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 of (29).

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 [19], and triaxial compression tests on soil [20], 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.