This paper provides a unified framework for minimal mass design of tensegrity systems. For any given configuration and any given set of external forces, we design force density (member force divided by length) and cross-section area to minimize the structural mass subject to an equilibrium condition and a maximum stress constraint. The answer is provided by a linear program. Stability is assured by a positive definite stiffness matrix. This condition is described by a linear matrix inequality. Numerical examples are shown to illustrate the proposed method.
The simplest form for the dynamics of constrained
tensegrity system is a non-minimal realization. This paper gives
a control law to force the tensegrity system to modify its shape
to a pre-specified shape, using the smallest control force. The
approach is similar to a multi-Lyapunov approach. We create a
vector of Lyapunov functions, chosen to force the desired shape
change, as well as other performance properties that may be
selected. This vector is forced by the control system to satisfy
a linear stable differential equation. Some tensegrity examples
illustrate the ideas.
The multi-body dynamics appear in a new form, as a matrix differential
equation, rather than the traditional vector differential equation. The
model has a constant mass matrix, and the equations are non-minimal.
A specific focus of this paper is tensegrity systems. A tensegrity system
requires prestress for stabilization of the configuration of rigid bodies and
tensile members. This paper provides an efficient model for both static
and dynamic behavior of such systems, specialized for the case when the
rigid bodies are axi-symmetric rods.
We report here on the optimization of 0.5 cm thick pixelated Orbotech CZT detectors with regards to the
best contacting materials and the use of steering grids. We evaluated the performance of different contacting
materials. Our study differs from earlier ones in that we investigated the performance of different anode and
cathode materials separately. We obtain the best performance with Au cathodes. For different anode materials
Ti and In give the best energy resolutions. The detector (2.0×2.0×0.5 cm3, 8×8 pixels) shows excellent 59 keV,
122 keV and 662 keV energy resolutions of 1.4 keV, 1.9 keV, and 7.4 keV, respectively. Furthermore, we report
on using steering grids to improve on the performance of the pixelated detectors. Previously, the benefit of
steering grids had been limited by additional electronic noise associated with currents between the negatively
biased steering grids and the anode pixels. We are currently exploring the possibility to isolate the steering grid
from the CZT substrates by a thin layer of Al2O3. We performed a series of measurements to determine by
how much the isolation layer reduces the grid-pixel currents. Comparing the currents between two Au contacts
before and after isolating one of the two contacts from the CZT with a 700 nm thick layer of Al2O3, we measure
that the isolation layer reduces the currents by a factor of about 10 at 500 V. We present some results from
a detector before and after deposition of an isolated steering grid. The grid indeed improves on the detectors
energy resolution and detection efficiency. We show that simulations can be used to model the anode to cathode
charge correlation in excellent agreement with the experimental results.
This paper introduces hydrodynamic loads for tensegrity structures, to examine their behavior in marine environments. Wave compliant structures are of general interest when considering large marine structures, and we are motivated by the aquaculture industry where new concepts are investigated in order to make offshore installations
for seafood production. This paper adds to the existing models and software simulations of tensegrity structures exposed to environmental loading from waves and current. A number of simulations are run to show behavior of the structure as a function of pretension level and string stiffness for a given loading condition.
This work investigates the design of a new class of three dimensional tensegrity tower structures with nodes lying on a cylinder. The novel aspect of the proposed topology is the fact that all bars in all stages are oriented in the same way, clockwise or counterclockwise. We investigate the existence of conditions for static equilibrium of such towers with an arbitrary number of stages and uniform force distribution.
This paper concerns optimization of prestress of a tensegrity
structure to achieve the optimal mixed dynamic and control
performance. A linearized dynamic model of the structure is
derived. The force density variables that parameterize prestress
of the structure appear linearly in the model. The feasible region
of these parameters is defined in terms of the extreme directions
of the prestress cone. Several properties of the problem are
established inside the feasible region of the parameters. The
problem is solved using a gradient method that provides a
monotonic decrease of the objective function inside the feasible
region. A numerical example of a cantilevered planar tensegrity
beam is shown.
This experimental study demonstrates the efficiency of simple control strategies to damp a 3-stage tensegrity tower structure. The tower is mounted on a moving support which is excited with a limited bandwidth random signal (filtered white noise) by a shaker. Our goal is to minimize the tansmissibility between base acceleration and top plate acceleration using piezoelectric displacement actuators and force sensors collocated at the bottom stage of vertical strings. Two types of controllers have been designed, namely, it local integral force feedback control and
acceleration feedback control. It can be shown that both controllers can effectively damp the first 2 bending modes by about 20 dB, and the acceleration feedback controller performs even better as it can also reduce the amplitude of the next 2
bending modes by about 5-10 dB.
This paper concerns open-loop control laws for reconfiguration of
tensegrity towers. By postulating the control strategy as an
equilibrium tracking control, very little control energy is
required. Several different reconfiguration scenarios are possible
for different string connectivity schemes. This includes unit
radius control, twist angle control and truncation parameter
control. All these control laws allow a nonuniform distribution of
the control parameters among units. By defining a wave--like
reference signal and injecting it in the open--loop control law,
we demonstrate the concept of self--propelled tensegrity structure
that are capable of locomotion.
Tensegrity structures are special truss structures composed of bars in compression and cables in tension. Most tensegrity structures under investigation, to date, have been of Class 1, where bars do not touch. In this article, however, we demonstrate the hardware implementation of a 2 stage symmetric Class 2 tensegrity structure, where bars do connect to each other at a pivot. The open loop control law for tendon lengths to accomplish the desired geometric reconfiguration are computed analytically. The velocity of the structure's height is chosen and reconfiguration is accomplished in a quasi-static manner, ignoring dynamic effects. The main goal of this research was to design, build, and test the capabilities of the Class 2 structure for deployment concepts and to further explore the possibilities of multiple stage structures using the same design and components.
A systematic design method is proposed for
the selecting of actuators and sensors in the structural control
in order to minimize the instrumental cost. With actuators and
sensors placed at all the admissible locations initially, an
iterative minimization algorithm is carried out to identify the
sensor/actuator that requires the least precision. By deleting the
roughest sensor/actuator each time till loss of feasibility, one
can conclude simultaneously the necessary number and type of
sensor/actuator, and the location and precision for each
sensor/actuator. A tensegrity structure example has
been solved as an application of the proposed algorithm.
This paper concerns the optimal mass-to-stiffness ratio design
of class-2 tensegrity towers. For different loading scenarios,
the procedure seeks the topology and geometry of the structure
that yields an optimal design satisfying common constraints. The
domain of feasible tensegrity geometries is defined by imposing
tensegrity equilibrium conditions on both unloaded and loaded
structure. Remaining constraints include strength constraints for
all elements of the structure and buckling constraints for bars.
The symmetry of the design is imposed by restricting the domain of
geometric variables and element parameters. The static response of
the structure is computed by using a nonlinear large displacement
model. The problem is cast in the form of a nonlinear program. The
influence of material parameters on the optimal shape of the
structure is investigated.
A new topology for a prestressed tensegrity plate, the unstable-unit tensegrity plate (UUTP), is introduced, together with a detailed algorithm for its design. The plate is a truss made of strings (flexible elements) and bars (rigid elements), which are loaded in tension and compression, respectively, where bars do not touch each other. Given the outline dimensions of the desired plate, and the number of bars along the plate's width and length, the algorithm solves for the nodes' positions and the prestress forces that make a plate in equilibrium. This is done by solving a non-linear matrix equation via Newton's method. This equation reflects static equilibrium conditions. We've designed several such plates, proving the feasibility of the proposed topology and the effectiveness of its design algorithm. Two such plates are characterized in detail, both statically and dynamically (via simulation). The proposed algorithm may be extended to solve for other tensegrity structures having different topologies and/or different shapes. The UUTP may be used as a building block of many types of structures, both uncontrolled and controlled, either large-scale or miniature-scale.
For a new class of tendon-driven robotic systems that is generalized to include tensegrity structures, this paper focuses on a method to jointly optimize the control law and the structural complexity for a given point-to-point maneuvering task. By fixing external geometry, the number of identical stages within the domain is varied until a minimal mass design is achieved. For the deployment phase, a new method is introduced which determines the tendon force inputs from a set of admissible, non-saturating inputs, that will reconfigure each kinematically invertible unit along its own path in minimum time. The approach utilizes the existence conditions and solution of a linear algebra problem that describe how the set of admissible tendon forces is mapped onto the set of path-dependent torques. Since this mapping is not one-to-one, free parameters in the control law always exist. An infinity-norm minimization with respect to these free parameters
is responsible for saturation avoidance. In addition to the required time to deploy, the expended control energy during the post-movement phase is also minimized with respect to the total number of stages. Conditions under which these independent minimizations yield the same robot illustrate the importance of considering control/structure interaction within this new robotics paradigm.
The economic simulation design problem is that given performance requirements, design the simulation of a linear system and distribute precision among the instruments such that the computational cost is minimized without violating the simulation accuracy. In this paper, we consider the simulation of a large-scale linear system in digital devices with fixed-point arithmetic and finite wordlength. Given the output variance upperbound, we focus on finding an optimal realization and the allocation of wordlength among A/D converter, computer and sensors. This problem is in general not convex because of the scaling constraint. By exploring the special structure of this
joint optimization problem and under reasonable assumptions, we simplify this problem and find the optimal coordinate transformation and the wordlength allocation scheme simultaneously by solving LMIs (linear matrix inequalities). Numerical results are given which compare this new realization with the balanced realization and random realizations.
Tensegrity structures consist of tendons (in tension) and bars (in
compression). Tendons are strong, light, and foldable, so
tensegrity structures have the potential to be light but strong
and deployable. Pulleys, NiTi wire, or other actuators to
selectively tighten some strings on a tensegrity structure can be
used to control its shape. This article describes the problem of
asymmetric reconfiguration of tensegrity structures and poses one
method of finding the open loop control law for tendon lengths to
accomplish the desired geometric reconfiguration. In addition, a
practical hardware experiment displays the readiness and
feasibility of the method to accomplish shape control of the
We are developing 10 mm thick CZT detectors with 3-D readout for ~100 keV to ~1.5 MeV gamma-rays. Multiple-site gamma-ray interactions are fully measured, i.e., the energy and 3-D position of each site are determined. Spatial resolution is 1 mm FWHM. Anode pixel readout with 1 mm pitch is used for x- and y-positions and charge drift times for z-positions. Drift time measurements are triggered by the cathode signal and end when each interaction site's charge cloud reaches an anode pixel. Post-event processing corrects for signal loss due to charge trapping and accurately determines gamma-ray energies, with a goal of 1% energy resolution at 662 keV. Compton kinematic analysis can identify the initial interaction site in most cases as well as constrain the incident gamma-ray direction. Tests were made with a prototype detector, measuring 10 x 10 x 10 mm3 and operated at 1000 V bias. The measured drift time resolution of 25 nsec FWHM at 662 keV and 60 nsec at 122 keV corresponds to z-position resolution of 0.25 and 0.60 mm FWHM, respectively. The technique is described and results of modeling and tests are presented.
One of the main properties of tensegrity structures, that sets them apart from most of structures, is that they are vary suitable for shape control. This can be accomplished by controlling lengths of string members. Tensegrity deployment is considered herein as a tracking control problem. Therefore, the required trajectories should be feasible for a given structure. For tensegrity structures, this means that in every desired configuration, the structure has to satisfy tensegrity conditions, which require strings to be in tension, and the structure to be stable. To define an open-loop deployment control law, geometry parameterization of those configurations and corresponding rest lengths of string elements guaranteeing equilibrium are defined first. By slowly varying desired geometry, an open-loop string rest length control is defined. This makes the structure track trajectories defined by the time dependent geometry parameters. Two examples are illustrated: 1. Deployment of planar tensegrity beams made of symmetric stable tensegrity units, $2)$ Deployment of plates made of stable symmetric shell class tensegrity units.
Cadmium Zinc Telluride (CZT) is a room temperature semiconductor detector well suited for high energy x-ray astronomy. We have developed a CZT detector with 500 micron crossed strip readout and an advanced electrode design that greatly improves energy resolution. We conducted two balloon flights from Fort Sumner, NM, to study the cross strip detector and a standard planar detector both sensitive in the energy range of 20-350 keV. The flights utilized a total of seven shielding schemes: 3 passive, 2 active and 2 hybrid passive-active. In the active shielding modes, the anti- coincidence shield pulse heights were telemetered for each CZT event, allowing us to study the effect of the shield's energy threshold on the spectral shape and magnitude of the background. We are also developing an energy-dependent background rejection technique based on the charge collection properties of the CZT detector. This technique employs the depth of interaction, as inferred by the ratio of cathode to anode pulse height, to reject events inconsistent with incident source x-rays. The long duration of the May flight enabled us to study activation effects. We present result of the effectiveness of each of the shielding schemes on both detectors, the rejection power of depth of interaction technique on the crossed strip detector, inferred aperture background flux and the level of activation after 22 hours as float.
Position-sensitive CZT detectors for research in astrophysics in the five - several hundred keV range are being developed by several groups. These are very promising for large area detector arrays in coded mask imagers and small-area focal plane detectors for focusing x-ray telescopes. We have developed detectors with crossed-strip readout and optimized strip widths and gaps to improve energy resolution. A 'steering electrode' is employed between the anode strips to improve charge collection. A model of charge drift in the detectors and charge induction on the electrodes has been developed to allow us to better understand these types of detectors and improve their design. The model presently accounts for the electric field within the detector, the charges' trajectories, mobility and trapping of holes and electrons, and charge induction on all electrodes including their time dependence. Additional effects are being added. The model is described and its predictions are compared with laboratory measurements. Results include (1) the dependence of anode, cathode and steering electrode signal on interaction depth, transverse position, electron and hole trapping, strip width and gap, and bias, (2) trajectories of charges for various anode and steering electrode bias voltages, (3) a method to improve energy resolution by sign depth of interaction information, and (4) an electrode geometry and bias optimized for the improved energy resolution. In general, the model provides good agreement with the measurements.
This paper proposes a new space telescope design in which the classical truss structure of the telescope is replaced by a tensegrity structure. A tensegrity structure is a prestressed structure whose structural shape is guaranteed by the interaction between elastic members in tension (tendons) and a set of rigid members (bars). A nonlinear dynamical model of a two stage tensegrity telescope is derived. Static analysis is performed for tensegrity telescopes composed of two stages. The performance specifications for the control system are formulated in terms of the peak value (L(infinity ) norm) of the pointing and alignment errors. The control system is designed to minimize a certain upper bound on this peak value subject to a peak value constraint on the external disturbances. Evaluations of the design are performed through numerical simulations of the closed loop system.
CdZnTe (CZT) is a room-temperature semiconductor well suited for high energy x-ray astronomy. Knowledge of its background properties is essential for optimizing CZT based instruments. To study its background, we flew two CZT detectors on dedicated high altitude balloon flights from Fort Sumner, NM, the first in October 1997 and the second in May 1998. The first detector is a 12 by 12 by 2 mm3 detector with orthogonal crossed strips and the second is a standard 12 by 12 by 2 mm 3 planar detector. The cross strip detector has 500 micron pitched electrodes plus anode 'steering electrodes' to improve the anode charge collection. The energy range for these flights was 20 to approximately 350 keV. We have found a preliminary background level in 20-40 keV of 8.6 by 10-3 cts/cm2-s-keV using passive 2 mm PbSnCu shielding and 6 by 10-4 cts/cm2-s-keV using active collimated schemes for the first position-sensitive CAT detector at balloon altitudes.
HEXIS is a MIDEX-class mission concept for x-ray astronomy. Its objectives are to improve our knowledge of the high energy x-ray sky by increasing the number of sources above 20 keV to > 2,000, discovering transient sources such as x-ray novae and gamma-ray bursts, and making spectral and temporal studies of the sources. With mission life > 3 years, a 1-year all-sky survey sensitivity of approximately 0.3 mCrab, and continuous monitoring of the entire visible sky, HEXIS will provide unprecedented capabilities. Source positions will be determined to accuracies of a few arcmin or better. Spectra will be determined with an energy resolution of a few keV and source variability will be studied on time scales from < 1 sec to years. In addition, 10 times more sensitive studies of limited fields will be performed at the same time. Gamma-ray bursts will be detected about 4 times/week at about the same sensitivity as BATSE and the sensitivity to nova-like x-ray transients will be approximately 6 mCrab in one day. HEXIS contains a set of coded mask imagers that use position-sensitive CZT detectors operating from approximately 5 keV to 200 keV. Detector planes are built with 41 cm2 CZT detector modules which employ crossed-strip readout to obtain a pixel size of 0.5 mm. Nine modules are grouped in a 369 cm2 array for each imager. In the past 2 years significant progress has been made on techniques requires for HEXIS: position-sensitive CZT detectors and ASIC readout, coded mask imaging, and background properties at balloon altitudes. Scientific and technical details of HEXIS are presented together with result form tests of detectors and a coded mask imager.
In this paper we propose a new motion simulator base don tendon controlled tensegrity structures. The simulator is equipped with a robust nonlinear controller which achieves robust tracking by the simulator of a desired motion. The controller parameters can be tuned to guarantee tracking to within a prespecified tolerance and with a prescribed rate of exponential convergence. The design is verified through numerical simulations for specific longitudinal motions of a symmetric aircraft.
In this paper we consider the problem of deployment of tensegrity structures. Our idea is to make use of a certain set of equilibria to which the undeployed and deployed configurations belong. In the state space this set is represented by a connected equilibrium manifold and can be completely characterized analytically. The deployment is conducted such that the deployment trajectory is close to the equilibrium manifold and the deployment time is minimized.
Tensegrity structures represent a special class of tendon space structures, whose members may simultaneously perform the functions of strength, sensing, actuating and feedback control. The paper exploits this advantage, proposing a smart tensegrity sensor for simultaneous measurement of six quantities: three orthogonal forces and three orthogonal moments. The paper shows how the static and dynamic characteristics of the device can be calibrated through pretension and damping adjustment. The external forces and torques of interest are estimated using the measurements provided by selected tendons. A state estimator, based on the linearized model, finalizes the design.
Coded mask imagers for future high energy x-ray astronomy missions will require detector planes with areas of hundreds to thousands of cm2 and position resolutions < 1 mm. Such detectors will enable coded mask imagers to discover and study thousands of high energy x-ray sources. The UCSD/WU/UCR/NOVA collaboration has been developing CZT detector systems with crossed-strip readout to meet these requirements. We report progress on a compact detector module with 41 cm2 area and 0.5 mm spatial resolution. The design includes the bias network and ASIC readout electronics, and allows modules to be combined in large area arrays with very high live-area factors. Results from laboratory and balloon flight tests are presented.
The scientific objectives, status, and future instrumental requirements of high energy X-ray astronomy (20 to 200 keV) are discussed. Two particularly compelling requirements are: (1) an improvement in sensitivity to a level of about 5 microCrab and (2) a survey of the sky at a sensitivity of about 0.1 milliCrab, which will discover and characterize about 10,000 new sources. The first requirement can be fulfilled by imaging telescopes that use large-area focusing X-ray mirrors, which are effective over 5-30 arcminute fields, and the second requirement can be met by arrays of large area coded mask imagers with wide fields, about 50 deg. Multilayer mirror and CdZnTe detector technology now in development offers the potential to meet these objectives. Position-sensitive CdZnTe detectors are well-suited to both of these imaging techniques, and instrument concepts that use these detectors are described. Detectors with pixel readout are better suited for focusing telescopes, and those with crossed-strip readout are better suited for coded mask imagers. Technical aspects of these detectors are discussed. Recent work at UCSD and WU on CdZnTe strip detectors is described in detail. Studies with small, 40 micron, X-ray beams have mapped a crossed-strip detector's spatial response with fine spatial resolution.
Smart structures include active control elements, hence the integration of structure design with control design in inevitable. This paper provides a step in this integration in order to achieve optimal performance. The approach presented here solves a mixed passive and active control problem with performances characterized by the so-called mixed H2/H(infinity ) performances.
Tensegrity structures represent a special class of tendon space structures, whose members may simultaneously perform the functions of strength, sensing, actuating and feedback control. Thus, these structures ideally match the definition of smart structures. This paper introduces the concept of controllable tensegrity as a new class of smart structures capable of large displacement. The kinematics and nonlinear dynamics of one element of this class is derived and analyzed. Pre-stressability conditions are given and a particular equilibrium identified. The equations of motion are then linearized about this equilibrium and linear parametric models generated. These are next used for controller design. For control system design some of the tendons are chosen as actuators and some as sensors and a family of dynamic controllers designed such that the control energy is minimized while requiring output variance constraints to be satisfied. Another family of controllers is designed such that the output variance is minimized while requiring input variance constraints to be satisfied. The performances of these controllers are evaluated.
With the properties of prestressability, deployability and simplicity of member connections, tensegrity structures can have wide applications. This paper presents an application of tensegrity structures to the NESTOR project. This paper presents a description of the geometry description and the static analysis algorithm. A deployment sequence is also presented.
The scientific objectives and future requirements of high energy x-ray astronomy are discussed and concepts for imaging instruments based on CdZnTe detectors and coded masks are reviewed. An instrument concept based on CdZnTe strip detectors, HEXIS, is described in detail. Technical requirements for large area CdZnTe strip detectors are discussed and recent work at UCSD and WU on the capabilities of CdZnTe strip detectors is described in detail. Studies with small, approximately 50 micron beams demonstrate that crossed strip detectors have good properties for both spatial and spectral measurements.
In this paper, an integrated identification and control procedure is studied. This integrated procedure seeks to find a high performance controller for real world systems. The procedure, which is inherently iterative, involves three steps in each iteration: (1) closed loop identification; (2) system model extraction from the closed loop experiment data; (3) controller design. The algorithm proposed in this paper uses weighted closed loop identification for deducing a model. The weight used for identification is obtained from the control design step and it provides a measure to evaluate the relative importance of each output channel in the closed loop behavior. Hence the weighted identification can capture models which are good for control design so as to achieve better closed loop performance. The procedure is demonstrated by controlling a smart structure under development at Purdue.
A dynamic system with satisfactory performance generally consists of a mechanical system (the plant) and a controller that drives the mechanical system to meet certain performance requirements. Traditionally the control engineer designs the controller only after the plant design is completed. This two-step approach to plant and controller design does not provide the best system design because the dynamics of the plant and the dynamics of the controller often oppose each other. This paper presents an application of the iterative system equivalent optimal mix algorithm to perform a smart design of a nine-member truss substructure and its accompanying controller. The objective of the design algorithm is to reduce the amount of energy used by the controller to maintain control performance, subject to the structure design constraints. Two unique features of the algorithm are that each iteration of the design problem is stated as a convex quadratic programming problem, and the control effort monotonically converges to its final value.
An integrated means for active controller design and structure redesign is presented. The techniques of covariance control are used to parametrize all possible combinations of active controllers/structure redesign parameters which can stabilize the plant, and achieve certain closed-loop performance.