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Publication# A geometric optimal control approach for imitation and generalization of manipulation skills

Abstract

Daily manipulation tasks are characterized by regular features associated with the task structure, which can be described by multiple geometric primitives related to actions and object shapes. Only using Cartesian coordinate systems cannot fully represent such geometric descriptors. In this article, we consider other candidate coordinate systems and propose a learning approach to extract the optimal representation of an observed movement/behavior from these coordinates. This is achieved by using an extension of Gaussian distributions on Riemannian manifolds, which is used to analyze a small set of user demonstrations statistically represented in different coordinate systems. We formulate the skill generalization as a general optimal control problem based on the (iterative) linear quadratic regulator ((i)LQR), where the Gaussian distribution in the proper coordinate systems is used to define the cost function. We apply our approach to object grasping and box-opening tasks in simulation and on a 7-axis Franka Emika robot using open-loop and feedback control, where precision matrices result in the automatic determination of feedback gains for the controller from very few demonstrations represented in multiple coordinate systems. The results show that the robot can exploit several geometries to execute the manipulation task and generalize it to new situations. The results show high variation along the do-not-matter direction, while maintaining the invariant characteristics of the task in the coordinate system(s) of interest. We then tested the approach in a human-robot shared control task. Results show that the robot can modify its grasping strategy based on the geometry of the object that the user decides to grasp.

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Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth.

Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring.

Geographic coordinate system

The geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various spatial reference systems that are in use, and forms the basis for most others. Although latitude and longitude form a coordinate tuple like a cartesian coordinate system, the geographic coordinate system is not cartesian because the measurements are angles and are not on a planar surface.

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