Rigid Body Dynamics and Beyond

Transcription

1 Rigid Body Dynamics and Beyond 1

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3 Rigid Bodies 3

4 A rigid body Collection of particles Distance between any two particles is always constant What types of motions preserve these constraints? Translation, rotation 4

5 Rigid Body Parameterization (reduced coordinates) x(t) 0 p 0 p(t) p t = x t + R t p 0 position orientation 5

6 Body Orientation Rotates vectors from body to world coordinates Columns of R t encode world coordinates of body axes 6

7 Center of Mass (COM) Geometric center of the body (0,0,0) in body coordinates x t in world coordinates Geometric center: σ m i p 0i (0,0,0) M = m i COM: σ m ip i (t) M = x(t) 0 p 0i

8 Body velocities How do the COM position and orientation change with time? Linear velocity: v t = dx(t) dt = x(t) ሶ Angular velocity: ω t =? ω t encodes spin direction and magnitude What is the relationship between ω t and R t? 8

9 Angular velocity Consider vector r t. What is rሶ t? rሶ t = ω(t) r(t) Relation to ሶ R t = dr t dt? 9

10 Angular Velocity R rotates vectors from body to world coords Columns of R t : world coordinates of body axes Columns of Rሶ t : change of body axes world coordinates wrt time x ሶ t = ω(t) x (t) y ሶ t = ω(t) y (t) z ሶ t = ω(t) z (t) Putting these all together: ሶ R t =ω t R(t) 10

11 Recall a 11

12 ሶ Summary Kinematics: how does the body move? p t, R t v t = dx(t) dt ሶ R t =ω t R(t) = x(t) ሶ p t = x t + R t p 0 p t = xሶ t + ω t R t p 0 = xሶ t + ω t (p t x(t)) Dynamics: what causes this motion? 12

13 Forces and Torques External forces: acting on individual particles Net force on body: F = σ F i Conservation of linear momentum (Newton s 2 nd law): p = Mv; p ሶ = F Analogous concepts for angular motion 13

14 Forces and Torques Forces on individual particles generate torques - (consequence of constant inter-particle distance) Net torque on body: τ = σ τ i = σ r i f i 14

15 Forces and Torques Forces on individual particles generate torques - (consequence of constant inter-particle distance) Net torque on body: τ = σ τ i = σ r i f i Conservation of angular momentum: L = Iω; ሶ L = τ What is I? Moment of Inertia tensor 15

16 The Inertia Tensor Analogous to mass, but for rotational motions - quantifies distribution of mass as a 2 nd order tensor I = RI b R T I b = m i (p T 0i p 0i 1 p 0i p T 0i ) i Body-coords MOI is constant, can be precomputed easy to look up for common shapes! World-cords MOI changes with time! 16

17 Conservation of Linear and Angular Linear Momentum: Momenta p = Mv; p ሶ = F; v ሶ = 1 M F Angular Momentum: L = Iω; ሶ L = τ; ω ሶ = I 1 (τ ω Iω) Note: they are decoupled! 17

18 Numerical Integration COM Acceleration Velocity Position Easy: v t+1 = v t + Δt ሶ v; x t+1 = x t + Δtv t+1 Angular Acceleration Angular Velocity Easy:ω t+1 = ω t + Δtωሶ Angular Velocity to Rotations? A bit trickier: R t+1 = R t + Δt ሶ R t+1? 18

19 Updating Rotations R t+1 = R t + ΔtRሶ t+1 = R t + Δtω t+1 R t = (I + Δtω t+1 ) R t No longer a rotation matrix! Option 1: orthonormalize (Gram Schmidt) Option 2: explicitly compute rotation R Δt due to spinning with angular speed ω t+1 for Δt seconds, apply incremental rotations: R t+1 = R Δt R t NOTE: same concept applies if other rotation parameterizations (i.e. quaternions) are employed 19

20 Computing forces Given a set of forces, you know how to compute the motion of a rigid body Where do forces come from? User interaction Gravity 20

21 Rigid Bodies 21

22 Computing forces Given a set of forces, you know how to compute the motion of a rigid body Where do forces come from? User interaction Gravity Collisions and contacts 22

23 Collision Response 23

24 Collision Response 24

25 Collision Response 25

26 Collision Response 26

27 Collision Response 27

28 Collision Response Nonconvex Rigid Bodies with Stacking, Guendelman et al., SIGGRAPH

29 Computing forces Given a set of forces, you know how to compute the motion of a rigid body Where do forces come from? User interaction Gravity Collisions and contacts Easiest way to model: spring penalty forces Articulation 29

30 Articulated Rigid Body Dynamics 30

31 Computing forces Given a set of forces, you know how to compute the motion of a rigid body Where do forces come from? User interaction Gravity Collisions and contacts Easiest way to model: spring penalty forces Articulation Easiest way to model: spring penalty forces 31

32 Artistic control over rigid body simulations Many-Worlds Browsing for Control of Multibody Dynamics Twigg and James, 2007

33 Many Worlds Browsing Sampling Plausible Worlds v t+1 ` v t compute and apply impulse [O Sullivan et al., 2003]

34 Many Worlds Browsing Interactive Browsing various criteria

35 Many Worlds Browsing

36 For more information An Introduction to Physically Based Modeling: -