Nonlinear Dynamics of Equine Locomotion and Robotic Systems

Abstract

Inherently most systems are nonlinear, whether they are biological systems in nature or mechanical systems in the industry. The analyses of nonlinear dynamic phenomena in these systems play an important role in monitoring and predicting their behavior. Tranquilization of horses with acepromazine has been used to suppress erratic head movements and increase the accuracy of a lameness examination. Some equine clinicians believe that tranquilization with acepromazine will make lameness more evident by causing the horse to focus on adjusting its gait to avoid limb pain rather than its surroundings. In this work, the effect of acepromazine on the Lyapunov exponents of lame horses was investigated. Ten lame horses were trotted straight for a minimum of 25 strides. Kinematic data created by head movement were analyzed with nonlinear dynamic methods. The local stability of horse gait remains unchanged after the administration of acepromazine.

To further investigate the critical phase of gait cycle for developing lameness, the impact of a horse hoof wall on three solid surfaces, steel, concrete, and asphalt, was studied. Impact experiments were conducted for different impact angles and various initial impact velocities. The effect of impact surfaces, impact angles, and initial impact velocities on the coefficient of restitution and the coefficient of friction were tested using one-way analyses of variance. Analytical and numerical modeling of the impact was developed. The impact interval was divided into two phases: compression and restitution. For compression, a contact force with a damping term was used. The restitution was characterized by an elastic contact force. The stiffness and damping coefficients of the contact force were estimated from the normal impacts. The simulated velocities after the oblique impacts were compared to those in the in vitro investigation. The coefficient of restitution varied significantly on different surfaces. The coefficient of friction was lower on steel than on concrete and asphalt. The presented model can be applied to refine the impact simulation of the equine hoof during locomotion.

The pendulum system is widely used in the dynamic models of bio-inspired quadruped robots. During impact, energy dissipation of some materials that are often used in quadruped robots, such as steel, is dominantly due to plastic deformation. To include this factor, a method to solve the impact of a kinematic chain in terms of a nonlinear contact force is presented. The nonlinear contact force has different expressions for elastic compression, elastoplastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the nonlinear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum, the kinetic energy of the non-impacting link increases, and the total kinetic energy of the impacting link decreases.

Actuation algorithm is another factor that affects the dynamics of a robot system. A two-link planar robotic arm with revolute and prismatic joints is considered. The control torque on the first link and the control force on the second link are periodic functions. The controllers are selected using generalized active forces. For the system’s dynamics, the phase plane is plotted. Entropy is used to quantitatively measure the regularity of the controlled periodic motion. The approximate entropy decreases with the magnitude of the actuation periods. A novel approach combining the continuous wavelet transform and a deep convolutional neural network is proposed to analyze the robotic system’s periodic motion automatically. The robustness of this approach is assessed by classifying the types of actuation controllers and the actuation period with an accuracy of 100%.

Additive manufacturing has rapidly developed as a highly flexible processing technique. In various engineering applications, additive manufactured structures suffer from vibrations during normal operations. However, the responses of additively manufactured parts are often studied only in the context of static loads. Here, the effects of fabrication parameters on the vibratory properties were examined. A total of 420 vibratory system identification experiments were conducted on 70 additive-manufactured cantilever beams. In addition, the Euler-Bernoulli beam theory and the finite element method were used to estimate the first natural frequency of the beams. It was found that build parameters change not only the stiffness of the beam but these build parameters also affect the damping ratio. Further, the frequency response of the beams is amplitude-dependent; this nonlinear effect is important in predicting the behavior of 3D printed structures. The complex relationship between the build parameters and the nonlinear vibratory response points to the possibility of creating tailored vibratory responses of 3D-printed structures.