M.S. Thesis Presentation 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

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

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

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 Due to spin acceleration 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, 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 6/34

The Mechanical Analogy Total Ang. Mom. Position IPAC-CMG Momentum Link Length Torque End Effector Velocity Steering Problem 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 Minimum 2-Norm Solution Singularity Avodiance Steering Logic Singularity Robust Inverses ? 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 9/34

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

IPAC-CMG Cluster Simulations Torque Command Power Command Min Ang.Mom.of each IPAC-CMG [Nms] 7.7 IPAC-CMG Flywh. Spin Interval [kRPM] 15 – 60 Initial Flywheel Spin Rates (kRPM) [40, 40, 40, 40] Initial Gimbal Angles (deg) [0, 0, 0, 0] 25/34

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

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

IPACS Simulations 28/34 Spacecraft Inertias [ kgm2 ] [15, 15, 10] Initial Orientation of S/C [deg] [0, 0, 0] IPAC-CMG Flywh. Spin Interval [kRPM] 15 - 60 Initial Flywheel Spin Rates [kRPM] [39, 40, 41, 42] Initial Gimbal Angles [deg] [-75, 0, 75, 0] 28/34

Spacecraft IPACS Simulation Model IPACS Simulations Spacecraft IPACS Simulation Model 29/34

IPACS Simulations Attitude Command Power Command 30/34

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

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

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

Future Work B-inverse in highly redundant robotic mechanisms Detail Design of IPAC-CMG Capabilities of B-inverse 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; det(J)= 0 ( or det(JJT)=0 ) Singularity Measure m=det(JJT) J-1 ( or (JJT)-1 ) becomes undefined #/30

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

Singularity Robust Solutions Singularity Robust Inverse : k = 0 for m > mcr k0(1-m/m0)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 13/40

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 Extends the jacobian matrix with additional functions, creating a well –conditioned one, belonging to a “virtual” system square matrix singularity Proposes to transform the 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  14/40

Thesis Objectives Blended Inverse on Redundant Robotic Manipulators Spacecraft Energy Storage & Attitude Control IPAC-CMG based IPACS Blended Inverse on IPAC-CMG clusters 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 4/40

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