Elsevier

Automatica

Volume 44, Issue 3, March 2008, Pages 785-791
Automatica

Brief paper
Periodic motions of the Pendubot via virtual holonomic constraints: Theory and experiments

https://doi.org/10.1016/j.automatica.2007.07.011Get rights and content

Abstract

This paper presents a new control strategy for an underactuated two-link robot, called the Pendubot. The goal is to create stable oscillations of the outer link of the Pendubot, which is not directly actuated. We exploit a recently proposed feedback control design strategy, based on motion planning via virtual holonomic constraints. This strategy is shown to be useful for design of regulators for achieving: stable oscillatory motions, a closed-loop-design-based swing-up, and propeller motions. The theoretical results are verified via successful experimental implementation.

Introduction

The Pendubot (see Fig. 1) is an underactuated mechanical system, broadly used for nonlinear control education (Spong & Block, 1995). The typical problem is to design a regulator, which brings the nonactuated link into the vertical open-loop unstable equilibrium. This problem can be solved, as suggested in Block (1996), using switching between two regulators. The first one, which typically is based on an open-loop, a heuristic closed-loop feedback or an energy-based control design technique, allows to move the robot close to the open-loop unstable configuration. The second one is traditionally linear and designed with the help of linearization around the upward equilibrium.

In this paper, we show how to use the virtual holonomic constraints approach, developed in Shiriaev, Perram, and Canudas de Wit (2005), in order to describe a class of stabilizable periodic motions in the system and a way to achieve them by feedback. The oscillations, generated in such a way, are locally orbitally exponentially stable (Khalil, 2002, Section 8.4). Correspondingly, we automatically obtain ‘for free’robustness with respect to sufficiently small disturbances (Khalil, 2002, Sections 9.1, 9.2) and a guaranteed convergence rate. In addition, unmeasured derivatives can be substituted with estimates of a high-gain observer (Atassi & Khalil, 2001; Khalil, 2002, Section 14.5) without loss of exponential stability of the corresponding set. We should notice that the same motion planning strategy have been recently exploited in Grognard and Canudas de Wit (2005), where it has been complemented with an ad hoc feedback design strategy. Another technique, that theoretically could be used for asymptotic stabilization of periodic motions for the Pendubot is an energy-based design of Fantoni, Lozano, and Spong (2000). A known disadvantage of this strategy is unavailability of tuning procedures for the controller parameters allowing to achieve acceptable convergence rates (Shiriaev, Kolesnichenko, & Paramonov, 2002). As an alternative, a completely different approach for generation of almost periodic motions (based on partial feedback linearization (Spong, 1994), modifications of the energy-based strategy mentioned above, and averaging technique), as suggested in Freidovich, Shiriaev, Gomez-Estern, Gordillo, and Aracil (2006), can be employed for the Pendubot as well. This design leads to another class of achievable, typically high frequency, oscillations. Some preliminary results, exploiting yet another design idea has been presented in Orlov, Aguilar, and Acho (2005). The approach, presented in Chemori and Alamir (2005), seems to be also applicable for the Pendubot. Describing function techniques in combination with second-order sliding-mode control is used in Aguilar, Boiko, Fridman, and Iriarte (2006). We should notice that the control torques generated according to Orlov et al. (2005), Chemori and Alamir (2005) and Aguilar et al. (2006) may have some hard-to-implement jumps. Note that one of the motions in the class, achievable via the approach of this paper, is a propeller-like rotation of the second link. It can be used as a methodologically well-founded substitute for the first controller for the classical swing-up strategy.

The theoretical contributions of this paper are as follows. We combine the approach of Shiriaev et al. (2005) and Shiriaev, Perram, Robertsson, and Sandberg (2006) with a high-gain observer design technique (Khalil, 2002, Section 14.5) and propose a simple modification allowing to improve the convergence rate. Both modifications are crucial for an implementation. As another contribution, we explicitly describe the set of trajectories which are feasible for the particular set of parameters related to our hardware set-up. Furthermore, we have theoretically shown a bound on achievable amplitudes of oscillations of the Pendubot. Note that the known result (Shiriaev et al., 2006) guarantees existence of only sufficiently small cycles. The main contribution, however, is reporting a successful experimental verification. It is important to realize that despite unavoidable parametric uncertainty, sampling of the control law, inaccuracy of the obtained numerical solution of a periodic Riccati equation, ignored hidden electrical dynamics, quantization of the measured outputs and other disturbances, a complex nonlinear feedback control law still works. This is an unambiguous proof of robustness.

The rest of the paper is organized as follows. We describe the system dynamics in Section 2, solve the motion planning problem through an appropriate choice of a virtual holonomic constraint in Section 3, design a stabilizing regulator in Section 4, formulate the main result in Section 5, show results of hardware experiments in Section 6, and conclude with some remarks in Section 7.

Section snippets

Pendubot dynamics

Dynamics of a planar two-link manipulating robot with actuated first link can be described (Block, 1996, Spong and Block, 1996) by the system of two differential equations, originating from Lagrangian mechanics(p1+p2+2p3cosθ)ϕ¨+(p2+p3cosθ)θ¨-p3(2θ˙ϕ˙+θ˙2)sinθ+gp4cosϕ+gp5cos(ϕ+θ)=u,(p2+p3cosθ)ϕ¨+p2θ¨+p3ϕ˙2sinθ+gp5cos(ϕ+θ)=0,where ϕ is the absolute angle of the first link, counted counterclockwise from the horizontal direction; θ is the relative angle between the two links; p1,p2,p3,p4,p5 are

Motion planning

A natural approach is to define a coordinate transformation and to plan the desired trajectory that corresponds to the situation when some of these new coordinates are regulated to zero. The main idea of the virtual holonomic constraints approach is the special way to define such a transformation imposing a relation between the position coordinates, ϕ and θ in our case. It is shown in Shiriaev et al. (2005) that proceeding in this way, that is, assuming that ϕ and θ are smooth functions of a

Controller design

A possible feedback control strategy is the one suggested in Shiriaev et al. (2005) and is based on the LQR design for periodic systems (Yakubovich, 1986). Here we are going to follow this procedure. The next step, after the desired trajectory is chosen, is to make the change of variables (ϕ,ϕ˙,θ,θ˙)(y,y˙,θ,θ˙), where y is the deviation from the constraint (5): y=ϕ-ϕ0-k(θ-θ0).After straightforward calculations, proceeding with partial feedback linearization (Spong, 1994) and a control

Main result

The state feedback control law (7), (4), (14) is not applicable in the case when the derivatives (angular velocities of the links) are not available. Therefore, we substitute the derivatives with the outputsϕ˙^=sat10(x^3)andθ˙^=sat15(x^4),where satL(·) is the standard saturation function with unit gain and saturation at the levels ±L, which corresponds to the reasonable highest value of each velocity (cf. Figs. 2 and 3), of the high-gain observer (Khalil, 2002, Section 8.4) ddtx^1x^2x^3x^4=x^3x^

Experimental results

To test the proposed procedure, motions have been planned and feedback controllers have been designed for several different cases; three of them are as follows:

Case 1: Take ϕ0=-π/2,θ0=0,k=0.5 and pick the desired periodic solution of (3) that passes through the point (0.5,0) (cf. Fig. 2).

Case 2: Take ϕ0=-π/2,θ0=0,k=0 and pick the desired solution of (3) that corresponds to a propeller motion and passes through the point (π,0.1) (cf. Fig. 3).

Case 3: Take ϕ0=-π/2,θ0=π,k=-2 and pick the desired

Conclusions

The subject of this paper has been the problem of generation of oscillatory or periodic motions of the not actuated link of the Pendubot via output feedback control. The virtual holonomic constraint technique has been our choice for motion planning, nontrivial due to the lack of full actuation. This technique is based on integrability of reduced dynamics. The controller ensures convergence of the conserved quantity to a desired value. In order to make this strategy implementable, we have

Leonid B. Freidovich received the M.Sc. in Mechanics and Engineering and the Kandidat of Physical and Mathematical Sciences degrees from Saint-Petersburg State Technical University, Russia, and the Ph.D. degree in Mathematics from Michigan State University, East Lansing, in 1996, 1999, and 2005, respectively. He is currently an Assistant Professor in Control Systems at the Department of Applied Physics and Electronics, UmeÅ University, Sweden. His research interests include nonlinear control

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Leonid B. Freidovich received the M.Sc. in Mechanics and Engineering and the Kandidat of Physical and Mathematical Sciences degrees from Saint-Petersburg State Technical University, Russia, and the Ph.D. degree in Mathematics from Michigan State University, East Lansing, in 1996, 1999, and 2005, respectively. He is currently an Assistant Professor in Control Systems at the Department of Applied Physics and Electronics, UmeÅ University, Sweden. His research interests include nonlinear control systems and robotics.

Anders Robertsson received the M.Sc. degree in electrical engineering and the Ph.D. degree in automatic control from the Lund Institute of Technology, Lund University, Sweden, in 1992 and 1999, respectively. He was appointed Docent in 2005. He is currently an Associate Professor at the Department of Automatic Control, LTH, Lund University. His research interests are in nonlinear control systems, robotics, observer-based control, real-time systems and different control issues in telecommunications and computing systems, such as admission and overload control in network nodes and server systems. The work on sensor-data integration and force control of industrial robots in collaboration with ABB Robotics was awarded the EURON Technology Tranfer Award in 2005. He has been guest lecturer at UmeÅ University during 2006–2007 and been a Visiting Professor at UPV, Valencia, Spain in 2007.

Anton S. Shiriaev received M.Sc. and Ph.D. degrees both in Applied Mathematics from St. Petersburg State University, Russia in 1993 and 1997, respectively. In 2003, he became a Professor in Automatic Control at the Department of Applied Physics and Electronics, UmeÅ University, Sweden. Since 2006, he holds a part-time Professor position at the Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway. Professor Shiriaev held visiting research positions at the Lund Institute of Technology, Sweden, the University of Seville, Spain, the Centre National de la Recherche Scientifique, France, the Mittag-Leffler Institute of the Royal Swedish Academy of Sciences, Sweden, the University of Aalborg, Denmark and the Sungkyunkwan University, Suwon, Republic of Korea. His research interests in systems and control theory, as well as in robotics and mechatronics subjects, are related to motion planning, stability and stabilization of nonlinear systems.

Rolf Johansson received the Master-of-Science degree in Technical Physics in 1977, the Bachelor-of-Medicine degree in 1980, the doctorate in control theory in 1983, and the Doctor-of-Medicine degree (M.D.) in 1986, all from Lund University, Lund, Scandinavia. He was appointed Docent in 1985. Since 1986, he has been with the Department of Automatic Control, Lund University, where he is currently Professor of control science. In his scientific work, he has been involved in research in adaptive system theory, mathematical modeling, system identification, robotics and signal processing. Since 1987, he has also participated in research and as a Graduate Advisor at the Faculty of Medicine, Lund University Hospital. He has had the following visiting appointments: Researcher, 1985, Centre National de la Recherche Scientifique (CNRS), Grenoble, France; Visiting Scientist CalTech, CA, June 1997, June 2001; Rice University, Houston, TX, May 1998, May 2001; Supélec, Paris, France, June 1998; University of Illinois at Urbana-Champaign, IL; UC Santa Barbara, CA, August 1999; University of Napoli Fed II, Italy, July 2000; Guest Professor, NTNU, Trondheim, NO, August 2001; Guest Professor at CIDAC, University of Newcastle, Australia, July 2003; Tsinghua University, Beijing, China, November 2003, August 2004; Honorary Visiting Professor, North China University of Science and Technology (NCUST), Taiyuan, Shanxi, China, 2003; Honorary Visiting Professor, Wuhan University of Science and Technology (WUST), Wuhan, Hubei, China, 2004; Russell S. Springer Visiting Professor 2004, UC Berkeley, Berkeley, CA; Universidad de Jaén, Jaén, Spain, July 2005.

Rolf Johansson was awarded the 1995 biomedical engineering prize (the Ebeling Prize) of the Swedish Society of Medicine for distinguished contribution to the study of human balance through application and development of system analysis and robotics. In 1993–2002, he was coordinating director in robotics research with participants from several departments of Lund University. He is an Associate Editor of International Journal of Adaptive Control & Signal Processing since 1999. In 1993, he published the book System Modeling and Identification, Prentice Hall, Englewood Cliffs, NJ (Information and System Sciences Series Ed. T. Kailath).

Preliminary version of this paper has been presented at the 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, July 19–21, 2006, Nagoya, Japan. This paper was recommended for publication in revised form by Associate Editor Antonio Loria under the direction of Editor Hassan Khalil. This work has been partly supported by Kempestiftelserna (the Kempe Foundation) and the Swedish Research Council under the Grant 2005-4182.

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