THE SIMPLEST KALMAN FILTER IN THE WORLD

2016-12-23

671

THE SIMPLEST KALMAN FILTER IN THE WORLD

Jack Mo

INTRODUCTION

Kalman filter is the optimal information processor in zero-mean gaussian pdf conditions. It makes use of a key feature of the gaussian distribution family: the product of two gaussian function is another gaussian function. The process to multiply two GDFs (gaussian distribution functions) together is also called data fusion.

THE FUSION OF TWO DATA SOURCES IN ONE DIMENSION

Suppose we have two temperature sensors whose measurements are both noisy and gaussian, then we can describe them use two GDFs.

(1)

(2)

The fusion function is given by the product of and .

(3)

The quadratic terms in this new function can be expanded and then the whole expression can be rewritten to be a new GDF

(4)

where

(5)

(6)

IMPLEMENTING THE SIMPLEST KALMAN FILTER

One thing should be pointed out is that the information we combine together does not necessarily come from measurements. As a matter of fact, in most of the cases we always have some previous knowledge about the system and we can predict how it will change according to the current state. For example, imagine we are in a moving train. With current position and speed given, we can approximately predict where the train will be in the next few seconds. But we are not so sure about our prediction, because we don’t know whether the driver will take an acceleration or a break during the period. That is to say, our prediction is noisy too. Consider it’s gaussian, then we can fuse it with sensor measurements. Here below, I just implemented a basic kalman filter that can be the easiest one in the world. It fuses current prediction and measurement together, and produces an estimation of the most probable state.

File: kalman.h

/* file: kalman.h */

#ifndef __KALMAN_H__
#define __KALMAN_H__

#ifdef __cplusplus
extern "C" {
#endif

#include 
#include 
#include 

typedef struct Kalman
{
  /* Critical Section */
  float q;    // process noise
  float r;    // measurement noise
  /* Critical Section */
  
  float e;    // estimation
  float d;    // difference
  float k;    // kalman gain
  float p;    // fused variance
}Kalman;

struct Kalman* KalmanCreate();
void KalmanReset(struct Kalman* kalman);
void KalmanSetE(struct Kalman* kalman, float e);
void KalmanSetD(struct Kalman* kalman, float d);
void KalmanSetQ(struct Kalman* kalman, float q);
void KalmanSetR(struct Kalman* kalman, float r);
float KalmanFilter(struct Kalman* kalman, float x);
void KalmanDestroy(struct Kalman* kalman);

#ifdef __cplusplus
}
#endif

#endif

File: kalman.c

/* file: kalman.c */

#include "kalman.h"

#define SQR(x) (x*x)

struct Kalman* KalmanCreate()
{
  Kalman* kalman = (Kalman*)malloc(sizeof(Kalman));
  if (kalman == NULL) return NULL;
  KalmanReset(kalman);
  return kalman;
}

void KalmanReset(struct Kalman* kalman)
{
  kalman->e = 0;
  kalman->d = 0;
  kalman->k = 0;
  kalman->p = 0;
}

void KalmanSetE(struct Kalman* kalman, float e)
{
  kalman->e = e;
}

void KalmanSetD(struct Kalman* kalman, float d)
{
  kalman->d = d;
}

void KalmanSetQ(struct Kalman* kalman, float q)
{
  kalman->q = q;
}

void KalmanSetR(struct Kalman* kalman, float r)
{
  kalman->r = r;
}

float KalmanFilter(struct Kalman* kalman, float x) {
  // update
  kalman->e += kalman->d;
  kalman->p += kalman->q;
  // fusion
  kalman->k = kalman->p / (kalman->p + kalman->r); // kalman gain
  kalman->d = (x - kalman->e) / kalman->k; // delta mean
  kalman->e += kalman->d;                  // fused mean
  kalman->p -= kalman->p * kalman->k; // fused variance
  return kalman->e;
}

void KalmanDestroy(struct Kalman* kalman)
{
  if (kalman != NULL) {
    free(kalman);
    kalman = NULL;
  }
}

PREVIOUS JOB TO DO BEFORE USING OUR KALMAN FILTER

In the implementation above, two parameters must be known before the filter process is actually put into use. One is the process noise, which is the uncertainty comes from the prediction step, the other is measurement noise, which is the uncertainty comes from the sensor. The former can be assigned a reasonable one, according to the system itself. The latter should be tested out carefully. To accomplish this, I wrote another helper functions which can calculate various of statistic features all together.

File: gauss.h

/* file: gauss.h */

#ifndef __GAUSS_H__
#define __GAUSS_H__

#ifdef __cplusplus
extern "C" {
#endif

#include 
#include 
#include 

typedef struct Gauss
{
  /* Critical Section */
  uint16_t N; // buffer length
  /* Critical Section */
  
  float* buf; // buffer
  uint16_t n; // initialization counter
  uint16_t i; // current enqueuing buffer index
  float sum;  // sum of buffer
  float diff; // diff of sum
  float res0; // residual of the number 0 parameter
  float resn; // residual of the number (n - 1) parameter
  float mean; // mean of buffer
  float last_mean; // last mean
  float delta_mean; // delta of mean
  float sse; // sum of square error
  float delta_sse; // delta of sse
  float mse; // mean square error
  float last_mse; // last mse
  float delta_mse; // delta mse
}Gauss;

struct Gauss* GaussCreate(uint16_t N);
void GaussReset(struct Gauss* gauss);
float GaussFilter(struct Gauss* gauss, float x);
void GaussDestroy(struct Gauss* gauss);

#ifdef __cplusplus
}
#endif

#endif

File: gauss.c

/* file: gauss.c */

#include "gauss.h"

#define SQR(x) (x*x)

struct Gauss* GaussCreate(uint16_t N)
{
  Gauss* gauss = (Gauss*)malloc(sizeof(Gauss));
  if (gauss == NULL) return NULL;
  gauss->buf = (float*)malloc(N * sizeof(float));
  if (gauss->buf == NULL) {
    free(gauss);
    gauss = NULL;
    return NULL;
  }
  gauss->N = N;
  GaussReset(gauss);
  return gauss;
}

void GaussReset(struct Gauss* gauss)
{
  memset(gauss->buf, 0, gauss->N * sizeof(float));
  gauss->n = 0;
  gauss->i = 0;
  gauss->sum = 0;
  gauss->diff = 0;
  gauss->res0 = 0;
  gauss->resn = 0;
  gauss->mean = 0;
  gauss->last_mean = 0;
  gauss->delta_mean = 0;
  gauss->sse = 0;
  gauss->delta_sse = 0;
  gauss->mse = 0;
  gauss->last_mse = 0;
  gauss->delta_mse = 0;
}

float GaussFilter(struct Gauss* gauss, float x) {
  gauss->last_mean = gauss->mean;
  gauss->last_mse = gauss->mse;
  if (gauss->n < gauss->N) {
    gauss->buf[gauss->n++] = x;
    gauss->sum += x;
    gauss->mean = gauss->sum / gauss->n;
    gauss->delta_mean = gauss->mean - gauss->last_mean;
    gauss->resn = x - gauss->mean;
    gauss->sse += SQR(gauss->resn);
    gauss->mse = gauss->sse / gauss->n;
    gauss->delta_mse = gauss->mse - gauss->last_mse;
  } else {
    if (gauss->i == gauss->N) gauss->i = 0;
    gauss->diff = x - gauss->buf[gauss->i];
    gauss->sum += gauss->diff;
    gauss->mean = gauss->sum / gauss->n;
    gauss->delta_mean = gauss->mean - gauss->last_mean;
    gauss->res0 = gauss->buf[gauss->i] - gauss->last_mean;
    gauss->resn = x - gauss->mean;
    gauss->delta_sse = (gauss->diff) * (gauss->resn + gauss->res0);
    gauss->sse += gauss->delta_sse;
    gauss->mse = gauss->sse / gauss->n;
    gauss->delta_mse = gauss->mse - gauss->last_mse;
    gauss->buf[gauss->i++] = x;
  }
  return gauss->mean;
}

void GaussDestroy(struct Gauss* gauss)
{
  if (gauss != NULL) {
    if (gauss->buf != NULL) {
      free(gauss->buf);
      gauss->buf = NULL;
    }
    free(gauss);
    gauss = NULL;
  }
}

USE OUR KALMAN FILTER IN AN INTEL CRURIE BOARD

To make our kalman filter take real effect and verify it’s performance, we must get it run. I happen to have an arduino101 board with an intel crurie core inside on hand. Other than most of the arduino boards, the arduino101 board integrates a 6-axis imu, so I decide to make use of it. I use the kalman filter implemented above to smooth the acceleration in x-axis. The code is directly modified from the samples given by the arduino IDE. To protect copyright, the notice is included. Hope it’ll not bother you.

File: CrurieIMUKalman.ino

/* file: CrurieIMUKalman.ino */

/*
  ===============================================
  Example sketch for CurieIMU library for Intel(R) Curie(TM) devices.
  Copyright (c) 2015 Intel Corporation.  All rights reserved.

  Based on I2C device class (I2Cdev) demonstration Arduino sketch for MPU6050
  class by Jeff Rowberg: https://github.com/jrowberg/i2cdevlib

  ===============================================
  I2Cdev device library code is placed under the MIT license
  Copyright (c) 2011 Jeff Rowberg

  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  copies of the Software, and to permit persons to whom the Software is
  furnished to do so, subject to the following conditions:

  The above copyright notice and this permission notice shall be included in
  all copies or substantial portions of the Software.

  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
  THE SOFTWARE.
  ===============================================
*/

#include "CurieIMU.h"
#include "gauss.h"
#include "kalman.h"

int ax, ay, az;         // accelerometer values
int gx, gy, gz;         // gyrometer values

const int ledPin = 13;      // activity LED pin
boolean blinkState = false; // state of the LED

int calibrateOffsets = 1; // int to determine whether calibration takes place or not

#define GAUSS_N 100
Gauss* gauss = NULL;
Kalman* kalman = NULL;
void setup() {
  Serial.begin(9600); // initialize Serial communication
  while (!Serial);    // wait for the serial port to open

if (gauss == NULL) {
    gauss = GaussCreate(GAUSS_N);
    if (gauss == NULL) {
      Serial.println("GaussCreate() returned NULL, system pending.");
      while(1);
    } else {
      Serial.println("GaussCreate() done.");
    }
  }
  
  if (kalman == NULL) {
    kalman = KalmanCreate();
    if (kalman == NULL) {
      GaussDestroy(gauss);
      Serial.println("KalmanCreate() returned NULL, system pending.");
      while(1);
    } else {
      Serial.println("KalmanCreate() done.");
    }
  }
  
  // initialize device
  Serial.println("Initializing IMU device...");
  CurieIMU.begin();

  // verify connection
  Serial.println("Testing device connections...");
  if (CurieIMU.begin()) {
    Serial.println("CurieIMU connection successful");
  } else {
    Serial.println("CurieIMU connection failed");
  }

  // use the code below to calibrate accel/gyro offset values
  if (calibrateOffsets == 1) {
    Serial.println("Internal sensor offsets BEFORE calibration...");
    Serial.print(CurieIMU.getAccelerometerOffset(X_AXIS));
    Serial.print("\t"); // -76
    Serial.print(CurieIMU.getAccelerometerOffset(Y_AXIS));
    Serial.print("\t"); // -235
    Serial.print(CurieIMU.getAccelerometerOffset(Z_AXIS));
    Serial.print("\t"); // 168
    Serial.print(CurieIMU.getGyroOffset(X_AXIS));
    Serial.print("\t"); // 0
    Serial.print(CurieIMU.getGyroOffset(Y_AXIS));
    Serial.print("\t"); // 0
    Serial.println(CurieIMU.getGyroOffset(Z_AXIS));

    // To manually configure offset compensation values,
    // use the following methods instead of the autoCalibrate...() methods below
    //CurieIMU.setAccelerometerOffset(X_AXIS,495.3);
    //CurieIMU.setAccelerometerOffset(Y_AXIS,-15.6);
    //CurieIMU.setAccelerometerOffset(Z_AXIS,491.4);
    //CurieIMU.setGyroOffset(X_AXIS,7.869);
    //CurieIMU.setGyroOffset(Y_AXIS,-0.061);
    //CurieIMU.setGyroOffset(Z_AXIS,15.494);

    Serial.println("About to calibrate. Make sure your board is stable and upright");
    delay(5000);

    // The board must be resting in a horizontal position for
    // the following calibration procedure to work correctly!
    Serial.print("Starting Gyroscope calibration and enabling offset compensation...");
    CurieIMU.autoCalibrateGyroOffset();
    Serial.println(" Done");

    Serial.print("Starting Acceleration calibration and enabling offset compensation...");
    CurieIMU.autoCalibrateAccelerometerOffset(X_AXIS, 0);
    CurieIMU.autoCalibrateAccelerometerOffset(Y_AXIS, 0);
    CurieIMU.autoCalibrateAccelerometerOffset(Z_AXIS, 1);
    Serial.println(" Done");

    Serial.println("Internal sensor offsets AFTER calibration...");
    Serial.print(CurieIMU.getAccelerometerOffset(X_AXIS));
    Serial.print("\t"); // -76
    Serial.print(CurieIMU.getAccelerometerOffset(Y_AXIS));
    Serial.print("\t"); // -2359
    Serial.print(CurieIMU.getAccelerometerOffset(Z_AXIS));
    Serial.print("\t"); // 1688
    Serial.print(CurieIMU.getGyroOffset(X_AXIS));
    Serial.print("\t"); // 0
    Serial.print(CurieIMU.getGyroOffset(Y_AXIS));
    Serial.print("\t"); // 0
    Serial.println(CurieIMU.getGyroOffset(Z_AXIS));
  }
  
  // configure Arduino LED for activity indicator
  pinMode(ledPin, OUTPUT);
}

#define GAUSS_PRECISION 0.5f
#define PROC_NOISE 0.1
bool calied = false;
void loop() {
  
  // read raw accel/gyro measurements from device
  CurieIMU.readMotionSensor(ax, ay, az, gx, gy, gz);

  // these methods (and a few others) are also available

  //CurieIMU.readAcceleration(ax, ay, az);
  //CurieIMU.readRotation(gx, gy, gz);

  //ax = CurieIMU.readAccelerometer(X_AXIS);
  //ay = CurieIMU.readAccelerometer(Y_AXIS);
  //az = CurieIMU.readAccelerometer(Z_AXIS);
  //gx = CurieIMU.readGyro(X_AXIS);
  //gy = CurieIMU.readGyro(Y_AXIS);
  //gz = CurieIMU.readGyro(Z_AXIS);

  // display tab-separated accel/gyro x/y/z values
  /*
  Serial.print("a/g:\t");
  Serial.print(ax);
  Serial.print("\t");
  Serial.print(ay);
  Serial.print("\t");
  Serial.print(az);
  Serial.print("\t");
  Serial.print(gx);
  Serial.print("\t");
  Serial.print(gy);
  Serial.print("\t");
  Serial.println(gz);
  */

  if (calied == false) {
    if (gauss->n < gauss->N || fabs(gauss->delta_mse) > GAUSS_PRECISION) {
      GaussFilter(gauss, ax);
      Serial.print("ax:\t");
      Serial.print(ax);
      Serial.print("\t");
      Serial.print(gauss->mean);
      Serial.print("\t");
      Serial.print(gauss->mse);
      Serial.print("\t");
      Serial.println(gauss->delta_mse);
    } else {
      KalmanSetE(kalman, gauss->mean);
      KalmanSetD(kalman, gauss->delta_mean);
      KalmanSetQ(kalman, PROC_NOISE);
      KalmanSetR(kalman, gauss->mse);
      calied = true;
      GaussDestroy(gauss);
    }
  }

  if (calied) {
    KalmanFilter(kalman, ax);
    Serial.print("ax:\t");
    Serial.print(ax);
    Serial.print("\t");
    Serial.println(kalman->e);
  }

  // blink LED to indicate activity
  blinkState = !blinkState;
  digitalWrite(ledPin, blinkState);
}

TESTING RESULT

The following two pictures show the effectiveness of the kalman filter we implemented above. It’s obvious that the curve looks better when filtered.

Figure 1.