Author: kodlab

This paper reviews certain theoretical results in robot task planning and control obtained with the support of NSF Research Initiation Grant DMC-8505160. The “natural control” paradigm is reviewed and detailed attention is focused upon the specific task of robot navigation in a cluttered environment. The paper concludes with some comments concerning the problems which remain before this approach to robot command and control can be made practicable in a real manufacturing environment. A summary of formal results presented in the paper now follows. The class of “navigation functions” simultaneously encodes the task of navigating amidst obstacles and automatically generates correct feedback control laws for this purpose as well. We show that the topology of navigation tasks precludes any stronger class of controllers (with respect to convergence properties). On the other hand we demonstrate that a member of this class must exist for any environment. Finally, we construct controllers for an increasingly realistic catalogue of environments by recourse to “deformation” of the true environment into a computationally simpler but topologically equivalent model.

Examination of total energy shows that the global limit behavior of a dissipative mechanical system is essentially equivalent to that of its constituent gradient vector field. The class of “navigation functions” is introduced and shown to result in “almost global” asymptotic stability for closed loop mechanical control systems upon which a navigation function has been imposed as an artifical potential energy. Two examples from the engineering literature – satellite attitude tracking and robot obstacle avoidance – are provided to demonstrate the utility of these observations.

For more information: Kod*Lab

There mingle in the contemporary field of robotics a great many disparate currents of thought from a large variety of disciplines. Nevertheless, a largely unspoken understanding seems to prevail in the field to the effect that certain topics are conceptually distinct. In general, methods of task planning are held to be unrelated to methods of control. The former belong to the realm of geometry and logic whereas the latter inhabit the earthier domain of engineering analysis; geometry is usually associated with off-line computation whereas everyone knows that control must be accomplished in real-time; the one is a “high level” activity whereas the other is at a “low level”. This article concerns one circle of ideas that, in contrast, intrinsically binds action and intention together in the description of the robot’s task. From the perspective of task planning, this point of view seeks to represent abstract goals via a geometric formalism which is guaranteed to furnish a correct control law as well. From the perspective of control theory, the methodology substitutes reference dynamical systems for reference trajectories. From the point of view of computation, less is required off-line, while more is demanded of the real-time controller. From the historical perspective, these techniques represent the effort of engineers to avail themselves of natural physical phenomena in the synthesis of unnatural machines.

For more information: Kod*Lab

The introduction of “error coordinates” and a “tracking potential” on the rotations affords a global nonlinear version of inverse dynamics for attitude tracking. The resulting algorithm produces “almost global” asymptotically exact tracking: this convergence behavior is as strong as the topology of the phase space can allow. A new family of strict global Lyapunov functions for mechanical systems is applied to achieve an adaptive version of the inverse dynamics algorithm in the case that the inertial parameters of the rigid body are not known a priori. The resulting closed loop adaptive system is shown to be stable, and the rigid body phase errors are shown to converge to the limit trajectories of the non-adaptive algorithm.

This article describes a modular computational engine that has become the workhorse for almost all real-time motion control tasks that we encounter in the Yale Robotics Laboratory. Roughly speaking, these tasks amount to the marshalling of various data – from external sensors; from user specified commands; from motor joint variables – and rapid computation involving their appropriate combinations all in time to produce motor commands that will ensure high performance coordinated movement of many coupled mechanical degrees of freedom…

We investigate the computational needs of advanced real-time robot control. First, sampling rate issues in the control of nonlinear systems are discussed. Second, a representative nonlinear robot control algorithm using an explicit robot dynamical model is derived. Some typical terms of the exact equations are given for two industrial robot arms. Third, we define some performance criteria of interest in realtime control. Finally, we compare a variety of implementations of the above control algorithm on a network of INMOS Transputers.

A navigation function is a scalar valued function on a robot configuration space which encodes the task of moving to a desired destination without hitting any obstacles. Our program of research concerns the construction of navigation functions on a family of configuration spaces whose “geometric expressiveness” is rich enough for navigation amidst real world obstacles. A sphere world is a compact connected subset of Eⁿ whose boundary is the finite union of disjoint (n-1)-spheres. In previous work we have constructed navigation functions for every sphere world. In this paper we embark upon the task of extending the construction of navigation function to “star worlds.” A star world is a compact connected subset of Eⁿ obtained by removing from a compact star shaped set a finite number of smaller disjoint open star shaped sets.This paper introduces a family of transformations from any star world into a suitable sphere world model, and demonstrates that these transformations are actually analytic diffeomorphisms. Since the defining properties of navigation functions are invariant under diffeomorphism, this construction, in composition with the previously developed navigation function on the corresponding model sphere world, immediately induces a navigation function on the star world.

For more information: Kod*Lab

We develop a formalism for describing and analyzing a very simple representative of a class of robotic tasks which require “dynamical dexterity,” among them the task of juggling. We introduce and report on our preliminary empirical experience with a new class of control algorithms for this task domain that we call “mirror algorithms.”

The introduction of “error coordinates” and a “tracking potential” on the rotations affords a global nonlinear version of inverse dynamics for attitude tracking. The resulting algorithm produces “almost global” asymptotically exact tracking: this convergence behavior is as strong as the topology of the phase space can allow. A new family of strict global Lyapunov functions for mechanical systems is applied to achieve an adaptive version of the inverse dynamics algorithm in the case that the inertial parameters of the rigid body are not known à priori. The resulting closed loop adaptive system is shown to be stable, and the rigid body phase errors are shown to converge to the limit trajectories of the non-adaptive algorithm.

For more information: Kod*Lab

A novel Lyapunov function is introduced. The author treats the PD compensated mechanical system, which is defined on a configuration space which admits a trivial tangent bundle. Previous results are derived in a simpler form, namely, that such systems are exponentially stable, hence BIBO stable, and steady-state ouptut magnitudes are proportional to input magnitudes with a constant of proportionality depending in a simple fashion on the PD gain magnitudes.

For more information: Kod*Lab