I extract the one-branch framework into the model and did a sketch. All the dimensions are measured from the real umbrella. Because some of the links can be bent slightly in the real situation, some of the dimensions come from the average of the folding state and the fully stretching state.

The general motion is shown in the picture. The input is the upward push of link 2 (the sliding link) by the user. The output is the stretching of all framework. Because the link 8, which locates outermost, has most representative but dangerous motion, I mainly focus on the motion of link 8 and the point J, which is at the outer end of link 8.

There are two four-bar-mechanism in the umbrella framework. The first one is CDGE which is a parallelogram according to dimension. The other one is FHIG which is the more important one, and it will control the rotation of link 8.

As we push the link 2 upward, link 3 and 5 will rotate clockwise, while the link 4,6 and 7 will rotate counterclockwise. Then the point F will draw the link 7 back, while the point G will push link 6 outward. These will result in the clockwise rotation of link 8.

Dimension:

L1=L1(t) (input, L1i=17.3cm, L1f=4.2cm)

L3=16.5cm

L4=9.3cm

L5=7.2cm

L6=20.6cm

L7=18.8cm

L8=0.95cm

DG=CE=L5=7.2cm

CD=EG=2.1cm

FG=1.15cm

IJ=15.7cm @-20degree

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