Trajectory Planning and Coordinated Control

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The problems that will be proposed here have to do with one of the basic problem of applied control theory: the problem of designing trajectories for controlled systems. These problems, as are discussed here, arose in response to the needs of air traffic control, but are more basic than any one application. The most basic of the problems is to be given a controlled system, The measurement y is to be thought of as measuring the position of the system. A set of way-points in the form (yi, ti) that must be attained by the plant is assumed to be given. The problem is to find a control law that will move the system in such a way that y(ti)=yi. For most systems there is no unique way of solving this problem so it is necessary to impose other conditions in order to ensure uniqueness. For the air traffic control problem there are d/dt x=f(x)+ug(x) with initial data x(O)=x-O and a measurement function y=h(x). obvious constraints concerning the dynamics of the aircraft and passenger comfort. One obvious condition that should be met is that the dynamics of x(t) be should be smooth. For passengers abrupt changes in velocity or acceleration are very noticeable and physically reality requires that x(t) be continuous. The additional conditions that are imposed make a great deal of difference on the final behavior of the system as does the nature of the system itself. The goal of this research is to develop a tool that will allow the design of trajectories that meet the needs within the context of general control design. In order to do this we must reexamine the basic problem as posed above and extend it to other more complicated tasks. The basic problem as posed here is intimately related to the construction of numerical splines. To extend this to more general problems in control theory it is necessary to enter the area of hybrid systems. Here the problem is related to the more general mathematical problem of Birkhoff splines and to a set of new problems which we choose to call problems of coordinated control. in Section 2 the basic assumptions which we impose on the system will be set forth and the solutions that have been obtained to the basic problem will be described. Here is outlined an elementary, but important basic procedure for solving this entire set of problems and will justify many of the assumptions which have been made. Here will also be described the contents of the several papers that have been published on this problem and they will be related to other work that is ongoing on the problem of trajectory design. In Section 3 there will be presented a set of new problems that will be related to classical problems of Birkhoff splines. These problems arise in aircraft control when it is desired to be able to set commanded velocities, accelerations, angular velocities etc. So in fact we have a set of measurements yk(t) and at various points in time some, but not all, of these must have prescribed values and the control must be designed in order to attain the required values. Here the problem is more basic than uniqueness and is related to the most basic of problems-existence. In Section 4 will be discussed what is really new ground. In many complicated systems the movement of the system is governed by subsystems that effect particular avenues of control. The basic example of this research is the muscular control of the human body. For the most part muscles occur in opposing pairs and only one of a pair can be active at any given time. In order to execute a movement a muscle contracts and causes one particular movement. To execute a complicated movement muscles are switched on and off in accordance with some higher control law and the body makes a smooth coordinated movement. One only need watch a professional gymnast to see the superb coordination that must be maintained for the body to move smoothly and quickly through a complicated maneuver as, for example, one would see on the pommel horse. There are no current plans to develop a control strategy for such a complicated maneuver, but control strategies will be developed that will perform other simpler tasks. Finally in Section 5 it is shown that there are basic problems in stability that must be considered with each of these problems. The problem of stability for systems which are being switched is relatively new. Here is discussed the current state of research and described a series of problems that must be solved to insure that the controls schemes that are being proposed are implementable.