8. Modeling of Rotary Inverted Pendulum

Transcription

1 NCU Deprtment of Eectric n Computer Engineering 5 Spring Course <Dynmic System Simution n Impementtion> by Prof. Yon-Ping Chen 8. oeing of otry Inverte Penuum z m (, y, z,) g y L Figure 8- he ynmic moe of rotry inverte penuum is shown in Figure 8-, which is so ce Furut penuum. he system structure is forme by motor, n rm with ength L n penuum with effective mss m n effective ength. he ngur position of the rm referring to -is is increg when it rottes bout the z-is in right-hn rue. he ngur position of the penuum referring to the upwr is is increg when it is rotting bout the is ong the rm, so in right-hn rue. L v s (t) i (t) v b (t) τ(t) (t) Figure 8- It is nown tht the motor is riven by the votge v s (t) n its ynmic eqution is given s i vs i L vb i vb (8-) where the rmture inuctnce L cn be negecte n torque generte by the motor is v b. Hence, the 8-

2 NCU Deprtment of Eectric n Computer Engineering 5 Spring Course <Dynmic System Simution n Impementtion> by Prof. Yon-Ping Chen s b s (8-) i ( v v ) v which is use to contro the system vi the foowing eqution τ (8-3) where is the rotor moment of inerti, is the friction coefficient n the torque τ(t) is require to rive the rotry inverte penuum. From (8-) n (8-3), we hve τ v s (8-4) For simpicity, we wi omit the time vribe t in the foowing equtions. Now, we wi erive the ynmic moe of the rotry inverte penuum bse on the Lgrnge s equtions, epresse s τ τ (8-5) (8-6) where LV is the Lgrngin, is the tot inetic energy n V is the tot potenti energy. It is esy to fin tht the inetic energy n the potenti energy V of the rm re (8-7) V (8-8) where is the moment of inerti rete to the rm. As for the penuum, its effective mss m is concentrte t (, y, z ) s shown in Figure 8-. he Crtesin coorintes cn be obtine s L (8-9) L (8-) y z n their erivtives re (8-) 8-

3 NCU Deprtment of Eectric n Computer Engineering 5 Spring Course <Dynmic System Simution n Impementtion> by Prof. Yon-Ping Chen Hence, we hve L (8-) L (8-3) y (8-4) z ( ) L (8-5) y ( ) L ( ) L (8-6) L ( ) z (8-7) he inetic energy n potenti energy of the penuum re then obtine s ( y z ) m (8-8) V m gz m g (8-9) where is the moment rete to the penuum. he tot inetic energy n the tot potenti energy re n V V V n the Lgrngin is epresse s L V (8-) V V ( ) ( ) m g Concerning the generize coorinte, we hve ( ) ( ) (8-) (8-) 8-3

4 NCU Deprtment of Eectric n Computer Engineering 5 Spring Course <Dynmic System Simution n Impementtion> by Prof. Yon-Ping Chen (8-3) From (8-5), it cn be obtine tht ( ) (8-4) m τ C where τ τ C, i.e., the generize force τ incues the torque τ ppie to the rm n the viscous friction C of the rm. As to the generize coorinte, it cn be foun tht ( ) ( ) (8-5) (8-6) L ml mg (8-7) From (8-6), we obtin ( m ) (8-8) m m g C where τ C, i.e., τ is the viscous friction C of the penuum. From (8-4) n (8-8), the ynmic eqution of the inverte penuum is epresse in mtri form s beow: ml m ml m L m m m ml m τ C m g C (8-9) 8-4

5 NCU Deprtment of Eectric n Computer Engineering 5 Spring Course <Dynmic System Simution n Impementtion> by Prof. Yon-Ping Chen 8-5 where is the inerti mtri n is the mtri rete to the centrifug forces n coriois forces. here re two importnt properties concerning n. First, the inerti mtri is symmetric n positive-efinite, i.e., n > for. Secon, is sew-symmetic, i.e., for. For the first property, it cn be seen from the eements of tht the symmetricity is true. Furthermore, by irect ccution we hve [ ] (8-3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) > L m L L m which shows > for. his proves the first property. For the secon property, et s ccute the mtri, which cn be obtine s φ φ (8-3) where φ m. For, it is esy to chec tht, i.e., the secon property is true. From (8-), we substitute the torque τ ppie to the rm into (8-9) n then the ynmic eqution is rewritten s (8-3)

6 NCU Deprtment of Eectric n Computer Engineering 5 Spring Course <Dynmic System Simution n Impementtion> by Prof. Yon-Ping Chen 8-6 s v m g C C where v s (t) is the votge source to rive the system.