Elsevier

Automatica

Volume 61, November 2015, Pages 289-301
Automatica

Mechanics, control and internal dynamics of quadrotor tool operation

https://doi.org/10.1016/j.automatica.2015.08.015Get rights and content

Abstract

We propose a novel control framework to enable a quadrotor to operate a tool attached on it. We first show that any Cartesian control at the tool-tip can be generated if and only if the tool-tip is located strictly above or below the quadrotor’s center-of-mass. We then fully characterize the internal dynamics of the spatial quadrotor tool operation, which arises due to the quadrotor’s under-actuation, and elucidate a seemingly counter-intuitive necessary condition for the internal stability, that is, the tool-tip should be located above the quadrotor’s center-of-mass. We further manifest that this internal dynamics can exhibit finite-time escape and propose a stabilizing action to prevent that. The theory is then illustrated for the problems of rotating tool operation and hybrid force/position control with relevant simulation results.

Introduction

With the recent advances in sensors, actuators, materials, computing and control technologies, quadrotor-type unmanned aerial vehicles (UAVs) have gained ability to realize many useful applications such as aerial photography and movie shooting, landscape survey, surveillance and reconnaissance, search of areas after disaster, to name just few. Due to this ability and promise, the quadrotor has been the focus of attention from research community and general public alike.

Numerous successful control techniques have been proposed for the motion control of the quadrotor’s center-of-mass position: trajectory tracking control (e.g., Aguiar & Hespanha, 2007, Frazzoli, Dahleh, & Feron, 2000, Ha, Zuo, Choi, & Lee, 2014 and Hua, Hamel, Morin, & Samson, 2009), formation control (e.g., Abdessameud & Tayebi, 2010, Lee, 2012 and Turpin, Michael, & Kumar, 2012), teleoperation (e.g., Franchi, Secchi, Son, Bulthoff, & Robuffo, 2012, Ha et al., 2014 and Lee et al., 2013), and even acrobatic flying (e.g., Mellinger, Michael, & Kumar, 2012 and Purwin & D’Andrea, 2009).

Different from these motion control results, the main concern of the current paper is aerial manipulation, an ability crucial to make the quadrotor truly a versatile aerial robotic platform. We particularly focus on the case where the quadrotor operates a simple rigid tool attached on it (e.g., screw-driver, contact probe, etc.) acting itself as the actuator for the tool. See Fig. 1. We envision this quadrotor-tool system to be useful for such applications as inspection and repair of remote infrastructure, operation on high-rise building exteriors, etc. We also believe this usage of simple tool would be more promising at least for some application than using a multi-link actuated robotic arm (with motors) attached to the quadrotor (e.g., Ghadiok, Goldin, & Ren, 2012, Korpela, Orsag, Oh, & Pekala, 2013 and Yang & Lee, 2014), given the limited payload of typical quadrotor platforms.

The problem of quadrotor tool operation control, however, turns out to be much more challenging than that of the motion control of the quadrotor, as a proper tool control (e.g., tool-tip position y3 of Fig. 1) would require simultaneous control of both the translation and rotation of the quadrotor, which, yet, is under-actuated only with four control inputs. More specifically, in this paper, we first show that we can generate any Cartesian control action at the tool-tip y, if and only if this tool-tip y is located either strictly above or strictly below the quadrotor’s center-of-mass x3 (i.e., d30—see Fig. 1). We then reveal that, due to the quadrotor’s under-actuation, this Cartesian control generation gives a rise to four-dimensional internal dynamics (Sepulchre et al., 1997, Slotine and Li, 1991) and elucidate a seemingly counter-intuitive necessary condition for its stability, that is, the tool-tip y should be installed above the quadrotor’s center-of-mass x (i.e., d3<0—see Fig. 1), and not below it as would likely be attempted and designed in practice. We further manifest that this internal dynamics can even exhibit finite-time escape (Sepulchre et al., 1997) due to some quadratic terms therein and propose a stabilizing control action to prevent finite-time escape thereof. Finally, relying on this theoretical finding, we propose novel control laws for the two perhaps practically most important tool operation tasks: rotating tool operation (i.e., screw-driver and vertical jack) and hybrid position/force control against a working surface.

In contrast to the plethora of quadrotor motion control results as mentioned above, that for the quadrotor aerial manipulation are much fewer. Among them, we believe the followings are the most related ones to our result in this paper.

The issue of stability with perturbed/changing center-of-mass (e.g., with payload) was considered in Palunko and Fierro (2011) and Pounds, Bersak, and Dollar (2012). These results, however, were limited only to the stabilization problem (e.g., stability of the linearized planar dynamics under the PID-stabilization Pounds et al., 2012; different output than ours, which produces different and always stable internal dynamics Palunko & Fierro, 2011), and not applicable to our problem of spatial quadrotor-tool operation, where we not only need to achieve the quadrotor stability but also to drive the tool in the three-dimensional space by recruiting the quadrotor’s translation and rotation motions.

In Brescianini, Hehn, and D’Andrea (2013) and Hehn and D’Andrea (2011), the authors considered the problem of a quadrotor manipulating a pole and two quadrotors achieving airborne-transfer of a pole between them. For this, they assumed that the pole motion is coupled only to the quadrotor center-of-mass position x, not to its rotation RSO(3), and also the control inputs are the quadrotor’s thrust force λ and its angular rate w3. This in fact is in a stark contrast to our result here, where the quadrotor’s rotation R must be recruited on top of its translation x to properly operate the tool, and the internal dynamics, which may exhibit even finite-time escape, arises with the angular torque input τ3.

Hybrid position/force control problem was studied for the quadrotor (Bellens et al., 2012, Darivianakis et al., 2014) and for the ducted-fan UAV (Marconi & Naldi, 2012), where, however, the quadrotor system was modeled only as a quasi-static “black box” wrench generator (Bellens et al., 2012) or simplified as a combination of the decoupled linearized north-pitch, east-roll, and down dynamics (Darivianakis et al., 2014); or the dynamics of the ducted-fan UAV was again restricted to the saggital plane and further linearized (Marconi & Naldi, 2012). Therefore, the complex (and potentially unstable) nonlinear and spatial dynamics of the quadrotor-tool operation (e.g., quadratic terms possibly triggering finite-time escape, and control action to prevent that by using pitch–roll–yaw coupling) could be neither revealed nor addressed in those works.

Another line of relevant research is Hauser et al., 1992, Martin et al., 1996, where the authors obtained similar internal dynamics and condition for its stability as ours for PVTOL (planar vertical take-off and landing) systems. These results (Hauser et al., 1992, Martin et al., 1996), however, (1) were limited only to the planar dynamics, whose extension to the spatial operation not only requires substantial development as done in this paper (from the planar result of Lee & Ha, 2012) but also turns out necessary for us to design the control action to prevent finite-time escape; (2) overlooked the issue of finite-time escape, which, different from the pure motion control of PVTOL (Hauser et al., 1992, Martin et al., 1996), is crucial for the aerial tool operation, where the quadrotor can experience sudden (escape-triggering) surges of its velocity after impacts with environments/objects (see Fig. 13); and (3) considered a different problem (i.e., pure motion control of PVTOL) with a focus on a different mechanism of non-minimum phase dynamics (i.e., “small force” due to slant wing-tip jets) and only the position tracking control by using a specific flat output (i.e., Huygens center of oscillation). Due to this reason, our results in this paper may be thought of as an extension (or re-finding) of Hauser et al. (1992) and Martin et al. (1996) to the quadrotor aerial tool operation with a complete analysis of its spatial/nonlinear (internal) dynamics and a novel control action to subdue its finite-time escape.

Portions of this paper were presented in Lee and Ha (2012) and Nguyen and Lee (2013). The current manuscript integrates, refines and completes these results (Lee and Ha, 2012, Nguyen and Lee, 2013) under the unifying theme of quadrotor tool operation, with new derivation of the internal dynamics, new simulation results, and significantly improved readability. The current manuscript also features a newly-designed finite-time escape prevention control of Nguyen and Lee (2013) and a complete proof of its effectiveness, presented here for the first time.

The rest of the paper is organized as follows. The dynamics of quadrotor-tool system is derived and the condition to generate Cartesian control at its tool-tip is established in Section  2. The internal dynamics of spatial quadrotor tool operation is fully characterized in Section  3, with the seemingly counter-intuitive necessary condition for its stability revealed (i.e., tool above the quadrotor) and the finite-time escape prevention action designed/analyzed therein as well. The obtained theoretical results are then applied in Section  4 to the problems of rotating tool operation (i.e., screw-driver, vertical-jack) and hybrid position/force control against a working surface. Section  5 summarizes the paper with comments on future research.

Section snippets

Dynamics modeling of quadrotor-tool system

Consider the quadrotor with a rigid tool as shown in Fig. 1. Similar to Brescianini et al. (2013) and Hehn and D’Andrea (2011), we assume that the tool is light enough (or counter-balanced with symmetric design) so that the center-of-mass of the total quadrotor-tool system is still close to that of the quadrotor. This assumption is adopted here only for simplicity: the results of this paper can be similarly derived when these two centers-of-mass are not coincident with each other.

With this

Internal dynamics

Although Eq. (6) contains three rows, all of them may not constitute the internal dynamics, since: (1) the arbitrarily-assignable thrust input λ may effectively eliminate one-dimension of the internal dynamics (6); and (2) rank[S(d)]=2, thus, only two-dimensional vector space among all possible ẇd3 are relevant to the internal dynamics. To overcome this geometric complication, instead of ω, we utilize the transformed angular velocity ν[ν1,ν2,ν3]T3 as defined by ω=1d̄[d30d10d̄0d10d3]νΣν

Illustrative examples

In this Section  4, we apply our theoretical results to the two perhaps most practically-important aerial tool operation tasks, namely, (1) rotating tool operation, where the quadrotor itself serves as the actuator for some rotating tool (e.g., screw-driver, vertical-jack); and (2) hybrid force/position control, where the quadrotor controls its tool-tip y to follow some desired trajectory on a surface while exerting certain desired force normal to it.

Summary and future work

In this paper, we investigate issues salient to the dynamics and control of the quadrotor, when it is operating a simple rigid tool attached on it. Some structural conditions are elucidated to generate arbitrary tool-tip Cartesian control action (Theorem 1) and also to avoid instability of the internal dynamics (Theorem 2), which inevitably arises due to the quadrotor’s under-actuation. This necessary condition for the internal stability (Theorem 2) is particularly interesting, since it

Hai-Nguyen Nguyen received the B.S. degree in Mechatronics and the M.S. degree in Applied Mechanics from the Hanoi University of Science & Technology, Vietnam, 2008 and 2010. From 2009 to 2012, he was a permanent researcher with the Institute of Mechanics, Vietnam Academy of Science & Technology. He is currently working toward the Ph.D. degree in Mechanical Engineering at Seoul National University. He research interests include dynamics and control problems related to aerial manipulation.

References (34)

  • Frazzoli, E., Dahleh, M. A., & Feron, E. (2000). Trajectory tracking control design for autonomous helicopters using a...
  • W.M. Getz et al.

    Sufficiency conditions for finite escape times in systems of quadratic differential equations

    IMA Journal of Applied Mathematics

    (1977)
  • V. Ghadiok et al.

    On the design and development of attitude stabilization, vision-based navigation, and aerial gripping for a low-cost quadrotor

    Autonomous Robots

    (2012)
  • Hehn, M., & D’Andrea, R. (2011). A flying inverted pendulum. In Proc. IEEE int’l conference on robotics & automation...
  • M.-D. Hua et al.

    A control approach for thrust-propelled underactuated vehicles and its application to vtol drones

    IEEE Transactions on Automatic Control

    (2009)
  • O. Khatib

    A unified approach for motion and force control of robot manipulators: the operational space formulation

    IEEE Journal of Robotics & Automation

    (1987)
  • Korpela, C., Orsag, M., Oh, P., & Pekala, M. (2013). Dynamic stability of a mobile manipulating unmanned aerial...
  • Cited by (0)

    Hai-Nguyen Nguyen received the B.S. degree in Mechatronics and the M.S. degree in Applied Mechanics from the Hanoi University of Science & Technology, Vietnam, 2008 and 2010. From 2009 to 2012, he was a permanent researcher with the Institute of Mechanics, Vietnam Academy of Science & Technology. He is currently working toward the Ph.D. degree in Mechanical Engineering at Seoul National University. He research interests include dynamics and control problems related to aerial manipulation.

    ChangSu Ha received the B.S. degree from the Sungkyunkwan University, Suwon, Korea, 2002, and the M.S. degree from the Seoul National University (SNU), Seoul, Korea, 2013, both in Mechanical Engineering. He is currently working toward his Ph.D. degree in Mechanical Engineering at the Seoul National University. His research interests include teleoperation and control of mobile robots.

    Dongjun Lee is an Associate Professor with the Department of Mechanical & Aerospace Engineering at the Seoul National University. He received the B.S. and M.S. degrees from KAIST, Korea, and the Ph.D. degree in Mechanical engineering from the University of Minnesota, respectively in 1995, 1997 and 2004. He was an Assistant Professor with the Department of Mechanical, Aerospace & Biomedical Engineering at the University of Tennessee, 2006–2011, and a Postdoctoral Researcher with the Coordinated Science Laboratory at the University of Illinois at Urbana–Champaign, 2004–2006. His main research interests are dynamics and control of robotic and mechatronic systems with emphasis on teleoperation/haptics, multirobot systems, aerial robots, and geometric mechanics control theory. Dr. Lee received the US NSF CAREER Award in 2009, the Best Paper Award from the IAS-2012, and the 2002–2003 Doctoral Dissertation Fellowship of the University of Minnesota. He was an Associate Editor of the IEEE Transactions on Robotics.

    Research supported in part by the Basic Science Research Program of the National Research Foundation (NRF) by the Korea Government (MEST) (2012-R1A2A2A0-1015797) and by the Global Frontier R&D Program on “Human-centered Interaction for Coexistence” of the NRF of Korea funded by MEST (NRF-2013M3A6A3079227). The material in this paper was partially presented at the ASME Dynamic Systems & Control Conference, October 17–19, 2012, Fort Lauderdale, FL, USA and at the IEEE/RSJ International Conference on Intelligent Robots and Systems, November 3–8, 2013, Tokyo, Japan. This paper was recommended for publication in revised form by Associate Editor Marco Lovera under the direction of Editor Toshiharu Sugie..

    View full text