Lyapunov-Based Control of Robotic Systems

Aman Behal, Warren Dixon, Darren M. Dawson, Bin Xian

• Publisher: CRC Press
• Year: 2009
• Print ISBN: 9780367452421
• eBook ISBN: 9781000654431
• Topic: Computer Science
• Subtopic: Computer Science General
• Index: Computer Science

Lyapunov-Based Control of Robotic Systems describes nonlinear control design solutions for problems that arise from robots required to interact with and manipulate their environments. Since most practical scenarios require the design of nonlinear controllers to work around uncertainty and measurement-related issues, the authors use Lyapunov's direc

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Introduction

1.1 History of Robotics

From time immemorial, human beings have been fascinated with building and employing machines with human-like capabilities. As far back as 300 B.C., Greek philosophers, mathematicians, and inventors were postulating or designing mechanical contraptions that had steam or water-based actuation as well as some level of autonomy. Around the middle of the last millennium, Leonardo da Vinci and others built mechanical devices primarily for amusement purposes. Cut to the industrial age, the great scientist Nikola Tesla built and demonstrated a remote controlled submersible robot boat at Madison Square Garden in the year 1898. The word “robot” (which actually means “forced labor” in Czech) was introduced into our vocabulary by playwright Karel Capek in a satirical 1920 play entitled Rossum’s Universal Robots. Science fiction fans are, of course, very familiar with the work of Isaac Asimov who first popularized the term “Robotics” and was responsible for proposing the three fundamental laws of robotics.

Robotics has always attracted the fancy of moviemakers, and robots have been an integral part of popular culture in the United States. The first robot to appear on film was “Maria” in Fritz Lang’s 1926 movie named Metropolis. In 1951, an alien and his robot appeared on the silver screen in the movie The Day the Earth Stood Still. Arthur Clarke’s novel 2001: A Space Odyssey was made into a movie in 1968 which featured a high functioning robot named HAL that turns rogue and is eventually disconnected. A movie that received a great deal of acclaim was Ridley Scott’s Blade Runner that was released in 1982 and which featured Harrison Ford as a hunter of illegal mutinous androids known as Replicants. Other movies like The Terminator and the Matrix series have dealt with sophisticated, high-functioning humanoids. Most recently, Pixar produced the smash hit animated robotics film WALL-E which features a sentimental robot of the same name that is designed to clean up the pollution created by mankind.

Space exploration has advantageously employed manipulator arms and mobile robots over the years. Lunokhod 1 and 2 were the first robotic exploration vehicles (rovers) to be launched to an extraterrestrial body, the moon, by the Soviets in 1970. After a long gap, the rover Sojourner landed on Mars in 1997 as part of the Pathfinder Mission; it had vision assisted autonomous navigation and was successful at obtaining and analyzing rock and soil samples. This was followed in 2004 by the rovers Spirit and Opportunity that are still active and continue to analyze the Martian geology, as well as its environment, to assess the possibility that life may have been supported on Mars in the past. Most recently, the Phoenix lander executed the first successful polar landing on the Martian surface and is currently exploring the possibility of water existing or having existed on the red planet.

The first commercial robotics company, named Unimation, was started in 1956 by George Devol and J. Engelberger. As a result of this venture, the first industrial robot was manufactured and marketed in the United States. Unimate began work in a General Motors automobile plant in New Jersey in 1961. This manipulator arm performed spot welding operations as well as unloading of die casts. This was followed in 1978 by the Programmable Universal Machine for Assembly, a.k.a. PUMA. Since that time, quite a few other robot manufacturers have come and gone with only a few achieving commercial success or longevity in the market. In recent years, personal and professional service robots have picked up steam. For example, Lego has achieved success with its Mindstorms Robotics Invention System as has Sony with its AIBO robot pets. Most recently, Honda’s humanoid robot ASIMO has hogged media limelight with its ability to perform a wide variety of service and human interaction tasks.

Robotics in research settings has steadily continued to experience an upward spiral since its inception. Robotics research got its academic start in 1959 with the inauguration of the Artificial Intelligence Laboratory at the Massachusetts Institute of Technology by John McCarthy and Marvin Minsky. Other inaugurations of note were the establishment of the Artificial Intelligence Laboratory at Stanford University in 1963, and the Robotics Institute at Carnegie Mellon University in 1979. The first computer-controlled mechanical hand was developed at MIT in 1961 followed by the creation of the Stanford Arm in the Stanford Artificial Intelligence Laboratory by Victor Scheinman in 1969. Early flexible robots of note were Minsky’s octopus-like Tentacle Arm (MIT, 1968) and Shigeo Hirose’s Soft Gripper (Tokyo Institute of Technology, 1976). An eight-legged walking robot named Dante was built at Carnegie Mellon University which was followed by a more robust Dante II that descended into the crater of the volcano Mt. Spurr in Alaska in 1994. Demonstrations of planning algorithms for robots began in the late 1960s when Stanford Research Institute’s Shakey was able to navigate structured indoor environments. A decade later, the Stanford cart attempted navigation of natural outdoor scenes as well as cluttered indoor environments. Modern robotics research is focused on higher dimensional robots, modular robots, and the planning issues associated with these types of devices. Simultaneously, robotics is making great strides in medicine and surgery as well as assistance for individuals with disabilities.

1.2 Lyapunov-Based Control Philosophy

The requirements for increasing levels of autonomy and precision in robots have necessitated the development of sophisticated control strategies. Multiple link robots have presented complex, coupled nonlinear dynamics that have inspired the design of numerous output and state feedback control designs, especially the global output feedback problem for robots has been very challenging. Wheeled mobile robots have inspired the design of setpoint and tracking controllers for nonholonomic systems. Other applications that have challenged control designers have been rigid link flexible joints in the late nineties as well as higher dimensional and continuum robots of late.

Linear control design is often inadequate outside narrow operating regimes where linearized system models are valid. Nonlinear control strategies can take advantage of full or partial knowledge of the structure and/or parameters of the system in order to craft techniques that are robust to exogenous disturbances, measurement noise, and unmodeled dynamics. Research investigators have utilized a variety of tools for analyzing nonlinear systems arising from nonlinear controllers, nonlinear models, or a combination thereof – singular perturbation, describing functions, and phase plane analysis are some of the popular tools. However, Lyapunov-based techniques (in particular, the so-called direct method of Lyapunov) offer the distinct advantage that they allow both design and analysis under a common framework with one stage motivating the other in an iterative fashion. Lyapunov theory and its derivatives are named after the Russian mathematician and engineer Aleksander Mikhailovich Lyapunov (1857–1918).

Lyapunov stability theory has two main directions — the linearization method and the aforementioned direct method of Lyapunov. The method of linearization provides the fundamental basis for the use of linear control methods [1]. It states that a nonlinear system is locally stable if all the eigenvalues of its linear approximation (via a Taylor series expansion about a nominal operating point) are in the open left half plane and is unstable if at least one is in the open right half plane. Furthermore, the stability cannot be determined without further analysis if the linearization is marginally stable. The direct method of Lyapunov relies on the physical property that a system whose total energy is continuously being dissipated must eventually end up at an equilibrium point [1, 2]. Given a scalar, non-negative energy (or energy-like) function V (t) for a system, it can be shown that if its time derivative $\dot{V}(t)\le 0$, the system is stable in the sense of Lyapunov in that the system states (energy) can be constrained for all future time to lie inside a ball that is directly related to the size of the initial states of the system.

While a lot of results have been derived in the last fifty years in order to deduce stability properties based on the structure of the Lyapunov function V (t) and its time derivative, we are no closer to understanding how one may choose an appropriate V (t), i.e., it is not clear how closely the scalar function V (t) should mimic the physical (kinetic and potential) energy of the system. What is clear is that the objectives of the control design and the constraints on the measurements lead to the definition of system states that often guide the development of the Lyapunov function. Furthermore, the control design itself is impacted by the need to constrain the time derivative of the Lyapunov function to be negative definite or semi-definite along the closed loop system trajectories. Thus, the control design and the development of the Lyapunov function are intertwined, even though the presentation may tend to indicate a monotonic trajectory from control design toward stability analysis. In the ensuing chapters, one will be able to gain an insight into the variety of choices for Lyapunov functions as well as the appearance of non-intuitive terms in the control input signals that will likely indicate an influence of the Lyapunov-based analysis method on the control design.

While Lyapunov’s direct method is good at characterizing the stability of equilibrium points for autonomous and nonautonomous systems alike, it works equally well in showing the boundedness and ultimate boundedness of solutions when no equilibrium points exist [3]. Furthermore, the analysis not only provides a guarantee of stability and the type of stability result (uniform asymptotic, exponential, semi-global ultimately bounded, etc.), it is also able to point out bounds on the regions where the results are guaranteed to be valid. This is in sharp contrast to linearization based approaches where regions of convergence are not easily obtained. Finally, Lyapunov-based design leads to faster identifiers and stronger controllers that are able to prevent catastrophic instabilities associated with traditional estimation based methods such as certainty equivalence [4]. While traditional methods work well with linear systems, they can lead to troubling results such as finite escape times in the case of nonlinear systems. A shortcoming of the Lyapunov-based analysis techniques is that the chosen parameters (while guaranteed to produced closed-loop stability) may be too conservative, thereby compromising the transient response of the system. Moreover, Lyapunov stability theorems only provide sufficient conditions for stability, i.e., without further work, it is not possible to say which of those conditions are also necessary [3].

1.3 The Real-Time Computer Revolution

As we will see in the ensuing chapters, the nonlinear control, estimation, and observation schemes emanating from an application of Lyapunov’s direct method tend to have a complex structure and are generally computationally intense compared to their linear counterparts [2]. Thus, there is a requirement for the use of microprocessors, microcontrollers, and/or computers to crunch the numbers. Furthermore, there is a requirement for fast interface hardware for allowing the physical (generally analog) world to interact bidirectionally with the digital domain without creating instabilities and uncertainties due to factors such as phase lags from slow computation, quantization noise due to finite precision, aliasing due to slow sampling, uncertain order of execution of various computation modules, etc. In the last decade, a multitude of control environments have been created in the academic and industrial research and development communities to serve this emerging need for reliable real-time computation.

A real-time implementation is different from a traditional implementation in that the worst-case performance of the hardware and the software is the most important consideration rather than the average performance. In real-time operation, the processing of external data arriving in the computer must be completed within a predetermined time window, failing which the results obtained are not useful even if they are functionally accurate [5]. In real-time applications, two types of predictability have been specified, namely, microscopic and macroscopic predictability [6]. Microscopic predictability is the idea that each layer of the application from data input to control output should operate deterministically and predictably with failure at one layer dooming the entire application. However, a more robust idea is macroscopic or overall or top-layer predictability which is more suited for complex applications — here, failure to meet a deadline for an internal layer is taken care of by specialized handling. As an example, visual processing schemes generally aggregate data (inliers) to reach a threshold of statistical significance, thus, the computation required may vary substantially between different control cycles. Under such variation, a robot running under an external visual-servoing control and an internal encoder based control is normally programmed to extrapolate a setpoint if a visual processing deadline is missed rather than shut down its entire operation — a correction to the prediction (if necessary) is made in the ensuing cycles when the visual information becomes available.

Traditionally, real-time prototyping has been performed on a heterogeneous system comprising a PC host and a DSP single-board computer (SBC) system, where the control executes on the DSP SBC while the host PC is used to provide plotting, data logging, and control parameter adjustments (online or offline) [7]. As explained in [8], the DSP board is designed to very rapidly execute small programs that contain many floating point operations. Moreover, since the DSP board is dedicated to executing the control program, the host computer is not required to perform fast and/or real-time processing — thus, in a heterogeneous architecture, the host computer can run a non real-time operating system such as MS-DOS, Windows, Linux, MacOS, etc. An example of such a system is the popular dSPACE^TM Controller Board based on PowerPC technology that sits in the PCI bus of a general purpose computer (GPC). Other examples of Host/DSP systems include the MS-Windows based Winmotor and the QNX based QMotor 1.0 that were developed by the Clemson University Controls and Robotics group in the 1990s. While still enormously popular, the...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1 Introduction
• 2 Robot Control
• 3 Vision-Based Systems
• 4 Path Planning and Control
• 5 Human Machine Interaction
• Appendices
• Index