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Unformatted text preview: CSCI 545 – Introduction to robotics Homework Assignment 1) (40 points) Consider the transfer function: ) ( ) )( ( ) ( 2 s u m k s m b s s m s x + + = α α a) The block diagram of this system b) The differential equation: The system is combined by 2 subsystems: ) ( ) ( ) ( ) ( 1 2 s u s H s H s x = Thus, we can get a differential equation: ) ( ) ( ) ( t u m m k m b m k m b x x x x α α α α = + + • • • • • • It has 3 states, x x x , , • • • , which mean velocity and position. x • • means the acceleration, x • means the velocity and x means the position. x • • • is the additional states of the system. Because it let the system’s input force becomes unstable and changes all the time, it is the jerk in the system. The system is like the picture showed below: F is the unstable force. 1 u(s) m α ) )( ( 1 2 m k s m b s s + + α x(s) c) Summarize all states in the state vector x and actions in vector u. x = = • • • x x x x x x 3 2 1 , u = [ ] ) ( t u d) Reformulate the system to ) ( ) ( ) ( t Bu t Ax t x + = • [ ] ) ( 1 1 ) ( t u m m b m k b m k t x x x x x x x +  = = • • • • • • • • • • α α α α [ ] ) ( , , ) ( , 1 1 , ) ( t u u m B t m b m k b m k A t x x x x x x x x = = =  = = • • • • • • • • • • • α α α α Because A is a 3 3 × matrix, B is a 1 3 × matrix and u is a 1 1 × matrix, we know n is equal to 3 and m is equal to 1 in this system....
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 Spring '08
 Schaal
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