THE MOVING FRAME METHOD

Transcription

1 THE MOVING FRAME METHOD To Infuse Subsea Engineering Analyses with A.I. THOMAS J. IMPELLUSO WESTERN NORWAY UNIVERSITY OF APPLIED SCIENCES

2 About myself Columbia University, New York City Art History B.A Columbia University, New York City Civil Engineering B.S Columbia University, New York City Bioengineering M.S University of California, San Diego Computational Mechanics Ph.D USGS Coded Seismic Data Acquisition, Visualization software. DARPA Battlefield Surgery Project Tenured, full professor: San Diego State University. Force Feedback VR Married a woman from Bergen, Norway. Professor HVL

3 About this presentation New mathematics for 3D Dynamic analysis of Subsea vehicles. New mathematics for 3D Dynamic analysis of cranes on ships. New mathematics for Gyroscopic wave energy converters (fish farms, rigs. New programming styles to endow machines with artificial intelligence. Dynamics is the underlying process.

4 What is Dynamics Rigid body dynamics is the study of the motion in which deformation is neglected. But it has gone off the rails. How do we put it back on track? Need to understand the history of the discipline.

5 Galileo: ( Formulated Galilean relativity: The laws of motion are the same in all inertial frames.

6 Newton: Vector Dynamics Formulated the laws of motion for particles in the inertial frame First law Second law Third law 0 = ma F = ma What happens when there is no force? What happens when there is a force (gravity? Free Body Diagrams Vector Dynamics

7 Leonard Euler:( Formulated the laws of motion for bodies in an inertial frame

8 Industrial Revolution: ( Flawed Pedagogy formed Limited to Planar Analyses Limited to Inertial Observer

9 Flawed Approach to Problem Solving P P / A + Ω rp / A + 2 ( vp / A ( P A xyz / xyz a ( t = Ω Ω r Ω + a Centripetal Angular Coriolis Linear ONLY 2D

10 Summary of the State of the Discipline Perspective: Inertial Frames as the formative element Math: Vectors and Cross product (not associative Dimension: 2D Motion; Angular velocity in a plane How do we put dynamics back on track?

11 Elie Cartan ( Élie Cartan studied the change of a frame in terms of the frame. The MFM extends Cartan s views by placing a moving frame of reference on every moving body. But how do we relate the frames?

12 Sophus Lie ( Lie Groups and Lie Algebra: SO(3 is the Group of Rotations so(3 the Algebra ofangular Velocities The MFM adopts the machinery of SO(3/so(3, distilled to matrices. It abandons vectors (which cannot easily model rotations in 3D. It simplifies the math to matrix multiplications (coding. But will this lead to complicated notation?

13 Theodore Frankel ( Frames to the left v1 ( t v( t = ( e1 ( t e2 ( t e 3 ( t v2 ( t v ( t = e( t v( t v ( t Linguistics Algebra ( t ( ( t 1 = ( 1 ( 2 ( 3 ( 2 ω( t e t e t e t t 3 3 Compact Notation 0 = Non-associative cross product Associative matrix multiplication v( t = ω( t r( t v( t = e( t ( t r( t

14 Assertion The Moving Frame method, poised to be a 3D textbook, models 3D Dynamics easily. We have undergraduate students doing the coding and mathematics for ship roll, pitch and yaw. The method is eminently programmable. It will open the door to AI coding.

15 ሶ ሶ Lie Group Theory of SO(3 e(t = (e 1 (t e 2 (t e 3 (t e I = e 1 (0 e 2 (0 e 3 (0 e(t = e I R(t Orthogonality of SO(3 e I = e(tr T (t Rate of change of frame e (t = e I R(t e I 3 e I 1 e 3 e 1 e 2 e I 2 Insert Orthogonality e ሶ (t = e(tr T (t ሶ R(t e ሶ (t = e(tω (t

16 The Fictious Forces (d'alembert forces P P / A + Ω rp / A + 2 ( vp / A ( P A xyz / xyz a ( t = Ω Ω r Ω + a Angular Centripetal ( a ( t = a + e( t s ( t + 2 ( t s ( t + ( t s ( t + ( t ( t s ( t B A B B B B One frame; Time Dependent (not «at this instant»; No Cross Product Matrices: easily programmed 2D and 3D Coriolis Linear

17 Kinetics M C ( ( m v ( t + ( tv ( t F( t = e( t C C ( J ( t +( t J ( ( t = e( t t C C Moment of Inertia explicitly stated in moving body frame Can construct laws for a frame at fixed point of rotation Can construct laws at an inertial frame Can apply Newton at one frame and Euler at another.

18 But can we make it more powerful? Yes!! We exploit the Special Euclidean Group: SE(3 We combine both translation and rotation into ONE data structure. The resulting data structure will act EXACTLY the same as the rotation data structure very little new learning!

19 SE(3 R T 0 v 1 3 R s E = T 0 1 R SO(3, E v R R R R s R R R s = R31 R32 R33 s As with the rotation matrix, the inverse of E, is analytically known. T T 1 R R s E = T 0 1 And so are the rates R s E = T 0 0

20 Frame Connection Rate: W W ( t E ( t E ( t ( ( 1 ( SE(3: for translation and rotation ( t R ( t R ( t ( ( 1 ( SO(3: for rotation

21 Analytical Dynamics Hamilton s Principle: Calculus ofvariations Minimize the Action in the Direction of a «test» variation Gateaux/Frechet Derivative δ Take derivatives in variational directions.

22 Restriction on the Variation Translational Velocity ( d ( xc ( t = xc ( t dt Restriction on the Variation Restriction Angular Velocity ( ( t = d dt ( ( T ( ( ( ( ( ( T ( R R t t R R ( t +

23 Virtual Rotation ( ( t = d dt ( ( T ( ( ( ( ( ( T ( R R t t R R ( t + e Moment vs. virtual rotation is a natural pair, which yields Euler s equation. This was the weakest point in the classical multibody dynamics. Wittenburg postulated the principle of virtual power to use the weighted form of Euler's equation by the virtual angular velocity. We must use moment and virtual rotation as the natural pair (( T ( t = e ( t R R ( t ( ( ( (

24 Get the Frame Connection Structures ( ( = ( ( ( ( 2 (3/ (2 (2 (3 (3 d I R h I t t t t c c r e r e ( ( ( ( 1 ( ( t E t E t W = W 0 0 ( ( ( ( ( ( T v C t t t ( ( (3 (3 (2 (2 (2/1 ( ( ( ( ( c c t t t t E t = e r e r

25 ROV Motion using Moving Frames

26 After the Variation Process (10 pages M ( t q + C( t q F = 0 * M ( t = B M B T T ( C( t = B M B+ D M B B

27 JOURNAL OF ENERGY RESOURCES TECHNOLOGY

28 JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING

29 JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS

30 JOURNAL OF DYNAMICS AND CONTROL

31 JOURNAL OF DYNAMIC SYSTEMS MEASUREMENTS AND CONTROL

32 Laying Cables Frobenius Theorem: topological manifolds

33 Swimming Robots

34 Modern Mathematics We abandon Newton s Vectorial Dynamics. We embrace Hamilton s Analytical Dynamics, supplemented with: Free Body Diagrams SO(3 as matrices for rotations so(3 as matrices for angular velocity SE(3 as matrices for structured data. Proper restriction on the angular velocity Moving Frames Notational Consistency across dynamics Class Structures (C++

35 So why should a company switch? Machines think (CPU. Machines communicate (IP Address. Machines learn: (Artficial Intelligence/Data Base Access. Old codes (ADAMS, DADS, etc. will not work. New codes must be written. Need modular coding. Need deployable coding. Need codes for multi-phase mechanics. Consistent notation.

36 Textbook pending: will change the discipline

37 MFM EMPOWERS MODULAR CODING HVL Server HVL Server A.I. Massively Parallel Accept( Listen( Bind( Socket( Fork( Exec( bash( Cell phone PM Client Web page Shared Memory Semaphore control Dynamics PG FEM PG Socket( Connect( A.I. Thermal PG CFD PG Learning PG

38 Thank you Thomas J. Impelluso