Minimum snap trajectory generation and control for quadrotors
09 May 2011-pp 2520-2525
TL;DR: An algorithm is developed 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.
Abstract: 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.
Citations
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TL;DR: In this article, a tutorial for modeling, estimation, and control for multi-rotor aerial vehicles that includes the common four-rotors or quadrotors case is presented.
Abstract: This article provides a tutorial introduction to modeling, estimation, and control for multirotor aerial vehicles that includes the common four-rotor or quadrotor case.
1,241 citations
TL;DR: It is shown that this approach permits the development of trajectories and controllers enabling such aggressive maneuvers as flying through narrow, vertical gaps and perching on inverted surfaces with high precision and repeatability.
Abstract: We study the problem of designing dynamically feasible trajectories and controllers that drive a quadrotor to a desired state in state space. We focus on the development of a family of trajectories defined as a sequence of segments, each with a controller parameterized by a goal state or region in state space. Each controller is developed from the dynamic model of the robot and then iteratively refined through successive experimental trials in an automated fashion to account for errors in the dynamic model and noise in the actuators and sensors. We show that this approach permits the development of trajectories and controllers enabling such aggressive maneuvers as flying through narrow, vertical gaps and perching on inverted surfaces with high precision and repeatability.
838 citations
Cites background from "Minimum snap trajectory generation ..."
...This is explored in Mellinger and Kumar (2011)....
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...This assumption is relaxed in Mellinger and Kumar (2011)....
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TL;DR: The main objective of this paper is to present a comprehensive survey of RUAS research that captures all seminal works and milestones in each GNC area, with a particular focus on practical methods and technologies that have been demonstrated in flight tests.
Abstract: Recently, there has been growing interest in developing unmanned aircraft systems (UAS) with advanced onboard autonomous capabilities. This paper describes the current state of the art in autonomous rotorcraft UAS (RUAS) and provides a detailed literature review of the last two decades of active research on RUAS. Three functional technology areas are identified as the core components of an autonomous RUAS. Guidance, navigation, and control (GNC) have received much attention from the research community, and have dominated the UAS literature from the nineties until now. This paper first presents the main research groups involved in the development of GNC systems for RUAS. Then it describes the development of a framework that provides standard definitions and metrics characterizing and measuring the autonomy level of a RUAS using GNC aspects. This framework is intended to facilitate the understanding and the organization of this survey paper, but it can also serve as a common reference for the UAS community. The main objective of this paper is to present a comprehensive survey of RUAS research that captures all seminal works and milestones in each GNC area, with a particular focus on practical methods and technologies that have been demonstrated in flight tests. These algorithms and systems have been classified into different categories and classes based on the autonomy level they provide and the algorithmic approach used. Finally, the paper discusses the RUAS literature in general and highlights challenges that need to be addressed in developing autonomous systems for unmanned rotorcraft. © 2012 Wiley Periodicals, Inc. © 2012 Wiley Periodicals, Inc.
605 citations
01 Apr 2016
TL;DR: This work presents a method of jointly optimizing polynomial path segments in an unconstrained quadratic program that is numerically stable for high-order polynomials and large numbers of segments, and is easily formulated for efficient sparse computation.
Abstract: We explore the challenges of planning trajectories for quadrotors through cluttered indoor environments. We extend the existing work on polynomial trajectory generation by presenting a method of jointly optimizing polynomial path segments in an unconstrained quadratic program that is numerically stable for high-order polynomials and large numbers of segments, and is easily formulated for efficient sparse computation. We also present a technique for automatically selecting the amount of time allocated to each segment, and hence the quadrotor speeds along the path, as a function of a single parameter determining aggressiveness, subject to actuator constraints. The use of polynomial trajectories, coupled with the differentially flat representation of the quadrotor, eliminates the need for computationally intensive sampling and simulation in the high dimensional state space of the vehicle during motion planning. Our approach generates high-quality trajecrtories much faster than purely sampling-based optimal kinodynamic planning methods, but sacrifices the guarantee of asymptotic convergence to the global optimum that those methods provide. We demonstrate the performance of our algorithm by efficiently generating trajectories through challenging indoor spaces and successfully traversing them at speeds up to 8 m/s. A demonstration of our algorithm and flight performance is available at: http://groups.csail.mit.edu/rrg/quad_polynomial_trajectory_planning.
578 citations
Cites background or methods or result from "Minimum snap trajectory generation ..."
...Several cost functions may be suitable candidates: [5] minimizes total time subject to constraints, while [19] fixes the total time by hand and minimizes snap (the original cost function) with the remaining degrees of freedom....
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...Differential flatness of this model was demonstrated in [19]....
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...Minimum-snap polynomial splines have proven very effective as quadrotor trajectories, since the motor commands and attitude accelerations of the vehicle are proportional to the snap, or fourth derivative, of the path [19]....
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...While the method above works well for single segments and small joint optimization problems as in [19], this formulation becomes ill-conditioned for more than several segments, polynomials of high order, and when widely varying segment times are involved....
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...[19], and show that their minimum-snap trajectory generation can be solved in a numerically stable unconstrained quadratic program (QP) for long-range trajectories composed of many segments....
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TL;DR: It is argued that the reduction in size leads to agility and the ability to operate in tight formations and experimental arguments in support of this claim are provided.
Abstract: We describe a prototype 75 g micro quadrotor with onboard attitude estimation and control that operates autonomously with an external localization system The motivation for designing quadrotors at this scale comes from two observations First, the agility of the robot increases with a reduction in size, a fact that is supported by experimental results in this paper Second, smaller robots are able to operate in tight formations in constrained, indoor environments We describe the hardware and software used to operate the vehicle as well our dynamic model We also discuss the aerodynamics of vertical flight and the contribution of ground effect to the vehicle performance Finally, we discuss architecture and algorithms to coordinate a team of these quadrotors, and provide experimental results for a team of 20 micro quadrotors
429 citations
References
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TL;DR: A mathematical model is formulated which is shown to predict both the qualitative features and the quantitative details observed experimentally in planar, multijoint arm movements, and is successful only when formulated in terms of the motion of the hand in extracorporal space.
Abstract: This paper presents studies of the coordination of voluntary human arm movements. A mathematical model is formulated which is shown to predict both the qualitative features and the quantitative details observed experimentally in planar, multijoint arm movements. Coordination is modeled mathematically by defining an objective function, a measure of performance for any possible movement. The unique trajectory which yields the best performance is determined using dynamic optimization theory. In the work presented here, the objective function is the square of the magnitude of jerk (rate of change of acceleration) of the hand integrated over the entire movement. This is equivalent to assuming that a major goal of motor coordination is the production of the smoothest possible movement of the hand. Experimental observations of human subjects performing voluntary unconstrained movements in a horizontal plane are presented. They confirm the following predictions of the mathematical model: unconstrained point-to-point motions are approximately straight with bell-shaped tangential velocity profiles; curved motions (through an intermediate point or around an obstacle) have portions of low curvature joined by portions of high curvature; at points of high curvature, the tangential velocity is reduced; the durations of the low-curvature portions are approximately equal. The theoretical analysis is based solely on the kinematics of movement independent of the dynamics of the musculoskeletal system and is successful only when formulated in terms of the motion of the hand in extracorporal space. The implications with respect to movement organization are discussed.
4,226 citations
"Minimum snap trajectory generation ..." refers background in this paper
...The trajectory stays within the dotted lines that illustrate the corridor....
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TL;DR: It is shown that this approach permits the development of trajectories and controllers enabling such aggressive maneuvers as flying through narrow, vertical gaps and perching on inverted surfaces with high precision and repeatability.
Abstract: We study the problem of designing dynamically feasible trajectories and controllers that drive a quadrotor to a desired state in state space. We focus on the development of a family of trajectories defined as a sequence of segments, each with a controller parameterized by a goal state or region in state space. Each controller is developed from the dynamic model of the robot and then iteratively refined through successive experimental trials in an automated fashion to account for errors in the dynamic model and noise in the actuators and sensors. We show that this approach permits the development of trajectories and controllers enabling such aggressive maneuvers as flying through narrow, vertical gaps and perching on inverted surfaces with high precision and repeatability.
838 citations
01 Dec 2010
TL;DR: New results for the tracking control of a quadrotor unmanned aerial vehicle (UAV) are provided and a nonlinear tracking controller is developed on the special Euclidean group SE(3), shown to have desirable closed loop properties that are almost global.
Abstract: This paper provides new results for the tracking control of a quadrotor unmanned aerial vehicle (UAV). The UAV has four input degrees of freedom, namely the magnitudes of the four rotor thrusts, that are used to control the six translational and rotational degrees of freedom, and to achieve asymptotic tracking of four outputs, namely, three position variables for the vehicle center of mass and the direction of one vehicle body-fixed axis. A globally defined model of the quadrotor UAV rigid body dynamics is introduced as a basis for the analysis. A nonlinear tracking controller is developed on the special Euclidean group SE(3) and it is shown to have desirable closed loop properties that are almost global. Several numerical examples, including an example in which the quadrotor recovers from being initially upside down, illustrate the versatility of the controller.
827 citations
"Minimum snap trajectory generation ..." refers background in this paper
...Note that spatial scaling also applies to the problem with multiple keyframes but the property is less useful as the positions of all keyframes must be scaled by the same factor....
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Posted Content•
10 Mar 2010
TL;DR: In this article, a nonlinear tracking controller is developed on the special Euclidean group for each flight mode, and the closed loop is shown to have desirable closed loop properties that are almost global in each case.
Abstract: This paper provides new results for control of complex flight maneuvers for a quadrotor unmanned aerial vehicle (UAV). The flight maneuvers are defined by a concatenation of flight modes or primitives, each of which is achieved by a nonlinear controller that solves an output tracking problem. A mathematical model of the quadrotor UAV rigid body dynamics, defined on the configuration space $\SE$, is introduced as a basis for the analysis. The quadrotor UAV has four input degrees of freedom, namely the magnitudes of the four rotor thrusts; each flight mode is defined by solving an asymptotic optimal tracking problem. Although many flight modes can be studied, we focus on three output tracking problems, namely (1) outputs given by the vehicle attitude, (2) outputs given by the three position variables for the vehicle center of mass, and (3) output given by the three velocity variables for the vehicle center of mass. A nonlinear tracking controller is developed on the special Euclidean group $\SE$ for each flight mode, and the closed loop is shown to have desirable closed loop properties that are almost global in each case. Several numerical examples, including one example in which the quadrotor recovers from being initially upside down and another example that includes switching and transitions between different flight modes, illustrate the versatility and generality of the proposed approach.
814 citations
TL;DR: In the last five years, advances in materials, electronics, sensors, and batteries havefueled a growth in the development of microunmanned aerial vehicles (MAVs) that are between 0.1 and 0.5 m in length and0.1-0.5 kg in mass.
Abstract: In the last five years, advances in materials, electronics, sensors, and batteries have fueled a growth in the development of microunmanned aerial vehicles (MAVs) that are between 0.1 and 0.5 m in length and 0.1-0.5 kg in mass [1]. A few groups have built and analyzed MAVs in the 10-cm range [2], [3]. One of the smallest MAV is the Picoftyer with a 60-mmpropellor diameter and a mass of 3.3 g [4]. Platforms in the 50-cm range are more prevalent with several groups having built and flown systems of this size [5]-[7]. In fact, there are severalcommercially available radiocontrolled (PvC) helicopters and research-grade helicopters in this size range [8].
806 citations
"Minimum snap trajectory generation ..." refers background or methods in this paper
...In between each keyframe there is a safe corridor that the quadrotor must stay within....
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...These cost functionals are meaningful since the input variables are algebraically related to the snap....
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...As α goes to infinity all the derivatives of position and yaw angle as well as the angular velocity go to zero which leads to u(t)→ [mg, 0, 0, 0]T ....
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...This is convenient because it is faster to analytically modify a solution than to solve a QP....
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