## control-oriented model for bend propagation

**Background**

Stereotypical maneuvers of octopus arms has elicited much recent interest from biologists and engineers alike. The most canonical of these maneuvers is the bend propagation where the octopus propagates (pushes) the bend from the base of the arm to the tip of the arm.

While the computational models are useful to investigate the physics of the problem, it is difficult to use these for the purposes of control design. In part due to the unavailability of suitable control-oriented models, there is no control-theoretic understanding of the traveling wave of muscle activation profile (e.g., whether it is optimal in some way). This motivates the work described in this paper which is to develop a control-oriented reduced order model.

A video sequence of an octopus reaching toward a target

**Biological Experiments**

We recover the smooth arm and curvature profiles from the recorded video frames using a data smoothing algorithm. The characteristics of the bend propagation movement is that the curvature has a bell shape which travels from the initial bend to the tip of the arm.

Arm and curvature reconstruction

**Full Order Model - Cosserat Rod**

We model the flexible octopus arm using the Cosserat rod theory. Such computational model can be expressed as a Hamiltonian control system. The dynamics take into account both the muscle actuation and drag forces from the fluid.

The challenges with the Cosserat rod model are that the dynamics are governed by highly nonlinear PDEs, making it difficult for both model-based control analysis and simulation. In this paper, we are interested in studying bend propagation, which leads to the question that can we find a simplified model to characterize such motion?

**Reduced Order Model**

A reduced-order model is constructed through parametrization of curvature using two variables, \mu and \epsilon. Intuitively, \mu is the location of the bend and \epsilon is the extent of the bend.

**Physics of the Problem**

1) Propagating stiffening wave of muscle activation;

2) Drag effects from the surrounding fluid.

**Simulation Results**

Bend propagation comparison:

Full order model

Reduced order model

Control parameter analysis:

The stiffening wave of muscle activation is given by the following function with two control parameters, the speed and magnitude.

We simulate the bend propagation with varying stiffening wave speed and magnitude. Then we record the minimum value of coordinate \epsilon over time for each simulation and obtain the following map. The corresponding control parameters at the top right corner of the map can generate the muscle activation that reproduces the bend propagation.

**Limitations**

The reduced order model has singularity point for mass term when the rod is straight. This is the limitation due to our parametrization of curvature.

**Presentations**

*Rapid interactive session *WeB16 - Biological and Biomedical Systems from 2022 American Control Conference - Atlanta, GA, USA, June 8-10, 2022.

**Publications**

T. Wang, U. Halder, E. Gribkova, M. Gazzola, and P.G., Mehta, "Control-oriented modeling of bend propagation in an octopus arm," in *202**2** American Control Conference (ACC)*. IEEE, 2022.

**Acknowledgements**

Financial support from ONR MURI N00014-19-1-2373, NSF/USDA #2019-67021-28989, and NSF EFRI C3 SoRo #1830881.