Uki-goshi (浮腰), the floating hip.

Nage-waza (throwing techniques), Koshi-waza (hip techniques), Gokyo 1st set (Dai-ikkyo).
Required for: 4th Kai (white belt with a green stripe, JudoScotland); 5th Kai (white belt with a blue stripe, JudoScotland); 6th Kai (white belt with a purple stripe, JudoScotland); 3rd Mon (red belt, British Judo Association); 6th Kyu (red belt, JudoScotland); 5th Dan (black belt, British Judo Association).

About
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Kano's favourite throw: a half-entry hip technique in which uke is caught on the side of the hip and floated over with a twist of the trunk.

Step by step (standard right-handed form)
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1. From a right-handed grip, release the lapel and slide your right arm around your partner's waist so your hand lies flat on the small of their back.
2. Step your right foot inside their right foot and turn, bringing your right hip only into the fold of their body: this is a half entry, hips side-on rather than fully across, and you stay tall rather than squatting.
3. Draw them tight against your right side with the arm at their back and pull the sleeve so their chest is fixed to you with no gap.
4. Twist your trunk sharply to your left, hip pressing back into them. There is no big lift in this throw: the twist alone floats them over the crest of your hip.
5. They roll around your hip and land on their back in front of you. Keep the sleeve to steady the fall.

Key points
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1. Enter with the hip only halfway across, not fully through
2. Hold uke tight to your side with the arm around the waist
3. Throw with the twist of the body, not a lift of the hips

Receiving it
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An arm comes around your waist and their hip presses into you; instead of a big lift you feel a sharp twist, and your feet skim the mat as you are turned over the hip. The fall is a forward roll around their body. Land on your back, left arm slapping, chin tucked; it arrives lower and quicker than a full hip throw.

Reference pages elsewhere
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Uki-goshi on the IJF technique reference (International Judo Federation): https://judo.ijf.org/techniques/Uki-goshi

From Waza Book: https://wazabook.com/techniques/uki-goshi/