# Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost

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1 Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost

2 Rigid body physics

3 Particle system Most simple instance of a physics system Each object (body) is a particle Each particle have forces acting upon it Constant, e.g. gravity Position dependent, e.g. force fields Velocity dependent, e.g. drag forces Event based, e.g. collision forces Restrictive, e.g. joint constraint So net force is a function F(p o, v, a, m, t, ) 3

4 Particle system Use the equations of motion to find the position of each particle at each frame At the start of each frame Sum up all of the forces for each particle From these forces compute the acceleration Integrate into velocity and position 4

5 Mass The mass is the measure of the amount of matter in the volume of an object m = ρ dv V where ρ is the density at each location in the object volume V We can also state that the mass is a measure of an object s resistance to motion or a change of motion the larger mass, the more difficult it is to set in or change the motion of an object 5

6 Mass For a 3D object, the mass is therefore the integral over its volume along the three dimensions m = ρ(x, y, z) dx dy dz For uniform density objects (rigid bodies usually are), the mass is then m = ρ V where ρ is the density of the object and V its volume 6

7 Center of mass The center of mass (COM) is the point at which all the mass can be considered to be concentrated obtained from the first moment, i.e. mass times distance point of balance of the object if uniform density, COM is also the centroid COM 7

8 Coordinate of the COM Center of mass COM = 1 ρ p p dv m V where p is the position at each location in V For a body made of particles COM = 1 m m i p i i=1 where m i is the mass of each particle p i in the body n 8

9 Center of mass Example for a body made of two particles in 1D x 2 x 1 x COM m 1 COM m 2 x COM = m 1x 1 + m 2 x 2 m 1 + m 2 9

10 Center of mass Quite easy to determine for primitive shapes 4.1 But what about complex surface based models? 10

11 Rigid body In rigid body physics, the motion of an object is not summarized to the translational motion of its center of mass A rigid body is considered as a system of particles remaining at fixed distances from each other with no relative translation or rotation among them 11

12 Rigid body A force can be applied anywhere on the object, producing also a rotational motion F COM α a 12

13 Moment of inertia The moment of inertia of a rigid body is a measure of how much the mass of the body is spread out It is a measure of the rigid body's ability to resist change in rotational motion It is defined with respect to a specific rotation axis We define r as the distance between any point in the object and the axis of rotation 13

14 For a mass point: I = m r 2 Moment of inertia r m For a collection of mass points: I = m i r i 2 i r 2 m 2 r 1 m 1 r 3 m 3 For a continuous mass distribution: I = r 2 dm M dm r 14

15 Momentum and inertia Remember that the angular momentum is L = I ω = r p = r mv = m(r (ω r)) If we look at the set of all elementary mass elements in the body, we have L = r ω r dm M 15

16 Let s define r = So we have Momentum and inertia x y z and ω = ω x ω y ω z L = y 2 + z 2 ω x xyω y xzω z yxω x + z 2 + x 2 ω y yzω z zxω x zyω y + (x 2 + y 2 )ω z dm 16

17 Let s define Momentum and inertia I xx = y 2 + z 2 dm I xy = I yx = xy dm I yy = z 2 + x 2 dm I xz = I zx = xz dm I zz = x 2 + y 2 dm I yz = I zy = yz dm 17

18 Momentum and inertia We then have the angular momentum L = I xx I xy I xz I yx I yy I yz I zx I zy I zz ω x ω y ω z = I ω 18

19 Inertia Finally the inertia can be expressed as the matrix y 2 + z 2 dm xy dm xz dm I = xy dm z 2 + x 2 dm yz dm xz dm yz dm x 2 + y 2 dm The diagonal elements are called the (principal) moment of inertia The off-diagonal elements are called products of inertia 19

20 Inertia Equivalently, we can define the inertia as I = V ρ x, y, z I = M r 2 dm = ρ r 2 dv V y 2 + z 2 xy xz xy z 2 + x 2 yz dx dy dz xz yz x 2 + y 2 you can notice that the diagonal elements are the distances to the respective principal axis and the non-diagonal elements the products of the perpendicular distances to the respective planes 20

21 Inertia of primitive shapes For primitive shapes, the inertia can be expressed with the parameters of the shape Illustration on a solid sphere we can calculate the inertia of the sphere by integration of the moment of inertia of thin discs along one axis (e.g. z) the surface of the sphere is defined by x 2 + y 2 + z 2 = R 2 21

22 Illustration on a solid sphere the distance to the axis of rotation is the radius of the disc at the cross section along z: r 2 = x 2 + y 2 = R 2 z 2 the inertia is given by the sum of moments of inertia of small cylinders of inertia I = r2 m along the z-axis: di = 1 2 r2 dm = 1 2 r2 ρdv = 1 2 r2 ρπr 2 dz so we have I = 1 ρπ R r 4 dz 2 1 Inertia of primitive shapes R 2 = 1 ρπ R R 2 z 2 2 dz 2 ρπ 2 R4 z 2R 2 z z 5 5 R R = ρπ R 5 as m = ρ 4 3 πr 3, we finally have I = 2 5 mr2 R = 22

23 Inertia of primitive shapes Solid sphere, radius r and mass m I = 2 5 mr mr mr2 z y x Hollow sphere, radius r and mass m I = 2 3 mr mr mr2 23

24 Inertia of primitive shapes Solid ellipsoid, semi-axes a, b, c and mass m I = 1 5 m(b2 +c 2 ) m(a2 +c 2 ) m(a2 +b 2 ) z y x Solid box, width w, height h, depth d and mass m h w d I = 1 12 m(h2 +d 2 ) m(w2 +d 2 ) m(w2 +h 2 ) 24

25 Inertia of primitive shapes Solid cylinder, radius r, height h and mass m h I = 1 12 m(3r2 +h 2 ) m(3r2 +h 2 ) mr2 y z x Hollow cylinder, radius r, height h and mass m h I = 1 12 m(6r2 +h 2 ) m(6r2 +h 2 ) mr 2 25

26 Parallel axis theorem But the object does not necessarily rotate around the center of mass To account for it, we need to modify the inertia matrix by applying the parallel axis theorem I v = I COM + mr 2 where I v is the inertia of the object about any axis v, I COM the inertia about an axis through the COM, m is the mass of the object and r is the distance between the axes 26

27 Parallel axis theorem So the elements of our inertia matrix become 4.4 I xx = y 2 + z 2 dm + md x 2 I xy = xy dm + md x d y I yy = z 2 + x 2 dm + md y 2 I xz = xz dm + md x d z I zz = x 2 + y 2 dm + md z 2 I yz = yz dm + md y d z 27

28 Perpendicular axis theorem For a planar 2D object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia of two perpendicular axes through the same point in the plane of the object z y z y x x I z = I x + I y for any planar object I z = 2I x = 2I y for symmetrical objects 28

29 Reference frame Do not forget that the inertia tensor is constant in body space but varies in world space So at each simulation frame, the inertia tensor in world space is calculated by I world t = R t I body t R(t) T where R is the rotation matrix describing the orientation of the body in the world space 29

30 Complex objects When an object consists of multiple primitive shapes 1. Calculate the individual inertia of each shape 2. Use parallel axis theorem to transform to inertia about an axis through the COM of the object 3. Add the inertia matrices together

31 Position on an object Remember that the object moves linearly as the COM moves Rotation add to the movement for points on the object Total motion of a point on the object is the sum of the two motions F 31

32 Sometimes a rigid body is not free to move around on its own, we want to constrain its movement wheels on a chair human body parts trigger of a gun opening door Motion constraint actually almost anything you can think of in a game... 32

33 To describe how a body can move in space you have to specify its degrees of freedom (DOF) Translational Rotational Degree of freedom 33

34 Kinematic pair A kinematic pair is a connection between two bodies that imposes constraints on their relative movement Lower pair, constraint on a point, line or plane Revolute pair, or hinged joint: 1 rotational DOF Prismatic joint, or slider: 1 translational DOF Screw pair: 1 coordinated rotation/translation DOF Cylindrical pair: 1 translational + 1 rotational DOF Spherical pair, or ball-and-socket joint: 3 rotational DOF Planar pair: 3 translational DOF Higher pair, constraint on a curve or surface 34

35 Integrate constraints A good enough approximation for game applications is to assume that the energy of the components of the net force acting on constrained degrees of freedom is converted into heat and sound (E o ) Therefore you can project the net force on the unconstrained degrees of freedom and forget about the other ones Not true for soft bodies 35

36 As solving a constraint for a body may influence the solving for another one, we need an iterative process while (!done) { } Integrate constraints for all constraints c do solve c; Convergence is usually ensured by reaching either a maximal amount of iterations or a minimal change in every constraints 36

37 Sequential impulses (SI) is a popular example of such iterative solver It applies impulses at each constraint to correct velocity SI is quite stable and converges to a global solution Why impulses? Sequential impulses Easier to deal with friction and collision Work with velocity rather than acceleration Given the time step, impulse and force are interchangeable But velocity constraints not precise, might produce position drift, breaking the constraint 37

38 Sequential impulses Step 1 Integrate applied forces, yielding to tentative velocities Step 2 Apply impulses sequentially for all constraints to correct the velocity errors Step 3 Use the new velocities to update the positions 38

39 Coordinate simulation Instead of adding impulses to enforce constraints, you can just not produce any force / torque along a specific DOF This is referred to as reduced coordinate simulation while impulse-based is a full coordinate simulation It allows for faster simulation (as less calculations are performed) and less tuning (notably of the magnitude of the impulses) But more vulnerable to numerical instability and might be difficult to parameterize the DOF system 39

40 Featherstone s Algorithm The Featherstone s Articulated Body Method (FABM) is an example of reduced coordinate constraint representation It computes the DOF accelerations given the articulated body current state and any external forces and torques made of three loops over the articulated body, so O(n) relies on the linear relation between the acceleration and the force f + F ext = I a + p where f is the test force, F ext are the external forces (e.g. gravity), I the inertia of the articulated body, a the DOF acceleration (unknown) and p the bias force 40

41 Featherstone s Algorithm Algorithm FABM (q, q, F ext, τ) // q is the state of the body (DOF values) v 0 0 for i 1 to n // compute velocities for all links j i 1 // j is the parent of link i v i R j j i v j + s i q i // R i is the transformation from local system j to i, si is the joint axis p i v i I i v i F ext,i c i v i s i q i for i n to 1 // compute inertia and bias force I accu i I i // start accumulation of inertia p accu i p i // start accumulation of bias force for j 1 to numchildren(i) // for each child of i of index j I i accu I i accu + R i j p i accu I i accu + R i j h i I accu i s i d i s T i h i u i τ i h T i c i s T accu i p i a 0 0 for i 1 to n j i 1 q i u i h i T Ri j aj d i a i R i j a j + c i + s i q i I j accu h jh j T d j R j i p j accu + I j accu c j + u j d j h j // compute acceleration for all joints // j is the parent of link i 41

42 Motion control Imagine you finally have rigid bodies moving and constrained correctly according to the forces you apply To let them live on their own will be fine for passive objects projectile, furniture, environmental objects etc. But not for living beings as they will just fall onto the ground at the beginning of the game 42

43 Motion control In current games, animated bodies are controlled using a mix of kinematics and dynamics Kinematics to replay and slightly adapt prerecorded motions Dynamics to passively animate objects reacting to external forces (e.g. human ragdoll) A real-time controller switches from one to the other according to events, forces, poses etc. 43

44 But imagine that you want to actively actuate rigid bodies using forces and torques not yet in games but will probably in a near future You need actuators (to generate motion from within the bodies) Joint torques External forces Virtual forces Muscle forces Motion control contractile elements activated by the brain... up to the source of motion! Where do we stop for a game? 44

45 Joint torques Most straightforward actuation model Joint torques directly generate torques for each actuated DOF We assume that there is a fake motor at the location of the joint that can produce torque For example if you want to increase a joint angle you will generate a positive torque and add it to the solving of the laws of motion Very useful to track pre-recorded motions or any other error-based poses (e.g. balance pose) Amount of applied torque depends directly on the error 45

46 External forces We can try to control the motion by applying external forces on the right object at the right local position for the right amount of time Difficult to produce realistic motions Do not really fit the real world neither as motion originates from inside Similar to a puppetry technique but need a master knowing how to control the system But very useful for the control of the global orientation and position of a complex system (root node) 46

47 Virtual forces Not really an actuation method Emulate the effect of applying an external force by computing the equivalent joint torque Use the relation between joint rotation and position where a force is applied (Jacobian of the system) 47

48 Muscle forces Motion actually comes from the contraction of muscles they also produce torques at joints, but not in a fake motor way By adding their position and physical behavior to the model, we can actuate rigid bodies with muscle activation commonly used in biomechanics / motion analysis the muscle model is usually a combination of two nonlinear springs and one damper 48

49 Controller design Read the survey paper Interactive Character Animation Using Simulated Physics: A State-ofthe-Art Review Joint-space motion control defines and tracks kinematics target Stimulus-response network control genetically evolves controllers according to objectives Constrained dynamics optimization control finds optimal torques through online optimization 49

50 PD controller If your goal is to actuate a joint by feeding targets to track over time, then updating the necessary torque is quite easy You can use a Proportional Derivative (PD) controller it can be used to compute joint torques linearly proportional to the difference between the current state and the target state it will be based on joint orientation and angular velocity 50

51 PD controller τ = k p θ d θ + k v θ d θ where τ is the generated joint torque, θ d and θ the desired and current joint angle, θ d and θ the desired and current joint angular velocity k p and k v are the controller gains, they regulate how responsive the controller is to deviations in orientation and angular velocity, respectively 51

52 PD controller A difficult and time-consuming task for the motion controller designer is to define the gain values too high k p will produce stiff unresponsive motions too low k p will not track correctly the target too high k v will converge too slowly to the target too low k v will produce oscillations Expect to spend time fine tuning gains 52

53 End of Rigid body physics Next Numerical Integration

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