Open loop vs Closed Loop. Example: Open Loop. Example: Feedforward Control. Advanced Control I

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1 Open loop vs Closed Loop Advanced I Moor Command Movemen Overview Open Loop vs Closed Loop Some examples Useful Open Loop lers Dynamical sysems CPG (biologically inspired ), Force Fields Feedback conrol PID design 2 nd order sysems Feedforward conrol Transfer funcions Dynamics esimaion To Do: Revise IVR noes on PID conrol Advanced I Advanced II IAR - Dr. Sehu Vijayakumar 1 IAR - Dr. Sehu Vijayakumar 2 Example: Open Loop Example: Feedforward chooses acions (ses inpus) based on requiremens sysem knows naural process dynamics Assumes disurbances can be negleced Law Inpus Process Disurbances Room Heaing θ emp = θ 0 θ Q= L( θ θ0) + C Q = L( θ ) θ = arge disurbance L = process dynamics (assumed perfec knowledge) Wha is he seady sae emperaure ignoring disurbance? IAR - Dr. Sehu Vijayakumar 3 Q sensed curren disurbance Inpus Acions chosen using arges + sensed (unconrolled) sae variables sysem knows dynamics Assumes accurae process model (including disurbance dynamics) Law 0 Process Disurbances Room Heaing emp = θ Q θ Q = L( θ Q= L( θ θ0) + C θ ˆ 0 ) ˆθ 0 = measured disurbance...compensaes for varying θ and assumes accurae L(dynamics) θ 0 IAR - Dr. Sehu Vijayakumar 4 1

2 Example: Feedback (I) Example: Feedback (II) Sensory feedback Inpus Process Disurbances Acions chosen using arges + oucomes (via sensory feedback) ler ofen jus compares and uses error signals Accurae process model is no a pre-requisie In mos cases, desired are used in conjuncion wih he feedback o generae commands (see below ) Targes Error Room Heaing emp = θ Q θ 0 Feedback conrol changes dynamics θ Q= L( θ θ0) + C Law : Q = k( θ θ) k = gain consan kθ Seady Sae : Lθ 0 θ = + L + k L + k Time Consan : C τ= (L+ k) A high gain k ensures ha he seady sae emperaure is nearly he desired one and also ha i is aained quickly. Verify? IAR - Dr. Sehu Vijayakumar 5 IAR - Dr. Sehu Vijayakumar 6 Where can i be useful? Open Loop In cases which require fas, reacive response In cases where you do repeiive moion which does no need modulaion from he sensory inpu Wha are examples? Cenral Paern Generaors, Force fields ec. Use none or minimal sensory feedback Used in reflex acions and movemen in simple life forms (and no so simple ones oo!!) We will use animal locomoion as a case sudy o explore hese ideas Movemen (locomoion) Large diversiy of differen ypes of locomoion: swimming, crawling, walking, hoping, burrowing, flying, IAR - Dr. Sehu Vijayakumar 7 IAR - Dr. Sehu Vijayakumar 8 2

3 Locomoion: A difficul & Ill-posed problem Wha Are Dynamical s? Due o large redundancies and muliple degrees of freedom: Many possible end-poin rajecories Many possible posures for a given end-poin Many possible muscle acivaions for a given posure Many possible moor uni aciviaions for a given muscle acivaions Addiionally: Efficien locomoion is only obained when all degrees of freedom (DOF) are correcly coordinaed Need for righ frequencies, phases, ampliudes and signal shapes for all DOFs A dynamical sysem is a deerminisic process in which a funcion's value changes over ime according o a rule ha is defined in erms of he funcion's curren value. This is ofen formalized as a differenial equaion Differenial equaion: an equaion ha describes how sae variables evolve over ime, for insance: = α( c y) The redundancy and coordinaion problems are solved using he concep of paern generaors, i.e. dynamical sysems ha produce complex paerns while being conrolled by simple inpu signals IAR - Dr. Sehu Vijayakumar 9 IAR - Dr. Sehu Vijayakumar 10 Ineresing Regimes of Differenial Equaions Unsable node From Srogaz 1994 Linear differenial equaion: differenial equaion in which he sae variables only appear in linear combinaions Nonlinear differenial equaion: differenial equaion in which some sae variables appear in nonlinear combinaions (e.g. producs, cosine, ) Fixed poin: poin a which all derivaives are zero (can be an aracor, a repeller, or a saddle poin, cf laer) Limi cycle: periodic isolaed closed rajecory (can only occur in nonlinear sysems) Firs order linear sysem: How o solve his equaion, for a given y(=0), c, and α? Two mehods: analyical soluion or numerical inegraion Analyical soluion: Firs Order Linear s = α( c y) y ( ) = ( y0 c)exp( α) + c saddles Numerical inegraion: Euler mehod, Runge-Kua, Aracor Aracors Limi cycles Chaos IAR - Dr. Sehu Vijayakumar 11 IAR - Dr. Sehu Vijayakumar 12 3

4 Firs Order Linear s = α( c y) Second Order Linear s = α(β ( c x) y) x& = y y ( ) = ( y0 c)exp( α) + c IAR - Dr. Sehu Vijayakumar 13 IAR - Dr. Sehu Vijayakumar 14 Second Order Linear s = α(β ( c x) y) x& = y α = 8 β = 8 Second Order Non-linear = 2cos x cos y x& = 2cos y cos x Repeller α = 8 β = 2 α = 8 β = 1 Aracor saddles IAR - Dr. Sehu Vijayakumar 15 IAR - Dr. Sehu Vijayakumar 16 4

5 Oupu signal: Basic oscillaor: τ r& = µ ( r r0 ) τ & φ = 1 τ z& = α( β ( y i τ = z + r T I = [ x, v] m N y) z) r r T ψ w I = 1 i N [ φ, x, v] i= 1 ψ A Basic CPG oscillaor i i IAR - Dr. Sehu Vijayakumar 17 Neuromechanical Simulaion of Robo Neuromechanical simulaion: simulaion of boh he body and he neural conroller. Two-dimensional biomechanical model (Ekeberg 1993): ariculaed rigid body wih spring-and-damper muscles Couresy: Dr. Auke Jan Ijspeer, EPFL IAR - Dr. Sehu Vijayakumar 18 Lamprey Segmenal Nework An evolved swimming conroller Brain sem Brain sem The swimming paern is induced by simple onic simulaion from he brainsem. Increasing he simulaion leads o an increase of he frequency, hence of he speed of swimming IAR - Dr. Sehu Vijayakumar 19 IAR - Dr. Sehu Vijayakumar 20 5

6 Direcion wih Sensory Inpu Feedforward componen in Open Loop Addiional informaion abou sae variables (dynamics) help solve he problem Problems wih changing environmen dynamics of direcion is obained by asymmerical simulaion beween lef and righ sides of he spinal cord IAR - Dr. Sehu Vijayakumar 21 Sensory feedback is crucial o coordinae he neural aciviy wih he body movemens, especially in non-saionary condiions IAR - Dr. Sehu Vijayakumar 22 Neural conrol of discree movemen Summary Conceps of force fields (Bizzi e al): Simulaion of he spinal cord movemen owards an equilibrium poin (from any iniial condiions) (Lemay e al 2001) Simulaion of wo sies linear superposiion of he wo force fields IAR - Dr. Sehu Vijayakumar 23 We have compared and conrased beween open loop and closed loop conrol We have looked a examples of: Open loop conrol Feedforward conrol Feedback conrol We have looked a open loop conrol in deail CPGs using dynamical sysems were reviewed Basics of dynamical sysems Applicaion o locomoion conrol The idea of force fields was inroduced We will look a Feedback and Feedforward conrol in more deail in he nex class (2 nd order sysems and ransfer funcions) IAR - Dr. Sehu Vijayakumar 24 6