Neural-network approximation of piecewise continuous functions: application to friction compensation

Abstract

One of the most important properties of neural nets (NNs) for control purposes is the universal approximation property. Unfortunately,, this property is generally proven for continuous functions. In most real industrial control systems there are nonsmooth functions (e.g., piecewise continuous) for which approximation results in the literature are sparse. Examples include friction, deadzone, backlash, and so on. It is found that attempts to approximate piecewise continuous functions using smooth activation functions require many NN nodes and many training iterations, and still do not yield very good results. Therefore, a novel neural-network structure is given for approximation of piecewise continuous functions of the sort that appear in friction, deadzone, backlash, and other motion control actuator nonlinearities. The novel NN consists of neurons having standard sigmoid activation functions, plus some additional neurons having a special class of nonsmooth activation functions termed "jump approximation basis function." Two types of nonsmooth jump approximation basis functions are determined- a polynomial-like basis and a sigmoid-like basis. This modified NN with additional neurons having "jump approximation" activation functions can approximate any piecewise continuous function with discontinuities at a finite number of known points. Applications of the new NN structure are made to rigid-link robotic systems with friction nonlinearities. Friction is a nonlinear effect that can limit the performance of industrial control systems; it occurs in all mechanical systems and therefore is unavoidable in control systems. It can cause tracking errors, limit cycles, and other undesirable effects. Often, inexact friction compensation is used with standard adaptive techniques that require models that are linear in the unknown parameters. It is shown here how a certain class of augmented NN, capable of approximating piecewise continuous functions, can be used for friction compensation.

Improving backdrivability in geared rehabilitation robots

“…In the given friction model, friction depends only on velocity. However, friction can also depend on position, but this dependence is negligible [13] and neglected here.…”

“…The friction model (2) is not continuous at zero, which can cause stability problems [13]. Therefore, a combination of a sigmoid and a linear function is selected to fit the measured data [12].…”

Constructive Approximation of Discontinuous Functions by Neural Networks

“…piecewise-continuous). Examples include friction, deadzones, backlash, and so on [28,29]. It is found that attempts to approximate piecewise continuous functions using smooth activation functions require many hidden nodes (neurons) and many training iterations, and still do not yield very good results [27].…”

“…The first example was studied in [27]. The test function f 1 is defined on [−3, 3] by The continuous function f 1c has been approximated by a SLFNN with 13 neurons.…”

Recursive model free controller: Application to friction compensation and trajectory tracking

“…Compensation of these disturbances, the most common way, is to derive one or several friction maps and use them for approximate cancellation of the friction forces [3]. In practice, the major difficulty is taking into consideration of various nonlinear effects like hysteresis, static and coulomb friction, presliding displacement, Dahl effect, frictional memory and Stribeck effect [4,5]. These nonlinear effects in model based approaches require the pre-identification of friction model [3,6] and thus lack of robustness of friction model uncertainties such as drive-to-drive variation, zone-to-zone variation or environmental changes.…”