This file has been edited by Andrew Conway, from posts by Masaki Nishikawa, Smokeless and Andrew Conway.
In Japanese traditional juggling performance "daikagura", there are acts with an umbrella and objects (ball, ring, etc.) on it.
I try to explain the Japanese trick, though I'm not sure if it is what you want to do or not.
I studied the trick through trial and error watching daikagura performance by a professional on TV, so I don't think my method is the best or the standard. Because the trick with umbrella is a traditional act in Japan, I hope no one calls me a thief.
My method is a little different from the one described by Andrew Conway. With my method, you can use larger umbrella because you hold the umbrella over your head. But you need light source on the ceiling to monitor the ball through the umbrella.
Use a Chinese parasol, not a western style umbrella. If you can find one use a cloth parasol, as you will rip up half a dozen of the paper ones while learning. The cloth ones last much longer.
Simon (Stapleton?) at Circus Space uses bamboo strip parasols which are both durable and stiff.
I use a Japanese traditional umbrella made from oilpaper and bamboo. (It is an umbrella, not a parasol, because it is supposed to be used as rain shield, not as sun shield.) It has 36 almost-straight spokes. Using usual umbrella would be much more difficult because the spokes are curved and the surface is bumpy as the number of spokes is far less than that of Japanese umbrella.
Choose one with thick shaft because the thickness makes spinning easy. Choose one with light color and less marking on it.
Japanese professionals use special umbrellas made from thin silk cloth and bamboo, because they are beautiful and the half-transparent silk enables a performer to see the object on the umbrella through it. But I heard that the special umbrellas are pretty expensive. They are primarily for Japanese traditional dance, not for real use in rain.
Availability of Japanese traditional umbrella outside Japan is a problem.
I've never tried Chinese parasols. I've heard that there is a similar act with umbrella/parasol in Chinese circus, but I've never seen it yet. I want to see it and find the difference from the Japanese act someday.
I like balls which weigh 100g to 200g and are almost 10cm in diameter. Dubé's 4 inch stage ball works well. Once you've mastered the trick, size and weight of ball isn't so important.
Suppose that you are on an escalator carrying you upstairs. If you try to walk down as fast as the steps go up, you stay at the same place forever.
The ball tries to roll down on the surface of the umbrella, but it can't, because the umbrella tries to carry it up at the same speed.
This assumes you are right handed:
There are two ways to hold the parasol - you can hold it and turn it with just the right hand, about half way up the shaft, or you can hold it loosely with the right hand at the top of the shaft and turn it with the left hand at the bottom of the shaft.
The parasol should turn anti-clockwise when viewed from above.
Hold the parasol so the top is a little below eye level, and it is to the right of you. Arrange the shaft so the spokes coming from the hub directly towards you are horizontal. This is where you will place the ball, and having the spokes horizontal prevents the ball from rolling off the edge or towards the hub. Tilt the whole parasol forwards slightly.
Now start the parasol rotating, and place the ball on it, about two thirds of the way out from the hub. The rest is balance. It the ball rolls forward, tilt the shaft back. If it rolls towards you, tilt the shaft away from you and so on. Practice for a few minutes every day and you should have it in a week.
BE CAREFUL NOT TO STICK YOUR EYE WITH A SPOKE! Wear eyeglasses or goggles if you need.
BE SURE THERE IS NO ONE AROUND YOU! Someone may rush at you chasing his lovely silicone balls.
I'm right-handed. (But Japanese professional performers use the mirror-imaged form of mine, because they are supposed to have their assistants at their right side for a traditional reason.)
Hold the shaft of the umbrella with both hands, left hand near the bottom end and right hand at the middle of the shaft. I use both hands alternatively(asynchronously) to spin the shaft.
At first, don't spin the umbrella. Hold the shaft in front of you and keep the spokes over your head. Tilt the shaft slightly right and forward. Choose one of the spokes and make it parallel to both the ground and your chest in front of you. Put a ball on the spoke (3/4 or 2/3 away from the hub) and see if it rolls down perpendicular from the spoke. Try it a few times to find the appropriate tilt angle to right. If everything is OK, the ball should roll straight forward from you.
[ASCII figure: seen from above] * * ^ * | * | *---x------ the spoke * you put the ball here * * * <- the rim of the umbrella YOU (under the umbrella)
By the way, how can you see the ball while your head is under the umbrella and the ball is on it? Remember that the umbrella is made from oilpaper or silk. If it is silk, you can see the ball through it. If it is oilpaper, you can see the shadow of the ball with help of the light from the ceiling.
If you put a ball on the spoke that is parallel to the ground and the surface is tilted forward, the ball goes forward and it doesn't curve to the hub or the rim.
Now, spin the umbrella counterclockwise (seen from above) as soon as you put the ball on it. Find the appropriate speed and forward tilt angle to keep the ball at the same position. You can find it through trial and error.
If you don't tilt the umbrella forward enough, the ball can't get enough rolling speed to be dynamic stable. If you tilt it too much, you can't kill the speed of the rolling ball.
It took me 3 weeks (30 min. a day) to become a little solid with it.
Simon (Stapleton?) at Circus Space uses bamboo strip parasols which are both durable and stiff. He manages two parasols and does stuff like curls and transfers. Then he moves on to two balls, one on each parasol. He uses 15cm balls but I have seen him use a 50p piece (3cm, 7 sides).
More advanced tricks include rolling rubber dog rings or bagels, coins, or even rectangular boxes! Incidentally, I use a 50p piece too. A large ring can be made to spiral inwards, and land flat around the spike of the umbrella.
As you throw a coin onto the umbrella, put some spin on it in the direction that it is going to be turning, so that it does not have to pick up spin once it lands.
It's possible to buy double (and triple?) decker umbrellas. That is, two or more umbrellas on a single shaft, a few inches apart, and getting smaller as they get near the top. You can make a prop jump from one level to the next and back.
Try ring. It isn't so difficult. Rectangular box is pretty difficult. You need to give it enough initial rolling speed to keep it rolling without wobbling. Rugby football, egg and thick paperback are also possible. Try anything that may roll.
Toss the rolling ball up and catch it by the umbrella. Try throw-up start.
Try a ball on a half-closed umbrella.
Without spinning the umbrella, move the umbrella around in big clockwise(seen from above) circular motion over your head and keep the ball rolling around on the umbrella.