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It is necessary to generate a smooth and feasible
path for an UAV. It must not take any sharp turn during its maneuvering.

In this section a smooth trajectory will be
generated by differential flatness when any obstacle is detected by the UAV
while it is moving through predefined path. So, the UAV will start movement
through the predefined path until it detects any obstacle. The path will be
generated using differential flatness. The definition of differential flatness according
to 13 is, a nonlinear system is differentially ?at, if there exists a
function A such that

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and the solution can be written as a function of output
z and its derivatives 13.

 

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The trajectory between two points, from initial
positionto final positionmust be generated in such a way so
that to ensure the smoothness of the function of the curve in a certain time
period using differentiable function. The
trajectory is defined for x(t) and y(t) as-

 

 
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 is the basis function picked from Andrews’
curve, which helps to eliminate the effect of growing magnitude of time. andare all coef?cients used to generate
trajectory along with Andrew’s curve basis function. According to the
assumption of differential flatness to generate smooth and feasible trajectory
of virtual leader equation must be agreed when –

 
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Now we wish to generate trajectory using equation –

 
 
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If the agent is differentially flat then-

 

 
 
 
 
 
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and
we see that the initial and ?nal condition in the full state space depends on just
the output z x and z and its
derivatives at the initial and ?nal times.
Thus, any trajectory for z that satis?es
these boundary conditions will be a feasible trajectory for the system. We can parameterize the ?at output
trajectory using a set of smooth basis functions
?(t)-

 
 
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We seek set of
co-efficient a (t) satis?es the boundary conditions. The derivatives of the ?at
output can be computed in terms of the derivatives of the basis functions-

 
 
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The coefficient is picked from Andrew’s curve and
can be expressed as below-

 
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The
boundary condition of the system output z is described as –

 
 
 
 
 
 
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The coefficients a and v can be determined by
taking inverse of both the side of equ () as follows-

 
 
 
 
 
 
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Similarly, coefficient b can be determined as-

 

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