An integral method for parameter identification of a nonlinear robot subject to quantization error

Abstract

This paper focuses on the parameter identification of a class of nonlinear mechanical dynamic systems subject to feedback quantization errors. A second-order integral method for the parameter identification of a nonlinear two-degrees-of-freedom robotic manipulator is studied. To obtain accurate parameter estimates, an optimal excitation trajectory subject to state constraints is proposed. The designated excitation trajectory is able to minimize estimation uncertainties, providing the controlled system trajectory reaches the desired optimal excitation trajectory. Moreover, for most of the nonlinear robotic systems, the output measurements are acquired from the encoders subject to apparent quantization error. Directly differentiating the position to estimate the velocity and acceleration trajectories is unrealistic and unsuitable due to the amplified quantization errors will cause considerable estimation errors. Instead, the finite Fourier series is introduced to perform optimal estimation of the position, velocity, and acceleration histories to capture the dominant frequencies of the system response. A weighted least-squares solution is further proposed to improve parameter identification precision. The comparative studies reveal that the integral method has superior parameter estimation performance than the direct difference model and the filtered regression model in the existing literature. Experimental simulations illustrate the effectiveness of the presented method.

Appendices

Appendix A Design strategy of integral factors

To illustrate how the proposed integral operator based regression model suppresses the influence of the initial condition, the first element of the matrix differential equation $\ddot{{\textbf {Y}}}_{2,2}(t) = -(\lambda _1+\lambda _2)\dot{{\textbf {Y}}}_{2,2}(t) - \lambda _1\lambda _2{\textbf {Y}}_{2,2}(t) + \lambda _1^2\varvec{\varphi }_2(t)$ in (42) is considered:

$$\begin{aligned} \ddot{y}_{22,1}(t) = -k_2{\dot{y}}_{22,1}(t) - k_1y_{22,1}(t) + \lambda ^2\theta _1(t) \end{aligned}$$

(95)

where $k_1=\lambda _1\lambda _2$, $k_2=\lambda _1+\lambda _2$, and $y_{22,1}(t)$ is the first element of the matrix ${\textbf {Y}}_{2,2}(t)$. The associated initial conditions are $y_{22,1}(0)=0$ and ${\dot{y}}_{22,1}(0)=\lambda _1\theta _1(0)-{\dot{\theta }}_1(0)$. Define the auxiliary state variables $\eta _1(t)=y_{22,1}(t)$ and $\eta _2(t)={\dot{y}}_{22,1}(t)$, the state-space equation in the controllable form can be expressed by

$$\begin{aligned} \begin{bmatrix} {\dot{\eta }}_1(t)\\ {\dot{\eta }}_2(t) \end{bmatrix}= & {} \begin{bmatrix} 0 &{}\quad 1\\ -k_1 &{}\quad -k_2 \end{bmatrix} \begin{bmatrix} {\eta }_1(t)\\ {\eta }_2(t) \end{bmatrix} + \begin{bmatrix} 0\\ \lambda ^2 \end{bmatrix}\theta _1(t)\nonumber \\ y(t)= & {} \eta _1(t) \end{aligned}$$

(96)

where the initial conditions are $\eta _1(0)=0$ and $\eta _2(0)=\lambda _1\theta _1(0)-{\dot{\theta }}_1(0)$. The analytic solution of (96) is

$$\begin{aligned} y(t) = {\textbf {C}}e^{{\textbf {A}}t}\varvec{\eta }(0) + {\textbf {C}}\int _{0}^{t}\!e^{{\textbf {A}}t}{} {\textbf {B}}{\theta }_1(t-\tau )\,d\tau \end{aligned}$$

(97)

where $\varvec{\eta }(0)=[\eta _1(0),\eta _2(0)]^T$, and

$$\begin{aligned} {\textbf {A}} = \begin{bmatrix} 0 &{}\quad 1\\ -k_1 &{}\quad -k_2 \end{bmatrix};\quad {\textbf {B}}=\begin{bmatrix} 0\\ \lambda ^2 \end{bmatrix};\quad {\textbf {C}}=\begin{bmatrix} 1&\quad 0 \end{bmatrix} \end{aligned}$$

(98)

For the analytic solution (97), the term ${\textbf {C}}e^{{\textbf {A}}t}\varvec{\eta }(0)$ represents the transient response of the auxiliary system (96). Another convolution term denotes the force response. If the initial condition of (96) is unknown, applying larger values of $\lambda _{1}$ and $\lambda _{2}$ could make the transient response ${\textbf {C}}e^{{\textbf {A}}t}\varvec{\eta }(0)$ decay faster, which is able to attenuate integration bias.

Appendix B Feasibility analysis of robust stability

The feasibility of the positive definite and symmetric matrices ${\textbf {P}}$ and ${\textbf {Q}}$ which satisfy (74) is discussed by introducing the linear matrix inequality (LMI) [6] as follows.

Factorizing $\Delta {\textbf {A}}$ in (70) yields

$$\begin{aligned} \Delta {\textbf {A}}= & {} \begin{bmatrix} {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}}\\ {\textbf {M}}_{0}^{-1} &{}\quad {\textbf {M}}_{0}^{-1} &{}\quad {\textbf {M}}_{0}^{-1} \end{bmatrix}\begin{bmatrix} \tilde{{\textbf {M}}} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}} \\ {\textbf {0}} &{}\quad \tilde{{\textbf {M}}} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad \tilde{{\textbf {M}}} \end{bmatrix}\nonumber \\{} & {} \begin{bmatrix} {\textbf {K}}_{I} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {K}}_{P} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad {\textbf {K}}_{D} \end{bmatrix} \nonumber \\\triangleq & {} {\textbf {E}}\varvec{\nabla }{} {\textbf {F}} \end{aligned}$$

(99)

where

$$\begin{aligned} {\textbf {E}}= & {} \begin{bmatrix} {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}}\\ {\textbf {M}}_{0}^{-1} &{}\quad {\textbf {M}}_{0}^{-1} &{}\quad {\textbf {M}}_{0}^{-1} \end{bmatrix};\quad \varvec{\nabla } = \begin{bmatrix} \tilde{{\textbf {M}}} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}} \\ {\textbf {0}} &{}\quad \tilde{{\textbf {M}}} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad \tilde{{\textbf {M}}} \end{bmatrix};\nonumber \\ {\textbf {F}}= & {} \begin{bmatrix} {\textbf {K}}_{I} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {K}}_{P} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad {\textbf {K}}_{D} \end{bmatrix} \end{aligned}$$

(100)

and

$$\begin{aligned} \tilde{{\textbf {M}}}= & {} \left( {\textbf {I}}_{2}+\Delta {\textbf {M}}{} {\textbf {M}}_{0}^{-1}\right) ^{-1}\Delta {\textbf {M}} \end{aligned}$$

(101)

Under the assumption that the uncertain matrix $\Delta {\textbf {M}}$ is bounded, there exists a known positive constant matrix $\varvec{\mu }$ such that

$$\begin{aligned}{} & {} \varvec{\nabla }\varvec{\nabla }^{T} = \begin{bmatrix} \tilde{{\textbf {M}}}\tilde{{\textbf {M}}}^T &{}\quad {\textbf {0}} &{}\quad {\textbf {0}} \\ {\textbf {0}} &{}\quad \tilde{{\textbf {M}}}\tilde{{\textbf {M}}}^T &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad \tilde{{\textbf {M}}}\tilde{{\textbf {M}}}^T \end{bmatrix} \nonumber \\ {}{} & {} < \begin{bmatrix} \varvec{\mu }^{+} &{}\quad {\textbf {0}} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad \varvec{\mu }^{+} &{}\quad {\textbf {0}}\\ {\textbf {0}} &{}\quad {\textbf {0}} &{}\quad \varvec{\mu }^{+} \end{bmatrix} \triangleq \varvec{\mu } \end{aligned}$$

(102)

where $\tilde{{\textbf {M}}}\tilde{{\textbf {M}}}^{T} < \varvec{\mu }^{+}$. Based on (99), the term $\Delta {\textbf {A}}{^T}{} {\textbf {P}} + {\textbf {P}}\Delta {\textbf {A}}$ can be factorized by

$$\begin{aligned} \Delta {\textbf {A}}{^T}{} {\textbf {P}} + {\textbf {P}}\Delta {\textbf {A}} = {\textbf {F}}^{T}\varvec{\nabla }^{T}{} {\textbf {E}}^{T}{} {\textbf {P}} + {\textbf {P}}{} {\textbf {E}}\varvec{\nabla }{} {\textbf {F}} \end{aligned}$$

(103)

Consider the well-known inequality

$$\begin{aligned} \textbf{X}^{T}\textbf{Y} + \textbf{Y}^{T}\textbf{X} \le \varepsilon \textbf{X}^{T}\textbf{X} + \frac{1}{\varepsilon }\textbf{Y}^{T}\textbf{Y} \end{aligned}$$

(104)

where $\varepsilon >0$, and $(\textbf{X},\textbf{Y})$ are square matrices with proper dimensions. On the basis of (102) and (104), (103) satisfies the following inequality [5]

$$\begin{aligned} {\textbf {F}}^{T}\varvec{\nabla }^{T}{} {\textbf {E}}^{T}{} {\textbf {P}} + {\textbf {P}}{} {\textbf {E}}\varvec{\nabla }{} {\textbf {F}}&\le \varepsilon {\textbf {F}}^{T}{} {\textbf {F}} + \frac{1}{\varepsilon }\left( {\textbf {P}}^{T}{} {\textbf {E}}\varvec{\nabla }\right) \left( \varvec{\nabla }^{T}{} {\textbf {E}}^{T}{} {\textbf {P}}\right) \nonumber \\&= \varepsilon {\textbf {F}}^{T}{} {\textbf {F}} + \frac{1}{\varepsilon }({\textbf {P}}{} {\textbf {E}}\varvec{\nabla }\varvec{\nabla }^{T}{} {\textbf {E}}^{T}{} {\textbf {P}})\nonumber \\&\le \varepsilon {\textbf {F}}^{T}{} {\textbf {F}} + \frac{1}{\varepsilon }{} {\textbf {P}}{} {\textbf {E}}\varvec{\mu }{} {\textbf {E}}^{T}{} {\textbf {P}} \end{aligned}$$

(105)

where $\varepsilon >0$ is a decision variable to be solved.

Thus, the robust stability requirement (74) is achieved if the following inequality is guaranteed:

$$\begin{aligned} {\textbf {A}}^{T}{} {\textbf {P}} + {\textbf {P}}{} {\textbf {A}} + \varepsilon {\textbf {F}}^{T}{} {\textbf {F}} + \frac{1}{\varepsilon }{} {\textbf {P}}{} {\textbf {E}}\varvec{\mu }{} {\textbf {E}}^{T}{} {\textbf {P}} + {\textbf {Q}} < 0 \end{aligned}$$

(106)

By Schur complement, (106) can be reformulated as the feasibility problem of the following LMIs:

$$\begin{aligned}{} & {} \begin{bmatrix} {\textbf {A}}^{T}{} {\textbf {P}} + {\textbf {P}}{} {\textbf {A}} + \varepsilon {\textbf {F}}^{T}{} {\textbf {F}} + {\textbf {Q}} &{}\quad {\textbf {P}}{} {\textbf {E}} \\ {\textbf {E}}^{T}{} {\textbf {P}} &{}\quad -\varepsilon \varvec{\mu }^{-1} \end{bmatrix} < 0,\nonumber \\{} & {} {\textbf {P}}^{T}={\textbf {P}}>0,\quad {\textbf {Q}}^{T}={\textbf {Q}}>0, \quad \varepsilon > 0. \end{aligned}$$

(107)

As a result, given $({\textbf {K}}_{P},{\textbf {K}}_{I},{\textbf {K}}_{D})$ and $\varvec{\mu }$, the robust stability of the closed-loop error dynamics (64) is guaranteed if the LMIs in (107) are satisfied.

Appendix C Analysis of quantization error

The quantization error $e_{Q}(t)$ is the error caused by using the quantized signal $x_{Q}(t)$ instead of the true signal x(t), defined as

$$\begin{aligned} e_{Q}(t) = x_{Q}(t) - x(t) \end{aligned}$$

(108)

The quantization interval is denoted as Q. The probability density function of the quantization error is given by

$$\begin{aligned} {\mathcal {P}}(e_{Q}) = {\left\{ \begin{array}{ll} \frac{1}{Q}, &{}\quad -\frac{Q}{2} \le e_{Q} \le \frac{Q}{2}\\ 0, &{}\quad \text{ otherwise } \end{array}\right. } \end{aligned}$$

(109)

By definition, the expectation of $e_{Q}$ is

$$\begin{aligned} {\mathcal {E}}\left[ e_{Q}\right]&= \int _{-\infty }^{\infty }\! e_{Q}{\mathcal {P}}(e_{Q})\,d e_{Q}\nonumber \\&= \int _{-Q/2}^{Q/2}\!e_{Q}\cdot \frac{1}{Q}\,de_{Q}\nonumber \\&= \frac{1}{2Q} \left. e^{2}\right| _{-Q/2}^{Q/2}\nonumber \\&= 0 \end{aligned}$$

(110)

and the variance is

$$\begin{aligned} {{\,\textrm{var}\,}}{(e_{Q})}&= {\mathcal {E}}\left[ e_{Q}^{2}\right] - {\mathcal {E}}\left[ e_{Q}\right] ^{2}\nonumber \\&= \int _{-\infty }^{\infty }\!e_{Q}^{2}{\mathcal {P}}(e_{Q})\,d e_{Q} - 0\nonumber \\&= \frac{1}{Q}\int _{-Q/2}^{Q/2}\!e_{Q}^{2}d e_{Q}\nonumber \\&= \frac{1}{3Q}\left. e_{Q}^{3}\right| _{Q/2}^{Q/2}\nonumber \\&= \frac{Q^{2}}{12} \end{aligned}$$

(111)

Hence, due to the zero-mean property, the root-mean-square error of $e_{Q}$ can be obtained by taking the square root of ${{\,\textrm{var}\,}}{(e_{Q})}$:

$$\begin{aligned} {{\,\textrm{RMS}\,}}{\{e_{Q}(t)\}} = \sqrt{{{\,\textrm{var}\,}}{(e_{Q})}} = \frac{Q}{\sqrt{12}} \end{aligned}$$

(112)

The property (112) will be used in the cost function of the optimal trajectory estimation presented in (85).

If the quantized signal is used in the traditional velocity estimation, it will induce significant estimation errors. For example, estimate the velocity ${\dot{x}}(t)$ at $t=kT$ by applying the backward difference formula:

$$\begin{aligned} {\dot{x}}(t)|_{t=kT}&= \frac{x_{Q}(kT)-x_{Q}(kT-T)}{T}\nonumber \\&= \frac{x(kT)-x(kT-T)}{T} + {\dot{e}}(kT) \end{aligned}$$

(113)

where T is the sample interval and

$$\begin{aligned} {\dot{e}}(kT) = \frac{e_{Q}(kT)-e_{Q}(kT-T)}{T} \end{aligned}$$

(114)

It can be seen that velocity estimation error ${\dot{e}}(kT)$ is amplified by a factor of 1/T due to the numerical differentiation. Hence, if the quantized signal is used in the acceleration estimation, the estimation error will be affected by a factor of $1/T^{2}$.

Appendix D Derivation of low-pass filter-based filtered regression model

The studies [27, 29, 34, 35] introduce the first-order low-pass filter to establish a filtered regression model to avoid differentiating the acceleration. The derivation is discussed as follows.

Rewrite (20) for the regression model as

$$\begin{aligned} \varvec{\uptau }(t) = \left[ \frac{d}{dt}\varvec{\varphi }_1(t) + \varvec{\varphi }_0(t)\right] \textbf{X} \end{aligned}$$

(115)

where

$$\begin{aligned}{} & {} \varvec{\varphi }_1(t) = \begin{bmatrix} {\dot{\theta }}_1 &{}\quad {\dot{\theta }}_1+{\dot{\theta }}_2 &{}\quad 0 &{}\quad \phi _{1,14} &{}\quad 0 &{}\quad \theta _1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad {\dot{\theta }}_1+{\dot{\theta }}_2 &{}\quad 0 &{}\quad \phi _{1,24} &{}\quad 0 &{}\quad 0 &{}\quad \theta _2 &{}\quad 0 &{}\quad 0 \end{bmatrix}\nonumber \\{} & {} \varvec{\varphi }_0(t)\nonumber \\{} & {} = \begin{bmatrix} 0 &{}\quad 0 &{}\quad gc_{\theta _1} &{}\quad \phi _{0,14} &{}\quad gc_{\theta _{12}} &{}\quad 0 &{}\quad 0 &{}\quad \phi _{0,18} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \phi _{0,24} &{}\quad gc_{\theta _{12}} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \phi _{0,29} \end{bmatrix}\nonumber \\ \end{aligned}$$

(116)

and $\phi _{1,14}$, $\phi _{0,14}$, $\phi _{0,24}$, $\phi _{0,18}$, $\phi _{0,29}$ are defined in (28). Taking the Laplace transform of (115) yields

$$\begin{aligned} \left[ s\varvec{\varphi }_1(s) - \varvec{\varphi }_1(0) + \varvec{\varphi }_0(s)\right] \textbf{X} = \varvec{\uptau }(s) \end{aligned}$$

(117)

where s is the Laplace operator and

$$\begin{aligned} \varvec{\varphi }_1(s)= & {} {\mathscr {L}}\left\{ \varvec{\varphi }_1(t)\right\} \nonumber \\ \varvec{\varphi }_0(s)= & {} {\mathscr {L}}\left\{ \varvec{\varphi }_0(t)\right\} \nonumber \\ \varvec{\uptau }(s)= & {} {\mathscr {L}}\left\{ \varvec{\uptau }(t)\right\} \end{aligned}$$

(118)

Introduce the first-order low-pass filter as

$$\begin{aligned} f(s)= \frac{\lambda }{s+\lambda } \end{aligned}$$

(119)

where $\lambda >0$ denotes the cut-off frequency of the filter. Multiplying both sides of (117) by (119) gives

$$\begin{aligned} \varvec{\uptau }^{1f}(s) = \varvec{\varphi }^{1f}(s)\textbf{X} \end{aligned}$$

(120)

where

$$\begin{aligned} \varvec{\uptau }^{1f}(s)= & {} \frac{\lambda }{s+\lambda }\varvec{\uptau }(s)\nonumber \\ \varvec{\varphi }^{1f}(s)= & {} \varvec{\varphi }_{1}^{1f}(s) + \varvec{\varphi }_0^{1f}(s)\nonumber \\ \varvec{\varphi }_{1}^{1f}(s)= & {} \frac{\lambda s}{s+\lambda }\varvec{\varphi }_1(s) - \frac{\lambda }{s+\lambda } \varvec{\varphi }_1(0)\nonumber \\ \varvec{\varphi }_{0}^{1f}(s)= & {} \frac{\lambda }{s+\lambda }\varvec{\varphi }_0(s) \end{aligned}$$

(121)

Reformulating $\varvec{\varphi }_1^{1f}(s)$ to eliminate $s\varvec{\varphi }_1(s)$ produces

$$\begin{aligned} \varvec{\varphi }_{1}^{1f}(s) ={}&\frac{\lambda s}{s+\lambda }\varvec{\varphi }_1(s) - \frac{\lambda }{s+\lambda } \varvec{\varphi }_1(0)\nonumber \\ ={}&\frac{\lambda (s+\lambda )}{{s+\lambda }}\varvec{\varphi }_1(s)\nonumber \\&+ \frac{1 }{s+\lambda }\left( -\lambda ^{2}\varvec{\varphi }_1(s)-\lambda \varvec{\varphi }_1(0)\right) \nonumber \\ ={}&\lambda \varvec{\varphi }_{1}(s) + \textbf{y}_1(s) \end{aligned}$$

(122)

where

$$\begin{aligned} {\textbf {y}}_1(s) = \frac{1 }{s+\lambda }\left( -\lambda ^{2}\varvec{\varphi }_1(s)-\lambda \varvec{\varphi }_1(0)\right) \end{aligned}$$

(123)

Taking the inverse Laplace transform of (120) gives the following filtered regression model:

$$\begin{aligned} \varvec{\uptau }^{1f}(t) = \varvec{\varphi }^{1f}(t)\textbf{X} \end{aligned}$$

(124)

where

$$\begin{aligned} \varvec{\uptau }^{1f}(t)&= {\mathscr {L}}^{-1}\left\{ \varvec{\uptau }^{1f}(s)\right\} \end{aligned}$$

(125)

$$\begin{aligned} \varvec{\varphi }^{1f}(t)&= {\mathscr {L}}^{-1}\left\{ \varvec{\varphi }^{1f}(s)\right\} \nonumber \\&=\varvec{\varphi }^{1f}_1(t)+\varvec{\varphi }^{1f}_0(t)\nonumber \\&= \lambda \varvec{\varphi }_1(t)+{\textbf {y}}_1(t) +\varvec{\varphi }^{1f}_0(t) \end{aligned}$$

(126)

The integral quantities $\varvec{\uptau }^{1f}(t)$, ${\textbf {y}}_1(t)$, and $\varvec{\varphi }_0^{1f}(t)$ are obtained by numerically integrating the following matrix differential equations:

$$\begin{aligned} \dot{\varvec{\uptau }}^{1f}(t)= & {} -\lambda \varvec{\uptau }^{1f}(t) + \lambda \varvec{\uptau }(t),\quad \varvec{\uptau }^{1f}(0)={\textbf {0}}\nonumber \\ \dot{\textbf{y}}_1(t)= & {} -\lambda \textbf{y}_1(t) - \lambda ^2 \varvec{\varphi }_1(t),\quad \textbf{y}_1(0)=-\lambda \varvec{\varphi }_1(0)\nonumber \\ \dot{\varvec{\varphi }}^{1f}_0(t)= & {} -\lambda \varvec{\varphi }_{0}^{1f}(t) + \lambda \varvec{\varphi }_0(t),\quad \varvec{\varphi }^{1f}_0(0)={\textbf {0}} \end{aligned}$$

(127)

The parameter estimation can be conducted by using (20) shown in Sect. 3.