Decentralized Integral Nested Sliding Mode Control for Time Varying Constrained Modular and Reconfigurable Robot

Abstract

This paper presents a novel decentralized integral nested sliding mode control approach for a time varying constrained modular and reconfigurable robot (MRR) and addresses the problem of joint trajectory tracking control of the MRR with strongly coupled model uncertainty. First, the dynamic model of the time varying constrained MRR is described as a synthesis of interconnected subsystems. Second, a decentralized control scheme is proposed based on only local dynamic information of each module. An integral nested sliding surface is implemented to reduce the chattering effect of the controller. Model uncertainties, including the unmodeled subsystem dynamics, the friction modeling error, and the interconnected dynamic coupling, are compensated by a variable gain supertwisting algorithm (VGSTA) based controller. The stability of the close-loop system is proved using the Lyapunov theory. Finally, simulations are performed for a time varying constrained 2-DOF MRR to study the effectiveness of the proposed method.

1. Introduction

Modular and reconfigurable robot consists of robot modules which incorporate power electronics, computing systems, sensors, and actuators. These modules can be serially connected with standard electromechanical interfaces and assembled to desirable configuration for satisfying the requirements of various tasks under different external environments and constraints. An MRR needs an appropriate control system to provide accuracy and stability after reconfigurations.

A number of researches have been made toward the dynamics and control method of MRR. A neurofuzzy hierarchical control architecture is proposed for an MRR which uses learning control to compensate unmodeled system dynamics due to reconfiguration [1]. The adaptive control parameters are updated using a skill module that is a part of the higher level of the control system hierarchy. A dynamic control of robots using a virtual decomposition-based control approach is presented in [2]; following this work, they propose a precision control method for MRR with embedded FPGA [3]. Based on a calculating torque control method, [4] proposes a neurofuzzy compensator to reduce the modeled and unmodeled uncertainty of the modular manipulators. However, these methods are concentrated on the centralized control. Indeed, due to high computation costs, robustness, and complexities, a centralized controller designed on the basis of an entire system may hardly be applicable for controlling an MRR.

Decentralized control is a particularly promising concept in implementing an MRR system. The decentralized control systems often arise from various complex situations where there exist physical limitations on information exchange among several subsystems for which there is insufficient capability of having a single central controller. Furthermore, based on only local dynamic information of the modules, the decentralized control architecture can not only provide the flexibility, which is applicable to all the configurations, but also reduce the computation time that is critical for real-time application to an MRR system. Therefore, both practically and theoretically, it is meaningful to devise an ideal decentralized control architecture for the MRR. A decentralized reinforcement learning robust optimal tracking control theory for time varying constrained MRR based on action-critic-identifier and state-action value function is presented to address the problem of the continuous time nonlinear optimal control policy for strongly coupled uncertainty robotic system [5]. A decentralized active disturbance rejection control method for reconfigurable modular robot is proposed with a high-precision extended state observer to estimate the dynamic model nonlinear terms and the interconnection terms of the subsystems [6]. A decentralized robust control algorithm based on Lyapunov stability analysis and backstepping technique is given for MRR with harmonic drive transmission [7]. A modular control method is proposed based on joint torque sensing technique; with this method, an MRR can be stabilized joint by joint [8]. A calculating torque based robust fuzzy neural network controller is proposed in [9], which is used to solve the problem of model uncertainty in the process of model generating. In [10], a design methodology of interval type-2 fuzzy controller is derived for MRR manipulators with uncertain dynamic parameters. An observer based decentralized adaptive fuzzy controller for reconfigurable manipulator is proposed in [11]. By designing the state observer, the adaptive fuzzy systems which are used to model the unknown dynamics of subsystem and interconnected coupling term can be constructed by using the state estimations.

Some researchers have studied the control method for an MRR that combined the sliding mode control method with the decentralized control scheme. In [12, 13], a stable decentralized adaptive fuzzy sliding mode control method is proposed for modular manipulators to satisfy the concept of modular software. In these researches, the fuzzy logic system is used to approximate the unknown dynamics of the subsystem, and a sliding mode controller with an adaptive scheme is designed to compensate both interconnected coupling term and fuzzy approximation error. However, the main drawback for these control strategies is the so-called “chattering effect,” which is introduced by the one-order sliding surface which is designed directly by using a Lyapunov function and may affect the control precision of the MRR system. Therefore, an excellent control approach for an MRR should be able to compensate the model uncertainty accurately and reduce the chattering effect of the controller.

In this paper, a novel decentralized integral nested sliding mode control approach is proposed to address the problem of joint trajectory tracking control of a time varying constrained MRR with strongly coupled model uncertainty. The dynamic model of the time varying constrained MRR is described as a synthesis of interconnected subsystems. With the decentralized control scheme, only the local dynamic information of each module is required in the controller of the module, so that the modules can be added or removed without the need to readjust the control parameters of the other modules. According to the concepts of integral sliding mode (ISM) [14 –17] and nested sliding mode (NSM) [18, 19], an integral nested sliding surface, which is designed by employing a pseudosliding surface block, is implemented to reduce the chattering effect of the controller. A sigmoidal function, which is used to design the sliding surface, is able to introduce robustness in each block, and the integral part will reduce the sliding surface gains. Model uncertainties, including unmodeled subsystem dynamics, friction modeling error, and the interconnected dynamic coupling, are compensated by a VGSTA based controller [20, 21], which can adjust the control gains according to the actual bound of the uncertainty. The Lyapunov theory is used to prove the stability of the close-loop system. Finally, simulations are performed for a time varying constrained 2-DOF MRR to illustrate the effectiveness of the proposed method.

2. Dynamic Model Formulation

Define the time varying external constraint for the end of MRR as follows:

Φ (q, t) = 0,

(1)

where $q \in R^{n}$ is the vector of joint position. In function $Φ : R^{n} \to R^{m}$ , m is the dimension of the external limiting conditions. Under the time varying constraint, the dynamic model of MRR with nth degree of freedom can be given as follows:

M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F (q, \dot{q}) = u + J_{Φ}^{T} (q, t) f,

(2)

where $M (q) \in R^{n \times n}$ is the positive definite inertia matrix, $C (q, \dot{q}) \dot{q} \in R^{n \times 1}$ is the Coriolis and centripetal torque, $G (q) \in R^{n \times 1}$ is the gravitational torque, $F (q, \dot{q}) \in R^{n \times 1}$ is the friction torque, $u \in R^{n \times 1}$ is the vector of control torque, and J_Φ^T(q, t)f is the contact force generated by the contact between the end-effector of MRR and time varying constraint. After introducing the mth constraints for the robot which worked in the free space, due to the limitation of (1), the system lost the mth degree of freedom. Therefore, the degree of freedom of the robot changed from n to n − m, so that only n − m independent joint positions can be used to describe the system with the restricted motion.

Define

q = [\begin{bmatrix} q_{1} \\ q_{2} \end{bmatrix}], q_{1} \in R^{n - m}, q_{2} \in R^{m} .

(3)

Substituting (3) into (1), one can obtain that

Φ (q_{1}, Θ (q_{1}, t), t) = 0,

(4)

where

q_{2} = Θ (q_{1}, t) .

(5)

Therefore, (3) can be described with independent joint position q₁, shown as

q = [\begin{bmatrix} q_{1} \\ Θ (q_{1}, t) \end{bmatrix}] .

(6)

The derivation of (6) is given as follows:

\dot{q} = [\begin{bmatrix} {\dot{q}}_{1} \\ \frac{\partial Θ (q_{1}, t)}{\partial q_{1}} {\dot{q}}_{1} + \frac{\partial Θ (q_{1}, t)}{\partial t} \end{bmatrix}] = [\begin{bmatrix} I_{n - m} & 0 \\ \frac{\partial Θ (q_{1}, t)}{\partial q_{1}} & I_{m} \end{bmatrix}] [\begin{bmatrix} {\dot{q}}_{1} \\ 0 \end{bmatrix}] + [\begin{bmatrix} 0 \\ \frac{\partial Θ (q_{1}, t)}{\partial t} \end{bmatrix}] = T \dot{θ} + H,

(7)

where

\begin{matrix} T = [\begin{bmatrix} I_{n - m} & 0 \\ \frac{\partial Θ (q_{1}, t)}{\partial q_{1}} & I_{m} \end{bmatrix}] \in R^{n \times n}, \\ \dot{θ} = [\begin{bmatrix} {\dot{q}}_{1} \\ 0 \end{bmatrix}] \in R^{n}, \\ H = [\begin{bmatrix} 0 \\ \frac{\partial Θ (q_{1}, t)}{\partial t} \end{bmatrix}] \in R^{n} . \end{matrix}

(8)

Therefore, the second-order derivative of q can be represented as follows:

\ddot{q} = T \ddot{θ} + \dot{T} \dot{θ} + \dot{H} .

(9)

Substituting (7) and (9) into (2), one can obtain that

M (q) (T \ddot{θ} + \dot{T} \dot{θ} + \dot{H}) + C (q, \dot{q}) (T \dot{θ} + \dot{H}) + G (q) + F (q, \dot{q}) = u + J_{Φ}^{T} (q, t) f .

(10)

Define

E = [\begin{bmatrix} I_{(n - m) \times (n - m)} \\ 0_{m \times (n - m)} \end{bmatrix}] \in R^{n \times (n - m)},

(11)

so that θ can be rewritten as follows:

θ = [\begin{bmatrix} q_{1} \\ 0 \end{bmatrix}] = E q_{1} .

(12)

Then, (2) can be decomposed into the following form:

\sum_{j = 1}^{n} M_{i j} (q) ({(T E {\ddot{q}}_{1})}_{j} + {(\dot{T} E {\dot{q}}_{1})}_{j} + {\dot{H}}_{j}) + F_{i} (q_{i}, {\dot{q}}_{i}) + \sum_{j = 1}^{n} C_{i j} (q, \dot{q}) ({(T E {\dot{q}}_{1})}_{j} + H_{j}) + {\bar{G}}_{i} (q) - f_{i} = u_{i} .

(13)

In (13), ${(T E {\ddot{q}}_{1})}_{j}$ , ${(\dot{T} E {\dot{q}}_{1})}_{j}$ , ${(T E {\dot{q}}_{1})}_{j}$ , and H_j are the jth element of $(T E {\ddot{q}}_{1})$ , $(\dot{T} E {\dot{q}}_{1})$ , $(T E {\dot{q}}_{1})$ , and H, respectively; ${\bar{G}}_{i} (q)$ , $F_{i} (q_{i}, {\dot{q}}_{i})$ , and u_i are the ith element of the G(q), $F (q, \dot{q})$ , and u; f_i is the constraint force of the ith joint. M_ij(q) and $C_{i j} (q, \dot{q})$ are the ijth element of M(q) and $C (q, \dot{q})$ , respectively. So the dynamic model of the ith subsystem can be formulated as follows:

M_{i} (q_{i}) {\ddot{q}}_{i} + C_{i} (q_{i}, {\dot{q}}_{i}) {\dot{q}}_{i} + G_{i} (q_{i}) + F_{i} (q_{i}, {\dot{q}}_{i}) + Z_{i} (q, \dot{q}, \ddot{q}) - f_{i} = u_{i},

(14)

where $Z_{i} (q, \dot{q}, \ddot{q})$ is given as follows:

Z_{i} (q, \dot{q}, \ddot{q}) = \sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} M_{i j} (q) ({(T E {\ddot{q}}_{1})}_{j} + {(\dot{T} E {\dot{q}}_{1})}_{j} + {\dot{H}}_{j}) + M_{i i} (q) ({(T E {\ddot{q}}_{1})}_{i} + {(\dot{T} E {\dot{q}}_{1})}_{i} + {\dot{H}}_{i}) - M_{i} (q_{i}) {\ddot{q}}_{i} + \sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} C_{i j} (q, \dot{q}) ({(T E {\dot{q}}_{1})}_{j} + H_{j}) + C_{i i} (q, \dot{q}) ({(T E {\dot{q}}_{1})}_{j} + H_{j}) - C_{i} (q_{i}, {\dot{q}}_{i}) {\dot{q}}_{i} + ({\bar{G}}_{i} (q) - G_{i} (q_{i})) .

(15)

It is noted that the joint friction $F_{i} (q_{i}, {\dot{q}}_{i})$ in (14) can be described as a function of joint position and velocity [22, 23], which is given as follows:

F_{i} (q_{i}, {\dot{q}}_{i}) = (f_{c i} + f_{s i} e^{(- f_{τ i} {\dot{q}}_{i}^{2})}) sgn ({\dot{q}}_{i}) + b_{f i} {\dot{q}}_{i} + f_{q i} (q_{i}, {\dot{q}}_{i}),

(16)

where f_ci denotes the Coulomb friction related parameter; f_si denotes the static friction related parameter; f_τi is a position dependency of friction and other friction modeling errors, $sgn ({\dot{q}}_{i})$ is a sign function that is defined as follows

sgn ({\dot{q}}_{i}) = \{\begin{cases} 1, & for {\dot{q}}_{i} > 0, \\ 0, & for {\dot{q}}_{i} = 0, \\ - 1, & for {\dot{q}}_{i} < 0, \end{cases})

(17)

and $f_{q i} (q_{i}, {\dot{q}}_{i})$ denotes the nonparametric friction term which is bounded as

|f_{q i} (q_{i}, {\dot{q}}_{i})| < ρ_{f i},

(18)

where ρ_fi is a known constant bound for any position and velocity q_i and ${\dot{q}}_{i}$ .

Therefore, we can obtain the acceleration of the ith joint ${\ddot{q}}_{i}$ , which is given as follows:

{\ddot{q}}_{i} = Ψ_{i} (q_{i}, {\dot{q}}_{i}) + F c_{i} (q, \dot{q}) + h_{i} (q, \dot{q}, \ddot{q}) - b_{i} u_{i},

(19)

where $Ψ_{i} (q_{i}, {\dot{q}}_{i})$ is the unmodeled subsystem dynamic term; $F c_{i} (q_{i}, {\dot{q}}_{i})$ is the friction term multiplied by the inertia matrix transpose; $h_{i} (q, \dot{q}, \ddot{q})$ is the interconnected dynamic coupling term; and b_i = − M_i⁻¹(q_i). It is noted that we need to design an appropriate controller to compensate the model uncertainties which are represented as follows:

\begin{matrix} Ψ_{i} (q_{i}, {\dot{q}}_{i}) = - M_{i}^{- 1} (q_{i}) (C_{i} (q_{i}, {\dot{q}}_{i}) {\dot{q}}_{i} + G_{i} (q_{i}) - f_{i}), \\ F c_{i} (q_{i}, {\dot{q}}_{i}) = - M_{i}^{- 1} (q_{i}) (F_{i} (q_{i}, {\dot{q}}_{i})), \\ h_{i} (q, \dot{q}, \ddot{q}) = - M_{i}^{- 1} (q_{i}) Z_{i} (q, \dot{q}, \ddot{q}) . \end{matrix}

(20)

Define $x_{i} = {[\begin{bmatrix} x_{i 1} & x_{i 2} \end{bmatrix}]}^{T} = {[\begin{bmatrix} q_{i} & {\dot{q}}_{i} \end{bmatrix}]}^{T}$ , for i = 1, …, n; then, the dynamic equation (14) can be represented as the following state equation:

S_{i} : \{\begin{cases} {\dot{x}}_{i 1} = x_{i 2}, \\ {\dot{x}}_{i 2} = Ψ_{i} (q_{i}, {\dot{q}}_{i}) + F c_{i} (q, \dot{q}) + h_{i} (q, \dot{q}, \ddot{q}) - b_{i} u_{i}, \\ y_{i} = x_{i 1}, \end{cases})

(21)

where x_i is the state vector of subsystem S_i and y_i is the output of subsystem S_i. The architecture of the time varying constrained MRR system is shown as Figure 1.

Figure 1:

The architecture of the time varying constrained reconfigurable modular robot system.

3. Decentralized Integral Nested Sliding Mode Control

In this section, a decentralized integral nested sliding mode control approach is proposed based on VGSTA, and the stability of the close-loop system is proofed by the Lyapunov theory.

Assumption 1. Desired trajectory y_ir1(t) is bounded.

Assumption 2. The interconnected dynamic coupling term $h_{i} (q, \dot{q}, \ddot{q})$ is bounded and satisfies

|h_{i} (q, \dot{q}, \ddot{q})| \leq g_{i 0} + \sum_{j = 1}^{n} g_{i j},

(22)

where g_i0 is a positive constant and g_ij is a smooth Lipschitz function.

Define the tracking position error as

e_{i 1} = x_{i 1} - y_{i r 1} (t) .

(23)

Then, we can obtain the derivative of ${\dot{e}}_{i 1}$ , which is given as follows:

{\dot{e}}_{i 1} = {\dot{x}}_{i 1} - {\dot{y}}_{i r 1} (t) = x_{i 2} - {\dot{y}}_{i r 1} (t) .

(24)

Define the pseudosliding surface s_i1 for the first block as

\begin{matrix} s_{i 1} = e_{i 1} + z_{i 1}, \\ z_{i 1} (0) = - e_{i 1} (0), \end{matrix}

(25)

and the derivative of s_i1 can be given as follows:

{\dot{s}}_{i 1} = {\dot{e}}_{i 1} + {\dot{z}}_{i 1} = x_{i 2} - {\dot{y}}_{i r 1} (t) + {\dot{z}}_{i 1} .

(26)

In (25) and (26), z_i1 is an integral variable which will be defined later. It is noted that, in order to substitute the designed control law into the sliding surface, it is of importance to introduce the second block of sliding surface, which is defined as follows:

\begin{matrix} s_{i 2} = e_{i 2} + z_{i 2}, \\ z_{i 2} (0) = - e_{i 2} (0), \end{matrix}

(27)

where z_i2 is an integral variable to be determined, and e_i2, which is used to design the second block of sliding surface, can be defined as

e_{i 2} = x_{i 2} - y_{i r 2} (t),

(28)

where y_ir2(t) is defined as

y_{i r 2} (t) = y_{i r 2,0} (t) + y_{i r 2,1} (t) .

(29)

In (29), y_ir2,0(t) is the nominal part of the control design, and y_ir2,1(t) is used to design the control law for the purpose of compensating the unmodeled subsystem dynamics.

Substituting (28) into (27), one can obtain that

x_{i 2} = s_{i 2} + y_{i r 2} (t) - z_{i 2} .

(30)

Combining (26) with (30), the derivative of the pseudosliding surface s_i1 can be rewritten as

{\dot{s}}_{i 1} = s_{i 2} - z_{i 2} + y_{i r 2,0} (t) + y_{i r 2,1} (t) - {\dot{y}}_{i r 1} (t) + {\dot{z}}_{i 1} .

(31)

Choose ${\dot{z}}_{i 1}$ in the form of

{\dot{z}}_{i 1} = - (s_{i 2} - z_{i 2} + y_{i r 2,0} (t)) .

(32)

Then, combining (27) and (31) with (32), the derivative of s_i1 and z_i1 can be represented as

\begin{matrix} {\dot{s}}_{i 1} = y_{i r 2,1} (t) - {\dot{y}}_{i r 1} (t), \\ {\dot{z}}_{i 1} = - e_{i 2} - y_{i r 2,0} (t) . \end{matrix}

(33)

With the initial condition z_i1(0) = − e_i1(0), define y_ir2,0(t) and y_ir2,1(t) as

\begin{matrix} y_{i r 2,0} (t) = - k_{1} e_{i 1}, \\ y_{i r 2,1} (t) = - σ_{1} sigm (ε_{1}, s_{i 1}), \end{matrix}

(34)

where k₁, σ₁, and ∊₁ are positive constants to be determined. sigm(·) is a sigmoid function which can be defined as follows:

sigm (ε_{1}, s_{i 1}) = \tanh (ε_{1} s_{i 1}) .

(35)

Figure 2 shows the approximation for various values of ∊₁, which defines the slope of the sigmoid function for every s_i1.

Figure 2:

Sigmoid function for various values of the slope σ₁.

Then, we can obtain the derivative of s_i2 as

{\dot{s}}_{i 2} = {\dot{e}}_{i 2} + {\dot{z}}_{i 2} = {\dot{x}}_{i 2} - {\dot{y}}_{i r 2} (t) + {\dot{z}}_{i 2} .

(36)

According to (34), one can obtain that ${\dot{y}}_{i r 2} (t)$ is represented as

{\dot{y}}_{i r 2} (t) = {\dot{y}}_{i r 2,0} (t) + {\dot{y}}_{i r 2,1} (t) = - k_{1} {\dot{e}}_{i 1} + σ_{1} ε_{1} (1 - \tanh^{2} (ε_{1} s_{i 1})) \cdot (σ_{1} sigm (ε_{1}, s_{i 1}) + {\dot{y}}_{i r 1} (t)) .

(37)

Substituting (21) into (36), we can rewrite ${\dot{s}}_{i 2}$ as

{\dot{s}}_{i 2} = Ψ_{i} (q_{i}, {\dot{q}}_{i}) + F c_{i} (q_{i}, {\dot{q}}_{i}) + h_{i} (q, \dot{q}, \ddot{q}) - b_{i} u_{i} - {\dot{y}}_{i r 2} (t) + {\dot{z}}_{i 2},

(38)

where ${\dot{z}}_{i 2}$ is chosen as

{\dot{z}}_{i 2} = - k_{2} {\dot{e}}_{i 1} + σ_{2} ε_{1} (1 - \tanh^{2} (ε_{1} s_{i 1})) \cdot (σ_{1} sigm (ε_{1}, s_{i 1}) + {\dot{y}}_{i r 1} (t)),

(39)

where k₂ and σ₂ are positive constants to be determined. Note that the integral variable z_i2, which is designed with a sigmoid function, is applicable for the chattering alleviation and attenuation of the sliding mode system. Then, for the purpose of compensating the different terms of the derivative of the sliding surface, we can design the control law in the form of u_i = u_i0 + u_i1 + u_i2. So (38) can be rewritten as

{\dot{s}}_{i 2} = F c_{i} (q_{i}, {\dot{q}}_{i}) - b_{i} u_{i 0} + Ψ_{i} (q_{i}, {\dot{q}}_{i}) - b_{i} u_{i 1} + h_{i} (q, \dot{q}, \ddot{q}) - b_{i} u_{i 2} - {\dot{y}}_{i r 2} (t) + {\dot{z}}_{i 2} .

(40)

First, design the control law u_i0 to compensate the effect of joint friction $F c_{i} (q_{i}, {\dot{q}}_{i})$ :

u_{i 0} = {\hat{b}}_{f i} {\dot{q}}_{i} + ({\hat{f}}_{c i} + {\hat{f}}_{s i} e^{({\hat{f}}_{τ i} {\dot{q}}_{i}^{2})}) sgn ({\dot{q}}_{i}) + u_{i 0}^{1} + Y ({\dot{q}}_{i}) (u_{i 0}^{2} + u_{i 0}^{3}),

(41)

where $Y ({\dot{q}}_{i})$ can be defined as follows:

Y ({\dot{q}}_{i}) = [{\dot{q}}_{i}, sgn ({\dot{q}}_{i}), e^{(- {\hat{f}}_{τ i} {\dot{q}}_{i})} sgn ({\dot{q}}_{i}), - {\hat{f}}_{s i} {\dot{q}}_{i}^{2} e^{(- {\hat{f}}_{τ i} {\dot{q}}_{i})} sgn ({\dot{q}}_{i})] .

(42)

In order to incorporate variable parametric model uncertainty compensation, we can design

{\tilde{F}}^{i} = {[{\hat{b}}_{f i} - b_{f i}, {\hat{f}}_{c i} - f_{c i}, {\hat{f}}_{s i} - f_{s i}, {\hat{f}}_{τ i} - f_{τ i}]}^{T} = {\tilde{F}}_{c}^{i} + {\tilde{F}}_{v}^{i},

(43)

where ${\tilde{F}}_{c}^{i}$ and ${\tilde{F}}_{v}^{i}$ are constant and variable unknown vectors, respectively. And ${\tilde{F}}_{v}^{i}$ is bounded as

|{\tilde{F}}_{v n}^{i}| < ρ_{n}^{i}, n = 1,2, 3,4 .

(44)

In (42), ${\hat{b}}_{f i}$ , ${\hat{f}}_{c i}$ , ${\hat{f}}_{s i}$ , and ${\hat{f}}_{τ i}$ are the normal friction parameters and u_i0¹ is the term designed to compensate the nonparametric uncertainty $f_{q i} (q, \dot{q})$ in (16). The terms u_i0² and u_i0³ are designed to compensate for the parametric uncertainties ${\tilde{F}}_{c}^{i}$ and ${\tilde{F}}_{v}^{i}$ , respectively. The friction compensation is of the same form for all the joints, and hence, for the ith joint, the compensators, u_i0¹, u_i0², and u_i0³, are defined by

\begin{matrix} u_{i 0}^{1} = \{\begin{cases} - ρ_{f i} \frac{s_{i 2}}{|s_{i 2}|}, & |s_{i 2}| > ε^{i}, \\ - ρ_{f i} \frac{s_{i 2}}{ε^{i}}, & |s_{i 2}| \leq ε^{i}, \end{cases}) \\ u_{i 0}^{2} = - k_{3} \int_{0}^{t} Y {({\dot{q}}_{i})}^{T} s_{i 2} d t, \\ u_{i 0}^{3} = \{\begin{cases} - ρ_{n}^{i} \frac{δ_{n}^{i}}{|δ_{n}^{i}|}, & |δ_{n}^{i}| > ε_{p n}^{i}, \\ - ρ_{n}^{i} \frac{δ_{n}^{i}}{ε_{p n}^{i}}, & |δ_{n}^{i}| \leq ε_{p n}^{i}, \end{cases}) \end{matrix}

(45)

where $δ^{i} = Y {({\dot{q}}_{i})}^{T} s_{i 2}$ and ρ_fi, ρ_nⁱ are the parametric uncertainty bounds. ∊ⁱ and ∊_pnⁱ are parameters to be determined.

Second, design the control law u_i1 to compensate the terms of $Ψ_{i} (q_{i}, {\dot{q}}_{i})$ , ${\dot{y}}_{i r 2} (t)$ , and ${\dot{z}}_{i 2}$ :

u_{i 1} = b_{i}^{- 1} ({\dot{y}}_{i r 2} (t) - {\dot{z}}_{i 2} + k_{3} sat (ε_{s}, s_{i 2})) = b_{i}^{- 1} ((k_{1} - k_{2}) {\dot{e}}_{i 1} + k_{3} sat (ε_{s}, s_{i 2})) (+ (σ_{1} - σ_{2}) (1 - \tanh^{2} (ε_{1} s_{i 1}))),

(46)

where ∊_s and k₃ are positive constants to be determined. sat ⁡ (·) is a saturation function which is defined as follows:

sat (ε_{s}, s_{i 2}) = \{\begin{cases} sgn (s_{i 2}), & |s_{i 2}| > ε_{s}, \\ \frac{s_{i 2}}{ε_{s}}, & |s_{i 2}| \leq ε_{s} . \end{cases})

(47)

Third, based on the VGSTA theory, design the control law u_i2 to compensate the interconnected dynamic coupling term $h_{i} (q, \dot{q}, \ddot{q})$ . According to Assumption 2, the interconnected dynamic coupling term, which is treated as an uncertainty term, is bounded by known continuous functions. To provide a description of the uncertainty term supported by the VGSTA, note that the interconnected dynamic coupling term $h_{i} (q, \dot{q}, \ddot{q})$ , which is treated as an uncertainty term, can be represented as the following form:

h_{i} (q, \dot{q}, \ddot{q}) = g_{i 1} (q_{i}, t) + g_{i 2} (q_{i}, t),

(48)

where g_i1(q_i, t) and g_i2(q_i, t) are bounded by

\begin{matrix} |g_{i 1} (q_{i}, t)| \leq ς_{i 1} (q_{i}, t) |ϕ_{i 1} (s_{i 2})| = ς_{i 1} (q_{i}, t) (1 + κ_{i 3} (t) {|s_{i 2}|}^{1 / 2}) {|s_{i 2}|}^{1 / 2}, \\ |\frac{d g_{i 2} (q_{i}, t)}{d t}| \leq ς_{i 2} (q_{i}, t) |ϕ_{i 2} (s_{i 2})| = \frac{1}{2} ς_{i 2} (q_{i}, t) + κ_{i 3} (t) ς_{i 2} (q_{i}, t) \cdot (\frac{3}{2} + κ_{i 3} (t) {|s_{i 2}|}^{1 / 2}) {|s_{i 2}|}^{1 / 2}, \end{matrix}

(49)

where ς_i1|q_i, t| > 0 and ς_i2|q_i, t| > 0 are known continuous functions. φ₁(s_i2) and φ₂(s_i2) are defined as

\begin{matrix} ϕ_{i 1} (s_{i 2}) = {|s_{i 2}|}^{1 / 2} sgn (s_{i 2}) + κ_{i 3} (t) s_{i 2}, \\ ϕ_{i 2} (s_{i 2}) = ϕ_{i 1}^{'} (s_{i 2}) ϕ_{i 1} (s_{i 2}) = \frac{1}{2} sgn (s_{i 2}) + \frac{3}{2} κ_{i 3} (t) {|s_{i 2}|}^{1 / 2} sgn (s_{i 2}) + κ_{i 3}^{2} (t) s_{i 2}, \end{matrix}

(50)

where κ_i3(t) is a positive gain to be determined. Substituting (48) into (40), one can obtain that

{\dot{s}}_{i 2} = F c_{i} (q_{i}, {\dot{q}}_{i}) - b_{i} u_{i 0} + Ψ_{i} (q_{i}, {\dot{q}}_{i}) - b_{i} u_{i 1} + g_{i 1} (q_{i}, t) + g_{i 2} (q_{i}, t) - b_{i} u_{i 2} - {\dot{y}}_{i r 2} (t) + {\dot{z}}_{i 2} .

(51)

Design the control law u_i2 to compensate the interconnected dynamic coupling term, shown as follows:

u_{i 2} = b_{i}^{- 1} (κ_{i 1} (t) ϕ_{i 1} (s_{i 2}) + \int_{0}^{t} κ_{i 2} (t) ϕ_{i 2} (s_{i 2}) d t),

(52)

where the variable gains κ_i1(t) and κ_i2(t) are chosen as

\begin{matrix} κ_{i 1} (t) = \frac{1}{ρ_{v}} \{\frac{1}{4 ε_{2}} {(2 ε_{2} ς_{i 1} + ς_{i 2})}^{2} + 2 ε_{2} ς_{i 2}) (+ ε_{2} + (2 ε_{2} + ς_{i 1}) (ρ_{v} + 4 ε_{2}^{2})\} + φ, \\ κ_{i 2} (t) = ρ_{v} + 4 ε_{2}^{2} + 2 ε_{2} κ_{i 1} (t), \end{matrix}

(53)

where ϕ, ρ_v, and ∊₂ are positive constants to be determined. Note that, for the control law u_i2, if κ_i1(t) and κ_i2(t) are constants and κ_i3(t) = 0, then it is a standard supertwisting control law. If κ_i3(t) > 0 and κ_i1(t) and κ_i2(t) are designed in the form of (53); then, the variable gains κ_i1(t) and κ_i2(t) make it possible to render the sliding surface insensitive to perturbations growing with bounds given by known functions, and the linear growth disturbance of the system can be reduced.

Therefore, combining (41) and (46) with (52), the decentralized integral nested sliding mode control law u_i for the ith joint can be defined as follows:

u_{i} = u_{i 0} + u_{i 1} + u_{i 2} = b_{i}^{- 1} (b_{i} ({\hat{b}}_{f i} {\dot{q}}_{i} + Y ({\dot{q}}_{i}) (u_{i 0}^{2} + u_{i 0}^{3}) + u_{i 0}^{1})) (+ ({\hat{f}}_{c i} + {\hat{f}}_{s i} e^{({\hat{f}}_{τ i} {\dot{q}}_{i}^{2})}) sgn ({\dot{q}}_{i})) + (k_{1} - k_{2}) {\dot{e}}_{i 1} + (σ_{1} - σ_{2}) (1 - \tanh^{2} (ε_{1} s_{i 1})) + k_{3} sat (ε_{s}, s_{i 2}) (+ κ_{i 1} (t) ϕ_{i 1} (s_{i 2}) + \int_{0}^{t} κ_{i 2} (t) ϕ_{i 2} (s_{i 2}) d t) .

(54)

Theorem 3. Given a time varying constrained n-DOF MRR, with the dynamic model as formulated in (14) and the model uncertainties that existed in (20), the close-loop system is stable under the decentralized integral nested sliding mode control law defined by (54).

Proof. The proof of Theorem 3 will be done by using two Lyapunov function candidates for the different block of sliding surface.

First, we can reformulate the derivative of second block of sliding surface (51) as follows:

{\dot{s}}_{i 2} = F c_{i} (q_{i}, {\dot{q}}_{i}) - b_{i} u_{i 0} + Ψ_{i} (q_{i}, {\dot{q}}_{i}) - {\dot{y}}_{i r 2} (t) + {\dot{z}}_{i 2} - b_{i} u_{i 1} + g_{i 1} (q_{i}, t) - κ_{i 1} (t) ϕ_{i 1} (s_{i 2}) + Z_{i},

(55)

where ${\dot{Z}}_{i}$ is defined as follows:

{\dot{Z}}_{i} = - κ_{i 2} (t) ϕ_{i 2} (s_{i 2}) + \frac{d}{d t} g_{i 2} (q_{i}, t) .

(56)

The Lyapunov function candidate of the second block can be defined as follows:

V_{i 2} (s_{i 2}, Z_{i}) = ξ_{i}^{T} P ξ_{i} .

(57)

In (57), ξ_i^T and P in V_i2(s_i2, Z_i) are defined as

ξ_{i}^{T} = [\begin{bmatrix} ϕ_{i 1} (s_{i 2}) & Z_{i} \end{bmatrix}],

(58)

P = [\begin{bmatrix} p_{1} & p_{3} \\ p_{3} & p_{2} \end{bmatrix}] = [\begin{bmatrix} ρ_{v} + 4 ε_{2}^{2} & - 2 ε_{2}^{2} \\ - 2 ε_{2}^{2} & 1 \end{bmatrix}] .

(59)

Note that the function V_i2(s_i2, Z_i) is positive definite, continuous, and differentiable everywhere. Rewriting the inequation of (49) yields that $g_{i 1} (q_{i}, t) = ϖ_{i 1} (q_{i}, t) ϕ_{i 1} (s_{i 2})$ and $d g_{i 2} (q_{i}, t) / d t = ϖ_{i 2} (q_{i}, t) ϕ_{i 2} (s_{i 2})$ for the functions of $|ϖ_{i 1} (q_{i}, t)| \leq ς_{i 1} (q_{i}, t)$ and $|ϖ_{i 2} (q_{i}, t)| \leq ς_{i 2} (q_{i}, t)$ . Using these functions and noting that φ_i2(s_i2) = φ_i1′(s_i2)φ_i1(s_i2), one can obtain the derivative of ξ_i, represented as follows:

{\dot{ξ}}_{i} = [ϕ_{i 1}^{'} (s_{i 2}) (- κ_{i 1} (t) ϕ_{i 1} (s_{i 2}) + Z_{i} + g_{i 1} (q_{i}, t))) (- κ_{i 2} (t) ϕ_{i 2} (s_{i 2}) + \frac{d g_{i 2} (q_{i}, t)}{d t}] = ϕ_{i 1}^{'} (s_{i 2}) [\begin{bmatrix} - κ_{i 1} (t) - ϖ_{i 1} (q_{i}, t) & 1 \\ - κ_{i 2} (t) - ϖ_{i 2} (q_{i}, t) & 0 \end{bmatrix}] ξ_{i} = ϕ_{i 1}^{'} (s_{i 2}) A (q_{i}, t) ξ_{i} .

(60)

So the derivative of V_i2(s_i2, Z_i) can be given as

{\dot{V}}_{i 2} (s_{i 2}, Z_{i}) = ϕ_{i 1}^{'} (s_{i 2}) ξ_{i}^{T} (A^{T} (q_{i}, t) P + P A (q_{i}, t)) ξ_{i} = - ϕ_{i 1}^{'} (s_{i 2}) ξ_{i}^{T} Ω (q_{i}, t) ξ_{i},

(61)

where Ω(q_i, t) is given as follows:

Ω (q_{i}, t) = [\begin{bmatrix} Ω_{1} & Ω_{3} \\ Ω_{3} & Ω_{2} \end{bmatrix}],

(62)

where Ω₁, Ω₂, and Ω₃ are designed as

\begin{matrix} Ω_{1} = 4 ε_{2} (ϖ_{i 2} (q_{i}, t) - κ_{i 2} (t)) - 2 (ϖ_{i 1} (q_{i}, t) - κ_{i 1} (t)) (4 ε_{2}^{2} + ρ_{v}), \\ Ω_{2} = 4 ε_{2}, \\ Ω_{3} = κ_{i 2} (t) - ϖ_{i 2} (q_{i}, t) - ρ_{v} + 2 ε_{2} (ϖ_{i 1} (q_{i}, t) - κ_{i 1} (t)) - 4 ε_{2}^{2} . \end{matrix}

(63)

Choosing P in (59) and the gains in (53), we can obtain that

Ω (q_{i}, t) - 2 ε_{2} = [\begin{bmatrix} (\begin{pmatrix} 4 ε_{2} ϖ_{i 2} + 2 ρ_{v} κ_{i 1} - 2 ε_{2} \\ - 2 (2 ε_{2} + ϖ_{i 1}) (ρ_{v} + 4 ε_{2}^{2}) \end{pmatrix}) & 2 ε_{2} ϖ_{i 1} - ϖ_{i 2} \\ 2 ε_{2} ϖ_{i 1} - ϖ_{i 2} & 2 ε_{2} \end{bmatrix}] .

(64)

It is obvious that (64) is positive definite. So we can obtain that

{\dot{V}}_{i 2} (s_{i 2}, Z_{i}) = - ϕ_{i 1}^{'} (s_{i 2}) ξ_{i}^{T} (A^{T} (q_{i}, t) P + P A (q_{i}, t)) ξ_{i} = - ϕ_{i 1}^{'} (s_{i 2}) ξ_{i}^{T} Ω (q_{i}, t) ξ_{i} \leq - 2 ε_{2} ϕ_{i 1}^{'} (s_{i 2}) ξ_{i}^{T} ξ_{i} = - 2 ε_{2} (\frac{1}{2 {|s_{i 2}|}^{1 / 2}} + κ_{i 3} (t)) ξ_{i}^{T} ξ_{i} .

(65)

Since

λ_{\min} \{P\} {‖ξ_{i}‖}_{2}^{2} \leq ξ_{i}^{T} P ξ_{i} \leq λ_{\max} \{P\} {‖ξ_{i}‖}_{2}^{2},

(66)

where $λ_{\min} \{P\}$ and $λ_{\max} \{P\}$ are the minimum and maximum eigenvalues of P,

{‖ξ_{i}‖}_{2}^{2} = ξ_{i 1}^{2} + ξ_{i 2}^{2} = |s_{i 2}| + 2 κ_{i 3} (t) {|s_{i 2}|}^{3 / 2} + κ_{i 3}^{2} (t) s_{i 2}^{2} + Z_{i}^{2},

(67)

where ${‖ξ_{i}‖}_{2}^{2}$ is the Euclidean norm of ξ_i, and

|ξ_{i 1}| \leq {‖ξ_{i}‖}_{2} \leq \frac{V_{i 2}^{1 / 2} (s_{i 2}, Z_{i})}{λ_{\min}^{1 / 2} \{P\}} .

(68)

From (65) to (68), we can get the following equation:

{\dot{V}}_{i 2} (s_{i 2}, Z_{i}) = - \frac{ε_{2}}{{|s_{i 2}|}^{1 / 2}} ξ_{i}^{T} ξ_{i} - 2 ε_{2} κ_{i 3} (t) ξ_{i}^{T} ξ_{i} = - \frac{ε_{2} P^{1 / 2}}{V_{i 2}^{1 / 2} (s_{i 2}, Z_{i})} ξ_{i}^{T} ξ_{i} - \frac{2 ε_{2} κ_{i 3} (t)}{P} V_{i 2} (s_{i 2}, Z_{i}) \leq - \frac{ε_{2} P^{1 / 2}}{P} V_{i 2}^{1 / 2} (s_{i 2}, Z_{i}) - \frac{2 ε_{2} κ_{i 3} (t)}{λ_{\max} \{P\}} V_{i 2} (s_{i 2}, Z_{i}) \leq - \frac{ε_{2} λ_{\min}^{1 / 2} \{P\}}{λ_{\max} \{P\}} - \frac{2 ε_{2} κ_{i 3} (t)}{λ_{\max} \{P\}} V_{i 2} (s_{i 2}, Z_{i}) \leq - χ_{1} V_{i 2}^{1 / 2} (s_{i 2}, Z_{i}) - χ_{2} V_{i 2} (s_{i 2}, Z_{i}),

(69)

where χ₁ and χ₂ are positive definite, shown as

\begin{matrix} χ_{1} = \frac{ε_{2} λ_{\min}^{1 / 2} \{P\}}{λ_{\max} \{P\}}, \\ χ_{2} = \frac{2 ε_{2} κ_{i 3} (t)}{λ_{\max} \{P\}} . \end{matrix}

(70)

The solution of the differential equation

\begin{matrix} v = - χ_{1} v^{1 / 2} - χ_{2} v, \\ v (0) = v_{0} \geq 0 \end{matrix}

(71)

is given by

v (t) = \exp (- χ_{2} t) {(v_{0}^{1 / 2} + \frac{χ_{1}}{χ_{2}} (1 - \exp (\frac{χ_{2}}{2} t)))}^{2} .

(72)

On the basis of the Comparison Lemma [24, Lemma 2.5, p. 85], one can obtain that V_i|t| ≤ v|t| when V_i|0| ≤ v₀. It follows that the equilibrium point |s_i2, Z_i| = 0 is reached in finite time and the reaching time can be estimated by

T = \frac{2}{χ_{2}} \ln (\frac{χ_{2}}{χ_{1}} V_{i 2}^{1 / 2} (0) + 1) .

(73)

Second, proceeding in similar way for the first block, we can design the Lyapunov function candidate as follows:

V_{i 1} (s_{i 1}) = \frac{1}{2} s_{i 1}^{T} s_{i 1} .

(74)

The derivative of V_i1(s_i1) can be obtained as

{\dot{V}}_{i 1} (s_{i 1}) = s_{i 1}^{T} (- σ_{1} sigm (ε_{1}, s_{i 1}) - {\dot{y}}_{i r 1} (t)) .

(75)

Design the following equation:

sigm (ε_{1}, s_{i 1}) = sgn (s_{i 1}) - Δ (ε_{1}, s_{i 1}),

(76)

where Δ(∊₁, s_i1) can be shown as the difference between the signum and sigmoid functions. Then, we can obtain the following equation by combining (75) with (76), shown as

{\dot{V}}_{i 1} (s_{i 1}) < ‖s_{i 1}‖ (- k_{1} ((1 - ‖Δ (ε_{1}, s_{i 1})‖) + ‖{\dot{y}}_{i r 1} (t)‖)) .

(77)

It is evident that Δ(∊₁, s_i1) is bounded while ∊₁ is given. There is a positive ϑ (0 < ϑ < 1) with the following equation:

‖Δ (ε_{1}, s_{i 1})‖ = ϑ .

(78)

Therefore, we can get the conclusion that

{\dot{V}}_{i 1} (s_{i 1}) < 0

(79)

and s_i1 converges to a region

‖s_{i 1}‖ \leq \frac{\ln ((2 - ϑ) / ϑ)}{2 ε_{1}} .

(80)

This concludes the proof of the theorem.

4. Simulations

4.1. Dynamic Model

In order to confirm that the proposed control method can be applied in different configurations, in this paper, two different configurations of the time varying constrained 2-DOF MRR, which are shown in Figures 3 and 4, are used to conduct simulations.

Figure 3:

Configuration A for time varying constrained MRR.

Figure 4:

Configuration B for time varying constrained MRR.

For the sake of the facilitation of the analysis of the configurations shown in Figures 3 and 4, we can transform them into a form of analytic charts, which are shown in Figures 5 and 6, where L₁, L₂, and L₄ are the length of the links. L₃ is the distance between the time varying constraint and the base module.

Figure 5:

The analytic chart of configuration A.

Figure 6:

The analytic chart of configuration B.

The time varying constraint can be defined as a kind of column with the length L₅ = 0.5 m and the mass m_c = 4 kg, which rotates with a certain degree of freedom. The constraint equations and the end-effector constrained force of configuration A and configuration B are shown as follows:

\begin{matrix} Φ_{A} (q, t) = L_{4} - (L_{1} \sin (q_{1}) + L_{2} \sin (q_{2})) - \tan α (t) (L_{1} \cos (q_{1}) + L_{2} \cos (q_{2}) - L_{3}), \\ Φ_{B} (q, t) = L_{4} - L_{2} \sin (q_{2}) - \tan α (t) (L_{1} + L_{2} \cos (q_{2}) - L_{3}), \\ J_{Φ A}^{T} (q, t) f = - \frac{1}{2} m_{c} g \frac{L_{5} \cos (α (t)) \cdot \sin (α (t)) \cdot \cos (α (t) + q_{2})}{L_{4} - L_{1} \sin (q_{1}) - L_{2} \sin (q_{2})}, \\ J_{Φ B}^{T} (q, t) f = - \frac{1}{2} m_{c} g \frac{L_{5} \cos (α (t)) \cdot \sin (α (t)) \cdot \cos (α (t) + q_{2})}{L_{4} - L_{2} \sin (q_{2})}, \end{matrix}

(81)

where g is the acceleration of gravity and α|t| is the angle between the time varying constraint and the x-axis, which can be defined as follows:

α (t) = 0.75 π + 0.2 \sin \frac{t}{2} .

(82)

The desired positions of configurations A and B are given as follows.

Configuration A:

\begin{matrix} y_{1 d} = 0.5 \cos (t) + 0.2 \sin (3 t), \\ y_{2 d} = Θ (y_{1 d}, t) = \arcsin (\frac{L_{3}}{L_{2}} \sin (α (t)) - \frac{L_{1}}{L_{2}} \sin (α (t) + q_{1})) (- \frac{L_{4}}{L_{2}} \sin (α (t) - \frac{π}{2})) - α (t) . \end{matrix}

(83)

Configuration B:

\begin{matrix} y_{1 d} = 0, \\ y_{2 d} = Θ (y_{1 d}, t) = \arcsin (\frac{L_{3}}{L_{2}} \sin (α (t)) - \frac{L_{1}}{L_{2}} \sin (α (t))) (- \frac{L_{4}}{L_{2}} \sin (α (t) - \frac{π}{2})) - α (t) . \end{matrix}

(84)

Note that, due to the limitation of the external time varying constraint, the position variety of joint 1 of configuration B is zero.

The initial positions of the joints are defined as q₁|0| = 0 and q₂|0| = − 2 for both configurations A and B. The dynamics of configurations A and B are designed as follows:

\begin{matrix} M_{A} (q) = [\begin{bmatrix} 0.36 \cos (q_{2}) + 0.6066 & 0.18 \cos (q_{2}) + 0.1233 \\ 0.18 \cos (q_{2}) + 0.1233 & 0.1233 \end{bmatrix}], \\ M_{B} (q) = [\begin{bmatrix} 0.17 - 0.1166 \cos^{2} (q_{2}) & - 0.06 \cos (q_{2}) \\ - 0.06 \cos (q_{2}) & 0.1233 \end{bmatrix}], \\ C_{A} (q, \dot{q}) = [\begin{bmatrix} - 0.36 \sin (q_{2}) {\dot{q}}_{2} & - 0.18 \sin (q_{2}) {\dot{q}}_{2} \\ 0.18 \sin (q_{2}) ({\dot{q}}_{1} - {\dot{q}}_{2}) & 0.18 \sin (q_{2}) {\dot{q}}_{1} \end{bmatrix}], \\ C_{B} (q, \dot{q}) = [\begin{bmatrix} 0.1166 \sin (2 q_{2}) {\dot{q}}_{2} & 0.06 \sin (q_{2}) {\dot{q}}_{2} \\ 0.06 \sin (q_{2}) {\dot{q}}_{2} & 0 \end{bmatrix}], \\ G_{A} (q) = [\begin{bmatrix} - 5.88 \sin (q_{1} + q_{2}) - 17.64 \sin (q_{1}) \\ - 5.88 \sin (q_{1} + q_{2}) \end{bmatrix}], \\ G_{B} (q) = [\begin{bmatrix} 0 \\ - 5.88 \cos (q_{2}) \end{bmatrix}], \\ F_{A} (q, \dot{q}) = [\begin{bmatrix} {\dot{q}}_{1} + 10 \sin (3 q_{1}) + 2 sgn ({\dot{q}}_{1}) \\ 1.2 {\dot{q}}_{2} + 5 \sin (2 q_{2}) + sgn ({\dot{q}}_{2}) \end{bmatrix}], \\ F_{B} (q, \dot{q}) = [\begin{bmatrix} 0 \\ 1.5 {\dot{q}}_{2} + \sin (q_{2}) + 1.2 sgn ({\dot{q}}_{2}) \end{bmatrix}] . \end{matrix}

(85)

The parameters of the friction model are given as follows:

\begin{matrix} f_{c i} = 3.5 + 0.7 \sin (10 q_{i}) Nm, \\ f_{s i} = 5 + \sin (10 q_{i}) Nm, \\ f_{τ i} = 100 + 20 \sin (10 q_{i}) s^{2} / ra d^{2}, \\ b_{f i} = 1.5 + 0.3 \sin (10 q_{i}) Nm s / rad . \end{matrix}

(86)

The parameters of the dynamic model and the controller are represented in Tables 1 and 2.

Table 1:

Parameters of the dynamic model.

Name	Value	Unit

ρ₁ⁱ	0.3	Nm s/rad
ρ₂ⁱ	1.0	Nm
ρ₃ⁱ	0.7	Nm
ρ₄ⁱ	20	s²/rad²
${\hat{f}}_{τ i}$	80	s²/rad²
${\hat{f}}_{c i}$	3	Nm
${\hat{b}}_{f i}$	1.2	Nm s/rad
${\hat{f}}_{s i}$	4.0	Nm

Table 2:

Parameters of the controller.

Name	Value	Name	Value

∊_pnⁱ	0.01	k ₁	5
∊ⁱ	0.1	k ₂	4
∊_s	0.001	k ₃	100
∊₁	2	∊_v	7.3
∊₂	0.15	∊_fi	0.5
σ₁	7	∊	0.05
σ₂	5	ϕ	2

4.2. Simulation Results

To verify the effectiveness and precision of the proposed control method, in this paper, the simulations are conducted with two different controllers, respectively, that include the standard supertwisting algorithm (STA) based one-order sliding mode controller (SMC) and the VGSTA based integral nested sliding mode controller (INSMC). The simulation results are shown in Figures 7 to 22.

Figure 7:

Joint 1 position tracking curve of configuration A under STA based SMC.

Figures 7, 8, 9, 10, 11, and 12 show the joint position tracking curves and the position tracking error curves of two different configurations under the STA based SMC. From these figures we can conclude that the chattering effects in the initial period are obvious under the STA based SMC. Besides, the standard STA based control method can ensure the convergence of the position error; however, because the control gains cannot be adjusted according to the actual bound of the uncertainty, the periodic position error is relatively obvious under this control.

Figure 8:

Joint 2 position tracking curve of configuration A under STA based SMC.

Figure 9:

Joint 1 position error curve of configuration A under STA based SMC.

Figure 10:

Joint 2 position error curve of configuration A under STA based SMC.

Figure 11:

Joint 2 position tracking curve of configuration B under STA based SMC.

Figure 12:

Joint 2 position error curve of configuration B under STA based SMC.

Figures 13, 14, 15, 16, 17, and 18 show the joint position tracking curves and the position tracking error curves of the two different configurations under VGSTA based INSMC. As the simulation results show in these figures, we can conclude that when the VGSTA based INSMC is used, the tracking performance of the joint positions is improved obviously because the model uncertainties have been compensated accurately. The chattering effect can be reduced with the proposed integral nested sliding surface and the sliding manifold can be reached in finite time; after that, the robot system can keep in sliding manifold.

Figure 13:

Joint 1 position tracking curve of configuration A under VGSTA based INSMC.

Figure 14:

Joint 2 position tracking curve of configuration A under VGSTA based INSMC.

Figure 15:

Joint 1 position error curve of configuration A under VGSTA based INSMC.

Figure 16:

Joint 2 position error curve of configuration A under VGSTA based INSMC.

Figure 17:

Joint 2 position tracking curve of configuration B under VGSTA based INSMC.

Figure 18:

Joint 2 position error curve of configuration B under VGSTA based INSMC.

Figures 19 and 20 have shown the end-effector constrained force curves, and Figures 21 and 22 have shown the 3D end-effector trajectory tracking curves of the two different configurations under the VGSTA based INSMC. From these figures, we can conclude that when the external constrained force is exerted on the end-effector of the MRR with different configurations, the end-effector desired trajectory can be tracked accurately under the proposed VGSTA based INSMC.

Figure 19:

The end-effector constrained force curve of configuration A.

Figure 20:

The end-effector constrained force curve of configuration B.

Figure 21:

The 3D end-effector trajectory tracking curve of configuration A under VGSTA based INSMC.

Figure 22:

The 3D end-effector trajectory tracking curve of configuration B under VGSTA based INSMC.

Comparing the simulation results, we can see that the MRR system can achieve the asymptotic attractiveness of the sliding manifold in finite time with the designed sliding surface, and the model uncertainty can be compensated accurately under the proposed VGSTA based INSMC.

5. Conclusions

This paper presents a novel decentralized integral nested sliding mode control approach to address the problem of joint trajectory tracking control of a time varying constrained MRR with strongly coupled model uncertainty. The dynamic model of the time constrained MRR is described as a synthesis of interconnected subsystems. Based on only local dynamic information of each module, a decentralized integral nested sliding mode control approach is proposed to reduce the chattering effect of the controller and compensate the model uncertainty. The stability of the close-loop system is proved using the Lyapunov theory. The simulation results have confirmed the effectiveness of the proposed method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

This work is financially supported by the National Natural Science Foundation of China (Grants no. 61374051 and 60974010) and The Scientific Technological Development Plan Project in Jilin Province of China (Grant no. 20110705). The first author is also funded by China Scholarship Council.

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