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Abstract

This paper presents an approach to distributed state estimation-based control of nonlinear MIMO systems, capable of incorporating delayed measurements in the estimation algorithm while also being robust to packet losses. First, the paper examines the problem of distributed nonlinear filtering over a communication/sensors network, and the use of the estimated state vector in a control loop. As a possible filtering approach, an extended information filter (EIF) is proposed. The extended information filter requires the computation of Jacobians which in the case of high order nonlinear dynamical systems can be a cumbersome procedure, while it also introduces cumulative errors to the state estimation due to the approximative linearization performed in the Taylor series expansion of the system's nonlinear model. To overcome the aforementioned weaknesses of the extended information filter, a derivative-free approach to extended information filtering has been proposed. Distributed filtering is now based on a derivative-free implementation of Kalman filtering which is shown to be applicable to MIMO nonlinear dynamical systems. In the proposed derivative-free extended information filtering, the system is first subject to a linearization transformation that makes use of the differential flatness theory. It is shown how the proposed distributed filtering method can succeed in compensation of random delays and packet drops which may appear during the transmission of measurements and of state vector estimates, thus assuring a reliable performance of the distributed filtering-based control scheme. Evaluation tests are carried out on benchmark MIMO nonlinear systems, such as multi-DOF robotic manipulators.

Keywords

Distributed Filtering Derivative-Free Nonlinear Kalman Filtering State Estimation-Based Control Nonlinear MIMO Dynamical Systems Robotic Manipulators Networked Control Systems Communication Delays Packet Drops

1. Introduction

This paper presents an approach to distributed state estimation-based control of nonlinear MIMO systems, capable of incorporating delayed measurements in the estimation algorithm while also being robust to packet losses. First, the paper examines the problem of distributed nonlinear filtering over a communication/sensors network, and the use of the estimated state vector in a control loop. As a possible filtering approach, the extended information filter is proposed (Rigatos, 2011a). However, the extended information filter introduces numerical errors due to the approximative linearization of the system's nonlinear model performed by the local extended Kalman filters. To overcome the aforementioned weaknesses of the extended information filter, a derivative-free approach to extended information filtering has been proposed (Rigatos et al., 2009b), (Rigatos, 2010a), (Rigatos, 2011b). Distributed filtering is now based on a derivative-free implementation of Kalman filtering which is shown to be applicable to MIMO nonlinear dynamical systems (Rigatos, 2012a), (Rigatos, 2012b)], (Rigatos, 2011c), (Rigatos, 2011d). In the proposed derivative-free Kalman filtering method, the system is first subject to a linearization transformation that is based on the differential flatness theory (Rigatos, 2011e), (Villagra et al., 2007), (Fliess & Mounier, 1999), (Sira-Ramirez & Agrawal, 2004). Next state estimation is performed by applying the standard Kalman filter recursion to the linearized model. Unlike EIF, the proposed method provides estimates of the state vector of the nonlinear system without the need for derivatives and Jacobian calculation. By avoiding linearization approximations, the proposed filtering method improves the accuracy of estimation of the system state variables, and results in smooth control signal variations and in minimization of the tracking error of the associated control loop. This paper extends the results of (Rigatos, 2012a), (Rigatos, 2010a), (Rigatos, 2011d), (Rigatos, 2011e) towards a networked control scheme that is subject to random delays and measurement packets drops.

This paper proposes a method for the compensation of random delays and packet drops which may appear during the transmission of measurements and of state vector estimates, and which can cause the deterioration of the performance of the distributed filtering-based control scheme (Xia et al., 2009), (Schenato, 2007), (Schenato, 2008). Two cases are distinguished: (i) there are time delays and packet drops in the transmission of information between the distributed local filters and the master filter, (ii) there are time delays and packet drops in the transmission of information from distributed sensors to each one of the local filters. In the first case, the structure and calculations of the master filter for estimating the aggregate state vector remain unchanged. In the second case, the effect of the random delays and packets drops has to be taken into account in the redesign of the local Kalman filters, which implies a modified Riccati equation for the computation of the covariance matrix of the state vector estimation error, as well as the use of a correction (smoothing) term in the update of the state vector's estimate so as to compensate for delayed measurements arriving at the local Kalman filters. Finally, this paper shows that the aggregate state vector produced by a derivative-free extended information filter, suitably modified to compensate for communication delays and packet drops, can be used for sensorless control and for robotic visual servoing (see Fig. 3).

The structure of this paper is as follows: in Section 2 Kalman filtering for nonlinear dynamical systems is analysed in the form of the extended Kalman filter. In Section 3 the concept of differential flatness for nonlinear dynamical systems is explained. In Section 4 derivative-free nonlinear Kalman filtering for MIMO dynamical systems is analysed, in accordance to the previously presented principles of differential flatness theory. In Section 5 a derivative-free implementation of the extended information filter is introduced. In Section 6, distributed nonlinear filtering is proposed capable of compensating for the case of delays and losses in the transmission of sensor data to the controller. In Section 7 an application of the proposed distributed filtering method is presented in the case of robotic virtual servoing. Finally, in Section 8 concluding remarks are stated.

2. Extended Kalman filtering for nonlinear dynamical systems

In the discrete-time case, a dynamical system is assumed to be expressed in the form of a discrete-time state model (Rigatos 2009a), (Rigatos, 2011a), (Rigatos & Tzafestas, 2007a):

$\begin{array}{l} x (k + 1) = A (k) x (k) + L (k) u (k) + w (k) \\ z (k) = C x (k) + v (k) \end{array}$ (1)

It is assumed that the process noise w(k) and the measurement noise v(k) are uncorrelated. Now the problem of interest is to estimate the state x(k) based on the sequence of output measurements z(1),z(2),…,z(k). For the linear model of Eq. (1), the standard Kalman filter equations are

measurement update:

$\begin{array}{l} K (k) = P^{-} (k) C^{T} {[C \cdot P^{-} (k) C^{T} + R]}^{- 1} \\ \overset{⌢}{x} (k) = {\overset{⌢}{x}}^{-} (k) + K (k) [z (k) = C {\hat{x}}^{-} (k)] \\ P (k) = P^{-} (k) - K (k) C P^{-} (k) \end{array}$ (2)

time update:

$\begin{array}{l} P^{-} (k + 1) = A (k) P (k) A^{T} (k) + Q (k) \\ {\overset{⌢}{x}}^{-} (k + 1) = A (k) \overset{⌢}{x} (k) + L (k) u (k) \end{array}$ (3)

State estimation can be also performed for nonlinear dynamical systems using the extended Kalman filter recursion. The following nonlinear state model is considered (Rigatos, 2011a), (Rigatos & Tzafestas, 2007a):

$\begin{array}{l} x (k + 1) = φ (x (k)) + L (k) u (k) + w (k) \\ z (k) = γ (x (k)) + v (k) \end{array}$ (4)

where x ∊ R^mx1 is the system's state vector and z ∊ R^px1 is the system's output, while w(k) and v(k) are uncorrelated, zero-mean, Gaussian noise processes with covariance matrices Q(k) and R(k) respectively. Taking the Taylor series expansion of φ and γ, and computing the associated Jacobian matrices J_φ and J_γ one obtains the linearized ^ version_^ of ^ the system: $x (k + 1) = φ (\hat{x}, (k)) + J_{φ} (\hat{x} (k)) [x (k) - \hat{x} (k)] + w (k)$ and $z (k) = γ ({\hat{x}}^{-} (k)) + J_{γ} ({\hat{x}}^{-} (k)) [x (k) - {\hat{x}}^{-} (k)] + v (k) .$ Now, the EKF recursion is as follows:

Measurement update. Acquire z(k) and compute:

$\begin{array}{l} K (k) = P^{-} (k) J_{γ}^{T} ({\overset{⌢}{x}}^{-} (k)) \cdot {[J_{γ} ({\overset{⌢}{x}}^{-} (k)) P^{-} (k) J_{γ}^{T} ({\overset{⌢}{x}}^{-} (k)) + R (k)]}^{- 1} \\ \overset{⌢}{x} (k) = {\overset{⌢}{x}}^{-} (k) + K (k) [z (k) - γ ({\overset{⌢}{x}}^{-} (k))] \\ P (k) = P^{-} (k) - K (k) J_{γ} ({\overset{⌢}{x}}^{-} (k)) P^{-} (k) \end{array}$ (5)

Time update. Compute:

$\begin{array}{l} P^{-} (k + 1) = J_{φ} (\overset{⌢}{x} (k)) P (k) J_{φ}^{T} (\overset{⌢}{x} (k)) + Q (k) \\ {\overset{⌢}{x}}^{-} (k + 1) = φ (\overset{⌢}{x} (k)) + L (k) u (k) \end{array}$ (6)

3. Differential flatness for nonlinear dynamical systems

3.1 Definition of differentially flat systems

A system is said to be differentially flat if there is a collection of $m$ functions $y = (y_{1}, \dots, y_{m})$ of the system variables and of their time-derivatives, i.e., $y_{i} = φ (w, \dot{w}, \ddot{w}, \dots, w^{(α_{i})}), i = 1, \dots, m$ satisfying the following two conditions (Villagra et al., 2007), (Fliess & Mounier, 1999), (Bououden et al., 2011): (1) No differential relation of the form $R (y, \dot{y}, \dots, y^{(β)}) = 0$ exists, which implies that the derivatives of the flat output are not coupled in the sense of an ODE, or equivalently it can be said that the flat output is differentially independent, (2) all system variables (i.e., the elements of the system's state vector $w$ and the control input) can be expressed using only the flat output $y$ and its time derivatives $w_{i} = ψ_{i} (y, \dot{y}, \dots, y^{(γ_{i})}), i = 1, \dots, s .$

3.2 Conditions for applying the differential flatness theory

The generic class of nonlinear systems ẋ = f(x,u) is considered. Such a system can be transformed to the form of an affine in the input system by adding an integrator to each input (Bououden et al., 2011)

$\dot{x} = f (x) + \sum_{i = 1}^{m} g_{i} (x) u_{i}$ (7)

If the system of Eq. (7) can be linearized by a diffeomorphismz = φ(x) and a static state feedback u = α(x) + β(x) v into the following form

$\begin{array}{l} {\dot{z}}_{i, j} = z_{i + 1, j} for 1 \leq j \leq m and 1 \leq i \leq v_{j} - 1 \\ {\dot{z}}_{v_{i, j}} = v_{j} \end{array}$ (8)

with ∊^m_j=1 v_j = n, then y_j = z_1,j for 1 ≤ j ≤ m are the 0-flat outputs which can be written as functions of only the elements of the state vector x. To define conditions for transforming the system of Eq. (7) into the canonical form described in Eq. (8), the following theorem holds (Bououden et al., 2011)

Theorem: For the nonlinear systems described by Eq. (7), the following variables are defined: (i)G₀ = span [g₁,…,g_m], (ii) G₁ = span [g₁,….g_m, ad_f g₁,…,ad_f g_m],… (k)G_k = span {ad^j_f g_i for 0 ≤ j ≤ k,1 ≤ i ≤ m}. Then, the linearization problem for the system of Eq. (7) can be solved if and only if: (1) the dimension of G_i, i = 1,…,k is constant for x ε X Rⁿ and for 1 ≤ i ≤ n – 1, (2) the dimension of G_n-1 is of ordern, (3) the distribution G_k is involutive for each 1 ≤ k ≤ n – 2.

3.3 Transformation of MIMO nonlinear systems into the Brunovsky form

It is assumed now that after defining the flat outputs of the initial MIMO nonlinear system, and after expressing the system state variables and control inputs as functions of the flat output and of the associated derivatives, the system can be transformed in the Brunovsky canonical form (Marino, 1990), (Marino & Tomei, 1992):

$\begin{array}{l} {\dot{x}}_{1} = x_{2} \\ \dots \\ {\dot{x}}_{r_{1} - 1} = x_{r_{1}} \\ {\dot{x}}_{r_{1}} = f_{1} (x) + \sum_{j = 1}^{p} g_{1_{j}} (x) u_{j} + d_{1} \\ \\ {\dot{x}}_{r_{1} + 1} = x_{r_{1} + 2} \\ \dots \\ {\dot{x}}_{p - 1} = x_{p} \\ {\dot{x}}_{p} = f_{p} (x) + \sum_{j = 1}^{p} g_{p_{j}} (x) u_{j} + d_{p} \\ y_{1} = x_{1} \\ y_{p} = x_{n - r_{p} + 1} \end{array}$ (9)

wherex = [x₁,…,x_n]^T is the state vector of the transformed system (according to the differential flatness formulation), u = [u₁,…,u_p]^T is the set of control inputs, y = [y₁,…,y_p]^T is the output vector, f_i are the drift functions and g_i,j, i,j = 1,2,…, p are smooth functions corresponding to the control input gains, while d_j is a variable associated to external disturbances. It holds that r₁ + r₂ +… + r_p = n. Having written the initial nonlinear system into the canonical (Brunovsky) form it holds

$y_{i}^{(r_{i})} = f_{i} (x) + \sum_{j = 1}^{p} g_{i j} (x) u_{j} + d_{j}$ (10)

Next, the following vectors and matrices can be defined f(x) = [f₁ (x),…,f_n (x)]^T, g(x)=[g₁ (x) … g_p (x)] with g_i (x) = [g_1i (x) … g_pi (x)]^T and also A = diag[A₁,…,A_p],B = diag[B₁,…,B_p], C = diag[C₁,…,C_p], d = [d₁,…,d_p]^T, where matrix A has the MIMO canonical form, i.e., with block-diagonal elements

$\begin{array}{l} A_{i} = {(\begin{matrix} 0 & 1 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & \dots & ⋮ \\ 0 & 0 & \dots & 1 \\ 0 & 0 & \dots & 0 \end{matrix})}_{r_{i} \times r_{i}} \\ \\ B_{i}^{T} = {(\begin{matrix} 0 & 0 & \dots 0 & 1 \end{matrix})}_{1 \times r_{i}} \\ C_{i} = {(\begin{matrix} 1 & 0 & \dots 0 & 0 \end{matrix})}_{1 \times r_{i}} \end{array}$ (11)

Thus, Eq. (10) can be written in state-space form (Rigatos & Tzafestas, 2006)

$\begin{array}{l} \dot{x} = A x + B v + B \tilde{d} \\ y = C x \end{array}$ (12)

where the control input is written as v = f (x) + g(x)u.

4. Derivative-free Kalman filtering for MIMO nonlinear systems

The proposed method for derivative-free Kalman filtering for MIMO nonlinear systems will be analysed through an application example. A 2-DOF rigid link robotic manipulator is considered. The dynamic model of the robot is given by

$(\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) (\begin{matrix} {\ddot{θ}}_{1} \\ {\ddot{θ}}_{2} \end{matrix}) + (\begin{matrix} F_{1} (θ, \dot{θ}) \\ F_{2} (θ, \dot{θ}) \end{matrix}) + (\begin{matrix} G_{1} (θ) \\ G_{2} (θ) \end{matrix}) = (\begin{matrix} T_{1} \\ T_{2} \end{matrix})$ (13)

where $F_{i} (θ, \dot{θ}), i = 1, 2$ are the Coriolis-centripetal terms and G_i (θ), i = 1,2 are the gravitational terms. Denoting the inverse of the inertia matrix as

${(\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix})}^{- 1} = (\begin{matrix} N_{11} & N_{12} \\ N_{21} & N_{22} \end{matrix})$ (14)

and defining the following state variables $x_{1} = θ_{1}, x_{2} = {\dot{θ}}_{1}, x_{3} = θ_{2}, x_{4} = {\dot{θ}}_{2},$ it holds that ${\ddot{x}}_{1} = f_{1} (x) + g_{1} (x) u$ and ${\ddot{x}}_{3} = f_{2} (x) + g_{2} (x) u,$ where $\begin{array}{l} f_{1} (x) = - N_{11} F_{1} (θ, \dot{θ}) - N_{12} F_{2} (θ, \dot{θ}) - N_{11} G_{1} (θ) - \\ - N_{12} G_{2} (θ) \in R^{1 \times 1}, g_{1} (x) = [N_{11}, N_{12}] \in R^{1 \times 2} \end{array}$ and $\begin{array}{l} f_{2} (x) = - N_{21} F_{2} (θ, \dot{θ}) - N_{22} F_{2} (θ, \dot{θ}) - N_{21} G_{1} (θ) - \\ - N_{22} G_{2} (θ) \in R^{1 \times 1}, g_{2} (x) = [N_{21}, N_{22}] \in R^{2 \times 2} . \end{array}$ The flat output is defined as

$y = [θ_{1}, θ_{2}] = [x_{1}, x_{3}]$ (15)

It holds that ${\dot{x}}_{1} = x_{2}, {\dot{x}}_{2} = f_{1} (x) + g_{1} (x) u, {\dot{x}}_{3} = {\dot{x}}_{4} = f_{2} (x) + g_{2} (x) u,$ therefore all system state variables can be written as functions of the flat output $y$ and its derivatives. The same holds for the control input $u$ , i.e., $x_{1} = [\begin{matrix} 1 & 0 \end{matrix}] y^{T}, x_{2} = [\begin{matrix} 1 & 0 \end{matrix}] {\dot{y}}^{T}, x_{3} = [\begin{matrix} 0 & 1 \end{matrix}] y^{T}$ and $x_{2} = [\begin{matrix} 0 & 1 \end{matrix}] {\dot{y}}^{T} .$ Moreover, from the robot's nonlinear dynamics it holds

$\begin{array}{l} (\begin{matrix} {\ddot{x}}_{1} \\ {\ddot{x}}_{3} \end{matrix}) = (\begin{matrix} f_{1} (x) \\ f_{2} (x) \end{matrix}) + (\begin{matrix} g_{1} (x) \\ g_{2} (x) \end{matrix}) u i . e . \\ \\ u = {(\begin{matrix} g_{1} (x) \\ g_{2} (x) \end{matrix})}^{- 1} {(\begin{matrix} {\ddot{x}}_{1} \\ {\ddot{x}}_{3} \end{matrix}) - (\begin{matrix} f_{1} (x) \\ f_{2} (x) \end{matrix})} \end{array}$ (16)

Thus, knowing that x = h(y,ẏ) one finally obtains

$u = {(\begin{matrix} g_{1} (h (y, \dot{y})) \\ g_{2} (h (y, \dot{y})) \end{matrix})}^{- 1} {(\begin{matrix} [\begin{matrix} 1 & 0 \end{matrix}] {\ddot{y}}^{T} \\ [\begin{matrix} 0 & 1 \end{matrix}] {\ddot{y}}^{T} \end{matrix}) - (\begin{matrix} f_{1} (h (y, \dot{y})) \\ f_{2} (h (y, \dot{y})) \end{matrix})}$ (17)

Therefore, the above robotic system is a differentially flat one. Next, considering also the effects of additiv disturbances, the dynamic model becomes

$(\begin{matrix} {\ddot{x}}_{1} \\ {\ddot{x}}_{3} \end{matrix}) = (\begin{matrix} f_{1} (x, t) \\ f_{2} (x, t) \end{matrix}) + (\begin{matrix} g_{1} (x, t) \\ g_{2} (x, t) \end{matrix}) u + (\begin{matrix} {\tilde{d}}_{1} \\ {\tilde{d}}_{2} \end{matrix})$ (18)

Moreover, the following control input is defined

$u = {(\begin{matrix} g_{1} (\overset{⌢}{x}, t) \\ g_{2} (\overset{⌢}{x}, t) \end{matrix})}^{- 1} {(\begin{matrix} {\ddot{x}}_{1}^{d} \\ {\ddot{x}}_{3}^{d} \end{matrix}) - (\begin{matrix} f_{1} (\overset{⌢}{x}, t) \\ f_{2} (\overset{⌢}{x}, t) \end{matrix}) - (\begin{matrix} K_{1}^{T} \\ K_{2}^{T} \end{matrix}) e + (\begin{matrix} u_{c_{1}} \\ u_{c_{2}} \end{matrix})}$ (19)

where [u_c1 u_c2]^T is a supervisory control term that is used for the compensation of the model's uncertainties, as well as of the external disturbances, and K^T = [k₁ ⁱ,k₂ ⁱ,…,kⁱ_n=1,kⁱ_n] are the rows of the error feedback gain matrix (Rigatos and Tzafestas, 2006), (Rigatos and Tzafestas, 2007b). Finally, the differentially flat robotic model is written in the Brunovsky (canonical) form. Considering the state vector x ∊ R^4×1, with the previously defined state variables defined as x_i, i = 1 … 4, the following matrices are defined

$A = (\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}), B = (\begin{matrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{matrix}), C^{T} = (\begin{matrix} 1 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix})$ (20)

Using the matrices of Eq. (20), the MIMO robot model dynamics is written in a controller and observer canonical form, ẋ = Ax + Bv and y = Cx, where the new input v is given by

$v = (\begin{matrix} f_{1} (x, t) \\ f_{2} (x, t) \end{matrix}) + (\begin{matrix} g_{1} (x, t) \\ g_{2} (x, t) \end{matrix}) u + (\begin{matrix} {\tilde{d}}_{1} \\ {\tilde{d}}_{2} \end{matrix})$ (21)

The robotic model of the form of Eq. (12) can be written in discrete-time form after applying common discretization methods, and next state estimation can be performed using the standard Kalman filter recursion, as described in Eq. (2) and Eq. (3).

5. Distributed state estimation using the extended information filter

5.1 Fusing estimations from local distributed filters

Again, the discrete-time nonlinear system of Eq. (4) is considered. The EIF performs fusion of the local state vector estimates which are provided by the local extended Kalman filters, using the Information matrix and the information state vector (Lee, 2008), (Makarenko et al., 2006), (Nettleton et al., 2003), (Rao et al., 1991). The information matrix is the inverse of the state vector covariance matrix and can also be associated to the Fisher information matrix (Rigatos & Zhang, 2009). The information state vector is the product between the information matrix and the local state vector estimate

$\begin{array}{l} Y (k) = P^{- 1} (k) = I (k) \\ \overset{⌢}{y} (k) = P^{-} {(k)}^{- 1} \overset{⌢}{x} (k) = Y (k) \overset{⌢}{x} (k) \end{array}$ (22)

The outputs of the local filters are treated as measurements which are fed into the aggregation fusion filter (see Fig. 1) (Lee, 2008). Then each local filter is expressed by its respective error covariance and estimate in terms of information contributions, and is described by

Figure 1.

Fusion of the distributed state estimates with the use of the extended information filter

$\begin{array}{l} P_{i}^{- 1} (k) = P_{i}^{-} {(k)}^{- 1} + J_{γ}^{T} (k) R {(k)}^{- 1} J_{γ} (k) {\overset{⌢}{x}}_{i} (k) = \\ = P_{i} (k) (P_{i}^{-} {(k)}^{- 1} {\overset{⌢}{x}}_{i}^{-} (k)) + J_{γ}^{T} (k) R {(k)}^{- 1} \\ [z^{i} (k) - γ^{i} (x (k)) + J_{γ}^{i} (k) {\overset{⌢}{x}}_{i}^{-} (k)] \end{array}$ (23)

The global estimate and the associated error covariance for the aggregate fusion filter can be rewritten in terms of the computed estimates and covariances from the local filters using the relations

$\begin{array}{l} J_{γ}^{T} (k) R {(k)}^{- 1} J_{γ} (k) = P_{i} {(k)}^{- 1} - P_{i}^{-} {(k)}^{- 1} \\ J_{γ}^{T} (k) R {(k)}^{- 1} [z^{i} (k) - γ^{i} (x (k)) + J_{γ}^{i} (k) {\overset{⌢}{x}}^{-} (k)] = \\ = P_{i} {(k)}^{- 1} {\overset{⌢}{x}}_{i} (k) - P_{i} {(k)}^{- 1} {\overset{⌢}{x}}_{i} (k - 1) \end{array}$ (24)

For the general case of N local filters i = 1,…,N, the distributed filtering architecture is described by the following equations

$\begin{array}{l} P {(k)}^{- 1} = P^{-} {(k)}^{- 1} + \sum_{i = 1}^{N} [P_{i} {(k)}^{- 1} - P_{i}^{-} {(k)}^{- 1}] \\ \overset{⌢}{x} (k) = P (k) [P^{-} {(k)}^{- 1} {\overset{⌢}{x}}^{-} (k) + \sum_{i = 1}^{N} (P_{i} {(k)}^{- 1} {\overset{⌢}{x}}_{i} (k) - P_{i}^{-} {(k)}^{- 1} {\overset{⌢}{x}}_{i}^{-} (k))] \end{array}$ (25)

The global state update equation in the above distributed filter can be written in terms of the information state vector and of the information matrix, i.e.,

$\begin{array}{l} \overset{⌢}{y} (k) = {\overset{⌢}{y}}^{-} (k) + \sum_{i = 1}^{N} ({\overset{⌢}{y}}_{i} (k) - {\overset{⌢}{y}}_{i}^{-} (k)) \\ \overset{⌢}{Y} (k) = {\overset{⌢}{Y}}^{-} (k) + \sum_{i = 1}^{N} ({\overset{⌢}{Y}}_{i} (k) - {\overset{⌢}{Y}}_{i}^{-} (k)) \end{array}$ (26)

From Eq. (25) it can be seen that if a local filter (processing station) fails, then the local covariance matrices and the local state estimates provided by the rest of the filters will enable an accurate computation of the target's state vector.

5.2 Derivative-free extended information filtering

As mentioned above, for the system of Eq. (20), 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. (2) and Eq. (3) is applied.

If the derivative-free Kalman filter is used in place of the extended Kalman filter then in the EIF equations the following matrix substitutions should be performed: J_φ (k) → A_d,J_γ (k) → C_d, where matrices A_d and C_d are the discrete-time equivalents of matrices A and C which have been defined in Eq. (20) and which appear also in the measurement and time update of the standard Kalman filter recursion. Matrices A_d and C_d can be computed using established discretization methods. Moreover, the covariance matrices P(k) and P⁻ (k) are the ones obtained from the linear Kalman filter update equations given in section 2.

6. Distributed nonlinear filtering under random delays and packet drops

6.1 Networked Kalman filtering for an autonomous system

The structure of networked Kalman filtering is shown in Fig. 2.

Figure 2.

Distributed filtering over sensors network with communication delays and packet drops

The problem of distributed filtering becomes more complicated if random delays and packet drops affect the transmission of information between the sensors and local processing units (filters), or between the local filters and the master filter where the fused state estimate is computed. First, results on the stability of the networked linear Kalman filter will be presented (Xia et al., 2009). The general state-space form of a linear autonomous time-variant dynamical system is given by

$x (k) = A x (k - 1) + w (k, k - 1)$ (27)

where x(k) ∊ R^mx1 is the system's state vector, A ∊ R^nxn is the system's state transition matrix, and w(k, k – 1)is the white process noise between time instants k and k – 1. The sensor measurements are received starting at time instant k ≥ 1 and are described by the measurement equation

$z (k) = C x (k) + v (k)$ (28)

where C ∊ R^pxm,z(k) ∊ R^px1 and v(k) is the white measurement noise. Measurements z(k) are assumed to be transmitted over a communication channel. To denote the arrival or loss of a measurement to the local Kalman filter, through the communication network, one can use variable γ_k ∊ {0,1}, where 1 stands for successful delivery of the packet, while 0 stands for loss of the packet. Thus, in the case of packet losses, the discrete time Kalman filter recursion that was described in section 2 (measurement update) is modified as

$K (k) = _{γ k} P ^{-} (k) C^{T} {[C P^{-} (k) C^{T} + R]}^{- 1}$ (29)

where γ_k ∊ {0,1}. This modification implies that the value of the estimated state vector $\overset{⌢}{x} (k)$ remains unchanged if a packet drop occurs, i.e., when γ_k = 0.

6.2 Smoothing estimation in the case of delayed measurements

An issue which is associated to the implementation of such networked control systems is how to compensate for random delays and packet losses so as to enhance the accuracy of estimation and consequently to improve the stability of the control loop. The idea of incorporating delayed measurements within a Kalman filter framework is a possible solution for the compensation of network-induced delays and packet losses, and is also known as an update with out-of-sequence measurements (Bar-Shalom, 2007). The solution proposed in (Bar-Shalom, 2007) is optimal under the assumption that the delayed measurement is processed within the last sampling interval (one-step-lag problem). There have also been some attempts to extend these results to nonlinear state estimation (Golapalakhrishnan et al., 2011), (Jia et al, 2008). More recently there has been research effort in the redesign of distributed Kalman filtering algorithms for linear systems so as to eliminate the effects of delays in measurement transmissions and packet drops, while also alleviating the one-step-lag assumption (Xia et al., 2009).

The smoothing approach that will be presented in the following is according to the results given in (Xia et al., 2009). The smoothing approach considers that for a linear time-invariant non-autonomous system

$\begin{array}{l} x (k) = A x (k - 1) + B u (k - 1) + w (k - 1) \\ z (k) = C x (k) + v (k) \end{array}$ (30)

it holds

$\begin{array}{l} x (k) = A^{N} x (k - N) + \sum_{j = 1}^{N} A^{N - j} B u (k - N + j - 1) + \\ + \sum_{j = 1}^{N} A^{N - j} w (k - N + j - 1) \end{array}$ (31)

Denoting

$A^{N} = Φ (k, k - N)$ (32)

one has

$\begin{array}{l} x (k) = Φ (k, k - N) x (k - N) + \\ + \sum_{j = 1}^{N} | A^{N - j} B u (k - N + j - 1) + \\ \sum_{j = 1}^{N} A^{N - j} w (k - N + j - 1) \end{array}$ (33)

Setting

$\begin{array}{l} w_{1} (k, k - N) = \sum_{j = 1}^{N} A^{N - j} B u (k - N + j - 1) + \\ + \sum_{j = 1}^{N} A^{N - j} w (k - N + j - 1), \end{array}$ (34)

one has that x(k) can be written in a compact form as

$x (k) = Φ (k, k - N) x (k - N) + w_{1} (k, k - N)$ (35)

Using that matrix Φ (k,k – N) is invertible, one has

$\begin{array}{l} x (k - N) = Φ {(k, k - N)}^{- 1} x (k) - \\ - Φ {(k, k - N)}^{- 1} w_{1} (k, k - N) . \end{array}$ (36)

The following notation is used

$Φ_{1} (k - N, k) = - Φ {(k, k - N)}^{- 1}$ (37)

while for the retrodiction of w₁ (k,k – N), it holds

$w_{a} (k - N, k | k) = - Φ {(k, k - N)}^{- 1} w_{1} (k, k - N)$ (38)

Then, to smooth the state estimation at time instant k – N, using the measurement of output z_i k – N) received at time instant k, one has the state equation

$\overset{⌢}{x} (k - N, k) = Φ_{1} (k - N, k) \overset{⌢}{x} (k | k) + {\overset{⌢}{w}}_{a} (k - N, k | k)$ (39)

while the associated measurement equation becomes

$z (k - N) = C x (k - N) + v (k - N)$ (40)

Substituting the above equation for x(k – N) into the equation for z(k – N) provides

$z (k - N) = C Φ_{1} (k - N, k) x (k) + C w_{a} (k - N, k) + v (k - N)$ (41)

and the associated estimated-output at time instant k – N is

$\overset{⌢}{z} (k - N) = C Φ_{1} (k - N, k) \overset{⌢}{x} (k | k) + C {\overset{⌢}{w}}_{a} (k - N, k)$ (42)

From Eq. (42) the innovation for the delayed measurement can be obtained

$\tilde{z} (k - N) = z (k - N) - \overset{\land}{z} (k - N)$ (43)

i.e.

$\tilde{z} (k - N) = C Φ_{1} (k - N) \tilde{x} (k | k) + C {\tilde{w}}_{a} (k - N),$ (44)

where x(k | k) = x(k) – x(k | k) is the state estimation error and w˜ _a (k – N, k|k) = w _a (k – N, k) – w˜_a (k – N, k) is the estimation error for w_α. With this innovation the estimation of the state vector x(k|k) at time instant k is corrected. The correction (smoothing) relation is (Xia et al., 2009)

${\overset{⌢}{x}}^{*} (k | k) = \overset{⌢}{x} (k | k) + M \tilde{z} (k - N, k)$ (45)

Therefore, again the basic problem for the implementation of the smoothing relation provided by Eq. (45) is the calculation of the term w _a (k – N,k), i.e.,

$w_{a} (k - N, k) = - Φ {(k, k - N)}^{- 1} w_{1} (k, k - N)$ (46)

The estimation of the term w _a (k – N,k) is provided by (Xia et al., 2009)

$\begin{array}{l} {\overset{\land}{w}}_{a} (k - N, k) = - Φ_{1} (k - N, k) \sum_{n = 0}^{N = 1} \cdot \tilde{C} (n) \\ {[C P (k - n | k - n - 1) C^{T} + R (k - n)]}^{- 1} \tilde{z} (k - n) \end{array}$ (47)

where x˜(k – n) = z(k – n) – z˜(k – n) is the innovation for time-instant k – n, while

$\begin{array}{l} \tilde{C} (n) = {A^{n} Q (k - n, k - n - 1) + \\ + \sum_{j = n + 2}^{N} A^{j - 1} Q (k - j + 1, k - j) . \\ \cdot [Π_{m = n + 1}^{j - 1} A [I - K_{i} (k - m) C^{T}]} C^{T} \end{array}$ (48)

with Q(k – n,k – n – 1) and R (k – n) denoting the process and measurement noise covariance matrices at k – n. Then, the smoothing filter for the processing of the delayed measurements at the i-th local filter is given by Eq. (45), i.e.,

${\overset{⌢}{x}}_{i}^{*} (k | k) = {\overset{⌢}{x}}_{i} (k | k) + M_{i} [z_{i} (k - N) - {\overset{⌢}{z}}_{i} (k - N | k)]$ (49)

With matrix M_i to be given by (Xia et al., 2009)

$M_{i} = [P (k | k) Φ_{1} {(k - N, k)}^{T} + P_{i}^{\tilde{x} \tilde{w}}] C_{i}^{T} W_{i}^{- 1}$ (50)

and matrices W_i and P_i^x˜w˜ to be defined as

$\begin{array}{l} W_{i} = C_{i} {Φ_{1} (k - N, k) P_{i} (k | k) Φ_{1} {(k - N, k)}^{T} \\ + Φ_{1} (k - N, k) P_{i}^{\tilde{x} \tilde{w}} + {[Φ_{1} (k - N, k) P_{i}^{\tilde{x} \tilde{w}}]}^{T} + \\ + Q_{i}^{*} (k - N, k)} \cdot C_{i}^{T} + R_{i} (k - N) \end{array}$ (51)

and

$\begin{array}{l} P_{i}^{\tilde{x} \tilde{w}} = A^{N} \sum_{n = 0}^{N - 1} P_{i} (k - N | k - N) D_{i} {(n)}^{T} \cdot \\ \cdot {[C_{i} P_{i} (k - n - 1) C_{i}^{T} + R_{i} (k - n)]}^{- 1} \cdot \\ \cdot {\tilde{C}}_{i} {(n)}^{T} Φ_{1} {(k - N, k)}^{T} - A^{N} Q_{i}^{*} (k - N, k) \end{array}$ (52)

while matrix D_i (n) is defined as

$\begin{array}{l} D_{i} (n) = C_{i} A if N = 1 and \\ D_{i} (n) = C_{i} A Π_{j = n}^{N - 2} [I - K_{i} (k - j - 1) C_{i}] \cdot A, if N > 1 \end{array}$ (53)

The covariance matrix of the estimated noise term w_αi (k – N,k) is calculated as

$\begin{array}{l} Q_{i}^{*} (k - N, k) = Q (k - N, k) - Φ_{1} (k - N, k) \cdot \\ \cdot \sum_{n = 0}^{N - 1} {\tilde{C}}_{i} (n) {[C_{i} P_{i} (k - n | k - n - 1) C_{i}^{T} + R_{i} (k - n)]}^{- 1} \cdot \\ \cdot {\tilde{C}}_{i} {(n)}^{T} Φ_{1} {(k - N, k)}^{T} \end{array}$ (54)

where

$\begin{array}{l} Q (k - N, k) = Φ_{1} (k - N, k) \\ [\sum_{j = 1}^{N} A^{j - 1} \cdot Q (k - j + 1, k - j) {(A^{j - 1})}^{T}] Φ_{1} {(k - N, k)}^{T} \end{array}$ (55)

After smoothing, the covariance matrix of the modified state estimation error becomes

$\begin{array}{l} P_{i}^{*} (k | k) = P_{i} (k | k) - [P_{i}^{\tilde{x} \tilde{w}} + P_{i} (k | k) Φ_{1} {(k - N, k)}^{T}] \times \\ \times C_{i} {(k - N)}^{T} W_{i}^{1} C_{i} (k - N) \times \\ \times {[P_{i}^{\tilde{x} \tilde{w}} + P_{i} (k | k) Φ_{1} {(k - N, k)}^{T}]}^{T} \end{array}$ (56)

6.3 Derivative-free EIF under delayed measurements

The description of the nonlinear robotic model in the canonical form of Eq. (12) with matrices A, B and C to be given in Eq. (20) enables the application of the previous analysis for the compensation of time-delays and packet drops through smoothing in the computation of the linear Kalman filter. The fact that matrices A, B and C described in Eq. (20) are time invariant, facilitates the computation of the smoothing Kalman Filter according to Eq. (49) and Eq. (50). Using the time invariant matrices defined in Eq. (20) and applying common discretization methods, one gets the discrete-time equivalents of matrices A,B and C which are defined as A_d,B_d and C_d, respectively. It is noted that due to the specific form of matrix B, the term Bu(k – 1) appearing in Eq. (30) is a variable of small magnitude with mean value close to zero. Thus the term w₁ k k,k – 1) = Bu(k – 1) + w(k,k – 1) differs little from w(k,k – 1).

6.4 Distributed filtering-based fusion of the robot's state estimates

Fusion of the local state estimates which are provided by filters running on the vision nodes can improve the accuracy and robustness of the performed state estimation, thus also improving the performance of the robot's control loop (Sun & Deng, 2005), (Sun et al., 2011). Under the assumption of Gaussian noise, a possible approach for fusing the state estimates from the distributed local filters is the derivative-free extended information filter (DEIF). As explained in Section 4, the DEIF provides an aggregate state estimate by weighting the state vectors produced by local Kalman filters with the inverse of the associated estimation error covariance matrices.

Visual servoing over the previously described camera network is considered for the nonlinear dynamic model of a single-link robotic manipulator. The robot can be programmed to execute a manufacturing task, such as painting or welding (Tzafestas et al., 1997). The position of the robot's end-effector in the Cartesian space (and consequently the angle for the robotic link) is measured by the aforementioned m distributed cameras. The proposed multi-camera based robotic control loop can be also useful in other vision-based industrial robotic applications where the vision is occluded or heavily disturbed by noise sources, e.g., cutting. In such applications there is need to fuse measurements from multiple cameras so as to obtain redundancy in the visual information and to permit the robot to complete safely and within the specified accuracy constraints its assigned tasks (Moon et al., 2006), (Yoshimoto et al., 2010). The considered 2-DOF robotic model consists of two rigid links which are rotated by DC motors, as shown in Fig. 3.

Figure 3.

Visual servoing based on fusion of state estimates provided by local derivative-free nonlinear Kalman filters

7. Simulation tests

The performance of the proposed derivative-free nonlinear Kalman filter was tested in state estimation-based control for a 2-DOF rigid-link robotic manipulator (Fig. 3). The considered visual servoing control can find applications in several robotic tasks requiring dexterous manipulations (e.g., packing, welding, cutting, etc.) (Schuurman & Capson, 2004). The differentially flat model of the robot has been analysed in Section 4. It was assumed that only measurements of the angle of the robot's joints could be obtained through the vision system. In the simulation tests presented in Fig. 4 and Fig. 5, the setpoints for the state variables are marked as red lines, the estimated state variables were denoted as green lines whereas the real state variables were denoted as blue lines.

The delayed measurements were processed by the Kalman filter recursion according to the stages explained in subsection 6.2. The smoothing of the delayed measurements that was performed by the Kalman filter was based on Eq. (45), i.e., ^* x (k\k) = x(k\k) + M z(k – N, k). As explained in Section 6, matrix M is a gain matrix calculated according to Eq. (50). The innovation is given by. ∼ ^ z(k – N) = z(k – N) – z(k – N). The tracking accuracy of the distributed filtering-based control loop is depicted in Fig. 4 to Fig. 5.

Figure 4.

(a) Estimation and control of the motion of the robotic manipulator under transmission delays, when tracking a sinusoidal trajectory (b) control torques on the robot's joints.

Figure 5.

(a) Estimation and control of the motion of the robotic manipulator under transmission delays, when tracking a seesaw trajectory (b) control torques on the robot's joints.

Additionally, some performance measures were used to evaluate the distributed filtering-based control scheme. Tables I shows the variation of the RMSE for state variables $x_{2} = {\dot{θ}}_{1}$ and $x_{4} = {\dot{θ}}_{2}$ with respect to delay levels (d₁,d₂ =k · T_s, i.e., multiples of the sampling period T_s), as well as with respect to the probability of delay occurrence in the transmission of the measurement packets (p ∊ [0,1]).

Moreover, the traces of the covariance error matrices P_i, i = 1,2 and of the smoothed covariance error matrices P_i ^*,i = 1,2 appearing both at the local filters, as well as the trace of the covariance error matrix P at the master (aggregation) filter, for various delay levels, are shown in Table 2.

Table 1.

RMSE for robot state variables under measurement delays

d₁	d₂	p	RMSE_x2	RMSE_x4
15 – 23	17 – 21	0.5	3.284e – 3	3.283e – 3
10 – 16	12 – 18	0.4	3.221e – 3	3.219e – 3
8 – 12	15 – 19	0.6	3.235e – 3	3.231e – 3
20 – 25	25 – 30	0.5	3.275e – 3	3.272e – 3
14 – 20	9 – 15	0.5	3.239e – 3	3.236e – 3
14 – 20	9 – 15	0.7	3.256e – 3	3.252e – 3

Table 2.

Traces of the covariance error matrices for various delay levels

d₁	d₂	p	Tr(P_i)	Tr (P_i ^*)	Tr(P)
15 – 23	17 – 21	0.5	2.118·10⁻¹	1.046·10⁻¹	1.699·10⁻³
10 – 16	12 – 18	0.4	2.398·10⁻¹	1.180·10⁻¹	1.743·10⁻³
8 – 12	15 – 19	0.6	2.277·10⁻¹	1.121·10⁻¹	1.719·10⁻³
20 – 25	25 – 30	0.5	2.212·10⁻¹	1.090·10⁻¹	1.709·10⁻³
14 – 20	9 – 15	0.5	2.081·10⁻¹	1.029·10⁻¹	1.697·10⁻³
14 – 20	9 – 15	0.7	2.098·10⁻¹	1.037·10⁻¹	1.698·10⁻³

It can be noticed that despite the rise of the delay levels in the transmission of measurements from the sensors (cameras) to the local information processing nodes (local derivative-free Kalman filters), only slight variations of the tracking errors for state variables x_i, i = 1,…,4 were observed. Similarly, the changes of the traces of the estimation error covariance matrices, both at the local filters and at the master filter, were small.

8. Conclusions

Distributed state-estimation under communication delays and packet drops was examined. First, the results on networked Kalman filtering were outlined. These results were generalized in the case of the derivative-free extended information filter. Distributed filtering has been based on a derivative-free implementation of Kalman filtering which was shown to be applicable to MIMO nonlinear dynamical systems. In the considered derivative-free extended information filtering, the system was first subject to a linearization transformation that made use of the differential flatness theory.

Next, the problem of communication delays and packet drops in the distributed filtering-based control scheme were examined. This has the following forms: (i) there are time delays and packet drops in the transmission of information between the distributed local filters and the master filter, (ii) there are time delays and packet drops in the transmission of information from distributed sensors to each one of the local filters. In the first case, the structure and calculations of the master filter for estimating the aggregate state vector remain unchanged. In the second case, the effect of the random delays and packets drops has to be taken into account in the redesign of the local Kalman filters, which implies (i) a modified Riccati equation for the computation of the covariance matrix of the state vector estimation error, (ii) the use of a correction term in the update of the state vector's estimate so as to compensate for delayed measurements arriving at the local Kalman filters. In the simulation experiments it was shown that the aggregate state vector produced by the derivative-free extended information filter can be used for sensorless control and robotic visual servoing based on the processing of measurements which are provided by a multi-camera system.

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