Rigid Bodies Simulation Homework • Build a particle system based either on F=ma or procedural simulation – Examples:Thursday Smoke, Fire, Water, Wind, Leaves, Cloth, Magnets, Flocks, Fish, Insects, Crowds, etc. • SimulateSimulation a Rigid Body & – Examples: Angry birds, Bodies tumbling, bouncing, moving around in a room and colliding, Explosions & Fracture, Drop theUnity camera, Etc… Rigid Body as “Particle Chunks” A small chunk of a rigid body Consider a rigid body It can be broken up into chunks (or elements), and each chunk can be treated as a single particle if it is small enough Center of Mass • A body composed of 푛 particle “chunks” with masses 푚푖 has total mass 푛 푀 = ෍ 푚푖 푖=1 • If the particles have positions 푥Ԧ푖, the center of mass is σ푛 푚 푥Ԧ 푥ҧ = 푖=1 푖 푖 푀 • The center of mass for a rigid body has position 푥ҧ and translational velocity 푣ҧ similar to a single particle • When we refer to the position and velocity of a rigid body, we are referring to the position and translational velocity of its center of mass • The center of mass obeys the same ODEs for position and velocity as a particle does, but using the TOTAL mass and the NET force Orientation Orientation • A rigid body can rotate or change its orientation while its center of mass is stationary • Different ways to keep track of the rotation 푅: – 3x3 Matrix, 3 Euler angles, 1 Quaternion • Place a coordinate system at the center of mass in object space • The rotation 푅 rotates the rigid body (and the object space coordinate system) into its world space orientation • Recall: the columns of 푅 are the three object space axes in their world space orientations Combining Position and Orientation • The rigid body has an intrinsic coordinate system in its object space, with its center of mass at the origin • It’s put into world space with a translation and a rotation 푅 푥ҧ t 푥푤 푥푏 Combining Position and Orientation • The translation of the origin of the object space coordinate system is given by the position of the center of mass 푥ҧ • The world space orientation of the object space coordinate system is given by 푅 – Assume 푅 is a matrix, equivalently expressed as a unit quaternion • A point 푥표 in object space has a world space location 푥푤 = 푥ҧ + 푅푥표 – Notice that the center of mass (at the origin) maps to 푥ҧ – Notice that (1,0,0), (0,1,0), (0,0,1) are all rotated by 푅 before being translated by 푥ҧ Angular Velocity Angular Velocity • Both 푥ҧ and 푅 are functions of time • The rate of change of the position of the center of mass 푥ҧ with respect to time is the translational velocity of the center of mass 푣ҧ • (From our quaternion discussion…) The orientation of the body is changing as it is rotated about some axis 푛ො emanating from the center of mass • The rate of change of the orientation 푅 is given by the world space angular velocity 휔 – its direction is the axis of rotation, 푛ො – Its magnitude is the speed of rotation • The pointwise velocity of any point x on the rigid body is given by 푣푝 = 푣ҧ + 휔 × 푥 − 푥ҧ = 푣ҧ + 휔 × 푟 where r is the moment arm and × is the cross-product Aside: Cross Product Matrix • Given vectors 푎 = [푎1, 푎2, 푎3] and 푏 = 푏1, 푏2, 푏3 , their cross product 푎 × 푏 can be written as matrix multiplication 푎∗푏 by converting 푎 to a cross product matrix 0 −푎3 푎2 ∗ • 푎 = 푎3 0 −푎1 −푎2 푎1 0 • 푎∗푇 = −푎∗ • 푎∗푏 = −푏∗푎 • Using this new notation, the pointwise velocity of any point x on the object is then given by: ∗ 푣푝 = 푣ҧ + 휔 푟 Linear and Angular Velocity 휔 푣ҧ 푥ҧ ODE for Orientation • The ODE for orientation (angular position) is given by 푅ሶ = 휔∗푅 • That is, 푅ሶ = 휔 × 푅 where the cross product is applied independently to each of the three columns of 푅 • Writing the 3x3 matrix 휔∗ and using matrix multiplication 휔∗푅 automatically performs these 3 cross products Inertia Tensor Inertia Tensor • Linear momentum is defined as the product of the mass times the translational velocity – Mass is something that resists change in velocity • Angular momentum 퐿 is defined as an “angular mass” times the angular velocity 휔 • The “angular mass” is called the moment of inertia (or inertia tensor) 퐼 of the rigid body • If you spin in your chair while extending your legs, and then suddenly pull your legs closer the chair spins faster – 퐼 reduces when you pull your legs closer. – Hence 휔 has to increase to keep angular momentum 퐿 = 퐼휔 constant Inertia Tensor • For a system of n particles, the angular momentum is 푤 ∗ ∗ ∗ ∗ 퐿 = ෍(푥푖 ) 푚푖푣푖 = ෍ 푥ҧ + 푟푖 푚푖푣푖 = 푥ҧ 푀푣ҧ + ෍ 푟푖 푚푖푣푖 푖 푖 푖 푡ℎ – where 푟푖 is the moment arm of the 푖 particle ∗ • 푣푖 can be written as 푣ҧ + 휔 푟푖 so ∗ ∗ ∗ ∗ 퐿 = 푥ҧ 푀푣ҧ + ෍ 푟푖 푚푖푣ҧ + ෍ 푟푖 푚푖휔 푟푖 푖 푖 ∗ ∗ 푇 • σ푖 푟푖 푚푖푣ҧ = (푣ҧ ) σ푖 푚푖푟푖 = 0 so ∗ ∗ 푇 ∗ ∗ 퐿 = 푥ҧ 푀푣ҧ + ෍ 푚푖(푟푖 ) 푟푖 휔 = 푥ҧ 푀푣ҧ + 퐼휔 푖 ∗ 푇 ∗ – where 퐼 = σ푖 푚푖(푟푖 ) 푟푖 Object Space Inertia Tensor • We can pre-compute the object space inertia tensor as a 3x3 matrix 퐼표 • Then, the world space inertia tensor is given by the 3x3 matrix 퐼 = 푅퐼표푅푇 • One can compute the SVD of the symmetric 3x3 matrix 퐼표 to obtain 퐼표 = 푈퐷푈푇 where 퐷 is a 3x3 diagonal matrix of 3 singular values • Then rotating the object space rest state of the rigid body by 푈−1 gives a new object space inertia tensor of 푈−1퐼표푈−푇 = 푈−1푈퐷푈푇푈−푇 = 퐷 • That is, properly orienting the rest pose of a rigid body in object space gives a diagonal object space inertia tensor 퐼표 Forces and Torques Forces and Torques • Newton’s second law for angular quantities • A force 퐹 changes both the linear momentum 푀푣ҧ and the rotational (angular) momentum 퐿 • The change in linear momentum is independent of the point on the rigid body 푥 where the force is applied • The change in angular momentum does depend on the point 푥 where the force is applied • The torque is defined as 휏 = 푥 − 푥ҧ × 퐹 = 푟 × 퐹 • The net change in angular momentum is given by the sum of all the external torques 퐿ሶ = 휏 ODEs Rigid Body: Equations of Motion • State vector for a rigid body 푥ҧ 푅 푋 = 푀푣ҧ 퐿 • Equations of motion 푥ҧ 푣ҧ 푑 푑 푅 휔∗푅 푋 = = 푑푡 푑푡 푀푣ҧ 퐹 퐿 휏 Rigid Body: Equations of Motion • State vector for a rigid body 푥ҧ 푅 푋 = 푣ҧ 퐿 • Equations of motion 푥ҧ 푣ҧ Equations of 푑 푑 푅 휔∗푅 motion for a 푋 = = particle at the 푑푡 푑푡 푣ҧ 퐹/푀 center of mass 퐿 휏 Forward Euler Update 푣ҧ 푛 휔∗푅 푋푛+1 = 푋푛 + Δ푡 퐹/푀 휏 • Newmark for better accuracy and stability, etc… • Better results are obtained on the second equation by rotating the columns of 푅 directly using the vector Δ푡휔 • Need to periodically re-orthonormalize the columns of 푅 to keep it a rotation matrix Rigid Body: Equations of Motion • State vector for a rigid body 푥ҧ 푞 푋 = 푣ҧ 퐿 • Equations of motion 푣ҧ 푥ҧ 1 푑 푑 푞 휔∗푞 푋 = = 2 푑푡 푑푡 푣ҧ 퐿 퐹/푀 휏 • Once again, preferable to rotate q by Δ푡휔 • Renormalize q using a square root Question #1 LONG FORM: • Summarize rigid body simulation. • Identify 10 rigid bodies in a typical room. SHORT FORM: • Identify 10 rigid bodies in this room. Geometry Rigid Body Modeling • Store an object space triangulated surface to represent the surface of the rigid body • Store an object space implicit surface to represent the interior volume of the rigid body • Collision detection between two rigid bodies can then be carried out by checking the surface of one body against the interior volume of another • Implicit surfaces can be used to model the interior volume of kinematic and static objects as well • Implicit surface representations of interior volumes can also be used for collisions with particles and particle systems Recall: Implicit Surfaces • Implicit surfaces represent a surface with a function 휙(푥) defined over the whole 3D space • The inside region Ω−, the outside region is Ω+, and the surface 휕Ω are all defined by the function 휙(푥) • 휙 푥 < 0 inside • 휙 푥 > 0 outside • 휙 푥 = 0 surface • Easy to check if a point is inside an object • Efficient to make topology changes to an object • Efficient boolean operations: Union, Difference, Intersection Analytic Implicit Surfaces • For simple functions, write down the function and analytically evaluate 휙(푥) to see if 휙 푥 < 0 and thus 푥 is inside the object • 2D circle • 3D ellipsoid Discrete Implicit Surfaces • Lay down a grid that spans the space you are trying to represent, e.g. a padded bounding box • Store values of the function at grid points • Then for arbitrary locations in space, interpolate from the nearby values on the grid to see if 휙 푥 < 0 • use trilinear interpolation, like 3D textures • Signed Distance Functions are implicit surfaces where the magnitude of the function gives the distance to the closet point on the surface Constructing Signed Distance Functions • Start with a triangulated surface • Place a grid inside a slightly padded bounding box of the object – This grid will contain point samples for the signed distance function – The resolution of the grid is based on heuristics • Place a sphere at every grid point and find all intersecting triangles – the radius of the sphere only needs to be a few grid cells wide, because we only care about grid points near the triangulated surface – If the sphere does not intersect any triangles, then the grid point is not near the triangulated surface – (An acceleration structure is useful, e.g.