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Kalman filter (simplified)

 
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wangine



Joined: 07 Jul 2009
Posts: 96
Location: Curtea de Arges, Romania

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Kalman filter (simplified)
PostPosted: Tue Nov 03, 2015 7:57 pm     Reply with quote

Long time ago i just spent 6 - 8 months to find and write a simple fast and precise kalman filter to fit in low PIC18 family. After some months i put in a dsPIC33F family and with days and days past, made a milion comparisons on scope, a analizer, PC, matlab, i dont remember really. I optimize the filter, after i research the math of kalman, comparing other C and C++ kalman filters, to decide with is the best choice.
I will put here my filter because i search the forum for a Differential Evolution algorithm, also kalman for curiosity and i didn't find any library, just some comments.
To explain simple, how accurate is chosen the update filter, knowing a frecqency processor, the reading time of each device (in my case the acc and gyro) than accurate the filter will be. Normally the
Code:
float dt = 0.00003680f;
variable should be a constant, but in my case was updated in other functions after each ADC read, with a timer help.
Code:

/*********************************************************************
  Writer Tanase Bogdan
  (c) copyright www.wangine.ro
  ALL RIGHTS RESERVED
 
********************************************************************
 Ver 0.6 release 29/iunie/2011
********************************************************************/

//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
//~~~~~~~~~~~~~~~~~ m  ,u  ,p  //
 float dt = 0.00003680f;       //time in seconds 0.0000368f with single gyro axis0.000246f;
//static float dt = 0.00016605;       //time in seconds  aprox measure of my scope 0.00016605;
//static float dt = 0.0002372;       //time in seconds  aprox measure of my scope
//static float dt = 0.0002802;       //time in seconds  aprox measure of my scope

#define      R_angle_PARAM      0.3      //03 how match trust to acc

#define      Q_angle_PARAM      0.001  // 0.001 //0.0009f angle parameter update every increase "dt"
#define      Q_gyro_PARAM      0.002   //data relative to gyro; originally 0.001 & 0.003 //0.0016
//      >>>> how match trust to gyro

static float   P[2][2] = {            //Covariance matrix
                    { 1, 0 },
                      { 0, 1 } };
           
static float Pdot[4];

float         angle = 0.0f;            //angle state //98.20f
float         q_bias = 470;            //gyro bias state 4492 ,, 936 read dupa init, nou 470
float         rate = 0;            //unbiased angular rate //0.83f

 static float   R_angle   = R_angle_PARAM;
 static float   Q_angle   = Q_angle_PARAM;   //Q matrix
 static float   Q_gyro   = Q_gyro_PARAM;

//********************State Update Routine********************
//Called every dt seconds with a biased gyro measurement, it updates the , no need bias gyro
//current angle and rate estimate.

void  state_update(float q_m)                 //gyro measurement
{

   if(angle <= -0.006 ^ angle <= 0.006)
   {   dt += 0.00001597f; }   // stabilize to angle

   if(angle <= -0.02 ^ angle <= 0.02)
   {   dt += 0.00000897f; }   // stabilize to angle



    float q = q_m - q_bias;            //unbias gyro measurement

   Pdot[0] = Q_angle - P[0][1] - P[1][0];      //Compute the derivative of the covariance matrix
   Pdot[1] = - P[1][1];
   Pdot[2] = - P[1][1];
   Pdot[3] = Q_gyro;
   
   rate = q;                           //Save unbiased gyro estimate

   angle += q * dt;                     //Update angle estimate

   P[0][0] += Pdot[0] * dt;               //Update covariance matirx
   P[0][1] += Pdot[1] * dt;
   P[1][0] += Pdot[2] * dt;
   P[1][1] += Pdot[3] * dt;
}


//********************Kalman Update Routine********************
//Chemata de cate ori data de la acc este transferata
// incrementeaza la fiecare dt

float      angle_err;
float      C_0;

void kalman_update(float angle_m)
{
    float C_0 = 1;
   /* const float      C_1 = 0; */            //not used
   
    float      PCt_0 = C_0 * P[0][0];      //+ C_1 * P[0][1] = 0
    float      PCt_1 = C_0 * P[1][0];      //+ C_1 * P[1][1] = 0
   
    float      E = R_angle + C_0 * PCt_0;   //Compute error estimate
   /*   + C_1 * PCt_1 = 0 */               //not used
   
    float      K_0 = PCt_0 / E;         //Compute Kalman filter gains
    float      K_1 = PCt_1 / E;
   
    float      t_0 = PCt_0;            //Compute measured angle
    float      t_1 = C_0 * P[0][1];      //and error in the estimate

   P[0][0] -= K_0 * t_0;
   P[0][1] -= K_0 * t_1;
   P[1][0] -= K_1 * t_0;
   P[1][1] -= K_1 * t_1;

   angle_err = angle_m - angle;
   
   angle   += K_0 * angle_err ;               //Update state estimate
   q_bias   += K_1 * angle_err ;          //Update bias estimate  not necesary
}

For math explanation here http://math.stackexchange.com/questions/840662/an-explanation-of-the-kalman-filter or other simply google search.
and here in action
http://www.220.ro/documentare/Self-Balancing-Robot/umHTJRCLns/
I want to mention: On first attempts during first 2 months, trying many filters, my robot with same battery ran only 6-10 min. After some months and learning hard the math of kalman my robot was able to run more than 3 hours continuously.
wangine



Joined: 07 Jul 2009
Posts: 96
Location: Curtea de Arges, Romania

View user's profile Send private message Send e-mail Yahoo Messenger

PostPosted: Thu Nov 05, 2015 11:43 am     Reply with quote

AND My research comes from :
http://www.mymathlib.com/quadrature/rectangle_rule.html
http://www.mymathlib.com/quadrature/simpsons_rule.html
https://github.com/shimniok/diyrovers/blob/master/DataBus/Estimation/kalman.cpp
http://www.mymathlib.com/quadrature/rombergs_method.html
http://www.seattlerobotics.org/encoder/200311/brown/building_a_directional_gyro.html
http://www.chiefdelphi.com/forums/showthread.php?t=39537&page=3
http://www.instructables.com/id/ArduinoGyroscopeProcessing/
one of the best source
http://www.cs.unc.edu/~welch/kalman/
one milions sources for inspiration
https://sites.google.com/site/onewheeledselfbalancing/Home/links-to-other-self-balancing-projects
http://www.starlino.com/imu_guide.html
http://forum.arduino.cc/index.php/topic,8652.0.html
http://aeroquad.com/content.php?s=382bef36515168450968e0389c3c5193
http://www.microchip.com/forums/tm.aspx?m=127772&high=constrain
http://www.advancedsourcecode.com/gaborfiltering.asp[/b]
and many many others , and take so many time just to compare and read all the information.
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