Abstract
A large number of both aerial and underwater mobile robots fall in the category of underactuated systems that are defined on a manifold, which is not isomorphic to Euclidean space. Traditional approaches to designing controllers for such systems include geometric approaches and local coordinate-based representations. In this paper, we propose a global parameterization of the special orthogonal group, denoted by $ \mathsf {SO}(3)$ , to design path-following controllers for underactuated systems. In particular, we present a nine-dimensional representation of $ \mathsf {SO}(3)$ that leads to controllers achieving path-invariance for a large class of both closed and non-closed embedded curves. On the one hand, this over-parameterization leads to a simple set of differential equations and provides a global non-ambiguous representation of systems as compared to other local or minimal parametric approaches. On the other hand, this over-parameterization also leads to uncontrolled internal dynamics, which we prove to be bounded and stable. The proposed controller, when applied to a quadrotor system, is capable of recovering the system from challenging situations such as initial upside-down orientation and also capable of performing multiple flips.
Highlights
We consider a class of underactuated systems that are equipped with a mechanism capable of producing a torque input about each body axis, and a thrust input about one of the body axes
We consider a path following problem: given a system belonging to the CV class of vehicles, as well as a smooth non-self intersecting curve in the 3D space, our goal is to design a novel control law using a global parameterization of the manifold understudy such that the system converges to the path and follows it
In this paper, we proposed a nine-dimensional parameterization for a class of underactuated systems that are defined on SO(3) ×Rn.The proposed representation is both global and unique and leads to a simple set of differential equations
Summary
We consider a class of underactuated systems that are equipped with a mechanism capable of producing a torque input about each body axis, and a thrust input about one of the body axes. We denote this underactuated class of vehicles by CV. As described in [1], a large class of systems including satellites, quadrotors, underwater vehicles, and tail-sitting robots belong to this class of underactuated systems. The rotational dynamics of each system belonging to CV class are defined on a smooth manifold SO(3) ×R3. The special orthogonal group, SO(3) can be represented by a set of three by three orthogonal matrices which form a group under matrix multiplication and has a smooth manifold structure and form a Lie group [2]
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