Friction dynamics identification based on quadratic approximation of LuGre model

Abstract

In mechanical transmission servo systems, where friction is the primary disturbance, such as robot joints, dynamic friction is considered the leading cause of control inaccuracy in low-speed regions and during velocity direction changes. The LuGre model is a well-established dynamic friction model that has proved successful in describing dynamic frictional phenomena in such systems. However, most identification methods of the LuGre model rely on the prior knowledge of the inner parameters, and the internal state is difficult to observe. These factors make the friction dynamics identification become very cumbersome. On the other hand, we have discovered that the LuGre model exhibits redundancy in predicting friction dynamics. Therefore, in this article, the LuGre model is remodeled as a dynamic neuron by its recurrent quadratic approximation to expose high-order hidden parameters. On the basis of this modeling approach, a direct adaptive control architecture is proposed to identify all unknown parameters without any prior knowledge. In this scheme, a self-tuning combined error neuron is designed whose signal–noise ratio is maximized by principle component analysis. Besides, a kernel function-based stabilizing term is introduced in the update laws to suppress the oscillation during transient response. The feasibility of the strategy is verified through stability analysis. Simulation results show that the estimated LuGre model correctly reveals the actual behavior of friction dynamics. Finally, comparative experiments are carried out on a rotary actuator. The results show that the proposed adaptive control-based learning strategy significantly enhances position tracking performance, especially in the processes of slow crawling and switching in speed direction.

Appendices

Appendix A The recurrent calculation of G and H

The recurrent progress of the gradient G and Hessian matrix H is presented here. There are 4 paces for one recurrent step calculation.

Pace 1: The first and second derivatives of the Stribeck term $d_s$ with respect to parameters $f_c$, $f_s$, and $\nu _s$.

$$\begin{aligned}&\frac{\partial \phi }{\partial \nu _s}=-\frac{\rho }{\nu _s}\phi \end{aligned}$$

(A1)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial \phi }{\partial \nu _s}\right) =\frac{\rho \left( 1+\rho \right) }{\nu _s^2}\phi \end{aligned}$$

(A2)

$$\begin{aligned}&\frac{\partial \kappa }{\partial \nu _s}=\kappa \frac{\partial \phi }{\partial \nu _s} \end{aligned}$$

(A3)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial \kappa }{\partial \nu _s}\right) =\frac{\partial \kappa }{\partial \nu _s}\frac{\partial \phi }{\partial \nu _s}+\kappa \frac{\partial }{\partial \nu _s}\left( \frac{\partial \phi }{\partial \nu _s}\right) \end{aligned}$$

(A4)

$$\begin{aligned}&\frac{\partial g}{\partial f_c} = 1-\kappa \end{aligned}$$

(A5)

$$\begin{aligned}&\frac{\partial g}{\partial f_s} = \kappa \end{aligned}$$

(A6)

$$\begin{aligned}&\frac{\partial g}{\partial \nu _s} = \left( f_s-f_c\right) \frac{\partial \kappa }{\partial \nu _s} \end{aligned}$$

(A7)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial f_c}\right) = \frac{\partial }{\partial f_c}\left( \frac{\partial g}{\partial \nu _s}\right) = -\frac{\partial \kappa }{\partial \nu _s} \end{aligned}$$

(A8)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial f_s}\right) = \frac{\partial }{\partial f_s}\left( \frac{\partial g}{\partial \nu _s}\right) = \frac{\partial \kappa }{\partial \nu _s} \end{aligned}$$

(A9)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial \nu _s}\right) = \left( f_s-f_c\right) \frac{\partial }{\partial \nu _s}\left( \frac{\partial \kappa }{\partial \nu _s}\right) \end{aligned}$$

(A10)

$$\begin{aligned}&\frac{\partial h}{\partial f_c} = -\frac{|{\dot{r}}|}{g^2}\frac{\partial g}{\partial f_c} \end{aligned}$$

(A11)

$$\begin{aligned}&\frac{\partial h}{\partial f_s} = -\frac{|{\dot{r}}|}{g^2}\frac{\partial g}{\partial f_s} \end{aligned}$$

(A12)

$$\begin{aligned}&\frac{\partial h}{\partial \nu _s} = -\frac{|{\dot{r}}|}{g^2}\frac{\partial g}{\partial \nu _s} \end{aligned}$$

(A13)

$$\begin{aligned}&\frac{\partial }{\partial f_c}\left( \frac{\partial h}{\partial f_c}\right) = 2\frac{\vert {\dot{r}}\vert }{g^3}\left( \frac{\partial g}{\partial f_c}\right) ^2 \end{aligned}$$

(A14)

$$\begin{aligned}&\frac{\partial }{\partial f_s}\left( \frac{\partial h}{\partial f_c}\right) = \frac{\partial }{\partial f_c}\left( \frac{\partial h}{\partial f_s}\right) = 2\frac{\vert {\dot{r}}\vert }{g^3}\frac{\partial g}{\partial f_s}\frac{\partial g}{\partial f_c} \end{aligned}$$

(A15)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial f_c}\right) = \frac{\partial }{\partial f_c}\left( \frac{\partial h}{\partial \nu _s}\right) \nonumber \\&= 2\frac{\vert {\dot{r}}\vert }{g^3}\frac{\partial g}{\partial \nu _s}\frac{\partial g}{\partial f_c} - \frac{\vert {\dot{r}}\vert }{g^2}\frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial f_c}\right) \end{aligned}$$

(A16)

$$\begin{aligned}&\frac{\partial }{\partial f_s}\left( \frac{\partial h}{\partial f_s}\right) = 2\frac{\vert {\dot{r}}\vert }{g^3}\left( \frac{\partial g}{\partial f_s}\right) ^2 \end{aligned}$$

(A17)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial f_s}\right) = \frac{\partial }{\partial f_s}\left( \frac{\partial h}{\partial \nu _s}\right) \nonumber \\&= 2\frac{\vert {\dot{r}}\vert }{g^3}\frac{\partial g}{\partial \nu _s}\frac{\partial g}{\partial f_s} - \frac{\vert {\dot{r}}\vert }{g^2}\frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial f_s}\right) \end{aligned}$$

(A18)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial \nu _s}\right) = 2\frac{\vert {\dot{r}}\vert }{g^3}\left( \frac{\partial g}{\partial \nu _s}\right) ^2 - \frac{\vert {\dot{r}}\vert }{g^2}\frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial \nu _s}\right) \end{aligned}$$

(A19)

$$\begin{aligned}&\frac{\partial d_s}{\partial f_c} = \frac{\partial g}{\partial f_c}\textrm{sgn}\left( {\dot{r}}\right) \end{aligned}$$

(A20)

$$\begin{aligned}&\frac{\partial d_s}{\partial f_s} = \frac{\partial g}{\partial f_s}\textrm{sgn}\left( {\dot{r}}\right) \end{aligned}$$

(A21)

$$\begin{aligned}&\frac{\partial d_s}{\partial \nu _s} = \frac{\partial g}{\partial \nu _s}\textrm{sgn}\left( {\dot{r}}\right) \end{aligned}$$

(A22)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_c}\right) = \frac{\partial }{\partial f_c}\left( \frac{\partial d_s}{\partial \nu _s}\right) = \frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial f_c}\right) \textrm{sgn}\left( {\dot{r}}\right) \end{aligned}$$

(A23)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_s}\right) = \frac{\partial }{\partial f_s}\left( \frac{\partial d_s}{\partial \nu _s}\right) = \frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial f_s}\right) \textrm{sgn}\left( {\dot{r}}\right) \end{aligned}$$

(A24)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial \nu _s}\right) = \frac{\partial }{\partial \nu _s}\left( \frac{\partial g}{\partial \nu _s}\right) \textrm{sgn}\left( {\dot{r}}\right) \end{aligned}$$

(A25)

Pace 2: The first and second derivatives of the state y with respect to parameters $\sigma _0$, $\sigma _1$, $f_c$, $f_s$, and $\nu _s$.

$$\begin{aligned}&\frac{\partial y}{\partial \sigma _0} = \int _{t_0}^{t}\left( h\left( d_s-y\right) -\sigma _0 h\frac{\partial y}{\partial \sigma _0}\right) \textrm{d}\tau \end{aligned}$$

(A26)

$$\begin{aligned}&\frac{\partial y}{\partial f_c} = a\int _{t_0}^{t}\left( \frac{\partial h}{\partial f_c}\left( d_s-y\right) + h\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \right) \textrm{d}\tau \end{aligned}$$

(A27)

$$\begin{aligned}&\frac{\partial y}{\partial f_s} = a\int _{t_0}^{t}\left( \frac{\partial h}{\partial f_s}\left( d_s-y\right) + h\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \right) \textrm{d}\tau \end{aligned}$$

(A28)

$$\begin{aligned}&\frac{\partial y}{\partial \nu _s} = a\int _{t_0}^{t}\left( \frac{\partial h}{\partial \nu _s}\left( d_s-y\right) + h\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \right) \textrm{d}\tau \end{aligned}$$

(A29)

$$\begin{aligned}&\frac{\partial }{\partial \sigma _0}\left( \frac{\partial y}{\partial \sigma _0}\right) = -\int _{t_0}^{t}\left( 2h\frac{\partial y}{\partial \sigma _0}+\sigma _0h\frac{\partial }{\partial \sigma _0}\left( \frac{\partial y}{\partial \sigma _0}\right) \right) \textrm{d}\tau \end{aligned}$$

(A30)

$$\begin{aligned}&\frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial \sigma _0}\right) = \frac{\partial }{\partial \sigma _0}\left( \frac{\partial y}{\partial f_c}\right) \nonumber \\&= \int _{t_0}^{t}\left( \frac{\partial h}{\partial f_c}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \right. \nonumber \\&\left. -a\frac{\partial h}{\partial f_c}\frac{\partial y}{\partial \sigma _0}-ah\frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial \sigma _0}\right) \right) \textrm{d}\tau \end{aligned}$$

(A31)

$$\begin{aligned}&\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial \sigma _0}\right) = \frac{\partial }{\partial \sigma _0}\left( \frac{\partial y}{\partial f_s}\right) \nonumber \\&= \int _{t_0}^{t}\left( \frac{\partial h}{\partial f_s}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \right. \nonumber \\&\left. -a\frac{\partial h}{\partial f_s}\frac{\partial y}{\partial \sigma _0}-ah\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial \sigma _0}\right) \right) \textrm{d}\tau \end{aligned}$$

(A32)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial \sigma _0}\right) = \frac{\partial }{\partial \sigma _0}\left( \frac{\partial y}{\partial \nu _s}\right) \nonumber \\&= \int _{t_0}^{t}\left( \frac{\partial h}{\partial \nu _s}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \right. \nonumber \\&\left. -a\frac{\partial h}{\partial \nu _s}\frac{\partial y}{\partial \sigma _0}-ah\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial \sigma _0}\right) \right) \textrm{d}\tau \end{aligned}$$

(A33)

$$\begin{aligned}&\frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial f_c}\right) = a\int _{t_0}^{t}\left( \frac{\partial }{\partial f_c}\left( \frac{\partial h}{\partial f_c}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +2\frac{\partial h}{\partial f_c}\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \right. \nonumber \\&\left. -h\frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial f_c}\right) \right) \textrm{d}\tau \end{aligned}$$

(A34)

$$\begin{aligned}&\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_c}\right) = \frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial f_s}\right) \nonumber \\&= a\int _{t_0}^{t}\left( \frac{\partial }{\partial f_s}\left( \frac{\partial h}{\partial f_c}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +\frac{\partial h}{\partial f_c}\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) +\frac{\partial h}{\partial f_s}\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \right. \nonumber \\&\quad \left. -h\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_c}\right) \right) \textrm{d}\tau \end{aligned}$$

(A35)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_c}\right) = \frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial \nu _s}\right) = a\nonumber \\&\int _{t_0}^{t}\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial f_c}\right) \left( d_s-y\right) +\frac{\partial h}{\partial f_c}\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \right. \nonumber \\&\left. +\frac{\partial h}{\partial f_s}\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \right. \nonumber \\ {}&\quad \left. +h\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_c}\right) -\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_c}\right) \right) \right) \textrm{d}\tau \end{aligned}$$

(A36)

$$\begin{aligned}&\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_s}\right) = a\int _{t_0}^{t}\left( \frac{\partial }{\partial f_s}\left( \frac{\partial h}{\partial f_s}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +2\frac{\partial h}{\partial f_s}\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) -h\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_s}\right) \right) \textrm{d}\tau \end{aligned}$$

(A37)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_s}\right) = \frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial \nu _s}\right) \nonumber \\&= a\int _{t_0}^{t}\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial f_s}\right) \left( d_s-y\right) +\frac{\partial h}{\partial f_s}\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \right. \nonumber \\&\left. +\frac{\partial h}{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \right. \nonumber \\ {}&\quad \left. +h\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_s}\right) -\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_s}\right) \right) \right) \textrm{d}\tau \end{aligned}$$

(A38)

$$\begin{aligned}&\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial \nu _s}\right) = a\int _{t_0}^{t}\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial \nu _s}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +2\frac{\partial h}{\partial \nu _s}\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \right. \nonumber \\&\left. +h\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial \nu _s}\right) -\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial \nu _s}\right) \right) \right) \textrm{d}\tau \end{aligned}$$

(A39)

Pace 3: The gradient G, i.e., the first derivatives of the nonlinear dynamic part $f_z$ with respect to parameters $\sigma _0$, $\sigma _1$, $f_c$, $f_s$, and $\nu _s$.

$$\begin{aligned} G&= \left[ \begin{array}{ccccc} \frac{\partial f_z}{\partial \sigma _0}&\frac{\partial f_z}{\partial \sigma _1}&\frac{\partial f_z}{\partial f_c}&\frac{\partial f_z}{\partial f_s}&\frac{\partial f_z}{\partial \nu _s} \end{array} \right] ^T \nonumber \\&\frac{\partial f_z}{\partial \sigma _0} = \left( 1-\sigma _1h\right) \frac{\partial y}{\partial \sigma _0} \end{aligned}$$

(A40)

$$\begin{aligned}&\frac{\partial f_z}{\partial \sigma _1} = h\left( d_s-y\right) \end{aligned}$$

(A41)

$$\begin{aligned}&\frac{\partial f_z}{\partial f_c} = \frac{\partial y}{\partial f_c}+\sigma _1\left( \frac{\partial h}{\partial f_c}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \right) \end{aligned}$$

(A42)

$$\begin{aligned}&\frac{\partial f_z}{\partial f_s} = \frac{\partial y}{\partial f_s}+\sigma _1\left( \frac{\partial h}{\partial f_s}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \right) \end{aligned}$$

(A43)

$$\begin{aligned}&\frac{\partial f_z}{\partial \nu _s} = \frac{\partial y}{\partial \nu _s}+\sigma _1\left( \frac{\partial h}{\partial \nu _s}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \right) \end{aligned}$$

(A44)

Pace 4: The Hessian matrix H, i.e., the second derivatives of the nonlinear dynamic part $f_z$ with respect to parameters $\sigma _0$, $\sigma _1$, $f_c$, $f_s$, and $\nu _s$.

Table 6 Parameter training results of six groups with different initial conditions

Table 7 RMSE of friction prediction of six groups within three speed intervals ($10^{-3}$ Nm)

Table 8 One-way ANOVA results

$$\begin{aligned} H&= \frac{\partial ^2 f_z}{\partial \theta \partial \theta ^T}\nonumber \\ {}&= \left[ \begin{array}{ccccc} \frac{\partial ^2 f_z}{\partial \sigma _0^2} &{}\quad \frac{\partial ^2 f_z}{\partial \sigma _1\partial \sigma _0}&{}\quad \frac{\partial ^2 f_z}{\partial f_c\partial \sigma _0}&{}\quad \frac{\partial ^2 f_z}{\partial f_s\partial \sigma _0}&{}\quad \frac{\partial ^2 f_z}{\partial \nu _s\partial \sigma _0} \\ \frac{\partial ^2 f_z}{\partial \sigma _0\partial \sigma _1}&{}\quad \frac{\partial ^2 f_z}{\partial \sigma _1^2}&{}\quad \frac{\partial ^2 f_z}{\partial f_c\partial \sigma _1}&{}\quad \frac{\partial ^2 f_z}{\partial f_s\partial \sigma _1}&{}\quad \frac{\partial ^2 f_z}{\partial \nu _s\partial \sigma _1} \\ \frac{\partial ^2 f_z}{\partial \sigma _0\partial f_c}&{}\quad \frac{\partial ^2 f_z}{\partial \sigma _1\partial f_c}&{}\quad \frac{\partial ^2 f_z}{\partial f_c^2}&{}\quad \frac{\partial ^2 f_z}{\partial f_s\partial f_c}&{}\quad \frac{\partial ^2 f_z}{\partial \nu _s\partial f_c} \\ \frac{\partial ^2 f_z}{\partial \sigma _0\partial f_s}&{}\quad \frac{\partial ^2 f_z}{\partial \sigma _1\partial f_s}&{}\quad \frac{\partial ^2 f_z}{\partial f_c\partial f_s}&{}\quad \frac{\partial ^2 f_z}{\partial f_s^2}&{}\quad \frac{\partial ^2 f_z}{\partial \nu _s\partial f_s} \\ \frac{\partial ^2 f_z}{\partial \sigma _0\partial \nu _s}&{}\quad \frac{\partial ^2 f_z}{\partial \sigma _1\partial \nu _s}&{}\quad \frac{\partial ^2 f_z}{\partial f_c\partial \nu _s}&{}\quad \frac{\partial ^2 f_z}{\partial f_s\partial \nu _s}&{}\quad \frac{\partial ^2 f_z}{\partial \nu _s^2} \end{array} \right] \nonumber \\&\frac{\partial ^2 f_z}{\partial \sigma _0^2} = \left( 1-\sigma _1h\right) \frac{\partial }{\partial \sigma _0}\left( \frac{\partial y}{\partial \sigma _0}\right) \end{aligned}$$

(A45)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial \sigma _1\partial \sigma _0} = \frac{\partial ^2 f_z}{\partial \sigma _0\partial \sigma _1} = -h\frac{\partial y}{\partial \sigma _0} \end{aligned}$$

(A46)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial f_c\partial \sigma _0} = \frac{\partial ^2 f_z}{\partial \sigma _0\partial f_c} = \left( 1-\sigma _1h\right) \frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial \sigma _0}\right) \nonumber \\ {}&\quad - \sigma _1\frac{\partial h}{\partial f_c}\frac{\partial y}{\partial \sigma _0} \end{aligned}$$

(A47)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial f_s\partial \sigma _0} = \frac{\partial ^2 f_z}{\partial \sigma _0\partial f_s} = \left( 1-\sigma _1h\right) \frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial \sigma _0}\right) \nonumber \\ {}&\quad - \sigma _1\frac{\partial h}{\partial f_s}\frac{\partial y}{\partial \sigma _0} \end{aligned}$$

(A48)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial \nu _s\partial \sigma _0} = \frac{\partial ^2 f_z}{\partial \sigma _0\partial \nu _s} = \left( 1-\sigma _1h\right) \frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial \sigma _0}\right) \nonumber \\ {}&\quad - \sigma _1\frac{\partial h}{\partial \nu _s}\frac{\partial y}{\partial \sigma _0} \end{aligned}$$

(A49)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial \sigma _1^2} = 0 \end{aligned}$$

(A50)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial f_c\partial \sigma _1} = \frac{\partial ^2 f_z}{\partial \sigma _1\partial f_c} = \frac{\partial h}{\partial f_c}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \end{aligned}$$

(A51)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial f_s\partial \sigma _1} = \frac{\partial ^2 f_z}{\partial \sigma _1\partial f_s} = \frac{\partial h}{\partial f_s}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \end{aligned}$$

(A52)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial \nu _s\partial \sigma _1} = \frac{\partial ^2 f_z}{\partial \sigma _1\partial \nu _s} = \frac{\partial h}{\partial \nu _s}\left( d_s-y\right) +h\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \end{aligned}$$

(A53)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial f_c^2} = \frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial f_c}\right) \nonumber \\ {}&\quad +\sigma _1\left( \frac{\partial }{\partial f_c}\left( \frac{\partial h}{\partial f_c}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +2\frac{\partial h}{\partial f_c}\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) -h\frac{\partial }{\partial f_c}\left( \frac{\partial y}{\partial f_c}\right) \right) \end{aligned}$$

(A54)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial f_s\partial f_c} = \frac{\partial ^2 f_z}{\partial f_c\partial f_s} = \frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_c}\right) \nonumber \\ {}&\quad +\sigma _1\left( \frac{\partial }{\partial f_s}\left( \frac{\partial h}{\partial f_c}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +\frac{\partial h}{\partial f_c}\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \right. \nonumber \\ {}&\quad \left. +\frac{\partial h}{\partial f_s}\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) -h\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_c}\right) \right) \end{aligned}$$

(A55)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial \nu _s\partial f_c} = \frac{\partial ^2 f_z}{\partial f_c\partial \nu _s} = \frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_c}\right) \nonumber \\&+\sigma _1\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial f_c}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +\frac{\partial h}{\partial f_c}\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) +\frac{\partial h}{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_c}-\frac{\partial y}{\partial f_c}\right) \right. \nonumber \\&\left. +h\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_c}\right) -\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_c}\right) \right) \right) \end{aligned}$$

(A56)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial f_s^2} = \frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_s}\right) +\sigma _1\left( \frac{\partial }{\partial f_s}\left( \frac{\partial h}{\partial f_s}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +2\frac{\partial h}{\partial f_s}\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \right. \nonumber \\&\left. -h\frac{\partial }{\partial f_s}\left( \frac{\partial y}{\partial f_s}\right) \right) \end{aligned}$$

(A57)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial \nu _s\partial f_s} = \frac{\partial ^2 f_z}{\partial f_s\partial \nu _s} = \frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_s}\right) \nonumber \\&\quad +\sigma _1\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial f_s}\right) \left( d_s-y\right) \right. \nonumber \\&\left. +\frac{\partial h}{\partial f_s}\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) +\frac{\partial h}{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_s}-\frac{\partial y}{\partial f_s}\right) \right. \nonumber \\&\quad \left. +h\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial f_s}\right) -\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial f_s}\right) \right) \right) \end{aligned}$$

(A58)

$$\begin{aligned}&\frac{\partial ^2 f_z}{\partial \nu _s^2} = \frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial \nu _s}\right) +\sigma _1\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial h}{\partial \nu _s}\right) \left( d_s-y\right) \right. \nonumber \\&\quad \left. +2\frac{\partial h}{\partial \nu _s}\left( \frac{\partial d_s}{\partial \nu _s}-\frac{\partial y}{\partial \nu _s}\right) \right. \nonumber \\&\left. +h\left( \frac{\partial }{\partial \nu _s}\left( \frac{\partial d_s}{\partial \nu _s}\right) -\frac{\partial }{\partial \nu _s}\left( \frac{\partial y}{\partial \nu _s}\right) \right) \right) \end{aligned}$$

(A59)

Appendix B Simulation results with different initial conditions

The model is trained with six different initial conditions, and other simulation conditions are the same as in Sect. 4. The results are shown in Table 6. To evaluate the training performance, the command trajectory after 120 s is divided into three intervals according to velocity command ${\dot{r}}$: high-speed region ($\vert {\dot{r}}\vert>$0.1 rad/s), low-speed region (0.05 rad/s$<\vert {\dot{r}}\vert \le$0.1 rad/s) and reversal region ($\vert {\dot{r}}\vert \le$0.05 rad/s). The root mean square errors (RMSEs) of friction prediction in each interval are calculated in Table 7.

We used a one-way analysis of variance (ANOVA) to verify whether there were significant differences in errors among the six groups. First, the equal variance assumption is guaranteed by Bartlett’s test ($p=0.86$). The results are shown in Table 8 and Fig. 13. Since P-value$>0.05$, there is no significant difference between the friction prediction errors of six groups of parameters.

Fig. 13

RMSE of friction prediction of six different parameter sets