1. Fundamentals of Computer Animation Rigid Body Simulation (1)

2. Rigid Body Simulation Realistic-looking motion Physically based reaction of rigid bodies to forces such as gravity, viscosity, friction, and collisions. Equations of Motion Animation System Automatic calculation of reactions

3. Rigid Body Simulation

4. Collision Detection

5. Collision Response

9. Rigid Body Simulation: forces due to: Relative positioning of objects –Gravity –Collisions, … Object velocity –Viscosity, … Absolute positions of objects in user- specified vector-fields –Wind, …

10. Forces applied to objects = Linear accelerations based on the mass of object Angular accelerations based on mass distribution of the object

11. Rigid body simulation update cycle Forces Linear and angular accelerations (time derivative of velocities) Integrated over a delta time step Changes in object velocities (linear and angular) Changes in object positions and orientations Integrated over a delta time step Time Step

12. Rigid Body Simulation Update Cycle

13. Equations of Motion Current State Position and velocity Accelerations Forces Velocities Equations of Motion Integrate

14. Using Equations of Motion Standard Physics –Analyzing for times at which significant events happen (i.e. objects hitting the ground) Computer Animation –Modeling the motion of objects at discrete time steps –The significant events and their aftermath –Discrete time sampling  numerical issues!

15. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

16. Bodies in Free Fall : Basics

17. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

18. Time Step #1 Time Step #2 Time Step #3

19. Time Step #1 Time Step #2 Time Step #3

20. Time Step #1 Time Step #2 Time Step #3

21. Bodies in Free Fall : Example

25. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

26. Numeric Approximation Euler Integration

27. Numeric Approximation Physically Based Modeling: Principles and Practice (SIGGRAPH '97 Course notes) ©1997 Andrew Witkin and David Baraff

28. A Canonical Differential Equation

36. Midpoint Method A.k.a. 2nd order Runga Kutta F (t i ) F ’ (t i+0.5 ) F (t i+1 ) = F (t i ) + F ‘ (t i+0.5 )*Dt F ‘ (t i ) F (t i+0.5 ) Magnitude of the error term

38. 4th Order Runga Kutta F (t i ) F (t i+1 ) = k 1 /6 + k 2 /3 + k 3 /3 + k 4 /6

41. Since x may be vectors, the solver must know their length, to allocate storage, perform vector arithmetic ops, etc.

42. The solver must be able to install new values at the end of a step. In addition, a multi-step method must set x and t to intermediate values in the course of performing derivative evaluations.

43. Evaluate f at the current x and t.

45. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

46. Equations of Motion Linear force Angular force: torque Linear momentum Angular momentum Resistance to linear force: mass Resistance to angular force: inertia tensor.

47. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

48. Orientation and Rotational Movement Linear attributes: –position, –velocity, –Acceleration 3D objects: –orientation, –angular velocity, –angular acceleration Object orientation  rotation matrix R(t)

49. Position and Orientation A rigid body occupies a volume of space and has a particular shape. Because a rigid body can undergo only rotation and translation, we define the shape of a rigid body in terms of a fixed and unchanging space called body space. Given a geometric description of the body in body space, we use x(t) and R(t) to transform the body-space description into world space

50. The center of mass is transformed to the point x(t) in world space, at time t. Center of mass  point in the rigid body that lies at the geometric center of the body. Objects are defined in body space and transformed by the position and orientation into world space

51. At time t, the columns of R(t) are the world-space directions that the body-space x, y, and z axes transform to. Physical interpretation of the orientation matrix R(t).

52. Linear Velocity For simplicity, we’ll call x(t) and R(t) the position and orientation of the body at time t. How the position and orientation change over time? If we imagine that the orientation of the body is fixed, then the only movement the body can undergo is a pure translation. The quantity v(t) gives the velocity of this translation. Velocity of the center of mass in world space.

53. Linear Velocity The change of the center of mass over time For a pure translation ( constant), all points move with velocity

54. Angular Velocity In addition to translating, a rigid body can also spin. Imagine however that we freeze the position of the center of mass in space. Any movement of the points of the body must therefore be due to the body spinning about some axis that passes through the center of mass. (Otherwise the center of mass would itself be moving).

55. Angular Velocity : Rate at which the object is rotating irrespective to its linear velocity Direction  orientation of the axis about which the object is rotating Magnitude  speed of the rotation in revolutions per unit of time

56. For a given number of rotations per unit of time, the angular velocity is the same whether the axis of rotation is near or far away.

57. At time t, the columns of R(t) are the world-space directions that the body-space x, y, and z axes transform to. Physical interpretation of the orientation matrix R(t).

58. Angular Velocity The change in orientation over time Encodes both the axis and speed of the rotation  direction encodes the axis  magnitude encodes the speed (rad/s) But, how are and related?

59. Let’s examine how the change of an arbitrary vector in a rigid body is related to the angular velocity Let’s remind ourselves of the physical meaning of R(t) The directions of the transformed x, y and z body axes at time t. Describe the velocity with which the x, y, and z axes are being transformed.

60. First in 2D  Linear Velocity from Angular Velocity Vector from the origin of our body O (i.e. center of mass) to point B The velocity of a point (V) on a rotating body (B) is calculated by scaling the perpendicular vector from the origin to the point (r OB ) by the angular velocity (omega). Perpendicular vector to r The velocity vector’s direction must be perpendicular to the radius vector. Intuitively: a point rotating around another fixed point can only move perpendicularly to the vector between the points at any instant; it can’t move closer or farther away, or the movement wouldn’t be a simple rotation.

61. Angular Velocity For a given vector The columns of represent the transformed axes

62. Now in 3D  The rate of change of a rotating vector. The rate of change of a vector At time t, we know that the direction of the x axis of the rigid body in world space is the first column of R(t) At time t, the derivative of the first column of R(t) is just the rate of change of this vector: using the cross product rule we just discovered, this change is

63. Let’s simplify things: If a and b are 3-vectors, then a X b is the vector Given the vector a, let us define a* to be the matrix This means that we can write

64. This construct can be derived by considering what’s known as infinitesimal rotations. Refer to chapter 4.8 of Herbert Goldstein. Classical Mechanics. Addison Wesley, Reading, MA, 1950. for a more complete derivation of the a* matrix. Using the * notation, we can rewrite Then

65. Mass of a Body We’ll need to (conceptually) perform some integrations over the volume of our rigid body. Let’s imagine that a rigid body is made up of a large number of small particles (indexed from 1 to N). The mass of the ith particle is m i Each particle has a (constant) location r 0i in body space. The location of the ith particle in world space at time t:

66. Since a point can be represented at any time by Velocity of a Point (or Particle) Total velocity can then be expressed as Which can be rewritten as

67. The velocity of the ith point (particle) of a rigid body in world space Angular termLinear term

68. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

69. Mass distribution: Typical approach  a mass value to each of the object’s vertices

70. Center of mass coordinate system  body space x(t) = location of the center of mass at time t

71. Rigid Body Simulation UpdateRigidBody(float time) { Force=ComputeTotalForce(); Torque=ComputeTotalTorque(); Momentum=Momentum+Force*time; Velocity=Momentum/Mass; Position=Position+Velocity*time; AngMomentum=AngMomentum+Torque*time; Matrix34 I=Matrix*RotInertia*Matrix.Inverse(); AngVelocity=I.Inverse()*AngMomentum; Matrix.Rotate(AngVelocity*time); }

72. Rigid Body Simulation UpdateRigidBody(float time) { Force=ComputeTotalForce(); Torque=ComputeTotalTorque(); Momentum=Momentum+Force*time; Velocity=Momentum/Mass; Position=Position+Velocity*time; AngMomentum=AngMomentum+Torque*time; Matrix34 I=Matrix*RotInertia*Matrix.Inverse(); AngVelocity=I.Inverse()*AngMomentum; Matrix.Rotate(AngVelocity*time); }

73. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

74. We can apply forces to the object at any point Force Total force on an object is simply No information about where the forces are applied

75. Force acting on a rigid body due to some external influence (e.g. gravity, wind, contact forces) Acceleration due to the earth's gravity can be approximated by a constant acceleration, g, that is 32 f/sec/sec. Given an object's initial position (i.e., height) and the assumption that it was at rest at time t=0, then for any point in the future time we can plug the time into the equations above to get its acceleration, velocity and distance that it's fallen.

76. Gravity between two objects is a known force. Given two objects with comparable masses, M1 and M2, the force of gravity between two masses is given by: f = G*M1*M2/D where G = 6.67x10-11, a universal constant D = distance between centers of mass

77. object 1 mass: 1.0x10**13 radius: 60 initial position: 0 0 0 initial velocity: 0 0 0 object 2 mass: 1.0x10**11 radius: 30 initial position: 400 -200 0 initial velocity: -1.5 -0.125 0 500 time slices were computed with every tenth time slice displayed (i.e., ten samples per frame). The motion was produced by, for each time slice for both objects, updating the force, computing acceleration, calculating average velocity, updating position, and then updating velocity. Gravity Animation This animation shows motion produced by setting initial positions and velocities of two bodies and then letting gravity do the rest.

78. Simulation Loop

79. Check course’s website for more details, including other simulation codes…

80. Rigid Body Simulation UpdateRigidBody(float time) { Force=ComputeTotalForce(); Torque=ComputeTotalTorque(); Momentum=Momentum+Force*time; Velocity=Momentum/Mass; Position=Position+Velocity*time; AngMomentum=AngMomentum+Torque*time; Matrix34 I=Matrix*RotInertia*Matrix.Inverse(); AngVelocity=I.Inverse()*AngMomentum; Matrix.Rotate(AngVelocity*time); }

81. Torque describes the “rotational force” Torque Total torque on an object is simply Tells us about the force distribution over the object

82. Forces & Torques τ=dL/dt A torque is a change in angular momentum (similar to a force which is a change in linear momentum)

83. Offset Forces Torque resulting from offset force: τ=r×f Total force: Total torque:

84. Rigid Body Simulation UpdateRigidBody(float time) { Force=ComputeTotalForce(); Torque=ComputeTotalTorque(); Momentum=Momentum+Force*time; Velocity=Momentum/Mass; Position=Position+Velocity*time; AngMomentum=AngMomentum+Torque*time; Matrix34 I=Matrix*RotInertia*Matrix.Inverse(); AngVelocity=I.Inverse()*AngMomentum; Matrix.Rotate(AngVelocity*time); }

85. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

86. Linear momentum of a particle is Linear Momentum Linear momentum of a rigid body is then density integration over the body

87. Linear momentum can be simplified as follows Linear Momentum Assuming constant mass gives

88. Angular momentum of a rigid body Angular Momentum Taking the time derivative inertia tensor Angular momentum is conserved for no torque

89. Angular Momentum L=Iω = AI 0 A -1 ω L=angular momentum I=rotational inertia ω=angular velocity A=3x3 orientation matrix

90. Rigid Body Simulation UpdateRigidBody(float time) { Force=ComputeTotalForce(); Torque=ComputeTotalTorque(); Momentum=Momentum+Force*time; Velocity=Momentum/Mass; Position=Position+Velocity*time; AngMomentum=AngMomentum+Torque*time; Matrix34 I=Matrix*RotInertia*Matrix.Inverse(); AngVelocity=I.Inverse()*AngMomentum; Matrix.Rotate(AngVelocity*time); }

91. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

92. Describes how mass is distributed in the body Inertia Tensor Analogous to mass in linear velocity Measures the preferred axis of rotation Expensive to compute this at every time step

93. Rotational Inertia

94. Bodies in Free Fall Basics Example Numeric Approximation Equations of Motion Orientation and Rotational Movement Center of Mass Forces Momentum Inertia Tensor Equations

95. Position Orientation Linear Momentum Angular Momentum Rigid Body Equations of Motion State of the Object (change over time)

96. Rigid Body Equations of Motion Other Attributes (do not change over time) Mass Inertia Tensor Time-varying Inertia Tensor Angular Velocity Linear Velocity

97. Combining the equations Rigid Body Equations of Motion Discretize these continuous equations and integrate Time Derivative of the Object’s State Vector

98. Use quaternions to represent orientation Using Quaternions The update rule is then

99. Using quaternions gives Rigid Body Equations of Motion Discretize these continuous equations and integrate

100. Algorithm with collisions and contact Simulation Algorithm compute new statedetect collisions and backtrack compute and apply impulses compute and apply constraint forces current state next state collision state post-collision state

101. Simulation Loop

102. Check course’s website for more details, including other simulation codes…