Model and control of tendon actuated robots

The use of tendons for the transmission of the forces and the movements in robotic devices has been investigated from several researchers all over the world. The interest in this kind of actuation modality is based on the possibility of optimizing the position of the actuators with respect to the moving part of the robot, in the reduced weight, high reliability, simplicity in the mechanic design and, finally, in the reduced cost of the resulting kinematic chain. After a brief discussion about the benefits that the use of tendons can introduce in the motion control of a robotic device, the design and control aspects of the UB Hand 3 anthropomorphic robotic hand are presented. In particular, the tendon-sheaths transmission system adopted in the UB Hand 3 is analyzed and the problem of force control and friction compensation is taken into account. The implementation of a tendon based antagonistic actuated robotic arm is then investigated. With this kind of actuation modality, and by using transmission elements with nonlinear force/compression characteristic, it is possible to achieve simultaneous stiffness and position control, improving in this way the safety of the device during the operation in unknown environments and in the case of interaction with other robots or with humans. The problem of modeling and control of this type of robotic devices is then considered and the stability analysis of proposed controller is reported. At the end, some tools for the realtime simulation of dynamic systems are presented. This realtime simulation environment has been developed with the aim of improving the reliability of the realtime control applications both for rapid prototyping of controllers and as teaching tools for the automatic control courses.

Interconnection systems for highly integrated computation devices

The sustained demand for faster,more powerful chips has beenmet by the availability of chip manufacturing processes allowing for the integration of increasing numbers of computation units onto a single die. The resulting outcome, especially in the embedded domain, has often been called SYSTEM-ON-CHIP (SOC) or MULTI-PROCESSOR SYSTEM-ON-CHIP (MPSOC). MPSoC design brings to the foreground a large number of challenges, one of the most prominent of which is the design of the chip interconnection. With a number of on-chip blocks presently ranging in the tens, and quickly approaching the hundreds, the novel issue of how to best provide on-chip communication resources is clearly felt. NETWORKS-ON-CHIPS (NOCS) are the most comprehensive and scalable answer to this design concern. By bringing large-scale networking concepts to the on-chip domain, they guarantee a structured answer to present and future communication requirements. The point-to-point connection and packet switching paradigms they involve are also of great help in minimizing wiring overhead and physical routing issues. However, as with any technology of recent inception, NoC design is still an evolving discipline. Several main areas of interest require deep investigation for NoCs to become viable solutions: • The design of the NoC architecture needs to strike the best tradeoff among performance, features and the tight area and power constraints of the on-chip domain. • Simulation and verification infrastructure must be put in place to explore, validate and optimize the NoC performance. • NoCs offer a huge design space, thanks to their extreme customizability in terms of topology and architectural parameters. Design tools are needed to prune this space and pick the best solutions. • Even more so given their global, distributed nature, it is essential to evaluate the physical implementation of NoCs to evaluate their suitability for next-generation designs and their area and power costs. This dissertation focuses on all of the above points, by describing a NoC architectural implementation called ×pipes; a NoC simulation environment within a cycle-accurate MPSoC emulator called MPARM; a NoC design flow consisting of a front-end tool for optimal NoC instantiation, called SunFloor, and a set of back-end facilities for the study of NoC physical implementations. This dissertation proves the viability of NoCs for current and upcoming designs, by outlining their advantages (alongwith a fewtradeoffs) and by providing a full NoC implementation framework. It also presents some examples of additional extensions of NoCs, allowing e.g. for increased fault tolerance, and outlines where NoCsmay find further application scenarios, such as in stacked chips.

TCAD approaches to multidimensional simulation of advanced semiconductor devices

Technology scaling increasingly emphasizes complexity and non-ideality of the electrical behavior of semiconductor devices and boosts interest on alternatives to the conventional planar MOSFET architecture. TCAD simulation tools are fundamental to the analysis and development of new technology generations. However, the increasing device complexity is reflected in an augmented dimensionality of the problems to be solved. The trade-off between accuracy and computational cost of the simulation is especially influenced by domain discretization: mesh generation is therefore one of the most critical steps and automatic approaches are sought. Moreover, the problem size is further increased by process variations, calling for a statistical representation of the single device through an ensemble of microscopically different instances. The aim of this thesis is to present multi-disciplinary approaches to handle this increasing problem dimensionality in a numerical simulation perspective. The topic of mesh generation is tackled by presenting a new Wavelet-based Adaptive Method (WAM) for the automatic refinement of 2D and 3D domain discretizations. Multiresolution techniques and efficient signal processing algorithms are exploited to increase grid resolution in the domain regions where relevant physical phenomena take place. Moreover, the grid is dynamically adapted to follow solution changes produced by bias variations and quality criteria are imposed on the produced meshes. The further dimensionality increase due to variability in extremely scaled devices is considered with reference to two increasingly critical phenomena, namely line-edge roughness (LER) and random dopant fluctuations (RD). The impact of such phenomena on FinFET devices, which represent a promising alternative to planar CMOS technology, is estimated through 2D and 3D TCAD simulations and statistical tools, taking into account matching performance of single devices as well as basic circuit blocks such as SRAMs. Several process options are compared, including resist- and spacer-defined fin patterning as well as different doping profile definitions. Combining statistical simulations with experimental data, potentialities and shortcomings of the FinFET architecture are analyzed and useful design guidelines are provided, which boost feasibility of this technology for mainstream applications in sub-45 nm generation integrated circuits.

Mobile TuCSoN: theoretical and technological requirements for TuCSoN's porting over Android mobile devices

Communication and coordination are two key-aspects in open distributed agent system, being both responsible for the system’s behaviour integrity. An infrastructure capable to handling these issues, like TuCSoN, should to be able to exploit modern technologies and tools provided by fast software engineering contexts. Thesis aims to demonstrate TuCSoN infrastructure’s abilities to cope new possibilities, hardware and software, offered by mobile technology. The scenarios are going to configure, are related to the distributed nature of multi-agent systems where an agent should be located and runned just on a mobile device. We deal new mobile technology frontiers concerned with smartphones using Android operating system by Google. Analysis and deployment of a distributed agent-based system so described go first to impact with quality and quantity considerations about available resources. Engineering issue at the base of our research is to use TuCSoN against to reduced memory and computing capability of a smartphone, without the loss of functionality, efficiency and integrity for the infrastructure. Thesis work is organized on two fronts simultaneously: the former is the rationalization process of the available hardware and software resources, the latter, totally orthogonal, is the adaptation and optimization process about TuCSoN architecture for an ad-hoc client side release.

Mixed Mode Fracture Behaviour of Piezoelectric Materials

Piezoelectrics present an interactive electromechanical behaviour that, especially in recent years, has generated much interest since it renders these materials adapt for use in a variety of electronic and industrial applications like sensors, actuators, transducers, smart structures. Both mechanical and electric loads are generally applied on these devices and can cause high concentrations of stress, particularly in proximity of defects or inhomogeneities, such as flaws, cavities or included particles. A thorough understanding of their fracture behaviour is crucial in order to improve their performances and avoid unexpected failures. Therefore, a considerable number of research works have addressed this topic in the last decades. Most of the theoretical studies on this subject find their analytical background in the complex variable formulation of plane anisotropic elasticity. This theoretical approach bases its main origins in the pioneering works of Muskelishvili and Lekhnitskii who obtained the solution of the elastic problem in terms of independent analytic functions of complex variables. In the present work, the expressions of stresses and elastic and electric displacements are obtained as functions of complex potentials through an analytical formulation which is the application to the piezoelectric static case of an approach introduced for orthotropic materials to solve elastodynamics problems. This method can be considered an alternative to other formalisms currently used, like the Stroh’s formalism. The equilibrium equations are reduced to a first order system involving a six-dimensional vector field. After that, a similarity transformation is induced to reach three independent Cauchy-Riemann systems, so justifying the introduction of the complex variable notation. Closed form expressions of near tip stress and displacement fields are therefore obtained. In the theoretical study of cracked piezoelectric bodies, the issue of assigning consistent electric boundary conditions on the crack faces is of central importance and has been addressed by many researchers. Three different boundary conditions are commonly accepted in literature: the permeable, the impermeable and the semipermeable (“exact”) crack model. This thesis takes into considerations all the three models, comparing the results obtained and analysing the effects of the boundary condition choice on the solution. The influence of load biaxiality and of the application of a remote electric field has been studied, pointing out that both can affect to a various extent the stress fields and the angle of initial crack extension, especially when non-singular terms are retained in the expressions of the electro-elastic solution. Furthermore, two different fracture criteria are applied to the piezoelectric case, and their outcomes are compared and discussed. The work is organized as follows: Chapter 1 briefly introduces the fundamental concepts of Fracture Mechanics. Chapter 2 describes plane elasticity formalisms for an anisotropic continuum (Eshelby-Read-Shockley and Stroh) and introduces for the simplified orthotropic case the alternative formalism we want to propose. Chapter 3 outlines the Linear Theory of Piezoelectricity, its basic relations and electro-elastic equations. Chapter 4 introduces the proposed method for obtaining the expressions of stresses and elastic and electric displacements, given as functions of complex potentials. The solution is obtained in close form and non-singular terms are retained as well. Chapter 5 presents several numerical applications aimed at estimating the effect of load biaxiality, electric field, considered permittivity of the crack. Through the application of fracture criteria the influence of the above listed conditions on the response of the system and in particular on the direction of crack branching is thoroughly discussed.

Multi Processor Systems On Chip with Configurable Hardware Acceleration

The evolution of the electronics embedded applications forces electronics systems designers to match their ever increasing requirements. This evolution pushes the computational power of digital signal processing systems, as well as the energy required to accomplish the computations, due to the increasing mobility of such applications. Current approaches used to match these requirements relies on the adoption of application specific signal processors. Such kind of devices exploits powerful accelerators, which are able to match both performance and energy requirements. On the other hand, the too high specificity of such accelerators often results in a lack of flexibility which affects non-recurrent engineering costs, time to market, and market volumes too. The state of the art mainly proposes two solutions to overcome these issues with the ambition of delivering reasonable performance and energy efficiency: reconfigurable computing and multi-processors computing. All of these solutions benefits from the post-fabrication programmability, that definitively results in an increased flexibility. Nevertheless, the gap between these approaches and dedicated hardware is still too high for many application domains, especially when targeting the mobile world. In this scenario, flexible and energy efficient acceleration can be achieved by merging these two computational paradigms, in order to address all the above introduced constraints. This thesis focuses on the exploration of the design and application spectrum of reconfigurable computing, exploited as application specific accelerators for multi-processors systems on chip. More specifically, it introduces a reconfigurable digital signal processor featuring a heterogeneous set of reconfigurable engines, and a homogeneous multi-core system, exploiting three different flavours of reconfigurable and mask-programmable technologies as implementation platform for applications specific accelerators. In this work, the various trade-offs concerning the utilization multi-core platforms and the different configuration technologies are explored, characterizing the design space of the proposed approach in terms of programmability, performance, energy efficiency and manufacturing costs.

Utilizzo di dispositivi isteretici per l'isolamento di piano: strategie per una progettazione sismica di tipo multi-prestazionale

Con la presente tesi si è inteso studiare le possibilità applicative di una particolare tipologia strutturale dotata di isolamento sismico “di piano”, intendendosi con ciò una struttura in cui l'intero piano terra, tramite l'inserimento di opportuni elementi dissipativi isteretici ed in analogia al consueto isolamento sismico di base, agisce da “strato” di protezione passiva per i piani sovrastanti. A riguardo, fra le possibili soluzioni per realizzare effettivamente tale isolamento "di piano" è stata considerata la disposizione di particolari elementi dissipativi isteretici di controvento detti Crecent-Shaped Braces, caratterizzati da una forma bilatera o curva, tale comunque da presentare un'eccentricità non nulla fra l'asse del controvento stesso e la linea congiungente gli estremi.

Advanced Beamforming for Distributed and Multi-Beam Networks

Beamforming entails joint processing of multiple signals received or transmitted by an array of antennas. This thesis addresses the implementation of beamforming in two distinct systems, namely a distributed network of independent sensors, and a broad-band multi-beam satellite network. With the rising popularity of wireless sensors, scientists are taking advantage of the flexibility of these devices, which come with very low implementation costs. Simplicity, however, is intertwined with scarce power resources, which must be carefully rationed to ensure successful measurement campaigns throughout the whole duration of the application. In this scenario, distributed beamforming is a cooperative communication technique, which allows nodes in the network to emulate a virtual antenna array seeking power gains in the order of the size of the network itself, when required to deliver a common message signal to the receiver. To achieve a desired beamforming configuration, however, all nodes in the network must agree upon the same phase reference, which is challenging in a distributed set-up where all devices are independent. The first part of this thesis presents new algorithms for phase alignment, which prove to be more energy efficient than existing solutions. With the ever-growing demand for broad-band connectivity, satellite systems have the great potential to guarantee service where terrestrial systems can not penetrate. In order to satisfy the constantly increasing demand for throughput, satellites are equipped with multi-fed reflector antennas to resolve spatially separated signals. However, incrementing the number of feeds on the payload corresponds to burdening the link between the satellite and the gateway with an extensive amount of signaling, and to possibly calling for much more expensive multiple-gateway infrastructures. This thesis focuses on an on-board non-adaptive signal processing scheme denoted as Coarse Beamforming, whose objective is to reduce the communication load on the link between the ground station and space segment.

Experimental and theoretical study of microstructure effect on piezoelectric property of one dimensional ZnO nanostructures

Emerging nanogenerators have attracted the attention of the research community, focusing on energy generation using piezoelectric nanomaterials. Nanogenerators can be utilized for powering NEMS/MEMS devices. Understanding the piezoelectric properties of ZnO one-dimensional materials such as ZnO nanobelts (NBs) and Nanowires (NWs) can have a significant impact on the design of new devices. The goal of this dissertation is to study the piezoelectric properties of one-dimensional ZnO nanostructures both experimentally and theoretically. First, the experimental procedure for producing the ZnO nanostructures is discussed. The produced ZnO nanostructures were characterized using an in-situ atomic force microscope and a piezoelectric force microscope. It is shown that the electrical conductivity of ZnO NBs is a function of applied mechanical force and its crystalline structure. This phenomenon was described in the context of formation of an electric field due to the piezoelectric property of ZnO NBs. In the PFM studies, it was shown that the piezoelectric response of the ZnO NBs depends on their production method and presence of defects in the NB. Second, a model was proposed for making nanocomposite electrical generators based on ZnO nanowires. The proposed model has advantages over the original configuration of nanogenerators which uses an AFM tip for bending the ZnO NWs. Higher stability of the electric source, capability for producing larger electric fields, and lower production costs are advantages of this configuration. Finally, piezoelectric properties of ZnO NBs were simulated using the molecular dynamics (MD) technique. The size-scale effect on piezoelectric properties of ZnO NBs was captured, and it is shown that the piezoelectric coefficient of ZnO NBs decreases by increasing their lateral dimensions. This phenomenon is attributed to the surface charge redistribution and compression of unit cells that are placed on the outer shell of ZnO NBs.

MULTISCALE EXAMINATION AND MODELING OF ELECTRON TRANSPORT IN NANOSCALE MATERIALS AND DEVICES

For half a century the integrated circuits (ICs) that make up the heart of electronic devices have been steadily improving by shrinking at an exponential rate. However, as the current crop of ICs get smaller and the insulating layers involved become thinner, electrons leak through due to quantum mechanical tunneling. This is one of several issues which will bring an end to this incredible streak of exponential improvement of this type of transistor device, after which future improvements will have to come from employing fundamentally different transistor architecture rather than fine tuning and miniaturizing the metal-oxide-semiconductor field effect transistors (MOSFETs) in use today. Several new transistor designs, some designed and built here at Michigan Tech, involve electrons tunneling their way through arrays of nanoparticles. We use a multi-scale approach to model these devices and study their behavior. For investigating the tunneling characteristics of the individual junctions, we use a first-principles approach to model conduction between sub-nanometer gold particles. To estimate the change in energy due to the movement of individual electrons, we use the finite element method to calculate electrostatic capacitances. The kinetic Monte Carlo method allows us to use our knowledge of these details to simulate the dynamics of an entire device— sometimes consisting of hundreds of individual particles—and watch as a device ‘turns on’ and starts conducting an electric current. Scanning tunneling microscopy (STM) and the closely related scanning tunneling spectroscopy (STS) are a family of powerful experimental techniques that allow for the probing and imaging of surfaces and molecules at atomic resolution. However, interpretation of the results often requires comparison with theoretical and computational models. We have developed a new method for calculating STM topographs and STS spectra. This method combines an established method for approximating the geometric variation of the electronic density of states, with a modern method for calculating spin-dependent tunneling currents, offering a unique balance between accuracy and accessibility.

Gesture-Based, Touch-Free Multi-User Gaming on Wall-Sized, High-Resolution Tiled Displays

Having to carry input devices can be inconvenient when interacting with wall-sized, high-resolution tiled displays. Such displays are typically driven by a cluster of computers. Running existing games on a cluster is non-trivial, and the performance attained using software solutions like Chromium is not good enough. This paper presents a touch-free, multi-user, humancomputer interface for wall-sized displays that enables completely device-free interaction. The interface is built using 16 cameras and a cluster of computers, and is integrated with the games Quake 3 Arena (Q3A) and Homeworld. The two games were parallelized using two different approaches in order to run on a 7x4 tile, 21 megapixel display wall with good performance. The touch-free interface enables interaction with a latency of 116 ms, where 81 ms are due to the camera hardware. The rendering performance of the games is compared to their sequential counterparts running on the display wall using Chromium. Parallel Q3A’s framerate is an order of magnitude higher compared to using Chromium. The parallel version of Homeworld performed on par with the sequential, which did not run at all using Chromium. Informal use of the touch-free interface indicates that it works better for controlling Q3A than Homeworld.

Quantitative multi-slice computed tomography assessment of the mitral valvular complex for transcatheter mitral valve interventions part 1: systematic measurement methodology and inter-observer variability

AIMS Transcatheter mitral valve replacement (TMVR) is an emerging technology with the potential to treat patients with severe mitral regurgitation at excessive risk for surgical mitral valve surgery. Multimodal imaging of the mitral valvular complex and surrounding structures will be an important component for patient selection for TMVR. Our aim was to describe and evaluate a systematic multi-slice computed tomography (MSCT) image analysis methodology that provides measurements relevant for transcatheter mitral valve replacement. METHODS AND RESULTS A systematic step-by-step measurement methodology is described for structures of the mitral valvular complex including: the mitral valve annulus, left ventricle, left atrium, papillary muscles and left ventricular outflow tract. To evaluate reproducibility, two observers applied this methodology to a retrospective series of 49 cardiac MSCT scans in patients with heart failure and significant mitral regurgitation. For each of 25 geometrical metrics, we evaluated inter-observer difference and intra-class correlation. The inter-observer difference was below 10% and the intra-class correlation was above 0.81 for measurements of critical importance in the sizing of TMVR devices: the mitral valve annulus diameters, area, perimeter, the inter-trigone distance, and the aorto-mitral angle. CONCLUSIONS MSCT can provide measurements that are important for patient selection and sizing of TMVR devices. These measurements have excellent inter-observer reproducibility in patients with functional mitral regurgitation.

Response profiles of murine spiral ganglion neurons on multi-electrode arrays.

OBJECTIVE Cochlear implants (CIs) have become the gold standard treatment for deafness. These neuroprosthetic devices feature a linear electrode array, surgically inserted into the cochlea, and function by directly stimulating the auditory neurons located within the spiral ganglion, bypassing lost or not-functioning hair cells. Despite their success, some limitations still remain, including poor frequency resolution and high-energy consumption. In both cases, the anatomical gap between the electrode array and the spiral ganglion neurons (SGNs) is believed to be an important limiting factor. The final goal of the study is to characterize response profiles of SGNs growing in intimate contact with an electrode array, in view of designing novel CI devices and stimulation protocols, featuring a gapless interface with auditory neurons. APPROACH We have characterized SGN responses to extracellular stimulation using multi-electrode arrays (MEAs). This setup allows, in our view, to optimize in vitro many of the limiting interface aspects between CIs and SGNs. MAIN RESULTS Early postnatal mouse SGN explants were analyzed after 6-18 days in culture. Different stimulation protocols were compared with the aim to lower the stimulation threshold and the energy needed to elicit a response. In the best case, a four-fold reduction of the energy was obtained by lengthening the biphasic stimulus from 40 μs to 160 μs. Similarly, quasi monophasic pulses were more effective than biphasic pulses and the insertion of an interphase gap moderately improved efficiency. Finally, the stimulation with an external electrode mounted on a micromanipulator showed that the energy needed to elicit a response could be reduced by a factor of five with decreasing its distance from 40 μm to 0 μm from the auditory neurons. SIGNIFICANCE This study is the first to show electrical activity of SGNs on MEAs. Our findings may help to improve stimulation by and to reduce energy consumption of CIs and thereby contribute to the development of fully implantable devices with better auditory resolution in the future.

The association between maternal use of spermicides, condoms, intra-uterine devices or progesterone and major structural birth defects

Birth defects occur in 1 of every 33 babies born in the United States, and are the leading cause of infant death. Mothers using contraceptives that become pregnant may continue to use their contraceptives after their first missed menstrual period, thus exposing their baby in utero to the contraceptive product. Progesterone is also sometimes prescribed during the first trimester of pregnancy to mothers with a history of miscarriages or infertility problems. To ensure the safety of these products, it is important to investigate whether there is an increased occurrence of babies born with birth defects to mothers using various contraceptive methods or progesterone in early pregnancy. Using data from the National Birth Defects Prevention Study (NBDPS), an ongoing multi-state, population based case-control study, this study assessed maternal exposures to IUDs, spermicides, condoms and progesterone in early pregnancy. ^ Progesterone used for threatened miscarriage during the first three months of pregnancy was associated with an increased occurrence of hypoplastic left heart (adjusted odds ratios (OR) 2.24, 95% CI 1.13-4.21), perimembranous ventricular septal defects (OR 1.64, 95% CI 1.10-2.41), septal associations (OR 2.52, 95% CI 1.45-4.24), esophageal atresia (OR 1.82, 95% CI 1.04-3.08), and hypospadias (OR 2.12, 95% CI 1.41-3.18). Mothers using progesterone for injectable contraception had increased (OR > 2.5), but insignificant odds ratios for anencephaly, septal associations, small intestinal atresias and omphalocel. Progesterone used for fertility was not associated with an increased occurrence of any birth defects examined. ^ Mothers using progesterone for fertility assistance and threatened miscarriage were very similar with respect to their demographics and pregnancy history. They also both reported similar types of progesterone. Thus, if progesterone was a causal risk factor for birth defects we would have expected to observe similar increases in risk among mothers using progesterone for both indications. Because we predominantly observed increased associations among mothers using progesterone for threatened miscarriage but not fertility assistance, it is possible the increased associations we observed were confounded by indication (i.e. progesterone was administered for vaginal bleeding which occurred as a sequelae to the formation of a congenital anomaly. ^ No significant increased associations were observed between maternal spermicide use during pregnancy and 26 of 27 types of structural malformations. While multiple statistical tests were performed we observed first trimester maternal spermicide use to be associated with a significant increased occurrence of perimembranous ventricular septal defects (OR 2.21, 95% CI 1.16-4.21). A decreased occurrence (OR < 1.0) was observed for several categories of birth defects among mothers who conceived in the first cycle after discontinuing the use of spermicides (22 of 28) or male condoms (23 of 33). ^ Overall the percent of IUD use was similar between mothers of controls and mothers of all cases in aggregate (crude OR 1.05, 95% CI 0.61-1.84). Power was limited to detect significant associations between IUD use and birth defects, however mothers using an IUD in the month immediately prior to conception or during pregnancy were not associated with an increase of birth defects. Limb defects and amniotic band sequence previously reported to be associated with IUD use during pregnancy were not found to occur among any mothers reporting the use of an IUD during pregnancy.^

A simulation study of the effects of small center enrollment in multi -center clinical trials

Multi-center clinical trials are very common in the development of new drugs and devices. One concern in such trials, is the effect of individual investigational sites enrolling small numbers of patients on the overall result. Can the presence of small centers cause an ineffective treatment to appear effective when treatment-by-center interaction is not statistically significant?^ In this research, simulations are used to study the effect that centers enrolling few patients may have on the analysis of clinical trial data. A multi-center clinical trial with 20 sites is simulated to investigate the effect of a new treatment in comparison to a placebo treatment. Twelve of these 20 investigational sites are considered small, each enrolling less than four patients per treatment group. Three clinical trials are simulated with sample sizes of 100, 170 and 300. The simulated data is generated with various characteristics, one in which treatment should be considered effective and another where treatment is not effective. Qualitative interactions are also produced within the small sites to further investigate the effect of small centers under various conditions.^ Standard analysis of variance methods and the "sometimes-pool" testing procedure are applied to the simulated data. One model investigates treatment and center effect and treatment-by-center interaction. Another model investigates treatment effect alone. These analyses are used to determine the power to detect treatment-by-center interactions, and the probability of type I error.^ We find it is difficult to detect treatment-by-center interactions when only a few investigational sites enrolling a limited number of patients participate in the interaction. However, we find no increased risk of type I error in these situations. In a pooled analysis, when the treatment is not effective, the probability of finding a significant treatment effect in the absence of significant treatment-by-center interaction is well within standard limits of type I error. ^