GPS World, April 2016

AUTONOMOUS NAV marizes the geometric relations where the primary body frame is labeled p frame and the secondary body frame is labeled s frame The body frame fixed to the primary P is shown by x p y p z p and body frame fixed to the secondary P P P s y S S is shown by x S s z S s The relative navigation equation is set up for the state of the secondary with respect to the state of the primary in the center of the body frame of the primary p frame 1 p is the primary position vector established in the where x P p is the secondary position vector defined p frame and x S in the p frame Note that these vectors can also be obtained from the primary secondary strapdown inertial navigation solutions after transferring to the reference eccentric point Equation 1 represents the fundamental equation from which the relative navigation equations are derived Once the relative kinematic model of the position and velocity are established the next step is to develop the relative attitude kinematic model The relative attitude denoted by the quaternion qp S is used to map vectors in the s frame to vectors in the p frame 2 where q p and q s are the quaternion attitudes of the primary and secondary with respect to the i frame q p is the conjugate of q p and is the quaternion multiplication operator RELATIVE EXTENDED KALMAN FILTER To establish the R EKF we must derive the relative inertial error equations The R EKF has 21 basic states including nine for relative position δΔxp PS relative velocity δΔvp PS and relative attitude Ψp ps and 12 to model the primarys gyro and accelerometer bias non constant and non linear scale factors Since the relative distance between the secondary and primary is small compared to the radius of the Earth the gravity terms are negligible Thus in the linearized terms the relative gravitational terms are ignored It should be noted that the secondary states are assumed to be known for retrieving the absolute primary TSPI information Since Equations 1 and 2 can only provide the general dynamic model for a nonlinear state model all these equations must be linearized using Taylor series about nominal values neglecting the higher order terms After perturbation state equations are established they should be discretized from a continuous time to a discrete time sequence The final solution to the state equation can be expressed as 20 GPS WORLD WWW GPSWORLD COM APRIL 2016 3 FIGURE 2 Relative observation model with 4 F p is the Jacobian matrix and the perturbation elements P S are all related to the primary 5 RELATIVE GPS MEASUREMENT MODEL When GPS is available high accuracy relative positions are derived from the use of carrier phase differential GPS a technique commonly used in static positioning applications such as surveying However unlike those applications in this case the reference receiver is not stationary it is located on a moving platform secondary creating a moving baseline The relative GPS measurement in our system is provided by epoch by epoch EBE differential carrier phase processing which measures accurate relative position between the secondary and primary systems The EBE relative position has a typical accuracy better than 3 cm 1 sigma horizontal and 6 cm 1 sigma vertical Testing of the relative measurement was conducted using two ground vehicles configured with 10 Hz dual frequency GPS sensors The mean difference was less than 5 cm As a conclusion the GPS relative mode was shown to provide accurate relative positions between the platforms Once the relative position is measured the R EKF observation model can be established as 6 The Δxp PS GPS term is the relative position measured by using GPS data and the term Δxp PS INS is the relative position which is predicted by using the last updated inertial solutions Note that in order to use this relative observation the lever arm vector between the GPS and IMU of both the