Taylor Models and Multibody Modeling in MOBILE E. Auer University of Duisburg-Essen, Germany December 2004
Page 1 of 21 1 – The Task 1 The Task Mechanical system y l l a u n a m Description e r a w t f o s MoFrame�K0,K1,K2; MoAngularVariable�phi; Model MoVector�l; MoElementaryJoint�R; MOBILE MoRigidLink�rod(K1,K2,l); MoReal�m; Sought MoMassElement�Tip(K2,m); Characteristics MoMapChain�Pendulum; Pendulum<<R<<rod<<Tip; U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 2 of 21 1 – The Task 1.1 Sought Characteristics • kinematic behavior of a system • dynamic behavior of a system • equilibrium of a system, etc. 1.2 Main Reasons for Using Interval Arithmetic in MOBILE • guaranteed correctness of results (verification) • through allowing uncertainty in parameters: – uniform study of system’s behavior for different parameters – more realistic models: MoElementaryJoint MoSlacknessJoint � i+1 � i+1 body i+1 − → body i+1 � i � i body i body i (see C. H¨ orsken, H. Traczinski, Modeling of Multibody Systems with Interval Arithmetic , 2001) U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 3 of 21 1 – The Task 1.3 Goals and Functions of the ext. MOBILE The goal: Given a system’s description in MOBILE , provide its validated characteristics Achievements: kinematics, dynamics, equilibrium validated Drawbacks: wrapping effect ↓ 1.4 The Task for This Talk Study the possibilities of wrapping effect reduction through using COSY in MOBILE U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 4 of 21 1 – The Task 1.5 The Topics of This Talk • Basics on MOBILE • Basics on the extended MOBILE : – our approach to verified modeling – notes on implementation – some results – notes on wrapping effect • Introduction of COSY into the ext. MOBILE : – implementation – kinematics: TMoSlacknessJoint – an outlook on dynamics U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
� � � � � � � � � � � � � � Page 5 of 21 2 – Basics on MOBILE 2 Basics on MOBILE 2.1 MOBILE’s Structure (Kecskem´ ethy, 1993) Transmission elements q ′ = φ ( q ) � � q ′ ˙ = J φ ˙ q � � MoMap � � � � q + ˙ q ′ ¨ = J φ ¨ J φ ˙ � � q � � T φ Q ′ = Q J MoRigidLink , . . . ↓ f Joint 1 : K 0 �→ K 1 Global kinematics g doMotion() Joint 2 : K 1 �→ K 2 ϕ : ϕ = g ◦ f K 0 �→ K 2 ↓ M ( q ; t ) ¨ q + b ( q, ˙ q ; t ) Equations of motion MoEqmBuilder = Q ( q, ˙ q ; t ) ↓ State-space form ˙ = f ( y ; t ) y MoMechanicalSystem = y ( t 0 ) y 0 ↓ Solution MoIntegrator y ( y 0 ; t ) U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 6 of 21 2 – Basics on MOBILE 2.2 Example: a Triple Pendulum The system its description and The MOBILE program MoFrame K0, K1, K2, K3, K4, K5 ; MoAngularVariable beta_1, beta_2, beta_3 ; MoElementaryJoint R1 ( K0, K1, beta_1, x_axis ) ; MoElementaryJoint R2 ( K2, K4, beta_2, x_axis ) ; MoElementaryJoint R3 ( K3, K5, beta_3, x_axis ) ; MoVector a_1, a_2, a_3, a_4 ; MoRigidLink arm_a ( K1, K2, a_1 ) ; MoRigidLink arm_b ( K1, K3, a_2 ) ; MoReal m1, m2 ; MoMassElement M1 ( K1, m1, a_3 ) ; MoMassElement M2 ( K4, m2, a_4 ) ; MoMassElement M3 ( K5, m2, a_4 ) ; MoMapChain pendulum ; pendulum << R1 << arm_a << arm_b << R2 << R3 << M1 << M2 << M3 ; MoVariableList q ; q << beta_1 << beta_2 << beta_3 ; MoMechanicalSystem sys ( q , pendulum , K0 , zAxis ) ; MoAdamsIntegrator SystemIntegrator(sys) ; for (int i=0;i<1000;i++) SystemIntegrator.doMotion() ; U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 7 of 21 3 – Extended MOBILE 3 Extended MOBILE 3.1 Choices for Modeling of Dynamics Simulation�of�dynamics needs ODEs’�solving numerical�methods validated�methods (FPA) (interval�arithmetic) need need no�derivatives derivatives methods�from recursive�methods mechanics from�CA algorithmic�differentiation automatic�differentiation implies implies program expression “symbolic” “numerical” Modeling�software U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 8 of 21 3 – Extended MOBILE 3.2 Verification Approach and Software Choices “numerical” “symbolic” interval�library PROFIL/BIAS algorithmic�differentiation automatic/alg.�differentiation FADBAD/TADIFF FADBAD/TADIFF interval�IVP�solver expressions VNODE intermediate adjusted�solver symbolic�solver MoMaple verifying MOBILE SYMKIN extension required�by verified�data produces U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 9 of 21 3 – Extended MOBILE 3.3 Design: Basic Data Type Required data types: – for ordinary values: INTERVAL – for Taylor coefficients (TC): TINTERVAL – for TC of the variational equation: TFINTERVAL ”The designing problem”: Given: f 1 to compute the right side Required: 2 types of DAGs Extra code: f 2 , f 3 that differ in data types only Our solution: class TMoInterval{ INTERVAL Enclosure; TINTERVAL TEnclosure; TFINTERVAL TFEnclosure; } U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 10 of 21 3 – Extended MOBILE 3.4 Design Aspects: Universality TMoAWAIntegrator TMoTaylor TMoEnclosure TMoIntegrator TMoDynamicSystem TMoEqmBuilder TMoConfig TMoMapChain TMoBase TMoRigidLink TMoElementaryJoint TMoMassElement TMoMap MoAxis TMoInertiaTensor TMoStack TMoVectorStack TMoXYZRotationMatrix TMoFrameList TMoVariableList MoNullState TMoRotationMatrix TMoFrame TMoStateVariable TMoMatrix TMoAngle TMoVector TMoInterval U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 11 of 21 3 – Extended MOBILE Goal: FP/Interval/... modeling in a single version Our solution: Replace MOBILE’s classes with template classes Example: typedef TMoInterval mytype; //typedef MoReal mytype; T MoFrame < mytype > K0, K1, K2, K3, K4, K5 ; T MoAngularVariable < mytype > beta 1, beta 2, beta 3 ; T MoElementaryJoint < mytype > R1 ( K0, K1, beta 1, x axis ) ; T MoElementaryJoint < mytype > R2 ( K2, K4, beta 2, x axis ) ; T MoElementaryJoint < mytype > R3 ( K3, K5, beta 3, x axis ) ; T MoVector < mytype > a 1, a 2, a 3, a 4 ; T MoRigidLink < mytype > arm a ( K1, K2, a 1 ) ; T MoRigidLink < mytype > arm b ( K1, K3, a 2 ) ; mytype m1, m2 ; T MoMassElement < mytype > M1 ( K1, m1, a 3 ) ; T MoMassElement < mytype > M2 ( K4, m2, a 4 ) ; T MoMassElement < mytype > M3 ( K5, m2, a 4 ) ; T MoMapChain < mytype > pendulum ; pendulum << R1 << arm a << arm b << R2 << R3 << M1 << M2 << M3; T MoVariableList < mytype > q ; q << beta 1 << beta 2 << beta 3; T MoMechanicalSystem < mytype > sys (q,pendulum,K0,zAxis) ; TMoAWAIntegrator SystemIntegrator(sys, 0.01,ITS QR,15 ) ; //MoAdamsIntegrator SystemIntegrator(sys) ; SystemIntegrator.doMotion() ; U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
Page 12 of 21 3 – Extended MOBILE 3.5 Example: the Triple Pendulum 0.6 ext. MOBILE position of M1 (rad) MOBILE 0.4 0.2 0 -0.2 -0.4 -0.6 Initial conditions: 0 2 4 6 8 10 time (sec) β 1 = 30 ◦ , rel. position of M1 (numerical) 10 -6 1.2 . 10 -8 rel. position of M1 (interval) ext. MOBILE β 2 = 10 ◦ , MOBILE VNODE+ β 3 = − 10 ◦ 5 . 10 -7 6 . 10 -9 Time: 646 s 0 0 Error: 1 . 61 e − 07 -5 . 10 -7 -6 . 10 -9 Break-down: 33 . 6 -10 -6 -1.2 . 10 -8 0 2.5 5 7.5 10 time (s) • ”symbolic” and ”numerical” solutions intersect • the trajectory is verified • the standard MOBILE ’s solution is inaccurate U�N�I�V�E�R�S�I T Ä T D��U��I��S��B��U��R��G Taylor Models and Multibody Modeling in MOBILE E��S��S��E��N
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