Design of a bond graph model for robot control.

Rezic, Snjezana ; Pehar, Slaven ; Crnokic, Boris 等

1. INTRODUCTION

Many current automation industry solutions focus on the ideas of broadening the market to include applications now covered by both man and machine. " The need for smaller, more precise assemblies will stimulate major changes in assembly technology during the year 2000 and beyond (Woodward, Michael S., 1999).

The control of robotic manipulators is partly limited by the contamination of measurements of link angular velocities. This paper explores how a bond graph model of a robotic manipulator may be used to create a model_based observer to improve the control of the manipulator by using estimated link angular velocities in a conventional proportional and derivative feedback controller.

The use of observers in this area is not new Canudas de Wit and Slotine (1989) used a sliding observer, a class of variable structure nonlinear system to estimate the joint speeds of rigid robots whilst Nicosia et al (1986) used a pseudolinearisation technique involving a dynamic state.

The contribution of the paper is twofold, to show that the generic methods of this study are applicable to a demanding robotics application and to present the results of a detailed experimental study.

2. MODELLING OF THE ROBOT'S STRUCTURE

Robots which are used in the industry must have good performances, e.g. big precision and the speed of response, compact and modular structure, and they must consume minimum of energy. For the robot programming we need to know the relation between its local coordinates and global coordinate system. From the known global coordinates we need to determine position of its kinematic structure-local robot coordinates. The first case is called the forward kinematic and the other one is the inverse kinematic. We start from robot forward kinematic structure in our research course. Bond graphs provide a graphical format for modelling dynamic energy exchanging systems in an unambiguous way which allows the dynamic equations of motion of the system to be derived automatically by computers for example. Further more as bond graphs deal with energy exchange more than one physical domain may be represented in the same bond graph. For instance, a single bond graph may represent the transformation of electrical energy into mechanical energy by a dc motor. This property makes bond graphs particularly suitable for modelling robotic manipulators which are predominantly electro-mechanical devices.

The bond graph modelling of generic rigid planar rotational joint manipulators was outlined by Gawthrop, and Gawthrop and Smithand is summarised here. The bond graph of a generic two_link manipulator is given in Fig 1 .The basis of this bond graph lies in creating the absolute velocities i.e. velocities de_ned in an inertial frame of all the relevant parts of the two link manipulator. Once the absolute velocities of all the relevant parts of the manipulator have been represented by junctions on the bond graph, the dynamics of the system may be incorporated by attaching inertial (I) elements to these junctions. The inertial elements define the constitutive law relevant to that particular junction. For example, for the l:wl junction the I:J1 element defines the law of angular momentum, h\ = j\LO\, where h\ is the angular momentum of the first link and j\ is its moment of inertia around the centre of mass. Junctions representing the absolute angular velocities of the links have now been augmented. The next step is to create junctions representing the absolute cartesian velocities of the link centres of mass. This can be done from algebraic combinations of u>\ and u>2 and the link lengths (see Fig. 2).

[FIGURE 1 OMITTED]

For example, the relative angular velocities of the first and second joints are represented by the vtrl and vtr2 junctions respectively on the bond graph. As the first joint is fixed in space, the absolute angular velocity of the first link wi (wl on the graph) is the same as the relative angular velocity vtrl so.

[FIGURE 2 OMITTED]

Similarly, for the cartesian velocity junction vx2, the LM2 element defines the law of linear momentum, p2 = rri2 [V.sub.x2], where p2 is the linear momentum in the x direction of the second link and ni2 is the mass of the second link.

2.1 Research course

The aim of this researches is development of bond graph model of robot 3D structure. The use of observers in this area is not new Canudas de Wit and Slotine (1989) used a sliding observer, a class of variable structure nonlinear system to estimate the joint speeds of rigid robots whilst Nicosia et al (1986) used a pseudolinearisation technique involving a dynamic state. The bond graph for the two link manipulator represents the forward system relating motor voltage inputs to the outputs of link angular velocities and velocities. By running this model in parallel with the experimental manipulator the outputs of the model known as the observed outputs may be used in the feedback control of the experimental manipulator in preference to the noise contaminated measured outputs. Before this can be done, however, the bond graph must be augmented into an observer format to incorporate feedback around the model to force the states of the observer to track the states of the experimental system.

For the two link manipulator bond graph the states are given by: two inertial elements representing link angular momenta and two compliance elements representing link relative positions.

2.2 Method used

An advantage of the bond graph modelling technique is that, once the generic bond graph for a system exists, it may be easily modified for a range of purposes. For example, the Source and Measurement elements may be re-arranged to give an inverse model or, as in this paper, extra Source elements may be added to create a bond graph observer. From any of these new formats, the mathematical equations describing the dynamics of the system in its new format may be derived automatically for a range of representations (e.g. the set of state space equations, linearised transfer functions, computer code, human readable equations, etc) by a package of Model Transformation Tools (MTT) developed at Glasgow (Gawthrop et al., 1991; Gawthrop 1995). It is this flexibility which makes the creation of the bond graph worthwhile.

2.3 Results

The specific bond graph for the direct drive two-link manipulator may be derived from the generic bond graph through addition of bond graphs representing d.c. motors together with the construction of the cartesian tip velocities at the end of the first link to allow the mass of the second motor to be incorporated into the bond graph. The resultant bond graph is given in Fig. 3.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

3. CONCLUSION

An important aspect of the bond graph observer is that it may be created quickly and easily from the bond graph of the normal system. The software required to implement the observer in practice may then be created automatically from this bond graph. Furthermore, the linearised state-space matrices can be produced for any state-point to allow the observer feedback gain matrix to be designed using standard linear observer theory.

4. REFERENCES

Canudas de Wit, C., Astrom, K. J. & Fixot, N. (1990). Computed torque control via a non-linear observer. International Journal of Adaptive Control and Signal Processing 4(6), 443-452

Gawthrop, P. J. (1991). Bond graphs: A representation for mechatronic systems. Mechatron-ics 1(2), 127-156

Gawthrop, P. J. (1995). Mtt: Model transformation tools. In Proceedings Of International Conference On Bond Graph Modeling And Simulation (ICBGM'95). Society for Computer Simulation. Las Vegas, pp 197-202

Karnopp, D. C. (1979). Bond graphs in control: Physical state variables and observers. J. Franklin Institute 308(3), 221-234

Paul, R.P. (1981). Robot Manipulators: Mathematics, Programming, and Control, Cambridge, Massachusetts, MIT press 1981

***Robotic Industry Trends Report (1998). Robotics International of Society of Manufacturing Engineers