Optimal control of a swimming robot based on Purcell's microswimmer model

Purcell's swimmer is a well-known planar model of a swimming microorganism, governed by low Reynolds number hydrodynamics, which is comprised of three rigid links connected by actuated rotary joints. This model has been analyzed as a robotic locomotion system governed by first-order nonlinear dynamics with a periodic input (gait) of the two joint angles. In this work, we present a robotic macro-scale realization of this three-link swimmer moving in a highly viscous fluid. We propose a simple variant of Purcell's theoretical model with non-slender links and a central rigid sphere which represents the added drag of the robot's central flotation block, and calibrate the model's parameters to fit experimental measurements. Next, we apply optimal control formulation based on Pontryagin's Maximum Principle (PMP) in order to find optimal gaits that maximize the displacement per cycle under bounds on the joint angles. Employing a differential geometric method that transforms the problem to area integral enclosed by the gait trajectory in the plane of joint angles, enables visual interpretation which explains topological changes in displacement-optimal gaits upon varying the bound on the joint angles. We then apply PMP formulation to the problem of maximizing Lighthill's energy efficiency in order to obtain a boundary value problem (BVP) whose solution gives efficiency-optimal gaits for Purcell's swimmer model, as well as its variant with a central sphere. Finally, we utilize numerical methods such as parameterizing the input gait as a truncated Fourier series, as well as GPOPS-II solver, to produce sufficient initial guess values for solving the BVPs and obtaining efficiency-optimal gaits.

Dynamics, stability, and energy efficiency of an energy-recycling rimless wheel with spring-clutch legs

This paper proposes an energy-recycling rimless wheel with spring-clutch legs. The proposed mechanism uses a lockable clutch to store part of the impact-induced elastic energy after foot contact and reinject it in the next gait cycle. First, we develop a hybrid dynamic model of the energy-recycling rimless wheel. Second, numerical simulations are used to examine the dynamics, local stability of periodic gaits, and the Cost of Transport (CoT) of the proposed mechanism. The simulation results show that the proposed mechanism reduces the CoT by up to 16.13% compared with a benchmark viscoelastic-legged rimless wheel with telescopic spring-damper legs. Compared with the rigid rimless wheel, the viscoelastic-legged and energy-recycling models reduce the CoT by more than 50%. The energy-recycling model also maintains locally stable periodic gaits over the tested slope and stiffness ranges. Finally, prototype experiments on an inclined plane are conducted to examine the feasibility of the proposed mechanism. The experimental results show that the proposed rimless wheel achieves passive walking on a shallow 1° slope, corresponding to a CoT of approximately 0.02. These results suggest that the proposed spring-clutch mechanism can improve the simulated walking efficiency of the energy-recycling rimless wheel, while the prototype experiments support the feasibility of passive walking with the mechanism.

Sample Efficient Learning of Body-Environment Interaction of an Under-Actuated System

Geometric mechanics provides valuable insights into how biological and robotic systems use changes in shape to move by mechanically interacting with their environment. In high-friction environments it provides that the entire interaction is captured by the ``motility map''. Here we compare methods for learning the motility map from motion tracking data of a physical robot created specifically to test these methods by having under-actuated degrees of freedom and a hard to model interaction with its substrate. We compared four modeling approaches in terms of their ability to predict body velocity from shape change within the same gait, across gaits, and across speeds. Our results show a trade-off between simpler methods which are superior on small training datasets, and more sophisticated methods, which are superior when more training data is available.

Analysis and experiments of the dissipative Twistcar: direction reversal and asymptotic approximations

Underactuated wheeled vehicles are commonly studied as nonholonomic systems with periodic actuation. Twistcar is a classical example inspired by a riding toy, which has been analyzed using a planar model of a dynamical system with nonholonomic constraints. Most of the previous analyses did not account for energy dissipation due to friction. In this work, we study a theoretical two-link model of the Twistcar while incorporating dissipation due to rolling resistance. We obtain asymptotic expressions for the system's small-amplitude steady-state periodic dynamics, which reveals the possibility of reversing the direction of motion upon varying the geometric and mass properties of the vehicle. Next, we design and construct a robotic prototype of the Twistcar whose center-of-mass position can be shifted by adding and removing a massive block, enabling demonstration of the Twistcar's direction reversal phenomenon. We also conduct parameter fitting for the frictional resistance in order to improve agreement with experiments.

Locomotion Dynamics of an Underactuated Three-Link Robotic Vehicle

The wheeled three-link snake robot is a well-known example of an underactuated system modelled using nonholonomic constraints, preventing lateral slippage (skid) of the wheels. A kinematically controlled configuration assumes that both joint angles are directly prescribed as phase-shifted periodic input. In another configuration of the robot, only one joint is periodically actuated while the second joint is passively governed by a visco-elastic torsion spring. In our work, we constructed the two configurations of the wheeled robot and conducted motion experiments under different actuation inputs. Analysis of the motion tracking measurements reveals a significant amount of wheels' skid, in contrast to the assumptions used in standard nonholonomic models. Therefore, we propose modified dynamic models which include wheels' skid and viscous friction forces, as well as rolling resistance. After parameter fitting, these dynamic models reach good agreement with the motion measurements, including effects of input's frequency on the mean speed and net displacement per period. This illustrates the importance of incorporating wheels' skid and friction into the system's model.

Multistable Physical Neural Networks

Artificial neural networks (ANNs), which are inspired by the brain, are a central pillar in the ongoing breakthrough in artificial intelligence. In recent years, researchers have examined mechanical implementations of ANNs, denoted as Physical Neural Networks (PNNs). PNNs offer the opportunity to view common materials and physical phenomena as networks, and to associate computational power with them. In this work, we incorporated mechanical bistability into PNNs, enabling memory and a direct link between computation and physical action. To achieve this, we consider an interconnected network of bistable liquid-filled chambers. We first map all possible equilibrium configurations or steady states, and then examine their stability. Building on these maps, both global and local algorithms for training multistable PNNs are implemented. These algorithms enable us to systematically examine the network's capability to achieve stable output states and thus the network's ability to perform computational tasks. By incorporating PNNs and multistability, we can design structures that mechanically perform tasks typically associated with electronic neural networks, while directly obtaining physical actuation. The insights gained from our study pave the way for the implementation of intelligent structures in smart tech, metamaterials, medical devices, soft robotics, and other fields.

Geometric analysis of gaits and optimal control for three-link kinematic swimmers

Many robotic systems locomote using gaits - periodic changes of internal shape, whose mechanical interaction with the robot`s environment generate characteristic net displacements. Prominent examples with two shape variables are the low Reynolds number 3-link "Purcell swimmer" with inputs of 2 joint angles and the "ideal fluid" swimmer. Gait analysis of these systems allows for intelligent decisions to be made about the swimmer`s locomotive properties, increasing the potential for robotic autonomy. In this work, we present comparative analysis of gait optimization using two different methods. The first method is variational approach of "Pontryagin`s maximum principle" (PMP) from optimal control theory. We apply PMP for several variants of 3-link swimmers, with and without incorporation of bounds on joint angles. The second method is differential-geometric analysis of the gaits based on curvature (total Lie bracket) of the local connection for 3-link swimmers. Using optimized body-motion coordinates, contour plots of the curvature in shape space gives visualization that enables identifying distance-optimal gaits as zero level sets. Combining and comparing results of the two methods enables better understanding of changes in existence, shape and topology of distance-optimal gait trajectories, depending on the swimmers' parameters.

Experimental Verification of Stability Theory for a Planar Rigid Body with Two Unilateral Frictional Contacts

Stability of equilibrium states in mechanical systems with multiple unilateral frictional contacts is an important practical requirement, with high relevance for robotic applications. In our previous work, we theoretically analyzed finite-time Lyapunov stability for a minimal model of planar rigid body with two frictional point contacts. Assuming inelastic impacts and Coulomb friction, conditions for stability and instability of an equilibrium configuration have been derived. In this work, we present for the first time an experimental demonstration of this stability theory, using a variable-structure rigid ''biped'' with frictional footpads on an inclined plane. By changing the biped's center-of-mass location, we attain different equilibrium states, which respond to small perturbations by divergence or convergence, showing remarkable agreement with the predictions of the stability theory. Using high-speed recording of video movies, good quantitative agreement between experiments and numerical simulations is obtained, and limitations of the rigid-body model and inelastic impact assumptions are also studied. The results prove the utility and practical value of our stability theory.

Inertial Inchworm Crawling

Inchworm crawling allows for both quasistatic and dynamic gaits at a wide range of actuation frequencies. This locomotion mechanism is common in nonskeletal animals and exploited extensively in the bio-inspired field of soft robotics. In this work we develop and simulate the hybrid dynamic crawling of a three-link robot, with passive frictional contacts. We fabricate and experimentally test such robot under periodic inputs of joints' angles, with good agreement to the theoretical predictions. This allows to comprehend and exploit the effects of inertia in order to find optimal performance in inputs' parameters. A simple criterion of robustness to uncertainties in friction is proposed. Tuning the inputs according to this criterion improves the robustness of low-frequency actuation, while increasing the frequency allows for gaits with both high advancement velocity and robustness. Finally, the advantages of uneven mass distribution are studied. Time-scaling technique is introduced to shape inputs that achieve similar effect without reassembling the robot. A machine-learning based optimization is applied to these inputs to further improve the robot's performance.

Understanding Legged Crawling for Soft-Robotics

Crawling is a common locomotion mechanism in soft robots and nonskeletal animals. In this work we propose modeling soft-robotic legged locomotion by approximating it with an equivalent articulated robot with elastic joints. For concreteness we study our soft robot with two bending actuators via an articulated three-link model. The solution of statically indeterminate systems with stick-slip contact transitions requires for a novel hybrid-quasitatic analysis. Then, we utilize our analysis to investigate the influence of phase-shifted harmonic inputs on performance of crawling gaits, including sensitivity analysis to friction uncertainties and energetic cost of transport. We achieve optimal values of gait parameters. Finally, we fabricate and test a fluid-driven soft robot. The experiments display remarkable agreement with the theoretical analysis, proving that our simple model correctly captures and explains the fundamental principles of inchworm crawling and can be applied to other soft-robotic legged robots.