PDF《Parameterized Maneuver Learning for》

90 degrees backwards and ?ies upwards until it reaches zero velocity. At that point it spins

180 degrees around its vertical axis, and descends nose down and levels out into forward ?ight. This is a standard aerobatic maneuver that can be useful for changing directions. maneuver within the class can be speci?ed using a small number of parameters. Throughout the paper, we use stall turns as a clarifying example of such a maneuver class. We are given a number of demonstration trajectories (likely of unequal durations) from a single maneuver class. Our goal is to generate a new trajectory from the same maneuver class that satis?es a given parameterization. In order to specify parameterizations for different maneuver classes, we ?rst de?ne the concepts of parameters and waypoints. A. Parameters We de?ne the parameters of a maneuver to be the de?ning attributes that one must specify when generating a trajectory from a maneuver class. In the case of stall turns, we use the trajectory'

s maximal altitude as the only parameter. B. Waypoints All trajectories within a single maneuver class share a similar shape and structure. This structure can be captured by a set of characteristic points that de?ne the major variations between different trajectories within the class. For example, all stall turns generally share the same shape;

however, two different stall turns are easily distinguished by their altitude at their peaks. For each maneuver class, we de?ne a set of characteristic points, which we refer to as waypoints. For stall turns, we de?ne a single waypoint at the top of the maneuver, when the helicopter is at its maximal altitude. Our trajectory learning algorithm speci?es what values certain state variables should take at the waypoints. These values are derived from the input parameters. Rather than specifying all of the state variables at each waypoint, we only specify the subset of the state variables required to capture the key characteristics of the maneuver at that waypoint. We refer to this subset of state variables and the speci?ed values as the waypoint constraints. At the waypoint of a stall turn, the waypoint constraints specify an altitude equal to the input parameter and a vertical velocity of zero. This enforces that the waypoint corresponds to the maximal altitude. Our trajectory learning algorithm attempts to ?nd a trajectory of the speci?ed maneuver class that satis?es these generated waypoint constraints. In addition to the de?ned waypoints for each maneuver class, we always include waypoints at the beginning and end of each trajectory. The constraints for these waypoints specify the position and orientation of the helicopter. III. TRAJECTORY LEARNING ALGORITHM Our trajectory learning algorithm estimates a target trajec- tory of a particular maneuver class from a set of demon- strations from the same maneuver class and parameters describing the desired trajectory. The algorithm consists of two core steps. After initializa- tion, the ?rst step is time alignment;

we align each of our demonstration trajectories to the current estimate of the target trajectory. The second step consists of a Kalman smoother Input: Demonstrations D from desired maneuver class c, parameters p Output: Estimate of target trajectory z Initialization: Find closest trajectory r;

w ← GenerateWaypointConstraints(c, p, r);

Initialize z ← r;

for i =

1 to numIters do foreach demonstration d ∈ D do Align(z, d);

end z ← KalmanSmoother(D, w);

end Algorithm 1: Outline of trajectory learning algorithm. used in conjunction with the EM algorithm to infer the target trajectory from the aligned demonstrations. These steps are summarized in Algorithm 1. A. Initialization Our algorithm requires three initialization steps. First, we ?nd the closest trajectory from the same maneuver class, where distance is calculated in the parameter space. For stall turns, we ?nd the trajectory of the closest altitude to the speci?ed input parameter. We then use this trajectory along with the input parameter to generate the waypoint constraints, as described in Section III-D. Lastly, we set the initial estimate of the target trajectory to be this same closest trajectory. B. Dynamic Time Warping The demonstration trajectories collected from our expert vary considerably in size and duration. Furthermore, the important points in each trajectory may not occur at the same time;