Proceedings of the ASME 2009 International Mechanical Engineering Congress & Exposition IMECE2009 November 13-19, Lake Buena Vista, Florida, USA

IMECE2009-10781 DYNAMICS-BASED DESIGN OF A SOFT ROBOT

Steven E. McHugh Tufts University Medford, MA, USA

Frank J. Saunders Tufts University Medford, MA, USA

ABSTRACT This paper develops a methodology for converting the results of coarse dynamic simulations into fully realized designs. In particular, we demonstrate how to convert a caterpillar-like soft-bodied robot from a lumped-parameter form into a CAD model that could be easily manufactured. To simplify this design problem, we propose a decomposition method involving three steps. The first step groups the elements of the lumped model into segments. This segmentation simplifies the second step, where rough CAD models are automatically synthesized for each segment from the union of many prisms. In the third step, a human designer combines and smoothes these segment models to create a fullbody CAD model. This approach simplifies the development of a high degree-of-freedom soft robot by guiding the designer to a final model with an almost fully automated process. I. INTRODUCTION In an attempt to meet the demand for more versatility in mobile robots, designers are increasingly integrating soft materials and structures in their designs. Potential applications of soft robots include: medical applications, search and rescue applications, and surveillance. Soft robots are ideal for human body contact, such as in medical applications, because their inherent compliance reduces the risk of damaging sensitive tissue [1, 2]. Soft robots would also be ideal for searching areas inaccessible by either humans or rigid robots, leveraging their flexibility to enter buildings through unconventional means. For example, a soft robot might traverse through a space several times smaller than its largest characteristic dimension and reconstitute its size, while also performing a function or task with an embedded payload [3]. Many soft robotic capabilities have already been demonstrated in the laboratory. For example, soft materials have been incorporated into very high degree-of-freedom robotic manipulators capable of adaptable gripping in the fashion of an elephant trunk or an octopus arm [4]. Soft

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Jason H. Rife Tufts University Medford, MA, USA

materials have also been used to create platforms that help understand soft-robot dynamics by studying passive and active deformations using electro-active polymers [5-7] and shapememory alloy actuators [8]. Specialized soft casing materials have even been used to perform electrochemical computation, in which the casing enables the robot to complete a maze without relying on a silicon processor or additional sensors [9]. Soft materials have additionally been used to develop compliant joints critical to the dynamic stability and speed of walking robots, such as the Sprawl and Rhex hexapods [10, 11]. However, soft robots have not yet transitioned from the research lab into commercial product development. One of the major limitations is coordination of dynamics for very high degree-of-freedom systems [12]. While rigid structures can often be visualized in static or quasi-static terms, dynamics must generally be considered in soft robot design since soft structures deform when loads are applied. Furthermore, the inherent flexibility in soft systems means that actuator placement is not obvious, nor are the motions that those actuators will create. These characteristics of soft robots undermine the conventional strategy of designing structure first and adding controls last. Rather, to obtain efficient movement, it is essential to design the form and the controls of a soft robot concurrently. Conceptually, design optimization techniques [13] can be applied to enable simultaneous tuning of the structure and controls of a soft robot. Design optimization methods have previously been applied to evolve rigid robots from libraries of fundamental building blocks, such as simple bars and linear actuators [14, 15]. Design optimization for soft robots is more challenging, however. Firstly, physics simulations for continuum materials, such as finite-element analysis, are computationally expensive, particularly when applied to compute the response to dynamic loading. Secondly, soft structures are particularly sensitive to geometry, so fundamental building blocks, such as rods, spheres and prisms, cannot be used to physically construct prototypes. Soft robots

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need to be designed as smooth, continuous and enclosed shapes to properly control the dynamics. For these reasons, we desire new design methods that (1) run fast enough to perform optimization of structures under dynamic loading and that (2) generate smooth structures with tunable dynamics. We propose an innovative approach for designing soft robots by starting with an abstracted model of robot motion and subsequently refining the abstract model to obtain a detailed three-dimensional design. We refer to this approach as dynamics-based design. Dynamics-based design consists of three steps. In the first two steps, we develop a lumpedparameter conceptual model for the robot and then apply genetic algorithms to optimize the robot’s structural and control parameters [16, 17]. In the final step, we refine the optimal lumped-parameter model to obtain a threedimensional CAD design with sufficient detail to enable fabrication. This paper will focus on the final step, that of converting a lumped-parameter model into a detailed design. Our approach will be to apply automation tools to streamline a human-centered design process. To provide context, the next section will describe the broader concept for dynamics-based design. Subsequent sections will describe the application of this approach to creating a caterpillar-like soft robot. Section III will briefly describe the definition and optimization of a lumped-element model for the caterpillar-like robot. Section IV will discuss our methods for converting the lumped model into a detailed CAD design. The discussion of Section V will describe lessons learned from the caterpillar-study. Section VI will conclude the paper with a brief summary. II. CONCEPTUAL APPROACH The approach described in this paper attempts to overcome the problems inherent to designing soft robots. Due to interdependence of controls and morphology for soft robots, we propose optimizing approximate system dynamics with a lumped-parameter model prior to building a detailed threedimensional CAD design. The principle challenge in designing a soft robot is coordinating the robot’s many degrees of freedom to generate efficient locomotion. Soft robots are highly deformable, continuum structures that exhibit an essentially infinite number of degrees of freedom. Since it is impractical to introduce an equivalent number of actuators into the robot, the number of actuated degrees of freedom is significantly lower than the total number of degrees of freedom of the robot. As underactuated structures, soft robots must therefore exploit tuned structural dynamics to enable efficient locomotion. Such tuning has been observed in nature. As an example, the caterpillar Manduca Sexta uses a remarkably simple neuromuscular system to control the motion of its highly deformable, essentially infinite-degree-of-freedom body. Our proposed dynamics-based design approach introduces a transformation that exchanges the difficult problem of coordinating the soft robot’s degrees-of-freedom for three easier problems. These three easier problems consist of converting the design concept into a lumped-parameter model, optimizing the structural and control parameters associated with that model, and finally transforming the lumped-

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parameter model into a fully resolved three-dimensional CAD design. A flow chart for the proposed dynamics-based design method is illustrated in Figure 1.

FIGURE 1: DYNAMICS-BASED DESIGN APPROACH

Conceptually, this transformation approach is not unlike the Eigenvalue transformation which can be used to simplify the solution for a set of coupled ordinary differential equations (ODEs). The Eigenvalue transformation also exchanges a difficult problem – solving a set of coupled ODEs – for three easier problems, which include a forward matrix transformation into a space in which the ODEs are decoupled, the solution of those decoupled ODEs, and the subsequent reverse transformation of the ODE solutions to recover the original set of system variables. In the lumped-parameter domain, the number of structural degrees of freedom is significantly reduced. The lumpedparameter model incorporates only a small subset of desirable degrees of freedom, which abstract the soft robot’s flexibility and compressibility. Consequently, the problem of optimally coordinating the robot’s morphology and control parameters is concentrated to a relatively low-dimensional search space. An example of model reduction and optimization for the case of a caterpillar-inspired robot will be described in the following section. To optimize locomotion, we implemented a geneticalgorithm approach that evolves morphology and control parameters to maximize the distance traveled by the robot in a fixed interval of time.   In simplifying the coordination of the robot’s degrees of freedom, the proposed dynamics-based design approach introduces the new problem of converting between the lumped-parameter and three-dimensional CAD representations of the robot. The process of converting between these two representations is a “projection” process, in which only the most important details of the fully resolved CAD design appear in the lumped-parameter model. In this sense, multiple solutions exist for converting the optimized lumped-parameter model into a three-dimensional CAD design, since this process must generate missing information that was not essential for simulation of the system’s dynamics. The bulk of this paper focuses on semi-automated methods that aid the human

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designer in the non-trivial process of generating these missing details. III. EVOLUTION OF A BIOMIMETIC ROBOT Our motivation is the design and fabrication of a caterpillar-like soft robot. Caterpillars are capable of crawling, climbing and burrowing to traverse a wide range of terrains. These capabilities make caterpillars an ideal biological inspiration for soft-robot applications such as surgical or search and rescue robotics, for which the ability to locomote in confined spaces is essential [18]. An early prototype based on Manduca sexta demonstrated a proof of concept for the fabrication of a soft robot [19]. A representative Manduca specimen is shown in Figure 2a alongside the early prototype robot shown in Figure 2b. Early attempts to locomote the soft robot prototype using shapememory alloy (SMA) spring actuators met with limited success.

FIGURE 2a: MANDUCA SEXTA

resulting lumped-parameter model. In the figure, each pair of “minor” elements makes up a segment (or “major” element, as labeled in the figure).

FIGURE 3: SIMULATION CATERPILLAR MODEL

The model is actuated with twelve SMA springs, four for each “major” element pair. Applying an electrical current to any SMA spring causes resistive heating. The resulting temperature increase, alters the phase of the nickel-tin alloy from which the SMA spring is fabricated. In its higher temperature phase, called Austenite, the SMA spring contracts, while in its lower temperature phase, called Martensite, the SMA spring relaxes. In the lumped-parameter model, the x-y-z locations of these springs are adjustable assuming dual bilateral symmetry in the front-view plane (four springs in all, based off of one attachment point). Springs are attached between every other “minor” element (specifically, between the anterior “minor” elements of successive “major” elements). Other adjustable model parameters include “minor” element mass and geometric parameters such as foot separation {w}, “minor” element length {L}, and “minor” element radius {R}, as depicted in Figure 4.

FIGURE 2b: AN EXAMPLE PROTOTYPE

To achieve more effective coordination of robot morphology and actuation, we are implementing the design concept summarized in Figure 1. For this purpose, a lumpedparameter model for the caterpillar robot was defined and evolved using a genetic algorithm [16]. Lumped-Parameter Model We hypothesized and later demonstrated that a lumpedparameter model could provide sufficient resolution of the physics of the caterpillar-like soft robot to enable optimization of actuator commands [17]. The lumped-parameter model was developed to capture lower-order flexibility in the robot structure. Because the caterpillar, seen in Figure 2a, is made up of several distinct segments, the prototype mimicked this characteristic, as seen in Figure 2b. In the lumped-parameter representation, segments were modeled as rigid bodies joined by spherical-prismatic joints to enable inter-segmental bending and compression. Linear viscoelastic forcing was applied to each degree of freedom of these joints to represent structural stiffness and internal damping. These segment representations were further subdivided into two rigid elements, also joined by spherical-prismatic joints, in order to model additional bending and compression within individual segments. Figure 3 illustrates the basic structure of the

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FIGURE 4: MORPHOLOGICAL PARAMETERS PASSED INTO SIMULATION

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Genetic Algorithm A genetic algorithm was implemented to evolve the morphological and actuation parameters that together define the soft robot’s gait [16]. The fitness function for the genetic algorithm optimization favored parameter combinations that caused the robot to travel a straight path on a level surface. Specifically, the fitness of the robot was measured by how far it traveled in an allotted time and how much energy it used. The distance traveled for each set of morphological and actuation parameters was determined using a rigid-body dynamics simulation based on Nvidia’s PhysX package. In the optimization, parameter values were restricted to a range of realizable values, inferred from our experience with the prototype robot illustrated in Figure 2b. Our subsequent design efforts are based on the highest fitness case identified by the genetic algorithm. This highperforming design is characterized by the parameters summarized in Table 1. Optimized values for morphological (or “form”) parameters are given in the table. Actuator timing parameters are not specified in the table, because these values (distinct on and off times for each of six actuator pairs) do not impact the design procedure described in the remainder of the paper. TABLE 1: PARAMETERS OF THE OPTIMIZED DESIGN Variable

Description

Rga

Radius of minor segment (units: meters)

0.0143

Lga

Length of minor segment (units: meters)

0.0193

Mass of minor segment (units: kilograms)

0.0103

Foot separation

0.54Rga

wga

Vertical distance from the bottom of the foot to the major segment’s center of mass

2Rga

xsga

SMA attachment point, x distance from center of mass

-0.08Lga

ysga

SMA attachment point, y distance from center of mass

0.04Rga

zsga

SMA attachment point, z distance from center of mass

0.9Rga

Inertia along the major segment x-axis, as derived for a solid prism (see Figure 4; units: kg/m3)

2.81e-6

Inertia along the major segment y-axis, as derived for a solid prism (see Figure 4; units: kg/m3)

3.96e-6

Inertia along the major segment z-axis, as derived for a solid prism (see Figure 4; units: kg/m3)

3.96e-6

Form

Axial Stiffness of a minor segment (units: N/m) --

Actuation start time

--

Actuation stop time

Control

4

Optimized Value

IV. CONVERSION OF LUMPED-PARAMETER MODEL TO FULLY RESOLVED ROBOT MODEL While using the lumped-element model improves the performance of the genetic algorithm, it necessitates a strategy for converting the simplified parameters into a detailed threedimensional CAD design. To implement this process, the last step of the dynamics-based design procedure illustrated in Figure 1, we propose an approach that leverages the strengths of both design automation and human-centric design. Specifically, we propose three sub-steps for generating a three-dimensional design. In the first sub-step, we group together related elements of the lumped-parameter model. This step reduces the number of three-dimensional components that must be designed. In a second sub-step, we apply an automated design method to develop rough threedimensional models for each element grouping. This automation leverages the ability of computer automation to compute rough shapes (such as rods and prisms) that embody the desired physics for each element grouping. In a third substep, a human designer smoothes and joins the rough threedimensional designs to create a manufacturable CAD representation of the full robot. Generating a smooth design, with fewer stress concentrations or corners that might snag in the field, is a challenging task for computer automation, but a simpler one for a human designer. The three sub-steps are summarized in Figure 5.

FIGURE 5: SEMI-AUTOMATED CONVERSION PROCESS

304 Six values for each actuator pair

The proposed conversion process assumes the soft-robot body can be constructed as a union of smaller threedimensional components. In general, slicing a soft structure into arbitrary components does not ensure that those components can be analyzed in a decoupled manner, since the distortion of the overall structure may result from interactions among neighboring components. As an example, it may not be possible to conduct an accurate finite-element analysis for a single component, since boundary conditions may depend on the deformation of neighboring components. To justify designing the full robot structure as a union of decoupled component requires additional justification. Element Grouping For the case of the caterpillar robot, it is natural to group together “minor” elements into “major” elements or “segments.” As noted above, a key issue is ensuring that these segments can indeed be joined together in a manner that ensures the appropriate behavior for the full soft-robot body. In particular, we desire that the stiffness parameters for the

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1

(1)

For comparison to this theoretical result, we performed a series of finite-element analyses using CosmosWorks™. The analyses were performed on two representative segment shapes (with elliptical and rectangular cross-sections as shown in Figure 6). For the purpose of this validation analysis, several different body lengths were considered, including integer lengths of one, two, three and four segments. The four-segment geometries are illustrated in Figure 7. In each finite element simulation, the end points of the soft structures were constrained to lie on a plane.

FIGURE 6: SEGMENTAL BASE SHAPES

computed the body stiffness ktotal for the four-segment geometries (those of Figure 7) and then applied the inverse of Equation (1) to compute the segment ki values, which were assumed equal for all segments, resulting in Equation (2). 4

(2)

Equation (2) was subsequently applied to compute a theoretical curve for stiffness as a function of body length. It should be noted that kseg was computed using the four-segment body (rather than the single-segment body) with the assumption that boundary conditions within the robot might differ from those at the ends of the robot. The impact of boundary conditions variations can be assessed by identifying at what point the theoretical springs-in-series curve diverges from the FEA results. For the case of perfect decoupling, the theoretical curve should not diverge from the FEA data. When the axial stiffnesses for the full-body analyses are graphed, we see that these stiffnesses are very close to theoretical values, as seen in Figure 8 for the elliptical case. The bending stiffness, however, was not entirely accurate and diverged for the single segment case, as seen in Figure 9. This divergence indicates that decoupling is not perfect. Although the error between the FEA analyses and the theoretical curve is relatively small (less than 20%), we have opted not to automate the design of bending stiffness, leaving the process of tuning bending stiffness to the human designer in the final design stage. Finding equations that accurately predict the stiffness behavior of flexure-to-flexure segments, as well as sub-segments, is planned for future work.

Spring Constant (N/m)

lumped-element model are faithfully recreated by the refined three-dimensional design. It is our hypothesis that, at least for caterpillar-like soft robots, it is possible to obtain the desired high-level stiffness properties (e.g. stiffness for the entire body) by applying design automation at the segment level. To justify this hypothesis, at least in a linearized sense, we demonstrate that the full-body stiffness predicted by a finite-element analysis matches that of a theoretical calculation for a series of decoupled components, modeled as springs in series. The following is the equation for the stiffness of a set of springs in series [20]. The stiffness of the set of N springs ktotal depends on the stiffness of each of individual spring, ki.

2500 2000 1500 FEA Simulation

1000

Interpolated k_total(N)

500 0 1

2

3

4

Number of Segments

FIGURE 8: AXIAL STIFFNESS FOR ELLIPTICAL SEGMENTS

FIGURE 7: CONCEPTUAL 3-D CAD DESIGNS

To relate the theoretical result for the springs in series to the finite element result, we first computed ki, both for segments of square and elliptical cross-section. To do this we

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Spring Constant (N/m)

0.3 0.25 0.2 0.15

FEA Simulation

0.1 Interpolated k_total(N)

0.05 0 1

2

3

4

with this model, so we then implemented a less constrained (detailed pyramidal) model by adding more prisms to the body. Optimization of this model slightly improved parameter values but the detailed pyramidal model still did not match the desired inertia and stiffness values simultaneously. To address this issue we introduced a third (flexible hitch) model structure to reduce coupling between the axial stiffness parameter and the inertia parameters. This third rough design configuration proved very effective at matching the desired dynamical parameters, as indicated by Table 2.

Number of Segments

FIGURE 9: BENDING STIFFNESS FOR ELLIPTICAL SEGMENTS

Using Approximate Models to Define Shape This section introduces an intermediate modeling step to streamline the conversion of the dynamical simulation results into a detailed design. This step is beneficial because it helps overcome the challenges encountered with nonlinear coupling among dynamical parameters (including mass, center of mass, contact points, moments of inertia, and component stiffness requirements). This coupling introduces two problems. First, it is difficult to develop simple rules for iterative design refinement, since design decisions affect many dynamical parameters at the same time. Second, many design configurations do not satisfy dynamical parameter requirements; consequently, the optimization of a parametrically dimensioned CAD design may fail to converge. Our approach to address both these issues is to enable designers to quickly define and test parameterized shapes to assess whether or not they satisfy dynamical requirements prior to developing a refined CAD design. Specifically, our approach allows the designer to evaluate and tune component configuration through the use of a rough model constructed solely from rectangular prisms. These prisms may be solid or hollow; hollow prisms are created by subtracting a second prism from the first to create a void. The simple structure of this modeling approach allows for quick model definition, because prisms are easy to visualize and place, and for quick optimization, because prism parameters can be described by straightforward closed form equations amenable to optimization. For the caterpillar design, we considered three rough models for the shape of a body segment. (All body segments were assumed to be identical, as they were in the genetic algorithm.) The progression of the models illustrates the utility of the prism approach in supporting the conversion of a dynamical simulation into a detailed design. The three models (seen in Figure 10) are labeled the simple pyramidal model, the detailed pyramidal model, and the flexible hitch model. Though the basic topology is the same for all models, consisting of a body and two feet, refinements in the prism layout resulted in progressive improvements in the ability of each model to match dynamical parameter specifications. We began by specifying the first (simple pyramidal) model, which coarsely represents the segment geometries illustrated in Figure 6. We were unable to match the optimized parameters

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FIGURE 10: SIMPLE PYRAMIDAL MODEL (LEFT), DETAILED PYRAMIDAL MODEL (CENTER), FLEXIBLE HITCH MODEL (RIGHT)

To demonstrate the method for specifying a rough model, we consider the simple pyramidal model. Figure 11 provides a detailed sketch of the model. The model consists of five hollow prisms (three for the main body plus one for each foot). Each prism is defined with a uniform wall thickness, with one exception. To help the designer place the vertical center of mass, the wall thickness of the large central prism was defined to be larger on top than on bottom, as shown in the Front View of Figure 11. The main body was assumed to be a hollow continuous tube capable of holding a payload. Consequently, the prisms of the body were open in the lengthwise direction (along the x-axis).

FIGURE 11: DETAILED SKETCH OF SIMPLE PYRAMIDAL MODEL

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We used an optimization routine to drive model parameters as close as possible to their specified values (as summarized in Table 1). Specifically, the optimization tuned five parameters for each prism. The basic parameters for each prism were its edge dimensions and centroid location: height {h}, width {w}, length {l}, x-centroid {xc}, y-centroid {yc}. Each prism was also defined by two parameters defined by the designer. The first of these was the z-centroid location of the prism {zc}, which was set to zero for all blocks except the feet (whose separation distance {wga} is defined as a parameter in Table 1). Also, each prism was given a sign {s}, which was positive for prisms which added additional material to the model and negative for prisms which removed material from the model. The length of the parameter vector {p} is 7n, where n is the total number of prisms in the model. The parameter vector has the following form:

… …

(3)

The optimization problem is formulated to tune this parameter vector (specifically, the first five parameters of each prism were tuned and the other two were set constant) in order to minimize the difference between the rough model and the dynamical simulation. This difference is evaluated by considering a subset of only six of the most highly coupled dynamical parameters: mass, vertical center of mass, axial stiffness, and the principal moments of inertia. The optimization seeks to minimize the cost function, , which is defined below as the sum of the relative errors between the rough model parameters and the desired parameter values (indicated by the "ga" subscript). Functions for each of the rough model parameters { , , , , , } are described in more detail below.

The detailed form of these constraints varied for each of the three rough designs considered. In all cases, these constraints included the following: Equality Constraints ƒ The sum of the lengths (lj) of the body segments is constrained by the dynamical simulation (to be 2Lga). ƒ Centroid locations and edge dimensions are constrained for prisms with adjacent surfaces. ƒ Prism lengths (lj) are constrained to ensure both foreaft symmetry about the y-z plane, and bilateral symmetry about the x-y plane. Inequality Relationships ƒ Wall thicknesses are constrained to exceed 2 mm. ƒ Prisms representing the segment body are ordered in a sequence of decreasing edge heights (hj) and widths (wj), in accordance with distance from the midline. The optimization cost function depends on closed-form equations for mass, center of mass, axial stiffness, and inertias derived for a union of rectangular prisms. The mass parameter, for example, is the sum of the masses of the n individual prisms.

(7)

The mass of an individual prism is volume multiplied by density { }, which was set to a representative value for our intended elastomer design materials (DragonSkin™ or a DragonSkin™ foam).

(8)

The y-location of the center of mass for a component can be found by summing the contributions of all the individual prisms' to the y-centroid location. (4) (9)

(10) The optimization is subject to inequality and equality constraints, with a general form given by equations (5) and (6).

(5)  

The axial stiffness parameter is computed by applying a series Hooke’s Law model in the axial direction. When crosssectional area is constant, Hooke’s Law can be used to define a spring constant of the form used in the dynamical simulation to represent stiffness (see Table 1).

(6)

7

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(11)

This formula is based on the baseline length of the part {l}, its cross-sectional area {A}, and its Young's Modulus {E}. The last parameter is set based on representative material properties for DragonSkin™. When the cross-sectional area changes in discrete steps along the length of the component, this formula for spring stiffness must be modified slightly. Specifically, the total spring stiffness for the component can be approximately modeled with the typical springs in series summation. This equation has the following form [20]:

(12)

This model is dependent on the number of unique crosssectional areas {q} perpendicular to the x-axis. The total area of each cross-section is calculated as the sum of the contributions (positive and negative) for each of the c prisms associated with that cross-section.

(13)

The feet are assumed to not contribute to axial stiffness, so their cross-sectional area is not evaluated. Lastly, the principal inertias are the summation of the individual prism inertias.

(14)

(15)

(16)

12 12

(18) (19)

We chose to solve the optimization problem defined by equations (4) – (19) using MATLAB™ (and specifically using the fmincon function), which is designed for solving general nonlinear constrained optimization problems. Final Synthesis by Human Designer Sub-step three relies on the designer to adapt the optimized geometry into a 3D CAD model. In addition to smoothing the simple-geometry segments into clean, continuous shapes (such as those in Figure 6) and connecting these segments together with flexures, we need to ensure the model matches all dynamical parameters. As noted in the previous section, the optimization only considered the most highly coupled dynamical parameters; thus, matching other parameters, such as bending stiffness and SMA actuator attachment points, is left to this final design step. To generate a detailed design, the designer drafted 2-D cross sections based on the rough model and lofted these together to form the segment. Specifically, the designer defined planes for each unique cross-section. Some interpolation was done to ensure the amount of material was approximately maintained and the constraining dimensions, such as maximum segment length, were respected. After the lofting process, the designer added attachment points for the SMA springs at the locations specified by the genetic algorithm and removed material that interfered with spring motion. Lastly, the segments were connected at their flexures to form the full model. In this process the model’s dynamical parameters changed slightly from those of the rough model. Therefore, further parameter manipulation was needed to bring these properties back to their target values. We used an FEA program (CosmosWorks™) to determine the resulting stiffness, inertias, mass and center of mass, and compared these properties to the respective values determined from the dynamic simulation. We then iterated through a process of minor modifications to the segment's morphology to tweak these parameters to better match the dynamical specifications. The final design for one caterpillar segment is illustrated in Figure 14. The full caterpillar system, consisting of four segments, is illustrated in Figure 15.

Equations for the principal inertia of each prism are based on formulae commonly derived in introductory dynamics textbooks [20], with an appropriate application of the parallel axis theorem.

12

8

(17)

FIGURE 14: SMOOTHED FLEXIBILE HITCH MODEL WITH SMA ATTACHMENT POINTS

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assumption introduced problems, in that it was very difficult to simultaneously achieve the desired vertical center of mass location and x-inertia. In future work, we will investigate methods for improving the genetic algorithm by constraining its search space. We propose that constraints similar to equations (5) – (19) be imposed on the genetic algorithm to simplify the subsequent process of generating rough three-dimensional designs. In concept, these two automation steps, the genetic algorithm and design generation, might be combined to streamline the design process further.

FIGURE 15: 4-SEGMENT FLEXIBLE HITCH MODEL

A comparison of the dynamical parameters for each design step is summarized in Table 2. The first line of the table lists the specifications generated by the genetic algorithm. The second line lists the parameter values obtained for the prism-based rough model. The third line lists the parameter values that were achieved for the final design illustrated in Figure 14. A percentage parameter-matching error is evaluated for each of the last two lines. This percentage error was computed as a mean of the relative errors for each of the six parameters listed in the table. TABLE 2: COMPARISON OF OPTIMIZED PARAMETER VALUES AND THOSE OF THE FINAL MODEL m (g) Target Rough Smooth

20.6 20.6 20.9

y k (mm) (N/m) 28.6 28.6 28.7

152 150 152

Inertia (kg/m2) xx yy zz 2.81e-6 2.82e-6 3.17e-6

3.96e-6 3.99e-6 3.74e-6

3.96e-6 3.97e-6 3.74e-6

Error (%) -0.5 4.3

V. DISCUSSION Obtaining a three-dimensional design which closely matched the desired lumped-parameter model was challenging. The final model matched the specifications within a mean relative error of 4.3%. In comparison, the mean relative error for the first model was 261% including axial stiffness and even 19.7% when ignoring axial stiffness. The final model's bending stiffness was lower than the target value, but this could be corrected for by adding thin lengthwise flanges to the flexible hitches in future iterations. This modification would increase bending stiffness while having almost no effect on the other parameters. In part, some of the difficulty in matching dynamical specifications can be traced back to the definition of the conceptual lumped-parameter model simulated by the genetic algorithm (Step 1 in Figure 1). For example, the conceptual model used by the genetic algorithm assumed the inertia and mass of the feet were negligible. In realizing a threedimensional design (as shown, for example, in Figure 10), this

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VI. SUMMARY This paper identifies a strategy for simultaneously designing the structure and control for a soft body robot, when combined with previous work on genetic algorithms [16]. In particular, we apply our methodology to the design of a soft caterpillar-like robot. A genetic algorithm was used to optimize a lumped-parameter dynamical model of the system. This paper proposes a methodology to subsequently convert the lumped-parameter model into a detailed threedimensional design using a three step process. The first step grouped the elements of the lumped-parameter model into aggregated components, called segments. It was shown empirically, for a caterpillar-like robot’s geometry, that the axial stiffness parameters for individual segments could be analyzed in a manner decoupled from the full body. The second step applied an automated design tool to create a rough model for individual caterpillar segments. The rough model was composed of a set of rectangular prisms, which enabled rapid model development and optimization to ensure close matching of the physical characteristics of the lumpedparameter model. In the third step, a human designer used the prismatic structure as the basis to design a smooth segment which defined a full three-dimensional model propelled by SMA springs actuators.

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[5] Otake, M., Kagami, Y., Inaba, M., and Inoue, H., 2001, “Dynamics of Gel Robots made of Electro-Active Polymer Gel,” IEEE International Conference on Robotics & Automation, pp. 1457-1462.

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[18] Trimmer, B., Rogers, C., Hake, D., and Rogers, D., 2006, “Caterpillar Locomotion: A New Model for Soft-Bodied Climbing and Burrowing Robots,” 7th International Symposium on Technology and the Mine Problem.

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[19] Trimmer, B., 2008, “New Challenges in Biorobotics: Incorporating Soft Tissue into Control Systems,” Applied Bionics and Biomechanics, pp. 119-126. [20] Norton, R., 2004 Design of Machinery: Third Edition, McGraw-Hill, Boston, USA, Chap. 10.

[8] Sugiyama, Y., and Hirai, S., 2004, “Crawling and Jumping by a Deformable Robot,” Experimental robotics IX: the 9th International Symposium on Experimental Robotics. pp. 281290. [9] Adamatzky, A., Costello, B., Melhuish, C., and Ratcliffe, N., 2003, “Experimental Reaction-Diffusion Chemical Processors for Robot Path Planning,” Journal of Intelligent and Robotic Systems, 37(3), pp. 233-249. [10] Clark, J., Cham, J., Bailey, S., Froehlich, E., Nahata, P., Full, R., and Cutkosky, M., 2001, “Biomimetic Design and Fabrication of a Hexapedal Running Robot,” IEEE International Conference on Robotics and Automation, 2001. Proceedings 2001 ICRA., 4, pp. 3643-3649. [11] Lin, P., Komsuoglu, H., and Koditschek, D., 2005, “A Leg Configuration Measurement System for Full-Body Pose Estimates in a Hexapod Robot,” IEEE Transactions on Robotics, 21(3), pp. 411-422. [12] Jones, B., McMahan, W., Walker, I., 2006, “Practical Kinematics for Real-Time Implementation of Continuum Robots,” Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006., pp. 1840-1847. [13] Zang, T., Hemsch, M., Hilburger, M., Kenny, S., Luckring, J., Maghami, P., Padula, S., and Stroud, W., 2002, “Needs and Opportunities for Uncertainty-Based Multidisciplinary Design Methods for Aerospace Vehicles,” NASA. [14] Lipson, H., 2006, “Evolutionary Robotics and OpenEnded Design Automation,” Biomimetics, CRC Press, pp. 129-156. [15] Pollack, J., and Lipson, H., 2000, “Automatic Design and Manufacture of Robotic Lifeforms,” Nature, 406, pp. 974-978. [16] Saunders, F., 2009, “Simulation of Soft Bodied Robots: Methods, Benefits, and Potential Applications,” Masters Thesis, Tufts University.

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Nov 13, 2009 - caterpillar-like soft-bodied robot from a lumped-parameter form into a CAD model ... continuum materials, such as finite-element analysis, are.

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