Constrained Target Motion Modeling Part I: Straight Line Track

Transcription

1 Constrained Target Motion Modeling Part I: Straight Line Trac Zhansheng Duan Center for Information Engineering Science Research Xi an Jiaotong University Xi an, Shaanxi 749, China X. Rong Li Department of Electrical Engineering University of New Orleans New Orleans, LA 748, USA Abstract Straight line motion is one of the most fundamental target motions. Its modeling has been well studied for unconstrained targets, e.g., air targets. However, the existing straight line motion models can not be directly used for the constrained case on straight line tracs, which has wide application, e.g., in ground target tracing. In this paper, modeling of the constrained target motion on a straight line trac is considered. First, the constraints imposed by the straight trac are explicitly set up. Then both direct elimination and projection along-trac are applied to obtain two forms of constrained motion models in the D Cartesian plane and 3D Cartesian space. The connections between these two forms are studied. It is found that the traditional linear Gaussian assumption is still valid for the nearly constant velocity models, the nearly constant acceleration models and the Singer model. Inspired by this, a general condition under which the traditional linear Gaussian assumption is valid for modeling of constrained motion on a straight trac is also discussed. Keywords: Unconstrained system, constrained system, straight line trac, target motion modeling. I. INTRODUCTION The state of many dynamic systems evolves subject to some equality constraints. For example, in ground target tracing 3, if we treat roads as curves without width, the road networs can then be described by equality constraints. In an airport, an aircraft moves on the runways or taxiways. In the quaternion-based attitude estimation problem, the attitude vector must have a unit norm 4. In a compartmental model with zero net inflow 5, mass is conserved. In undamped mechanical systems, such as one with Hamiltonian dynamics, the energy conservation law holds. Liewise, in electric circuit analysis, Kirchhoff s current and voltage laws hold. There are many types of research problems concerning equality constrained dynamic systems. Two of them are more concerned with estimation. In the first type, the goal is to develop state estimation algorithms for a given constrained system. This is the focus of most existing wor (see, e.g., 6 5). In the second type, the goal is to design and analyze equality constrained dynamic systems, that is, to study how to specify building blocs of the system so that it can be Research supported in part by National 973 project of China through grant 3CB3945, National Natural Science Foundation of China through grant 67438, NASA/LEQSF(3-5)-Phase3-6 through grant NNX3AD9A, ONR-DEPSCoR through grant N and the Fundamental Research Funds for the Central Universities of China. guaranteed that the system state satisfies the constraints. This type of wor is scarce only and 6 are nown to us. For linear equality constrained (LEC) state estimation, numerous results are available 6. For example, the dimensionality reduction method equivalently converts a constrained state estimation problem to a reduced dimensional unconstrained one. The equivalence can be achieved by representing part of the state vector as a linear function of the remaining part maing use of the deterministic linear equality constraint 7. It can also be achieved through null space decomposition as in 8. Another popular method, the projection method 6, 9,, projects the unconstrained estimate onto the constraint subspace by applying classical constrained optimization techniques. Specifically in 6, after the unconstrained estimate has been obtained, the problem is formulated as one of weighted least-squares estimation, in which the unconstrained estimate is treated as data and the inverse of its error covariance matrix is used as a weighting matrix. The pseudo measurement method 3 5 has also been applied to equality constrained state estimation. Its ey idea is to treat the equality constraints as noise-free pseudo measurements. Thus the LEC state estimation problem is converted to a regular filtering problem with two types of measurements. For nonlinear equality constrained state estimation, by the Taylor series expansion (TSE) based linearization, the result of 7 was claimed to be approximately generalized maximum lielihood-optimal, and 6 extended their LEC state estimation results to the case with nonlinear equality constraints. The second-order TSE was utilized in 3, to obtain better estimation results. To guarantee that the mean (i.e., estimate) of the constrained distribution also satisfies the nonlinear EC, 9 proposed a two-step projection strategy. It is important to now how to design LEC systems and analyze their properties. For example, when evaluating performance of state estimation algorithms for LEC systems, how can we generate the ground truth for an LEC system which meets the assumption in the estimator development? When applying developed LEC state estimators, how to model the LEC system is critically important as well. The ey idea of the design and analysis for an LEC linear system in and 6 is to convert an unconstrained system to a constrained one. For example, the conversion is done in by premultiplying an unconstrained system by

2 an orthogonal projection matrix. The conversion-based design has the advantage that once we are given an unconstrained system, we can find a dynamic system which satisfies the required equality constraint. And this resultant LEC system can serve the performance evaluation purpose. The disadvantage is that the resultant LEC dynamic system is not necessarily the one we expected for the applications we are dealing with. For conversion-based design, to achieve an expected LEC dynamic system, to now what ind of unconstrained system should be designed is not necessarily easy in general. Inspired by the direct elimination method for linear least squares parameter estimation subject to linear equality constraints, a systematic way to design and analyze LEC linear systems was proposed in 7. The difference is that before design and analysis, the desired model class is given. That is, the state transition matrix, the deterministic input transition matrix, and the deterministic input are given. What needs to be designed is just the distribution of the initial state and process noise and possibly their cross-correlation. It was also found that the existing formulations of linear equality constrained linear systems only cover a small part of the whole class. Straight line motion modeling for unconstrained targets, e.g, air targets, has been well studied. A lot of models are available, e.g, the nearly constant velocity model, the nearly constant acceleration model, and the Singer model. However, the straight line motion may be limited to certain straight line tracs, e.g., in ground target tracing. For such cases, the existing straight line motion models are not ready to be directly used because their tracs are either different from realization to realization or may not even be a straight line. In this paper, we consider modeling of the constrained target motion on a straight line trac in both the D Cartesian plane and the 3D Cartesian space. First, the linear equality constraints imposed by the straight line trac are determined. Second, the direct elimination based LEC system design technique is applied to obtain a form of the constrained target motion model. An along-trac motion projection based design technique is also applied to get another form of the constrained target motion model. Connections between the two forms are discussed in detail. Besides the nearly constant velocity model, the nearly constant acceleration model and the Singer model are considered as the desired model classes. It is found that for all three constrained motion models, the traditional linear Gaussian assumptions are still valid. To generalize, the conditions under which the traditional linear Gaussian assumption is valid for constrained motion modeling on a straight line trac is also discussed. Turing motion is also a very fundamental target motion. As a continuation of the wor here, modeling of the constrained target motion on a circular trac is discussed in part II. This paper is organized as follows. Sec. II summarizes the LEC system design techniques based on direct elimination. Sec. III presents a constrained nearly constant velocity model on a straight line trac in the D Cartesian coordinate plane. Sec. IV presents a constrained nearly constant acceleration model and a Singer model on a straight line trac also in the D Cartesian coordinate plane. Sec. V presents constrained motion models on a straight line trac in the 3D Cartesian space. Sec. VI gives conclusions. II. SUMMARY OF LEC SYSTEM DESIGN BASED ON DIRECT ELIMINATION The goal of LEC system design is to construct a linear stochastic system of the form X + = F X +G u +w so that at any time, all realizations of its state satisfy the following linear equality constraint C X = d () X R n, u R nu, and C R m n and d R m are both nown completely, C is of full row ran, m < n. To achieve the above goal, the direct elimination based design 7 can be applied. It assumes that we are given a desired class of models for system evolution, represented by F, G, u, and we only need to design the distribution of X and {w } and possibly their cross-correlation. The ey to this design is to partition the desired constraint equation as X C C X = d without loss of generality, it is assumed that X has been reshuffled such that C is nonsingular and C = C C, X = (X ) (X ) Accordingly, F, G and w are partitioned into F F = F G F F, G = w G, w = w Then, the design procedure can be decomposed into the following two steps. First, design an unconstrained dynamic sub-system according to X + = (F F A )X +F (C ) d +G u +w A = (C ) C and the distributions of X and w can be arbitrarily specified. Second, determine the remaing part according to X = (C ) d A X () w = D (G +A +G )u B X A +w (3) D = (C+) d + (F (C) +A + F (C) )d B = A + (F F A )+F F A A linear stochastic system designed this way is always guaranteed to satisfy the desired LEC (). It can be clearly seen that the ey to this design is to find the linear relationships () between X and X, and (3) between w and w. This method is used below to design target motion models on a straight line trac.

3 III. CV MODELS IN D PLANE Let the slope and y-intercept of the desired straight line trac in the D Cartesian plane be a and b, respectively. Then its equation can be easily written as y = ax+b Let x(t) and y(t) be the target position along x and y axes in continuous time. Then the constraint imposed on the position components by the straight line trac is y(t) = ax(t)+b and if the velocity components are also involved, the constraint is ẏ(t) = aẋ(t) Correspondingly, in a matrix form the constraints in discrete time are C X = d (4) C = a a X = x ẋ y ẏ b, d = If we desire that the vehicle shall tae a constant velocity motion on the straight line trac, probably we can use the nearly constant velocity (NCV) motion model in the D Cartesian plane X + = F X +w (5) F = diag{f CV,F CV }, F CV = w = w x wẋ w y T However, it is easy to see that arbitrarily specified distributions of X and {w } can not guarantee that LEC (4) holds. As discussed in 5, 7, their statistical characterization (e.g, the moments) must have some special structure if the LEC (4) is desired. In the following, we discuss how to disclose these underlying structures. A. D Constrained NCV Form Given the desired model class, the D NCV model (5), and the desired LEC (4), for the direct elimination based design, we have C = ai, C = I X = x ẋ, X = y ẏ w = w x wẋ, w = w y, u = nu F = F =, F = F = F CV I n is an n n identity matrix, n is an n zero vector, and n n is an n n zero matrix. Then, A = a I, D =, B = and X = a (X +d ) w = a w (6) Remar : Since both D and B are zero matrices, if w is a zero-mean white noise sequence, it is guaranteed that w is also zero-mean white and uncorrelated with the system state. This means that the linear Gaussian assumption about the linear system is still valid when the NCV model is constrained on the straight line trac. Suppose that X N( X,C X ) Then for the initial state X, we have X N( X,P ) X = a ( X +d ) ( X ) P = a C X a C X C X a C X It can be easily seen that P is singular. That is, X follows a singular Gaussian distribution. For the process noisew, there are two popular forms in the literature. One is used in the discrete white noise acceleration (DWNA) model, and the other is used in the continuous white noise acceleration (CWNA) model. In the DWNA model, w = G CV w ax, w = G CV w ay G CV = T / T and w ax and way are the white noise acceleration along x and y axes, respectively. From the general relationship (6) between the process noise along the two axes for the constrained NCV motion, it follows that for the constrained DWNA model w ax = a way which agrees with the intuition. As a result, for the constrained DWNA model, we have (σ ay cov(w ) = Q DWNA w ) = a Q (σay w ) a Q (σ ay w ) Q (σw ay ) Q In the CWNA model, a Q = G CV (G CV ) w N(, q x Q) w N(, q y Q)

4 T Q = 3 /3 T / T / T and q x and q y are the process noise intensity along x and y axes, respectively. From the linear relationship (6) between w and w, it follows that for the constrained CWNA model q x Q = a qy Q which leads to q x = a qy This is intuitively correct since qx T a = q y T = qy T T T a T and q x T T and q y T T can be thought of as the acceleration along x and y axes, respectively. Correspondingly, for the constrained CWNA model, we have q y cov(w ) = Q CWNA = Q qy Q a a q y Q a q y Q B. D Constrained NCV Form Along the straight line trac y = ax+b, denote as s the traveled distance of the vehicle from the point (,b) at and as ṡ its change rate. Model the evolution of s and ṡ as S + = F CV S +w S (7) S = s ṡ, w S = ws wṡ That is, the traveled distance along trac and its change rate evolve according to the NCV model. Suppose that the angle between the straight line trac and the positive x axis is θ. Then x = s cosθ, y = s sinθ +b ẋ = ṡ cosθ, ẏ = ṡ sinθ which can be rewritten in the following matrix form X = T(θ)S +δ (8) T(θ) = I cosθ I sinθ, δ = b Similarly at +, we have x + = s + cosθ = (s +Tṡ +w s )cosθ = x +Tẋ +w s cosθ ẋ + = ṡ + cosθ = (ṡ +wṡ )cosθ = ẋ +wṡ cosθ y + = s + sinθ+b = (s +Tṡ +w s )sinθ+b = y +Tẏ +w s sinθ ẏ + = ṡ + sinθ = (ṡ +wṡ )sinθ = ẏ +wṡ sinθ They can be rewritten in a matrix form as X + = diag{f CV,F CV }X +T(θ)w S (9) Remar : By comparing the two constrained NCV models (5) and (9) both represented in the D Cartesian coordinate plane, we can see that in (5), the process for the y axis can be arbitrarily designed and it controls the process noise for the x axis. In (9), however, the process noise for both x and y axes are controlled by arbitrarily designed process noise w S. Assume that Then, from (8) we have S N( S,P S ) X = T(θ) S +δ P P = S cos θ P S sinθcosθ X = S cosθ X b = S sinθ + P S sinθcosθ P S sin θ It can be easily seen that we have the following form: X = S cosθ sinθ sinθ = a (X +d ) which is in accordance with the first form (5) (6). By comparing the two D constrained NCV models (5) and (9), we can see that w = T(θ)w S which can be equivalently rewritten as w = w S cosθ, w = w S sinθ That is, (w x,wẋ ) and (wy, ) are projections of (ws,wṡ ) onto the x and y axes, respectively. Furthermore, we have w = cosθ ws sinθ sinθ = a w which is also in accordance with the first form (5). Assume that w S N(,QS ) Then Q = w N(,Q ) Q S cos θ Q S sinθcosθ Q S sinθcosθ QS sin θ Again, for the process noise part in (7), we may use two different popular models: the DWNA model and the CWNA model. Correspondingly, the second form of the D constrained NCV model (9) is slightly different. then If the DWNA model is used, that is, w S = G CV w as w = GCV w as cosθ, w = GCV w as sinθ which is Form for the DWNA case with w ax = was cosθ, way = was sinθ

5 That is, w ax and w ay are the projections of w as onto x and y axes, respectively. This agrees with the intuition. It can be easily seen that w ax = cosθ was sinθ sinθ = a way which agrees with the first form (5). Suppose that Then Q = (σ as (σ as w as N(,(σas w ) ) w ) Qcos θ w ) Qsinθcosθ It can be easily seen that w ) Qsinθcosθ w ) Qsin θ (σ as (σ as (σ ax w ) = (σ as w ) cos θ, (σ ay w ) = (σ as w ) sin θ Furthermore, we have (σw ax ) = (σw as ) cos θ sin θ sin θ = a (σay w ) (σw as ) sinθcosθ = (σw as ) cosθ sinθ sin θ = a (σay w ) which agree with the first form (5). then If the CWNA model is used, that is, Q = w S N(, q s Q) q s Qcos θ q s Qsinθcosθ q s Qsinθcosθ q s Qsin θ which is the first form for the CWNA case with It can be easily seen that q x = q s cos θ, q y = q s sin θ q x = q scos θ sin θ sin θ = a qy sinθcosθ q s = q scosθ sinθ sin θ = a qy which is in accordance with the first form (5). IV. CA AND SINGER MODELS IN D PLANE A nearly constant acceleration (NCA) model is represented by (5) with F = diag{f CA,F CA } () F CA = T T / T And for the Singer model representation of (5), F Singer = F = diag{f Singer,F Singer } () T (αt +e αt )/α ( e αt )/α e αt and α is the reciprocal of the maneuver time constant. For both cases, X = x ẋ ẍ y ẏ ÿ Note that the acceleration components along bothx and y axes are introduced. Thus the constraint imposed by the straight line trac is C X = d () C = a a a, d = b Given the desired model class, the D NCA model () or the Singer model (), and the desired LEC (), for the direct elimination based design, we have Then and C = ai 3, C = I 3 X = x ẋ ẍ, X = y ẏ ÿ w = wx wẋ wẍ, w = wy u =, F = F = 3 3 F = F = F CA or F = F = F Singer A = a I 3, D = 3, B = 3 3 X = a (X +d ) w = a w wÿ In the above, although F CA differs from F Singer, for both cases we have D = 3, B = 3 3 (3) This leads to w = a w and maes the traditional linear Gaussian assumption valid, which greatly simplifies the development of the corresponding state estimation. So (3) is a desired property for LEC system design. It is then of interest to now that under what conditions (3) holds. This is discussed next. To mae the following discussion easier, let us assume F = F = 3 3, i.e., no coupling between the x axis and the y axis when the state transits. This assumption is quite natural and widely used in many target motion models. It follows that which means that if and only if B = A + F F A = a (F F ) B = 3 3 F = F

6 From F = F = 3 3, it also follows that D = (C+ ) d + F (C ) d = (I F )(C ) d = F (,) F (,) F (,3) F (,) F (,) F (,3) F (3,) F (3,) F (3,3) b/a which means that if and only if or D = 3 b = F (:,) = for case, and λ C = y d = y x y z for case, and λ C = y d = z x z y for case 3, and = x x, = y y, = z z V. MOTION MODELS IN 3D SPACE In the above, we discussed the constrained target motion modeling on a straight line trac in the D Cartesian plane. Modeling in the 3D Cartesian space is also of interest in many applications. This is discussed next. The equation of a straight line in the 3D Cartesian space can be written as x x x x = y y y y = z z z z (x,y,z ) and (x,y,z ) are two arbitrary but different points on the straight line. Then, cosα = x x, cosβ = y y, cosγ = z z d d d α, β, and γ are the angles between the straight line and positive x-axis, y-axis, and z-axis, respectively, and d = (x x )+(y y ) +(z z ) Let x(t), y(t), and z(t) be the target position along x, y, and z axes in continuous time. Then the constraint imposed on the position components by the 3D straight line trac is x(t) x x x = y(t) y y y = z(t) z z z and if the velocity components are also involved, the constraint on them is ẋ(t) x x = ẏ(t) y y = ż(t) z z Correspondingly, in a matrix form the constraints in discrete time are C X = d (4) To guarantee C to be of full row ran, three equivalent cases can be obtained, X = x ẋ y ẏ z ż λ C = x d = x y x z Remar 3: Although the forms of the desired LEC (4) loo different in all three cases, they are equivalent in essence. Lie in the D case, let us first consider the case when the desired model class is the NCV motion model in the 3D Cartesian space X + = F X +w (5) F = diag{f CV,F CV,F CV } w = w x wẋ w y w z wż Similarly as in the D case, we can either use the direct elimination based design or the along-trac motion projection based design to obtain the 3D constrained motion models. This is discussed next. A. 3D Constrained NCV Form Given the desired model class, the 3D NCV model (5), and the desired LEC (4) in case, for the direct elimination based design, we have C =, C = X = x ẋ y ẏ, X = z ż w = w x wẋ w y, w = w z wż u =, F = 4, F = 4 F = diag{f CV,F CV }, F = F CV Then (C) = A = I I D = (I F )(C) d = 4 B = A + F F A = 4

7 and X = I I X + x λz z y λz z w = A + w = I B. 3D Constrained NCV Form I w Let s be the traveled distance of the vehicle along the straight line trac from the point (x,y,z ) and ṡ its change rate. Then, x = x +s cosα, y = y +s cosβ, z = z +s cosγ ẋ = ṡ cosα, ẏ = ṡ cosβ, ż = ṡ cosγ which can be rewritten in the following matrix form X = T(α,β,γ)S +δ (6) T(α,β,γ) = I cosα I cosβ I cosγ S = s ṡ, δ = x y z Similarly as in the D case, modeling the evolution of s and ṡ according to the NCV model as in (7) yields that at +, x + = x +s + cosα = x +(s +Tṡ +w s )cosα = x +Tẋ +w s cosα ẋ + = ṡ + cosα = (ṡ +wṡ)cosα = ẋ +wṡcosα y + = y +s + cosβ = y +(s +Tṡ +w s )cosβ = y +Tẏ +w s cosβ ẏ + = ṡ + cosβ = (ṡ +wṡ)cosβ = ẏ +wṡcosβ z + = z +s + cosγ = z +(s +Tṡ +w s )cosγ = z +Tż +w s cosγ ż + = ṡ + cosγ = (ṡ +wṡ )cosγ = ż +wṡ cosγ which can be rewritten in a matrix form as and w S = ws X + = F X +T(α,β,γ)w S (7) F = diag{f CV,F CV,F CV } wṡ can have an arbitrary distribution. Now let us see the connection between the above two 3D constrained NCV models (5) and (7). From (6), we have X = I cosα I cosβ S + x y X = (cosγ)i S + z Then it can be easily seen that X = I cosα cosγ + x y I cosβ cosγ (X z ) cosα cosβ = I cosγ I cosγ X + x cosα cosγ z y cosβ cosγ z = I I X + x λz z y λz z which agrees with the first form (5). By comparing the two constrained 3D NCV models (5) and (7), we can see that w = T(α,β,γ)w S which can be equivalently written as w = I cosα I cosβ w S w = (cosγ)i w S Furthermore, it follows that w = I cosα = I cosγ cosβ I cosγ w I w which also agrees with the first form (5). If the desired model class is the NCA motion or the Singer model, the corresponding 3D models can be obtained similarly as in the D case. VI. CONCLUSIONS Straight line motion modeling for unconstrained targets, e.g, air targets, has been well studied. A lot of models are available, e.g, the nearly constant velocity model, the nearly constant acceleration model, and the Singer model. However, the straight line motion may be limited to certain straight line tracs, e.g., in ground target tracing. For such cases, the existing straight line motion models are not ready to be directly used because their tracs are either different from realization to realization or may not even be a straight line. In this paper, we consider constrained target motion modeling on a straight line trac. The constraints imposed by the straight trac are established explicitly first. Then by using both the direct elimination and the along-trac motion projection, two closely related forms of the constrained models on a straight line trac are obtained. It is found that the traditional linear Gaussian assumption on target motion modeling, a desired property, is still valid for the constrained nearly constant velocity motion model, the constrained nearly constant acceleration motion model and the constrained Singer model. Inspired by this, general conditions under which the traditional linear Gaussian assumption holds for constrained motion modeling on a straight line trac are discussed. REFERENCES T. Kirubarajan, Y. Bar-Shalom, K. R. Pattipati, and I. Kadar, Ground target tracing with variable structure IMM estimator, IEEE Transactions on Aerospace and Electronic Systems, vol. 36, no., pp. 6 46, January. C. Yang, M. Baich, and E. Blasch, Nonlinear constrained tracing of targets on road, in Proceedings of 5 International Conference on Information Fusion, Philadelphia, PA, July 5-9 5, pp C. Yang and E. Blasch, Kalman filtering with nonlinear state constraints, IEEE Transactions on Aerospace and Electronic Systems, vol. 45, no., pp. 7 84, January 9. 4 J. L. Crassidis and F. L. Marley, Unscented filtering for spacecraft attitude estimation, AIAA Journal of Guidance, Control, and Dynamics, vol. 6, no. 4, pp , 3. 5 D. S. Bernstein and D. C. Hyland, Compartmental modelling and second-moment analysis of state space systems, SIAM Journal on Matrix Analysis and Applications, vol. 4, no. 3, pp. 88 9, 993.

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