@ 2004-11-23 @

  °ÔÀÓ¿¡¼­ÀÇ ¹°¸®ÇÐ (±è¼º¿Ï »ù °¨»ç^^)

1 Çϵå¿þ¾î °¡¼Ó 2 °ÔÀÓ¿¡¼­ÀÇ ¹°¸®ÇÐ 2.1 Euler's Method 2.2 Hook's law 2.3 Verlet Intergrator 3 Chaos 4 Jump ±âº»·ÎÁ÷

* Ãæµ¹Âü°í

1 Çϵå¿þ¾î °¡¼Ó #

#.PPU(¹°¸®¿¬»ê ÇÁ·Î¼¼½º À¯´Ö) ; CPU°¡ ´ã´çÇÏ´Â À̹ÌÁö Àç»ý°ú °ü·ÃµÈ ¹°¸® ¿¬»êÀ» Àü´ãÇÏ´Â ÁÖº¯±â±â
#.GPU (Graphics Processing Unit)´Â
Áß¾Óó¸®ÀåÄ¡ÀÎ CPU¸¸À¸·Î °íÁ¤¹ÐµµÀÇ ±×·¡ÇÈÀ» ´Üµ¶À¸·Î ó¸®Çϱ⿡ ¹«¸®°¡ Àֱ⠶§¹®¿¡, À̸¦ º¸Á¶Çϱâ À§ÇÏ¿© 3D±×·¡ÇÈ ¿¬»ê Àü¿ëÀÇ ÇÁ·Î¼¼¼­ GPU°¡ °³¹ß µÇ¾ú½À´Ï´Ù.

GPU¶ó´Â ¿ë¾î´Â ¿£ºñµð¾Æ(NVIDIA)»ç¿¡¼­ 1999³â¿¡ ¡®ÁöÆ÷½º(GeForce)¡¯¶ó´Â À̸§ÀÇ »õ·Î¿î ±×·¡ÇÈ ÄÁÆ®·Ñ·¯(Graphics Controller: ±×·¡ÇÈÄ«µå¿ë Ĩ)À» ³»³õÀ¸¸ç óÀ½ Á¦Ã¢ÇÑ °ÍÀÌ´Ù. ÁöÆ÷½º´Â CPUÀÇ µµ¿ò ¾øÀÌ ÀÚüÀûÀ¸·Î Æú¸®°ï(Polygon: 3D ±×·¡ÇÈÀ» ±¸¼ºÇÏ´Â µµÇü)ÀÇ º¯Çü(Transform) ¹× ±¤¿ø(Lighting)È¿°ú¸¦ ±¸»çÇÏ´Â ±â´É, À̸¥¹Ù ¡®Çϵå¿þ¾î(Hardware) T&L¡¯À» °®Ãß°í ÀÖ¾ú´Ù. ÀÌ´Â ÀÌÀü±îÁö »ç¿ëÇß´ø ±×·¡ÇÈ ÄÁÆ®·Ñ·¯¿Í´Â È®¿¬È÷ ´Ù¸¥ °³³äÀ̾ú±â ¶§¹®¿¡ À̸¦ ±¸ºÐÇÏ°íÀÚ GPU¶ó´Â À̸§ÀÌ ºÙ°Ô µÇ¾ú´Ù. ±×¸®°í ÁöÆ÷½ºÀÇ Ãâ½Ã ÈÄ 1³âÀÌ Áö³­ 2000³â, ATi(ÇöÀçÀÇ AMD)»ç¿¡¼­ ¡®¶óµ¥¿Â(Radeon)¡¯À̶ó´Â GPU¸¦ Ãâ½ÃÇÏ°Ô µÇ¸é¼­ ¾ç»çÀÇ °æÀïÀÌ º»°ÝÈ­ µÈ´Ù.

#.ÀÎÅÚ > Havok - ¹°¸® °è»êÀ» CPU¿¡¼­ º¸Á¶ ¿ªÇÒÀ» ´ã´ç
-. Çï°ÔÀÌÆ®´Â Á¤»óÀûÀÎ °æ¿ì, ½Ì±Û ÇÁ·Î¼¼½º·Î ÀÛµ¿À» ÇÏÁö¸¸
#.NVIDIA > (AGEIA) PhysX - ¹°¸® °è»êÀ» ±×·¡ÇÈÄ«µå¿¡¼­ º¸Á¶ ¿ªÇÒÀ» ´ã´ç

2 °ÔÀÓ¿¡¼­ÀÇ ¹°¸®ÇÐ #

• F = ma¿¡¼­ °¡¼Óµµ(a)¡êÀ§Ä¡¿Í ½Ã°£ÀÇ °³³ä
  
  • Â÷¿ø¿¡ ¿µÇâÀ» ¹ÞÁö ¾Ê´Â´Ù.(´ÜÀ§ ¹«½Ã)
    
  • Áú·® °³³äÀÌ ¾ø´Ù.(ÀÓÀÇÀûÀÎ settingÀÌ °¡´É)
    
  • ½Ç½Ã°£À¸·Î ó¸®ÇØ¾ß ÇÑ´Ù.
    
• Âü°í »çÀÌÆ®
  
  • Open Dynamics Engine http://ode.org/ode.html
    
  • Verlet Intergrator(À¯¿ëÇÑ ¹æ½Ä) http://www.ioi.dk/Homepages/thomasj/publications/gdc2001.htm
    
  • Physics of Racing Series http://www.miata.net/sport/Physics/
    
  • Rigid Body Dynamics http://www-2.cs.cmu.edu/~baraff/ (SIGGRAPH '97 course)
    
  • ¼­¿ï´ëÇб³ Graphics & Media Lab http://graphics.snu.ac.kr/~kjchoi/
    
  • ȸÀü¿îµ¿ÀÇ °ü¼º¸ð¿ì¸àÆ® °è»ê½Ä http://www.powerindex.net/Machine/gd2/gd2.htm#
    
  • http://developer.nvidia.com/page/opengl.html
    
    

2.1 Euler's Method #

• ÈûF(t) = m*A ( m´Â Áú·®, A´Â °¡¼Óµµ )
  
  F(t)¸¦ ¾Ë°í ÀÖÀ» ¶§, ¹ÌºÐ¹æÁ¤½ÄÀ» À§Ä¡¿¡ ´ëÇؼ­ Ǭ´Ù¸é ½Ã°£¿¡ µû¸¥ À§Ä¡°ªR(t)À» ¾Ë¼ö ÀÖ´Ù.
  
• ¥Ät<<1 ( ¥Ät°¡ ÃæºÐÈ÷ ÀÛ´Ù¸é )
  
  ¦£ Rnew = Rold + Vnew£ª¥Ät ( RÀº À§Ä¡, Rold´Â ÃʱâÀ§Ä¡ ) À§Ä¡ÀÇ °ªÀÌ Èû¿¡ ¿µÇâÀ» ¹ÌÄ£´Ù.
  ¦¦ Vnew = Vold + F/m£ª¥Ät ( V´Â ¼Óµµ, Vold´Â Ãʱâ¼Óµµ )
  
  

2.2 Hook's law #

• ½ºÇÁ¸µ°è¼ö(ks); ÆÄƼŬ °£ÀÇ ÀÀÁý·Â ¿îµ¿À» À¯¹ß½ÃÅ´
  
  -1) °è¼ö°¡ Ŭ¼ö·Ï ¸ð¾çÀ» ±×´ë·Î À¯ÁöÇÒ·Á´Â ÈûÀ¸·Î ÀÛ¿ë
  

  --> ÀÓÀÇÀÇ ÇÑ°èÄ¡¸¦ ³ÑÀ» °æ¿ì´Â ¾ðÁ¦µçÁ¦ Æø¹ßÀÇ À§Ç輺À» °¡Áö°í ÀÖ´Ù.
  

  -2) ½ºÇÁ¸µÀº »ó´ëÀûÀÎ À§Ä¡ÀÇ (°Å¸®¿¡ µû¸¥)º¯È­¿¡ ÀÇÇØ Åº¼ºÀÌ ¹ß»ý
  
• °¨¼è°è¼ö(kd); ÀúÇ×À̳ª ¹°¼ºÄ¡¿¡ ÀÇÇÑ º¯È­¸¦ °¨¼Ò½ÃÅ°´Â ¿ä¼Ò
  
  -1) °¨¼è°è¼ö°¡ Å©¸é ¹Ý¹ßÇÏ´Â ÇÏ´Â ÈûÀÇ ÀÛ¾ÆÁø´Ù. -> µüµüÇÑ ¹°Ã¼ Ç¥Çö
  

  °¨¼è°è¼ö°¡ ÀÛÀ¸¸é À¯µ¿¼ºÀÌ Å« Á©¸®¿Í °°Àº ¹°ÁúÀ» Ç¥Çö ÇÒ ¼ö ÀÖ´Ù.
  
• ÇÑÂÊÀÌ °íÁ¤µÇ¾î ÀÖ´Â ½ºÇÁ¸µ¿¡ Áú·® mÀÎ ¹°Ã¼°¡ ´Ù¸¥ÂÊ ³¡¿¡ ÀÖÀ» ¶§,
  
  ¡ÅFs = -ks(l - lo)
  

  ( ks´Â ½ºÇÁ¸µ»ó¼ö, lÀº ½ºÇÁ¸µÀÌ ´Ã¾î³­ ±æÀÌ, loÀº ½ºÇÁ¸µÀÇ ¿ø·¡±æÀÌ )
  

  ¡ÅFd = kd(v1 - v2)
  

  ( kd´Â °¨¼è°è¼ö, v1Àº Ãʱ⠼ӵµ, v2´Â ³ªÁß¼Óµµ, (v1 - v2)´Â »ó´ë¼Óµµ )
  

  ¡ÅF = Fs + Fd Euler's Method Àû¿ëÇؼ­ À§Ä¡°ªRnew¸¦ ±¸ÇÑ´Ù.
  
¡Ø ±¸¼Ó Á¶°Ç¿¡¼­ ½ºÇÁ¸µ »ó¼ö°¡ Å« µüµüÇÑ ¹°Ã¼¿¡ ½ÉÇÑ Áøµ¿À» °¡ÇÒ °æ¿ì Æø¹ßÀÇ À§Ç輺Àº ¾ðÁ¦³ª ÀÖ´Ù.

2.3 Verlet Intergrator #

• °³º°ÀûÀÎ ÆÄƼŬÀÇ ¿òÁ÷ÀÓÀ» °­Ã¼ ¿îµ¿À¸·Î Ç¥ÇöÇÒ ¼ö ÀÖ°í, ¿¡³ÊÁö°¡ º¸Á¸ µÊÀ¸·Î½á ¾ÈÁ¤ÀûÀÌ´Ù.
  
  ¡Å R( t+¥Ät ) = 2R(t) - R(t-¥Ät) + A*¥Ät©÷
  
-1) ¿ÀÀÏ·¯ ÇعýÀÇ V(¼Óµµ)¿Í R(À§Ä¡)ÀÇ Ç׸ñÀÌ 2°³ÀÇ R(À§Ä¡)·Î ´ëü

-2) ÃʱâÄ¡ R(t-¥Ät) = R(t) - V(t)*¥Ät ·Î °è»ê
-3) ¿¹»óÇØ´ø °á°ú°¡ ³ª¿À¸é 2~3¹ø ·çÇÁ¸¦ µ¹·Á¼­ ÇØ´ç ÇÁ·¹ÀÓ¿¡¼­ °íÁ¤½Ãų ¼ö ÀÖ´Ù.

3 Chaos #

• Ä«¿À½º¶ó´Â ¸»Àº ±×¸®½º¾î khaos¿¡¼­ ³ª¿Â °ÍÀÌÁö¸¸ ÇöÀç´Â ÀϹÝÀûÀ¸·Î Chaos(Ä«¿À½º, È¥µ·À̶ó´Â ÀǹÌ)·Î Ç¥±âÇÑ´Ù.
  
  • ¾î¶² ½Ã½ºÅÛÀÇ ¾î´À ½ÃÁ¡¿¡¼­ÀÇ »óÅÂ(=ÃʱⰪ)°¡ Á¤ÇØÁö¸é ±× ÈÄÀÇ »óÅ´ ¿ø¸®ÀûÀ¸·Î ¸ðµÎ °áÁ¤µÈ´Ù°í ÇÏ´Â °áÁ¤·ÐÀû ¹ýÄ¢À» µû¸£°í ÀÖÀ½¿¡µµ ºÒ±¸ÇÏ°í, ¾ÆÁÖ º¹ÀâÇؼ­ ºÒ±ÔÄ¢ÇÏ°í ºÒ¾ÈÁ¤ÇÑ ¸Õ Àå·¡ÀÇ »óŸ¦ ¿¹ÃøÇÏ´Â °ÍÀÌ ºÒ°¡´ÉÇÑ Çö»ó
    
• ÀοëµÇ´Â ½ÄÀÇ ¿¹;
  
  y = ax(1-x) a´Â »ó¼ö, xÀÇ ÃʱâÄ¡¸¦ 0.1À̶ó°í °¡Á¤ÇÒ ¶§,
  ´ÙÀ½¿¡ ½Ä¿¡ Àû¿ëÇÒ xÀÇ °ªÀº ÇöÀç °è»êµÈ yÀÇ °ªÀ» »ç¿ëÇÑ´Ù.(0°ú 1»çÀÌ °ª)
  

  
  
  

4 Jump ±âº»·ÎÁ÷ #

• Á¡ÇÁ´Â Å©°Ô 3°³ÀÇ ¿µ¿ªÀ¸·Î ±¸ºÐ
  
  -1.À­ÂÊÀ¸·Î ÈûÀÌ ÀÛ¿ëÇÏ´Â ºÎºÐ
  -2.Á߷¿¡ ÀÇÇØ À­ÂÊÀ¸·Î ¿Ã¶ó°¡´Â ÈûÀÌ »ç¶óÁö´Â ºÎºÐ(²À´ë±â ºÎºÐ)
  -3.Á߷¿¡ ÀÇÇØ ³«ÇÏÇÏ´Â ºÎºÐ
  
  
• ¾Æ·¡ ÄÚµåÀÇ ¹®Á¦´Â float ŸÀÔÀ» ¼­·Î ºñ±³ÇÏ´Â ±âº»ÀûÀÎ ¿À·ù¸¦ °¡Áö°í ÀÖÀ½
  
  -.m_fGravity += (JumpGHIGHTRAVITY * (fDelta*fDelta)*0.5f ); ¿¡¼­
  

  (fDelta*fDelta)*0.5f´Â ÇÁ·¹ÀÓ º¯È­¿¡ µû¸¥ Áøµ¿À» °¨¼èÇÏ´Â È¿°ú°¡ ÀÖ´Ù.
  

  -.¹®Á¦Á¡; ½Ã°£ÀÇ º¯È­(µ¨Å¸time)°¡ ÀÏÁ¤ÇÏÁö ¾ÊÀ» ¶§, ¸Å²ô·¯¿î ±ËÀûÀ» ¸¸µéÁö ¸øÇÑ´Ù.
  -.oldTrans; ÇöÀçÀÇ À§Ä¡
  

  kNewTrans; ¹Ì·¡ÀÇ À̵¿ À§Ä¡
  

const float JumpGHIGHTRAVITY 1800.0f
const float SAFEHEIGHT = 2.0f;
const float STARTJUMP  = 100.0f;
static float fUp = 0.0f;	

// Á¡ÇÁ ¿©ºÎ
if( !Á¡ÇÁÁß && ÇöÀç ¾Ö´Ï¸ÞÀ̼ÇÀÌ Á¡ÇÁÀΰ¡? )
{
  m_bJumpstart = true;
  m_bJump = true;
  fUp = STARTJUMP;
  m_fGravity = 0.0f;
}

if( m_bJump )  // »ó½Â ºÎºÐ
{
  m_fGravity += (JumpGHIGHTRAVITY * (fDelta*fDelta)*0.5f );
  fUp -= m_fGravity;

  if( fUp < 0.0f )
  {
    m_fGravity = 0.0f;
    m_bJump = false;
  }
  else
  {
    oldTrans.z += fUp;		
  }
}
else	// ³«ÇÏ ¿©ºÎ
{
  if( (oldTrans.z - kNewTrans.z) > SAFEHEIGHT )
  {
    m_fGravity += (JumpGHIGHTRAVITY * (fDelta*fDelta)*0.5f );
    oldTrans.z -= m_fGravity;
  }
  else
  {
    oldTrans = kNewTrans;
    m_fGravity = 0.0f;
    m_bJumpstart = false;
  }
}

• À§ÀÇ Äڵ忡¼­ Á¡ÇÁ¸¦ °ËÃâÇÏ´Â ºÎºÐÀÌ ·ÎÁ÷ ¾È¿¡ Àֱ⠶§¹®¿¡ ÇÁ·¹ÀÓ¸¶´Ù ÆǺ°ÇÏ´Â ¹®Á¦°¡ À־
  
  ¿ÜºÎ¿¡¼­ Å° ÀÔ·ÂÀÌ µÇ´Â ºÎºÐ¿¡¼­ jump start¸¦ ÇÔ¼ö·Î È£ÃâÇϵµ·Ï ¼öÁ¤
  -.±ËÀûÀ» ÀÏÁ¤Çϱâ À¯Áö Çϱâ À§ÇØ Sin°î¼±À» µû¶ó¼­ »ó½Â ÈÄ Á¤Á¡¿¡¼­ ÇÔ¼ö »ç¿ëÇؼ­ ³«ÇÏ
  -.Á¡ÇÁ À̺¥Æ® ¹ß»ý½Ã ÇöÀç À§Ä¡(m_pSavePosition)¸¦ ÀúÀå ÈÄ »ó½Â±¸¿ª¿¡¼­ ±âÁØÀ¸·Î »ç¿ë
  -.Á¤»óºÎºÐ¿¡¼­ m_bJump02 = false, m_bJumpstart = false¿¡¼­ ´Ù½Ã Á¡ÇÁ°¡´É
  

class className
{
public:
  void StartJump();	

private:
  Point3	m_vSavePosition;
  bool	m_bJump02;	// Á¡ÇÁ À̺¥Æ®¸¦ ¹Þ¾Æ¼­ ij¸¯ÅÍ°¡ ÇöÀç »ó½Â ÁßÀÎÁö¸¦ Ç¥½Ã	
  bool 	m_bJumpstart;	// Á¡ÇÁ¸¦ ÇÒ ¼ö ÀÖ´Â »óÅ ÀÎÁö¸¦ Ç¥½Ã

  float 	m_fGravity;
  float 	m_fHightPos;
  float	m_fJumpStartTime;
  float 	m_fJumpTime;
  float	m_fJumpTimeMAX;
}
-----------------------------------------------------------------------------------
const float DEF_JumpGHIGHTRAVITY = 1600.0f;
const float DEF_JumpRAD = PI * 0.5;	// 90µµ
const float DEF_JumpMAX = 12.0f;	// jump½Ã ÃÖ°í Á¤Á¡ÀÇ ³ôÀÌ

void className::StartJump()	
{ 
  if( m_bJumpstart == true )	
    return;

  CPlayer * pkPlayer = GetPlayer();
  _TKASSERT( pkPlayer );
  
  m_pSavePosition = pkPlayer->GetWorldTranslate();

  ControllerSequence* pkSequence = pkPlayer->GetSequence(Animation_jump);
  m_fJumpTimeMAX = pkSequence->GetLength() * 1.1f;
  m_fJumpTime = m_fJumpTimeMAX * 0.5f;
  m_fJumpStartTime = GetCurTime();

  m_bJumpstart = true;
  m_bJump02 = true;
}

bool className::UpDate()
{
  ...
  // Á¡ÇÁ ¹× ³«ÇÏ ºÎºÐ - Sin°î¼±À» µû¶ó¼­ »ó½Â ÈÄ Á¤Á¡¿¡¼­ ÇÔ¼ö »ç¿ëÇؼ­ ³«ÇÏ
  //=========================================================================================
  float fjumptime = GetCurTime() - m_fJumpStartTime;

  if( m_bJump02 )	  // Á¡ÇÁ½Ã »ó½Â ±¸°£
  {
    float fjumpRAD = (fjumptime/m_fJumpTime) * DEF_JumpRAD;

    if( fjumptime < m_fJumpTime )	// »ó½Â Áß
    {
      m_fHightPos = Sin(fjumpRAD) * DEF_JumpMAX;
      m_fHightPos += m_vSavePosition.z;
    }
    else		// ÃÖ°í Á¤Á¡ºÎºÐ
    {
      m_fHightPos = m_vSavePosition.z + DEF_JumpMAX;
      m_bJump02 = false;
    }		
  }
  else  //Á¡ÇÁ°¡ ¾Æ´Ò¶§ ³«ÇÏ
  {
    if((oldTrans.z - kNewTrans.z) < 2.0f)
    {
      m_fHightPos = kNewTrans.z;
      oldTrans = kNewTrans;
    }

    if(m_fHightPos > kNewTrans.z)
    {
      m_fGravity += (DEF_JumpGHIGHTRAVITY * ((fDelta*fDelta)*0.5));
      m_fHightPos  -= m_fGravity;
    }
    else	// if(m_fHightPos <= kNewTrans.z)
    {
      m_fHightPos = kNewTrans.z;
      m_fGravity = 0.0f;
      m_bJumpstart = false;
      m_bJump02 = false;
    }
  }

  oldTrans.z = m_fHightPos;
}