Character Animation with Direct3D- P7

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Character Animation with Direct3D- P7:This book is primarily aimed at teaching indie and hobby game developers how to create character animation with Direct3D. Also, the seasoned professional game developer may find some interesting things in this book. You will need a solid understanding of the C++ programming language as well as general object-oriented programming skills.

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106 Character Animation with Direct3D FIGURE 5.9 Mocap with actor jumping. How is the difference in size between actors and models handled? JN: Basically, there are no limits there at all. We have even had a human actor do the motions for a T-Rex once. You will, of course, have to make adjustments if you record a very small person and use the recorded motions for a large game character. Often small people and large people move completely different. Another funny thing is that you can really see from the motion whether the actor is a woman or a man. One time we tried to capture motion for a 3D character, a woman, and I was acting. That’s actually somewhat of an inside joke here now. An outsider came to look at the final animation and he immediately said that it has got to be a man. JM: That’s the cool accuracy of this system. If you know the actor personally, you can see that it is him or her just from watching the motion capture data. There’s more emotion in movements than you might think, and you can really see the difference when you compare motion capture animation to traditional artist created animation. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 5 Advanced Skeletal Animation Techniques 107 C ONCLUSIONS In this chapter you learned about some more advanced topics of character animation. You learned about animation blending, which is something you’ll definitely need if you ever aim to create a realistic character. With it, you can create new animations by blending two others together. You can also create a transition between two animations by blending between them. Another advanced topic covered in this chapter was animation callbacks, with which you can time events in your game to when a specific animation occurs. If you made a fighting game for instance, you might want to time the “punch” sound to a specific time in the punch animation. Lastly, the topic of motion capture and its variations were briefly covered. That about wraps up the skeletal animation part of this book, which the last three chapters have focused on. Now it’s time to turn to dynamic animation and look at implementing a ragdoll system. C HAPTER 5 E XERCISES Try modifying Example 5.1 so that you run the same animation in two different tracks, with different weights and slightly different speeds. See what happens? Create animation callbacks for the footsteps in the Soldier’s walk animation. F URTHER R EADING [1] http://www.organicmotion.com [2] http://www.mova.com [3] http://www.moven.com [4] http://www.vicon.com ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
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6 Physics Primer As is common practice with most programming books that include only one chapter covering physics, I’ll start this one with a disclaimer. This book is not about physics! There are books focusing solely on the topic of game physics. However, since I will be covering ragdoll animations in this book, I need cover the “bare bones,” so to speak, of creating a physics engine. The physics engine demonstrated in this chapter builds on simulating particles connected with springs. In this chapter you’ll learn about the following topics: 109 ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
110 Character Animation with Direct3D Basics of rigid body physics Quaternions Oriented bounding boxes Intersection tests Simulating a particle Simulating a spring In the beginning of this chapter, many basic physical concepts will be covered. If you are already comfortable with Newton’s three laws of motion, gravity, quaternions, etc., feel free to skip ahead to the implementation part at the end of this chapter. INTRODUCTION TO RIGID BODY PHYSICS If you’ve never come across the topic of rigid bodies before, don’t worry; it is quite easy to understand (even though it can be quite hard to simulate). A rigid body is by definition a solid mass in space that does not change shape. A real-life basketball is an example of a non-rigid body. The fact that the basketball is not rigid is what causes it to bounce once it hits the floor. Instead, imagine a sphere the size of a basketball in solid steel hitting a steel floor. Very likely there wouldn’t be a very large bounce from a collision like this. This is because the steel sphere won’t change its shape. It is rigid! In computer graphics there are accurate mathematical systems where rigid bodies are simulated colliding with the environment and with each other. In games, however, this simulation has to be able to run in real time. This essentially means several shortcuts need to be taken, and some serious optimizations must be done in order to get it fast enough for real-time applications. So when making physics engines for games, things like speed, stability, and appearance take prece- dence over accuracy and realism. Before I dive into rigid body physics, I’ll cover the basics of physics in general. Over 300 years ago a fellow named Isaac Newton published a three-volume work called Philosophiae Naturalis Principia Mathematica, or Principia for short. In plain English, the title was “Mathematical Principles of Natural Philosophy” and contained, as you may already know, Newton’s three laws of motion. The first of the three volumes was called De motu corporum (“On the motion of bodies”), and now, more than 300 years later, these laws are still used when simulating physics in games. Table 6.1 shows a simplified version of Newton’s laws of motion. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 111 TABLE 6.1 NEWTON’S LAWS OF MOTION 1st Law: An object’s velocity will not change unless affected by an external force. 2nd Law: The acceleration of an object is proportional to the magnitude of the force acting on the object and inversely proportional to its mass. 3rd Law: Every action has an equal and opposite reaction. The second law can also be described with the famous formula: F F = ma or a= m The force equals the mass of an object times its acceleration. However, more often you are interested in the acceleration resulting from an external force, in which case the acceleration a is the force F divided by an object’s mass m. Later on I’ll dis- cuss how the acceleration of an object affects its velocity and its position. But first it is time to cover some important concepts needed to create your own physics engine. FORCES As you may have seen in Newton’s three laws, there was a lot of talk about forces. A force has both a magnitude and a direction. Imagine, for example, two equally strong men pushing a box from opposite sides. Since the directions of their efforts are opposing, the box won’t move an inch. However, if the two men were pushing from the same side, the two forces would combine and the box would move in the direction they are pushing. This little thought-experiment proves the fact that two opposing forces cancel each other out. The most common force you face on a daily basis is gravity (unless you happen to be an astronaut). In his Principia, Newton also defined the law of gravity: m1 m2 FG = G d2 FG is the resulting gravitational force, G is the universal gravitational constant [Wikipedia], and m1 and m2 are the masses of the two objects attracting each other. Finally, d is the distance between the objects. Simply put: Two objects attract each other with a force proportional to the product of their masses divided by the square of the distance between them (quite a mouthful). Take the simple example of the Sun and the Earth, as shown in Figure 6.1. ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
112 Character Animation with Direct3D FIGURE 6.1 Gravitational pull between the Earth and the Sun. It might be hard to see from the numbers representing the mass of the Sun and the Earth. But the Sun has 333,000 times more mass than the Earth and it represents 98% of all the mass in our solar system. This means that whatever the gravitational force is between the Sun and the Earth, 99.9997% of that force affects the earth, and 0.0003% of it affects the sun. This gravitational force is the only thing keeping the Earth in orbit around the Sun. Now turn your attention to the Earth itself, where there’s a similar gravitational force affecting all smaller objects on the Earth’s surface (for instance, an apple). Just like in the example of the Sun and the Earth, each small object actually also attracts the Earth toward it. However, this force is so small that it is negligible. This leaves us with one force pulling all objects toward the Earth’s surface. In games, this is almost always represented as a constant force in the negative Y direction (and the curvature of the Earth is completely dismissed). Just as in real life, games try to simulate the Earth’s gravitational pull (9.8 m/s2), making objects behave as realistically as possible. Later on I’ll cover how we apply gravity to game objects, updating their velocity and position over time. That pretty much covers the gravitational force. However, there are plenty of other non-constant forces in the world, such as wind, collision impacts, etc.—for example, in an action game when a bullet hits an object, the bullet will affect the object with a force proportional to its speed and mass. This brings us to the different ways a force can affect an object. THE EFFECT OF FORCES ON A RIGID BODY Okay, so you know that there are different forces affecting an object (i.e., a rigid body). The next thing to cover is how these bodies react when affected by an external force. First, consider what happens to a rigid body’s position when you push or pull it with an external force. Figure 6.2 shows an example. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 113 FIGURE 6.2 How a force affects the position of a rigid body. As you can see in Figure 6.2A, the body is being affected by two external forces pushing at the object from different directions. Just like with the analogy of the two men pushing a box, the rigid body will move over time in the combined direction of the forces as long as the forces are applied. In this example you can see the center of the rigid body’s mass represented as a cross. The forces in Figure 6.2 are pointing straight at the object’s center of mass, resulting in the energy being used 100% to move the object. However, this is of course not always the case; sometimes forces are applied to an object causing it to spin, rather than to move linearly. Try it yourself. Find a small object (like a pen) and poke it close to its center of mass. No doubt the object will move in the direction you poked it. Try it again, but this time poke it far from its center of gravity. This time the object will spin instead. This phenomenon is shown Figure 6.3. FIGURE 6.3 How a force affects the orientation of a rigid body. ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
114 Character Animation with Direct3D Just like in the previous case, when there are several forces affecting the orien- tation of an object, the forces are all summed up before applying the final rotation of the object. So far I have only been talking about objects in 2D space, but the same fundamental reactions naturally occur with objects in 3D space as well. To describe the orientation of an object in a physics simulation, you basically have three different options: Euler angles Rotation matrix Quaternions If you have covered the basics of 3D math, you have probably come in contact with Euler angles. Euler angles are easy to understand; you have one rotation value for the yaw, pitch, and roll of an object. Even though the Euler angles are easy to grasp, they come with some limitations such as Gimbal lock (covered in Chapter 4). Another option is to use a rotation transformation matrix. Although this is an often used option, it suffers from the fact that small imperfections creep in (due to rounding of float values) and the matrix becomes skewed over time. This pretty much leaves us with quaternions! QUATERNIONS Before continuing with the implementation of the physics engine, and later the ragdoll system, you need to understand the concept of quaternions. Quaternions are stored as four values; the first value is a scalar value, and the three following values describe a vector. Since a quaternion is a four-dimensional entity, it’s close to impossible to create a mental image of it. Instead, the best approach for any non-mathematician is just to learn how to create an arbitrary orientation using quaternions, and then get more comfortable with them by using them in your code. The word quaternion comes from Latin’s “Quaternio,” which means “set of four.” A quaternion is defined as the following four values: q = [w, x, y, z] Quaternions were invented by an Irish mathematician named Sir William Hamilton some 200 years ago. His motivation was to come up with a method to describe the transformation required to transform a vector V1 into a vector V2. In the case of two points A and B, there exists a vector V that transforms point A into point B, as shown in Figure 6.4. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 115 FIGURE 6.4 Transforming one point into another. A quaternion performs the same operation but on two vectors. A quaternion can transform one specific vector into another specific vector. Usually a vector is described as its x, y, and z components, but a vector can also be described with a direction and a length. Here’s the general theory of how you can transform one vector into another when they are described as a direction and a length: FIGURE 6.5 Vectors V1 and V2 differ both in magnitude and orientation. Figure 6.5 shows the two vectors V1 and V2. The first step of changing V1 into V2 is to make sure their length (or magnitude) is the same. For this you simply ignore their orientation and calculate the scale factor S you need to apply to V1 so that its length matches that of V2. V2 S= V1 The scale factor S, is calculated by simply dividing the length of vector V2 with the length of vector V1. Next you need a way of transforming a direction into another, and for this you need to use versors. A versor describes the difference in orientation of two vectors of equal length, as shown in Figure 6.6. ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
116 Character Animation with Direct3D FIGURE 6.6 Transforming the direction of a vector to another. All you need to reorient V1 into V2 is the angle between the vectors and the axis around which to rotate. The angle is obtained by calculating the inverse cosine value of the vectors’ dot product: a = a cos(V1•V2) The axis around which the rotation should take place is simply the cross product of the vectors: A = V1 V2 The quaternion that combines the scaling and rotating of V1 into V2 can then be created using one value for scale, one for the angle, and two values for the plane on which V1 and V2 lie. When I talk about orientations and rotations using quaternions throughout this book, only a small subset of all possible quaternions are of particular interest. This subset is all the unit quaternions (i.e., quaternions with a length of 1). All possible unit quaternions form a hyper-sphere, which basically is a four-dimensional sphere (something quite hard to visualize). Here’s a practical problem—one which you will no doubt run into once you switch to quaternions. Have a look at the problem shown in Figure 6.7. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 117 FIGURE 6.7 Transforming the unit vector p1 into p2 using quaternions. In Figure 6.7, a rotation of -radians around Axis A is described. This rotation will transform the vector p1 into p2. Remember that the axis around which you want to rotate can be any arbitrary axis. Here’s how you can create a quaternion to define this rotation. Remember that quaternions were defined as: q = [w, x, y, z] In the example of rotating -radians around Axis A (x, y, z), the quaternion for this would be: cox(a/2), x•sin(a/2), q= y•sin(a/2), z*sin(a/2) In DirectX, a quaternion is stored using the D3DXQUATERNION structure: struct D3DXQUATERNION { FLOAT x; FLOAT y; FLOAT z; FLOAT w; }; ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
118 Character Animation with Direct3D ... //Previous example in code D3DXVECTOR3 A(0.2f, 0.6f, -0.3f); //Any arbitrary axis D3DXVec3Normalize(&A, &A); //Normalized float angle = 0.342f; //Arbitrary angle //Quaternion is then defined as D3DXQUATERNION q; q.w = cos(angle * 0.5f); q.x = sin(angle * 0.5f) * A.x; q.y = sin(angle * 0.5f) * A.y; q.z = sin(angle * 0.5f) * A.z; Luckily the D3DX library is full of quaternion helper functions. Table 6.2 lists the names of the most useful quaternion functions. These will be used throughout the following chapters, but if you want to familiarize yourself with them now, check out the DirectX Documentation. TABLE 6.2 D3DX QUATERNION FUNCTIONS D3DXQuaternionRotationAxis(): This function creates a quaternion that rotates around an arbitrary axis. D3DXQuaternionRotationMatrix(): This function creates a quaternion from a (rotation) matrix. D3DXQuaternionNormalize(): This function normalizes a quaternion making it a unit quaternion of length 1. D3DXQuaternionInverse(): This function creates the inverse quaternion. D3DXMatrixRotationQuaternion(): This function creates a rotation matrix from a quaternion. For a more in-depth explanation of quaternions and quaternion math, check out [Ibanez01] or [Svarovsky00]. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 119 Matrices have the identity matrix, which when applied to an object leaves the translation, rotation, and scale unaffected: 1 0 0 0 01 00 Mi = 00 10 00 01 In the same way, quaternions have the identity quaternion, which does not af- fect the rotation when applied to an object: D ESCRIBING THE W ORLD If you have ever implemented things like viewport culling or picking, you may have encountered the Axis-Aligned Bounding Box (AABB). This bounding volume has two vectors, one containing the minimum corner (x, y, z) and once containing the maximum corner (x, y, z) of the box. Although it is fast to perform different inter- section tests, etc. with this volume, it is quite inaccurate in many cases. To describe the world your objects will interact with in the physics simulation, Oriented Bounding Boxes (OBB) will be used instead. By combining several of these Oriented Bounding Boxes, complex-enough worlds can be built for the purpose of this book. Figure 6.8 shows an image comparing the AABB and the OBB in two dimensions: As shown in Figure 6.8, the OBB has one great advantage over the AABB: It fits much closer to the object (especially when the object is rotated, as is the case in Figure 6.8). With the OBB you can build up a much more accurate representation of the game world. As explained earlier, an AABB is often described as a MAX and a MIN vector. These hold the maximum and minimum coordinates of the box in the x, y, and z dimensions. With an OBB, however, you need one vector with the size of the box, one vector for its position, and one vector for its orientation. However, in most physics applications, vectors (Euler angles) are avoided for orientation because of the Gimbal lock. So instead I’ll use a quaternion for the orientation as covered in the previous section. ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
120 Character Animation with Direct3D FIGURE 6.8 Axis-Aligned Bounding Box (AABB) versus Oriented Bounding Box (OBB). THE ORIENTED BOUNDING BOX CLASS The Oriented Bounding Box (OBB) is a structure that describes a volume with width, height, length, and orientation. For it to be useful in physic simulations, you need to be able to perform a number of different intersection tests. Often you also need to know how far an intersecting object is into the OBB so the physics engine can handle the collision accordingly. The OBB class is defined as follows: class OBB { public: OBB(); OBB(D3DXVECTOR3 size); bool Intersect(D3DXVECTOR3 &point); bool Intersect(OBB &b); public: D3DXVECTOR3 m_size, m_pos; D3DXQUATERNION m_rot; }; Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 121 At the moment I’ve only defined two intersection tests: one Point-OBB and one OBB-OBB test. The OBB-OBB intersection test is a bit beyond the scope of this book. You’ll find a good article on intersection tests online [Gomez99]. You can also find the code for the OBB-OBB test in Example 6.1 on the accompanying CD-ROM. However, you need to have a closer look at the more important Point-OBB test. First consider the point intersection test of an Axis-Aligned Bounding Box (AABB): class AABB { public: AABB(D3DXVECTOR3 max, D3DXVECTOR3 min) { m_max = max; m_min = min; } bool Intersect(D3DXVECTOR3 &p) { if(p.x < m_min.x || p.x > m_max.x)return false; if(p.y < m_min.y || p.y > m_max.y)return false; if(p.z < m_min.z || p.z > m_max.z)return false; return true; } public: D3DXVECTOR3 m_max, m_min; }; The Point-AABB intersection test is very simple; you simply check for each dimension if the point is smaller than the minimum or larger than the maximum value of the bounding box. If so, the point must be outside the box. If after check- ing all the dimensions (x, y, z) none of these tests failed, then the point must be inside the box. To do this same test for an OBB is not quite as simple because it also has an orientation. Figure 6.9 shows this problem. ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
122 Character Animation with Direct3D FIGURE 6.9 Point intersection, AABB versus OBB. Look at Figure 6.9 and perform the intersection test listed earlier for the AABB. Now if you try to do the same for the OBB, you would soon realize that it is not quite as easy to tell whether the points are inside or outside the box using this method. However, with one simple trick it becomes just as easy to do for the OBB as for the AABB. You simply need to look at it from a different point of view. Tilt your head until the OBB becomes an AABB and then check the points using the same algorithm used for the Point-AABB intersection test. With this in mind, here’s the code for the Point-OBB intersection test: bool OBB::Intersect(D3DXVECTOR3 &point) { //Calculate the translation & rotation matrices D3DXMATRIX p, r, world; D3DXMatrixTranslation(&p, m_pos.x, m_pos.y, m_pos.z); D3DXMatrixRotationQuaternion(&r, &m_rot); //Calculate the world matrix for the OBB D3DXMatrixMultiply(&world, &r, &p); //Calculate the inverse world matrix D3DXMatrixInverse(&world, NULL, &world); //Transform the point with the inverse world matrix D3DXVECTOR4 pnt; D3DXVec3Transform(&pnt, &point, &world); Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 123 //Check the point just as if it was an AABB if(abs(pnt.x) > m_size.x)return false; if(abs(pnt.y) > m_size.y)return false; if(abs(pnt.z) > m_size.z)return false; return true; } With the OBB structure you can describe some fairly complex (although an- gular) environments for us to run physics simulations in. In Figure 6.10 you can see an example of this: on the left side, the mesh used in the rendering pipeline, and on the right, the physical representation of the same mesh using OBBs. FIGURE 6.10 Example of an environment described using Oriented Bounding Boxes. The mesh being rendered can have thousands of polygons and would therefore be very expensive to do collision detection with. The OBB representation, on the other hand, is very fast, without losing too much detail. ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
124 Character Animation with Direct3D EXAMPLE 6.1 This chapter has now gone on long enough without a proper code example. This example implements the OBB class and also shows you an OBB-OBB intersection test. Try to construct a representation of a cabin like the one in Figure 6.10 using this OBB class. P HYSICS S IMULATION Remember that the long-term goal is to create a ragdoll animation and have a character fall/collide with objects in a realistic way. There’s still a long way to go before realizing this, although a lot of things have now been covered. You know how to represent the environment (or at least how to create a simplified version of it). Now it is time to start looking at the interaction bit. I guess a quick recap is in order. So far you have learned the theory of rigid bod- ies, how forces affect an object, the basics of quaternions, and how to describe the environment using Oriented Bounding Boxes. You will now finally get to put these lessons into practice. Now it is time to look at how to implement a few physical prop- erties of a rigid body and how to update (i.e., simulate) these in a realistic manner. In this book I’ll use the following interface to describe objects in the physics simulation. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Chapter 6 Physics Primer 125 class PHYSICS_OBJECT { public: virtual void Update(float deltaTime) = 0; virtual void Render() = 0; virtual void AddForces() = 0; virtual void SatisfyConstraints(vector &obstacles) = 0; }; The Update() function updates the position of the object, the Render() function renders it, and the AddForces() function adds forces to the objects such as gravity, wind, etc. The SatisfyConstraints() function takes a list of Oriented Bounding Boxes as a parameter. The OBBs are the physical representation of the world. The SatisfyContraints() function handles the object’s collision with the world. Next, there’s the PHYSICS_ENGINE class, which takes care of a collection of PHYSICS_OBJECT objects and runs a simulation with these in the physical representation of the world. class PHYSICS_ENGINE { public: PHYSICS_ENGINE(); void Init(); void Update(float deltaTime); void AddForces(); void SatisfyConstraints(); void Render(); void Reset(); private: vector m_physicsObjects; vector m_obstacles; float m_time; }; As you can see, this class contains a vector of physics objects and a list of OBBs. Each time step, every object in the physics simulation is updated in the following manner: 1. Add forces (gravity, wind). 2. Update position. 3. Satisfy constraints (check for collisions, and move object to a legal position). ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.