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Sigma-point Kalman Filter with Nonlinear Unknown Input Estimation via Optimization and Data-driven Approach for Dynamic Systems

Junn Yong Loo, Ze Yang Ding, Vishnu Monn Baskaran, Surya Girinatha Nurzaman, Chee Pin Tan

TL;DR

A derivative-free UI sigma-point Kalman filter (SPKF-nUI), where the SPKF is interconnected with a general nonlinear UI estimator that can be implemented via nonlinear optimization and data-driven approaches.

Abstract

Most works on joint state and unknown input (UI) estimation require the assumption that the UIs are linear; this is potentially restrictive as it does not hold in many intelligent autonomous systems. To overcome this restriction and circumvent the need to linearize the system, we propose a derivative-free Unknown Input Sigma-point Kalman Filter (SPKF-nUI) where the SPKF is interconnected with a general nonlinear UI estimator that can be implemented via nonlinear optimization and data-driven approaches. The nonlinear UI estimator uses the posterior state estimate which is less susceptible to state prediction error. In addition, we introduce a joint sigma-point transformation scheme to incorporate both the state and UI uncertainties in the estimation of SPKF-nUI. An in-depth stochastic stability analysis proves that the proposed SPKF-nUI yields exponentially converging estimation error bounds under reasonable assumptions. Finally, two case studies are carried out on a simulation-based rigid robot and a physical soft robot, i.e., robots made of soft materials with complex dynamics to validate effectiveness of the proposed filter on nonlinear dynamic systems. Our results demonstrate that the proposed SPKF-nUI achieves the lowest state and UI estimation errors when compared to the existing nonlinear state-UI filters.

Sigma-point Kalman Filter with Nonlinear Unknown Input Estimation via Optimization and Data-driven Approach for Dynamic Systems

TL;DR

A derivative-free UI sigma-point Kalman filter (SPKF-nUI), where the SPKF is interconnected with a general nonlinear UI estimator that can be implemented via nonlinear optimization and data-driven approaches.

Abstract

Most works on joint state and unknown input (UI) estimation require the assumption that the UIs are linear; this is potentially restrictive as it does not hold in many intelligent autonomous systems. To overcome this restriction and circumvent the need to linearize the system, we propose a derivative-free Unknown Input Sigma-point Kalman Filter (SPKF-nUI) where the SPKF is interconnected with a general nonlinear UI estimator that can be implemented via nonlinear optimization and data-driven approaches. The nonlinear UI estimator uses the posterior state estimate which is less susceptible to state prediction error. In addition, we introduce a joint sigma-point transformation scheme to incorporate both the state and UI uncertainties in the estimation of SPKF-nUI. An in-depth stochastic stability analysis proves that the proposed SPKF-nUI yields exponentially converging estimation error bounds under reasonable assumptions. Finally, two case studies are carried out on a simulation-based rigid robot and a physical soft robot, i.e., robots made of soft materials with complex dynamics to validate effectiveness of the proposed filter on nonlinear dynamic systems. Our results demonstrate that the proposed SPKF-nUI achieves the lowest state and UI estimation errors when compared to the existing nonlinear state-UI filters.
Paper Structure (16 sections, 6 theorems, 68 equations, 5 figures, 3 tables)

This paper contains 16 sections, 6 theorems, 68 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Unscented Kalman Filter with Nonlinear Unknown Input Estimation
  3. Stochastic Stability Analysis
  4. State Error Propagation
  5. Kalman Gain and Covariance Updates
  6. Stochastic Stability Results
  7. Nonlinear Remainder Errors
  8. Illustrative Examples
  9. Case Study 1: Rigid-link robot
  10. Case Study 2: Pneumatic Soft Actuator (PSA)
  11. State Error Bound Computation
  12. Results and Discussions
  13. Baselines
  14. Case Study 1: Rigid-link robot
  15. Case Study 2: Pneumatic Soft Actuator (PSA)
  16. ...and 1 more sections

Key Result

Lemma 1

folland Suppose $f := ^T :\mathbb{R}^n \rightarrow \mathbb{R}^m$ and each $f_i :\mathbb{R}^n \rightarrow \mathbb{R}$ is of class $C^{k}$, i.e., $k$-times differentiable on an open convex set $S$. If $\hat{x} \in S$ and $x = \hat{x} + \tilde{x} \in S$ and, then

Figures (5)

  • Figure 1: Illustration of the proposed SPKF-nUI. A block diagram is used to illustrate and summarize the proposed SPKF-nUI filter (\ref{['eq:init']})-(\ref{['eq:Pcovariance_predict']}).
  • Figure 2: Case study 1: Rigid-link robot.${\theta}$ is the angular displacement, and the two-axis force $f = f_Xf_Y^T$ is applied at the link tip.
  • Figure 3: Case study 2: Pneumatic soft actuator (PSA). (A, Left) Using a contact bulb attached on top of a multi axis load cell (Axia80, ATI Industrial Automation Inc.), contact forces are applied on the PSA at its fingertip (i.e., Tip Contact). (A, Right) Contact forces are applied on the PSA along its surface (i.e., Surface Contact). $f_Xf_Y^T$ is the two-axis reaction force from the PSA (with direction opposite to the contact forces) measured by the load cell attached to the contact bulb. 10 reflective camera markers are placed evenly along the inextensible base layer to capture the PSA motion. (B) Three cameras are used to track the marker coordinates. A rigid robot arm is used to maneuver the position of PSA. (C) Planar line segment model to characterize PSA bending, where $X_{i} \quad Y_{i}^T$ are the coordinates of the $i^{th}$ camera marker and $\theta_i$ is the $i^{th}$ segmental bending angle.
  • Figure 4: Estimation results of Case Study 1. (A) and (B) show the ground truths $x_{1_t}$, $x_{2_t}$ and estimates $\hat{x}_{1_t}, \hat{x}_{2_t}$ of the states. (C) shows the $u_{1_t}$ (ground truth) and estimate $\hat{u}_{1_t}$ of the non-zero UI. These results in (A-C) are obtained from the first MC simulation. (D) shows the NMSE $\mathop{\mathrm{\mathbb{E}}}\nolimits[\|\tilde{x}_{t}\|^2]$ of the state estimates, the state error bounds (ERBs) of the SPKF-nUI (Section \ref{['ssect:OERB']}), and the norm of the theoretical CRLB (benchmark). (E) shows the NMSE $\mathop{\mathrm{\mathbb{E}}}\nolimits[\|\tilde{u}_{t}\|]$ of the UI estimates. These results in (D-E) are obtained via averaging across the 50 MC simulations.
  • Figure 5: Estimation results of Case Study 2. The results of the four combinations: (Tip Contact, Oscillatory Actuation), (Tip Contact, Random Actuation), (Surface Contact, Oscillatory Actuation) and (Surface Contact, Random Actuation) are shown in (A), (B), (C), (D), respectively. On each combination, (I) shows the ground truth $\|x_{t}\|_1$ and the mean state estimate $\mathop{\mathrm{\mathbb{E}}}\nolimits[\|\hat{x}_{t}\|_1]$ of SPKF-nUI and EKF-nUI. The 1-norm $\|x_{t}\|_1$ (physically) represents the aggregate bending angle. (II) shows the ground truth $\|u_{t}\|$ and the mean UI estimate $\mathop{\mathrm{\mathbb{E}}}\nolimits[\|\hat{u}_{t}\|]$ of SPKF-nUI and EKF-nUI. The 2-norm $\|u_{t}\|$ (physically) represents the magnitude of the resultant contact force. These results are obtained via averaging across the 10 MC simulations. Legend in (B-I) applies to all plots.

Theorems & Definitions (10)

  • Lemma 1: Multivariate Taylor Series Expansion
  • Lemma 2
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof