Result and conclusion

From matlab program, I get these results:

for the first opening mode (v=const)

2. For the second opening mode ( v=Asinwt)

3. Conclusion

We can easily find that the behavior of direction( theta) is similar between these two modes, however the velocity and acceleration is quite different. The reason is that in the first mode, the velocity is a normal constant. In the beginning and the end, when the umbrella is folded or fully stretched, the four-bar-mechanism FGIH is close to its toggle position. Little change of the input angle will cause big change of output. So we can see the magnitude of velocity and acceleration is extremely big. Actually, this is the part of the reason why we feel it stuck when opening the umbrella. However, the second opening mode avoids this situation by making the velocity start from zero and end with zero.

In the video we can find the second opening mode makes it more smooth to open, and this will lessen the danger of opening umbrella.