There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper presents a tracking system for ego-motion estimation which fuses vision and inertial measurements using EKF and UKF (Extended and Unscented Kalman Filters), where a comparison of their performance has been done. It also considers the multi-rate nature of the sensors: inertial sensing is sampled at a fast sampling frequency while the sampling frequency of vision is lower. the proposed approach uses a constant linear acceleration model and constant angular velocity model based on quaternions, which yields a non-linear model for states and a linear model in measurement equations. Results show that a significant improvement is obtained on the estimation when fusing both measurements with respect to just vision or just inertial measurements. It is also shown that the proposed system can estimate fast-motions even when vision system fails. Moreover, a study of the influence of the noise covariance is also performed, which aims to select their appropriate values at the tuning process. The setup is an end-effector mounted camera, which allow us to pre-define basic rotational and translational motions for validating results.

Fast Ego-motion Estimation with Multi-rate Fusion of Inertial and Vision

In many industrial control applications the control action updating can be faster than the output measurement, leading to multirate controllers. In this paper, some dual rate operations are used to model the controller as well as the controlled plant. The controller design is model-based and depends on the input to be tracked. The controller is split into two parts acting at different sampling rates and its design is approached based on the characteristics of each available sampling rate. The control target is to reach the similar performances to those the faster single rate controller would achieve. Model-based cancellation controllers are designed using this approach and promising results are obtained.

Model-based multirate controllers design

http://hflab.k.u-tokyo.ac.jp/fujimoto/papers/cep02.pdf

High-performance servo systems based on multirate sampling control $

In this paper, novel multirate two-degree-of-freedom controllers are proposed for digital control systems, where it is restricted that the speed of the A/D converters are slower than that of the D/A converters. The proposed feedforward controller assures the perfect tracking at M inter-sampling points. Moreover, it is shown that the structure of the proposed perfect tracking controller is very simple and clear. Next, the proposed method is extended to systems with time delay. The proposed scheme is applied to the seeking control for hard disk drive, and advantages of this approach are demonstrated by simulations and experiments.

/pdf/proposal-of-seeking-control-of-hard-disk-drives-based-on-578lv62h2w.pdf

Proposal of seeking control of hard disk drives based on perfect tracking control using multirate feedforward control

The paper is concerned with the analysis of multi-rate controllers, which update the controller output faster than the measurement sampling frequency by a factor of N. The controlled system is continuous, and the discrete time controller is obtained as its zero order hold equivalent. After deriving a set of equations necessary for analysis and design of multi-rate systems, the open loop characteristics of the multi-rate controller and the closed loop characteristics are examined. Several physical meanings can be derived from the formulation of the multi-rate controller.

Analysis of multi-rate discrete equivalent of continuous controller

This paper presents the digital redesign of continuous time controller when the measurement sampling period is relatively large and standard discretization methods are most likely not able to provide a suitable digital controller from either stability or performance point of view. An analysis tool is provided to improve the performance of digital control systems by the following two measures: 1) update the controller output N times faster than the measurement sampling frequency; and 2) utilize past sampled outputs to predict the evolution of the output error between two consecutive sampling instances. The analysis tool is developed by regarding the overall system as a multirate sampled data system. An illustrative example is given to show the utilization of the developed analysis tool.

Digital redesign of continuous time controller by multirate sampling and high order holds

High performance tracking control system under measurement constraints by multi-rate control

This paper deals with performance enhancement of tracking control systems by multi-rate control. The feedback controller is updated at the same rate as the sampling rate of the output measurements. The feedforward controller processes the desired output signal for high accuracy tracking, and its output is updated at a rate N-times faster than the sampling rate of the output measurements. The discrete time model of the controlled plant may possess unstable zeros, and the zero phase error tracking controller is used as a feedforward controller. Inter-sample behavior of the plant is included in evaluating the tracking performance of the multi-rate system. Illustrative examples are given to show advantages of the proposed multi-rate feedback/feedforward control scheme.

Multi-Rate Feedforward Tracking Control for Plants With Nonminimum Phase Discrete Time Models

https://ieeexplore.ieee.org/iel5/6343/16976/00786375.pdf

Yuping Gu

Papers

Analysis of multi-rate discrete equivalent of continuous controller

Digital redesign of continuous time controller by multirate sampling and high order holds

High performance tracking control system under measurement constraints by multi-rate control

Multi-Rate Feedforward Tracking Control for Plants With Nonminimum Phase Discrete Time Models

Multi-rate feedforward tracking control for plants with nonminimum phase discrete time models