Elsevier

Neurocomputing

Volume 37, Issues 1–4, April 2001, Pages 31-49
Neurocomputing

Polyhedral mixture of linear experts for many-to-one mapping inversion and multiple controllers

https://doi.org/10.1016/S0925-2312(00)00306-4Get rights and content

Abstract

Feed-forward control schemes require an inverse mapping of the controlled system. In adaptive systems this inverse mapping is learned from examples. The biological motor control is very redundant, as are many robotic systems, therefore the mapping is many-to-one and the inverse problem is ill posed. In this paper we present a novel architecture and algorithms for the approximation and inversion of many-to-one functions. The proposed architecture retains all the possible solutions available to the controller in real time. This is done by a modified mixture of experts architecture, where each expert is linear and more than a single expert may be assigned to the same input region. The learning is implemented by the hinging hyperplanes algorithm. The proposed architecture is described and its operation is illustrated for some simple cases. Finally, the virtue of redundancy and its exploitation by multiple controllers are discussed.

Introduction

One of the salient characteristics of the biological motor control system is its apparent redundancy (see e.g. [1], [18]). The human arm consists of seven degrees of freedom, which is more than needed to obtain a particular position or configuration of the hand in the workspace. Most of the joints are surrounded by more muscles than needed to produce any desired moment. The muscles themselves are composed of many motor units that enable many possibilities of producing the same force at the tendon. In the presence of redundancy the controller has to act on a many-to-one (MTO) system and has to choose one of the many possible actions to obtain the same desired target.

Due to the delays in the nervous system, simple feedback cannot offer a proper explanation for the control of fast movement. Thus, it was suggested that the nervous system contains an inverse model of the musculo-skeletal system that is contextually being updated [7]. For reviews of recent modeling with artificial neural networks, see [10], [14]. Two methods for learning this inverse model are distal supervised learning [11] and feedback error learning [16]. These methods do not confront the MTO problem. They choose an arbitrary solution that is the closest to the training set and to the initial conditions of the network. In some cases these architectures may incorporate a smoothness criterion to choose a biologically plausible solution, but they still learn just one solution.

The same MTO problem occurs in the robot inverse kinematics problem for manipulators with excess degrees-of-freedom. This problem can be separated into global and local ill-posedness, and therefore a two-fold solution can be pursued: first global regularization, that is identifying and labeling the solution branches and then local regularization corresponding to parameterization of the solution manifolds. DeMers [4] investigated this problem by describing the topological properties of the systems. He suggested such a two-fold learning method. Lu and Ito [20] tried to solve the inverse kinematics of a redundant arm with a modular neural network, where each network learned part of the configuration space; but as they admit, the regions can overlap.

In this paper a simpler and more tractable and analyzable method is suggested. The main idea is to construct a piecewise invertiable approximation that can be then inverted to produce a multiple controller. We present a novel architecture that divides the input space into polyhedral regions, which are convex regions that can cover the whole space. We call this special architecture polyhedral mixture of linear experts (PMLE) because it can be viewed as a special case of the mixture of experts (ME) architecture proposed by Jacobs and Jordan [8]. The PMLE has the advantage of being a piecewise invertiable function and therefore the multiple inverse PMLE can serve as a multiple controller in order to exploit the virtue of redundancy. We use the hinging hyperplanes (HH) method proposed by Breiman [3] in order to learn the piecewise linear approximation. Then we present a new algorithm to transform the parameters of the HH to the parameters of the PMLE. We further prove the ability of the PMLE to estimate inverse functions. Part of this work was presented as a short conference paper [12] and further details are available in Chapters 5 and 6 of [15].

The reminder of the paper is organized as follows: Section 2 describes the inverse problem and its ill-posedness. In the next two sections the HH algorithm and the PMLE architecture are described. The PMLE is shown to be capable of approximating any inverse function. In Section 5 the transformation and inversion algorithms are outlined and in Section 6, a few examples of its performance are given. Finally, the role of the PMLE in utilizing the virtue of redundancy in biological and artificial systems is briefly discussed and conclusions are drawn.

Section snippets

Learning to invert many-to-one mappings

The problem of finding an inverse mapping is described in Fig. 1. Given a desired output yd, find x such that F(x)=yd. For example, given the desired position of the hand, what should be the neural excitations to the muscles in order to bring the hand to the desired position. In a robotic system, the question is what should be the currents in each motor, in order to bring the manipulator to the desired position. When such a problem is given many questions can be formalized, for example: (I) Is

Hinging hyperplanes

The problem of finding a model for an unknown system by observing a set of input–output examples has many solutions in the adaptive control and neural networks literature (see [23] for a unified overview). In order to be able to invert the approximate system, an estimation with a kernel of an invertible function is needed. A linear function is an appropriate one and the hinging hyperplanes (HH) method proposed by Breiman [3] is an elegant and efficient way of identifying piecewise linear models

The architecture

In the mixture of experts (ME) architecture of Jacobs et al. [9] the input is fed to a group of experts, and the output is a weighted sum of the experts’ output. These weights are also a function of the input through the gate (see Fig. 4 and Eq. (2)).y=igi(x,θ)·fi(x,w),igi(x,θ)=1,gi(x,θ)≥0.The polyhedral mixture of linear experts (PMLE) is a special case of the ME architecture where each expert is a linear function, that is a weighted sum of the inputs, and the gate function is an indicator

Parametrization via hinging hyperplanes

In this section the relationship between the parameters of the PMLE (, ) and the HH function approximation (1) are derived, that is, given the number of hinge functions K, the hinges Δk, and the hyperplanes βk,β+k, the parameters of the PMLE, θ and w, and the structure of the gate functions gi are derived.

In order to make the description compact and readily programmable with MATLAB, the parameters are written in vector and matrix notation as follows:X=1x(1)x(M),D=Δ1(1)Δ2(1)ΔK(1)Δ1(2)Δ1

Simulations

In this section we will illustrate the algorithm by simulation for two examples. The first example is of a smooth function that is not injective, to demonstrate the construction of the MI-PMLE. The second example is of a smooth function in two dimensions.

Discussion — The virtue of redundancy

The problem of redundancy in the biological system has been known for many years and a large volume of literature has been dedicated to finding the optimization criterion to choose the best single solution (e.g. [5]). In many other problems, the formulation of the question is half the way to the answer. We believe that redundancy should be regarded as a virtue rather than a problem and therefore the biological system has to find an optimal way to exploit this virtue rather than to “solve this

Conclusions

A new architecture for learning the inverse of a redundant system was proposed. This architecture is the polyhedral mixture of linear experts (PMLE), which can learn from examples a piecewise linear approximation of the system and then be easily inverted. The structure of the architecture was presented, its ability to approximate any inverse function was proven, and an algorithm to learn its parameters from examples was described. Finally, the PMLE learning algorithm was demonstrated and the

Amir Karniel was born in 1967 in Jerusalem, Israel. He received the B.Sc. degree (Cum Laude) in 1993 the M.Sc. degree in 1996, and the Ph.D. degree in 2000, all in Electrical Engineering from the Technion-Israel Institute of Technology, Haifa, Israel. He served four years in the Israeli Navy as an electronics technician and worked during his undergraduate studies at Intel Corporation, Haifa, Israel. From 1993 to 1999 he had been a teaching assistant (projects supervisor, tutor and lecturer) in

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  • Cited by (0)

    Amir Karniel was born in 1967 in Jerusalem, Israel. He received the B.Sc. degree (Cum Laude) in 1993 the M.Sc. degree in 1996, and the Ph.D. degree in 2000, all in Electrical Engineering from the Technion-Israel Institute of Technology, Haifa, Israel. He served four years in the Israeli Navy as an electronics technician and worked during his undergraduate studies at Intel Corporation, Haifa, Israel. From 1993 to 1999 he had been a teaching assistant (projects supervisor, tutor and lecturer) in the faculty of Electrical Engineering at the Technion. Since February 2000, he has been a post doctoral fellow at the department of physiology, Northwestern University Medical School, Chicago, Illinois. Dr. Karniel received prizes as distinguished instructor, the E. I. Jury award for excellent students in the area of systems theory, and the Wolf scholarship award for excellent research students. His current research interests are Brain Theory, Neural Networks, Human Motor Control and Motor Learning.

    Ron Meir received the B.Sc. degree in Physics and Mathematics from the Hebrew University in Jerusalem in 1982, and the M.Sc. and Ph.D. degrees from the Weizmann Institute in 1984 and 1988, respectively. After two years as a Weizmann research fellow at Caltech he spent a year and a half working at Bellcore on various aspects of neural network design and analysis. He joined the Department of Electrical Engineering at the Technion in 1992. His main research areas include the statistical theory of learning, pattern recognition, time series modeling and neural network design and analysis.

    Gideon F. Inbar received the B.Sc. degree from the Technion-Israel Institute of Technology, Haifa, Israel, in 1959, the M.Sc. degree from Yale University, New Haven, CT, in 1963, and the Ph.D. degree from the University of California, Davis, in 1969, all in electrical engineering. In 1970 he joined the Faculty of the Department of Electrical Engineering at the Technion, where he is now Professor and holds the Otto Barth Chair in Biomedical Sciences. From January 1986 he served as Dean of the department of Electrical Engineering for four years. He spent an extended sabbatical at the Harvard Division of Applied Science and School of Public Health, 1977–1978, and shorter periods at Göttingen University, West Germany, at the Centro de Investigacion Del IPN, in Mexico, at the University der BW in Munich and 1991–92 at the Beckman Institute, University of Illinois at Urbana. His major interests are in the areas of biocybernetics and biomedical signal analysis with an emphasis on the neuromuscular system. Dr. Inbar is a member of the Israel Association for Automatic Control, the Israeli Society for Physiology and Pharmacology, and the Israeli Society of Biomedical and Medical Engineering. He is a member and Fellow of the IEEE.

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    This study was done while Dr. Karniel was at the Department of Electrical Engineering, Technion - Israel Institute of Technology, Haifa, Israel.

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