constrained near-optimal control using a numerical kinetic solver alan l. jennings & ra úl...

Constrained Near-Optimal Control Using a Numerical Kinetic Solver Alan L. Jennings & Raúl Ordóñez,

ajennings1, [email protected] and Computer Engineering, University of Dayton

Frederick G. Harmon, [email protected] 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

http://www.actapress.com/Abstract.aspx?paperId=41525



mailto:[email protected]?subject=Numerical%20Optimal%20Robotic%20Control



IASTED Robotics and Applications 2 of 15

The 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, 2010

http://www.autoblog.com/2010/05/11/volkswagen-and-stanfords-junior-autonomously-power-slides-int/

http://www.wired.com/images_blogs/dangerroom/2010/08/OE_in_orbit-660x443.jpg


The Method

1. Draw solid model describing the object.

2. Import into a kinetic model and verify.

3. Add outputs and inputs to interface to kinetic model.

4. Compose optimal control problem.

5. Run optimization.6. Inspect results.

• Optima

l Control

• Dynamics

• Mass &

joints

• Set up DIDO

• Draft project

• Set up Simulin

k

• What

does it look

like

• What are the controls

• What is

trying to be done


x(t), u(t) → g(t)

ψo

ϕ

J ψf

xo

xf

Xf

Xo


The Solid Model

• Draft pieces• As complex as desired• Assemble linkages• Scale density to match total weight, if individual inertia is not available• Provides visualization


2) Face constraintCo-axial constraint

Rotary joint

1) Draw parts

3) Repeat as needed


The Kinetic Model

• Generated from solid model assembly• Each rigid body has

MassMoment of inertia matrixRigid coordinate systems

• Joint relate adjacent CS’sRotary -> anglePrismatic ->

translationHybrid -> relation

• Sensors measure States or derivativesForces

• Actuators driveStates Forces


Added from importing

Add input and output sensors

Moving Link

Rotary Joint

Base

Animation of solid model

Many extra blocks available


Problem Scope

• Free initial & final states• Path constraints• Bolza problem• Rigid body linkages• Optimal solution exists

Limitations• Known system• Nonsingular• Only simple joints tested


x(t), u(t) → g(t)

ψo

ϕ

J ψf

xo

xf

Xf

Xo

General Optimal Control Problem

Rigid Body DynamicsSingular Example


0 0.5 10

2

4

u

u

u+ u

0 0.5 10

2

4

u

u

0 0.5 10

2

4

u

u

u+ u

0 0.5 10

2

4

u

u

0 0.5 10

2

4

u

0 0.5 10

2

4

u

Numeric Optimal Control


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:


Verify 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 reasonable

Propagate results and check deviationAdd more nodes orrescale problem



Example: Pendulum

• Suspended or inverted• Move from initial angle to equilibrium in fixed time• Minimum energy problem


www.mathworks.com/matlabcentral/fileexchange/28597

Equations of Motion

Cost function

The Truth

LQ Path controller

LQR Feedback controller

http://www.mathworks.com/matlabcentral/fileexchange/28597


Example: Pendulum

• DIDO has lowest cost• Suspended was harder for LQR• LQR can fail to reach final state





Example: Pendulum






Example: 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


http://www.mathworks.com/matlabcentral/fileexchange/28596Initial Pose

FinalPose

_√J from 45.7 Nm to 19.5 Nm,

57% reduction



Example: 4 DOF Arm

•Optimized Path: Lower gravity -> U Low inertia -> B Combining Torque -> R, θ Much more complex




IASTED Robotics and Applications

Thank 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 controller

Tuesday, Nov 2, 2010 15 of 15

constrained near-optimal control using a numerical kinetic solver alan l. jennings & ra úl...

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