Iterative Learning Control of Perspective Dynamic Systems

1
Iterative Learning Control of Perspective Dynamic
Systems

• Lili Ma, Kevin L. Moore, and YangQuan Chen
• Virginia Tech Center for Autonomous Systems
  (VaCAS)
• G.A. Dobelman Distinguished Professor of
  Engineering
• Colorado School of Mines
• 1610 Illinois Street, Golden, CO 80401
• Center for Self-Organizing and Intelligent
  Systems (CSOIS)
• Dept. of Electrical and Computer Engineering
• 4160 Old Main Hill, Utah State University, Logan,
  UT 84322-4160, USA

2006 CCA/CACSD/ISIC Technische University
Munchen, Munich, Germany, October 4-6, 2006
TexPoint fonts used in EMF AAAAAAAAAAAAAA
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Colorado School of Mines Located in Golden,
Colorado, USA 10 miles West of Denver
CSM sits in the foothills of the Rocky Mountains

• CSM has about 300 faculty and 4000 students
• CSM is a public research institution devoted to
  engineering and applied science, especially
• Discovery and recovery of resources
• Conversion of resources to materials and energy
• Utilization in advanced processes and products
• Economic and social systems necessary to ensure
  
• prudent and provident use of resources in a
• sustainable global society

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Motivation

• To use visual feedback measurements instead of
  encoder information for ILC.
• The motion dynamics of the object are assumed to
  be linear.
• The visual measurements are homogeneous since a
  perspective camera model is used.
• Notations
• ILC Iterative learning control
• PDS Perspective dynamic system

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ILC Consider Systems that Execute the Same
Trajectory Over and Over
Step 1 Robot at rest, waiting for workpiece.
Step 2 Workpiece moved into position.
Step 3 Robot moves to desired location
and executes its task.
Step 4 Robot returns to rest and
waits for next workpiece.
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Errors are Repeated WhenTrajectories are
Repeated

• A typical joint angle trajectory for the
  example might look like this
• Each time the system is operated it will see
  the same overshoot, rise time,
• settling time, and steady-state error.
• Iterative learning control attempts to improve
  the transient response by
• adjusting the input to the plant during future
  system operation based on the
• errors observed during past operation.

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Iterative Learning Control
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An ILC Example
Target Path
Laser Pointer
Camera
Image Capture ILC Algorithm Motor Control
(two computers)
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Gimbal Motion (k 1)
Gimbal Motion (k 4)
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Introduction PDS
A perspective dynamic system is described
by where the projective observation function
is defined as where is the homogeneous
line spanned by the nonzero vector
. The set is defined as
A PDS is a linear system with homogeneous
observation.
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Introduction PDS (cont.)
An example of the PDS when the object is moving
according to an affine motion can be described
as with and
the outputs defined to be where
3-D position of the moving
object in the camera-centered 3D
space. projection in the camera
frame that can be derived from
observations on the image plane.
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Problem of Concern

• Consider an object that is moving along a 3-D
  trajectory in camera-centered 3-D space whose
  motion is observed via a camera.
• Suppose that the motion is iterative.
• THE QUESTION OF CONCERN is whether the
  observations on the image plane suffice to refine
  the 3-D motion trajectory from trial-to-trial.

Our preliminary study shows that the homogeneous
output from the image plane can help to improve
the system performance under the IIC precondition.
IIC Identical Initial Condition
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Problem Formulation

• Given
• Object moving according to a known, iterative
  motion.
• Desired observed trajectory on the image plane of
  a stationary camera.
• Desired actual trajectory in the object space.
• Calibrated camera whose parameters, such as the
  camera's intrinsic parameters, distortion
  coefficients, and focal length, are known.
• Task
• Track desired trajectories (both in 2-D and 3-D)
  in the presence of repeatable uncertainties,
  where the uncertainties could arise from
  calibration of the camera, measurements on the
  image plane, and about controlling the plant.
• Additional Condition
• With or without the identical initial condition
  (IIC).

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Preliminary Study

• Consider a 2-D ILC-PDS system described as
  follows
• Motion Dynamics
• Homogeneous Output
• where the index denotes the trial number.
• Control Problem Control the plant to track a
  desired 2-D trajectory,
• by adjusting the input
• from trial to trial, using information
• which is the information available from the
  image plane.

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Intuitive Idea
Before theoretically proving the feasibility, the
following simulations are conducted to give an
intuitive idea of the problem.

• Motion Dynamics
• ILC Updating Law
• Other Selections
• Notice that
• The learning rates are chosen by trial and error.
• No knowledge of the plant dynamics ( ) are
  used.
• Derivative of the error is used in the updating
  law.

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Simulation Results
With IIC
Without IIC
1-D Observation
Tracking in 2-D object space is achieved.
Tracking in 2-D object space is NOT achieved.
2-D Object Space
Results presented are after 41th iterations.
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Simulation Results (cont.)
With IIC
Without IIC
Error does not go to zero.
shows a monotone convergence in both
cases. The error does not go to zero when the IIC
condition is not satisfied.
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Corollary
References
1 I. Toshiharu Sugie Toshiro Ono, An
Iterative Learning Control Law for Dynamical
Systems, Automatica, vol. 27, no. 4, pp.
729-732, 1991
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Theory
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Sketched Proof
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Conclusions

• Conclusions
• The problem of iterative learning control for
  perspective dynamic systems is formulated.
• For 2-D motion with 1-D perspective measurements,
  ILC can be used to force the system to a desired
  motion using only the perspective measurement,
  provided that the identical initial condition
  (IIC) is satisfied.
• When the IIC condition is violated, the above
  statement is no longer true.
• Question
• If the IIC condition can be relaxed and at what
  expense?
• Next Step
• Experiment

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References

• Kevin L. Moore, Iterative Learning Control An
  Expository Overview'', Applied and Computational,
  Controls, Signal Processing, and Circuits, vol.
  1, no.1, pp. 425-488, 1998.
• Bijoy K. Ghosh and Clyde F. Martin, Homogeneous
  Dynamical Systems Theory'', IEEE Transactions on
  Automatic Control, vol. 47, no. 3, pp. 462-472,
  2002.
• Bijoy K. Ghosh and Hiroshi Inaba and Satoru
  Takahashi, Identification of Riccati Dynamics
  Under Perspective and Orthographic
  Observations'', IEEE Transactions on Automatic
  Control, vol. 45, no. 7, pp. 1267-1278, 2000.
• Mrdjan Jankovic and Bijoy K. Ghosh, Visually
  Guided Ranging from Observations of Points, Lines
  and Curves via An Identifier Based nonlinear
  Observer'', vol. 25, pp. 63-73, 1995.
• Joao P. Hespanha, State Estimation and Control
  for Systems with Perspective Outputs'', CDC,
  2002.
• Lili Ma, Vision-Based Measurements for Dynamic
  Systems and Control, Ph.D. dissertation, Utah
  State University, Dec. 2004.
• I. Toshiharu Sugie Toshiro Ono, An Iterative
  Learning Control Law for Dynamical Systems,
  Automatica, vol. 27, no. 4, pp. 729-732, 1991.