Strict Lyapunov functions are constructed for a class of nonlinear feedback compensated mechanical systems, requiring no à priori information concerning the initial conditions of the closed-looped system. These Lyapunov functions may be used to design a stable adaptive version of the “computed-torque” algorithm for tracking a reference trajectory. A particular Lyapunov function is then generalized to permit an adaptive version of control scheme forced by reference dynamics rather than reference trajectory.
The “mechanical systems” define a large and important class of highly nonlinear dynamical equations which, for example, encompasses all robots. In this report it is shown that a strict Lyapunov Function suggested by the simplest examplar of the class – a one degree of freedom linear time invariant dynamical system – may be generalized over the entire class. The report lists a number of standard but useful consequences of this discovery. The analysis suggests that the input-output properties of the entire class of nonlinear systems share many characteristics in common with those of a second order, phase canonical, linear time invariant differential equation.
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The limits in global navigation capability of potential function based robot control algorithms are explored. Elementary tools of algebraic and differential topology are used to advance arguments suggesting the existence of potential functions over a bounded planar region with arbitrary fixed obstacles possessed of a unique local minimum. A class of such potential functions is constructed for certain cases of a planar disk region with an arbitrary number of smaller disks removed.
The practice of automatic control has its origins in antiquity. It is only recently – within the middle decades of this century – that a body of scientific theory has been developed inform and improve that practice. Control theorists tend to divide their history into two periods. A “classical” period, prior to the sixties witnessed the systematization of feedback techniques based upon frequency domain analysis dominated by applications to electronics and telephony. A “modern” period in the sixties and seventies was characterized by a growing concern with formal analytical techniques pursued within the time domain motivated by the more stringent constraints posed by space applications and the enhanced processing capability of digital technology. The hallmark of control theory has been, by and large, a systematic exploitation of the properties of linear dynamical systems whether in the frequency or time domain. The dynamical behavior of mechanical systems appears to depart dynamically from the familiar linear case. The intent of this article is to show that a systematic application of Lyapunov Theory affords qualitative understanding of certain aspects of the input/output properties of a broad class of nonlinear systems (which includes all robots) analogous to that available for linear time invariant systems. For concreteness, this discussion is limited to robot arms – open kinematic chains with rigid links.
Asymptotically stable linear time invariant systems are capable of tracking arbitrary reference signals with a bounded error proportional to the magnitude of the reference signal (and its derivatives). It is shown that a similar property holds for a general class of nonlinear dynamical systems which includes all robots. As in the linear case, the error bound may be made “arbitrarily” small by increasing the magnitude of the feedback gains which stabilize the system.
A feedback control strategy for the command of robot motion includes some limited automatic planning capabilities. These may be seen as sequential solution algorithms implemented by the robot arm interpreted as a mechanical analog computer. This perspective lends additional insight into the manner in which such control techniques may fail, and motivates a fresh look at requisite sensory capabilities.
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This paper introduces a class of linearizing coordinate transformations for mechanical systems whose moment of inertia matrix has a square root which is a jacobian. The transformations, when they exist, define a local isometry from joint space to euclidean space, hence, may afford further insight into the transient behavior of robot motion. It remains to be seen whether any appreciably large class of robots admit such linearizing isometries.
Earlier results of this author and others demonstrate that a broad range of robotic tasks can be commanded through relatively simple feedback controllers with a guarantee of global asymptotic stability. A weakness of such methods is the requirement that exact values of all dynamical parameters be available, since they are used to cancel the disturbance torques introduced by gravity. An adaptive strategy is reported here which guarantees stability and global boundedness of a natural controller in the absence of à priori information regarding dynamical parameters. The present results, however, are not yet satisfactory since they cannot assure convergence to the correct spatial position.