Conference Paper

# Geometric control and differential flatness of a quadrotor UAV with a cable-suspended load

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## Abstract

A quadrotor with a cable-suspended load with eight degrees of freedom and four degrees underactuation is considered and a coordinate-free dynamic model, defined on the configuration space SE(3)×S2, is obtained by taking variations on manifolds. The quadrotor-load system is established to be a differentially-flat hybrid system with the load position and the quadrotor yaw serving as the flat outputs. A nonlinear geometric control design is developed, that enables tracking of outputs defined by (a) quadrotor attitude, (b) load attitude, and (c) position of the load. In each case, the closed-loop system exhibits almost-global properties. Stability proofs for the controller design, as well as simulations of the proposed controller are presented.

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... In aerial transportation, a cable establishes a physical connection between the UAV and the cargo. Geometric nonlinear controllers of multiple quadrotors with a suspended point-mass load were studied in Sreenath et al. (2013) and with a rigid body load in Lee (2014), Wu and Sreenath (2014)-where in particular the cables were modeled as inflexible, inelastic, and massless. In Goodarzi and Lee (2015), the authors model the cables as flexible chains comprised of inflexible, inelastic links with mass. ...
... Remark 3 As discussed in Kotaru et al. (2017), Sreenath et al. (2013), it is more physically realistic to model our system by a hybrid dynamical system which transitions between the cases of a taut cable (with positive tension magnitude) and a cable with zero tension. However, such a consideration will not play a role in the control design and subsequent analysis, so we will omit such a development. ...
... The differential flatness of the under-actuated system comprised of one quadrotor carrying a cable-suspended point-mass load is discussed, for instance, in Sreenath et al. (2013). In that case, the 4 flat outputs are given by (x L , ψ) ∈ R 3 × R, where x L is the load position and ψ is the yaw angle of the quadrotor. ...
Article
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This paper presents the design of a geometric trajectory tracking controller for an underactuated multi-body system describing the cooperative task of two quadrotor UAVs (unmanned aerial vehicles) carrying and transporting a rigid bar, which is attached to the quadrotors via inflexible elastic cables. The elasticity of the cables together with techniques of singular perturbation allows a reduction in the model to that of a similar model with inelastic cables. In this reduced model, we design a controller such that the rod exponentially tracks a given desired trajectory for its position and attitude, under some assumptions on initial error. We then show that exponential tracking in the reduced model corresponds to exponential tracking of the original elastic model. We also show that the previously defined control scheme provides uniform ultimate boundedness in the presence of unstructured bounded disturbances.
... A backstepping approach is combined with an integral action to design the adaptive controller. The quadrotor-slung-load system is differentially flat [19] that is, the states and the inputs can be written as algebraic functions of carefully selected flat outputs, which are load position and quadrotor yaw, and their derivatives. Consequently, accurate trajectory tracking implies that the system states follow known functions of the reference and its derivatives. ...
... The load is assumed to be connected to the center of mass of the quadrotor. This assumption is reasonable and mild, it is also used in [19,21] and comes with negligible impact to the system dynamics. The cable connecting the quadrotor and the load is a massless, which has a fixed length, and is always taut during flight. ...
... is calculated from taking partial derivative of (19) with respect to v L . ...
Article
In aerial load transportation applications, knowing the mass of the load in advance is not always possible. The load dynamics depend on its mass and using a high-performance model-based controller with an inaccurate mass will introduce unmodeled disturbances to the system that will negatively affect the closed-loop performance. This paper addresses the design of a trajectory tracking controller for a quadrotor with a slung-load that has an unknown mass. The proposed solution is an adaptive controller with online estimation designed using the backstepping technique. Nonlinear control laws for thrust and angular velocity, and an adaption law for mass estimation are proposed, which guarantee the convergence of the trajectory tracking and the estimation errors to zero, and are robust to variations in load mass. Simulation and experimental results are presented to assess the validity and performance of the proposed controller.
... Adaptive control based on backstepping is derived in [16] *This work was supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) and the Swedish Research Council, Knut and Alice Wallenberg Foundation (KAW). 1 christos.verginis@angstrom.uu.se and L 1 adaptive control is used for aggressive flight maneuvers in [17]. Flatness-based control is employed in [18], [19], and H ∞ controller in [20]. With the increase of available computational power in embedded devices, Model predictive control (MPC) became very popular due its ability to handle state and input constraints and optimize the trajectory online [21], [22]. ...
... is needed for the boundedness of the intermediate signal (19). Intuitively, it requires the UAV thrust, set in (18), to be always positive and compensate for the gravitational force g. ...
... for t ∈ [0, τ max ). By using T φθ = e φθ + T φθ,r , (19), (18), the boundedness ofv r ,ρ v , and (30), and the continuity and boundedness of F d (χ,χ, t) in (χ,χ) and t, respectively, we arrive aṫ ...
Preprint
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We propose a control protocol based on the prescribed performance control (PPC) methodology for a quadrotor unmanned aerial vehicle (UAV). Quadrotor systems belong to the class of underactuated systems for which the original PPC methodology cannot be directly applied. We introduce the necessary design modifications to stabilize the considered system with prescribed performance. The proposed control protocol does not use any information of dynamic model parameters or exogenous disturbances. Furthermore, the stability analysis guarantees that the tracking errors remain inside of designer-specified time-varying functions, achieving prescribed performance independent from the control gains' selection. Finally, simulation results verify the theoretical results.
... Other works instead directly model the hybrid dynamics. For instance, [18], [35], [36] suggest modeling collision between quadrotors and the suspended payloads as perfectly inelastic collision but do not account for the transient state transition. [37] provides a complete modeling of collision between a quadrotor and a point-mass payload. ...
... According to [35], by applying the Lagrange-d'Alembert principle, we obtain the equations of motion of the system ...
... We describe the controllers for the systems in our simulator. 1) Single Quadrotor: We present the geometric controller for cable-suspended payload with a single quadrotor [35]. The desired force acting on the payload is ...
Preprint
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Low-cost autonomous Micro Aerial Vehicles (MAVs) have the potential to help humans by simplifying and speeding up complex tasks that require their interaction with the environment such as construction, package delivery, and search and rescue. These systems, composed of single or multiple vehicles, can be endowed with passive connection mechanisms such as rigid links or cables to perform transportation and manipulation tasks. However, they are inherently complex since they are often underactuated, and evolve on nonlinear manifold configuration spaces. In addition, the complexity of systems with cable-suspended load is further increased by the hybrid dynamics depending on the cables' varying tension conditions. In this paper, we present the first aerial transportation and manipulation simulator incorporating different payloads and passive connection mechanisms with full system dynamics as well as planning and control algorithms. Furthermore, it includes a novel model accounting for the transient hybrid dynamics for aerial systems with cable-suspended load to closely mimic real-world systems. The availability of a flexible and intuitive interface further contributes to its usability and versatility. Comparisons between simulations and real-world experiments with different vehicles' configurations show the fidelity of the simulator results with respect to real-world settings and its benefit for rapid prototyping and transitioning of aerial transportation and manipulation systems to real-world deployment.
... L (t k )∆t k to minimize the energy consumption, since the torque of the quadrotor depends on the 6 th derivative of the payload position through differentialflatness property, as proved in [13]. Additionally, we have ∑ N k=1R 0 T (t k )∆t k and ∑ N k=1R 1 (l 0 − l(t k ))∆t k representing the cost on the taut and slack modes, respectively. ...
... Introducing the slack variable s as the penetration distance in (13) would make this nonlinear optimization smoother. ...
... The control-loop is closed around this trajectory, i.e, we implement trajectory tracking control on the quadrotor. Note, this is due to our limitations in estimating the load-cable attitude and its angular velocity accurately, as the load attitude states are vital to achieve load position control [13]. For the path planning optimization problems, we use open source solver IPOPT [1] in Matlab with modeling language Yalmip [10]. ...
Preprint
Full-text available
Generating agile maneuvers for a quadrotor with a cable-suspended load is a challenging problem. State-of-the-art approaches often need significant computation time and complex parameter tuning. We use a coordinate-free geometric formulation and exploit a differential flatness based hybrid model of a quadrotor with a cable-suspended payload. We perform direct collocation on the differentially-flat hybrid system, and use complementarity constraints to avoid specifying hybrid mode sequences. The non-differentiable obstacle avoidance constraints are reformulated using dual variables, resulting in smooth constraints. We show that our approach has lower computational time than the state-of-the-art and guarantees feasibility of the trajectory with respect to both the system dynamics and input constraints without the need to tune lots of parameters. We validate our approach on a variety of tasks in both simulations and experiments, including navigation through waypoints and obstacle avoidance.
... Raffo and de Almeida (2016) employed a  ∞ based controller to transport the load along a predefined trajectory. Sreenath, Lee, and Kumar (2013) proposed a nonlinear geometric control strategy to actively control the load's position. Qian and Liu (2020) developed a translational control law based on uncertainty and disturbance estimators to asymptotically stabilize a QSL along a given path under wind disturbances. ...
... To demonstrate its effectiveness and performance, the proposed controller is implemented and embedded on a newly constructed testbed, and a series of experimental tests are performed. The QSL testbed employs a gyroscope to replace the optical sensors, as seen in Foehn et al. (2017), Palunko et al. (2012) and Sreenath et al. (2013), to measure the cable's swing angle, such that the issue of visual-blocking in optical sensors can be overcome. To the best of our knowledge, this is the first finite-time control strategy reported in the field of QSL, and the proposed controller is specially adjusted to the newly constructed experimental QSL. ...
... Remark 2. The virtual input in Eq. (2c) is = 2̈. It implies that the dynamics of (2c) contains no explicit physical input, which is different from other works (Qian & Liu, 2020;Raffo & de Almeida, 2016;Sreenath et al., 2013). Then, the model (2) can be rewritten as (4) to facilitate the cascade control design in the following section. ...
Article
In this paper, we propose a nonlinear finite-time control strategy to solve the motion control problem for a quadrotor with a slung load (QSL). This work aims to realize high-performance motion control for the QSL, even in perturbations. To improve the dynamic performance and the robustness of the QSL system, a novel nonlinear controller is designed with cascade structure. The finite-time stability of the resulting closed-loop system is theoretically analyzed in this work. Furthermore, the advantages of the proposed control strategy are demonstrated by comparison results with different strategies through simulations in MATLAB/SimMechanics. Finally, the effectiveness of the proposed controller is verified in actual experiments on a specially designed experimental QSL.
... where f is the thrust magnitude, M is the moment vector; W 1 = R T δR and W 2 = δx Q = δx L − lδq are the virtual work of the system. To sum up, the dynamics equation for the quadrotor with a suspended payload [12] is given as ...
... The cable-suspended load is a hybrid system in which two states are determined by the tension in the cable, being either 9635 taut or slack [17]. A payload controller [12] that can stabilize the payload oscillation is utilized in the paper. ...
... where orientation error functions on S 2 are defined as e q = q 2 q d , eq =q − (q d ×q d ) × q. Then replace the desired quadrotor and load attitude by their computed values, R c , Ω c and q c respectively [12]. Define quadrotor thrust f as: ...
... In the aerial transportation task with multiple quadrotor UAVs considered in this work, a cable establishes a physical connection between the UAV and the cargo. Geometric control of multiple quadrotors with a suspended point-mass load with inelastic cables has been studied in [13], and with a rigid body load in [14], [15]. In [16], the authors model the cables as flexible chains comprised of inflexible links with mass. ...
... We begin by further simplifying the dynamical system (10)- (13). In particular, we find an equation forq j that we will substitute into equations (11) and (12). ...
... By differentiatingq j = ω j ×q j and expanding it with the vector triple product identity, it can be shown thatq j =ω j × q j − ||ω j || 2 q j . Now we may substitute (13) in forω j to find that ...
Preprint
Groups of unmanned aerial vehicles (UAVs) are increasingly utilized in transportation task as the combined strength allows to increase the maximum payload. However, the resulting mechanical coupling of the UAVs impose new challenges in terms of the tracking control. Thus, we design a geometric trajectory tracking controller for the cooperative task of four quadrotor UAVs carrying and transporting a rigid body, which is attached to the quadrotors via inflexible elastic cables. The elasticity of the cables together with techniques of singular perturbation allows a reduction in the model to that of a similar model with inelastic cables. In this reduced model, we design a controller such that the position and attitude of the load exponentially converges to a given desired trajectory. We then show that this result leads to an uniformly converging tracking error for the original elastic model under some assumptions. Furthermore, under the presence of unstructured disturbances on the system, we show that the error is ultimately bounded with an arbitrarily small bound. Finally, a simulation illustrates the theoretical results.
... A planar aerial manipulator with any number of rigid or elastic joints was proven to be flat in [11], and such result was generalized to any number of protocentric manipulators in [12]; however, the overall CoM of the system must be fixed, else there are unaccounted Coriolis terms. In [13], valid flat outputs for rotorcraft with cable-suspended loads were given. But this result does not generalize to aerial manipulators, as passive cable dynamics cannot model active dynamic coupling in manipulation. ...
... Inverting the yaw rotation by ψ, the directional vector parallel to gravity in body frame gives us the roll and pitch angle as shown in (22) and (23). A key difference arises in our use of the reconstruction equation (13), which models the coupling between the arm dynamics and multi-rotor base dynamics in symbolically compact way. ...
Preprint
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This paper shows that the dynamics of a general class of aerial manipulators, consist of an underactuated multi-rotor base with an arbitrary k-linked articulated manipulator, are differentially flat. Methods of Lagrangian Reduction under broken symmetries produce reduced equations of motion whose key variables: center-of-mass linear momentum, vehicle yaw angle, and manipulator relative joint angles become the flat outputs. Utilizing flatness theory and a second-order dynamic extension of the thrust input, we transform the mechanics of aerial manipulators to their equivalent trivial form with a valid relative degree. Using this flatness transformation, a quadratic programming-based controller is proposed within a Control Lyapunov Function (CLF-QP) framework, and its performance is verified in simulation.
... The orientation of the MAV R WB also affects Eq. 2d. Considering the differential flatness of the MAV system Sreenath et al. (2013), the controller uses a constant desired yaw angle ψ des = 0 while the desired roll ϕ des and pitch θ des angles will be calculated based on the desired and actual states of the robot. ...
Article
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This work focuses on catching safely an aerial micro-robot in mid-air using another aerial robot that is equipped with a universal soft gripper. To avoid aerodynamic disturbances such as downwash, that would push the target robot away, we follow a horizontal grasping approach. To this end, the article introduces a gripper design based on soft actuators that can stay horizontally straight with a single fixture and maintain sufficiently compliance in order to bend when air pressure is applied. Further, we develop the Soft Aerial Gripper (SoAG), an open-source aerial robot equipped with the developed soft end-effector and that features an onboard pneumatic regulation system. Experimental results show that the developed low-cost soft gripper has fast opening and closing responses despite being powered by lightweight air pumps, responses that are comparable to those of a commercially available end-effector tested we test against. Static grasping tests study the soft gripper’s robustness in capturing aerial micro-robots under aerodynamic disturbances. We experimentally demonstrated the feasibility of using the SoAG robot to catch a hovering micro-robot with or without propeller guards. The feasibility of dynamic catching is also shown by capturing a moving aerial micro-robot with a velocity of 0.2 m/s. The free flight performance of the SoAG robot is studied against a conventional quadrotor and in different gripper and payload status.
Article
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... When the bundle is nontrivial, global sections do not exist, providing an upper bound on the domain of geometric flat outputs. However, as Example 3 indicates, the approach provides a principled means of generating a global atlas of overlapping local flat outputs generated from local sections, yielding a differentially flat hybrid system [15] and enabling planning and control over the entire configuration manifold [14]. ...
Preprint
Mechanical systems naturally evolve on principal bundles describing their inherent symmetries. The ensuing factorization of the configuration manifold into a symmetry group and an internal shape space has provided deep insights into the locomotion of many robotic and biological systems. On the other hand, the property of differential flatness has enabled efficient, effective planning and control algorithms for various robotic systems. Yet, a practical means of finding a flat output for an arbitrary robotic system remains an open question. In this work, we demonstrate surprising new connections between these two domains, for the first time employing symmetry directly to construct a flat output. We provide sufficient conditions for the existence of a trivialization of the bundle in which the group variables themselves are a flat output. We call this a geometric flat output, since it is equivariant (i.e. maintains the symmetry) and is often global or almost-global, properties not typically enjoyed by other flat outputs. In such a trivialization, the motion planning problem is easily solved, since a given trajectory for the group variables will fully determine the trajectory for the shape variables that exactly achieves this motion. We provide a partial catalog of robotic systems with geometric flat outputs and worked examples for the planar rocket, planar aerial manipulator, and quadrotor.
... In the relevant USV path tracking literature [5], the kinematic model of USV is added to path generation to make the navigation more suitable for USV tracking control, but the computational burden and algorithm stability become tough problems. Referring to [19], [20], [21], this paper proposes the differential flatness property of the USV motion system to simplify the kinematic model as an algebraic expression of the position and its higher order derivatives. It effectively connects the navigation motion planning with the USV tracking control problem, so that the USV can more stably follow the trajectory and improve the stability and successful rate. ...
Preprint
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Unmanned surface vessels (USVs) are widely used in ocean exploration and environmental protection fields. To ensure that USV can successfully perform its mission, trajectory planning and motion tracking are the two most critical technologies. In this paper, we propose a novel trajectory generation and tracking method for USV based on optimization theory. Specifically, the USV dynamic model is described with differential flatness, so that the trajectory can be generated by dynamic RRT* in a linear invariant system expression form under the objective of optimal boundary value. To reduce the sample number and improve efficiency, we adjust the trajectory through local optimization. The dynamic constraints are considered in the optimization process so that the generated trajectory conforms to the kinematic characteristics of the under-actuated hull, and makes it easier to be tracked. Finally, motion tracking is added with model predictive control under a sequential quadratic programming problem. Experimental results show the planned trajectory is more in line with the kinematic characteristics of USV, and the tracking accuracy remains a higher level.
... Aerial payload transportation and manipulation using a single quadcopter was studied [20,[22][23][24][25]. Stabilization of a single quadcopter carrying a single payload was also analyzed [26,27], Researcher have proposed H ∞ control [24], and PID control [28] for a quadcopter carrying a suspended payload. A quadcopter carrying payload with varying length cable was studied [25]. ...
Preprint
This paper studies the problem of fast and safe aerial payload transport by a single quadcopter in urban areas. The quadcopter payload system (QPS) is considered as a rigid body and modeled with a nonlinear dynamics. The urban area is modeled as an obstacle-laden environment with obstacle geometries obtained by incorporating realistic LIDAR data. Our approach for payload transport is decomposed into high-level motion planning and low-level trajectory control. For the low-level trajectory tracking, a feedback linearization control is applied to stably track the desired trajectory of the quadcopter. For high-level motion planning, we integrate A* search and polynomial planning to define a safe trajectory for the quadcopter assuring collision avoidance, boundedness of the quadcopter rotor speeds and tracking error, and fast arrival to a target destination from an arbitrary initial location.
... The structure of (2c) deduced by Lagrangi an approach differs from other works, like [15], [30], and contains no explicit physical input, the virtual input is u σ = M 2ξ . ...
Article
This paper is concerned with the motion control for a quadrotor with a cable-suspended load (QCSL). A fixed-time control strategy is presented to improve the transient response and robustness of the QCSL with external disturbance. The overall control scheme is designed with a cascade structure to better cope with the underactuated property of the QCSL and the indirect effect of the control force on the load’s velocity through the tensile force on the cable. The simulation results are given to demonstrate the performance of the proposed scheme. Furthermore, actual flight tests were performed on a new experimental QCSL to validate the effectiveness of the proposed control strategy.
... Regulation of dynamical systems on state manifolds appears to be an appealing research field in non-linear control theory which is gaining increasing interest in the scientific community, especially in the field of mechanical systems control [1][2][3][4][5][6]. Depending on the application at hand, one may be facing two cases [7]: positional control, arising in the regulation of first-order as well as second-order systems, and velocity control, arising in the regulation of second-order systems on manifolds. ...
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The present tutorial paper constitutes the second of a series of tutorials on manifold calculus with applications in system theory and control. The aim of the present tutorial, in particular, is to explain and illustrate some key concepts in manifold calculus such as covariant derivation and manifold curvature. Such key concepts are then applied to the formulation, to the control, and to the analysis of non-linear dynamical systems whose state-space are smooth (Riemannian) manifolds. The main flow of exposition is enriched by a number of examples whose aim is to clarify the notation used and the main theoretical findings through practical calculations.
... Geometric control techniques that incorporate differential geometry [1] with control theory have been applied to many robotics systems, e.g., legged robots [2]- [5] and unmanned aerial vehicles (UAV) [6], [7]. For systems on Lie groups, geometric thinking enables a better choice of coordinates. ...
Preprint
This paper presents a control framework on Lie groups by designing the control objective in its Lie algebra. Control on Lie groups is challenging due to its nonlinear nature and difficulties in system parameterization. Existing methods to design the control objective on a Lie group and then derive the gradient for controller design are non-trivial and can result in slow convergence in tracking control. We show that with a proper left-invariant metric, setting the gradient of the cost function as the tracking error in the Lie algebra leads to a quadratic Lyapunov function that enables globally exponential convergence. In the PD control case, we show that our controller can maintain an exponential convergence rate even when the initial error is approaching $\pi$ in SO(3). We also show the merit of this proposed framework in trajectory optimization. The proposed cost function enables the iterative Linear Quadratic Regulator (iLQR) to converge much faster than the Differential Dynamic Programming (DDP) with a well-adopted cost function when the initial trajectory is poorly initialized on SO(3).
... Interaction control Impedance control [92,117,118,135,138], Admittance control [11,120,135,138], Force control [17,110], Contact inspection [6] Load transport Slung load transport [19,129], Aerial manipulators [4,43] Reconfiguration Whole-body manipulation [156], Full controllability [24,121,120] Groups of robots Manipulation Transport [91,93], Grasping [45], Aerial Manipulators [13,95] Assembly / Construction ...
Thesis
Model-Based Control of Flying Robots for Robust Interaction under Wind Influence The main goal of this thesis is to bridge the gap between trajectory tracking and interaction control for flying robots in order to allow physical interaction under wind influence by making aerial robots aware of the disturbance, interaction, and faults acting on them. This is accomplished by reasoning about the external wrench (force and torque) acting on the robot, and discriminating (distinguishing) between wind, interactions, and collisions. This poses the following research questions. First, is discrimination between the external wrench components even possible in a continuous real-time fashion for control purposes? Second, given the individual wrench components, what are effective control schemes for interaction and trajectory tracking control under wind influence? Third, how can unexpected faults, such as collisions with the environment, be detected and handled efficiently and effectively? In the interest of the first question, a fourth can be posed: is it possible to obtain a measurement of the wind speed that is independent of the external wrench? In this thesis, model-based methods are applied in the pursuit of answers to these questions. This requires a good dynamics model of the robot, as well as accurately identified parameters. Therefore, a systematic parameter identification procedure for aerial robots is developed and applied. Furthermore, external wrench estimation techniques from the field of robot manipulators are extended to be suitable for aerial robots without the need of velocity measurements, which are difficult to obtain in this context. Based on the external wrench estimate, interaction control techniques (impedance and admittance control) are extended and applied to flying robots, and a thorough stability proof is provided. Similarly, the wrench estimate is applied in a geometric trajectory tracking controller to compensate external disturbances, to provide zero steady-state error under wind influence without the need of integral control action. The controllers are finally combined into a novel compensated impedance controller, to facilitate the main goal of the thesis. Collision detection is applied to flying robots, providing a low level reflex reaction that increases safety of these autonomous robots. In order to identify aerodynamic models for wind speed estimation, flight experiments in a three-dimensional wind tunnel were performed using a custom-built hexacopter. This data is used to investigate wind speed estimation using different data-driven aerodynamic models. It is shown that good performance can be obtained using relatively simple linear regression models. In this context, the propeller aerodynamic power model is used to obtain information about wind speed from available motor power measurements. Leveraging the wind tunnel data, it is shown that power can be used to obtain the wind speed. Furthermore, a novel optimization-based method that leverages the propeller aerodynamics model is developed to estimate the wind speed. Essentially, these two methods use the propellers as wind speed sensors, thereby providing an additional measurement independent of the external force. Finally, the novel topic of simultaneously discriminating between aerodynamic, interaction, and fault wrenches is opened up. This enables the implementation of novel types of controllers that are e.g. compliant to physical interaction, while compensating wind disturbances at the same time. The previously unexplored force discrimination topic has the potential to even open a new research avenue for flying robots.
... Such novel regulation scheme will be referred to as M-PID regulation. Control of systems on state manifolds is a relatively new research branch in non-linear control theory which is gaining increasing interest in applied sciences and in engineering, especially in the field of mechanical systems regulation [2,3,13,22,30]. ...
Article
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The present document outlines a non-linear control theory, based on the PID regulation scheme, to synchronize two second-order dynamical systems insisting on a Riemannian manifold. The devised extended PID scheme, referred to as M-PID, includes an unconventional component, termed 'canceling component', whose purpose is to cancel the natural dynamics of a system and to replace it with a desired dynamics. In addition, this document presents numerical recipes to implement such systems, as well as the devised control scheme, on a computing platform and a large number of numerical simulation results focused on the synchronization of Duffing-like non-linear oscillators on the unit sphere. Detailed numerical evaluations show that the canceling contribution of the M-PID control scheme is not critical to the synchronization of two oscillators, however, it possesses the beneficial effect of speeding up their synchronization. Simulation results obtained in non-ideal conditions, namely in the presence of additive disturbances and delays, reveal that the devised synchronization scheme is robust against high-frequency additive disturbances as well as against observation delays.
... The dynamics of the quadrotor-slung-load system have been proven to be differentially flat (Sreenath et al. 2013), with the flat output given by the load position p L and the quadrotor yaw angle. This property is exploited to generate smooth polynomial trajectories directly in the flat output space so as to leverage the swing dynamics of the load. ...
Article
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... Because it can perform rapid vertical takeoff and landing and free hovering in the air and other maneuvers that cannot be performed by ordinary aircraft, it has an increasingly wide range of applications. Initially, it was used in the military field, such as controlling aircraft to perform various tasks in dangerous environments, ensuring the safety of human life and property and even participating in rescue operations [1][2][3]. The use of quadrotors for transporting supplies in rescue activities is one of the important directions for quadrotor application research. ...
Article
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This paper assumes that considering the unknown and time-varying nature of different strong and weak wind field disturbances and considering the nonlinear, under-driven, strongly coupled quadrotor carrying, a load is disturbed by the complex and variable wind field and unmodeled parts when flying in the real external environment, which will reduce the control effect of the nonlinear controller and make the vehicle fail to affect the flight effect. In order to ensure that the quadrotor carrying a load can carry supplies in the harsh environment for stable trajectory tracking, a neural network adaptive control algorithm is introduced in the article. The neural network algorithm has the role of online dynamic approximation, the compensation of arbitrary external disturbance and the compensation of external disturbance. Its structure is simple and low computation. In the article, the Lyapunov method is used to design the adaptive weight and estimate the weight of the online neural network, and the stability of the system is proved. Finally, the comparison of three algorithms verified by simulation proves that the above interference problem can be solved effectively by the proposed algorithm.
... O1 There is only one pose (load and quadrotor position and cable and quadrotor attitude) compatible with driving the load position and yaw tracking errors to zero. This is a direct consequence of the so-called differentially-flat property of the slungload dynamic system [33]. In a differentiallyflat system, the states and inputs can be directly expressed in terms of the flat outputs (typically the load position and quadrotor yaw angle) and a finite number of their derivatives. ...
Article
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... It is possible to examine the literature on manipulating slung-load unmanned aerial vehicles, mainly in two categories. Studies in the first category are concerned with designing controllers for tracking desired payload trajectories [5], [7], [8], [9], and [17]. On the other hand, studies in the second category are trajectory planning studies to optimize a particular objective function [3], [6], [15], [18], and [19]. ...
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Chapter
This paper presents the design and dynamic modelling of a soft pneumatic actuator that can be used to mimic snake or worm-like locomotion. The bond graph technique is used to derive the dynamics of the actuator. To validate the accuracy of the derived dynamic model, we conduct numerical simulations using 20-sim$$^{\text{\textregistered} }$$ software. Experimental results demonstrate that the soft actuator achieves bidirectional bending and linear displacement, which is essential for mimicking snake or worm-like locomotion.KeywordsBond graphControlDynamicsSnake robotsSoft robots
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Sandwich constructions with a honeycomb core have recently become popular for high strength and dynamic load. The purpose of this study is to look at the shock wave resistance performance of two distinct types of honeycomb core sandwich structures in terms of face plate deflection and energy absorption when subjected to a blast load. This study employed square and hexagonal honeycomb core structures to determine the minimal face deflection under blast conditions. The honeycomb sandwich panel is composed of steel that is very ductile. In the sandwich construction, both the front and rear plates are solid, and the core structure is of the shell type. To administer the air-blast loads of 1, 2, and 3 kg TNT, a 10-cm stand-off distance is used from the front face. The dynamic response of the sandwich constructions is determined using the ABAQUS/explicit finite element method (FEM). For both square honeycomb and hexagonal honeycomb core sandwich panels, the front and rear face plate deflections were measured. Under comparable blast loading circumstances, the front and rear plates of the hexagonal sandwich panel showed less deformation than the square honeycomb core sandwich panel.
Preprint
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Glossary Definition of the Subject Introduction Discrete Lagrangian and Hamiltonian Mechanics Optimal Control of Discrete Lagrangian and Hamiltonian Systems Controlled Lagrangian Method for Discrete Lagrangian Systems Future Directions Acknowledgments Bibliography
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In this paper we present an overview of techniques and approaches used for a load transportation system based on small size unmanned helicopters. The focus is on the control approach and on the movement of the rope connecting helicopters and load. The proposed approach is based on two control loops: an outer loop to control the translation of each helicopter in compound and an inner loop to control the orientation of helicopters. The challenge here is that in both loops the dynamics of the whole system - all helicopters and load - should be accounted for. It is shown, that for designing the outer loop controller a complex model of the helicopters and load can be replaced by a simplified model based on interconnected mass points. For designing the inner loop controller, the complete dynamics of the whole system are considered. The usage of force sensors in the ropes is proposed in order to simplify the inner loop controller and to make it robust against variations of system parameters. The presented inner loop controller is independent of the number of coupled helicopters. The outer loop controller depends on the number of helicopters. The problem of oscillations in the flexible ropes due to external disturbancies (e.g. wind gusts) is discussed and a solution based on load state observer is presented. The performance of the presented system was verified in simulations and in real flight experiments with one and three helicopters transporting the load. The worldwide first demonstration of a slung load transportation using three helicopters was performed in December 2007.
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We address the controller design and the trajectory generation for a quadrotor maneuvering in three dimensions in a tightly constrained setting typical of indoor environments. In such settings, it is necessary to allow for significant excursions of the attitude from the hover state and small angle approximations cannot be justified for the roll and pitch. We develop an algorithm that enables the real-time generation of optimal trajectories through a sequence of 3-D positions and yaw angles, while ensuring safe passage through specified corridors and satisfying constraints on velocities, accelerations and inputs. A nonlinear controller ensures the faithful tracking of these trajectories. Experimental results illustrate the application of the method to fast motion (5-10 body lengths/second) in three-dimensional slalom courses.
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This paper discusses a feedback control method for incompletely restrained wire-suspended mechanisms. The incompletely restrained wire-suspended mechanism has a merit that it enables three dimensional positioning and orientating of suspended object with simple mechanism using a small number of wires. However it also has a drawback that the manipulated object is easy to swing as seen in overhead crane which is one of the simplest incompletely restrained mechanism. For this reason, anti-sway control method for incompletely restrained type mechanisms is needed. A key point of our method is exact linearization using inverse dynamics of the system. Feedback control method with exact linearization and measuring method for the suspended object are presented, and effectiveness of proposed methods are shown by an experiment.
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Considers a nonlinear dynamical model for a gantry crane to design a control input that transports a load along a specified transport trajectory in such a way that the load angle oscillation is suppressed as quickly as possible. A nonlinear tracking controller for the load position and velocity is designed that gives guaranteed performance. The controller has two loops: an outer tracking loop, and an inner loop that stabilizes the internal oscillatory dynamics using a singular perturbation design. The model and control scheme are simulated on a digital computer and the results prove that the feedback control works well for fast load positioning without sway or oscillation
Geometric Tracking Control of a
• T Lee
• M Leok
• N H Mcclamroch
T. Lee, M. Leok, and N. H. McClamroch, " Geometric Tracking Control of a Quadrotor UAV on SE ( 3 ), " in IEEE Conference on Decision and Control, Atlanta, GA, 2010, pp. 5420–5425.