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5 TRAJECTORY
A trajectory is a path that an object follows through space. In robotics the object could be the
end-effector for a serial robot or the TCP for a Delta-3 robot. A trajectory can be described
mathematically either by the geometry of the path, or as the position of the object over time.
There are some parameters that have to be decided when a trajectory will be calculated. For
example if the motion should be linear, quadratic or cubic, also if the motion should be done
with some predefined start and end velocity, constant velocity or for example with minimum
time. Then with this information a trajectory is calculated with help of interpolation. This
trajectory contains then information about the speed, acceleration and position.
5.1 TRAJECTORY WITH DESIRED START AND END VELOCITY
One way to generate a smooth trajectory for an object, for example a time history of desired
end-effector coordinates is by a polynomial (5). In this case there are some constraints such as
the desired start and end velocities. If the variable X is a vector that describes the coordinates
(x,y,z) for the end-effectors position. Then suppose that the object coordinates at time t 0
satisfies
and the final values for time t f satisfies
X ( t0
) = X
X&
( t ) = X&
0
Eq. 5.1
f
X&
( t ) = X&
f
Eq. 5.2
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Public Report ELAU GmbH, Marktheidenfeld
28
0
0
X ( t ) = X
Because there is four constraints the generated polynomial has to contain four independent
coefficients that can be chosen to satisfy these four constraints. This will give a cubic trajectory
of the form
X d
( t)
= a + a t + a t + a t
0 1
Eq. 5.3
where Xd (t)
is the desired (x,y,z) positions at time t.
Then the generated velocity that will satisfy the start and end constraints will be given by the
derivative of Xdesired as
f
f
2
2
3
3