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Showing papers on "Autonomous system (mathematics) published in 2007"


Journal ArticleDOI
TL;DR: The effectiveness of the proposed MPC formulation is demonstrated by simulation and experimental tests up to 21 m/s on icy roads, and two approaches with different computational complexities are presented.
Abstract: In this paper, a model predictive control (MPC) approach for controlling an active front steering system in an autonomous vehicle is presented. At each time step, a trajectory is assumed to be known over a finite horizon, and an MPC controller computes the front steering angle in order to follow the trajectory on slippery roads at the highest possible entry speed. We present two approaches with different computational complexities. In the first approach, we formulate the MPC problem by using a nonlinear vehicle model. The second approach is based on successive online linearization of the vehicle model. Discussions on computational complexity and performance of the two schemes are presented. The effectiveness of the proposed MPC formulation is demonstrated by simulation and experimental tests up to 21 m/s on icy roads

1,184 citations


Journal ArticleDOI
TL;DR: This work develops an algorithm, called FM*, to efficiently extract a 2-D continuous path from a discrete representation of the environment and takes underwater currents into account thanks to an anisotropic extension of the original FM algorithm.
Abstract: Efficient path-planning algorithms are a crucial issue for modern autonomous underwater vehicles. Classical path-planning algorithms in artificial intelligence are not designed to deal with wide continuous environments prone to currents. We present a novel Fast Marching (FM)-based approach to address the following issues. First, we develop an algorithm we call FM* to efficiently extract a 2-D continuous path from a discrete representation of the environment. Second, we take underwater currents into account thanks to an anisotropic extension of the original FM algorithm. Third, the vehicle turning radius is introduced as a constraint on the optimal path curvature for both isotropic and anisotropic media. Finally, a multiresolution method is introduced to speed up the overall path-planning process

438 citations


Journal ArticleDOI
TL;DR: A broad survey of the various paradigms of cognition, addressing cognitivist (physical symbol systems) approaches, emergent systems approaches, encompassing connectionist, dynamical, and enactive systems, and also efforts to combine the two in hybrid systems.
Abstract: This survey presents an overview of the autonomous development of mental capabilities in computational agents. It does so based on a characterization of cognitive systems as systems which exhibit adaptive, anticipatory, and purposive goal-directed behavior. We present a broad survey of the various paradigms of cognition, addressing cognitivist (physical symbol systems) approaches, emergent systems approaches, encompassing connectionist, dynamical, and enactive systems, and also efforts to combine the two in hybrid systems. We then review several cognitive architectures drawn from these paradigms. In each of these areas, we highlight the implications and attendant problems of adopting a developmental approach, both from phylogenetic and ontogenetic points of view. We conclude with a summary of the key architectural features that systems capable of autonomous development of mental capabilities should exhibit

423 citations


Journal ArticleDOI
TL;DR: In this article, a game-theoretical approach is proposed to solve the problem of autonomous vehicle-target assignment, where a group of vehicles are expected to optimally assign themselves to a set of targets.
Abstract: We consider an autonomous vehicle-target assignment problem where a group of vehicles are expected to optimally assign themselves to a set of targets. We introduce a game-theoretical formulation of the problem in which the vehicles are viewed as self-interested decision makers. Thus, we seek the optimization of a global utility function through autonomous vehicles that are capable of making individually rational decisions to optimize their own utility functions. The first important aspect of the problem is to choose the utility functions of the vehicles in such a way that the objectives of the vehicles are localized to each vehicle yet aligned with a global utility function. The second important aspect of the problem is to equip the vehicles with an appropriate negotiation mechanism by which each vehicle pursues the optimization of its own utility function. We present several design procedures and accompanying caveats for vehicle utility design. We present two new negotiation mechanisms, namely, "generalized regret monitoring with fading memory and inertia" and "selective spatial adaptive play," and provide accompanying proofs of their convergence. Finally, we present simulations that illustrate how vehicle negotiations can consistently lead to near-optimal assignments provided that the utilities of the vehicles are designed appropriately.

420 citations


Journal ArticleDOI
TL;DR: In this paper, a nonlinear feedback control law is developed to steer an AUV along a desired path, which overcomes initial condition constraints that are present in a number of path-following control strategies described in the literature.

314 citations


Journal ArticleDOI
TL;DR: In this paper, a leader-follower control is proposed to make the following robot keep a relative position with respect to the leader and avoid collisions in the presence of obstacles, and graph theory is introduced to formalise specified formation patterns in a simple but effective way.
Abstract: A framework for controlling groups of autonomous mobile vehicles to achieve predetermined formations based on a leader-following approach is presented. A three-level hybrid control architecture is proposed to implement both centralised and decentralised cooperative control. Under such architecture, the global-level formation control problem of n vehicles is decomposed into decentralised control problems between n-1 pairs of follower and their designated leader. In the leader-follower control level, two basic controllers are proposed to make the following robot keep a relative position with respect to the leader and avoid collisions in the presence of obstacles. Then, graph theory is introduced to formalise specified formation patterns in a simple but effective way, and two types of switching between these formations are also proposed. Numerical simulations and physical robot experiments show the effectiveness of our approach

242 citations


Journal ArticleDOI
TL;DR: A large space of possible robots capable of autonomous self-reproduction is demonstrated, composed of actuated modules equipped with electromagnets to selectively control the morphology of the robotic assembly.
Abstract: Long-term physical survivability of most robotic systems today is achieved through durable hardware. In contrast, most biological systems are not made of robust materials; long-term sustainability and evolutionary adaptation in nature are provided through processes of self-repair and, ultimately, self-reproduction. Here we demonstrate a large space of possible robots capable of autonomous self-reproduction. These robots are composed of actuated modules equipped with electromagnets to selectively control the morphology of the robotic assembly. We show a variety of 2-D and 3-D machines from 3 to 2n modules, and two physical implementations that each achieves two generations of reproduction. We show both automatically generated and manually designed morphologies

195 citations


Journal ArticleDOI
TL;DR: In this paper, a two-dimensional path following control system for autonomous marine surface vessels is presented, which is obtained through a way-point guidance scheme based on line-of-sight projection algorithm and the speed controller is achieved through state feedback linearization.

192 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present a system for autonomous mobile robot navigation with only an omnidirectional camera as sensor, which is able to build automatically and robustly accurate topologically organized environment maps of a complex, natural environment.
Abstract: In this work we present a novel system for autonomous mobile robot navigation. With only an omnidirectional camera as sensor, this system is able to build automatically and robustly accurate topologically organised environment maps of a complex, natural environment. It can localise itself using such a map at each moment, including both at startup (kidnapped robot) or using knowledge of former localisations. The topological nature of the map is similar to the intuitive maps humans use, is memory-efficient and enables fast and simple path planning towards a specified goal. We developed a real-time visual servoing technique to steer the system along the computed path. A key technology making this all possible is the novel fast wide baseline feature matching, which yields an efficient description of the scene, with a focus on man-made environments.

189 citations


Journal ArticleDOI
TL;DR: In this article, the authors present the general principles of autonomy and the proposed concepts, methods and technologies to realize autonomous processes in assembly systems, different approaches for design and autonomous operation of assembly, and future trends towards fully autonomous components of an assembly system as well as autonomous parts and products.

139 citations


Journal ArticleDOI
TL;DR: In this article, a nested-saturation-based nonlinear controller for the stabilization of a rotary-wing aircraft is proposed, which allows the incorporation of actuator magnitude saturation and has satisfactory dynamic performance.
Abstract: Control system design of aerospace vehicles with actuator saturation is an important practical design problem that many previous approaches to nonlinear autopilot design did not consider. In particular, small unmanned aerial vehicle rotorcraft actuators often have physical limitations such as a restricted onboard power supply. Disregard of actuator saturation can affect the final performance, but the reduction in performance can be mitigated if actuator saturation is included in the controller design. In this paper, we propose a nested-saturation-based nonlinear controller for the stabilization of a rotary-wing aircraft. This control strategy allows the incorporation of actuator magnitude saturation and has satisfactory dynamic performance. The nested-saturation technique enables the controller to ensure the global asymptotic stability of a quadrotor helicopter while improving the performance of the closed-loop system. By using Lyapunov analysis, the convergence property is established for the complete nonlinear model of the quadrotor rotorcraft. Simulation results show the performance of the proposed control strategy. Using embedded sensors and onboard control, we performed a real-time autonomous flight. Indeed, experimental results have shown that the proposed control strategy is able to autonomously perform the tasks of taking off, hovering, and landing.

Journal ArticleDOI
TL;DR: A variation of a learning algorithm called "adaptive play" that has desirable convergence properties in potential games with reduced computation is introduced that helps improve the efficiency of an equilibrium with respect to the total mutual information.
Abstract: We consider a multi-link MIMO interference system in which each link wishes to maximize its own mutual information by choosing its own signal vector, which leads to a multi-player game. We show the existence of a Nash equilibrium and obtain sufficient conditions for the uniqueness of equilibrium. We consider two decentralized link adjustment algorithms called best-response process (a.k.a. iterative water- filling) and gradient-play (an autonomous and non-cooperative version of the well-known gradient ascent algorithm). Under our uniqueness conditions, we establish the convergence of these algorithms to the unique equilibrium provided that the links use some inertia. To improve the efficiency of an equilibrium with respect to the total mutual information by imposing limits on the number of independent data streams, we present a stream control approach using linear transformation of the link covariance matrices. We then show how to decentralize our stream control approach by allowing the links to negotiate the limits on the number of independent data streams that they are willing to impose upon themselves. To achieve this, we introduce a variation of a learning algorithm called "adaptive play" that has desirable convergence properties in potential games with reduced computation.

Journal ArticleDOI
TL;DR: Initial insight of autonomous navigation for mobile robots is provided, a description of the sensors used to detect obstacles and a descriptions of the genetic algorithms used for path planning are provided.
Abstract: Engineers and scientists use instrumentation and measurement equipment to obtain information for specific environments, such as temperature and pressure. This task can be performed manually using portable gauges. However, there are many instances in which this approach may be impractical; when gathering data from remote sites or from potentially hostile environments. In these applications, autonomous navigation methods allow a mobile robot to explore an environment independent of human presence or intervention. The mobile robot contains the measurement device and records the data then either transmits it or brings it back to the operator. Sensors are required for the robot to detect obstacles in the navigation environment, and machine intelligence is required for the robot to plan a path around these obstacles. The use of genetic algorithms is an example of machine intelligence applications to modern robot navigation. Genetic algorithms are heuristic optimization methods, which have mechanisms analogous to biological evolution. This article provides initial insight of autonomous navigation for mobile robots, a description of the sensors used to detect obstacles and a description of the genetic algorithms used for path planning.

Journal ArticleDOI
TL;DR: A complex vision system, able to provide the two basic sensorial capabilities needed by autonomous vehicle navigation in extreme environments: obstacle detection and path detection, is presented.
Abstract: Autonomous driving in off-road environments requires an exceptionally capable sensor system, particularly given that the unstructured environment does not provide many of the cues available in on-road environments. This paper presents a complex vision system, which is able to provide the two basic sensorial capabilities needed by autonomous vehicle navigation in extreme environments: obstacle detection and path detection. A variable-width-baseline (up to 1.5 m) single-frame stereo system is used for pitch estimation and obstacle detection, whereas a decision-network approach is used to detect the drivable path by a monocular vision system. The system has been field tested on the TerraMax vehicle, which is one of the only five vehicles to complete the 2005 Defense Advanced Research Projects Agency (DARPA) Grand Challenge course.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a method for the economic evaluation of an autonomous independent network of distributed energy resources called Microgrid Microgrid is a new framework of power delivery system that is formed by small, modular generation systems connected to each other to create a small autonomous grid.

Journal ArticleDOI
TL;DR: Using the framework, autonomous functions were constructed for maintaining postural stability constraint while satisfying the desired trajectory of operation points, including the feet, while interacting with the environment.
Abstract: This paper introduces a framework for whole-body motion generation integrating operator's control and robot's autonomous functions during online control of humanoid robots. Humanoid robots are biped machines that usually possess multiple degrees of freedom (DOF). The complexity of their structure and the difficulty in maintaining postural stability make the whole-body control of humanoid robots fundamentally different from fixed-base manipulators. Getting hints from human conscious and subconscious motion generations, the authors propose a method of generating whole-body motions that integrates the operator's command input and the robot's autonomous functions. Instead of giving commands to all the joints all the time, the operator selects only the necessary points of the humanoid robot's body for manipulation. This paper first explains the concept of the system and the framework for integrating operator's command and autonomous functions in whole-body motion generation. Using the framework, autonomous functions were constructed for maintaining postural stability constraint while satisfying the desired trajectory of operation points, including the feet, while interacting with the environment. Finally, this paper reports on the implementation of the proposed method to teleoperate two 30-DOF humanoid robots, HRP-1S and HRP-2, by using only two 3-DOF joysticks. Experiments teleoperating the two robots are reported to verify the effectiveness of the proposed method.

Proceedings ArticleDOI
27 Mar 2007
TL;DR: The way to design the aspects of a hybrid power system that will target remote users with the optimisation of the components size and the improvement of the capital cost is shown.
Abstract: This paper shows the way to design the aspects of a hybrid power system that will target remote users. It emphasizes on the hydrogen hybrid power system to obtain a reliable autonomous system with the optimisation of the components size and the improvement of the capital cost. This system is chosen to provide electricity for a small and remote located community. A methodology is developed for calculating the correct size of the system and for optimizing the management. The main power for the hybrid system comes from the photovoltaic panels, while the fuel cell and secondary batteries are used as backup units. The optimization software used for this paper is the hybrid optimization model for electric renewables (HOMER). HOMER is a design model that determines the optimal architecture and control strategy of the hybrid system. It can also determine the sensitivity of the outputs to changes in the inputs. It performs an hourly time series analysis on each of hundreds or thousands of different system configurations

Journal ArticleDOI
TL;DR: This paper proposes a method for limiting the social costs of cascading failures by eliminating violations before dependent outages occur, and places one autonomous software agents at each bus of a power network, each of which is tasked with solving the global control problem with limited data and communication.
Abstract: Cascading failures in electricity networks often result in large blackouts with severe social consequences. A cascading failure typically begins with one or more equipment outages that cause operating constraint violations. When violations persist in a network, they can trigger additional outages which in turn may cause further violations. This paper proposes a method for limiting the social costs of cascading failures by eliminating violations before dependent outages occur. Specifically, our approach places one autonomous software agents at each bus of a power network, each of which is tasked with solving the global control problem with limited data and communication. Each agent builds a simplified model of the network based on locally available data and solves its local problem using model predictive control and cooperation. Through extensive simulations with IEEE test networks, we find that the autonomous agent design meets its goals with limited communication. Experiments also demonstrate that allowing agents to cooperate can vastly improve system performance.

Journal ArticleDOI
TL;DR: This contribution shows that the recently emerged paradigm of Reservoir Computing (RC) is very well suited to solve all of the above mentioned problems, namely learning by example, robot localization, map and path generation.
Abstract: Autonomous mobile robots form an important research topic in the field of robotics due to their near-term applicability in the real world as domestic service robots. These robots must be designed in an efficient way using training sequences. They need to be aware of their position in the environment and also need to create models of it for deliberative planning. These tasks have to be performed using a limited number of sensors with low accuracy, as well as with a restricted amount of computational power. In this contribution we show that the recently emerged paradigm of Reservoir Computing (RC) is very well suited to solve all of the above mentioned problems, namely learning by example, robot localization, map and path generation. Reservoir Computing is a technique which enables a system to learn any time-invariant filter of the input by training a simple linear regressor that acts on the states of a high-dimensional but random dynamic system excited by the inputs. In addition, RC is a simple technique featuring ease of training, and low computational and memory demands.

Patent
30 Apr 2007
TL;DR: In this article, the authors present a technique to determine whether a component failure in a distributed computing system is genuine by analyzing monitoring data from other application nodes to determine if the component failure is genuine.
Abstract: In general, the techniques of this invention are directed to determining whether a component failure in a distributed computing system is genuine. In particular, embodiments of this invention analyze monitoring data from other application nodes in a distributed computing system to determine whether the component failure is genuine. If the component failure is not genuine, the embodiments may adjust a fault tolerance parameter that caused the component failure to be perceived.

Journal ArticleDOI
TL;DR: A case study of a robotic excavator is introduced to explore a series of issues around system development, navigation in unstructured environments, autonomous decision making and changing the behaviour of autonomous machines to suit the prevailing demands of users.
Abstract: The development of autonomous mobile machines to perform useful tasks in real work environments is currently being impeded by concerns over effectiveness, commercial viability and, above all, safety. This paper introduces a case study of a robotic excavator to explore a series of issues around system development, navigation in unstructured environments, autonomous decision making and changing the behaviour of autonomous machines to suit the prevailing demands of users. The adoption of the Real-Time Control Systems (RCS) architecture (Albus, 1991) is proposed as a universal framework for the development of intelligent systems. In addition it is explained how the use of Partially Observable Markov Decision Processes (POMDP) (Kaelbling et al., 1998) can form the basis of decision making in the face of uncertainty and how the technique can be effectively incorporated into the RCS architecture. Particular emphasis is placed on ensuring that the resulting behaviour is both task effective and adequately safe, and it is recognised that these two objectives may be in opposition and that the desired relative balance between them may change. The concept of an autonomous system having "values" is introduced through the use of utility theory. Limited simulation results of experiments are reported which demonstrate that these techniques can create intelligent systems capable of modifying their behaviour to exhibit either `safety conscious' or `task achieving' personalities.

Journal ArticleDOI
TL;DR: The range of autonomous behavior that is relevant and useful in space exploration is explored and the range of possible behaviors is illustrated by presenting four case studies in space-exploration systems, each differing from the others in the degree of autonomy exemplified.
Abstract: This article provides an overview of the nature and role of autonomy for space exploration, with a bias in focus towards describing the relevance of AI technologies. It explores the range of autonomous behavior that is relevant and useful in space exploration and illustrates the range of possible behaviors by presenting four case studies in space-exploration systems, each differing from the others in the degree of autonomy exemplified. Three core requirements are defined for autonomous space systems, and the architectures for integrating capabilities into an autonomous system are described. The article concludes with a discussion of the challenges that are faced currently in developing and deploying autonomy technologies for space.

Journal ArticleDOI
TL;DR: The impact of "agent relatedness," a measure of common descent between two agents based on their underlying manufacturers, on the choices in scenarios 1) and 2) is shown and the approach provides a framework to study design choices that escape systematic analysis in other frameworks.
Abstract: An analogy between an agent (e.g., an autonomous vehicle) and a biological forager is extended to a social environment by viewing a communication network as implementing interagent sociality. We first describe engineering design within an evolutionary game-theoretic framework. We then explain why sociality may emerge in some environments and for some agent objectives. Next, we derive the evolutionarily stable design strategy for an agent manufacturer: 1) choosing whether the agent it produces should cooperate with other agents in a search problem and 2) choosing the group size of a multiagent system tasked with a cooperative search problem. We show the impact of "agent relatedness," a measure of common descent between two agents based on their underlying manufacturers, on the choices in scenarios 1) and 2). Our predictions are evaluated in an autonomous vehicle simulation testbed. The results illustrate a new methodology for manufacturers to make robust, optimal choices for multiagent system design for a given set of objectives and domain of operation. Note to Practitioners-The design of autonomous multirobot systems with various applications, such as in parts production or search and destroy operations in a military environment, is of growing importance. Here, we integrate economic and technical issues into an unified engineering design framework for the manufacturers of robots. Our approach leads to manufacturer design decisions that are robust relative to the market for a manufacturer's products. Robot component aspects, such as sensors and communications as well as mission performance aspects, can be captured and coupled into the design process. We use the design of intervehicle cooperation and robot group size to illustrate this approach. The practical significance lies in the fact that we take a broad perspective on engineering design, one closer to the real world, due to the considerations of marketplace economics. Moreover, the approach provides a framework to study design choices that escape systematic analysis in other frameworks (e.g., group size)

15 May 2007
TL;DR: The Autonomy Levels for Unmanned Systems (ALFUS) workshop series as discussed by the authors was organized to address the autonomous nature of unmanned, robotic systems, or unmanned systems (UMS).
Abstract: The Autonomy Levels for Unmanned Systems (ALFUS) workshop series was convened to address the autonomous nature of unmanned, robotic systems, or unmanned systems (UMS). Practitioners have different perceptions or different expectations for these systems. The requirements on human interactions, the types of tasks, the teaming of the UMSs and the humans, and the operating environment are just a few of the issues that need to be clarified. Also needed is a set of definitions and a model with which the autonomous capability of the UMS can be described. This paper reports the current results and status of the ALFUS framework, which practitioners can apply to analyze the autonomy requirements and to evaluate the performance of their robotic programs.

Journal ArticleDOI
TL;DR: Hardware results of the autonomous control system where the trajectory tracker applies two velocity controllers accounting for fixed-wing UAV-like input constraints, are compared to simulation results of dynamic controllers that are based on non-smooth backstepping to demonstrate the effectiveness of the overall system.
Abstract: An autonomous control system designed for a non-holonomic wheeled mobile robot that is programmed to emulate a fixed-wing unmanned air vehicle (UAV) flying at constant altitude is experimentally validated. The overall system is capable of waypoint navigation, threat avoidance, real-time trajectory generation and trajectory tracking. Both the wheeled mobile robot experimental platform and the hierarchical autonomous control software architecture are introduced. Programmed to emulate a fixed-wing UAV flying at constant altitude, a non-holonomic mobile robot is assigned to follow a desired time-parameterised trajectory generated by a real-time trajectory generator to transition through a sequence of targets in the presence of static and popup threats. Hardware results of the autonomous control system where the trajectory tracker applies two velocity controllers accounting for fixed-wing UAV-like input constraints, are compared to simulation results of dynamic controllers that are based on non-smooth backstepping to demonstrate the effectiveness of the overall system.

Proceedings ArticleDOI
12 Nov 2007
TL;DR: The latest version of an integrated hardware and software framework developed for autonomous operation of rescue robots is described, with an emphasis on extendibility and the specific requirements of a typical unstructured rescue scenario.
Abstract: This paper describes the latest version of an integrated hardware and software framework developed for autonomous operation of rescue robots. The successful operation of an autonomous rugbot -short for "rugged robot -was especially demonstrated during several runs at the RoboCup world championship 2006 in Bremen. The design of the autonomous system is described in detail with an emphasis on extendibility and the specific requirements of a typical unstructured rescue scenario.

Book ChapterDOI
TL;DR: This chapter presents an overview on the modeling and model-based control of autonomous helicopters, introducing some of the platforms and control architectures that has been developed in the last 15 years and considering the modeling of the helicopter and the identification techniques.
Abstract: This chapter presents an overview on the modeling and model-based control of autonomous helicopters. Firstly it introduces some of the platforms and control architectures that has been developed in the last 15 years. Later, the Chapter considers the modeling of the helicopter and the identification techniques. Then, it overviews different linear and non-linear model-based control approaches. This section also includes experiments on the control of the helicopter vertical motion that illustrate the presented techniques and point out the interest of nonlinear analysis methods to study the dynamic behavior of the helicopter. Finally, the Chapter presents open research lines coming from two challenging applications: the autonomous landing in oscillating platforms and the lifting and transporting of a single load with several helicopters.

Journal ArticleDOI
TL;DR: This article describes some of the current challenges in autonomous sensor and robot networks, with special focus on issues concerning the fusion of information from several sensors to improve the accuracy in the cooperative localization of objects or to help the wide-sense networked sensors to self-localize.
Abstract: This article describes some of the current challenges in autonomous sensor and robot networks, with special focus on issues concerning the fusion of information from several sensors to improve the accuracy in the cooperative localization of objects or to help the wide-sense networked sensors to self-localize. Although many other approaches exist, the focus is on Bayesian approaches to sensor fusion and state estimation.

Book ChapterDOI
TL;DR: This paper introduces an efficient sphere representation method for environment representation, collision detection and distance queries for an autonomous bridge maintenance system and shows that this method is effective in environment representation and collision detection.
Abstract: When applying autonomous industrial robotic systems in an unknown/partially known or cluttered environment, mapping and representation of the environment as well as collision detection becomes crucial. Existing techniques in these areas are generally complex and computationally expensive to implement. In this paper an efficient sphere representation method is introduced for environment representation, collision detection and distance queries. In particular, this method is designed for the application in an autonomous bridge maintenance system. Simulation results show that this method is effective in environment representation and collision detection. Furthermore, the proposed method is also computationally efficient for real-time implementation.

Journal ArticleDOI
TL;DR: In this article, a method for studying isochronous oscillations in some systems of ODE reducible to the equation is described and applied to obtain the necessary and sufficient conditions for isochronicity of a cubic two-dimensional autonomous system depending on six parameters.
Abstract: A method for studying isochronous oscillations in some systems of ODE reducible to the equation is described. It is applied to obtain the necessary and sufficient conditions for isochronicity of a cubic two-dimensional autonomous system depending on six parameters. For all isochronous systems in this family the Urabe function is explicitly constructed.