Here is linkage to simulate a real folding umbrella. This structure is more complicated than the one simulating umbrella. Click right picture to see the animation. The following is a detailed construction process. 

Step of construction

 Illustration

 1. Start with an segment OA on z-axis, and an regular octagon around z-axis in xy-plane.

　

 2. Create a plane P containing z-axis and one vertex of the regular octagon.

　

 3. Take a movable point M on segment OA.　

 4. Translate segment OA mapping O to M to get segment O'A', and Translate segment O'A' mapping O to M to get segment O''A".

　

 5. Translate point M mapping O to O'' to get point M'.　

 6. Change color of segments and hide useless label. Take a point B on z-axis such that  = 0.378

　

 7. In plane P, draw circle centered on M' of radius = 0.681  and circle centered on B of radius = 0.698 . Point C is the intersection of two circles.　

 8. Hide previous circles. Create a line though points B and C. Point D is on the line such that = 0.641 .

　

 9. Connect segment CM. Point E is on the segment such that  = 0.146 .

　

 10. In plane P, draw circle centered on E of radius = 0.632  and circle centered on D of radius = 0.137 . Point F is the intersection of two circles.　

 11. Hide previous circles. Create a line  though points D and F. Connect segment BD, EF, and FD.

　

 12. Rotate line  clockwise 6.75 degrees around the line perpendicular to plane P and though point F to obtain .　

 13. Hide line . Point G is on line  such that = 1.128 . Point H is on segment CD such that  = 0.208

　

 14. In plane P, draw circle centered on H of radius = 1.245  and circle centered on G of radius= 0.238 . Point I is the intersection of two circles.

　

 15. Hide previous circles. Create a line  through points G and I. Connect segment DG, HI, and  IG.

　

 16. Rotate line  counterclockwise 6.75 degrees around the line perpendicular to plane P and though point G to obtain .

　

 17. Point J is on  such that  =

　

 18. Hide all objects created at steps 8~17 except segment M'C, BD, DG, and GJ. Also hide plane P.

　

 19. Rotate segment M'C, BD, DG, and GJ around z-axis mapping one vertex of regular octagon towards adjacent vertex to get segment M'C', BD', D'G', and G'J'.

　

 20. Draw convex polygon BDD', DGG'D', and GJJ'G'.　

 21. Repeat steps 19 and 20.

　

 22. Reflect segment OA in xy-plane.

　

 23. Hide z-axis, xy-plane, regular octagon, and segments in convex polygon. Change point size and border radius of convex polygons to smallest.