high level of performance, their controllers should be able to address nonlinearities in the closed-loop system, such as input nonlinearities. Meanwhile, computational efficiency is also required for real-time implementation. In this paper, an unknown input Bouc–Wen hysteresis control problem is investigated for robotic manipulators using adaptive control and a dynamical gain-based approach. The dynamics of hysteresis are modeled as an additional control unit in the closed-loop system and are integrated with the robotic manipulators. Two adaptive parameters are developed for improving the computational efficiency of the proposed control scheme, based on which the outputs of robotic manipulators are driven to track desired trajectories. Lyapunov theory is adopted to prove the effectiveness of the proposed method. Moreover, the tracking error is improved from ultimately bounded to asymptotic tracking compared to most of the existing results. This is of important significance to improve the control quality of robotic manipulators with unknown input Bouc–Wen hysteresis. Numerical examples including fixed-point and trajectory controls are provided to show the validity of our method.


Introduction
It is well-known that robotic manipulators are a class of important systems in industrial and academic research [1]. Based on their widespread use in engineering fields, the control of robotic manipulators has attracted much attention of researchers of robotic systems and control science [2][3][4][5][6][7][8]. The modern demand for electronics requires robotic manipulators to be operated in a high-demanding status to reject possible nonlinearities in the closed-loop systems. One of the current research topics is to investigate unknown input nonlinearities in the robotic manipulators.
In practical systems, control inputs are one of the essential units in the closed-loop system and play a key role in maintaining performance and quality [9]. As for the nonlinearities on the input signal, backlash nonlinearity is considered for output feedback control of uncertain nonlinear systems in [10] through backlash inverse. Fu and Xie [11] considered a quantized control problem using a sector bound approach and quantized output feedback systems using a dynamic scaling method [12]. A system with a hysteretic quantizer is considered by Hayakawa et al. [13] to cancel the chattering caused by the logarithmic quantizer. Zhou et al. [14] considered a quantization control

Problem Formulation
A class of robotic manipulators, such as robotic manipulators, are formulated as the following differential equation [1][2][3][4]: where q ∈ R L×1 is a system state vector, D(q) ∈ R L×L denotes an inertial matrix, H(q,q) ∈ R L×L represents the Coriolis and centrifugal matrix of the ith robotic arm, W(q) ∈ R L×1 denotes the gravitational force vector, v ∈ R L×1 means the input of the robotic manipulator and will drive the joint space variable q to a predetermined trajectory. Note that the robotic manipulator governed by (1) is capable of modeling jet engines and aircraft.
To summarize the design purpose, we give the closed-loop system after applying the control scheme to the robotic manipulators in Figure 2.

Trajectory Tracking Design for Robotic Manipulators with Unknown Input Hysteresis
In this section, we specify the control method, control design, and the main result for robotic manipulators with unknown input hysteresis. We show that trajectory tracking control is ensured using the proposed adaptive control in the sense of Lyapunov theory.

Control Method
In this subsection, we review the dynamical gain-based approach [42], which will be combined with the adaptive control technique to handle unknown coefficients caused by input hysteresis.
Here, the dynamical gain is given as [42] N where χ is a real variable. Recalling the result in [42], one has the following result: Lemma 1. Let functions V(t) and χ(t) smooth over [0, t s ) with V(t) ≥ 0 and χ(0) bounded. Moreover, χ(t) is a monotonic function. The dynamic loop gain function N is as (8). If one has where β is a bounded variable and and g M are positive constants, then the boundedness of V(t) and χ(t) are derived over [0, t s ).
In what follows, we show how to use dynamical gain (8) to handle unknown input hysteresis in robotic manipulators and how to use one parameter to adaptively tune control coefficients for multiple inputs.

Controller Design
In this subsection, we show the control design to handle input hysteresis in robotic manipulators. Recalling the work in [3], we defineq where K r ∈ R L×L > 0 is a positive-definite matrix. Now, a controller for robotic manipulators to reject input hysteresis is given as with whereλ is an estimate of λ to be detailed later, K e is a positive-definite matrix, and I denotes an identity matrix with the dimension of R L×L . The adaptive laws for (13) and (14) are given aṡ where , Φ, and Φ 1 denote positive constants and || · || 2 denotes a norm operator. The initial values of λ(t) and χ(t) are set to be non-negative, i.e., λ(0) ≥ 0 and χ(0) ≥ 0. To summarize the design purpose, we show the designed control scheme in Figure 3.

Stability Analysis
Based on the control design in the previous subsection, we use Lyapunov theory to analyze the stability of the proposed adaptive control with a focus on handling unknown input Bouc-Wen hysteresis. Our main result is summarized as follows.
Theorem 1. Supposing that the robotic manipulators are modeled as (1) with input hysteresis as (4) and (5), the controller is given as (13), and adaptive mechanisms are as (15) and (16), the asymptotic tracking performance in terms of q(t) andq(t) in (1) where Ψ is a regression matrix and θ is a constant vector with an appropriate dimension. Define a function whereλ is defined asλ with λ being defined later and D(q) being a positive definite matrix following [2]. From (18), one haṡ where λ = ||θ|| F . Following [2,4], one has thatḊ(q) − 2H(q,q) is a skew-symmetric matrix.
Substituting (14) and (16) into (20) yieldṡ where the first inequality is derived after using Young's inequality. Now, −λλ in the right-hand side of (21) is changed into where both the result in (19) and Young's inequality are used. From (15) and (22), (21) is further changed intoV It is clear that , µ min , and β 0 are positive constants.

Remark 1.
Here, (23) specifies how to transform the unknown input Bouc-Wen hysteresis control problem into a problem of handling an unknown control coefficient µ min and an unknown variable β 0 , where µ min is given in (29) and β 0 is given in (25). We use the dynamical loop gain (8) and the designed control scheme (14) to make sure that the multiplication in (13) is non-positive. Based on the non-positiveness of N (χ(t))u N , one now finds a upper bound governed by the minimum (29). Please note that even though robotic manipulators are modeled as multiple-input and multiple-output systems, one only needs to handle two scalars, µ min and β 0 , in (23). This further prompts our adaptive method that uses two adaptive laws to achieve asymptotic control.

From (23), one has
where Considering that V(0) is predetermined to be bounded and and β 0 are bounded, one obtains that β is also bounded. Now, we obtain that (32) is structurally the same as (9). Therefore, the result in Lemma 1 will hold for (32). That is, from the result in Lemma 1, one obtains the boundedness of V(t) and χ(t) [37]. As an immediate result from (14) and (15), one has Note that the boundedness of χ(t) and χ(0) has been ensured and is a predetermined constant. It is clear that t 0 e T (τ)e(τ)dτ exists and is finite. From Lemma 2 (Barbalat's Lemma), one has lim t→∞ e T e = 0, (35) so that lim t→∞ e(t) = 0.
Therefore, the convergence in (6) and (7) results. Thus, the proof is completed.

Remark 2.
In Theorem 1, we have proven that even though multiple inputs coexist in the considered robotic manipulators, as shown in (3), the control objective is achieved with computational efficiency using two parameters to be tuned online. In particular, one is in (15) and is responsible for handling unknown input coefficients from hysteresis, and the other one is in (16) and is responsible for addressing the regression matrix from robotic manipulators. This two-parameter adaptive control scheme is feasible due to our unique control design as in (14) and the dynamical gain as in (8). Furthermore, we employ the adaptive control technique to achieve stability for the trajectory tracking control of robotic manipulators with unknown input Bouc-Wen hysteresis.

Simulation Example
A two-link articulated robotic manipulator is used for the simulation, which follows the work of [3]. The proposed method is employed to testify to the validity of the proposed control scheme. The manipulator is simulated to move in a horizontal plane and is described as in [3]: where D 11 = θ 1 + 2θ 3 cos(q 2 ), D 12 = H 21 = θ 2 + θ 3 cos(q 2 ) + θ 4 sin(q 2 ), with the physical parameters being θ = [θ 1 , θ 2 , θ 3 , θ 4 ] T . Here, it is noted that the considered system (37) has two inputs and two outputs. As for the system inputs, we specify the input Bouc-Wen hysteresis for each torque as in (4) and (5) with hysteresis parameters µ 1 = 4.5, µ 2 = 4, = 1, β = 0, and n = 1.
Note that the parameters of hysteresis nonlinearity for each torque can be different according to our result in Section 3. Here, we choose the same hysteresis parameters for simplification. The initial states for the robotic manipulators are also chosen randomly. Note that we need two adaptive laws to implement our method. Here, we set the initials of these two adaptive laws in (15) and (16) as zeroes.
That is,λ(0) = 0 and χ(0) = 0. Note that the physical parameters to be estimated are vectorized as θ = [θ 1 , θ 2 , θ 3 , θ 4 ] T . Here, we we use the adaptive law (16) to estimate the scalar λ = ||θ|| F , not the vector θ. Therefore, the number of estimators drops significantly to only one when compared to the traditional adaptive method. As a result, the computational efficiency is ensured by our method.
In what follows, we give two scenarios that frequently happen in the motion control of robotic manipulators.

Fixed-Point Control Using the Proposed Adaptive Control
In this scenario, we predetermine the desired trajectory as the predetermined points Simulation results are given in Figures 4-9. From the observation of Figures 4-7, the adaptive variables including χ, N (χ), u, andλ are bounded under unknown input hysteresis and the proposed control method. The outputs of the considered robotic manipulators q andq, as well as the predetermined ones q d andq d , are shown in Figures 8  and 9, where outputs q 1 (t) and q 2 (t) are, respectively, driven to the predetermined points 0.2 and 0.5 in the presence of the proposed control, while the velocitiesq 1 (t) andq 2 (t) are regularized to zeroes. Therefore, it is clear that the proposed method is effective in handling input hysteresis in robotic manipulators for the fixed-point control.

Tracking Control Using the Proposed Adaptive Control
In this tracking control scenario, we set the desired trajectory to be a sine wave. Simulation results for this scenario including the adaptive signals χ, N (χ),λ, and control signal u, are given in Figures 10-15. In particular, the signal of χ and its dynamical gain N (χ) are given in Figures 10 and  11. The control signal u is given in Figure 12. The adaptive law ofλ is given in Figure 13. The results in Figures 10-13 show that our method is effective in ensuring all the signals in the closed-loop robotic manipulator are bounded. Finally, the outputs q andq are provided in Figures 14 and 15, where the proposed method drives the outputs of robotic manipulators to converge to the desired trajectories. This guarantees the effectiveness of the proposed method in achieving tracking control in the presence of unknown input hysteresis.      Moreover, we give the tracking performance under a traditional controller without compensating the hysteresis nonlinearities. For the comparison, we consider the same two-link robotic manipulator as in the previous case. To be specific, a proportional plus derivative controller is applied with v = − 1 2 (q −q d ) − 5 2 (q − q d ). The tracking performance in the presence of the traditional controller is given in Figures 16 and 17. From Figures 14-17, it is clear that our method provides a better tracking performance compared to the traditional controller.

Conclusions
In this paper, the problem of input hysteresis is addressed for robotic manipulators. We utilize the adaptive control technique and a dynamical gain-based approach to handle input hysteresis. We use two adaptive parameters to address input hysteresis in robotic manipulators so that computational efficiency is ensured for real-time implementation. Therefore, the proposed adaptive method may be feasible for the purpose of applications. Moreover, we drive the outputs of robotic manipulators to the desired trajectories with zero errors, which guarantees a high level of control quality for robotic manipulators even in presence of unknown input hysteresis. We adopt Lyapunov theory to validate the stability of our method and to prove that all the states and adaptive variables in the closed-loop systems are bounded. In addition, we provide a numerical example including fixed-point and trajectory controls so that the validity of our method is ensured. Future works may extend the proposed method and combine it with advanced learning methods such as those in [43][44][45][46][47][48][49].
Author Contributions: K.X. and W.L. conceived of the original idea of the paper. K.X. and Y.L. performed the experiments. K.X., Y.L., and W.L. wrote the paper.