Topics in Computer AnimationKinematicsDynamics Translational RotationalKey FramingMotion Capture Lecture 22 Slide 1 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide01.html [12/7/2000 12:47:45 PM]Different Approaches to Computer AnimationPhysically Based Animations -All about physics.● Assign masses to our objects, establish initial and reaction forces,then run simulations.● Results look real● Lack of control● Hand Tweaked Motions -Establish the positions and orientations of objects at "key" time steps● Interpolate the positions of objects in-between● Very good control● Making it "look" real is an art, and sometimes we don't want things to look real● Motion Capture -Can capture style and nuance● Looks real● Good Control● Hard to edit● Lecture 22 Slide 2 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide02.html [12/7/2000 12:47:50 PM]KinematicsKinematics is that branch of mechanics that describesthe motions of bodies without considering the forcesrequired to produce and maintain the motion.We start with the time varying motions of points as afunction of time:Consider a point undergoing a constant acceleration kLecture 22 Slide 3 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide03.html [12/7/2000 12:47:51 PM]Newtonian PhysicsKinematics describes the motion of objects in equilibrium. Dynamics (or Kinetics) describesthe change in an object's kinematics due to a change in the object's mass or the application offorces. To understand dynamics we'll need to review Newtonian physics. A moving mass has momentum. Forces are needed tochange the momentum of a mass.We can also define aggregate properties of point masses.Lecture 22 Slide 4 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide04.html [12/7/2000 12:47:53 PM]Total MomentumThe total momentum of an aggregate object (set of point masses) is given by:Since force is the time derivative of momentum. We can also define the total force.This means that we can treat all forces acting on a given rigidbody as if their vector sum was acting on a single point at thebody's center of mass with the same mass as the entire body.Lecture 22 Slide 5 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide05.html [12/7/2000 12:47:55 PM]Rotational KinematicsRecall our approach to specifying rotations at the originaround an axis a by an angle θ from Lecture 10.Assuming that θ is a function of time, we can determine the component of linear velocitythat is induced by the rotation by differentiating our rotation expression. We can also orientour frame of reference so that θ = 0 at t = 0.The instantaneous angular velocity vector is often used to simplify this equation:Lecture 22 Slide 6 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide06.html [12/7/2000 12:47:57 PM]Rotating Away from the OriginGenerally, our center of rotation is not at the origin. We can simply fix our equation tohandle this case. We introduce the center of rotation, o.This equation states that the total velocity at a point is the sum of the point's angular velocityand the velocity seen at the center of rotation. The total acceleration is given bydifferentiating once more.Often, you will see the angular velocity vector defined as:What's going on with that third term in the total acceleration expression?Lecture 22 Slide 7 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide07.html [12/7/2000 12:47:59 PM]Angular DynamicsAngular momentum is the component ofmomentum due to rotation. Angular momentummust be specified relative to a center of rotation.Angular momentum describes the rotationalmotion of the vector from o to x due to the motionat x (p). In other words, the fraction of x'smomentum rotating around o. This rotationalmotion will be about an axis perpendicular to boththe vector and p. The angular momentum capturesthis axis of rotation (centered at o). The time derivative of angular momentum is called torque.Lecture 22 Slide 8 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide08.html [12/7/2000 12:48:01 PM]Total Angular MomentumWe can compute the total angular momentum of a body relative to the point o as follows:Substituting in the angular velocity term from two slides ago gives:Now the only term that we have that varies with time is the angular velocity vector. We canfactor it out of the summation and the resulting summation is fixed for a given o. We callthis term the Inertial tensor (it's just a 3 by 3 matrix, however).For rigid bodies we will find it convenient to specify the Inertial tensor relative to theobject's center of mass.Lecture 22 Slide 9 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide09.html [12/7/2000 12:48:03 PM]What Now?Generally, we will be given rigid bodies with forces applied to them. Those forces cause theobjects to move (animate). We now know everything we need to simulate this.Lecture 22 Slide 10 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide10.html [12/7/2000 12:48:04 PM]The Key to DynamicsThe motion of an object is changed by theapplication of Forces. How the object'stranslation changes can be determined bysumming up all of the Forces applied to theobject. The object's rotation is changedaccording to where the forces are appliedrelative to the center of mass.We only need to know a few more things.We'll need a method for solving first-orderdifferential equations. Here we'll use thesimplest solution technique, Euler Integration.Lecture 22 Slide 11 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide11.html [12/7/2000 12:48:06 PM]The Simulation LoopLecture 22 Slide 12 6.837 Fall '00Lecture 22 --- 6.837 Fall '00http://graphics.lcs.mit.edu/classes/6.837/F00/Lecture22/Slide12.html [12/7/2000 12:48:07 PM]Key FramingAn alternative approach to simulation is "key framing". This is the animation approach usedin traditional cel animation. The animator starts by specifying the positions and
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