Large Effective-Strain Piezoelectric Actuators Using Nested Cellular Architecture with Exponential Strain Amplification Mechanisms

Transcription

1 Large Eective-Strain Piezoelectric Actuators Using Nested Cellular Architecture with Exponential Strain Ampliication Mechanisms Jun Ueda, Member, IEEE, Thomas Secord and H. Harry Asada, Member, IEEE, Fellow, ASME Abstract Design and analysis o piezoelectric actuators having over % eective strain using an exponential strain ampliication mechanism are presented in this paper. Piezoelectric ceramic material, such as Lead Zirconate Titanate (PZT), has large stress and bandwidth, but its extremely small strain, i.e., only.%, has been a major bottleneck or broad applications. This paper presents a new strain ampliication design, called a nested rhombus multi-layer mechanism, that increases strain exponentially through its hierarchical cellular structure. This allows or over % eective strain. In order to design the whole actuator structure, not only the compliance o piezoelectric material but also the compliance o the ampliication structures need to be taken into account. This paper addresses how the output orce and displacement are attenuated by the compliance involved in the strain ampliication mechanism through kinematic and static analysis. An insightul lumped parameter model is proposed to quantiy the perormance degradation and acilitate design trade-os. A prototype nested PZT cellular actuator that weighs only 5 g has produced % eective strain (.5 mm displacement rom mm actuator length and 3 mm width) and.7 N blocking orce. Index Terms Piezoelectric transducers, Actuators, Flexible structures. I. INTRODUCTION Actuators play key roles in mechatronic and robotic devices. When a new robotics-related technology area emerges the demand or new actuation technologies strengthens. The authors have presented a unique cellular actuator concept, which in turn has a potential to be a novel approach to synthesize biologically-inspired robot actuators [] [] [3] [4] [5]. The concept is to connect many small actuator units in series or in parallel, and compose in totality a macro-size linear actuator array similar to skeletal muscles. Promising applications include human assistive technologies [6]. In these applications, actuators are required to be ast, have zero backlash, be silent, energy eicient, and compact. Compliance may also be required or the sake o human saety. A large strain beyond % in actuation direction is desirable that is comparable to natural skeletal muscles [7]. It is expected that a large strain cellular actuator array can be used in a manner similar to biological muscles directly attached to skeletal structures. Another salient eature o the cellular actuator concept is modularity. The multitude o reconigurable modular Jun Ueda is with the George W. Woodru School o Mechanical Engineering, Georgia Institute o Technology, Atlanta, GA , USA. Tel: , Fax: , jun.ueda@me.gatech.edu. Thomas Secord and H. Harry Asada are with Department o Mechanical Engineering, Massachusetts Institute o Technology, Cambridge, MA, USA. actuator units are connected in series and parallel to build various actuators with diverse stroke, orce, and impedance characteristics. Simple ON-OFF controls [3] [4] will suice to drive individual actuator units, since the aggregate outputs will be smooth and approximately continuous i a large number o modules are involved. ON-OFF controls are eective to overcome prominent hysteresis and nonlinearity o actuator materials [8]. Moreover, expensive analogue drive ampliiers are not required. The basic module o this actuator array is a compact PZT stack actuator. Piezoelectric ceramics, such as PZT, have a high power density, high bandwidth, and high eiciency. PZT outperorms other actuator materials, including shape memory alloy (SMA), conducting polymers, and electostrictive elastomers, with respect to speed o response and bandwidth. Its maximum stress is as large as SMA, and the eiciency is comparable to electrostrictive elastomers. Furthermore, PZT is a stable and reliable material that is usable in diverse, harsh environments. The most critical drawback o PZT is its extremely small strain, i.e., only.%. Over the last several decades eorts have been taken to generate displacements out o PZT that are large enough to drive robotic and mechatronics systems [9] [] [] [] [3] [4] [5] [6] [7] [8] [9]. These can be classiied into a) inching motion or periodic wave generation, b) bimetal-type bending, and c) lextensional mechanisms. Inching motion provides ininite stroke and bimetal-type mechanism [8] [] can produce large displacement and strain, applicable to various industrial applications when used as a single actuator unit; however, unortunately, the reconigurability by using these types may be limited due to the diiculty in arbitrarily connecting a large number o actuator units in series and/or in parallel to increase the total stroke and orce respectively. In contrast, lextensional mechanisms such as Moonie [9] [], Cymbal [4], Rainbow [], and others [6] are considered suitable or the reconigurable cellular actuator design. An individual actuator can be stacked in series to increase the total displacement. Note that this simple stacking also increases the length o the overall mechanism and does not improve the strain in actuation direction, which is known to be up to 3%. In this paper a new approach to ampliying PZT displacement will be presented [], [] to achieve over % eective strain. The key idea is hierarchical nested architecture that encloses smaller lextensional actuators with larger ampliying structures. A large ampliication gain on the order o several hundreds can be obtained with this method. This structure

2 is undamentally dierent rom traditional layered structures, such as telescoping cylindrical units [7] or stacking multiple plates connected by actuator wires []. Unlike these traditional stacking mechanism, where the gain α is proportional to the dimension o the lever or number o stacks, the ampliication gain o the new mechanism increases exponentially as the number o layers increases. Suppose that strain is ampliied α times at each layer o the hierarchical structure. For K layers o hierarchical mechanism, the resultant gain is given by α K, the power o the number o layers. This nesting method allows us to gain a large strain in a compact body, appropriate or many robotic applications. This nesting approach may look rather straightorward; however, to the best o the authors knowledge, there is no work that clearly ormulated this exponential eect. In the ollowing the design concept o exponential strain ampliication with a nested structure will be presented, ollowed by an idealized orce-displacement analysis and a easibility study or achieving % strain using PZT stack actuators. A static lumped parameter model including mechanical compliance o the nested strain ampliier will be proposed to investigate how the orces and displacements generated by the individual PZT actuators are transmitted through the hierarchical mechanism, resulting in aggregate orce and displacement at the output port. Design trade-os between the admissible and constrained motion spaces will be discussed based on the insightul lumped model. The lumped parameter model o lower layers can be recursively nested into a higher lumped parameter model to evaluate complex nested mechanisms without developing a ull numerical model such as inite element method (FEM) []. The validity o the proposed concept will be conirmed through the design o a prototype actuator producing a displacement o.53 [mm] rom [mm] actuator length, which is equivalent to % eective strain, 3 [mm] o width,.69 [N] o blocking orce, and 5 [g] o body mass. II. EXPONENTIAL STRAIN AMPLIFICATION USING NESTED RHOMBUS MECHANISMS A. Preliminary Kinematic Analysis o Flextensional Mechanisms and Eective Strain This subsection briely describes the kinematics o traditional Moonie lextensional mechanisms [9] that will be used or the basis o our proposed nested rhombus multilayer mechanism. As shown in Fig. (a), the main part o the mechanism is a rhombus-like hexagon that contracts vertically as the internal unit shown in grey expands. The vertical displacement, that is, the output o the mechanism, is ampliied i the angle o the oblique beams to the horizontal line is less than 45 degrees. Figure (b) illustrates how the strain is ampliied with this mechanism. Let h, w, and ɛ be, respectively, the height, width, and strain o the internal unit. Also let d be the initial gap between the surace o the internal unit and the apex o the rhombus mechanism. In this section it is assumed that all the joints are purely rotational and that all the beams are completely rigid. Also assume that the internal unit is extensible; it can be extended to a contractive case with Internal unit (extensible) OFF output (a) Actuation o lextensional mechanism ON input d d h w / θ (+ε ) w / (b) Ampliication o eective strain Fig.. Ampliication Principle o Flextensional Mechanisms [9] minor changes to the ollowing ormulation. As the internal unit, e.g., a PZT stack actuator, expands, the gap d contracts to d : d d = (ɛ +ɛ )w /4. () Then, the ampliication gain a o the displacement is given by a = Δx () ɛ w where Δx = d d.forsmallɛ this can be approximated to a = w d = cotθ where θ is the angle o the oblique beam to the horizontal line, as shown in the igure. Note that this instantaneous ampliication gain does not apply to large strain because o the nonlinearity in (). A smaller value or the angle o the oblique beams θ gives a larger ampliication gain. However, the angle θ needs to be careully determined to avoid buckling o the beams due to unexpected external orces. As will be explained the later section, buckling o the beams cannot be neglected in the mechanical design since the working direction alternates rom layer to layer in the proposed nested structure. Typically this ampliication gain alone can increase displacement to only 3 5 times larger. The initial length o the rhombus mechanism measures d +h along the output axis. Since the displacement created in this output direction is Δx, the eective strain along the output axis can be deined as Δx ɛ =. (3) d + h Note that the lateral size, perpendicular to the output direction, is not included in the deinition o the eective strain. Comparing this to the input strain ɛ yields the strain ampliication gain given by ɛ w α = = a. (4) ɛ d + h where w /(d + h ) is the ratio o the width to the height o the rhombus, i.e., the aspect ratio o the mechanism. Note that both the displacement ampliication and the aspect ratio o the mechanism contribute to the resultant strain ampliication α. Strictly speaking, the aspect ratio is not a strain ampliier. However, since ) the eective strain ampliication is deined to be the ratio o output displacement to the natural body length in the same direction as the output, and ) the direction o input strain and that o the output displacement are perpendicular to each other, the eective gain α is apparently ampliied by the aspect ratio. Increasing the aspect

3 3 PZT stack actuator (Actuation layer) Actuation voltage V i,j Output First layer rhombus (i-th unit) j N i N Third layer rhombus Second layer rhombus (j-th unit) Fig.. Proposed Nested Structure or Exponential Strain Ampliication: The strain is ampliied by three layers o rhombus strain ampliication mechanisms, with the irst layer, called an actuator layer, consisting o the smallest rhombi directly attached to the individual PZT stack actuators. ratio increases the strain ampliication gain α. However, space constraints must be considered since a larger aspect ratio increases the lateral size o the actuator that would aect on orce capacity. B. Nested Rhombus Structure The above mechanism or ampliying small displacements o PZT actuators have already been developed both in macro [9] and micro scale [9] and have been applied to commercial products [3]. Our method is to extend this technique to: ) Gain an order-o-magnitude larger strain ampliication, and ) Build a modular structure that is lexible and extensible. Figure illustrates a new mechanism, called a nested rhombus strain ampliier, which consists o the multitude o rhombus mechanisms arranged in a hierarchical structure. The inner-most unit, i.e., the building block o the hierarchical system, is the standard rhombus mechanism, or conventional lextensional mechanism, described above. These units are connected in series to increase the output displacement. Also these units can be arranged in parallel to increase the output orce. The salient eature o this hierarchical mechanism is that these rhombus units are enclosed with a larger rhombus mechanism that ampliies the total displacement o the smaller rhombus units. These larger rhombus units are connected together and enclosed with an even larger rhombus structure to urther ampliy the total displacement. Note that the working direction alternates rom layer to layer; e.g., the second layer rhombus extends when the inner-most irst layer units contract as shown in Fig. (b). As this enclosure and ampliication process is repeated, a multi-layer strain-ampliication mechanism is constructed, and the resultant displacement increases exponentially. Let K be the number o ampliication layers. Assuming that each layer ampliies the strain α times, the resultant ampliication gain is given by α to the power o K: α total = α K. (5) For α =5the gain becomes α total = 5 by nesting two rhombus layers and α total = 3375 with three rhombus layers. The nested rhombus mechanism with this hierarchical structure is a powerul tool or gaining an order-o-magnitude larger ampliication o strain. As described beore, our immediate goal is to produce % strain. This goal can be accomplished with α =5and K =:.% 5 5 =.5%. This nested rhombus mechanism has a number o variations, depending on the numbers o serial and parallel units arranged in each layer and the eective gain in each layer. In general the resultant ampliication gain is given by the multiplication o each layer gain: α total = K k= α k where α k = ɛk /ɛ k is the k-th layer s eective gain o strain ampliication computed recursively with the ollowing ormula, d k 4d k (ɛ k +ɛ k )wk ɛ k = (k =,,K). (6) d k + h k Another important eature o the nested rhombus mechanism is that two planes o rhombi in dierent layers may be arranged perpendicular to each other. This allows us to construct three-dimensional structures with diverse conigurations. For simplicity, the schematic diagram in Figure shows only a two-dimensional coniguration, but the actual mechanism is three-dimensional, with output axes being perpendicular to the plane. Three-dimensional arrangement o nested rhombus mechanisms allows us to densely enclose many rhombus units in a limited space. For example, Figure 3 illustrates a 3- dimensional structure. Note that the serially connected irstlayer rhombus units are rotated 9 degrees about their output axis x. This makes the rhombus mechanism at the second layer more compact; the length in the x direction is reduced. Namely, the height h in Figure (b), which is a non-unctional dimension or strain ampliication, can be reduced. These size reductions allow us not only to pack many PZT units densely but also increase the eective strain along the output axis, ɛ, since h is involved in the denominator o (3). III. PROPERTIES OF IDEAL NESTED RHOMBUS PZT ACTUATORS A. Aggregate Force and Displacement In the nested rhombus PZT actuator, displacements o the individual PZT actuators are aggregated and transmitted through the multiple layers o strain ampliication mechanisms, resulting in an output displacement at the inal layer. Similarly, the output orce is the resultant orce o many PZT actuators. In this section, these aggregate orce and displacement are analyzed in relation to the individual PZT actuator outputs based on an ideal kinematic and static model o the nested rhombus mechanism. Consider a PZT stack actuator shown in Fig. 4. Let l, w,andh be the length, width, and height o a PZT stack actuator respectively. The x-axis is deined as the actuation direction. Choice o y and z axes is arbitrary. For descriptive purposes, the y axis is chosen to the direction o w as shown

4 4 Fig. 3. x z y x z PZT stack 3 Dimensional Nesting or % Strain l (actuation direction) z y x y z z y x z h w Note that the equivalent stiness o the PZT stack viewed rom the output side o the rhombus mechanism is attenuated by a actor o /(a ). Suppose that N units o this irst layer are connected in series and are enclosed with a second-layer mechanism. Each unit is numbered rom to N. Parallel connection in a layer is not considered since it orms a closed kinematic chain or ideal rhombus mechanisms and solving the kinematic chain problem is not essential. Let V i, i,andδxi ( i =,,N ), respectively, be the voltage, orce, and displacement o the i- th unit in the serial connection o the irst layers. The orce is common to all the N units: From (9), we have = = = N = com. () Fig. 4. Actuator Coordinate System o PZT Stack Actuator in Fig. 4. The displacement o this PZT stack actuator when no load is connected to the actuator is given by Δx = N ilm d 33 V, (7) where N ilm is the number o PZT ilms along the actuation direction, d 33 is piezoelectric coeicient, and V (> ) is voltage applied to each PZT ilm. Strictly speaking, the piezoelectric coeicient d 33 is not a constant; according to [4] it may vary signiicantly as strain gets larger. In this paper, however, it is assumed to be constant. The inherent stiness o this actuator is given by k = Ehw l, where E is the elastic modulus o PZT material. The no-load displacement given by (7) results rom the balance between the net orce produced by the PZT and the restoring orce due to the stiness k, which is proportional to Δx. Unlike standard electromagnetic actuators, e.g., DC and AC motors, PZT and other actuator materials cannot produce orce independent o its displacement. Due to the inherent structural stiness the net output orce o these actuator materials gets substantially lower when producing a displacement at the same time. Consider the ollowing orce-displacement relationship; the orce generated by the PZT stack actuator while producing displacement Δx is given by = k (βv Δx ) (8) where β = N ilm d 33. As this PZT stack is imbedded in a irst rhombus mechanism, the orce is reduced to /a and the displacement is ampliied a times, i.e., = /a and Δx = a Δx. Assuming that the rhombus mechanism is loss-less and that the beams are completely rigid and are connected with ree joints, the orce-displacement relationship at the output axis o the irst layer rhombus mechanism is given by = a = k βv k a (a ) Δx (9) k βv i k a (a ) Δxi = com, (i =,,N ). () Suppose that the second rhombus mechanism ampliies displacement and attenuates orce a times. The resultant displacement at this layer is given by Δx + +Δx N = Δx. () a From () and (), the relationship between the output orce and displacement or the second layer is given by = com = βk a N a a N i= V i k N (a ) (a ) Δx. (3) Repeating the same process yields the relation between the aggregate displacement and orce along the K-th layer output axis: K = N K βk K K a k N k k= k= k K K N k k= k= N N j= i= i, j, V (a k ) Δx K (4) Totally N K N K N PZTunitsareinvolvedinthe i, j, system, and V in the above equation represents the voltage applied to each individual PZT actuator. See Figure ork =3where V i, j is applied to the i-th PZT unit in the irst layer involved in the j-th unit (j =,,N ) o the second layer. From the above results we can note that: ) Given applied voltages, the maximum o the aggregate displacement is obtained when no orce is generated, i.e., ree load. This aggregate ree-load displacement Δx ree K is proportional to the total sum o the inputs N N N K i, j, V ampliied by a actor o K a k, k= ) The maximum o the aggregate orce is obtained when the output displacement is totally blocked. This aggregate

5 5 blocking orce K block is proportional to the average o the N K entire inputs: N N i, K V j,. k= N k I the total number o PZT actuators is very large, the individual PZT stack actuators can be driven with simple ON- OFF controls [3] [4], since the net eect upon the output displacement and orce is the summation and average o many PZT actuators. Expensive analogue drivers and controllers are unnecessary or the cellular actuators. As the number o PZT actuator units increases, discretization error becomes small and smooth output displacement and orce can be expected. B. Feasibility Check or % Strain Figure 3 also illustrates a design example aiming at % eective strain. In this design example, 6 irst-layer rhombus units are connected in series. As described in the previous section, 3-dimensional structure plays a key role or large strain. The serially connected units are rotated 9 degrees and inserted into the second layer rhombus. By this design the second layer rhombus extends when the PZT stack actuators are turned ON since they are extensible and the number o ampliying layers is. Let the size o the PZT stack actuator be.8[mm] (l ) 6 [mm] (w ).5 [mm] (h ). The initial gap, d, between the surace o the PZT stack actuator and the apex o the irst rhombus mechanism is. [mm]. We apply typical values o PZT-ceramics or Young s modulus and strain, i.e., E = 55. [GPa], and ɛ (= ɛ ) =.%. These dimensional parameters have been determined according to a commercially available PZT actuator, Cedrat APA5XS [3], as the irst layer unit. The size o the second layer is.[mm] (length, actuation direction) 8.[mm] (width).8[mm] (height). The thicknesses o the ampliying mechanisms and connection parts between the units have been neglected or simplicity. From iterative calculations o (9) and (3), the ampliied strain and reduced blocking orce are obtained as shown in Fig. 5. The prospective displacement is.8 [mm] or the actuator length o [mm], which is equivalent to ɛ =3.9%. This result implies that over % stain is easible by the proposed nested structure. Also, the resultant blocking orce is predicted to be 5. [N]. IV. NESTED RHOMBUS MECHANISMS WITH STRUCTURE FLEXIBILITY The above initial design was conducted based on the ideal kinematic model having rigid beams and ree joints at the strain ampliication mechanism. Actual mechanisms, however, inevitably have some compliance at the structure, which may degrade the aggregate orce and displacement. Not only the compliance o piezoelectric stack actuators but also the compliance o the ampliication structures need to be taken into account in designing the nested strain ampliication mechanism to minimize its adverse eect. In this paper, we ocus on static analysis in order to better understand physical limitations and design trade-os o the nested rhombus mechanisms. Meeting a static perormance Strain (absolute value) % eective strain (.8mm displacement) ε ε ε Blocking orce [N] % 5.N block (a) Strain (without load) block (b) Blocking orce block Fig. 5. Idealized Analysis: Compliance o the ampliication mechanism is not considered. Fig. 6. PZT stack actuator k Δx k load Load (pure spring) Model o PZT stack actuator connected to a spring load goal is the primary concern in designing a strain ampliier o this kind. Dynamic analysis and synthesis will be reported in a companion paper [5]. We assume that the linear characteristic in (8) holds or simplicity since the hysteresis compensation o piezoelectric materials has long been studied [6] and is not the main aim o this paper. A. Eects o Joint Stiness and Beam Compliance Consider a spring load serially connected to a PZT stack actuator, as shown in Fig. 6. Let k load be a spring constant o the load, and Δx be the displacement o the load. The ollowing equations hold: = (k load + k )Δx (5) = k load Δx, (6) where is the actuator output orce applied to the load. The ree-load displacement Δx ree is calculated by letting k load =. Eliminating Δx yields k load = (7) k + k load Note that the actuator output orce becomes signiicantly lower than the original PZT orce when k gets larger or k load gets lower. Similarly, the output displacement Δx also gets attenuated: k Δx = Δx ree (8) k + k load The simple model described above shows that both output orce and displacement are attenuated due to the compliance o the connected load as well as the stiness o the actuator itsel. When this PZT stack actuator is connected to a rhombus strain ampliication mechanism, an external load having properties similar to the above k load and k will be imposed on the PZT actuator. As many layers o the ampliication mechanism

6 6 are attached to the PZT stack actuator, these structural eects will be even more prominent. In the ideal mechanism shown in Fig. (b), we have assumed that the our beams o the rhombus are completely rigid and that all the joints are ree to rotate and purely revolving. However, these assumptions do not hold in real structures. Note that abrication o ree joints is diicult in small scale due to mechanical tolerance and play. For the irst and second layers, in particular, where the displacement is extremely small, the displacement created by the PZT is likely to diminish due to play at the joints. Thereore, lexural pivots and lexible beams [9] [3] [6] have been used or ampliying PZT displacement. Figure 7 shows an example embodiment o the rhombus mechanism. These lexural joints and beams inevitably bring undesirable properties to the system. There are three types o undesirable properties: ). First, the joints are no longer ree joints, but they impose a spring load that the PZT has to overcome. Figure 8 depicts this parasitic eect o joint stiness; some raction o the PZT orce is wasted or coping with the joint stiness. This results in reduction in ree-load displacement. This implies that the joint stiness has an equivalent eect to that o the PZT stiness k in Fig. 6. The stiness o the joints brings increased stiness k or the PZT to overcome. ). Second, lexibility at the beams may attenuate the displacement and orce created by the PZT. Consider the case where the output displacement is blocked as shown in Fig. 9. As the PZT generates a displacement, the beams are deormed and thereby the transmitted orce becomes lower; at least it does not reach the same level as that o the rigid beams. Similarly, i the output axis is coupled to another compliant load, the output orce and displacement will be prorated between the load compliance and the beam compliance. As the beam stiness becomes lower, the output orce and displacement decrease. 3). Third, lexural joints not only create pure rotational displacements but also oten cause unwanted translational displacements. These elastic deormations at the joint along the direction o the beam incur the same problem as the beam compliance; the orce and displacement created by the PZT tend to diminish at the joints. It is important to distinguish two dierent types o compliance in the above cases: One is to take place in the kinematically admissible space o the ideal rhombus mechanism, and the other is in the orthogonal complement to the ormer, termed the constrained space [7]. The joint stiness described in ) is in the admissible motion space, while ) and 3) are in the constrained space. Curved beams, such as seen in Moonies, contain compliance in both constrained and admissible spaces. The distributed compliance can be approximated into the two types o lumped compliant elements. To minimize the adverse eects o the nested rhombus mechanism, the stiness in the admissible space must be minimized and the stiness in the constrained space must be maximized. As multiple layers o strain ampliication mechanisms are used, the compliances in the admissible and constrained spaces become more intricate. In the ollowing sections the kinematic and static characteristics o multi-layer compliant rhombus mechanisms will be Fig. 7. Fig. 8. Fig. 9. Compliant joint by thin beam Internal Unit Embodiment o Rhombus Mechanism Free joint (a) Ideal Rhombus Rigid beam Elastic joint Compliant beam (b) Rhombus with Elastic Joints Eect o Joint Compliance on Free-load Displacement Free joint analyzed. (a) Ideal Rhombus Rigid beam Free joint (b) Rhombus with Elastic Beams Eect o Beam Compliance on Blocking Force Rigid beam Elastic beam B. Modeling o Single-Layer Flexible Rhombus Mechanisms Consider the case shown in Fig. (a) where a rhombus mechanism, including Moonies, is connected to a spring load. k load is an elastic modulus o the load, and k is an elastic modulus o the internal unit such as a PZT stack actuator. Δx is the displacement o the internal unit, and is the orce applied to the ampliication mechanism rom the internal unit. is the orce applied to the load rom the actuator, and Δx is the displacement o the load. In this igure, we assume that the internal unit is contractive or later convenience. The rhombus strain ampliication mechanism is a two-port compliance element, whose constitutive law is deined by a x stiness matrix: [ ] [ ] I Δx = S (9) O Δx [ ] where S s s = 3 is a stiness matrix. s 3 s I is the net orce applied to the mechanism rom the internal unit, and O is the reaction orce rom the external load. Note that the stiness matrix S is non-singular, symmetric, and positivedeinite; s >, s > and s s s 3 >. The symmetric nature o the stiness matrix ollows Castigliano s theorems. When the input port is connected to a PZT stack actuator producing orce with inherent stiness k and the output port is connected to a load o stiness k load,wehave I = k Δx = s Δx + s 3 Δx () O = = k load Δx = s 3 Δx + s Δx. ()

7 7 Eliminating Δx rom the above equations yields: ( k + s = k load + s (k + s ) s ) 3 Δx () s 3 s 3 Deining = s 3 k + s (3) k = s (k + s ) s 3 = s k + dets >, (4) k + s k + s the above equation () reduces to =(k load + k)δx (5) Force and stiness k represent the eective PZT orce and the resultant stiness o the PZT stack all viewed rom the output port o the ampliication mechanism. A drawback with the above two-port model representation is that it is hard to gain physical insights as to which elements degrade actuator perormance and how to improve it through design. In the previous section two distinct compliances were introduced, one in the admissible motion space and the other in the constrained space. To improve perormance with respect to output orce and displacement, the stiness in the admissible motion space must be minimized, while the one in the constrained space must be maximized. To maniest these structural compliances, we propose a lumped parameter model shown in Fig. (b) with 3 spring elements, k J, k BI and k BO, and one ampliication leverage a. As the spring constants, k BI and k BO, tend to ininity, the system reduces to the one consisting o all rigid links, where the output Δx is directly proportional to the input displacement Δx. Stiness k J impedes this rigid body motion, representing the stiness in the admissible motion space. Elastic deormation at k BI and k BO represent deviation rom the rigid body motion. From Figure (b), + k BI (Δx c Δx ) k Δx = (6) a k BO (a Δx c Δx )+k J Δx c +k BI (Δx c Δx )=(7) = k load Δx = k BO (a Δx c Δx ), (8) where Δx c is the displacement at the connecting point between the leverage and springs; however this point is virtual and Δx c does not correspond to a physical displacement. This model is applicable to a wide variety o rhombus-type ampliication mechanisms including Moonies. See Appendix or more detail about the validation o the model and parameter calibration. Consider the blocking orce when the PZT stack actuator generates its maximum orce, max, given as ollows: block ak BI k BO = (a k BI k BO +k BI k J )+k (a k BO +k J +k BI ) max. (9) Similarly, the ree-load displacement or this rhombus mechanism, where k load, isgivenby Δx ree ak BI = max. (3) k (k BI + k J )+k J k BI Fig.. PZT stack actuator Δx Δx k PZT Δx k load (a) Rhombus Mechanism with Structural Flexibilities Δx k BO k k J k BI Δx c a Δx k load (b) Proposed Lumped Parameter Model a k load Load (c) Model o Idealized Rhombus (k BI,k BO, k J ) Model o Rhombus Mechanism with Flexibility As addressed above, these equations imply that the blocking orce will be maximized by k BI,k BO. Similarly, k J maximizes Δx ree. The other advantage is that the 3-spring model is able to represent the ideal rhombus shown in Fig. 9 (a) as a special case as shown in Fig. (c). See (4) and conirm that the stiness matrix S cannot be deined or both k BI,k BO and k J. As described in Section V, the number o unknown parameters becomes 4 as the rigid ampliication leverage is explicitly included, which makes the calibration problem ill-posed; however, this ampliication leverage is necessary to include the ideal case. In addition, 3 lumped springs are considered minimum to satisy the input-output bidirectionality, which is a basic requirement o Castigliano s theorems. C. Model Simpliication From (6) to (8), the relationship between and Δx is given by (a k BI k BO ) = [ kload {a k BI k BO + k BI k J + k (a k BO + k J + k BI )} +k BO (k BI k J + k k J + k BI k )] Δx. (3) The above equation can be written as =(k load + k )Δx (3)

8 8 where k BO (k BI k J +k k J +k BI k ) k = (a k BI k BO +k BI k J )+k (a (33) k BO +k J +k BI ) ak BI k BO = (a k BI k BO +k BI k J )+k (a k BO +k J +k BI ).(34) This implies that the proposed lumped parameter model shown in Fig. (b) can be urther simpliied to Fig.. Note that the direction o is opposite to due to the ampliication leverage. This simpliied model has a similar orm as in (5). As will be described in the ollowing section, this simpliication enables perormance evaluation o complex nested mechanisms simply by nesting a simpliied model o lower layers into a higher lumped parameter model. As a result, the perormance o the overall mechanism such as aggregate displacement and orce can be predicted in a recursive manner without developing a ull FEM model [] or solving continuum elastic equations [8]; parameter calibration at the layerlevel is suicient. Fig.. ~ k ~ PZT stack actuator Fig.. k ~ ~ Δx k load Simpliied Representation o Lumped Parameter Model k BI k J st layer k BO a Equivalent model o each nested unit N serial connection k BI k J nd layer k BO a N serial connection Recursive Formula or Nested Rhombus Model k J 3 k BI 3 3 rd layer a 3 k BO3 D. Recursive Formula o Aggregate Force and Displacement o Flexible Nested Mechanisms The goal o this section is to describe a recursive ormula to obtain an equivalent model or a general nested mechanism. Figure shows a nested rhombus structure. As addressed in the previous sections, each nested layer can be represented by its equivalent model, which enables us to describe the orcedisplacement property or the nested structure in an iterative manner. Let K be the number o nesting layers. Also, let k Jk, k BIk, k BOk, N k be the joint compliance, beam compliances, and the number o serial connection or the k-th (k =, K) layer. This mechanism involves N K N K N PZT stack actuators. By applying (33) and (34), the equivalent model or the k-th layer can be represented by k BOk (k BIk k Jk + k k kk N k k = k k Jk +k BIk N k ) (a k k BIkk BOk +k BIk k Jk )+ k (35) k N k (a k k BOk+k Jk +k BIk ) a k k BIk k BOk k = (a k k BIkk BOk +k BIk k Jk )+ k k N k (a k k BOk+k Jk +k BIk ) N k N k, i (36) k i= where k i is the equivalent orce o the i-th unit in the (k )- th layer. Recall that k is proportional to the average o the entire orces at the (k )-th layer as described in (III-A). Assume that all actuators in the (k )-th layer are controlled in a binary manner [4], i.e., Vk i = {V max (ON), (OFF)}. Also assume that all units are uni- block k orm and each unit generates as its blocking orce. Thereore, when n units out o N k actuators are ON, the last term o (36) can be replaced as N k N k i= i k n N k block k (37) Fig. 3. (a) Structure (b) Structure Example Ampliication Mechanisms Δx Δx (c) Displacement when connected to a ixed beam The ree-load displacement changes accordingly. Both the aggregate ree-load displacement and the blocking orce are proportional to the number o ON units. V. VERIFICATION AND CALIBRATION OF 3-SPRING LUMPED PARAMETER MODEL The validity o the proposed lumped parameter model is conirmed by FEM (Finite Element Method). Consider the two ampliication mechanisms shown infig.3orexample.the size o the both mechanisms is: 4 [mm] (length, actuation direction) 96 [mm] (width) 5 [mm] (thickness). Brass (Young s modulus =. [GPa]) is used. The our structural lumped parameters, i.e., a, k BI, k BO,andk J, are calibrated by the displacements and orces rom two dierent conditions; one case is blocked case where the output displacement is totally constrained (or letting k load ), and the other one is ree-load case (or letting k load =). By applying an input orce,, Δx block and block are measured or the blocked case, and Δx ree and Δx ree are measured or the ree-load case. From (6) to (8) we have block Δx block block Δx block Δx ree Δx ree = a k BI k BO + k J k BI a k BO + k J + k BI = X (38) = = ak BI k BO a k BO + k J + k BI = X (39) ak BI k J + k BI = X3 (4)

9 9 ree Δx ree = k J a = X 4. (4) Note that the actual number o independent equations described above is 3, which can be conirmed by X = X 3 (X + X 4 ). This implies that the calibration o the our structural parameters, [a, k BI, k BO, k J ], is an ill-posed problem. This can be conirmed by the two-port model representation in (9). The stiness matrix S is generally given as: [ X X S = X X 3 X ]. (4) Recall S = S T and it ully represents the relation between the displacements and orces. Thereore, the number o independent elements is 3 by calibrating S. Unlike the ideal rhombus mechanism consisting o all rigid links, the displacement ampliication gain a cannot be deined uniquely as long as the stiness in the constrained space is inite, i.e., k J >. Note that the choice o a does not change S or the characteristics o the estimated model; however, a nominal gain â should be determined to have a physically easible lumped parameter model, that is, k BI,k BO,k J >. One way o determining â is based on ree-displacement characteristics and kinematic characteristics o the structure such as the angle o the oblique beam θ, i.e., X 3 < â<cotθ, to satisy the requirement. X 3 can be assumed as a lower bound o â since X 3 is always lower than the actual a i k J is positive. In addition, cotθ can be assumed as an upper bound o â since this gain is =. The ollowing steps estimate realized only when k J the remaining parameters: ˆk J = â X 4, ˆk BI = X3ˆk J â X 3,and ˆk BO = (ˆk BI+ˆk J )X. âˆk BI â X Table I shows the observed values rom FEM when applying = [N]. The structural lumped parameters are calculated as shown in Table II. The nominal ampliication gains are determined accordingly based on the observed X 3 s and kinematic characteristics to keep all spring constants positive. As shown in Table I, Structure provides approximately 3 times larger ree-load displacement than Structure, while the blocking orces o the two structures are almost the same magnitude. This observation suggests that Structure has a more avorable structure than Structure as an ampliication mechanism. This can be explained based on the estimated lumped parameters: The eective stiness in the constrained space k B viewed rom the input port is calculated as k B = k BI +. (43) a k BO This expression gives kb =6.8e+6 or Structure and k B =.3e+7 or Structure. As a result, Structure has a smaller stiness in the admissible space, k J, and a larger stiness in the constrained space, kb, compared with Structure. Although Structure provides relatively good perormances, it could also have problems in development due to its complex shape and in strength due to stress concentration at thin sections having large deormation. Maximum stress when producing the ree-load displacement is shown in Table II. The validity o the calibrated models is conirmed by examining Δx and Δx when connecting the mechanism TABLE I OBSERVED VALUES FROM FEM Structure Structure Δx ree [m] (@ =).95e-5.6e-4 block [N] (@ =).73.5 X 6.64e+6.5e+7 X.8e+6 3.4e+6 X X 4 5.3e e+4 TABLE II ESTIMATED LUMPED PARAMETERS Structure Structure â ˆk J [N/m].64e+6.5e+5 ˆk BI [N/m].46e+7 4.5e+7 ˆk BO [N/m].5e+6.3e+6 σ max [N/m ] 5.35e+6.75e+7 TABLE III ESTIMATED DISPLACEMENTS Structure Structure t Disp. FEM Model Error FEM Model Error [mm] [μm] [%] [%] Δx Δx Δx Δx Δx Δx Ave. [%]..884 to a spring load realized by a ixed beam shown in Fig. 3 (c). The length o the beam is L = [mm] and Brass is used as material. There is a small protrusion on the top o the actuator so that the ampliication mechanism and the ixed beam are assumed to be in point contact. Three dierent thicknesses, rom [mm], [mm], to 3 [mm], are used to vary the stiness. Table III shows the comparison o the estimated displacements rom the proposed lumped parameter model and the true values rom FEM analysis. As can be observed in the table, the estimated values agree well with the true values, conirming the validity o the model. Consider two constrained cases shown in Fig. 4 (a) and (b) to conirm the necessity o 3 lumped springs. For simplicity, the ampliication mechanism is designed to have a square shape resulting in a =. Δx =Δx must hold or the displacements when applying the same magnitude o orce, which is a basic requirement o Castigliano s theorems. By using the proposed lumped parameter model, these constrained cases are represented by the igures shown below. The condition Δx =Δx is satisied i k BI = k BO. This can also be conirmed in (4) where the o-diagonal elements are the same. As described in Section V, this lumped parameter model has a redundancy in parameter calibration; however, 3 spring elements are minimally required to satisy this condition. VI. PROTOTYPE ACTUATOR A. Mechanical Design o Rhombus Mechanism A prototype nested actuator with over % eective strain is designed based on the structural compliance analysis. Consider a nested structure with ampliication layers as shown in Fig.

10 Δx Δx Δ a = a = x k BO k BO Δx TABLE IV CHARACTERISTICS OF THE APA5XS ACTUATOR [3] FOR THE FIRST LAYER(DEFINITION OF THE DIMENSIONS HAS BEEN MODIFIED.) Displacement 8 [μm] Blocking Force block 8. [N] Stiness k.5 [N/μm] Voltage range - 5 [V] Length (output actuation direction) 4.7 [mm] Width ( stack actuation direction).8 [mm] Height 9. [mm] Mass. [g] k BI k BI k J (a) Output constrained k J (b) Input constrained.mm.3mm 3.5mm h J b J L J Fig. 4. Requirement o Input-Output Bidirectionality mm 3.5mm 4.97 deg φ 3mm k φ 3. The APA5XS Moonie piezoelectric actuators developed by Cedrat Inc. [3] are adopted or the irst layer. According to the preliminary design in Section II, over % o eective strain can be obtained by a two-layer mechanism; K =and α =5. By stacking 6 APA5XS actuators or the irst layer, i.e., N =6, this large strain may be achieved with a proper design o the second layer. From Table IV, we have k =.5 6 block [N/m], and =8. [N] or the irst layer units. The remainder o this section ocuses on the design o the secondlayer rhombus mechanism. From (35) and (36) we obtain an equivalent model or the second layer by substituting (33) and (34), which provide a design guideline in terms o k BI, k BO and k J or the target eective strain, i.e., %. As described in the previous section, the stiness in the admissible space, i.e., k J, must be minimized. The compliant joint may be represented as shown in Fig. 5. The rotational stiness o this structure is given by k φ = EbJh3 J L J where E is Young s modulus o the material. In order to reduce this stiness, either the width b J or thickness h J must be reduced, or length o the gap L J must be increased. Note that the reduction o h J is the most eective or reducing k J since it is proportional to h 3 J ; however, the thickness must be careully determined considering manuacturing process. The maximum stress must be lower than the yield stress o material. In addition, in order to increase the stiness in the constrained space, i.e., k BI and k BO, the oblique beam need to have a suicient thickness except the thin part or the compliant joint. Figure 5 shows the designed rhombus mechanism or the second layer. Phosphor bronze (C544, H8) is used or the material. The length o the mechanism in actuation direction is [mm], and the width is 3 [mm]. The thickness o. [mm] is given to h J or electrical discharge machining. The thickness o.3 [mm] is given to the oblique beams or suicient stiness. The oblique angle o the beams is 4.97 [deg] that gives the displacement ampliication ratio o.5 assuming the mechanism is ideal. For the second layer mechanism shown in Fig. 5, X = , X =.95 5, X 3 =.33, and X 4 = are obtained rom FEM analysis. The range o Fig. 5. Design o Rhombus Mechanism or the nd Layer a that makes all spring constants positive is shown in Fig. 6. Finally â =.4 is chosen, which is between X 3 and cotθ (=.5). Finally, the parameters or the lumped parameter model are calculated as ˆk BI =6.7 6 [N/m], ˆk BO = 5. 4 [N/m], ˆkJ = [N/m] by determining the ampliication gain as â =.4. The analysis using the lumped parameter model predicts that the maximum ree-load displacement is.64 [mm], which is equivalent to % eective strain. Also, the maximum blocking orce is.56 [N]. B. Development and Perormance Evaluation Figure 7 (a) shows the developed rhombus mechanism or the second layer and (b) shows the assembled actuator with 6 irst-layer units. The second layer mechanism weighs approximately 3 [g], resulting in the total weight o 5 [g]. The connected irst-layer units were inserted in the second layer mechanism and the both ends were manually bonded to the inside walls. Note that we noticed that the tolerances on the interaces were important or satisactory actuator perormances. Figure 8 shows snapshots o ree-load displacement where rhombus mechanisms are connected in series. This actuator extends since the irst layer units are contractive. The perormance o this prototype is evaluated by measuring ree-load displacement and blocking orce. Figure 9 (a) shows the maximum ree-load displacement measured using a laser displacement sensor (Micro-Epsilon optoncdt 4) when all 6 units are ON by applying 5 [V] actuation voltage. The measured displacement is.53 [mm] that is equivalent to.% eective strain. Figure 9 (b) shows the blocking orce where a sinusoidal wave input ranging rom 5 [V] is applied. The maximum blocking orce measured using a compact load cell (Transducer Techniques MLP) is.69 [N]. As shown in Table. V, the experimental values by using the lumped parameter model agree well with the estimated values, which conirm the validity o the approach.

11 Positive stiness constants k BI 5 3 x k BI k BO kj kj x7 k BO 3 (a) Connected irst layer units -3 aˆ Fig. 6. Choice o a ˆ or Positive Spring Constants (b) Second layer ampliication mechanism (c) Assembled actuator Fig. 7. Prototype Actuator: 6 CEDRAT actuators are used or the irst layer TABLE V C OMPARISON BETWEEN I DEALIZED MODEL, PROPOSED LUMPED PARAMETER MODEL, AND EXPERIMENTAL MEASUREMENT Eective strain Free displacement Blocking orce Analysis Idealized model Analysis Lumped model 3.9 [%]. [%].8 [mm].64 [mm] 5. [N].56 [N] Experiment [%].53 [mm] (std ).69 [N] (std ) OFF Fig. 8. ON ON Free-load Displacement when all o the 6 irst-layer actuators are The comparison between Fig. 5 and Fig. 9 suggests that the aggregated orce has been considerably attenuated, while the aggregated displacement or strain is as large as predicted by the idealized analysis. Note again that the result in Fig. 5 will never be achieved unless completely rigid beams and purely rotational joints without play are utilized. One o the diiculties in mechanical design is that physical structural parameters are intricately related to lumped parameters. For example, the increase o the gap LJ in Fig. 5 contributes to reducing the joint stiness but it also reduces the beam stiness by having a long thin gap in the longitudinal direction. This gap may be reduced i the design ocus is more on producing a larger blocking orce. Displacement [mm] Time [s] 5 (a) Free-load Displacement (Step response) Force [N] Figure shows the aggregate displacements when ONOFF control is applied to the internal 6 units by applying a constant actuation voltage when ON. For convenience, the measured displacements are normalized by the maximum displacement when all 6 units are turned on. As described in Section III-A, and IV-D, the distribution o the ON units in a layer does not theoretically aect on the aggregate displacement i an ampliication mechanism encloses seriallyconnected internal units. As can be observed in Fig., the measured displacement is not largely aected by the combination o ON units. For example, there are (=6 C3 ) combinations when the number o ON units is 3; however, the standard variation is at most.7, showing a suicient repeatability. Another observation is that the increment o the displacement is not uniorm, which is considered due to the nonlinearity o (). In Section II-A, we have neglected this issue or simplicity; however, this characteristic should be urther relected to the design and control i rhombus mechanisms are used or creating large strain Time [s] 8 (b) Blocking-orce (Sinusoidal wave input) Fig. 9. Experimental Result C. Modular Design o Cellular Actuators The authors have proposed a new control method to control vast number o cellular units or the cellular actuator concept inspired by the muscle behavior [3] [4] [5]. Instead o wiring many control lines to each individual cell, each cellular actuator has a stochastic local control unit that receives the broadcasted signal rom the central control unit, and turns its state in a simple ON-OFF manner as described in Section III-A. A array structure using polyvinylideneluoride (PVdF) ilm actuators has been abricated that produced. mm ree displacement and 5.5 mn blocking orce rom

12 Fig.. Normalized Displacement Measurement Average.8 (.6).43 (.7).43 (.7).596 (.5).794 (.3) Ave Average normalized value (SD) Standard deviation Number o Units Turned On Binary ON-OFF Control Experiment Fig.. Modularity o Cellular Actuators: Mock-up Cellular Actuators with stacks and 4 bundles (let) and 6 stacks and 7 bundles (right). mm mm ootprint [], whose orce may be insuicient to drive macro-size robots. Figure shows two example conigurations where cells are connected in series (stack) and in parallel (bundle). A wide variety o sizes and shapes is conigurable using the designed actuator as a building-block. For example, the coniguration shown in the right is expected to produce.8 [N] (.67N 7 bundles) o blocking orce and 5. [mm] (.53mm 6 stacks) o ree displacement i the developed units are applied. The macro actuator array in the igure corresponds to a single muscle, and each o the prototype units (cells) corresponds to Sarcomere known to be controlled in an ON-OFF manner. For example, a pair o the actuator arrays will be attached to a link mechanism in an antagonistic arrangement. VII. CONCLUSIONS AND FUTURE WORKS This paper has presented a nested rhombus multi-layer mechanism or PZT actuators. The idealized analysis has been given or undamental design o the nested structure. Through kinematic and static analysis this paper has addressed how the output orce and displacement are attenuated by the structural compliances involved in the strain ampliication mechanism. A lumped parameter model has been proposed to quantiy the perormance degradation. A prototype nested PZT cellular actuator that weighs only 5 [g] has produced % eective strain (.53 [mm] displacement rom [mm] actuator length and 3 [mm] width) and.69 [N] blocking orce. A modular design concept has been presented or building reconigurable cellular actuators with matched stroke and orce requirements. Future work includes () nonlinear and dynamic modeling such as requency response, () analysis o a closed kinematic chain ormed by serial-parallel mixed coniguration, (3) design optimization and eiciency analysis, and (4) application to practical systems such as robot hands and rehabilitation robots. Especially, dynamic analysis is important or high-speed actuation. Based on our preliminary work [5], the dynamics modeling will be discussed in our uture publication. The uture work also includes (5) investigation o manuacturing aspects or accuracy improvement and uture cost reduction since the number o actuators as well as the complexity increase as the number o layer increases. REFERENCES [] J. Ueda, T. Secord, and H. Asada, Static lumped parameter model or nested PZT cellular actuators with exponential strain ampliication mechanisms, in IEEE International Conerence on Robotics and Automation, 8. 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