Lecture 5: Basic Dynamical Systems CS 344R/393R: Robotics - PDF document

Lecture 5: Basic Dynamical Systems CS 344R/393R: Robotics Benjamin Kuipers Dynamical Systems • A dynamical system changes continuously (almost always) according to n ˙ x = F ( x ) where x � � • A controller is defined to change the coupled robot and environment into a desired dynamical system. ˙ x = F ( x , u ) ˙ x = F ( x , H i ( G ( x ))) y = G ( x ) ˙ x = � ( x ) u = H i ( y ) 1

Two views of dynamic behavior • Time plot ( t,x ) • Phase portrait ( x,v ) Phase Portrait: ( x,v ) space • Shows the trajectory ( x ( t ), v ( t )) of the system – Stable attractor here 2

Interesting Phase Portrait • The van der Pol equation has a limit cycle. x � µ (1 � x 2 ) ˙ ˙ ˙ x + x = 0 In One Dimension ˙ • Simple linear system x ˙ x = kx • Fixed point ˙ x x = 0 � x = 0 • Solution kt x ( t ) = x 0 e – Stable if k < 0 – Unstable if k > 0 3

In Two Dimensions • Often, we have position and velocity: T v = ˙ x = ( x , v ) where x • If we model actions as forces, which cause acceleration, then we get: � ˙ � � ˙ � � � x x v ˙ x = � � � � � � � = � = � � � � ˙ ˙ ˙ v x forces � � � � � � The Damped Spring • The spring is defined by Hooke’s Law: F ma m x k x & & = = = � 1 • Include damping friction m x k x k x & & & = � 1 � 2 • Rearrange and redefine constants x b x cx 0 & & & + + = � � � � � � ˙ ˙ x x v ˙ x = � � � � � � � = � = � � � � ˙ ˙ ˙ � b ˙ v x x � cx � � � � � � 4

The Linear Spring Model ˙ x + b ˙ ˙ x + cx = 0 c � 0 • Solutions are: 1 t + Be r r 2 t x ( t ) = Ae • Where r 1 , r 2 are roots of the characteristic 2 + b � + c = 0 equation � b 2 � 4 c 1 , r 2 = � b ± r 2 Qualitative Behaviors b 2 � 4 c c 1 , r 2 = � b ± r unstable stable 2 spiral • Re( r 1 ), Re( r 2 ) < 0 nodal nodal b means stable. • Re( r 1 ), Re( r 2 ) > 0 unstable means unstable. • b 2 - 4 c < 0 means complex roots, means oscillations. 5

It’s the same in higher dimensions ˙ x = A x • The characteristic equation for generalizes to det( A � � I ) = 0 – This means that there is a vector v such that A v = � v • The solutions are called eigenvalues . � • The related vectors v are eigenvectors . 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. 6

Node Behavior Focus Behavior 7

Saddle Behavior Spiral Behavior (stable attractor) 8

Center Behavior (undamped oscillator) The Wall Follower ( x,y ) � 9

The Wall Follower • Our robot model: � � � � ˙ x v cos � � � � � ˙ ˙ x = y = F ( x , u ) = v sin � � � � � � � � � ˙ � � � � � � u = ( v ω ) T y =( y θ ) T θ ≈ 0. • We set the control law u = ( v ω ) T = H i ( y ) e = ˙ ˙ ˙ e = ˙ ˙ ˙ e = y � y set so y and y The Wall Follower • Assume constant forward velocity v = v 0 – approximately parallel to the wall: θ ≈ 0. • Desired distance from wall defines error: e = ˙ ˙ ˙ e = ˙ ˙ ˙ e = y � y set so y and y • We set the control law u = ( v ω ) T = H i ( y ) – We want e to act like a “damped spring” e + k 1 ˙ ˙ ˙ e + k 2 e = 0 10

The Wall Follower ˙ e + k 1 ˙ ˙ e + k 2 e = 0 • We want • For small values of θ ˙ ˙ e = y = v sin � v � � v cos � ˙ ˙ ˙ ˙ ˙ e = y = v � � � • Assume v=v 0 is constant. Solve for ω � � v 0 � � u = v � � � k 1 � � k 2 � = H i ( e , � ) � = � e � � � � v 0 � � – This makes the wall-follower a PD controller. Tuning the Wall Follower ˙ e + k 1 ˙ ˙ • The system is e + k 2 e = 0 2 � 4 k 2 = 0 • Critical damping requires k 1 k 4 k 2 1 = • Slightly underdamped performs better. – Set k 2 by experience. – Set k 1 a bit less than 4 k 2 11

An 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 PI control law, to adapt to motion. • Try different tuning values for parameters. – This is a simple model. – Unmodeled effects might be significant. 12

Ziegler-Nichols Tuning • Open-loop response to a unit step increase. • d is deadtime. T is the process time constant. • K is the process gain. K d T Tuning the PID Controller • We have described it as: t � � k D ˙ u ( t ) = � k P e ( t ) � k I edt e ( t ) 0 • Another standard form is: � t � � + T D ˙ u ( t ) = � P e ( t ) + T I edt e ( t ) � � � � 0 • Ziegler-Nichols says: P = 1.5 � T T I = 2.5 � d T D = 0.4 � d K � d 13

Ziegler-Nichols Closed Loop 1. Disable D and I action (pure P control). 2. Make a step change to the setpoint. 3. Repeat, adjusting controller gain until achieving a stable oscillation. • This gain is the “ultimate gain” K u . • The period is the “ultimate period” P u . Closed-Loop Z-N PID Tuning • A standard form of PID is: � t � � + T D ˙ u ( t ) = � P e ( t ) + T I edt e ( t ) � � � � 0 • For a PI controller: T I = P P = 0.45 � K u u 1.2 • For a PID controller: T I = P T D = P u u P = 0.6 � K u 2 8 14

Summary of Concepts • Dynamical systems and phase portraits • Qualitative types of behavior – Stable vs unstable; nodal vs saddle vs spiral – Boundary values of parameters • Designing the wall-following control law • Tuning the PI, PD, or PID controller – Ziegler-Nichols tuning rules – For more, Google: controller tuning 15