trajectory generation solution

When we get a one path(or trajectory) in the joint space, we sometimes use linear function with parabolic blends with via points. And then How to make one trajectory in the below figure.1

figure.1

First, we must aware that start segment and last segment are different with interior segments because of a region of end blend segment. For example, when we make path in the interior points(j, k), we will use only region between tj and tk. However, when we make  one trajectory in the start , we use region between ti and tk as well as t0(time 0).

End segment trajectory can be taken by similar concept.

 Therefore, we use next equation set for interior regions

 eq.4                               eq.5

eq.6                               eq.7

In the start and last segments, we use following equations

......eq.1

First equation means that the velocity of the end of the blend region must have same velocity of the linear section.

......eq.2

Equation.2 shows that when we make blend region with second order polynomial equation and constant acceleration value.

.......eq.3

From eq,1 and eq.2, we can get eq.3 with some skills.

By combining eq.3 and the concepts of interior equation set, we can get equations to make trajectories for the start and last segments. Next equations show that.