Concurrent Learning-Based Adaptive Control of an Uncertain Robot Manipulator With Guaranteed Safety and Performance

This article investigates the tracking problem of an uncertain $n$ -link robot manipulator with guaranteed safety and performance. To tackle parametric uncertainties, the torque filtering-augmented concurrent learning (CL) method is introduced for online identification of the unknown system without requirements of joints acceleration. By using CL, the parameter convergence is guaranteed by exploiting the current and historical data simultaneously. This technique enjoys practicability compared with common methods that need to incorporate external noises to satisfy the persistence of excitation condition for the parameter convergence. Based on the estimated model, we design a barrier Lyapunov function (BLF)-based adaptive control law by the backstepping technique and Lyapunov analysis. By ensuring the boundness of the BLF, the system output and the tracking error are proved to lie in the safety set and performance set, respectively. Numerical simulation results and experiment tests validate the proposed strategy.

works do not fully consider requirements of task performance during the controller design process. Model predictive control (MPC) [7] is an effective method that considers objectives of both safety and performance, whereas it suffers from high computational complexity, which limits its application in realtime scenarios [8]. Given the task performance, methods based on Bellman's principle of optimality [9] or Pontryagin's maximum principle [10] are common. Nonetheless, based on these existing optimization frameworks [11], safety issues are difficult to be tackled. As an alternative, learning-based methods proposed in [2] and [12] can tackle safe optimization problems but with no stability guarantee. Clearly, it is nontrivial work to achieve the objectives of safety and performance simultaneously. To this end, we integrate the requirements of both safety and performance together and formulate a comprehensive constrained adaptive control problem.
To deal with constraints, the barrier Lyapunov function (BLF), which enjoys simplicity and low computational complexity for practical applications [13], emerges as an efficient tool based on the backstepping technique [14], [15]. BLF can be categorized as symmetric BLF (SBLF) and asymmetric BLF (ABLF). Symmetric and asymmetric constraints can be effectively addressed by SBLF and ABLF-based controllers, respectively. SBLFs are originally proposed in [16] and extended in [13] to deal with symmetric output constraints. In addition, multiple SBLFs in a weighted form are used to construct ABLFs to deal with asymmetric output constraints [13]. For further extensions (see [17]- [20]).
Although BLF-based controllers have been used for various aims, their effectiveness relies on an explicit model that is not always available in real applications [21]. For example, even for a well-structured robot manipulator, parametric uncertainties may still occur [22]. To tackle model uncertainties, function approximation-based methods are widely utilized. The present function approximation-based methods can be categorized in terms of different approximation schemes, such as polynomials [23], trigonometric series [24], orthogonal functions [25], splines [26], fuzzy systems [27], and neural networks (NNs) [28]. Among these approximation schemes, NNs play a vital role in learning-based control methods [29]- [31]. Normally, the NN approximation scheme is first adopted to approximate an unknown dynamics and then, a BLF-based control law is designed based on this approximated model to deal with corresponding constraints. See [20], [32], [33] for its applications. Note that the adopted This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ NNs are in essence linearly parameterized two-layer NNs with a hidden layer and an output layer. The hidden layer performs a fixed nonlinear transformation of the input, and NN weights appear linearly between the hidden layer and the output layer. The radial basis function (RBF) NN approximation scheme reported in [20] is a local approximation scheme where each basis function only contributes to the output locally. In this regard, the approximation ability of the RBF NNs depends on the valid region of each chosen RBF. The valid region of the RBF is determined by their centers and widths. These hyperparameters should be determined a priori by considering the whole working space of the investigated system. However, the working space is often unknown, especially when there exist uncertainties. To give full play of the effectiveness of the RBF NN approximation scheme, the choice of suitable RBFs needs trials and fails.
Among all the aforementioned applications of BLF-based controllers via the NN approximation scheme, the fundamental problem about weight convergence of NNs is yet to be discussed. In general, weight (parameter) convergence is not required in adaptive control, and it has been illustrated in [20], [21], [34], and [35] that parameter estimation errors may not converge to 0 (indeed it may not converge at all) even though an acceptable tracking performance is achieved. However, it has been proved that robustness and transient performance of adaptive control methods can be improved with guaranteed parameter convergence [36]- [38]. For the aforementioned literature based on the NN approximation scheme [20], [32], [33], the problem of guaranteed parameter convergence to the actual value, nonetheless, is out of consideration. Conventionally, the persistence of excitation (PE) condition is used to check the parameter convergence. Parameter convergence to the actual value is guaranteed if the PE condition is satisfied [35], [39]. Among the existing works [21], [40], although the PE condition can be satisfied by incorporating external noises to control inputs, this method is restrictive to practical applications in terms of the following aspects: 1) as for the robot control, it may degrade control performance and cause nuisance, waste of energy, etc.; 2) since the satisfaction of the PE condition is a problem relying on future signals, it is often impractical to monitor online whether the PE condition is satisfied or not; and 3) there is no generic method to construct the appropriate form of external noises to satisfy the PE condition since it correlates with scenarios [41]. Thus, instead of using external noises to satisfy the PE condition, we adopt the concurrent learning (CL) method by using the current and historical data simultaneously to provide the required excitation for the parameter convergence. The CL technique is first introduced in [42] to tackle parametric uncertainties of the model reference adaptive control (MRAC). Then, CL is extended to identify parameters of an Euler-Lagrange (E-L) system [43]. However, information of joints acceleration that is sensitive to noises is required in [43] for parameter identification. To control a robot manipulator efficiently for real applications, we need to develop a kind of CL technique that avoids using the information of joints acceleration directly. Closely related to CL is the experience replay technique, which replays experience data to accelerate the online learning process, adopted in the reinforcement learning field [44]. However, its usage accompanies with extensive parameter tuning efforts and the subsequent way of data usage suffers from a data deficiency problem.

B. Contributions
The main contribution of this article is to develop a stable adaptive control strategy that can deal with the following problems simultaneously: 1) safety issues in terms of output constraints; 2) guaranteed performance concerning tracking errors; and 3) parametric uncertainties of a partially unknown robot manipulator. By integrating requirements of safety and performance together, a simple BLF is first designed to account for the safety and performance-related constraints simultaneously. Then, a novel double regressor matrix technique is developed to enable the combination of the BLF-based control strategy and the torque filtering augmented CL (TF-CL) aided online system identification feasible. The torque filtering technique is integrated into CL to get rid of joints acceleration for the parameter estimation, and the guaranteed parameter convergence to the actual value is achieved for the function approximation-based BLF control strategy. Additionally, an efficient history stack management algorithm is proposed to collect sufficient rich data to enable the estimated parameters to converge to the desired values fast.

C. Organization of This Article
The remainder of this article is organized as follows. Section II introduces preliminaries and the problem formulation of the stable adaptive control of a partially unknown robot manipulator with guaranteed safety and performance. The torque filtering technique is illustrated in Section III, which serves to Section IV to construct a TF-CL aided online parameter estimation update law based on the double regressor matrix technique and the history stack management algorithm. Section V elucidates the recursive design process of the BLFbased controller by the backstepping technique and presents stability proofs, compact sets of both tracking errors and system outputs. Numerical simulation and experimental results shown in Section VI illustrate the effectiveness of the proposed control strategy. The conclusion is provided in Section VII.
Notations: Throughout this article, R denotes the set of real numbers, R n is the Euclidean space of the n-dimensional real vector, R n×m is the Euclidean space of n × m real matrices, λ max (M) and λ min (M) are the maximum and minimum eigenvalues of a symmetric real matrix M, respectively, and · represents the Euclidean norm for vectors and induced norm for matrices. Denoting x i as the ith entry of a vector x ∈ R n , diag(x) is the n × n diagonal matrix with the ith diagonal entry equals x i . For any two real vectors a, b ∈ R n , a ≤ ( )b is the component-wise comparison, i.e., a i ≤ (<)b i for ∀i ∈ {1, . . . , n}.

II. PRELIMINARIES AND PROBLEM FORMULATION
The dynamics of an n-link robot manipulator is represented by the following E-L equation: where M(q) : R n → R n×n is the symmetric positive-definite inertia matrix, C(q,q) : R n × R n → R n×n is the matrix of centrifugal and Coriolis terms, G(q) : R n → R n represents the gravitational terms, F ∈ R n×n denotes values of viscous friction, q ∈ R n ,q ∈ R n , andq ∈ R n are the vectors of joint angles, velocities, and accelerations, respectively, and τ ∈ R n represents the vector of input torques applied at each joint. Property 1 [22]: The left-hand side of system equation (1) can be written as the following linear in parameter (LIP) form: where Y(q,q,q) : R n × R n × R n → R n×m is the regressor matrix and θ * ∈ R m is the desired coefficient vector of the E-L system dynamics. Remark 1: Property 1 exploits the known model properties at hands to construct the regression matrix Y(q,q,q).
Considering the E-L equation (1), model uncertainties include varying masses or lengths of joints, varying friction parameters, and unknown payloads. The aforementioned model uncertainties can all be incorporated into the coefficient vector θ * of (2). Note that for the model-free NN approximation scheme [28], the known physical structure of the investigated system is abandoned, which usually inevitably suffers from the well known poor sample complexity problem.
The E-L equation (1) can be written in the following statespace form by setting x 1 = q and x 2 =q: where y ∈ R n is the system output that denotes joint angles of the n-link robot manipulator, and assuming that it lies in the following set: Here, we consider a trajectory tracking problem, where the robot manipulator is driven to track the desired trajectory y d precisely. Throughout this article, we confine ourselves that the desired trajectory y d satisfies the following assumption.
Assumption 1: The desired trajectory y d ∈ R n satisfies −y d ≤ y d ≤ y d , where y d and y d are positive constant vectors. Based on the system output y of (3) and the desired trajectory y d , we define the tracking error e 1 ∈ R n as For the safety issues considered during the trajectory tracking process, following the definition of barrier functions illustrated in [45], here, the safety set regarding the system output y is defined as: where h(y) : R n → R is a continuous function. The explicit form of h(y) is determined by considering various safety issues during the tracking process. As for the investigated n-link robot manipulator, considering the human-robot interactions or limited spaces, its safety set is usually defined as an allowable operation region [46], [47] that follows: where k c = [k c 1 , . . . , k c n ] ∈ R n and k d = [k d 1 , . . . , k d n ] ∈ R n are known constant vectors determined by controller designers. The safety setS in (7) is a representative and explicit form of S in (6). Note that k c −y d and y d k d hold, i.e., y d lies in the safety setS.
For the performance issues, we demand that the tracking error e 1 (5) lies in the following performance set: where k a = [k a 1 , . . . , k a n ] ∈ R n and k b = [k b 1 , . . . , k b n ] ∈ R n are the predefined constant vectors. According to (5), the resulting working space based on the desired tracking error bound (8) would bē To counter the constraints concerning safety in (7) and performance in (8), BLF emerges as an efficient tool. To deal with both symmetric and asymmetric constraints, a simple indicator function-based BLF is proposed as where z ∈ R, k l ∈ R and k u ∈ R are constraint bounds. When z → k l or z → k u , V(z) → ∞, and p(z) is an indicator function that follows: According to [13,Definition 2], the proposed BLF in (10) is an effective BLF. Note that when the value of p(z) in (10) is always set as 0 or 1 regardless of the value of z, (10) is an SBLF, and it can only deal with the symmetric constraint of which the constraint boundaries are ±k l or ±k u . When the value of p(z) alternates between 0 and 1 as defined in (11), (10) is an ABLF that can counter an asymmetric constraint. For the ABLF case, we set k l < 0 and k u > 0, where k l and k u are the lower bound and upper bound of the asymmetric constraint set, respectively.
Remark 2: From the perspective of the guaranteed performance represented by (8), prescribed performance control (PPC) [48] is closely related to our work, which exploits the prescribed performance function (PPF)-based system transformation technique to guarantee that the tracking error converges to an explicit residual set, the convergence rate is no less than a predefined value, and a maximum overshoot is less than a prespecified constant. However, although multiple performance criteria could be provided by PPC, we found in practice that its efficient application requires extensive parameter tuning efforts because the adopted PPF is sensitive and easier lead to singularity. Moreover, the system transformation process results in additional complexity. Thus, a simple BLF is chosen here to achieve guaranteed performance and safety. Although no explicit values of the final residual set and convergence rate are provided by our designed BLF-based control strategy in Section V, both simulation and experimental results in Section VI have shown satisfying performance comparable to the PPC-based work [48].
For the investigated tracking problem, the priority of safety is over performance. To improve the tracking performance while always guaranteeing safety, C ⊆S and C ⊆P should be satisfied together. For the purpose of achieving these considerations simultaneously, we could first choose the values of −k a and k b to meetP ⊆S, and then design a tracking controller to enforce C ⊆P. For example, consider a scenario that a robot manipulator works close to humans, where the preplanned y d for given tasks ensures collision free with humans. To guarantee the collision avoidance while accomplishing the predefined tasks, we need to restrict the operation range of the robot manipulator such that C ⊆S and enable the robot manipulator to track y d precisely to satisfy C ⊆P, respectively. The aforementioned safety and performance requirements can be integrated together by choosing the explicit values (i.e., −k a and k b ) of the guaranteed tracking performance such that P ⊆S holds. Then, a BLF-based controller that drives the robot manipulator to track y d with the guaranteed tracking performance also enforces the executed trajectory to lie in the restricted operation range at the same time. Although it seems to be a conservative approach, comparing to works that can only consider partial objectives of performance [1] or safety [49], the resulting BLF-based controller can drive the robot manipulator to track the desired trajectory while satisfying the requirements of both safety and performance together.
Based on the aforementioned settings, the tracking problem with guaranteed safety and performance is formulated as follows.
Problem 1: Given the uncertain robot manipulator (1), the desired trajectory y d is within the prior known safety setS (7). Choose appropriate bounds for the performance set P (8) and design a stable adaptive control strategy based on the proposed BLF (10) to drive the uncertain robot manipulator to track the desired trajectory y d while satisfying requirements of safety characterized byS and performance denoted as P together.

III. TORQUE FILTERING TECHNIQUE
For the LIP form of the E-L equation given in (2), measurements of the accelerationq are required to construct the regressor matrix Y(q,q,q). Since the information of joint acceleration is sensitive to measurement noises, it is not applicable to use it directly to design a controller. To eliminate the need for this information, the torque filtering technique is adopted here to reformulate the original LIP form (2) to get a new equivalent LIP form without requirements for the information of joints acceleration. Comparing to the common Kalman filter that highly depends on prior knowledge (e.g., noises to be filtered) and requires extensive parameter tuning efforts [50], the adopted torque filtering technique is a simple and easily implemented method for real applications.
To introduce the torque filtering technique, two auxiliary vectors h(q,q) : where Y 1 (q,q) : R n × R n → R n×m and Y 2 (q,q) : R n × R n → R n×m are two new regressor matrices without incorporating the information ofq.
Based on the auxiliary vectors in (12), system equation (1) can be rewritten aṡ The advantage of writing the robot manipulator model in the form (13) is that this new equivalent form of (1) has been separated in a way that allowsq to be filtered out. To filter outq existing inẎ 1 (q,q), a linear stable filter is introduced as where s is the Laplace operator and k ∈ R is a time constant. By filtering (13) based on (14), we can get the filtered version of (13) asḣ are the filtered versions ofḣ(q,q) and g(q,q), respectively. τ f ∈ R n is the filtered version of τ . Based on (13), the corresponding LIP form of the filtered system (15) is given as where Y 1 f (q,q) : R n × R n → R n×m and Y 2 f (q,q) : R n × R n → R n×m are the filtered versions of the regressor matrices Y 1 (q,q) and Y 2 (q,q), respectively. For the filter given in (14), the filtered variables and their original forms satisfy the following equations: Substituting the first equation of (17) into (16), finally, we get the filtered LIP form of the E-L equation (1) as is the new filtered regressor matrix without requirements for information of joints acceleration. Now, the new filtered regressor matrix Y f (q,q) and the resulting filtered LIP form (18) can be adopted to identify the unknown coefficient vector θ * without requirements for information of joint acceleration, which is clarified in detail in the next section.

IV. CONCURRENT LEARNING AIDED SYSTEM IDENTIFICATION
Since the ideal coefficient vector θ * in (18) is unknown and only the estimated parameter vectorθ is available, the identification problem to be addressed here is to obtainθ online based on the system input u, the output state y, and the filtered regression matrix Y f . Online identification ofθ is an adaptive parameter estimation problem, where the PE condition needs to be satisfied before estimated parameters converge to the desired values. Unlike common methods that introduce external noises to satisfy the PE condition, based on the LIP form in (18), the TF-CL method is adopted here to guarantee the parameter convergence by utilizing both the current and historical data.

A. Parameter Estimation Update Law
Denoting the parameter estimation error asθ =θ − θ * , the corresponding model approximation error follows: Define the quadratic cost of the approximation error as V e f = 1/2e f e f , by following the common gradient descent method to minimize V e f , the adaptive parameter estimation update law is derived as:θ where ∈ R m×m is a constant positive-definite matrix. It is well known that the estimatedθ converges to the desired θ * , iff the regression matrix Y f satisfies the PE condition [51] t+T where γ , T ∈ R are appropriate positive constants. The PE condition in (21) can be interpreted as requirements for a degree of data richness: when the regressor matrix Y f varies sufficiently enough over the time interval T so that the entire γ dimension parameter space is spanned, the estimated parameters are guaranteed to converge to the desired values. Common methods usually adopt the parameter estimation update law in (20) and introduce external noises, e.g., signals in sin or cos form, to satisfy the PE condition shown as (21). However, the PE condition in (21) is hard to check online whether it is satisfied or not. An online verification condition is desirable to tell controller designers that under this condition, the estimated parameters are guaranteed to converge to the desired values. Observing that the PE condition is, in essence, a condition about data richness, and only current data contribute to the common parameter estimation update law (20); therefore, to get rich enough data, historical data will also be exploited here to construct the parameter estimation update law.
In this section, a parameter estimation update law is proposed by using the current and historical data simultaneously. The need for adding external noises to satisfy the PE condition (21) can be avoided with the benefit of the recorded historical data. Based on the TF-CL method, the parameter estimation update law for the unknown coefficient vector θ * is designed asθ (22) where k t , k h ∈ R are the positive constant gains to tradeoff the relative importance between the current and historical data to the parameter estimation update law. P is a positive natural number denoting the volume of the history stacks H and E.
The history stacks H and E are collections of historical data, where the filtered regressor matrix Y f j and the filtered approximation error e f j denote the jth collected data of the history stack H and E, respectively. The parameter estimation update law in (22) contains two parts. The first part − k t Y f e f relates to the current data, which is a common gradient descent update law to minimize the quadratic model approximation error V ef , as like (20). However, an update law only with the first part cannot guarantee parameter convergence. Thus, the second part − P j=1 k h Y f j e f j , which is constructed by historical data, is introduced to provide the sufficient excitation required for the parameter convergence. To analyze the parameter convergence problem based on the parameter estimation update law (22), a rank condition is first clarified in Assumption 2.
Note that comparing to the traditional PE condition in (21), the rank condition of a history stack H in Assumption 2 provides an index about the richness of the historical data that can be checked online. If the rank condition is satisfied, it guarantees that the estimated parameters will converge to the desired values and vice versa. Proofs for parameter convergence are given as follows.
Theorem 1: Given Assumption 2 and the parameter estimation update law in (22), the parameter estimation errorθ converges to 0 asymptotically.
Proof: See Appendix A.

B. History Stack Management Algorithm
The parameter estimation update law and the corresponding convergence proof have been provided in Theorem 1. The premise of Theorem 1 is the satisfaction of the rank condition in Assumption 2, i.e., a history stack H containing sufficiently different data is needed. Besides, according to (56) and Q = HH , the convergence rate of the estimated parameters is related to the minimum eigenvalues of the history stack H, i.e., λ min (HH ). With the above analysis, we know that the convergence of the estimated parameters to the desired values with a fast speed equals: 1) the satisfaction of the rank condition in Assumption 2 and 2) the enlargement of the minimum eigenvalues λ min (HH ). Thus, to achieve parameter convergence with a fast speed, in our algorithm, the history stacks H and E are updated with new data points based on two criteria: one is the data threshold ε that acts as a criterion for data difference, and guides the algorithm to collect different enough data to if V max ≥ V then 15: H(:, nI − n + 1:nI) = Y f in (18) 16: (19) 17: else 18: end if 20: end if 21: end if satisfy the rank condition; the other is the minimum eigenvalues of the history stack H that relates to the convergence rate of the estimated parameters. Note that for computation simplicity, the minimum singular value σ min (HH ) replaces with λ min (HH ) to act as a criterion for data storage given that σ min (HH ) = λ min (HH ).
The details of Algorithm 1 are as follows. First, the hyperparameter data threshold ε ensures that only new data that is sufficiently different from the latest collected data will be incorporated into the history stacks H and E. Second, to improve the parameter convergence speed, when H reaches its volume limit P, only data points that lead to an increment of the minimum singular values of the history stack H will be collected. As for the method proposed in [42], the same data might be used multiple times in the history stack H (data richness deteriorates), and the monotonic increment of the minimum singular values cannot be guaranteed (convergence rate of the estimated parameter is discouraged). To ensure monotonic increment of the minimum singular values, in our algorithm, the newly coming data always compares with the latest data inserted into the history stack H. Note that the history stack volume P is a hyperparameter that requires careful tuning, which requires P ≥ m to satisfy the rank condition in Assumption 2, where m is the dimension of the desired coefficient vector θ * . The pseudocode of the history stack management algorithm is shown as Algorithm 1.
Remark 3: In very special cases, it is still possible that the collected historical data from one single trajectory might not be rich enough to satisfy the rank condition in Assumption 2.
To counter this potential data deficiency problem, in the initial learning period, a random noise can be incorporated into the regressor matrix, i.e., Y f ← Y f + , within a short time to Fig. 1. Schematic of the proposed method that consists of the TF-CL aided system identification process followed by the BLF-based controller design.
enable Algorithm 1 to collect the historical data that the real system does not experience. The random noise is abandoned once the rank condition in Assumption 2 is satisfied.

V. CONTROLLER DESIGN FOR ROBOT MANIPULATOR
In this section, based on the identified system from Section III, a recursive controller design process is clarified to yield a stable adaptive control strategy by the backstepping technique and Lyapunov analysis, as shown in Fig. 1.
The resulting control strategy renders the time derivative of the BLF (10) to be always negative semidefinite, which guarantees that with a finite initial value of the BLF, then the BLF value will always be bounded during the tracking process. The boundness of the BLF implies that the safety set (7) and the performance set (8) will not be transgressed, i.e., requirements of safety and performance are both satisfied. Note that since an SBLF is a special case of an ABLF, the following derivation is based on an ABLF, which makes the controller design more challenging.
Recall the tracking error e 1 = x 1 − y d in (5) and define the error e 2 = x 2 − α, where α ∈ R n is a stabilizing function to be designed.
Step 1: For the n-link robot manipulator, the following ABLF candidate is chosen to design a controller: Taking the time derivative of V 1 yieldṡ The time derivative of e 1 iṡ

Substituting (25) into (24) yieldṡ
where α i andẏ d i are the ith dimension of α andẏ d , respectively. In order to make (26) be negative semidefinite, the stabilizing function α is designed as where k 1 ∈ R n×n is a diagonal matrix of positive constants, and its ith diagonal entry is denoted as k 1i . Since asymmetric constraints are considered in this article, the last two terms of (27) are designed to characterize the upper and lower constraint boundaries, i.e., k b and −k a , respectively. For simplicity, denoting Then, (26) can be rewritten as the following form based on (27): where Step 2: We define and choose The time derivative of V blf iṡ Considering (3), the time derivative of e 2 follows: Invoking (28), (31), and (32) yieldṡ where the stabilizing function α is defined in (27). If an accurate model is available, a stabilizing control law can be directly designed as where k 2 ∈ R n×n is a matrix of positive constants to be designed, (e 2 ) + L is a stabilizing term, and + stands for the Moore-Penrose inverse. Accurate model information is required in (34) to design a stabilizing control law, but it is not always available. To provide an approximation of the unknown model information existing in the right-hand side of (34), comparing the difference between (2) and (34), a double regressor matrices technique for the TF-CL is introduced here. Specifically speaking, like the regressor matrix Y(q,q,q) that is proposed for approximation of the system (2), a new regressor matrix X(x 1 , x 2 , α,α) is formulated to approximate the unknown model in (34). Based on the newly designed X (x 1 , x 2 , α,α), the following approximation equation establishes: (35) where X(x 1 , x 2 , α,α) ∈ R n×m is a regressor matrix constructed based on the information of x 1 , x 2 , α, andα. We defer a detailed discussion of the relationship between regressor matrices X(x 1 , x 2 , α,α) and Y(q,q,q) in Remark 4 and focus now on the design of the parameter estimation update law for the BLF-based controller with help of the new regressor matrix X(x 1 , x 2 , α,α). Based on (35), the model based control law (34) can be reformulated as Since θ * is unknown and onlyθ is available, based on the double regressor matrices technique, a TF-CL-aided parameter estimation update law for the BLF-based controller is designed aṡ Comparing the difference between (22) and (37), the first term of (37) is designed as a stabilizing term, which serves to provide the stability proof in Theorem 2. By adjusting the values of , k t , and k h , the importance of each part to the parameter estimation update law is traded off. Finally, based on the estimated parameter vectorθ from the TF-CL method, the stabilizing control law (36) can be rewritten as Remark 4: Observing (2) and (35), we find that these two equations share the same coefficient vector θ * but with different regressor matrices. The double regressor matrices technique illustrated here makes a combination of the TF-CL method and the BLF-based control strategy feasible. Y(q,q,q) is a regressor matrix fully dependents on the model structure. X(x 1 , x 2 , α,α) is a regressor matrix constructed based on both model properties and the stabilizing function α. In the reaming part of this section, the main conclusions of this article and the corresponding proofs are given based on the parameter estimation update law (37) and the stabilizing control law (38).
Theorem 2: Consider an n-link robot manipulator in (1), the parameter estimation update law (37) and the control policy (38). Given Assumptions 1 and 2, for initial values of the system output and the tracking error lying in the predefined safety set (7) and performance set (8), the following properties hold.
1) The tracking error e 1 , error e 2 , and parameter estimation errorθ are stable and converge to 0 asymptotically. 2) The tracking error e 1 is bounded by e 1 , where e 1 = e 1 ∈ R n : − U e 1 ≤ e 1 ≤ U e 1 ∈ P (39) where is the value of the BLF at t = 0. The error e 2 remains in the compact set e 2 e 2 = e 2 ∈ R n : e 2 ≤ 3) For all t > 0, there holds y(t) ∈ y , where Proof: See Appendix B.

VI. SIMULATION AND EXPERIMENTAL RESULTS
In this section, both numerical simulations and experiments are implemented to show the effectiveness of the proposed parameter estimation update law (37) and the BLF-based control strategy (38). Numerical simulations are first conducted on a popular 2-DoF robot manipulator [40], [52] in two cases: 1) a static coefficient vector and 2) a time-varying coefficient vector. The 2-DoF robot manipulator model can serve as a benchmark problem to test whether the TF-CL method can enable the estimated parameters to converge to the desired values. Then, experiments are applied to a more complex 3-DoF robot manipulator to show the superiority of our method. Unlike the numerical simulations presented in Section VI-A, the desired values of θ * is unknown for the 3-DoF robot manipulator case. By using the collected historical data, the parameter convergence result is shown, and the satisfaction of both safety and performance criteria are achieved.
As for the ABLF-based control law (38), the control parameters are set as k 1 = diag (40,30) and k 2 = diag (30,40). y d = [ sin 0.5t, 2 cos 0.5t] is chosen as the desired trajectory. The required safety and performance issues for the two joints are set as follows. For joint 1, the safety set is chosen asS 1 = {−1.17 < q 1 < 1.2}, the performance set follows  The parameter estimation update law (37) is adopted for the online estimation of the unknown coefficient vector θ * . In Fig. 2, without incorporating external noises to satisfy the PE condition, the estimated parameters converge to the desired values. The corresponding parameter estimation errors are shown in Fig. 3, where they finally converge to a small neighborhood around 0, which means that an estimated model with high quality is gotten. However, even though a good tracking performance is achieved in [20], the norm of estimated weights does not converge.The proposed control law (38) is applied to the system and the simulation results are shown as follows. Fig. 4(a) and (b) shows that trajectories of q 1 and q 2 can follow their desired trajectories q 1r and q 2r precisely. The safety setS 1 for q 1 with upper bound k d 1 and lower bound k c 1 , and safety setS 2 for q 2 with upper bound k d 2 Fig. 6. Trajectories of the control τ under the proposed control strategy (38). and lower bound k c 2 are never be violated during the tracking process.The tracking errors of the two joints are shown in Fig. 5. Tracking errors of the two joints finally converge to 0 and always lie in the required performance set P 1 and P 2 , respectively. Note that in simulation, P 1 and P 2 share the same upper bound k b and lower bound −k a . The control trajectory is shown in Fig. 6, where τ oscillates when the estimated parameter vector is in the converging process. From the above analysis, it is concluded that the proposed parameter estimation update law (37) guarantees that the estimated parameters converge to their desired values. The control strategy given in (38) can drive the 2-DoF robot manipulator to track the reference trajectory precisely and satisfy the requirements of safety and performance together.
2) 2-DoF Robot Manipulator With Time-Varying Coefficient Vector: For the 2-DoF robot manipulator, to test the effectiveness of the proposed parameter estimation update law to counter time-varying uncertainties in terms of unknown payloads or friction parameters, the initial desired coefficient vector θ * = [3.473, 0.196, 0.242, 5.3, 1.1] is randomly reset as a new desired parameter vector θ * d = [4, 0.2, 0.3, 5.6, 0.8] at time t = 50s. In practice, only parts of the coefficient vector will change due to environmental effects on the mass, length, or friction parameters, a hard disturbance is chosen here  Fig. 7 shows trajectories of the estimated parameters, where an additional disturbance is incorporated at t = 50 s. As displayed in Fig. 7, in the first 50 s, the estimated parameters converge to the initial desired values of θ * . When an additional disturbance is added at t = 50 s, the TF-CL method collects new data and enables the estimated parameters to finally converge to the new desired values of θ * d . As shown in Fig. 8, the trajectories of parameter estimation errors abruptly change at t = 50s when an additional disturbance is added. Then, the parameter estimation errors still converge to a small neighborhood around zero. Fig. 9(a) and (b) demonstrates that under the proposed control strategy (38), two joints track their reference trajectories efficiently and will not violate their corresponding predefined safety setsS 1 andS 2 even when additional disturbance is added. The tracking errors of the two joints are shown in Fig. 10. It can be observed that the tracking errors e 1 1 and e 1 2 oscillate when the disturbance is added at time t = 50 s, then they finally converge to 0. The tracking errors always lie in the given performance set defined by lower bound −k a and upper bound k b . The control trajectories given in Fig. 11 could provide additional information about the influence of the time-varying uncertainties on the control strategy. When the disturbance is added at t = 50 s, the magnitude of control τ 1 increases, and the magnitude of control τ 2 decreases in order to drive the robot to track the desired trajectory.

B. Experimental Validation on 3-DoF Robot Manipulator
Here, the proposed control strategy and the parameter estimation update law are applied to a 3-DoF robot manipulator to show their effectiveness. The 3-DoF robot manipulator (see Fig. 12  Euler solver at the sampling rate of 1 kHz. More detailed information about the manipulator kinematics and dynamics can be found in [53] and [54].
During the experiment, the 3-DoF robot manipulator is driven to track the desired sinusoidal trajectory q r ∈ R 3 that is designed as where k amp = [0.2, 0.5, 0.8] is the coefficient vector to distribute different amplitudes to each joint. The desired trajectories of the three joints are displayed in Fig. 13. Considering the required safety and performance issues, for joint 1, the safety set is set asS 1 = {−0.1 < q 1 < 0.52}, the performance set is chosen as P 1 = {−0.1 < e 1 1 < 0.12}; for joint 2, the safety set is designed asS 2 = {−0.1 < q 2 < 1.15}, the performance set follows P 2 = {−0.  To verify the robustness of the proposed method, the experiment is conducted with different payloads under the above same parameter settings. The payloads are installed to the end effector of the manipulator. The tracking errors of the three joints with different payloads are displayed in Fig. 14(a)-(c), respectively. It can be observed that during the experiment, even under different payloads, the robot manipulator can track the desired trajectory well, tracking errors always lie in the required performance set P i , i = 1, 2, 3.For the robot manipulator without the payload case, the convergence result of the estimated parameters is shown in Fig. 15. As displayed in Fig. 15, in the beginning, the estimated parameters reach undesirable values with limited data. When rich enough data are recorded, the TF-CL technique effectively identifies the unknown parameters and parameter convergence is achieved. Note that the regular peaks of the estimated parameters are caused by encoder position drifting.

VII. CONCLUSION
This article presented a stable adaptive control strategy with guaranteed safety and performance based on ABLF, TF-CL, and backstepping methods. The TF-CL-based parameter estimation update law guarantees that without incorporating external noises to satisfy the PE condition, the estimated parameters converge to the desired values fast. The information of joints acceleration is avoided by the torque filtering technique. Based on the estimated model, the proposed control strategy can drive the uncertain n-link robot manipulator to track the desired trajectory efficiently, while satisfying the requirements of safety and performance simultaneously. It is proven that the system output always remains in the predefined safety set, the tracking error is bounded by the performance set, and the parameter estimation error finally converges to 0 asymptotically. Both simulation and experimental results show the effectiveness of the proposed parameter estimation update law and the control strategy. To improve the generality and practicability of our method, the future work aims to extend the developed method to provide guaranteed performance and safety on full states even under the consideration of input saturation. Besides, the considered safety issue regarding the restricted operation range in this article allows us to integrate objectives of both safety and performance. The future work aims to extend the proposed method to tackle general safety concepts, e.g., collision avoidance with dangerous regions.

APPENDIX A PROOF OF THEOREM 1
Let V cl : R m → R be a candidate continuously differential Lyapunov function defined as The bound of the Lyapunov function is Calculating time derivative of V cl and substituting (22) into it yieldsV where Q = P j=1 k h Y f j Y f j ∈ R m×m . According to Assumption 2, Q is positive definite and λ min (Q) is a positive constant. Thus, the following inequality holds: It is concluded that the parameter estimation error will converge to zero asymptotically. This completes the proof.

APPENDIX B PROOF OF THEOREM 2
Proof of 1): For the stability proof, let Z = [e 1 , e 2 ,θ ] ∈ R 2n+m and consider the following Lyapunov function: The time derivative of (57) based on (55) and (33) yieldṡ Substituting (35), (37) and (38) Let N = diag(K 1 , k 2 , Q) ∈ R (2n+m)×(2n+m) with Q = P j=1 k h Y f j Y f j ∈ R m×m , (59) can be rewritten aṡ where λ min (N) = min(λ min (K 1 ), λ min (k 2 ), λ min (Q)). Finally, it is concluded that the tracking error e 1 , the error e 2 , and the parameter estimation errorθ converge to zero asymptotically. Proof of 2): Since V(Z) is positive definite andV(Z) < 0 according to (60), V(Z) ≤ V(Z(0)) establishes. From V(Z) = V 1 (e 1 )+V 2 (e 2 )+V cl (θ) and the fact that V 2 (e 2 ) and V cl (θ) are positive functions, it is concluded that V 1 (e 1 ) < V(Z(0)), i.e., V 1 (e 1 ) is bounded. From characteristics of the ABLF (23), when e 1 → −k a or e 1 → k b , V 1 (e 1 ) → ∞. The boundness of V 1 (e 1 ) means that e 1 = −k a or e 1 = k b . Given that −k a e 1 (0) k b , it is concluded that −k a e 1 (t) k b , ∀t > 0, which means that the tracking error always lies in the required performance set (8). Besides, from the above analysis, we know that V 1 (e 1 ) < V(0). To get the bound of e 1 , first we take the ith element of e 1 as an example. For e 1 i , the following inequalities establish: It can be represented as the following equivalent form: From above it is concluded that for e 1 i > 0,  (8), (39) establishes.
Proof of 3): The output y = x 1 = e 1 +y d . According to (39), −U e 1 ≤ e 1 ≤ U e 1 establishes. We know that −y d ≤ y d ≤ y d from Assumption 1, it is easy to get that −U e 1 − y d ≤ y ≤ U e 1 + y d . Since U e 1 k a and U e 1 k b , −k a − y d −U e 1 − y d 0 and 0 U e 1 + y d k b + y d establishes, i.e., y ∈P. Since −k a and k b are chosen to satisfyP ⊆S, y ∈S also establishes, i.e., system outputs will not transgress the predefined safety set (7). This completes the proof.