Distributed Control of Unmanned Surface Vessels Using the Derivative-free Nonlinear Kalman Filter
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Abstract
Intelligence and autonomy is becoming a prerequisite for maritime transportation systems. In this paper a distributed control problem for unmanned surface vessels (USVs) is formulated as follows: there are N USVs which pursue another vessel (moving target). Each USV can be equipped with various sensors, such as IMU, cameras and non-imaging sensors such as sonar, radar and thermal signature sensors. At each time instant each USV can obtain measurements of the target’s cartesian coordinates. Additionally, each USV is aware of the target’s distance from a reference monitoring station (coastal or satellite monitoring units). The objective is to make the USVs converge in a synchronized manner towards the target, while avoiding collisions between them and avoiding collisions with obstacles in their motion plane. A distributed control law is developed for the USVs which enables not only convergence of the USVs to the goal position, but also makes possible to maintain the cohesion of the multi-USV system. Moreover, distributed filtering is performed, so as to obtain an estimate of the target vessel’s state vector. This provides the desirable state vector to be tracked by each one of the USVs. To this end, a new distributed nonlinear filtering method of improved accuracy and computation speed is introduced. This filtering approach, under the name Derivative-free distributed nonlinear Kalman filter is based on differential flatness theory and on an exact linearization of the target vessel’s dynamic/kinematic model. The performance of the proposed distributed filtering scheme is compared against the Extended and the Unscented Information filter.
Keywords
Distributed control Lyapunov stability Unmanned surface vessels Target tracking Distributed nonlinear Kalman filteringIntroduction
Distributed control of unmanned surface and underwater vessels has received significant attention during the last years [1, 2, 3, 4, 5, 6]. In this paper a solution is developed for the problem of distributed control of cooperating unmanned surface vessels (USVs) which pursue a target vessel. The distributed control aims at achieving the synchronized convergence of the USVs towards the target and at maintaining the cohesion of the USVs swarm, while also avoiding collisions between the individuals USVs and collisions between the USVs and obstacles in their motion plane. To estimate the motion characteristics of the target vessel, distributed filtering is performed. Actually, each vessel is supplied with equipment which permits to measure the coordinates of the target vessel, such as IMU and cameras, as well as with sonar, radar and thermal signature sensors. Besides each USV receives information about the distance of the target vessel from a reference surface, the latter being provided by a coastal or a satellite-based measurement unit. By fusing the aforementioned measurements through a filtering procedure an estimate of the state vector of the target vessel is obtained. Next, to obtain an estimate of improved precision about the motion characteristics of the target vessel distributed filtering is performed which fuses the distributed state vector estimates into one single estimate.
To treat the distributed control problem for the cooperating USVs a Lyapunov theory-based method is introduced. Motion planning for the individual USVs is determined by the minimization of a Lyapunov function which comprises a quadratic term associated with the distance of each USV from the target vessel, as well as quadratic terms which are associated with the distance of the USVs between each other. By applying LaSalle’s theorem it is proven that the USVs will follow the target’s motion while remaining within a small area that encircles the target.
To treat the distributed filtering and state estimation in the multi-USV system one can apply established methods for decentralized state estimation, such as the extended information filter (EIF) and the Unscented Information Filter (UIF). EIF stands for the distributed implementation of the Extended Kalman Filter while UIF stands for the distributed implementation of the Unscented Kalman Filter. Moreover, to obtain a distributed filtering scheme in this paper the Derivative-free Extended Information Filter (DEIF) is implemented. This stands for the distributed implementation of a differential flatness theory-based filtering method under the name Derivative-free distributed nonlinear Kalman Filter [7, 8, 9, 10, 11]. The improved performance of DEIF comparing to the EIF and UIF is confirmed both in terms of higher estimation accuracy and in terms of elevated speed of computation.
The structure of the paper is as follows: In “Target Tracking Mobile Sensor Networks” section the problem of tracking of a target ship by multiple USVs is formulated. In “Distributed Motion Planning for the Multi-USV System” section a solution is provided to the problem of distributed control and distributed motion planning of the system of the multiple USVs. In “Distributed State Eestimation Using the Extended Information Filter” section the EIF is proposed as a solution of the problem of distributed state estimation for the multi-USVs system. In “Distributed State Estimation Using the Unscented Information Filter” section the UIF is analyzed and proposed as an alternative solution for the problem of distributed state estimation for the multi-USVs model. In “Filtering Using Differential Flatness Theory and Canonical Forms” section differential flatness theory and diffeomorphisms lead ing to canonical state space forms are used to develop a distributed filtering method of improved performance, under the name Derivative-free Extended Information filtering (DEIF). In “Simulation Tests” section simulation tests are provided to confirm the stability of the distributed control method for tracking of the target-ship by the fleet of the multiple UAVs and also to confirm the improved performance of DEIF against the EIF and UIF. Finally, in “Conclusions” section concluding remarks are stated.
Target Tracking in Mobile Sensor Networks
The Problem of Distributed Target Tracking
where \(x{\in }R^{m{\times }1}\) is the target’s state vector and \(z{\in }R^{p{\times }1}\) is the measured output, while w(k) and v(k) are uncorrelated, zero-mean, Gaussian zero-mean noise processes with covariance matrices Q(k) and R(k) respectively. The operators \(\phi (x)\) and \(\gamma (x)\) are defined as \(\phi (x)=[{\phi _1}(x),{\phi _2}(x),\ldots ,{\phi _m}(x)]^T\), and \(\gamma (x)=[{\gamma _1}(x),{\gamma _2}(x),\ldots ,{\gamma _p}(x)]^T\), respectively.
The exact position and orientation of the target can be obtained through distributed filtering. Actually, distributed filtering provides a two-level fusion of the distributed sensor measurements. At the first level, local filters running at each USV provide an estimate of the target’s state vector by fusing the cartesian coordinates of the target with the target’s distance from a reference surface which is measured in an inertial coordinates system [14]. At a second level, fusion of the local estimates is performed with the use of the EIF and the UIF. It is also assumed that the time taken for communication between USVs is small, and that time delays, packet losses and out-of-sequence measurement problems in communication do not distort significantly the flow of the exchanged data.
Comparing to the traditional centralized or hierarchical fusion architecture, the network-centric architectures for the considered multi-USV system has the following advantages: (i) Scalability: since there are no limits imposed by centralized computation bottlenecks or lack of communication bandwidth, every USV can easily join or quit the system, (ii) Robustness: in a decentralized fusion architecture no element of the system is mission-critical, so that the system is survivable in the event of on-line loss of part of its partial entities (USVs), (iii) Modularity: every partial entity is coordinated and does not need to possess a global knowledge of the network topology. However, these benefits are possible only if the sensor data can be fused and synthesized for distribution within the constraints of the available bandwidth.
Tracking of the Reference Path by the Target
Theorem
Distributed Motion Planning for the Multi-USV System
Kinematic Model of the Multi-USV System
The objective is to lead the fleet of N USVs, with different initial positions on the 2-D plane, to converge to the target’s position, and at the same time to avoid collisions between the USVs, as well as collisions with obstacles in the motion plane. An approach for doing this is the potential fields theory, in which the individual USVs are steered towards an equilibrium by the gradient of an harmonic potential [7, 20, 21, 22]. Variances of this method use nonlinear anisotropic harmonic potential fields which introduce to the USVs’ motion directional and regional avoidance constraints [23, 24, 25, 26, 27, 28]. In the examined coordinated target-tracking problem the equilibrium is the target’s position, which is not a-priori known and has to be estimated with the use of distributed filtering.
- (i)
The cohesion of the USVs ensemble should be maintained, i.e. the norm \(||x^i-x^j||\) should remain upper bounded \(||x^i-x^j||<\epsilon ^h\),
- (ii)
Collisions between the USVs should be avoided, i.e. \(||x^i-x^j||>\epsilon ^l\),
- (iii)
Convergence to the target’s position should be succeeded for each USV through the negative definiteness of the associated Lyapunov function \(\dot{V}^i(x^i)={\dot{e}^i(t)}^T{e^i(t)}<0\), where \(e=x-x^{*} \) is the distance of the i-th USV from the target’s position.
Stability of the Multi-USV System
- (i)
\(lim_{t \rightarrow \infty } \bar{x}=x^{*}\), i.e. the center of the multi-USV system converges to the target’s position,
- (ii)
\(lim_{t \rightarrow \infty } {x^i}=\bar{x}\), i.e. the i-th USV converges to the center of the multi-USV system,
- (iii)
\(lim_{t \rightarrow \infty } \dot{\bar{x}}=\dot{x}^{*}\), i.e. the center of the multi-USV system stabilizes at the target’s position.
Convergence of the Mean of the Multi-USV System
- (i)The target is not moving, i.e. \(\dot{x}^{*}=0\). In that case Eq. (21) results in an homogeneous differential equation, the solution of which is given byKnowing that \(A>0\) results into \(lim_{t \rightarrow \infty }e_{\sigma }(t)=0\), thus \(lim_{t \rightarrow \infty }{\bar{x}}(t)=x^*\).$$\begin{aligned} \quad {\epsilon _{\sigma }(t)}={\epsilon _{\sigma }(0)}e^{-At} \end{aligned}$$(22)
- (ii)the target is moving at constant velocity, i.e. \(\dot{x^{*}}=a\), where \(a>0\) is a constant parameter. Then the error between the mean position of the multi-USV formation and the target becomeswhere the exponential term vanishes as \(t{\rightarrow }\infty \).$$\begin{aligned} \quad {\epsilon _{\sigma }(t)} = \left[ \epsilon _{\sigma }(0)+{a \over A}\right] e^{-At}-{a \over A} \end{aligned}$$(23)
- (iii)
the target’s velocity is described by a sinusoidal signal or a superposition of sinusoidal signals, as in the case of function approximation by Fourier series expansion. For instance consider the case that \(\dot{x^{*}}=b{\cdot }sin(at)\), where \(a,b>0\) are constant parameters. Then the nonhomogeneous differential equation Eq. (21) admits a sinusoidal solution. Therefore the distance \(\epsilon _{\sigma }(t)\) between the center of the multi-USV formation \(\bar{x}(t)\) and the target’s position \(x^{*}(t)\) will be also a bounded sinusoidal signal.
Convergence Analysis Using La Salle’s Theorem
From Eq. (17) it has been shown that each USV will stay in a cycle C of center \(\bar{x}\) and radius \(\epsilon \) given by Eq. (18). The Lyapunov function given by Eq. (14) is negative semi-definite, therefore asymptotic stability cannot be guaranteed. It remains to make precise the area of convergence of each USV in the cycle C of center \(\bar{x}\) and radius \(\epsilon \). To this end, La Salle’s theorem can be employed [29, 33] (Fig. 3).
La Salle’s Theorem
Assume the autonomous system \(\dot{x}=f(x)\) where \(f: D \rightarrow R^n\). Assume \(C \subset D\) a compact set which is positively invariant with respect to \(\dot{x}=f(x)\), i.e. if \(x(0) \in C \Rightarrow x(t) \in C \ \forall \ t\). Assume that \(V(x): D \rightarrow R\) is a continuous and differentiable Lyapunov function such that \(\dot{V}(x) \le 0\) for \(x \in C\), i.e. V(x) is negative semi-definite in C. Denote by E the set of all points in C such that \(\dot{V}(x)=0\). Denote by M the largest invariant set in E and its boundary by \(L^+\), i.e. for \(x(t) \in E: lim_{t \rightarrow \infty }x(t)=L^+\), or in other words \(L^+\) is the positive limit set of E. Then every solution \(x(t) \in C\) will converge to M as \(t \rightarrow \infty \).
Distributed State Estimation Using the Extended Information Filter
Extended Kalman Filtering at Local Processing Units
- Measurement update Acquire z(k) and compute:$$\begin{aligned} K(k)= & {} P^{-}(k)J^{T}_{\gamma }(\hat{x}^{-}(k)){\cdot }[J_{\gamma }(\hat{x}^{-}(k))P^{-}(k)J^T_{\gamma }(\hat{x}^{-}(k))\nonumber \\&+R(k)]^{-1}\nonumber \\ \hat{x}(k)= & {} \hat{x}^{-}(k)+K(k)[z(k)-{\gamma }(\hat{x}^{-}(k))] \nonumber \\ P(k)= & {} P^{-}(k)-K(k)J_{\gamma }(\hat{x}^{-}(k)){P^{-}(k)} \end{aligned}$$(31)
- Time update Compute:$$\begin{aligned} P^{-}(k+1)= & {} J_{\phi }(\hat{x}(k))P(k){J_{\phi }^T(\hat{x}(k))}+Q(k)\nonumber \\ \hat{x}^{-}(k+1)= & {} {\phi }(\hat{x}(k))+L(k)u(k) \end{aligned}$$(32)
Calculation of Local Estimations in Terms of EIF Information Contributions
As in the case of the extended Kalman filter the local filters which constitute the EIF can be written in terms of time update and a measurement update equation.
Extended Information Filtering for State Estimates Fusion
Distributed State Estimation Using the Unscented Information Filter
Unscented Kalman Filtering at Local Processing Units
- 1.
A set of weighted samples (sigma-points) are deterministically calculated using the mean and square-root decomposition of the covariance matrix of the system’s state vector. As a minimal requirement the sigma-point set must completely capture the first and second order moments of the prior random variable. Higher order moments can be captured at the cost of using more sigma-points.
- 2.
The sigma-points are propagated through the true nonlinear function using functional evaluations alone, i.e. no analytical derivatives are used, in order to generate a posterior sigma-point set.
- 3.
The posterior statistics are calculated (approximated) using tractable functions of the propagated sigma-points and weights. Typically, these take on the form of a simple weighted sample mean and covariance calculations of the posterior sigma points.
- 1)Denoting the current state mean as \(\hat{x}\), a set of \(2n+1\) sigma points is taken from the columns of the \(n{\times }n\) matrix \(\sqrt{(n+\lambda )P_{xx}}\) as follows:and the associate weights are computed:$$\begin{aligned} x^{0}= & {} \hat{x}\nonumber \\ x^{i}= & {} \hat{x}+\left[ \sqrt{(n+\lambda )P_{xx}}\right] _i, \quad i=1,\ldots ,n \nonumber \\ x^{i}= & {} \hat{x}-\left[ \sqrt{(n+\lambda )P_{xx}}\right] _i, \quad i=n+1,\ldots ,2n \end{aligned}$$(51)where \(i=1,2,\ldots ,2n\) and \(\lambda =\alpha ^2(n+\kappa )-n\) is a scaling parameter, while \(\alpha , \beta \) and \(\kappa \) are constant parameters. Matrix \(P_{xx}\) is the covariance matrix of the state x.$$\begin{aligned} W_0^{(m)}= & {} {\lambda \over {(n+\lambda )}} \quad W_0^{(c)}={\lambda \over {{(n+\lambda )}+(1-\alpha ^2+b)}} \nonumber \\ W_i^{(m)}= & {} {1 \over {2(n+\lambda )}}, \quad W_i^{(c)}={1 \over {2(n+\lambda )}}\nonumber \\ \end{aligned}$$(52)
- 2)Transform each of the sigma points as$$\begin{aligned} z^{i}=h(x^{i}) \quad i=0,\ldots ,2n \end{aligned}$$(53)
- 3)Mean and covariance estimates for z can be computed as$$\begin{aligned}&\hat{z}\,\simeq \, {\sum \limits _{i=0}^{2n}}{W_{i}^{(m)}}z^{i} \nonumber \\&P_{zz}={\sum \limits _{i=0}^{2n}}{W_{i}^{(c)}}(z^{i}-\hat{z})(z^{i}-\hat{z})^T \end{aligned}$$(54)
- 4)The cross-covariance of x and z is estimated asThe matrix square root of positive definite matrix \(P_{xx}\) means a matrix \(A=\sqrt{P_{xx}}\) such that \(P_{xx}=A{A^T}\) and a possible way for calculation is SVD.$$\begin{aligned} P_{xz}={\sum \limits _{i=0}^{2n}}{W_{i}^{(c)}}(x^{i}-\hat{x})(z^{i}-\hat{z})^T \end{aligned}$$(55)
As in the case of the extended Kalman filter, the UKF also consists of prediction stage (time update) and correction stage (measurement update) [44, 45].
Unscented Information Filtering
Calculation of Local Estimations in Terms of UIF Information Contributions
As in the case of the Unscented Kalman Filter, the UIF running at the i-th measurement processing unit can be written in terms of measurement update and time update equations
Distributed Unscented Information Filtering for State Estimates Fusion
Filtering Using Differential Theory and Canonical Forms
Conditions for Applying the Differential Flatness Theory
- (i)
Lie derivative: \({L_f}h(x)\) stands for the Lie derivative \({L_f}h(x)=({\nabla }h)f\) and the repeated Lie derivatives are recursively defined as \({L_f^0}h=h \ \ \text {for} \ i=0, {L_f^i}h={L_f}{{L_f^{i-1}}h}=\nabla {L_f^{i-1}h}f \ \ \text {for} \ i=1,2,\cdots \).
- (ii)
Lie Bracket: \({ad_f^i}g\) stands for a Lie Bracket which is defined recursively as \({ad_f^i}g=[f,{ad_f^{i-1}}g]\) with \({ad_f^0}g=g\) and \({ad_f}g=[f,g]={{\nabla }g}f-{{\nabla }f}g\).
Theorem
For nonlinear systems described by Eq. (83) the following variables are defined: (i) \(G_0=\text {span}[g_1,\ldots ,g_m]\), (ii) \(G_1=\text {span}[g_1,\ldots ,g_m,ad_f{g_1},\ldots ,ad_f{g_m}], \cdots \) (k) \(G_k=\text {span}\{{ad_f^j}{g_i} \ \text {for} \ 0{\le }j{\le }k, \ 1{\le }i{\le }m \}\). Then, the linearization problem for the system of Eq. (83) can be solved if and only if: (1). The dimension of \(G_i, \ i=1,\ldots ,k\) is constant for \(x{\in }X{\subseteq }R^n\) and for \(1{\le }i{\le }n-1\), (2). The dimension of \(G_{n-1}\) if of order n, (3). The distribution \(G_k\) is involutive for each \(1{\le }k{\le }{n-2}\).
Transformation of MIMO Systems into Canonical Forms
Canonical Forms for the USV Model
Derivative-free Extended Information Filtering
As mentioned above, for the system of Eq. (89), state estimation is possible by applying the standard Kalman Filter. The system is first turned into discrete-time form using common discretization methods and then the recursion of the linear Kalman Filter described in Eq. (90) and Eq. (91) is applied.
Simulation Tests
Estimation of Target’s Position with the use of the Extended Information Filter
The number of USVs used for target tracking in the simulation experiments was \(N=10\). However, since the USVs fleet (mobile sensor network) is modular a larger number of USVs could have been also considered. It is assumed that each USV can obtain an estimation of the target’s cartesian coordinates and bearing, i.e. the target’s cartesian coordinates [x, y] as well as the target’s orientation \(\theta \). To improve the accuracy of the target’s localization, the target’s coordinates and bearing are fused with the distance of the target from a reference surface measured in an inertial coordinates system (see Figs. 2, 9).
To obtain the Extended Kalman Filter (EKF), the kinematic model of the target described in Eq. (2) is discretized and written in the discrete-time state-space form of Eq.(25) [16, 46].
The tracking of the target by the fleet of the USVs was tested in the case of several reference trajectories, both for motion in an environment without obstacles as well as for motion in a plane containing obstacles. The proposed distributed filtering scheme enabled accurate estimation of the target’s state vector \([x,y,\theta ]^T\) through the fusion of the measurements of the target’s coordinates and orientation obtained by each USV with the measurement of the distance from a reference surface in an inertial coordinates frame. The state estimates provided by the Extended Kalman Filters running at each mobile sensor were fused into one single state estimate using Extended Information filtering. The aggregate estimated coordinates of the target \((\hat{x}^{*},\hat{y}^{*})\) provided the reference setpoint for the distributed motion planning algorithm. Each mobile sensor was made to move along the path defined by \((\hat{x}^{*},\hat{y}^{*})\). The convergence properties of the distributed motion planning algorithm were described in “Distributed Motion Planning for the Multi-USV System” section. The tracking of the target’s trajectory by the USVs as well as the accuracy of the two-level sensor fusion-based estimation of the target’s coordinates is depicted in Figs. 10, 11, 12, 13 and 14. The target is marked as a thick-line rectangle and the associated reference trajectory is plotted as a thick line.
It is noted that using distributed EKFs and fusion through the EIF is more robust comparing to the centralized EKF since (i) if a local processing unit is subject to a fault then state estimation is still possible and can be used for accurate localization of the target, as well as for tracking of target’s trajectory by the individual mobile sensors (unmanned surface vessels), (ii) communication overhead remains low even in the case of a large number of distributed mobile sensors, because the greatest part of state estimation procedure is performed locally and only information matrices and state vectors are communicated between the local processing units, (iii) the aggregation performed on the local EKF also compensates for deviations in state estimates of local filters (which can be due to linearization errors).
Estimation of Target’s Position with the use of Unscented Information Filtering
Next, the estimation of the target’s state vector was performed using the UIF. Again, the proposed distributed filtering enabled precise estimation of the target’s state vector \([x,y,\theta ]^T\) through the fusion of measurements of the target’s coordinates and bearing obtained by each mobile sensor with the distance of the target from a reference surface measured in an inertial coordinates system. The state estimates of the local Unscented Kalman Filters running at each mobile sensor (USV) were aggregated into a single estimation by the UIF. The estimated coordinates \([\hat{x}^{*},\hat{y}^{*}]\) of the target were used to generate the reference path which was followed by the mobile sensors. The tracking of the target’s trajectory by the USVs ensemble as well as the accuracy of the two-level sensor fusion-based estimation of the target’s position is shown in Figs. 15, 16, 17, 18 and 19.
Estimation of the Target’s Position with the Derivative-free Distributed Nonlinear Kalman Filter
The DEIF is also used to solve the problem of the synchronized USVs navigation based on distributed state estimation. In the latter case, local Derivative-free Kalman Filters perform fusion of the target’s coordinates measurements \((x_i,y_i)\) with the distance \(d_i\) of the target from a reference surface, as follows:
Indicative results about the accuracy of estimation provided by the considered nonlinear filtering algorithms (i.e. EIF, UIF and DEIF), as well as about the accuracy of tracking succeeded by the associated state estimation-based control loop are given in Table 1. It can be noticed that the DEIF is significantly more accurate and robust than the Extended Information Filter. Its accuracy is comparable to the one of the Unscented Information Filter. Results on the total runtime and the cycle time of the aforementioned distributed filtering algorithms are given in Table 2 (using the Matlab platform on a PC with an Intel i7 processor at 1.6GHz).
Conclusions
RMSE of tracking with nonlinear filtering (Gaussian noise)
\(\mathrm{{RMSE}}_x\) | \(\mathrm{{RMSE}}_y\) | \(\mathrm{{RMSE}}_{\theta }\) | |
---|---|---|---|
UIF | 0.0088 | 0.0104 | 0.0013 |
EIF | 0.0123 | 0.0167 | 0.0019 |
DEIF | 0.0087 | 0.0093 | 0.0013 |
Run time of nonlinear estimation algorithms
UIF | EIF | DEIF | |
---|---|---|---|
Total runtime (sec) | 203.97 | 181.04 | 162.65 |
Cycle time (sec) | 0.0410 | 0.0366 | 0.0325 |
Another major problem in the design of the proposed distributed controller for the USVs has been the localization of the target vessel and the estimation of its motion characteristics. To this end a new nonlinear distributed filtering approach under the name Derivative-free distributed nonlinear Kalman Filter has been introduced. Actually, this filter consists of fusion states estimates about the target’s motion provided by local nonlinear Kalman filters running on the individual USVs. Each one of these local Kalman filters is based on differential flatness-theory and on the transformation of the target’s kinematic model into a canonical state-space description. It has been demonstrated that comparing to the EIF and to UIF, the proposed nonlinear Kalman Filter has improved performance both in terms of accuracy of estimation and in terms of speed of computation. These parameters provide a significant advantage for the design of efficient multi-USVs systems for target tracking applications and intelligent maritime transportation systems.
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