Differential-Flatness and Control of Quadrotor(s) with a Payload Suspended through Flexible Cable(s)

We present the coordinate-free dynamics of three different quadrotor systems : (a) single quadrotor with a point-mass payload suspended through a flexible cable; (b) multiple quadrotors with a shared point-mass payload suspended through flexible cables; and (c) multiple quadrotors with a shared rigid-body payload suspended through flexible cables. We model the flexible cable(s) as a finite series of links with spherical joints with mass concentrated at the end of each link. The resulting systems are thus high-dimensional with high degree-of-underactuation. For each of these systems, we show that the dynamics are differentially-flat, enabling planning of dynamically feasible trajectories. For the single quadrotor with a point-mass payload suspended through a flexible cable with five links (16 degrees-of-freedom and 12 degrees-of-underactuation), we use the coordinate-free dynamics to develop a geometric variation-based linearized equations of motion about a desired trajectory. We show that a finite-horizon linear quadratic regulator can be used to track a desired trajectory with a relatively large region of attraction

Controlling a Quadrotor Carrying a Cable-Suspended Load to Pass Through a Window

In this paper, we design an optimal control system for a quadrotor to carry a cable-suspended load flying through a window. As the window is narrower than the length of the cable, it is very challenging to design a practical control system to pass through it. Our solution includes a system identification component, a trajectory generation component, and a trajectory tracking control component. The exact dynamic model that usually derived from the first principles is assumed to be unavailable. Instead, a model identification approach is adopted, which relies on a simple but effective low order equivalent system (LOES) to describe the core dynamical characteristics of the system. After being excited by some specifically designed manoeuvres, the unknown parameters in the LOES are obtained by using a frequency based least square estimation algorithm. Based on the estimated LOES, a numerical optimization algorithm is then utilized for aggressive trajectory generation when relevant constraints are given. The generated trajectory can lead to the quadrotor and load system passing through a narrow window with a cascade PD trajectory tracking controller. Finally, a practical flight test based on an Astec Hummingbird quadrotor is demonstrated and the result validates the proposed approach

Agile load transportation systems using aerial robots

In this dissertation, we address problems that can occur during load transport using aerial robots, i.e., small scale quadrotors. First, detailed models of such transportation system are derived. These models include nonlinear models of a quadrotor, a model of a quadrotor carrying a fixed load and a model of a quadrotor carrying a suspended load. Second, the problem of quadrotor stabilization and trajectory tracking with changes of the center of gravity of the transportation system is addressed. This problem is solved using model reference adaptive control based on output feedback linearization that compensates for dynamical changes in the center of gravity of the quadrotor. The third problem we address is a problem of a swing-free transport of suspended load using quadrotors. Flying with a suspended load can be a very challenging and sometimes hazardous task as the suspended load significantly alters the flight characteristics of the quadrotor. In order to deal with suspended load flight, we present a method based on dynamic programming which is a model based offline method. The second investigated method we use is based on the Nelder-Mead algorithm which is an optimization technique used for nonlinear unconstrained optimization problems. This method is model free and it can be used for offline or online generation of the swing-free trajectories for the suspended load. Besides the swing-free maneuvers with suspended load, load trajectory tracking is another problem we solve in this dissertation. In order to solve this problem we use a Nelder-Mead based algorithm. In addition, we use an online least square policy iteration algorithm. At the end, we propose a high level algorithm for navigation in cluttered environments considering a quadrotor with suspended load. Furthermore, distributed control of multiple quadrotors with suspended load is addressed too. The proposed hierarchical architecture presented in this doctoral dissertation is an important step towards developing the next generation of agile autonomous aerial vehicles. These control algorithms enable quadrotors to display agile maneuvers while reconfiguring in real time whenever a change in the center of gravity occurs. This enables a swing-free load transport or trajectory tracking of the load in urban environments in a decentralized fashion

AutoTrans: A Complete Planning and Control Framework for Autonomous UAV Payload Transportation

The robotics community is increasingly interested in autonomous aerial transportation. Unmanned aerial vehicles with suspended payloads have advantages over other systems, including mechanical simplicity and agility, but pose great challenges in planning and control. To realize fully autonomous aerial transportation, this paper presents a systematic solution to address these difficulties. First, we present a real-time planning method that generates smooth trajectories considering the time-varying shape and non-linear dynamics of the system, ensuring whole-body safety and dynamic feasibility. Additionally, an adaptive NMPC with a hierarchical disturbance compensation strategy is designed to overcome unknown external perturbations and inaccurate model parameters. Extensive experiments show that our method is capable of generating high-quality trajectories online, even in highly constrained environments, and tracking aggressive flight trajectories accurately, even under significant uncertainty. We plan to release our code to benefit the community.Comment: Accepted by IEEE Robotics and Automation Letter

Multi-rotor with suspended load: System Dynamics and Control Toolbox

There is an increasing demand for Unmanned Aerial Systems (UAS) to carry suspended loads as this can provide significant benefits to several applications in agriculture, law enforcement and construction. The load impact on the underlying system dynamics should not be neglected as significant feedback forces may be induced on the vehicle during certain flight manoeuvres. The constant variation in operating point induced by the slung load also causes conventional controllers to demand increased control effort. Much research has focused on standard multi-rotor position and attitude control with and without a slung load. However, predictive control schemes, such as Nonlinear Model Predictive Control (NMPC), have not yet been fully explored. To this end, we present a novel controller for safe and precise operation of multi-rotors with heavy slung load in three dimensions. The paper describes a System Dynamics and Control Simulation Toolbox for use with MATLAB/SIMULINK which includes a detailed simulation of the multi-rotor and slung load as well as a predictive controller to manage the nonlinear dynamics whilst accounting for system constraints. It is demonstrated that the controller simultaneously tracks specified waypoints and actively damps large slung load oscillations. A linear-quadratic regulator (LQR) is derived and control performance is compared. Results show the improved performance of the predictive controller for a larger flight envelope, including aggressive manoeuvres and large slung load displacements. The computational cost remains relatively small, amenable to practical implementations

비선형 최적화를 이용한 멀티로터 현수 운송의 경로 계획 및 제어 기법

학위논문(박사) -- 서울대학교대학원 : 공과대학 기계항공공학부, 2021.8. 김현진.경로 계획과 제어는 안전하고 안정적으로 멀티로터를 운용하기 위해서 필수적인 요소이다. 충돌을 회피하며 효율적인 경로를 생성하고 이를 실제로 추종하기 위해서는 동역학 모델이 고려되어야 한다. 일반 멀티로터의 동역학 모델은 높은 차원을 가진 비선형식으로 표현되는데, 현수 운송 물체를 추가할 경우 계산이 더욱 복잡해진다. 본 논문은 멀티로터를 이용한 현수 운송에 있어 경로 계획과 제어에 대한 효율적인 기법을 제안한다. 첫 번째로 단일 멀티로터를 이용한 현수 운송을 다룬다. 물체가 별도의 엑츄에이터 없이 운송될 경우 물체는 기체의 움직임에 의해서만 제어가 가능하다. 하지만, 동역학식의 높은 비선형성으로 운용에 어려움이 존재한다. 이를 경감시키기 위해서 회전 동역학식의 비선형성을 줄이고 자세 제어에 존재하는 시간 지연을 고려하여 동역학식을 간소화한다. 경로 계획에 있어서는 충돌 회피를 위해 기체, 케이블, 그리고 운송 물체를 다른 크기와 모양을 가진 타원체들로 감싸며, 효과적이면서도 덜 보수적인 방식으로 충돌 회피 구속조건을 부과한다. Augmented Lagrangian 방법을 이용하여 비선형 구속조건이 부과된 비선형 문제를 실시간 최적화하여 경로를 생성한다. 생성된 경로를 추종하기 위해서 Sequential linear quadratic 솔버를 이용한 모델 예측 제어기로 최적 제어 입력을 계산한다. 제안된 기법은 여러 시뮬레이션과 실험을 통해 검증한다. 다음으로, 다중 멀티로터를 이용한 협업 현수 운송 시스템을 다룬다. 해당 시스템의 상태 변수나 동역학식에서 연결된(coupled) 항의 개수는 기체의 수에 비례하여 증가하기 때문에, 효과적인 기법 없이는 최적화에 많은 시간이 소요된다. 높은 비선형성을 가진 동역학식의 복잡성을 낮추기 위하여 미분 평탄성을 사용한다. 경로 또한 piece-wise Bernstein 다항식을 이용하여 매개변수화하여 최적화 변수의 개수를 줄인다. 최적화 문제를 분해하고 충돌 회피 구속조건들에 대해 볼록화(convexification)를 수행하여 운송 물체의 경로와 장력의 경로에 대한 볼록한(convex) 하위문제들이 만들어진다. 첫 번째 하위문제인 물체 경로 생성에서는, 장애물 회피와 멀티로터의 공간을 확보하기 위하여 안전 비행 통로(safe flight corridor, SFC)와 여유 간격 구속조건을 고려하여 최적화한다. 다음으로, 장력 벡터들의 경로는 장애물 회피와 상호 충돌을 방지하기 위하여 안전 비행 섹터(safe flight sector, SFS)와 상대 안전 비행 섹터(relative safe flight sector, RSFS) 구속조건을 부과하여 최적화한다. 시뮬레이션과 실험으로 복잡한 환경에서 효율적인 경로 계획 기법을 시연하며 검증한다.Trajectory generation and control are fundamental requirements for safe and stable operation of multi-rotors. The dynamic model should be considered to generate efficient and collision-free trajectories with feasibility. While the dynamic model of a bare multi-rotor is expressed non-linearly with high dimensions which results in computational loads, the suspended load increases the complexity further. This dissertation presents efficient algorithms for trajectory generation and control of multi-rotors with a suspended load. A single multi-rotor with a suspended load is addressed first. Since the load is suspended through a cable without any actuator, movement of the load must be controlled via maneuvers of the multi-rotor. However, the highly non-linear dynamics of the system results in difficulties. To relive them, the rotational dynamics is simplified to reduce the non-linearity and consider the delay in attitude control. For trajectory generation, the vehicle, cable, and load are considered as ellipsoids with different sizes and shapes, and collision-free constraints are expressed in an efficient and less-conservative way. The augmented Lagrangian method is applied to solve a nonlinear optimization problem with nonlinear constraints in real-time. Model predictive control with the sequential linear quadratic solver is used to track the generated trajectories. The proposed algorithm is validated with several simulations and experiment. A system with multiple multi-rotors for cooperative transportation of a suspended load is addressed next. As the system has more state variables and coupling terms in the dynamic equation than the system with a single multi-rotor, optimization takes a long time without an efficient method. The differential flatness of the system is used to reduce the complexity of the highly non-linear dynamic equation. The trajectories are also parameterized using piece-wise Bernstein polynomials to decrease the number of optimization variables. By decomposing an optimization problem and performing convexification, convex sub-problems are formulated for the load and the tension trajectories optimization, respectively. In each sub-problem, a light-weight sampling method is used to find a feasible and low-cost trajectory as initialization. In the first sub-problem, the load trajectory is optimized with safe flight corridor (SFC) and clearance constraints for collision avoidance and security of space for the multi-rotors. Then, the tension histories are optimized with safe flight sector (SFS) and relative safe flight sector (RSFS) constraints for obstacle and inter-agent collision avoidance. Simulations and experiments are conducted to demonstrate efficient trajectory generation in a cluttered environment and validate the proposed algorithms.Chapter 1 Introduction 1 1.1 Literature Survey 5 1.2 Contributions 9 1.3 Outline 10 Chapter 2 Single Multi-rotor with a Suspended Load 11 2.1 Dynamics 11 2.2 Trajectory Generation 23 2.3 Optimal Control 31 Chapter 3 Multiple Multi-rotors with a Suspended Load 36 3.1 Problem Setting 36 3.2 Load Trajectory Generation 45 3.3 Tension History Generation 54 Chapter 4 Experimental Validation 68 4.1 Single Multi-rotor with a Suspended Load 68 4.2 Multiple Multi-rotors with a Suspended Load 79 Chapter 5 Conclusion 100 Appendix A Detailed Derivation of Dierential Flatness 102 B Preliminaries of Bernstein Polynomials 108 B.1 Denition of a Bernstein Polynomial 108 B.2 Convex hull property of a Bernstein Polynomial 110 B.3 Representation of a General Polynomial with Bernstein Basis Polynomials 111 B.4 Representation of the Derivative of a Bernstein Polynomial with Bernstein Basis Polynomials 112 References 113 Abstract (in Korean) 119박

Suspended Load Path Tracking Control Using a Tilt-rotor UAV Based on Zonotopic State Estimation

This work addresses the problem of path tracking control of a suspended load using a tilt-rotor UAV. The main challenge in controlling this kind of system arises from the dynamic behavior imposed by the load, which is usually coupled to the UAV by means of a rope, adding unactuated degrees of freedom to the whole system. Furthermore, to perform the load transportation it is often needed the knowledge of the load position to accomplish the task. Since available sensors are commonly embedded in the mobile platform, information on the load position may not be directly available. To solve this problem in this work, initially, the kinematics of the multi-body mechanical system are formulated from the load's perspective, from which a detailed dynamic model is derived using the Euler-Lagrange approach, yielding a highly coupled, nonlinear state-space representation of the system, affine in the inputs, with the load's position and orientation directly represented by state variables. A zonotopic state estimator is proposed to solve the problem of estimating the load position and orientation, which is formulated based on sensors located at the aircraft, with different sampling times, and unknown-but-bounded measurement noise. To solve the path tracking problem, a discrete-time mixed $\mathcal{H}_2/\mathcal{H}_\infty$ controller with pole-placement constraints is designed with guaranteed time-response properties and robust to unmodeled dynamics, parametric uncertainties, and external disturbances. Results from numerical experiments, performed in a platform based on the Gazebo simulator and on a Computer Aided Design (CAD) model of the system, are presented to corroborate the performance of the zonotopic state estimator along with the designed controller

Reinforcement Learning and Planning for Preference Balancing Tasks

Robots are often highly non-linear dynamical systems with many degrees of freedom, making solving motion problems computationally challenging. One solution has been reinforcement learning (RL), which learns through experimentation to automatically perform the near-optimal motions that complete a task. However, high-dimensional problems and task formulation often prove challenging for RL. We address these problems with PrEference Appraisal Reinforcement Learning (PEARL), which solves Preference Balancing Tasks (PBTs). PBTs define a problem as a set of preferences that the system must balance to achieve a goal. The method is appropriate for acceleration-controlled systems with continuous state-space and either discrete or continuous action spaces with unknown system dynamics. We show that PEARL learns a sub-optimal policy on a subset of states and actions, and transfers the policy to the expanded domain to produce a more refined plan on a class of robotic problems. We establish convergence to task goal conditions, and even when preconditions are not verifiable, show that this is a valuable method to use before other more expensive approaches. Evaluation is done on several robotic problems, such as Aerial Cargo Delivery, Multi-Agent Pursuit, Rendezvous, and Inverted Flying Pendulum both in simulation and experimentally. Additionally, PEARL is leveraged outside of robotics as an array sorting agent. The results demonstrate high accuracy and fast learning times on a large set of practical applications

Optimal Control of Multiple Quadrotors for Transporting a Cable Suspended Payload

In this thesis, the main aim is to improve the flight control performance for a cable suspended payload with single and two quadrotors based on optimised control techniques. The study utilised optimal controllers, such as the Linear Quadratic Regulator LQR, the Iterative based LQR (ILQR), the Model Predictive Control MPC and the dynamic game controller to solve tracking control problems in terms of stabilisation, accuracy, constraints and collision avoidance. The LQR control was applied to the system as the first control method and compared with the classical Proportional-Derivative controller PD. It was used to achieve the load path tracking performance for single and two quadrotors with a cable slung load. The second controller was ILQR, which was developed based on the LQR control method to deal with the model nonlinearity. The MPC technique was also applied to the linearised nonlinear model LMPC of two quadrotors with a payload suspended by cables and compared with a nonlinear MPC (NMPC). Both MPC controllers LMPC and NMPC considered the constraints imposed on the system states and control inputs. The dynamic game control method was developed based on an incentive strategy for a leader-follower framework with the consideration of different optimal cost functions. It was applied to the linearised nonlinear model. Selecting these control techniques led to a number of achievements. Firstly, they improved the system performance in terms of achieving the system stability and reducing the steady-state errors. Secondly, the system parameter uncertainties were taken into consideration by utilising the ILQR controller. Thirdly, the MPC controllers guaranteed the handling of constraints and external disturbances in linear and nonlinear systems. Finally, avoiding collision between the leader and follower robots was achieved by applying the dynamic game controller. The controllers were tested in MATLAB simulation and verified for various desired predefined trajectories. In real experiments, these controllers were used as high-level controllers, which produce the optimised trajectory points. Then a low-level controller (PD controller) was used to follow the optimised trajectory points