Ergodic Trajectory Planning with Dynamic Sensor Footprints

TL;DR

This work extends ergodic trajectory planning by incorporating dynamic sensor footprints, capturing how footprint size and resolution change with robot state. It introduces the Footprint Ergodic Metric and Footprint Ergodic Optimization (FEO/ETO-DF), derives local optimality conditions, and develops a sampling-based numerical method for computation. The approach enables simultaneous trajectory and footprint optimization, extends to multi-robot and 3D object coverage, and demonstrates substantial improvements in ergodicity in simulations and real quadrotor experiments. The results show the practical value of adapting sensing footprint during exploration to achieve more efficient information gathering in complex environments.

Abstract

This paper addresses the problem of trajectory planning for information gathering with a dynamic and resolution-varying sensor footprint. Ergodic planning offers a principled framework that balances exploration (visiting all areas) and exploitation (focusing on high-information regions) by planning trajectories such that the time spent in a region is proportional to the amount of information in that region. Existing ergodic planning often oversimplifies the sensing model by assuming a point sensor or a footprint with constant shape and resolution. In practice, the sensor footprint can drastically change over time as the robot moves, such as aerial robots equipped with downward-facing cameras, whose field of view depends on the orientation and altitude. To overcome this limitation, we propose a new metric that accounts for dynamic sensor footprints, analyze the theoretic local optimality conditions, and propose numerical trajectory optimization algorithms. Experimental results show that the proposed approach can simultaneously optimize both the trajectories and sensor footprints, with up to an order of magnitude better ergodicity than conventional methods. We also deploy our approach in a multi-drone system to ergodically cover an object in 3D space.

Figures (12)

• Figure 1: Demonstration with a drone. (a) and (b) show the trajectory planned by our planner and the corresponding height values (z coordinate) of the robot along the trajectory. (c) shows the execution of the trajectory with a real drone equipped with a downward pointing LED to visualize the dynamic sensor footprint whose size depends on the flying height of the robot. With our approach, the robot flies high over regions with widespread information and flies low over regions with concentrated information. The orange shadow on the ground in (c) visualizes the time-averaged statistics of the dynamic sensor footprint, which is similar to the information map to be covered as shown in (a).
• Figure 2: Illustration of sensor footprint and footprint trajectory of a flying drone with downwards pointing camera over a 2D information map.
• Figure 3: Numerical computation of the footprint ergodic metric. (a) shows a trajectory of the robot and the corresponding footprint trajectory $\gamma(w,x(t))$, which is a random process. The orange dots in the footprint at each time point represent the set of sampled points at that time. All these points together form $W_s$, a set of $M$ realizations of the random process $\gamma(w,x(t))$. (b) shows the sampling method we used in this paper as explained in Example \ref{['example2']}.
• Figure 4: Illustration of sampling from surface in 3D. The pink surface shows the surface $\partial O$ of the object to be searched. The green cone shows the sensor footprint $\gamma(w,x(t))$ at some time point $t$, and a set of rays within the sensor footprint (green cone) are sampled and ray-traced to intersect with the object surface $\partial O$. The green dots in the yellow area show $I_R$, the set of intersected points of the rays and the object surface.
• Figure 5: The experimental results in four different information maps. The first row (I) shows four information maps with various regions with widespread and concentrated information. Row (II) visualizes the planned trajectories. Row (III) shows the corresponding height value ($z$ coordinates) of the robot along the planned trajectories by our FEO. Our FEO can adapt the sensor footprint and the flying height of the robot based on the information map.

Theorems & Definitions (10)

• Definition 1: Sensor Footprint
• Example 1
• Definition 2: Footprint Trajectory (FT)
• Definition 3: Time-Averaged Statistics of FT
• Definition 4: Footprint Ergodic Metric
• Theorem 1
• Definition 5
• Theorem 2
• proof
• Example 2