In (7), as defined above, [Pos.sup.t.sub.i] is the position vector
and [Vel.sup.t.sub.i] is the velocity vector of particle i; [Pos.sup.pbest.sub.i] represents the local best position and [f.sup.pbest.sub.i] represents the local best fitness value found by particle i up to the current time t.
Because we bound index 4 of the position vector
with customer 4's normalized time window [[e.sup.nor.sub.k], [l.sup.nor.sub.k], index 4 will always be larger than the other variables during the search process.
As shown in Figure 3, the instantaneous slant range R([r.sub.P],[r.sub.T]) = [square root of [r.sup.2.sub.P] + [r.sup.2.sub.T] - 2[r.sub.P][r.sub.T] cos [beta]], where [beta] is the angle between position vector
[r.sub.P]([t.sub.m]) and [r.sub.T],
Let [phi] - [phi](s) be a unit speed curve and first component of position vector
on Frenet axis be constant in [E.sup.3]:
where [mathematical expression not reproducible] is the transform matrix from the aircraft body frame to the ECEF frame and [r.sup.(i).sub.cnd/aft] is the constant position vector
of the ith contact node with respect to the aircraft body frame.
The r = (%, y) position vector
is sent to the other microcontroller, which solves discrete-time equations (3), (4), and (5).
Algorithm 2 Pseudo-code of the updates mechanism 1 Inputs: [X.sub.i]= ([x.sub.1], [x.sub.2], ..., [x.sub.d]) /* A current position vector
of a particle i*/ RFV = ([r.sub.1], [r.sub.2], ..., [r.sub.d]) /* A relevant features vector */ Pbest = ([p.sub.1], [p.sub.2], ..., [p.sub.d]) /* A Pbest vector of a Current particle*/ Gbest = ([g.sub.1], [g.sub.2], ....
, [X.sub.4] are the position vectors
(coordinates) of element nodes 1, 2, 3, and 4, respectively, referred to the global X, Y, Z axes.
To write the position vector
of the surface, it is useful to define the new variable S in terms of X, Y as follows:
The position vector
r = [(x y z).sup.T] for a given reference point A can be determined by (10) for the corresponding attitude angle and the corresponding moving platform relative to the rotation table R of the base platform, and the length of each limb can be obtained by
The position vector
P and rotation matrix R are expressed as
Thus when fulfilling the above assumptions, we can write for the vectors of observed field [B.sub.Obs] = [[B.sub.x], [B.sub.y], [B.sub.z]] and vector of car disturbance field [B.sub.car] = [[B.sub.cx], [B.sub.cy], [B.sub.cz]] utilizing the well-known equation for magnetic field of a magnetic dipole with magnetic moment m [[m.sub.x], [m.sub.y], [m.sub.z]] position vector
r [[r.sub.x], [r.sub.y], [r.sub.z]] and an (orthogonal) rotation matrix R: