optimized binary modular reconfigurable robotic devices

18 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 8, NO. 1, MARCH 2003

Optimized Binary Modular ReconfigurableRobotic Devices

Moustapha Hafez, Matthew D. Lichter, and Steven Dubowsky

Abstract—Binary robotic devices with large degrees of freedomhave been proposed by a number of researchers. However, experi-mental implementations of these concepts have been built with con-ventional components. These physical systems are heavy, complexand far from being practical devices. In this paper, a lightweight,compliant mechanism driven by optimized magnet-coil actuatorsis proposed and developed as an element for modular hyper-redun-dant degrees of freedom robotic systems. Such elements could beused in a number of applications and would replace conventional,complex, and heavy components. The device has a parallel kine-matic structure. Its binary actuation simplifies its control architec-ture.

Index Terms—Binary robotics, bistable mechanisms, digitalmechatronics, electromagnetic actuators, flexures.

I. INTRODUCTION

CHALLENGING applications are being proposed forrobotic systems such as robots for surgery, service in the

home, and space explorations [1]. Researchers have proposedrobotic systems based on binary movements [2]–[5]. Thesedevices, which can be called digital mechanisms, are able toperform precise, discrete motions without need for sensing,complex electronics or feedback control. Traditional mecha-nisms have a small number of continuous degrees of freedom(DOF). A digital mechanism approximates this motion byusing a large number of binary DOF. The larger the numberof DOF, the smaller the position and orientation error [6],[7]. Recent studies have considered the kinematic motion ofthese devices in more depth [8]. However, most experimentalimplementations of the concept have been done with conven-tional components using elements such as bearings and gears.Since a large number of DOF are required, they are heavy,expensive and not practical. Therefore, new design paradigmsare required for such systems.

An optimized binary modular reconfigurable device is pro-posed in this paper. It has potential applications in the biomed-ical field such as camera placement and light positioning forsurgeon assistance. Moreover, such elements could be used askey components for self-transforming robots for planetary ex-ploration [6], [7], [9], [10]. The overall design is based on theassembly of modular parallel platforms. This allows the device

Manuscript received November 4, 2002; revised December 18, 2002. Thework was supported by the National Aeronautics and Space Administration(NASA) Institute for Advanced Concepts (NIAC).

M. Hafez is with the French Atomic Energy Commission (CEA), 92265Fontenay-Aux-Roses, France (e-mail: [email protected]).

M. D. Lichter and S. Dubowsky are with the Department of Mechanical En-gineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.

Digital Object Identifier 10.1109/TMECH.2003.809156

Fig. 1. BRAID has five-DOF which allows for precise camera placement.

to reconfigure itself to achieve mechanisms of different charac-teristics. A magnet-coil actuator is used to drive the structure.Based on electromagnetic theory, an analytical model of the ac-tuator is introduced to optimize its geometrical parameters andmaximize its efficiency. The discrete nature of the mechanismis ensured by the use of bistable mechanisms. Deployable struc-tures that are stable at a discrete set of configurations could alsobe used [11]. The binary deployable structures discussed in thispaper are given the name of binary robotic articulated integrateddevices (BRAID). A BRAID (see Fig. 1) is a network of flex-ible and rigid members with binary embedded actuators. It isa lightweight, simple, and robust mechanism that is fault-tol-erant. The BRAID is composed of a large number of parallelstages, which have three DOF each (two rotations and one trans-lation) mounted in a serial configuration. Each single stage canachieve 2 8 possible configurations. The end stage has fiveDOF (two rotations and three translations). The BRAID con-cept was based on polymer actuators which have yet to realizetheir projected performance [12]. A first BRAID version wasdeveloped based on shape memory alloy (SMA) actuators [10].However for the BRAID, such actuators are slow, inefficient,and have poor thermal properties as they dissipate a lot of heatand can be triggered by changing environmental conditions. Thesecond-generation BRAID presented here has electromagneticactuators. This design is intended to demonstrate that it is fea-sible to construct practical binary devices. This paper addressesdesign issues such as flexure design and bistable mechanismsand discusses the detailed actuator modeling and optimization.Simulations [7] and experimental results suggest that practicalbinary devices with large number of DOF can be achieved withcurrent technology.

1083-4435/03$17.00 © 2003 IEEE

HAFEZ et al.: OPTIMIZED BINARY MODULAR RECONFIGURABLE ROBOTIC DEVICES 19

Fig. 2. Coordinate frames of each module within a binary device.

II. BRAID FORWARD KINEMATICS

One interesting aspect of binary robotic devices is thatforward kinematics computations can be done without repeateduse of computationally costly transcendental functions duringrun-time [13], [14]. Whereas continuous robots have infinitesolution spaces requiring complex geometric calculations beperformed during run-time, binary-actuated robot geometriescan be computed offline ahead of time since there are only afinite number of states involved. The solution of the modulekinematics ( ) may of course requiretrigonometric or more complex mathematics, but these needonly be solved once. During run-time, forward kinematiccomputations are derived from (1) based on values stored inmemory. No computationally costly mathematics are requiredat run-time; however, with high-DOF systems the memoryrequirements for this can become quite large. For example,a five-stage BRAID can be viewed as five identical 3-DOFmodules, requiring 2 floating point numbersof storage.

For binary robotic systems, it is sometimes convenient toformulate the forward kinematics using four-by-four homoge-neous transformation matrices [15]. For example, the transfor-mation matrix describing the position and orientation ofthe end-effector relative to the base can be viewed as the productof the intermediate transformations from module tomodule within the structure (see Fig. 2). In other words

(1)

where is the number of intermediate modules. This methodof solution decomposes the kinematics of a complex structure

Fig. 3. One stage of the BRAID, showing its eight binary configurations.

Fig. 4. Coordinate frames for a single stage.

into a series of smaller, simpler structures that are easier andfaster to solve [13], [14].

A BRAID device is composed of a serial stack of parallelstages. Each of these stages has three binary-actuated linkages,thus yielding three binary DOF per stage. This results in 2different binary configurations for a single stage (see Fig. 3). Itis convenient to use homogeneous transformation matrices thatdescribe the position and orientation of the top ring of the stage(the end frame) relative to the bottom ring (the base frame). Theconvention used in the following equations implies that a baseframe is situated at the center of the base ring (see Fig. 4).The

plane is coplanar with the base ring, with theaxis normalto the ring. The end frame is situated at the center of the topring, with the top ring in the - plane and the axis normalto the top ring. The binary-actuated links are numbered in coun-terclockwise order when viewed from above, with link 1 beingthe link in the plane. In a single BRAID stage, there are 15rotational hinges (five at each link). The axes of revolution ofthe five hinges in a single link are shown in Fig. 5. Axes 1, 2,and 3 are parallel to theaxis, with axis 1 being in the plane.Axis 5 is parallel to the axis. Axis 4 is coincident with theaxis. The derivation that follows will be carried out for one statethe (100) knowing that the seven other derivations follow thesame procedure (see Fig. 6). In state 100, link one is open andthe others are closed. The homogeneous transformation matrixdescribing the top ring relative to the base can be described as atranslation in the directions followed by a rotation about the

axis as follows:

translation

rotation (2)


(a)

(b)

Fig. 5. (a) Axes of revolution of the five flexural hinges in one link of a stage.(b) Hinge angle definitions.

(3)

where

(4)

(5)

The angles of each of the hinge joints in link 1 can be computedas follows:

(6)

(7)

(a)

(b)

Fig. 6. Geometry of stage in state 100. (a) Front view. (b) Side view.

Fig. 7. Workspace of ten-stage BRIAD element (BRAID element basecenter=origin).

(8)

(9)

Fig. 7 shows the workspace generated for a ten-stage BRAIDelement. The workspace for this example consists of 2uniquestates. It is clear that when the number of stages increases, thebinary mechanisms approximate the behavior of continuousrobots.


Fig. 8. Exploded view of a single stage.

III. BRAID D ESIGN

BRAID structures are made largely of polymer materials(plastics) to make them light. Fig. 8 shows an exploded view ofa single stage. The three legs each have two flexural bearingswith one DOF, which can rotate by25 and 17 , respec-tively. This eliminates the need for heavy conventional bearingswith their friction, backlash, and added weight. The systemalso decreases the demands on the actuators. One cone-ballsliding bearing with three rotational DOF is used for eachleg. This sliding bearing satisfies the large angles of tilt upto 60 , while keeping a high stiffness of the structure. Thisis difficult to achieve with flexural bearings. The stages aremounted on each other using magnetic preload forces betweenpermanent magnets and steel parts. This magnetic connectionenables a modular design, which is of prime importance forreconfigurability. The structure is driven by electromagneticactuators composed of curved magnets and coils which min-imize the air gap between the actuator components. Finally,bistable mechanisms are integrated in the design to enforce thediscrete binary motion. These also eliminate the need to powerthe device to hold it in position.

The material selected for the BRAID structure, whichincorporates both flexural and sliding bearings, is Delrin100 (PolyOxyMethylene with 20% Teflon). Delrin has hightensile strength, impact resistance, and stiffness in addition tooutstanding fatigue endurance. These properties make it a goodchoice for flexural hinges. Moreover, Delrin has excellent wearand friction behavior due to its natural lubricity. Such propertieslead to highly efficient sliding bearings. Furthermore, Delrinis easily machinable using water-jet cutting technology andtraditional machining processes. The key elements of theBRAID, which are discussed in detail below, are its bearings,actuators, and bistable mechanisms.

A. Design of Compliant Bearings

Flexural bearings are wear-free and their motions are smoothand continuous. The accuracy of a flexural bearing depends onhow well the bearing is assembled and machined. Moreover,they have high repeatability and resolution. However they havesome disadvantages. The stiffness of such bearings is inverselyproportional to the range of motion. Furthermore, they are sen-

(a)

(b)

Fig. 9. Flexural bearings. (a) Cross-flexural hinge. (b) BRAID stage.

sitive to thermal variations, especially when made from poly-mers. In addition, they cannot tolerate large loads, in whichcase buckling occurs. Finally, a flexible connection might re-sult in vibration problems in some high-speed applications. Theflexible pivot configuration chosen for the BRAID is based oncross-flexural hinges [see Fig. 9(a)]. It consists of a pair ofcrossed plastic leaf springs, of uniform length (), width ( ),and thickness (). Compared to other types of flexural bearings,cross-flexural hinges greatly improve fatigue life, range of mo-tion, and out-of-plane stiffness.

Angular Stiffness:For small angular deflections of thebearing ( ), it is convenient to assume that the springs deformin a circular arc shape. In this case, the angular stiffness ()of the cross-springs is twice the stiffness of a single leaf springin pure bending

where (10)

where ( ) is the second moment of area and () the Young’smodulus of elasticity. For larger angular deflections such as re-quired for the BRAID ( ), the strip does not deformexactly in a circular arc and its stiffness is higher. For such an-gular deflections, stiffness increases by less than 10% [16], [17].

Angular Deflection Limits: If the springs are assumed to bedeformed in a circular arc shape, then the allowed deflection( ) of the cross-spring hinge is equal to the deflection of asingle leaf spring and is given by

(11)

For larger angles, the allowable stresses in the springs () in-crease due to a noncircular arc deformation. For an angular de-


flection of 45 , the stress is 30% greater for the same deflectionsgiven by (11) [17].

The Delrin has a 10cycles fatigue stress ( ) of 30 MPaand its Young’s modulus is 2600 MPa [18]. Using these valuesand a safety factor, the leaf springs in the BRAID [see Fig. 9(b)]have the following dimensions of mm (0.424”),

mm (0.283”), mm (0.125”), and mm(0.02”) to achieve 10cycles of 25 and 17 for hinge oneand hinge two, respectively. A life of 10cycles is quite ade-quate for most robotic applications.

B. BRAID Sliding Bearing Design

Because sliding contact bearings often distribute loads overa large area, contact stresses and space requirements are oftenlow while stiffness and damping are usually high. A sphericalbearing, which is composed of a ball that slides in a cone, isappropriate for the kinematics of the BRAID structure as it pro-vides three rotational DOF. The materials for this bearing wereselected to give minimum wear and friction. Polymers in contactwith a hard material with very low surface roughness meet thisobjective. The result is low adhesive interaction at the contactpoints and leads to low friction, minimal stick and slip effect,and relatively low wear. A cone made of Delrin and a ruby ballis an almost ideal combination [20]. However, to preload thejoint by using a magnet beneath the cone, a corrosion resistantmagnetic steel ball is used instead. This bearing has a life thatapproaches 10cycles. In order to ensure an accurate tilt, al-most no wear is allowed on the ball and just slight wear on thecone. Comprehensive wear and friction analyses for this type ofsliding bearing have been done previously [19], [20].

IV. ELECTROMAGNETIC ACTUATOR DESIGN

Electromagnetic actuators (also called voice coils) are fre-quently used in high performance and high-precision devicessuch as in disk drive head positioning. They were selected forthe BRAID where the main actuator design criteria are the forcedelivered, the power dissipation, and the total volume and massof the actuator. The choice of magnet and coil geometry witha rectangular cross section was made mainly on the availablespace in the BRAID structure and on the manufacturing tech-nologies offered. Closed-form expressions have been derivedfor the levitation forces between two circular magnetic discs andtwo noncoaxial circular coils [21], [22]. While an optimized cir-cular magnet-coil force actuator and its application to precisionelastic mechanisms has been presented [23], almost no litera-ture is available for coils with rectangular cross-sections. Basedon the laws of magnetostatics and the magnetic vector poten-

tial ( ), an analytical model was developed in this study to de-termine the force between the curved magnets and coils with arectangular cross section and is briefly presented below.

A. Analytical Model of a Magnet-Coil Actuator WithRectangular Cross Section

The -component of the magnetic vector potential () ina wire segment of length ( ) [see Fig. 10(a)] going in the

(a) (b) (c)

Fig. 10. Electromagnetic actuator. (a) Parallel wire segments. (b) Actuatormodel composed of a magnet and a coil. (c) Equivalent wire model.

direction, with its middle point at the origin and in which a cur-rent ( )is flowing is given by [24]

(12)

The magnetic field ( ) created by this current is derived from

. In the case of a straight wire positioned

along the axis, the component of the magnetic field ()is zero and the two other components are given in the followingequations:

(13)

For two wire parallel segments [see Fig. 10(a)], one of length( ) and with a current ( ) and the other wire of length ( )with a current ( ), there is an attractive or repelling force be-tween them. The direction of the force depends on the currents’directions. An attractive force is created if the two currents flowin the same direction. The and components of the forceacting on wire two due to the current flowing in wire one can bedetermined from the general equation of Lorentz force:

(14)

where ( ) is an element along the length of the conductor. The( ) and ( ) are expressed as follows:

and (15)

These formulas can be applied to calculate the resulting forcebetween a magnet and a coil. In fact, it is possible to representa permanent magnet by a coil, which has one single layer of aconducting material with a certain number of turns. The curvedaxis of the coil corresponds to the magnetization direction of thepermanent magnet. Therefore, the following relation applies:

(16)

where ( ) is the number of turns, () the current flowing in thecoil, ( ) the remanence of the magnet, and ( ) is the height


Fig. 11. Calculated magnetic force.

along magnetization. This equation is valid in the case where theBH-curve is a straight line in the second quadrant. On the otherhand, when the material has a nonlinear behavior, the right handside of (16) should be divided by the relative recoil permeability( ). The actuator can be represented by an equivalent analyt-ical model [see Fig. 10(b)]. The two components of the forcebetween each single wire and all the other parallel wires in boththe and directions are calculated from (15). In the case of fullsymmetry, which means that the two coils are coaxial, the forcesacting in the direction between two parallel wires cancel. Thesum of forces along the axis ( ) gives the force generatedby the actuator

(17)

Because of the BRAID binary action, actuators are not judgedon linearity, accuracy, or resolution but on the force delivered.Fig. 11 shows a plot of the calculated force delivered bythe linear actuator as a function of the distance between thecenter of gravity of both the magnet and the coil (). TheNd-Fe-B magnet dimension is 106 6 mm , and the coil is15 15 14 mm with a thickness of 4.2 mm. The numberof turns ( ) is 250, and a peak current of 3 A is used. Theposition of the magnet with respect to the coil is very importantas the force varies significantly from one position to another. Itshould be noted that when the two centers of gravity coincide( 0), the force delivered by the actuator is zero. The forceincreases to a maximum value when the magnet is partially inand partially out (positions and ). The force then decreasesexponentially as the two components move apart.

In the BRAID, actuators act in parallel with bistable mecha-nisms that hold the structure in either of the two desired posi-tions even when the power isOFF. Power to the actuators is onlyrequired to move from one state to another. Using impulses inthe millisecond range as input signals to the actuator allow theuse of peak currents which are relatively high (several amperes)and the power dissipation in the coil will still remain admissibleas no dc current is used in a BRAID application, which does notrequire a heavy duty cycle.

If just one side of the coil is used, it is quite difficult to get anefficient binary actuator although it is still possible to achieve

Fig. 12. Working principle of a binary actuator based on two magnets ofinverted polarities.

(a)

(b)

Fig. 13. Calculated magnetic force (a) for a dual push–pull actuator and (b)combined force.

some relatively good deflection. The two binary states will lieon each side of the peak shown in Fig. 11 as the two points indi-cated and . A more efficient way is to use both ends of thecoil where the magnetic field is at maximum. This is achievedwith two magnets with inverted polarities (see Fig. 12). The con-tributions of the two magnets sum together as shown in Fig. 13.The advantage of such a design is that the amplitude of the strokecan be increased without significantly increasing inertia of themobile part. The force remains high over the actuator stroke.Fig. 13(a) shows the calculated force from the model as a func-tion of the distance between the different components: magnet1,magnet2, relative to the coil (same dimension as previously in-dicated). The second curve indicated in Fig. 13(b) is a close-upof the area of interest. The two magnets’ curves are added andthe resulting force of the actuator has roughly a sinusoidal shape.The maximum points on the curve are chosen to be the two states


Fig. 14. Bistable detent that locks each leg of the BRAID in two discretepositions.

Fig. 15. Bistable compliant mechanism.

of the binary actuator. It should also be noted that the force isrelatively high over the entire stroke.

V. BISTABLE MECHANISMS DESIGN

Many existing binary actuation technologies require power tomaintain a fixed state, even though there is no useful output ofwork. This leads to poor energy efficiency in slow-moving bi-nary systems, whose actuators spend most of their time in a fixedstate. A system that spends a great deal of energy maintaining itsrigidity is not acceptable for applications where power is a verylimited resource. In physical implementation, it is often advan-tageous to make use of bistable joints. Bistable mechanisms arecommonly used as switches, closures, hinges, shampoo bottlecaps, bicycle kickstands, tape measures, and retractable pens.They provide accurate and repeatable motion. Bistability can beachieved in a variety of ways, such as through the use of detentsor with an over-throw design. Bistable mechanisms will allowa robotic device to turnOFF power to those actuators that aresimply maintaining a desired state. In addition, well-designedbistable mechanisms give a robotic device high repeatability,and the robot will maintain the desired binary configuration inthe presence of disturbances and environmental variations.

Many mechanisms exhibit bistability. Two designs seemedappropriate for incorporation into a rotary joint-based binarydevice. One was a detent-based latch, with small interlockingknobs that prevent rotation of the joint (see Fig. 14). The accu-racy of the motion is achieved through the flexural hinge, whichis located within the bistable mechanism. The second alterna-tive for bistability can be achieved through the snap-through ofa buckled beam to maintain two distinct states (see Fig. 15). Atwo-stage prototype with different configurations is shown inFig. 16

(a)

(b)

Fig. 16. Two-stage electromagnetically actuated BRAID prototype in variousconfigurations. (a) State 000 000. (b) State 100 011.

VI. CONCLUSION AND PERSPECTIVES

The BRAID presented in this paper is a concept module forrobotic systems that are capable of accomplishing tasks of sub-stantial complexity, with flexibility and robustness. The electro-magnetic actuator proposed provides considerable force to drivethe structure and provides large deflections that lead to a largeworkspace of the manipulator. The bistable mechanisms intro-duced in this paper lock the structure into discrete states to con-serve power and provide high accuracy and repeatability. For amore efficient design that allows a larger number of stages, eachmodule will be designed according to the mass it is required todisplace. Thus, the stage located at the base will be larger thanthe end stage used for tool manipulation, since it will requirelarger and more powerful actuators. The BRAID will have morean obelisk shape rather than a rectangular shape, which will alsoincrease the resolution of the manipulator. Currently applica-tions of the concepts being studied for BRAID devices usingthe optimized structure include assistance devices for operatingrooms and camera positioning mechanisms mounted on mobilerobots.

ACKNOWLEDGMENT

The authors would like to acknowledge A. Wingert, V. Sujan,P. Weiss, and E. Fontaine from MIT for their significant contri-butions to the project.


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Moustapha Hafez received the B.S. degree inmechanical engineering from the American Uni-versity, Cairo, Egypt, the M.S. degree in roboticsfrom l’ENSAM, Paris, France, and the Ph.D. degreefrom the Institute of Applied Optics, Swiss FederalInstitute of Technology, Lausanne, Switzerland,in 1993, 1995, and 2000, respectively. During hisPh.D. research, he designed and developed a highlyinnovative fast-steering compact laser scanner forhigh-power material processing applications.

In 2001, he was a Postdoctoral Researcher at theField and Space Robotics Laboratory, Massachusetts Institute of Technology(MIT), Cambridge, where he was the Lead Researcher of the NASA Institutefor Advanced Concepts (NIAC) project on self-transforming robotic planetaryexplorers. Currently, he is the Head of the Mechatronics Group at the FrenchAtomic Energy Commission (CEA), Fontenay-aux-Roses, France. His researchinterests lie in the field of mechatronics, dynamics, microrobotics, and smartmaterials.

Matthew D. Lichter received the B.S. degree fromPennsylvania State University (PSU), UniversityPark, in 1999 and the M.S. degree from the Massa-chusetts Institute of Technology (MIT), Cambridge,in 2001, where he is currently working toward thePh.D. degree in mechanical engineering.

He has also worked on research projects at theNational Aeronautics and Space Administration(NASA) John H. Glenn Research Center, Cleveland,OH and the Applied Research Laboratory at PSU.His current interests include sensing, planning, and

control for orbital service robots and binary robotic systems in general.

Steven Dubowskyreceived the bachelor’s degreefrom Rensselaer Polytechnic Institute of Troy, Troy,NY in 1963, and the M.S. and Sc.D. degrees fromColumbia University, New York, in 1964 and 1971,respectively.

From 1963 to 1971, he was employed by thePerkin-Elmer Corporation, the General DynamicsCorporation, and the American Electric PowerService Corporation. He has been a Professor ofengineering and applied science at the University ofCalifornia, Los Angeles, and a Visiting Professor at

Cambridge University, Cambridge, U.K., California Institute of Technology,Pasadena, the University of Paris (VI), Paris, France, and Stanford University,Stanford, CA. He has also served as an Advisor and Consultant to the NationalScience Foundation, the National Academy of Science/Engineering, theDepartment of Energy, the U.S. Army, and industry. Currently, he is a Professorof mechanical engineering at the Massachusetts Institute of Technology (MIT),Cambridge. He has also made important contributions to the areas of field andspace robotics. He has authored or coauthored over 100 papers in the area ofthe dynamics, control, and design of high performance mechanical, electro-mechanical, and robotic systems. His research has included the developmentof modeling techniques for manipulator flexibility and the development ofoptimal and self-learning adaptive control procedures for rigid and flexiblerobotic manipulators.

Dr. Dubowsky is a Registered Professional Engineer in the State of California,has been elected a Fellow of the American Society of Mechanical Engineers(ASME), and is a Member of Sigma Xi and Tau Beta Pi.

optimized binary modular reconfigurable robotic devices

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