Differential Flatness Research Articles

Differential Flatness of Lifting-Wing Quadcopters Subject to Drag and Lift for Accurate Tracking

Differential Flatness of Lifting-Wing Quadcopters Subject to Drag and Lift for Accurate Tracking

Optimal Trajectory Planning for Wheeled Robots (OTPWR): A Globally and Dynamically Optimal Trajectory Planning Method for Wheeled Mobile Robots

In recent years, with the widespread application of indoor inspection robots, efficient motion planning has become crucial. Addressing the issue of discontinuous and suboptimal robot trajectories resulting from the independent nature of global and local planning, we propose a novel optimal path-planning method for wheeled mobile robots. This method leverages differential flatness to reduce dimensionality and decouple the problem, achieving globally optimal, collision-free paths in a two-dimensional flat output space through diagonal search and polynomial trajectory optimization. Comparative experiments in a simulated environment demonstrate that the proposed improved path search algorithm reduces search time by 46.6% and decreases the number of visited nodes by 43.1% compared to the original algorithm. This method not only ensures the optimal path and efficient planning but also ensures that the robot’s motion trajectory satisfies the dynamic constraints, verifying the effectiveness of the proposed optimal path planning algorithm for wheeled mobile robots.

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.

Improved dynamic performance of triple active bridge DC-DC converter using differential flatness control for more electric aircraft applications

The triple active bridge (TAB) converter has been proposed to improve power density and availability in more electric aircraft applications. However, the conventional proportional-integral (PI) controller used for TAB voltage control often exhibits slow response and significant overshoot, which can impact dynamic performance. Moreover, achieving decoupling control among ports introduces computational complexity in controller design. To address these issues, this paper introduces an effective diagonal matrix decoupling strategy based on differential flatness control with single-phase shift modulation applied to a TAB converter. The proposed differential flatness control offers superior transient performance, control flexibility, and precision, ensuring compliance with DC voltage regulations while achieving optimal decoupling among ports. The comparative analysis assesses various criteria, including steady-state and dynamic performance, control complexity, and regulation and tracking properties, against PI control, unit matrix decoupling control, and the proposed differential flatness control. Hardware-in-loop (HIL) experimentation verifies the performance, confirming faster dynamic characteristics and robust port power decoupling. The results show a significant improvement in efficiency, reaching a maximum of 95.5 %, indicating reduced switching losses and enhanced overall system performance. The study concludes with a discussion on the implications of these findings and potential future research directions, such as integrating advanced optimization algorithms further to enhance the control strategy's robustness and adaptability.

Theory and experiment on distributed output formation tracking of unmanned aerial and ground vehicle swarm systems over jointly connected digraphs

This paper studies the distributed output formation tracking control problem of the unmanned aerial vehicle (UAV) and unmanned ground vehicle (UGV) swarm systems, where the UAV swarm cooperatively tracks the output trajectory of the UGV in a formation under directed jointly connected communication networks. A three-layer formation tracking protocol is proposed. Firstly, a novel distributed observer using neighbor interactions is designed for each UAV to estimate the states of the UGV with parameterized inputs over periodic jointly connected digraphs. Next, based on the observation result and output regulation theory, a virtual reference system that tracks the trajectory of the UGV in the desired formation is constructed to generate reference states for each UAV. Then based on the differential flatness of UAVs, a geometric controller is utilized for UAVs to track the reference states and form the formation. An algorithm to determine the gain matrices of the protocol is also presented while the convergence of the system is analyzed. Finally, an experiment platform with three quadrotor UAVs and one UGV is built. The effectiveness of the proposed protocol is validated both by the simulation and experiment.

Differential Flatness Based Unmanned Surface Vehicle Control: Planning and Conditional Disturbance-Compensation

To achieve precise control of the symmetrical unmanned surface vehicle (USV) under strong external disturbances, we propose a disturbance estimation and conditional disturbance compensation control (CDCC) scheme. First, the differential flatness method is applied to convert the underactuated model into a fully actuated one, simplifying the controller design. Then, a nonlinear disturbance observer (NDOB) is designed to estimate the lumped disturbance. Subsequently, a continuous disturbance characterization index (CDCI) is proposed, which not only indicates whether the disturbance is beneficial to the system stability but also makes the controller switch smoothly and suppresses the chattering phenomenon greatly. Indicated by the CDCI, the proposed CDCC method can not only utilize the beneficial disturbance but also compensate for the detrimental disturbance, which improves the USV’s control performance under strong external disturbances. Moreover, a trajectory-planning method is designed to generate an obstacle avoidance reference trajectory for the controller. Finally, simulations verify the feasibility of applying the proposed control method to USV.

Time Optimal Altitude-Hold Flight Mode Transition Strategy for a Class of Ducted Fan Tail Sitter UAV

For special tail sitter configurations such as the ducted fan tail sitter unmanned aerial vehicle (UAV), the widely used trajectory planning methodology based on differential flatness might not be applicable due to complex aerodynamic coupling effects. As a result, the flight mode transition remains a challenging task. In this paper, we address the time optimal altitude-hold flight mode transition issue for a class of ducted fan tail sitter UAV. The foundation of the framework is the dynamic transition corridor in which the limitation of jerk is particularly considered, aiming to thoroughly reflect the dynamic feature of aggressive maneuvers. Based on this, we propose a time optimal strategy to generate feasible altitude-hold transition trajectories. Simultaneous, by fully utilizing the manifestation of time optimal altitude-hold flight behavior revealed by the transition corridor, we try to tackle the time optimal altitude-hold transition by means of a novel model-free control scheme. Comparative simulations show that both of the transition strategies achieve satisfactory performance on time optimal altitude-hold transition in the absence of disturbance, while the model-free control scheme exhibits better robustness under external disturbance.

Differential Flatness of Single-Input Commensurate Delay Systems With Applications to Trajectory Planning, Tracking, and Transformation to Fully Actuated Systems

Differential Flatness of Single-Input Commensurate Delay Systems With Applications to Trajectory Planning, Tracking, and Transformation to Fully Actuated Systems

Efficient feedforward sloshing suppression strategy for liquid transport

This paper introduces an advanced feedforward control method to suppress sloshing in high-speed liquid container transfers in robotic systems. Three key concepts are combined: (1.) A virtual pendulum tray is envisaged to suspend the liquid container. By moving only the virtual pivot, the resulting container motion robustly reduces sloshing. (2.) A geometric condition is found for the virtual tray/container assembly to fully suppress one specific sloshing mode, allowing rapid transfers in a way that the dominant sloshing dynamics is fully suppressed. (3.) An analytic feedforward trajectory control based on differential flatness is formulated to design fast and efficient rest-to-rest transport manoeuvres. Any sloshing of modelled higher modes which is excited during the manoeuvre is accurately stopped at the end of such a manoeuvre. The resulting control law is computationally simple and achieves excellent sloshing-suppression performance. The concept is validated via simulations, finite-element analyses, and experimental tests, proving excellent theoretical and real-world performance and showing high robustness even with variations in the liquid filling level.

An Advanced IBVS-Flatness Approach for Real-Time Quadrotor Navigation: A Full Control Scheme in the Image Plane

This article presents an innovative method for planning and tracking the trajectory in the image plane for the visual control of a quadrotor. The community of researchers working on 2D control widely recognizes this challenge as complex, because a trajectory defined in image space can lead to unpredictable movements of the robot in Cartesian space. While researchers have addressed this problem for mobile robots, quadrotors continue to face significant challenges. To tackle this issue, the adopted approach involves considering the separation of altitude control from the other variables, thus reducing the workspace. Furthermore, the movements of the quadrotor (pitch, roll, and yaw) are interdependent. Consequently, the connection between the inputs and outputs cannot be reversed. The task complexity becomes significant. To address this issue, we propose the following scenario: When the quadrotor is equipped with a downward-facing camera, flying at high altitude is sensible to spot a target. However, to minimize disturbances and conserve energy, the quadrotor needs to descend in altitude. This can result in the target being lost. The solution to this problem is a new methodology based on the principle of differential flatness, allowing the separation of altitude control from the other variables. The system first detects the target at high altitude, then plots a trajectory in the image coordinate system between the acquired image and the desired image. It is crucial to emphasize that this step is performed offline, ensuring that the image processing time does not affect the control frequency. Through the proposed trajectory planning, complying with the constraints of differential flatness, the quadrotor can follow the imposed dynamics. To ensure the tracking of the target while following the generated trajectory, the proposed control law takes the form of an Image Based Visual Servoing (IBVS) scheme. We validated this method using the RVCTOOLS environment in MATLAB. The DJI Phantom 1 quadrotor served as a testbed to evaluate, under real conditions, the effectiveness of the proposed control law. We specifically designed an electronic card to transfer calculated commands to the DJI Phantom 1 control joystick via Bluetooth. This card integrates a PIC18F2520 microcontroller, a DAC8564 digital-to-analogue converter, and an RN42 Bluetooth module. The experimental results demonstrate the effectiveness of this method, ensuring the precise tracking of the target as well as the accurate tracking of the path generated in the image coordinate system.

Identification of linear flat outputs using neural networks—Examples of two-degree-of-freedom underactuated mechanical systems

This paper proposes a neural networks-based approach of finding flat output of linearized underactuated mechanical systems (UMS). Given that differential flatness and controllability are equivalent for linear systems, the problem is equivalent to finding the Brunovsky canonical form of linearized UMSs. We use a two degree-of-freedom (2DOF) system to illustrate the theoretical development. The proposed method identifies the local flat output of nonlinear mechanical systems from the measurements only, without a detailed mathematical model. The identification allows us to combine the well-known active disturbance rejection control (ADRC) and differential flatness control. A time-domain direct identification (TDDI) algorithm and its variant based on the algebraic method are proposed for flat output identification (FOID). The neural networks for implementing the TDDI algorithm called FOID-NN are created to evaluate flat output candidates and to use the flat output to reconstruct the system states in terms of a linear mapping. The neural networks are trained with the loss functions defined by reconstruction errors. Two special layers, namely tracking differentiator (TD) and algebraic layer, are inserted in the FOID-NN to handle noisy signal differentiations. Simulations of a cart–pole system and a stable 2DOF nonlinear UMS are carried out to show the range of applications of the FOID-NN. The experimental data of an underactuated rotary crane are used to validate the identified flat output.

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Multi-Agent Path Planning for Level Set Estimation Using B-Splines and Differential Flatness

Multi-Agent Path Planning for Level Set Estimation Using B-Splines and Differential Flatness

Robust Tracking Control of Wheeled Mobile Robot Based on Differential Flatness and Sliding Active Disturbance Rejection Control: Simulations and Experiments.

This paper proposes a robust tracking control method for wheeled mobile robot (WMR) against uncertainties, including wind disturbances and slipping. Through the application of the differential flatness methodology, the under-actuated WMR model is transformed into a linear canonical form, simplifying the design of a stabilizing feedback controller. To handle uncertainties from wheel slip and wind disturbances, the proposed feedback controller uses sliding mode control (SMC). However, increased uncertainties lead to chattering in the SMC approach due to higher control inputs. To mitigate this, a boundary layer around the switching surface is introduced, implementing a continuous control law to reduce chattering. Although increasing the boundary layer thickness reduces chattering, it may compromise the robustness achieved by SMC. To address this challenge, an active disturbance rejection control (ADRC) is integrated with boundary layer sliding mode control. ADRC estimates lumped uncertainties via an extended state observer and eliminates them within the feedback loop. This combined feedback control method aims to achieve practical control and robust tracking performance. Stability properties of the closed-loop system are established using the Lyapunov theory. Finally, simulations and experimental results are conducted to compare and evaluate the efficiency of the proposed robust tracking controller against other existing control methods.

Event-triggered trajectory tracking control for USV with prescribed performance and time delay based on differential flatness

This paper presents a trajectory tracking control scheme for underactuated surface vessels (USVs) with input delay. Firstly, the underactuated surface vessel system is transformed into a fully actuated system using differential flatness theory. To estimate the unknown nonlinear terms introduced in the transformation process, a fuzzy neural network (FNN) is employed. Secondly, to conserve control resources and communication bandwidth, the controller of the system under prescribed performance is designed using the backstepping method. This method updates the controller according to an event-triggered condition that is designed using a Lyapunov function. Finally, theoretical proof and simulation experiments are conducted to demonstrate the convergence and effectiveness of the proposed method.

Dynamic compensation of the threading speed drop in rolling processes

This paper presents an approach for the dynamic speed drop compensation during threading in rolling processes. The feedforward control design exploits the differential flatness of the mechanical model and accelerates both the rolls and the drive train in a manner such that the acceleration torque is equal to the rolling torque during threading, while simultaneously maintaining the roll at the desired target speed. Ideally, this prevents the speed drop and enhances the quality and stability of the rolling process. The flatness-based feedforward trajectories are optimized in an online fashion to determine the optimal initial roll speed and duration of the acceleration process. An extensive experimental validation on a hot strip finishing mill shows superior performance in terms of various key performance indicators in comparison with a standard overspeed approach.

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.

Energy management of electric vehicle using a new strategy based on slap swarm optimization and differential flatness control

Optimal energy management of electric vehicles using slap swarm optimization and differential flatness control has been proposed. A battery–supercapacitor power system is adopted. Each source is connected in parallel to the DC-bus using DC–DC bidirectional converters and supplies a synchronous reluctance motor (SynRM) based drive. The proposed EMS fundamental forces lie in using a combination of complementary proprieties of two approaches, a Slap Swarm optimization Algorithm and Differential Flatness (DF). With a fast optimization mechanism, the Slap Swarm optimization algorithm allows adapting in real-time conditions the DF gains to optimize the system performances. On its side, DF uses predefined trajectories respecting the physical proprieties of the system, which is a powerful tool to guarantee the dynamic constraints of the sources when ensuring desired robust control proprieties. To check the feasibility and performance of the suggested EMS, comprehensive processor-in-the-loop co-simulations of the electric vehicle were carried out using the C2000 launchxl-f28379d DSP board. The main goal of the proposed EMS is to guarantee the DC-bus stabilization, reducing the DC-bus voltage ripples (Δv = 5 V) and the voltage overshoots 15 V (3.2%), respect the source dynamics, and satisfy the SynRM motor power demand. Furthermore, the algorithm minimizes induced harmonics by the drive (10.49%), reducing the battery current ripple by 17.15A, thereby enhancing the battery lifecycle.

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.