M.S. Thesis Presentation on

1. MIDDLE EAST TECHNICAL UNIVERSITY Aerospace Engineering Department M.S. Thesis Presentation on Steering of Redundant Robotic Manipulators and Spacecraft Integrated Power and Attitude Control-Control Moment Gyroscopes Presentation By : Alkan Altay Thesis Supervisor : Assoc. Prof. Dr. Ozan Tekinalp

2. Presentation Outline • Redundant Actuator Systems • Robotic Manipulator Simulations • IPAC-CMG Cluster & IPACS Simulations • IPAC-CMG Systems • Robotic Manipulators • Mechanical Analogy • Steering of Redundant Actuators • Inverse Kinematics Problem & Solutions • Blended Inverse Steering Logic • Thesis Work and Results • Conclusion & Future Work 2/34

3. IPAC – CMG Cluster A Variable Speed CMG That Stores Energy IPACS Integrated Power and Attitude Control System (IPACS) 3/34

4. Due to gimbal velocity Due to spin acceleration Integrated Power and Attitude Control - Control Moment Gyroscope (IPAC-CMG) • A CMG variant, whose flywheel spin rate is altered by a motor/generator 4/34

5. IPAC-CMG Cluster • Single IPAC-CMG, single direction • At least 3 IPAC-CMGs for 3-axis attitude control PYRAMID CONFIGURATION • 1 redundancy • Nearly spherical momentum envelope with β= 54.73 deg, 5/34

6. Robotic Manipulators • An actuator system composed of joints and series of segments • Tasked to travel its end-effector on a certain trajectory • Redundancy Applied To Increase Motion Capability • Mechanically analog to CMG cluster 6/34

7. The Mechanical Analogy 7/34

8. ? Inverse Kinematics Calculations Steering Laws Steer the actuator through the desired path Calculate the angular speed of each actuator Invert a rectangular matrix ? What if singular ? • Steering Laws For Redundant Systems • Minimum 2-Norm Solution • Singularity Avodiance Steering Logic • Singularity Robust Inverses 8/34

9. Moore Penrose Pseudo Inverse (Minimum 2-Norm Solution) • Minimum normed vector; the solution that requires minimum energy • Singularity is a problem • Most steering laws are variants of this pseudo inverse • OTHER SOLUTIONS : • Singularity Avoidance Steering Logic • Singularity Robust Inverse, Damped Least Squares Method • Extended Jacobian Method, Normal Form Approach, Modified Jacobian Method 9/34

10. where, and Q and R are symmetric positive definite weighting matrices Blended Inverse Satisfy two objectives; realize the desired path in desired configuration PROBLEM SOLUTION The proper desired quantity is injected through this term Pre-planned Steering 10/34

11. 3-link planar robot manipulatordynamics : Robotic Manipulator Simulations Direct Kinematical Relationship Steering Logic 11/34

12. Robotic Manipulator Simulations (Test Case I) • AIMS : • Repeatability performance of B-inverse on a routinely followed closed path • Tracking performance of B-inverse, when supplied with false 12/34

13. Robotic Manipulator Simulations (Test Case I –MP-inverse Results) 13/34

14. Robotic Manipulator Simulations (Test Case I –B-inverse Results) 14/34

15. Robotic Manipulator Simulations (Test Case II) • AIM : • The singularity avoidance performance of B-inverse • MP-inverse drives the system close to an escapable singularity at [ x1 , x2 ] = [-2 , 0 ] Escapable Singularity 15/34

16. Robotic Manipulator Simulations (Test Case II –MP-inverse Results) 16/34

17. Robotic Manipulator Simulations (Test Case II –B-inverse Results) 17/34

18. Robotic Manipulator Simulations (Test Case II – Results) Escapable Singularity Simulations Steering with B-inverse Steering with MP-inverse 18/34

19. Robotic Manipulator Simulations (Test Case III) • AIM : • Singularity transition performance of B-inverse • The path passes an inescapable singularity at [ x1 , x2 ] = [ 0 , 0 ] Inescapable Singularity 19/34

20. Robotic Manipulator Simulations (Test Case III –MP-inverse Results) 20/34

21. Robotic Manipulator Simulations (Test Case III –B-inverse Results) 21/34

22. Robotic Manipulator Simulations (Test Case III – Results) Inescapable Singularity Simulations Steering with B-inverse 22/34

23. IPAC-CMG Cluster Simulations Rate Command to each IPAC-CMG Torque and Power Commands Realized Torque and Power STEERING ALGORITHMS IPAC-CMG Cluster • AIMS : • Investigate the performance of IPAC-CMG cluster • Investigate the performance of B-inverse 23/34

24. IPAC-CMG Cluster Simulations Two different simulation models are employed to steer IPAC-CMG cluster Generic simulation model ( used in MP-inverse simulations ) B-inverse simulation model 24/34

25. IPAC-CMG Cluster Simulations Torque Command Power Command 25/34

26. Torque & Angular Momentum Realized Energy and Power Profiles Gimbal Angle History Flywheel Spin Rates Singularity Measure IPAC-CMG Cluster Simulations – MP-inverse Results 26/34

27. Energy and Power Profiles Singularity Measure Flywheel Spin Rates Gimbal Angle History Torque Error & Ang. Mom. Profile IPAC-CMG Cluster Simulations – B-inverse Results 27/34

28. IPACS Simulations 28/34

29. IPACS Simulations Spacecraft IPACS Simulation Model 29/34

30. IPACS Simulations Attitude Command Power Command 30/34

31. Gimbal Angles Attitude Profile IPAC-CMG Flywheel Spin Rates Energy and Power Profile Singularity Measure Torque and Angular Momentum History IPACS Simulations – MP-inverse Results 31/34

32. Gimbal Angles Attitude Profile IPAC-CMG Flywheel Spin Rates Singularity Measure Torque Error and Ang.Mom. Profile Energy and Power Profiles IPACS Simulations – B-inverse Results 32/34

33. Conclusion • B-inverse is employed in robotic manipulators : • Singularity Avoidance • Singularity Transition • Repeatability • IPACS is discussed : • Comparison to Current Technologies • Algorithm Construction • Theoretical Performance • B-inverse is employed in IPACS : • In IPAC-CMG Clusters & S/C IPACS • Singularity Avoidance & Multi Steering 33/34

34. Future Work B-inverse in highly redundant robotic mechanisms Capabilities of B-inverse Detail Design of IPAC-CMG 34/34

35. Singularity in Robotic Manipulators and CMG Systems • Physically, no end effector velocity (torque) can be produced in a certain direction • Controllability in that direction is lost. • Mathematically, Jacobian Matrix loses its rank.Thus; • det(J)= 0 ( or det(JJT)=0 ) • Singularity Measure m=det(JJT) • J-1 ( or (JJT)-1 ) becomes undefined #/30

36. Singularity Avoidance Steering Logic Particular Solution Homogeneous Solution Addition of null motion, n, in the proper amount (determined by γ) 12/40

37. 0 for m > mcr k0(1-m/m0)2 for m < mcr k = Singularity Robust Solutions Singularity Robust Inverse : • Disturbsthe pseudo solutionnear singularitiesto artificially generate a well –conditionedmatrix • Increases the tracking error, causes sharp velocity changes around singularities • Another example may be the Damped Least Squares Method 13/40

38. Extends the jacobian matrix with additional functions, creating a well –conditioned one, belonging to a “virtual” system square matrix singularity Proposes to transformthe kinematics to its quadratic normal form,employing equivalence transformation, around singularities  Proposes to replace the linearly dependent row of Jacobian Matrix, to remove the singularity, with a derivative of a configuration dependent function Singularity Robust Solutions New generation of solutions, offering accurate and smooth singularity transitions, not mature yet • Extended Jacobian Method • Normal Form Approach • Modified Jacobian Method 14/40

39. Thesis Objectives • Blended Inverse on Redundant Robotic Manipulators • Blended Inverse on IPAC-CMG clusters • Spacecraft Energy Storage & Attitude Control • IPAC-CMG based IPACS 3/40

40. Electrochemical Batteries vs. Flywheel Energy Storage Systems (FES) Spacecraft Energy Storage and Attitude Control • Rotating flywheels for smooth attitude control • Spacecraft store & drain energy periodically. • Integrate energy storage & attitude control 4/40

41. How to select ? Blended Inverse Pre-planned Steering 11/40