A Probabilistic Hybrid Automata SLDS nonlinear dynamics, guarded transitions Hofbaur williams 2002 Continuous dynami 1,62,B1,B2 O. =e +OLa 62k4=621k+621a+ nose ball tre false 0.=8 +8.8+noise 62k4=621k+624+ noise a=fm(B1x,61,62,6214,W)+ noise 62k4=2(1,a ar)+noise Observations Continuous x2) -an Discrete transition now depends on the continuous state scree states A(d,,ld r(also, the observation function g depends on discrete state) Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16412/6.834 Lecture,15 March2004 Discrete prediction in PHa tue)tmue)物e(d、M121,a2-) ()(i) Main idea t-15t-1 p(d1…d-1d1a11…a-1z1-) compute transition probability from continuous estimate p(d1…d1|a121…a121) Cannot simplify as easily as before p(d1…d-1|a121a1-21)p(d1|d1,d1-1,a121…a1-1z1-) Instead Lpld, Id, -1 ,x p(x-11 dod,- @ 24.,,-dx,-=Po constant for all x. that satisfy a given guard E p(transition t) p(guard for t satisfied previous estimate) Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 6. 412/6.834 Lecture, 15 March 2004Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 23 Probabilistic Hybrid Automata z SLDS + nonlinear dynamics, guarded transitions Hofbaur & Williams 2002 has-ball true false T1  0.7 0.1 0.9 T1 ! 0.7 0.5 0.7 T1 ! 0.5 T1  0.7 1.0 0.5 0.5 1 2 1 2 T ,T ,T ,T   Continuous dynamics: f t noise f t noise t noise t noise k yes k k k k k yes k k k k k k k k k k           ( , , , , ) ( , , , , ) 2, 1 2, 1, 1, 2, 2, 1, 1 1, 1, 1, 2, 2, 2, 1 2, 2, 1, 1 1, 1, T T T T T G T T T T T G T T T G T T T G         f t noise f t noise t noise t noise k no k k k k k no k k k k k k k k k k           ( , , , , ) ( , , , , ) 2, 1 2, 1, 1, 2, 2, 1, 1 1, 1, 1, 2, 2, 2, 1 2, 2, 1, 1 1, 1, T T T T T G T T T T T G T T T G T T T G         µ µ T ball: a1 b1 Z1 x1 b2 Z2 x2 ( | , ) t 1 t p x a x  Actions Beliefs Observations Continuous states Observable Hidd en ( | ) t t p z x d1 d2 Discrete states Discrete transition now depends on the continuous state ( | , ) t 1 t t p d d x  (also, the observation function g depends on discrete state) Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 24 Discrete prediction in PHA Main idea: compute transition probability from continuous estimate constant for all xt-1 that satisfy a given guard ( ) 1 ( ) 1 ˆ , i t i xt C  true true true true false ( ... | ... ) ( | ... , ... ) ( ... | ... ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1         t t t t t t t t t t p d d a z a z p d d d a z a z p d d a z a z ( ... | ... ) 1 t1 1 1 t1 t1 p d d a z a z ( ... | ... ) 1 t1 t 1 1 t1 t1 p d d d a z a z O def t t t t t X p dt dt xt p x  d d  a z a  z  dx  P ³ 1 1 1 1 1 1 1 1 1 1 ( | , ) ( | ... , ... ) Cannot simplify as easily as before Instead: = 6 p(transition W) p(guard for W satisfied | previous estimate)