Jacobian Estimation with Adaptive Kalman Filter for Uncalibrated Visual Servoing

Abstract

An uncalibrated visual servo method based on Jacobian estimation with adaptive Kalman filter (AKF) is proposed in this paper. With less or no priori knowledge of the parameters of robotic manipulator and camera, the presented method introduces the projective Jacobian matrix estimated by an adaptive Kalman filter for noise covariance recursive estimation. By doing this, the Jacobian estimation adaptability can be greatly improved to achieve better tracking performance in UVS system. Finally, the simulation experiments are performed to evaluate the performance of the proposed method, indicating better performance compared with state-of-the-art UVS methods using standard Kalman filter (SKF).

Supported by China Postdoctoral Project (No. 2021M692960).

Appendix

According to the definition of mean value, when $E(\mathbf {\tau \tau }^{T})<\infty$, we have:

$$\begin{aligned} \mathbf {E}_{k}(\mathbf {\tau \tau }^{T})=\underset{k\rightarrow \infty }{lim}\frac{1}{k}\underset{i=0}{\overset{k}{\sum }}\mathbf {\tau }_{i}\mathbf {\tau }^{T}_{i} \end{aligned}$$

(26)

Equation (26) can be transformed into

$$\begin{aligned} E_{k}(\mathbf {\tau \tau }^{T})&= \frac{1}{k}(\underset{i=0}{\overset{k-1}{\sum }}\mathbf {\tau }_{i}\mathbf {\tau }_{i}^{T}+\mathbf {\tau }_{k}\mathbf {\tau }_{k}^{T})\nonumber \\&= 1/k((k-1)E_{k-1}(\mathbf {\tau \tau }^{T})+\mathbf {\tau }_{k}\mathbf {\tau }_{k}^{T}) \end{aligned}$$

(27)

According to (21), we have:

$$\begin{aligned} E_{k-1}[\mathbf {\tau \tau }^{T}]=\mathbf {C}_{k-1}\mathbf {P}_{k-1}^{p}\mathbf {C}_{k-1}^{T}+\mathbf {V}_{k-1} \end{aligned}$$

(28)

Substituting (28) into (27), $E_{k}(\mathbf {\tau \tau }^{T})$ will be

$$\begin{aligned} (k+1)/k(\mathbf {C}_{k-1}\mathbf {P}_{k-1}^{p}\mathbf {C}_{k-1}^{T}+\mathbf {V}_{k-1})+1/k\mathbf {\tau }_{k}\mathbf {\tau }_{k}^{T} \end{aligned}$$

(29)

Define $\mathbf {L}_{k} = \mathbf {C}_{k}\mathbf {P}_{k}^{p}\mathbf {C}_{k}^{T}$, then we will have:

$$\begin{aligned} \mathbf {L}_{k}+\mathbf {V}_{k}=\mathbf {L}_{k-1}-1/k\mathbf {L}_{k-1}+(k-1)/k\mathbf {V}_{k-1}+1/k\mathbf {\tau }_{k}\mathbf {\tau }_{k}^{T} \end{aligned}$$

(30)

Finally, (30) can be written as:

$$\begin{aligned} \mathbf {V}_{k}=(k+1)/k\mathbf {V}_{k-1}+1/k\mathbf {\tau }_{k}\mathbf {\tau }_{k}^{T}-\varDelta \mathbf {L}_{k}-1/k\mathbf {L}_{k-1} \end{aligned}$$

(31)

With the increase of time, $\mathbf {V}_{k}$ depends on $\mathbf {V}_{k-1}$ much more than $\mathbf {L}_{k-1}$, the estimator of $\mathbf {V}_{k}$ can be rewritten as:

$$\begin{aligned} \mathbf {V}_{k}=(k-1)/k\mathbf {V}_{k-1}+1/k\mathbf {\tau }_{k}\mathbf {\tau }_{k}^{T}-\varDelta \mathbf {L}_{k} \end{aligned}$$

(32)