1. Constrained Near-Optimal Control Using a Numerical Kinetic Solver Alan L. Jennings & Ra úl Ordóñez, ajennings1, raul.ordonez@notes.udayton.edu ajennings1raul.ordonez@notes.udayton.edu Electrical and Computer Engineering, University of Dayton Frederick G. Harmon, frederick.harmon@afit.edufrederick.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, 2010IASTED Robotics and Applications: 706-21

2. 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, 2010IASTED Robotics and Applications2

3. 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. 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 3IASTED Robotics and ApplicationsTuesday, Nov 2, 2010 x(t), u(t) → g(t) ψoψo ϕ J ψfψf xoxo xfxf XfXf XoXo

4. 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 4IASTED Robotics and ApplicationsTuesday, Nov 2, 2010 2) Face constraint Co-axial constraint Rotary joint 1) Draw parts 3) Repeat as needed

5. The Kinetic Model Generated from solid model assembly Each rigid body has Mass Moment of inertia matrix Rigid coordinate systems Joint relate adjacent CS’s Rotary-> angle Prismatic -> translation Hybrid -> relation Sensors measure States or derivatives Forces Actuators drive States Forces 5IASTED 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

6. Problem Scope Free initial & final states Path constraints Bolza problem Rigid body linkages Optimal solution exists Limitations Known system Nonsingular Only simple joints tested 6IASTED Robotics and ApplicationsTuesday, Nov 2, 2010 x(t), u(t) → g(t) ψoψo ϕ J ψfψf xoxo xfxf XfXf XoXo General Optimal Control Problem Rigid Body Dynamics Singular Example

7. Numeric Optimal Control 7IASTED Robotics and ApplicationsTuesday, 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:

8. Verify Results Should make sense Exploit some system aspect Verify 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 deviation Add more nodes or rescale problem 8IASTED Robotics and ApplicationsTuesday, Nov 2, 2010

9. Example: Pendulum Suspended or inverted Move from initial angle to equilibrium in fixed time Minimum energy problem 9IASTED 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

10. Example: Pendulum DIDO has lowest cost Suspended was harder for LQR LQR can fail to reach final state 10IASTED Robotics and ApplicationsTuesday, Nov 2, 2010 www.mathworks.com/matlabcentral/fileexchange/28597

11. Example: Pendulum 11IASTED Robotics and ApplicationsTuesday, 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

12. Example: Pendulum 12IASTED Robotics and ApplicationsTuesday, 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

13. 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 13IASTED Robotics and ApplicationsTuesday, Nov 2, 2010 http://www.mathworks.com/matlabcentral/fileexchange/28596 Initial Pose Final Pose _ √J from 45.7 Nm to 19.5 Nm, 57% reduction

14. Example: 4 DOF Arm Optimized Path: Lower gravity -> U Low inertia -> B Combining Torque -> R, θ Much more complex 14IASTED Robotics and ApplicationsTuesday, Nov 2, 2010 http://www.mathworks.com/matlabcentral/fileexchange/28596

15. 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, 2010IASTED Robotics and Applications15 of 15