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.

## Adaptive strategies for the control of natural motion

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.

## The Controllability of Planar Bilinear Systems

## Natural motion for robot arms

This paper describes some initial steps toward the development of more natural control strategies for free motion of robot arms. The standard lumped parameter dynamical model of an open kinematic chain is shown to be stabilizable by linear feedback, after nonlinear gravitational terms have been cancelled. A new control algorithm is proposed and is shown to drive robot joint positions and velocities asymptotically toward arbitrary time-varying reference trajectories.

## Limit Cycles of Planar Quadratic Differential Equations

Since Hilbert posed the problem of systematically counting and locating lhe limit cycle of polynomial systems on the plane in 1900, much efTort has been expended in its investigation. A large body of literature – chiefly by Chinese and Soviet authors – has addressed this question in the context of differential equations whose field is specified by quadratic polynomials, In this paper we consider the class of quadratic differential equations which admit a unique equilibrium state, and establish sufficient conditions, algebraic in system coefficients, for the existence and uniqueness of a limit cycles. The work is based upon insights and techniques developed in an earlier analysis of such systems [1] motivated by questions from mathematical control theory.

## Stabilizability of Second Order Bilinear Systems

## The Stability of Second Order Quadratic Differential Equations

This paper investigates the stability properties of second-order systems, *x ^{.}* =

*ƒ(x)*, where

*ƒ(x)*contains either quadratic terms-system (1)-or linear and quadratic terms-system (2)-in

*x*. The principal contributions are summarized in two theorems which give necessary and sufficient conditions for stability and asymptotic stability in the large of systems (1) and (2), respectively.

## Fixed Structure Automata in a Multi-Teacher Environment

The concept of an automaton operating in a multi-teacher environment is introduced, and several interesting questions that arise in this context are examined. In particular, we concentrate on the consequences of adding a new teacher to an existing n-teacher set as it affects the choice of a switching strategy. The effect of this choice on expediency and speed of convergence is presented for a specific automaton structure.