Optimal control for cyberoctopus
Background
Over the past few decades, the optimal control paradigm has been increasingly used to explain and understand dynamic phenomena in biological systems. Examples range from game-theoretic models of population dynamics, optimality principles for collective motion in starling murmuration, the minimum-jerk hypothesis for motion planning. Through a mixture of experimental data analysis and theoretical modeling, these approaches often reveal deep insights into the underlying mechanisms at play. In this work, we take a similar route to examine the problem of octopus arm movement.
Flexible octopus arms are excellent candidates for studying the intricate interplay between continuum mechanics and sensorimotor control. As opposed to articulated limbs in humans, octopus arms are soft and possess a complex muscular architecture that provides exquisite manipulation control. The muscles are independently innervated by motor neurons along the arm enabling a rich repertoire of deformations – stretch, shear, bend, and twist. However, despite their virtually infinite degrees of freedom – and thus having many options to carry out a single task – octopuses are observed to engage in certain (task-specific) stereotypical movement strategies. In experimental studies, these strategies are broadly categorized into two groups: reaching with bend propagation and fetching with pseudo-joints.
Bend propagation
Sumbre, Hochner et al. (2001)
Pseudo-joints formation
Sumbre, Hochner et al. (2005)
Dynamic Model - Cosserat Rod
The dynamics of a soft arm are modeled using the Cosserat rod theory. The system is a Hamiltonian system with the Hamiltonian being the total energy, the kinetic energy T and potential energy V. Then the dynamics are described by the Hamilton’s equations, six PDEs in total. Internal muscle forces and couples, when considered as control inputs, give rise to a control system in an infinite-dimensional state space setting.
Optimal Control Methodology - Maximum Principle
The Pontryagin’s Maximum Principle (PMP) is used to derive the six adjoint PDEs for the costate variables. The resulting two-point boundary value problem is numerically solved in an iterative manner, referred to here as the forward-backward algorithm. A custom solver is implemented to simulate the backward path or the costate equations. The control is updated in the direction of the steepest gradient ascent so as maximize the pre-Hamiltonian.
Forward-backward algorithm
Simulation Results
Reaching task:
Fetching task:
Shooting task:
Limitations
The forward-backward algorithm is facing numerical instability and other challenges when the model and dynamics become too complicated, especially when the muscle actuation is considered. Also, the simulation is very slow and the optimal control solution is not online.
Publications
T. Wang, U. Halder, H.S. Chang, M. Gazzola, and P.G., Mehta, "Optimal control of a soft cyberoctopus arm," in 2021 American Control Conference (ACC). IEEE, 2021, pp. 4757-4764.
Acknowledgements
Financial support from ONR MURI N00014-19-1-2373, NSF/USDA #2019-67021-28989, and NSF EFRI C3 SoRo #1830881.