Trajectory control of a two DOF rigid–flexible space robot by a virtual space vehicle

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Abstract

Model based control schemes use inverse dynamics of the robot arm to produce the main torque component necessary for trajectory tracking. For a model-based controller one is required to know the model parameters accurately. This is a very difficult job especially if the manipulator is flexible. This paper presents a control scheme for trajectory control of the tip of a two arm rigid–flexible space robot, with the help of a virtual space vehicle. The flexible link is modeled as an Euler–Bernoulli beam. The developed controller uses the inertial parameters of the base of the space robot only. Bond graph modeling is used to model the dynamics of the system and to devise the control strategy. The efficacy of the controller is shown through simulated and animation results.

Highlights

► A two arm rigid–flexible space robot is considered. ► Flexible link is modeled as an Euler–Bernoulli beam. ► Bond graph is used as a modeling tool and for controller design. ► A virtual space vehicle and overwhelming controller have been designed. ► Controller uses the inertial parameters of space vehicle.

Introduction

The flexible space robot will be useful for space application due to their light weight, less power requirement, ease of maneuverability and ease of transportation. Because of the light weight, they can be operated at high speed. For flexible manipulators flexibility of the manipulator have a considerable influence on its dynamic behaviors. It is observed by many researchers that fixed-gain linear controllers alone do not provide adequate dynamic performance at high speeds for multi-degrees-of-freedom robot manipulators. Out of various schemes studied so far, those involving the calculation of the actuator torque (force) using an inverse dynamics model (the computed torque methods), and those applying adaptive control techniques have been extensively studied, and show the greatest promise. In literature inverse dynamics models based on the trajectory pattern method have been reported [1]. Model based control schemes generally use the inverse dynamics of the robot arm to produce the main torque component necessary for trajectory tracking [2]. Thus, the nonlinearities are effectively compensated, and a linear controller, usually PD or PID, is used to provide any additional corrective torque needed for tracking. The inverse dynamics based control schemes have proven to be very effective in reducing tracking errors in robot manipulators [3]. The real time implementation of the model based control schemes has been possible in a powerful computational environment, especially when lightweight manipulators and high speed motions are involved. In order to reduce the computational delays associated with the model based control schemes, researchers have focused firstly on the development of methods allowing efficient and high speed use of the inverse dynamics model [2], and secondly on customization of the dynamics equations [4]. The customized nonlinear dynamic models are specific to a given manipulator; therefore the controllers using such models require significant modification when it is used with different robots.

The model based computed torque method, also known as nonlinear feedback control is based upon exact knowledge of the dynamic model of a robot manipulator, and therefore it may not be robust enough in the presence of modeling/parameter errors and external disturbances [5], [6]. Zhu et al. [7] used an additional model-based parallel-compensator with a conventional model-based computed torque controller which is in the form of a serial compensator to enhance the robustness of robot manipulator control. Reyes and Kelly [8] described the experimental comparison among four model-based control algorithms on a direct-drive robotic arm. Fawaz et al. [9] presented a model based real-time virtual simulator of an industrial robot in order to detect the eventual external collision. The implemented method concerns a model based Fault Detection and Isolation. It is used to determine any lack of motion from an actuated robot joint after contact with static obstacles.

Researchers [10] have reported the inverse dynamics model-based control for flexible-link robots, based on model analysis. It is based on the assumption that the deformation of the flexible-link can be written as a finite series expansion containing the elementary vibration modes [10]. However, this inverse dynamics model-based control may result in unsatisfactory performance when an accurate model is unavailable, due to parameters uncertainty or truncation of high order vibration modes. Rigatos [11] presented a comparative study on representative methods for model-based and model-free control of flexible-link robots. Sueur and Dauphin-Tanguy [12] have elaborated the bond-graph modeling of flexible robots. They proposed a solution for improvement of the slow subsystem. Tso et al. [13] worked on a model-based control scheme for robot manipulators employing a variable structure control law in which the actuator dynamics is taken into consideration. Ho-Hoon Lee [14] developed a new trajectory control of a flexible link robot based on a distributed parameter dynamic model. Majda et al. [15] have provided a good comparison of different formulations of 2D beam elements based on the Bond Graph technique. Masoudi and Mahzoon [16] studied a free-floating space robot with four linkages, two flexible arms and a rigid end-effector that are mounted on a rigid spacecraft. Touati et al. [17] developed a procedure of fault detection and isolation by considering the presence of both parameter and measurement uncertainties for all linear time invariants and some classes of nonlinear systems using the bond graph approach. Borutzky and Dauphin-Tanguy [18] studied the incremental bond graph process as a starting point for setting up symbolically the canonical form as well as the standard interconnection form of state equations used for robustness.

The work is motivated with the aim of designing a controller for a rigid–flexible space vehicle, where the controller does not requires the flexible or rigid link parameters information. This paper presents the control of a rigid–flexible space robot with the help of a virtual space vehicle. The controller requires the information of the base of the space robot and tip velocity information of the manipulator. A robust overwhelming controller is used along with a virtual space vehicle to control the tip velocity of the rigid–flexible space manipulator. To illustrate the methodology, an example of a two DOF rigid–flexible space robot is considered. Bond graphs are used to represent both rigid and flexible body dynamics of the link in a unified manner.

Section snippets

The two arm rigid–flexible space robot

The modeling of the space robot involves the modeling for linear and rotational dynamics of the links and the base of the space robot. For modeling it is assumed that the space robot system has a single manipulator with revolute joints and is in an open kinematic chain configuration.

Fig. 1 shows the schematic sketch of a two arm rigid–flexible space robot. In this figure {A} represents the absolute frame, {V} represents the vehicle frame, {0} frame is located in the space vehicle at the base of

Bond graph modeling

The word bond graph of the two arm rigid–flexible space robot is shown in Fig. 2. The kinematic analysis of flexible and rigid links is performed in order to draw the bond graph as shown in Fig. 3(a). From the kinematic relations the different transformer moduli used in the bond graph are derived. The flexible link of the space robot has been modeled as an Euler–Bernoulli beam. In the Euler–Bernoulli beam model lumped inertia of the beam is taken into account though rotary inertia and shear

Controller design using virtual space vehicle

In work space control the robot is required to track a given end-effector trajectory, this is achieved by using a model based controller. The bond graph model of a two DOF rigid–flexible planar space robot controller is shown in Fig. 3(b). The model is represented in the inertial frame. In the controller design the flexible link is considered as rigid. The controller consists of three parts: (i) the space robot base model, (ii) the overwhelming controller [19], and (iii) the Jacobian.

  • (i)

    Space

Simulation and animation results

To validate the control strategy two types of end effector trajectory are considered. The first trajectory is a polynomial of third degree while the second one is a polynomial of fifth degree. Fig. 4 shows the initial configuration of the space robot. It is assumed that initially base rotation, first and second joint angles are 0 rad, 0.1 rad and 0 rad respectively.

Case I

Let us assume that the reference trajectory in the X and Y directions is assumed to be a polynomial of third degree (i.e. a

Conclusions

In this paper a control scheme for tip trajectory control of a flexible space manipulator has been demonstrated with the help of a virtual space vehicle base. The Euler–Bernoulli beam model is used to model the flexible link. The controller consists of (i) the model of the base of the space vehicle, (ii) an overwhelming controller and (iii) a Jacobian evaluation to evaluate joint torques. The advantage of the controller is that it does not require the link dynamics as compared to other

Amit Kumar is an Assistant Professor in the Department of Mathematics, S.M.P. Government Girls PG College Meerut, Uttar Pradesh, India. He obtained his M.Sc (2002) and M.Phil (2005) degrees in mathematics from Chaudhary Charan Singh University, Campus Meerut, India. He completed his Ph.D. in flexible space robots from IIT, Roorkee in 2010. His areas of interest are bond graph modeling and flexible space robots.

References (19)

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Amit Kumar is an Assistant Professor in the Department of Mathematics, S.M.P. Government Girls PG College Meerut, Uttar Pradesh, India. He obtained his M.Sc (2002) and M.Phil (2005) degrees in mathematics from Chaudhary Charan Singh University, Campus Meerut, India. He completed his Ph.D. in flexible space robots from IIT, Roorkee in 2010. His areas of interest are bond graph modeling and flexible space robots.

Pushparaj Mani Pathak is an Associate Professor at the Mechanical & Industrial Engineering Department, Indian Institute of Technology, Roorkee, since 2006. He obtained the B.Tech from NIT Calicut, the M.Tech in solid mechanics and design from IIT Kanpur and the Ph.D. in the area of space robotics from IIT Kharagpur. His research interests are space robotics, walking robots, in vivo robots dynamics and control.

N. Sukavanam is Professor in the Department of Mathematics, IIT Roorkee. He did his M.Sc from the Government Arts College, Salem, University of Madras, India in 1979. He completed his Ph.D. in integral operators from the Indian Institute of Science, Bangalore in 1985. He worked as Scientist B at NSTL, DRDO, Vizag from 1984 to 1986. He was a Research Scientist in the Department of Mathematics, IIT, Bombay from 1987 to 1990. He worked as a lecturer at BITS, Pilani, Rajasthan from 1990 to 1996. He joined the Department of Mathematics as an Assistant Professor at IIT, Roorkee in 1996 and since then he has published over 40 research papers in journals and conferences of repute. His areas of research includes nonlinear analysis (control theory), robotics and control.

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