A novel impedance-based robust fuzzy sliding mode compliance control for the dexterous robot finger with uncertainties

Abstract

To promote the precision and flexibility of the dexterous robot finger, a novel impedance-based robust fuzzy sliding mode control approach is developed. In the proposed scheme, an impedance control part constructed aims to regulate the contact force; while the robust fuzzy sliding mode controller proposed accounts for enhancing the anti-interference of this uncertain robotic system. Specifically, by analyzing the forward and inverse kinematics of the finger, its dynamical model with unknown and uncertain force disturbances can be established, and based on this model, a novel robust sliding mode impedance force controller has been designed, also, several critical impedance control parameters can be quickly optimized by invoking the fuzzy logic system. Ultimately, the Lyapunov stability of the proposed control is strictly demonstrated in mathematics. The simulations present the efficacy of the proposed scheme in both constrained and unconstrained spaces.

Keywords

Robust sliding mode control dexterous robot finger compliance control system uncertainties fuzzy logic control

Introduction

Stable and effective controls are critical for the dexterous robot finger to work successfully. To accomplish the predefined tasks precisely, dexterous robot finger not only needs to track the desired trajectories but also regulates the contact force between the finger and the environment properly.¹ Given that the dexterous robot finger often operates in a restricted environment, the position tracking deviations of constrained motion usually lead to excessive contact forces and then impair the finger itself. As a result, a certain flexibility is required for dexterous hands in performing contact control tasks.

Currently, compliance control can be classified into two categories, that is, active and passive control modes. For the passive control mode, the kernel of which is installing several flexible devices in the terminal joints of the dexterous finger; however, as the weight of the finger increases, the dexterity will be reduced. Compared to the passive control mode, the active compliance mode utilizes the external control inputs to adjust contact forces, thus it has a higher flexibility and recently extensive research studies were working on this aspect. The most frequently employed compliances are the force/position hybrid control and impedance control. Despite the force/position hybrid control scheme has been successfully applied to many robotic fields,^2,3 the shortages of this approach are noticeable, for instance, considering the choice of stiffness matrix and the existing force control loop, the force/position hybrid control method will be highly dependent on the specific tasks and environments, besides, the desired forces should be known a priori and which incurs the higher requirements for control structure.

Distinct from the mechanism of force/position hybrid control, by establishing a mapping relationship for the target impedance between contact forces and positions, the impedance control can implement better flexibility and robustness to parameter perturbations⁴; thus it gains wide applications in the compliance control of industrial robots since this scheme was pioneered by Hogan. For instances, Chiu et al.⁵ proposed a joint position-based impedance control scheme for load compensation; Focchi et al.⁶ put forward a design approach for joint impedance controllers based on the inner torque loop and the positive feedback loop; Roveda et al.⁷ proposed an model-based reinforcement learning (MBRL) variable impedance controller to minimize the interaction forces in human–robot collaborations; Ji et al.⁸ proposed an adaptive impedance control scheme for an apple harvesting robot to reduce the damages to fruits; Perez-Ibarra et al.⁹ proposed an assistive–resistive approach to promote the active participation of stroke patients during rehabilitation therapy; considering the practical compliant requirements in operating robot dexterous hands in uncertain environments, Ott et al.¹⁰ developed a passivity-based framework for controlling the flexible joint manipulators. Yang et al.¹¹ proposed the Cartesian impedance controller for a dexterous robot hand called HIT. Furthermore, to compensate for the unknown uncertainties or disturbances, recently several adaptive and fuzzy logic control schemes have been developed for the compliance purpose.^12
–14

From the detailed overview aforementioned, although it seems that all the compliance control methods have been proposed for dexterous hands, actually, most of them only verify the control performance in joint space, without considering the influence of modeling error and nonlinear coupling between joints on the system performance. In terms of control algorithm, the system adopts impedance control or adaptive impedance control, which can ensure contact flexibility, but it is difficult to ensure tracking accuracy and anti-interference capability. Moreover, many articles do not strictly analyze the stability of the control strategy.

Since the robot dexterous fingers often adopt the rope transmission,¹⁵ the high transmission ratio and multi-freedoms usually lead to flexibility, coupling, and nonlinearity. As a result, system uncertainties are unavoidable in the dynamics of dexterous hands.^16,17 Moreover, considering the uncertainties are possibly time-varying and fast time-varying, the compliance control and uncertainty suppression problems for dexterous hands need to be further explored. To ensure the control performance of the dexterous finger before and after contact, motivated by these existing investigations, and to improve the tracking precision and robustness, this article integrates the impedance control, the fuzzy logic system, and the robust sliding mode control seamlessly. The major contributions of current research are twofold.

Aiming for the grasp characteristics of the dexterous hand, a novel impedance controller is developed to increase the flexibility of the finger operation. For several critical impedance parameters, the fuzzy set theory can well optimize them.

By combining the impedance control and robust sliding mode control design, not only the time-varying uncertainties, such as nonlinear frictions and unknown disturbances, can be well compensated but also the tracking precision in unconstrained space and the flexible contact control in constrained space can be both guaranteed. The Lyapunov stability of the proposed control is strictly demonstrated in mathematics.

System description

Kinematical modeling of dexterous robot finger

Before the dynamical modeling, it is necessary to analyze the forward and inverse kinematics of the dexterous finger. This article mainly investigates the index finger (or ring finger) with three degrees of freedom (see Figure 1). The motion of this dexterous finger includes the lateral swing of the lower knuckle, and the flexion of the middle knuckle and the upper knuckle, where the lateral swing axis is perpendicular to the flexion axis. To better describe the kinematics of this robot, the D-H parameters are given in Table 1.

Figure 1.

Diagram of the finger.

Table 1.

D-H parameters for index finger.

i	$α_{i - 1} (degrees)$	$l_{i - 1} (mm)$	$q_{i} (degrees)$
1	0°	0	q ₁
2	90°	0	q ₂
3	0°	l ₂	q ₃
4	0°	l ₃	0

According to the D-H parameters and kinematics principle, the position and orientation of each joint frame can be obtained as follows

_{1}^{0} T = [\begin{array}{l} c_{1} & - s_{1} & 0 & 0 \\ s_{1} & c_{1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}],_{2}^{1} T = [\begin{array}{l} c_{2} & - s_{2} & 0 & l_{1} \\ 0 & 0 & 1 & 0 \\ s_{2} & c_{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}]_{3}^{2} T = [\begin{array}{l} c_{3} & - s_{3} & 0 & l_{2} \\ s_{3} & c_{3} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}],_{4}^{3} T = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]

then, the total transformation matrix $_{4}^{0} T$ turns into

_{4}^{0} T =_{1}^{0} T_{2}^{1} T_{3}^{2} T_{4}^{3} T = [\begin{matrix} c_{1} c_{23} & - c_{1} s_{23} & - s_{1} & l_{3} c_{1} c_{23} + l_{2} c_{1} c_{2} + l_{1} c_{1} \\ s_{1} c_{23} & - s_{1} s_{23} & c_{1} & l_{3} s_{1} c_{23} + l_{2} s_{1} c_{2} + l_{1} s_{1} \\ s_{23} & c_{23} & 1 & l_{3} s_{23} + l_{2} s_{2} \\ 0 & 0 & 0 & 1 \end{matrix}]

where $s_{i} = sin (q_{i})$ , $c_{i} = cos (q_{i})$ , $s_{i j} = sin (q_{i} + q_{j})$ , $c_{i j} = cos (q_{i} + q_{j})$ , and $i, j = 1, \dots,3$ . For the position of fingertip relative to the reference frame 0, we have

\begin{array}{l} x = l_{3} c_{1} c_{23} + l_{2} c_{1} c_{2} + l_{1} c_{1} \\ y = l_{3} s_{1} c_{23} + l_{2} s_{1} c_{2} + l_{1} s_{1} \\ z = l_{3} s_{23} + l_{2} s_{2} \end{array}

Subject to equation (1), the solutions to joint positions $q_{1}, q_{2}, q_{3}$ are given as follows: Dividing y by x, we can obtain the joint angle q ₁, that is

q_{1} = arctan (y / x)

Next, by substituting equation (1) into equation (2), the joint position q ₂ can be acquired as

q_{2} = \pm arctan (\frac{C}{\sqrt{A^{2} + B^{2} - C^{2}}}) - arctan (\frac{A}{B})

where $A = 2 l_{2} (x / c_{1} - l_{1})$ , $B = 2 z l_{2}$ , and $C = (x / c_{1} - l_{1})^{2} + l_{2}^{2} + z^{2} - l_{3}^{2}$ . Similarly, by substituting equation (3) into z in equation (1), we can get q ₃ as follows

q_{2} = \pm arctan (\frac{F}{\sqrt{D^{2} + E^{2} - F^{2}}}) - arctan (\frac{D}{E})

where $D = l_{3} s_{2}$ , $E = l_{3} c_{2}$ , $F = z - l_{2} s_{2}$ , and the velocity mapping between the joint space and operation space can be expressed as

\dot{x} (\dot{q}, q, t) = J (q, t) \dot{q}

In view of the mapping relationship $\dot{x} = J (q) \cdot \dot{q}$ between the joint space and operation space, where $\dot{x} = [x, y, z]^{T} \in ℝ^{3 \times 1}$ , $\dot{q} = [{\dot{q}}_{1}, {\dot{q}}_{2}, {\dot{q}}_{3}]^{T} \in ℝ^{3 \times 1}$ , and $J (q) \in ℝ^{3 \times 3}$ (Jacobian matrix). Besides, the Jacobian matrix $J (q) \in ℝ^{3 \times 3}$ is given as

J (q) = [\begin{matrix} - l_{2} s_{1} c_{2} - l_{3} s_{1} c_{23} & - l_{2} s_{2} c_{1} - l_{3} c_{1} s_{23} & - l_{3} c_{1} s_{23} \\ l_{2} c_{1} c_{2} + l_{3} c_{1} c_{23} & - l_{2} s_{1} s_{2} - l_{3} s_{1} s_{23} & - l_{3} s_{1} s_{23} \\ 0 & l_{2} c_{2} + l_{3} c_{23} & l_{3} c_{23} \end{matrix}]

by representing the velocity mapping from joint space to workspace.

Dynamical modeling of dexterous finger

Considering the following dynamical equation for dexterous finger

\begin{array}{l} M (q (t), t) \ddot{q} (t) + C (q (t), \dot{q} (t), t) \dot{q} (t) + G (q (t), t) \\ + τ_{f} (\dot{q} (t), t) = τ (t) \end{array}

where $q, \dot{q}, \ddot{q} \in ℝ^{3 \times 1}$ denote the joint position, velocity, and acceleration; $M (q, t)$ , $C (q (t), \dot{q} (t), t)$ , and $G (q (t), t)$ represent the inertia matrix, the force matrix, and the gravity vector, respectively. τ is the control system input and $τ_{f}$ is the joint friction torque.

Given that the dexterous finger generally adopts the tendon rope transmission, the nonlinear frictions that mainly encompass coulomb and viscous friction are modeled as $τ_{f} (\dot{q}, t) = sgn (\dot{q}) (k_{1} | \dot{q} | + k_{2})$ , where $sgn (\dot{q})$ is on behalf of a symbolic function of joint velocity, $k_{1} and k_{2}$ are the coulomb and viscous friction coefficients, respectively. Under the Newton–Euler equation,¹⁸ several system parameters can be determined, that is

\begin{array}{l} M (q, t) = [\begin{matrix} M_{11} & 0 & 0 \\ 0 & M_{22} & M_{23} \\ 0 & M_{32} & M_{33} \end{matrix}], C (q, \dot{q}, t) \dot{q} = [\begin{matrix} C_{1} \\ C_{2} \\ C_{3} \end{matrix}] \\ G (q, t) = [\begin{matrix} 0 \\ m_{3} g l_{3} c_{23} + (m_{2} + m_{3}) g l_{2} c_{2} \\ m_{3} g l_{3} c_{23} \end{matrix}] \end{array}

where $I_{2}, I_{3}$ denote the moments of inertia of links and

M_{11} = I_{2} + I_{3} + m_{2} l_{2}^{2} c_{2}^{2} + 2 m_{3} l_{3} l_{2} c_{2} c_{23} + m_{3} l_{3}^{2} c_{23}^{2} + m_{3} l_{2}^{2} c_{2}^{2}

M_{22} = I_{2} + I_{3} + m_{2} l_{2}^{2} + 2 m_{3} l_{3} l_{2} c_{3} + m_{3} l_{3}^{2} + m_{3} l_{2}^{2}

M_{23} = M_{32} = I_{3} + m_{3} l_{3}^{2} + m_{3} l_{2} l_{3} c_{3}

M_{33} = I_{3} + m_{3} l_{3}^{2}

\begin{array}{l} C_{1} = - (2 m_{3} l_{2} l_{3} s_{2} c_{23} + 2 m_{3} l_{3}^{2} c_{23} s_{23} + 2 m_{3} l_{2} l_{3} c_{2} s_{23} \\ + 2 m_{3} l_{2}^{2} c_{2} s_{2}) {\dot{q}}_{1} {\dot{q}}_{2} - (2 m_{3} l_{3}^{2} c_{23} s_{23} + 2 m_{3} l_{2} l_{3} c_{2} s_{23}) {\dot{q}}_{1} {\dot{q}}_{3} \\ C_{2} = (m_{2} l_{2}^{2} s_{2} c_{2} + m_{3} l_{2} l_{3} c_{2} s_{23} + m_{3} l_{3}^{2} c_{23} s_{23} + m_{3} l_{2}^{2} c_{2} s_{2} \\ + m_{3} l_{2} l_{3} c_{23} s_{2}) {\dot{q}}_{1}^{2} - m_{3} l_{2} l_{3} s_{3} {\dot{q}}_{3}^{2} - 2 m_{3} l_{2} l_{3} s_{3} {\dot{q}}_{2} {\dot{q}}_{3} \\ C_{3} = (m_{3} l_{3}^{2} c_{23} s_{23} + m_{3} l_{2} l_{3} c_{2} s_{23}) {\dot{q}}_{1}^{2} + m_{3} l_{2} l_{3} s_{3} {\dot{q}}_{2}^{2} \end{array}

For the sake of compliance control, it is necessary to transform the dynamical model of the dexterous finger from joint space to Cartesian space. Under several static equilibrium conditions, there has a linear mapping between the translational force F_x transmitted to the fingertip and the joint torque,¹⁸ that is

F_{x} (q, t) = J^{- T} (q, t) τ (t)

Taking the derivative of equation (5), it yields

\ddot{q} (t) = J^{- 1} (q, t) (\ddot{x} (t) - \dot{J} (q, t) J^{- 1} (q, t) \dot{x} (t))

substituting equation (11) into equation (7) and reorganizing, we can obtain the dynamical model of the dexterous finger in the workspace as follows

\begin{array}{l} M_{x} (q (t), t) \ddot{x} (t) + C_{x} (q (t), \dot{q} (t), t) \dot{x} (t) + G_{x} (q (t), t) \\ + τ_{x f} (\dot{q} (t), t) + Δ (q (t), \dot{q} (t), \ddot{q} (t), t) = F_{x} (q, t) - F_{e} (t) \end{array}

where

\begin{array}{l} G_{x} (q, t) = J^{- T} (q) G (q, t) \\ τ_{x f} (\dot{q}, t) = J^{- T} (q) τ_{f} (\dot{q}, t) \\ M_{x} (q, t) = J^{- T} (q) M (q, t) J^{- 1} (q) \\ C_{x} (q, \dot{q}, t) = J^{- T} (q) (C (q, \dot{q}, t) - M (q, t) J^{- 1} (q) \dot{J} (q) \dot{q}) \end{array}

and $Δ (q, \dot{q}, \ddot{q}, t)$ ( $| | Δ (q, \dot{q}, \ddot{q}, t) | | \leq δ$ ) denotes the lumped uncertainty, $F_{e} (t)$ represents the contact resistance between the fingertip and the object. To benefit for the later analysis of stability, two assumptions are given as follows.

Assumption 1

For any $(q, t) \in ℝ^{n} \times ℝ$ , the inertia matrix $M_{x} (q, t)$ is symmetric and positive definite.

Assumption 2

For any $(q, \dot{q}, t) \in ℝ^{n} \times ℝ^{n} \times ℝ$ , the inertia matrix ${\dot{M}}_{x} (q, t) - 2 C_{x} (q, \dot{q}, t)$ is skew symmetric.

Remark 1

Assumptions 1 and 2 are the inherent properties rather than the hypotheses, since for most of articulated robots, both Assumptions 1 and 2 can be satisfied.

Impedance-based robust fuzzy sliding mode controller design

Aiming for the position and impedance control targets of robot dexterous finger, the following design requirements shall be met. (1) The trajectory following control: When the robot finger moves in the free space, the system uncertainties and frictions should be well attenuated, so as to track the reference trajectories swiftly and precisely. (2) The impedance control: To render the contact forces to be within an expected range, the controller should meticulously manipulate the contact forces between the fingertip and the object, with the purpose that guaranteeing the flexibility of operation and avoiding the rigid impact.

Subject to the above requirements,¹⁹ the proposed control block for the finger is shown in Figure 2. First, given the workspace of the fingertip, the reference trajectory x_c is planned, then by invoking the impedance controller proposed, the transformation from the fingertip position to impedance force can be implemented, besides, the practical position of the fingertip x can track the desired impedance trajectory x_d , and the contact resistance F_e is acquired. Next, via analyzing the inverse kinematics of the finger, its spatial position has been converted to the joint angle q, and then, both the position tracking error e and its derivative $\dot{e}$ can be deduced. Eventually, by constructing the robust sliding mode controller, not only the position tracking control can be achieved, but also the compliance control and uncertainty compensation will be well settled.

Figure 2.

Proposed control block for the dexterous robot finger.

Impedance control design and fuzzy optimization

In this section, we define the impedance control (i.e. the interaction between the fingertip and environment) as a mass–damping–spring system, which describes the majority of the dynamic mapping between the forces and positions. Let the desired trajectory be x_c , the contact resistance between the fingertip and physical object be F_e , as shown in Figure 3, the goal of impedance control is as follows: The practical position x of the fingertip can track the desired impedance trajectory x_d when the fingertip acts on the physical object, and x_d can be derived by the following differential equation

M_{t} ({\ddot{x}}_{c} - {\ddot{x}}_{d}) + B_{t} ({\dot{x}}_{c} - {\dot{x}}_{d}) + K_{t} (x_{c} - x_{d}) = F_{e}

where $x_{d} (0) = x_{c} (0)$ , ${\dot{x}}_{d} (0) = {\dot{x}}_{c} (0)$ , and $M_{t}, B_{t}, K_{t}$ represent the mass, the damping, and the stiffness coefficients, respectively.

Figure 3.

Schematic of the impedance control for the dexterous robot finger.

In general, the stiffness of robot finger contacting with the environment is uncertain, and this causes a significant change in the contact forces between the finger and the environment, thus the control efficacy is very difficult to guarantee. By regulating the impedance parameters appropriately, the force/position mapping between the fingertip and the actual environment can be changed, furthermore, to avoid the complex adjustment of impedance parameters and promote the estimation accuracy of contact forces, the fuzzy logic system is developed to perform the fuzzy self-tuning for impedance parameters.

The specific regulating steps are as follows: First, we establish the fuzzy mapping between the position deviation e and the changing rate $\dot{e}$ , the impedance control parameters $M_{t}, B_{t}, K_{t}$ . Second, based on the rules of fuzzy formulations, the desired increment of impedance parameters (i.e. $Δ M_{t}, Δ B_{t}, Δ K_{t}$ ) can be finally determined.

Let $e, \dot{e}$ be the input variables, $Δ M_{t}, Δ B_{t}, Δ K_{t}$ be the output variables, taking the output $Δ M_{t}$ for instance, the fuzzy sets for system input/output can be built via the following triangular membership functions, that is

\begin{array}{l} e = {NB NM NS ZO PS PM PB} \\ \dot{e} = {NB NM ZO PM PB} \\ Δ M_{t} = {NB NM ZO PM PB} \end{array}

where $NB, NM, NS, ZO$ represent the negative numbers, that is, the negative biggest, the negative medium, the negative small, and the negative zero, respectively, while $PS, PM, PB$ represent the positive numbers, that is, the positive small, the positive medium, and the positive biggest, respectively. The fuzzy rules constructed are shown in Table 2.

Table 2.

Fuzzy logic rules.

$\dot{e} \ Δ M_{t} \ e$	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	PS	PB	PB
NM	NB	NM	NM	NS	PS	PM	PB
ZO	NB	NM	NS	ZO	PM	PM	PB
PM	NB	NM	NS	PS	PM	PM	PB
PB	NB	NM	NS	NM	PM	PM	PB

NB: negative biggest; NM: negative medium; NS: negative small; ZO: negative zero; PS: positive small; PM: positive medium; PB: positive biggest.

To solve the fuzzy parameters (e.g. $Δ M_{t}$ ), the parallel method is employed for fuzzy reasoning and the centroid algorithm is for defuzzification, that is

Δ M_{t} = \frac{\sum_{i = 1}^{n} μ_{i} R (e, \dot{e})}{\sum_{i = 1}^{n} R (e, \dot{e})}

where $μ_{i}$ denotes the degree of membership of ith fuzzy output, $R (e, \dot{e})$ denotes the fuzzy output obtained on basis of the fuzzy rules. As a result, we can obtain an updated M_t by the equation ${M^{'}}_{t} = M_{t} + γ [\begin{matrix} Δ M (t) & 0 \\ 0 & Δ M (t) \end{matrix}]$ , and matrix γ is the quantization factor diagonal matrix. when the variation range of tracking error is [-e,e], and the corresponding fuzzy domain is given as {2 1 0 1 2}, then, gamma=diag $(\frac{5}{e}, \frac{5}{e})$ . As a result, when the elements in diagonal increase, the variation range of tracking error will decrease, and vice versa, which improves the sensitivity of error control, and vice versa. Generally, its value is determined by experiences.

Similarly, the fuzzy treatment of damping and stiffness coefficients (i.e. $Δ B_{t}$ and $Δ K_{t}$ ) can reference $Δ M_{t}$ as above and will not be repeated anymore.

Robust sliding mode position control design

Before presenting the robust sliding mode controller, it is necessary to choose the proper switching function and reaching law. Supposing the desired trajectory of the fingertip is x_d , and x_d is C ² continuous, then, the tracking error can be expressed as

e (t) = x_{d} (t) - x (t)

For position tracking control of fingertips, the sliding function is given below

s (t) = \dot{e} (t) + Λ e (t)

where $e (t)$ and $\dot{e} (t)$ represent the tracking error and its change rate, respectively, $Λ$ is a positive definite matrix that satisfies the Hurwitz condition, that is, each eigenvalue is located at the left-half plane.

Since the oscillation will arise if the symbolic function is applied in the reaching law of sliding mode control, in this article, we adopt the exponential reaching law instead of the switching or other functions to remove the discontinuous characteristics, this can not only reduce the chattering in sliding mode control but also improve the dynamic response of sliding mode control. The reaching law is designed as

\dot{s} = - K s - δ tanh (s / ε)

where $\dot{s} = - K s$ belongs to the exponential approaching term, and which can ensure the system states converge to the sliding surfaces rapidly in the initial stage; while $\dot{s} = - δ tanh (s / ε)$ denotes a constant approaching term, as s drops down to around zero, the approaching speed is close to δ; therefore, the system states reach the sliding manifold in finite time.

Let

{\dot{x}}_{s} (t) = {\dot{x}}_{d} (t) + Λ e (t)

$x_{s} (t)$ is an intermediate variable to facilitate the expression of equation (19), according to the dynamic model of finger of the robot hand in Cartesian space, the complete robust sliding mode controller of the robot finger for tracking control is proposed, that is

\begin{matrix} F_{x} (q, \dot{q}, t) = M_{x} (q, t) {\ddot{x}}_{s} (t) + C_{x} (q, \dot{q}, t) {\dot{x}}_{s} + G_{x} (q, t) \\ + τ_{x f} (\dot{q}, t) + F_{e} (t) + K s + δ tanh (s / ε) \end{matrix}

where $K, ε > 0$ denotes two constant scalars. In the next section, we will show the stability analysis of the proposed control method (19).

Stability analysis for the proposed control

Theorem 1

Consider the uncertain mechanical system (7) subjected to Assumptions 1 and 2, the uniform boundedness (UB) and uniform ultimate boundedness (UUB) can be ensured by the control (19).

UB: For any $r > 0$ and $| | s (t_{0}) | | \leq r$ , there is always a finite $η (r)$ such that $| | s (t) | | \leq η (r)$ when $t > t_{0}$ .

UUB: For any $r > 0$ and $| | s (t_{0}) | | \leq r$ , there is always a finite $T (\bar{η}, r)$ such that $| | s (t) | | \leq \bar{η}$ when $t \geq t_{0} + T (\bar{η}, r)$ .

Proof

By substituting equation (19) into equation (12), equation (12) can be rewritten as

\begin{array}{l} M_{x} (q (t), t) \ddot{x} (t) + C_{x} (q (t), \dot{q} (t), t) \dot{x} (t) + G_{x} (q (t), t) \\ + τ_{x f} (\dot{q} (t), t) + Δ (q (t), \dot{q} (t), \ddot{q} (t), t) \\ = M_{x} (q, t) {\ddot{x}}_{s} (t) + C_{x} (q, \dot{q}, t) {\dot{x}}_{s} + G_{x} (q, t) \\ + τ_{x f} (\dot{q}, t) + F_{e} (t) + K s + δ tanh (s / ε) - F_{e} (t) \end{array}

With equations (15), (16), and (18), we have

\begin{array}{l} \dot{x} (t) = {\dot{x}}_{s} (t) - s (t) \\ \ddot{x} (t) = {\ddot{x}}_{s} (t) - \dot{s} (t) \end{array}

next, by substituting equation (21) into equation (20), it leads to

\begin{array}{l} M_{x} (q, t) \dot{s} + C_{x} (q, \dot{q}, t) s + K s + δ tanh (s / ε) \\ - Δ (q, \dot{q}, \ddot{q}, t) = 0 \end{array}

Here, recalling Assumption 1, $M_{x} (q, t)$ is a symmetric and positive definite matrix for any $(q, t) \in ℝ^{n} \times ℝ$ , thus we select a legitimate Lyapunov function $V (s, t)$ as follows

V = s^{T} M_{x} (q, t) s / 2

differentiating equation (23) along the system trajectory (7), we can obtain

\dot{V} = s^{T} M_{x} (q, t) \dot{s} + s^{T} {\dot{M}}_{x} (q, t) s / 2

as shown in Assumption 2, the matrix ${\dot{M}}_{x} (q, t) - 2 C_{x} (q, \dot{q}, t)$ is skew symmetric, then, $s^{T} ({\dot{M}}_{x} (q, t) - 2 C_{x} (q, \dot{q}, t)) s = 0$ and equation (24) can be rewritten as

\dot{V} = s^{T} M_{x} (q, t) \dot{s} + s^{T} C_{x} (q, \dot{q}, t) s

by combining equations (25) and (22), equation (24) further turns into

\dot{V} = s^{T} (- \hat{K} s - δ tanh (s / ε) + Δ (q, \dot{q}, \ddot{q}, t))

Subject to the claim in Polycarpou and Ioannou,²⁰ we have

\begin{array}{l} s^{T} (- δ tanh (s / ε) + Δ (q, \dot{q}, \ddot{q}, t)) \\ = - δ s^{T} tanh (s / ε) + s^{T} Δ (q, \dot{q}, \ddot{q}, t) \\ = - δ | | δ | | + δ μ ε + s^{T} Δ (q, \dot{q}, \ddot{q}, t) \\ \leq δ μ ε \end{array}

where $μ = e^{- (μ + 1)},$ that is, $μ = 0.2785$ , and therefore

\begin{array}{l} \dot{V} \leq - s^{T} K s + δ μ ε \\ \leq - λ_{min} (K) s^{T} s + δ μ ε \\ = \frac{2 λ_{min} (\hat{K})}{λ_{max} (M_{x})} \cdot λ_{max} (M_{x}) s^{T} s / 2 + δ μ ε \\ \leq - 2 λ V + b \end{array}

where $λ_{min} (\cdot)$ and $λ_{max} (\cdot)$ represent the minimum and maximum eigenvalues of a matrix, respectively. Furthermore, $λ = λ_{min} (\hat{K}) / λ_{max} (M_{x})$ and $b = δ μ ε$ .

In the light of the lemmas in Petros and Jing²¹ and Yang et al.,²² it can be inferred that $V (q, t)$ is uniformly bounded, that is

\begin{matrix} V (t) \leq e^{- 2 λ (t - t_{0})} V (t_{0}) + b e^{- 2 λ t} \int_{t_{0}}^{t} e^{2 λ τ} d τ \\ = e^{- 2 λ (t - t_{0})} V (t_{0}) + \frac{b}{2 λ} (1 - e^{- 2 λ (t - t_{0})}) \end{matrix}

and equation (29) implies that

lim_{t \to \infty} V (t) \leq \frac{δ μ ε}{2 λ}

According to the above formula, the tracking error and error derivative converge gradually, the convergence accuracy depends on the δ, ε, and λ, when ε and δ are smaller, λ is larger, the error is smaller, and the convergence effect is better.

This accomplishes the proof of Theorem 1.

Numerical simulations

In this section, to testify the feasibility and effectiveness of the proposed control scheme, a dexterous finger with three joints (see Figure 1) is taken into account. Assuming that once the finger contacts the physical object, the normal direction of the contact surface is blocked, joints 2 and 3 perform the flex and extend operations, while joint 1 is static. As a result, the length of joint 1 is set as zero with a purpose of regulating the operation plane of the fingertip.

In simulations, the parameters involved are summarized as follows: $m_{2} = 2.5$ , $m_{3} = 1.75$ , $l_{2} = 2$ , $l_{3} = 1$ , $τ_{x f} = [5 {\dot{q}}_{2} + 3 sgn ({\dot{q}}_{2}),5 {\dot{q}}_{3} + 3 sgn ({\dot{q}}_{3})]^{T}$ , $x_{d} (0) = x_{c} (0)$ , ${\dot{x}}_{d} (0) = {\dot{x}}_{c} (0)$ , $M_{t} = diag (2, 2)$ , $B_{t} = diag (5, 5)$ , $K_{t} = diag (50, 50)$ , $K = diag (15, 15)$ , $Λ = diag (15, 15)$ , $δ = 1.2$ , and $ε = 0.5$ ; the system uncertainty and disturbances $Δ (q, \dot{q}, \ddot{q}, t) = sin (\frac{1}{2} t)$ ; the desired trajectory in Cartesian space: $x = sin (\frac{Π}{4} t)$ , $y = 2 + cos (\frac{Π}{4} t)$ ; the constraint position for contact is given as $x = 0.5$ , if $x < 0.5$ , the fingertip will not contact the physical object, and the contact resistances $F_{e} = 0$ ; but when $x \geq 0.5$ , the fingertip will contact with the physical object, and the contact resistance can be computed by equation (13). Finally, Figures 4 to 8 have shown the numerical results.

Figure 4.

Position tracking using fuzzy-impedance-sliding mode control.

Figure 5.

Position tracking using PD control.

Figure 6.

Position tracking using sliding mode control.

Figure 7.

Position tracking effect in X and Y directions.

Figure 8.

External forces acting on the fingertip.

Figure 4 shows the practical tracking for the desired trajectories in Cartesian space, from which it can be seen that, when $x < 0.5$ (i.e. in free space), the practical position of the fingertip can well coincide with the desired trajectories, and the tracking error is around 0.25%; but when $x \geq 0.5$ (i.e. in the constraint space), due to the contact force, the fingertip has taken the control flexibility while tracking the impedance trajectories generated by the damping force.

From the error distribution in Figure 4, the tracking error of spatial position is small in the free motion phase. After being constrained, the tracking error increases and finally stabilizes around 0.042, with a maximum error of 0.053, which indicates the maximum tracking error is 5.3%. The increase of the error occurs in the damping movement stage, which indicates that the error is mainly caused by the damping motion, and it is a loss of tracking accuracy to improve the flexibility of the system.

Compared with Figure 4, when the constraint of the fingertip is not considered, the system only adopts proportional derivative (PD) control and sliding mode control, and the control effect is shown in Figures 5 and 6. Figure 5 shows the effect of PD control. It is obvious that the effect of PD control is poor, the tracking error fluctuates around 0.22, which shows the tracking error reaches 22%, and there is no compliance control effect. Figure 6 is the effect diagram of sliding mode control, and its relevant design parameters are consistent with those in the text. It can be seen that if the system operates without constraints in Cartesian space, the sliding mode control performance is very good, with the highest accuracy, and the tracking error is stable at 0.0006, which shows the anti-interference ability and high-precision tracking ability of the sliding mode control, but there is also no compliance control effect. These two control effects cannot meet the requirements of the position accuracy and compliance control of finger movement at the same time.

Figure 7 shows the position tracking effect in X and Y directions; this figure shows that the impedance mainly acts in the X-direction, which is the normal direction of the contact surface. Besides, Figure 7 implies that the external contact force begins to act on the fingertip after $t = 0.67 s$ .

Figure 8 presents the external contact force acting on fingertip, as the fingertip and the physical object begin to contact (i.e. $t = 0.67 s$ ), the fingertip is constrained by the external contact forces. Figure 8 indicates that the maximum of external contact forces in the X-direction is around 20. On the other hand, since the constraint surface is perpendicular to the X-direction, the external force acting on the X-direction is much larger than that of the Y-direction. The desired and impedance trajectories are shown in Figure 9, from this figure it is not difficult to find that, the impedance trajectory in the Y-direction coincides with the desired trajectory very well, in other words, when the fingertip is blocked in the X-direction, the impedance trajectory x_d is close to 0.5, which means that the goal of compliance control is achieved. This demonstrates the high tracking precision of our proposed control from another perspective.

Figure 9.

Desired and practical trajectories.

Through fuzzy optimization, under the condition of the extreme value of position error $e_{max} = \pm 1$ , the impedance parameters converge $M_{t} = diag (1, 1)$ ; $B_{t} = diag (9, 9)$ ; $K_{t} = diag (40, 40)$ . Compared with the impedance control with traditional constant parameters, the trajectory x_d caused by the fuzzy impedance parameters optimized is represented in Figure 10. It is obvious from it, after the optimization of fuzzy control, the simulation results meet the requirements of the system setting and converge to the stable value faster.

Figure 10.

Effect of fuzzy optimization of impedance parameters.

Conclusion

By analyzing the forward and inverse kinematics of the dexterous finger, the dynamical models in joint space and Cartesian space are established for this finger, on basis of these, the proposed robust fuzzy sliding mode impedance controller can not only force the fingertip to track the desired trajectory in unconstrained space but also implement the compliance control in constrained environment, moreover, the nonlinear friction and uncertainties can be well compensated. The simulation results demonstrate the effectiveness and feasibility of our proposed control approach, in the future, the experimental verification and multi-fingered coordinated control will be further considered.

Footnotes

Author contributions

Jiufang Pei: Methodology, Formal analysis, and Writing—Original Draft.

Siyang Yang: Methodology, Review, and Editing.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the research project of 2020 Anhui Polytechnic University research program (No. Xjky2020002).

ORCID iD

Jiufang Pei

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