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Rigid Body Dynamics (I) An Introduction

Rigid Body Dynamics (I) An Introduction. Algorithm Overview. 0 Initialize(); 1 for t = 0; t < t f ; t += h do 2 Read_State_From_Bodies(S); 3 Compute_Time_Step(S,t,h); 4 Compute_New_Body_States(S,t,h); 5 Write_State_To_Bodies(S); 6 Zero_Forces(); 7 Apply_Env_Forces();

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Rigid Body Dynamics (I) An Introduction

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  1. Rigid Body Dynamics (I)An Introduction M. C. Lin

  2. Algorithm Overview 0 Initialize(); 1 for t = 0; t < tf; t += h do 2 Read_State_From_Bodies(S); 3 Compute_Time_Step(S,t,h); 4 Compute_New_Body_States(S,t,h); 5 Write_State_To_Bodies(S); 6 Zero_Forces(); 7 Apply_Env_Forces(); 8 Apply_BB_Forces(); M. C. Lin

  3. Particles No rotations Linear velocity v only Rigid bodies Body rotations Linear velocity v Angular velocity ω From Particles to Rigid Bodies M. C. Lin

  4. Outline • Rigid Body Preliminaries • Coordinate system, velocity, acceleration, and inertia • State and Evolution • Quaternions • Collision Detection and Contact Determination • Colliding Contact Response M. C. Lin

  5. Coordinate Systems • Body Space (Local Coordinate System) • bodies are specified relative to this system • center of mass is the origin (for convenience) • World Space • bodies are transformed to this common system: p(t) = R(t) p0 + x(t) • R(t) represents the orientation • x(t) represents the position of the body center M. C. Lin

  6. Coordinate Systems Meaning of R(t): columns represent the coordinates of the body space base vectors (1,0,0), (0,1,0), (0,0,1) in world space. M. C. Lin

  7. Velocities • How do x(t) and R(t) change over time? • v(t) = dx(t)/dt • Let (t) be the angular velocity vector • Direction is the axis of rotation • Magnitude is the angular velocity about the axis M. C. Lin

  8. Velocities M. C. Lin

  9. Angular Velocities M. C. Lin

  10. Dynamics: Accelerations • How do v(t) and dR(t)/dt change over time? • First we need some more machinery • Inertia Tensor • Forces and Torques • Momentums • Actually formulate in terms of momentum derivatives instead of velocity derivatives M. C. Lin

  11. Inertia Tensor • 3x3 matrix describing how the shape and mass distribution of the body affects the relationship between the angular velocity and the angular momentum I(t) • Analogous to mass – rotational mass • We actually want the inverse I-1(t) M. C. Lin

  12. Inertia Tensor Ixx = Iyy = Izz = Iyz = Izy = Ixy = Iyx = Ixz = Izx = M. C. Lin

  13. Inertia Tensor • Compute I in body space Ibodyand then transformed to world space as required • I vary in World Space, but Ibody is constant in body space for the entire simulation • Transformation only depends on R(t) -- I(t) is translation invariant • I(t)= R(t) Ibody R-1(t)= R(t) Ibody RT(t) • I-1(t)= R(t) Ibody-1 R-1(t)= R(t) Ibody-1 RT(t) M. C. Lin

  14. Computing Ibody-1 • There exists an orientation in body space which causes Ixy, Ixz, Iyz to all vanish • increased efficiency and trivial inverse • Point sampling within the bounding box • Projection and evaluation of Greene’s thm. • Code implementing this method exists • Refer to Mirtich’s paper at http://www.acm.org/jgt/papers/Mirtich96 M. C. Lin

  15. Approximation w/ Point Sampling • Pros: Simple, fairly accurate, no B-rep needed. • Cons: Expensive, requires volume test. M. C. Lin

  16. Use of Green’s Theorem • Pros: Simple, exact, no volumes needed. • Cons: Requires boundary representation. M. C. Lin

  17. Forces and Torques • Environment and contacts tell us what forces are applied to a body: F(t) =  Fi(t) (t) =  ( ri(t) x Fi(t) ) where ri(t) is the vector from the center of mass to the point on surface of the object that the force is applied at, ri(t)= pi - x(t) M. C. Lin

  18. Momentums • Linear momentum • P(t) = m v(t) • dP(t)/dt = m a(t) = F(t) • Angular Momentum • L(t) = I(t) (t) • (t) = I(t)-1 L(t) • It can be shown that dL(t)/dt = (t) M. C. Lin

  19. Outline • Rigid Body Preliminaries • State and Evolution • Variables and derivatives • Quaternions • Collision Detection and Contact Determination • Colliding Contact Response M. C. Lin

  20. Spatial information Velocity information New State Space v(t) replaced by linear momentum P(t) (t) replaced by angular momentum L(t) Size of the vector: (3+9+3+3)N = 18N M. C. Lin

  21. Rigid Body Dynamics M. C. Lin

  22. State of a Body • Y(t) = ( x(t), R(t), P(t), L(t) ) • We use P(t) and L(t) because of conservation • From Y(t) certain quantities are computed • I-1(t) = R(t) Ibody-1 RT(t) • v(t) = P(t) / M • ω(t) = I-1(t) L(t) • d Y(t) / dt = ( v(t), dR(t)/dt, F(t), (t) ) d(x(t),R(t),P(t),L(t))/dt =(v(t), dR(t)/dt, F(t), (t)) M. C. Lin

  23. Simulate: next state computation • From X(t) certain quantities are computed I-1(t) = R(t) Ibody-1 RT(t) v(t) = P(t) / M ω(t) = I-1(t) L(t) • We cannot compute the state of a body at all times but must be content with a finite number of discrete time points, assuming that the acceleration is continuous • Use your favorite ODE solver to solve for the new state X(t), given previous state X(t-t) and applied forces F(t) and (t)X(t) Ã Solver::Step(X(t- t), F(t),  (t)) M. C. Lin

  24. Simple simulation algorithm X à InitializeState() For t=t0 to tfinal with step  t ClearForces(F(t), (t)) AddExternalForces(F(t), (t)) Xnewà Solver::Step(X, F(t), (t)) X à Xnew t à t + t End for M. C. Lin

  25. Outline • Rigid Body Preliminaries • State and Evolution • Quaternions • Merits, drift, and re-normalization • Collision Detection and Contact Determination • Colliding Contact Response M. C. Lin

  26. Unit Quaternion Merits • A rotation in 3-space involves 3 DOF • Rotation matrices describe a rotation using 9 parameters • Unit quaternions can do it with 4 • Rotation matrices buildup error faster and more severely than unit quaternions • Drift is easier to fix with quaternions • renormalize M. C. Lin

  27. Unit Quaternion Definition • [s,v] -- s is a scalar, vis vector • A rotation ofθ about a unit axis u can be represented by the unit quaternion: [cos(θ/2), sin(θ /2) * u] • || [s,v] || = 1 -- the length is taken to be the Euclidean distance treating [s,v] as a 4-tuple or a vector in 4-space M. C. Lin

  28. Unit Quaternion Operations • Multiplication is given by: • dq(t)/dt = [0, w(t)/2]q(t) • R = M. C. Lin

  29. Unit Quaternion Usage • To use quaternions instead of rotation matrices, just substitute them into the state as the orientation (save 5 variables) • d (x(t), q(t), P(t), L(t) ) / dt = ( v(t), [0,ω(t)/2]q(t), F(t), (t) ) = ( P(t)/m, [0, I-1(t)L(t)/2]q(t), F(t), (t) ) where I-1(t) = (q(t).R) Ibody-1 (q(t).RT) M. C. Lin

  30. Outline • Rigid Body Preliminaries • State and Evolution • Quaternions • Collision Detection and Contact Determination • Intersection testing, bisection, and nearest features • Colliding Contact Response M. C. Lin

  31. What happens when bodies collide? • Colliding • Bodies bounce off each other • Elasticity governs ‘bounciness’ • Motion of bodies changes discontinuously within a discrete time step • ‘Before’ and ‘After’ states need to be computed • In contact • Resting • Sliding • Friction M. C. Lin

  32. Detecting collisions and response • Several choices • Collision detection: which algorithm? • Response: Backtrack or allow penetration? • Two primitives to find out if response is necessary: • Distance(A,B): cheap, no contact information, fast intersection query • Contact(A,B): expensive, with contact information M. C. Lin

  33. Algorithm Overview 0 Initialize(); 1 for t = 0; t < tf; t += h do 2 Read_State_From_Bodies(S); 3 Compute_Time_Step(S,t,h); 4 Compute_New_Body_States(S,t,h); 5 Write_State_To_Bodies(S); 6 Zero_Forces(); 7 Apply_Env_Forces(); 8 Apply_BB_Forces(); M. C. Lin

  34. Collision Detection and Contact Determination • Discreteness of a simulation prohibits the computation of a state producing exact touching • We wish to compute when two bodies are “close enough” and then apply contact forces • This can be quite a sticky issue. • Are bodies allowed to be penetrating when the forces are applied? • What if contact region is larger than a single point? • Did we miss a collision? M. C. Lin

  35. Collision Detection and Contact Determination • Response parameters can be derived from the state and from the identity of the contacting features • Define two primitives that we use to figure out body-body response parameters • Distance(A,B) (cheaper) • Contacts(A,B) (more expensive) M. C. Lin

  36. Distance(A,B) • Returns a value which is the minimum distance between two bodies • Approximate may be ok • Negative if the bodies intersect • Convex polyhedra • Lin-Canny and GJK -- 2 classes of algorithms • Non-convex polyhedra • much more useful but hard to get distance fast • PQP/RAPID/SWIFT++ M. C. Lin

  37. Contacts(A,B) • Returns the set of features that are nearest for disjoint bodies or intersecting for penetrating bodies • Convex polyhedra • LC & GJK give the nearest features as a bi-product of their computation – only a single pair. Others that are equally distant may not be returned. • Non-convex polyhedra • much more useful but much harder problem especially contact determination for disjoint bodies • Convex decomposition M. C. Lin

  38. Compute_Time_Step(S,t,h) • Let’s recall a particle colliding with a plane M. C. Lin

  39. Compute_Time_Step(S,t,h) • We wish to compute tc to some tolerance M. C. Lin

  40. Compute_Time_Step(S,t,h) • A common method is to use bisection search until the distance is positive but less than the tolerance • This can be improved by using the ratio (disjoint distance) : (disjoint distance + penetration depth) to figure out the new time to try -- faster convergence M. C. Lin

  41. Compute_Time_Step(S,t,h) 0 for each pair of bodies (A,B) do 1 Compute_New_Body_States(Scopy, t, H); 2 hs(A,B) = H; // H is the target timestep 3 if Distance(A,B) < 0 then 4 try_h = H/2; try_t = t + try_h; 5 while TRUE do 6 Compute_New_Body_States(Scopy, t, try_t - t); 7 if Distance(A,B) < 0 then 8 try_h /= 2; try_t -= try_h; 9 else if Distance(A,B) <  then 10 break; 11 else 12 try_h /= 2; try_t += try_h; 13 hs(A,B) = try_t – t; 14 h = min( hs ); M. C. Lin

  42. Penalty Methods • If Compute_Time_Step does not affect the time step (h) then we have a simulation based on penalty methods • The objects are allowed to intersect and their penetration depth is used to compute a spring constant which forces them apart M. C. Lin

  43. Outline • Rigid Body Preliminaries • State and Evolution • Quaternions • Collision Detection and Contact Determination • Colliding Contact Response • Normal vector, restitution, and force application M. C. Lin

  44. What happens upon collision • Impulses provide instantaneous changes to velocity, unlike forces(P) = J • We apply impulses to the colliding objects, at the point of collision • For frictionless bodies, the direction will be the same as the normal direction: J = jTn M. C. Lin

  45. Colliding Contact Response • Assumptions: • Convex bodies • Non-penetrating • Non-degenerate configuration • edge-edge or vertex-face • appropriate set of rules can handle the others • Need a contact unit normal vector • Face-vertex case: use the normal of the face • Edge-edge case: use the cross-product of the direction vectors of the two edges M. C. Lin

  46. Colliding Contact Response • Point velocities at the nearest points: • Relative contact normal velocity: M. C. Lin

  47. Colliding Contact Response • If vrel > 0 then • the bodies are separating and we don’t compute anything • Else • the bodies are colliding and we must apply an impulse to keep them from penetrating • The impulse is in the normal direction: M. C. Lin

  48. Colliding Contact Response • We will use the empirical law of frictionless collisions: • Coefficient of restitution є [0,1] • є = 0 -- bodies stick together • є = 1 – loss-less rebound • After some manipulation of equations... M. C. Lin

  49. Apply_BB_Forces() • For colliding contact, the computation can be local 0 for each pair of bodies (A,B) do 1 if Distance(A,B) <  then 2 Cs = Contacts(A,B); 3 Apply_Impulses(A,B,Cs); M. C. Lin

  50. Apply_Impulses(A,B,Cs) • The impulse is an instantaneous force – it changes the velocities of the bodies instantaneously: Δv = J / M 0 for each contact in Cs do 1 Compute n 2 Compute j 3 P(A) += J 4 L(A) += (p - x(t)) x J 5 P(B) -= J 6 L(B) -= (p - x(t)) x J M. C. Lin

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