Optimal Gaits

Objectives and Significance

Inspirations from animal locomotion can provide a framework for designing robotic vehicles capable of robustly maintaining velocity by adaptively changing propulsion strategy as the surrounding environment changes. Animal locomotion may be viewed as a process of mechanical rectification in which a periodic body motion is converted to sustained thrust force through dynamic interactions with the environment. A specific motion pattern (or gait) is chosen by each animal, depending upon the given environment, desired locomotion speed and range, disability conditions on the body, and other factors. This research aims to develop an optimal gait theory that allows for computation of a gait with minimal cost or maximal performance. The result will be useful for robotic locomotor designs as well as for understanding animal locomotion mechanisms.

Results

The optimal gait theory can explain biologically observed gaits in terms of optimality with respect to specific cost functions. The theory goes beyond mimicking animal locomotion by generating nonintuitive gaits for robotic locomotion systems whose body shapes are not inspired by biological observations.

We have developed a functional model for a general class of mechanical rectifiers in 3-D space - a multibody mechanical system with full (six) degrees of freedom (DOF) for position and orientation within the inertial frame, in addition to arbitrary finite degrees of freedom for body shape deformation. The body is assumed to be in continuous contact with its environment, receiving environmental forces that are proportional to the relative velocities with directional preference. The equations of motion are derived within the inertial frame and are transformed into body coordinates in order to provide a comprehensible description of the dynamics for arbitrary 3-D maneuvers. Assuming small body deformation around a nominal posture, the equations of motion are reduced to a simplified dynamical model with quadratic nonlinearity that is tractable and captures the essential rectifying dynamics.

An optimal gait problem was formulated for the quadratic rectifier model as a minimization of a general quadratic cost function over the set of periodic functions subject to a constraint on the average locomotion velocity. The cost function can be specified in a variety of ways; power consumption, body curvature, joint (muscle) torque, their time derivatives, etc. We have proved that a globally optimal solution is given by a harmonic gait (i.e. joint angles oscillate sinusoidally) that can be found by generalized eigenvalue computation with a line search over cycle frequencies. The gait pattern is encoded in the generalized eigenvector of a matrix pair specified from the body-environment dynamics.

The optimal gaits are calculated for a chain of rigid links driven by joint torque actuators with motion restricted to a plane. Three cases are considered for the cost function : a) power consumption, b) time derivative of the body shape (joint angles), and c) time derivative of the joint torques. The optimal gaits turned out to be different for these cost functions as shown below. While the shape-derivative optimal gait looks like a jelly fish with pulsing motion, the power optimal gait undulates the two "arms" in a symmetric manner. The torque-rate optimal gait also undulates, but the arm motions are asymmetric, which would reduce the torque at the center joint.

Our theory provides an optimal gait without prejudice from biological observations -- the optimization does not have to be initialized with a gait expected from animal locomotion. To demonstrate this feature, we created an "H-swimmer" that has a body shape not observed in nature. The optimal gaits with a prescribed turning rate (angular velocity) and locomotion speed are calculated for three cost functions: a) power consumption, b) time derivative of body curvature, and c) time derivative of joint torques.

Thrust generation in the power optimal gait is achieved by the arms sending asymmetrical waves down the right and left sides of the body, while the legs oscillate from side to side with small amplitudes. The oscillation amplitudes are larger on the left-hand side to achieve the desired turning. This asymmetrical gait has a lower power cost than a gait with symmetrical waves traveling down the right and left sides of the locomotor. The shape-derivative cost is minimized when the legs use a flapping motion to propel the locomotor forward, similar to the gait in jellyfish swimming. This gait is desirable because it allows the main body to remain steady without large oscillations about its orientation. However, it comes at a price of very large torque and power costs. In the optimal torque rate gait, the legs synchronously oscillate from side to side at a relatively high frequency, similar to the motion of caudal fins in fish. This gait saves input torque magnitude, but sacrifices large body shape changes and yawing motion. The result shows that it is optimal to use either arms or legs, but not both simultaneously, to generate thrust for locomotion. This is a counterintuitive result that one would not be able to predict without using the optimal gait theory.

Fundamental questions in understanding locomotion mechanisms include whether animals optimize the oscillation pattern (or gait) to achieve “efficient” locomotion, and if so, what is the optimality criterion. We addressed these questions in the context of swimming batoids, which propel the body by flapping their large pectoral fins. The flapping pattern varies from one species to another; batoids with round-shaped fins tend to send traveling waves across the fins and are called undulators, while batoids with triangular-shaped fins tend to flap the fins up and down and are called oscillators. We have developed mathematical models of Dasyatis sabina (Atlantic Stingray) and Rhinoptera bonasus (Cownose Ray) as representatives of undulators and oscillators, respectively, to analyze optimal gaits with respect to various cost functions, and compare with observed kinematic data. Four cost functions are considered - power, body curvature, muscle tension, and metabolic cost. The analysis shows that gaits similar to observed (animated below) result from minimization of a metabolic cost, and a body resonance is exploited to reduce cost associated with negative work.

References

Optimal Gaits for Mechanical Rectifier Systems

J. Blair and T. Iwasaki, IEEE Transactions on Automatic Control, vol.56, no.1, pp.59-71, 2011, accepted version

Dynamical Model and Optimal Turning Gait for Mechanical Rectifier Systems

S. Kohannim and T. Iwasaki, IEEE Transactions on Automatic Control, vol.62, no.2, pp.682-693, 2017, accepted version

Exploiting natural dynamics for gait generation in undulatory locomotion

D.T. Ludeke and T. Iwasaki, International Journal of Control, 2019 (To appear)

Analytical insights into optimality and resonance in fish swimming

S. Kohannim and T. Iwasaki, Journal of Royal Society Interface, vol.11, 20131073, 2014.

Modeling and Optimality Analysis of Pectoral Fin Locomotion

X. Liu, F. Fish, R. Russo, S. Blemker, and T. Iwasaki, Chapter 9, pp.309-332, Neuromechanical Modeling of Posture and Locomotion, Eds. B.I. Prilutsky, D.H. Edwards Eds., Springer 2016.

Optimal Oscillations and Chaos Generation in Biologically-Inspired Systems

Saba Kohannim, Ph.D Dissertation, UCLA, June 2016

Optimal locomotion of mechanical rectifier systems

Justin T. Blair, Ph.D Dissertation, University of Virginia, August 2011

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