Dynamic modeling and adaptive fuzzy terminal sliding mode controller with nonlinear observer of a 2-(PUU) 2 R hybrid mechanism

Abstract

This article proposes a new spatial 2-(PUU)²R hybrid mechanism that can perform the three degrees of freedom translation and one degree of freedom rotation and presents an analysis of the dynamics of the mechanism. An adaptive fuzzy terminal sliding mode controller with nonlinear observer for the hybrid mechanism is proposed to achieve a precise trajectory tracking, which could be utilized in solving the problems of the hybrid mechanism caused by model uncertainties, varying payloads, and external disturbances. Firstly, through the interrelation between the constraints, the 6 × 6 Jacobian matrix and 6 × 6 × 6 Hessian matrix for the mechanism are derived. Furthermore, dynamic modeling is established based on the virtual work principle, through which the characteristics of dynamic modeling can be proved. To achieve high-precision position tracking, a nonlinear observer was introduced to feed into the terminal sliding mode control which had improved the mechanism’s ability to resist the external disturbances. In addition, the chattering caused by the terminal sliding mode control was eliminated by approximating the switching gain with the usage of adaptive fuzzy logic in a finite time. Finally, a series of numerical simulations are carried out to prove the validity of the proposed approach, and the results verify better robustness and higher precision for the trajectory tracking than proportional–integral–derivative and sliding mode control.

Keywords

Hybrid mechanism dynamic fuzzy logic terminal sliding mode control nonlinear observer

Introduction

In recent years, hybrid mechanisms have attracted much attention among researchers and engineers. Their structure can be considered as the integration of serial and parallel mechanisms, which combine the advantages of these two arrangements. Thus, hybrid mechanisms exhibit better stiffness and more precise positioning compared to serial mechanisms as well as larger workspace compared to parallel mechanisms. Hybrid mechanisms have some potential applications such as in robot arms, sensors, machine tools, surgical manipulators, and satellite surveillance platforms.^1
–3

The H4 mechanism is the most typical hybrid mechanism. It is widely used in many practical applications including manufacturing, medical, and food industries. Pierrot and coworkers presented the concept of designing for the H4 mechanism.^4
–6 To meet the industrial requirement of different applications, a family of H4 mechanisms was developed.⁷ Generally, the mechanisms suffer from a variety of drawbacks, such as a complex structure as well as low stiffness and accuracy. To overcome these drawbacks, this article presents and analyzes an improved 2-(PUU)²R mechanism in which all actuators are orthogonally arranged in the normal plane component. The hybrid mechanism has only 14 joints and can be driven by four actuators on the fixed platform. Its fixed linear actuators and hybrid structure contribute to accuracy and high stiffness as well as good dynamic responses because of the minimum number of driving type and joints, reduced movable mass, gap width between joints, and heavy servomotors. The proposed hybrid mechanism has potential for industrial applications. (The notations of R, P, and U denote revolute joint, prismatic joint, and universal joint, respectively.)

To realize a high-performance controller design, an effective dynamic model is necessary for decreasing the influence of the dynamic characteristics on the motion precision. In general, four main approaches can be applied to derive the dynamic model of a parallel mechanism: the Lagrangian formulation with Lagrange multipliers,^8,9 Newton–Euler laws,^10,11 the Kane equation,^12,13 and virtual work.^14,15 Whichever method is employed, it is crucial to derive an effective dynamic model in order to achieve a high-performance controller design. In fact, numerical solutions of mechanism dynamics can be used to analyze the dynamics, but because dynamic characteristics cannot be derived, the approaches are not directly used for controller design. Hence, it is important to utilize a parallel mechanism to derive dynamic characteristics, especially for the complicated hybrid mechanisms.

Traditional controllers such as proportional–integral–derivative (PID)^16,17 and computed torque control^18,19 that are based on motion controlling methods can easily achieve excellent performance but are not suitable for compensating model uncertainties and external disturbances. Hence, the traditional controllers cannot maintain a high better performance. Compared with serial or parallel mechanisms, the hybrid mechanisms have stronger coupling and higher nonlinearity in the control system. To improve the trajectory tracking performance, some advanced control strategies have been developed, such as adaptive control,^20,21 robust control,^22,23 fuzzy control,^24,25 neural network control,²⁶ and sliding mode control (SMC).^27

–33

To deal with modeling uncertainties, time-varying properties, and external disturbances, SMC is to be considered as an effective control scheme, having some superior properties such as fast transient response and high robustness to uncertainties. In the past few years, a great deal of SMC for parallel mechanisms have been successfully implemented. Yakhlef and Hamerlain²⁷ investigated a diagram of control based on SMC and applied iterative learning control to delta parallel mechanism. Pi and Wang²⁸ studied an SMC with discontinuous projection-based adaptation laws to improve the tracking performance of a Stewart parallel mechanism with uncertain load disturbances. Zhao et al.²⁹ used a discrete sliding mode controller with fuzzy adaptive reaching law for the trajectory tracking of a 6-PRRS parallel mechanism. Achili et al.³⁰ adopted a robust controller for external disturbances to improve the trajectory tracking of a parallel mechanism by combining neural network and SMC. Moreover, Tang et al.³¹ proposed chattering-free SMC and integral-surface adaptive SMC for the two degrees of freedom (DOF) and 6-DOF parallel mechanisms. Yu and Weng³² proposed an H_∞ tracking adaptive fuzzy integral SMC algorithm for controlling 6R parallel mechanisms with unmodeled dynamics, limb-to-limb couples, and external disturbances. Singh and Santhakumar³³ presented a disturbance estimation and rectification algorithm with an improvised SMC which adapts itself to modeling uncertainties and nonlinearities for a 2PRP-PPR parallel mechanism. (The notations of R, P, S, and U denote revolute joint, prismatic joint, sphere, and universal joint, respectively).

To achieve higher control precision and faster converging speed, high control gain is usually needed in asymptotic stability controller design. However, it is undesirable to use high gains in practice. Therefore, it is important to develop a fast convergent control approach for parallel mechanism tracking control.

Terminal sliding mode control (TSMC) can make the position error converge to zero in a finite time, which is particularly suitable for high precision control without using high gains. Different aspects of the conventional TSMC have been studied in the literature,^34
–36 including chattering problems and singularity. In addition, parameters and perturbation upper bounds are often uncertain in many time-variable systems. Hence, the switching control gain should be high enough to guarantee stability. To solve those problems, Li³⁷ proposed an adaptive fuzzy logic sliding mode controller that has the ability to successfully approximate nonlinear systems. The approach can be used to n-link robotic manipulators with unmodeled dynamics and external disturbances. Tao³⁸ proposed a fuzzy terminal sliding mode controller for linear systems with time-varying uncertainties, and the parameters of the output fuzzy sets in the fuzzy terminal sliding mode controller were online adapted online. Lotfi³⁹ studied a fuzzy sliding mode controller design based on distributed compensation and using a scalar sign function. The fuzzy logic combined with the TSMC theory has been proved to be an effective approach for fast robot manipulators. Nekoukar⁴⁰ proposed an adaptive continuous non-singular SMC for a class of nonlinear uncertain systems without relying on a prior knowledge about the dynamics of the system. The controller guarantees not only the boundedness of all signals but also the convergence to the equilibrium in a finite time. In this article, an adaptive fuzzy terminal sliding mode controller with nonlinear observer (AFTSMCO) for the hybrid mechanism is proposed to achieve a precise trajectory tracking in the task space, which could be applied to solve the uncertainties, as well as the singularity and chattering problems.

The rest of this article is organized as follows. In the second section, the 6 × 6 Jacobian matrix and 6 × 6 × 6 Hessian matrix for the mechanism are derived. In the third section, dynamic modeling is established, and the dynamic characteristics are proved. In the fourth section, AFTSMCO is designed for the hybrid mechanism to achieve a precise trajectory tracking in the task space. In the fifth section, numerical simulations are carried out to illustrate the proposed model’s robustness and higher precision compared with SMC and PID. In the sixth section, some concluding remarks are given.

Kinematic analysis

Structure description

The 2-(PUU)²R mechanism consists of a fixed platform, a moving platform, four subchains, and two lateral bars, as shown in Figure 1. Each subchain is composed of a P joint and two U joints in sequence, where the P joint is driven by linear actuators assembled on the fixed platform. The first and second subchains are connected by the first lateral bar, which form one closed loop. The third and fourth subchains are connected by the second lateral bar, which form the other closed loop. The two lateral bars are connected to the moving platform by two R joints.

Figure 1.

Solid modeling of the 2-(PUU)²R mechanism.

Inverse kinematics

To facilitate the analysis, as shown in Figure 2, the base frame O-XYZ and moving frame p-xyz are attached to the fixed platform and moving platform at the centered point O and the centered point p, respectively. Let the z-axis and Z-axis be perpendicular to the platform, and let the y-axis and Y-axis be parallel to the ${\vec{o a}}_{2}$ and $\vec{d_{1} d_{2}}$ . Here, $a_{i} (i = 1, 2, 3, 4)$ is a centered point on the axis of the ith P joint, while c_i and b_i are the centered points of the upper and lower U joints of the ith subchain, respectively. $d_{j} (j = 1, 2)$ denotes the centered point of the axes of the jth R joint. q_i denotes the length between point O and point a_i . H denotes the distance from point a_i to point b_i . l denotes the distance from point b_i to point c_j . h stands for the distance from point p to point d_j .

Figure 2.

Kinematic model of the 2-(PUU)²R mechanism. (a) Three-dimensional (3-D) view and (b) top view.

Let $P = {[\begin{matrix} x & y & z & 0 & 0 & α \end{matrix}]}^{T}$ be the position and pose of point p in the base frame. Then, the position vector ${}^{o}b _{i}$ in the base frame can be written as

{}^{o}b ​_{1} = {[\begin{matrix} 0 & q_{1} & H \end{matrix}]}^{T}, {}^{o}b ​_{2} = {[\begin{matrix} q_{2} & 0 & H \end{matrix}]}^{T}, {}^{o}b ​_{3} = {[\begin{matrix} 0 & q_{3} & H \end{matrix}]}^{T}, {}^{o}b ​_{4} = {[\begin{matrix} q_{4} & 0 & H \end{matrix}]}^{T}

The position vector ${}^{o}c _{i}$ is expressed as

{}^{o}c ​_{i} = {}^{o}p + {}_{p}^{o}R {}^{p}d ​_{j} + {}^{o} \vec{d_{j} c_{i}}

where ${}_{p}^{o}R$ is the constant homogeneous transfer matrix from moving frame to base frame, ${}^{o}p$ is the position vector of point p in the base frame, ${}^{o} \vec{d_{j} c_{i}}$ is the position vectors of lateral bar $d_{j} c_{i}$ , and ${}^{p}d _{j}$ is the position vector between point p and point d_j in the moving frame.

The position vectors ${}^{o} \vec{d_{j} c_{i}}$ can be expressed as

{}^{o} \vec{d_{1} c_{1}} = - {}^{o} \vec{d_{2} c_{3}} = [\begin{matrix} - μ & - s & 0 \end{matrix}], {}^{o} \vec{d_{1} c_{2}} = - {}^{o} \vec{d_{2} c_{4}} = [\begin{matrix} s & μ & 0 \end{matrix}]

where r and s are the X and Y components of the length between the upper joint of arm and the center of the adjacent R joint on the moving platform in the base frame, respectively.

The position vector ${}^{p}d _{j}$ in the base frame can be written as

{}^{p}d ​_{1} = {[\begin{matrix} 0 & - h & 0 \end{matrix}]}^{T}, {}^{p}d ​_{2} = {[\begin{matrix} 0 & h & 0 \end{matrix}]}^{T}

The closure of each subchain is given by

{({}^{o}c ​_{i} - {}^{o}b ​_{i})}^{T} ({}^{o}c ​_{i} - {}^{o}b ​_{i}) = L_{i}^{2}

Solving the abovementioned equations, the inverse kinematic solutions can be obtained as

{\begin{matrix} q_{1} = \pm \sqrt{l_{1}^{2} - {(x - r + h sin θ)}^{2} - {(z - H)}^{2}} + y - s - h cos θ \\ q_{2} = \pm \sqrt{l_{2}^{2} - {(y + r - h cos θ)}^{2} - {(z - H)}^{2}} + x + s + h sin θ \\ q_{3} = \pm \sqrt{l_{3}^{2} - {(x + r - h sin θ)}^{2} - {(z - H)}^{2}} + y + s + h cos θ \\ q_{4} = \pm \sqrt{l_{4}^{2} - {(y - r + h cos θ)}^{2} - {(z - H)}^{2}} + x - s - h sin θ \end{matrix}

From equation (6), there are eight inverse kinematics solutions for a given pose of the hybrid mechanism. To obtain the inverse configuration, the first and the last of the signs “±” in equation (6) should be “−,” while the other signs “±” should be “+.”

Jacobian matrix

The velocity and acceleration of point p are given by

\begin{array}{l} V = [\begin{matrix} v \\ ω \end{matrix}] = {[\begin{matrix} \dot{x} & \dot{y} & \dot{z} & 0 & 0 & \dot{α} \end{matrix}]}^{T} \\ A = [\begin{matrix} a \\ ε \end{matrix}] = {[\begin{matrix} \ddot{x} & \ddot{y} & \ddot{z} & 0 & 0 & \ddot{α} \end{matrix}]}^{T} \end{array}

Differentiating equation (2), the velocity $v_{c i}$ of point c_i is expressed as

v_{c i} = ω_{l i} \times n_{i} l_{i} + v_{a i} e_{i}

where $v_{a i}$ and $e_{i}$ are the value and unit vector of velocity of the ith P joint, respectively, and $ω_{l i}$ denotes the vector of the angular velocity of point H_i .

From equation (8), the velocity $v_{a i}$ can be written as

v_{a i} = \frac{n_{i}^{T} {(v + ω \times r)}_{i}}{n_{i}^{T} e_{i}} = [\begin{matrix} \frac{n_{i}^{T}}{n_{i}^{T} e_{i}} & \frac{{(r_{i} \times n_{i})}^{T}}{n_{i}^{T} e_{i}} \end{matrix}] [\begin{matrix} v \\ ω \end{matrix}]

where $r_{1} = r_{2} = {}^{p}d _{1}$ and $r_{3} = r_{4} = {}^{p}d _{2}$ .

Furthermore, the actuation velocities can be expressed as

v_{a} = [\begin{matrix} v_{a 1} \\ ⋮ \\ v_{a n} \end{matrix}] = J_{α} \dot{P} = [\begin{matrix} \begin{matrix} \frac{n_{1}^{T}}{n_{1}^{T} e_{1}} \\ ⋮ \\ \frac{n_{n}^{T}}{n_{n}^{T} e_{n}} \end{matrix} & \begin{matrix} \frac{{(r_{1} \times n_{1})}^{T}}{n_{1}^{T} e_{1}} \\ ⋮ \\ \frac{{(r_{n} \times n_{n})}^{T}}{n_{n}^{T} e_{n}} \end{matrix} \end{matrix}] [\begin{matrix} v \\ ω \end{matrix}]

where $J_{α}$ stands for the velocity Jacobian matrix.

To derive the 6 × 6 Jacobian matrix and 6 × 6 × 6 Hessian matrix, the poses of the constrained wrench should be determined, there are two constrained couples on moving platform. A constrained couple of each subchain can be determined by the geometrical approach,⁴¹ the direction unit vectors of two constrained couples of the moving platform are $(n_{1} \times n_{2}) \times S_{1}$ and $(n_{3} \times n_{4}) \times S_{1}$ , where $n_{i}$ is the direction unit vectors of constrained couple of the ith subchain.

Since the constrained torque limits the movement of platform, it does not provide any power to the moving platform. Thus, an augmented velocity equation is derived as

v_{v} = [\begin{matrix} v_{5} \\ v_{6} \end{matrix}] = J_{α} V = [\begin{matrix} \begin{matrix} 0_{1 \times 3} \\ 0_{1 \times 3} \end{matrix} & \begin{matrix} {((n_{1} \times n_{2}) \times S_{1})}^{T} \\ {((n_{3} \times n_{4}) \times S_{1})}^{T} \end{matrix} \end{matrix}] [\begin{matrix} v \\ ω \end{matrix}]

The inverse velocities of the mechanism can be solved from equation (11) as follows

[\begin{matrix} v_{a} \\ 0_{2 \times 1} \end{matrix}] = J \dot{P}, J = [\begin{matrix} v_{a} \\ v_{ν} \end{matrix}]

where $J$ is a 6 × 6 Jacobian martix.

The angular velocity of point $H_{i}$ can be derived as

ω_{l i} = J_{ω i} \dot{P}

where $J_{ω i}$ is the angular velocity of the Jacobian matrix which maps the angular velocity of the ith arm to the velocity of point p.

The velocity of the point $H_{i}$ in the base frame is derived as

[\begin{matrix} v_{l i} \\ ω_{l i} \end{matrix}] = J_{l i} V, J_{l i} = [\begin{matrix} J_{v i} \\ J_{ω i} \end{matrix}] = [\begin{matrix} [\begin{matrix} E_{3 \times 3} & - {\hat{r}}_{i} \end{matrix}] + \frac{l}{2} {\hat{n}}_{i} J_{ω i} \\ J_{ω i} \end{matrix}]

where ${\hat{r}}_{i}$ and ${\hat{n}}_{i}$ are defined as the cross-product factors, which makes $r_{i} \times n_{i} = {\hat{r}}_{i} n_{i} = - {\hat{n}}_{i} r_{i}$ , and $J_{l i}$ is the velocity Jacobian matrix of the ith arm, which maps the velocity of point $H_{i}$ to the velocity of point p.

Similarly, the velocity of the lateral bar can be derived as

[\begin{matrix} v_{d j} \\ ω_{d j} \end{matrix}] = J_{d j} \dot{P}, J_{d j} = [\begin{matrix} E_{3 \times 3} \\ - {\hat{r}}_{i} \end{matrix}]

where $J_{d j}$ is the velocity Jacobian matrix which maps the velocity of the jth lateral bar to the velocity of point p.

Hessian matrix

Differentiating equation (9), the acceleration $α_{a i}$ of the point a_i can be obtained as

α_{a i} = J_{α i} \ddot{P} + {\dot{q}}^{T} H_{a i} \dot{P}

where $H_{α i}$ is a 6 × 6 matrix.

The actuation accelerations can be expressed as

α_{a} = J_{α} \ddot{P} + {\dot{q}}^{T} H_{α} \dot{P}, H_{α} = [\begin{matrix} H_{α 1} \\ ⋮ \\ H_{α n} \end{matrix}]

where $H_{α}$ is an n × 6 × 6 matrix.

By differentiating equation (19), the inverse/forward accelerations of the mechanism can be solved as follows

[\begin{matrix} α_{a} \\ 0_{2 \times 1} \end{matrix}] = J \ddot{P} + {\dot{P}}^{T} H \dot{P}, H = [\begin{matrix} H_{α} \\ H_{ν} \end{matrix}]

where $H$ is a 6 × 6 × 6 cubical Hessian matrix. $H_{ν}$ is a 2 × 6 × 6 augmented matrix.

Differentiating equation (13), the acceleration of point $H_{i}$ can be obtained as

a_{l i} = a + ε \times r_{i} + ω \times (ω \times r_{i}) - ε_{l i} \times \frac{l}{2} n_{i} - ω_{l i} \times (ω_{l i} \times \frac{l}{2} n_{i})

Differentiating equation (13), the angular acceleration of point $H_{i}$ can be obtained as

ε_{l i} = J_{ω i} A + {\dot{J}}_{ω i} V

Dynamics modeling

Using the Newton–Euler equation, the inertial force and moment of each moving part can be determined by neglecting the friction force. Let m ₁,m ₂,m ₃, and m ₄be the masses of the moving platform, lateral bar, arm, and slider, respectively.

Based on the virtual work principle, the dynamic model of the hybrid mechanism can be expressed as

F_{q} = - {(J^{- 1})}^{T} (\sum_{i = 1}^{i = 4} (J_{l i}^{T} [\begin{matrix} m_{3} g - m_{3} {\dot{v}}_{l i} \\ - {}^{B}I ​_{l i} {\dot{ω}}_{l i} - ω_{l i} \times ({}^{B}I ​_{l i} ω_{l i}) \end{matrix}] + J^{T} [\begin{matrix} m_{4} g - m_{4} {\dot{v}}_{a i} \\ 0_{3 \times 3} \end{matrix}]) + \sum_{j = 1}^{2} J_{d j}^{T} [\begin{matrix} m_{2} g - m_{2} {\dot{v}}_{d j} \\ 0_{3 \times 3} \end{matrix}] + [\begin{matrix} m_{1} g - m_{1} \dot{v} \\ - {}^{B}I ​_{p} \dot{ω} - ω \times ({}^{B}I ​_{p} ω) \end{matrix}])

where ${}^{B}I _{p}$ is the moment of inertial of the moving platform about the point p, ${}^{B}I _{l i}$ is the moment of inertial of the ith arms about the point H_i , and $g$ is the gravitational acceleration vector.

Equation (21) can be simplified as

F_{q} = {(J^{- 1})}^{T} (M (P) + C (P, \dot{P}) + G (P) + τ_{ext})

where

M (P) = \sum_{i = 1}^{4} ​ (J_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I ​_{l i} \end{matrix}] J_{l i} + J^{T} [\begin{matrix} m_{4} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}] J) + \sum_{j = 1}^{2} J_{d j}^{T} [\begin{matrix} m_{2} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}] J_{d j} + [\begin{matrix} m_{1} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I ​_{p} \end{matrix}]

\begin{matrix} C (P, \dot{P}) = \sum_{i = 1}^{4} ​ (J_{l i}^{T} [\begin{matrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {\hat{ω}}_{l i} {}^{B}I ​_{l i} \end{matrix}] J_{l i} + J_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I ​_{l i} \end{matrix}] {\dot{J}}_{l i} + J^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}] \dot{J}) \\ + \sum_{j = 1}^{2} J_{d j}^{T} [\begin{matrix} m_{2} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}] {\dot{J}}_{d j} + [\begin{matrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & \hat{ω} {}^{B}I ​_{p} \end{matrix}] \end{matrix}

G (P) = \sum_{i = 1}^{4} ​ (- J_{l i}^{T} [\begin{matrix} m_{3} g \\ 0_{3 \times 1} \end{matrix}] - J^{T} [\begin{matrix} m_{4} g \\ 0_{3 \times 1} \end{matrix}]) - \sum_{j = 1}^{2} J_{d j}^{T} [\begin{matrix} m_{2} g \\ 0_{3 \times 1} \end{matrix}] - [\begin{matrix} m_{1} g \\ 0_{3 \times 1} \end{matrix}]

where $M (P)$ is a 6 × 1 inertia matrix, $C (P, \dot{P})$ is a 6 × 6 Coriolis and centrifugal force terms, $G (P)$ is a 6 × 1 gravity force terms, $M, C, and G$ are the gravity force terms, and $\hat{ω}$ is defined as cross-product factor that makes $ω \times ({}^{B}I _{p} ω) = \hat{ω} {}^{B}I _{p} ω$ .

To simplify, $M (P), C (P, \dot{P}), and G (P)$ are written as $M, C, and G$ . This dynamic model has the following properties that will be used in controller design.

Property 1

$M$ is a positive symmetric definite matrix, and its inverse matrix $M^{- 1}$ is valid. $M$ is uniformly bounded by $M_{min} \leq M \leq M_{max}$ , where $M_{min}$ and $M_{max}$ are positive constants that depend on the mass properties of the hybrid mechanism.

Property 2

$\dot{M} - 2 C$ is a skew-symmetric matrix.

To prove $M$ to be symmetric and a positive definite value, $M$ can be written as

$M = M_{1} + \sum_{j = 1}^{2} M_{2 j} + \sum_{i = 1}^{4} M_{3 i} + \sum_{i = 1}^{4} M_{4 i}$
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where

$M_{1} = J^{T} [\begin{matrix} m_{1} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I _{p} \end{matrix}] J, M_{2 j} = J_{d j}^{T} [\begin{matrix} m_{2} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}] J_{d j}, M_{3 i} = J_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I _{l i} \end{matrix}] J_{l i}, M_{4 i} = [\begin{matrix} m_{4} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}]$

It is obvious that $M_{1}, M_{2 j}, M_{3 i}, and M_{4 i}$ are a skew-symmetric matrix. Therefore, $M$ is symmetric and a positive definite value.

Then, $C$ can be written as

$C = C_{1} + \sum_{j = 1}^{2} C_{2 j} + \sum_{i = 1}^{4} C_{3 i} + \sum_{i = 1}^{4} C_{4 i}$
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where

$\begin{array}{l} C_{1} = J^{T} [\begin{matrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & \hat{ω} I_{p} \end{matrix}] \dot{J} C_{2 j} = J_{d j}^{T} [\begin{matrix} m_{2} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}] {\dot{J}}_{d j} C_{3 i} = J_{l i}^{T} [\begin{matrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {\hat{ω}}_{l i} I_{l i} \end{matrix}] J_{l i} + J_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & I_{l i} \end{matrix}] {\dot{J}}_{l i} \\ C_{4 i} = [\begin{matrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & \hat{ω} {}^{B}I _{p} \end{matrix}] \end{array}$

In order to prove $\dot{M} - 2 C$ to be their symmetric and a positive definite value, $\dot{M} - 2 C$ is divided into four parts. One part ${\dot{M}}_{3 i} - 2 C_{3 i}$ can be expressed as

$\begin{array}{l} {\dot{M}}_{3 i} - 2 C_{3 i} = {\dot{J}}_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I _{l i} \end{matrix}] J_{l i} + J_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I _{l i} \end{matrix}] {\dot{J}}_{l i} + J_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {\hat{ω}}_{l i} {}^{B}I _{l i} - {}^{B}I _{l i} {\hat{ω}}_{l i} \end{matrix}] J_{l i} \\ - 2 J_{l i}^{T} [\begin{matrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {\hat{ω}}_{l i} {}^{B}I _{l i} \end{matrix}] J_{l i} - 2 J_{l i}^{T} [\begin{matrix} m_{3} E_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & {}^{B}I _{l i} \end{matrix}] {\dot{J}}_{l i} = [\begin{matrix} Q_{1} & 0_{3 \times 3} \\ 0_{3 \times 3} & Q_{2} \end{matrix}] \end{array}$
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where $Q_{1} = {\dot{J}}_{v i}^{T} m_{3} J_{ν i} - J_{ν i}^{T} m_{3} {\dot{J}}_{ν i}, Q_{2} = {\dot{J}}_{ω i}^{T} {}^{B}I _{l i} J_{ω i} - J_{ω i}^{T} {}^{B}I _{l i} {\dot{J}}_{ω i} - J_{ω i}^{T} {\hat{ω}}_{l i} {}^{B}I _{l i} J_{ω i} - J_{ω i}^{T} {}^{B}I _{l i} {\hat{ω}}_{l i} J_{ω i}$ .

Since ${\dot{J}}_{ν i}^{T} m_{3} J_{ν i} - J_{ν i}^{T} m_{3} {\dot{J}}_{ν i} + {({\dot{J}}_{ν i}^{T} m_{3} J_{ν i} - J_{ν i}^{T} m_{3} {\dot{J}}_{ν i})}^{T} = 0$ , it is obvious that ${\dot{J}}_{ω i}^{T} {}^{B}I _{l i} J_{ω i} - J_{ω i}^{T} {}^{B}I _{l i} {\dot{J}}_{ω i}$ is a skew-symmetric matrix. Therefore, $Q_{1}$ is a skew-symmetric matrix.

Similarly, since ${\dot{J}}_{ω i}^{T} {}^{B}I _{l i} J_{ω i} - J_{ω i}^{T} {}^{B}I _{l i} {\dot{J}}_{ω i} + {({\dot{J}}_{ω i}^{T} {}^{B}I _{l i} J_{ω i} - J_{ω i}^{T} {}^{B}I _{l i} {\dot{J}}_{ω i})}^{T} = 0$ , it is obvious that ${\dot{J}}_{ω i}^{T} {}^{B}I _{l i} J_{ω i} - J_{ω i}^{T} {}^{B}I _{l i} {\dot{J}}_{ω i}$ is a skew-symmetric matrix.

Furthermore, as $J_{ω i}^{T} {\hat{ω}}_{l i} {}^{B}I _{l i} J_{ω i} + J_{ω i}^{T} {}^{B}I _{l i} {\hat{ω}}_{l i} J_{ω i} + {(J_{ω i}^{T} {\hat{ω}}_{l i} {}^{B}I _{l i} J_{ω i} + J_{ω i}^{T} {}^{B}I _{l i} {\hat{ω}}_{l i} J_{ω i})}^{T} = 0$ , $J_{ω i}^{T} {\hat{ω}}_{l i} {}^{B}I _{l i} J_{ω i} + J_{ω i}^{T} {}^{B}I _{l i} {\hat{ω}}_{l i} J_{ω i}$ is also a skew-symmetric matrix.

Therefore, $\dot{M} - 2 C$ is a skew-symmetric matrix.

Similarly, we can prove ${\dot{M}}_{1} - 2 C_{1}$ , ${\dot{M}}_{2 j} - 2 C_{2 j}$ , and ${\dot{M}}_{4 i} - 2 C_{4 i}$ to also be skew symmetric too. Thus, $\dot{M} - 2 C$ is also a skew-symmetric matrix.

Controller design

The proposed strategy is composed of TSMC, adaptive fuzzy logic, and a nonlinear observer. The feedback of the nonlinear observer is used as an input in the TSMC. In addition, the adaptive fuzzy logic is used to update the switching gain of the TSMC.

The block diagram of the proposed controller is shown in Figure 3, which is composed of three subsections including a user input part, a controller part, and robotic system. The user input part is used to compute the three signals ${\dot{P}}_{}^{r}$ , ${\ddot{P}}_{}^{r}$ , and $s$ with the desired trajectory $P^{d}$ and the signals from the signal processing part. The controller part includes dynamics compensation, STMC, adaptive fuzzy, and a nonlinear observer. This part guarantees that the positions and velocities are converged to their corresponding desired states. Based on the proposed controller, the force and torque of the moving platform about point p is obtained, and then, the actuator torque can be computed along with the inverse of the force Jacobian matrix. The robotic system part consists of two subblocks, including a manipulator block and sensor block. The manipulator block denotes the actual system. The sensor block measures the position $q$ of the active joints. The velocity of the active joints is calculated with the differentiation of the joint positions, and then, the velocity of the moving platform is calculated with the velocity Jacobian. Furthermore, the position $P$ of the moving platform can be calculated with its desired trajectory according to the forward kinematics. Comparing the differences between the actual and desired trajectory of the moving platform, the trajectory tracking errors can be obtained.

Figure 3.
The block diagram of the proposed controller.

TSMC design

The tracking control problem in the task space is to find a control law so that the position tracking error $e$ and its derivative $\dot{e}$ tend to zero as soon as possible with uncertainties. The position tracking error and its derivative are expressed as

$e = P^{d} - P$
29

$\dot{e} = {\dot{P}}^{d} - \dot{P}$
30

where $P^{d}$ and ${\dot{P}}^{d}$ denote a desired trajectory and its derivative, respectively.

We can define command vectors ${\dot{P}}^{r}$ and ${\ddot{P}}^{r}$ as follows

${\dot{P}}_{}^{r} = \dot{P} - s$
31

${\ddot{P}}_{}^{r} = \ddot{P} - \dot{s}$
32

The sliding surface can be defined as

$s = \dot{e} + Λ e + Γ e^{p_{1} / p_{2}}$
33

where $Λ, Γ \in ℝ^{6 \times 6}$ are positive diagonal gain matrices, and $p_{1}$ and $p_{2}$ are positive odd numbers and $0 \leq p_{1} \leq p_{2} \leq 2 p_{1}$ .

The time derivative of equation (33) is

$\dot{s} = \ddot{e} + Λ \dot{e} + Γ \frac{p_{1}}{p_{2}} {\dot{e}}^{\frac{p_{1}}{p_{2}} - 1}$
34

If system states come up to the fast terminal sliding mode, that is, $s = 0$ , according to equation (33), this leads to

$\dot{e} + Λ e + Γ e^{p_{1} / p_{2}} = 0$
35

The ith element of equation (35) can be expressed as

${\dot{e}}_{i} + Λ_{i i} e + Γ_{i i} e_{i} ^{p_{1} / p_{2}} = 0$
36

where $Λ_{i i}$ and $Γ_{i i}$ are the ii-th terms of matrix $Λ$ and $Γ$ .

If equation (36) has an equilibrium point at $e_{i} = 0$ with any given initial condition $e_{i} (0) \neq 0$ , the convergence time $t_{s i}$ for the solution is finite

$t_{s i} = \frac{p_{2}}{Λ_{i i} (p_{2} - p_{1})} ln \frac{{| e_{i} (0) |}^{(p_{2} - p_{1}) / p_{2}} + Γ_{i i}}{Γ_{i i}}$
37

Substituting equations (33) and (34) into equation (22) gives

$M \dot{s} + C s = - \overset{⌢}{M} {\ddot{q}}_{r} - \overset{⌢}{C} {\dot{q}}_{r} - \overset{⌢}{G} + τ - f_{e} - ζ_{edis}$
38

where $M = \tilde{M} + Δ M$ , $C = \tilde{C} + Δ C$ , $G = \tilde{G} + Δ G$ . $\tilde{M}$ , $\tilde{C}$ , and $\tilde{G}$ are the estimation value of $M$ , $C$ , and $G$ . $Δ M$ , $Δ C$ , and $Δ G$ stand for the model uncertainty. $f_{e}$ and $ζ_{edis}$ are the vector of the model uncertainties and external disturbance, respectively. The model uncertainties can be written as

$f_{e} = Δ M {\ddot{q}}_{r} + Δ C {\dot{q}}_{r} + Δ G$
39

Thus, equation (38) becomes

$\tilde{M} \ddot{q} + \tilde{C} \dot{q} + \tilde{G} + f_{e} + ζ_{edis} = τ$
40

To make the position error converge to zero in a finite time, the following control law gives

$τ = - K sgn {(s)}^{β} - R s - W + \overset{⌢}{M} \ddot{r} + \overset{⌢}{C} \dot{r} + \overset{⌢}{G} + {\overset{⌢}{ζ}}_{edis}$
41

where $K, R, W \in ℝ^{6 \times 6}$ are the position definite diagonal matrices. $s i g {(s)}^{β} = {[\begin{matrix} {| (s_{1}) |}^{β} s i g n (s_{1}) \dots {| (s_{6}) |}^{β} s i g n (s_{6}) \end{matrix}]}^{T}$ . $β = p_{1} / p_{2}$ . ${\overset{⌢}{ζ}}_{edis}$ is an estimation of the $ζ_{edis}$ .

Remark 1

$- K sgn {(s)}^{β} - R s$ is the TSMC reaching law, which can make system states arrive at the sliding surface in a finite time.

Remark 2

$\overset{⌢}{M} {\ddot{q}}_{r} + \overset{⌢}{C} {\dot{q}}_{r} + \overset{⌢}{G}$ stands for the feedforward part, which is used to compensate for the effect caused by dynamic modeling.

Remark 3

$- W$ is a generalized gain for the fast TSMC, which is used to compensate for the effect caused by the model uncertainties. The chattering can also be avoided in the operation.

Remark 4

${\overset{⌢}{ζ}}_{edis}$ is used to estimate external disturbances by the nonlinear observer.

Proof

The Lyapunov function for TSMC is

$V_{1} = \frac{1}{2} s^{T} M s$
42

Differentiating $V_{1}$ with respect to time, yields

${\dot{V}}_{1} = \frac{1}{2} s^{T} \dot{M} s + s^{T} M s = s^{T} (\frac{1}{2} \dot{M} - C) s + s^{T} C s + s^{T} M \dot{s}$
43

Applying property (P2) and substituting equation (38) into equation (43), the following equation leads to

$\begin{array}{l} {\dot{V}}_{1} = s^{T} C s + s^{T} M \dot{s} \\ = s^{T} C s + s^{T} (- \overset{⌢}{M} {\ddot{q}}_{r} - \overset{⌢}{C} {\dot{q}}_{r} - \overset{⌢}{G} (q) + τ - f_{e} - ζ_{edis} - C s) \\ = - s^{T} (- K sgn {(s)}^{β} - R s + W - f_{e} - ζ_{edis} + {\overset{⌢}{ζ}}_{edis}) \end{array}$
44

Observer design

To design a nonlinear observer to observe external disturbances, the following parameters are defined as

${\tilde{ζ}}_{edis} = ζ_{edis} - {\overset{⌢}{ζ}}_{edis}$
45

$\dot{\tilde{e}} = \dot{e} - \dot{\overset{⌢}{e}}$
46

where $\dot{\overset{⌢}{e}}$ is the estimation of $\dot{e}$ , ${\tilde{ζ}}_{edis}$ is the estimation error between $ζ_{edis}$ and ${\overset{⌢}{ζ}}_{edis}$ , and $\dot{\tilde{e}}$ is the estimation error between $\dot{e}$ and $\dot{\overset{⌢}{e}}$ .

An observer can be designed as

${\begin{matrix} {\dot{\overset{⌢}{ζ}}}_{edis} = K_{1} \dot{\tilde{e}} \\ \ddot{\overset{⌢}{e}} = - [{\ddot{q}}^{d} + M^{- 1} C \dot{q} + M^{- 1} G + M^{- 1} {\overset{⌢}{ζ}}_{edis} - M^{- 1} τ] + K_{2} \dot{\tilde{e}} \end{matrix}$
47

where $K_{1} \in ℝ^{6 \times 6}$ and $K_{2} \in ℝ^{6 \times 6}$ are the diagonal and positive definite matrices.

The Lyapunov function for the observer is

$V_{2} = \frac{1}{2} {\dot{\tilde{e}}}^{T} M \dot{\tilde{e}} + \frac{1}{2} {\tilde{ζ}}_{edis}^{T} K_{1}^{- 1} {\tilde{ζ}}_{edis}$
48

Assuming that the disturbance signal is slow time varying, that is to say $\dot{ζ} = 0$ , the derivative of the Lyapunov function with respect to time can be obtained as

${\dot{V}}_{2} = {\tilde{ζ}}_{edis}^{T} K_{1}^{- 1} ({\dot{ζ}}_{edis} - {\dot{\overset{⌢}{ζ}}}_{edis}) + {\dot{\tilde{e}}}^{T} M (\ddot{e} - \ddot{\overset{⌢}{e}}) + \frac{1}{2} {\dot{\tilde{e}}}^{T} \dot{M} \dot{\tilde{e}}$
49

Substituting equation (47) into equation (49) yields

$\begin{array}{l} {\dot{V}}_{2} = - {\dot{\tilde{e}}}^{T} {\dot{\tilde{ζ}}}_{edis} + {\dot{\tilde{e}}}^{T} M (M^{- 1} {\tilde{ζ}}_{edis} - K_{2} \dot{\tilde{e}}) + {\dot{\tilde{e}}}^{T} C \dot{\tilde{e}} = - {\dot{\tilde{e}}}^{T} K_{2} \dot{\tilde{e}} + {\dot{\tilde{e}}}^{T} C \dot{\tilde{e}} \\ \leq - λ_{min} (K_{2}) {‖ \dot{\tilde{e}} ‖}^{2} + λ_{max} (C) {‖ \dot{\tilde{e}} ‖}^{2} \end{array}$
50

where $λ_{min} (K_{2})$ is the minimum of $K_{2}$ , $λ_{max} (C)$ is the maximum of $C$ .

To make sure that ${\dot{V}}_{2} \leq 0$ , the following inequality can be satisfied as

$λ_{min} (K_{2}) \geq λ_{max} (C)$
51

Through the abovementioned analysis, the stability of the observer can be proved, the observer is asymptotic convergence to zero.

Adaptive fuzzy algorithm design

An adaptive fuzzy logic system (AFLS) is considered as an effective method to eliminate the chattering in the literature.⁴² Hence, AFLS is introduced here to approximate the switching gain and to eliminate the chattering. AFLS consists of a knowledge base, a fuzzifier, a fuzzy inference engine, and a defuzzifier. The knowledge base for the AFLS is comprised of a series of fuzzy If-Then inference rules as follows:

Rule i: IF $s (i)$ is $A_{i}^{m}$ , THEN $w (i)$ is $B_{i}^{m}$ ;

where m is the rule number. $s (i)$ and w(i) are the variables of input and output, respectively. Both $A_{i}^{m}$ and $B_{i}^{m}$ are partitioned into seven fuzzy subsets: Negative Big, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, and Positive Big.

Both $A_{i}^{m}$ and $B_{i}^{m}$ are defined as Gaussian membership functions:

$u (x_{k}) = exp [- {(\frac{x_{k} - α}{σ})}^{2}]$
52

where α represents the membership function’s center and σ determines its width.

The ith-output of fuzzy logic system can be defined as

$w_{i} = \frac{\sum_{m = 1}^{M} θ_{w_{i}}^{m} u_{A^{m}} (d_{i})}{\sum_{m = 1}^{M} u_{A^{m}} (d_{i})} = θ_{w_{i}}^{T} Ψ_{w_{i}} (d_{i})$
53

where $θ_{w_{i}} = {[θ_{w_{i}}^{1}, \dots, θ_{w_{i}}^{m}, \dots, θ_{w_{i}}^{M}]}^{T}$ , $Ψ_{w_{i}} (d_{i}) = {[φ_{w_{i}}^{1} (d_{i}), \dots, φ_{w_{i}}^{m} (d_{i}), \dots, φ_{w_{i}}^{M} (d_{i})]}^{T}$ , and $φ^{m} (x) = \prod_{j = 1}^{N} u_{A_{j}^{m}} (x) / \sum_{m = 1}^{M} \prod_{j = 1}^{N} u_{A_{j}^{m}} (x)$ .

Lemma 1

Let $τ_{d i}$ be a continuous function, for any constant $ε_{i} > 0$ , there exists³⁸

$| f_{e i} - θ_{ω_{i d}}^{T} Ψ_{ω_{i}} (s_{i}) | \leq ε_{i}$
54

Stability analysis

Proof

Choose a Lyapunov function as

$V_{3} = \frac{1}{2} s^{T} M s + \frac{1}{2} \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\tilde{θ}}_{ω_{i}}$
55

Applying property (P2), the time derivative of the Lyapunov function is

${\dot{V}}_{3} = \frac{1}{2} s^{T} \dot{M} s + s^{T} M s + \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\dot{\tilde{θ}}}_{ω_{i}} = s^{T} C s + s^{T} M \dot{s} + \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\dot{\tilde{θ}}}_{ω_{i}}$
56

Substituting equation (44) into equation (56) gives

$\begin{array}{l} {\dot{V}}_{3} = s^{T} C s + s^{T} (- \overset{⌢}{M} {\ddot{q}}_{r} - \overset{⌢}{C} {\dot{q}}_{r} - \overset{⌢}{G} (q) + τ - f_{e} - ζ_{edis} - C s) + \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\dot{\tilde{θ}}}_{ω_{i}} \\ = s^{T} (- K sgn {(s)}^{β} - R s + W - f_{e} - ζ_{edis} + {\overset{⌢}{ζ}}_{edis}) + \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\dot{\tilde{θ}}}_{ω_{i}} \end{array}$
57

where $w_{i} = {\tilde{ϑ}}_{ω_{i}}^{T} Ψ_{ω_{i}} (s_{i}) + ϑ_{ω_{i d}}^{T} Ψ_{ω_{i}} (s_{i})$ , equation (57) can be written as

$\begin{array}{l} {\dot{V}}_{3} = s^{T} (- K sgn {(s)}^{β} - R s) - s^{T} W + s^{T} f_{e} + \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\dot{\tilde{θ}}}_{ω_{i}} \\ = s^{T} (- K sgn {(s)}^{β} - R s) + \sum_{i = 1}^{n} s_{i} (f_{e i} - ({\tilde{θ}}_{ω_{i}}^{T} Ψ_{ω_{i}} (s_{i}) + θ_{ω_{i d}}^{T} Ψ_{ω_{i}} (s_{i}))) + \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\dot{\tilde{θ}}}_{ω_{i}} \\ = s^{T} (- K sgn {(s)}^{β} - R s) + \sum_{i = 1}^{n} s_{i} (f_{e i} - θ_{ω_{i d}}^{T} Ψ_{ω_{i}} (s_{i})) - \sum_{i = 1}^{n} s_{i} {\tilde{θ}}_{ω_{i}}^{T} Ψ_{ω_{i}} (s_{i}) + \sum_{i = 1}^{n} {\tilde{θ}}_{ω_{i}}^{T} {\dot{\tilde{θ}}}_{ω_{i}} \end{array}$
58

To satisfy the inequality ${\dot{V}}_{3} \leq 0$ , the parameter adaptation laws can be selected as

${\dot{\tilde{θ}}}_{ω_{i}} = s_{i} Ψ_{w_{i}} (s_{i})$
59

Substituting equation (59) into equation (58) yields

${\dot{V}}_{3} = s^{T} (- K sgn {(s)}^{β} - R s) + \sum_{i = 1}^{n} s_{i} (f_{e i} - θ_{ω_{i d}}^{T} Ψ_{ω_{i}} (s_{i}))$
60

Applying Lemma 1, we obtain

$s_{i} | f_{e} _{i} - θ_{ω_{i d}}^{T} Ψ_{ω_{i}} (s_{i}) | \leq s_{i} ε_{i} \leq ζ_{i} s_{i}^{2}$
61

Substituting equation (61) into equation (60) gives

${\dot{V}}_{3} \leq s^{T} (- K sgn {(s)}^{β} - R s) + \sum_{i = 1}^{n} ζ_{i} s_{i}^{2} = - s^{T} K sgn {(s)}^{β} - \sum_{i = 1}^{n} (χ_{i} - ζ_{i}) s_{i}^{2}$
62

where $χ_{i} > ζ_{i}$ , the following inequality holds

${\dot{V}}_{3} \leq - s^{T} K sgn {(s)}^{β} = - \sum_{i = 1}^{n} k_{i} sgn {(s)}^{β + 1} = - \sum_{i = 1}^{4} k_{i} {| s_{i} |}^{β + 1}$
63

Since

$\sum_{i = 1}^{4} k_{i} {| s_{i} |}^{β + 1} \geq k_{min} \sum_{i = 1}^{4} {| s_{i} |}^{β + 1} \geq k_{min} {(\frac{2}{\bar{m}})}^{(β + 1) / 2} {(\sum_{i = 1}^{4} \frac{\bar{m}}{2} s_{i}^{2})}^{(β + 1) / 2} \geq λ V^{η}$
64

where $\bar{m}$ represents the minimum eigenvalues of $M$ , $λ = k_{min} {(\frac{2}{\bar{m}})}^{(β + 1) / 2} > 0$ , $0 < η = (1 + β) / 2 < 1$ .

The continuous AFSTMCO will be reached in finite time

$\tilde{t} = t_{0} + \frac{V^{1 - η} (t_{0})}{λ (1 - η)}$
65

From equations (37) and (65), it follows that $e_{i} = 0 and {\dot{e}}_{i} = 0$ in a finite time $t \geq \tilde{t}$ with

$t_{i} = \tilde{t} + t_{s i}$
66

Thus, the tracking errors and its derivatives converge to zero in finite time.

Simulation

To demonstrate the performance of the proposed algorithm, simulations of trajectory tracking on the 2-(PUU)²R mechanism are performed using MATLAB [version: MATLAB 2014a] software. The related geometrical and inertial parameters of the mechanism are given in Tables 1 and 2.

Table 1.
Related geometrical parameters of the 2-(PUU)²R mechanism.

Parameter Description Values Unit

$l_{i}$ Length of the arm 500 mm

h Distance from the center of an R joints to point p 95 mm

H Distance between the center of down joint on arm and the center of joint on slider −135 mm

s The X-component of the length from the upper joint of an arm to the adjacent R joint on the moving platform 63.63 mm

μ The Y-component of the length from the upper joint of an arm to the adjacent R joint on the moving platform 196.15 mm

Table 2.
Related inertial parameters of the 2-(PUU)²R mechanism.

Parameter Description Values Unit

m ₁ Mass of sliders 2.85 kg

m ₂ Mass of arm 1.43 kg

m ₃ Mass of lateral bar 3.95 kg

m ₄ Mass of moving platform 2.34 kg

${}^{B}I _{p}$ Inertia moment of moving platform diag{0, 0, 1285} kg.mm²

$^{B} I_{l i}$ Inertia moment of arm diag{38422, 38377, 116} kg.mm²

To compare the simulation results, two trajectory tracking experiments were conducted. In addition, three control algorithms for the experiments are examined. The other popular control algorithms for the simulations such as PID and SMC are compared with the proposed algorithm, and their control vectors are given by:

${\begin{array}{l} τ = K_{P} e + K_{I} \int e d t + K_{D} \dot{e} + \overset{⌢}{G} : PID \\ τ = - K sign (s) - R s + \overset{⌢}{M} \ddot{r} + \overset{⌢}{C} \dot{r} + \overset{⌢}{G} :SMC \end{array}$
67

where $K_{P}$ , $K_{I}$ , and $K_{D}$ denote the proportional, integral, and derivative gain matrices of the PID controller.

For the purposes of comparison, three control algorithms including tradition PID, SMC, and the proposed approach were used to control the system, respectively. The control gains of the three schemes are listed in Table 3. Based on the abovementioned parameters, the simulations are shown in Figures 4 to 6. All of the controller gains are tuned in such a way that the three controllers’ performances are quite satisfactory in the ideal situation and almost equal in terms of their tracking errors.

Table 3.
Control gains of the three controllers.

Controllers Values

The proposed control $\overset{⌢}{M} = 0.8 M$ , $\overset{⌢}{C} = 0.8 C$ , $\overset{⌢}{G} = 0.8 G$ , $K_{1} = diag {500, 500, 500, 500, 500, 500}$ , $K = diag {100, 100, 100, 100, 100, 100}$ , $K_{2} = diag {200, 200, 200, 200, 200, 200}$ , $Γ = diag {10, 10, 10, 10, 10, 10}$ , $A = diag {400, 400, 400, 400, 400, 400}$ , $R = diag {100, 100, 100, 100, 100, 100}$ , $Λ = diag {10, 10, 10, 10, 10, 10}$

SMC $\overset{⌢}{M} = 0.8 M$ , $\overset{⌢}{C} = 0.8 C$ , $\overset{⌢}{G} = 0.8 G$ , $Λ = diag {10, 100, 10, 10, 10, 10}$ , $P = diag {100, 100, 100, 100, 100, 100}$ , $A = diag {400, 400, 400, 400, 400, 400}$

PID $K_{P} = diag {1500, 1500, 1500, 1500, 1500, 1500}$ , $K_{D} = diag {800, 800, 800, 800, 800, 800}$ , $K_{I} = diag {200, 200, 200, 200, 200, 200}$

SMC: sliding mode control; PID: proportional–integral–derivative.

Figure 4.
The circular trajectory tracking result. PID: proportional–integral–derivative; SMC: sliding mode control.

Figure 5.
The circular tracking error of the first experiment. (a) X-axis and (b) Y-axis. PID: proportional–integral–derivative; SMC: sliding mode control.

Figure 6.
Joint driving force of the first experiment. (a) The first joint, (b) the second joint, (c) the third joint, and (d) the fourth joint. PID: proportional–integral–derivative; SMC: sliding mode control.

In the first experiment, considering the task space and singularities of the mechanism, the moving platform is required to move along with a circular path, and the external disturbances are assumed as to be $ζ_{edis} = diag {0.5 sin (20 t), 0.5 sin (20 t), 0, 0, 0, 0} N$ . The following desired trajectory is chosen as

${\begin{matrix} x (t) = 10 cos (t) & (mm) & 0 \leq t \leq 7 \\ y (t) = 10 sin (t) & (mm) & 0 \leq t \leq 7 \\ z (t) = - 500 & (mm) & 0 \leq t \leq 7 \\ α (t) = 0 & (rad) & 0 \leq t \leq 7 \end{matrix}$
68

where x(t), y(t), and z(t) denote the displacement of the moving platform along the X, Y, and Z axes, respectively. $α (t)$ denotes the rotational angle around the Z-axis.

The tracking trajectory of point p of the moving platform is shown in Figure 4. For comparison purpose in different controllers, obviously, the tradition PID and SMC obtains a terrible tracking performance because of the constant switching gain and they did not consider the dynamic effect of varying payload. The error of the tracking trajectory by the AFSTMCO could become smaller.

The actual trajectory tracking errors along the x and y directional movements are shown in Figure 5. Compared with the other two controllers, the proposed controller has an ability to decrease the error of trajectory tracking of the moving platform of the hybrid mechanism effectively, which can keep the error in the range of 0.3 mm. It is observed that both the maximum control error and the steady-state error tolerance have been significantly reduced, and the settling time has been significantly shortened. In addition, a smoother trajectory tracking can be obtained by the AFSTMCO than the other two controllers.

The AFSTMCO is used to approximate the switching gain of SMC. As shown in Figure 6, joint force generated from the PID and SMC are compared, and a smooth input force signal can be obtained by the AFSTMCO than the other two controllers. Also, the chatter generated by the switching gain can be eliminated effectively.

Based on the simulations in Figures 4 to 6, the AFSTMCO can achieve a better trajectory tracking performance and eliminate the chattering of the SMC.

In the second experiment, the moving platform is required to move along with a circular with a high frequency. The following desired trajectory is chosen as

${\begin{matrix} x (t) = 10 cos (3 t) & (mm) & 0 \leq t \leq 7 \\ y (t) = 10 sin (3 t) & (mm) & 0 \leq t \leq 7 \\ z (t) = - 500 & (mm) & 0 \leq t \leq 7 \\ α (t) = 0 & (rad) & 0 \leq t \leq 7 \end{matrix}$
69

The circular tracking error of the second experiment is shown in Figure 7. Compared with the other two controllers, the proposed controller exhibited the ability to decrease the error of the high-frequency reference trajectory. It is observed that both the maximum control error and the steady-state error tolerance have been significantly reduced, and the settling time has been significantly shortened. Moreover, the joint forces generated from three controllers are shown in Figure 8, which shows that a smooth input force signal could be obtained by the AFSTMCO.

Figure 7.
The circular tracking error of the second experiment. (a) X-axis and (b) Y-axis. PID: proportional–integral–derivative; SMC: sliding mode control.

Figure 8.
Joint driving force of the second experiment. (a) The first joint, (b) the second joint, (c) the third joint, and (d) the fourth joint. PID: proportional–integral–derivative; SMC: sliding mode control.

The errors between simulation results were computed for all experiments. The cost function for the tracking position error is defined as

$E = \frac{1}{N} \sum_{i = 1}^{4} (| e_{x} (n) | + | e_{y} (n) | + | e_{z} (n) | + | e_{α} (n) |)$
70

where e_x (n), e_y (n), and e_z (n) are the displacement errors along X-, Y- and Z-axis of the nth sample obtained from the simulation results, restrictively. e_α (n) is the rotational error around the Z-axis. N is the number of the sample.

Table 4 gives the cost function values obtained from the two experiments. We can observe that the cost function for the tracking position error of the proposed controller is even smaller than that of SMC, as well the high frequency reference trajectory. As a result, the proposed control method provides a superior tracking performance.

Table 4.
The cost function values for the two experiments.

Experiment E

PID SMC AFSTMCO

1-EXP 0.3275 0.7709 0.2770

2-EXP 0.2993 0.8271 0.4130

AFSTMCO: adaptive fuzzy terminal sliding mode controller with nonlinear observer; PID: proportional–integral–derivative; SMC: sliding mode control.

Conclusion

This article presented an improved 2-(PUU)²R hybrid mechanism, whose actuators have orthogonally arranged in the normal plane component, which renders the motions of the plane more easier to perform. Based on the kinematic relation, the 6 × 6 Jacobian matrix and 6 × 6 × 6 Hessian matrix are derived. The inverse/forward velocity and acceleration of the hybrid mechanism are solved. Moreover, the dynamic characteristics are determined so that numerous control algorithms applied for the serial mechanisms can be easily extended for the hybrid mechanism.

The AFSTMCO for the hybrid mechanism to achieve a precise trajectory tracking is designed. The simulation results show that the proposed control method provides a superior tracking performance in the presence of model uncertainties, varying payloads, and external disturbances. Compared to PID and SMC, both the maximum control error and steady-state error tolerance of the AFSTMCO have been significantly reduced, and the settling time has been significantly shortened. In addition, the proposed control method can be used for pick-and-place robots to track the high-frequency reference trajectory.

In the future work, the design and validation methodology proposed in this article could extend to other types of hybrid mechanisms as well.

Parameter	Description	Values	Unit
$l_{i}$	Length of the arm	500	mm
h	Distance from the center of an R joints to point p	95	mm
H	Distance between the center of down joint on arm and the center of joint on slider	−135	mm
s	The X-component of the length from the upper joint of an arm to the adjacent R joint on the moving platform	63.63	mm
μ	The Y-component of the length from the upper joint of an arm to the adjacent R joint on the moving platform	196.15	mm

Parameter	Description	Values	Unit
m ₁	Mass of sliders	2.85	kg
m ₂	Mass of arm	1.43	kg
m ₃	Mass of lateral bar	3.95	kg
m ₄	Mass of moving platform	2.34	kg
${}^{B}I _{p}$	Inertia moment of moving platform	diag{0, 0, 1285}	kg.mm²
$^{B} I_{l i}$	Inertia moment of arm	diag{38422, 38377, 116}	kg.mm²

Controllers	Values
The proposed control	$\overset{⌢}{M} = 0.8 M$ , $\overset{⌢}{C} = 0.8 C$ , $\overset{⌢}{G} = 0.8 G$ , $K_{1} = diag {500, 500, 500, 500, 500, 500}$ , $K = diag {100, 100, 100, 100, 100, 100}$ , $K_{2} = diag {200, 200, 200, 200, 200, 200}$ , $Γ = diag {10, 10, 10, 10, 10, 10}$ , $A = diag {400, 400, 400, 400, 400, 400}$ , $R = diag {100, 100, 100, 100, 100, 100}$ , $Λ = diag {10, 10, 10, 10, 10, 10}$
SMC	$\overset{⌢}{M} = 0.8 M$ , $\overset{⌢}{C} = 0.8 C$ , $\overset{⌢}{G} = 0.8 G$ , $Λ = diag {10, 100, 10, 10, 10, 10}$ , $P = diag {100, 100, 100, 100, 100, 100}$ , $A = diag {400, 400, 400, 400, 400, 400}$
PID	$K_{P} = diag {1500, 1500, 1500, 1500, 1500, 1500}$ , $K_{D} = diag {800, 800, 800, 800, 800, 800}$ , $K_{I} = diag {200, 200, 200, 200, 200, 200}$

Experiment	E
1-EXP	0.3275	0.7709	0.2770
2-EXP	0.2993	0.8271	0.4130

Footnotes

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 or the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 61473112).

ORCID iD

Xing Zhang

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Experiment	E
Experiment	PID	SMC	AFSTMCO
1-EXP	0.3275	0.7709	0.2770
2-EXP	0.2993	0.8271	0.4130