246 Character Animation with Direct3D TWO-JOINT INVERSE KINEMATICS Now I’ll show you how to attack the Two-Joint “Reach” IK problem. To solve this problem easier, you must take the information you know about people in general and put it to good use. For example, in games the elbow joint is treated like a hinge joint with only one degree of freedom (1-DoF), while the shoulder joint is treated like a ball joint (3-DoF). The fact that you treat the elbow (or knee) joint as a hinge makes this a whole lot simpler. You know that the arm can be fully extended, completely bent, or something in between. So, in other words, you know that the angle between the upper and lower arm has to be between 0 and 180 degrees. This in turn makes it pretty easy for you to calculate the reach of an arm when you know the length of the upper and lower arm. Consider Figure 11.7, for example. EXAMPLE 11.1 This is the first inverse kinematics example featuring a simple Look-At example. The soldier will look at the mouse cursor just like in the earlier examples with the eyeballs, except in this example the head bone is manipulated to turn the whole head to face the cursor. As you can see in this example, the IK are applied on top of normal keyframed animation. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. The black line in Figure 11.7 defines all the points that this arm can reach, assuming that the elbow joint can bend from 0 to 180 degrees. Let’s say that you’re trying to make your character reach a certain point with his arm. Your first task is to figure out the angle of the elbow joint given the distance to the target. Using the Law of Cosines, this becomes a pretty straightforward task, since you know the length of all sides of the triangle. The formula for the Law of Cosines is: C 2 = A 2 + B 2 – 2ABcos(x) Trivia: You might recognize part of the Law of Cosines as the Pythagorean Theorem. Actually, the Pythagorean Theorem is a special case of the Law of Cosines where the angle x is 90 degrees. Since the cosine for 90 degrees is zero, the term -2ABcos(x) can be removed. Chapter 11 Inverse Kinematics 247 FIGURE 11.7 Within an arm’s reach? Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. Figure 11.7 shows the Law of Cosines applied to the elbow problem. In Figure 11.8, C is known because it is the length from the shoulder to the IK target. A and B are also known because they are simply the length of the upper and lower arm. So to solve the angle x, you just need to reorganize the Law of Cosines as follows: x = acos A 2 + B 2 – C 2 ͩ 2AB ͪ First you have to bend the elbow to the angle that gives you the right “length.” Then you just rotate the shoulder (a ball joint, remember?) using the same simple Look-At IK approach covered in the previous example. The ApplyArmIK() function has been added to the InverseKinematics class to do all this: void InverseKinematics::ApplyArmIK(D3DXVECTOR3 &hingeAxis, D3DXVECTOR3 &target) { // Set up some vectors and positions D3DXVECTOR3 startPosition = D3DXVECTOR3( m_pShoulderBone->CombinedTransformationMatrix._41, m_pShoulderBone->CombinedTransformationMatrix._42, m_pShoulderBone->CombinedTransformationMatrix._43); D3DXVECTOR3 jointPosition = D3DXVECTOR3( m_pElbowBone->CombinedTransformationMatrix._41, m_pElbowBone->CombinedTransformationMatrix._42, m_pElbowBone->CombinedTransformationMatrix._43); 248 Character Animation with Direct3D FIGURE 11.8 The Law of Cosines. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. D3DXVECTOR3 endPosition = D3DXVECTOR3( m_pHandBone->CombinedTransformationMatrix._41, m_pHandBone->CombinedTransformationMatrix._42, m_pHandBone->CombinedTransformationMatrix._43); D3DXVECTOR3 startToTarget = target - startPosition; D3DXVECTOR3 startToJoint = jointPosition - startPosition; D3DXVECTOR3 jointToEnd = endPosition - jointPosition; float distStartToTarget = D3DXVec3Length(&startToTarget); float distStartToJoint = D3DXVec3Length(&startToJoint); float distJointToEnd = D3DXVec3Length(&jointToEnd); // Calculate joint bone rotation // Calculate current angle and wanted angle float wantedJointAngle = 0.0f; if(distStartToTarget >= distStartToJoint + distJointToEnd) { // Target out of reach wantedJointAngle = D3DXToRadian(180.0f); } else { //Calculate wanted joint angle (using the Law of Cosines) float cosAngle = (distStartToJoint * distStartToJoint + distJointToEnd * distJointToEnd – distStartToTarget * distStartToTarget) / (2.0f * distStartToJoint * distJointToEnd); wantedJointAngle = acosf(cosAngle); } //Normalize vectors D3DXVECTOR3 nmlStartToJoint = startToJoint; D3DXVECTOR3 nmlJointToEnd = jointToEnd; D3DXVec3Normalize(&nmlStartToJoint, &nmlStartToJoint); D3DXVec3Normalize(&nmlJointToEnd, &nmlJointToEnd); //Calculate the current joint angle float currentJointAngle = acosf(D3DXVec3Dot(&(-nmlStartToJoint), &nmlJointToEnd)); Chapter 11 Inverse Kinematics 249 Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. //Calculate rotation matrix float diffJointAngle = wantedJointAngle - currentJointAngle; D3DXMATRIX rotation; D3DXMatrixRotationAxis(&rotation, &hingeAxis, diffJointAngle); //Apply elbow transformation m_pElbowBone->TransformationMatrix = rotation * m_pElbowBone->TransformationMatrix; //Now the elbow “bending” has been done. Next you just //need to rotate the shoulder using the Look-at IK algorithm //Calcuate new end position //Calculate this in world position and transform //it later to start bones local space D3DXMATRIX tempMatrix; tempMatrix = m_pElbowBone->CombinedTransformationMatrix; tempMatrix._41 = 0.0f; tempMatrix._42 = 0.0f; tempMatrix._43 = 0.0f; tempMatrix._44 = 1.0f; D3DXVECTOR3 worldHingeAxis; D3DXVECTOR3 newJointToEnd; D3DXVec3TransformCoord(&worldHingeAxis, &hingeAxis, &tempMatrix); D3DXMatrixRotationAxis(&rotation,&worldHingeAxis,diffJointAngle); D3DXVec3TransformCoord(&newJointToEnd, &jointToEnd, &rotation); D3DXVECTOR3 newEndPosition; D3DXVec3Add(&newEndPosition, &newJointToEnd, &jointPosition); // Calculate start bone rotation D3DXMATRIX mtxToLocal; D3DXMatrixInverse(&mtxToLocal, NULL, &m_pShoulderBone->CombinedTransformationMatrix); D3DXVECTOR3 localNewEnd; //Current end point D3DXVECTOR3 localTarget; //IK target in local space D3DXVec3TransformCoord(&localNewEnd,&newEndPosition,&mtxToLocal); D3DXVec3TransformCoord(&localTarget, &target, &mtxToLocal); D3DXVec3Normalize(&localNewEnd, &localNewEnd); D3DXVec3Normalize(&localTarget, &localTarget); 250 Character Animation with Direct3D Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. D3DXVECTOR3 localAxis; D3DXVec3Cross(&localAxis, &localNewEnd, &localTarget); if(D3DXVec3Length(&localAxis) == 0.0f) return; D3DXVec3Normalize(&localAxis, &localAxis); float localAngle = acosf(D3DXVec3Dot(&localNewEnd, &localTarget)); // Apply the rotation that makes the bone turn D3DXMatrixRotationAxis(&rotation, &localAxis, localAngle); m_pShoulderBone->CombinedTransformationMatrix = rotation * m_pShoulderBone->CombinedTransformationMatrix; m_pShoulderBone->TransformationMatrix = rotation * m_pShoulderBone->TransformationMatrix; // Update matrices of child bones. if(m_pShoulderBone->pFrameFirstChild) m_pSkinnedMesh->UpdateMatrices( (BONE*)m_pShoulderBone->pFrameFirstChild, &m_pShoulderBone->CombinedTransformationMatrix); } There! This humongous piece of code implements the concept of Two-Joint IK as explained earlier. As you can see in this function we apply any rotation of the joints both to the transformation matrix and the combined transformation matrix of the bone. This is because the SkinnedMesh class recalculates the combined transformation matrix whenever the UpdateMatrices() function is called. So if you haven’t applied the IK rotation to both matrices it would be lost when the UpdateMatrices() function is called. Chapter 11 Inverse Kinematics 251 Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. CONCLUSIONS This chapter covered the basics of inverse kinematics (IK) and explained that as a general problem it is quite tough to solve (even though there are quite a few approaches to doing so). I covered two specific IK applications for character animation: Look-At and Two-Joint “Reach” IK. The Two-Joint IK can also be used for placing legs on uneven terrain, making a character reach for a game- world object, and much more. You would also need IK to make a character hold an object (such as a staff, for example) with both hands. This could, of course, be done with normal keyframe animation as well, but it then often results in one hand not “holding on” perfectly and sometimes floating through the staff (due to interpolation between keyframes). 252 Character Animation with Direct3D EXAMPLE 11.2 Example 11.2 has all the code for the Two-Joint IK solution covered in this section. You move the target point around with the mouse, and the character will attempt to reach it with one arm. Try to modify this example by limiting the freedom of the shoulder joint so that the arm can’t move through the rest of the body. Also, see if you can apply Two-Joint IK to the other limbs (legs and other arm) as well. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. Hopefully this chapter served as a good IK primer for you to start implementing your own “hands-on” characters. This chapter essentially wraps up the many individual parts of character ani- mation in this book. CHAPTER 11 EXERCISES Add weights to the IK functions enabling you to blend between keyframed animation and IK animation. A good next step for you would be to combine a keyframed animation such as opening a door with IK. As the animation is in the state of holding the door handle, blend in the Two-Joint IK with the door handle as the IK target. The soldier is holding the rifle with two hands. Glue the other hand (the one that is not the parent of the rifle) to it using IK. Implement IK for the legs and make the character walk on uneven terrain. Implement aiming for the soldier. FURTHER READING [Melax00] Melax, Stan, “The Shortest Arc Quaternion,” Game Programming Gems. Charles River Media, 2000. Chapter 11 Inverse Kinematics 253 Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. This page intentionally left blank Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. 255 Wrinkle Maps12 I’ll admit that this chapter is a bit of a tangent and it won’t involve much animation code. This chapter will cover the fairly recent invention of wrinkle maps. In order to make your future characters meet the high expectations of the average gamer out there, you need to know, at the very least, how to create and apply standard normal maps to your characters. Wrinkle maps take the concept of normal maps one step further and add wrinkles to your characters as they talk, smile, or frown, etc. Albeit this is a pretty subtle effect, it still adds that extra little thing missing to make your character seem more alive. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. [...]... little lighting information Figure 12.1 demonstrates the problem with vertex-based lighting: FIGURE 12.1 The problem with vertex-based lighting ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark 258 Character Animation with Direct3D As you can see in Figure 12.1, the concept of vertex lighting can become a problem in areas where triangles are sparse As the light moves over the... tangent-space normal map is calculated as an offset of the surface normal, it remains largely uniform in color (depicted in the image as gray), where ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark 260 Character Animation with Direct3D the object-space normal map uses the entire range of colors (from white to black in this example) Figure 12.3 shows a real comparison of the appearances... have a brick wall with a tiled diffuse texture You wouldn’t be able to tile this object-space normal map over the wall without getting lighting artifacts Luckily, tangent-space normal maps don’t have this restriction, because with tangent-space normal maps, the final normal is calculated on-the-fly (instead of at the normal map creation time Please purchase PDF Split-Merge on www.verypdf.com to remove... shader, these three lines can be conveniently baked together into the following line: float3 normal = 2.0f * tex2D(NormalSampler, IN.tex0).rgb - 1.0f; ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark 262 Character Animation with Direct3D In this line of code, a single pixel is sampled from the normal map using the texture coordinates of the model, and then decoded back into a normal... the pixel shader, it is interpolated, giving you a correct world-to-pixel vector for each of the pixels This means that the light vector is in the ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark 264 Character Animation with Direct3D same coordinate system as the normal in a normal map, which in turn means that it is okay to perform the light calculation Figure 12.5 shows a 2D... this Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark Chapter 12 Wrinkle Maps 257 means each triangle is affected by three vertices and their normals This also means that for large triangles there’s a lot of surface that shares relatively little lighting information Figure 12.1 demonstrates the problem with vertex-based lighting: FIGURE 12.1 The problem with vertex-based lighting...256 Character Animation with Direct3D Before you get in contact with the wrinkle maps you need to have a solid understanding of how the more basic normal mapping technique works Even though normal mapping is a very common technique in... NORMAL MAPPING So far, we’ve dealt with vertices having position, normals, texture coordinates, bone indices, and bone weights Next, we need to be able to add a tangent component and a binormal component As you may remember, a vertex can be defined with the Fixed Vertex Format (FVF), but for more advanced things you need to create an ease purchase PDF Split-Merge on www.verypdf.com to remove this watermark... wherever they can get away with it This is because any given lighting scheme usually runs faster in a vertex shader compared to a pixel shader, since you usually deal with fewer vertices than you do pixels (The exception of this rule is, of course, when objects are far away from the camera, in which case some form of level of detail (LOD) scheme is used.) So in the case of character rendering, when... relative to the surface normal of the object that they describe Figure 12.2 attempts to show this somewhat fuzzy concept Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark Chapter 12 Wrinkle Maps 259 FIGURE 12.2 An object-space normal map compared with a tangent-space normal map This picture is not really mathematically correct since it would need two channels for the normals (X . to make your character seem more alive. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. 256 Character Animation with Direct3D Before you get in contact with the wrinkle. problem with vertex-based lighting: FIGURE 12.1 The problem with vertex-based lighting. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. 258 Character Animation with. object-space normal map compared with a tangent-space normal map. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. 260 Character Animation with Direct3D the object-space