Physical Simulation. October 19, 2005
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1 Physical Simulation October 19, 2005
2 Objects So now that we have objects, lets make them behave like real objects Want to simulate properties of matter Physical properties (dynamics, kinematics) [today s task] Collisions [next week] We will deal with rigid bodies (only)
3 If we want to simulate our reality We have to simulate the physical reality of our world. We have to simulate the PHYSICS of our world.
4 Why? Critical requirement for realistic simulation of complex environments. Critical requirement for realistic training.
5 Newton s Laws of Motion Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. The relationship between an object s mass m, its acceleration a, and the applied force F is F=ma. For every action there is an equal and opposite reaction.
6 Some definitions kinematics - the study of the motion of bodies without regard to the forces acting upon them dynamics - the study of the forces that cause the motion of bodies kinetics - study of the motion of bodies
7 Physics of Motion On the plane
8 Mass Fundamental property of matter need to simulate this for realistic VR Can be useful to model an object as a collection of point masses rigidly connected
9 Centre of Mass It will often be useful to collapse an object to a point (to its centre of mass).
10 And the distribution of Mass Moment of Inertia The distribution of mass around an axis (typically choose an axis passing through the centre of mass)
11 Newton s Laws for a point Point with mass m (kg) acceleration a (m/s/s) some external force F (N=kg.m/s/s)
12 Theory An applied force -> acceleration Integrate velocity and position vectors
13 Reality Need to numerically integrate for velocity and position need to compute time step
14 Time Step Easy choice is that time step is constant Issues with collisions (we may intrude into obstacles). Issues with numerical accuracy. Dynamic steps size -- computational overhead
15 Numerical Integration Euler s method is easiest Others: Improved Euler, Runge-Kutta,...
16 Some Forces Can be defined in body-centric or worldcentric coordinate systems Examples Thrusters (jet engines, rockets, propellers) Drag Gravity Buoyancy Friction
17 Where we are now Can simulate a point object, with forces acting on it Forces have a direction and magnitude Observe local forces vs. external forces.
18 Non-point objects Objects have both a position and orientation. Objects have both a velocity and a rotational velocity. Objects have both an acceleration and an angular acceleration
19 Angular and Translational Motion When a force is applied to may contribute to both an object s linear velocity and its rotational velocity. Basic approach is to take the forces and (i) apply them to the centre of mass of the object to compute estimate on position and (ii) apply them to the object to compute the effect on its orientation.
20 Torque
21 Rotational Motion Torque Angular acceleration Angular velocity Orientation
22 A simple 2D example w q h F A rectangular solid object, uniform density, one external (controllable force)
23 Mass and Moment of Inertia Uniform mass (m, so density is m/(wh)) Centre of mass is in the centre (define as origin)
24 Complete Example In vehicle coordinates Control inputs are (f,q) force f being applied at angle q at position r=(0,h/2). F=-f(cos(q),sin(q)) a=f/m tau=(0,h/2)xf -> (fh/2)sin(q)
25 Convince Yourself Try q=0 (no rotational motion) Try q=90 (both translational and rotational motion)
26 What s left These are all in a local coordinate frame Will have to map to a global coordinate frame to be useful. Adding more forces (e.g. friction, wind)
27 Physics of Motion Three dimensional Motion
28 In 3D Many similarities, but the math gets more complex (rotations) Same basic approach,
29 Mass Compute Centre of Mass and Inertia Tensor (a 3x3 matrix)
30 Motion of Centre of Mass An applied force -> acceleration Integrate velocity and position vectors
31 Rotational Motion Again, sum up torques and apply to update the orientation of the rigid body
32 Local vs. External As in 2D forces must be in the appropriate frame. Process is more tricky due to the nature of 3D rotations
33 Applications Essentially all of your favorite flight simulators, car racing games, etc., follow this mechanism Building accurate physical models is difficult hard to model Often a tradeoff between fidelity and performance.
34 Summary Physical simulation of rigid bodies. You are responsible for 2D version (should understand 3D) Many papers on the web site to help. May want to look at your old physics texts
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