Geometric Motion Planning for Systems With Toroidal and Cylindrical Shape Spaces

Abstract

Kinematic motion planning using geometric mechanics tends to prescribe a trajectory in a parameterization of a shape space and determine its displacement in a position space. Often this trajectory is called a gait. Previous works assumed that the shape space is Euclidean when often it is not, either because the robotic joints can spin around forever (i.e., has an S 1 configuration space component, or its parameterization has an S 1 dimension). Consider a shape space that is a torus; gaits that "wrap" around the full range of a shape variable and return to its starting configuration are valid gaits in the shape space yet appear as line segments in the parameterization. Since such a gait does not form a closed loop in the parameterization, existing geometric mechanics methods cannot properly consider them. By explicitly analyzing the topology of the underlying shape space, we derive geometric tools to consider systems with toroidal and cylindrical shape spaces.

Coordination of lateral body bending and leg movements for sprawled posture quadrupedal locomotion

“…Geometric mechanics techniques then seek to establish a relationship between the velocities of these spaces; this relationship is often called a connection and it shares many properties with a robot manipulator’s Jacobian. In this section, we provide a concise overview of the geometric tools needed for this article, but for a more detailed and comprehensive review, we refer readers to Kelly and Murray (1995), Ostrowski and Burdick (1998), Shapere and Wilczek (1989), Wilczek and Shapere (1989), Bloch et al (1996), Marsden and Ratiu (2013), Batterman (2003), Hatton and Choset (2015), and Gong et al (2018).…”

“…The net displacement can be approximated by the path integral along the assistive line ∂ ϕ 0 plus the surface integral of the area enclosed by the gait path ∂ ϕ and the assistive line (Gong et al, 2018):…”

“…A gait then maps a periodic path in the shape space to a displacement in the position space. One of the advantages of this approach is that it can readily generate, as well as evaluate, the displacement of gaits; this generative tool is based on a height function (sometimes called a connection vector field) which is essentially the curl of the connection (Gong et al, 2016(Gong et al, , 2018Hatton et al, 2013). In the last decade, our group has successfully advanced this scheme to enable gait design for limbless undulatory swimming systems in highly damped situations such as granular media, where the full contact between the system and the environment is maintained (Dai et al, 2016;Gong et al, 2016;Hatton et al, 2013).…”

“…To form an enclosed area in the Euclidean parameterization of the periodic shape space, we introduce the notion of an assistive line (Gong et al, 2018). The assistive line is defined to be a path in shape space where only one shape variable changes and is used to form a closed loop with the gait path in the shape space.…”

“…We refer the reader to Gong et al (2018) for a detailed derivation and proof of motion planning in a cylindrical shape spaces. In this way, the process for the gait design of legged systems is simplified to drawing a path in cylindrical shape space.…”