Lecture 5:Basic Dynamical SystemsCS 344R: RoboticsBenjamin KuipersDynamical Systems• A dynamical system changes continuously(almost always) according to• A controller is defined to change thecoupled robot and environment into adesired dynamical system.! ˙ x = F(x) where x " #n! ˙ x = F (x,u)y = G(x)u = Hi(y)! ˙ x = F (x,Hi(G(x)))In One Dimension• Simple linear system• Fixed point• Solution– Stable if k < 0– Unstable if k > 0! ˙ x = kx! x = 0 "˙ x = 0! x! ˙ x ! x(t) = x0ektIn Two Dimensions• Often, position and velocity:• If actions are forces, causing acceleration:! x = (x,v)Twhere v =˙ x ! ˙ x =˙ x ˙ v " # $ $ % & ' ' =˙ x ˙ ˙ x " # $ $ % & ' ' =vforces" # $ $ % & ' ' !The Damped Spring• The spring is defined by Hooke’s Law:• Include damping friction• Rearrange and redefine constantsxkxmmaF1!===&&xkxkxm&&&21!!=0=++ cxxbx&&&! ˙ x =˙ x ˙ v " # $ $ % & ' ' =˙ x ˙ ˙ x " # $ $ % & ' ' =v(b˙ x ( cx" # $ $ % & ' 'The Linear Spring Model• Solutions are:• Where r1, r2 are roots of the characteristicequation! ˙ ˙ x + b˙ x + cx = 0! "2+ b"+ c = 0! r1,r2="b ± b2" 4c2! x(t) = Aer1t+ Ber2t! c " 0Qualitative Behaviors• Re(r1), Re(r2) < 0means stable.• Re(r1), Re(r2) > 0means unstable.• b2-4c < 0 meanscomplex roots,means oscillations.bcunstablestablespiralnodalnodalunstable! r1,r2="b ± b2" 4c2Generalize to Higher Dimensions• The characteristic equation forgeneralizes to– This means that there is a vector v such that• The solutions are called eigenvalues.• The related vectors v are eigenvectors.! ˙ x = Ax! det( A "#I ) = 0! Av ="v! "Qualitative Behavior, Again• For a dynamical system to be stable:– The real parts of all eigenvalues must be negative.– All eigenvalues lie in the left half complex plane.• Terminology:– Underdamped = spiral (some complex eigenvalue)– Overdamped = nodal (all eigenvalues real)– Critically damped = the boundary between.NodeBehaviorFocusBehaviorSaddleBehaviorSpiralBehavior(stableattractor)CenterBehavior(undampedoscillator)The Wall Follower(x,y)! "The Wall Follower• Our robot model: u = (v ω)T y=(y θ)T θ ≈ 0.• We set the control law u = (v ω)T = Hi(y)! ˙ x =˙ x ˙ y ˙ " # $ % % % & ' ( ( ( = F (x, u) =v cos"v sin")# $ % % % & ' ( ( ( ! e = y " ysetso˙ e =˙ y and˙ ˙ e =˙ ˙ yThe Wall Follower• Assume constant forward velocity v = v0– approximately parallel to the wall: θ ≈ 0.• Desired distance from wall defines error:• We set the control law u = (v ω)T = Hi(y)– We want e to act like a “damped spring”! e = y " ysetso˙ e =˙ y and˙ ˙ e =˙ ˙ y ! ˙ ˙ e + k1˙ e + k2e = 0The Wall Follower• We want• For small values of θ• Assume v=v0 is constant. Solve for ω– This makes the wall-follower a PD controller.! ˙ ˙ e + k1˙ e + k2e = 0! ˙ e =˙ y = v sin"# v"˙ ˙ e =˙ ˙ y = v cos"˙ " # v$! u =v"# $ % & ' ( =v0)k1*)k2v0e# $ % % & ' ( ( = Hi(e,*)Tuning the Wall Follower• The system is• Critically damped is• Slightly underdamped performs better.– Set k2 by experience.– Set k1 a bit less than! ˙ ˙ e + k1˙ e + k2e = 0! k12" 4k2= 0! k1= 4k2! 4k2An Observer for Distance to Wall• Short sonar returns are reliable.– They are likely to be perpendicular reflections.Experiment with Alternatives• The wall follower is a PD control law.• A target seeker should probably be a PIcontrol law, to adapt to motion.• Try different tuning values for parameters.– This is a simple model.– Unmodeled effects might be significant.Ziegler-Nichols Tuning• Open-loop response to a step increase.d TKZiegler-Nichols Parameters• K is the process gain.• T is the process time constant.• d is the deadtime.Tuning the PID Controller• We have described it as:• Another standard form is:• Ziegler-Nichols says:! u(t) = "kPe(t) " kIe dt0t#" kD˙ e (t)! u(t) = "P e(t) + TIe dt0t#+ TD˙ e (t)$ % & ' ( ) ! P =1.5 " TK " dTI= 2.5 " d TD= 0.4 " dZiegler-Nichols Closed Loop1. Disable D and I action (pure P control).2. Make a step change to the setpoint.3. Repeat, adjusting controller gain untilachieving a stable oscillation.• This gain is the “ultimate gain” Ku.• The period is the “ultimate period” Pu.Closed-Loop Z-N PID Tuning• A standard form of PID is:• For a PI controller:• For a PID controller:! u(t) = "P e(t) + TIe dt0t#+ TD˙ e (t)$ % & ' ( ) ! P = 0.45 " KuTI=Pu1.2! P = 0.6 "
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