A tool for generating a self-excited oscillations for an inertia wheel pendulum by means of a variable structure controller is proposed. The original system is transformed into the normal form for exact linearization. The design procedure, based on Describing Function (DF) method, allows for finding the explicit expressions of the two-relays controller gain parameters in terms of the desired frequency and amplitude. Necessary condition for orbital asymptotic stability of the output of the exactly linearized system is derived. Performance issues of the system with self-excited oscillations are validated with experiments.
The problem of generating oscillations of the inertia wheel pendulum is considered. We combine exact feedback linearization with two-relay controller, tuned using frequency-domain tools, such as computing the locus of a perturbed relay system. Explicit expressions for the parameters of the controller in terms of the desired frequency and amplitude are derived. Sufficient conditions for orbital asymptotic stability of the closed-loop system are obtained with the help of the Poincare map. Performance is validated via experiments. The approach can be easily applied for a minimum phase system, provided the behavior of the states of the zero dynamics is of no concern. Copyright (C) 2011 John Wiley & Sons, Ltd.
The purpose of the paper is to analyze the performance of several global position regulators for robot manipulators with Coulomb friction. All the controllers include a proportional-differential part and a switched part whereas the difference between the controllers is in the way of compensation of the gravitational forces. Stability analysis is also revisited within the nonsmooth Lyapunov function framework for the controllers with and without gravity pre-compensation. Performance issues of the proposed controllers are evaluated in an experimental study of a five degrees-of-freedom robot manipulator. In the experiments, we choose two criteria for performance analysis. In the first set of experiments, we set the same gains to all the controllers. In the second set of experiments, the gains of the controller were chosen such that the work done by the manipulator is similar.
This work presents an extension of a design procedure for dynamic output feedback design for systems with nonlinearities satisfying quadratic constraints. In this work we used an axial gas compressor model described by the 3-state Moore-Greitzer compressor model (MG) that has some challenges for output feedback control design (Planovsky and Nikolaev 1990), (Rubanova 2013). The more general constraints for the investigation of the robustness with respect to parametric uncertainties and measurement noise are shown.
An intuitive solution for the problem of adaptive attenuation of a disturbance formed as a finite sum of unknown sinusoidal signals is proposed for an internally stable discrete-time plant. The compensator is formed as a weighted sum of stable filters. An identification-based procedure for adaptive tuning of the coefficients is proposed for the case of unknown disturbance. We also propose a time-invariant compensator that provides perfect attenuation of a disturbance for the case when a model identification error is sufficiently small and disturbance frequencies are known. The technique is applied to a case study on a challenging benchmark example in the field of active vibration control. Attenuation of a disturbance formed as a sum of up to three sinusoidal signals with unknown/time-varying frequencies is demonstrated via simulation and experimental studies. (C) 2013 European Control Association. Published by Elsevier Ltd. All rights reserved.
MODELING AND IDENTIFICATION OF DYNAMICS OF A HYDRAULIC ACTUATOR WITH A SPOOL VALVE. PART I. MODELING
Approaches to the problem of modeling of hydraulic actuator are considered. A model for the spool dynamics consisting of a linear block and two static nonlinear subsystems is presented.
MODELING AND IDENTIFICATION OF DYNAMICS OF A HYDRAULIC ACTUATOR WITH A SPOOL VALVE. PART II: IDENTIFICATION
A method for identification of dynamic model parameters is suggested for the case when only pressures in the hydraulic cylinders but not the valve displacements are measured. The approach is verified by presented results of experiments.
The problem of control system design for longitudinal axis of a small-size flying wing is studied. The new controller proposed is comprised of two controllers working together to provide robust stability and step reference tracking for the nonlinear dynamics of SmartFly UAV. More precisely, Linear Quadratic Regulator (LQR) is used together with Proportional, Integral, Derivative (PID) controller. The inspiration comes from the fact that each of the mentioned controllers have advantages and disadvantages that cannot be neglected. LQR, as an optimal in terms of energy-like regulator, provides robust stability with a minimized energy-like performance index. It is also very computationally efficient. But, when it comes to the transient of particular output, LQR is not the best solution. On the other hand, PID has the advantage of a possibility to tune the gains for optimized transient behavior, especially for well-behaving plants. Furthermore, PID controller is particularly useful for tracking problems. However, PID is often not robust (in terms of parameter uncertainties) and it is also difficult to tune PID parameters for unstable systems. By differentiating between system stability and performance in the controller design process, it is possible to benefit from both controllers, using them along side together in one system. Functionality of this method was verified through computer simulation in MATLAB/SIMULINK for a nonlinear model of SmartFly UAV. Closed-loop system performance was evaluated in terms of robustness and step reference tracking.
A linear law of control of a mechanical system through drives nonrigidly connected to the system is suggested. The law involves measurement of the positions of links and angular velocities of drives and features a nonminimal-phase character of the transfer function. Stability conditions that remain valid with an increase in the gain are set zip. Results of the numerical modeling in solving problems of positioning anal tracking for robots (manipulators) are described.
Two models of an elastic controller are studied. A control rule with feedback for members and rotor velocities is proposed. The global stability of the equilibrium of a controlled system is proved within the frame-work of a simplified model. Numerical experiments demonstrate that the stability area includes all reasonable initial data, even taking into account discarded small factors.
It is required to position a Lagrangian system whose free and controllable degrees of freedom are elastically linked. The equations of motion of such systems describe, in particular, the dynamics of a robot manipulator with elastic joints. The proposed control laws enable restrictions on the value of the control impulse to be taken into account. Zn particular, attention is given to the situation in which the velocities are not accessible to measurement. The analysis of the proposed control laws is based on Lyapunov’s direct method or, more specifically, on the Barbashin-Krasovskii theorem on asymptotic stability in the large. The proof uses an original method to verify that an auxiliary non-linear function, analogous to the total mechanical energy of a system, closed by a control law, is positive-definite. (C) 1997 Elsevier Science Ltd. All rights reserved.
Nowadays, forest harvesting is highly mechanized. The commercially available forestry machines are equipped with knuckleboom cranes that are hydraulically actuated, and manually controlled through a set of joysticks and buttons. A common problem that human operators face during manipulation of such knuckleboom cranes, are the crane structure oscillations created by non-smooth or too aggressive manual joystick-based commands. These oscillations not only contribute to actuator wear, but are also dangerous for operators and the environment as well. The current paper investigates the oscillation attenuation induced by the motion of the inner boom actuator and is based on H2-optimal controller synthesis active in the pressure feedback loop. Furthermore, the controller robustness is verified experimentally considering different working conditions of the reference machine, which also verifies the effectiveness of the approach.
In this paper we study two approaches of static friction identification in a pendulum system using specific controlled motions. We use two models: the well-known Tustin model and a polynomial model. Both models are identified using the least squares approximation method. For the identification generated point-to-point trajectories are designed in such a way that the closed-loop system guaranties constant velocity and torque regimes needed for capturing the friction points.
Simple nominal controllers (PI, PD, PID) are used in the closed-loop during the friction measurement process. A model based compensation method is implemented for tracking improvement. The identification and compensation methods used in this paper can be implemented in a straightforward way for precision improvement of industrial mechatronic systems.
Simplifying the operation of forestry machines with operator-centered semi-automation is needed in the modern timber harvesting industry in order to increase operator productivity and comfort, to reduce learning time of novice operators and to ensure safer manipulation of the cranes. In this paper, useful tools towards operator-centered semi-automation of the base joint actuator of a forwarder crane are proposed. The main goal is to allow comfortable automated motions that do not excite dangerous oscillations of the freely-hanging grapple. Moreover, operator commands are used interactively with a closed-loop position control scheme to assure automated slewing motions. Smooth reference trajectories are provided for the position controller with an on-line trajectory generation algorithm that is developed by combining properties of two standard trajectory generation methods. A practical algorithm based on experiments is introduced to find the trajectory that guaranties minimal grapple oscillations within a set of relatively fast trajectories. Further on, the log loading/unloading tasks are discussed and verified experimentally using the proposed approach on a forwarder crane prototype.
The modern timber harvesting industry would be ineffective without heavy duty advanced machinery used for logging. However, with benefits of mechanization comes the operation complexity. Introducing automation is expected to reduce the mental and physical load on the operator and improve the machine use efficiency. Nonetheless, with current technology fully autonomous timber harvesting is impossible. In this paper a semi-automation scenario is presented using the base joint actuator of a forestry forwarder crane taking into consideration the need to attenuate unwanted oscillations of its hanging grapple. We address the necessary motion planning and motion stabilization tasks. To reduce oscillations along a nominal trajectory, we design smooth reference profiles based on experiments. Meanwhile, a practical structure for a feedback controller is proposed and tested. In this process, actuator nonlinearities are dealt with feasible identification and compensation techniques.
Working with forestry cranes is not easy due to their complex mechanical structure, non-linear behavior of the hydraulic actuation system, and non-intuitive joint-based control; however, with automation, the level of manipulation difficulty can be reduced. This is potentially useful for the operators since they are prone to be more productive if semi-automation functions are introduced to a certain level. In this paper, a semi-automation function for the base joint actuator of a forestry forwarder crane is proposed. The semi-automation function is based on a design of an interactive on-line trajectory generation algorithm with variable final time that acts as a reference signal to a closed-loop position controller. Moreover, the advantage of this scheme is that the operators are kept in the loop by directly being in charge of controlling the final time for the on-line trajectory generation algorithm. Experiments with a downsized industry-standard forwarder crane verify the applicability and advantage of the proposed scheme.
Smooth operation of heavy-duty forestry cranes is not an easy task for the operators with the current joystick-based control method that is complex and non-intuitive. Moreover, abrupt movements of the same joysticks provoke aggressive signals that can lead to oscillatory motions in the actuators and in the entire crane. These oscillations, not only contribute to wear of the joint actuators but also can cause damage to both the operators and the environment; therefore, they must be attenuated. The proposed approach in this paper uses the popular input shaping control technique combined with a practical switching logic to deal with different frequency payload oscillations induced by the motion of the inner boom actuator of a forwarder crane. The results show a significant improvement in terms of visible oscillation reduction monitored through their appearance in the torque signal computed from pressure measurements. Experiments performed on a down-sized forestry crane verifies the effectiveness of the approach.
A class of globally asymptotically stable regulators for a finite-dimensional model of robot arm with flexible links under gravity is presented. The control law is formed as the sum of static compensation of gravity at the desired position and constrained state feedback. Only some of generalized coordinates (joint positions and velocities) are assumed available for measurement and saturation in amplifier characteristic curves is taken into account. Copyright (C) 2000 IFAC.
The design of robust output and state feedback controllers for practical track-ing and stabilization for nonlinear systems with large-scale parametric uncertainty is considered. We consider a class of single-input single-output systems that can be transformed into a special form, where unavailable states (if any) are derivatives of the measured outputs. We consider a large class of nonlinear systems that could be stabilized via a Lyapunov-based technique if the level of parametric uncertainty was much lower and if the unmeasured states were available for feedback. We propose and investigate a new approach based on subdividing the set of parameters into smaller subsets, designing candidate controllers for each subset, and implementing a logic-based switching between them. We use the output of an extended-order high-gain observer to substitute the unavailable states in the candidate controllers, check the inequality for the derivative of the Lyapunov function and switch if it is not satisfied, and to approximately identify the parameters. We discuss the issues of digital implementation and measurement noise and illustrate our design procedure on several examples.
В диссертации получены следующие новые научные результаты.
Для общей нелинейной Лагранжевой системы прямого управления и, в частности, для робототехнического манипулятора, податливостью элементов конструкции которого можно пренебречь, впервые показано:
пропорционально-интегрально-дифференциальный (ПИД) регулятор, широко распространённый в промышленности, гарантирует асимптотическую устойчивость "в целом" желаемого положения равновесия при естественных ограничениях на ПД-коэффициенты усиления и при достаточной малости коэффициента усиления интегральной (И) обратной связи;
тот же результат справедлив при замене дифференциальной (Д) - составляющей обратной связью по выходу некоторой линейной системы дифференциальных уравнений, вносящей диссипацию энергии и идейно близкой к простейшему и очень неточному оценивателю скоростей;
при удовлетворении некоторых несложных неравенств для матриц ПИД - коэффициентов имеет место экспоненциальная устойчивость "в большом", то есть удается оценить границы области притяжения и гарантированную скорость сходимости процесса;
тот же результат справедлив при замене Д--составляющей обратной связью по выходу некоторой линейной системы дифференциальных уравнений (решаемых параллельно с движением и вносящих диссипацию энергии);
ПИД - регулятор удовлетворительно решает задачу слежения, если движение по желаемой траектории происходит достаточно медленно;
Для математической модели многозвенного пространственного робототехнического манипулятора, учитывающей нежёсткость конструкции (путём введения в модель линейных податливых элементов в сочленениях), показано:
регулятор, состоящий из пропорционально-дифференциальной обратной связи по положению роторов управляющих двигателей и постоянной добавки компенсирующей статический прогиб ("ПД+"), обеспечивает асимптотическую устойчивость в целом замкнутой системы даже при учёте наличия насыщения в характеристиках усилителей обратных связей;
тот же результат справедлив при замене Д - составляющей с насыщением обратной связью по выходу некоторой нелинейной системы дифференциальных уравнений (с нелинейностью в точности совпадающей с нелинейностью характеристик усилителей), вносящей диссипацию энергии;
для псевдо "ПД+" регулятора, а именно, для случая наиболее распространённой и легче всего реализуемой на практике обратной связи по положениям звеньев и скоростям роторов двигателей, имеет место асимптотическая устойчивость замкнутой системы при достаточно естественных условиях;
некоторые робастные линейные законы управления, основывающиеся на той же наиболее разумной комбинации измерений (датчиков), и, в частности, два варианта реализации "псевдо ПИД" - регуляторов гарантируют асимптотическую устойчивость в целом при выполнении некоторых условий на матрицы коэффициентов усиления;
вместо интегральной обратной связи можно использовать также несложную итеративную процедуру обучения управления нужной компенсационной добавке.
Некоторые, но далеко не все, идеи по поводу управления роботами могут быть обобщены на случай общих нелинейных систем управления. В частности, удалось ввести известное из линейной теории систем понятие астатизма и проанализировать свойства астатических систем, такие как ограниченность реакции на линейно растущее возмущение.
We consider a tracking problem for a partially feedback linearizable nonlinear system with stable zero dynamics. The system is uncertain and only the output is measured. We use an extended high-gain observer of dimension n+1, where n is the relative degree. The observer estimates n derivatives of the tracking error, of which the first (n-1) derivatives are states of the plant in the normal form and the $n$th derivative estimates the perturbation due to model uncertainty and disturbance. The controller cancels the perturbation estimate and implements a feedback control law, designed for the nominal linear model that would have been obtained by feedback linearization had all the nonlinearities been known and the signals been available. We prove that the closed-loop system under the observer-based controller recovers the performance of the nominal linear model as the observer gain becomes sufficiently high. Moreover, we prove that the controller has an integral action property in that it ensures regulation of the tracking error to zero in the presence of constant nonvanishing perturbation.
We consider a partially feedback linearizable system with stable zero dynamics. The system is uncertain and only the output is measured. Consequently, exact feedback linearization is not applicable. We propose to design an extended high-gain observer to recover unmeasured derivatives of the output and an extra one, which contains information about the uncertainty. The observer can be stabilized via feedback linearization followed by a linear control design, such as pole placement or LQR. After a short peaking period, a partial state vector, which includes the output and its derivatives, will be in a small neighborhood of the state of the observer; therefore, the performance achievable under exact feedback linearization will be recovered.
We consider an underactuated two-link robot called the inertia wheel pendulum. The system consists of a free planar rotational pendulum and a symmetric disk attached to its end, which is directly controlled by a DC-motor. The goal is to create stable oscillations of the pendulum, which is not directly actuated. We exploit a recently proposed feedback-control design strategy based on motion planning via virtual holonomic constraints. This strategy is shown to be useful for design of regulators for achieving orbitally exponentially stable oscillatory motions. The main contribution is a step-by-step procedure on how to achieve oscillations with pre-specified amplitude from a given range and an arbitrary independently chosen period. The theoretical results are verified via experiments with a real hardware setup.
We consider a class of mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom. In addition to control forces, we take into account viscous and Coulomb friction forces and impacts with the environment modeled as impulsive updates of the states. We assume that a motion planning task is solved and a feasible forced periodic motion is described in terms of piece-wise smooth virtual holonomic constraints. The main contribution is an analytical method for computing coefficients of an impulsive linear control system, solutions of which approximate dynamics transversal to the preplanned trajectory. This linear system is shown to be useful for stability analysis and for design of feedback controllers orbitally stabilizing forced periodic motions in the hybrid mechanical system. As an illustration, we apply the obtained theoretical results providing a rigorous proof of orbital exponential stability of the periodic tumbling motion for a model of a descending strip of paper in a still air.
Recently, a new technique for generating periodic motions in mechanical systems which have less actuators than degrees of freedom has been proposed. A motivating example for studying such motions is a dynamically stabilized walking robot, where the target trajectory is periodic, and one of the joints - the ankle joint - is unactuated, or weakly actuated. In this paper, the technique is implemented on the Furuta pendulum, an experimental testbed that is simpler than a walking robot but retains many of the key challenges - it is underactuated, open-loop unstable, and practical problems such as friction and velocity estimation must be overcome. We present a detailed description of the practical implementation of the controller. The experiments show that the technique is sufficiently robust to be useful in practice.
The paper by Chaillet, Loría, and Kelly is devoted to study robustness of mechanical systems controlled by proportional integral-differential (PID) regulators. These control strategies are classical and are the most frequently used in industrial applications of robotic manipulators despite various other available techniques. There is a number of results on properties of PID-controlled mechanical systems, see references in the paper and [1,2,5–7,11–13] to mention a few.
We consider output feedback stabilization of uniformly observable uncertain nonlinear systems when the uncertain parameters belong to a known but comparably large compact set. In a previous paper, we proposed a new logic-based switching control to improve the performance of continuous high-gain-observer-based sliding mode controllers. Our main goal here is to show that similar techniques can be exploited for solving challenging control problems for a more general class of uncertain nonlinear systems. We require neither the sign of the high-frequency gain to be known nor the system to be minimum-phase. The key idea is to split the set of parameters into smaller subsets, design a controller for each of them, and switch the controller if, after a dwell-time period, the derivative of the Lyapunov function does not satisfy a certain inequality. A high-gain observer is used to estimate the derivatives of the output as well as the derivative of the Lyapunov function. Another goal of this paper is to introduce a switching strategy that uses on-line information to decide on the controller to switch to, instead of using a pre-sorted list as in our previous work. The new strategy can improve the transient performance of the system.
We consider a single-input–single-output minimum-phase nonlinear system with large parametric uncertainty. The system can be represented globally in the normal form and our goal is to find a dynamic output feedback control law to ensure that the output (practically) asymptotically tracks a bounded smooth reference signal. Earlier work used high-gain observers with saturation to derive adaptive as well as robust control laws for this problem. The adaptive control law requires the nonlinear functions to be linearly parameterized in the unknown parameters and could have unsatisfactory transient performance for a large parameter set. The robust control law is based on a worst-case design and could be overly conservative. High gain feedback is needed to implement both controllers in the case when the parameter set is large. As a result, the robust and adaptive controllers may perform poorly in the presence of unmodeled dynamics and measurement noise. In order to reduce the controller gain and improve performance we propose a new approach based on partitioning the set of uncertain parameters into smaller subsets. Robust control laws are designed for each subset and logic based switching is used to choose the appropriate control law. The switching rule uses an estimate of the derivative of a Lyapunov function, which is provided by a high-gain observer.
We consider a nonlinear single-input-single-output minimum phase system with large parametric uncertainty. The system can be represented globally in the normal form and our goal is to find an output dynamic feedback control law to ensure that the output (practically) asymptotically tracks a bounded smooth reference signal. Earlier work used high-gain observers with saturation to derive adaptive as well as robust control laws for this problem. The adaptive control law requires the nonlinear functions to be linearly parameterized in the unknown parameters and could have unsatisfactory transient performance for a large parameter set. The robust control law is based on a worst-case design and could be overly conservative for a large parameter set. In this paper we propose a new approach based on partitioning the set of uncertain parameters into smaller subsets. Robust control laws are designed for each subset and logic based switching is used to choose the appropriate control law. The switching rule uses an estimate of the derivative of a Lyapunov function, which is estimated using a high-gain observer.
We consider an example of a second order nonlinear system with large parametric uncertainties. The two parameters of the system are assumed to belong to a finite set. The goal is to guarantee (practical) convergence of the system output to a given constant reference signal. Feedback linearization-based candidate controllers with pole placement are designed for each possible set of parameters. After that, we consider the design of a high-level logic-based supervisor to organize switching between these candidate controllers. Three different approaches are used and compared.
We consider dynamic output feedback practical stabilization of uniformly observable nonlinear systems, based on high-gain observers with saturation. We assume that uncertain parameters and initial conditions belong to known but comparably large compact sets. In this situation, designs based on traditional robust or adaptive techniques, if applicable, would lead to high controller, observer, and adaptation gains. High gains may excite unmodeled dynamics and significantly amplify measurement noise. Moreover, they could be impossible or too costly to implement. In order to reduce the control efforts and improve robustness of a continuous high-gain-observer-based sliding mode control with respect to these non-ideal operational conditions, we have recently proposed a new logic-based switching design strategy. In this paper, we generalize our technique and apply it to a wider class of nonlinear systems and more general Lyapunov-function-based state and output feedback designs. It is important to notice, in particular, that we require neither the sign of the high-frequency gain to be known nor the system to be minimum-phase. The key idea is to split the set of parameters into smaller subsets, design a controller for each of them, and switch the controller if the derivative of the Lyapunov function does not satisfy a certain inequality, after a dwell-time period. We do not order the candidate controllers in advance, as in our earlier work. Instead, we use estimates of the derivatives of the states, provided by an extended order high-gain observer, to calculate instantaneous performance indices. When the controller is falsified, we switch to a new controller that corresponds to the smallest index among the controllers that have not been falsified yet. This modification is important when the number of candidate controllers is high and pre-routed search may lead to an unacceptable transient performance.
We consider output feedback stabilization of uniformly observable uncertain nonlinear systems when the uncertain parameters belong to a known but comparably large compact set. In a previous paper, we proposed a new logic-based switching control to improve the performance of continuous high-gain-observer-based sliding mode controllers. Our main goal here is to show that similar techniques can be exploited for solving challenging control problems for a more general class of uncertain nonlinear systems. We require neither the sign of the high-frequency gain to be known nor the system to be minimum-phase. The key idea is to split the set of parameters into smaller subsets, design a controller for each of them, and switch the controller if, after a dwell-time period, the derivative of the Lyapunov function does not satisfy a certain inequality. A high-gain observer is used to estimate the derivatives of the output as well as the derivative of the Lyapunov function. Another on-line information to decide on the controller to switch to, instead of using a goal of this paper is to introduce a switching strategy that uses on pre-sorted list as in our previous work. The new strategy can improve the transient performance of the system. (c) 2006 Elsevier Ltd. All rights reserved.
As a standard model for a rigid multilink robotic manipulator, we consider a MIMO nonlinear system of uniform vector relative degree {2,...,2}, which has a globally defined normal form with no zero dynamics. We consider a trajectory-following problem for a class of smooth bounded time-varying vector reference signals. We design an output feedback integral controller that ensures ultimately bounded tracking error which could be made as small as required. In addition, if the reference signal has a constant limit, the output approaches this limit. The controller is robust with respect to parameter uncertainties, decentralized, saturated, and simple to implement. Locally, it is a PID regulator with derivatives estimated via a linear high-gain observer. Regional (and semiglobal) practical (and asymptotic) stability is shown with the help of a vector Lyapunov function and a technique, typical for "continuous" sliding mode control. Copyright (C) 2001 IFAC.
A planar compass-like biped on a shallow slope is the simplest model of a passive walker. It is a two-degrees-of-freedom impulsive mechanical system known to possess periodic solutions reminiscent to human walking. Finding such solutions is a challenging task. We propose a new approach to obtain stable as well as unstable hybrid limit cycles without integrating the full set of differential equations. The procedure is based on exploring the idea of parameterizing a possible periodic solution via virtual holonomic constraints. We also show that a 2-dimensional manifold, defining the hybrid zero dynamics associated with a stable hybrid cycle, in general, is not invariant for the dynamics of the model of the compass-gait walker.
A planar compass-like biped on a shallow slope is one of the simplest models of a passive walker. It is a 2-degree-of-freedom (DOF) impulsive mechanical system that is known to possess periodic solutions reminiscent of human walking. Finding such solutions is a challenging computational task that has attracted many researchers who are motivated by various aspects of passive and active dynamic walking. We propose a new approach to find stable as well as unstable hybrid limit cycles without integrating the full set of differential equations and, at the same time, without approximating the dynamics. The procedure exploits a time-independent representation of a possible periodic solution via a virtual holonomic constraint. The description of the limit cycle obtained in this way is useful for the analysis and characterization of passive gaits as well as for design of regulators to achieve gaits with the smallest required control efforts. Some insights into the notion of hybrid zero dynamics, which are related to such a description, are presented as well.
A class of robotic manipulators having prismatic and revolute joints is considered. It is supposed that feedback control laws have proportional-integral-differential (PID) or PID-like forms. It ensures the astatism property of the closed-loop systems. The following results are presented and proved. There are simple inequalities for feedback gains ensuring exponential stability of a desired position under all initial deviations which are small enough. The stability domain can be estimated as well as the stability degree. Under a coordinated growth of coefficients, the stability region is enlarged but the stability degree goes down. The main properties of PID regulated system are not changed if a "dirty" derivatives are used instead of pure ones. PID regulators ensure the tracking with a bounded error under any desired movements which are changed slowly enough.
We consider a tracking problem for mechanical systems. It is assumed that feedback controller is designed neglecting some disturbances, which could be approximately modeled by a dynamic LuGre friction model. We are interested to derive an additive observer-based compensator to annihilate or reduce the influence of such a disturbance. We exploit a recently suggested approach for observer design for LuGre-friction-model-based compensation. In order to follow this technique, it is necessary to know the Lyapunov function for the unperturbed system, parameters of the dynamic friction model, and to have certain structural property satisfied. The case when this property is passivity with respect to the matching disturbance related to the given Lyapunov function is illustrated in the paper with an example of a DC-motor. The main contribution is some new insights into numerical real time implementation of friction compensators for various LuGre-type models. The other contribution, built upon the main one, is experimental verification of the suggested observer design procedure.
A tracking problem for a mechanical system is considered. We start with a feedback controller that is designed without attention to disturbances, which are assumed to be adequately described by a dynamic LuGre friction model. We are interested in deriving a superimposed observer-based compensator to annihilate or reduce the influence of such a disturbance. We exploit a recently suggested approach for observer design for LuGre-friction-model-based compensation. In order to apply this technique, it is necessary to know the Lyapunov function for the unperturbed system, as well as the parameters of the dynamic friction model, and to verify that a certain structural property satisfied. The case when the system is passive with respect to the matching disturbance related to the given Lyapunov function is illustrated in this brief with a DC-motor example. The main contribution is some new insights into the numerical real-time implementation of a compensator for disturbances describable by one of various LuGre-type models. The other contribution, which is built upon the main one, is experimental verification of the suggested model-based observer design procedure.
This paper presents a new control strategy for an underactuated two-link robot, called the Pendubot. The goal is to create stable oscillations of the outer link of the Pendubot, which is not directly actuated. We exploit a recently proposed feedback control design strategy, based on motion planning via virtual holonomic constraints. This strategy is shown to be useful for design of regulators for achieving: stable oscillatory motions, a closed-loop-design-based swing-up, and propeller motions. The theoretical results are verified via successful experimental implementation.
We consider the problem of creating oscillations with respect to a part of state variables in underactuated mechanical systems. The main contribution is a modification of a recently proposed control strategy, exploiting passivity to shape the energy of the system with respect to a subset of the state variables and neglecting the dynamics of the other ones. We propose a way to design an additional control action, which guarantees boundedness of the motion of these other degrees of freedom, which otherwise almost always evolve without any bound. The idea is presented on the well-known two-degrees-of-freedom benchmark example of inverted pendulum on a cart.
We consider the problem of creating oscillations of the Furuta pendulum around the open-loop unstable equilibrium. Following a recently proposed technique, we start with shaping the energy of the unactuated link. An dissipativity-based controller is designed to create oscillations, neglecting possibility of unbounded motion of the directly actuated link. After that, an auxiliary linear feedback action is added to the control law, stabilizing a desired level of the reshaped energy. Parameters of the controller are tuned to approximately keep the originally created oscillations but ensuring bounded motion of both links. The analysis is valid only for oscillations of sufficiently high frequency and is based on higher order averaging technique. Performance of the designed controller is verified using numerical simulations and experiments.
We consider the problem of creating oscillations of the Furuta pendulum around the open-loop unstable equilibrium. We start with a control transformation shaping the energy of the passive link. Then, a dissipativity-based controller is designed to create oscillations, neglecting the possibility of unbounded motion of the directly actuated link. After that, an auxiliary linear feedback action is added to the control law stabilizing a desired level of the reshaped energy. Parameters of the controller are tuned to approximately keep the originally created oscillations but ensuring bounded motion of both links. The analysis is valid only for oscillations of sufficiently high frequency and is based on higher order averaging technique. The performance of the designed controller is verified using numerical simulations as well as experimentally.
We consider the challenging problem of creating oscillations in underactuated mechanical systems. Target oscillatory motions of the indirectly actuated degree of freedom of a mechanical system can often be achieved via a straightforward to design feedback transformation. Moreover, the corresponding part of the dynamics can be forced to match a desired second-order system possessing the target periodic solution (Aracil, J., Gordillo, F., and Acosta, J.A. (2002), 'Stabilization of Oscillations in the Inverted Pendulum', in Proceedings of the 15th IFAC World Congress, Barcelona, Spain; Canudas-de-Wit, C., Espiau, B., and Urrea, C. (2002), 'Orbital Stabilisation of Underactuated Mechanical Systems', in Proceedings of the 15th IFAC World Congress, Barcelona, Spain). Sometimes, it is possible to establish the presence of periodic or bounded motions for the remaining degrees of freedom in the transformed system. However, typically this motion planning procedure leads to an open-loop unstable orbit and by necessity should be followed by a feedback control design. We propose a new approach for synthesis of a (practically) stabilising feedback controller, which ensures convergence of the solutions of the closed-loop system into a narrow tube around the preplanned orbit. The method is illustrated in detail by shaping oscillations in the inverted pendulum on a cart around its upright equilibrium. The complete analysis is based on application of a non-standard higher-order averaging technique assuming sufficiently high frequency of oscillations and is presented for this particular example.
The problem is to create a hybrid periodic motion, reminiscent of walking, for amodel of an underactuated biped robot. We show how to construct a transverse linearization analytically and how to use it for stability analysis and for design of an exponentially orbitally stabilizing controller. In doing so, we extend a technique recently developed for continuous-time controlled mechanical systems with degree of underactuation one. All derivations are shown on an example of a three-link walking robot, modeled as a system with impulse effects.
This work addresses the stabilization of dynamical systems in presence of uncertain bounded perturbations using epsilon-invariance theory. Under some assumptions, the problem is reduced to the stabilization of a chain of integrators subject to a perturbation and is treated in two steps. The evaluation of the disturbance and its compensation. Homogeneous observer and control [5], [19] are the tools utilized to achieve a global asymptotic stability and robustness. The result is formally proven and, to validate the theory, it is applied to the control of the telescopic link of a hydraulic actuated industrial crane used in forestry. Experimental results and a comparison with a standard PI controller are presented.
In this work an interval observer is proposed for on-line estimation of differentiation errors in some class of high order differentiators (like a high-gain differentiator from [26], or homogeneous nonlinear differentiator from [24 or super twisting differentiator [15]). The results are verified and validated on the telescopic link of a robotic arm for forestry applications in which the mentioned approaches are used to estimate the extension velocity while the interval observer gives bounds to this estimation.
Abstract: The paper presents a new method for numerical solution of matrixRiccati equation with periodic coeﬃcients. The method is based on approximationof stabilizing solution of the Riccati equation by trigonometric polynomials.
Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.