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To achieve precise trajectory tracking of robotic manipulators in complex environment, the precise dynamic model, parameters identification, nonlinear characteristics, and disturbances are the factors that should be solved. Although parameters identification and adaptive estimate method were proposed for robotic control in many literature studies, the essential factors, such as coupling and friction, are rarely mentioned as it is difficult to build the precise dynamic model of the robotic manipulator. An adaptive backstepping sliding mode control is proposed to solve the precise trajectory tracking under external disturbances with complex environment, and the dynamic response characteristics of a two-link robotic manipulator are described in this paper. First, the Lagrange kinetic method is used to derive the precise dynamic model which includes the nonlinear factor with friction and coupling. Moreover, the dynamic model of two-link robotic manipulator is built. Second, the estimate function for the nonlinear part is selected, and backstepping algorithm is used for analyzing the stabilities of the sliding mode controller by using Lyapunov theory. Furthermore, the convergence of the proposed controller is verified subject to the external disturbance. At last, numerical simulation results are reported to demonstrate the effectiveness of the proposed method.

Nowadays, with the development of modern industrial technology, robotic manipulators are widely used in automobile manufacturing, aerospace, electronic assembly, precision medical operation, and other fields. To obtain steady state accuracy and fast dynamical response, it is necessary for high precision of the trajectory tracking ability of robotic manipulator. Unfortunately, it is difficult to be satisfied on account of the nonlinear characteristics in complex environment like clearance of joints, friction, external disturbance, strong couple, and so on. The first stage of trajectory tracking is to establish the precise mathematical model of the robotic manipulator. However, the nonlinear part of the dynamic model of the robotic manipulator is ignored in many literatures [

Furthermore, the design of intelligence controllers for nonlinear systems affected by disturbances is a topic that has been studied by several authors, and many different approaches have been proposed for this problem [

For the multi-input-multi-output (MIMO) nonlinear system with uncertainties and disturbances, sliding mode PI control with backstepping approach [

By using Lagrange energy function, the precise dynamic model of the robotic manipulator is built, and the nonlinear characteristics and uncertainties are analyzed. Furthermore, dynamic model of a two-link robotic manipulator is derived.

According to the precision dynamic model of the two-link robotic manipulator, an estimate function of the nonlinear and coupling parts is proposed. Backstepping algorithm is used to construct the equivalent control law of sliding mode control through three steps, and the stability of the proposed controller is convergent by using Lyapunov theory.

This paper is organized in the following manner: Section

To make the dynamic model recursive, the Newton–Euler method is used to establish all force balance between the links of the robotic manipulator, and the dynamic equation can be derived. Forward recursion is used for speed and acceleration transfer between all of the links and backward recursion is used for force transfer from the end-effector to each link of the robotic manipulator.

The dynamic parameters which describe the dynamic model are important for the control algorithms, effective simulation results, and accurate trajectory tracking algorithms. Dynamic equation of the robotic manipulator with

However, quadratic velocity terms and dynamic coupling terms are not taken into account. So, the problem of model accuracy cannot be solved essentially only through parameter identification and compensation methods. The kinematic description of the

Kinematic description of the

An infinitesimal element

The kinetic energy component

The position vector relationship between

Differentiating equation (

Substituting equation (

Equation (

Translational kinetic energy is expressed as

Implicated motion kinetic energy is expressed as

Rotation kinetic energy, combining with equation (

where

where

Combining equation (

The kinetic energy component of the motor of

With the law of angular velocity synthesis, the total angular velocity is derived as follows:

The linear and angular velocities of the rotor center of mass can be expressed as

If the rotor turns around its center, then,

So, the kinetic energy component

Summing different components of a single link in (

The potential energy of the

So, the total potential energy

According to (

The dynamic equation is derived by using Lagrange function, yielding

The dynamic equation of robotic manipulator with

Comparing (

Considering the viscous friction and Coulomb friction, equation (

The dynamic mathematical model for a rigid planar robotic manipulator having two links and a contact surface with the external force acting on the surface is shown in Figure

Two-link robotic manipulator plant with contact force at tip.

The Lagrange method is used to build the precise dynamic model of a two-link robotic manipulator with their nominal values as listed in Table

Variables description of the two-link robotic manipulator.

Description | Nominal value |
---|---|

Center of mass link 1: | 0.1 |

Length of link 1: | 0.8 |

Length of center of mass link 1: | 0.4 |

Centroid inertia of link 1: ^{2}) | 0.064 |

Center of mass link 2: | 0.1 |

Length of link 2: | 0.4 |

Length of center of mass link 2: | 0.2 |

Centroid inertia of link 2: ^{2}) | 0.016 |

The total kinetic energy of the two-link robotic manipulator is

The total potential energy of the two-link robotic manipulator is

By using (

Meanwhile, the torque of joint 2

Combining (

Equation (

Furthermore, to consider the effect of the contact force acting on the end-effector, the right side of (

In this section, an adaptive backstepping sliding mode controller is presented which achieves precision trajectory tracking property by guaranteeing the robustness and stability of the closed-loop system of robotic manipulator. The uncertainties included in the system are required for compensating the external disturbances and nonlinear dynamics in terms described as (

The inverse dynamic equation can be expressed as

Defining vectors

Assuming that

Differentiating (

By using backstepping algorithm, let

Consider a robotic manipulator with n-DOF with the dynamic in (

Select the Lyapunov function of the first step as

Differentiating (

If

Differentiating (

An adaptive algorithm can be assigned for nonlinear function

The Lyapunov function of the third step is selected as

Differentiating (

So, the adaptive control law is derived as

Considering the boundary of

By giving the proper values of

In this section, a two-link robotic manipulator is utilized to verify the effectiveness of the proposed control strategy. The structural parameters are described in Table

The parameter values used in the adaptive backstepping sliding mode control system are ^{@} software is shown in Figure

Schematic for backstepping algorithm by using Matlab^{@} software.

Schematic of the proposed control strategy.

Figure

Trajectory tracking of two-link robotic manipulator. (a)

Actuated torque in the joint space as control signal. (a)

The trajectory tracking errors of the two joints are shown in Figure

Trajectory tracking errors of the two-link robotic manipulator in joint space.

The trajectory tracking performance in end-effector space (operating space) is shown in Figure

Trajectory tracking performance in the operating space of the end-effector.

Mean square error of the estimation function

To verify the robustness of the proposed control system, an external disturbance is added to the system in 6 s. The trajectory tracking of the two-link robotic manipulator under external disturbances is shown in Figure

Trajectory tracking of the two-link robotic manipulator under external disturbances.

Tracking errors of the two-link robotic manipulator under external disturbances.

In this article, an adaptive backstepping sliding mode control subject to external disturbance is proposed. The dynamic model of the robotic manipulator is built by considering the coupling and nonlinear characteristics, and the estimate function of these nonlinear factors is proposed and used for the equivalent control law of sliding mode control. The control system is designed by the backstepping algorithm, and the stability and robustness of the two-link robotic manipulator are analyzed. Simulation results show that the proposed control system has good tracking performance and strong robustness for the external disturbance.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant no. 51165009) and Innovation School Project of Education Department of Guangdong Province, China (Grant nos. 2017KZDXM060 and 2018KCXTD023).

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