This paper entails the application of the energy shaping methodology to control a flexible, elastic Cosserat rod model of a single octopus arm. The principal focus and novel contribution of this work is two-fold: (i) reduced order controloriented modeling of the realistic internal muscular architecture in an octopus arm; and (ii) incorporation of such models into the energy shaping methodology, extending our prior work by formally accounting for muscle constraints. Extension of the control scheme to the under-actuated muscle control case involves two steps: (i) design of a desired potential energy function whose static minimizer solves a given control task; and (ii) implementing the resulting energy shaping control input into the dynamic model. Due to the muscle actuator constraints, the desired potential energy function may not be arbitrarily chosen. Indeed, the desired energy must now satisfy a partial differential equation, known as the matching condition, which is derived for the infinite dimensional Hamiltonian control system. A particular solution to those matching conditions is described, paving the way to the application of energy shaping methodology. The overall control design methodology including muscle models is implemented and demonstrated in a dynamic simulation environment. Results of several bio-inspired numerical experiments involving the control of octopus arms are reported.
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.
In this paper, we use the optimal control methodology to control a flexible, elastic Cosserat rod. An inspiration comes from stereotypical movement patterns in octopus arms, which are observed in a variety of manipulation tasks, such as reaching or fetching. To help uncover the mechanisms underlying these observed behaviors, we outline an optimal control-based framework. A single octopus arm is modeled as a Hamiltonian control system, where the continuum mechanics of the arm is captured by the Cosserat rod theory, and internal, distributed muscle forces and couples are considered as controls. First order necessary optimality conditions are derived for an optimal control problem formulated for this infinite dimensional system. Solutions to this problem are obtained numerically by an iterative forward-backward algorithm. The state and adjoint equations are solved in a dynamic simulation environment, setting the stage for studying a broader class of optimal control problems. Trajectories that minimize control effort are demonstrated and qualitatively compared with observed behaviors.
Inspired by the unique neurophysiology of the octopus, a hierarchical framework is proposed that simplifies the coordination of multiple soft arms by decomposing control into high‐level decision‐making, low‐level motor activation, and local reflexive behaviors via sensory feedback. When evaluated in the illustrative problem of a model octopus foraging for food, this hierarchical decomposition results in significant improvements relative to end‐to‐end methods. Performance is achieved through a mixed‐modes approach, whereby qualitatively different tasks are addressed via complementary control schemes. Herein, model‐free reinforcement learning is employed for high‐level decision‐making, while model‐based energy shaping takes care of arm‐level motor execution. To render the pairing computationally tenable, a novel neural network energy shaping (NN‐ES) controller is developed, achieving accurate motions with time‐to‐solutions 200 times faster than previous attempts. The hierarchical framework is then successfully deployed in increasingly challenging foraging scenarios, including an arena littered with obstacles in 3D space, demonstrating the viability of the approach.
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