A Flying Inverted Pendulum Markus Hehn and Raffaello D’Andrea Abstract— We extend the classic control problem of the inverted pendulum by placing the pendulum on top of a quadrotor aerial vehicle. Both static and dynamic equilib- V ria of the system are investigated to find nominal states of the system at standstill and on circular trajectories. Control laws are designed around these nominal trajectories. A yaw- independent description of quadrotor dynamics is introduced, z using a ‘Virtual Body Frame’. This allows for the time-invariant description of curved trajectories. The balancing performance y of the controller is demonstrated in the ETH Zurich Flying O Machine Arena testbed. Development potential for the future is highlighted, with a focus on applying learning methodology x to increase performance by eliminating systematic errors that were seen in experiments. Fig. 1. The inertial coordinate system O and the vehicle coordinate system V. I. INTRODUCTION The inverted pendulum is a classic control problem, of- in Section VI and conclusions are drawn in Section VII, fering one of the most intuitive, easily describable and re- where an outlook is also presented. alizable nonlinear unstable systems. It has been investigated for several decades (see, for example, [11], and references II. DYNAMICS therein). It is frequently used as a demonstrator to show- We derive the equations of motion of the quadrotor and case theoretical advances, e.g. in reinforcement learning [6], the inverted pendulum for the trajectory-independent general neural networks [14], and fuzzy control [13]. case. In this paper, we develop a control strategy that enables Given that the mass of the pendulum is small compared to an inverted pendulum to balance on top of a quadrotor. the mass of the quadrotor, it is reasonable to assume that the Besides being a highly visual demonstration of the dynamic pendulum’s reactive forces on the quadrotor are negligible. capabilities of modern quadrotors, the solution to such a The dynamics of the quadrotor, then, do not depend on complex control problem offers insight into quadrotor control the pendulum, whereas the dynamics of the pendulum are strategies, and could be adapted to other tasks. influenced by the motion of the quadrotor. This assumption Quadrotors offer exceptional agility. Thanks to the off- is justified by the experimental setup, with the weight of center mounting of the propellers, extraordinarily fast rota- the pendulum being less than 5% of that of the quadrotor tional dynamics can be achieved. This is combined with typ- vehicle. ically high thrust-to-weight ratios, resulting in large achiev- able translational accelerations when not carrying a payload. A. Quadrotor While most early work on quadrotors focused on near- The quadrotor is described by six degrees of freedom: The hover operation (e.g. [5], and references therein), a growing translational position (x, y, z) is measured in the inertial community is working on using the full dynamical potential coordinate system O as shown in Figure 1. The vehicle of these vehicles. Flips have been executed by several groups, attitude V is defined by three Euler angles. From the inertial some focusing on speed and autonomous learning [7] and coordinate system, we first rotate around the z-axis by the some on safety guarantees [3]. Other complex maneuvers, yaw angle α. The coordinate system is then rotated around including flight through windows and perching have been the new y-axis by the pitch angle β, and finally rotated about demonstrated [8]. the new x-axis by the roll angle γ: In Section II, we introduce the dynamic models used in the controller design. Section III presents static and O , (1) dynamic nominal trajectories for the quadrotor to follow. V R(α,β,γ)= Rz (α) Ry (β) Rx (γ) The dynamics are then linearized around these trajectories in where Section IV, and linear state feedback controllers are designed in Section V. The experimental setup and results are shown 1 0 0 The authors are with the Institute for Dynamic Systems and Control, ETH Rx (γ)= 0 cos γ − sin γ , (2) Zurich, Switzerland. {hehnm, rdandrea}@ethz.ch 0 sin γ cos γ cos β 0 sin β the y-axis). For notational simplicity, we describe the relative Ry (β)= 0 1 0 , (3) position of the pendulum along the z-axis as − sin β 0 cos β 2 2 2 cos α − sin α 0 ζ := L − r − s , (8) p Rz (α)= sin α cos α 0 . (4) where L to denotes the length from the base of the pendulum 0 0 1 to its center of mass. We model the pendulum as an iner- The translational acceleration of the vehicle is dictated by tialess point mass that is rigidly attached to the mass center the attitude of the vehicle and the total thrust produced by of the quadrotor, such that rotations of the vehicle do not the four propellers. With a representing the mass-normalized cause a motion of the pendulum base. In the experimental collective thrust, the translational acceleration in the inertial setup, the point that the pendulum is attached to is mounted frame is off-center by about 10% of the length of the pendulum. While this assumption causes modeling errors, it simplifies x¨ 0 0 O the dynamics to such a great extent that the problem becomes y¨ = V R(α,β,γ) 0 + 0 . (5) much more tractable. The Lagrangian [9] of the pendulum z¨ a −g can be written as The vehicle attitude is not directly controllable, but it is subject to dynamics. The control inputs are the desired 1 rr˙ + ss˙ L = (˙x +r ˙)2 +(˙y +s ˙)2 +(˙z − )2 rotational rates about the vehicle body axes, (ωx, ωy, ωz), 2 ζ (9) and the mass-normalized collective thrust, a, as shown in − g (z + ζ) , Figure 2. High-bandwidth controllers on the vehicle track the desired rates using feedback from gyroscopes. The quadrotor where we assume unit pendulum mass without loss of has very low rotational inertia, and can produce high torques generality. The first term represents the kinetic energy of due to the outward mounting of the propellers, resulting in the pendulum, and the second the potential energy. The full, L very high achievable rotational accelerations on the order of nonlinear dynamic equations can be derived from using 200 rad/s2. The vehicle has a fast response time to changes in conventional Lagrangian mechanics: the desired rotational rate (experimental results have shown d ∂L ∂L − = 0 (10) time constants on the order of 20 ms). We will therefore dt ∂r˙ ∂r assume that we can directly control the vehicle body rates d ∂L ∂L and ignore rotational acceleration dynamics. As with the − = 0 , (11) dt ∂s˙ ∂s vehicle body rates, we assume that the thrust can be changed instantaneously. Experimental results have shown that the resulting in a system of equations of the form true thrust dynamics are about as fast as the rotational r¨ dynamics, with propeller spin-up being noticeably faster than = f (r,s, r,˙ s,˙ x,¨ y,¨ z¨) , (12) s¨ spin-down. The rates of the Euler angles are converted to the vehicle where f are the nonlinear equations (13) and (14). body coordinate system V through their respective transfor- C. Combined dynamics mations: The full dynamics of the combined system are described ωx γ˙ 0 0 −1 −1 −1 entirely by Equations (5), (7), and (12). The three body ωy = 0 + R (γ) β˙ + R (γ) R (β) 0 . x x y rate control inputs (ωx,ωy,ωz) control the attitude V of ωz 0 0 α˙ the vehicle in a nonlinear fashion. This attitude, combined (6) The above can be written more compactly by combining the Euler rates into a single vector, calculating the relevant a rows of the rotation matrices, and solving for the Euler angle rates: ωy ωz −1 γ˙ cos β cos γ − sin γ 0 ωx ωx β˙ = cos β sin γ cos γ 0 ωy . (7) α˙ − sin β 0 1 ωz B. Inverted Pendulum The pendulum has two degrees of freedom, which we describe by the translational position of the pendulum center Fig. 2. The control inputs of the quadrotor: The rotational rates ωx, ωy, of mass relative to its base in O (r along the x-axis, s along and ωz are tracked by an on-board controller, using gyroscope feedback. 1 2 r¨ = −r4x¨ − L2 − s2 x¨ − 2r2 sr˙s˙ + −L2 + s2 x¨ + r3 s˙2 + ss¨ − ζ (g +z ¨) + (L2 − s2)ζ2 r −L2ss¨ + s3s¨ + s2 r˙2 − ζ (g +z ¨) + L2 −r˙2 − s˙2 + ζ (g +z ¨) (13) 1 2 s¨ = −s4y¨ − L2 − r2 y¨ − 2s2 rr˙s˙ + −L2 + r2 y¨ + s3 r˙2 + rr¨ − ζ (g +z ¨) + (L2 − r2)ζ2 s −L2rr¨ + r3r¨ + r2 s˙2 − ζ (g +z ¨) + L2 −r˙2 − s˙2 + ζ (g +z ¨) (14) with the thrust a, controls the translational acceleration of To describe the vehicle position, the following coordinate the vehicle. While the acceleration drives the translational system C is introduced, with (u,v,w) describing the position motion of the vehicle linearly, it also drives the motion of in C: the pendulum through nonlinear equations. The combined x u cos Ωt − sinΩt 0 u system consists of thirteen states (three rotational and six y =: Rz (Ωt) v = sinΩt cos Ωt 0 v . translational states of the quadrotor, and four states of the z w 0 0 1 w pendulum), and four control inputs (three body rates, and (19) the thrust). To describe the vehicle attitude, a second set of Euler angles is introduced, describing the ‘virtual body frame’ W and III.