AN APPLICATION OF MOTION CORRECTION METHODS TO THE ALIGNMENT PROBLEM IN NAVIGATION1

In this paper, we apply some motion correction methods to the alignment problem in navigation. This problem consists in matching two coordinate systems having the common origins. As a rule, one of the systems named as basic coordinate system is located at a ship or airplane. The dependent coordinate system belongs to another object (e.g. missile ) that starts from the ship. The problem is considered with incomplete information on state coordinates which can be measured with disturbances without statistical description.


Introduction
Alignment is the process whereby the orientation of the axes of an inertial navigation system is determined with respect to the reference axis system. The basic concept of aligning an inertial navigation system is quite simple and straightforward. However, there are many complications that make alignment both time consuming and complex. Consider a simulated transport ship-airplane system. Suppose that the base coordinate system (BCS) of the ship is correct. Let − → Ω 1 be the absolute angular velocity of the BCS in the motionless coordinate system η 1 , η 2 , η 3 . The projection Ω 2 1 on vertical 2 1 equals zero. This system is shown on Fig. 1. The axis 1 1 is directed along the  Projecting the equality ω =˙ θ 1 +˙ θ 3 +˙ θ 2 for the angular velocities on the axes of the DCS, we obtain the kinematic Krylov equationṡ where ω i are the projections of the relative angular velocity. These projections are related with the absolute velocities by the formulas where ε i are the projections of an uncertain drift. For measurements, the differences of accelerometer readings in the DCS and BCS are used. These accelerometers are on the axes and gage the nongravity acceleration a = − → w M − g. Let a i be accelerometers readings in the DCS and a i 1 be gage readings in the BCS. Therefore, the measurement equations are of the form where w i are uncertain leavings of zero. About drifts ε i in (0.2), the assumption is accepted that they are constant but unknown. Uncertain functions in relations (0.3) satisfy the integral inequalities Let i 1 , i 2 , i 3 be the unit direction vectors of the BCS. The velocity of point M equals where R is the radius of Earth. From here we find the projections of velocity on the BCS axes: in the form of the sum of relative and translation accelerations. So, the accelerometers readings in BCS are of the form: where v is the velocity magnitude. As R = 6370 km and the velocity of the ship on water is no more than 20 m/c, we assume a 2 1 = g. Further we consider some approaches from motion correction for solving the alignment problem. This problem in inertial navigation was first in detail considered in [1]. Russian books devoted to this topic are [2][3][4][5]. The alignment problem was mostly solved in [1][2][3][4][5] by statistical methods with the help of Kalman filter or its modifications. On the other hand, in [2,6] it was noted that the statistics of disturbances often happens incomplete or completely absent. Therefore, it is natural to use here the minimax methods from books [7,8]. Thus, all the disturbances in our paper are deterministic.
Consider only the case of small angular deviations (no more than several degrees). Equations (0.1) are replaced by the follwing ones: Here, u i = Ω i − Ω i 1 , i ∈ 1 : 3. In the linear approximation, the differences of accelerometer readings in (0.3) are equal to Equations (0.6) contain the multiplications of controls and state variables, but, in the case of small angles and angular velocities these terms may be neglected. In the specific case of movement on the equator under condition θ 1 = θ 2 ≡ 0, we assume θ = θ 3 as shown on Fig. 3. The angular velocity Ω 1 = Ω 3 1 = 0 under given movement and the rest projections of absolute angular velocity are equal to zero. We haveθ where the first equation from (0.7) is taken as the output.

Set-membership background
So, we consider a determinate n-dimensional linear system of the forṁ assuming that the initial state x 0 of system (1.1) is completely unknown, the matrices A(t), B(t), C(t), and G(t) below are continuous. In comparison with equations (0.6), the term with disturbance v in where the symbol |x| 2 P equals x ′ P x, prime ′ means the transposition, Q(t), R(t) are symmetrical, positive-defined, and continuous matrices having suitable dimension. Constraint (1.3) involves that the elements of vector functions v(·) and w(·) belong to the space L 2 [0, T ]. We need the following where X(t, s) is the fundamental matrix of system (1.1).
We use piecewise-constant functions u(t), for which where P is a compact convex set. Constraint (1.4) is more realistic than integral constraints in [9]. The aim of the control is to minimize the terminal function |Dx(T )|, where | · | is the Euclidean norm and D ∈ R d×n is a matrix. The choice of uncertain parameters {x 0 , v(·), w(·)} may impede the minimization.

Informational and compatible sets
At first, let us consider a set-membership estimation scheme for system (1.1), (1.2) under constraint (1.3).
To describe the informational set, we introduce the Bellman function The Bellman equation for V (t, x) is of the form: If the solution of equation (1.5) in any sense is found, the informational set X(t, y, u) is written as the inequality where P (t) is a positive definite and continuously differentiable matrix, d(t) and g(t) are a continuously differentiable vector function and a function respectively. Substituting (1.6) into (1.5), we get Therefore, the parameters of (1.6) must satisfy the equationṡ It is known [10] that the matrix P (t) is non-singular for any t > 0 under Assumption 1. Then the ellipsoid (informational set) is bounded for any t > 0 with the centerx(t) = P −1 (t)d(t) and the function h(t) = g(t)−|d(t)| 2 P −1 (t) . Differentiating thex(t) and h(t), we obtain the equationṡ (1.9) Let us introduce the function that is similar to the innovation process in theory of Kalman filtering [10].
On the other hand, let the instant τ ∈ (0, T ) and f (·) be any function from L m Then we obtain Here, we set Note that the sets X(t, y, u) and V(t, y, u) depend only on y t (·) and u t (·). Suppose that we have the compatible set V(t, y, u), and on the interval [t, s] a signal y s t (·) and a control u s t (·) are realized. Similarly to Definitions 1 and 2, we can define the sets X(s, y s t , u s t | V(t, y, u)) and V(s, y s t , u s t | V(t, y, u)). The following assertion seems to be obvious.

Lemma 2.
The relation between compatible and information sets is given by the equality X(t, y, u) = proj R n V(t, y, u). The compatible set is described by the formula T ] when t ∈ (0, T ). Moreover, compatible sets posses the semigroup property: V(s, y s t , u s t | V(t, y, u)) = V(s, y, u), where 0 < t < s ≤ T . As a consequence, we have X(s, y s t , u s t | V(t, y, u)) = X(s, y, u). The final reachable set of system (1.1) from the compatible set V (t, y, u) is denoted further by X T (u t |V(t, y, u)). This set consists of all vectors x(T ) under searching in (1.12) for the set V (t, y, u) with w t = 0.

Problems formulation
Let λ : 0 < t 1 < · · · < t N +1 = T be a partition of the interval [0, T ]. The times t i are called the instants of control correction. It is easily seen that the compatible set V(t, y, u) depends only on the pair (x(t), h(t)) which is called the position at the instant t. The transition between two adjacent positions (x(t i ), h(t i )) and (x(t i+1 ), h(t i+1 )) depends on the control u i (·) and the innovation function f i (·) on the interval [t i , t i+1 ) according to Lemma 1. Consider two problems.
. Otherwise, we pass to the control u i+1 * i that minimizes value (2.5). Of course, the controls may be not unique. If so, we choose any minimizers.

Minimax solutions
For brevity we denotex(t i ) =x i and h(t i ) = h i . Introduce the function of next losses for Problem 1. It is easily seen that the functions W i (x i , h i ) satisfy the following recurrent relations

Relations (3.1) have the boundary condition
Consider the last stage of relations (3.1) when i = N . Using boundary condition, we obtain Here, the term with integral must be replaced on where A ≥ 0, B ≥ 0, and the maximum is achieved at r * = (1 − h N )A 2 (A 2 + B 2 ) −1/2 . Besides, the optimization over f (·) is fulfilled under the constraint T t N |f (s)| 2 R(s) ds = k. If λ(s) is the maximal eigenvalue of the matrix DP (T, s)D ′ , we use the fact that conc|l| Q on unite ball is equal to λ max (1 − |l| 2 ) + |l| 2 Q 1/2 , see [7]. Hereinafter, the symbol concϕ(l) means a minimal concave function majorizing ϕ(l) on unite ball. At last, we apply the minimax theorem.
Continuing calculations on the subsequent stages, we come to the conclusion.
Theorem 1 (Conditions of the optimality in Problem 1). On the stage i, we have Here The optimal controls necessarily satisfy the relation To solve Problem 2, we need to calculate values (2.4), (2.5). Doing as above we get Theorem 2 (Properties of controls in Problem 2). The control procedure in Problem 2 begins from i = 1 and leads to a sequence of positions, where j 1 (y) ≥ j 2 (y) ≥ · · · ≥ j N (y). P r o o f. Let us compare the values j i (y) and j i+1 (y). If J i (y, u T * i ) < j i (y), we get j i (y) > j i+1 (y). Otherwise, we use the control u i+1 * i that minimizes the value J i (y, u i ). Therefore, as λ(t i+1 ) ≤ λ(t i ). The last inequality implies the relation whence the norm of the matrix P (T, s) decreases on s.
Remark 3. The procedure of calculation of optimal controls in Problem 1 is more difficult than in Problem 2. But we can simplify it if by a slight increase of the function of future losses. Namely, we have W i (x i , h i ) ≤ j i (y). This inequality follows by induction from relations (3.3)-(3.5). One can find the controls in this simplified procedure by formulas (3.4).
To illustrate the different approaches to optimal control, consider a simple Example. Given the one-dimensional systemẋ = u + v, 0 ≤ t ≤ 3, with the measurement y(t) = x(t) + w and the constraints 3]. Let t 1 = 1, t 2 = 2 be two correction instants. Here, we add the limitation on initial state for simplicity. We have P Here, the choice of control is not unique. At the next stage W 2 (x 2 , h 2 ) = 1.0091 and the optimal control on [2, 3] equals u 2 = −x 2 = e −2 − 1/2 = −0.3647. In Problem 2, we have j 1 (y) = 1.3050 and we obtain the same sequence of optimal controls. At last, consider the partition of [1,3] with step 0.25, N = 8, and we use the procedure of Remark 3. This procedure leads us to the sequence of control u i = −0.5 at each step. The final value of the functional equals 0.7577.

Numerical simulation of alignment process
We restrict ourself by the consideration of the simple case of system (0.8) and the procedure of Remark 3. The qualitative sense does not change in the common case.

Conclusion
In this paper, we consider the application of motion correction methods to the alignment problem in inertial navigation. We use the deterministic approach with set-membership description of uncertainty. The Theorems 1, 2 and the procedure in Remark 3 are new. The investigation of the influence of ship movement on the accuracy of alignment was not performed. It will be done in subsequent papers.