Parameter estimation, data compression and stochastic noise elimination in robotics: a wavelet domain-based integrated approach

Abstract

Accurate parameter estimation in presence of stochastic noise is the essential part of almost all control hardware of the plants. However, the optimal design of the control hardware depends on processing power and installed memory. The proposed research investigation focuses on precise parameter estimation from compressed temporal data of error dynamics with exiguous susceptibility to the robot’s controllability. Instead of using Maximum Likelihood estimation (MLE) and Least Squares (LS) estimation in the time domain, the proposed method exploits the recursive wavelet domain’s properties to selectively store the error data coefficients negating the data related to noise. As a result, data compression is achieved. The proposed algorithm may be directly implemented on any scalable “Very Large Scale Integration” (VLSI) circuit due to the recursive implementation. For the evaluation of robustness, dynamic parameter variation is considered. The variation in scalar & vector-valued error is considered to evaluate the performance of the stochastic system. The proposed algorithm implementation is demonstrated experimentally on commercially available Omni Bundle robot.

Appendix A Recursive implementation

To compare the RLS in time domain with RLS in wavelet domain we first recast our model as follows: The model (31) that relates measurement data to parameters can be cast in following general form

$$\begin{aligned} \xi (t) = H(t)\theta +W(t),t=1,2... \end{aligned}$$

(101)

where we denote $\delta \theta$ by $\theta$, $\alpha _t+K_d\beta _t+K_p\rho _t$ by $\xi (t)$, $\Gamma _t$ by H(t) and W(t) is the noise. In the time domain, the LS estimate of $\theta$ based on temporal data upto time T is

$$\begin{aligned} {\hat{\theta }}(T)&=\underset{\theta }{\arg \min }\sum _{t=1}^T\parallel \xi (t)-H(t)\theta \parallel ^2 \end{aligned}$$

(102)

$$\begin{aligned}&=\left( \sum _{t=1}^TH(t)^TH(t)\right) ^{-1}.\left( \sum _{t=1}^TH(t)^T\xi (t)\right) \end{aligned}$$

(103)

which is easily cast in time recursive form by using the relations

$$\begin{aligned}&\left( \sum _{t=1}^{T+1}TH(t)^TH(t)\right) \nonumber \\&\quad =\left( \sum _{t=1}^TH(t)^TH(t)\right) +H(T+1)^TH(T+1)\nonumber \\ \end{aligned}$$

(104)

$$\begin{aligned}&\left( \sum _{t=1}^{T+1}TH(t)^T\xi (t)\right) \nonumber \\&\quad =\left( \sum _{t=1}^TH(t)^T\xi (t)\right) \!+\!H(T+1)^T\xi (T\!+\!1)\qquad \end{aligned}$$

(105)

and making use of the matrix inversion lemma [30]. On the other hand, if the same is attempted using the WT, by defining

$$\begin{aligned} W_{\xi }[T,n,k]&=\sum _{t=1}^T\xi (t)\psi _{n,k}(t) \end{aligned}$$

(106)

$$\begin{aligned} W_H[T,n,k]&=\sum _{t=1}^TH(t)\psi _{n,k}(t) \end{aligned}$$

(107)

then the model will be

$$\begin{aligned} W_{\xi }[T,n,k]=W_H[T,n,k]\theta +W_W(T,n,K) \end{aligned}$$

(108)

and our LS algorithm would be

$$\begin{aligned} {\hat{\theta }}(T)= & {} \left( \sum _{n,k}W_H[T,n,k]^TW_H[T,n,k]\right) ^{-1}\nonumber \\&\times \left( \sum _{n,k}W_H[T,n,k]^TW_{\xi }[T,n,k]\right) \end{aligned}$$

(109)

which can also be cast in time recursive form using

$$\begin{aligned} W_H[T+1,n,k]&=W_H[T,n,k]+H(T+1)\psi _{n,k}(T+1) \end{aligned}$$

(110)

$$\begin{aligned} W_{\xi }[T+1,n,k]&=W_{\xi }[T,n,k]+\xi (T+1)\psi _{n,k}(T+1) \end{aligned}$$

(111)

The reduction in the complexity using the WT method is apparent once we note that the number of wavelet indices (n, k) is much smaller than the total number of time samples. It should be noted that if the noise is not Gaussian white, then the ML algorithm in the time domain is not recursively implementable and of course not also recursively implementable in wavelet domain. In such a case, both ML in the time domain and ML in the wavelet domain must be implemented using a block processing approach and since we are using fewer wavelet coefficients then time samples, the wavelet based ML approach is far superior then time domain based ML approach. Specifically consider the model from Eq. (101) with W(t) as non-Gaussian white. Then, the ML estimate is given by

$$\begin{aligned} p(\xi (t), 1\le t\le N|\theta ) = \underset{\theta }{\max }\prod _{t=1}^{T} P_W(\xi (t)-H(t)\theta )\nonumber \\ \end{aligned}$$

(112)

and we have to solve the following, which is not implementable recursively

$$\begin{aligned} \frac{\partial }{\partial \theta }\sum _{t=1}^{T}\log P_W(\xi (t)-H(t)\theta )=0 \end{aligned}$$

(113)

Now for the Wavelet domain we can write Eq. (10) as

$$\begin{aligned} W_\xi (n,k)= W_H(n,k)\theta + W_W(n,k) \end{aligned}$$

(114)

and the ML estimate is given by

$$\begin{aligned} \underset{\theta }{\max }\prod _{n,k}^{}P_{wW}(W_\xi -W_H(n,k|\theta )) \end{aligned}$$

(115)

Equation (115) is not implementable recursively.