Nonlinear position and stiffness Backstepping controller for a two Degrees of Freedom pneumatic robot

doi:10.1016/j.conengprac.2017.12.007

Control Engineering Practice

Volume 73, April 2018, Pages 26-39

https://doi.org/10.1016/j.conengprac.2017.12.007 Get rights and content

Highlights

•
A Backstepping position and stiffness controller for a 2 DOF pneumatic robot is given.
•
The closed loop stability of the controller is proved.
•
A closed loop stiffness and damping tuning strategy is presented.
•
The performances of the controller are compared with a linear impedance controller.

Abstract

This paper presents an architecture of a 2 Degrees of Freedom pneumatic robot which can be used as a haptic interface. To improve the haptic rendering of this device, a nonlinear position and stiffness controller without force measurement based on a Backstepping synthesis is presented. Thus, the robot can follow a targeted trajectory in Cartesian position with a variable compliant behavior when disturbance forces are applied. An appropriate tuning methodology of the closed-loop stiffness and closed-loop damping of the robot is given to obtain a desired disturbance response. The models, the synthesis and the stability analysis of this controller are described in this paper. Two models are presented in this paper, the first one is an accurate simulation model which describes the mechanical behavior of the robot, the thermodynamics phenomena in the pneumatic actuators, and the servovalves characteristics. The second model is the model used to synthesize the controller. This control model is obtained by simplifying the simulation model to obtain a MIMO strict feedback form. Finally, some simulation and experimental results are given and the controller performances are discussed and compared with a classical linear impedance controller.

Introduction

Many robotic applications require an interaction between the end-effector of the robot and an uncertain environment. For instance, for human rehabilitation, for haptic interfaces, or for prosthetic devices, human–robot interactions are necessary. When these interactions occur, most of the time, a compliant behavior of the robot is required in order to avoid human injuries or to avoid damaging the robot itself. But on the other hand, these robots have to be stiff for some tasks. Therefore it is necessary to control the stiffness and damping of the robots. To ensure a compliant behavior of a robot, various Variable Stiffness Actuators (VSAs) or Variable Impedance Actuators (VIAs) have been developed during last decades. These actuators allow the equilibrium position and the stiffness to be tuned independently. Van Ham, Sugar, Vanderborght, Hollander, and Lefeber (2009) present a state of the art in the design of VSAs. Most of these actuators are designed with two internal motors and passive compliant elements. An advantage of this design is that the position and stiffness control of the VSA is obtained by controlling the position of two electric motors. The main drawbacks of this kind of VSAs are the cost and the stiffness range. Indeed, these actuator are often expensive because two electric actuators are needed to control one Degree Of Freedom (DOF). The range of the stiffness is also often limited (Huang et al., 2013) due to the use of passive stiffness components.

Another approach to obtain a compliant behavior for the robot is based on control strategies such as stiffness control (Salisbury, 1980), impedance control (Hogan, 1987) or hybrid force position control (Hayati, 1986). Most of these strategies have been developed for electromechanically actuated robots. The disadvantages of the electromechanical actuation are that, in order to implement these control strategies, a force/torque sensor is needed. This sensor is required to measure the environment interaction which implies knowing where this interaction will occur. Moreover, these sensors are often expensive and fragile. If force/torque sensors are not used, the actuators have to be backdrivable which mean reducing gear ratio and, consequently, the torque or force range of the robot.

On the other hand, due to their nonlinear behaviors, pneumatic cylinders were traditionally only use as bi-stable position actuators. The recent development of new servovalves and modern robust nonlinear control laws based on sliding mode and Backstepping allowed the development of position or force controller. Thus, since pneumatic cylinders are inexpensive and have a good power to weight ratio, there has been a recent surge of interest for this technology. If the independent force/stiffness or position/stiffness nonlinear controls of one pneumatic actuator have been addressed in literature Abry et al. (2015), Shen and Goldfarb (2007), Taheri et al. (2014), the extension of these nonlinear control strategies to multi DOF has not yet been studied. Thus, this article presents an nonlinear position/stiffness control strategy for a 2 DOF pneumatic robot adapted from the Abry et al. position and stiffness controller developed for a pneumatic cylinder (Abry et al., 2015). The synthesis of this controller is based on the Backstepping method and a gain tuning strategy which allows to reach a desired behavior of stiffness and damping.

The presented 2 DOF pneumatic robot is a part of a haptic interface. This haptic device will be used to develop a childbirth simulator. Herzig, Moreau, and Redarce (2014) and Herzig, Moreau, Redarce, Abry, and Brun (2015) give more details about the interest of using this kind of haptic interface to simulate a childbirth delivery.

This paper is structured as follows: In Section 2 the hardware architecture of the 2 DOF actuated robot is given. Then the models used for simulations and for control synthesis are described respectively in Sections 3 Simulation model, 4 Control model. The controller synthesis based on the Backstepping method is described in Section 5. In Section 6 response to an external disturbance force and a strategy to ensure a desired closed-loop stiffness by control gains tuning are discussed. Simulation results and a comparison with a classical linear impedance controller without force sensor are presented in Section 7. Section 8 deals with the experimental results to compare performances of the two controllers for position tracking and disturbance rejection. Finally, Section 9 provides a conclusion and describes future works.

Section snippets

Robot hardware design

The 2 DOF robot studied in this paper is illustrated in Fig. 1. Its architecture is based on the BirthSIM Herzig et al. (2014), Herzig et al. (2015) design, which is composed of two pneumatic cylinders. The main characteristics of these two cylinders, respectively denoted cylinder 1 and cylinder 2 for the vertical one and the horizontal one, are given in Table 1. The second cylinder has been chosen with a square rod in order to prevent the inner rotation.

Four Festo MPYE-5-M5-010-B proportional

Simulation model

This section presents the models which are used to test the control law in simulation. To describe the behavior of the robot, mechanical and thermodynamic models have to be defined.

Control model

The model described in the previous section is not adapted to apply the Multi-Input Multi-Output (MIMO) Backstepping method. Indeed, the latter is based on a recursive control design Freeman and Kokotovic (1993), Yao and Tomizuka (2001). To apply this method, it is suitable to rewrite the state model in a strict-feedback form. The strict-feedback form MIMO $n$ order system can be described by $\{\begin{matrix} \begin{matrix} {\dot{x}}_{i_{k}} = f_{i_{k}} (x_{1}, \dots, x_{i_{k}}, u_{1}, \dots, u_{k - 1}) + g_{i_{k}} (x_{1}, \dots, x_{i_{k}}, \\ u_{1}, \dots, u_{k - 1}) x_{i_{k} + 1} + δ_{i_{k}, j_{k}} g_{i_{k}, k} (x_{1}, \dots, x_{i_{k}}, \\ u_{1}, \dots, u_{k - 1}) u_{k} \end{matrix} \\ y_{k} = h_{k} (x_{1}, \dots, x_{j_{k}}) \end{matrix}$

Controller synthesis

The model obtained previously is now in a strict feedback form. The Backstepping method can be, therefore, applied to synthesize the control laws. The presented method is based on Abry et al. works (Abry et al., 2015) but has been adapted to the 2 DOF robot presented in Section 2. The four virtual mass flow rates are the control inputs. The two active mass flow rates $q_{m A 1}$ and $q_{m A 2}$ will be designed to track the desired position of the pistons $y_{d 1}$ and $y_{d 2}$ respectively for cylinder 1 and 2. To

Disturbance rejection and closed-loop stiffness

The controller synthesis method has been chosen because Abry et al. have shown that the tuning of some gains allows to control the system disturbance response (Abry et al., 2015). Indeed, it is possible to tune the closed-loop stiffness and damping of each actuator by adapting the control gains. It is important to distinguish the pneumatic stiffness and the closed-loop stiffness. Indeed, the pneumatic stiffness described in (15) is a state of the system. This state represents the actuator

Simulation results

The aim of this section is to compare the performances of the control law defined in Section 5 with a classical linear impedance controller (see Fig. 7). Indeed, two simulations with different objectives are presented in the following subsections. These simulations have been obtained using the simulation model presented in Section 3. The first simulation have been set in order to compare the accuracy of the controllers while tracking desired position and pneumatic stiffness trajectories. The

Experimental results

This section gives some experimental results to compare the two controllers presented in this paper. As in Section 7, this section will be divided into two subsections. The Section 8.1 gives the results for a reference position tracking. The Section 8.2 illustrates the behavior of the system when a disturbance force is applied.

Conclusions and future works

In this study, a two DOF pneumatic robot design and model are proposed. The assumptions and transformations to synthesize a position controller with the Backstepping method are provided. Then a strategy of gain tuning, which leads to a closed-loop stiffness and damping control, is presented. Finally, the performances of this controller are illustrated with some simulation and experimental results. These performances are discussed and compared to a classical linear impedance controller with

Acknowledgments

This work was funded by the ANR French National Research Agency[ANR-12-MONU-0006]. This work was partially supported by the Engineering and Physical Sciences Research Council[EP/N03211X/2 Morphological computation of perception and action].

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Nonlinear position and stiffness Backstepping controller for a two Degrees of Freedom pneumatic robot

Highlights

Abstract

Introduction

Section snippets

Robot hardware design

Simulation model

Control model

Controller synthesis

Disturbance rejection and closed-loop stiffness

Simulation results

Experimental results

Conclusions and future works

Acknowledgments

Automatica

Automatica

Automatica

Electropneumatic cylinder backstepping position controller design with real-time closed-loop stiffness and damping tuning

IEEE Transactions on Control Systems Technology

The analysis and design of pneumatic systems

Modeling, identification, and control of a pneumatically actuated, force controllable robot

IEEE Transactions on Robotics and Automation

Conservative congruence transformation for joint and cartesian stiffness matrices of robotic hands and fingers

International Journal of Robotics Research

Hybrid position/force control of multi-arm cooperating robots