![Performance Evaluation of Adaptive
Dither Control Frameworks
for Nonlinear Piezoactuator
Sandip Jana, Saikat Kumar Shome, Arpita Mukherjee,
and Partha Bhattacharjee
Abstract Piezoelectric actuators are one of the promising industrial actuators in the
precision positioning industry. However, the high quotient of nonlinearity inherently
present degrades their performance. Dither control is specifically used for control
of nonlinear applications through relatively simple and computationally inexpensive
logic. This research focuses on performance evaluation of different adaptive dither
control frameworks in the parlance of piezoelectric actuation. The results are then
compared with well-established control frameworks like feed forward and cascaded
feed forward and feedback topology. The observations suggest dither control as a
promising controller for piezoelectric actuation under several disturbances.
1 Introduction
Piezoelectric actuators are a class of flagship actuators which has been used in
micro/nano-manipulation-related industrial applications such as biological cell oper-
ator, precision engineering in PCB industry, MEMS manufacturing, space optics,
etc. due to prominent advantages like precision positioning [1], micropositioning
[2], nano-positioning [3], speed of response, resistance to electromagnetic interfer-
ence,hightorquehandlingcapabilityandhighforcecapabilityin[4–6].Ditheringofa
nonlinear system is used in different industry applications like several areas of science
and technology, including geophysics, biomedical applications, microelectronics and
multidisciplinary mechatronics.
In [7], dithering is introduced in robotics nonlinear manipulator to enhance the
input signal with the aid of outside noise. The occurrence of dithering is mostly
observed in nonlinear dynamic systems wherein the introduction of a weak input,
S. Jana · S. K. Shome (B) · A. Mukherjee
Academy of Scientific and Innovative Research (AcSIR), CSIR- Central Mechanical Engineering
Research Institute (CSIR-CMERI) Campus, Durgapur 713209, India
e-mail: saikatkshome@cmeri.res.in
S. Jana · S. K. Shome · A. Mukherjee · P. Bhattacharjee
I. T. Group, CSIR-Central Mechanical Engineering Research Institute, Govt of India, Durgapur
713209, India
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022
J. Gu et al. (eds.), Communication and Control for Robotic Systems, Smart Innovation,
Systems and Technologies 229, https://doi.org/10.1007/978-981-16-1777-5_1
3](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-23-320.jpg)
![4 S. Jana et al.
mostly considered as noise, leads to enhanced SNR of the process, which was first
reported by [8]. Three basic criteria are responsible for this occurrence—(i) an energy
activation barrier or threshold, (ii) presence of a weak periodic input and (iii) a noise
source which can either be inherent in the system or added externally. Provided
a system has these features, a manifestation of increased signal-to-noise ratio is
observed in correspondence to the disturbance which is referred to as stochastic
resonance. Accordingly, in a nutshell, dithering may be considered as purposefully
varying the intensity of noise to raise the performance of the nonlinear system.
Dithering has been used in several industrial applications, for example, reduction
of quantization error in analog to digital conversions, geographical applications like
accounting for periodicity of earth ice age and even in microelectronics semicon-
ductor industry [9, 10]. Piezoelectric actuators have been recently explored in the
domain of stochastic resonance toward improving its performance and this research
put forward a comparison of different adaptive dither control approaches along with
their advantages and disadvantages. In [11], a controller centered on adaptive dither
in voltage mode on spectral analysis of piezoactuator is introduced and controller
adaptively adjusted the dither amplitude to get optimum value and improved the
system performance. The proposed controller has better sinusoidal path tracking,
mixed-amplitude mixed-frequency input signal, outside disturbances like Gaussian,
impulse, step response compared with the classical feedback and feed forward
control.
The contribution of the paper is as follows: Different controllers been modeled
in MATLAB Simulink on the nonlinear piezoelectric actuator. The studied adap-
tive dither controllers have then been evaluated by exposing the plant model to
different real-time disturbances like error comparison, parameter sensitivity test,
mixed-amplitude mixed-frequency input signal, additional disturbances with set
pointtrackinglikesetpointtrackinginconjunctiontoimpulsedisturbances(delay1s,
sampletime20s), stepinput withimpulsedisturbanceandzeroset point trackingwith
Gaussian noise which are discussed subsequently. The different controllers studied
in this study are as follows: C1: Feed forward traditional control (FF), C2: Cascaded
feed traditional forward and feed back (FF + FB), C3: Cascaded feed traditional
forward and feed back—displacement dithered, C4: Cascaded feed forward and feed
back—voltage dithered, C5: Cascaded feed forward and feed back—adaptive voltage
dithered.
2 RMS Error Comparison
The study has been carried using a sinusoidal wave signal as an input amplitude of 10
µm, and frequency range of signal 1 Hz which is to be followed by the piezoactuator.
The RMS error of the corresponding control schemes is placed in Table 1.
It is observed that order of performance of controllers is as per the following order
Adaptive Voltage Dither, FF + FB, FF shown in Fig. 1.](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-24-320.jpg)
![Performance Evaluation of Adaptive Dither Control Frameworks … 5
Table 1 RMS error of various controller
Controllers C1 C2 C3 C4 C5
Error(nm) 6.67 0.26315 0.23277 0.23627 0.23628
Time(S)
Displacement
(m)
Zoomed view
Fig. 1 Tracking output response of different controller with sinusoidal input
3 Parameter Sensitivity Test
As presented in Ref [9], the main reason for occurrence of mismatch in plant and
model is accountable to the approximations while transforming a physical plant to its
mathematical model. This mismatch is quite common and takes place due to several
reasons like improper identification of system, or inappropriate order consideration
of the system. Besides, nonlinear properties also add up to this error in approximation.
In this section considers a plant model parameter mismatch of ± 2% and ± 5% in
plant parameters as compared to the classical model and the RMS errors are presented
in Tables 2 and 3, respectively. From Tables 2 and 3, it can be concluded that adaptive
voltage dither have the better compensation of the parametric uncertainty than FF
+ FB. The efficiency of adaptive voltage dither over FF + FB is clearly seen as the
parametric variation is increased to 5%
Table 2 RMS error for plant parameter variation of ± 2%
Variation (%) Controller M D K T
−2 C2 0.12933 0.24121 0.17788 0.15667
2 C2 0.16073 0.13105 0.21483 0.18924
−2 C5 0.06678 0.07269 0.07067 0.07657
2 C5 0.07380 0.06841 0.07035 0.07484](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-25-320.jpg)
![Comparative Performance Study
of Different Controllers for Nonlinear
Piezoelectric Stack Actuator
Sandip Jana, Saikat Kumar Shome, Arpita Mukherjee,
and Partha Bhattacharjee
Abstract Robust and adaptive nonlinear controllers play an indispensable role in
industrial applications targeting nanopositioning. Internal model control has been
widely used in compensation of hysteresis, creep and vibration for piezoelectric
actuators along with least-square estimator regimes. However, as real-time scenarios
and presence of uncertainties, disturbances play a crucial factor in determining the
efficacy of the chosen controller, a comparative analysis is very effective in this situ-
ation. This research focuses on an extensive comparative analysis of two controllers,
namely, modified-internal model control (M-IMC) and recursive least-square esti-
mator (RLSE) under several test cases. Results put forward which controller is effec-
tive under what scenario that enhances the overall precision positioning efficiency
of the system.
1 Introduction
Piezoelectric actuators are a class of flagship actuators which has been used in
micro/nano-manipulation-related industrial applications such as biological cell oper-
ator, precision engineering in PCB industry, MEMS manufacturing, space optics,
etc. due to prominent advantages like precision positioning, speed of response, resis-
tance to electromagnetic interference, high torque handling capability and high force
capability in [1–3]. In [4] paper, adaptive feed forward control and recursive least-
squareestimator(RLSE)withadaptivefeedforward(FF)controllerarepresentedthat
performs better for parameter uncertainty of the system. The PSO and BFO-based
redesigned internal model control (M-IMC) with PID controller is proposed in [5]
for nonlinear piezoactuator which allows good set point tracking and disturbance
S. Jana · S. K. Shome (B) · A. Mukherjee
Academy of Scientific and Innovative Research (AcSIR), CSIR- Central Mechanical Engineering
Research Institute (CSIR-CMERI) Campus, Durgapur 713209, India
e-mail: saikatkshome@cmeri.res.in
S. Jana · S. K. Shome · A. Mukherjee · P. Bhattacharjee
Ministry of Science and Technology, CSIR-Central Mechanical Engineering Research Institute,
Government of India, Durgapur 713209, India
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022
J. Gu et al. (eds.), Communication and Control for Robotic Systems, Smart Innovation,
Systems and Technologies 229, https://doi.org/10.1007/978-981-16-1777-5_2
13](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-33-320.jpg)
![14 S. Jana et al.
rejection properties. Internal model control is one of the simplest control strategies
being applied for several applications. PID-based IMC control strategy which is the
most practically used controller is one of the reasons of its widespread acceptance
and has hardware implementation reliability. A tuning method of IMC to control
overshoot in a speed controlling system—a crucial subsystem in the hydro-electric
power plant is elaborated in [6]. In this research, PSO-based M-IMC controller has
improved path tracking and disturbance rejection performance compare with the clas-
sical IMC controller. Piezoactuators are widely used for precision and positioning
control at micro-nanometer level in several industrial applications and have different
advantages like higher accuracy, quick response, and enhanced torque. An improved
PID-based IMC controller is proposed for nonlinear hysteretic piezoelectric actuator
in [7] and result proved that controller performance improved over the traditional
control. The piezoactuator is modeled as a second-order system by experimentally
identifiedparameterandM-IMCcontrollerisimplementedintimedelayenvironment
approximated using first-order Padé expansion. A Smith predictor-based M-IMC is
also proposed in [8] where M-IMC controller decreased overshoot and settling time
of the response compared to traditional IMC and PID designs.
Contributions and problems addressed in the paper are as follows: Different
controllershavebeenperformedinMATLABSimulinkonthenonlinearpiezoelectric
actuator mentioned in Sect. 1.2. Presented controller has been explored by exposing
the plant model to different disturbances like error comparison, parameter sensitivity
test, mixed-amplitude, mixed-frequency input signal, external disturbances with set
point tracking like zero path tracking with impulse disturbances (delay 1 s, sample
time 20 s), zero path tracking with step input conjunction with impulse disturbances
and zero set point tracking with Gaussian noise. The controller performance result
suggests that controller’s performance depends upon the various application of the
system.
2 RMS Error Comparison
The study has been carried using a sinusoidal wave signal as an input amplitude of
10 µm, and frequency range of signal 1 Hz which is to be followed by the piezoactu-
ator. The RMS error of the corresponding control schemes is placed in Table 1. C1:
Feed forward traditional control (FF), C2: Cascaded feed traditional forward and feed
back (FF + FB), C3: Recursive least-square estimator (RLSE), C4: Modified-internal
model control (M-IMC).
Table 1 Tracking RMS error of different control paradigms
Controllers C1 C2 C3 C4
Error(nm) 6.67 0.26315 0.19405 0.16577](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-34-320.jpg)
![Comparative Performance Study of Different Controllers … 15
Fig. 1 Tracking performance plot of different adaptive controller with sinusoidal input response
It is observed that order of performance of controllers is as per the following order
M-IMC, RLSE, FF + FB, FF as shown in Fig. 1.
3 Parameter Sensitivity Test
As illustrated in Ref [6], the main reason for occurrence of mismatch in plant and
model is accountable to the approximations while transforming a physical plant to its
mathematical model. This mismatch is quite common and takes place due to several
reasons like improper identification of system, or inappropriate order consideration
of the system. Besides, nonlinear properties also add up to this error in approximation.
The following section considers a plant model parameter mismatch of ± 2% and ±
5% in plant parameters as compared to the traditional model and the RMS errors are
presented in Tables 2 and 3, respectively. It has also been observed earlier that M-IMC
offers the best RMS tracking error (0.16577 nm) among the observed controllers.
Accordingly, to determine the deviation from this value, Tables 2 and 3 are presented
which contains the difference of the RMS errors for plant model mismatch with the
M-IMC controller value. From Tables 2 and 3, it can be concluded that M-IMC and
RLSE have the better compensation of the parametric uncertainty, which is followed
Table 2 RMS error for plant parameter variation of ± 2%
Variation (%) Controller M D K T
−2 C2 0.12933 0.24121 0.17788 0.15667
2 C2 0.16073 0.13105 0.21483 0.18924
−2 C4 0.0209 0.0553 0.00947 0.00453
2 C4 0.00368 0.01266 0.001 0.0047
−2 C3 0.02309 0.02365 0.02660 0.02917
2 C3 0.03375 0.03312 0.02993 0.02748](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-35-320.jpg)
![Comparative Performance Study of Different Controllers … 23
3. Shengdong, Y., Jinyu, M., Hongtao, W., Shengzheng, K.: Robust precision motion control of
piezoelectric actuators using fast nonsingular terminal sliding mode with time delay estimation.
Measure. Control 52, 11–19, (2019)
4. Shome, S.K., Mukherjee, A., Karmakar, P., Datta, U.: Adaptive Feedforward controller of piezo-
electric actuator for micronano positioning” in Sadhana, Indian Academy of Science, Springer,
2018. Issue—Academy proceedings in Engineering Science, Vol. 43(10) (2018)
5. Shome,S.K.,Jana,S.,Mukherjee,A.,Bhattarcharjee,P.,Datta,U.:Bioinspiredmodifiedinternal
model control approach for improved disturbance rejection of piezo micro manipulator. Stud.
Inf. Control 27, 295–306 (2018). https://doi.org/10.24846/v27i3y201805
6. Naik, K.A., Srikanth, P., Negi, P.: IMC tuned PID governor controller for hydro power plant
with water hammer effect[J]”. Procedia Technol. 4(4) (2012)
7. Shome, S.K., Jana, S., Mukherjee, A., Bhattacharjee, P., Datta, U.: Improved internal model
control based closed loopcontroller design for second order piezoelectric system with dead
time. In: Proceedings of 2018 8th IEEE India International Conference on Power Electronics;
Jaipur, India. pp. 1–6 (2010)
8. Shome, S., Jana, S., Mukherjee, A., Bhattarcharjee, P.: Model based control for second order
piezo actuator system with hysteresis in time delay environment. Turk. J. Elec. Eng. Comp. Sci.
https://doi.org/10.3906/elk-1907-28 (2019)](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-43-320.jpg)
![PID Controller Design for Reference
Tracking of Single Link Manipulator
Sayan Das and Naiwrita Dey
Abstract This paper is aimed to design a PID controller for reference signal
tracking. Design problem is formulated to control the angular position of a single-
link manipulator at any particular desired set value. Implementation of the overall
feedback control system and the proposed controller has been carried out in Python
language. Additive uncertainty is considered with the transfer function model. Simu-
lation results illustrate the usefulness of the proposed controller for angular velocity
tracking in presence of uncertainty. Average mean square error and settling time have
been calculated. Overall stability of the feedback system has been tested by obtaining
the eigenvalues of the closed-loop system.
1 Introduction
Robotics is getting more and more popular in industrial field as it reduces manpower
required for doing heavy jobs. Main types of robots used in industrial applications are
“multiple-linkmanipulators”withdifferentdegreesoffreedom.Thesekindsofrobots
can move according to their DOF and complete many tasks. Single-link manipulator
model is made out of one fixed base, on which the whole system resides, a revolute
joint and an arm, which can move freely on the x–y plane. DC motor or servomotors
are mainly used to rotate the arms of manipulators. Here, a simple DC motor with high
torque constant is used to move the arm of the manipulator. First, the dynamic model
of that motor is designed [1]. Then, control logic is applied on the motor to move
the arm to a specific angular position [2–4]. For tuning control parameters, simple
tuning method is used [5]. Nowadays, in many industrial applications, rigid link
manipulators cannot satisfy the needs. So, “Flexible link manipulators” have become
an option. In comparison with rigid body manipulators, flexible manipulators have
many advantages including higher efficiency, more flexibility, high load capacity,
S. Das
Department of AEIE, RCCIIT, Kolkata, India
N. Dey (B)
Department of ECE, RCCIIT, Kolkata, India
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022
J. Gu et al. (eds.), Communication and Control for Robotic Systems, Smart Innovation,
Systems and Technologies 229, https://doi.org/10.1007/978-981-16-1777-5_3
25](https://image.slidesharecdn.com/11468-250311081840-9cf5d76b/85/Communication-and-Control-for-Robotic-Systems-Smart-Innovation-Systems-and-Technologies-229-Jason-Gu-Editor-44-320.jpg)
Communication and Control for Robotic Systems Smart Innovation Systems and Technologies 229 Jason Gu (Editor)
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- 5. 123 Smart Innovation, Systems andTechnologies 229 Jason Gu Rajeeb Dey Nabanita Adhikary Editors Communication and Control for Robotic Systems
- 6. Smart Innovation, Systems and Technologies Volume 229 Series Editors Robert J. Howlett, Bournemouth University and KES International, Shoreham-by-Sea, UK Lakhmi C. Jain, KES International, Shoreham-by-Sea, UK
- 7. The Smart Innovation, Systems and Technologies book series encompasses the topics of knowledge, intelligence, innovation and sustainability. The aim of the series is to make available a platform for the publication of books on all aspects of single and multi-disciplinary research on these themes in order to make the latest results avail- able in a readily-accessible form. Volumes on interdisciplinary research combining two or more of these areas is particularly sought. The series covers systems and paradigms that employ knowledge and intelligence in a broad sense. Its scope is systems having embedded knowledge and intelligence, which may be applied to the solution of world problems in industry, the environment and the community. It also focusses on the knowledge-transfer methodologies and innovation strategies employed to make this happen effectively. The combination of intelligent systems tools and a broad range of applications introduces a need for a synergy of disciplines from science, technology, business and the humanities. The series will include conference proceedings, edited collections, monographs, handbooks, reference books, and other relevant types of book in areas of science and technology where smart systems and technologies can offer innovative solutions. High quality content is an essential feature for all book proposals accepted for the series. It is expected that editors of all accepted volumes will ensure that contributions are subjected to an appropriate level of reviewing process and adhere to KES quality principles. Indexed by SCOPUS, EI Compendex, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST), SCImago, DBLP. All books published in the series are submitted for consideration in Web of Science. More information about this series at http://www.springer.com/series/8767
- 8. Jason Gu · Rajeeb Dey · Nabanita Adhikary Editors Communication and Control for Robotic Systems
- 9. Editors Jason Gu Department of Electrical and Computer Engineering Dalhousie University Halifax, NS, Canada Nabanita Adhikary Department of Electrical Engineering National Institute of Technology Silchar Silchar, Assam, India Rajeeb Dey Department of Electrical Engineering National Institute of Technology Silchar Silchar, Assam, India ISSN 2190-3018 ISSN 2190-3026 (electronic) Smart Innovation, Systems and Technologies ISBN 978-981-16-1776-8 ISBN 978-981-16-1777-5 (eBook) https://doi.org/10.1007/978-981-16-1777-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
- 10. Preface This monograph is a collection of high-quality research articles disseminated as a research outcome of multiple authors in the areas of communication, computation and control of robotic systems. Through this monograph one of the objective of the project grant under the Scheme for Promotion of Academic and Research Collabo- ration (SPARC) funded by the Minister of Human Resource Development (MHRD) Government of India has been accomplished. This monograph has helped editors to identify and establish research collaboration within India and abroad. The Editors of the book acknowledge thanks to the MHRD for funding the collaborative research between National Institute of Technology Silchar, India and Dalhousie University, Canada. The book includes topics specific to the theme of the symposium, viz. emerging control for robotic systems, wireless communication and development of embedded systems for robotic applications. Robot control technology is widely used for space, surgery, rehabilitation, micro-machine, entertainment, underwater explo- ration, civil engineering, professional and domestic services, security, etc. Conse- quently, control will continue to play an increasing role in the areas of robotics including robot–robot and human–robot cooperation in various dynamic scenarios. Subsequently, due to changing communication infrastructure and technologies, it is highly desirable these days to propagate control commands through wireless commu- nication channels. Thus, the inclusion of wireless communication for control of robotic systems poses serious challenges to the performance of the control system, thereby necessitating the use of cutting-edge technologies for the development and testing of control algorithms for robotic systems under such a communication envi- ronment. Moreover, real-time implementation of the control through present-day communication networks requires realizable and scalable embedded design. Contri- butions on basic research of control over the network as well as on relevant robotics applications are also included in this book. This book uniquely provides a unified framework for analysis, design and deploy- ment of the robotic applications across various engineering and non-engineering disciplines including the three primary aspects focal to the symposium, i.e. control, communication and embedded systems. The entire volume consists of thirty chapters divided into five parts dedicated respectively to Control Theories for Robotic Systems, Communication and Control for Robotic Systems, Computation v
- 11. vi Preface Paradigm for Robotic Systems, Power Electronics Application for Transportation and Innovations in Robotic Systems. The first part includes six chapters on emerging control strategies for robotic applications. The articles describe modelling aspects of the robots specific to the application environment and thereby discuss the challenges involved in the modelling and the control theory. The validity of the developed theories is supported by the results obtained from the software or hardware implementation of the robotic system under consideration. The second part consists of six chapters on emerging communication technology at the different levels of robotics and automation systems. These articles provide an easy and structured explanation on the need of integral communication technology in an attempt to achieve the challenging control performance for various robotic applications. The specific challenges in the interface of communication technology and control which cannot be solved with conventional control theory are included in the articles. This section also includes articles from networked control system with due emphasis on robotic applications. The third part includes eight chapters on the use of intelligent- and knowledge- based computational techniques for solving highly complex, nonlinear and non- traditional robotic environment pertaining to control, communication or combination of both. The chapters give a clear theoretical integration of the computational tech- niques with the communication and control theories such that the readers can have a good comprehension of the theoretical and implementation basis for developing robotic applications. The fourth part of the book includes six chapters on the recent advances on power supplies and power electronics pertaining to autonomous and robotic transportation systems with a focus on electric vehicles. The fifth part of the book consists of four chapters describing recent innovations in various areas of robotics. These chapters mostly deal with application and devel- opment of robotic systems for agriculture, autonomous vehicles, launch vehicles, etc. We are thankful to all the authors who helped in keeping the quality of the mono- graph at a high level by their contributed papers as well the reviewers for giving their time to rigorously review the papers. The book is addressed to the research community including professors and students in robotics, control systems as well as power electronics. We hope that the book will provide useful information and inspiration to its intended readers. Halifax, Canada Silchar, India Silchar, India Jason Gu Rajeeb Dey Nabanita Adhikary
- 12. Contents Control Theories for Robotic Systems Performance Evaluation of Adaptive Dither Control Frameworks for Nonlinear Piezoactuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Sandip Jana, Saikat Kumar Shome, Arpita Mukherjee, and Partha Bhattacharjee Comparative Performance Study of Different Controllers for Nonlinear Piezoelectric Stack Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Sandip Jana, Saikat Kumar Shome, Arpita Mukherjee, and Partha Bhattacharjee PID Controller Design for Reference Tracking of Single Link Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Sayan Das and Naiwrita Dey Performance Comparison Between Higher-Order Sliding Mode and Fixed Boundary Layer Sliding Mode Controller for a 10-DoF Bipedal Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Koceila Cherfouh, Jason Gu, Umar Farooq, Muhammad Usman Asad, Rajeeb Dey, Nabanita Adhikary, and Chunqi Chang Adaptive Backstepping-Based Non-singular Finite-Time Sliding Mode Controller for Suspension of Maglev Platforms . . . . . . . . . . . . . . . . . 63 Nabanita Adhikary and Jobin Mathew Backstepping-Based Nonlinear Control of Underactuated 2-DoF Gyroscope for Robust Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 89 Manish Patel and Bhanu Pratap Communication and Control for Robotic Systems Adaptive Backstepping Control of Multiple Mobile Robots Under Limited Communication; An Event-Triggered Approach . . . . . . . . . . . . . . 107 Sami Al Issa and Indrani Kar vii
- 13. viii Contents Adaptive Robust Control of Tele-operated Master-Slave Manipulators with Communication Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Nabanita Adhikary, Rajeeb Dey, Muhammad Usman Asad, Jason Gu, Umar Farooq, and Rupak Dutta A Multi-Master Single-Slave Teleoperation System Through Composite State Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Muhammad Usman Asad, Jason Gu, Umar Farooq, Rajeeb Dey, Nabanita Adhikary, Rupak Datta, and Chunqi Chang Event-Triggered Integral Sliding Mode Control for an Uncertain Euler–Lagrange System with Actuator Saturation . . . . . . . . . . . . . . . . . . . . 155 Krishanu Nath, Asifa Yesmin, and Manas Kumar Bera AKIRA—A Voice Based Virtual Assistant with Authentication . . . . . . . . 177 Vishnu Sai Bhonsle, Sailaja Thota, and Surekha Thota Survey on Internet of Things Based Intelligent Wireless Sensor Network for Fire Detection System in Building . . . . . . . . . . . . . . . . . . . . . . . 193 Arpita Mukherjee, Saikat Kumar Shome, and Partha Bhattacharjee Computation Paradigm for Robotic Systems Nonlinear State Estimation Using Adaptive Gaussian Filters with One-Step Randomly Delayed Measurements . . . . . . . . . . . . . . . . . . . . 203 Poluri Sri Mannarayana and Aritro Dey Extended Kalman Filter-based Attitude Estimation using Magnetometer- and Sun Sensor-Aided MEMS Gyros . . . . . . . . . . . . . . . . . 221 Jiljo K. Moncy and Kesavabrahmaji Karuturi Characterisation of Multi-sensor 6D Pose Determination System for Underslung Winged Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 K. L. N. Sai Nitish, Jiljo K. Moncy, M. Dinesh Kumar, B. Karthik, V. T. Basker, and E. S. Padma Kumar Tracking and Interception of a Ballistic Target on Reentry Using Adaptive Gaussian Sum Quadrature Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Rahul Radhakrishnan, Manika Saha, Shovan Bhaumik, and Nutan Kumar Tomar Three-Dimensional Bearings-Only Target Tracking: Comparison of Few Sigma Point Kalman Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 U. Asfia, R. Radhakrishnan, and S. N. Sharma Performance Comparison of EKF and UKF for Offshore Boom Crane System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Manash Jyoti Deori, Nabanita Adhikary, and Krishna Jyothi Pallacherla
- 14. Contents ix Performance Analysis of Speech Command Recognition Using Support Vector Machine Classifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 M. Venkata Subbarao, Akhendra Kumar Padavala, and Kudupudi Durga Harika Parameter Identification of Coulomb Oscillator from Noisy Sensor Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Guddu Kumar, Vikash Kumar Mishra, R. Swaminathan, and Abhinoy Kumar Singh Power Electronics Application for Transportation Parameter Extraction of Dye-Sensitized Solar Cell: Improved Method and Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Soumik Chakraborty, Ranjith G. Nair, and Lalu Seban Role of Battery Management System (BMS) in Sustainable Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 B. S. Sagar, Rajashekar P. Mandi, B. P. Divakar, and Santoshkumar Hampannavar A Fundamental Study on Electric Vehicle Model for Longitudinal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 K. Indu and M. Aswatha Kumar Supercapacitor-Based Automated Fast Charging System for Electric Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Saswati Pattnaik, Mano Ranjan Kumar, and Sunil Kumar Mishra Different Control Mechanisms of a PMSM Drive for Electrified Transportation—A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Chiranjit Sain, Atanu Banerjee, Pabitra Kumar Biswas, Sudhakar Babu Thanikanti, and Karthik Balasubramanian Direct Torque Control of BOOST-VSI Fed Induction Motor Using SVPWM-PAM Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Sudharani Potturi and Rajashekar P. Mandi Innovations in Robotic Systems Launch Vehicle Autopilot Design Using H-Infinity Control Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Chintapalli Vaishnavi and M. V. Dhekane Design and Development of the Pineapple Harvesting Robotic Gripper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Francis Kurbah, Shemphang Marwein, Teiborlin Marngar, and Bikash Kumar Sarkar
- 15. x Contents Three-Axis Wireless Gesture-Controlled Car . . . . . . . . . . . . . . . . . . . . . . . . . 455 Naiwrita Dey, Tridib Dey, Rishav Chakraborty, Sibam Jana, and Abul Hassan Design of Robust Controller for Enhanced Performance of 2-DOF Torsion System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Priyam Rai and Bhanu Pratap
- 16. Editors and Contributors About the Editors Jason Gu is currently Professor in the Department of Electrical Engineering at Dalhousie University, Canada. Professor Jason Gu received the bachelor’s degree in electrical engineering and information science from the University of Science and Technology of China, in 1992, the master’s degree in biomedical engineering from Shanghai Jiao Tong University, in 1995, and the Ph.D. degree from the Univer- sity of Alberta, Canada, in 2001. He is currently Full Professor of electrical and computer engineering at Dalhousie University, Canada. He is also Cross-Appointed Professor with the School of Biomedical Engineering for his multidisciplinary research work. His research interests include robotics, biomedical engineering, reha- bilitation engineering, neural networks, and control. He is Fellow of the Engineering Institute of Canada. He is IEEE Canada President-elect, 2018–2019, and IEEE Canada President, 2020–2021. Rajeeb Dey is currently Assistant Professor in the Department of Electrical Engineering at the National Institute of Technology, Silchar, India. He holds M.Tech. degree in control system engineering, from the Indian Institute of Technology, Kharagpur, India, and Ph.D. degree also in control system engineering from Jadavpur University, Kolkata, India. He is Senior Member of IEEE Control System Society, Member Institution of Engineers (India), Executive Committee Member of ACDOS (Advanced Control and Dynamic Optimization Society), and Life Member of System Society of India. His research interest includes design of robust control, optimiza- tion based on LMI techniques, time-delay systems, intelligent control, decentralized control and control applications, and biomedical control applications. Nabanita Adhikary is currently Assistant Professor in the Department of Elec- trical Engineering at the National Institute of Technology, Silchar, India. She holds Ph.D. degree in control system engineering, from the Indian Institute of Technology Guwahati, India. She is Member of the IEEE Control System Society and IEEE Robotics and Automation Society. Her research interests include nonlinear control xi
- 17. xii Editors and Contributors system design, robust control, time-delay control, intelligent control, robotics, and networked control systems. Contributors Nabanita Adhikary Department of Electrical Engineering, National Institute of Technology Silchar, Silchar, India Sami Al Issa Department of Electronics and Electrical Engineering, IIT Guwahati, Guwahati, India; Department of Computer Engineering and Automation, Damascus University, Damascus, Syria Muhammad Usman Asad Department of Electrical and Computer Engineering, Dalhousie University, Halifax, Canada U. Asfia Department of Electrical Engineering, SV National Institute of Technology Surat, Surat, India M. Aswatha Kumar Christ (Deemed to be University), Bengaluru, Karnataka, India Karthik Balasubramanian Design and Engineering (Electrical), Offshore Tech- nology Development Pte., Ltd., Keppel Offshore and Marine Ltd., Singapore, Singapore Atanu Banerjee Department of Electrical Engineering, National Institute of Tech- nology Meghalaya, Bijni Complex, Laitumukhrah, Shillong, Meghalaya, India V. T. Basker Vikram Sarabhai Space Centre, I.S.R.O., Thiruvananthapuram, India Manas Kumar Bera Department of Electronics and Instrumentation Engineering, NIT Silchar, Silchar, India Partha Bhattacharjee I. T. Group, Ministry of Science and Technology, CSIR- Central Mechanical Engineering Research Institute, Government of India, Durgapur, India Shovan Bhaumik Department of Electrical Engineering, Indian Institute of Tech- nology Patna, Patna, India Vishnu Sai Bhonsle REVA University, Bangalore, Karnataka, India Pabitra Kumar Biswas Department of Electrical and Electronics Engineering, National Institute of Technology Mizoram, Aizawl, India Rishav Chakraborty Applied Electronics & Instrumentation Engineering, RCC Institute of Information Technology, Kolkata, West Bengal, India
- 18. Editors and Contributors xiii Soumik Chakraborty Department of Electronics and Instrumentation Engineering, National Institute of Technology, Silchar, Cachar, Assam, India; Solar Energy Materials Research and Testing Laboratory, (Smart Lab) Department of Physics, National Institute of Technology, Silchar, Cachar, Assam, India Chunqi Chang School of Biomedical Engineering, Health Sciences Center, Shenzhen University, Shenzhen, China Koceila Cherfouh Department of Electrical & Computer Engineering, Dalhousie University, Halifax, Canada Sayan Das Department of AEIE, RCCIIT, Kolkata, India Rupak Datta Department of Mathematics, National Institute of Technology Agar- tala, Agartala, India Manash Jyoti Deori Department of Electrical Engineering, National Institute of Technology Silchar, Silchar, India Aritro Dey Department of Electrical Engineering, National Institute of Technology Durgapur, Durgapur, West Bengal, India Naiwrita Dey Electronics and Communication Engineering, RCC Institute of Information Technology, Kolkata, West Bengal, India Rajeeb Dey Department of Electrical Engineering, NIT, Silchar, Assam, India Tridib Dey Applied Electronics & Instrumentation Engineering, RCC Institute of Information Technology, Kolkata, West Bengal, India M. V. Dhekane IIST, Thiruvananthapuram, India M. Dinesh Kumar Vikram Sarabhai Space Centre, I.S.R.O., Thiruvananthapuram, India B. P. Divakar REVA Universtiy, Bengaluru, India Rupak Dutta Mathematics Department, NIT Agartala, Agartala, India Umar Farooq Department of Electrical & Computer Engineering, Dalhousie University, Halifax, Canada Jason Gu Department of Electrical & Computer Engineering, Dalhousie Univer- sity, Halifax, Canada Santoshkumar Hampannavar REVA Universtiy, Bengaluru, India Kudupudi Durga Harika Department of Electronics and Communication Engineering, Shri Vishnu Engineering College for Women (Autonomous), Bhimavaram, AP, India Abul Hassan Applied Electronics & Instrumentation Engineering, RCC Institute of Information Technology, Kolkata, West Bengal, India
- 19. xiv Editors and Contributors K. Indu Christ (Deemed to be University), Bengaluru, Karnataka, India Sibam Jana Applied Electronics & Instrumentation Engineering, RCC Institute of Information Technology, Kolkata, West Bengal, India Sandip Jana Academy of Scientific and Innovative Research (AcSIR), CSIR- Central Mechanical Engineering Research Institute (CSIR-CMERI) Campus, Durgapur, India; I. T. Group, CSIR-Central Mechanical Engineering Research Institute, Govt of India, Durgapur, India Indrani Kar Department of Electronics and Electrical Engineering, IIT Guwahati, Guwahati, India B. Karthik Vikram Sarabhai Space Centre, I.S.R.O., Thiruvananthapuram, India Kesavabrahmaji Karuturi ISRO Inertial Systems Unit, Thiruvananthapuram, India Guddu Kumar Discipline of Electrical Engineering, Indian Institute of Technology Indore, Indore, India Mano Ranjan Kumar School of Electronics Engineering, Kalinga Institute of Industrial Technology, Bhubaneswar, India Francis Kurbah NIT Meghalaya, Shillong, India Rajashekar P. Mandi REVA University, Bangalore, India Poluri Sri Mannarayana Department of Electrical Engineering, National Institute of Technology Durgapur, Durgapur, West Bengal, India Teiborlin Marngar NIT Meghalaya, Shillong, India Shemphang Marwein NIT Meghalaya, Shillong, India Jobin Mathew Department of Electrical Engineering, Assam Engineering College, Guwahati, India Sunil Kumar Mishra School of Electronics Engineering, Kalinga Institute of Industrial Technology, Bhubaneswar, India Vikash Kumar Mishra Electrical and Electronics Engineering, Government Engineering College Raipur, Raipur, India Jiljo K. Moncy Vikram Sarabhai Space Centre, I.S.R.O., Thiruvananthapuram, India Arpita Mukherjee CSIR-Central Mechanical Engineering Research Institute, Ministry of Science and Technology, Government of India, Durgapur, India Ranjith G. Nair Solar Energy Materials Research and Testing Laboratory, (Smart Lab) Department of Physics, National Institute of Technology, Silchar, Cachar, Assam, India
- 20. Editors and Contributors xv Krishanu Nath Department of Electronics and Instrumentation Engineering, NIT Silchar, Silchar, India Akhendra Kumar Padavala Department of Electronics and Communication Engineering, Shri Vishnu Engineering College for Women (Autonomous), Bhimavaram, AP, India E. S. Padma Kumar Vikram Sarabhai Space Centre, I.S.R.O., Thiruvanantha- puram, India Krishna Jyothi Pallacherla Department of Electrical Engineering, National Insti- tute of Technology Silchar, Silchar, India Manish Patel Department of Electrical Engineering, National Institute of Tech- nology Kurukshetra, Kurukshetra, Haryana, India Saswati Pattnaik School of Electronics Engineering, Kalinga Institute of Industrial Technology, Bhubaneswar, India Sudharani Potturi REVA University, Bangalore, India Bhanu Pratap Department of Electrical Engineering, National Institute of Tech- nology Kurukshetra, Kurukshetra, Haryana, India R. Radhakrishnan Department of Electrical Engineering, SV National Institute of Technology Surat, Surat, India Rahul Radhakrishnan Department of Electrical Engineering, SVNIT, Surat, India Priyam Rai DepartmentofElectricalEngineering,NationalInstituteofTechnology Kurukshetra, Haryana, India B. S. Sagar REVA Universtiy, Bengaluru, India Manika Saha Department of Electrical Engineering, Meghnad Saha Institute of Technology, Kolkata, India K. L. N. Sai Nitish Vikram Sarabhai Space Centre, I.S.R.O., Thiruvananthapuram, India Chiranjit Sain Department of Electrical Engineering, Siliguri Institute of Tech- nology, Sukna, Siliguri, Darjeeling, India Bikash Kumar Sarkar NIT Meghalaya, Shillong, India Lalu Seban Department of Electronics and Instrumentation Engineering, National Institute of Technology, Silchar, Cachar, Assam, India S. N. Sharma Department of Electrical Engineering, SV National Institute of Technology Surat, Surat, India Saikat Kumar Shome CSIR-Central Mechanical Engineering Research Institute, Ministry of Science and Technology, Government of India, Durgapur, India
- 21. xvi Editors and Contributors Abhinoy Kumar Singh Discipline of Electrical Engineering, Indian Institute of Technology Indore, Indore, India M. Venkata Subbarao Department of Electronics and Communication Engineering, Shri Vishnu Engineering College for Women (Autonomous), Bhimavaram, AP, India R. Swaminathan Discipline of Electrical Engineering, Indian Institute of Tech- nology Indore, Indore, India Sudhakar Babu Thanikanti Department of Electrical and Electronics Engineering, Chaitanya Bharathi Institute of Technology (CBIT), Hyderabad, India Sailaja Thota REVA University, Bangalore, Karnataka, India Surekha Thota REVA University, Bangalore, Karnataka, India Nutan Kumar Tomar Department of Mathematics, Indian Institute of Technology Patna, Patna, India Chintapalli Vaishnavi IIST, Thiruvananthapuram, India Asifa Yesmin Department of Electronics and Instrumentation Engineering, NIT Silchar, Silchar, India
- 22. Control Theories for Robotic Systems
- 23. Performance Evaluation of Adaptive Dither Control Frameworks for Nonlinear Piezoactuator Sandip Jana, Saikat Kumar Shome, Arpita Mukherjee, and Partha Bhattacharjee Abstract Piezoelectric actuators are one of the promising industrial actuators in the precision positioning industry. However, the high quotient of nonlinearity inherently present degrades their performance. Dither control is specifically used for control of nonlinear applications through relatively simple and computationally inexpensive logic. This research focuses on performance evaluation of different adaptive dither control frameworks in the parlance of piezoelectric actuation. The results are then compared with well-established control frameworks like feed forward and cascaded feed forward and feedback topology. The observations suggest dither control as a promising controller for piezoelectric actuation under several disturbances. 1 Introduction Piezoelectric actuators are a class of flagship actuators which has been used in micro/nano-manipulation-related industrial applications such as biological cell oper- ator, precision engineering in PCB industry, MEMS manufacturing, space optics, etc. due to prominent advantages like precision positioning [1], micropositioning [2], nano-positioning [3], speed of response, resistance to electromagnetic interfer- ence,hightorquehandlingcapabilityandhighforcecapabilityin[4–6].Ditheringofa nonlinear system is used in different industry applications like several areas of science and technology, including geophysics, biomedical applications, microelectronics and multidisciplinary mechatronics. In [7], dithering is introduced in robotics nonlinear manipulator to enhance the input signal with the aid of outside noise. The occurrence of dithering is mostly observed in nonlinear dynamic systems wherein the introduction of a weak input, S. Jana · S. K. Shome (B) · A. Mukherjee Academy of Scientific and Innovative Research (AcSIR), CSIR- Central Mechanical Engineering Research Institute (CSIR-CMERI) Campus, Durgapur 713209, India e-mail: saikatkshome@cmeri.res.in S. Jana · S. K. Shome · A. Mukherjee · P. Bhattacharjee I. T. Group, CSIR-Central Mechanical Engineering Research Institute, Govt of India, Durgapur 713209, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gu et al. (eds.), Communication and Control for Robotic Systems, Smart Innovation, Systems and Technologies 229, https://doi.org/10.1007/978-981-16-1777-5_1 3
- 24. 4 S. Jana et al. mostly considered as noise, leads to enhanced SNR of the process, which was first reported by [8]. Three basic criteria are responsible for this occurrence—(i) an energy activation barrier or threshold, (ii) presence of a weak periodic input and (iii) a noise source which can either be inherent in the system or added externally. Provided a system has these features, a manifestation of increased signal-to-noise ratio is observed in correspondence to the disturbance which is referred to as stochastic resonance. Accordingly, in a nutshell, dithering may be considered as purposefully varying the intensity of noise to raise the performance of the nonlinear system. Dithering has been used in several industrial applications, for example, reduction of quantization error in analog to digital conversions, geographical applications like accounting for periodicity of earth ice age and even in microelectronics semicon- ductor industry [9, 10]. Piezoelectric actuators have been recently explored in the domain of stochastic resonance toward improving its performance and this research put forward a comparison of different adaptive dither control approaches along with their advantages and disadvantages. In [11], a controller centered on adaptive dither in voltage mode on spectral analysis of piezoactuator is introduced and controller adaptively adjusted the dither amplitude to get optimum value and improved the system performance. The proposed controller has better sinusoidal path tracking, mixed-amplitude mixed-frequency input signal, outside disturbances like Gaussian, impulse, step response compared with the classical feedback and feed forward control. The contribution of the paper is as follows: Different controllers been modeled in MATLAB Simulink on the nonlinear piezoelectric actuator. The studied adap- tive dither controllers have then been evaluated by exposing the plant model to different real-time disturbances like error comparison, parameter sensitivity test, mixed-amplitude mixed-frequency input signal, additional disturbances with set pointtrackinglikesetpointtrackinginconjunctiontoimpulsedisturbances(delay1s, sampletime20s), stepinput withimpulsedisturbanceandzeroset point trackingwith Gaussian noise which are discussed subsequently. The different controllers studied in this study are as follows: C1: Feed forward traditional control (FF), C2: Cascaded feed traditional forward and feed back (FF + FB), C3: Cascaded feed traditional forward and feed back—displacement dithered, C4: Cascaded feed forward and feed back—voltage dithered, C5: Cascaded feed forward and feed back—adaptive voltage dithered. 2 RMS Error Comparison The study has been carried using a sinusoidal wave signal as an input amplitude of 10 µm, and frequency range of signal 1 Hz which is to be followed by the piezoactuator. The RMS error of the corresponding control schemes is placed in Table 1. It is observed that order of performance of controllers is as per the following order Adaptive Voltage Dither, FF + FB, FF shown in Fig. 1.
- 25. Performance Evaluation of Adaptive Dither Control Frameworks … 5 Table 1 RMS error of various controller Controllers C1 C2 C3 C4 C5 Error(nm) 6.67 0.26315 0.23277 0.23627 0.23628 Time(S) Displacement (m) Zoomed view Fig. 1 Tracking output response of different controller with sinusoidal input 3 Parameter Sensitivity Test As presented in Ref [9], the main reason for occurrence of mismatch in plant and model is accountable to the approximations while transforming a physical plant to its mathematical model. This mismatch is quite common and takes place due to several reasons like improper identification of system, or inappropriate order consideration of the system. Besides, nonlinear properties also add up to this error in approximation. In this section considers a plant model parameter mismatch of ± 2% and ± 5% in plant parameters as compared to the classical model and the RMS errors are presented in Tables 2 and 3, respectively. From Tables 2 and 3, it can be concluded that adaptive voltage dither have the better compensation of the parametric uncertainty than FF + FB. The efficiency of adaptive voltage dither over FF + FB is clearly seen as the parametric variation is increased to 5% Table 2 RMS error for plant parameter variation of ± 2% Variation (%) Controller M D K T −2 C2 0.12933 0.24121 0.17788 0.15667 2 C2 0.16073 0.13105 0.21483 0.18924 −2 C5 0.06678 0.07269 0.07067 0.07657 2 C5 0.07380 0.06841 0.07035 0.07484
- 26. 6 S. Jana et al. Table 3 RMS error for plant parameter variation of ± 5% Variation (%) Controller M D K T −5 C2 0.12855 0.20083 0.15985 0.11177 5 C2 0.09108 0.12610 0.16552 0.1882 −5 C5 0.06193 0.07403 0.07092 0.07598 5 C5 0.07869 0.06539 0.07012 0.07088 4 Performance Against External Disturbances In real-life scenario, the disturbances of the transfer model are not enough to support for the random disturbances which is effected to the plant of the system. The external disturbance has been represented as variance of a Gaussian noise in the system, first at the input side as a process noise, and second, as a measurement noise toward the plant output. The Gaussian error values of each of the following controller are shown in Tables 4 and 5, respectively, and can be assessed through two metrics, Table 4 RMS error for external process disturbance Gaussian noise variance C5(nm) C2 (nm) 10e-22 0.25435 0.38004 10e-23 0.22815 0.40858 10e-24 0.23325 0.38373 10e-25 0.23556 0.27314 10e-26 0.23611 0.36425 10e-27 0.23623 0.26052 10e-28 0.23628 0.25366 10e-29 0.23628 0.27080 Max–min 0.0262 0.15492 Average error 0.23702 0.32434 Table 5 RMS error for external measurement disturbance Gaussian noise (variance) C5 (nm) C2 (nm) 10e-22 0.24092 0.29815 10e-23 0.23688 0.27197 10e-24 0.23641 0.28242 10e-25 0.23631 0.37162 10e-26 0.23629 0.35077 10e-27 0.23629 0.38303 10e-28 0.23628 0.33925 (a) Max. value−min. value 0.00464 0.11106 Average error 0.23705 0.32817
- 27. Performance Evaluation of Adaptive Dither Control Frameworks … 7 Fig. 2 Error plot with sinusoidal wave input and external measurement noise (Gaussian, variance 10e-15) (a) variation induced due to the presence of the noise which is represented by a difference of maximum and minimum error value and (b) deviation of the average error due to noise with the best attained tracking error. From the table, it is observed that using the second metric, the adaptive voltage dither controller performance is better than FF + FB controller. A schematic of the error plot suggesting similar controller performance is shown in Fig. 2. 4.1 Gaussian Noise as Process Noise Disturbance See Table 4. 4.2 Gaussian Noise as Measurement Noise Disturbance See Table 5. 5 Mixed-Amplitude, Mixed-Frequency Input Test 5.1 Performance of the Controllers to Different Input Wave Signals—of Alternating Frequency and Amplitude In this section, the piezoactuator has been subjected to different input wave signals and alteration of frequency (0.25 Hz, 0.5 Hz and 1 Hz) and alteration of amplitude (2.5 µm, 5 µm and 10 µm). For all the controllers, a common phenomenon is observed—the tracking error increases with increase in input frequency and also with an increase in input wave signal amplitude. Among the comparison of different
- 28. 8 S. Jana et al. Table 6 RMS tracking error of the controller for mixed-amplitude, mixed-frequency signal Controllers C1 C2 C3 C4 C5 Error(nm) 6.3145 0.33506 0.30774 0.30705 0.30703 controllers has been shown in Table 6, it can be concluded that for a majority of input signals, the order of controller performance is as follows: Adaptive voltage dither, cascaded feed traditional forward and feed back—voltage dithered, cascaded feed traditional forward and feed back—displacement dithered, cascaded feed forward and feed back (FF + FB), and feed forward traditional control (FF). 5.2 Performance Evaluation of the Controllers to a Mixed-Amplitude, Mixed-Frequency Input Wave Signal For a single mixed-amplitude, mixed-frequency signal of as shown in Fig. 3, the RMS tracking error has been tabulated in Table 6, in increasing order of controller effectiveness. Fig. 3 Mixed-amplitude, mixed-frequency input wave signal
- 29. Performance Evaluation of Adaptive Dither Control Frameworks … 9 6 Closed-Loop Set Point Tracking with Impulse Disturbances (Delay 1 s, Sample Time 20 s) The response of the controllers to set point tracking against an impulse disturbance of width 20 s at t = 20 s is shown in Fig. 4. Adaptive voltage dither system is noticed to have the least settling time of 20.44 s. Traditional FF + FB control settles at Fig. 4 a Impulse disturbance of 20 to 40 s, b set point traking performance of the controllers for impulse disturbances
- 30. 10 S. Jana et al. Fig. 5 PZA system response for step input conjunction with impulse disturbance 21.31 s; however, the response has a prominent undershoot (magnitude 0.27) along with presence of oscillation. 7 Step Input with Impulse Disturbances Similar nature of controller performance is observed for the system when subject to step input with an impulse disturbance shown in Fig. 5. The controllers can be arranged in order of least settling time as adaptive voltage dither, FF + FB (presence of undershoot). 8 Zero Set Point Tracking with Gaussian Noise For a zero set point tracking of the piezoplant, it is subjected to Gaussian noise. Fig. 6 shows the response of the different controllers with unity noise variance. Analyzing the response, it can be observed that the tracking performance adaptive voltage dither controller in noisy environment is better than FF + FB. 9 Conclusion A comparison of performance of the different dither-based controllers has been carried out under similar conditions. Performance evaluation suggests that a
- 31. Performance Evaluation of Adaptive Dither Control Frameworks … 11 Fig. 6 PZA system response for zero set point tracking with Gaussian response of variance one controller performs better under certain circumstances like parameter sensitivity test, performance against external disturbances, set point tracking with disturbances outperforming the others and can be chosen depending upon the application. The adaptive voltage dither shows best performance over parametric sensitivity and external disturbances with the other controller, and it is also noticed to have the least settling time with respect to set point tracking with impulse disturbance. Acknowledgements The authors gratefully acknowledge the funding received from Indo French Center for the Promotion of Advanced Research (IFCPAR/CEFIPRA) for carrying out the project work. References 1. Nguyen, M.L., Chen, X.: “MPC Inspired Dynamical output feedback and adaptive feedforward control applied to piezo-actuated positioning systems.” IEEE Trans. Ind. Electron. 67(5), 3921– 3931, doi: https://doi.org/10.1109/TIE.2019.2916356(2020) 2. Nafea, M., Mohamed, Z., Ali, M.S.M., Mehranzamir, K., Rehman, T.: “Hybrid PSO-Tuned PID and hysteresis-observer based control for piezoelectric micropositioning stages.” IEEE International Conference on Smart Instrumentation, Measurement and Application (ICSIMA), Kuala Lumpur, Malaysia, 2019, pp. 1–6 (2019) 3. Xie, S., Ren, J.: “Linearization of recurrent-neural-network- based models for predictive control of nano-positioning systems using data-driven koopman operators.” IEEE Access 8, 147077– 147088, doi: https://doi.org/10.1109/ACCESS.2020.3013935 (2020). 4. Lambert, P., Valentini, A., Lagrange, B.: Delcambre P.D Design and performances of a one- degree-of-freedom guided nano-actuator. Robot. Comput. Integr. Manuf. 19, 89–98 (2003) 5. Hii, K.F., Vallance, R.R., Mengu, M.P.: Design, operation, and motion characteristics of a precise piezoelectric linear motor. Precis. Eng. 34, 231–241 (2010)
- 32. 12 S. Jana et al. 6. Shengdong, Y., Jinyu, M., Hongtao, W., Shengzheng, K.: Robust precision motion control of piezoelectric actuators using fast nonsingular terminal sliding mode with time delay estimation. Measure. Control 52, 11–19 (2019) 7. Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, Stochastic Resonance: A remarkable idea that changed our perception of noise. Euro. Phys. J. B 69, 1–3, (2009) 8. Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70(1), 223–287 (1998) 9. Benzi, R., Parisi, G., Sutera, A., Vulpiani, A.: Stochastic Resonancein Climatic Change. Hamburg, Tellus (1982) 10. Priplata, A.A., Patritti, B.L., Niemi, J.B., et al.: Noise-enhanced balance control in patients with diabetes and patients with stroke. Ann. Neurol. 59, 4–12 (2006) 11. Shome, S.K., Jana, S., Mukherjee, A., et al.: Design of adaptive voltage dither control frame- work based on spectral analysis for nonlinear piezoelectric actuator. J Control Autom. Electr. Syst. 30, 954–969 (2019)
- 33. Comparative Performance Study of Different Controllers for Nonlinear Piezoelectric Stack Actuator Sandip Jana, Saikat Kumar Shome, Arpita Mukherjee, and Partha Bhattacharjee Abstract Robust and adaptive nonlinear controllers play an indispensable role in industrial applications targeting nanopositioning. Internal model control has been widely used in compensation of hysteresis, creep and vibration for piezoelectric actuators along with least-square estimator regimes. However, as real-time scenarios and presence of uncertainties, disturbances play a crucial factor in determining the efficacy of the chosen controller, a comparative analysis is very effective in this situ- ation. This research focuses on an extensive comparative analysis of two controllers, namely, modified-internal model control (M-IMC) and recursive least-square esti- mator (RLSE) under several test cases. Results put forward which controller is effec- tive under what scenario that enhances the overall precision positioning efficiency of the system. 1 Introduction Piezoelectric actuators are a class of flagship actuators which has been used in micro/nano-manipulation-related industrial applications such as biological cell oper- ator, precision engineering in PCB industry, MEMS manufacturing, space optics, etc. due to prominent advantages like precision positioning, speed of response, resis- tance to electromagnetic interference, high torque handling capability and high force capability in [1–3]. In [4] paper, adaptive feed forward control and recursive least- squareestimator(RLSE)withadaptivefeedforward(FF)controllerarepresentedthat performs better for parameter uncertainty of the system. The PSO and BFO-based redesigned internal model control (M-IMC) with PID controller is proposed in [5] for nonlinear piezoactuator which allows good set point tracking and disturbance S. Jana · S. K. Shome (B) · A. Mukherjee Academy of Scientific and Innovative Research (AcSIR), CSIR- Central Mechanical Engineering Research Institute (CSIR-CMERI) Campus, Durgapur 713209, India e-mail: saikatkshome@cmeri.res.in S. Jana · S. K. Shome · A. Mukherjee · P. Bhattacharjee Ministry of Science and Technology, CSIR-Central Mechanical Engineering Research Institute, Government of India, Durgapur 713209, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gu et al. (eds.), Communication and Control for Robotic Systems, Smart Innovation, Systems and Technologies 229, https://doi.org/10.1007/978-981-16-1777-5_2 13
- 34. 14 S. Jana et al. rejection properties. Internal model control is one of the simplest control strategies being applied for several applications. PID-based IMC control strategy which is the most practically used controller is one of the reasons of its widespread acceptance and has hardware implementation reliability. A tuning method of IMC to control overshoot in a speed controlling system—a crucial subsystem in the hydro-electric power plant is elaborated in [6]. In this research, PSO-based M-IMC controller has improved path tracking and disturbance rejection performance compare with the clas- sical IMC controller. Piezoactuators are widely used for precision and positioning control at micro-nanometer level in several industrial applications and have different advantages like higher accuracy, quick response, and enhanced torque. An improved PID-based IMC controller is proposed for nonlinear hysteretic piezoelectric actuator in [7] and result proved that controller performance improved over the traditional control. The piezoactuator is modeled as a second-order system by experimentally identifiedparameterandM-IMCcontrollerisimplementedintimedelayenvironment approximated using first-order Padé expansion. A Smith predictor-based M-IMC is also proposed in [8] where M-IMC controller decreased overshoot and settling time of the response compared to traditional IMC and PID designs. Contributions and problems addressed in the paper are as follows: Different controllershavebeenperformedinMATLABSimulinkonthenonlinearpiezoelectric actuator mentioned in Sect. 1.2. Presented controller has been explored by exposing the plant model to different disturbances like error comparison, parameter sensitivity test, mixed-amplitude, mixed-frequency input signal, external disturbances with set point tracking like zero path tracking with impulse disturbances (delay 1 s, sample time 20 s), zero path tracking with step input conjunction with impulse disturbances and zero set point tracking with Gaussian noise. The controller performance result suggests that controller’s performance depends upon the various application of the system. 2 RMS Error Comparison The study has been carried using a sinusoidal wave signal as an input amplitude of 10 µm, and frequency range of signal 1 Hz which is to be followed by the piezoactu- ator. The RMS error of the corresponding control schemes is placed in Table 1. C1: Feed forward traditional control (FF), C2: Cascaded feed traditional forward and feed back (FF + FB), C3: Recursive least-square estimator (RLSE), C4: Modified-internal model control (M-IMC). Table 1 Tracking RMS error of different control paradigms Controllers C1 C2 C3 C4 Error(nm) 6.67 0.26315 0.19405 0.16577
- 35. Comparative Performance Study of Different Controllers … 15 Fig. 1 Tracking performance plot of different adaptive controller with sinusoidal input response It is observed that order of performance of controllers is as per the following order M-IMC, RLSE, FF + FB, FF as shown in Fig. 1. 3 Parameter Sensitivity Test As illustrated in Ref [6], the main reason for occurrence of mismatch in plant and model is accountable to the approximations while transforming a physical plant to its mathematical model. This mismatch is quite common and takes place due to several reasons like improper identification of system, or inappropriate order consideration of the system. Besides, nonlinear properties also add up to this error in approximation. The following section considers a plant model parameter mismatch of ± 2% and ± 5% in plant parameters as compared to the traditional model and the RMS errors are presented in Tables 2 and 3, respectively. It has also been observed earlier that M-IMC offers the best RMS tracking error (0.16577 nm) among the observed controllers. Accordingly, to determine the deviation from this value, Tables 2 and 3 are presented which contains the difference of the RMS errors for plant model mismatch with the M-IMC controller value. From Tables 2 and 3, it can be concluded that M-IMC and RLSE have the better compensation of the parametric uncertainty, which is followed Table 2 RMS error for plant parameter variation of ± 2% Variation (%) Controller M D K T −2 C2 0.12933 0.24121 0.17788 0.15667 2 C2 0.16073 0.13105 0.21483 0.18924 −2 C4 0.0209 0.0553 0.00947 0.00453 2 C4 0.00368 0.01266 0.001 0.0047 −2 C3 0.02309 0.02365 0.02660 0.02917 2 C3 0.03375 0.03312 0.02993 0.02748
- 36. 16 S. Jana et al. Table 3 RMS error for plant parameter variation of ± 5% Variation (%) Controller M D K T −5 C2 0.12855 0.20083 0.15985 0.11177 5 C2 0.09108 0.12610 0.16552 0.1882 −5 C4 0.01868 0.00259 0.00480 0.01096 5 C4 0.01881 0.01178 0.00407 0.02133 −5 C3 0.01588 0.01716 0.02398 0.03050 5 C3 0.04259 0.04091 0.03231 0.02612 by FF + FB and FF. The efficiency of M-IMC over RLSE is clearly seen as the parametric variation is increased to 5% 4 Performance Against External Disturbances Transfer model of disturbances is not enough to withstand for the disturbances which is effect the plant of the system in real life environment. The external disturbance has been represented as variance of a Gaussian noise in the system, first at the input side as a process noise, and second, as a measurement noise toward the plant output. The error values of Gaussian noise in each of the controller are shown in Tables 4 and 5, respectively, and can be assessed through two metrics, a) variation induced due to the presence of the noise which is represented by a difference of maximum and minimum error value and b) deviation of the average error due to noise with the best attained tracking error of M-IMC controller is 0.16577 nm. From the table, it is observed that using the second metric, the performance of the controllers toward eliminating noise is as follows RLSE, M-IMC, FF + FB and FF. A schematic of the error plot suggesting similar controller performance is shown in Fig. 2. Table 4 RMS error for external process disturbance Gaussian noise variance C4 (nm) C3 (nm) C2 (nm) 10e-22 0.19762 0.19406 0.3800 10e-23 0.19316 0.19406 0.4085 10e-24 0.20276 0.19406 0.3837 10e-25 0.20666 0.19406 0.2731 10e-26 0.21949 0.19406 0.3642 10e-27 0.21455 0.19406 0.2605 10e-28 0.20431 0.19406 0.2536 10e-29 0.19686 0.19406 0.2708 Max–min 0.02633 - 0.1549 average 0.2044 0.19406 0.3243
- 37. Comparative Performance Study of Different Controllers … 17 Table 5 RMS error for external measurement disturbance Gaussian noise (variance) C4 (nm) C3 (nm) C2(nm) 10e-22 0.21649 0.19419 0.2981 10e-23 0.18903 0.19404 0.2719 10e-24 0.20680 0.19405 0.2824 10e-25 0.19391 0.19406 0.3716 10e-26 0.20290 0.19400 0.3507 10e-27 0.19158 0.19400 0.3830 10e-28 0.19154 0.19405 0.3392 (a) Max. value – Min. value 0.02746 0.000190 0.1110 Average 0.19889 0.19405 0.3281 Fig. 2 Analysis of error plot with sinusoidal input and external measurement noise environment (Gaussian, Variance 10e-15)
- 38. 18 S. Jana et al. 4.1 Gaussian Noise as Process Noise Disturbance See Table 4. 4.2 Gaussian Noise as Measurement Noise Disturbance See Table 5. 5 Mixed-Amplitude, Mixed-Frequency Input Test In this section, different input wave signals with varying amplitude and frequency have been applied to the PZA system and the correlative RMS tracking error is shown in Table 6. The amplitude of the input wave signal has been varied from 5 to 20 µm with the frequency varying from 0.25 to 1.0 Hz. An improvement in the error value is noticed with different adaptive controllers compared to feed forward and feedback control; it is also seen that for a particular amplitude, error increases with the increase in the frequency of the input. 5.1 Performance of the Controllers to Different Input Wave Signals—of Alternating Frequency and Amplitude The piezoactuator has been subjected to different input wave signals and alteration of frequency (0.25 Hz, 0.5 Hz and 1 Hz) and alteration of amplitude (2.5 µm, 5 µm and 10 µm). For all the controllers, a common phenomenon is observed—the tracking error increases with increase in input frequency, and also with an increment in input wave signal amplitude. Among the comparison of different controllers, M-IMC is seen to offer the least error value for all frequencies, apart from 1 Hz. For input signal of frequency 1 Hz, RLSE is seen to has better performance as compared to other controllers. As seen in Table 6, it can be concluded that for a majority of input signals, the order of controller performance is as follows: M-IMC, RLSE, FF + FB and FF. Table 6 RMS error of the controller a multi-amplitude, multi-frequency signal C1(nm) C2(nm) C4(nm) C3(nm) 6.3145 0.33506 0.16118 0.14479
- 39. Comparative Performance Study of Different Controllers … 19 Fig. 3 Mixed-amplitude, mixed-frequency input sinusoidal wave signal 5.2 Performance Evaluation of the Controllers to a Mixed-Amplitudes, Mixed-Frequency Input Wave Signal For a single mixed-amplitude, mixed-frequency signal of as shown in Fig. 3, the RMS tracking error has been tabulated in Table 6, in increasing order of controller effectiveness. 6 Zero Path Tracking Conjunction with Impulse Disturbances (Delay 1 s, Sample Time 20 s) The response of the controllers to set point tracking against an impulse disturbance of width 20 s at t = 20 s is shown in Fig. 4. FF + FB control settles at 21.31 s; however, the response has a prominent undershoot (magnitude 0.27) along with presence of oscillation. The overshoot value for RLSE starts from a magnitude of 0.8 (less than one) and the response settles at 26.9 s, without any oscillation. M-IMC is noticed to have the broad settling time of 32.5 s.
- 40. 20 S. Jana et al. Fig. 4 a Impulse disturbance of 20 to 40 s, b zero path traking performance of the RSLE,M-IMC and FF + FB controllers for impulse disturbances 7 Zero Path Tracking with Step Input Conjunction with Impulse Disturbances Similar nature of controller performance is observed for the system when subject to step input with an impulse disturbance shown in Fig. 5. The controllers can be arranged in order of least settling time as FF + FB (presence of undershoot), RLSE and M-IMC.
- 41. Comparative Performance Study of Different Controllers … 21 Fig. 5 System response of step input with impulse disturbance for piezoelectric stack actuator 8 Zero Set Point Tracking with Gaussian Noise For a zero set point tracking of the piezoplant, it is subjected to Gaussian noise. Figure 6 shows the response of the different controllers with unity noise variance. Analyzing the response, it can be observed that the controller tracking performance with Gaussian noise can be arranged in the following order RLSE, M-IMC, FF + FB. 9 Conclusion This study is focused on comparative analysis of two predominant classes of piezoelectric stack actuator and the results have been considered for different test cases. Result indicates that a specific type of controller outperforms the other under certain circumstances and needs to be chosen depending upon the application. Modi- fied IMC is observed to lend considerable improvement in performance for sinusoidal motion tracking and parametric uncertainty while RLSE shows a very good perfor- mance for disturbance rejection of external Gaussian noise, both as process and as measurement noise, and FF + FB controller has the least settling time with respect to set point tracking with impulse disturbance.
- 42. 22 S. Jana et al. Fig. 6 PZA system response for zero set point tracking with Gaussian response of variance one References 1. Lambert, P.,. Valentini, A., Lagrange, B.: Delcambre P.D design and performances of a one- degree-of-freedom guided nano-actuator. Robot. Comput. Integr. Manuf. 19, 89–98 (2003) 2. Hii, K.F., Vallance, R.R., Mengu, M.P.: Design, operation, and motion characteristics of a precise piezoelectric linear motor. Precis. Eng. 34, 231–241 (2010)
- 43. Comparative Performance Study of Different Controllers … 23 3. Shengdong, Y., Jinyu, M., Hongtao, W., Shengzheng, K.: Robust precision motion control of piezoelectric actuators using fast nonsingular terminal sliding mode with time delay estimation. Measure. Control 52, 11–19, (2019) 4. Shome, S.K., Mukherjee, A., Karmakar, P., Datta, U.: Adaptive Feedforward controller of piezo- electric actuator for micronano positioning” in Sadhana, Indian Academy of Science, Springer, 2018. Issue—Academy proceedings in Engineering Science, Vol. 43(10) (2018) 5. Shome,S.K.,Jana,S.,Mukherjee,A.,Bhattarcharjee,P.,Datta,U.:Bioinspiredmodifiedinternal model control approach for improved disturbance rejection of piezo micro manipulator. Stud. Inf. Control 27, 295–306 (2018). https://doi.org/10.24846/v27i3y201805 6. Naik, K.A., Srikanth, P., Negi, P.: IMC tuned PID governor controller for hydro power plant with water hammer effect[J]”. Procedia Technol. 4(4) (2012) 7. Shome, S.K., Jana, S., Mukherjee, A., Bhattacharjee, P., Datta, U.: Improved internal model control based closed loopcontroller design for second order piezoelectric system with dead time. In: Proceedings of 2018 8th IEEE India International Conference on Power Electronics; Jaipur, India. pp. 1–6 (2010) 8. Shome, S., Jana, S., Mukherjee, A., Bhattarcharjee, P.: Model based control for second order piezo actuator system with hysteresis in time delay environment. Turk. J. Elec. Eng. Comp. Sci. https://doi.org/10.3906/elk-1907-28 (2019)
- 44. PID Controller Design for Reference Tracking of Single Link Manipulator Sayan Das and Naiwrita Dey Abstract This paper is aimed to design a PID controller for reference signal tracking. Design problem is formulated to control the angular position of a single- link manipulator at any particular desired set value. Implementation of the overall feedback control system and the proposed controller has been carried out in Python language. Additive uncertainty is considered with the transfer function model. Simu- lation results illustrate the usefulness of the proposed controller for angular velocity tracking in presence of uncertainty. Average mean square error and settling time have been calculated. Overall stability of the feedback system has been tested by obtaining the eigenvalues of the closed-loop system. 1 Introduction Robotics is getting more and more popular in industrial field as it reduces manpower required for doing heavy jobs. Main types of robots used in industrial applications are “multiple-linkmanipulators”withdifferentdegreesoffreedom.Thesekindsofrobots can move according to their DOF and complete many tasks. Single-link manipulator model is made out of one fixed base, on which the whole system resides, a revolute joint and an arm, which can move freely on the x–y plane. DC motor or servomotors are mainly used to rotate the arms of manipulators. Here, a simple DC motor with high torque constant is used to move the arm of the manipulator. First, the dynamic model of that motor is designed [1]. Then, control logic is applied on the motor to move the arm to a specific angular position [2–4]. For tuning control parameters, simple tuning method is used [5]. Nowadays, in many industrial applications, rigid link manipulators cannot satisfy the needs. So, “Flexible link manipulators” have become an option. In comparison with rigid body manipulators, flexible manipulators have many advantages including higher efficiency, more flexibility, high load capacity, S. Das Department of AEIE, RCCIIT, Kolkata, India N. Dey (B) Department of ECE, RCCIIT, Kolkata, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gu et al. (eds.), Communication and Control for Robotic Systems, Smart Innovation, Systems and Technologies 229, https://doi.org/10.1007/978-981-16-1777-5_3 25
- 45. Exploring the Variety of Random Documents with Different Content
- 46. to Rickard's companion he directed him to make use of his cant-hook and dislodge the timbers. His request was made in a quiet tone and without anything offensive in his manner, and he stepped away from the men and started round to the other side of the horses to watch the work. As he did so he heard Rickard muttering something that was meant for his companion, though he did not conceal the fact that he cared very little whether King heard it or not. King stopped and came back. "Just now, Rickard, this is a one man's job," he said. "You get that straight." Rickard's mouth curled up into a sneer. He seemed on the point of making a reply, but he looked at King's face and shrugged his shoulders contemptuously without speaking. King then turned to Rickard's companion and stood by until the logs were cleared. Then he gave Rickard orders to go ahead. Letting loose a string of oaths, Rickard struck the horses with the knotted ends of the lines, and continued lashing them as he drove them at a mad pace down the trail and round the corner to where the men were working. King stood in the trail and watched Rickard abusing his team until the blood was hot in his veins. He made a quick start to overtake him—and then suddenly checked himself. Stepping back a little among the trees he waited. In a few minutes Rickard returned for another load. King waited until he came opposite him in the trail, and then stepped out. Rickard's companion had not come back as yet and he was alone.
- 47. "Whoa!" King said to the horses, and he stepped before them in the trail. Then he faced Rickard. "Tie up here a minute," he said, indicating with his hand a tree conveniently near, to which the team could be made secure. Rickard looked at King quickly and again gave a shrug of contempt. "Rickard," King said, "that won't get you anywhere. Tie up— here!" "I will—like——" Rickard never finished his sentence. King was beside him with one step and had seized him by the shoulder. "Rickard!" he said, sharply. Rickard looked at him for a moment, and then going to the heads of the horses, led his team over to the tree and made them fast. "Go in there," King commanded, and pointed into the woods in the direction of the river. Rickard did not turn to look this time, but picked his way through the underbrush, with King close at his heels. When they came within a yard or two of the bank of the river King spoke again. "This will do," he said. "I'm going to talk to you for about one minute, and I want you to listen." All the quietness had vanished both from King's voice and from his manner. He was shaking with passion and his face was almost white. He laid one hand on Rickard's shoulder and closed his fingers in a vice-like grip.
- 48. "Ten minutes ago, Rickard," he said, "by God, I'd have killed you. Just now, you dirty whelp—I'll give you about thirty seconds to make up your mind to get out. Leave that team where it is and get back out of the way till this job's done. If you're in town by Monday night I'll take my own way of putting you out. A little better than two days—that's enough time to square up and hit the trail. Are you ready?" Rickard squirmed under King's hand, but King pulled him up suddenly. "Are you ready?" he repeated. Rickard nodded. "Then move!" King waited until he had gone a few yards before he followed him. They had not retraced more than half the distance they had come when they heard a great splash in the river behind them. They turned at once and looked back. A large section of the river bank, undermined by the action of the water, had fallen and had taken away the very ground on which they had been standing only a moment before. King paused in silent contemplation of how petty, after all, are the things that vex us most. Only a moment did he allow his mind to wander from the business he had in hand; then he faced Rickard again, and without a word the two went off together. King took the team back and gave it into the keeping of one of the men. He never left Rickard's side, however, until he had seen him safely away from the workers. Then he returned and went on with his work.
- 49. That evening the task was completed and King, after taking supper at MacMurray's and chatting a moment with Anne, walked over to Hurley's to talk with Cherry a little before he went to his shack. All day his mind had reverted time and time again to the incident with Rickard, and more especially to what seemed like a miraculous escape from what might have meant death to both. Now that the work was over and his mind was free, the whole affair came back upon him with renewed freshness. He told it all to Cherry and Mrs. Hurley, and when he had finished, Cherry, who had listened throughout without speaking a word, turned a serious face to King and put her hand upon his arm. "It looks almost—as if God himself were helping us," she said. She did not speak fervently, nor with any emotion. Her voice was quiet and her tone matter-of-fact. And yet King was struck by the simplicity of her manner. She evidently believed implicitly in what she had said—and King found himself impelled to share somewhat in her faith. It was the last thought that lingered in his mind that night before he went to sleep to the sound of the rain falling upon the roof of his shack. Hugh Hurley and Keith McBain sat together in the land office very late that night. No one in town was in any mood for going to bed, and the sounds that came from Cheney's and MacMurray's bore ample evidence to the fact that the men were apparently preparing to make a night of it. Old Gabe Smith dropped in when it was very late and stayed long enough to observe, among other things, that if
- 50. the rain didn't soon cease in the hills the water in the river would be over the top of the bank. After Gabe had gone, the two men decided upon taking a walk down to the river to look at the rising water. What they saw when they got there struck fear into their hearts at once. Since it had grown dark the stream had risen a full foot, and was now rushing with terrific force around the bend, about the outer angle of which clustered the huts and cabins of the little town. Already the current had swept away large portions of the high bank, in which there was no rock or stone of any account to offer any resistance to the enormous weight of water that swept down like a vicious cataract out of the hills. "Look yonder," Hurley said, suddenly. Keith McBain turned to look in the direction indicated. Further up stream a little shack stood, with one corner already projecting over the edge of the bank. In a few hours at most the ground upon which it stood would be swept away and the shack with it. Without losing a moment they hurried back to MacMurray's and called out the men who had not yet retired for the night. In less than five minutes, more than a score were at work, and before another half hour had passed, the shack had been moved back upon safe ground. By the time the excitement was over there was not a man left in either MacMurray's or Cheney's. Everyone was out, either to help or look on. Keith McBain had left and gone back with Hurley to the office when the immediate danger was past. They were not in the crowd when Gabe Smith came running excitedly to the men to
- 51. announce that the bank was falling away just above the place where the corral and equipment sheds had been built during the week. At once the men hurried toward the corral. For a few minutes there was much excited and aimless running about on the part of the men, without any organization, and without any plan. Soon, however, there emerged certain unfailing indications that a part of the gang, at any rate, were under direction. Gabe Smith was probably the first to observe it, and his suspicions were confirmed when he saw McCartney's huge frame moving among the men. There was organization, but designed to frustrate all efforts to save the buildings, rather than to assist. Gabe left the crowd of men, who were already wrangling among themselves, and hurried to find Keith McBain. He had his hand upon the door of the office and was about to open it, when he felt himself seized by the shoulder and hurled back so violently that he stumbled and fell to the ground. He looked up and saw McCartney standing over him. "Stay out of the way, you old crust," McCartney said, "an' you won't get hurt." In a moment the office door was opened and Hurley was standing in the lighted doorway, with McBain behind him. "What's wrong?" demanded Hurley. For reply McCartney stepped into the office, pushing Hurley before him, and closed the door behind him. "This ain't an old man's town—that's what's wrong," he said. Hurley expressed his astonishment. "Well, but—an old man can live here as well as anywhere else, can't he?" he protested.
- 52. "All depends," McCartney replied, smiling cynically. "We'll settle that some other time. Just now I have business with Keith McBain." "It's time to settle," he said, looking at McBain who, for a moment, seemed beaten in the struggle that was raging within him. Suddenly he stood up and looked at McCartney, his eyes burning with the fierce hate that was in his soul. When he spoke his voice seemed a little uncertain, as if he were struggling to keep back the tears from his eyes. But almost immediately he mastered himself and spoke deliberately enough, if not quietly. "What is it, McCartney?" he asked. "Gabe Smith was here to announce to you that the new buildings an' the outfit is all goin' down stream before daybreak unless they're moved," McCartney replied. "And is nothing going to be done?" asked McBain. "That's just what I'm here for," returned McCartney. "It'll be done if you're ready to come through." "Well—what will settle it?" Keith McBain asked in a voice that had almost a touch of weariness in it. "We've talked about all that before—there's no change," McCartney replied. Hurley looked from one man to the other in bewilderment. "And if I refuse?" asked McBain. "You're wastin' time," McCartney snapped. Keith McBain raised his voice a little, but spoke with much the same deliberateness as before. "For two years, McCartney, I've been in hell expecting this time to arrive any day. I'm past that now. I've settled it—and I'm going to
- 53. see it to the end. Don't think you can frighten me—I'm old, but—I'll pay." The words seemed to strike McCartney almost dumb. "You'll pay?" he asked. "Yes—go ahead—tell all you know!" "By God, then, you will pay," McCartney exclaimed, and throwing the door open, went out. Hurley stepped over and, closing the door, turned to McBain. "What is this—this bargain, Keith?" he asked. "For two years he has kept a secret that has held me bound to him—because I have been afraid to die." "Die?" Hurley exclaimed. "Hugh—I have killed a man." For a moment they stood in silence and did not look at each other. Then Keith McBain moved wearily towards the door. Before he went out he turned and looked back at Hurley. "Hugh," he said, quietly, "look after the men—I'm going to the girl." Then he opened the door slowly and went out. McCartney stood alone in the darkness by the river and waited for Rickard, whose form was faintly visible a few yards up the river. When Rickard had joined him, McCartney caught him by the arm. "Well?" he asked. "All smooth," Rickard replied. "Nothin' rough?" McCartney prompted. "I said—all smooth," Rickard returned, a little impatiently.
- 54. They walked together to within a few yards of the men and stood looking at them. McCartney's group were in the majority, and stood near the corral. Some distance back the others stood about in small groups, talking angrily among themselves. A bit of the bank dropped away and fell with a dull splash into the water. McCartney put a cigarette into his mouth and applied a match leisurely. "I ain't much on religion, Rick," he said, jocularly, "but the Almighty sure looks friendly to-night." CHAPTER SEVENTEEN King awoke with a start. He had been sleeping very soundly, and at first, after he had opened his eyes, he had difficulty in bringing his senses to bear directly on what had disturbed him. The faint grey dawn was already at the window. Somewhere there had been a thumping and—the sound of a voice that, even to his sleep-fogged consciousness, was vaguely familiar. For a moment he waited, sitting up in his bunk and rubbing the sleep from his eyes. Suddenly the thumping was repeated—someone was at the door. Then he heard his name called and the sound of the voice brought him to his senses at once. It was the voice of Cherry McBain. In an instant he was at the door. "What's wrong?" he asked excitedly.
- 55. Cherry's voice was full of alarm. "Get dressed quickly, King," she replied. "We want you." King hurried into his clothes, and going to the door again shot the wooden bar back from its socket and threw the door open. A very light drizzling rain was still falling, and Cherry shook the wet wrap from her head and shoulders as she stepped through the doorway. In his hurry King had not taken time to light the lamp, but even in the darkness he could see the expression of fear on her face. Without waiting to close the door he placed an arm about her shoulders and drew her towards him. "Oh, King!" she cried, "it's come—it's come!" He did not need to ask what had come. He knew. Leading her gently to a seat he left her, and sitting down on the edge of his bunk, drew on his boots and laced them hurriedly. Then he got up quickly and throwing on his coat, took his hat and turned to Cherry. "All right—I'm ready," he announced. Cherry got up from her seat and moved towards the door. She had not spoken while King was completing his preparations to go out, and he knew that she had been weeping silently. When she got as far as the open doorway she paused and turned to him. "King—King—" she began, but her voice failed her. King stepped close to her and took her arm. "Tell me about it as we go," he said. She moved towards him, and reaching up placed her hands on his shoulders. King looked down at her face, white and tense in the darkness.
- 56. "You must fight, King," she said, with an emphasis that to King seemed almost pathetic. He pressed her closer for reply. "And you must win," she added. He smiled faintly. "I'm ready," he said. Her hands crept slowly about his neck, and King, with a suddenness that swept her off her feet, caught her to him and pressed a kiss upon her mouth, a kiss in which all the pent-up passion of weeks found expression at last. When he released her he stood with his arms about her for a brief moment, trembling before her. "I don't deserve it," he said, his voice trembling with emotion. "I guess I'll never deserve that—but I wanted to win first—to win for you." She leaned a little closer to him and then drew herself up and clung tightly with her arms about his neck. "King," she said, breathlessly, "I love you—I love you!" Again he put his lips to hers quickly, passionately—and then put her back from him. "We must get along down now," he said. Cherry drew her wrap around her and they went out together. A few minutes' walking brought them within sight of the town, apparently peaceful in the cold grey glimmer of light just breaking in the east. So quiet was it that King began to wonder if the disturbances of which Cherry had been telling him as they came along had not been settled. Then suddenly there arose a shout from the further side of the town, near the river, and King quickened his pace almost to a run, giving Cherry all she could do to keep up. At
- 57. last his eagerness mastered him, and leaving Cherry with a last warning to go back to Hurley's cottage and not to stir until he should come for her, he left her and went off at a run in the direction of the shouting. What King saw when he reached the point in the street where it turned and ran along the bank of the river made him stand a moment aghast. Back against the trees the buildings stood, huddling together closely in the cold light of the early morning. The water in the river was almost level with the ground on which he was standing, and large sections of the bank had been swept away during the night, until the corral in which the horses were placed before King left town the night before, was now standing on the very brink of the flood. This was in itself enough to strike fear into King's heart, but the movements of the men were what concerned him most. Half- drunken still from their night's debauch they seemed to be rolling about in a kind of ridiculous orgy, stumbling and falling and scrambling to their feet again, shouting and cursing and grappling each other in frenzied disorder. A glance was enough for King to realize fully what was wrong. He could not see McCartney anywhere among the men, but Cherry had told him enough—if telling had been at all necessary. Back a little from the struggling mass stood six or eight men, looking on quietly and talking among themselves. King recognized them as some of his own men, upon whom he thought he could rely for support. In a moment he was standing in the middle of the group. "What are you standing here for?" he asked. "Come on—get into it!"
- 58. In a flash they were into the struggle, King leading them as they bored their way through in an effort to reach the corral. King's plan was clear in his own mind. Once with his back to the walls of the corral, he could call his men one by one about him, and having displaced their opponents, drive them off by united effort, break up their organization, and beat them into submission. The plan, easily enough conceived, was not so easily carried into effect. King's appearance, it is true, had raised the spirits of the men who were fighting together to settle the scores they had accumulated during weeks of growing hatred for McCartney and his crowd. But as their spirits rose, the determination of their opponents became more grim as they saw themselves faced with possible defeat where they had never dreamed of anything but an easy victory. The fight became more and more furious every minute. Whereas before King's coming they had fought without much bad temper and with little evidence of losing control of themselves, now they struck out madly and grappled with the fierceness of men in a battle where life and death depended upon the outcome. They had fought only with their fists before. Now sticks and clubs began to make their appearance as if by magic, and in many cases the fight was for the possession of weapons. Once King saw the flash of a knife between two men who were struggling near him. Turning quickly he struck the fellow who held it, sending him to the ground, where he sprawled clumsily in an effort to escape being trampled under the feet of the fighters. The knife had fallen to the ground, and King, placing his foot on it for a moment, waited while he beat back a struggling pair who were close to him. Then stooping quickly he picked up the knife and threw it
- 59. into the river. No sooner had he thrown it away than the owner pushed his way towards King and accosted him for having attacked him. He was one of King's men. King pushed him back angrily. "Let them start that," he cried in a voice that rose above the din. "Get in there!" He pointed to where a group of his men were now massed against their opponents and were driving them back slowly from the corral. Then his eyes shifted suddenly in a new direction. Pushing his way through the crowd towards King, was McCartney, his huge shoulders towering above the other men, his dark face serious and totally divested of its usual cynical smile. Not far behind him, on the outskirts of the crowd, stood Old Silent. King wasted no time on the men about him. If McCartney's anxiety to reach him were greater than his own, there was no indication of the fact in the eagerness with which King pressed towards him, pushing first one and then another out of the way as he went forward. When the two men faced each other at last they paused a moment, and their eyes met in a long look in which there was something more than mere hatred. In fact, an observer might have refused to believe that the look was one of hate. There was grim resolve and unwavering determination to settle an account of long standing. But, for a moment at least, there arose in King's heart a feeling of something like admiration for the embodiment of sheer brute strength that stood before him. King did not pause long enough to ask what lingered in the look McCartney gave him. He
- 60. saw only that the tense seriousness that had darkened the face of McCartney was gradually giving place to the old sneer that had always played about one corner of his mouth—and the sight stung him to madness. He thought of Cherry McBain—he thought of the man whose life for two years had been one long curse to him—he thought of the woman who had died of a broken heart—and he stepped quickly and struck out at the sneering face before him. The dawn in the east had spread upward from the horizon and filled the sky, still clouded, with a thin grey light. There was light enough, however, to make every movement easily discernible, and King watched his opponent from the beginning with an alertness that rendered him proof against any foul play. He was not going to be taken unawares, at any rate. If he were beaten it would be because he had matched himself against a better man. Gradually the other men fell away from them and left the ground clear. McCartney's men had been driven back and were beaten. But friend and foe alike came round to watch what they rightly guessed was to be the last scene in a play that had been running for many weeks. Keith McBain himself stood off to one side, his face ashen white, his eyes set immovably upon the men who were settling once and for all, he hoped, not only their own accounts, but his as well. Old Gabe Smith stood directly behind King, calling out words of encouragement in his little piping voice, and totally oblivious to the existence of anyone else in the world. For fully five minutes the two men walked cautiously about each other, striking out quickly but lightly, and stepping back immediately to recover themselves after each advance. Though the sneer never left McCartney's face, there was behind it a deep seriousness that
- 61. expressed well the fact that he was fully conscious of the magnitude of the task before him. King's face was tense, set, terribly earnest. Only once was there any interference from the bystanders. Mike Cheney, who had been an interested spectator during the whole struggle, pushed his way to the inner part of the circle of men and voiced a feeble protest. The men near him laughed and jostled him out of the way. He was content to remain where he was, though he no doubt felt there was something incongruous in the fact that when he looked round he was standing next to Hugh Hurley. After some time had passed in which the men had remained wholly on the defensive, McCartney began to advance persistently against King, who stepped back out of reach whenever he found McCartney pressing him too closely. King's wary tactics were testing the patience of his opponent. With an agility that was surprising in a man of his size, he stepped about the enclosure, keeping just out of reach of McCartney, and starting forward, snapping out his left hand when an opportunity presented itself. His blows were not heavy, but he was reaching McCartney's face and body almost every time he struck. McCartney swung and lunged heavily every time he struck at King, but his blows were without control. Growing impatient at last with following King from place to place, he closed quickly and seized King about the body. This time, however, he had misjudged his man. As he came forward King stepped in and met him with a blow from the shoulder that struck McCartney on the chin. His full weight was behind the blow and McCartney's head went back from the force of it. Then his arms went round King and he hung on dazedly in an attempt to gain a little more time for recovery. But King was determined to make his
- 62. recovery as difficult as possible. With McCartney's full weight bearing him down, he sent half a dozen quick, short blows to the body that made his opponent gasp for breath. But McCartney kept his hold and tightened it, so that King found himself in a grip that made striking impossible. It was just this situation that King had tried to avoid. He knew McCartney's strength was probably more than a match for his own, and he had hoped that he might be able to keep him at a distance. As he felt the powerful arms closing more and more tightly about him he struggled to break the hold. After a few moments, however, he knew that his efforts were in vain. McCartney had him in a grip that reduced his effectiveness and made any attempt to break it simply a waste of reserve strength. He locked his arms about McCartney's shoulders and threw his whole weight upon him. His change of tactics was so sudden that McCartney staggered for a moment under his weight, and in that moment King's foot shot out suddenly and the two men went to the ground together, locked in each other's arms. Once, twice, three times, they rolled over, each attempting to gain the advantage of position without success. Then suddenly they broke apart and scrambled to their feet again, crouching at opposite sides of the circle. For some seconds the men faced each other without attacking, both apparently taking advantage of even a brief breathing spell. Those who were anxious for McCartney's defeat began to express their impatience at King's failure to assume the aggressive. McCartney was plainly weakening under the punishment that King was inflicting. The fact that his aggressive tactics had not already brought the fight to an end had taken the heart out of McCartney.
- 63. The face that during the earlier stages of the struggle had borne a sneer was now painfully serious. Even Hugh Hurley caught some of the excitement of the crowd as he saw that a well-directed aggressive on King's part would bring an end to the fight in a few minutes. Keith McBain's eyes were fixed upon King's face. Once or twice during the short lull in the struggle they exchanged glances. Keith McBain's heart sank within him, and he moved round to get closer to King. There was a look in King's eyes that he could not understand. When he found a place directly behind him he stepped in a little and put one hand on King's shoulder. "Just a bit more, boy," he said, encouragingly. "He's nearly done." King seemed on the point of turning his head to reply, but just then McCartney started towards him. This time King took a half step towards him and met the rush without attempting to step aside. Both men struck at the same moment, and both blows went home. McCartney's rush was checked, but the full force of his rush was behind the blow that caught King on the point of the chin. For a moment King was almost overcome by a sickening dizziness that set the world spinning about him. His mind went suddenly back to the night in McBain's camp when he had been hit on the head, and there started within him a terrible fear that the darkness that had overcome him then was creeping upon him now and blotting out his senses. For fully a minute—it seemed an hour—he fought to keep his eyes open and his attention centred on McCartney. He threw his weight against him blindly and gripped him in sheer desperation.
- 64. Gradually his legs steadied under him and his sight cleared. Still he clung to his man. Had McCartney had enough strength in reserve to deliver one more blow with any weight behind it, he could have finished the fight in another second. He knew as much himself, and he paused just a moment to muster what little strength he had left. Then he broke away suddenly and sent his right hand over as he stepped away. King's head went back and his arms went out before him helplessly. His men shouted to him in that one sickening moment when the sense of utter defeat was forcing itself upon him. Hurley and McBain called his name frantically, but he seemed not to hear them. He sank to the ground on one knee, holding himself as erect as possible in a last effort to meet the rush that he knew was bound to come. McCartney's men went wild with excitement. They called on him to bore in and finish it. Those behind stepped up and pushed him forward. When he didn't move they cursed him for a fool. But he stood swaying unsteadily, waiting, apparently, for King to fall to the ground. Behind King there was a sudden commotion in the crowd. Gabe Smith's thin voice was giving commands to the men to make way for him. He pushed his way to the front, leading behind him Cherry McBain. "Fight—you—fight!" he cried at the top of his voice. King glanced quickly about at the sound of Gabe's voice and his eyes fell upon Cherry's face. Her look was one of pathos and appeal —but she was smiling.
- 65. At once a change passed over King's countenance. Getting up he brushed his hand impatiently across his face and stepped towards McCartney. As he did so McCartney came forward and the two men met at the centre of the enclosure. From that moment neither man gave an inch of ground. Fighting furiously at close quarters they seemed both to have gained sudden strength and renewed powers of endurance. There was little attempt at defense, each man trying to inflict as much punishment as possible upon his opponent, and caring little how much he received himself. Fighting as they were, they could not hope to last much longer. The end came very suddenly. Stepping back quickly, King crouched a moment and waited for McCartney to advance. He had not a second to wait. When he saw him start he leaned far back and swung his right hand from his hip with all the strength he could command. The blow went straight and true, landing squarely on the side of McCartney's jaw, and the big foreman went down in a heap to the ground. For a moment King stood above him—but the struggle was over. Then the sickening sensation returned suddenly. He turned to Cherry, who was now at his side. "Take—me—away," he said, giving her his hand. The next moment the arms of Hugh Hurley and Keith McBain were about him, and he staggered out of the crowd with Cherry and old Gabe leading the way before him. It was not until they had gone some distance that they noticed King beginning to limp badly. At every step he took his face winced with pain. Finally he asked them to let him stand for a minute.
- 66. "It's my foot," he said, in answer to Hurley's question. "My ankle —something happened when we fell—just wait a little—it'll be all right in a minute." After a moment's pause they started off again, but King found walking impossible. Keith McBain called a couple of men and they carried him to Hurley's cottage, where they laid him on a couch and left him in the care of Cherry and Mrs. Hurley. McBain and Hurley went off at once to the scene of the early morning struggle. Gabe lingered a little while with King, busying himself with such odd jobs as Cherry and Mrs. Hurley found for him. In a short time King had recovered sufficiently from the first ill- effects of his battle with McCartney to give some thought to what was going on outside. He called Gabe to him. "Have they gone back—McBain and Hurley?" he asked. Gabe replied in the affirmative. "An' they'll handle it, too—don't you worry!" he added. King thought seriously for a moment. "Gabe," he said. Gabe took the hand that King extended to him and waited. "Get Anne—and bring her here," he said. Gabe went out at once and King looked at Cherry, who was standing above him, her hand resting lightly upon his head. "I want to tell Anne," he said quietly. "I want her to know I didn't want to do this. I want her to understand—it had to come." "Then she told you, too?" Cherry asked. King nodded in reply. Then he reached up and took her hand.
- 67. "Come down here beside me," he said, and his face was very serious. Cherry knelt on the floor beside the couch. "Cherry," he whispered, drawing her towards him, "I don't deserve it—but I want to kiss you." She leaned forward and King's arms went round her as their lips met. Keith McBain and his men went to work as if nothing had occurred for days to disturb the quiet, work-a-day life they had been living for months. Only one building was in imminent danger of being swept away by the flood, and in less than ten minutes after the close of the fight the men were busily engaged removing the camp equipment preparatory to taking the logs down and shifting the buildings back from the water's edge. Gabe came upon the old contractor giving orders and directing the work in his customary way. "Where is she?" asked Gabe, excitedly, as he came up with McBain. "She—who?" "Anne—she's gone!" Gabe replied. McBain left the men and accompanied Gabe back to MacMurray's. They found McCartney lying on a bench where his men had placed him. Rickard was standing beside him talking with MacMurray. "Where's the girl—Anne?" McBain asked MacMurray. He replied by looking at McCartney and then at Rickard. McCartney turned and looked at McBain and then allowed his eyes to
- 68. rest on Rickard. "Rick," he said, "get her and bring her here. You can tell her I want her." Rickard was gone less than ten minutes when he returned, preceded by Anne, who came quickly through the door and stopped suddenly before what she saw. She looked at the men standing about and then paused before Keith McBain. She did not ask the question, but McBain knew what was in her mind. His reply was brief. "Howden," he said, and Anne's slow smile proved that she understood. Then she went over to McCartney's side and looked down at him. "You always were a damn fool," she said very deliberately, and very slowly—and her voice had a strangely deep note of pity in it. Scattering the men before her, she hurried to the kitchen and came back with water in a basin and set about bathing McCartney's swollen face and washing the blood from his lips and chin. She was very silent and very gentle, and McCartney spoke no word to her as she worked over him. The men looked on only for a moment and then went out one by one, until the two were left alone. Later that morning Cherry went to MacMurray's to see if she could not prevail upon Anne to come over to Hurley's cottage to see King. She found Anne seated beside McCartney, who had fallen asleep. Anne was bending low over him, tears streaming down her cheeks. When she saw Cherry she got up quickly and brushed the
- 69. tears impatiently from her eyes. Then she came to Cherry, where she was standing in the doorway. "Anne—Anne," Cherry said, her voice soft with pity. But Anne was mistress of herself now. "How is King?" she asked, in a most matter-of-fact tone that expressed quite clearly how little she wanted anyone's sympathy. "He's all right now," Cherry replied. "He has a bad ankle and can't walk, but it will be all right in a day or two. He asked me to bring you over." "What does he want?" Cherry found it hard to reply to Anne's question—it was asked with such cold directness. "I think he wants to explain to you what he feels about——." Anne stopped her abruptly. "Tell him it's all right. I ain't goin' to worry over a thing that I've been expectin' for weeks. Tell him it's all right." Cherry turned to go. "Wait a minute," Anne called, and vanished into the house. She was gone a long time and Cherry waited patiently for her return. When she appeared again she held a folded paper in her hand and her hair was in disorder about her face. "I had a time gettin' it," she said, coming towards Cherry and holding the paper before her. "I had to wake him up to tell me where it was. But he told me. One thing about Bill—he knows when he's beat—an' that's sayin' something for a man that was never beat before—ain't it?" She smiled comically, and Cherry could not help smiling at her in reply.
- 70. "Anyhow, here it is," she said, giving the paper to Cherry. "I thought of takin' it over myself—I like that boy—but you'd better give it to him." Cherry knew little or nothing about official documents, but she could not help guessing the meaning of the paper she held in her hand. She opened it and glanced quickly over the written record of a timber claim in the hills, interjected between the lines of legally phrased printed matter. "Take it to him," Anne continued after a pause. "He'll know what to do with it. If he don't—ask old man Hurley." "But Anne——" Cherry protested, only to be interrupted again. "Don't worry—I ain't stealin' it. Ain't I his wife?" she asked with a laugh. "Anyhow there's something else. I had a claim once out west—a good claim, too—never mind!" She broke off abruptly and gave Cherry a little push. "Give it to him an' tell him 'God bless him' for me," she added. Cherry walked off slowly and Anne stood in the doorway watching her. When she had gone a few yards she stopped and came back. "But father——" she began and paused awkwardly. Anne's face took on a strange look. She stepped down from the doorway and confronted Cherry. "Say—did Bill spring that man-killin' joke on Old Silent?" she asked. Cherry nodded. "Well, I'm blistered!" she exclaimed. "Leave it with me—I'll make him straighten that out himself." And Cherry went off with a light heart.
- 71. That night Keith McBain came into the room where Cherry and King were sitting. King was preparing to leave for his shack—in spite of the protests of Mrs. Hurley—confident that he was able to get about and look after himself quite well with the help of old Gabe, who was going to stay with him. McBain came upon the two somewhat abruptly. When they looked up he was standing within a few feet of them, his old face beaming with a light that had not shone there for months. "Cherry, girl," he said, coming towards her and holding his arms out to her, "it's all right!" "What, father?" she asked, jumping up and going to him. "McCartney lied—he has told me everything. The man is alive— Anne nursed him back—it's all right!" Cherry threw her arms about her father's neck and kissed him. "Father, father, father!" she cried; and suddenly her voice broke. "If we had only known." "If we had only known!" repeated Old Silent; and his mind went back to a pile of stones and a little wooden cross that stood miles back beside, the right-of-way. CHAPTER EIGHTEEN King dropped his scythe upon the windrow of freshly-cut hay and stood a moment while he wiped the sweat from his brow. It was July, and the day had been very hot, and King had cut a very wide swath in the tall, wild grass. A little way off on the higher ground of
- 72. the ridge stood his first crop of growing wheat, the soft green shoots stretching upward from the new soil and bending before a gently moving breeze. Between the meadow and the wheat lay a stretch of newly-broken land where, only the day before, King had driven the plough through long furrows of rich mould. Even yet the mellow odor of freshly-turned soil came to him, mingled with the cool fragrance of the meadow. King looked about him until his eyes fell upon Sal, where she was working half-buried in a hole she had dug in a futile attempt to follow a gopher to its place of hiding under the ground. He gave a sharp whistle and crouched low, holding out his hands as the dog came bounding towards him. Taking her in his arms he lifted her from the ground and then rolled her over playfully on the hay. Getting up, he strolled off along the edge of the standing grass, Sal running before him in a zig-zag search for gophers. When they came to the edge of a small slough the dog pounced at once into the water, almost on top of a wild duck and her brood of half-grown ducklings. They started up suddenly with much splashing of water and beating of wings and loud quacking. "Back you—lie down!" King cried, and Sal retreated from the edge of the slough and came towards King wriggling and twisting her shaggy body in an effort to appear apologetic. It was a great day, and now that the afternoon was wearing on, King was strongly tempted to be lazy. He had worked hard during the past weeks. The land he had prepared for crop had been sown broadcast by hand. He had cut his hay with a scythe and would have to rake it by hand—though Cherry was longing for the hay to cure so
- 73. that she could get into the field with King and rake the long windrows into coils. Oh, yes—Cherry was King's helper now. One day in spring, just before the men had gone out to begin work on the railway construction again, there had been a final gathering from the whole valley. Cherry and King might have left it until midsummer. King wanted to get his land into shape and his first crop in—and Cherry wanted to see her father started once again on his right-of-way contracts. At least, so they said. The fact of the matter was that Old Silent wanted to keep his daughter by him for just a few weeks more, and King and Cherry had both agreed, to humor him a little until the work was well under way. But the men had settled it. McCartney and his crowd—or such of them as felt themselves unable to face Keith McBain again—had withdrawn before the snow was on the ground. The season in the camps had been highly successful in every sense, a fact, by the way, that reflected much credit upon King Howden, who had handled the men and had taken the responsibility of conducting the camp during the winter. The work on the grade was waiting, and when the men went out to the right-of-way and the young settlers went to their land, The Town would be no more. There had not been a wedding in the place since the first hut had been built. The men—through a committee duly chosen and given full powers—made known to Keith McBain their feelings on the matter. For once the old contractor allowed himself to be persuaded against his will. He made only one condition, namely, that he himself should announce to King and Cherry the decision that The Town had come to. The men agreed,
- 74. and withdrew from the presence of Old Silent to begin preparations for the great day. And it had been a day for all to remember. King thought of it now as he walked back to where his scythe lay, and picking it up stood it on its haft while he applied his whet-stone to the blade, and sent the rhythmic tune of the hay-maker ringing across the meadow. The Town was gone. There were a few old unfilled wells and the tumbled foundations of cabins, and a winding street grown over with grass and weeds—but that was all. Farther up the valley its ambitious successor was already thriving beside the right-of-way, waiting for the coming of the steel. Soon it would be linked up with the outside world, it would be given a name and placed on the map by someone who probably had never seen it—and the world's outer edge would have been pushed a little farther westward, and a little farther northward. King tossed his stone aside upon the coat that lay on the hay near him, and taking his scythe in his hands, stepped forward and swung it through the grass. From behind him came a clear call, and pausing at the end of his stroke he turned with a smile and waved his hand to Cherry, who was tripping along down the meadow towards him. King dropped his scythe and went to meet her. When they met he caught her by the arms, and lifting her from the ground, kissed her on the lips. "Leave the hay, King," she said, as soon as he had set her upon her feet again, "and let's go to the camp for supper. It's not four o'clock yet—we have more than two hours." King glanced at the hay waiting to be put into coils and then at Cherry, whose face was full of fresh girlish expectancy. Her eyes
- 75. were as roguish as they had been in those first days of their meeting, nearly a year ago. She caught his sidelong glance and read its meaning at once. "Ah, King," she pleaded, "it won't rain—see, there isn't a cloud in the sky! Besides—if it does—let it. There's lots and lots of hay— and there's only a little—just so much summer." She pinched the end of a slender finger to give point to her last statement, and looked at King with a smile brightening in her eyes. "You little scamp," he said, going to her and taking her head between his hands, "what's the use of a man making up his mind to anything where you are?" He kissed her again and started towards the little cabin on the ridge, with Cherry dancing along beside him, clinging to his arm and chattering as she went. When they came to the cabin they went in for a few moments to prepare for their trip. The cabin was larger and more comfortable than the shack in which King had lived during the previous summer —and infinitely cleaner. King had brought the logs from the hills during the winter, and had built the cabin with the assistance of a half dozen of Keith McBain's men. Cherry did the rest—and the place was as neat and snug as the heart could wish. In a moment King was out again and was gone to the corral among the willows below the ridge. When he returned and stood before the door of the cabin he led the horses, saddled and bridled and champing their bits. King called and Cherry emerged ready for the road. Sal leaped about them until they had got into the saddles, and then all went off together.
- 76. Keith McBain's camp lay some twelve or fifteen miles up the valley to the north and west. With two hours to make the trip they had ample time, without much loitering, to reach camp before the men should leave the grade for supper. They followed the freighters' trail that wound in and out, now skirting the edge of the right-of- way, now heading into the standing poplars, or running out across open reaches of green plain. Before the summer's end the steel gang would have laid the rails and the first trains would have steamed into the valley from beyond the hills. Even now the gang of engineers and levellers were close upon the heels of the graders, giving the road-bed its final touches before the steel was laid. Cherry and King rode along easily, without hurrying their horses, King listening while Cherry did most of the talking. Here and there new beauties came to meet them in the curving trail and waving grass and tall white poplars with glistening leaves and white powdered trunks. They crossed a half-dozen little streams of clear water rippling over gravel and shale. Frequently they came out where they caught a distant view of the hills that lay to the north of the valley, pale blue and lying low upon the horizon, like a fringe of dark cloud. To-day they were a very pale blue, and Cherry smiled as she pointed to them and reminded King of what she had told him in the meadow. "You see—it isn't going to rain for days," she said. "See how smoky the hills are." King extended his hand and leaned towards her. The horses moved closer together in instinct born of training at the hands of practised riders, and King's arm went about Cherry as he drew her close to him.
- 77. He seemed about to speak, but kissed her instead. The next moment they were off at a brisk run along a stretch of open trail. It was not yet six o'clock when the trail took them out upon the right-of-way a scant half mile from where Keith McBain's men were still at work on the grade. King drew his horse in and stood for some time gazing down the open right-of-way towards the workers, and then turned to look behind him, where the grade stretched far into the distance and was lost in the closing perspective. "I like this," he said to Cherry, who had drawn rein beside him. "There's something about it all that makes a man glad he has lived and taken some little part in it. If we could see the world in the making—I think it would be something like this." He stretched out his arm and swept it about him as he spoke. Cherry looked into his face, in contemplation, not so much of what he was saying but rather of what she saw in his eyes. All that made him a man—all that made him the man she loved—all that made him the man that men loved—was there in the simple gravity and the deep seriousness of his face. A few moments later they rode down among the men to where Keith McBain was standing alone smoking his pipe and watching a line of teamsters swinging about, an endless chain of "slushers" moving the earth from the side of the right-of-way to the grade in the middle. They were met on all sides by greetings from the men, who paused in their work to give them a welcome. When they came to Keith McBain, Cherry sprang to the ground and kissed her father, and King, swinging down from the saddle, came forward and shook hands with the old contractor. In Keith