MIDDLE EAST TECHNICAL UNIVERSITY Aerospace Engineering Department M

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 M. S. Seminar – METU Aerospace Engineering Department January 2006

Presentation Outline n n Redundant Actuator Systems n IPAC-CMG Systems n Robotic Manipulators n Mechanical Analogy Steering of Redundant Actuators n Inverse Kinematics Problem & Solutions n Blended Inverse Steering Logic Thesis Work and Results n Robotic Manipulator Simulations n IPAC-CMG Cluster & IPACS Simulations Conclusion & Future Work M. S. Seminar – METU Aerospace Engineering Department January 2006 2/34

Integrated Power and Attitude Control System (IPACS) IPAC – CMG Cluster A Variable Speed CMG That Stores Energy IPACS M. S. Seminar – METU Aerospace Engineering Department January 2006 3/34

Integrated Power and Attitude Control Moment Gyroscope (IPAC-CMG) • A CMG variant, whose flywheel spin rate is altered by a motor/generator Due to spin acceleration M. S. Seminar – METU Aerospace Engineering Department January 2006 Due to gimbal velocity 4/34

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, M. S. Seminar – METU Aerospace Engineering Department January 2006 5/34

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 M. S. Seminar – METU Aerospace Engineering Department January 2006 6/34

The Mechanical Analogy Total Ang. Mom. Position IPAC-CMG Momentum Link Length Torque End Effector Velocity Steering Problem M. S. Seminar – METU Aerospace Engineering Department January 2006 7/34

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 ? Ø Minimum 2 -Norm Solution Ø Singularity Avodiance Steering Logic Ø Singularity Robust Inverses M. S. Seminar – METU Aerospace Engineering Department January 2006 8/34

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 M. S. Seminar – METU Aerospace Engineering Department January 2006 9/34

Blended Inverse Satisfy two objectives; realize the desired path in desired configuration PROBLEM SOLUTION where, and Q and R are symmetric positive definite weighting matrices The proper desired quantity is injected through this term Pre-planned Steering M. S. Seminar – METU Aerospace Engineering Department January 2006 10/34

Robotic Manipulator Simulations 3 -link planar robot manipulator dynamics : Direct Kinematical Relationship Steering Logic M. S. Seminar – METU Aerospace Engineering Department January 2006 11/34

Robotic Manipulator Simulations Case I) (Test AIMS : • Repeatability performance of B-inverse on a routinely followed closed path • Tracking performance of B-inverse, when supplied with false M. S. Seminar – METU Aerospace Engineering Department January 2006 12/34

Robotic Manipulator Simulations Case I –MP-inverse Results) M. S. Seminar – METU Aerospace Engineering Department January 2006 (Test 13/34

Robotic Manipulator Simulations Case I –B-inverse Results) M. S. Seminar – METU Aerospace Engineering Department January 2006 (Test 14/34

Robotic Manipulator Simulations Case II) (Test AIM : • The singularity avoidance performance of B-inverse • MP-inverse drives the system close to an escapable singularity at [ x 1 , x 2 ] = [-2 , 0 ] Escapable Singularity M. S. Seminar – METU Aerospace Engineering Department January 2006 15/34

Robotic Manipulator Simulations Case II –MP-inverse Results) M. S. Seminar – METU Aerospace Engineering Department January 2006 (Test 16/34

Robotic Manipulator Simulations Case II –B-inverse Results) M. S. Seminar – METU Aerospace Engineering Department January 2006 (Test 17/34

Robotic Manipulator Simulations Case II – Results) (Test Escapable Singularity Simulations Steering with MP-inverse Steering with B-inverse M. S. Seminar – METU Aerospace Engineering Department January 2006 18/34

Robotic Manipulator Simulations Case III) (Test AIM : • Singularity transition performance of B-inverse • The path passes an inescapable singularity at [ x 1 , x 2 ] = [ 0 , 0 ] Inescapable Singularity M. S. Seminar – METU Aerospace Engineering Department January 2006 19/34

Robotic Manipulator Simulations Case III –MP-inverse Results) M. S. Seminar – METU Aerospace Engineering Department January 2006 (Test 20/34

Robotic Manipulator Simulations Case III –B-inverse Results) M. S. Seminar – METU Aerospace Engineering Department January 2006 (Test 21/34

Robotic Manipulator Simulations Case III – Results) (Test Inescapable Singularity Simulations Steering with B-inverse M. S. Seminar – METU Aerospace Engineering Department January 2006 22/34

IPAC-CMG Cluster Simulations Torque and Power Commands Rate Command to each IPAC-CMG Realized Torque and Power STEERING ALGORITHMS IPAC-CMG Cluster AIMS : • Investigate the performance of IPAC-CMG cluster • Investigate the performance of B-inverse M. S. Seminar – METU Aerospace Engineering Department January 2006 23/34

IPAC-CMG Cluster Simulations Two different simulation models are employed to steer IPAC-CMG cluster Generic simulation model B-inverse simulation model ( used in MP-inverse simulations ) M. S. Seminar – METU Aerospace Engineering Department January 2006 24/34

IPAC-CMG Cluster Simulations Torque Command Power Command Min Ang. Mom. of each IPAC-CMG [Nms] IPAC-CMG Flywh. Spin Interval [k. RPM] Initial Flywheel Spin Rates (k. RPM) Initial Gimbal Angles (deg) M. S. Seminar – METU Aerospace Engineering Department January 2006 7. 7 15 – 60 [40, 40, 40] [0, 0, 0, 0] 25/34

IPAC-CMG Cluster Simulations – inverse Results MP- Flywheel Spin Rates and Angle Power Profiles Torque. Energy & Angular Momentum Realized Gimbal History Singularity Measure M. S. Seminar – METU Aerospace Engineering Department January 2006 26/34

IPAC-CMG Cluster Simulations – inverse Results B- Torque Singularity Error & Ang. Measure Mom. Profile Energy and Power Profiles Gimbal Flywheel Angle Spin History Rates M. S. Seminar – METU Aerospace Engineering Department January 2006 27/34

IPACS Simulations Spacecraft Inertias [ kgm 2 ] [15, 10] Initial Orientation of S/C [deg] [0, 0, 0] IPAC-CMG Flywh. Spin Interval [k. RPM] 15 - 60 Initial Flywheel Spin Rates [k. RPM] Initial Gimbal Angles [deg] [39, 40, 41, 42] [-75, 0, 75, 0] M. S. Seminar – METU Aerospace Engineering Department January 2006 28/34

IPACS Simulations Spacecraft IPACS Simulation Model M. S. Seminar – METU Aerospace Engineering Department January 2006 29/34

IPACS Simulations Attitude Command Power Command M. S. Seminar – METU Aerospace Engineering Department January 2006 30/34

IPACS Simulations – -inverse Results MP Energy IPAC-CMG and Attitude Power Flywheel Profile Spin Rates Gimbal Angles Torque and. Singularity Angular Momentum History Measure M. S. Seminar – METU Aerospace Engineering Department January 2006 31/34

IPACS Simulations – inverse Results B- IPAC-CMG Flywheel Spin Rates Torque Energy Error Gimbal and Attitude Power Ang. Mom. Angles Profiles Profile Singularity Measure M. S. Seminar – METU Aerospace Engineering Department January 2006 32/34

Conclusion • B-inverse is employed in robotic manipulators : ü ü ü • Singularity Transition Repeatability IPACS is discussed : ü ü ü • Singularity Avoidance Comparison to Current Technologies Algorithm Construction Theoretical Performance B-inverse is employed in IPACS : ü ü In IPAC-CMG Clusters & S/C IPACS Singularity Avoidance & Multi Steering M. S. Seminar – METU Aerospace Engineering Department January 2006 33/34

Future Work B-inverse in highly redundant robotic mechanisms Detail Design of IPAC-CMG Capabilities of B-inverse M. S. Seminar – METU Aerospace Engineering Department January 2006 34/34

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; 1. det(J)= 0 ( or det(JJT)=0 ) 2. Singularity Measure m=det(JJT) 3. J-1 ( or (JJT)-1 ) becomes undefined M. S. Seminar – METU Aerospace Engineering Department January 2006 #/30

Singularity Avoidance Steering Logic Particular Solution Homogeneous Solution Addition of null motion, n, in the proper amount (determined by γ) M. S. Seminar – METU Aerospace Engineering Department January 2006 12/40

Singularity Robust Solutions Singularity Robust Inverse : 0 k= for m > mcr k 0(1 -m/m 0)2 for m < mcr • Disturbs the pseudo solution near singularities to artificially generate a well –conditioned matrix • Increases the tracking error, causes sharp velocity changes around singularities • Another example may be the Damped Least Squares Method M. S. Seminar – METU Aerospace Engineering Department January 2006 13/40

Singularity Robust Solutions New generation of solutions, offering accurate and smooth singularity transitions, not mature yet • Extended Jacobian Method Extends the jacobian matrix with additional functions, creating a well – conditioned one, belonging to a “virtual” system • singularity Normal Form Approach Proposes to transform the kinematics to its quadratic normal form, employing equivalence transformation, around singularities • square matrix Modified Jacobian Method Proposes to replace the linearly dependent row of Jacobian Matrix, to remove the singularity, with a derivative of a configuration dependent function M. S. Seminar – METU Aerospace Engineering Department January 2006 14/40

Thesis Objectives n Blended Inverse on Redundant Robotic Manipulators n Spacecraft Energy Storage & Attitude Control n IPAC-CMG based IPACS n Blended Inverse on IPACCMG clusters M. S. Seminar – METU Aerospace Engineering Department January 2006 3/40

Spacecraft Energy Storage and Attitude Control • Rotating flywheels for smooth attitude control • Spacecraft store & drain energy periodically. Electrochemical Batteries vs. Flywheel Energy Storage Systems (FES) • Integrate energy storage & attitude control M. S. Seminar – METU Aerospace Engineering Department January 2006 4/40

Blended Inverse How to select ? Pre-planned Steering M. S. Seminar – METU Aerospace Engineering Department January 2006 11/40