Brief paperCartesian control of robots without dynamic model and observer design☆
Introduction
Robot manipulators are high nonlinear systems. To achieve accurate tracking control, many schemes have been proposed in the last decades. Among the main problems which have to be solved, two of the most important are the lack of velocity measurements and of an exact robot model. One of the earliest schemes designed to work only with joint measurements is the one by Nicosia and Tomei (1990). In that paper, a nonlinear observer is presented which can be used to achieve tracking control. On the other hand, to handle an inaccurate model, some adaptive and robust schemes have been proposed (Slotine and Li, 1987, Spong, 1992). When using a robust technique, an extra term is introduced in the control law to achieve ultimately boundedness of the trajectory errors. An adaptive algorithm may be much more complex and is aimed at estimating on line the unknown parameters to get exact tracking. More recently, Parra-Vega, Arimoto, Liu, Hirzinger, and Akella (2003) proposed a sliding PID control, which is able to achieve an exact tracking without any knowledge of the model for implementation.
Of course, it is not enough to solve these problems separately. Thus, some robust control schemes have been proposed, which need only position measurements for control and do not require an accurate robot model. In Canudas de Wit and Fixot (1991), a sliding observer is developed. In Berghuis and Nijmeijer (1994), a quite simple control-observer scheme is proposed which notably does not need any knowledge of the robot model parameters nor of its structure to achieve uniform ultimate boundedness of the tracking and observation errors. Also, a robust estimator is presented in Qu, Dawson, Dorsey, and Duffie (1995) which achieves uniform ultimate boundedness as well. In Arteaga-Pérez and Kelly (2004) a quite simple linear observer is proposed together with a robust control scheme. The experimental results suggest that numerical differentiation may decrease the system performance, so that a (digitalized) observer should be used whenever possible. There are less adaptive laws in conjunction with observers. A result is given in Arteaga-Pérez (2003), although only uniform ultimate boundedness is achieved.
All the algorithms for rigid robots mentioned before are given in joint coordinates. However, since the task to be accomplished is usually expressed in Cartesian coordinates, inverse kinematics has to be computed for implementation. Alternatively, one can develop the necessary theory directly in workspace coordinates (Murray, Li, & Sastry, 1994). Since the new model owns basically the same properties than the original one, control schemes only have to be rewritten in task–space form to be used. For instance, in Caccavale, Natale, and Villani (1999) and Xian, de Queiroz, Dawson, and Walker (2004), control approaches designed in task–space coordinates and which do not require velocity measurements are proposed. Both schemes use the unit quaternion to deal with the orientation tracking. However, they also have the disadvantage of using the robot model both for the controller and the (nonlinear) observer. On the other hand, in Takegaki and Arimoto (1981), a PD control approach is presented which does not require the robot model nor inverse kinematics. However, it only guarantees position control and velocity measurements are needed.
In this paper, a control scheme based on that given in Parra-Vega et al. (2003) is presented. Thus, no robot model is needed for implementation, while the design is carried out directly in task–space coordinates. Furthermore, a linear observer is employed to avoid velocity measurements. While the control-observer scheme is quite simple, exact tracking control is achieved. The orientation problem is dealt using the analytical Jacobian instead of the geometric one. Experimental results are given to support the theory.
The paper is organized as follows. The robot model is given in Section 2. The tracking controller with linear observer scheme is proposed in Section 3. Section 4 presents experimental results. The paper concludes in Section 5.
Section snippets
Robot model and some properties
The dynamics of a rigid robot arm with revolute joints can adequately be described using the Euler–Lagrange equations of motion (Sciavicco & Siciliano, 2000), resulting inwhere is the vector of generalized joint coordinates, is the symmetric positive definite inertia matrix, is the vector of Coriolis and centrifugal torques, is the vector of gravitational torques, is the positive semidefinite diagonal matrix accounting for
Control-observer design
Consider the following well-known relationshipwhere is the analytical Jacobian, is the end-effector position and is a minimal representation of the orientation of the end-effector. Usually, and . Whenever the robot is not in a singularity, one has also the following relationship: Assumption 1 The robot does not reach any singularity.
Experimental results
In this section, the theory of Section 3 is tested. The test bed consists of the rigid robot A465 of CRS Robotics (Fig. 2). Although this manipulator has six degrees of freedom, for simplicity, we use only joints , and 5 to have a planar manipulator with three degrees of freedom. Furthermore, we rename them as , and 3, respectively.
Note that, since the robot model is not needed for control implementation, the motor dynamics of the actuators must not be computed. As pointed out in
Conclusions
The tracking control problem for rigid robots without model available nor velocity measurements has been studied in this paper. Furthermore, the proposed scheme is designed in Cartesian coordinates, so that inverse kinematics is not necessary to carry out a given task. The observer computes the not measurable velocities directly in Cartesian coordinates as well. It has been shown that tracking and observation errors tend to zero, under the condition that no singularity is reached. This
Acknowledgements
This work is based on research supported by the DGAPA-UNAM under Grants IN119003 and IN116804 and by the CONACYT.
Marco Antonio Arteaga-Pérez was born in Huejutla de Reyes, Hidalgo, Mexico, on September 9, 1967. He received the B.S. degree in computer engineering and the M.S. degree in electrical engineering from the Universidad Nacional Autónoma de México (UNAM), in 1991, and 1993, respectively. In 1998, he received the Dr.-Ing. degree from the Gerhard-Mercator University, Duisburg, Germany. Since 1998 he is a Professor at the Department of Electrical Engineering of the School of Engineering at the
References (16)
On the properties of a dynamic model of flexible robot manipulators
ASME Journal of Dynamic Systems, Measurement, and Control
(1998)Robot control and parameter estimation with only joint position measurements
Automatica
(2003)- et al.
Robot control without velocity measurements: New theory and experimental results
IEEE Transactions on Robotics and Automation
(2004) - et al.
Robust control of robots via linear estimated state feedback
IEEE Transactions on Automatic Control
(1994) - Caccavale, F., Natale, C., & Villani, L., 1999. Task–space tracking control without velocity measurements. In...
- et al.
Robot control via robust estimated state feedback
IEEE Transactions on Automatic Control
(1991) Nonlinear systems
(2002)- et al.
A mathematical introduction to robotic manipulation
(1994)
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Marco Antonio Arteaga-Pérez was born in Huejutla de Reyes, Hidalgo, Mexico, on September 9, 1967. He received the B.S. degree in computer engineering and the M.S. degree in electrical engineering from the Universidad Nacional Autónoma de México (UNAM), in 1991, and 1993, respectively. In 1998, he received the Dr.-Ing. degree from the Gerhard-Mercator University, Duisburg, Germany. Since 1998 he is a Professor at the Department of Electrical Engineering of the School of Engineering at the National University of Mexico. His main research interests are robotics and control.
Adrián Castillo-Sánchez received the Engineer's Mechanical-Electrician title and the master degree in Electric Engineering with specialty in Control in 2003 and 2004, respectively, at the Faculty of Engineering of the Universidad Nacional Autónoma de México (UNAM). Since 2002, he is professor at the Faculty of Engineering of the UNAM and since 2004 he is also professor at the Universidad Autónoma de la Ciudad de México in Mexico City.
Vicente Parra-Vega carried out doctoral studies in the Tokyo University, Japan, in 1995 under the supervision of Prof. Suguru Arimoto; and a postdoctoral leave in the Institute of Robotics and Mechatronics of the German Aerospace Agency in 2000 under the supervision of Prof. Gerd Hirzinger. He has published more than 100 journal and conference articles in the areas of robot control, haptic interfaces, visual servoing and mechatronics. Since 1995, he has been in the Mechatronics Division of the Research Center for Advanced Studies in Mexico City.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Masaki Yamakita under the direction of Editor Mituhiko Araki.