In the computational analysis of the mobile robot trajectory, some specific and fast tools of MAPLE11 soft are used. These tools are generally based on a wide spread numeric method - the Fehlberg fourth-fifth order Runge-Kutta method - the so-called ‘rkf45’ method - with degree four interpolant [3].

A widespread and fast graphical tool is the ‘phase-portrait’ tool. The ‘phase-portrait’ is a plot builder which realizes the phase-portrait for a system of differential equations. It is a fast procedure based on specific numeric methods for approximating the solution of the studied differential system. For the present computational aim, the classical method of Euler has been chosen, in fact, the ‘forward Euler’ method.

The calling sequence has the following form:

$\text{phaseportrait}(\mathbf{\text{deqns,}}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{vars,}}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{trange,}}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{inits,}}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{options}})$

The parameters are the following:

deqns - a list or a set of first-order ordinary differential equations, or a single differential equation of any order;

vars - dependent variable, or a list or a set of dependent variables;

trange - range of the independent variable;

inits - a set or a list of lists; initial conditions for solution curves;

options - (optional) equations of the form keyword = value.

The default method of integration is $\mathbf{method}\mathbf{=}\mathbf{\text{classical[rk4]}}$. Other methods can be specified in the optional equations. Note that because numerical methods are used to generate plots, the output is subject to the characteristics of the numerical method in use. In particular, unusual output may occur when dealing with asymptotes of solution curves. This also means that the initial conditions of the problem must be given in a standard form, that is, the function values and all derivatives up to the differential order of the differential equation at the same point minus one.

By default, plots are produced with boxed axes. In contrast with *DEplot* tool, this tool does not produce a direction field constituted by field arrows. This is not possible in 3D case, but its role is taken by the ‘scene’ parameter. Below some of the basic parameters of the procedure are detailed.

The

**inits** parameter must take the form of

$[[x(t0)=x0,y(t0)=y0,z(t0)=z0,\dots ],[x(t1)=x1,y(t1)=y1,z(t1)=z1,\dots ],\dots ]$

**inits** is a list (or a set) of lists, each sublist specifying one group of initial conditions (for one solution curve).

The ‘scene’ parameter has the following form:

$\mathbf{scene}\mathbf{=}\mathbf{\text{name,}}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{name,}}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{name}}$

**scene** specifies the plot to be viewed. For example, $\mathbf{\text{scene}}\mathbf{=}\mathbf{[}\mathbf{x}\mathbf{,}\phantom{\rule{0.25em}{0ex}}\mathbf{y}\mathbf{,}\phantom{\rule{0.25em}{0ex}}\mathbf{z}\mathbf{]}$ indicates that the plot of **x** versus **y** versus **z** is to be plotted, with **t** implicit, while $\mathbf{\text{scene}}\mathbf{=}\mathbf{[}\mathbf{t}\mathbf{,}\phantom{\rule{0.25em}{0ex}}\mathbf{y}\mathbf{,}\phantom{\rule{0.25em}{0ex}}\mathbf{z}\mathbf{]}$ plots **t** versus **y** versus **z** (**t** explicit). This option can also be used to change the order in which to plot the variables. If **vars** is entered as a set, there is no default ordering; if entered as a list, the given ordering is used.

The parameter ‘stepsize’ has the following form:

$\mathbf{\text{stepsize}}\mathbf{=}\mathbf{\text{real}}$

and specifies the distance between mesh points to be used in generating the graph. For $\mathbf{\text{trange}}\mathbf{=}\mathbf{a}\mathbf{\dots}\mathbf{b}$, the default **stepsize** value is $\mathbf{abs}\mathbf{(}\mathbf{(}\mathbf{b}\mathbf{-}\mathbf{a}\mathbf{)}\mathbf{)}\mathbf{/}\mathbf{20}$. If the **stepsize** specified is too large, the default is used.

In this paper the ‘phase-portrait’ graphical tool is tested in order to get comparative graphical analysis with other MAPLE tools. This plot tool is appropriate for the proposed model as it produces an appropriate representation of the trajectory of the studied mobile.