Parameter_estimation_demo_by_UKF

Communication

Language

Parameter estimation Demo by Unscented Kalman Filter (UKF) ¹². Example of dynamics is for the 1 DoF arm with spring, damping, and frictional forces at a joint. Joint angle is measured, then, unknown damping and frictional coefficients are identified by UKF ³⁴. In spite of discontinuous model of frictional force, UKF can be applied dislike Extended Kalman Filter (EKF).

Overview

Dynamics for the 1 DoF arm with spring, damping, and frictional forces at a joint is denoted as follows,

$$ J \ddot{\theta} + c \dot{\theta} + \mu {\rm sgn}{\dot{\theta}} + k \theta = u, $$

where damping and friction coefficients; $c$ and $\mu$ are unknown variables. From measured time series of data $y_m := \theta_m(t)$, these variables are identified by UKF.

Then, state space representation is denoted as,

$$ \begin{array}{c} {\bf \dot{x}} = f( {\bf x}, {\bf u}), \\ y = {\bf C} {\bf x}, \end{array} $$

where

$$ \begin{array}{c} {\bf x} = \left[ \begin{array}{c} x_1 \\ x_2 \\ \end{array} \right] := \left[ \begin{array}{c} \theta \\ \dot{\theta} \\ \end{array} \right], \\ f( {\bf x}, {\bf u}) := \left[ \begin{array}{c} x_2 \\ -J^{-1} c x_2 - J^{-1} \mu {\rm sgn} x_2 - J^{-1}k x_1 \\ \end{array} \right] + {\bf B} u, \\ \bf{B} := \left[ \begin{array}{c} 0 \\ J^{-1} \\ \end{array} \right], {\bf C} := \left[ \begin{array}{c} 1 \\ 0 \\ \end{array} \right]^T. \end{array} $$

Unknown damping and frictional coefficients are added to the expanded state vector as

$$ {\bf x_E} = \left[ \begin{array}{c} {\bf x} \\ x_3 \\ x_4 \\ \end{array} \right] := \left[ \begin{array}{c} {\bf x} \\ c \\ \mu \\ \end{array} \right]. $$

After that, the dynamics is discretized for time to apply UKF ("./functions/c2d_func.m" [3]).

Usages

[Step 1] Edit parameters

Edit code in "param_setting.m".

[Step 2] Start analysis

Execute "demo.m".

Images

Time series of state $x_1 := \theta (t)$.

Time series of output $y(t) := x_1(t)$

Identified damping and friction coefficients; $c$ and $\mu$, and deviations; $\sigma_c$ and $\sigma_\mu$, respectively.

References

Footnotes

1. Julier, S.J., Uhlmann, J.K., 2004. Unscented filtering and nonlinear estimation. Proc. IEEE 92 (3), 401–422. ↩

2. Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H.F., 1995. A new approach for filtering nonlinear systems. In: Proceedings of 1995 American Control Conference. ACC’95 3, pp. 1628–1632. http://dx.doi.org/10.1109/ACC.1995.529783. ↩

3. 足立 修一，丸田 一郎，「カルマンフィルタの基礎」，東京電機大学出版局． ↩

4. Fluid force identification acting on snake-like robots swimming in viscous fluids, Journal of Fluids and Structures, Vol. 106 (2021).
   https://doi.org/10.1016/j.jfluidstructs.2021.103351 ↩