C语言正余弦函数定点查表算法原理及实现

2023-05-16

如果用数学库中的sin和cos函数计算，可能涉及浮点、乘法、除法运算，运行速率较低。这里介绍一种正余弦查表算法。

参考代码：SMT32 FOC电机库的mc_math.c。

首先我们知道正弦和余弦的函数值差了90°，所以查表数据只需要保存正弦或者余弦的结果即可。待计算的角度范围为[0,2π]，我们只需要保存 $[0,\frac{\pi}{2}]$ 的角度，其它象限的角度通过三角函数公式变换一下即可。

1 原理与实现步骤

（1）正余弦表格生成

第一步当然是保存 $[0,\frac{\pi}{2}]$ 范围内正余弦函数的值，保存的越多计算结果越精确，但又消耗更多的存储空间。在STM32电机库中采用int16类型表示角度，其中(0,32767]表示(0°,180°]，[-32768,0]表示(180°,360°]，同时为了避免浮点运算，这里采用Q15格式表示角度。之所以是Q15是因为保存的是 $[0,\frac{\pi}{2}]$ 范围内正余弦函数的值，后续转换到别的象限还有正负号，故最后求出来的正余弦值也可以用int16范围变量表示。

这里将 $[0,\frac{\pi}{2}]$ 范围内的角度细分为256份，整个周期就细分了1024份。故将 $\sin(\frac{\pi}{2} * \frac{i}{256})\times 2^{15}$ 保存为表格，正余弦结果表格生成程序如下

#include <stdio.h>
#include <math.h>
#define NUM  256
#define PI   3.1415926535898f

int main()
{
    int i;
    for(i = 0; i < NUM; i++)
    {
        printf("0x%04X,", (unsigned short)((sin(PI/2*i/NUM)) * 32768));
        if((i+1) % 8==0)
        {
            putchar('\n');
        }
    }
    return 0;
}

（2）索引获取

输入的角度为int16范围的hAngle，作如下处理，求对应角度的索引：

int32_t shindex;
uint16_t uhindex;
shindex = ( ( int32_t )32768 + ( int32_t )hAngle );
uhindex = ( uint16_t )shindex;
uhindex /= ( uint16_t )64;

shindex是一个包含四个象限范围的值，将四个象限共细分为 $2^{16}$ =65536份，每个象限有 $2^{12}$ =4096个数。但基于存储空间考虑，在前面生成的正余弦表中，我们将一个象限细分为256份，则四个象限共细分为1024份，所以除以 $2^{6}=64$ 。此时可以根据最高两位判断角度所在的象限，同时将这个数取余256，就是该角度所对应在正余弦表中的索引。

（3）判断角度范围

前面代码中，将int16的角度转为正数，此时[0,32768)表示(180°,360°],(32768，65535]表示[0,180°) ，前面又除以了一个64，故十进制数范围和其对应的角度范围如下表所示：

十进制数范围	角度范围
(0,256)	(180°,270°)
(256,512)	(270°,360°)
(512,768)	(0°,90°)
(768,1024)	(90°,180°)

故角度和SIN_MASK相与，判断最高两位就可以知道输入角度的范围。

#define SIN_MASK        0x0300u
#define U0_90           0x0200u
#define U90_180         0x0300u
#define U180_270        0x0000u
#define U270_360        0x0100u

（4）查表求角度

这里的索引将uint16类型的变量uhindex强制转化为uint8范围，实际上就是对256取余，等价于uhindex % 256。然后就是直接查表了，由于前面建立的表中角度是(0°,90°)范围内的，所以根据三角函数公式转换其它象限的角度到这个范围内即可求出。

switch ( ( uint16_t )( uhindex ) & SIN_MASK )
  {
    case U0_90:
      Local_Components.hSin = hSin_Cos_Table[( uint8_t )( uhindex )];
      Local_Components.hCos = hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      break;

    case U90_180:
      Local_Components.hSin = hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      Local_Components.hCos = -hSin_Cos_Table[( uint8_t )( uhindex )];
      break;

    case U180_270:
      Local_Components.hSin = -hSin_Cos_Table[( uint8_t )( uhindex )];
      Local_Components.hCos = -hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      break;

    case U270_360:
      Local_Components.hSin =  -hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      Local_Components.hCos =  hSin_Cos_Table[( uint8_t )( uhindex )];
      break;
    default:
      break;
  }

2 完整代码

#define int16_t short
#define uint8_t unsigned char
#define uint16_t unsigned short
#define int32_t unsigned int

#define SIN_COS_TABLE {\
    0x0000,0x00C9,0x0192,0x025B,0x0324,0x03ED,0x04B6,0x057F,\
    0x0648,0x0711,0x07D9,0x08A2,0x096A,0x0A33,0x0AFB,0x0BC4,\
    0x0C8C,0x0D54,0x0E1C,0x0EE3,0x0FAB,0x1072,0x113A,0x1201,\
    0x12C8,0x138F,0x1455,0x151C,0x15E2,0x16A8,0x176E,0x1833,\
    0x18F9,0x19BE,0x1A82,0x1B47,0x1C0B,0x1CCF,0x1D93,0x1E57,\
    0x1F1A,0x1FDD,0x209F,0x2161,0x2223,0x22E5,0x23A6,0x2467,\
    0x2528,0x25E8,0x26A8,0x2767,0x2826,0x28E5,0x29A3,0x2A61,\
    0x2B1F,0x2BDC,0x2C99,0x2D55,0x2E11,0x2ECC,0x2F87,0x3041,\
    0x30FB,0x31B5,0x326E,0x3326,0x33DF,0x3496,0x354D,0x3604,\
    0x36BA,0x376F,0x3824,0x38D9,0x398C,0x3A40,0x3AF2,0x3BA5,\
    0x3C56,0x3D07,0x3DB8,0x3E68,0x3F17,0x3FC5,0x4073,0x4121,\
    0x41CE,0x427A,0x4325,0x43D0,0x447A,0x4524,0x45CD,0x4675,\
    0x471C,0x47C3,0x4869,0x490F,0x49B4,0x4A58,0x4AFB,0x4B9D,\
    0x4C3F,0x4CE0,0x4D81,0x4E20,0x4EBF,0x4F5D,0x4FFB,0x5097,\
    0x5133,0x51CE,0x5268,0x5302,0x539B,0x5432,0x54C9,0x5560,\
    0x55F5,0x568A,0x571D,0x57B0,0x5842,0x58D3,0x5964,0x59F3,\
    0x5A82,0x5B0F,0x5B9C,0x5C28,0x5CB3,0x5D3E,0x5DC7,0x5E4F,\
    0x5ED7,0x5F5D,0x5FE3,0x6068,0x60EB,0x616E,0x61F0,0x6271,\
    0x62F1,0x6370,0x63EE,0x646C,0x64E8,0x6563,0x65DD,0x6656,\
    0x66CF,0x6746,0x67BC,0x6832,0x68A6,0x6919,0x698B,0x69FD,\
    0x6A6D,0x6ADC,0x6B4A,0x6BB7,0x6C23,0x6C8E,0x6CF8,0x6D61,\
    0x6DC9,0x6E30,0x6E96,0x6EFB,0x6F5E,0x6FC1,0x7022,0x7083,\
    0x70E2,0x7140,0x719D,0x71F9,0x7254,0x72AE,0x7307,0x735E,\
    0x73B5,0x740A,0x745F,0x74B2,0x7504,0x7555,0x75A5,0x75F3,\
    0x7641,0x768D,0x76D8,0x7722,0x776B,0x77B3,0x77FA,0x783F,\
    0x7884,0x78C7,0x7909,0x794A,0x7989,0x79C8,0x7A05,0x7A41,\
    0x7A7C,0x7AB6,0x7AEE,0x7B26,0x7B5C,0x7B91,0x7BC5,0x7BF8,\
    0x7C29,0x7C59,0x7C88,0x7CB6,0x7CE3,0x7D0E,0x7D39,0x7D62,\
    0x7D89,0x7DB0,0x7DD5,0x7DFA,0x7E1D,0x7E3E,0x7E5F,0x7E7E,\
    0x7E9C,0x7EB9,0x7ED5,0x7EEF,0x7F09,0x7F21,0x7F37,0x7F4D,\
    0x7F61,0x7F74,0x7F86,0x7F97,0x7FA6,0x7FB4,0x7FC1,0x7FCD,\
    0x7FD8,0x7FE1,0x7FE9,0x7FF0,0x7FF5,0x7FF9,0x7FFD,0x7FFE}
const int16_t hSin_Cos_Table[256] = SIN_COS_TABLE;

#define SIN_MASK        0x0300u
#define U0_90           0x0200u
#define U90_180         0x0300u
#define U180_270        0x0000u
#define U270_360        0x0100u


typedef struct
{
  int16_t hCos;
  int16_t hSin;
} Trig_Components;


Trig_Components MCM_Trig_Functions( int16_t hAngle )
{

  int32_t shindex;
  uint16_t uhindex;
  Trig_Components Local_Components;
  /* 10 bit index computation  */
  shindex = ( ( int32_t )32768 + ( int32_t )hAngle );
  uhindex = ( uint16_t )shindex;
  uhindex /= ( uint16_t )64;

  switch ( ( uint16_t )( uhindex ) & SIN_MASK )
  {
    case U0_90:
      Local_Components.hSin = hSin_Cos_Table[( uint8_t )( uhindex )];
      Local_Components.hCos = hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      break;

    case U90_180:
      Local_Components.hSin = hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      Local_Components.hCos = -hSin_Cos_Table[( uint8_t )( uhindex )];
      break;

    case U180_270:
      Local_Components.hSin = -hSin_Cos_Table[( uint8_t )( uhindex )];
      Local_Components.hCos = -hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      break;

    case U270_360:
      Local_Components.hSin =  -hSin_Cos_Table[( uint8_t )( 0xFFu - ( uint8_t )( uhindex ) )];
      Local_Components.hCos =  hSin_Cos_Table[( uint8_t )( uhindex )];
      break;
    default:
      break;
  }
  return ( Local_Components );
}

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语言正余弦函数定点查表算法原理及实现