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姿态解算

姿态解算，网上用得最多的就是先求四元数，再转化成角度了。资料显示，相对于另两种旋转表示法（矩阵和欧拉角），四元数具有某些方面的优势，如速度更快、提供平滑插值、有效避免万向锁问题、存储空间较小等等 。

下面列出几个常用的算法，都是用C语言的格式。

1、让四轴飞的匿名四轴版

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//默认参数  10 0.008 
#define Kp 10.0f                        // proportional gain governs rate of convergence to accelerometer/magnetometer
#define Ki 0.008f                       // integral gain governs rate of convergence of gyroscope biases
#define halfT 0.001f                    // half the sample period采样周期的一半

float q0 = 1, q1 = 0, q2 = 0, q3 = 0;    // quaternion elements representing the estimated orientation
float exInt = 0, eyInt = 0, ezInt = 0;    // scaled integral error
void IMUupdate(float gx, float gy, float gz, float ax, float ay, float az)
{
  float norm;
//  float hx, hy, hz, bx, bz;
  float vx, vy, vz;// wx, wy, wz;
  float ex, ey, ez;
	
  // 先把这些用得到的值算好
  float q0q0 = q0*q0;
  float q0q1 = q0*q1;
  float q0q2 = q0*q2;
//  float q0q3 = q0*q3;
  float q1q1 = q1*q1;
//  float q1q2 = q1*q2;
  float q1q3 = q1*q3;
  float q2q2 = q2*q2;
  float q2q3 = q2*q3;
  float q3q3 = q3*q3;
	
	if(ax*ay*az==0)
 		return;
		
  norm = sqrt(ax*ax + ay*ay + az*az);       //acc数据归一化
  ax = ax / norm;
  ay = ay / norm;
  az = az / norm;

  // estimated direction of gravity and flux (v and w)      估计重力方向和流量/变迁
  vx = 2*(q1q3 - q0q2);												//四元素中xyz的表示
  vy = 2*(q0q1 + q2q3);
  vz = q0q0 - q1q1 - q2q2 + q3q3 ;

  // error is sum of cross product between reference direction of fields and direction measured by sensors
  ex = (ay*vz - az*vy) ;                           					 //向量外积在相减得到差分就是误差
  ey = (az*vx - ax*vz) ;
  ez = (ax*vy - ay*vx) ;

  exInt = exInt + ex * Ki;								  //对误差进行积分
  eyInt = eyInt + ey * Ki;
  ezInt = ezInt + ez * Ki;

  // adjusted gyroscope measurements
  gx = gx + Kp*ex + exInt;					   							//将误差PI后补偿到陀螺仪，即补偿零点漂移
  gy = gy + Kp*ey + eyInt;
  gz = gz + Kp*ez + ezInt;				   							//这里的gz由于没有观测者进行矫正会产生漂移，表现出来的就是积分自增或自减

  // integrate quaternion rate and normalise						   //四元素的微分方程
  q0 = q0 + (-q1*gx - q2*gy - q3*gz)*halfT;
  q1 = q1 + (q0*gx + q2*gz - q3*gy)*halfT;
  q2 = q2 + (q0*gy - q1*gz + q3*gx)*halfT;
  q3 = q3 + (q0*gz + q1*gy - q2*gx)*halfT;

  // normalise quaternion
  norm = sqrt(q0*q0 + q1*q1 + q2*q2 + q3*q3);
  q0 = q0 / norm;
  q1 = q1 / norm;
  q2 = q2 / norm;
  q3 = q3 / norm;

  Q_ANGLE.Z = GYRO_I.Z;//atan2(2 * q1 * q2 + 2 * q0 * q3, -2 * q2*q2 - 2 * q3* q3 + 1)* 57.3; // yaw
  Q_ANGLE.Y = asin(-2 * q1 * q3 + 2 * q0* q2)* 57.3; // pitch
  Q_ANGLE.X = atan2(2 * q2 * q3 + 2 * q0 * q1, -2 * q1 * q1 - 2 * q2* q2 + 1)* 57.3; // roll
}

*************************************************

下面两个算法为crazyflie使用的算法

变量定义如下：

//#define MADWICK_QUATERNION_IMU

#ifdef MADWICK_QUATERNION_IMU
  #define BETA_DEF     0.01f    // 2 * proportional gain
#else // MAHONY_QUATERNION_IMU
    #define TWO_KP_DEF  (2.0f * 0.4f) // 2 * proportional gain
    #define TWO_KI_DEF  (2.0f * 0.001f) // 2 * integral gain
#endif

#ifdef MADWICK_QUATERNION_IMU
  float beta = BETA_DEF;     // 2 * proportional gain (Kp)
#else // MAHONY_QUATERNION_IMU
  float twoKp = TWO_KP_DEF;    // 2 * proportional gain (Kp)
  float twoKi = TWO_KI_DEF;    // 2 * integral gain (Ki)
  float integralFBx = 0.0f;
  float integralFBy = 0.0f;
  float integralFBz = 0.0f;  // integral error terms scaled by Ki
#endif

float q0 = 1.0f;
float q1 = 0.0f;
float q2 = 0.0f;
float q3 = 0.0f;  // quaternion of sensor frame relative to auxiliary frame

*************************************************

2、crazyflie的Madgwick's IMU and AHRS algorithms

*************************************************

// Implementation of Madgwick's IMU and AHRS algorithms.
// See: http://www.x-io.co.uk/node/8#open_source_ahrs_and_imu_algorithms
//
// Date     Author          Notes
// 29/09/2011 SOH Madgwick    Initial release
// 02/10/2011 SOH Madgwick  Optimised for reduced CPU load
void sensfusion6UpdateQ(float gx, float gy, float gz, float ax, float ay, float az, float dt)
{
  float recipNorm;
  float s0, s1, s2, s3;
  float qDot1, qDot2, qDot3, qDot4;
  float _2q0, _2q1, _2q2, _2q3, _4q0, _4q1, _4q2 ,_8q1, _8q2, q0q0, q1q1, q2q2, q3q3;

  // Rate of change of quaternion from gyroscope
  qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
  qDot2 = 0.5f * (q0 * gx + q2 * gz - q3 * gy);
  qDot3 = 0.5f * (q0 * gy - q1 * gz + q3 * gx);
  qDot4 = 0.5f * (q0 * gz + q1 * gy - q2 * gx);

  // Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
  if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f)))
  {
    // Normalise accelerometer measurement
    recipNorm = invSqrt(ax * ax + ay * ay + az * az);
    ax *= recipNorm;
    ay *= recipNorm;
    az *= recipNorm;

    // Auxiliary variables to avoid repeated arithmetic
    _2q0 = 2.0f * q0;
    _2q1 = 2.0f * q1;
    _2q2 = 2.0f * q2;
    _2q3 = 2.0f * q3;
    _4q0 = 4.0f * q0;
    _4q1 = 4.0f * q1;
    _4q2 = 4.0f * q2;
    _8q1 = 8.0f * q1;
    _8q2 = 8.0f * q2;
    q0q0 = q0 * q0;
    q1q1 = q1 * q1;
    q2q2 = q2 * q2;
    q3q3 = q3 * q3;

    // Gradient decent algorithm corrective step
    s0 = _4q0 * q2q2 + _2q2 * ax + _4q0 * q1q1 - _2q1 * ay;
    s1 = _4q1 * q3q3 - _2q3 * ax + 4.0f * q0q0 * q1 - _2q0 * ay - _4q1 + _8q1 * q1q1 + _8q1 * q2q2 + _4q1 * az;
    s2 = 4.0f * q0q0 * q2 + _2q0 * ax + _4q2 * q3q3 - _2q3 * ay - _4q2 + _8q2 * q1q1 + _8q2 * q2q2 + _4q2 * az;
    s3 = 4.0f * q1q1 * q3 - _2q1 * ax + 4.0f * q2q2 * q3 - _2q2 * ay;
    recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
    s0 *= recipNorm;
    s1 *= recipNorm;
    s2 *= recipNorm;
    s3 *= recipNorm;

    // Apply feedback step
    qDot1 -= beta * s0;
    qDot2 -= beta * s1;
    qDot3 -= beta * s2;
    qDot4 -= beta * s3;
  }

  // Integrate rate of change of quaternion to yield quaternion
  q0 += qDot1 * dt;
  q1 += qDot2 * dt;
  q2 += qDot3 * dt;
  q3 += qDot4 * dt;

  // Normalise quaternion
  recipNorm = invSqrt(q0*q0 + q1*q1 + q2*q2 + q3*q3);
  q0 *= recipNorm;
  q1 *= recipNorm;
  q2 *= recipNorm;
  q3 *= recipNorm;
}

*************************************************

3、crazyflie的Mayhony's AHRS algorithm

*************************************************

// Madgwick's implementation of Mayhony's AHRS algorithm.
// See: http://www.x-io.co.uk/node/8#open_source_ahrs_and_imu_algorithms
//
// Date     Author      Notes
// 29/09/2011 SOH Madgwick    Initial release
// 02/10/2011 SOH Madgwick  Optimised for reduced CPU load
void sensfusion6UpdateQ(float gx, float gy, float gz, float ax, float ay, float az, float dt)
{
  float recipNorm;
  float halfvx, halfvy, halfvz;
  float halfex, halfey, halfez;
  float qa, qb, qc;

  gx = gx * M_PI / 180;
  gy = gy * M_PI / 180;
  gz = gz * M_PI / 180;

  // Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
  if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f)))
  {
    // Normalise accelerometer measurement
    recipNorm = invSqrt(ax * ax + ay * ay + az * az);
    ax *= recipNorm;
    ay *= recipNorm;
    az *= recipNorm;

    // Estimated direction of gravity and vector perpendicular to magnetic flux
    halfvx = q1 * q3 - q0 * q2;
    halfvy = q0 * q1 + q2 * q3;
    halfvz = q0 * q0 - 0.5f + q3 * q3;

    // Error is sum of cross product between estimated and measured direction of gravity
    halfex = (ay * halfvz - az * halfvy);
    halfey = (az * halfvx - ax * halfvz);
    halfez = (ax * halfvy - ay * halfvx);

    // Compute and apply integral feedback if enabled
    if(twoKi > 0.0f)
    {
      integralFBx += twoKi * halfex * dt;  // integral error scaled by Ki
      integralFBy += twoKi * halfey * dt;
      integralFBz += twoKi * halfez * dt;
      gx += integralFBx;  // apply integral feedback
      gy += integralFBy;
      gz += integralFBz;
    }
    else
    {
      integralFBx = 0.0f; // prevent integral windup
      integralFBy = 0.0f;
      integralFBz = 0.0f;
    }

    // Apply proportional feedback
    gx += twoKp * halfex;
    gy += twoKp * halfey;
    gz += twoKp * halfez;
  }

  // Integrate rate of change of quaternion
  gx *= (0.5f * dt);   // pre-multiply common factors
  gy *= (0.5f * dt);
  gz *= (0.5f * dt);
  qa = q0;
  qb = q1;
  qc = q2;
  q0 += (-qb * gx - qc * gy - q3 * gz);
  q1 += (qa * gx + qc * gz - q3 * gy);
  q2 += (qa * gy - qb * gz + q3 * gx);
  q3 += (qa * gz + qb * gy - qc * gx);

  // Normalise quaternion
  recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
  q0 *= recipNorm;
  q1 *= recipNorm;
  q2 *= recipNorm;
  q3 *= recipNorm;
}

*************************************************

匿名的程序在更新四元数的同时更新了姿态角，crazyflie使用单独的函数获得姿态角，同时使用了一个快速求平方根的逆的函数invSqrt.代码如下：

void sensfusion6GetEulerRPY(float* roll, float* pitch, float* yaw)
{
  float gx, gy, gz; // estimated gravity direction

  gx = 2 * (q1*q3 - q0*q2);
  gy = 2 * (q0*q1 + q2*q3);
  gz = q0*q0 - q1*q1 - q2*q2 + q3*q3;

  *yaw = atan2(2*q1*q2 - 2*q0*q3, 2*q0*q0 + 2*q1*q1 - 1) * 180 / M_PI;
  *pitch = atan(gx / sqrt(gy*gy + gz*gz)) * 180 / M_PI;
  *roll = atan(gy / sqrt(gx*gx + gz*gz)) * 180 / M_PI;
}

//---------------------------------------------------------------------------------------------------
// Fast inverse square-root
// See: http://en.wikipedia.org/wiki/Fast_inverse_square_root
float invSqrt(float x)
{
  float halfx = 0.5f * x;
  float y = x;
  long i = *(long*)&y;
  i = 0x5f3759df - (i>>1);
  y = *(float*)&i;
  y = y * (1.5f - (halfx * y * y));
  return y;
}

*************************************************

本帖就贴些代码了，下一帖再分析代码。

楼主的帖子就是教程类的帖子典范啊~~

先送上10个积分 以资鼓励

今天总算有点进展了，以前总觉得定时不准，终于查出原因了。在原来的main()函数中(可在《PWM电机驱动》章看到)，在配置了外部系统时钟后，又重新调用了sysInit()函数，该函数仍然是配置时钟，这样，开始配置的时钟不再有效，贴源码：

/* Personal configs */
#include "FreeRTOSConfig.h"

/* FreeRtos includes */
#include "FreeRTOS.h"
#include "task.h"

/* Project includes */
#include "config.h"
#include "system.h"
#include "nvic.h"

#include "usec_time.h"

#include "led.h"
#include "uart.h"
#include "motors.h"
#include "adc.h"
#include "imu.h"

/* ST includes */
#include "stm32f10x.h"

/* Private functions */
static void prvClockInit(void);
static void prvSetupHardware( void );

int main() 
{
  //Low level init: Clock and Interrupt controller
  prvSetupHardware();

  nvicInit();

  uartInit();
  xTaskCreate(vUartSendTask,(const signed char *)"UartSend", configMINIMAL_STACK_SIZE, NULL, tskIDLE_PRIORITY+3, NULL);

  ledInit();
  xTaskCreate(vLEDTask,(const signed char *)"LED", configMINIMAL_STACK_SIZE, NULL, tskIDLE_PRIORITY+3, NULL );

  motorsInit();
//  xTaskCreate(motorsTestTask,(const signed  char *)"MotorTest",configMINIMAL_STACK_SIZE,NULL,tskIDLE_PRIORITY+3,NULL);

  adcInit();
	
	
 // imu6Test();
	
  //Launch the system task that will initialize and start everything
//  systemLaunch();
//  xTaskCreate(vImu6ReadTask,(const signed char *)"Imu6Read", configMINIMAL_STACK_SIZE, NULL, tskIDLE_PRIORITY+3, NULL);
  
  //Start the FreeRTOS scheduler
  vTaskStartScheduler();

  //Should never reach this point!
  while(1);

 // return 0;
}

static void prvSetupHardware( void )
{
  SystemInit(); 
  prvClockInit();
}

//Clock configuration
static void prvClockInit(void)
{
  ErrorStatus HSEStartUpStatus;

  RCC_DeInit();
  /* Enable HSE */
  RCC_HSEConfig(RCC_HSE_ON);
  /* Wait till HSE is ready */
  HSEStartUpStatus = RCC_WaitForHSEStartUp();

	if (HSEStartUpStatus == SUCCESS)
  {
    /* Enable Prefetch Buffer */
    FLASH_PrefetchBufferCmd(FLASH_PrefetchBuffer_Enable);

    /* Flash 2 wait state */
    FLASH_SetLatency(FLASH_Latency_2);

    /* HCLK = SYSCLK */
    RCC_HCLKConfig(RCC_SYSCLK_Div1);

    /* PCLK2 = HCLK */
    RCC_PCLK2Config(RCC_HCLK_Div1);

    /* PCLK1 = HCLK/2 */
    RCC_PCLK1Config(RCC_HCLK_Div2);

    /* ADCCLK = PCLK2/6 = 72 / 6 = 12 MHz*/
    RCC_ADCCLKConfig(RCC_PCLK2_Div6);

    /* PLLCLK = 16MHz/2 * 9 = 72 MHz */
    RCC_PLLConfig(RCC_PLLSource_HSE_Div2, RCC_PLLMul_9);

    /* Enable PLL */
    RCC_PLLCmd(ENABLE);

    /* Wait till PLL is ready */
    while (RCC_GetFlagStatus(RCC_FLAG_PLLRDY) == RESET);

    /* Select PLL as system clock source */
    RCC_SYSCLKConfig(RCC_SYSCLKSource_PLLCLK);

    /* Wait till PLL is used as system clock source */
    while(RCC_GetSYSCLKSource() != 0x08);
  } else {
      GPIO_InitTypeDef GPIO_InitStructure;
  
    //Cannot start the main oscillator: LED of death...
    GPIO_InitStructure.GPIO_Pin = LED2_GPIO;
    GPIO_InitStructure.GPIO_Mode = GPIO_Mode_Out_PP;
    GPIO_InitStructure.GPIO_Speed = GPIO_Speed_10MHz;

    GPIO_Init(LED2_GPIO_PORT, &GPIO_InitStructure);
    
    GPIO_InitStructure.GPIO_Pin = LED3_GPIO;
    GPIO_InitStructure.GPIO_Mode = GPIO_Mode_Out_PP;
    GPIO_InitStructure.GPIO_Speed = GPIO_Speed_10MHz;

    GPIO_Init(LED3_GPIO_PORT, &GPIO_InitStructure);

    GPIO_ResetBits(LED2_GPIO_PORT, LED2);
    GPIO_ResetBits(LED3_GPIO_PORT, LED3);

    //Cannot start xtal oscillator!
    while(1); 
  }
}

运行后调用原始函数vTaskDelay()后延时准确了。

重新修改了ADC的程序模块，沿用了crazyflie的程序，修改情况见《ADC电压监测》

前段时间忙了一阵，最近工作不太忙，继续试验。今天调通了MPU6050的程序，但还是不太满意。原来一直想用硬件I2C来实现的，但是一直没成功，今天就退而求其次，用了软件模拟，幸好还算成功。有图有真相：

工程下载地址放到百度网盘了，分享一下：

  http://pan.baidu.com/s/1EklYI 

不知道是不是违规，违规就请斑竹删了吧。

应该编译之后就能用了吧，Keil 5.10的开发环境，st-link下载器，有兴趣的可以试试。