Oliver M. O'Reilly

• B.E. (Mech.), M.S. and Ph.D.
• Professor at University of California, Berkeley

Many recent designs of soft robots and nano robots feature locomotion mechanisms that cleverly exploit slipping and sticking phenomena. These mechanisms have many features in common with peristaltic locomotion found in the animal world. The purpose of the present paper is to examine the energy efficiency of a locomotion mechanism that exploits fric...

Geodesics on SO(3) are characterized by constant angular velocity motions and as great circles on a three-sphere. The former interpretation is widely used in optometry and the latter features in the interpolation of rotations in computer graphics. The simplicity of these two disparate interpretations belies the complexity of the corresponding rotat...

Accurate movement analysis systems are prohibitive in cost and size to be accessible to the general population while commercially available, affordable systems lack the accuracy needed for clinical relevance. To address these limitations, we have developed a Depth Camera Movement Assessment System (DCMAS) that features an affordable, widely availab...

The accidental untying of a shoelace while walking often occurs without warning. In this paper, we discuss the series of events that lead to a shoelace knot becoming untied. First, the repeated impact of the shoe on the floor during walking serves to loosen the knot. Then, the whipping motions of the free ends of the laces caused by the leg swing p...

Modeling soft robots that move on surfaces is challenging from a variety of perspectives. A recent formulation by Bergou et al. of a rod theory that exploits new developments in discrete differential geometry offers an attractive, numerically efficient avenue to help overcome some of these challenges. Their formulation is an example of a discrete e...

The dynamics of a collapsing stack of rigid blocks is explored in the present paper. Stacks of this type are ubiquitous mechanical systems that can be used to model boxes stacked in warehouses, containers on ships, and stacks of boxes being transported by robots. Although studies on the dynamics of a single block moving on a horizontal plane are we...

A common, yet hazardous, method of transporting cylindrical tanks used to carry compressed gas involves rolling both tanks at opposite angles of inclination to the vertical. By propelling one of the tanks while maintaining point contact between the tanks, both tanks can be moved such that their centers of mass move in a straight line. The purpose o...

Consider a rigid body rolling with one point in contact with a fixed surface. Now suppose that the instantaneous point of contact traces out a closed path. As a demonstration of a phenomenon known as holonomy, the body will typically not return to its original orientation. The simplest demonstration of this phenomenon in rigid body dynamics occurs...

A balance law for material momentum in shells and plates is proposed. The implications of this balance law for the propagation of defects, phase transformations, and shocks are explored. The developments are presented using a purely mechanical theory of a Cosserat (or directed) shell and specialized to the cases of a Kirchhoff–Love shell theory and...

Microelectromechanical systems (MEMS) often make use of a type of plate-based electrostatic actuator that can, under certain conditions, come into contact with a dielectric-coated electrode. After this “touchdown,” the presence of a free boundary between the contacting and non-contacting regions of the plate induces strong nonlinearities in the sys...

Consider two points P and Q on a surface. Modulo rotations about the normal vector to the surface at P and the normal vector to the surface at Q, a rotation can be defined that maps the unit normal vector to the surface at Q to the corresponding unit normal vector at P. With the help of Weingarten’s formulae, new representations are established for...

The equations of motion for the simplest non-holonomically constrained system of particles are formulated using six methods: Newton–Euler, Lagrange, Maggi, Gibbs–Appell, Kane, and Boltzmann–Hamel. The challenging tasks of exploring and explaining the relationships and equivalences between these formulations is accomplished by constructing a single...

In this technical brief, a simple concise derivation of the Gibbs-Appell equations for the dynamics of a constrained rigid body is presented.

A continuous model for the peristaltic locomotion of compressible and incompressible rod-like bodies is presented. Using Green and Naghdi’s theory of a directed rod, incompressibility is enforced as an internal constraint. A discussion on muscle actuation models for a single continuum is included. The resulting theory is demonstrated in a simulatio...

Simple mechanical contact between a deformable body and a rigid obstacle can lead to rich nonlinear behavior even when the equations governing the body’s motion are otherwise linear. We study the nonlinear dynamics of contact both numerically and analytically in a model problem of an Euler–Bernoulli beam and a flat substrate. This problem is releva...

We present a systematic method for analyzing the vibrations of elastic rods whose effective length is variable, with particular emphasis on rods in unilateral contact with rigid surfaces. Problems of this type abound in engineering applications at all length scales, from the laying of submarine pipelines to the stiction of cantilevers in microelect...

A discrete model for elastic rods undergoing planar motions based on the theory of a directed (or Cosserat) rod is presented. Edge vectors and a director are used to capture cross section deformations including stretch, stretch gradients, shear, shear gradients, and the Poisson effect. In addition, deformations such as longitudinal stretch and bend...

Cambridge Core - Fluid Dynamics and Solid Mechanics - Intermediate Dynamics for Engineers - by Oliver M. O'Reilly

In this article, we propose a minimal model for the cooking-induced deformation of spaghetti and related food products. Our approach has parallels to the use of rod theories for the mechanics of slender bodies undergoing growth and is inspired by a wealth of experimental data from the food science literature. We use our model to investigate the coo...

The variation in the dynamic characteristics of a flexible riser as the riser transitions from a vertical riser to a catenary-type riser is investigated. It is well known that the straight configuration of a flexible vertical riser conveying fluid destablizes in a divergence-type instability once the velocity of the transporting fluid exceeds a cri...

The celebrated mathematician John E. Littlewood noted that a hoop with an attached mass rolling on a ground plane may exhibit self-induced jumping. Subsequent works showed that his analysis was flawed and revealed paradoxical behavior that can be resolved by incorporating the inertia of the hoop. A comprehensive analysis of this problem is presente...

This primer is intended to provide the theoretical background for the standard undergraduate, mechanical engineering course in dynamics. Representative problems are discussed and simulated throughout the book to illustrate fundamental concepts and explore the development of mathematical models for mechanical systems. The text grew out of the author...

As water depths for oil and gas exploration and extraction increase, structures such as flexible risers, mooring lines, and umbilical cables are increasingly being used for subsea environments. Compared to conventional fixed-type structures and vertical risers, the dynamics of flexible risers is significantly more complex. In particular, the flexib...

Representations for conservative and nonconservative moments in classical mechanics are discussed in this expository article. When the rotation is parameterized by a set of Euler angles, a particularly transparent representation can be found which has ties to classic works in mechanics dating to Lagrange in 1780 and joint coordinate systems that ar...

As a prelude to the discussion of a system of particles, the linear and angular momenta of a single particle are introduced in this chapter. In particular, conditions for the conservation of these kinematical quantities are established. This is followed by a discussion of impact problems where particles are used as models for the impacting bodies.

In this chapter, we continue the process of extending several results pertaining to a single particle to a system of particles. We start by defining the linear momentum, angular momenta, and kinetic energy for a system of particles. Next, we introduce a new concept, the center of mass C of a system of particles. A discussion of the conservation of...

This chapter is the culmination of the primer. To start, the linear momentum of a system of K particles and N rigid bodies is discussed. Similarly, the angular momenta and kinetic energy of such a system are developed. We then turn to the balance laws for a system and demonstrate how the balance laws can be used to determine the equations of motion...

We discuss the differential geometry of space curves (a curve embedded in Euclidean three-space) in this chapter. In particular, we introduce the Serret-Frenet basis vectors. This is followed by the derivation of an elegant set of relations describing the rate of change of the unit tangent, unit principal normal, and unit binormal vectors. Several...

The chapter starts with a discussion of the notions of power and work. Subsequently, we make these ideas more precise by defining the mechanical power of a force and, from this, the work done by the force during the motion of a particle. Next, the work-energy theorem is derived from the balance of linear momentum. It is then appropriate to discuss...

In this chapter, we discuss the cylindrical polar coordinate system and how it can be used in particle mechanics. This coordinate system and its associated basis vectors find application in a range of problems including particles moving on circular arcs and helical curves. To illustrate applications of the cylindrical polar coordinate system to par...

We start by discussing Euler’s laws for a rigid body. These laws are known as the balances of linear and angular momenta. An alternative form of these laws is also presented that is useful for solving many classes of problems. We then discuss the kinetic energy of a rigid body and establish the Koenig decomposition. This decomposition, combined wit...

Formulations of two ubiquitous forces are discussed in this chapter: friction forces and spring forces. We start with the former and consider a simple classic experiment. Based on this experiment, general (coordinate-free) expressions for friction forces are obtained. The chapter closes with the corresponding developments for a spring force. Severa...

In this first chapter, we cover the basics on kinematics and kinetics of particles with particular emphasis on the Cartesian coordinate system. Euler’s first law (also known as Newton’s second law or the balance of linear momentum) is used to relate the kinematics of the particle to the forces acting on it. This law provides a set of differential e...

Background material on the planar kinematics of rigid bodies is presented in this chapter. In particular, we show how to establish certain useful representations for the velocity and acceleration vectors of any material point of a rigid body. We also discuss the angular velocity vector of a rigid body. These concepts are illustrated using three imp...

Spherical robots have a wide range of self-propulsion mechanisms. Of particular interest in this paper, are propulsion systems where wheels are placed in contact with the inner surface of the spherical shell of the robot. Here, locomotion is achieved by a combination of the actions of the motors along with the rolling constraints at the point of co...

Coordinate singularities and gimbal lock are two phenomena that present themselves in models for the dynamics of mechanical systems. The former phenomenon pertains to the coordinates used to parameterize the configuration manifold of the system, while the latter phenomenon has a distinctive physical manifestation. In the present paper, we use tools...

Relevant background material on approximating a continuous space curve using a discrete set of lines connected at vertices is assembled in this chapter. The formulation uses concepts from the nascent field of discrete differential geometry. The resulting discretized curve is a central component of the discrete elastic rod formulation. In particular...

The method by which a component of the rotation of the cross-section is computed in the discrete elastic rod formulation is exceptional and exploits a phenomenon in differential geometry known as a holonomy. In this chapter, relevant background from differential geometry and spherical geometry are presented so the reader can understand how the refe...

The material frames associated with an edge of a discretized curve is discussed in this chapter. For the discrete curve, a pair of material vectors play the role of directors in Kirchhoff’s rod theory. The motion of these vectors is calibrated using either a Bishop frame or a reference frame that is continually being updated. The pair of parallel t...

Two of the three frames associated with an edge of a discretized curve are discussed in this chapter: a Bishop frame and a reference frame. Both of these frames are continually updated as the rod deforms and a pair of rotation operators are used to perform these updates. The operators are known as a space-parallel transport operator and a time-para...

In the discrete elastic rod formulation, expressions for the variations, gradients, and Hessians of kinematic variables induced by changes to the vertices are required. The present chapter provides the background and intermediate computations that are needed to establish the desired representations for these gradients and Hessians. Some of resultin...

The purpose of this final chapter is to illuminate how the kinematic results presented in the earlier chapters are used to formulate the governing equations for the discrete elastic rod. In particular, representations of the kinetic and potential energies for the discrete elastic rod formulation are discussed. The gradients of the elastic energies...

By way of background, a rapid review of Kirchhoff’s theory of an inextensible, unshearable elastic rod is discussed in this chapter. The formulation of the theory incorporates a pair of deformable vectors (or directors) associated with each point of a flexible material curve. In addition, the Frenet and Bishop framings of the material curve are int...

Interest in soft robots is motivated by their enormous potential in applications with close contact to humans or unpredictable situations for which their soft structure and versatility is beneficial. New mathematical models and control strategies are required to optimize the geometry and performance of the soft components of these robots. The autho...

With the goal of robustly designing and fabricating a soft robot based on a caterpillar featuring shape memory alloy (SMA) actuators, analytical and numerical models for a soft robot were created based on the forward crawling motion of the Manduca sexta caterpillar. The analytical model features a rod theory and the mechanics of undulation were ana...

While soft robots have many attractive features compared to their hard counterparts, developing tractable models for these highly deformable, nonlinear, systems is challenging. In a recent paper, the authors published a non-classic, five-parameter constitutive relation for a rod-based model of a widely used, pneumatically actuated soft robot arm. I...

This book presents theories of deformable elastic strings and rods and their application to broad classes of problems. Readers will gain insights into the formulation and analysis of models for mechanical and biological systems. Emphasis is placed on how the balance laws interplay with constitutive relations to form a set of governing equations. Fo...

We now turn to applications of the treatment in the previous chapter and develop mechanical models for the dynamics of a string. Many of the systems we consider in this chapter are classical but the analyses of their equations of motion is heavily influenced by treatments of, and novel insights into, these problems that have appeared during the pas...

The theory of an elastic rod whose centerline is inextensible and whose cross sections remain plane and normal to the centerline is discussed. This theory, which is known as Kirchhoff rod theory, is presented in the modern context of a Cosserat rod theory. The governing equations for this widely used theory result in a set of equations to determine...

We now consider a generalization of Kirchhoff’s rod theory to a theory which can accommodate extensibility of the centerline and transverse shear of the cross sections. The theory is constructed by allowing the rod to extend and shear. Consequently, the only significant change to the balance laws lies in the strain energy function. Theories of this...

We review the concepts of writhe, twist, and linking as applied to space curves and ribbons. The application of these concepts to DNA is also discussed.

The rod theory discussed in this chapter originated in a paper by Green and Laws in 1966 [128]. In this theory, the material curve is extensible, and the directors d α can change their length and relative orientation. This theory was further developed in a series of papers by Green, Naghdi, and several of their coworkers. They also showed how it co...

The theory of the elastica is discussed in this chapter. In addition to classical buckling problems, several applications of this theory to rod-like bodies adhering to rigid substrates are discussed.

In this supplemental chapter, we review several elements of continuum mechanics. The main topics we cover are curvilinear coordinates, stress tensors, balance laws, constitutive equations, and kinematical constraints. The material covered is background needed for the development of rod and string theories from three-dimensional considerations that...

We start this book with arguably the simplest theory of a deformable elastic body - that of a string or, as it is also known, a one-dimensional continuum. Our motivation is the development of a theory that can accommodate a range of effects such as gravity, spatial discontinuities in velocity, applied forces which are concentrated at a point, large...

Several applications of methods from the calculus of variations to rod theories are presented in this chapter.

Coordinate singularities and gimbal lock are two phenomena that feature in models for the dynamics of mechanical systems. The coordinates used to describe the motion of these systems induce a basis for the tangent space to the configuration manifold of the system. In this paper, we demonstrate how coordinate singularities manifest themselves as the...

Biomechanics software programs, such as Visual3D, Nexus, Cortex, and OpenSim, have the capability of generating several distinct component representations for joint moments and forces from motion capture data. These representations include those for orthonormal proximal and distal coordinate systems and a non-orthogonal joint coordinate system. In...

Soft robots are bio-inspired, highly deformable robots with the ability to interact with workpieces in a manner that complements their hard robot counterparts. To develop practical applications and reproducible designs of soft robots, new models are necessary to describe their kinematics and dynamics. In the present work, we describe experimental a...

The recent surge of interest in soft robotics has led to interesting designs and fabrication of flexible actuators composed of soft matter. Modeling these actuators to obtain quantitative estimates of their dynamics is challenging. In the present paper, a rod-based model for a popular pneumatically activated soft robot arm is developed. The model i...

In a recent series of works, mass-modulation schemes have been proposed for a class of ocean wave energy converter (WEC). The goal of the schemes is to improve the energy harvesting capabilities of these devices by taking advantage of the ambient water. However this improvement comes at the cost of increased system complexity and possible impulse l...

Many recently synthesized materials feature aligned arrays or bundles of carbon nanotubes (CNTs) whose mechanical properties are partially determined by the van der Waals interactions between adjacent tubes. Of particular interest in this paper are instances where the resulting interaction between a pair of CNTs often produces a fork-like structure...

Geodesics on SO(3) are characterized by constant angular velocity motions and as great circles on a three-sphere. The former interpretation is widely used in optometry and the latter features in the interpolation of rotations in computer graphics. The simplicity of these two disparate interpretations belies the complexity of the corresponding rotat...

In contrast to their more rigid counterparts, soft robots have the ability to gently grip and maneuver objects with open-loop kinematic control. Guided by several recent designs and implementations of soft robot hands, the present paper analyzes a rod-based model for the fingers in the hand of a soft robot. We show precisely how gripping is achieve...

Several recent designs of soft robots feature locomotion mechanisms that entail orchestrating changes to intrinsic curvature to enable the robot's limbs to either stick, adhere, or slip on the robot's workspace. The resulting locomotion mechanism has several features in common with peristaltic locomotion that can be found in the animal world. The p...

The simplest model with which to examine the dynamics of the human eye consists of a rigid body which is free to rotate about a fixed point. Two classical laws governing monocular vision, which are known as Listing's law and Donders' law, can be enforced in this model using a single holonomic constraint. While there has been considerable attention...

In a recent paper by Bosi et al. (2014 Proc. R. Soc. A 470, 20140232. (doi:10.1098/rspa.2014.0232)) an ingenious deformable arm scale was designed and developed. The scale's operation relies on the presence of Eshelby-like forces. In this paper, we gain new insight into the operation of the arm scale by using a material (configurational) force bala...

In the context of controlling the attitude of a rigid body, this communique uses recent results on representations of torques (moments) to establish cost functions. The resulting cost functions are both properly invariant under whatever choice of Euler angles is used to parameterize the rotation of the rigid body and have transparent physical inter...

In this expository article, a simple concise treatment of Lagrange's prescription for constraint forces and constraint moments in the dynamics of rigid bodies is presented. The treatment is suited to both Newton-Euler and Lagrangian treatments of rigid body dynamics and is illuminated with a range of examples from classical mechanics and orthopedic...

This supplemental work to Christophy et al. ``On the modeling of the intervertebral joint in multibody models for the spine" (Multibody System Dynamics, Vol. 30, No. 4, 413-432 (2013).) reviews various formulations of stiffness matrices that can be used in conjunction with the relative motion of a pair of rigid bodies. Particular attention is paid...

The need to develop feasible computational musculoskeletal models of the spine has led to the development of several multibody models. Central features in these works are models for the ligaments, muscles, and intervertebral joint. The purpose of the present paper is to show how experimental measurements of joint stiffnesses can be properly incorpo...

Electronic supplementary material for On the modeling of the intervertebral joint in multibody models for the spine: Review of Stiffness Matrices for the Intervertebral Joint

In a recent paper by Orazov et al. [On the dynamics of a novel ocean wave energy converter. Journal of Sound and Vibration329 (24) (2010) 5058-5069], a wave energy converter (WEC) was proposed. The converter features a mass modulation scheme and a simple model was used to examine its efficacy. The simple model did not adequately account for the mom...

In studies of the biomechanics of joints, the representation of moments using the joint coordinate system has been discussed by several authors. The primary purpose of this technical brief is to emphasize that there are two distinct, albeit related, representations for moment vectors using the joint coordinate system. These distinct representations...

Motivated by applications such as gecko-inspired adhesives and microdevices featuring slender rod-like bodies, there has been an increase in interest in the deformed shapes of elastic rods adhering to rigid surfaces. A central issue in analyses of the rod-based models for these systems is the stability of the predicted equilibrium configurations. S...

An oscillator of the form q¨(t)+2ζq˙(t)+q(t)=−κ[q(t)−q(t−r)] is unstable when the strength of the feedback (κκ) is greater than a critical value (κcκc). Oscillations of constant amplitude persist when κ=κcκ=κc. We study the almost-sure asymptotic stability of the oscillator when κ=κcκ=κc and the system is excited by a two-state Markov noise. For sm...

This paper presents a treatment of material symmetry for hyperelastic rods. The rod theory of interest is based on a Cosserat (or directed) curve with two director fields, and was developed in a series of works by Green, Naghdi and several of their co-workers. The treatment is based on Murdoch and Cohen's work on material symmetry of Cosserat surfa...

Using variational methods, we establish conditions for the nonlinear stability of adhesive states between an elastica and a rigid halfspace. The treatment produces coupled criteria for adhesion and buckling instabilities by exploiting classical techniques from Legendre and Jacobi. Three examples that arise in a broad range of engineered systems, fr...

The human spine has an elaborate system of muscles and ligaments which serve to actuate this complex biomechanical system. In this paper, the buckling instabilities of the ligamentous spine are explored using a model based on Euler's elastica. The model features the intrinsic curvature and self-weight of the spine. With the help of nonlinear stabil...

The dynamics of a simple model for an ocean wave energy converter is discussed. The model for the converter is a hybrid system consisting of a pair of harmonically excited mass–spring–dashpot systems and a set of four state-dependent switching rules. Of particular interest is the response of the model to a wide spectrum of harmonic excitations. Pa...

It is common practice in analyses of the configurations of an elastica to use Jacobi’s necessary condition to establish conditions for stability. Analyses of this type date to Born’s seminal work on the elastica in 1906 and continue to the present day. Legendre developed a treatment of the second variation which predates Jacobi’s. The purpose of th...

Chronic low back pain (CLBP) is one of the most common health problems in humans. To better understand the mechanics and dynamics of the spine, six thoracolumbar spines (T12 - Pelvis) were mounted into a custom mechanical testing apparatus, which closely approximated in vivo conditions. This testing apparatus was mainly composed of three components...

New necessary conditions for the nonlinear stability of branched tree-like structures composed of elastic rods are presented. The conditions, which are found by examining the second variation of an energy functional, are established by extending Legendre' classical work on this topic. For the branched tree-like structures of interest in this paper,...

Motivated by studies on a type of brain injury known as diffuse axonal injury, the dynamic azimuthal shearing of a mixture of a transversely isotropic viscoelastic material which is surrounded by a softer isotropic viscoelastic material is considered. It is demonstrated how the states of maximum strain and strain rate occur near the interface betwe...

Back pain is a well known health issue and several approaches to understanding the mechanics and dynamics of the spine, including mechanical testing, are used. Presently, several spine testing devices, such as those described in [1,2], are available which impose physiologic loads for various postures of the spine. Although these devices are success...

In many applications of rod theories as models for plant stem growth and development, it is necessary to allow the intrinsic curvature and flexural stiffness of the rod to evolve. In the present paper, the application of evolution equations for these quantities is examined and a new evolution equation for the intrinsic curvature is proposed. To ill...

Structures featuring rod-like bodies connected in a branched ensemble are ubiquitous. They appear in antennae, replicating DNA strands, and, most visibly, in the plant kingdom. In spite of this, modeling these branched structures using rod theories have received little attention in the literature. With the help of a general rod theory, which was or...

A new musculoskeletal model for the lumbar spine is described in this paper. This model features a rigid pelvis and sacrum, the five lumbar vertebrae, and a rigid torso consisting of a lumped thoracic spine and ribcage. The motion of the individual lumbar vertebrae was defined as a fraction of the net lumbar movement about the three rotational degr...

Several Cartesian stiffness matrices for a single rigid body subject to a conservative force field are developed in this paper. The treatment is based on energetic arguments and an Euler angle parameterization of the rotation of the rigid body is employed. Several new representations for the stiffness matrix are obtained and the relation to other w...

Posterolateral rotatory instability (PLRI) of the elbow results from injury to the lateral collateral ligament complex from trauma or iatrogenic injury. The lateral pivot-shift test (PST) is standard for diagnosing PLRI, but its subjectivity affects diagnosis and makes it difficult to train young surgeons. A well-controlled investigation has not be...

Buoy-type ocean wave energy converters are designed to exhibit resonant responses when subject to excitation by ocean waves. A novel excitation scheme is proposed which has the potential to improve the energy harvesting capabilities of these converters. The scheme uses the incident waves to modulate the mass of the device in a manner which amplifie...

One of the more common comparative tools used to quantify the motion of the vertebral joint is the orientation and position of the (finite) helical axis of motion as well as the amount of translation along, and rotation about, this axis. A survey of recent studies that utilize the helical axis of motion to compare motion before and after total disc...

For convenience in this appendix we use the following abbreviations: BF, Bedford 5 and Fowler [6]; BJ, Beer and Johnston [7]; H, Hibbeler [36]; MK, Meriam and 6 Kraige [48]; RS, Riley and Sturges [63]; and S, Shames [69].

In this chapter we discuss the differential geometry of space curves (a curve embedded in Euclidean three-space \mathbbE3 \mathbb{E}^3 ). In particular, we introduce the Serret-Frenet basis vectors { et, en, eb } \left\{ {{\bf e}_t, {\bf e}_n, {\bf e}_b } \right\} . This is followed by the derivation of an elegant set of relations describing t...