A Test for Differential Flatness by Reduction to Single Input Systems

For nonlinear control systems (p inputs), we present a test for flatness. The method consists of making an initial guess for p-1 of the flat outputs, which may involve parameters still to be determined. A choice of functions of time for the p-1 outputs reduce the system to one with a single input. For single input systems the problem of flatness has been solved and thus leads to the identification of the last flat output, or to obstructions to flatness under the hypotheses. We demonstrate the method for a coupled rigid body in ℝ2 and for a single rigid body in ℝ3

Configuration Flatness of Lagrangian Systems Underactuated by One Control

Lagrangian control systems that are differentially flat with flat outputs that only depend on configuration variables are said to be configuration flat. We provide a complete characterisation of configuration flatness for systems with n degrees of freedom and n - 1 controls whose range of control forces only depends on configuration and whose Lagrangian has the form of kinetic energy minus potential. The method presented allows us to determine if such a system is configuration flat and, if so provides a constructive method for finding all possible configuration flat outputs. Our characterisation relates configuration flatness to Riemannian geometry. We illustrate the method by two examples

Flat control of industrial robotic manipulators

Published ArticleA new approach to tracking control of industrial robot manipulators is presented in this paper. The highly coupled nonlinear dynamics of a six degrees of freedom (6-DOF) serial robot is decoupled by expressing its variables as a function of a flat output and a finite number of its derivatives. Hence the derivation of the flat output for the 6-DOF robot is presented. With the flat output, trajectories for each of the generalized coordinates are easily designed and open loop control is made possible. Using MATLAB/Simulink Sfunctions combined with the differential flatness property of the robot, trajectory tracking is carried out in closed loop by using a linear flat controller. The merit of this approach reduces the computational complexity of the robot dynamics by allowing online computation of a high order system at a lower computational cost. Using the same processor, the run time for tracking arbitrary trajectories is reduced significantly to about 10 s as compared to 30 min in the original study (Hoifodt, 2011). The design is taken further by including a Jacobian transformation for tracking of trajectories in cartesian space. Simulations using the ABB IRB140 industrial robot with full dynamics are used to validate the study

Experimental Validation on Flatness based Control of Flexible Robot Arm

Published Conference ProceedingsThis paper discusses the practical implementation of a flatness based control for a flexible joint robot arm. Using differential flatness theory, reference trajectories are generated for a flexible joint robot and then a tracking controller is implemented. The vibrations experienced by the robot arm are sufficiently damped and nonminimum phase behaviour is eliminated. The control shows fast transcient response as desired for flexible robots. Experimental results proves the effectiveness of the flatness based control approach

Coordination control of robot manipulators using flat outputs

Published ArticleThis paper focuses on the synchronizing control of multiple interconnected flexible robotic manipulators using differential flatness theory. The flatness theory has the advantage of simplifying trajectory tracking tasks of complex mechanical systems. Using this theory, we propose a new synchronization scheme whereby a formation of flatness based systems can be stabilized using their respective flat outputs. Using the flat outputs, we eliminate the need for cross coupling laws and communication protocols associated with such formations. The problem of robot coordination is reduced to synchronizing the flat outputs between the respective robot manipulators. Furthermore, the selection of the flat output used for the synchronizing control is not restricted as any system variable can be used. The problem of unmeasured states used in the control is also solved by reconstructing the missing states using flatness based interpolation. The proposed control law is less computationally intensive when compared to earlier reported work as integration of the differential equations is not required. Simulations using a formation of single link flexible joint robots are used to validate the proposed synchronizing control

Coordination control of robot manipulators using flat outputs

Published ArticleThis paper focuses on the synchronizing control of multiple interconnected flexible robotic manipulators using differential flatness theory. The flatness theory has the advantage of simplifying trajectory tracking tasks of complex mechanical systems. Using this theory, we propose a new synchronization scheme whereby a formation of flatness based systems can be stabilized using their respective flat outputs. Using the flat outputs, we eliminate the need for cross coupling laws and communication protocols associated with such formations. The problem of robot coordination is reduced to synchronizing the flat outputs between the respective robot manipulators. Furthermore, the selection of the flat output used for the synchronizing control is not restricted as any system variable can be used. The problem of unmeasured states used in the control is also solved by reconstructing the missing states using flatness based interpolation. The proposed control law is less computationally intensive when compared to earlier reported work as integration of the differential equations is not required. Simulations using a formation of single link flexible joint robots are used to validate the proposed synchronizing control

Observability singularities and observer design: dual immersion approach

It is well-known that, for nonlinear systems, the observability is often only a local property and depends on the input. Moreover, it is often required that the observer be of the same dimension as the original system. A direct consequence of this requirement is that it enlarges the set of observability singularities. If, on one hand, it is impossible to observe the state variables that are structurally unobservable, it is, however, possible to overcome the observability singularities introduced by the constraints on the observer design. In this paper, we propose a novel dual immersion method which allows to reduce the set of observability singularities. In addition, a step by step design of a high order sliding mode observer based on the proposed dual immersion approach is presented. Finally, a thorough analysis and discussion on the simulation results with respect to a non-autonomous system is given

Differential flatness and absolute equivalence

In this paper we give a formulation of differential flatness-a concept originally introduced by Fliess, Levine, Martin, and Rouchon (1992)-in terms of absolute equivalence between exterior differential systems. Systems which are differentially flat have several useful properties which can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was given in the context of differential algebra, and required that all mappings be meromorphic functions. Our formulation of flatness does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, we show that in the case of single input control systems (i.e., codimension 2 Pfaffian systems), a system is differentially flat if and only if it is feedback linearizable via static state feedback. However, in higher codimensions feedback linearizability and flatness are not equivalent: one must be careful with the role of time as well the use of prolongations which may not be realizable as dynamic feedbacks in a control setting. Applications of differential flatness to nonlinear control systems and open questions are also discussed

Bàn về khả năng ứng dụng lý thuyết hệ phẳng vào phân tích và điều khiển hệ phi tuyến

Theory of flat systems has provided many opportunities, but not few challenges for solving of analysis and control problems. This article focuses on appreciations of these opportunities and from this ahead on some open theory problems to be carried out. With these appreciations, the paper provides also an objective view of applicability of flat systems theory in analysis and control of nonlinear systems.Lý thuyết hệ phẳng đã mang lại nhiều cơ hội song cũngkhông ít thách thức cho việc thực hiện các bài toán điều khiển. Bài báo này tập trung vào việc bàn luận về các cơ hội đó cũng như những vấn đề mở cần phải giải quyết của lý thuyết hệ phẳng, để từ đó có được một cái nhìn khách quan hơn về khả năng ứng dụng hiệu quả lý thuyết hệ phẳng vào phân tích và điều khiển hệ phi tuyến