// q11 = su^2.dt
//
- // variance of integrated output at 1/dT Hz
- // (random drift)
- const float q00 = gyroVAR * dT;
+ const float dT2 = dT*dT;
+ const float dT3 = dT2*dT;
+
+ // variance of integrated output at 1/dT Hz (random drift)
+ const float q00 = gyroVAR * dT + 0.33333f * biasVAR * dT3;
// variance of drift rate ramp
const float q11 = biasVAR * dT;
-
- const float u = q11 / dT;
- const float q10 = 0.5f*u*dT*dT;
+ const float q10 = 0.5f * biasVAR * dT2;
const float q01 = q10;
GQGt[0][0] = q00; // rad^2
// initial covariance: Var{ x(t0) }
// TODO: initialize P correctly
P = 0;
+
+ // it is unclear how to set the initial covariance. It does affect
+ // how quickly the fusion converges. Experimentally it would take
+ // about 10 seconds at 200 Hz to estimate the gyro-drift with an
+ // initial covariance of 0, and about a second with an initial covariance
+ // of about 1 deg/s.
+ const float covv = 0;
+ const float covu = 0.5f * (float(M_PI) / 180);
+ mat33_t& Pv = P[0][0];
+ Pv[0][0] = covv;
+ Pv[1][1] = covv;
+ Pv[2][2] = covv;
+ mat33_t& Pu = P[1][1];
+ Pu[0][0] = covu;
+ Pu[1][1] = covu;
+ Pu[2][2] = covu;
}
bool Fusion::hasEstimate() const {
mat34_t Fusion::getF(const vec4_t& q) {
mat34_t F;
+
+ // This is used to compute the derivative of q
+ // F = | [q.xyz]x |
+ // | -q.xyz |
+
F[0].x = q.w; F[1].x =-q.z; F[2].x = q.y;
F[0].y = q.z; F[1].y = q.w; F[2].y =-q.x;
F[0].z =-q.y; F[1].z = q.x; F[2].z = q.w;
const vec4_t q = x0;
const vec3_t b = x1;
const vec3_t we = w - b;
- const vec4_t dq = getF(q)*((0.5f*dT)*we);
- x0 = normalize_quat(q + dq);
+ // q(k+1) = O(we)*q(k)
+ // --------------------
+ //
+ // O(w) = | cos(0.5*||w||*dT)*I33 - [psi]x psi |
+ // | -psi' cos(0.5*||w||*dT) |
+ //
+ // psi = sin(0.5*||w||*dT)*w / ||w||
+ //
+ //
// P(k+1) = Phi(k)*P(k)*Phi(k)' + G*Q(k)*G'
+ // ----------------------------------------
//
// G = | -I33 0 |
// | 0 I33 |
const mat33_t wx(crossMatrix(we, 0));
const mat33_t wx2(wx*wx);
const float lwedT = length(we)*dT;
+ const float hlwedT = 0.5f*lwedT;
const float ilwe = 1/length(we);
const float k0 = (1-cosf(lwedT))*(ilwe*ilwe);
const float k1 = sinf(lwedT);
+ const float k2 = cosf(hlwedT);
+ const vec3_t psi(sinf(hlwedT)*ilwe*we);
+ const mat33_t O33(crossMatrix(-psi, k2));
+ mat44_t O;
+ O[0].xyz = O33[0]; O[0].w = -psi.x;
+ O[1].xyz = O33[1]; O[1].w = -psi.y;
+ O[2].xyz = O33[2]; O[2].w = -psi.z;
+ O[3].xyz = psi; O[3].w = k2;
Phi[0][0] = I33 - wx*(k1*ilwe) + wx2*k0;
Phi[1][0] = wx*k0 - I33dT - wx2*(ilwe*ilwe*ilwe)*(lwedT-k1);
+ x0 = O*q;
+ if (x0.w < 0)
+ x0 = -x0;
+
P = Phi*P*transpose(Phi) + GQGt;
checkState();
K[1] = transpose(P[1][0])*LtSi;
// update...
- // P -= K*H*P;
+ // P = (I-K*H) * P
+ // P -= K*H*P
+ // | K0 | * | L 0 | * P = | K0*L 0 | * | P00 P10 | = | K0*L*P00 K0*L*P10 |
+ // | K1 | | K1*L 0 | | P01 P11 | | K1*L*P00 K1*L*P10 |
+ // Note: the Joseph form is numerically more stable and given by:
+ // P = (I-KH) * P * (I-KH)' + K*R*R'
const mat33_t K0L(K[0] * L);
const mat33_t K1L(K[1] * L);
P[0][0] -= K0L*P[0][0];