Concept Of Differential Flatness Research Articles

Quadrotor UAV Dynamic Visual Servoing Based on Differential Flatness Theory

In this paper, we propose 2D dynamic visual servoing (Dynamic IBVS), where a quadrotor UAV tries to track a moving target using a single facing-down perspective camera. As an application, we propose the tracking of a car-type vehicle. In this case, data related to the altitude and the lateral angles have no importance for the visual system. Indeed, to perform the tracking, we only need to know the longitudinal displacements (along the x and y axes) and the orientation along the z-axis. However, those data are necessary for the quadrotor’s guidance problem. Thanks to the concept of differential flatness, we demonstrate that if we manage to extract the displacements according to the three axes and the orientation according to the yaw angle (the vertical axis) of the quadrotor, we can control all the other variables of the system. For this, we consider a camera equipped with a vertical stabilizer that keeps it in a vertical position during its movement (a gimbaled camera). Other specialized sensors measure information regarding altitude and lateral angles. In the case of classic 2D visual servoing, the elaboration of the kinematic torsor of the quadrotor in no way guarantees the physical realization of instructions, given that the quadrotor is an under-actuated system. Indeed, the setpoint has a dimension equal to six, while the quadrotor is controlled only by four inputs. In addition, the dynamics of a quadrotor are generally very fast, which requires a high-frequency control law. Furthermore, the complexity of the image processing stage can cause delays in motion control, which can lead to target loss. A new dynamic 2D visual servoing method (Dynamic IBVS) is proposed. This method makes it possible to generate in real time the necessary movements for the quadrotor in order to carry out the tracking of the target (vehicle) using a single point of this target as visual information. This point can represent the center of gravity of the target or any other part of it. A control by flatness has been proposed, which guarantees the controllability of the system and ensures the asymptotic convergence of the generated trajectory in the image plane. Numerical simulations are presented to show the effectiveness of the proposed control strategy.

Robust Flatness-Based Tracking Control for a “Full-Bridge Buck Inverter–DC Motor” System

By developing a robust control strategy based on the differential flatness concept, this paper presents a solution for the bidirectional trajectory tracking task in the “full-bridge Buck inverter–DC motor” system. The robustness of the proposed control is achieved by taking advantage of the differential flatness property related to the mathematical model of the system. The performance of the control, designed via the flatness concept, is verified in two ways. The first is by implementing experimentally the flatness control and proposing different shapes for the desired angular velocity profiles. For this aim, a built prototype of the “full-bridge Buck inverter–DC motor” system, along with Matlab–Simulink and a DS1104 board from dSPACE are used. The second is via simulation results, i.e., by programming the system in closed-loop with the proposed control algorithm through Matlab–Simulink. The experimental and the simulation results are similar, thus demonstrating the effectiveness of the designed robust control even when abrupt electrical variations are considered in the system.

Aggressive maneuvering of a quadcopter via differential flatness-based fuzzy controllers: From tuning to experiments

In recent years, the rapid development in contemporary technology has brought nano quadcopters with high agility. This paper presents a new differential flatness-based Single Input Fuzzy Logic Controller (SFLC) structure for aggressive maneuvering control alongside its real-world application on Crazyflie 2.1 nano quadcopter. We propose both Type-1 and Interval Type-2 SFLCs as the primary controllers in the flight control system, which are built on the concept of differential flatness. We investigate how the design parameters of SFLCs shape the characteristics of the fuzzy mapping through a geometric approach by analyzing the region and level of aggressiveness/smoothness. Based on the analysis, we present simple tuning guidelines and then design fuzzy logic-based flight control systems, which were implemented as onboard real-time controllers. Finally, we evaluate the performance of SFLCs in comparison with their crisp differential flatness-based nonlinear counterparts for four trajectories with distinct dynamics and shapes in the real world. The presented comparative experimental results clearly show the performance improvements when the proposed T1 and IT2 SFLCs are deployed for real-time aggressive maneuvering.

Nonlinear Position Control Using Differential Flatness Concept with Load Torque Observer for Electro Hydraulic Actuators with Sinusoidal Load Torque

We propose a nonlinear position control using a differential flatness concept with a load torque observer to compensate for the sinusoidal load torque in electro-hydraulic actuators (EHAs) EHAs. In an EHA with a rotational joint, the load torque is a sinusoidal disturbance, whose magnitude can be estimated via a load torque observer. In the proposed load torque observer, the load torque can be estimated without requiring its maximum frequency to be known. The position controller tracks position and comprises an inner-loop load pressure controller and an outer-loop position controller. The former tracks the desired pressure via near input–output linearization. The desired pressure is developed using the differential flatness of the mechanical system in the EHA. The feedback plus feedforward outer-loop position controller is designed to track the desired position and to compensate for the load torque. The stability of the closed-loop system is mathematically proven using the input-to-state stability property.

Periodic Control of Unmanned Aerial Vehicles Based on Differential Flatness

Unmanned aerial vehicles (UAVs) are making increasingly long flights today with significantly longer mission times. This requires the UAVs to have long endurance as well as have long range capabilities. Motivated by locomotory patterns in birds and marine animals which demonstrate a powered-coasting-powered periodic locomotory behavior, an optimal control problem is formulated to study UAV trajectory planning. The concept of differential flatness is used to reformulate the optimal control problem as a nonlinear programing problem where the flat outputs are parameterized using Fourier series. The Π test is also used to verify the existence of a periodic solution which outperforms the steady-state motion. An example of an Aerosonde UAV is used to illustrate the improvement in endurance and range costs of the periodic control solutions relative to the equilibrium flight.

Multi-vehicle Cooperative Control for Load Transportation

Abstract This work proposes a cooperative control solution to the problem of transporting a suspended load using multiple quadrotor vehicles. The problem is addressed for two quadrotors, with a methodology that can be generalized for any number of quadrotors. A dynamic model of the system is developed considering a point-mass load, rigid massless cables, and neglecting aerodynamic effects of the cables. The concept of differential flatness is explored and a new set of flat outputs, which can be used to fully characterize the state of the system, is proposed. A nonlinear Lyapunov-based controller in cascaded form is derived, by defining adequate mappings between the cable tension vectors and the quadrotor thrust vectors and exploring the analogy with the problem of controlling a single quadrotor. Simulation results are presented for tracking of load trajectories. Comparisons are made with a free-flying quadrotor control scheme to highlight the enhanced performance of the proposed scheme.

Flatness-Based Design of Experiments for Model Selection

High product standards and cost-efficient robust process designs are key issues in the pharmaceutical industry and require thorough system understanding of the underlying physical phenomena. Model-based approaches have been proven favorable to gain valuable system insights. However, qualitative inferences can be drawn only if the model is adequately derived and validated by - at best - optimally designed experiments. In this paper, we present a new model inversion strategy to solve model-based design of experiments (MBDoE) problems for model selection; i.e., we maximize the difference of competing model candidates by using the differential flatness concept. Differentially flat systems can represent their states and controls analytically by the so-called flat output and its derivatives. By maximizing the difference of the flat outputs of all considered model candidates, we efficiently obtain analytical model-revealing control actions for the MBDoE problem. When these optimized control actions are implemented, the resulting experimental data are expected to invalidate incorrect models more reliably. We demonstrate the effectiveness of the proposed MBDoE concept with potential model candidates that describe different realizations of a mass action reaction system.

Decentralized Control of dynamic supply chain systems with parametric uncertainties

Dynamic supply chain control is tackled in this paper using the concept of differential flatness where we focus on operational activities and algebraic estimation techniques for the uncertain parameters. The main objective is to maintain the inventory level at a desired one according to the customer demand while synchronizing the flows within a set of physical boundaries and constraints. In addition, we consider uncertainties on delay and adaptation times and we use the new setting of the algebraic estimation techniques to adapt and update the controller when any change occurs. These two methods lead to a simple design of the control algorithm without integration of any differential equation and without a need to any model to estimate, which yield to an efficient control action. Convincing numerical simulations using a case study of a crude oil blending and distribution system demonstrate the effectiveness of the proposed approach. Copyright 2017 IFAC.

Nonlinear State Estimation and Control for Freeway On‐Ramp Metering

AbstractThe work presented in this paper deals with freeway ramp metering using a differential flatness concept. Such an approach is deployed in the cases when the traffic data provided by loop detectors (or any measurements device), are partially unknown or missed, and/or the downstream measurement station, located at the vicinity of the controlled ramp, is faulty. The proposed solution rests on the estimation of the main variables using the “revised” method of numerical differentiation, i.e., estimation of the derivative of noisy time signals. Such an algebraic approach, which does not need any integration of differential equations, turns out to be quite robust with respect to perturbations inputs and measurements noises. Convincing numerical simulations are provided using a modified second‐order continuum macroscopic model. Furthermore, a comparative study using the Extended Kalman Filter (EKF), which is the most used in the area of traffic state estimators, is proposed in order to demonstrate the effectiveness of the strategy.

Optimal Trajectory Generation for Car-type Mobile Robot using Spline Interpolation∗∗This work is carried out under the research project grant sanctioned under the WOS-A scheme by Department of Science and Technology (DST), Govt. of India.

Abstract This paper focuses on the spline based trajectory generation and motion planning in a 1:10 scale down prototype of a car. The car type vehicle can be modelled as a nonholonomic system with constraints on sideways movement. The spline based trajectory generation gives us continuous, smooth and optimized path trajectories. The motion planning algorithm is based on the nonholonomic model of a car-type robot which is differentially flat in nature. Our contribution lies in the development of spline based trajectories which overcome the parametric singularities arising from the differential flatness based approach. The spline trajectories also take care of the singularities which are otherwise observed in the optimization process if the cubic polynomial based interpolation is implemented. The hardware implementation for a prototype of a car type model is also presented. Using the concept of differential flatness, we derive control inputs which steer the vehicle along this trajectory.

A Hybrid Differential Flatness and Sliding Modes Controller for Dynamical Structural Testing on Lower Limp Prostheses

The standard ISO 10328 specifies the procedure for structural tests on lower limb prosthetic devices, as a method for quality control. Having the standard as reference, a machine is designed and implemented to follow the test routine, which if succeeded, shows the quality concept for the devices tested. Furthermore, it is shown the controller design process, which manage the application of a sinusoid force with specified amplitude, frequency and offset, in order to realize the dynamic test, by means of a double effect cylinder and proportional valves for pressure and flow.The dynamic model was calculated through a parametric identification technique in a pre-stabilized closed loop. Later, a hybrid control strategy was set using the differential flatness concept and a sliding modes controller, so that both identified model and real system could be properly controlled.

Nonlinear single-loop control of the parallel converters for a fuel cell power source used in DC grid applications

Abstract This paper presents an innovative control law for a multiphase interleaved converter in the distribution of power supply in fuel cell (FC) generators. Traditionally, to control the DC output power, voltage, or current in a converter, a linear multiple-loop feedback control technique is used. The nonlinear multiple-loop feedback control approach offers several techniques that help to improve the system response. In this paper, an alternative nonlinear single-loop feedback control scheme is proposed. This scheme is based on the differential flatness concept, which provides a solution to achieve the preferred response using a less sophisticated algorithm. To validate the proposed technique, a prototype of a FC power converter (a 600-W two-phase interleaved boost DC–DC converter) was constructed in the laboratory, and the control algorithm was implemented to control the prototype using a dSPACE 1104 controller card. The control scheme exhibited excellent experimental results for use with a 1.2-kW Nexa Ballard polymer electrolyte membrane fuel cell (PEMFC) regarding the steady state and dynamic responses as well as the control robustness.

Parameter identification for ordinary and delay differential equations by using flat inputs

The concept of differential flatness has been widely used for nonlinear controller design. In this contribution, it is shown that flatness may also be a very useful property for parameter identification. An identification method based on flat inputs is introduced. The treatment of noisy measurements and the extension of the method to delay differential equations are discussed. The method is illustrated by two case studies: the well-known FitzHugh-Nagumo equations and a virus replication model with delays.

Optimal Guidance Based on Receding Horizon Control and Online Trajectory Optimization

In this paper, a general guidance method based on a receding horizon control technique and online trajectory optimization is proposed for various interception problems. In this method, the interception problem is defined as an optimal control problem by full dynamics of the interceptor and the current position and velocity of the target in the three-dimensional space. By solving the interception optimal control problem for various objective and constraint functions, integrated guidance and control commands are generated online during interception. To find the closed-loop and state-dependent solution, the interception optimal control problem is defined in a receding horizon control framework. To quickly solve the interception problem, the direct trajectory optimization method, which is a combination of a differential flatness concept, b-spline curves, and nonlinear programming, is used. To demonstrate the implementation of the proposed method, a numerical example of interception of a maneuvering target by a missile is presented. The proposed optimal guidance method can be applied for integrated guidance and nonlinear control of the interceptors in various interception problems in aerospace engineering and robotics.

Flat controller design for hardware-cost reduction in polar-axis photovoltaic systems

Abstract Initial costs of solar-photovoltaics have remained a significant factor contending against the wide-spread utilization of solar energy. The paper demonstrates that the concept of differential flatness could be used to design controllers for a polar-axis solar tracker that would enable reduction in hardware costs. Key results show that, the implementation of trajectories of motion in the controller can reduce the drive power requirement and associated photovoltaic (PV) cost by about 31%, and enable the use of smaller drive motors with a possible 25% reduction in drive motor costs; and that the implementation of sensorless control in the polar-axis tracking system is able to further lead to sensor cost elimination.

A FLAT MODEL PREDICTIVE CONTROLLER FOR TRAJECTORY TRACKING IN IMAGE BASED VISUAL SERVOING

Image-Based Visual Servoing (IBVS) is a control strategy using visual information to control the motion of robotic systems. Classical IBVS can not take into account either the mechanical constraints (joint and actuator limitations) or the visibility constraints, very important in this scheme. Model Predictive Control (MPC) is well adapted to deal with these drawbacks. However, applied to fast systems (e.g. mobile robots), the computational time is a great challenge for real time applications. One way to reduce this time is to use the concept of differential flatness. In this paper, a new IBVS strategy based on a flat MPC approach is proposed. The capabilities of this approach in terms of trajectory tracking and obstacle avoidance are pointed out. Applied to mobile robot trajectory tracking, a simulation experiment shows the efficiency and the robustness of this new control scheme. The computational time required by the proposed solution is compared with the nonlinear solution and easily enables a real-time application.

Some Steps Towards Autonomous Cars

Developing new Intelligent Transportation Systems which take into consideration the social-economical, environmental, and safety factors of the modern human society, is one of the grand challenges of the 21th century. This paper presents the motion autonomy capabilities which have been developed for future cars in the scope of the French “Praxitele” and “Automated Road” projects. These new capabilities relies onto effective solutions that we have developed for contributing to solve three main problems: autonomous maneuvering for a car-like vehicle using the concept of Sensor-Based Maneuver, planning and controling the motions of a double-steered vehicle (the CyCab) using the concept of Differential Flatness, and obstacle avoidance in a dynamic environment using the concept of Non-Linear Velocity Obstacle. Experimental results obtained with real vehicles are also presented and discussed.

A Generalization of Flatness to Nonlinear Systems of Partial Differential Equations. Application to the Command of a Flexible Rod

We introduce a concept of differential flatness for systems described by nonlinear partial differential equations. It generalizes the now classical notion of differential flatness for finite differential systems and its recent extensions to linear partial differential equations. We apply it to the motion planning of a very flexible rod, outside of the linear approximation range.