  By generating control policies that create natural search behaviors in
autonomous systems, ergodic control provides a principled solution to address
tasks that require exploration. A large class of ergodic control algorithms
relies on spectral analysis, which suffers from the curse of dimensionality,
both in storage and computation. This drawback has prohibited the application
of ergodic control in robot manipulation since it often requires exploration in
state space with more than 2 dimensions. Indeed, the original ergodic control
formulation will typically not allow exploratory behaviors to be generated for
a complete 6D end-effector pose. In this paper, we propose a solution for
ergodic exploration based on the spectral analysis in multidimensional spaces
using low-rank tensor approximation techniques. We rely on tensor train
decomposition, a recent approach from multilinear algebra for low-rank
approximation and efficient computation of multidimensional arrays. The
proposed solution is efficient both computationally and storage-wise, hence
making it suitable for its online implementation in robotic systems. The
approach is applied to a peg-in-hole insertion task using a 7-axis Franka Emika
Panda robot, where ergodic exploration allows the task to be achieved without
requiring the use of force/torque sensors.
