Differential Flatness Property Research Articles

A Control System Design and Implementation for Autonomous Quadrotors with Real-Time Re-Planning Capability

Real-time (re-)planning is crucial for autonomous quadrotors to navigate in uncertain environments where obstacles may be detected and trajectory plans must be adjusted on-the-fly to avoid collision. In this paper, we present a control system design for autonomous quadrotors that has real-time re-planning capability, including the hardware pipeline for the hardware–software integration to realize the proposed real-time re-planning algorithm. The framework is based on a modified version of the PX4 Autopilot and a Raspberry Pi 5 companion computer. The planning algorithm utilizes minimum-snap trajectory generation, taking advantage of the differential flatness property of quadrotors, to realize computationally light, real-time re-planning using an onboard computer. We first verify the control system and the planning algorithm through simulation experiments, followed by implementing and demonstrating the system on hardware using a quadcopter.

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear Optimal Control for a PMLSG-VSC Wave Energy Conversion Unit

This article aims to treat the nonlinear control problem for the complex dynamics of a wave energy unit (WEC) that consists of a Permanent Magnet Linear Synchronous Generator (PMLSG) and a Voltage Source Converter (VSC). The article has developed a globally stable nonlinear optimal control method for this wave power generation unit. The new method avoids complicated state-space model transformations and minimizes the energy dispersion by the control loop. A novel nonlinear optimal control method is proposed for the dynamic model of a wave energy conversion system, which includes a Permanent Magnet Linear Synchronous Generator (PMLSG) serially connected with an AC/DC three-phase voltage source converter (VSC). The dynamic model of this renewable energy system is formulated and differential flatness properties are proven about it. To apply the proposed nonlinear optimal control, the state-space model of the PMLSG-VSC wave energy conversion unit undergoes an approximate linearization process at each sampling instance. The linearization procedure relies on a first-order Taylor-series expansion and involves the computation of the system’s Jacobian matrices. It takes place at each sampling interval around a temporary operating point, which is defined by the present value of the wave energy conversion unit’s state vector and by the last sampled value of the control inputs vector. An H-infinity feedback controller is designed for the linearized model of the wave energy conversion unit. To compute the feedback gains of this controller, an algebraic Riccati equation is repetitively solved at each time step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis.

Cooperative Safe Trajectory Planning for Quadrotor Swarms.

In this paper, we propose a novel distributed algorithm based on model predictive control and alternating direction multiplier method (DMPC-ADMM) for cooperative trajectory planning of quadrotor swarms. First, a receding horizon trajectory planning optimization problem is constructed, in which the differential flatness property is used to deal with the nonlinear dynamics of quadrotors while we design a relaxed form of the discrete-time control barrier function (DCBF) constraint to balance feasibility and safety. Then, we decompose the original trajectory planning problem by ADMM and solve it in a fully distributed manner with peer-to-peer communication, which induces the quadrotors within the communication range to reach a consensus on their future trajectories to enhance safety. In addition, an event-triggered mechanism is designed to reduce the communication overhead. The simulation results verify that the trajectories generated by our method are real-time, safe, and smooth. A comprehensive comparison with the centralized strategy and several other distributed strategies in terms of real-time, safety, and feasibility verifies that our method is more suitable for the trajectory planning of large-scale quadrotor swarms.

Nonlinear optimal control for the multi-variable tumor-growth dynamics

The multivariable tumor-growth dynamic model has been widely used to describe the inhibition of tumor-cells proliferation under the simultaneous infusion of multiple chemotherapeutic drugs. In this article, a nonlinear optimal (H-infinity) control method is developed for the multi-variable tumor-growth model. First, differential flatness properties are proven for the associated state-space description. Next, the state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instant around a time-varying operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the system a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach.

Differential flatness properties and multivariable adaptive fuzzy control of hormonal system dynamics

Computer-based control for hormonal (ovarian) system dynamics and for administering in a computationally optimized manner the associated pharmaceutical treatment has been little investigated up to now. Although there has been prior work on the modeling of the ovarian system dynamics and its stability features, there have been hardly any results for applying to the ovarian system exogenous computer-based control. This is due to the complexity, nonlinearity and the high dimensionality of this system. Actually, the ovarian system exhibits nonlinear dynamics which is modeled by a set of thirteen coupled nonlinear differential equations, while receiving only two control inputs. Besides, the fact that the parameters of the model cannot be precisely identified and the fact that any mathematical model for such a biosystem is likely to be subjected to uncertainties and external perturbations, make the application of model-based control schemes for this system be questionable. To overcome this, the paper proposes model-free adaptive fuzzy control based on differential flatness theory for the complex dynamics of the ovarian system. It is proven that the dynamic model of the ovarian system, having as state variables the LH and the FSH hormones and their derivatives, is a differentially flat one. This means that all its state variables and its control inputs can be written as differential functions of the flat output and its derivatives. By exploiting differential flatness properties the system's dynamic model is written in the multivariable linear canonical (Brunovsky) form, for which the design of a state feedback controller becomes possible. After this transformation, the new control inputs of the system contain unknown nonlinear parts, which are identified with the use of neurofuzzy approximators. Lyapunov stability analysis shows that H-infinity tracking performance is achieved for the feedback control loop and this assures improved robustness to the aforementioned model uncertainty as well as to external perturbations.

Optimal control of differentially flat systems is surprisingly easy

As we move to increasingly complex cyber–physical systems (CPS), new approaches are needed to plan efficient state trajectories in real-time. In this paper, we propose an approach to significantly reduce the complexity of solving optimal control problems for a class of CPS with nonlinear dynamics. We exploit the property of differential flatness to simplify the Euler–Lagrange equations that arise during optimization, and this simplification eliminates the numerical instabilities that plague optimal control in general. We also present an explicit differential equation that describes the evolution of the optimal state trajectory, and we extend our results to consider both the unconstrained and constrained cases. Furthermore, we demonstrate the performance of our approach by generating the optimal trajectory for a planar manipulator with two revolute joints. We show in simulation that our approach is able to generate the constrained optimal trajectory in 4.5ms while respecting workspace constraints and switching between a ‘left’ and ‘right’ bend in the elbow joint.

Trajectory generation and tracking control for aggressive tail-sitter flights

We address the theoretical and practical problems related to the trajectory generation and tracking control of tail-sitter UAVs. Theoretically, we focus on the differential flatness property with full exploitation of actual UAV aerodynamic models, which lays a foundation for generating dynamically feasible trajectory and achieving high-performance tracking control. We have found that a tail-sitter is differentially flat with accurate (not simplified) aerodynamic models within the entire flight envelope, by specifying coordinate flight condition and choosing the vehicle position as the flat output. This fundamental property allows us to fully exploit the high-fidelity aerodynamic models in the trajectory planning and tracking control to achieve accurate tail-sitter flights. Particularly, an optimization-based trajectory planner for tail-sitters is proposed to design high-quality, smooth trajectories with consideration of kinodynamic constraints, singularity-free constraints, and actuator saturation. The planned trajectory of flat output is transformed into state trajectory in real time with optional consideration of wind in environments. To track the state trajectory, a global, singularity-free, and minimally parameterized on-manifold MPC is developed, which fully leverages the accurate aerodynamic model to achieve high-accuracy trajectory tracking within the whole flight envelope. The proposed algorithms are implemented on our quadrotor tail-sitter prototype, “Hong Hu,” and their effectiveness are demonstrated through extensive real-world experiments in both indoor and outdoor field tests, including agile SE(3) flight through consecutive narrow windows requiring specific attitude and with speed up to 10 m/s, typical tail-sitter maneuvers (transition, level flight, and loiter) with speed up to 20 m/s, and extremely aggressive aerobatic maneuvers (Wingover, Loop, Vertical Eight, and Cuban Eight) with acceleration up to 2.5 g. The video demonstration is available at https://youtu.be/2x_bLbVuyrk.

Flatness-Based Quadcopter Trajectory Planning and Tracking With Continuous-Time Safety Guarantees

This work proposes a convex optimization-based framework for the trajectory planning and tracking of quadcopters that ensures continuous-time safety guarantees. Using the convexity property of B-spline curves and the differential flatness property of quadcopters, a second-order cone program is formulated to generate an optimal nominal trajectory that respects state and input constraints, including position, linear velocity, angle, angular velocity, thrust, waypoints, and obstacle avoidance constraints, rigorously in the continuous-time sense. To ensure safe trajectory tracking, a convex quadratic program is proposed based on control barrier functions, which guarantees that the actual trajectory of the quadcopter remains within a prescribed safe tube of the nominal trajectory in continuous time. Furthermore, conditions that ensure the safe tracking controller respects thrust, roll, and pitch constraints are also presented. Both the planning and control approaches are suitable for online implementation, and the effectiveness of the proposed framework is demonstrated through simulations and experiments with a Crazyflie2.1 nano quadcopter.

Synchronization and Consensus of a Group of Direct Current Servo Motors Using the Differential Flatness Control Approach

This article deals with a differential flatness control leader-follower for a group of DC servo motors. Graph theory uses to design the form of connection and communication between servo motors. Each servo motor includes an integral action in its local control. In contrast, the input control law of the leader incorporates the speed desired reference trajectory, which will track each servo motor connected to the leader. The speed desired reference trajectory build employing a high-order Bezier polynomial-the experimental setup using the dSPACE equipment DS1104 model. The experimental results show the effectiveness and robustness of the synchronization and consensus of the control based on the differential flatness property for the group of DC servomotors.

A numerically-stable trajectory generation and optimization algorithm for autonomous quadrotor UAVs

This paper introduces a novel trajectory generation and optimization algorithm (TGO) that enables agile and aggressive flight of quadrotor UAVs while considering various constraints associated with robot dynamics, actuator inputs, and flight environment. The TGO algorithm employs time-parametrized polynomial trajectories based on a predetermined sequence of waypoints to produce dynamically feasible and collision-free trajectories. This approach extends previous work that utilized differential flatness property and polynomial based trajectories by eliminating the need for iterative searching and computationally intensive sampling. One of the significant advantages of the TGO algorithm is its numerical stability for large number of waypoints and high-order polynomials. To address the ill-conditioned problem of Quadratic Programming (QP) based methods, the TGO algorithm reformulates the trajectory generation and optimization problem into an unconstrained quadratic programming (UCP) using the numerically stable null-space factorization method. The TGO algorithm produces minimum derivative trajectories and minimum waypoints arrival times, generating a wide range of aggressive trajectories that can leverage the full maneuvering capabilities of quadrotor robots. The proposed algorithm’s numerical stability and computational advantages are demonstrated through various scenarios and comparisons. An animated simulation of the TGO algorithm is available at: https://youtu.be/MvvhBG14iIg.

Nonlinear optimal control for permanent magnet synchronous spherical motors

Nonlinear optimal control for permanent magnet synchronous spherical motors

Nonlinear optimal control of a multi-rotor wind power unit with PMSGs and AC/DC converters)

A nonlinear optimal (H-infinity) control method is developed for a wind power unit that comprises twin turbines, permanent magnet synchronous generators (PMSGs) and AC/DC converters. By proving differential flatness properties for this system the associated setpoints definition problem is solved. The dynamic model of the wind power unit being initially expressed in a nonlinear and multivariable state-space form, undergoes approximate linearisation around a temporary operating point that is recomputed at each time-step of the control method. The linearisation relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearised state-space model of the wind power unit, a stabilising optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem of the wind power unit under model uncertainty and external perturbations. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control of the wind power unit, without the need to measure its entire state vector, the H-infinity Kalman Filter is used as a robust state estimator.

Nonlinear optimal control for a gas compressor driven by an induction motor

The article proposes a nonlinear optimal control approach for the dynamic model of a gas centrifugal compressor being driven by an induction motor (IM). The dynamic model of the integrated compressor-induction motor system, being initially expressed in a nonlinear and multivariable state-space form, undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the compressor-IM a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem of the compressor-IM under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control of the compressor-IM, without the need to measure its entire state vector, the H-infinity Kalman Filter is used as a robust state estimator. The differential flatness properties of the compressor-IM system are proven. The article’s method provides one of the few algorithmically simple and computationally efficient solutions for the nonlinear optimal control problem of the compressor-IM system.

Analysis of DC/DC Boost Converter–Full-Bridge Buck Inverter System for AC Generation

This paper presents an analysis and simulation of the mathematical model associated to the DC/DC Boost converter–full-bridge Buck inverter system to regulate the voltage output of the DC/DC Boost converter allowing bipolar voltages higher than the input voltage via the full-bridge Buck inverter. To validate the model, the differential flatness property is applied via the flat outputs of the system (energy for the DC/DC Boost converter and the voltage of the full-bridge Buck inverter) considering the complete dynamics, in conjunction with fixed and time-variant trajectory planning. In the simulation results, it is observed that the error signals of the states versus the reference trajectories are acceptable. Regarding the validation of the model, this is performed with open-loop simulations at the circuit level using the SimPowerSystems Toolbox of Matlab-Simulink. The simulation results validate the good performance of the system under study. In this way, the main contribution of this work is that for the first time in the literature, the analysis of a complete dynamics for a conversion system from DC to AC without the use of a transformer and taking advantage of differential flatness is reported; thus, the system analyzed could be represented as an alternative in applications of renewable energies that require conversion from DC to AC.

Transient analysis of electrical drives: Applications to the permanent magnet synchronous generator

This paper establishes that differential flatness, a property shared by most of the model of electrical drives, allows to perform transient analysis on the models of electrical machines, and especially electric generators which are so important in renewable energy systems (wind turbine, tidal turbine...). Transient analysis of electrical machines is indeed a difficult subject not completely solved today. The possibility to parameterize the trajectories of a system from the differential flatness property gives the possibility to perform simple transient analysis in various operation modes. The example of the Permanent Magnet Synchronous Generator (PMSG) illustrates the paper.

Hierarchical Flatness-Based Control for Velocity Trajectory Tracking of the “DC/DC Boost Converter–DC Motor” System Powered by Renewable Energy

In this investigation, a tracking control is designed for the angular velocity of the DC/DC Boost converter–DC motor system. To this end, the dynamics of the power supply, generated through a renewable energy power source, is considered in both the mathematical model and the designed control. This latter is proposed by using a two-level hierarchical approach, where the dynamics of the DC/DC Boost converter and the one associated with the DC motor not only are treated as two independent subsystems, but also they exploit their differential flatness property. For the DC/DC Boost converter, an alternative mathematical model of first order is obtained for designing the low-level voltage control. Whereas, the well known second order mathematical model of the DC motor is used for developing the high-level angular velocity control. The robustness and performance of the hierarchical tracking control are verified via realistic numerical simulations and experimental results by using Matlab-Simulink, a prototype of the system, the DS1104 board, and the renewable energy emulator TDK-Lambda G100-17. The results demonstrate and validate the effectiveness of the proposed approach.

Continuous-Time Trajectory Estimation for Differentially Flat Systems

Continuous-time estimation using splines on Lie groups has been gaining traction in the literature due to the ability to incorporate high-frequency sensor data without introducing new optimization parameters. However, evaluating time derivatives and Jacobians of Lie group splines is computationally expensive, limiting their use mainly to offline applications. Motivated by the trajectory planning literature, we develop a new estimation technique that leverages the differential flatness property of many dynamic systems to define the spline in the system's flat output space, which is often Euclidean. Doing so has the added benefit of providing a simple and effective way to include system inputs in the estimation process. We show an example of flatness-based estimation for the unicycle dynamic model. We then show that this new method can achieve similar performance as Lie group spline estimation with significantly less computation time, and validate its use in hardware using a differential-drive robot.

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.

A Time-Optimal Trajectory Planning Strategy for an Aircraft With a Suspended Payload via Optimization and Learning Approaches

In this article, a real-time interactive time-optimal trajectory planning (TOTP) strategy is proposed for aircraft slung-load systems, in which the practical performance is guaranteed by taking advantage of convex optimization and reinforcement learning (RL) techniques. To the best of our knowledge, it is the first TOTP solution for an aircraft suspension system subject to path, velocity, acceleration, and cable tension constraints, where a practice-oriented RL policy is designed to interact with the system online. Specifically, by exploring the differential flatness property of the system, the states are projected into path coordinate space, where all the physical constraints are formulated in the TOTP with convex forms. Then, the TOTP is transformed into a large sparse optimization problem after discretization, which can be efficiently solved using convex solvers. Subsequently, a deep RL network is designed to optimize the TOTP results, where efficient learning is achieved in the solution space, thereby guaranteeing practical reliability and strong robustness via online interaction in applications. In addition, by incorporating convex optimization and learning techniques, the gap between simulation and experiment is successfully bridged, where the physical feasibility and efficient learning are ensured in the framework. Finally, comparative simulations and experimental results are included to show the effectiveness of the proposed strategy.