Differential Flatness Property Research Articles

Optimal Control for Kinematic Bicycle Model With Continuous-Time Safety Guarantees: A Sequential Second-Order Cone Programming Approach

The optimal control problem for the kinematic bicycle model is considered where the trajectories are required to satisfy the safety constraints in the continuous-time sense. Based on the differential flatness property of the model, necessary and sufficient conditions in the flat space are provided to guarantee safety in the state space. The optimal control problem is relaxed to the problem of solving three second-order cone programs (SOCPs) sequentially, which find the safe path, the trajectory duration, and the speed profile, respectively. Solutions of the three SOCPs together provide a sub-optimal solution to the original optimal control problem. Simulation examples and comparisons with state-of-the-art optimal control solvers are presented to demonstrate the effectiveness of the proposed approach.

Design and Trajectory Tracking Control of a New Bi-Copter UAV

A pair of vectored thrusts brings versatility and power efficiency to bi-copter unmanned aerial vehicles (UAVs). One common difficulty confronted by state-of-the-art bi-copter UAVs is the non-minimum phase nature, which puts fundamental limitations in the system and challenges the control. This letter introduces a new minimum phase bi-copter UAV that outperforms the conventional bi-copter UAV on attitude control with faster angular response and halved settling time. The comparison is made based on the optimal performance found by a particle swarm optimization (PSO)-based attitude controller on SO(3). The differential flatness property of generic bi-copter UAVs considering servo dynamics is analyzed. Flatness-based trajectory generation and tracking control of the new bi-copter UAV are described in detail together with flight experiments demonstrating the performance of the bi-copter flying through a drone race gate following a four-dimensional trajectory.

A general real-time optimization framework for polynomial-based trajectory planning of autonomous flying robots

This paper presents a general real-time, numerically stable optimization framework for time polynomial-based trajectory generation of autonomous aerial robots. The proposed general optimization framework (GOF) allows various optimization criteria for trajectory generation cost-function, such as minimizing the trajectory total length, time, and position derivatives. Minimizing position derivatives includes velocity, acceleration, jerk, and snap, or any combination of them. This study considers the quadrotor as the test platform. By exploiting tools from the calculus of variations, differential flatness property, and polynomial-based trajectories, the developed algorithm finds feasible trajectories without extensive computational sampling and iterative searching in the high-dimensional state space of quadrotor dynamics. The GOF includes a segment-wise gradient descent-like algorithm to iteratively decrease the allowed time of each segment individually so as to avoid getting stuck at a local minimum. The comparison analysis with existing methods validated the numerical stability and computational speed advantages of the proposed approach. It also shows that the algorithm is suitable for the real-time generation of high-performance long-range trajectories consisting of a large number of waypoints and high-order piecewise polynomials. An animated simulation of this work is available at https://youtu.be/E1AC1vyPqOE

Feedback control of social distancing for COVID-19 via elementary formulae

Social distancing has been enacted in order to mitigate the spread of COVID-19. Like many authors, we adopt the classic epidemic SIR model, where the infection rate is the control variable. Its differential flatness property yields elementary closed-form formulae for open-loop social distancing scenarios, where, for instance, the increase of the number of uninfected people may be taken into account. Those formulae might therefore be useful to decision makers. A feedback loop stemming from model-free control leads to a remarkable robustness with respect to severe uncertainties and mismatches. Although an identification procedure is presented, a good knowledge of the recovery rate is not necessary for our control strategy.

A Hierarchical Control Approach for Power Loss Minimization and Optimal Power Flow within a Meshed DC Microgrid

This work considers the DC part of a hybrid AC/DC microgrid with a meshed topology. We address cost minimization, battery scheduling and the power loss minimization within the power distribution network through constrained optimization. The novelty comes from applying differential flatness properties to the microgrid components and formulating the cost and constraints in terms of the associated B-splines parametrization of the flat outputs (the voltages and currents of the system). This allows us to obtain optimal power profiles to minimize the power dissipation and the cost of the electricity purchase from the external grid. These profiles are tracked by a model predictive controller at the higher level, while at a a lower level a controller deals with the operation of the switches within the DC/DC converters. Extensive simulations under nominal and fault-affected scenarios using realistic data validate the proposed approach.

Nonlinear optimal control for multi‐DOF robotic manipulators with flexible joints

AbstractA main problem in the control of flexible‐joint robots and in the precise positioning of their end‐effector, is synchronization between the turn angles of the robots' joints and the turn angles of the motors that provide actuation to these joints. In this article, a nonlinear optimal (H‐infinity) control method is proposed for multi‐DOF robotic manipulators with flexible joints. The dynamic model of these robotic manipulators is shown to be underactuated and to satisfy differential flatness properties. The state‐space description of these robots undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure makes use of first‐order Taylor‐series expansion and relies on the computation of Jacobian matrices. For the approximately linearized model of these multi‐DOF robotic systems an H‐infinity feedback controller is designed. This controller stands for the solution of the robots' optimal control problem under model uncertainty and external disturbances. The computation of the controller's feedback gain requires the solution of an algebraic Riccati equation, which is performed again at each time‐step of the control algorithm. The stability properties of the control scheme are proven trough Lyapunov analysis. First, it is confirmed that the controller satisfies the H‐infinity tracking performance criterion which ascertains its robustness. Moreover, it is proven that the control loop is globally asymptotically stable. Finally, to implement sensorless control for such robotic systems the H‐infinity Kalman Filter is used as a robust state estimator. The proposed control method for multi‐DOF robotic manipulators with flexible joints retains the advantages of linear optimal control, that is fast and accurate tracking of its reference setpoints under moderate variations of the control inputs.

Data-driven robust tracking control of underactuated mechanical systems using identified flat output and active disturbance rejection control

This paper presents a data-driven control design framework to achieve robust tracking without exploiting mathematical model of nonlinear underactuated mechanical systems (UMS). The method leverages differential flatness property of linearised systems, online estimation and compensation of disturbances by active disturbance rejection control. The differentially flat output is derived directly from measured data with unknown dynamics and parameters by the flat output identification algorithm. A reduced nominal model of UMS is proposed to simplify the process of finding flat output and trajectory planning. Technique of sparse regression is applied to identify relationships between flat output and system states, which reduce the order of the well-known extended state observer (ESO) and thereby make the ESO more effective for trajectory planning and tracking in terms of the flat output. The proposed control is validated by experimental studies of a rotary crane system in which a rest-to-rest tracking control is implemented.

Active disturbance rejection control of a permanent magnet synchronous generator for wind turbine applications

With sight on maximizing the amount of energy that can be extracted, by a wind turbine, from the wind, this article solves the maximum power point tracking problem for a permanent magnet synchronous generator-based horizontal wind turbine connected to the electrical grid. A three-phase back-to-back converter, which allows a decoupling between the electrical grid and the generator, is employed as an interphase between the wind turbine and the utility grid. Based on the mathematical model in the synchronous reference frame and taking advantage of the differential flatness property the system exhibits, controllers based on the active disturbance rejection methodology are designed, in this work, to track the curve of maximum extracted power from the wind and manage the generated electricity into the grid. At the same time, the phase angle of the electricity generated is synchronized with the phase angle of the electrical grid. Numerical simulations are performed to support the controllers presented in this work.

Differential Flatness-Based, Full-Order Nonlinear Control of a Modular Multilevel Converter (MMC)

Modular multilevel converter (MMC) is an attractive topology in industrial applications such as high-voltage direct current (HVdc) transmission system where fast power control is required. The most conventional control method for MMC uses cascaded structures, where the power transmission and internal energy dynamic (outer loops) is limited by the “frequency separation” constraints since the dynamic choice of the currents (inner loops) must consider the tradeoff between rapidity and robustness. In this article, we present a new control method based on a differential flatness theory. The main interest of the proposed control is the possibility to obtain a very high dynamic performance of the MMC power flow, even under noisy measurements and parametric disturbances. Using the well-known average model of the MMC, a flat output is proposed, proving the differential flatness property of the MMC. Trajectory tracking controllers of ac grid active and reactive power are proposed using input-state full-order linearization. Simulation results, including losses corresponding to a realistic case of on-shore, point-to-point HVdc interconnection, are presented to show the performance of the proposed control scheme even with strong measurement noise.

SwarmCCO: Probabilistic Reactive Collision Avoidance for Quadrotor Swarms Under Uncertainty

We present decentralized collision avoidance algorithms for quadrotor swarms operating under uncertain state estimation. Our approach exploits the differential flatness property and feedforward linearization to approximate the quadrotor dynamics and performs reciprocal collision avoidance. We account for the uncertainty in position and velocity by formulating the collision constraints as chance constraints, which describe a set of velocities that avoid collisions with a specified confidence level. We present two different methods for formulating and solving the chance constraints: our first method assumes a Gaussian noise distribution. Our second method is its extension to the non-Gaussian case by using a Gaussian Mixture Model (GMM). We reformulate the linear chance constraints into equivalent deterministic constraints, which are used with an MPC framework to compute a local collision-free trajectory for each quadrotor. We evaluate the proposed algorithm in simulations on benchmark scenarios and highlight its benefits over prior methods. We observe that both the Gaussian and non-Gaussian methods provide improved collision avoidance performance over the deterministic method. On average, the Gaussian method requires ~ 5 ms to compute a local collision-free trajectory, while our non-Gaussian method is computationally more expensive and requires ~ 9 ms on average in scenarios with 4 agents.

Hierarchical Control of DC Motor Coupled with Cuk Converter Combining Differential Flatness and Sliding Mode Control

This paper proposes a hierarchical control law for DC motor fed by DC–DC power Cuk Converter. The control is divided into two parts: Firstly, the property of differential flatness associated with the mathematical model of the DC motor is studied to design a robust control that achieves the task of tracking the reference angular speed trajectory for the motor. It also gives the voltage profile $$\vartheta $$ which must be followed by the Cuk converter. The second independent controller, based on cascade control, is proposed for the Cuk converter, which allows the converter output voltage to follow the specified trajectory $$\vartheta $$ . Sliding mode control is used in the inner loop, whereas proportional integral control is used in the proposed cascade controller’s outer loop. Numerical simulation of the hierarchical control technique is carried out in MATLAB/Simulink, and results under parametric variation show robustness. Finally, a comparison is drawn for speed trajectory tracking by the DC motor coupled with DC–DC buck and Cuk converter to show the performance improvement in case of using Cuk converter for angular speed trajectory tacking of a DC motor.

Active Disturbance Rejection Control for Reference Trajectory Tracking Tasks in the Pendubot System

In this article, an Extended State Observer (ESO) based Active Disturbance Rejection Control (ADRC) scheme is applied to the Pendubot system for a trajectory tracking tasks. The tangent linearization of the system allows to implement the control scheme taking advantage of the differential flatness property, while including a cascade configuration. The proposed method assumes a limited knowledge of the underactuated system where the control input gain and the order of the system are the only needed data. The scheme is experimentally tested leading to accurate tracking results.

Exploiting Differential Flatness for Robust Learning-Based Tracking Control Using Gaussian Processes

Learning-based control has shown to outperform conventional model-based techniques in the presence of model uncertainties and systematic disturbances. However, most state-of-the-art learning-based nonlinear trajectory tracking controllers still lack any formal guarantees. In this letter, we exploit the property of differential flatness to design an online, robust learning-based controller to achieve both high tracking performance and probabilistically guarantee a uniform ultimate bound on the tracking error. A common control approach for differentially flat systems is to try to linearize the system by using a feedback (FB) linearization controller designed based on a nominal system model. Performance and safety are limited by the mismatch between the nominal model and the actual system. Our proposed approach uses a nonparametric Gaussian Process (GP) to both improve FB linearization and quantify, probabilistically, the uncertainty in our FB linearization. We use this probabilistic bound in a robust linear quadratic regulator (LQR) framework. Through simulation, we highlight that our proposed approach significantly outperforms alternative learning-based strategies that use differential flatness.

Differential flatness theory-based control and filtering for a mobile manipulator

The article proposes a differential flatness theory based control and filtering method for the model of a mobile manipulator. This is a difficult control and robotics problem due to the system’s strong nonlinearities and due to its underactuation. Using the Euler-Lagrange approach, the dynamic model of the mobile manipulator is obtained. This is proven to be a differentially flat one, thus confirming that it can be transformed into an input-output linearized form. Through a change of state and control inputs variables the dynamic model of the manipulator is finally written into the linear canonical (Brunovsky) form. For the latter representation of the system’s dynamics the solution of both the control and filtering problems becomes possible. The global asymptotic stability properties of the control loop are proven. Moreover, a differential flatness theory-based state estimator, under the name of Derivative-free nonlinear Kalman Filter, is developed. This comprises (i) the standard Kalman Filter recursion on the linearized equivalent model of the mobile manipulator and (ii) an inverse transformation, relying on the differential flatness properties of the system which allows for estimating the state variables of the initial nonlinear model. Finally, by redesigning the aforementioned Kalman Filter as a disturbance observer one can achieve estimation and compensation of the disturbance inputs that affect the model of the mobile manipulator.

Intercepting a Superior Missile: Trajectory Optimization Approach to a Pursuit-Evasion Game

This paper handles the problem of intercepting a superior missile using a formation of lower-capabilities pursuers. The problem is formulated as multi-player differential game and solved using optimal control techniques. This approach makes use of the Apollonius circles as a final path constraint of the problem in order to achieve the capturing conditions. Additionally, a dimensionality reduction technique based on differential flatness property of the formation is investigated.

On the tracking of fast trajectories of a 3DOF torsional plant: A flatness based ADRC approach

AbstractIn this article, a highly oscillatory 3 degrees of freedom (DOF) torsional plant is used to demonstrate the effectiveness of the combined differential flatness property and active disturbance rejection control (ADRC) methodology in the output tracking of fast reference trajectory and vibration suppression problems. The controller design is synthesized in the absence of detailed knowledge of the system model, which means that the resonant modes are essentially unknown to the designer. Experimental results confirm that the excellent rest‐to‐rest trajectory tracking performance is attained while permanently attenuating remaining vibrations in all the three disks.

Development of Auto-parking and Collision Avoidance Algorithms on Car type Autonomous Mobile Robots

This paper discusses the development and implementation of a motion planning algorithm for a Nonholonomic Car type Wheeled Mobile Robot (WMR) through a specified via point. This technique will prove useful specifically; when the WMR needs to avoid obstacles which are present on the predefined trajectory and when the WMR is to be parked autonomously in the parking slot. A via-point is specified by the user and a trajectory passing through this via-point is generated using the property of differential flatness. Local optimization is employed for optimal trajectory planning. This generated trajectory is always smooth with feasible minimum length satisfying the nonholonomic and holonomic constraints on the car type vehicle and passes through the via-point specified by the user. The contribution of the work lies in reducing complexity of the problem by introducing via-point and local optimization, which results into easy implementation of the algorithm on the hardware setup. The experimentation is carried out on 1:10 reduced model of Cheverolet Camero model to show efficacy of the proposed scheme.

Trajectory Tracking for a Multicopter under a Quaternion Representation

This paper proposes a two-layer hierarchical control scheme for trajectory tracking of a multicopter system using attitude quaternion. We first present the differential flatness properties of the system and then, exploit them to design the feedback linearization laws for the position controller at the high control level. Next, the computed-torque control method is employed for stabilizing the attitude quaternion. The whole control scheme is illustrated through simulations while the position controller is further tested under experiments over a nano-drone quadcopter platform.

Acceleration control of a multi-rotor UAV towards achieving microgravity

In this paper, we present control and parameter estimation strategies with theoretical guarantees to turn a hex-rotor unmanned aerial vehicle (UAV) into a microgravity enabling platform. We make the UAV to maintain a constant acceleration equal to free-fall acceleration for it and any payload on-board to experience microgravity. Towards this, we derive a feedback linearisation-based acceleration control law exploiting the differential flatness property of our system. The proposed control law requires the estimates of the system parameters. Therefore, ancillary to this control law, we propose a parameter estimation scheme and prove that the proposed control law along with the parameter estimation scheme ensures convergence of acceleration to the desired value under certain conditions. We also characterize these conditions that guarantee convergence. The flight tests that we have performed employing the proposed control and parameter estimation schemes gave microgravity levels of the order of $$10^{-3}g$$ for 1.6 s. To our knowledge, our hex-rotor UAV is the first multi-rotor UAV to achieve microgravity, and the first UAV—fixed-wing or rotary—to attain and maintain such levels of microgravity.

Control descentralizado basado en eventos para el consenso de múltiples robots tipo péndulo invertido en el esquema líder-seguidor

<p>El trabajo presenta el diseno de una estrategia de control distribuido con comunicación activada por eventos, que resuelve el problema de consenso líder-seguidor, de un conjunto de robots móviles tipo péndulo invertido (RMPI). La linealización de las ecuaciones de movimiento de los RMPI, alrededor del punto de equilibrio, permiten explotar las propiedades de planitud diferencial, dando lugar a una reparametrizacion del sistema mediante la salida plana. Asumiendo que los vehículos se comunican mediante una red, cuya topología es representada por un grafo no dirigido y fuertemente conectado, se disena una ley de control distribuido y una funcion de evento que indica el instante en el que el i-ésimo vehículo debe transmitir informacion (su estado) a sus vecinos. El resultado es un intercambio asíncrono de informacion entre vehículos y donde el tiempo entre eventos no es equidistante. El analisis de estabilidad se lleva a cabo en el sentido de Lyapunov y en el sentido entrada-estado ISS (Input-to-State Stability). Los resultados en simulacion numérica muestran el buen desempeño del consenso de la red de vehículos en dos escenarios representativos: regulacion y seguimiento de trayectoria.</p>