The objective of this paper is to present the design and analysis of a new explicit force control family with a term driven by a large class of saturated-type hyperbolic functions
to handle the force error for the problem of force regulation in interaction tasks.
This is done as relatively rapid changes in indifference points are observed at relatively shorter delay values, whereas changes become less pronounced out at relatively longer delay values, which is characteristic of hyperbolic functions
Substituting the values of [a.sub.i] (i = 0, ..., N), [b.sub.i] (i = 1, ..., N), V, [lambda], [mu], [A.sub.1], and [A.sub.2] obtained into (30), one can obtain the travelling wave solutions expressed by the hyperbolic functions
This idea comes from the vast applications of hyperbolic functions
to solve differential equations.
The resulting traveling wave solutions generated by this step with the transformation in (11) are expressed by hyperbolic functions
By using the method, we have obtained new exact solutions in terms of the hyperbolic functions
, the trigonometric functions, the exponential functions, and the rational functions.
For solving (2.1) without convection [r.sub.ix] = [r.sub.iy] = [r.sub.i]z = 0 we consider hyperbolic functions
in the z-direction with parameters [a.sub.iz1], [a.sub.iz2], i = [bar.1, N]:
Hyperbolic models have also been used in geomechanics problems to model interfaces [17, 18], soil reinforcement pullout , and triaxial compression tests on soil , It should be noted that hyperbolic functions
have provided components of tensile load-strain models reported in the geosynthetics literature [1, 21-25].
The algorithm can be configured to operate in vectoring or rotation mode in several coordinate systems, providing the possibility to calculate hyperbolic functions
. In rotation mode, a vector is iteratively rotated by an angle, to calculate a final vector corresponding to the sine and cosine functions of an input angle.
Properties on Hyperbolic Functions
. Some properties on hyperbolic functions
will be used.
Right and skew-right circulant matrices remains invariant on their type when evaluated on trigonometric and hyperbolic functions
that are used in this paper.
The hyperbolic functions
with [delta] = 0.2 and [delta] = 1 are shown in Figure 4, in which the [l.sub.1] norm is also demonstrated for comparison.