IEEE Robotics & Automation Magazine - December 2013 - 98

In the following sections, we explain how we arrive at a
choreographed quadrocopter dance performance. We first
present our motion library, from which we extract viable
motions for the performance. We then introduce a basic
description of music that serves as a reference when designing
a new choreography. Finally, we explain the strategies we use
for linking the motion to the music.

beat and to achieve precise trajectory tracking (see the "Synchronization" section). The overall system setup is outlined
in Figure 3.
Three different kinds of motions can be used to describe
the motion trajectories in this article: 1) periodic motions,
2) acrobatic motions, and 3) transition motions. Periodic
motions are strictly linked to the music beat, whereas acrobatic motions are thought to be aesthetic highlights. Finally,
transition motions allow for smooth transitions between different motion primitives.

Motion Library
Motion primitives are the key building blocks of our choreographies: they represent well-defined motions of a quadrocopter during a finite period of time. The concatenation of
motion primitives defines the vehicles' trajectories during the
whole performance. The motion primitives are parameterized, categorized, and stored in the motion library to be later
combined into a variety of choreographic designs.
Quadrocopters possess six degrees of freedom. However,
since their dynamics are differentially flat, all states and
inputs of the system can be expressed in terms of a set of outputs and their derivatives [12]. One of these sets consists of
the three-dimensional (3-D) position of the vehicle and its
heading, given by the yaw angle [13]. Therefore, the motion
primitives considered in this article are described by their
position trajectory in 3-D space and by a yaw trajectory profile, which can be independently specified. These terms fully
specify the vehicle's position and attitude.
The trajectories are input to an underlying trajectory-following controller that computes the necessary inputs to the
vehicle based on its current state. Adaptation techniques are
used to synchronize the quadrocopter trajectory to the music

Periodic Motion Primitives
Periodic motion is a natural human response to a recurring
beat: we often clap, sway, or tap our feet when we hear music.
Thus, we want the flying vehicle to mimic this behavior. For
our purpose, the trajectory of a periodic motion primitive is
represented by a sum of sines and cosines
N

x 0 + / c k cos ^k ^Xt + {hh + d k sin ^k ^Xt + {hh,
k =1

where the frequency X is related to the music beat. The center point x 0, the amplitudes c k and d k (all vectors in R 3 ), and
the phase { are design parameters. Trajectories with different
frequency components are combined if N is chosen to be
greater than 1.
The above formulation is mathematically redundant but
allows us to easily define a multitude of periodic motion
primitives using a set of intuitive design parameters. For
example, a circle of radius r in the horizontal plane can be

Choreography Description
1)

Circle

Radius, Phase,
Center Point, Beats

Periodic