Adaptive Fuzzy Finite-Time Command Filtered Impedance Control for Robotic Manipulators

In order to improve the security and compliance of physical human-robot interaction (pHRI), an adaptive fuzzy impedance control for robotic manipulators based on finite-time command filtered method is proposed in this paper. Firstly, robots usually encounter system uncertainties in practical applications, and the adaptive fuzzy control is introduced to approximate the system uncertainties. Secondly, the finite-time control method is used to improve the interaction performance of the system. Then, the command filtered control technique is used to deal with the “computational complexity ” of traditional backstepping. Finally, simulations are conducted to illustrate the effectiveness of the proposed control method in physical human-robot interaction.


I. INTRODUCTION
In recent years, robots have been widely applied in social services, such as rehabilitation [1], home service [2], education and entertainment [3]. Physical human-robot interaction has become one of the active research fields in social service robots [4], [5]. It should be concerned that security and compliance should be guaranteed in pHRI. Hence, researchers pay more attention to how to design more effective control strategies to achieve better interaction effects. To regulate the physical interaction between humans and robots, impedancebased controllers have been widely used [6]- [8]. In addition, robots usually encounter system uncertainties in practical applications [9], [10], such as sensor error, and parameters change, which will affect the performance of the robot control if they are not handled properly. Therefore, it is crucial to study the effective impedance control approaches for robots, which ensure the physical interaction performance by dealing with robot system uncertainties.
During the past years, many research results show that fuzzy logic control plays a significant role in estimating the dynamic model of complex nonlinear systems [11], [12], and The associate editor coordinating the review of this manuscript and approving it for publication was Choon Ki Ahn .
various impedance control approaches based on the adaptive control [13], [14] and fuzzy logic control [15]- [17] have been proposed for uncertain manipulators. Among these works, an impedance sliding mode control with adaptive fuzzy compensation scheme was proposed in [15] which employing adaptive fuzzy to estimate uncertain model. In [17], an adaptive fuzzy impedance control method where the fuzzy logic system (FLS) was used to approximate unknown nonlinear dynamics was exploited for pHRI. To improve interaction performance in pHRI, finite-time control [18]- [20] has been used in robotic manipulators. In [18], S. Yu introduced finitetime control into the robotic manipulator system, combined with terminal sliding mode technique to achieve a higher precision control performance and converge to the equilibrium in finite time. Up till now, the finite-time impedance control of uncertain manipulators has been seldom studied because it is extremely tough to eliminate the influence of manipulator uncertainty in the design of the finite-time impedance controller. Hence, it is a challenging work how to extend the finite-time impedance control to the uncertain manipulator to ensure the finite-time convergence of the control error.
In another research field, backstepping is one of the most effective control techniques for strict-feedback nonlinear systems, but the repeated derivatives of the virtual control VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ law in the backstepping control increase the "computational complexity " [21], [22]. For the sake of solving this problem, the dynamic surface control (DSC) was proposed by Swaroop et al. [23], Zhang and Ge [24], Yu et al. [25], but it doesn't take into account the problem of approximation errors generated by the first-order filters, and the control quality of the system may be affected. Therefore, Farrell et al.
proposed command filtered control (CFC) [26]- [28], which solves approximation errors problem by some compensated signals, and reduces the complexity of the controller design. However, how to design the finite-time command filtered impedance control for the robotic manipulator systems is still a problem to be solved. Based on the observations above, this paper proposes an adaptive fuzzy finite-time command filtered impedance control (AFFTCFIC) method for robotic manipulators to improve interaction performance. Adaptive fuzzy control is used to approximate the uncertain dynamics. The finite-time control was used to improve pHRI performance. CFC technique can deal with the issue of the "computational complexity " in the backstepping design, and overcome approximation errors problem for DSC by designing compensated signals. Lyapunov stability criterion is used for stability analysis. The primary contributions of the proposed control method can be summarized as follows: • In the face of pHRI system, an AFFTCFIC scheme is proposed for the first time, which can achieve desired tracking performance in finite time. Hence, it expands the application scope for practical pHRI systems.
• Command filtered control technique with compensation signals is adopted to solve the problem of the "computational complexity " in the process of classical backstepping impedance controller design in [17].
• Compared with [15] and [17], the finite-time control can ensure the impedance control of the robot system with higher control accuracy and faster convergence speed. Therefore, it is introduced into the impedance control of the robotic manipulator, which can improve the interaction performance in pHRI system. Notation: To facilitate the design of the AFFTCFIC. Let v The Euclidean norm of a vector is denoted by * . The maximum and minimum eigenvalues of the matrix * are denoted by λ max ( * ) and λ min ( * ), respectively.
The rest of this study is arranged as follows. The mathematical model and preliminaries are given in Section II . AFFTCFIC controller design and stability analysis are presented in Sections III and IV . Simulink results and conclusions are shown in Sections V and VI .

A. SYSTEM DESCRIPTION
Consider a physical human-robot interaction system, including a P-link manipulator and the force sensor mounted on the end-effector (P ≥ 1). A P-link manipulator dynamics [29] in Cartesian space is described as where x,ẋ,ẍ ∈ R n are the position, velocity, acceleration vectors at the end-effector of the manipulator in Cartesian space, D (x)ẍ ∈ R n denotes the inertia force vector of the manipulator in Cartesian space, C (x,ẋ)ẋ ∈ R n denotes the Centripetal and Coriolis force vector of the manipulator in Cartesian space, G (x) ∈ R n denotes the gravitational force vector of the manipulator in Cartesian space, τ ∈ R n denotes the control input vector, τ e ∈ R n denotes the vector of constraining force on robotic end-effector in Cartesian space, which is 0 when the robotic manipulator is no contact with human or environment. n denotes the dimension of the operational space. Property 1 ( [30], [31]): The matrix D (x) is symmetric positive definite.
Property 2 ( [30], [31]): For the convenience, D, C and G represent D (x), C (x,ẋ) and G (x), respectively. Let x 1 = x and x 2 =ẋ 1 . Equation (1) becomeṡ When the manipulator comes into contact with human or environment, an interaction force will be generated based on the user-defined dynamics, the target impedance. The desired impedance dynamics in the workspace [32] is expressed as where e = x d − x r , x r denotes the commanded trajectory, x d denotes the desired trajectory. the desired inertia, damping, and stiffness matrices are denoted by M d , B d , and K d specified by the user, respectively. If the manipulator moves in free space, there has x r = x d and τ e = 0. However, when the manipulator is in contact with the environment, the contact force of the end-effector is defined by the desired impedance dynamics (3). If x tracks x r precisely, (3) becomes It should be mentioned that τ e can be measured from force sensor on robotic end-effector, M d , B d , K d and x d are defined by the user. Therefore, x r can be calculated from (3).
Consider l fuzzy IF-THEN rules R (s) , s = 1, . . . , l, where R (s) represents the sth rule. The fuzzifier maps the input point x i in the input space U ⊂ R n to a fuzzy set A s i in the input space, and x = [x 1 , x 2 , . . . , x n ] T ∈ U is the input vector of the fuzzy system. Membership functions of linguistic variable x i for i = 1, . . . , n are used to represent fuzzy sets. The fuzzy inference engine implements a mapping from fuzzy sets in the 50918 VOLUME 9, 2021 input space to fuzzy sets in the output space V ⊂ R m based on fuzzy rules, and y = [y 1 , y 2 , . . . , y m ] T ∈ V is the output vector of the fuzzy system. Finally, the defuzzifier maps fuzzy sets in the output space V into a crisp output value. The fuzzy logic system is where For the sake of clarity, the fuzzy basis function vector and weight vector are defined can be described as FLS can approximate any given continuous function f j (x), j = 1, 2, . . . , m, to arbitrary accuracy on a compact set . Hence, for any constant where θ * j is an actual weight vector, ε j is the approximation error, which on page satisfies max x∈ ε j ≤ δ j .
ϕ 1 = α 0 , ι 1 =α 0 , and the differentiator's solutions have finite-time stability. When the differentiator's input signal is given to be unaffected by noise, that is α = α 0 . Consider the input noise satisfying the inequation |α − α 0 | ≤ κ. Thence, the following inequations completely dependent on differentiator parameters R 1 and R 2 hold in finite time: where ϑ 1 and ζ 1 are normal numbers determined by the firstorder Levant differentiator design parameters.ω 1 andω 2 are normal numbers.

III. CONTROL DESIGN
According to the principle of backstepping method, the error variables are defined as follows: where x 1,c is the first-order Levant differentiator's output signal when virtual control law α is the input signal. The error compensation signals are defined as The specific structure of virtual control law and the error compensation signals are given in the following design.
Step1: Selecting a Lyapunov function as Differentiating V 1 with respect to time yieldṡ 10) Designing virtual control law α and compensation signal ξ 1 as follows: where the gain matrix Substituting equations (11) and (12) into equation (10) yieldṡ Step2: Then, selecting the Lyapunov function as and taking its time derivative yieldṡ Since there are uncertainties in D, C and G, FLSs are used to approximate the uncertainties in D, C, and G. The fuzzy-approximation-based adaptive impedance control will be designed to approach the uncertain dynamics of the robotic manipulator and to adjust the interaction between human and manipulator.
Designing the control law τ as The updating laws are designed aṡ where Dk > 0, Ck > 0, Gk > 0, and σ Dk , σ Ck , σ Gk are positive constants for improving the robustness.

IV. STABILITY ANALYSIS
For the stability analysis, the Lyapunov function is selected as Substituting (17)- (19) and (23) into the time derivative of (24) yieldṡ Based on Young's inequality, there holds Substituting (26) Applying Lemma 3 to the terms where Rewrite (41) as followṡ From (42), selecting parameters can obtain a > c . By Lemma 1, v j (j = 1, 2) will be within Remark 1: Control parameters determine the radius of the tracking error domain, that is, the smaller radius of the tracking error domain can be ensured by the larger parameters λ max (K i ) and λ max (S i ).
Remark 2: In the control law, the K r is chosen as K rii ≥ E ri . For stability, K r is chosen to be properly large. This is not very ideal due to the introduction of the chattering. Therefore, the control parameter K r can be changed as Theorem 1: Consider manipulator dynamics (1) with Property 1, 2 and impedance dynamics (3). If the finitetime command filter and the error compensation mechanism are chosen as (7), (12), and the adaptive FLS control law (16) with updating laws (17)- (19) are chosen, the tracking error z 1 converges to a small enough region with the radius max √ c/a, 2 c 2b The proof of Theorem 1 is given in the Appendix. Remark 3: In the proof of Theorem 1, the result x 1,c − α ≤w 1 from Lemma 4 will be used. Note that if the α of the finte-time command filter (7) is not influenced by the noise, there hasw 1 = 0. Therefore, the conclusion of Theorem 1 can be obtained when the noise is bounded.
Remark 4: Note that the manipulator is a highly nonlinear system. The finite-time convergence speed in the nonlinear  system cannot be guaranteed by the traditional PID control. When the nonlinear system contains unknown dynamics, the excellent tracking performance cannot be guaranteed by the PID control. Although the excellent robustness and fast convergence capability of the unknown manipulator system can be guaranteed by the sliding mode control, the sliding mode control can only deal with the matched unknown dynamics, and the control effect cannot be ensured when the nonlinear system contains the unmatched unknown dynamics. When the robot system contains an unmatched unknown dynamics, the AFFTCFIC can ensure that the control error variable z 1 converges to a small enough domain of the origin in finite time, and that all signals in the robot system can be kept in the appropriate region in finite time.

V. SIMULATION RESULT
In this section, a 2-degrees of freedom (2-DOF) robotic manipulator on a vertical plane, which is shown in Fig.3, is considered, simulations of pHRIs verify the validity of the proposed method and the robotic manipulator system of two rotary degrees of freedom is defined by (2), it can be described as follows: c1 + m 2 l 2 1 + l 2 c2 + 2l 1 l c2 cos q 2 + I 1 + I 2 m 2 l 2 c2 + l 1 l c2 cos q 2 + I 2 m 2 l 2 c2 + l 1 l c2 cos q 2 + I 2 m 2 l 2 c2 + I 2   C * = −m 2 l 1 l c2q2 sin q 2 −m 2 l 1 l c2 (q 1 +q 2 ) sin q 2 m 2 l 1 l c2q1 sin q 2 0 G * = (m 1 l c2 + m 2 l 1 ) g cos q 1 + m 2 l c2 g cos (q 1 + q 2 ) m 2 l c2 g cos (q 1 + q 2 ) The Jacobian matrix of 2-DOF robotic manipulator is shown as follows: J = − (l 1 sin q 1 + l 2 sin (q 1 + q 2 )) −l 2 sin (q 1 + q 2 ) l 1 cos q 1 + l 2 cos (q 1 + q 2 ) l 2 cos (q 1 + q 2 ) Some formulas for robotic system are D = J −T D * J −1 , The parameters of the 2-DOF manipulator are shown as the length of link 1 and link 2 are 1.00m, the mass of link 1 and link 2 are 1.00kg, the moment of inertia of link 1 and link 2 are 0.25kg · m 2 . The initial parameters of the robotic manipulator are x 11 (0) = 0.4, x 12 (0) = 1m,ẋ 11 (0) = x 12 (0) = 0, q 1 (0) = π 2 , q 2 (0) = − π 2 , the desired trajectory of the 2-DOF robotic manipulator is shown as The updating law parameters are Dk = Ck = Gk = diag [20,20] In this part, the PD control method, the adaptive fuzzy impedance control (AFIC) method in [17] and the proposed AFFTCFIC method in this paper are compared. The robotic manipulator system control parameters are chosen as follows: A). For PD method of the 2-DOF robotic manipulator, the control law is given as τ = −K 1 z 1 − K 2 z 2 and control parameters are chosen as K 1 = diag [800, 800], K 2 = diag [200,200]; B). For AFIC method of the 2-DOF robotic manipulator in [17], control parameters are given as K 1 = diag [6,6], [8,8]; C). For AFFTCFIC method of the 2-DOF robotic manipulator, control the control parameters are given as K 1 = diag [6,6], K 2 = diag [8,8], S 1 = diag [2,2], S 2 = diag [2, 2], γ = 0.6, h 1 = 1, R 1 = 20, R 2 = 0.6.    Fig.4(a) and Fig.4(b) show the position tracking curves of the manipulator end-effector on the X-axis and Y-axis under three schemes. Fig.5 shows the position tracking error curves of the manipulator end-effector on the X-axis and Y-axis under three schemes. Indicated from Fig.4 to Fig.5, it can be seen that the tracking performances are better under the proposed control method whether the manipulator contact with the wall or not, and the proposed AFFTCFIC scheme has better control accuracy and faster convergence speed than the AFIC algorithm in [17] and PD algorithm. Figs.6 shows contact forces from the wall on X-axis. When the manipulator contacts with the solid wall, it can be seen that the desired impedance model can be obtained more smoothly and quickly under the AFFTCFIC method and PD control has a relatively large collision force in Fig.6. Fig.7 shows the control inputs are in proper bounds, but the PD control input is relatively large, which is not conducive to practical application. Fig.8 shows that the filtered signal x 1,c has an excellent tracking approximation to the virtual signal α.

VI. CONCLUSION
In this paper, the adaptive fuzzy impedance controller that combines the finite-time control and the CFC technique has been proposed to improve the security and compliance of pHRI. The manipulator tracking quality has been improved by the finite-time control technique. Simultaneously, the combination of the CFC technique and the backstepping can solve the "computational complexity" issue in the backstepping controller design. Through the Lyapunov stability analysis and simulations, the validity of the proposed method is proven. Our future research is to design a novel finite-time state constraint control scheme of robotic manipulators, which can ensure that the manipulator moves in a finite space and achieves the desired performance in finite time.