Fuzzy-based variable impedance control of uncertain robot manipulator in the flexible environment: A nonlinear force contact model-based approach

Abstract

In this paper, a variable impedance control method is proposed for uncertain robotic systems based on a nonlinear force contact-based flexible environmental model. First, a nonlinear force contact model between the rigid manipulator and flexible environment is applied to the compliant control of the manipulator, which can avoid excessive force overshoot that usually exists in the traditional spring-damping environmental model. Then, to achieve better force/position tracking performances, a fuzzy-based adaptive variable impedance controller is designed based on the force contact-based flexible environmental model, where the impedance parameters are adjusted online through the force and position feedback of the robotic system, and the fuzzy logic system is used to compensate the uncertainties. Moreover, the stability of the adaptive variable impedance control scheme is proved by the Routh stability criterion, and the boundness of all the signals in the closed-loop control system is guaranteed by the Lyapunov stability theorem. Finally, the effectiveness of the proposed method is verified by the simulation of a two-link manipulator, and the results demonstrate that the performances of position tracking are improved, while the force overshoot and oscillation time are reduced.

Keywords

Variable impedance control flexible environment fuzzy logic system force contact model robotic system

1 Introduction

Industrial robots have been widely applied to assembling, testing, polishing, welding, and other operations, which require direct interaction between the manipulator and the environment [1]. The control of the interaction force of the manipulator is crucial in performing tasks [2 –4]. The application scenarios of robots are not limited to rigid environment, such as picking robots, cleaning robots, folding robots, etc. However, the existing force control methods mainly focused on the interaction between the manipulator and the rigid environment. Due to the rapid development of the industrial level and the increasing demand for intelligence, the interaction force control between the robots and the flexible environments is still a challenging research direction. Therefore, the compliance of the manipulator is strongly required to ensure the safety and stability of the robotic system [5 –7].

In practical applications, most objects in contact with manipulators are nonrigid [8]. It is worth noting that there are many modeling methods involving contact force with a flexible environment, where the contact force is usually modeled as a linear structure, such as a spring model or spring-damping mechanical system. Since many materials have nonlinear stiffness, the traditional linear structure has the limitations of only describing the linear contact force of objects and insufficient physical accuracy. A non-rigid environment usually has nonlinear characteristics, and the nonlinear model of the contact surface should be considered [9]. Luo et al. [10] derived the nonlinear force model of a deformed object based on the Duffing equation, and proposed a unique method to simulate the force and deformation between rigid and elastic objects in complex contact. Omar et al. [11] used a tapered spring method to simulate any linear and nonlinear behavior of soft tissue. Felix-Rendon et al. [12] realized a non-rigid body deformation control scheme by using finite element methods to simulate deformation dynamics. Selecting a sufficiently simple and powerful flexible environment model is a prerequisite for solving the compliant control problem in flexible environments, which is still a topic worthy of in-depth research.

At present, there are two mainstream compliance control methods: the impedance control [13] and the hybrid position/force control [14]. The impedance control takes motion and force into consideration, which selects different impedance parameters to adjust the relationship between the contact force and position [15] and has been widely used in contact force tracking [16 –19]. Albu-Schaffer et al. [20] improved the compliance of impedance controller in Cartesian space by local stiffness control. By adding integral terms to the traditional impedance model, Chen et al. [21] eliminated tracking errors and improved the performance of the impedance controller. However, due to friction, external disturbances, unknown joint velocities, and other uncertainties, traditional impedance control is hard to achieve satisfied control performances in the actual operation [22]. Since fuzzy logic and neural networks can deal with uncertain systems and adapt to the human decision-making process [23, 24], the intelligent-based control methods that combine fuzzy logic or neural networks with traditional impedance control have been widely investigated [25 –30]. He et al. [31] designed a neural network-based adaptive impedance controller, which not only considered the system uncertainties but also solved the input saturation. Subsequently, He et al. [32] proposed an adaptive fuzzy neural network learning algorithm and an impedance learning strategy to improve the interaction between the manipulator and the environment. In addition, Sun et al. [33] proposed a composite learning impedance controller for robots with parameter uncertainties. To realize contact force tracking of manipulators in the flexible environment, the above compliance control method has also been applied in the flexible environment. Baptista et al. [34] studied the application of a neural network impedance control scheme combining trajectory a prediction algorithm with force error compensation. To enhance the robustness of the manipulator when interacting in a flexible environment, Jafari et al. [35] and Wu et al. [36] introduced adaptive hybrid control methods. The above methods can achieve good performances for flexible environment models. However, when the environments are uncertain or complicated, an adaptive controller is usually required to meet the force control performances.

Considering the shortcomings of traditional impedance control to the uncertainty of environmental stiffness, the variable impedance control schemes have been studied [37 –41]. Jung et al. [37] proposed a variable impedance control strategy, which minimized force error through the adaptive method. Variable impedance control can compensate for the poor adaptability of constant impedance control. However, the standard stability analysis is not appropriate for variable impedance control. It is crucial to ensure the stable execution of the task. To prove the stability of the system, Kronander et al. [42] proposed a state-independent stability constraint that related stiffness and its time derivative to damping. Duan et al. [43] first proposed a control method based on tracking error to adjust impedance parameters online, which could compensate for environmental uncertainties. Roveda et al. [44] proposed a sensorless model-based force control method, which improved the performance of force tracking by adjusting stiffness and damping parameters. Compared with traditional control methods, adaptive variable impedance control methods have strong adaptability to unknown environments, therefore, the force control of robots in flexible environments is significance.

In this paper, a nonlinear force contact environment model is applied to avoid the limitations of the traditional linear spring model on force tracking, such as excessive force overshoot and long oscillation time. And a fuzzy adaptive variable impedance controller is then designed to achieve satisfied contact force tracking performance based on the nonlinear force contact model. The main contributions are summarized as follows:

(1) Compared with the traditional spring model [37, 43], the nonlinear force contact model between rigid and elastic bodies is derived from the Duffing equation, which can reflect the nonlinear characteristics of flexible objects and significantly reduce the overshoot of contact force in the initial contact stage. Based on the proposed force contact environmental model, the position-based adaptive variable impedance trajectory generator (PBAVITG) is designed to obtain the reference trajectory. The stability of PBAVITG is proved by the Routh stability criterion, and the steady-state error of force tracking is also proved to be zero.

(2) Different from the existing variable impedance control methods [43, 44], we consider the existence of space conversion of the manipulator, as well as the uncertainties and disturbance in the practical robotic system. Based on the environmental model, a fuzzy-based adaptive variable impedance control (FBAVIC) scheme is proposed to track the reference trajectory and interaction force, where the impedance parameters are adjusted online by force feedback error, and the fuzzy logic system (FLS) is used to compensate for the uncertainties. All the signals in the closed-loop control system are ensured to be bounded by the Lyapunov stability theorem.

This paper is organized as follows, in Section 2, the description of FLS, the force contact model of the flexible environment and the dynamic model of the robotic manipulator are introduced. In Section 3, the derivation and stability analysis of the fuzzy adaptive variable impedance controller are given. The simulation tests are conduct in Section 4 to demonstrate the performances of the proposed method, and the conclusions are drawn in Section 5.

2 Problem statement and preliminaries

In this paper, $R$ is defined as the real number set, $R^{n}$ is defined as the n-dimensional real vector space, and $R^{n \times n}$ is defined as the n × n real matrix space. I_n×n is the n × n identity matrix. The norm of vector $x \in R^{n}$ and that of matrix $A \in R^{n \times n}$ are defined as $∥ x ∥ = \sqrt{x^{T} x}$ and ∥x ∥ = tr (A^TA), respectively. If y is a scalar, then ∥y∥ denotes the absolute value. And |x| denotes the absolute value.

2.1 Description of fuzzy logic system

In general, fuzzy logic system (FLS) can approximate any real continuous function over a compact set to arbitrary accuracy. According to the rules of fuzzy control, the fuzzy inference engine performs fuzzy reasoning on the fuzzy input x = (x₁, x₂, . . . , x_m) ^T to solve the fuzzy relational equation and get the fuzzy output $y \in R$ . The jth fuzzy rule is written as,

R^j: If x₁ is $A_{1}^{l_{1}}$ and . . . x_m is $A_{m}^{l_{m}}$ , Then y is B^j

where l_i = 1, 2, . . . , q_i (i = 1, 2, . . . , m), q_i is the number of fuzzy set of the ith input x_i, $A_{i}^{l_{i}}$ and $B^{j} (j = \prod_{i = 1}^{m} l_{i})$ are the fuzzy sets of input and output. The center of the fuzzy sets B^j (j = 1, 2 . . . p) is defined as ${\bar{y}}^{j}$ . By using a product inference machine, single value fuzzier and center average fuzzier, the output of FLS is,

$y (x) = \frac{\sum_{j = 1}^{p} {\bar{y}}^{j} (\prod_{i = 1}^{m} μ_{A_{i}^{l_{i}}} (x_{i}))}{\sum_{l_{i} = 1}^{q_{i}} (\prod_{i = 1}^{m} μ_{A_{i}^{l_{i}}} (x_{i}))}$ (1) where $μ_{A_{i}^{l_{i}}} (x_{i})$ is the membership function of x_i.

The fuzzy system basis function vector ξ (x) = (ξ₁ (x) , ξ₂ (x) , . . . , ξ_p (x)) ^T can be introduced, Equation (1) can be rewritten as,

$y (x) = {\hat{Θ}}^{T} ξ (x)$ (2) where $\hat{Θ} = [{\bar{y}}^{1}, {\bar{y}}^{2}, . . ., {\bar{y}}^{p}]^{T}$ is the parameter vector, and $ξ_{j} (x) = \frac{\prod_{i = 1}^{m} μ_{A_{i}^{l_{i}}} (x_{i})}{\sum_{l_{i} = 1}^{q_{i}} (\prod_{i = 1}^{m} μ_{A_{i}^{l_{i}}} (x_{i}))}$ .

In this paper, we use FLS to approximate the uncertain nonlinear function, yields,

$Ψ (x) = Θ^{* T} ξ (x) + o (x)$ (3) where Θ^* is ideal adaptive adjustment parameter and o (x) is the minimum reconstruction error. Assuming exist a compact set $Ω_{Θ}^{*} = {Θ^{*} \in R^{p} : ∥ Θ^{*} ∥ \leq E_{Θ}}$ in which ideal parameters exist. The ideal parameter is expressed as,

$Θ^{*} = arg min_{Θ^{*} \in Ω_{Θ}^{*}} {sup | Ψ (x) - {\hat{Θ}}^{T} ξ (x) |}$ (4)

2.2 Modeling of flexible environment

Generally, flexible objects are made of isotropic elastic materials that are hard to be described by using the traditional spring-damper model. The Duffing equation is one of the standard models for nonlinear systems under external forces, which essentially defines a nonlinear spring damp-restorer model [10]. Figure 1 shows a single-point contact with normal compression, and x_e indicates the original point of contact, which means the position before deformation, and x indicates the maximum deformation point. The normal contact force between the rigid manipulator and flexible environment is defined as follows,

$F_{e} = m (ω_{0}^{2} D + \frac{3 β_{0}^{2} ɛ}{4} D^{3})$ (5) where m is the mass of the rigid manipulator, ω₀ denotes the linear restoring force, $β_{0}^{2} ɛ$ is the parameter of nonlinear restoring term, D = |x - x_e| is the deformation displacement. This model is sufficient to simulate complex behavior, and the contact forces of different flexible objects can be simulated using different relevant parameters. Moreover, since a nonlinear recovery term is added to this model, which represents the nonlinear recovery of the contact point of the flexible object, the model may reduce overshoot and oscillation when the end-effector is transforming from free space to contact space.

It should be noted that the model of nonlinear contact force was presented in [10]. However, no work has been reported to apply the model to the interaction compliance control for robots with flexible environment. Since this model has a nonlinear characteristic and is closer to the real interaction behavior than the traditional spring model, the control process may be unstable, and the traditional compliance control methods may thus not show a good tracking performance. Therefore, it is necessary to design an improved control method for the nonlinear force contact model.

Fig. 1

Model of vertical contact force with flexible environment.

2.3 Dynamic modeling and properties of robotic manipulator

The dynamic equation of n-link rigid robot obtained from the Lagrange equation is expressed as follows,

$M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + τ_{d} = τ - τ_{e}$ (6) where q, $\dot{q}$ , and $\ddot{q} \in R^{n}$ are the vectors of joint position, velocity, and acceleration of the manipulator, respectively. M (q) and $C (q, \dot{q})$ $\in R^{n \times n}$ are the inertia matrix and the centrifugal and Coriolis forces, respectively. G (q), τ_d, τ, and $τ_{e} \in R^{n}$ denote the gravity vector, bounded unknown disturbances, torque input vector, and interaction torque vector between manipulator and environment, respectively.

Considering the measurement errors, environment, and payload factors, it is difficult to obtain the accurate physical parameters of the manipulator and the dynamic parameter matrices M (q), $C (q, \dot{q})$ and G (q). Therefore, we express the actual value M (q), $C (q, \dot{q})$ and G (q) as the nominal parts M₀ (q), $C_{0} (q, \dot{q})$ and G₀ (q) and the uncertain parts ΔM (q), $Δ C (q, \dot{q})$ and ΔG ((q), where M (q) = M₀ (q) + ΔM (q), $C (q, \dot{q}) = C_{0} (q, \dot{q})$ + $Δ C (q, \dot{q})$ , G (q) = G₀ (q) + ΔG (q). Then, the dynamic equation of the robot can be rewritten as follows,

$\begin{matrix} M_{0} (q) \ddot{q} + C_{0} (q, \dot{q}) \dot{q} + G_{0} (q) + Y (q, \dot{q}, τ) + τ_{d} \\ = τ - τ_{e} \end{matrix}$ (7) where $Y (q, \dot{q}, τ) = Δ M (q) \ddot{q} + Δ C (q, \dot{q}) \dot{q} + Δ G (q) = Δ M (q) [M^{- 1} (q) (τ - C (q, \dot{q}) \dot{q} - G (q) - τ_{d} - τ_{e})] + Δ C (q, \dot{q}) \dot{q} + Δ G (q)$ .

The following properties and assumptions are required for the subsequent development.

Property 1. The inertia matrix M (q) is positive definite, symmetric, and satisfies, $0 \leq M_{m} I_{n \times n} \leq M_{0} (q) \leq M_{M} I_{n \times n}, \forall q \in R^{n}$ (8) where M_m and M_M are positive constants.

Property 2. $\dot{M} (q) - 2 C (q, \dot{q})$ is a skew symmetric matrix,i.e. $ζ^{T} ({\dot{M}}_{0} (q) - 2 C_{0} (q, \dot{q})) ζ = 0, \forall ζ \in R^{n}$ (9)

Property 3. The norm $C (q, \dot{q})$ is linear as regard $\dot{q}$ such that, $\begin{matrix} C_{0} (q, ζ + ς) = C_{0} (q, ζ) + C_{0} (q, ς) \\ C_{0} (q, ζ) ς = C_{0} (q, ς) ζ \\ ∥ C_{0} (q, \dot{q}) ∥ \leq C_{m} ∥ \dot{q} ∥, for q, ζ, ς \in R^{n} \end{matrix}$ (10) where C_m is a positive constant.

Assumption 1. The disturbance term is bounded, i.e $∥ τ_{d} ∥ \leq τ_{D}$ (11) where τ_D is a positive constant.

The position and velocity vectors of the robot end-effector in the Cartesian coordinate system are denoted by $X \in R^{m}$ and $\dot{X}$ , respectively, and the relationship between the Cartesian and joint coordinate systems can be obtained as follows,

$X = L (q), \dot{X} = J (q) \dot{q}$ (12) where L (q) is the kinematic function of the robot, $J (q) \in R^{m \times n}$ is the Jacobian matrix from the joint coordinate system to the Cartesian coordinate system.

The dynamic equation Equation (6) is converted to Cartesian coordinate system and expressed as,

$M_{0 x} \ddot{X} + C_{0 x} \dot{X} + G_{0 x} + Y_{x} + F_{τ_{d}} = F - F_{e}$ (13) where X, $\dot{X}$ and $\ddot{X}$ denote the position, velocity, and acceleration vectors of the end-effector in Cartesian space, respectively, and $\ddot{X} = J \ddot{q} + \dot{J} \dot{q}$ , M_0x = J^-TM₀ (q) J^-1, $C_{0 x} = J^{- T} C_{0} (q, \dot{q}) J^{- 1}$ , G_0x = J^-TG₀ (q), $Y_{x} = J^{- T} Y (q, \dot{q}, τ)$ , F_{τ_d} = J^-T τ_d, F = J^-T τ, F_e = J^-T τ_e.

The objective of this paper is to design a trajectory generator and an adaptive variable impedance controller of end-effector in the flexible environment, where the nonlinear force contact model is used to simulate the force contact response, and the nonlinear force recovery characteristic of the model is used to reduce the force overshoot and oscillation. And the adaptive variable impedance trajectory generator can obtain a smooth reference trajectory, which avoids excessive force overshooting by adjusting the impedance parameters online when the manipulator transforms from free space to nonrigid contact space.

3 Fuzzy-based variable impedance controller and stability analysis

In this section, the motion spaces of the robot are divided into flexible contact space and free space, where the contact force with the flexible environment is obtained by Duffing equation. According to the force contact model, the proposed PBAVITG is used to obtain the reference trajectory by adjusting the impedance parameters adaptively. And the FBAVIC is designed to realize the trajectory and desired force tracking of the manipulator, the uncertainties are approximated by FLS and a robust term is designed to estimate the approximation errors and external disturbances.

3.1 Position-based adaptive variable impedance trajectory generator

PBAVITG contains an internal position control loop and an external force control loop, which can transform force feedback into position trajectory correction error and modify the reference trajectory input of the manipulator by adjusting impedance three parameters (inertia M_d, damping B_d and stiffness K_d) of the impedance controller. Then, the target impedance equation of the robotic system is given as follows,

$M_{d} \ddot{E} + B_{d} \dot{E} + K_{d} E = Δ F$ (14) where E = X_d - X is the position error between the desired trajectory and the actual trajectory, X_d, and X respectively represent the desired and actual position trajectory, $\dot{E}$ and $\ddot{E}$ represent velocity trajectory and acceleration trajectory error, respectively, and ΔF = F_e - F_d.

According to the contact force model Equation (5) with the flexible object, we have,

$\begin{matrix} {\begin{matrix} f_{e} = a_{1} (x - x_{e}) + a_{2} (x - x_{e})^{3} \\ {\dot{f}}_{e} = a_{1} \dot{x} + 3 a_{2} (x - x_{e})^{2} \dot{x} \\ {\ddot{f}}_{e} = a_{1} \ddot{x} + 3 a_{2} (x - x_{e})^{2} \ddot{x} + 6 a_{2} (x - x_{e}) {\dot{x}}^{2} \end{matrix} \end{matrix}$ (15) where $a_{1} = 10^{2} m ω_{0}^{2}$ , $a_{2} = 10^{6} m \frac{3 β_{0}^{2} ɛ}{4}$ .

In free space, the end-effector does not exert force on the flexible environment, and the constant impedance controller is designed to track the desired position. In contact space, the variable impedance controller is used to track the desired force of the manipulator in the force direction. For convenience, we consider one-dimensional force in the vertical direction, the impedance relation can be written as,

$\begin{matrix} {\begin{matrix} m_{d} \ddot{e} + b_{d} \dot{e} + k_{d} e = 0 (in free space) \\ (m_{d} + Δ m_{d} (t)) \ddot{e} (t) + (b_{d} + Δ b_{d} (t)) \dot{e} (t) \\ + (k_{d} + Δ k_{d} (t)) e (t) = Δ f (t) (in contact space) \end{matrix} \end{matrix}$ (16) where e = x_d - x, Δf = f_e - f_d, Δm_d (t), Δb_d (t) and Δk_d (t) are time-varying to decrease the force tracking error Δf. The adaptation is defined as,

$\begin{matrix} {\begin{matrix} Δ m_{d} (t) = \frac{Δ f (t)}{\ddot{e} (t)} + \frac{m_{d}}{g (x, t) \ddot{e} (t)} [- 6 a_{2} (x (t) - x_{e}) {\dot{x}}^{2} (t) \\ + 2 \dot{g} (x, t) {\dot{x}}_{d} (t) + \ddot{g} (x, t) x_{d} (t)] \\ Δ b_{d} (t) = b_{d} \frac{\dot{g} (x, t) x_{d} (t)}{g (x, t) \dot{e} (t)} \\ Δ k_{d} (t) = k_{d} [\frac{x (t)}{g (x, t) e (t)} - \frac{f_{d} (t)}{g (x, t) e (t)} - \frac{Δ f (t)}{g (x, t) e (t)} + \frac{Φ (t)}{g (x, t) e (t)}] \\ Φ (t) = Φ (t - λ) - α Δ f (t - λ) \\ g (x, t) = a_{1} + 3 a_{2} [x (t) - x_{e}]^{2} \end{matrix} \end{matrix}$ (17)

where λ represents the sampling time of the controller, α is a positive constant. The discrete form of the reference trajectory is expressed as follows,

$\begin{matrix} {\begin{matrix} {\ddot{X}}_{r} (t) = {\ddot{X}}_{d} (t) - \frac{1}{M_{d} (t)} [Δ F (t) - B_{d} (t) ({\dot{X}}_{r} (t - 1) \\ - {\dot{X}}_{d} (t)) - k_{d} (t) (X_{r} (t - 1) - X_{d} (t))] \\ {\dot{X}}_{r} (t) = {\dot{X}}_{r} (t - 1) + {\ddot{X}}_{r} (t) λ \\ X_{r} (t) = X_{r} (t - 1) + {\dot{X}}_{r} (t) λ \end{matrix} \end{matrix}$ (18) where X_r represents the reference trajectory.

According to Equation (15), substituting Equation (17) into Equation (16), yields,

$\begin{matrix} Δ f (t) \\ = [m_{d} + Δ m_{d} (t)] \ddot{e} (t) + [b_{d} + Δ b_{d} (t)] \dot{e} (t) \\ + [k_{d} + Δ k_{d} (t)] e (t) \\ = m_{d} [{\ddot{x}}_{d} (t) - \frac{{\ddot{f}}_{e} (t)}{g (x, t)} + \frac{2 \dot{g} (x, t) {\dot{x}}_{d} (t) + \ddot{g} (x, t) x_{d} (t)}{g (x, t)}] \\ + Δ f (t) + b_{d} [{\dot{x}}_{d} (t) - \frac{{\dot{f}}_{e} (t)}{g (x, t)} + \frac{\dot{g} (x, t) x_{d} (t)}{g (x, t) \dot{e} (t)}] \\ + k_{d} [x_{d} (t) - \frac{f_{d} (t)}{g (x, t)}] + k_{d} \frac{Φ (t - λ)}{g (x, t)} \\ - k_{d} α \frac{Δ f (t - λ)}{g (x, t)} - k_{d} \frac{Δ f (t)}{g (x, t)} \end{matrix}$ (19)

Simplifying and multiplying both sides of Equation (19) by g (x, t), we obtain,

$\begin{matrix} m_{d} [g (x, t) {\ddot{x}}_{d} (t) + 2 \dot{g} (x, t) x_{d} (t) + \ddot{g} (x, t) x_{d} (t) - {\ddot{f}}_{d} (t)] \\ + b_{d} [g (x, t) {\dot{x}}_{d} (t) - {\dot{f}}_{d} (t) + \dot{g} (x, t) x_{d} (t)] \\ + k_{d} [g (x, t) x_{d} (t) - f_{d} (t)] \\ = m_{d} Δ \ddot{f} (t) + b_{d} Δ \dot{f} (t) + k_{d} Δ f (t) + k_{d} α Δ f (t - λ) \\ - k_{d} Φ (t - λ) \end{matrix}$ (20)

Defining c (t) = Δf and r (t) = g (x, t) x_d - f_d, Equation (20) can be rewritten as,

$\begin{matrix} m_{d} \ddot{r} + b_{d} \dot{r} + k_{d} r & = m_{d} \ddot{c} + b_{d} \dot{c} + k_{d} c - k_{d} Φ (t - λ) \\ + α k_{d} c (t - λ) \end{matrix}$ (21)

Basing on the principle of dispersion, n elements of Φ series can be expanded as,

$\begin{matrix} k_{d} Φ (t - λ) = & k_{d} (Φ (t - (n + 1) λ) - α c (t - (n + 1) λ) \\ - \dots - α c (t - 2 λ)) \end{matrix}$ (22)

The initial value of Φ (t - (n + 1) λ) is generally set to 0, Equation (21) can be rewritten as,

$\begin{matrix} m_{d} \ddot{r} + b_{d} \dot{r} + k_{d} r = & m_{d} \ddot{c} + b_{d} \dot{c} + k_{d} c \\ + k_{d} (α c (t - (n + 1) λ) + \dots \\ + α c (t - 2 λ) + α c (t - λ)) \end{matrix}$ (23)

According to the Laplace transform, the transfer function can be obtained as,

$\frac{c (s)}{r (s)} = \frac{m_{d} s^{2} + b_{d} s}{T (s)}$ (24) where T (s) = m_ds² + b_ds + k_d + k_d (α (e^-(n+1)λs + ⋯ + e^-λs).

The characteristic equation of the system is as follows,

$\begin{matrix} m_{d} s^{2} + b_{d} s + k_{d} + + k_{d} α (e^{- (n + 1) λ s} + \dots + e^{- λ s}) = 0 \end{matrix}$ (25)

Assuming that n is a large enough number, we have $\sum_{n = 1}^{\infty} e^{- λ ns} = \frac{e^{- λ s}}{1 - e^{- λ s}}$ .

Then, the characteristic equation can be expanded by the Taylor series when the sampling rate is sufficient,

$λ m_{d} s^{3} + λ b_{d} s^{2} + λ k_{d} (1 - α) s + k_{d} α = 0$ (26)

The stability conditions of the system can be obtained according to Routh stability criterion, yields,

$0 < α < \frac{λ b_{d}}{λ b_{d} + m_{d}}$ (27)

Therefore, when the input is a step function and denoted as r (s) =1/s, the steady-state error in the frequency domain is as follows,

$\begin{matrix} e_{ss} & = lim_{s \to 0} sE (s) = lim_{s \to 0} s (c (s) - r (s)) = - 1 \end{matrix}$ (28)

Considering the Equation (28), we can obtain,

$\begin{matrix} lim_{s \to 0} sc (s) = 0, lim_{t \to 0} c (t) = 0 \end{matrix}$ (29)

Therefore, when t→ ∞, we have Δf → 0. The error between the actual contact force and the desired contact force converges to zero.

Unlike most adaptive variable impedance trajectory generators, we proposes a PBAVITG based on Equation (5). The impedance parameters can be updated adaptively to reduce the error of force tracking on the premise of ensuring the stability of PBAVITG.

3.2 Adaptive fuzzy-based controller design

Defining the position and velocity tracking errors between actual trajectory and reference trajectory,

$E_{z} = X_{r} - x_{1}, {\dot{E}}_{z} = {\dot{X}}_{r} - x_{2}$ (30) where x₁ = X and $x_{2} = \dot{X}$ .

Defining a composite error as follows,

$z = {\dot{E}}_{z} + \land e^{*}$ (31) where ∧ = ∧ ^T represents a diagonal positive definite matrix, and e^∗ is expressed as follows,

$e^{*} = {\begin{matrix} E_{z} (in free space) \\ Δ F (in contact space) \end{matrix}$ (32)

Differentiating Equation (31) and multiplying M₀ (x₁) on both sides, we can obtain,

$\begin{matrix} M_{0 x} \dot{z} & = M_{0 x} ({\ddot{E}}_{z} + \land {\dot{e}}^{*}) \\ = M_{0 x} ({\ddot{X}}_{r} + \land {\dot{e}}^{*}) - M_{0 x} {\dot{x}}_{2} \\ = M_{0 x} ({\ddot{X}}_{r} + \land {\dot{e}}^{*}) - (F - F_{e} - F_{τ_{d}} \\ - Y_{x} - C_{0 x} x_{2} - G_{0 x}) \\ = F_{e} + F_{τ_{d}} - F - C_{0 x} z + Ψ (x_{1}, x_{2}, e^{*}) \end{matrix}$ (33) where Ψ (x₁, x₂, e^∗) is an unknown nonlinear function and defined as follows,

$\begin{matrix} Ψ (x_{1}, x_{2}, e^{*}) \\ = M_{0 x} ({\ddot{X}}_{r} + \land {\dot{e}}^{*}) + G_{0 x} + C_{0 x} ({\dot{X}}_{r} + \land e^{*}) + Y_{x} \end{matrix}$ (34)

In this section, we use FLS to approximate the uncertain nonlinear function, i.e.,

$Ψ (x_{1}, x_{2}, e^{*}) = Θ^{* T} ξ (x_{1}, x_{2}, e^{*}) + o (x_{1}, x_{2}, e^{*})$ (35) where the definitions of Θ^*T, ξ (x₁, x₂, e^∗) and o (x₁, x₂, e^∗) are shown in the Equation (3). According to the above derivation, variable impedance control law based on FLS can be designed as,

$τ = J^{T} [k_{z} z + F_{e} + {\hat{Θ}}^{T} ξ (x_{1}, x_{2}, e^{*}) + u]$ (36) where k_z > 0 is the controller gain, u is a robust term that is used to approximation error of FLS and the external disturbances.

Substituting Equation (36) into (33), the system error can be expressed as follows,

$\begin{matrix} M_{0 x} \dot{z} = & F_{τ_{d}} - C_{0 x} z - (k_{z} z + {\hat{Θ}}^{T} ξ (x_{1}, x_{2}, e^{*}) + u) \\ + Ψ (x_{1}, x_{2}, e^{*}) \end{matrix}$ (37)

Noted that $Ψ (x_{1}, x_{2}, e^{*}) - {\hat{Θ}}^{T} (x_{1}, x_{2}, e^{*}) = Θ^{* T} ξ (x_{1}, x_{2}, e^{*}) + o (x_{1}, x_{2}, e^{*}) - {\hat{Θ}}^{T} ξ (x_{1}, x_{2}, e^{*})$ $= {\tilde{Θ}}^{T} ξ (x_{1}, x_{2}, e^{*}) + o (x_{1}, x_{2}, e^{*})$ , and $\tilde{Θ} = Θ^{*} - \hat{Θ}$ represents the weight estimated error.

Assumption 2. The reconstruction error of FLS is bounded and satisfies, $∥ o (x_{1}, x_{2}, e^{*}) ∥ \leq ρ_{o}$ (38) where ρ_o is a positive constant.

Then, according to Assumptions 1 and 2, the minimum reconstruction error of FLS and the external disturbance is bounded as,

$∥ o (x_{1}, x_{2}, e^{*}) + τ_{d} ∥ \leq ρ_{o} + τ_{D} ≜ ϖ_{0}$ (39) where ϖ₀ is a positive constant.

3.3 Stability analysis

Theorem 1. Based on the robot dynamic model Equation (13), and supposing that Assumptions 1 and 2 are satisfied, the fuzzy-based adaptive impedance control law is designed as Equation (36), where the robust compensation term is expressed as, $u = ϖ_{0} sgn (z)$ (40)

The adaptive law of FLS is designed as,

$\dot{\hat{Θ}} = Γ [ξ (x_{1}, x_{2}, e^{*}) z - η \hat{Θ}]$ (41) where Γ and η are positive constants, then closed-loop control system is stable, and the composite error s and the weight estimation error $\tilde{Θ}$ are bounded. It means that the position trajectory tracking error E_z and the force tracking error Δf respectively in free and contact space are bounded and can be made as small as possible.

Proof. Choosing a Lyapunov function candidate as,

$V = \frac{1}{2} z^{T} M_{0 x} z + \frac{1}{2} tr {{\tilde{Θ}}^{T} Γ^{- 1} \tilde{Θ}}$ (42) Differentiating the above Lyapunov function and substituting it to Equation (33), and according to Property 2, we have,

$\begin{matrix} \dot{V} & = z^{T} M_{0 x} \dot{z} + \frac{1}{2} z^{T} {\dot{M}}_{0 x} z + tr {{\tilde{Θ}}^{T} Γ^{- 1} \dot{\tilde{Θ}}} \\ = z^{T} [F_{τ_{d}} - k_{z} z - C_{0 x} z - u + {\tilde{Θ}}^{T} ξ (x_{1}, x_{2}, e^{*}) \\ + o (x_{1}, x_{2}, e^{*})] + \frac{1}{2} z^{T} {\dot{M}}_{0 x} z + tr {{\tilde{Θ}}^{T} Γ^{- 1} \dot{\tilde{Θ}}} \\ = z^{T} [F_{τ_{d}} - k_{z} z - u + {\tilde{Θ}}^{T} ξ (x_{1}, x_{2}, e^{*}) + o (x_{1}, x_{2}, e^{*})] \\ + tr {{\tilde{Θ}}^{T} Γ^{- 1} \dot{\tilde{Θ}}} \end{matrix}$ (43)

Considering Equation (40), we can obtain,

$z^{T} [F_{τ_{d}} - u + o (x_{1}, x_{2}, e^{*})] \leq 0$ (44)

According to the updating law Equation (41), we can obtain,

$z^{T} {\tilde{Θ}}^{T} ξ (x_{1}, x_{2}, e^{*}) + tr {{\tilde{Θ}}^{T} Γ^{- 1} \dot{\tilde{Θ}}} = η tr {{\tilde{Θ}}^{T} \hat{Θ}}$ (45)

Therefore, Equation (43) is bounded as,

$\begin{matrix} \dot{V} \leq & - z^{T} k_{z} z + η tr {{\tilde{Θ}}^{T} \hat{Θ}} \leq - z^{T} k_{z} z - \frac{η}{2} tr {{\tilde{Θ}}^{T} \tilde{Θ}} \\ + \frac{η}{2} tr {Θ^{* T} Θ^{*}} \end{matrix}$ (46)

The above Equation (46) can be rewritten as,

$\dot{V} \leq - κ V + γ$ (47) where $κ = \frac{min (λ_{min} (k_{z}), η / 2)}{max (λ_{max} (M_{0 x}), λ_{max} (Γ^{- 1}))}$ and $\frac{η}{2} tr {Θ^{* T} Θ^{*}} \leq \frac{η}{2} E_{Θ}^{2} ≜ γ$ , and γ is a positive constant.

Then, the closed-loop system is stable based on the Lyapunov stability theorem. And solving the inequality Equation (47) yields,

$0 \leq V (t) \leq [V (t_{0}) - \frac{γ}{κ}] e^{- κ t} + \frac{γ}{κ}$ (48) where t₀ denotes the initial time. Equation (48) means that the error signals z and $\tilde{Θ}$ are bounded. Moreover, for arbitrary z (t₀), as long as t > t₀, we have,

$∥ z ∥ \leq \sqrt{\frac{V (t_{0}) - γ / κ}{p} ∥ r (t_{0}) ∥^{2} e^{- κ t} + \frac{2 γ}{w}}$ (49) where p = min(λ_min (M_0x) , λ_min (Γ^-1)), w = pκ. Since the first term within the square root in Equation (49) will converge to zero, it means that the composite error $∥ z ∥ \leq \sqrt{2 γ / w}$ as t→ + ∞. Therefore, when the parameters are selected appropriately, the composite error can be minimized, and the position tracking error E_z in the free space and the force tracking error Δf in the contact space are bounded. This completes the proof.

According to the above analysis, the control structure of the closed-loop control system can be shown in Fig. 2.

Fig. 2

The schematic of the fuzzy-based variable impedance control for robotic system.

4 Simulation studies

To verify the proposed method, simulation tests are conducted on a two-link manipulator as shown in Fig. 3. In practical applications, the controller receives the sensor signal and calculates the control torque based on the control program. By sending a torque control command, the motor of each joint drives the joint motion of the manipulator. The parameters of the two-link manipulator are described as follows,

Fig. 3

Diagram of a two-link robot manipulator.

$M (q) = [\begin{matrix} χ_{1} & χ_{2} \\ χ_{3} & χ_{4} \end{matrix}]$ (50)

$C (q) = [\begin{matrix} - 2 m_{2} l_{1} l_{2} {\dot{q}}_{2} s_{2} & m_{2} l_{1} l_{2} {\dot{q}}_{2} s_{2} \\ m_{2} l_{1} l_{2} {\dot{q}}_{2} s_{2} & 0 \end{matrix}]$ (51)

$G (q) = [\begin{matrix} (m_{1} + m_{2}) l_{1} g c_{1} + m_{2} l_{2} g c_{12} \\ m_{2} l_{2} g c_{12} \end{matrix}]$ (52)

where $χ_{1} = (m_{1} + m_{2}) l_{1}^{2} + m_{2} (l_{2}^{2} + 2 l_{1} l_{2} c_{2})$ , $χ_{2} = m_{2} l_{2}^{2} + m_{2} l_{1} l_{2} c_{2}$ , $χ_{3} = m_{2} l_{2}^{2} + m_{2} l_{1} l_{2} c_{2}$ and $χ_{4} = m_{2} l_{2}^{2}$ , and the mass of link 1 and link 2 are denoted as m₁ and m₂, respectively; The length of link 1 and link 2 are denoted as l₁ and l₂, respectively; sin(q_i), cos(q_i), and cos(q_i + q_j) can be noted to s_i, c_i and c_ij, respectively, for i = 1, 2 and j = 1, 2, g is the acceleration of gravity.

4.1 Design procedure

To verify the proposed method in Section 3, the step-by-step procedures of the FBAVIC are outlined as follows,

Step 1. Construct PBAVITG: Select the appropriate impedance parameters, m_d = 1, b_d = 50, k_d = 500 in free space, and m_d = 1, b_d = 1, k_d = 1, α = 0.0009 in contact space.

Step 2. Construct the FLS: Set the inputs $x = [x_{1}^{T}, x_{2}^{T}, {\land e^{*}}^{T}]^{T}$ , and choose 5 Gaussian relationship functions as, $μ_{A_{i}}^{l} (x_{i}) = \exp {- \frac{x_{i} + π / 6 - (l - 1) π / 12}{π / 24}}^{2}$ (53)

where i = 1, 2, 3, l = 1, 2, 3, 4, 5. And select the learning parameter Γ= 100 in Equation (41).

Step 3. Construct the FBAVIC: Choose controller gain K_s = 500I_2×2 and set ϖ₀ = 2I_2×2 in Equation (40), ∧=2I_2×2 in free space and ∧=0.1I_2×2 in contact space in Equation (31). The adaptive variable impedance controller based on FLS can be obtained from Theorem 1.

4.2 Simulation results

In the subsection, two tests are conducted on the two-link manipulator. The actual parameters of the robot are m₁ = m₂ = 1 kg and l₁ = l₂ = 1 m, while the nominal parameters of the robot are chosen as m₁ = 0.8 kg, m₂ = 0.9 kg, l₁ = 1.1 m, l₂ = 0.9 m and g = 9.8 m/s² to introduce the parameters uncertainties. Choosing the contact force Equation (5) in the x direction, where ω₀ = 0.2 and $β_{0}^{2} ɛ = 0.4$ , the parameters were selected according to the literature [10]. And the desired position trajectory is X_d (t) = [1.6 sin(0.4t + π/6) , 1.6 sin(0.4t + π/3)] ^T m. To test the robustness and stability of this designed method, choosing external disturbances as τ_d = [-2 cos(2t) , 2 sin(2t)] ^T N/m.

4.2.1 Example 1

Assuming that the desired force F_d = [5, 0] ^T N is exerted on the manipulator as x ≥ x_e ≥ 1 m in x direction. To test the performances of the force contact environmental model and the proposed PBAVITG, the adaptive variable impedance trajectory generator (AVITG) [43] is compared with the proposed method. In the AVITG, the traditional spring environmental model is used, and only the damping coefficient is adaptively changed by the force feedback error. Figure 4(a) and (b) show the force tracking results and errors for invariable force. The results of tracking the desired trajectory in the direction x and y of Cartesian space are shown in Figs. 6(a), respectively. Figured 5(b) and 6(b) illustrate the desired position tracking errors between desired and actual positions in the direction x and y of Cartesian space, respectively.

Fig. 4

Force tracking and errors of PBAVITG and AVITG.

Fig. 5

Actual position and errors of PBAVITG and AVITG in the x axis.

Fig. 6

Actual position and errors of PBAVITG and AVITG in the y axis.

From Fig. 4, it can be found that the proposed PBAVITG has smaller error and less overshoot in the initial stage of force tracking in comparison to the AVITG, and the oscillation time is also longer in the AVITG. From Fig. 5(a), it can be seen that the proposed PBAVITG has a smoother trajectory in the direction x of Cartesian space, and the generated trajectory of AVITG in the force direction cannot track the desired force. The reason is that the model adopted in PBAVITG is derived from the nonlinear damping restorer model, and the nonlinear recovery term is added to characterize the contact response in comparison to the spring model adopted in the AVITG, which can reduce the impact force when the manipulator contacts with the flexible environment. Moreover, the behavior of the contact force in a flexible environment is only expressed by the stiffness coefficient, which is not only inaccurate but also too stiff in the conversion process. The spring model may produce a larger overshoot and long-term oscillation at the instant of contact. Therefore, when the manipulator transforms from the free space to contact space, the overshoot of PBAVITG is smaller than that of AVITG, and the oscillation time of PBAVITG is shorter than that of AVITG in the initial stage of force tracking. In Fig. 6, it can be seen that the two methods have good track tracking performances in the direction y of Cartesian space. Although the AVITG can obtain a smooth trajectory and has a good control performance in the case of direct contact with the environmental model, it is not applicable in the case of spatial transformation in the simulation test. The reason is that the transformation of space and the nonlinear contact force requires high adaptability of the control method. Therefore, the above results indicate that the performance of PBAVITG is better than that of AVITG, and the superiority of the nonlinear force contact-based environmental model and the adaptability of the variable impedance control method are verified.

4.2.2 Example 2

The desired force F_d = [50, 0] ^T N is exerted on the end-effector to verify the performance of the proposed FBAVIC. The two groups of methods adopt the nonlinear environmental model mentioned in this paper, but the control methods are different. The proposed FBAVIC is compared with fuzzy-based impedance control (FBIC) for specific flexible environmental models considering the uncertainties and disturbances of the manipulator. In this experiment, Fig. 7(a) and (b) show the force tracking results and errors of FBAVIC and FBIC. Figures 9(a) show the results of tracking the desired trajectory by two methods in the direction x and y of Cartesian space, respectively. Figures 9(b) show position tracking errors between the desired and actual trajectories in the direction x and y of Cartesian space, respectively. Figure 10 shows the variation of adaptive impedance parameters in contact space. Figure 11 shows the motion space and position tracking of the end-effector. Figure 12 shows the update rate of adaptive adjustment parameters of FLS.

Fig. 7

Force tracking and errors of FBAVIC and FBIC.

Fig. 8

Actual position and errors of FBAVIC and FBIC in the x axis.

Fig. 9

Actual position and errors of FBAVIC and FBIC in the y axis.

Fig. 10

Position tracking of end-effector.

Fig. 11

Adaptive impedance parameter variation diagram.

Fig. 12

Adaptive adjustment parameters of FLS.

From Fig. 7, it can be found that the overshoot and the force tracking error of the proposed FBAVIC are significantly reduced in comparison to those of the FBIC. The reason is that the impedance parameters in the FBAVIC are updated adaptively, and the variable impedance adaptive law is relevant to the feedback errors of force and position. Due to the real-time feedback of force error in the process of force tracking, the adaptability of the proposed FBAVIC can achieve better robustness than that of FBIC. Therefore, the force tracking performance of the proposed FBAVIC is better than that of FBIC. From Figs. 8-9, it can be seen that the position tracking performance of the proposed FBAVIC is similar to that of FBIC. This is because both of them use the same nonlinear contact force model, which has a better response in characterizing the manipulator’s contact with the flexible environment. From Fig. 10, it can be seen that the position tracking performance of the proposed FBAVIC is better at the moment of contact because the variable impedance control improves the force tracking performance. Since there is no interaction force in the y direction and only position tracking is involved, both FBAVIC and FBIC show good position tracking performances. Figure 11 shows the adaptive impedance parameters that are determined by the adaptive law. Figure 12 shows that the adaptive parameters of FLS are bounded and converged to the optimal values in a finite time.

According to the above analysis, it can be concluded that the proposed FBAVIC not only reduces the force overshot and oscillation during contact but also make the force tracking error smaller than FBIC, which meets the design requirements and improves the adaptability and robustness of the system.

5 Conclusion

In this paper, a fuzzy-based adaptive variable impedance control is proposed for the robotic system in the flexible environment. In this control scheme, according to the nonlinear force contact-based environmental model, an adaptive fuzzy variable impedance controller is designed to track the desired contact force trajectory of the manipulator. And the impedance parameters are adjusted adaptively according to force feedback. The boundness of force/position tracking errors and the stability of the controller are proved by the Routh stability criterion and Lyapunov stability theory. And the FLS is used to compensate for the uncertainties of the robotic system. Finally, the effectiveness and feasibility of the proposed control strategy are verified by simulation on a two-link manipulator. In future work, the proposed method will be proven experimentally and applied to real robotic systems.

Footnotes

Acknowledgments

The authors would like to acknowledge the funding supported by the National Natural Science Foundation of China (NSFC) (62273311, 61773351, 61733004) and the Key Specialized Research and Development Breakthrough in Henan Province, China (222102220117).

References

Gao

, Lu

and Wang

, Motion path planning of 6-dof industrial robot based on fuzzy control algorithm, Journal of Intelligent & Fuzzy Systems 38(4) (2020), 3773–3782.

Evjemo

L.D.

, Gjerstad

, Grøtli

E.I.

and Sziebig

, Trends in smart manufacturing: Role of humans and industrial robots in smart factories, Current Robotics Reports 1(2) (2020), 35–41.

, Li

and Luo

, An overview of calibration technology of industrial robots, IEEE/CAA Journal of Automatica Sinica 8(1) (2021), 23–36.

Babarahmati

K.K.

, Tiseo

, Smith

, Lin

H.-C.

, Erden

M.S.

and Mistry

, Fractal impedance for passive controllers: a framework for interaction robotics, Nonlinear Dynamics, pp. 1–17, 2022.

Schumacher

, Wojtusch

, Beckerle

and von

, Stryk, An introductory review of active compliant control, Robotics and Autonomous Systems 119 (2019), 185–200.

H.-Y.

, Paranawithana

, Yang

, Lim

T.S.K.

, Foong

, Ng

F.C.

and Tan

U.-X.

, Stable and compliant motion of physical human–robot interaction coupled with a moving environment using variable admittance and adaptive control, IEEE Robotics and Automation Letters 3(3) (2018), 2493–2500.

, Lai

, Chen

, Ma

and Yao

, Desired compensation adaptive robust repetitive control of a multi-dofs industrial robot, ISA Transactions 128 (2022), 556–564.

Gierlak

and Szuster

, Adaptive position/force control for robot manipulator in contact with a flexible environment, Robotics and Autonomous Systems 95 (2017), 80–101.

Baptista

, Sousa

and da Costa

J.S.

, Predictive force control of robot manipulators in nonrigid environments, 2006.

10.

Luo

and Xiao

, Contact and deformation modeling for interactive environments, IEEE Transactions on Robotics 23(3) (2007), 416–430.

11.

Omar

M.N.

and Zhong

, Flexible mass spring method for modelling soft tissue deformation, International Journal of Engineering Technology and Sciences 7(2) (2020), 24–41.

12.

Felix-Rendon

, Bello-Robles

J.C.

and Fuentes-Aguilar

R.Q.

, Control of differential-drive mobile robots for soft object deformation, ISA Transactions 117 (2021), 221–233.

13.

Hogan

, Impedance control: An approach to manipulation, in 1984 American control conference, pp. 304–313, 1984.

14.

Reibert

, Hybrid position/force control of manipulators, ASME, Journal of Dynamic Systems, Measurement, and Control 103 (1981), 2–12.

15.

Wang

, Peng

, Zhang

and Wang

, High-order control barrier functions-based impedance control of a robotic manipulator with time-varying output constraints, ISA Transactions 129 (2022), 361–369.

16.

Nazmara

, Fateh

M.M.

and Ahmadi

S.M.

, A robust adaptive impedance control of robots, in 2018 6th RSI International Conference on Robotics and Mechatronics (IcRoM), pp. 40–45, 2018.

17.

Saldarriaga

, Chakraborty

and Kao

, Damping selection for cartesian impedance control with dynamic response modulation, IEEE Transactions on Robotics 38(3) (2021), 1915–1924.

18.

Kang

S.H.

, Jin

and Chang

P.H.

, A solution to the accuracy/robustness dilemma in impedance control, IEEE/ASME Transactions on Mechatronics 14(3) (2009), 282–294.

19.

Fateh

M.M.

and Babaghasabha

, Impedance control of robots using voltage control strategy, Nonlinear Dynamics 74(1) (2013), 277–286.

20.

Albu-Schaffer

and Hirzinger

, Cartesian impedance control techniques for torque controlled light-weight robots, in Proceedings 2002 IEEE International Conference on Robotics and Automation 1 (2002), 657–663.

21.

Chen

, Guo

, Hou

, Wang

and Wang

, Fractional order impedance control, in Nonlinear Dynamics and Control 2 (2020), 167–175.

22.

Izadbakhsh

, Khorashadizadeh

and Ghandali

, Robust adaptive impedance control of robot manipulators using szász–mirakyan operator as universal approximator, ISA Transactions 106 (2020), 1–11.

23.

Agitha

and Sivarani

, Deep neural fuzzy based fractional order pid controller for level control applications in quadruple tank system, Journal of Intelligent & Fuzzy Systems no. Preprint (2023), 1–15.

24.

Bharathi

and Sakthivel

, Unmanned mobile robot in unknown obstacle environments for multi switching control tracking using adaptive nonlinear sliding mode control method, Journal of Intelligent & Fuzzy Systems no. Preprint (2022), 1–13.

25.

Fanaei

and Farrokhi

, Robust adaptive neuro-fuzzy controller for hybrid position/force control of robot manipulators in contact with unknown environment, Journal of Intelligent & Fuzzy Systems 17(2) (2006), 125–144.

26.

Khoshdel

, Akbarzadeh

, Naghavi

, Sharifnezhad

and Souzanchi-Kashani

, semg-based impedance control for lower-limb rehabilitation robot, Intelligent Service Robotics 11(1) (2018), 97–108.

27.

, Ge

S.S.

and Wang

, Impedance control for human-robot interaction with an adaptive fuzzy approach, in 2017 29th Chinese Control and Decision Conference (CCDC), pp. 5889–5894, 2017.

28.

Yang

, Peng

and Liu

, Adaptive neural network force tracking impedance control for uncertain robotic manipulator based on nonlinear velocity observer, Neurocomputing 331 (2019), 263–280.

29.

Ding

, Peng

, Zhang

and Wang

, Neural network-based adaptive hybrid impedance control for electrically driven flexible-joint robotic manipulators with input saturation, Neurocomputing 458 (2021), 99–111.

30.

Peng

, Ding

, Yang

and Xin

, Adaptive neural impedance control for electrically driven robotic systems based on a neuro-adaptive observer, Nonlinear Dynamics 100(2) (2020), 1359–1378.

31.

, Dong

and Sun

, Adaptive neural impedance control of a robotic manipulator with input saturation, IEEE Transactions on Systems, Man, and Cybernetics: Systems 46(3) (2015), 334–344.

32.

and Dong

, Adaptive fuzzy neural network control for a constrained robot using impedance learning, IEEE Transactions on Neural Networks and Learning Systems 29(4) (2017), 1174–1186.

33.

Sun

, Peng

, Cheng

, Hou

Z.-G.

and Pan

, Composite learning enhanced robot impedance control, IEEE Transactions on Neural Networks and Learning Systems 31(3) (2019), 1052–1059.

34.

Baptista

L.F.

and da Costa

J.M.S.

, A force control approach of robotic manipulators in non-rigid environments, in 1999 European Control Conference (ECC), pp. 2777–2782, 1999.

35.

Jafari

, Ryu

J.-H.

, Rezaei

, Monfaredi

, Talebi

and Ghidary

S.S.

, An adaptive hybrid force/motion control design for robot manipulators interacting in constrained motion with unknown non-rigid environments, in IEEE ISR 2013, pp. 1–5, 2013.

36.

Jianqing

, Zhiwei

, Yamakita

and Ito

, Adaptive hybrid control of manipulators on uncertain flexible objects, Advanced Robotics 10(5) (1995), 469–485.

37.

Jung

, Hsia

T.C.

and Bonitz

R.G.

, Force tracking impedance control of robot manipulators under unknown environment, IEEE Transactions on Control Systems Technology 12(3) (2004), 474–483.

38.

Lee

and Buss

, Force tracking impedance control with variable target stiffness, IFAC Proceedings Volumes 41(2) (2008), 6751–6756.

39.

and Cao

, Adaptive variable impedance control of dual-arm robots for slabstone installation, ISA Transactions 2021. doi: https://doi.org/10.1016/j.isatra.2021.10.020

40.

Jinjun

, Yahui

, Ming

and Xianzhong

, Symmetrical adaptive variable admittance control for position/force tracking of dual-arm cooperative manipulators with unknown trajectory deviations, Robotics and Computer-Integrated Manufacturing 57 (2019), 357–369.

41.

Zhang

, Sun

and Deng

, Neural approximation-based adaptive variable impedance control of robots, Transactions of the Institute of Measurement and Control 42(13) (2020), 2589–2598.

42.

Kronander

and Billard

, Stability considerations for variable impedance control, IEEE Transactions on Robotics 32(5) (2016), 1298–1305.

43.

Duan

, Gan

, Chen

and Dai

, Adaptive variable impedance control for dynamic contact force tracking in uncertain environment, Robotics and Autonomous Systems 102 (2018), 54–65.

44.

Roveda

and Piga

, Robust state dependent riccati equation variable impedance control for robotic force-tracking tasks, International Journal of Intelligent Robotics and Applications 4(4) (2020), 507–519.