Numerical Methods for Rigid Multibody Dynamics

Transcription

1 Numerical Methods for Rigid Multibody Dynamics Claus Führer Centre for Mathematical Sciences Lund University Lappenranta 2012

2 Unit 0: Preface These notes serve as a skeleton for the compact course. They document together with the assignments the course outline and course content. All references in the notes refer to the textbook by Eich-Soellner/Führer if not otherwise stated. The notes are a guide to read the textbook. They are no textbook. 1

3 Unit 1: Equations of Motion Unconstrained system: 2nd order ODEs Constrained system: 2nd order DAEs Linear Implicitness: O(n)-methods. 2

4 1.1 Multibody System A multibody system (MBS) consist of bodies: mass, inertia, rigid or elastic degrees of freedom (dofs) interconnections: force elements, e.g. springs, dampers, controlers joints: constraints which reduce the dofs 3

5 1.2 The MBS Simulation Loop mechanical system engineer modelling, abstraction multibody model formalism establish equations of motion M p = f(p; _p) G(p) T equations of motion 0 = g(p) numerics solution of equations of motion 25.9d d d d07 85.d0d d d d d d07 85.d0d d d d d d07 85.d0d d+03 simulation results 4

6 1.3 Basic Mathematical Tasks in Multibody Dynamics Kinematic analysis Given a robot and its hand position. How must the joint angles be chosen? Nonlinear equation system. Static equilibrium position How must the forces be chosen such that the system is in equilibrium? Nonlinear equation system. Dynamic simulation How does the system behave dynamically? Numerically Solving ODEs/DAEs. Linear system analysis Stability? Input/output behavior? Linearization, Eigenanalysis. 5

7 Design, optimal control How can the behavior of the system be optimized? Parameter Identification, System Optimization, Optimal Control. 6

8 1.4 Equations of Motion: Unconstrained System or in first order format M p = f a (t, p, ṗ). ṗ = v M v = f a (t, p, v) p positions, v velocities t time M n p n p mass matrix f a applied forces 7

9 1.5 Model of Unconstrained Truck p p p p CM5 6 CM4 p 7 loading area (body 5) cabin (body 4) chassis (body 2) p 3 p 2 CM2 p rear wheel (body 1) p 1 4 CM1 front wheel (body 3) CM3 inertial frame (body 0) 8

10 1.6 Coordinates p p p p CM5 6 CM4 p 7 loading area (body 5) cabin (body 4) chassis (body 2) p 2 p 3 CM2 p 1 rear wheel (body 1) CM1 front wheel (body 3) p 4 inertial frame (body 0) CM3 p 1 Vert motion of the rear wheel (body 1) p 2 Vert motion of the truck chassis (body 2) p 3 Rot of the truck chassis (body 2) p 4 Vert motion of the front wheel (body 3) p 5 Vert motion of the driver cabin (body 4) p 6 Rot of the driver cabin (body 4) p 7 Horiz motion of the loading area (body 5) p 8 Vert motion of the loading area (body 5) p 9 Rot of the loading area (body 5) 9

11 1.7 Equations of motion: Unconstr. Truck m 1 p 1 = f f 122 m 1 g gr m 2 p 2 = f 122 f f f f f 1d2 + f 2d2 m 2 g gr l 2 p 3 = ( a 23 f 232 a 12 f 122 h 1 (f f 121 )) cos p 3 ( a 23 f 231 a 12 f 121 h 1 (f f 122 )) sin p 3 + (a 25 f a 52 (f 1d2 + f 2d2 ) + h 2 (f f 1d1 + f 2d1 )) cos p 3 (a 25 f a 52 (f 1d1 + f 2d1 ) + h 2 (f f 1d2 + f 2d2 )) sin p 3 (a 24 f a 42 f h 2 (f f 421 )) cos p 3 (a 24 f a 42 f h 2 (f f 422 )) sin p 3 m 3 p 4 =.. f f 232 m 3 g gr 10

12 1.8 Unconstr. Truck: Geometry and Forces c d1 c 25 b 24 b 42 c d2 h 3 f 1d f 2d f 25 f 24 f 42 f 1d f 2d f 12 f 25 f 24 f 42 f 23 h 2 a 52 a 24 a 12 a 25 a 23 a 42 h 1 f 12 f 23 f 10 f 30 11

13 1.9 Constrained Multibody Systems M p = f a (t, p, ṗ) G T (p)λ 0 = g(p) with g(p) being n c constraints G(p) := d dp g(p) constraint Jacobian (n c n p ) λ Lagrange multipliers 12

14 1.10 Constrained Truck p 8 p 9 p 5 p 6 p 7 loading area (body 5) cabin (body 4) chassis (body 2) p 3 p 2 p 1 rear wheel (body 1) p 4 front wheel (body 3) reference frame (body 0) 13

15 1.11 Constrained Truck: Typical Constraint p p p p 6 p 7 loading area (body 5) cabin (body 4) p 2 chassis (body 2) p 3 p rear wheel (body 1) p 1 4 front wheel (body 3) reference frame (body 0) ρ 52 = ( ) ( ) [( p7 cc1 0 +S(p p 9 ) 8 c c2 p2) + S(p 3 ) ( ac1 a c2 )] = 0. 14

16 1.12 Different Types of constraints holonomic constraints non-holonomic constraints rhenomic constraints g(p, t) skleronomic constraints g(p). 15

17 1.13 3D-Rotation: Kardan Angles S(α, β, γ) = S(0, 0, γ)s(0, β, 0)S(α, 0, 0) with the elementary rotation matrices S, e.g. S(α, 0, 0) = cos α sin α. 0 sin α cos α α is the angle of the rotation about the x-axis, β the angle of rotation about the (new) y-axis and γ the angle about the (even newer) z-axis. 16

18 1.14 3D-Rotation: angular velocity Poisson equation: with ω: Ṡ(α, β, γ) = ωs(α, β, γ). ω := 0 ω 3 ω 2 ω 3 0 ω 1 ω 2 ω 1 0 This defines the angular velocities ω 1, ω 2, ω 3. 17

19 1.15 3D-Rotation: angular velocity vs ṗ α β = γ cos γ/ cos β sin γ/ cos β 0 sin γ cos γ 0 ω 1 ω 2. cos γ tan β sin γ tan β 1 ω 3 Singularities! 18

20 1.16 3D-Rotation: complete system ṗ = Z(p)v M v = f a (t, p, v, s) Z(p) T G(p) T λ 0 = g(p) (see also Exercise 1). 19

21 1.17 Relative vs Absolute coordinates Absolute coordinates Mass matrix constant and block-diagonal. Constraints DAEs. Relative coordinates: Mass matrix dense and position dependent. For tree structured systems the constraints can be directly eliminated. 20

22 1.18 Tree structured systems

23 1.19 Truck: relative coordinates p 9 p 8 p 6 p 5 p 7 loading area (body 5) cabin (body 4) chassis (body 2) p 2 p 3 p 1 rear wheel (body 1) p 4 front wheel (body 3) q 1 = ρ 10 q 2 = ρ 12 q 3 = p 3 q 4 = ρ 23 q 5 = ρ 24 q 6 = p 6 p 3 q 7 = p 9 p 3. 7 degrees of freedom (dofs) reference frame (body 0) 22

24 1.20 Relation between absolute/relative positions Absolute relative: n q = n dof equations 0 = g(p, q), Relative absolute: n p = n c + n q equations 0 = ( ) g(p) g(p, q) =: γ(p, q). 23

25 1.21 Relation between absolute/relative velocities Derivative assumed to be regular (Grübler condition): Γ p (p, q) = γ(p, q) p Velocities: and 0 = γ(p, q) } p {{} =:Γ p (p,q) ṗ + γ(p, q) } q {{} =:Γ q (p,q) ṗ = Γ p (p, q) 1 Γ q (p, q) }{{} =:V (p,q) q, q, 24

26 1.22 Relation between absolute/relative accelerations p = V (p, q) q + ζ(p, q, ṗ, q). with ζ collecting all terms with lower derivatives of p and q. 25

27 1.23 State Space Form in relative coordinates We note G(p)V (p, q) = 0. From this we obtain V (p, q) T MV (p, q) }{{} =: M(p,q) q = V (p, q) T (f a (p, ṗ) Mζ(p, q, ṗ, q)). Note: Mass matrix looses its structure and becomes state dependent. 26

28 1.24 Mixed Coordinate Formulation for Tree Structured Systems For chain like strucktures, mixed coordinate formulations are much more efficient. block diagonal matrices to invert. k k 1 i

29 1.25 Mixed Coordinate Formulation for Tree Structured Systems ( ) M 0 0 0) ( p q = ( ) fa (t, p, ṗ) 0 0 = γ(p, q). ( ) Γp (p, q) T Γ q (p, q) T µ We note the i-th interconnection has the form γ i (p, q) = γ i (p i, p i 1, q i ). Resorting of components gives... 28

30 1.26 Mixed Coordinate Formulation for... (Cont.) Resorting:x = (..., ( p i ) T, ( q i ) T, (µ i ) T, ( p i 1 ) T, ( q i 1 ) T, (µ i 1 ) T,...) T A k C T k C k A k 1 Ck 1 T C k 1 A k 2 Ck 2 T C k 2... C T 3 C 3 A 2 C T 2 C 2 A 1 C T 1 C 1 A 0 with x i = (( p i ) T, ( q i ) T, (µ i ) T ) T and b i = ((f i a) T, 0, (z i ) T ) T. x k x k 1 x k 2.. x 2 x 1 x 0 = b k b k 1 b k 2.. b 2 b 1 b 0 29

31 1.27 Mixed Coordinate Formulation for... (Cont.) A i := M i 0 Γ T p i,i 0 0 Γ T q i,i Γ pi,i Γ qi,i 0. (1) A i maximal dimension 18 = 3 6. C T i := Γ pi,i

32 1.28 Mixed Coordinate Formulation for... (Cont.) a.) Downward recursion Initialization: Recursion (i=k-1:-1:0)  k := A k ˆbk := b k. b.) Upward recursion: Solve for x 0 and then for x i :  i := A i C i+1  1 ˆbi := b i C i  1 i i+1 CT i+1 ˆbi+1.  0 x 0 = ˆb 0  i x i = ˆb i Ci T xi 1 with i = 1 : k. 31

33 1.29 Linearization Some assumptions f a (t, p, v) = f a (p, v, u(t)) u(t) plays in the sequel the role of a given input function Time dependency of the constraint = kinematic excitation g(t, p) = g(p) z(t). 32

34 1.30 Linearization: Nominal Solution Nominal Solutions and p N (t), v N (t) and λ N (t) Nominal input u N (t) = 0 and z N (t) = 0 33

35 1.31 Linearization: Taylor ṗ = v M(t) v = K(t) p D(t) v G(t) λ + B(t)u(t) 0 = G(t) p z(t) Here: Numerical Linearization, see Lab 1 34

36 1.32 Linearization: Matrices mass matrix M(t) := M(p N (t)) stiffness matrix K(t) := ( v T N p M(p) p f a(p, v N (t), 0) + p G(p)T λ N ) p=pn (t) damping matrix D(t) := ( v f a(p N (t), v, 0)) v=vn (t) constraint matrix G(t) := ( d dp g(p)) p=p N (t) input matrix B(t) := ( u f a(p N (t), v N (t), u)) u=0 35