Energy-shaping control of a muscular octopus arm moving in three dimensions

Abstract

Flexible octopus arms exhibit an exceptional ability to coordinate large numbers of degrees of freedom and perform complex manipulation tasks. As a consequence, these systems continue to attract the attention of biologists and roboticists alike. In this article, we develop a three-dimensional model of a soft octopus arm, equipped with biomechanically realistic muscle actuation. Internal forces and couples exerted by all major muscle groups are considered. An energy-shaping control method is described to coordinate muscle activity so as to grasp and reach in three-dimensional space. Key contributions of this article are as follows: (i) modelling of major muscle groups to elicit three-dimensional movements; (ii) a mathematical formulation for muscle activations based on a stored energy function; and (iii) a computationally efficient procedure to design task-specific equilibrium configurations, obtained by solving an optimization problem in the Special Euclidean group SE ( 3 ) . Muscle controls are then iteratively computed based on the co-state variable arising from the solution of the optimization problem. The approach is numerically demonstrated in the physically accurate software environment Elastica . Results of numerical experiments mimicking observed octopus behaviours are reported.

Hierarchical Control and Learning of a Foraging CyberOctopus

“…To effect this motor primitive, we employ the energy shaping methodology. [50][51][52] As developed in our prior work, [25,35,49] an energy-shaping control law is derived to determine the static muscle activations α ¼ fα m g m∈M that cause the tip of the arm to reach a target location. The equilibrium arm configuration that achieves this goal is obtained by solving an optimization problem that minimizes the tip-to-target distance δðα, qÞ ¼ jq À xðL 0 Þj along with the muscle activation cost [25] Energy Shaping ðESÞ where α 0 are the initial muscle activations, and μ tip is a constant (regularization) coefficient.…”

“…While we previously demonstrated the use of ES for muscle coordination in a soft arm, [25,35,49] the solution to the above optimization problem is reliant on a computationally expensive forward-backward iterative scheme. Here, in a hierarchical context where energy shaping will be frequently called upon by a high-level controller, fast solutions are instead imperative.…”

“…By concatenating sequences of this motor primitive, a variety of tasks, such as reaching to a food target, fetching food to the mouth, or crawling can be attained. While in the present context this single motor primitive is sufficient to coordinate planar arm deformations, in general more than one primitive may be desirable (e.g., 3D deformations), [ 38,49 ] in which case our approach can be readily extended.…”

“…To effect this motor primitive, we employ the energy shaping methodology. [ 50–52 ] As developed in our prior work, [ 25,35,49 ] an energy‐shaping control law is derived to determine the static muscle activations that cause the tip of the arm to reach a target location.…”

Robust and Adaptive Sampled-Data Control of Twisted and Coiled Artificial Muscles

“…This model takes the form of a set of partial differential equations (PDEs), leading to a number of challenges including a need for numerical solution methods and complex controller derivation. Nevertheless, this model has been used in the formulation of several controllers, including energy shaping control [16], [17], infinite dimensional state feedback control [18], sliding mode control [19], and others [20], [21]. On the other hand, path planning with Cosserat is not well explored and only a few examples can be found in literature.…”