Alan L Jennings amp Ra úl Ordóñez ajennings1 raulordoneznotesudaytonedu Electrical and Computer Engineering University of Dayton Frederick G Harmon frederickharmonafitedu ID: 801564
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Slide1
Constrained Near-Optimal Control Using a Numerical Kinetic Solver
Alan L. Jennings & Raúl Ordóñez, ajennings1, raul.ordonez@notes.udayton.eduElectrical and Computer Engineering, University of DaytonFrederick G. Harmon, frederick.harmon@afit.edu Dept. of Aeronautic and Astronautics, Air Force Institute of Technology
The views expressed in this article are those of the authors and
do not
reflect the official policy or position of the United States Air Force, Department of Defense, or the U. S. Government.
Tuesday, Nov 2, 2010
IASTED Robotics and Applications: 706-21
Slide2The Challenge
Multiple coordinate system transforms and degrees of freedom make robotic control via equations confusing and error prone.Optimal control equations are difficult to solve due to boundary conditions.
Desire higher energy efficiency.
Tuesday, Nov 2, 2010IASTED Robotics and Applications
2
Slide3The Method
Draw solid model describing the object.Import into a kinetic model and verify. Add outputs and inputs to interface to kinetic model.Compose optimal control problem. Run optimization.Inspect results.
Optimal Control
Dynamics
Mass & joints
Set up DIDO
Draft project
Set up Simulink
What does it look like
What are the controls
What is trying to be done
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IASTED Robotics and Applications
Tuesday, Nov 2, 2010
x(t), u(t) → g(t)
ψ
o
ϕ
J
ψ
f
x
o
x
f
X
f
X
o
Slide4The Solid Model
Draft pieces As complex as desired Assemble linkages Scale density to match total weight, if individual inertia is not available Provides visualization4IASTED Robotics and Applications
Tuesday, Nov 2, 2010
2) Face constraint
C
o-axial constraint
Rotary joint
1) Draw parts
3) Repeat as needed
Slide5The Kinetic Model
Generated from solid model assembly Each rigid body hasMassMoment of inertia matrixRigid coordinate systems Joint relate adjacent CS’sRotary -> anglePrismatic -> translationHybrid -> relation Sensors measure
States or derivativesForces Actuators driveStates
Forces 5
IASTED Robotics and ApplicationsTuesday, Nov 2, 2010
Added from importing
Add input and output sensors
Moving Link
Rotary Joint
Base
Animation of solid model
Many extra blocks available
Slide6Problem Scope
Free initial & final states Path constraints Bolza problem Rigid body linkages Optimal solution existsLimitations Known system Nonsingular Only simple joints tested
6IASTED Robotics and Applications
Tuesday, Nov 2, 2010
x(t), u(t) → g(t)
ψ
o
ϕ
J
ψ
f
x
o
x
f
X
f
X
o
General Optimal Control Problem
Rigid Body Dynamics
Singular Example
Slide7Numeric Optimal Control
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IASTED Robotics and Applications
Tuesday, Nov 2, 2010
the addition results in a higher cost.
The field of Calculus of variations
The Hamiltonian
Optimality conditions
States
Co-States
Control
For any function,
and any other function,
Discretize for:
Numeric, Constrained Nonlinear Optimization
The Link:
Slide8Verify Results
Should make senseExploit some system aspectVerify it is not maximum Not violate constraints Check for constraints that should be added or cost function revised Discretization and numeric error should be reasonablePropagate results and check deviationAdd more nodes orrescale problem
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IASTED Robotics and ApplicationsTuesday, Nov 2, 2010
Slide9Example: Pendulum
Suspended or inverted Move from initial angle to equilibrium in fixed time Minimum energy problem9
IASTED Robotics and ApplicationsTuesday, Nov 2, 2010
www.mathworks.com/matlabcentral/fileexchange/28597
Equations of Motion
Cost function
The Truth
LQ
Path controller
LQR
Feedback controller
Slide10Example: Pendulum
DIDO has lowest cost Suspended was harder for LQR LQR can fail to reach final state10IASTED Robotics and ApplicationsTuesday, Nov 2, 2010
www.mathworks.com/matlabcentral/fileexchange/28597
Slide11Example: Pendulum
11IASTED Robotics and ApplicationsTuesday, Nov 2, 2010www.mathworks.com/matlabcentral/fileexchange/28597
DIDO has lowest cost Suspended was harder for LQR LQR can fail to reach final state
Slide12Example: Pendulum
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IASTED Robotics and Applications
Tuesday, Nov 2, 2010
www.mathworks.com/matlabcentral/fileexchange/28597 DIDO has lowest cost Suspended was harder for LQR LQR can fail to reach final state
Slide13Example: 4 DOF Arm
Based on Motoman SIA-20D Traditional Method: Ramp to constant velocity Optimized Path: Move to low gravity, low inertia pose Use low torque maneuvers Much more complex
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IASTED Robotics and Applications
Tuesday, Nov 2, 2010http://www.mathworks.com/matlabcentral/fileexchange/28596
Initial
Pose
Final
Pose
_
√J from 45.7 Nm to 19.5 Nm,
57% reduction
Slide14Example: 4 DOF Arm
Optimized Path: Lower gravity -> U Low inertia -> B Combining Torque -> R, θ Much more complex14
IASTED Robotics and Applications
Tuesday, Nov 2, 2010
http://www.mathworks.com/matlabcentral/fileexchange/28596
Slide15Thank you for your Attention!
Optimized paths without specific robotic analysis or optimal control specialty______________________ ______________________Able to handle nonlinearities and stable or unstable systems ______________________ ______________________Offers improvement over path, feedback and another traditional controllerTuesday, Nov 2, 2010
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