Ayumu means to walk. Hence, simply said you have to take a step forward. Do not step with the feet, but walk with the body. Push the hips forward slightly as you do hiza no nuki. During Ukimi stage you have to bring the back hip forward connecting to your back leg, to move with the whole body.
It is said that in Karate you use the whole body. Now that's a great saying but what does that mean practically? One key is to let your upper body move independently from your lower body and vice versa. When you do Ayumi Ashi (lower body) your Kamae (upper body) can be utilised effectively as you'll see in this video.
Tsugi Ashi is essentially the same as Ayumi Ashi:
The difference is mainly NO BODY ROTATION...
...and of course the fact that your back leg doesn't pass your front leg.
It's almost half an Ayumi Ashi and then advance forward with your front foot.
Have you ever analysed how humans walk? Maybe you have to be a professional in a certain field or a bit crazy (like me).
Either way, if you analyse it you will see that naturally you NEVER lead with your foot when you walk. Instead, feet smoothly follow.
The funny thing is that when it comes to Karate there are specific ways to move and with incorrect instruction this leads to leading with the feet. Let me stress once more to NEVER do that.
You telegraph, you lose power, it's slow.
So what to do?
Sakagami sensei says "body - leg - arm". You'll learn exactly how in the video.
Surikomi ashi is the same as Tsugi ashi, but this time cross the supporting leg. The distance of the surikomi ashi depends on the situation. However, if you don’t cross the supporting leg it is called tsugi ashi. A variation of this movement is called chidori ashi, which is a style of walking on the edge of the feet. This can be seen in Naihanchi and Seishan.
It is called surikomi ashi from hanmi dachi and chidori ashi from mahanmi dachi. In case of chidori ashi, the edge of the foot touches first naturally. Surikomu means to rub in, this might give you an idea of the movement. Hence, lower the waist as you move and keep the foot close to the ground.
Nijiri ashi is the art of pulling the body forward, stealing Maai inch by inch.
The meaning is to feel Maai and Kiai, getting accustomed to the feeling of a closing Maai and a closing threat. Learn how it feels to have your opponent close the distance. When he does - you and him learn how to measure intention. To force a response from your opponent.
This video tells you what to do and what not to do about Nijiri Ashi. Make sure to avoid these mistakes.
The stealing of Maai is a good way to feel Maai and Mazakai. Mazakai is the limit of the distance - as thin as a piece of paper. Mazakai is the place where you have to attack. If you don't, your opponent will. Mazakai is easily crossed using methods like Yori Ashi - that's why you approach Mazakai with Nijiri Ashi carefully. Never hurry to execute the first attack. Learning to close Maai and feel the opponent is just as important as the lesson learned from the first attack.
Of course, there are other methods that help to develop a sense of Maai and your reach. To approach and judge Mazakai however, Nijiri Ashi is one of the best methods - because the line is so incredibly thin.