Lecture 5: Basic Dynamical Systems

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Lecture 5Basic Dynamical Systems

• CS 344R Robotics
• Benjamin Kuipers

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Dynamical Systems

• A dynamical system changes continuously (almost
  always) according to
• A controller is defined to change the coupled
  robot and environment into a desired dynamical
  system.

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In One Dimension

• Simple linear system
• Fixed point
• Solution
• Stable if k lt 0
• Unstable if k gt 0

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In Two Dimensions

• Often, position and velocity
• If actions are forces, causing acceleration

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The Damped Spring

• The spring is defined by Hookes Law
• Include damping friction
• Rearrange and redefine constants

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The Linear Spring Model

• Solutions are
• Where r1, r2 are roots of the characteristic
  equation

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Qualitative Behaviors
c
unstable
stable

• Re(r1), Re(r2) lt 0 means stable.
• Re(r1), Re(r2) gt 0 means unstable.
• b2-4c lt 0 means complex roots, means oscillations.

spiral
nodal
nodal
b
unstable
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Generalize to Higher Dimensions

• The characteristic equation for
  generalizes to
• This means that there is a vector v such that
• The solutions are called eigenvalues.
• The related vectors v are eigenvectors.

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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.

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Node Behavior
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Focus Behavior
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Saddle Behavior
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Spiral Behavior(stable attractor)
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Center Behavior(undamped oscillator)
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The Wall Follower
(x,y)
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The Wall Follower

• Our robot model
• u (v ?)T y(y ?)T ? ? 0.
• We set the control law u (v ?)T Hi(y)

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The Wall Follower

• Assume constant forward velocity v v0
• approximately parallel to the wall ? ? 0.
• Desired distance from wall defines error
• We set the control law u (v ?)T Hi(y)
• We want e to act like a damped spring

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The Wall Follower

• We want
• For small values of ?
• Assume vv0 is constant. Solve for ?
• This makes the wall-follower a PD controller.

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Tuning the Wall Follower

• The system is
• Critically damped is
• Slightly underdamped performs better.
• Set k2 by experience.
• Set k1 a bit less than

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An Observer for Distance to Wall

• Short sonar returns are reliable.
• They are likely to be perpendicular reflections.

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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.

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Ziegler-Nichols Tuning

• Open-loop response to a step increase.

K
d
T
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Ziegler-Nichols Parameters

• K is the process gain.
• T is the process time constant.
• d is the deadtime.

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Tuning the PID Controller

• We have described it as
• Another standard form is
• Ziegler-Nichols says

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Ziegler-Nichols Closed Loop

• Disable D and I action (pure P control).
• Make a step change to the setpoint.
• Repeat, adjusting controller gain until achieving
  a stable oscillation.
• This gain is the ultimate gain Ku.
• The period is the ultimate period Pu.

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Closed-Loop Z-N PID Tuning

• A standard form of PID is
• For a PI controller
• For a PID controller