Trajectory tracking control of a pneumatically actuated 6-DOF Gough–Stewart parallel robot using Backstepping-Sliding Mode controller and geometry-based quasi forward kinematic method

https://doi.org/10.1016/j.rcim.2018.06.001Get rights and content

Highlights

  • The dynamic model of pneumatic system of the 6-DoF HexaTaar robot is extracted.

  • Unknown parameters consisting friction force of pneumatic cylinder are identified.

  • Backstepping-Sliding Mode controller is proposed for control of the HexaTaar robot.

  • A novel approach is proposed for obtaining the position of the end-effector.

  • A closed-loop control is performed for trajectory tracking of the HexaTaar robot.

Abstract

In this paper, the trajectory tracking control of a 6-DoF pneumatically actuated Gough–Stewart parallel robot is investigated. The dynamic model of each link, comprising of a pneumatic actuator and a proportional electrical valve is extracted with the aim of obtaining the corresponding state space representation of the pneumatic system. Unknown parameters of the dynamic model consisting friction force of the cylinder and parameters of the proportional valve are identified by employing genetic algorithm. Position control of the pneumatic actuator is performed based on Back-Stepping Sliding Mode controller according to the dynamic model of the system. As such trajectory tracking control is performed for different trajectories by employing a rotation sensor and calculated position based on joint space and task space simultaneously. Desired sinusoidal trajectories with pure motions are tracked with root mean square error of the pure translations and rotations lower than 0.85 (cm) and 1.9 (deg), respectively. The results reveal that the trajectory is tracked by the Back-Stepping Sliding Mode controller properly. This shows the efficiency of the control strategy and the proposed method for calculating the position of the end-effector.

Introduction

Parallel robots entail many advantages compared to their counterparts, serial robots, such as higher rigidity, higher load-to-weight ratio, higher precision and can reach to a higher speed and acceleration [1]. The Gough–Stewart Platform (GSP) a 6 °-of-freedom (6-DoF) parallel robot has a movable platform, called End-Effector (E-E), connected to a fixed base platform via 6 expandable actuators [2]. The GSP is used in various applications and manufacturing, e.g., flight simulators and aeronautics applications [3], radio telescopes [4], tire test machines [5], multi-dimensional vibration isolation mechanisms [6] and rehabilitation [7], [8].

Trajectory tracking is one of the challenging problems in manufacturing for this kind of parallel robot according to its complicated dynamic and kinematic modeling [9], [10]. Two strategies have been proposed to control a GSP, namely joint space control and task space control. In [11], the joint space control has been developed by measuring desired lengths of the GSP’s actuators, which has been obtained based on solving the inverse kinematic problem of the desired trajectory in task space. Although, the task space control is more precise and it is more complicated than the joint space control. Task space control can be performed based on: (1) Position and rotation of the GSP’s E-E using vision, position and orientation sensors [12], (2) Solving forward kinematic problem (FKP) using the measured lengths of the GSP’s links [13]. The first approach might be very expensive and the latter method deals with computational effort. In [14], a vision-based control has been investigated for control of a 6-DoF parallel robot in which the pose of E-E was measured by a motion-tracking system. In the foregoing study, a H controller is implemented in order to tune the control signal. In [15], a novel method has been proposed for solving nonlinear equations in FKP of a parallel robot based on Homotopy Continuation method. Their method has been implemented in solving FKP of a 3-RPR parallel robot with high precision in results. Moreover, a real-time solution for solving froward kinematic problem of a GSP has been proposed in [16]. In [17], a comparative study has been investigated in differences between the results of conventional and the proposed methods.

Complexity or simplicity of the trajectory tracking problem depends on the type of actuators. There are three types of actuators for driving a GSP which are electric, hydraulic and pneumatic. Using a pneumatic actuator has some drawbacks which mainly originate from compressibility of the air, different regimes of air flow through valves, and dynamic behavior of friction force of a pneumatic cylinder. On the other hand, it has many advantages in comparison with other types of driving systems, such as a cost-effective actuation, clean for the working environment, easy to maintenance, and rapid movement and reactions.

Different control strategies have been developed by researchers in control of a 6-DoF GSP such as LQG controller, PID controller, Robust approaches, Adaptive control strategy, and Sliding-Mode controller. In [18], the Linear Quadratic Gaussian (LQG) controller based on reference tracking has been designed for motion control of a pneumatically-driven 6-DoF GSP. The LQG controller has been designed by combining a Linear Quadratic Estimator (LQE) with a Linear Quadratic Regulator (LQR). In the foregoing study, the authors evaluated the robustness of the control under different external loads in experimental tests. In [19], the PID controller with feedback linearization have been designed for trajectory tracking of a 6-DoF pneumatically GSP. In the foregoing study, the authors preferred using heuristic algorithm to solve the non-convex problems for tuning the controller’s parameters. The inner control loop has been used for controlling the work pressure and the outer loop has been related to displacement of the load under various external forces. Their analysis was performed in the Simulink of the MatLab software. In [20], a pneumatic wearable walking assistance GSP has been developed for ankle-foot rehabilitation system. Double-acting pneumatic actuators have been used and an I-PD controller has been designed to control the platform at a desired position. They proved that overshoot of an I-PD controller is smaller than usual PID controller.

In [21], an adaptive controller has been proposed for the position tracking of a 6-DoF SGP driving simulator in numerical simulation based on a linearized dynamic equations model. The authors have employed two inner and outer control loops and the combined PD controller and adaptive compensation has been applied for the control design. In [22], a robust adaptive controller of a 6-DoF parallel robot has been investigated without requiring inverse dynamic model. Besides, they used a hybrid method based on Sliding Mode and Neural Networks in order to control the robot. In [23], an experimental study on an accurate model-based controller of a 6-DoF parallel robot has been carried out in order to improve the tracking accuracy in both joint and task spaces. Robust control design has been applied for a single actuator developed by a feed-forward dynamic compensator and a feedback-observer controller. In [24], a robust adaptive controller has been proposed based on task space dynamic equations. To this end, a 6-DoF GSP was built in Simulink environment of MatLab software. The authors evaluated performance of the proposed controller for sinusoidal trajectory tracking. Sliding Mode control strategy has the potential to be used for control of robotic systems with large uncertainties [25]. In [26], a simulation-based study has been investigated in position control of a typical SGP based on smooth integral sliding mode controller. The discontinuous controller has been replaced by a twisting algorithm based on combination of two continuous controls. Their method is robust to external matched disturbances of the system. Moreover, vision-based methods have been widely explored in the literature for controlling parallel robots. In [27], a visual-based kinematic model of a GSP has been investigated through the observation of its legs which has been inspired by geometry of lines. They claimed that the proposed method simplified both identification and control of the GSP. In [28], a general concept of visual servoing of parallel robots has been proposed which is based on hidden robot model based on observing the leg directions.

The main contribution of this paper is proposing a closed-loop control strategy for a 6-DoF pneumatically actuated Gough-Stewart platform, namely HexaTaar, with the aim of solving the trajectory tracking problem. The HexaTaar is actuated by 6 pneumatic systems (PneuSys) which each of them consists a proportional directional electrical valve and a pneumatic actuator. The unknown parameters of the PneuSys including mass flow rate through the proportional valve, viscosity coefficient and friction force of the pneumatic actuator are identified based on evolutionary algorithm, the so-called Genetic Algorithm (GA). The PneuSys are considered independent and the effects of other actuators on each one are considered as an external disturbance and uncertainties. Therefore, the identified PneuSys is controlled by employing Backstepping-Sliding Mode controller, separately. In this paper, the trajectory tracking control is performed based on combination of the joint space and task space control, simultaneously. Moreover, a novel approach is proposed, namely Geometry-based Quasi Forward Kinematic (GQFK) method, in order to calculate the position of the HexaTaar’s E-E without using position sensors and vision-based observations. In the GQFK method, position of the E-E along the x, y and z axes are calculated using the angles measured by the IMU sensor and the potentiometers.

The remainder of this paper is organized as follows. The experimental setup of the 6-DoF pneumatically actuated robot and mechanical parts are described in Section 2. Moreover, mathematical model of the PneuSys including cylinder and pressure dynamic with the aim of obtaining the state space representation of the PneuSys are extracted. Section 3 is devoted to identification procedure of the PneuSys’s unknown parameters using GA, and the proposed controller, BS-SMC, is designed for position control of the pneumatic actuator. The proposed closed-loop control strategy including outer and inner control loops for trajectory tracking control of the HexaTaar robot using the nonlinear controller BS-SMC and GQFK method are explained in Section 4. Finally, the paper concludes with some hints and remarks as ongoing works.

Section snippets

Experimental setup of the 6-Dof hexataar parallel robot

In this section, geometry configuration and size of different mechanical parts of the HexaTaar robot are introduced. Moreover, the PneuSys of each link of the robot is explained in details.

Identification of the pneusys and position control of the pneumatic actuator

The obtained dynamic model in Section 2.2.1 has unknown parameters, e.g., friction force and viscose coefficient of the pneumatic actuator, mass flow rate through the proportional valve. In this section, the unknown parameters of the PneuSys is identified according to the extracted dynamic model in Eq. (8). Moreover, position control of the pneumatic actuator is performed by designing a nonlinear controller, namely Backstepping-Sliding Mode controller.

The PneuSys of each link of the robot which

Trajectory tracking control of the hexataar robot based on Back-Stepping Sliding Mode controller and geometry-based quasi-forward kinematic method

In this section, the control strategy is described which employs the task space and joint space control, simultaneously. The proposed closed-loop control method is based on employing BS-SMC and using GQFK method. Furthermore, trajectory tracking control of the HexaTaar robot is performed for different trajectories including pure sinusoidal motions and more complicated sinusoidal motion. Besides, data obtained from experimental results are reported in tables and figures.

Conclusion

This paper addressed the trajectory tracking control of a 6-DOF pneumatically actuated Gough-Stewart parallel robot, namely HexaTaar. To this end, dynamic model of the pneumatic system of each link of the HexaTaar robot were extracted including dynamic equations of the pneumatic cylinders and the proportional electrical valves. The obtained dynamic model includes unknown parameters which were identified using Genetic Algorithm. Moreover, position control of the pneumatic actuator was performed

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