The Pendubot: A Mechatronic Systemfor Control Research and Education

Mark W. Spong Daniel J. Block

Coordinated Science LaboratoryUniversity of Illinois at Urbana{Champaign

m-spong@uiuc.edu

Abstract1

In this paper we describe the Pendubot, a mechatronicdevice for use in control engineering education and forresearch in nonlinear control and robotics. This de-vice is a two{link planar robot with an actuator at theshoulder but no actuator at the elbow. With this sys-tem, a number of fundamental concepts in nonlineardynamics and control theory may be illustrated. Thependubot complements previous mechatronic systems,such as the Acrobot [3] and the inverted pendulum ofFuruta [4].

1 Introduction

In this paper we discuss the design, and control ofthe Pendubot, a two{link, underactuated robotic mech-anism that we are using for research in nonlinear controland to educate students in various concepts in nonlin-ear dynamics, robotics, and control system design. Thenovelty of our system lies in the ease with which we areable to demonstrate advanced concepts such as partialfeedback linearization and zero dynamics, both as a ve-hicle for research, and as an instructional device.

2 Description of the Hardware

Figure 1 shows a drawing of the Pendubot.

Figure 1: a. Front and Side Perspective Drawings ofThe Pendubot

1Research partially supported by the National Science Foun-

dation under grants CMS-9402229 and IRI-9216428

The Pendubot consists of two rigid aluminum linksof lengths 14in and 8in, respectively. Link 1 is directlycoupled to the shaft of a 90V permanent magnet DCmotor mounted to the end of a table. The motor mountand bearings are then the support for the entire system.Link 1 also includes the bearing housing for joint two.Needle roller bearings riding on a ground shaft wereused to construct the revolute joint for link 2. Theshaft extends out both directions of the housing allow-ing coupling to the second link and to an optical encodermounted on link one. The design gives both links full360o of rotational motion. Link 2 is constructed of a1

4{inch (0.635 cm) thick length of aluminumwith a cou-

pling that attaches to the shaft of joint two.All of our control computations are performed on a

Dell 486DX2/50 PC workstation with a D/A card andan encoder interface card. Using the standard softwarelibrary routines supplied with these interface cards weare able to program control algorithms directly in C.

3 Dynamics

Since our device is a two link robot (with only oneactuator) its dynamic equations can be found in numer-ous robotics textbooks as

d11�q1 + d12�q2 + h1 + �1 = � (1)

d21�q1 + d22�q2 + h2 + �2 = 0 (2)

where q1, q2 are the joint angles and � is the inputtorque. The important distinction then between thesystem (1){(2) and a standard two{link robot is, ofcourse, the absence of a control input torque to the sec-ond equation (2). Underactuated mechanical systemsgenerally have equilibria which depend on both theirkinematic and dynamic parameters. If the Pendubotis mounted so that the joint axes are perpendicular togravity, then there will be a continuum of equilibriumcon�gurations, each corresponding to a constant value,�� , of the input torque � . These equilibria are charac-terized by the second link vertical for any position ofthe �rst link.

4 Identi�cation

The dynamic parameters of the Pendubot were iden-ti�ed two ways: using the AutoCAD drawings, and on{line using the Hamiltonian based approach of Gautierand Khalil [2]. Because of the low friction in our device

the Hamilton based approach to identi�cation worksremarkably well.

4 The Pendubot as an Inverted Pendu-lum

In this section we discuss the application of the Pen-dubot as an inverted pendulum in which the motionof the actuated �rst link is used to balance the secondlink. This system is distinct from the more classicalcart{pole system in which the linear motion of the cartis used to balance a pendulum. Our system may alsobe contrasted with the recent and very elegant varia-tion due to Furuta [4] which mounts the pendulum atthe end of a horizontally rotating �rst link. One maythink of Furuta's device as a two link underactuatedrobot arm with the two joint axes orthogonal, whileour Pendubot has the joint axes parallel.

4.1 Controllability and Balancing

The balancing problem for the Pendubot may besolved by linearizing the equations of motion aboutan operating point and designing a linear state feed-back controller, very similar to the classical cart{poleproblem, so we will only give a few remarks here. Onevery interesting distinction of the Pendubot over boththe classical cart{pole system and Furuta's system isthe continuum of balancing positions. This feature ofthe Pendubot is pedagogically useful in several ways,to show students how the Taylor series linearization isoperating point dependent and for teaching controllerswitching and gain scheduling. Students can also eas-ily understand physically how the linearized system be-comes uncontrollable at q1 = 0;��. As the Pendubotapproaches this uncontrollable con�guration, the con-trollabilitymatrix becomes increasingly ill{conditioned.

4.2 Swing Up Control

The problem of swinging the Pendubot up from thedownward con�guration to the inverted con�guration isan interesting and challenging nonlinear control prob-lem. With this problem one may illustrate the non-linear control ideas of nonlinear relative degree, partialfeedback linearization and zero dynamics.We have used the method of partial feedback lin-

earization to swing up the Pendubot, i.e., to move fromthe stable equilibrium q1 = ��=2, q2 = 0, to the in-verted position q1 = �=2, q2 = 0. The same approachcan easily be modi�ed to swing the system to any pointon its equilibrium manifold. The control is switched toa second control to balance the Pendubot about theequilibrium whenever the swing up controller moves itinto the basin of attraction of the balancing controller.The actual control design and analysis is similar to ourprevious work with the Acrobot [3].The experimental results have been very good. As

a balancing controller we have used linear quadratic

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t (Seconds)

Po

sitio

n

q1

q2

Figure 2: Swing-Up and Balance Control at the TopPosition

methods. Figure 2 shows the Pendubot swinging andbalancing at its top position using a partial feedbacklinearization swingup strategy.

5 Conclusions

This paper presents our concept of a two link under-actuated planar revolute robot, named the Pendubot.This system is useful both in research and for instruc-tion in controls. Students at all levels may bene�t fromthis system. Various aspects of control theory from lo-cal linear state feedback to balance the Pendubot in oneposition to more complex global nonlinear controllersto swing up the Pendubot are easily illustrated. Thereader is referred to [1] for additional details. In ad-dition the full version of this paper is available uponrequest.

References

[1] Block, D.J., \Mechanical Design & Control of thePendubot," SAE Earthmoving Industry Confer-

ence, Peoria, IL, April 4-5, 1995.

[2] Gautier, M., and Khalil, W., \On the Identi�ca-tion of the Inertial Parameters of Robots," Pro-

ceedings of 27th IEEE CDC, pages 2264-2269,1988.

[3] Spong, M.W., \Swing Up Control of the Acrobot,"IEEE Control Systems Magazine, Feb. 1995.

[4] Furuta, K., and Yamakita, M., \Swing up Controlof Inverted Pendulum," In IECON'91, pp. 2193-2198, 1991.