Kuk Mu I

1. Left hand block
2. Left hand chop
3. Right hand block
4. Right hand chop
5. Left lower block
6. Right hand rising arm block
7. Left hand rising arm block
8. Right hand punch(middle target)

(Uhtz)

1. Left hand turn to Left hand block
2. Left hand chop

1. Right hand block
2. Right hand chop
3. Left lower block
4. Right hand rising arm block
5. Left hand rising arm block
6. Right hand punch(Middle  target)
   (Uhtz) 
7. Left hand turn to Left hand block
8. Left hand block
9. Right hand block
10. Right hand chop (good horse stance looking over right hand)

Video

Background:

The name is spelled Kuk Mu and pronounced gugmu based on my translation of "국무". It has something to do the skills to defend your nation.

Found in dictionary: Korean > English.

• 국무國務

1.     state affairs[business];the affairs[business, matters] of state

§  국무를 맡아보다 administer the affairs of state

§  국무를 처리하다 transact[conduct, attend to] state affairs

§  국무에 다망하다 be busy with state affairs

무 武

1.             military affairs;military[martial] arts;the art of war

1.  
   • 무를 숭상하다 pursue the policy of militarism;glorify the military
   • 무를 닦다 train oneself in military[martial] arts
   • 문을 버리고 무를 배우다 give up the pen for the sword

So in our context it should translate to  "Korean(referring to their  State)" + "Martial Arts"