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ekf_filter.py
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# -*- coding: utf-8 -*-
from math import sqrt,atan2,asin,sqrt
from numpy import pi ,array
import numpy as np
def FastSqrtI(n=1):
if n ==0:
#print "norm ==0 !!"
return 1
else:
return 1.0/sqrt(n)
def Sqr(n):
return n**2
def qa_to_erula_angle(q):
q0,q1,q2,q3=q
RollAngle = atan2(2.0*q2*q3 - 2.0*q0*q1, 2.0*Sqr(q0) + 2.0*Sqr(q3) - 1.0);
num = (2.0*q1*q2 - 2.0*q0*q2);
if (num<-1):
#print num
num=-1.0
elif (num>1) :
#print num
num=1.0
PitchAngle = asin(num)
YawAngle = atan2(2.0*q1*q2 - 2.0*q0*q3, 2.0*Sqr(q0) + 2.0*Sqr(q1) - 1.0);
return np.array([PitchAngle,RollAngle,YawAngle])
def EKF_IMUGetAngle(X):
CBn=np.zeros(9)
rpy =np.zeros(3)
CBn[0] = 2.0 * (X[0] * X[0] + X[1] * X[1]) - 1.0
CBn[1] = 2.0 * (X[1] * X[2] + X[0] * X[3])
CBn[2] = 2.0 * (X[1] * X[3] - X[0] * X[2])
# //CBn[3] = 2.0f * (X[1] * X[2] - X[0] * X[3]);
# //CBn[4] = 2.0f * (q0q0 + X[2] * X[2]) - 1.0f;
#CBn[5] = 2.0 * (X[2] * X[3] + X[0] * X[1])
CBn[5] = 2.0 * (X[2] * X[3] + X[0] * X[1])
# //CBn[6] = 2.0f * (X[1] * X[3] + X[0] * X[2]);
# //CBn[7] = 2.0f * (X[2] * X[3] - X[0] * X[1]);
CBn[8] = 2.0 * (X[0] * X[0] + X[3] * X[3]) - 1.0
#//roll
RollAngle = atan2(CBn[5], CBn[8])
RollAngle -=np.pi
if(RollAngle<-np.pi):
RollAngle+=2*np.pi
RollAngle=-RollAngle
#//pitch
if (CBn[2] >= 1.0):
PitchAngle = -np.pi/2.0
elif (CBn[2] <= -1.0):
PitchAngle = np.pi/2.0
else:
PitchAngle = asin(-CBn[2])
#//yaw
YawAngle = atan2(CBn[1], CBn[0])
if (YawAngle < 0.0):
YawAngle += np.pi*2
if (YawAngle >= np.pi*2):
YawAngle = 0.0
return np.array([PitchAngle,RollAngle,YawAngle])
class ekf_filter(object):
def __init__(self):
self.status_dm =7
self.measure_dm =6
self.init_flag = 0
self.qq=0.0045 ##
self.qw=0.025
self.ra= 0.07
self.rm=0.105
self.pq = 0.000001
self.pw =0.000010
###
### 对应陀螺漂移误差
self.Q = np.array([
[self.qq, 0, 0, 0, 0, 0, 0],
[0, self.qq, 0, 0, 0, 0, 0],
[0, 0, self.qq, 0, 0, 0, 0],
[0, 0, 0, self.qq, 0, 0, 0],
[0, 0, 0, 0, self.qw, 0, 0],
[0, 0, 0, 0, 0, self.qw, 0],
[0, 0, 0, 0, 0, 0, self.qw],
])
### 重力和磁场误差
self.R = np.array([
[self.ra, 0, 0, 0, 0, 0],
[0, self.ra, 0, 0, 0, 0],
[0, 0, self.ra, 0, 0, 0],
[0, 0, 0, self.rm, 0, 0],
[0, 0, 0, 0, self.rm, 0],
[0, 0, 0, 0, 0, self.rm],
]) ##6,6
###
self.I =np.eye(self.status_dm)
self.F =np.eye(self.status_dm)
self.H =np.zeros((self.measure_dm,self.status_dm))
#self.H =np.zeros((self.status_dm,self.status_dm)) ##7,6
self.P =np.array([
[self.pq, 0, 0, 0, 0, 0, 0],
[0, self.pq, 0, 0, 0, 0, 0],
[0, 0, self.pq, 0, 0, 0, 0],
[0, 0, 0, self.pq, 0, 0, 0],
[0, 0, 0, 0, self.pw, 0, 0],
[0, 0, 0, 0, 0, self.pw, 0],
[0, 0, 0, 0, 0, 0, self.pw],
])
self.X=np.zeros(self.status_dm)
self.Y=np.zeros(self.measure_dm)
self.K=np.zeros((self.status_dm,self.measure_dm))
self.factor = 16.4*180.0/np.pi
def filter_init(self,gyro,accel):
nedVector = np.array([0, 0 , -1.0])
accelVector=np.zeros(3)
crossVector=np.zeros(3)
#//unit accel
norm = FastSqrtI(accel[0] * accel[0] + accel[1] * accel[1] + accel[2] * accel[2])
## or use norm = np.linalg.norm(accel)
accelVector[0] = accel[0] * norm
accelVector[1] = accel[1] * norm
accelVector[2] = accel[2] * norm
#cross product between accel and reference
crossVector[0] = accelVector[1] * nedVector[2] - accelVector[2] * nedVector[1]
crossVector[1] = accelVector[2] * nedVector[0] - accelVector[0] * nedVector[2]
crossVector[2] = accelVector[0] * nedVector[1] - accelVector[1] * nedVector[0]
sinwi = FastSqrtI(crossVector[0] * crossVector[0] + crossVector[1] * crossVector[1] + crossVector[2] * crossVector[2])
crossVector[0] *= sinwi
crossVector[1] *= sinwi
crossVector[2] *= sinwi
#the angle between accel and reference is the dot product of the two vectors
cosw = accelVector[0] * nedVector[0] + accelVector[1] * nedVector[1] + accelVector[2] * nedVector[2]
# coshalfw = sqrt((1.0- cosw) * 0.5)
# sinhalfw = sqrt((1.0 +cosw) * 0.5)
coshalfw = sqrt((1.0 + cosw) * 0.5)
sinhalfw = sqrt((1.0 -cosw) * 0.5)
self.X[0] = coshalfw
self.X[1] = crossVector[0] * sinhalfw
self.X[2] = crossVector[1] * sinhalfw
self.X[3] = crossVector[2] * sinhalfw
norm = FastSqrtI(self.X[0] * self.X[0] + self.X[1] * self.X[1] + self.X[2] * self.X[2] + self.X[3] * self.X[3])
self.X[0] *= norm
self.X[1] *= norm
self.X[2] *= norm
self.X[3] *= norm
def filter_update(self,gyro,accel,dt=0.002):
halfdt =dt*0.5
### 0.5*t(w - wbias)
##self.X[4:6]存储的是w的
halfdx = halfdt * (gyro[0]/self.factor - self.X[4])
halfdy = halfdt * (gyro[1]/self.factor - self.X[5])
halfdz = halfdt * (gyro[2]/self.factor - self.X[6])
neghalfdx = -halfdx
neghalfdy = -halfdy
neghalfdz = -halfdz
q0 = self.X[0]
q1 = self.X[1]
q2 = self.X[2]
q3 = self.X[3]
# //////////////////////////////////////////////////////////////////////////
# //Extended Kalman Filter: Prediction Step
# //state time propagation
# //Update Quaternion with the new gyroscope measurements
#x = x +∫f(x)dt 四元素方程微分形式
#这里相当于执行 xk=A*xk-1
self.X[0] = q0 - halfdx * q1 - halfdy * q2 - halfdz * q3
self.X[1] = q1 + halfdx * q0 + halfdz * q2 - halfdy * q3
self.X[2] = q2 + halfdy * q0 - halfdz * q1 + halfdx * q3
self.X[3] = q3 + halfdz * q0 + halfdy * q1 - halfdx * q2
halfdtq0 = halfdt * q0
neghalfdtq0 = -halfdtq0
halfdtq1 = halfdt * q1
neghalfdtq1 = -halfdtq1
halfdtq2 = halfdt * q2
neghalfdtq2 = -halfdtq2
halfdtq3 = halfdt * q3
neghalfdtq3 = -halfdtq3
#生成预测矩阵 F
#有关系x = F*x
#对陀螺输出值进行积分有次计算姿态角
# q0 = q0 - wx*q1/2 - wy*q2/2 -（wz-wbias)*q3/2
#self.F[0][0] =1
self.F[0][1] = neghalfdx
self.F[0][2] = neghalfdy
self.F[0][3] = neghalfdz
self.F[0][4] = halfdtq1
self.F[0][5] = halfdtq2
self.F[0][6] = halfdtq3
self.F[1][0] = halfdx
#self.F[1][1] =1.0
self.F[1][2] = halfdz
self.F[1][3] = neghalfdy
self.F[1][4] = neghalfdtq0
self.F[1][5] = halfdtq3
self.F[1][6] = neghalfdtq2
self.F[2][0] = halfdy
self.F[2][1] = neghalfdz
#self.F[2][2]=1
self.F[2][3] = halfdx
self.F[2][4] = neghalfdtq3
self.F[2][5] = neghalfdtq0
self.F[2][6] = halfdtq1
self.F[3][0] = halfdz
self.F[3][1] = halfdy
self.F[3][2] = neghalfdx
# self.F[3][3] = 1
self.F[3][4] = halfdtq2
self.F[3][5] = neghalfdtq1
self.F[3][6] = neghalfdtq0
# //P = F*P*F' + Q
self.P= np.dot(np.dot(self.F,self.P),self.F.T) + self.Q
# //////////////////////////////////////////////////////////////////////////
# //measurement update
# //kalman gain calculation
# //K = P * H' / (R + H * P * H')
# print self.H
_2q0,_2q1,_2q2,_2q3 = 2.0 * self.X[:4]
## 更新观察矩阵
### H=J(y,x) Jacobian
# y = h(x)=Cbn*x
#计算Cbn 对X的Jacbian方程即可。H阵为6,7
#即计算 四元素矩阵的Jacobin 公式
self.H[0][0] = _2q2
self.H[0][1] = -_2q3
self.H[0][2] = _2q0
self.H[0][3] = -_2q1
self.H[1][0] = -_2q1
self.H[1][1] = -_2q0
self.H[1][2] = -_2q3
self.H[1][3] = -_2q2
self.H[2][0] = -_2q0
self.H[2][1] = _2q1
self.H[2][2] = _2q2
self.H[2][3] = -_2q3
# //K=P*H'/(H*P*H'+R)
S=np.dot(np.dot(self.H,self.P),self.H.T) + self.R
SI=np.linalg.inv(S)
self.K=np.dot(self.P,np.dot(self.H.T,SI))
# //state measurement update
# //Y = Z-h(x) //h(x) is just
# //X = X + K * Y;
# //acccel on earth [0 0 1] is standard g vector
# //计算重力在机体上的投影
# 为何为负值 对应H中负值
self.Y[0] = -2.0 * (self.X[1] * self.X[3] - self.X[0] * self.X[2])
self.Y[1] = -2.0 * (self.X[2] * self.X[3] + self.X[0] * self.X[1])
self.Y[2] = 1.0 -2.0 * (self.X[0] * self.X[0] + self.X[3] * self.X[3])
#//normalize accel
#计算实际观测到的重力坐标
norm = FastSqrtI(accel[0] * accel[0] + accel[1] * accel[1] + accel[2] * accel[2])
acc=np.array(accel)
acc = acc*norm
# accel[0] *= norm;
# accel[1] *= norm;
# accel[2] *= norm;
# y-h(x) 算作误观测误差，这里本质上为用重力来校准陀螺误差
# 对于磁力有类似的算法
#Y阵当前为[accx,accy,accy,0,0,0]
#待添加磁校准形成观察阵[accx,accy,accy,mx,my,mz]
self.Y[:3] = acc - self.Y[0:3]
# // Update State Vector
# X=X + K(Y-h(X))
self.X = self.X + np.dot(self.K,self.Y)
#//normalize quaternion
norm=FastSqrtI(self.X[0]**2 + self.X[1]**2+ self.X[2]**2 + self.X[3]**2 )
self.X[0] *= norm
self.X[1] *= norm
self.X[2] *= norm
self.X[3] *= norm
# #
# //covariance measurement update
# //P = (I - K * H) * P
# //P = P - K * H * P
# //or
# //P=(I - K*H)*P*(I - K*H)' + K*R*K'
self.P = np.dot(self.I - np.dot(self.K,self.H),self.P)
q=[self.X[0],self.X[1] ,self.X[2] ,self.X[3]]
#print q
return EKF_IMUGetAngle(q)
#return qa_to_erula_angle(q)
if __name__ == '__main__':
filter=ekf_filter()
gyro=np.array([0,0,0])
acc=np.array([0,0,1])
a=filter.filter_update(gyro,acc,0.002)
erual= np.array([a])
print erual*180/np.pi