CS 685 notes, J. Koˇseck´a
1
Trajectory Generation
The material for these notes has been adopted from:
John J. Craig:
Robotics:
Mechanics and Control
.
This example assumes that we have a starting position and goal pose of the
end effector and we are asked to move the joint angles to move the end effector
from one pose to another. Here we describe a strategy how to do so by designing a
trajectory in joint space from one end point to another. Assuming that we know the
inverse kinematics of the system, we can compute the desired joint angle for goal
position of the end effector. This example shows how to design a trajectory of a
single joint
θ
(
t
)
as function of time. Suppose that we have following constrains of
out trajectory: we have desired position at the beginning and end of the trajectory
and we the velocity at the beging and end has to be zero.
Hence our desired
trajectory has to satisfy the following constraints:
θ
(
0
) =
θ
0
;
θ
(
t
f
) =
θ
d
˙
θ
(
0
) =
0
˙
θ
(
t
f
) =
θ
d
(1)
Cubic polynomials
In order to satisfy the above constraints, our trajectory has
to be at least polynomial of the 3
r
d
order, which has four coefficients, and hence
can satisfy the above 4 constraints.
This can be achieved by third order cubic
polynomial which has the following form
θ
(
t
) =
a
0
+
a
1
t
+
a
2
t
2
+
a
3
t
3
. Given the above form the joint velocity and acceleration will have the following
forms
˙
θ
=
a
1
+
2
a
2
t
+
3
a
3
t
2
(2)
¨
θ
=
2
a
2
+
6
a
2
(3)
1
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Using the above equations and instantiating the constraints we can solve for the
coefficients of the cubic polynomial and obtain
a
0
=
θ
0
(4)
a
1
=
0
(5)
a
2
=
3
t
2
f
(
θ
f

θ
0
)
(6)
a
3
=
2
t
3
f
(
θ
f

θ
0
)
(7)
Now given a particular instance of the problem, we can substitute to the above
equations the desired parameters
θ
0
,
θ
f
,
t
f
and obtain different trajectories.
Linear functions with parabolic blends
If we were to simply just connect the
desired position with a linear function, it would cause the velocity to be discon
tinuous at the beginning and end of the motion. Also note that of the shape of the
part in the joint space is linear, that does not mean that the shape of the path in
the end effector space is linear. Hence what can be done is to take a linear path
in the end effector space and interpolate it linearly. We would like to do it in a
way that the velocities at the would not be discontinuous at the places where the
pieces meet. One way to achieve this is to add a parabolic blend region, such
that the we will create a smooth and continuous path. During the blend portion
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 Fall '08
 Luke,S
 Acceleration

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