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

Aerial navigation in obstructed environments with embedded nonlinear model predictive control

We propose a methodology for autonomous aerial navigation and obstacle avoidance of micro aerial vehicles (MAV) using nonlinear model predictive control (NMPC) and we demonstrate its effectiveness with laboratory experiments. The proposed methodology can accommodate obstacles of arbitrary, potentially non-convex, geometry. The NMPC problem is solved using PANOC: a fast numerical optimization method which is completely matrix-free, is not sensitive to ill conditioning, involves only simple algebraic operations and is suitable for embedded NMPC. A C89 implementation of PANOC solves the NMPC problem at a rate of 20Hz on board a lab-scale MAV. The MAV performs smooth maneuvers moving around an obstacle. For increased autonomy, we propose a simple method to compensate for the reduction of thrust over time, which comes from the depletion of the MAV's battery, by estimating the thrust constant

Model Predictive Control for Micro Aerial Vehicles: A Survey

This paper presents a review of the design and application of model predictive control strategies for Micro Aerial Vehicles and specifically multirotor configurations such as quadrotors. The diverse set of works in the domain is organized based on the control law being optimized over linear or nonlinear dynamics, the integration of state and input constraints, possible fault-tolerant design, if reinforcement learning methods have been utilized and if the controller refers to free-flight or other tasks such as physical interaction or load transportation. A selected set of comparison results are also presented and serve to provide insight for the selection between linear and nonlinear schemes, the tuning of the prediction horizon, the importance of disturbance observer-based offset-free tracking and the intrinsic robustness of such methods to parameter uncertainty. Furthermore, an overview of recent research trends on the combined application of modern deep reinforcement learning techniques and model predictive control for multirotor vehicles is presented. Finally, this review concludes with explicit discussion regarding selected open-source software packages that deliver off-the-shelf model predictive control functionality applicable to a wide variety of Micro Aerial Vehicle configurations

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

모션 프리머티브를 이용한 복잡한 로봇 임무 학습 및 일반화 기법

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 항공우주공학과, 2020. 8. 김현진.Learning from demonstrations (LfD) is a promising approach that enables robots to perform a specific movement. As robotic manipulations are substituting a variety of tasks, LfD algorithms are widely used and studied for specifying the robot configurations for the various types of movements. This dissertation presents an approach based on parametric dynamic movement primitives (PDMP) as a motion representation algorithm which is one of relevant LfD techniques. Unlike existing motion representation algorithms, this work not only represents a prescribed motion but also computes the new behavior through a generalization of multiple demonstrations in the actual environment. The generalization process uses Gaussian process regression (GPR) by representing the nonlinear relationship between the PDMP parameters that determine motion and the corresponding environmental variables. The proposed algorithm shows that it serves as a powerful optimal and real-time motion planner among the existing planning algorithms when optimal demonstrations are provided as dataset. In this dissertation, the safety of motion is also considered. Here, safety refers to keeping the system away from certain configurations that are unsafe. The safety criterion of the PDMP internal parameters are computed to check the safety. This safety criterion reflects the new behavior computed through the generalization process, as well as the individual motion safety of the demonstration set. The demonstrations causing unsafe movement are identified and removed. Also, the demolished demonstrations are replaced by proven demonstrations upon this criterion. This work also presents an extension approach reducing the number of required demonstrations for the PDMP framework. This approach is effective where a single mission consists of multiple sub-tasks and requires numerous demonstrations in generalizing them. The whole trajectories in provided demonstrations are segmented into multiple sub-tasks representing unit motions. Then, multiple PDMPs are formed independently for correlated-segments. The phase-decision process determines which sub-task and associated PDMPs to be executed online, allowing multiple PDMPs to be autonomously configured within an integrated framework. GPR formulations are applied to obtain execution time and regional goal configuration for each sub-task. Finally, the proposed approach and its extension are validated with the actual experiments of mobile manipulators. The first two scenarios regarding cooperative aerial transportation demonstrate the excellence of the proposed technique in terms of quick computation, generation of efficient movement, and safety assurance. The last scenario deals with two mobile manipulations using ground vehicles and shows the effectiveness of the proposed extension in executing complex missions.시연 학습 기법(Learning from demonstrations, LfD)은 로봇이 특정 동작을 수행할 수 있도록 하는 유망한 동작 생성 기법이다. 로봇 조작기가 인간 사회에서 다양한 업무를 대체해 감에 따라, 다양한 임무를 수행하는 로봇의 동작을 생성하기 위해 LfD 알고리즘들은 널리 연구되고, 사용되고 있다. 본 논문은 LfD 기법 중 모션 프리머티브 기반의 동작 재생성 알고리즘인 Parametric dynamic movement primitives(PDMP)에 기초한 알고리즘을 제시하며, 이를 통해 다양한 임무를 수행하는 모바일 조작기의 궤적을 생성한다. 기존의 동작 재생성 알고리즘과 달리, 이 연구는 제공된 시연에서 표현된 동작을 단순히 재생성하는 것에 그치지 않고, 새로운 환경에 맞게 일반화 하는 과정을 포함한다. 이 논문에서 제시하는 일반화 과정은 PDMPs의 내부 파라미터 값인 스타일 파라미터와 환경 변수 사이의 비선형 관계를 가우스 회귀 기법 (Gaussian process regression, GPR)을 이용하여 수식적으로 표현한다. 제안된 기법은 또한 최적 시연를 학습하는 방식을 통해 강력한 최적 실시간 경로 계획 기법으로도 응용될 수 있다. 본 논문에서는 또한 로봇의 구동 안전성도 고려한다. 기존 연구들에서 다루어진 시연 관리 기술이 로봇의 구동 효율성을 개선하는 방향으로 제시된 것과 달리, 이 연구는 강한 구속조건으로 로봇의 구동 안전성을 확보하는 시연 관리 기술을 통해 안정성을 고려하는 새로운 방식을 제시한다. 제안된 방식은 스타일 파라미터 값 상에서 안전성 기준을 계산하며, 이 안전 기준을 통해 시연을 제거하는 일련의 작업을 수행한다. 또한, 제거된 시위를 안전 기준에 따라 입증된 시위로 대체하여 일반화 성능을 저하시키지 않도록 시위를 관리한다. 이를 통해 다수의 시연 각각 개별 동작 안전성 뿐 아니라 온라인 동작의 안전성까지 고려할 수 있으며, 실시간 로봇 조작기 운용시 안전성이 확보될 수 있다. 제안된 안정성을 고려한 시연 관리 기술은 또한 환경의 정적 설정이 변경되어 모든 시연을 교체해야 할 수 있는 상황에서 사용할 수 있는 시연들을 판별하고, 효율적으로 재사용하는 데 응용할 수 있다. 또한 본 논문은 복잡한 임무에서 적용될 수 있는 PDMPs의 확장 기법인 seg-PDMPs를 제시한다. 이 접근방식은 복잡한 임무가 일반적으로 복수개의 간단한 하위 작업으로 구성된다고 가정한다. 기존 PDMPs와 달리 seg-PDMPs는 전체 궤적을 하위 작업을 나타내는 여러 개의 단위 동작으로 분할하고, 각 단위동작에 대해 여러개의 PDMPs를 구성한다. 각 단위 동작 별로 생성된 PDMPs는 통합된 프레임워크내에서 단계 결정 프로세스를 통해 자동적으로 호출된다. 각 단계 별로 단위 동작을 수행하기 위한 시간 및 하위 목표점은 가우스 공정 회귀(GPR)를 이용한 환경변수와의의 관계식을 통해 얻는다. 결과적으로, 이 연구는 전체적으로 요구되는 시연의 수를 효과적으로 줄일 뿐 아니라, 각 단위동작의 표현 성능을 개선한다. 제안된 알고리즘은 협동 모바일 로봇 조작기 실험을 통하여 검증된다. 세 가지의 시나리오가 본 논문에서 다루어지며, 항공 운송과 관련된 첫 두 가지 시나리오는 PDMPs 기법이 로봇 조작기에서 빠른 적응성, 임무 효율성과 안전성 모두 만족하는 것을 입증한다. 마지막 시나리오는 지상 차량을 이용한 두 개의 로봇 조작기에 대한 실험으로 복잡한 임무 수행을 하기 위해 확장된 기법인 seg-PDMPs가 효과적으로 변화하는 환경에서 일반화된 동작을 생성함을 검증한다.1 Introduction 1 1.1 Motivations 1 1.2 Literature Survey 3 1.2.1 Conventional Motion Planning in Mobile Manipulations 3 1.2.2 Motion Representation Algorithms 5 1.2.3 Safety-guaranteed Motion Representation Algorithms 7 1.3 Research Objectives and Contributions 7 1.3.1 Motion Generalization in Motion Representation Algorithm 9 1.3.2 Motion Generalization with Safety Guarantee 9 1.3.3 Motion Generalization for Complex Missions 10 1.4 Thesis Organization 11 2 Background 12 2.1 DMPs 12 2.2 Mobile Manipulation Systems 13 2.2.1 Single Mobile Manipulation 14 2.2.2 Cooperative Mobile Manipulations 14 2.3 Experimental Setup 17 2.3.1 Test-beds for Aerial Manipulators 17 2.3.2 Test-beds for Robot Manipulators with Ground Vehicles 17 3 Motion Generalization in Motion Representation Algorithm 22 3.1 Parametric Dynamic Movement Primitives 22 3.2 Generalization Process in PDMPs 26 3.2.1 Environmental Parameters 26 3.2.2 Mapping Function 26 3.3 Simulation Results 29 3.3.1 Two-dimensional Hurdling Motion 29 3.3.2 Cooperative Aerial Transportation 30 4 Motion Generalization with Safety Guarantee 36 4.1 Safety Criterion in Style Parameter 36 4.2 Demonstration Management 39 4.3 Simulation Validation 42 4.3.1 Two-dimensional Hurdling Motion 46 4.3.2 Cooperative Aerial Transportation 47 5 Motion Generalization for Complex Missions 51 5.1 Overall Structure of Seg-PDMPs 51 5.2 Motion Segments 53 5.3 Phase-decision Process 54 5.4 Seg-PDMPs for Single Phase 54 5.5 Simulation Results 55 5.5.1 Initial/terminal Offsets 56 5.5.2 Style Generalization 59 5.5.3 Recombination 61 6 Experimental Validation and Results 63 6.1 Cooperative Aerial Transportation 63 6.2 Cooperative Mobile Hang-dry Mission 70 6.2.1 Demonstrations 70 6.2.2 Simulation Validation 72 6.2.3 Experimental Results 78 7 Conclusions 82 Abstract (in Korean) 93Docto

Data-driven system identification and model predictive control of a multirotor with an unknown suspended payload

Thesis (MEng)--Stellenbosch University, 2022.ENGLISH ABSTRACT: This thesis considers the problem of stabilised control for a multirotor with an unknown suspended payload. The swinging payload negatively affects the multirotor flight dynamics by inducing oscillations in the system. An adaptive control architecture is proposed to damp these oscillations and produce stable flight with different unknown payloads. The architecture includes a data-driven system identification method that assumes no prior knowledge of the payload dynamics. This method is demonstrated in simulation and with practical flight data. Model Predictive Control (MPC) is applied for swing damping control and is verified with Hardware-in-the-Loop (HITL) simulations. A parameter estimator and Linear Quadratic Regulator (LQR) is used as a baseline control architecture. The LQR uses a predetermined model of the system, which is completed with estimates of the payload mass and cable length. The newly proposed architecture uses Dynamic Mode Decomposition with Control (DMDc) to estimate a linear state-space model and approximate the dynamics without using a predetermined model. The architecture was also tested with a Hankel Alternative View Of Koopman (HAVOK) algorithm which was extended in this work to account for control. An MPC uses the data-driven model to control the multirotor and damp the payload oscillations. A Simulink™ simulator was designed and verified with practical data. Within simulations both the baseline and proposed architectures produced near swing-free control with different payload masses and cable lengths. Even with a dynamic payload producing irregular oscillations, both methods achieved stabilised control. Both architectures also showed effective disturbance rejection. Despite the baseline method using an accurate predetermined model, the proposed method produced equal performances without prior knowledge of the dynamics. The baseline performance degraded significantly with a changed multirotor mass because this parameter was not considered as an unknown. In contrast, the proposed method consistently produced good performances. The accuracy of the DMDc models was verified with practical flight data. The proposed control architecture was also demonstrated in HITL simulations. The hardware executed the MPC at the desired frequency, producing near swing-free control within a Gazebo simulator. Overall, it was shown that the proposed control architecture is practically feasible. Without knowledge of the payload dynamics, a data-driven model can be used with MPC for effective swing damping control with a multirotor.AFRIKAANSE OPSOMMING: Hierdie tesis hanteer die probleem van gestabiliseerde beheer vir ’n multirotor hommeltuig met ’n onbekende hangende loonvrag. Die swaaiende loonvrag be¨ınvloed die vlugdin amika deur ossillasies in die stelsel te veroorsaak. ’n Aanpasbare beheerargitektuur word voorgestel om hierdie ossillasies te demp vir stabiele vlugte met verskillende onbekende loonvragte. Die argitektuur maak gebruik van ’n datagedrewe stelsel-identifikasiemetode wat geen voorafkennis van die loonvragdinamika gebruik nie. Hierdie metode word in simulasies en met praktiese vlugdata gedemonstreer. Model Voorspellende Beheer (MVB) word toegepas vir swaaidempingsbeheer en word geverifieer met Hardeware-in-die-Lus (HIDL) simulasies. ’n Parameter-afskatter en Lineˆere Kwadratiese Gaussiese (LKG) word in die basislyn beheerargitektuur gebruik. Die LKG gebruik ’n voorafbepaalde model van die sisteem wat voltooi word met afskattings van die loonvragmassa en kabellengte. Die nuwe voorgestelde argitektuur gebruik Dinamiese Modus Ontbinding met beheer (DMOb) om ’n lineˆere toestand-ruimte model te bereken en die dinamika af te skat sonder ’n voorafbepaalde model. Die argitektuur is ook getoets met ’n Hankel Alternatiewe Siening van Koopman (HASK)-algoritme wat in hierdie werk uitgebrei is om beheer in te sluit. ’n MVB gebruik die data-gedrewe model om die multirotor te beheer en die loonvrag se ossillasies te demp. ’n Simulink™-simululeerder is ontwerp en geverifieer met praktiese data. In simulasies het beide die basislyn en voorgestelde argitekture byna-swaaivrye beheer met verskillende loon vragmassas en kabellengtes geproduseer. Selfs met ’n dinamiese loonvrag wat onre¨elmatige ossillasies voortbring, het beide metodes gestabiliseerde beheer tot gevolg gehad. Beide ar gitekture het ook effektiewe versteuringsverwerping getoon. Al gebruik die basislynmetode ’n akkurate voorafbepaalde model, het die voorgestelde metode gelyke prestasies gelewer sonder voorafkennis van die dinamika. Die basislyn prestasie het aansienlik afgeneem vir ’n aangepaste multirotormassa omdat hierdie parameter nie as ’n onbekende beskou is nie. Daarteenoor het die voorgestelde metode deurgaans goeie prestasies gelewer. Die akkuraatheid van die DMOb modelle is geverifieer met praktiese vlugdata. Die voorgestelde beheerargitektuur is ook in HIDL-simulasies gedemonstreer. MVB is teen die verlangde frekwensie uitgevoer en het byna-swaaivrye beheer in ’n Gazebo-simululeerder gelewer. In die geheel is dit gewys dat die voorgestelde beheerargitektuur prakties uitvoerbaar is. Sonder kennis van die loonvragdinamika kan ’n data-gedrewe model met MVB gebruik word vir effektiewe swaaidempingsbeheer met ’n multirotor.Master