Adaptive Neural Task Space Control for Robot Manipulators with Unknown and Closed Control Architecture Under Random Vibrations

Robot manipulators are now used in various domains and environments, where they can be subjected to random vibrations. Random vibrations mainly affect the torque control signal, and a torque controller is therefore required to be designed for stabilization purposes. However, for security or intellectual property protection reasons, most commercialized robots are manufactured with unknown and inaccessible torque controller interface such that the user can only design a position/velocity controller. This paper proposes an adaptive task-space velocity controller free from the inner controller’s structure and exhibiting stochastic and deterministic disturbances rejection to deal with these issues. To deal with the unknown inner controller, the paper exploits the fact that most torque controllers use a velocity feedback term, and it considers the other terms as an unknown functions vector. To cope with random disturbances, it is demonstrated that the random excitation matrix can be linearly parameterized, and therefore, a direct adaptive method is constructed. Using radial basis function neural network (RBF NN), an indirect adaptive method is developed to cope with deterministic uncertainties. Through Lyapunov theory, the paper proves that all the closed-loop signals are bounded in probability. The effectiveness of the proposed approach is further demonstrated through simulation comparisons.


I. INTRODUCTION
T HE first use of robot manipulators was handling objects in industrial manufacturing lines. In the industrial domain, manipulators evolve in a static and secure environment, where their interactions with this environment are predictable. Nowadays, robot manipulators are used in many domains, such as shipping areas for ship hull maintenance [1], [2], and air transportation for wall painting, fault inspection, or drugs delivery [3], [4]. In most of these usages, it is difficult to stabilize the robot's environment or predict the different interactions between the robot and the environment due to stochastic phenomenons. A robot evolving in a stochastic environment is mainly subject to random vibrations. For example, a robot manipulator used in a boat is subject to boat random movements generated by the waves, whereas an aerial manipulator is exposed to random movements generated by the wind. Therefore, the modeling and control of robot manipulators subject to random vibrations have gained interest for control practitioners, and several solutions have been proposed.
Note that there are two stochastic modeling techniques in the literature, and all existing control schemes use one of these models. The first stochastic modeling approach describes the robot dynamics using Itô-type stochastic differen-tial equations systems (SDEs) and considers random perturbations as white noises. Several control techniques based on an Itô-type stochastic model were proposed in the literature. Cui et al. [5] tackled the problem of robot control in a random vibration environment and proposed a vector form backstepping approach to ensure the convergence of the tracking error mean square to a neighborhood close to zero. The authors in [6] proposed an adaptive dynamic surface control approach to deal with stochastic disturbances. Hui et al. [7] used a neural network-based approximation approach to develop an output feedback control for robot systems subjected to stochastic disturbances. The proposed controller achieves the semiglobally and ultimately uniformly boundedness of the mean square error. Wu et al. [8] investigated the issue of tracking control for a benchmark system under random vibrations and developed a vectorial backstepping control approach. Sun et al. [9] investigated the tracking control for robot manipulators subjected to state constraints and random disturbances. They achieved system stability through a backstepping controller with a tangent-type constraints term. Sun et al. [10] tackle the problem of controlling flexible joint robots subjected to random disturbances from the external environment. They proposed a finite-time adaptive fuzzy command filtered backstepping control approach. Su et al. [11] proposed an adaptive neural network controller with unknown control gains for a nonlinear system to achieve stochastic disturbances rejection and ensure full state constraints. The design procedure uses a backstepping technic, and the control gains of the virtual and the main control laws are updated online to achieve the tracking requirements. However, the stochastic Itô-type dynamic model has a drawback related to the fact that it does not represent a natural physical system, and therefore, control approaches based on such a model are difficult to be used in engineering. Indeed, white noise is rarely encountered in a natural environment as it has infinite power. Moreover, some terms in Itô-type modeling, such as Hessian terms, have no physical meaning [12].
To introduce the second stochastic modeling approach, Wu et al. [12] show that an engineering system evolving in a stochastic environment is ideally represented by a system of random differential equations (RDEs) with random perturbations viewed as colored noises and stationary processes. Attractive control schemes using RDEs stochastic model are found in the literature. Cui et al. [13] developed a vectorial backstepping for the control of flexible joint electrically driven robot manipulators subjected to mechanical and electrical random disturbances. Cui et al. [14] investigated the global output feedback tracking control of robot manipulators in random vibrating environments without velocity measurement. However, all these valuable works proposed torque-based controllers, therefore, cannot been used for robot manipulators with inaccessible torque control interface.
Most robot manipulators are manufactured so that the inner controller is designed and embedded by the manufacturer, and the structure is not revealed to the end-user. Indeed, the robot's inner controller structure is not disclosed to the user for security and intellectual property protection reasons, and the torque control interface is not accessible. The user can only design position or velocity commands (outer controller) in this context. Therefore the following problem arises: How to achieve disturbances rejection and high tracking accuracy using only velocity commands? Some research has addressed this issue, and good results have been obtained. For instance, Wang et al. [15] developed two adaptive task-space controllers that incorporate a dynamic compensator, where the inner controller is assumed to be a PI (proportional-integral) velocity controller with uncertain design parameters. However, only deterministic disturbances are considered, which means that this solution may yield poor tracking performances for stochastic disturbances. Furthermore, the controller cannot achieve good tracking performances if the actual inner controller is not a PI controller. Khan et al. [16] proposed a joint velocity controller which is free from the structure of the inner loop. They deal with the closed structure of the inner controller by considering that a torque controller can be written as a combination of velocity feedback terms and an unknown vector representing the unknown structure. However, this work only consider deterministic uncertainties; therefore, the controller may yield poor tracking performances in a stochastic environment.
One can observe from this brief literature review that there is a need to propose a control strategy for robot manipulators with closed and unknown inner controller working in a random vibrating environment. This paper aims to cope with this problem. Two challenges emerge from this problem, namely: How to deal with the unknown structure and the inaccessibility of the robot's inner controller? How to perform both stochastic and deterministic disturbances rejection? To cope with the first challenge, this paper takes advantage of the fact that for stability purposes, most torque controllers use the velocity feedback term K d (q −q c ), with K d an unknown gain matrix,q the joints velocity vector, andq c the joint velocity commands (the outputs of the outer controller). From this observation, it is considered that the inner controller has the form τ = −K d (q −q c ) + Φ q,q, q c , t 0 q(s)ds, t 0 q c (s)ds , with Φ an unknown part of the inner controller, whose upper bound is approximated using RBF NN. Therefore,q c is designed such that the unknown vector Φ is canceled in the closed-loop.
For the second challenge, which is performing random disturbances rejection, note that two approaches exist in the literature. In order to present these approaches, let us consider that random excitation dynamics are represented by the matrix Γ (q) ∈ R n×r , where q is the vector of the n angular positions of the robot joints, and r is the dimension of the task space. The first approach is to separate random disturbances from the robot dynamics and obtain ∥Γ (q)∥ 2 F , representing the Frobenius norm of Γ (q). In the end, the upper bound (which can be a constant or a linear function) of ∥Γ (q)∥ 2 F is used to perform random vibrations rejection [5], [6], [8], [14]. The second approach separates random distur- VOLUME 4, 2016 3 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ bances from the robot dynamics to obtain a term dependent on Γ (q) Γ T (q); this term is then used to perform random disturbances rejection [13]. However, since Γ (q) depends on the robot parameters, these two approaches are sensitive to modeling errors. Therefore, this paper proposes a direct adaptive random disturbances rejection method to overcome this problem. The proposed adaptive approach is based on the fact that Young's inequality can separate random disturbances from the robot dynamics to obtain the term Γ (q) Γ T (q) y with y ∈ R n . This term is then used to construct the random disturbances rejection. To design an adaptive approach, this paper demonstrates that the vector Γ (q) Γ T (q) y can be linearly parametrizable such that Γ (q) Γ T (q) y = Q (q, y) Θ, with Q (q, y) ∈ R n×m the regressor, and Θ ∈ Rm the robot parameters vector. The vector Θ is updated online to improve the accuracy of random disturbances rejection.
Compared with the existing control approaches, the proposed control technique has advantages that are summarized below: (1) Unlike [5]- [10], this paper considers real practical problems the robot may encounter in various environments. Therefore, the control law developed in this paper has more substantial applicability compared with the previous methods. Furthermore, the proposed control strategy can be used for a robot manipulator with a closed and unknown torque control interface. (2) Compared with control methods in [13], [14], the proposed adaptive control method is more accurate and less conservative because no assumption is made about the Frobenius norm of random excitation matrix; instead, a direct adaptive method is used. Moreover, the proposed controller has more substantial applicability since it can be applied for robot manipulators with closed or opened torque control interfaces. (3) In contrast with the control laws in [15], [16], the proposed control technique deals with both deterministic and stochastic uncertainties; therefore, it is adequate for the control of robot manipulators in complex environments. Furthermore, the proposed control approach allows the user to specify the desired tracking performances to improve accuracy.
The rest of this article is organized as follows. In Section II, the preliminaries and problem statement are presented. Section III shows the design procedure. The stability of the closed-loop system is evaluated in Section IV. In Section V, simulations are carried out to verify the effectiveness of the proposed control scheme. Section VI concludes the paper.
Notations: For a vector y, ∥y∥ stands for its Euclidean norm, y T denotes its transpose, and diag (y) transforms the vector into a diagonal matrix; For a matrix Y , Y −1 is inverse matrix, λ min (Y ) and λ max (Y ) stands for the minimum and the maximum eigenvalue of Y , respectively; For a scalar α, Eα is its mathematical expectation; R n represents the real ndimensional space; R n×r denotes the real n×r matrix space; I n×r represents the n×r unity matrix; 0 n×r represents a n×r matrix with all entries being equal to zero; 1 n×r represents a n × r matrix with all entries being equal to 1. For the sake of simplicity, the function arguments are sometimes omitted.

II. PRELIMINARIES AND PROBLEM FORMULATION A. PROBLEM FORMULATION
Consider an n-DOF revolute joints robot manipulator shown in Fig. 1, with x ∈ R r its end-effector (EE) position in the world frame. The robot is connected to O 0 on the floor and is affected by the environment's random vibrations. Consider that these vibrations result in random accelerations ξ ∈ R r of point O 0 . In order to get closer to the real-world context, we consider that these accelerations are stationary and independent processes. With the help of Lagrange's and relative motion theories, the following random model is obtained [13], [14] M (q)q + C (q,q)q + G (q) = τ + Γ (q) ξ, whereq ∈ R n , M (q) ∈ R n×n , C(q,q) ∈ R n×n , g(q) ∈ R n , τ ∈ R n , and ξ = [ξ 1 , ξ 2 , ..., ξ r ] T are the joint acceleration vector, the inertia matrix, the Coriolis matrix, the gravitational torque, the torque control, and random accelerations vector, respectively. Γ (q) = (Γ ij (q)) n×r is a random excitation matrix. The underlying complete probability space is taken to be a quartet (Ω, F, F t , P ), the filtration F t being increasing and right continuous while F 0 contains all P -null sets.
Most commercialized robots have an unknown and inaccessible torque-control interface (inner controller). For security and intellectual property protection purposes, the manufacturer designs the torque controller τ , and the structure is not revealed to the user. The only way to specify joint inputs is through position or velocity commands (outer controller) in such a context. However, when the robot operates in an environment subject to random vibrations, these vibrations affect the actuator torques, as shown in (1). Torque control is the appropriate approach to achieve torque-disturbance rejection, as shown by the many techniques proposed in the literature. However, the user cannot design a torque control law when the robot has a closed and unknown torque-control interface. Therefore, it is tricky to control a robot manipulator with a closed and unknown inner controller and subjected to random vibrations. This paper aims to design a task-space velocity commandq c for the random system (1), which is independent of the embedded torque controller, such that the end-effector trajectory x (t) can track a given bounded reference signal x des (t) (which is assumed at least firstorder differentiable) as close as possible while ensuring all the closed-loop signals are bounded in probability.
Before getting into the detailed control scheme, let us summarize the key idea. The general approach for the taskspace control is the design of two controllers: a task-space controller (outer controller)q r = f (q, x, x des ) for the task-space tracking error compensation, and a torque controller (inner controller) τ = −K d (q −q r ) + g (q,q, q r , ...) for the joint-space tracking error compensation, with K d a control gain matrix, f (q, x, x des ) a smooth function, and g (q,q, q r , ...) a smooth function ensuring the gravity or the modeling error compensation, or the stiffness compensation, and so forth [15]. According to the tasks to be executed by the robot, the user can modify f , g, and K d . However, for robot manipulators with closed-form architecture, τ is already designed by the manufacturer, and the enduser does not know its structure, and he cannot modify it. Furthermore, for the sake of simplicity, the manufacturer does not consider stochastic disturbances during the design of τ . Therefore, for a robot manipulator with closed-form architecture, K d and g (q,q, q r , ...) are unknown, and the random disturbances are not considered. The key idea of the proposed method is the design of a task-space controllerq c = q r −K −1 d h (q,q, q r , ...), whereK −1 d is the online estimation of K −1 d , and h (q,q, q r , ...) is a smooth function ensuring the kinematic and dynamic modeling errors compensation, the rejection of stochastic disturbances, and the compensation of the unknown structure g (q,q, q r , ...). Therefore when the inner controller takesq c as input, it is modified as follows: The key difficulty of the proposed approach is the design of K −1 d , and h (q,q, q r , ...).

B. RADIAL BASIS FUNCTION NEURAL NETWORK
Nowadays, neural networks (NN) are widely used when approximating uncertain dynamics. The RBF NN is a particular type of NN that uses Gaussian radial basis functions as basis functions. It is well-known that for a continuous function f (y) defined on a compact set ℑ ⊂ R n , there exists a RBF NN W T f S(y) such that for a given positive scalar δ f (y) and a positive constant ϵ f , and a sufficient number of nodes ι, we have [17] with |δ f (y) | < ϵ f , and W f = [w 1 , w 2 , ..., w ι ] T ∈ R ι being the weight vector. Furthermore, y ∈ ℑ ⊂ R l is the input vector and S(y) = [s 1 (y), s 2 (y), ..., s ι (y)] T is the basis function vector, with s i (y) being a Gaussian function given as The vector µ i = [µ i1 , µ i2 , ..., µ il ] T is the vector of Gaussian centers, and η the width of Gaussian functions. This paper uses a RBF NN to approximate the upper bound of the robot's nonlinear dynamics.

C. USEFUL PROPERTIES, ASSUMPTIONS AND LEMMAS
The following properties, assumptions, and lemmas are helpful for the ease of controller design. Property 1: The inertia matrix is positive definite, symmetric, and satisfy the following inequality for all y ∈ R n [18] Property 2: The inertia and the Coriolis matrix satisfy the following property known as skew symmetry [18] whereṀ (q) is the derivative of the inertia matrix. Property 3: Let J (q) ∈ R r×n be the Jacobian matrix of a n-DOF revolute joint robot manipulator. For any vector y ∈ R n , the vector J(q)y is linearly parametrizable in the sense that [16] J(q)y = Y (q, y)θ, with Y (q, y) ∈ R r×p the kinematic regressor matrix, and θ ∈ R p the kinematic parameters vector. Assumption 1: The process ξ (t) is continuous, stationary, and F t -adapted. There exist a constant α > 0 such that for all t ≥ t 0 , [12] sup t0≤s≤t E∥ξ (s)∥ 2 ≤ α.
Lemma 1: For a given n-DOF revolute joint robot manipulator under random vibrations as shown in Fig. 1, the random excitation matrix Γ (q) ∈ R n×r is derived such that for all y ∈ R n , the vector Γ (q) Γ T (q) y is linearly parametrizable in the sense that with Θ ∈ Rm a vector of kinematic and dynamic parameters, and Q (q, y) ∈ R n×m a regressor matrix. The proof of Lemma 1, is given in APPENDIX A.
For all k and l chosen such that 0 < k ≤ l, the following inequality holds [17]  This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Lemma 3: For any y ∈ R and ε > 0, the following inequality holds [17] Lemma 4: Given two vectors x, y ∈ R n and scalars a > 0, b > 1 and c = b b−1 , the following inequality holds [19] x Remark 1: So far, random excitation dynamics are canceled in the closed-loop system using the upper bound of the norm of the random excitation matrix [5], [14], [20], [21]. In contrast to this approach and for an accurate stochastic disturbances rejection, the method proposed in this paper provides a direct adaptive method using Lemma 1.

III. CONTROL DESIGN
This section presents the task-space velocity controller design procedure. Note that this paper deals with robot manipulators with a closed and unknown inner controller. The latter means that the torque vector τ is designed and already embedded by the manufacturer, but its structure is not revealed to the user and cannot be modified. To overcome this difficulty, we take advantage of the fact that most inner controllers use a velocity feedback term K d (q −q c ) for stability issues [15], [16]. Therefore, we consider that the inner controller has the form withq c the joint velocity command (outer controller), K d the unknown feedback gain matrix, and Φ ∈ R n×n the unknown part of the inner controller.
the task-space tracking error and the pseudo inverse of the Jacobian matrix J (q), respectively. The computation of the velocity commandq c requires the following joint reference velocitẏ wherex j is the j-th element ofx, the parameters ϖ j , j = 1, 2, ..., r, are the user specified tracking performances, and ϱ 1 a positive design parameter. The parameters ϖ j represent the maximum task-space tracking errors that the controller can tolerate; therefore, the physical unit of each of them is a mater (m). When the tracking errorx j is out of the expected range [−ϖ j , ϖ j ] (that isx 2 j > ϖ 2 j ), from the increasing property of the exponential function, the term ϱ 0 ψ (x) increases to enforce the tracking error's convergence to a neighborhood close to zero. As a result, the convergence rate increases when the tracking performance constraints are violated. In addition, define the joint velocity tracking error as Taking the derivative of (15) and substituting into (1), and using (12), it results in the following joint error dynamic with The task-space error dynamic is also required for the controller design. To this end, we use the fact that the task-space velocityẋ and the joint velocityq are related asẋ = J (q)q. Furthermore, from Property 3, one has J (q)q = Y (q,q) θ. Therefore, taking the derivative ofx yields to the following task-space error dynamiċ withθ = θ −θ, andθ the estimate of the ideal kinematic parameters vector θ. This paper considers that kinematic and dynamic parameters are uncertain; as a result, the vector Ψ (Z) is uncertain. Therefore, Ψ (Z) represents the deterministic uncertainties while Γ (q) ξ represents the stochastic uncertainties. Furthermore, K d is unknown because the user does not know the structure of the inner controller. The latter means that K d cannot be taken as a diagonal matrix as it is considered in [15]. Therefore, findingq c which copes with the uncertainties mentioned above while ensuring the stability of the closedloop system (represented by (16) and (18)) is a challenge. To solve this problem, the following task-space velocity control law is proposed:q with Υ ∈ R n×n the estimation of K −1 d , ϱ 2 a positive design parameter, andβ ∈ R the estimation of β the parameter used to build the RBF NN model of the upper bound of z T Ψ (Z). β is defined as follows This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and  The vectorŪ ∈ R n is chosen as with ε i > 0 a design parameter, and z i the i-th element of z, i = 1, 2, ..., n. The function U i is defined as The estimation of adaptive parameters θ, β, Θ, and Υ are obtained according to the following lawṡ whereΥ i stands for the i-th column of the matrixΥ and ϑ i the i-th element of ϑ, i = 1, 2, ..., n. The parameters λ j , γ j , λ 4i , and γ 4i , j = 1, 2, 3, are positive design parameters. For the sake of clarity, Fig. 2 shows the control scheme proposed in this paper.

IV. STABILITY ANALYSIS
In order to evaluate the stability of the controller, let us formulate the following theorem. Theorem 1: Consider the random model (1) for a n-DOF revolute joint manipulator. Given a bounded reference signal x des (t) being at least first order differentiable, under Assumption 1, the controller (19) with update laws (24), (25), (26), and (27) can ensure that: (1) The closed-loop system has a unique global solution, and its state has an asymptotic gain in the second moment.
(2) All closed-loop signals are bounded in probability.
The proof of Theorem 1 is given in APPENDIX B. Remark 2: It is essential to know how to choose the design parameters to reduce the bound 2(ϱ2α+ρ2)

ρ3
. Note that this bound becomes smaller for ρ 3 taken significant enough, and ϱ 2 and ρ 2 taken small enough. The parameter ρ 2 is small for γ j and γ 4i chosen as mall as possible, and λ j and λ 4i chosen significant enough, j = 1, 2, 3, i = 1, 2, ..., n. From the latter, it is tricky to find a significant value for ρ 3 that fulfills the constraints (60), (61), and (62). Furthermore, tacking a small value for ϱ 2 can result in a large control signal as shown in (20). Therefore, a trade-off must be found between the tracking performance requirements and the fulfillment of constraints through the trial and error method.

V. SIMULATION RESULTS
To demonstrate the effectiveness of the proposed control scheme, we carry out simulations tests using a 5-DOF revolute joint robot manipulator with kinematic, dynamic, and Denavit-Hartenberg (DH) parameters given in Tab. V. d 1 and d 5 are the link 1 and link 5 offset, α 1 and α 4 are the link 1 and link 4 twist, and l i is the link length, i = 1, 2, 3, 4. In this study, two groups of simulation studies are carried out. The first group aims to study the effect of the main design parameters ϖ j and ϱ 2 on the tracking performances, j = 1, 2, 3. The second group aims to show the superiority of the proposed controller over the observer-based adaptive control developed in [15]. For the two groups of simulation tests, the desired EE trajectory is given as x des = 0.2 + 0.1 cos( π 12.5 t), 0.2 + 0.1 sin( π 12.5 t), 0.4 + 0.2 sin( π 12.5 t) T .

1) Comparative simulations with different design parameters
Among the design parameters that can improve the tracking performances, ϖ j and ϱ 2 are the most important. Indeed, ϖ j can improve the transient performances of the closed-loop system, whereas ϱ 2 can ameliorate the steady-state performances as stated in Remark 2. This section carries out simulations tests using different ϖ j and ϱ 2 values to verify these properties. During these tests, the other design parameters are chosen as: Initial values are taken as: For the first group of simulation tests, we choose ϖ j = 0.001, j = 1, 2, 3, whereas three different values for ϱ 2 are used, that is ϱ 2 = {0.01, 0.5, 1}. Simulations results are given in Fig. 3 and Fig. 4. Fig. 3 depicts the EE trajectory tracking errors for different values of ϱ 2 . It is observed that during the transient phase (that is t ≤ 2Sec), the lowest settling time is obtained for ϱ 2 = 1, the highest settling time is obtained for ϱ 2 = 0.01, and the highest bound for the tracking error (that is |x j (t)| ≤ 0.103m) is obtained for ϱ 2 = 1. At the steady-state phase, it is observed that |x j (t)| ≤ 2 × 10 −3 m for ϱ 2 = 1, and |x j (t)| ≤ 2 × 10 −4 m for ϱ 2 = 0.01. From Fig. 4, it is observed that at the transient phase, the highest bound of the control signals (that is |q ci (t)| ≤ 1.1rad/s) is obtained for ϱ 2 = 0.01, whereas for ϱ 2 = 1, the bound is 0.86rad/s. These observations show that the lowest the parameter ϱ 2 is, the better are the tracking performances, but with a high control demand at the transient stage. These results are compatible with expectations because from (28) a low value for ϱ 2 results in a low value for E ∥x (t)∥ 2 , whereas from (20) a low value of ϱ 2 can lead to high control demands. Therefore, ϱ 2 must be chosen according to the admissible maximum actuator velocity. For the last group of simulation tests, we choose ϱ 2 = 0.5, whereas three different values for ϖ j are used, that is, ϖ j = {10 −3 , 0.2, 2}. Fig. 5 and Fig. 6 give the simulation results. From Fig. 5, it is observed that the lowest tracking error (that is |x j (t)| ≤ 1.2 × 10 −3 m) is obtained for ϖ j = 0.001. From Fig. 6, the control signals are higher for ϖ j = 2 at the steady-state, but the highest control signal is observed for ϖ j = 0.001 at the transient phase. These 8 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and   results show that a low value for ϖ j leads to good steadystate and transient performances and high control demand at the transient phase. These results confirm the effectiveness of the disturbances rejection due to ψ (x). Indeed, when the constraintx 2 j ≤ ϖ 2 j is not fulfilled (mainly encountered at the transient phase), ψ (x) increases the control gain ϱ 1 ϱ 0j to improve the disturbances rejection. However, ϖ j must be chosen according to the admissible maximum actuator velocity.
2) Comparative simulations with the observer-based adaptive tracking control scheme developed in [15] This section aims to prove the superiority of the proposed control scheme over the controller developed in [15]. The design parameters for the proposed controller are chosen as ϖ j = 0.001, j = 1, 2, 3, and ϱ 2 = 0.01. For the controller in (rad/s)   Fig. 7 depicts the random accelerations chosen for this study. Note that these random accelerations are the worstcase encountered in the real world. Fig. 8 presents the EE trajectory and the EE trajectory tracking errors for the two controllers. It is observed that the two controllers achieve the EE trajectory tracking. The proposed controller performs the lowest settling time at the transient phase (t ≤ 4Sec). At the steady-state phase, tracking error is bounded as |x j (t)| ≤ 1.5 × 10 −4 m for the proposed controller, and as |x j (t)| ≤ 2 × 10 −3 m for the controller in [15]. Control demands for the two controllers are given in Fig. 9. It is observed that |q ci (t)| ≤ 1.1rad/s for the proposed controller, and |q ci (t)| ≤ 5rad/s for the controller in [15]. Therefore, the proposed controller performs the best tracking performances with low control signals. Two reasons can explain these observations. The first reason is that in contrast to the observer-based adaptive tracking control scheme [15] our controller exhibits both deterministic and stochastic disturbances rejection. The second reason is that, for the proposed controller, ψ (x) allows the designer to impose the required tracking performances. Note that, in contrast to the method in [15], the proposed controller is free from the structure of the inner controller; consequently it can be used for any robot manipulator.
At the end of these simulation studies, it appears that, in contrast to the literature, the proposed control approach presents better tracking performances, with the lowest control signals. These good performances can be explained by the ability of the controller to achieve both deterministic and stochastic disturbances rejection. Moreover, through the ψ (x) vector and particularly the ϖ j parameters, the controller succeeds in maintaining the constraintx j (t) 2 ≤ ϖ 2 j , which improves the tracking accuracy even in the presence of significant disturbances. Note that the designer does not need to know the robot's inner controller structure to implement the proposed controller.

VI. CONCLUSION
This paper considers the problem of tracking control of a robot manipulator with a closed and unknown inner controller under random vibrations. A direct adaptive method is proposed to achieve a stochastic disturbances rejection. An indirect RBF NN adaptive method is used to deal with deterministic uncertainties. Based on the fact that most inner controllers use a velocity feedback term, an adaptive velocity controller is designed to be free from the structure of the inner controller. Through Lyapunov theory, the paper shows that the tracking error is bounded in probability. Simulation comparisons demonstrate the effectiveness of the proposed control strategy. .

APPENDIX A PROOF OF LEMMA 1
Before proceeding to the proof of the Lemma 1, let us recall the modeling procedure of the effect of random vibrations on a planar robot manipulator, developed by Cui et al. [5]. Indeed, they consider that due to the random vibrations of the robot's environment, the robot's connection point to the ground O 0 acquires an acceleration ξ ∈ R r . This acceleration propagates along the robot arms and generates in each arm's center of mass, a stochastic inertial force i F ci = −m i i a ci2 , where m i is the i-th link mass, and i a ci2 the projection of ξ in the Y-direction (if the link evolve in the vertical plane) or in the Z-direction (if the link evolve in the horizontal plane) of the i-th link frame. Therefore, the stochastic torque τ c = Γ (q) ξ is the generalized stochastic inertial force. Now let us find the explicit formula for the random excitation matrix Γ (q) for any revolute joint robot manipulator. For this purpose, definex where i x ci ∈ R r is the position of the i-th link center of mass in the i-th link frame. Define 0 H i ∈ R (r+1)×(r+1) , the homogeneous transformation describing the relative position between the base frame and the i-th link frame. The generalized stochastic force along the j-th generalized coordinates is given as [5] where F ci (s) and x ci (s) are the s-th element of F ci and x ci , respectively. F ci and x ci are the stochastic inertial force, and the position of i-th link center for mass in the base frame, respectively. Using the homogeneous transformation 0 H i , (34) can be rewritten as From the definition of i a ci2 , there exist a matrix Ω i (q) ∈ R (r+1)×(r+1) such thať Note that for a revolute joint robot one has ∂ ∂qj 0 H i = 0 (r+1)×(r+1) , for j > i. Using the latter and (36) in (35) it follows with and Γ j×r the j-line of Γ (q).

VOLUME 4, 2016
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
with 1 ≤ j ≤ n, and 1 ≤ s ≤ r. It is known that for a revolute joint robot, 0 H i depends linearly on kinematic parameters. Therefore, from (39), we can conclude that Γ js depends linearly on dynamic and kinematic parameters. Now define v, y ∈ R n such that v = Γ (q) Γ T (q) y. The j-th element of v is given as with y s the s-th element of y. Since Γ js depends linearly on dynamic and kinematic parameters, from (40) we can easily conclude that v j depends linearly on kinematic and dynamic parameters. The latter means that v can be linearly parameterizable. In order words, we can found a matrix Q (q, y) ∈ R n×m and a vector of robot's dynamic and kinematic parameters Θ ∈ Rm such that Γ (q) Γ T (q) y = Q (q, y) Θ, this ends the proof □
Substituting (27) and (51) into (50) yieldṡ (52) Note that K d K −1 di is a vector with the i-th element equal to 1, whereas all the other elements are zero. Therefore, the following equality holds From (53) and Lemma 4 it follows Substituting (54) into (52) results iṅ with ρ 2 = n i=1 γ4i 2λ4i + ρ 1 . In this study, K d is assumed to be unknown for the user. Therefore, the sign of K d − 1 2 I n×n is not known a priori. Nevertheless, we should notice that the feedback gains K dii are chosen significantly enough to reduce the settling time and the tracking error for most torque controllers. Therefore, for most practical cases, K dii ≫ 1 2 such that the matrices K d and K d − 1 2 I n×n are positive definite [22], [23]. The latter means that for all y ∈ R n , the following inequalities hold: From the increasing property of the function exp (x), it follows −ϱ 0x T ψ (x) ≤ −x T diag ϱ 01 ϱ 1 exp −ϖ 2 1 , ..., ϱ 0r ϱ 1 exp −ϖ 2 r x.