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Controlling a CyberOctopus Soft Arm with Muscle-like Actuation
Heng-Sheng Chang, Udit Halder, Ekaterina Gribkova, Arman Tekinalp, Noel Naughton, Mattia Gazzola, Prashant G. Mehta
TL;DR
Energy shaping for octopus-arm control must account for muscle constraints, which limit the energy landscape and make matching conditions difficult to solve. This paper models muscle forces with stored energy functions and embeds them in a bilevel optimization controller. The resulting method is applied in simulation to reaching and grasping, and constant muscle controls stabilize the arm.
Problem
Prior energy shaping for an octopus-arm Cosserat rod ignored muscle actuator constraints, which limit the energy landscape and make matching conditions difficult to solve.
Source: §I · p. 1Method
The paper models conservative muscle forces with stored energy functions and formulates muscle activation as a bilevel optimization problem.
Source: §I · p. 1Results
Constant muscle controls stabilize the arm, and numerical simulations demonstrate reaching and grasping tasks.
Source: §V · p. 6, §VI · p. 7, 2) Grasping an object · p. 7Takeaways & Limitations
The total closed-loop energy can be expressed by augmenting the rod’s elastic energy with muscle stored energy functions.
Source: §VI · p. 7
Abstract
from arXiv · showhide
This paper presents an application of the energy shaping methodology to control a flexible, elastic Cosserat rod model of a single octopus arm. The novel contributions of this work are two-fold: (i) a control-oriented modeling of the anatomically realistic internal muscular architecture of an octopus arm; and (ii) the integration of these muscle models into the energy shaping control methodology. The control-oriented modeling takes inspiration in equal parts from theories of nonlinear elasticity and energy shaping control. By introducing a stored energy function for muscles, the difficulties associated with explicitly solving the matching conditions of the energy shaping methodology are avoided. The overall control design problem is posed as a bilevel optimization problem. Its solution is obtained through iterative algorithms. The methodology is numerically implemented and demonstrated in a full-scale dynamic simulation environment Elastica. Two bio-inspired numerical experiments involving the control of octopus arms are reported.
I. INTRODUCTION
This paper extends energy shaping for an octopus-arm Cosserat rod by incorporating anatomically realistic muscle constraints through stored-energy models. It formulates muscle activation as a bilevel control problem and applies the approach to reaching and grasping.
I. INTRODUCTION
- The paper builds on prior energy-shaping work for a fully actuated arm model, whose muscle actuator constraints were ignored.The present work incorporates anatomically correct longitudinal, transverse, and oblique muscle architecture.Source: §I · p. 1
- The muscle constraints complicate energy shaping because they restrict achievable energy landscapes, while explicitly solving the matching conditions is difficult.The paper therefore models conservative muscle forces through stored energy functions, avoiding the need to solve those conditions explicitly.Source: §I · p. 1
- The muscle model separates conservative force-length behavior from dissipative force-velocity behavior and derives stored-energy expressions for longitudinal and transverse muscles.The model draws on Hill’s model and published force-length characteristics.Source: §I · p. 1
2. Application of energy shaping for muscular arm
- The control design is posed as a bilevel optimization problem, with optimal muscle activations subject to equilibrium constraints from a lower-level optimization.Necessary conditions are derived using the maximum principle, leading to iterative algorithms for computing control.Source: §2 · p. 1
A. Physiology of a muscular octopus arm
- The planar model focuses on longitudinal muscles, which bend and shorten the arm, and transverse muscles, which lengthen it and support active bending.The paper identifies oblique muscles as providing twist, but focuses on planar movement.Source: A. Physiology of a muscular octopus arm · p. 2, Figure 1 · p. 2
B. Modeling a single arm as a Cosserat rod
- Muscle activation is treated as the control input in a damped Hamiltonian Cosserat-rod model, with internal muscle actuation mapped to resultant forces and couples.The arm is represented as a planar rod with stretch, shear, and curvature deformations.Source: B. Modeling a single arm as a Cosserat… · pp. 2–3
III. CONTROL-ORIENTED MODELING OF MUSCLES
This section presents the paper’s control-oriented modeling of internal muscles and identifies its principal results as Proposition 3.1 and Corollary 3.1.
III. CONTROL-ORIENTED MODELING OF MUSCLES
- The section’s main results are Proposition 3.1 and Corollary 3.1 on control-oriented modeling of internal muscles.Readers focused on the control problem may skip to Sec. III-E and then Sec. IV.Source: §III · p. 3
A. Muscle geometry
The planar muscle geometry identifies the longitudinal and transverse groups used in the model, then describes how their activation produces forces and, for off-center longitudinal muscles, couples.
A. Muscle geometry
- The planar model considers the top and bottom longitudinal muscles and the transverse muscles in the middle.Each muscle’s position is specified relative to the rod centerline.Source: A. Muscle geometry · p. 3
B. Forces from muscle actuation
- Activated muscles provide an internal distributed contraction force whose dependence on activation remains to be prescribed.The model treats a generic muscle group through its position relative to the centerline.Source: B. Forces from muscle actuation · p. 3, A. Muscle geometry · p. 3
- The off-centerline arrangement of the two longitudinal muscles also produces a couple.The muscle model relates the couple to the muscle position and contraction force.Source: B. Forces from muscle actuation · p. 3
C. Hill’s lumped model of a muscle
The Hill model separates active contractile tension from passive elasticity and damping. The paper adapts its muscle model to a Cosserat rod by representing active muscle forces through stored energy and deriving longitudinal and transverse muscle effects.
C. Hill’s lumped model of a muscle
- Hill’s model uses parallel active contractile, passive spring, and damping elements; active tension scales with activation and varies with muscle length.Maximum active force occurs at rest length and decreases as the muscle contracts or elongates.Source: C. Hill’s lumped model of a muscle · pp. 3–4
- The force-length relationship can be experimentally specified; this paper uses a third-degree polynomial fit to squid tentacle muscle data.The form of the relationship is not essential to the modeling and control design.Source: C. Hill’s lumped model of a muscle · p. 4
- The passive spring and dashpot are not modeled separately; their effects are assigned to the rod’s inherent elasticity and damping.Source: C. Hill’s lumped model of a muscle · p. 4
D. Muscle model adaptation to Cosserat rod formalism
- For each muscle, active internal force depends on its occupied cross-sectional area, maximum force per unit area, activation, and force-length function of muscle stretch strain.The paper treats muscle stretch strain as a function of rod deformation.Source: D. Muscle model adaptation to Cosserat… · p. 4
- Longitudinal muscles sit away from the rod centerline and generate both contraction force and a couple that bends the arm locally.The top and bottom muscles are modeled symmetrically in their area and maximum-force parameters.Source: D. Muscle model adaptation to Cosserat… · p. 4
- Transverse muscle contraction shrinks the cross section and lengthens the arm under the model’s tissue-incompressibility assumption.The model also represents the inverse relationship between arm stretch and transverse-muscle stretch.Source: D. Muscle model adaptation to Cosserat… · p. 4
E. Muscle energy function
The paper represents muscle forces and torques using a stored-energy function for a prescribed activation. This representation also expresses the muscle-actuated arm as a Hamiltonian control system.
E. Muscle energy function
- For a given muscle activation, a stored-energy function yields the muscle’s internal forces and torques.The proof of this proposition is given in Appendix I.Source: E. Muscle energy function · p. 4
- The stored-energy functions for the two longitudinal muscles and the transverse muscle are reported in Table I.Source: E. Muscle energy function · p. 4, Table I · p. 5
- The total stored energy for the arm is formed by integrating the activation-weighted muscle energy along the rod.Source: E. Muscle energy function · pp. 4–5
- With a prescribed muscle activation, the muscle-actuated arm is a Hamiltonian control system with the stated Hamiltonian.The paper obtains muscle forces from that Hamiltonian; it omits the corollary proof, noting its similarity to the passive elasticity argument.Source: E. Muscle energy function · p. 5
IV. ENERGY SHAPING CONTROL DESIGN
The energy-shaping design selects static muscle controls to shape potential energy and stabilize an equilibrium. Under the stated local conditions, the control law yields local asymptotic stability.
IV. ENERGY SHAPING CONTROL DESIGN
- Under a potential-energy minimum and a locally positive-definite Hessian, the constant muscle control renders the corresponding equilibrium locally asymptotically stable.Source: A. Energy shaping control law · p. 5
- The stability argument uses the closed-loop Hamiltonian as a Lyapunov functional, with total energy non-increasing along trajectories.A complete proof would require rigorous infinite-dimensional LaSalle arguments, which are beyond the paper’s scope.Source: A. Energy shaping control law · p. 5
- Modeling muscle control terms as Hamiltonian vector fields makes the constant control law a solution to the matching condition, which generally requires solving a PDE.Source: A. Energy shaping control law · p. 5
B. Design of potential energy: The static problem
The static design is posed as a bilevel optimization: muscle activations are chosen to minimize muscle and task costs while satisfying rod-equilibrium and task constraints. The lower-level equilibrium problem simplifies for a fixed-base, free-tip arm, and an iterative algorithm solves the two levels.
Upper-level optimization
- The upper-level problem chooses admissible muscle activations to minimize muscle-use and task costs, subject to rod equilibrium and task constraints such as obstacle avoidance.Tip costs target a point, grasping costs apply along the arm, and obstacle constraints are expressed as state inequalities.Source: B. Design of potential energy: The… · pp. 5–6
Lower-level optimization
- For fixed activations, the lower-level forward problem finds the rod’s equilibrium by minimizing total potential energy subject to the rod kinematics and boundary conditions.The CyberOctopus case uses a fixed base and free tip.Source: B. Design of potential energy: The… · p. 6
- With a free tip, transversality makes the costate zero throughout the rod, simplifying the equilibrium constraint to equation (13).Equation (13) is the equilibrium constraint for a given activation.Source: B. Design of potential energy: The… · p. 6
Algorithms
- An iterative algorithm solves the equilibrium equations at each iteration, then uses a forward-backward method to update activations within [0, 1].The final activations are the energy-shaping controls.Source: B. Design of potential energy: The… · p. 6
V. NUMERICAL SIMULATION
The numerical simulation demonstrates reaching and grasping behaviors with the controlled soft octopus arm. Figure 3 depicts sequential target-reaching and wrapping around a static sphere.
Numerical experiments
- The simulation demonstrates two octopus-inspired behaviors: reaching and grasping.The experiments mimic reaching and grasping motion of an octopus arm.Source: §V · p. 6
Reaching task
- In the reaching task, the arm visits three targets in sequence; Figure 3 marks them at (12, 14), (16, 6), and (2, −2) cm.The targets mimic fetching motions.Source: Figure 3 · p. 7
Grasping task
- In the grasping task, the arm wraps around a static sphere of radius 2 cm centered at (12, 12) cm.Figure 3 also depicts activations at 2.5 s for this task.Source: Figure 3 · p. 7
A. Simulation setup
The simulation represents the arm as a discretized Cosserat rod in Elastica, with a straight rest configuration and a Hill-model force–length curve fitted to experimental data.
Simulation setup
- The planar Cosserat rod’s dynamic equations are discretized into connected cylindrical segments and solved in the open-source Elastica simulation framework.The paper describes Elastica as a dynamic, three-dimensional simulation framework.Source: A. Simulation setup · p. 7
- The rod’s rest configuration is straight, has length L0, and has intrinsic strains (ν°1, ν°2, κ°) = (1, 0, 0).Biologically realistic parameter values are listed in Tables I and II and reference.Source: A. Simulation setup · p. 7
Muscle model
- The model is fitted from experimental data [25, Fig. 6], with a specified polynomial of 3.06z^3 − 13.64z^2 + 18.01z − 6.44 for 0.6 ≤ z ≤ 1.6 and 0 otherwise.Source: A. Simulation setup · p. 7
B. Experiments
The experiments demonstrate sequential reaching and sphere grasping under muscle actuation. The conclusion describes the energy-shaping formulation and identifies broader 3D and neuromuscular control as future directions.
1) Reaching multiple static targets:
- The reaching experiment directs the arm tip to multiple static targets one by one, with the tip cost independent of tip angle.The tip cost targets position rather than pose.Source: 1) Reaching multiple static targets · p. 7
2) Grasping an object:
- The distal arm, from s = 0.4L0, grasps the sphere by approaching its boundary without penetrating it.The grasping cost encourages proximity to the boundary, while an inequality constraint prevents penetration.Source: 2) Grasping an object · p. 7
VI. CONCLUSION AND FUTURE WORK
- The conclusion models the arm as a planar Cosserat rod and muscle mechanisms as distributed internal forces and couples; it identifies general 3D control and neuronal or sensor-based activation as future work.The closed-loop energy combines rod elastic energy with muscle stored-energy functions, and constant muscle controls stabilize the arm.Source: §VI · pp. 7–8
APPENDIX I PROOF OF PROPOSITION 3.1
The proof defines a quantity for the top longitudinal muscle and states a derivative property shared by the other muscles. The PMP conditions augment the state for the constrained problem and require pointwise Hamiltonian maximization.
APPENDIX I PROOF OF PROPOSITION 3.1
- For the top longitudinal muscle LMt, the proof defines W_LMt using muscle parameters and a function of ν1 and κ.Source: Appendix I · p. 8
- The proof states that the derivative of n_LMt with respect to ν2 is zero and says similar arguments apply to the other two muscles.Source: Appendix I · p. 8
APPENDIX II PMP CONDITIONS FOR PROBLEM (14)
- For problem (14), the state is augmented with Nobs additional states whose evolution uses cj(q) = max(Ψj(q), 0), with each added state initialized to zero.Source: APPENDIX II PMP CONDITIONS FOR PROBLEM… · p. 8
- The optimal α is required to maximize the modified Hamiltonian pointwise over [0, L0].Source: APPENDIX II PMP CONDITIONS FOR PROBLEM… · p. 8