Actuator placement optimization in an active lattice structure using generalized pattern search and verification

Shape morphing structures are actively used in the aerospace and automotive industry. By adapting their shape to a stimulus such as heat, light, or pressure, a design can be optimized to achieve a broader band of functionality over its lifetime. The quality of a structure with respect to shape-morphing can be assessed using five criteria: weight, load-carrying capacity, energy consumption, accuracy of the controlled deformation, and the range and number of achievable target shapes. This work focuses on the use of lightweight and stiff active lattice structures, where the layout of actuators within the structure determines the final deformation. It uses a statically and kinematically determinate Kagome lattice pattern that has been shown to deform the most accurately with the least energy. The use of a determinate structure justifies the implementation of a simplified deformation model. The deformation resulting from a given actuator layout can be expressed as a linear combination of the deformation of individual actuators, which are all computed in a pre-processing step and expressed with an influence matrix. The actuator layout is thus optimized for several target shapes. The linear combination model is shown to replicate FEM simulations with an average of 94.8% accuracy for all target shapes. The actuator layouts in one-level lattices are tested using a novel design for a 3D printed modular Kagome pneumatic lattice structure. The experimental results replicate the target shapes with an average accuracy of 79.9%. The resulting actuator layouts are shown to form more target shapes with a similar deformation range as similar publications.


Introduction
The permanent push to achieve ever better performance of structures with as little cost, energy use, and waste as possible, has increased interest in shape-morphing structures. The concept of a structure adapting its shape is seen often in * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. nature: plants and trees constantly adapt to their environment, flowers open, close, and move to increase sunlight exposure. Enabling a structure to morph and adapt to a given set of stimuli broadens its functionality band and optimizes its behavior over its lifetime.
Morphing structures have been used both in macroscopic and microscopic applications, effectively providing products ranging from plane wings [1][2][3] to micro-electrical mechanical systems (MEMS) [4,5] with shape changing abilities. The space, aeronautic, and automobile industry are investigating their use for vibration and shape control, most ubiquitously in morphing airplane wings but also in deployable satellite antennas [5][6][7][8].
Given a morphing structure's application, there are five main criteria to evaluate its design: weight, load-carrying capacity (stiffness), energy consumption, accuracy of the controlled deformation, and the range and number of deformation modes [9]. One effective approach to reduce weight is the use of lattice structures. Lattice structures can be extremely light and stiff, they can be easily parametrized, and their topology can support a whole range of load conditions [3,10]. The use of lattices can be categorized in two main approaches: compliance-controlled deformation and actuator-controlled deformation.
Compliant lattices rely on external stimuli that cause the whole structure to deform [3,11]. They are designed to deform under specific external loading conditions, such as a pressure field or a given displacement input by a single actuator on one or several of its edge nodes. Actuator-controlled lattices rely on actuators placed within the structure for deformation. The layout of these actuators can be optimized for a range of target deformations. Their number can be constrained to meet weight, stiffness, and energy requirements. The energy consumption and controllability of these lattice structures are directly influenced by the concept of static and kinematic determinacy. Symons et al [12] showed that if an overdetermined structure contains states of self-stress it will require more work from the actuators to deform. On the other hand, if the structure is statically and kinematically determinate, it will require minimum work and achieve complete controllability while remaining stiff [13]. Morphing wings have often been designed using determinate structures [1,2,14], although the models usually consist of 2D lattices or a small number of elements due to the difficulty of creating a finite three dimensional determinate pattern [2,15]. Tetrahedral [2,10,12] and octahedral [15] cells are often used for actuator controlled lattices.
Determining the layout of sets of actuators in a lattice structure to deform it into a target shape is a non-linear and discrete design problem whose complexity increases exponentially with the total number of elements in the lattice. It is possible to determine the actuator layout for certain simple target displacements experimentally through trial and error if the lattice is small enough [1,14,16]. However, this approach is not scalable and can only achieve an approximation of a target shape. Due to the non-linear and nonconvex nature of the actuator layout problem, it is commonly solved using Meta-Heuristic (MH) methods or exhaustive search methods [2]. Ramrkahyani deduces the necessary cell locations and orientations from the target shape to achieve a desired displacement in a morphing wing [1]. Baker and Friswell compare the efficiency of a genetic algorithm to stepwise forward selection (SFS), a regression method, in the layout of linear actuators to control the camber of a morphing wing [2]. Guo and Liu use genetic algorithms, swarm searches, and simulated annealing to determine the optimal layout of actuators and sensors respectively [17,18]. dos Santos e Lucato [10,17] determine the optimal layout of actuators in a statically and kinematically determinate Kagome face using a genetic algorithm. Although these approaches deliver satisfactory results for a relatively small lattice, they become highly inefficient as the problem is scaled up.
This work addresses the complex problem of actuator layout within an active lattice structure in three steps. First, building on several previous publications on the topic of active Kagome lattices, a kinematically and statically determinate lattice design is produced using the patching concept presented by Symons et al [12]. The determinacy of the lattice means the displacement at control nodes triggered by a set of actuators can be calculated as a linear combination of the displacement caused by single actuators from the set. This approach presented by dos Santos e Lucato et al [9,17] uses an influence coefficient matrix that is built in FEM by replacing single elements of the structure by actuators one after the other. Then, the actuator layout problem is solved using only the influence coefficient matrix. To increase the efficiency of the optimization, the actuator layout problem is solved by using generalized pattern search (GPS) with a controlled MH component called variable neighborhood search (VNS). This deterministic approach produces robust and repeatable results and could scale with larger lattices since it can handle hundreds of variables without reducing its efficiency.
Second, the main contribution of this work is that it increases the range of motion that can be achieved given the same underlying shape morphing structure that can accommodate many different actuator combinations. This is achieved by efficiently solving the actuator layout problem exactly for several key target shapes: hinging, twisting, concave up and concave down curvatures along the x-axis, i.e. the length of the lattice, and the transition from one to the other. Where previous work has focused on maximizing one or two types of deformation for a given number of actuators [15,17], this work minimizes the number of actuators for controlled deformation.
Third, the modeling and optimization results are verified using a novel fabrication method to produce quasi-pin-joined lattice structures with a modular 3D printed design. The prototype presented here is fabricated using a multi-material printer, which allows printing the lattice structure all at once while varying the material at different key locations. It is designed so its joints are soft and members are stiff; the stark difference in hardness replicates the behavior of a pin-joined lattice structure. Soft pneumatic linear actuators are used to deform the structure and can be placed to implement the optimization results for hinging, linear deformation, and other shapes. This is in contrast to related work that either does not verify the results experimentally or use a welded aluminum frame actuated by stepper motors [10,19].
The structure of this paper is as follows. After an overview of past work on the topic of load-carrying Kagome active lattice structures and the determinacy theory behind it, the simplification assumptions of the linear combination model are explained in detail. Then, the numerical validation methods for the model are introduced. The design and fabrication of the 3D printed active lattice prototype for physical validation is presented. The model is simulated and tested for several target deformations, control point locations, and lattice sizes. The desired, simulated, and measured displacements are compared to validate the model's accuracy and flexibility with regards to the range of deformation. The results are discussed first in the context of this work and then in the context of shape morphing structures in general.

Background
Load-carrying Kagome active lattice structures have already been researched extensively [12,[19][20][21][22]. The main motivation of using the Kagome pattern is that it contains no mechanisms, can undergo arbitrary displacement, and has the least resistance to deformation in the form of self-stress [9,13]. The infinite, pin-joined version of the pattern has been shown to satisfy most of the requirements for static determinacy, which means it stores the least energy, and thus needs the least work to deform. Symons et al and Hutchinson et al also showed that it is possible to build a statically and kinematically determinate Kagome lattice by adding patches to the structure's edges [12,20]. They use Maxwell's equation to determine how many mechanisms the structure contains and add that number of patching bars along the edges. The resulting structure is a good candidate for all five shape morphing criteria, and is thus used in this work.
Previous work has focused both on in-plane [12] and outof-plane [19,21] deformation of a Kagome lattice. Assuming static and kinematic determinacy in a pin-joined lattice structure as defined by Calladine and Pellegrino [23], the deformation triggered by a set of actuators can be modelled as the linear combination of individual actuator deformation [9,17] as shown in figure 1: Thus, each displacement u i at a predefined control node i ∈ [1; n] can be determined by using the influence coefficient matrix A and the member actuation vector ε. Control nodes are a predefined set of nodes where the target deformation is prescribed. The influence coefficient matrix A is constructed using FEM simulation, collecting the deformation at the i th control node from a unit actuation of each possible actuator location m. A thus has the dimension n × m and needs n simulations to be computed. The advantage of computing A in preprocessing is that FEM is not necessary to determine displacements for each objective function call in the optimization loop. As long as the structure and control nodes remain the same, it is possible to determine the overall deformation of the structure as a linear combination of ε.
This model has been implemented in actuator layout optimization mainly by using MH methods such as genetic algorithms and simulated annealing [17,19,21]. Dos Santos e Lucato et al use an embedded optimization loop. They first run MH methods for 10,000 iterations and from 10 different starting points to select which of a predetermined number of actuators contribute the most to the target deformation. Then, they manually reduce the problem to these actuators and optimize ε to determine the necessary strain in each actuator to achieve the target deformation [17]. The structure is modeled as a construction of welded aluminum beams, which compromises the assumption of a statically and kinematically determinate structure and of linear deformation. They complement their model with considerable effort to compensate the force and energy effects of having welded joints rather than a pin-joined lattice structure [10,17,19]. The authors made these assumptions because they plan to verify their optimization results on an aluminum lattice paired with linear stepper motors and a thin aluminum face sheet [19,21]. In two separate publications, they compare their predicted objective to maximize hinging and twisting at the edge of the Kagome structure given a predetermined number of actuators to their measured deformation.
Another important aspect of shape morphing applications is the type of actuation used. There are two categories of actuators used most frequently in shape-morphing. First are smart materials, understood here as piezoelectric elements, viscoelastic materials, and shape memory alloys or polymers, among others. They can be activated by an electric current, heat, or light. Their intrinsic ability to deform is an advantage in terms of weight since they require no additional hardware. However, most of these materials can only apply a small force while deforming, which impairs their load-carrying capacity in the deformed state. They also require large amounts of energy for a limited deformation range [11,14], although they have been shown to exhibit precise control for shape-morphing applications [24][25][26].
Kagome lattice prototypes use mechanical actuators such as stepper motors, tendons, or pneumatic/hydraulic actuators, the second category. They provide large and accurate amounts of force and deformation [1,27] but they are bulky and can also require large amounts of energy.
A compromise between these two categories is found in the field of soft robotics (SR). In SR, deformation and control are directly embedded into an actuator's design [28]. The compliant materials are generally light, highly deformable, and can be pressurized to carry significant loads [16]. Previously limited in design complexity by the available fabrication methods, due to advances in additive manufacturing (AM) these soft pneumatic actuators can now be 3D printed to meet the requirements of shape morphing structures.

Method
The method section is structured as shown in figure 2. First, the theory behind the determinate structure design is reviewed and applied to design a one-level and a two-level Kagome lattice. The procedure to calculate the influence matrix is explained. The GPS optimization method to determine the optimized actuator layout is outlined in detail. The strategy to verify the results' validity both numerically and with a prototype is defined.

Lattice structure design
The static and kinematic determinacy of a structure are calculated using the method of Calladine and Pellegrino [23]. They define two equations with which the states of self-stress s and mechanisms m can be calculated:

One level
Two levels r B is the rank of B which is the structure's equilibrium matrix. r B is computed using a singular value decomposition (SVD) of B and the identity matrix [23]. The design process for the lattices and necessary patches to make the structure determinate is manual and iterative since there currently exists no statically and kinematically determinate lattice generator. They are iteratively redesigned until they contain no states of self-stress and mechanisms. One-and two-level lattices are considered to assess the impact of the number of elements and structure complexity on the range of motion, the efficiency of the FEM simulation, and the optimization algorithm. In the one-level design, there are 2.5 Kagome cells along the x-axis, and two along the y-axis. In the two-level design, there are 3.5 Kagome cells along the x-axis, and two along the y-axis. Each cell is composed of regular triangular pyramids with a length of 20 mm.
As shown in figure 3, two sets of control nodes are considered separately, namely a set along the y-axis (in yellow), the width of the lattice, and a set along the x-axis (in red), the length of the lattice. The boundary conditions are indicated by blue squares. These configurations are chosen so as to achieve target deformations such as hinging or twist at the free edge of the structure and linear, parabolic, or sine-shaped deformations along its central axis. The FEM simulation calculates exact outputs for each control node, so the number of control nodes chosen is a compromise between computational effort and accuracy of representation. The load applied is heat expansion, explained in detail in section 3.2.
The model geometric parameters are given in table 1. Only elements within the XY-plane, shown in figure 3 in green, can be replaced by actuators. These can be oriented at either ±30 • or 90 • of the x-axis.

Influence matrix A
A FEM model is designed to compute the displacements at the i control nodes as a function of a single actuator by replacing each possible member n by an actuator one by one. Each run generates a new row in the coefficient matrix, A. A thus has the dimension i × n.
Since it is assumed that the actuators deform linearly, heat expansion can be used to simulate the actuator elongation. The expansion coefficient of the actuators is determined using the equation, ε = α∆T, with ε defined as the measured soft pneumatic actuator maximum elongation and ∆T set in the FEM simulation to 100 for each actuator. ∆T is applied using a predefined temperature field. The expansion coefficient for elongating members, α, is set to 10 −3 , which sets the elongation to 10% (as shown in table 1).
The FEM model calculates the displacements at the i control nodes as a function of a single actuator expansion by replacing each possible member n by an actuator one by one. Each run generates a new row in the matrix.

Optimization of linear-combination model
The assumption of kinematic and static determinacy allows the displacement at control nodes to be calculated with equation (1) as a linear combination of the displacements caused by a single actuator expanding. ε is a binary vector in which each row represents one possible actuator location. If ε j = 1, the member's length will increase; if ε j = 0, the member's length is fixed.
As shown in equation (4), the objective of the optimization is to minimize the sum of the elements of ε, and thus minimize the number of actuators in the structure. The deformation at the control nodes is controlled by an inequality constraint. A solution is only considered feasible if the least square difference between the desired and optimized deformation at the control nodes is smaller than a given error tolerance threshold (equation (5)): Sinusoidal curve:

Implementation details
In this work, the method is implemented in a MATLAB code that verifies the determinacy of a structure by calculating the values of s and m. The influence matrix necessary to calculate the out-of-plane displacements triggered by the actuators is built using Abaqus CAE 6.14-1. The implementation of GPS used is from NOMAD [29,30], a derivative-free optimization environment that also offers the possibility to include MH characteristics in the search using the variable neighborhood search (VNS) method to avoid local optima [31]. GPS results in the NOMAD environment depend on the seed it uses. The seed determines the pseudo-random number sequence that control its steps. The interface between the NOMAD and MATLAB is the OPTI Toolbox. MATLAB 2019a is used for this work.

Case studies
The quality of the out-of-plane deformation model for an active Kagome lattice structure is verified for one-and two-level lattices, as well as for control nodes along the structure edge (y-axis) and the structure length (x-axis). As shown in table 2, each category is tested for two deformation modes. The twolevel lattices can take on both positive, negative, convex, and concave shapes thanks to the antagonistic effect of the two actuated layers.
The quality of the model is assessed first numerically and then verified with the prototype.

Numerical validation
The target deformation along the z-axis z 0 , the optimized results for the linear-combination model z lc , and the simulated deformation z sim are compared to verify the quality of the linear-combination assumption presented in equation (1).
For the one-level lattices, the quality of the model is also verified using the measured deformation z meas . They are compared using the following parameters: -Normalized accuracy: is one minus the sum of the error between either the optimized, simulated or measured deformation normalized by the desired deformation for each control node, divided by the number of control nodes. This value determines how accurately the target deformation is achieved. -Number and layout of actuators: results are assessed by the number of actuators, since they inherently weaken the structure and are to be minimized. The layout of actuators for different deformations is compared to evaluate its impact on certain given target shapes.

Prototyping
The prototype for lattice structures presented in this work is meant to replicate a pin-joined lattice structure. It is designed to be modular, so that different actuator configurations can be tested using the same setup. A detailed description of its components is given below.

Soft linear actuator.
The design of the soft pneumatic linear actuator is shown in figure 5. Each actuator is comprised of three parts: the actuator body, an air inlet, and a rigid cage. The actuator geometry is inspired by existing soft linear actuators; its four bellows allow it to extend beyond the elasticity of its material. The inlet is a rigid interface to connect to the pressure source. The cage allows the actuator to expand linearly without buckling under self-weight. Previous testing determined that the cage does not influence the inflation behavior: the elliptical shape of the ribs prevent the bellows from catching on them.
All parts are printed on a Form2 from Formlabs, a stereolithography printer, using Elastic resin for the actuator body and Tough resin for the inlet and the cage. All three parts are assembled after printing using liquid Elastic resin as an adhesive and are then cured using the FormCure for the advised 60 min at 60 • C. The actuators are inflated to 75 kPa each using a network of inlet tubes.

Beams.
For all non-actuated members, 20 mm beams are printed using a Form2 and Elastic resin from Formlabs. They interface with the lattice with the same press-fit connection as the actuators. Their geometry is shown in the bottom right of figure 5.

Pin-joined lattice.
The lattice used in this work (shown in figure 4) is designed to replicate a pin-joined structure. It is fabricated using a Connex Objet500 multi-material polyjet printer. The lattice beams and nodes are printed using two different materials: RGD8530 and FLX9870. They have a Shore hardness of 76-81D and 70 A respectively. This difference in hardness at the joints replicates the behavior of an ideal pin-joint.

Lattice-actuator interface.
The modularity of the lattice design creates a need for an airtight connection at each possible actuator location. This connection is made with a barbed-fitting. Actuators are press fit into place. To maximize the range of the printed actuators, they are pre-compressed by 20% (from 25 to 20 mm) of their printed length. As visible on the actuator images in figure 5, stoppers placed after each barbed-fitting ensure this pre-compression is even for all actuators and that they push the lattice apart without slipping further down the barbed fitting. angle. The coordinates of the deformed control points are determined using imageJ; a ruler is placed in the video frame to verify the scale. All tests and measurements are performed at approximatively 15 MPa. The structure is fixed at the constrained nodes. It is tested on its side (XY-plane perpendicular to the ground) so as to neglect the effect of gravity on deformation along the z-axis.

Results
The optimization parameters used to obtain the results are as follows. Preliminary work determined that the minimization of the number of actuators works best by starting each run with ε = 0 meaning that the structure contains no actuators. GPS is applied with default settings, so that the algorithm can select from 2 · n variables different coordinate search directions where n variables is the number of variables. For efficiency reasons, the algorithm stops once it achieves an error smaller than 0.125% of the maximum deformation for one-level lattices, and 0.4% for two-level lattices. It has an upper limit of 10,000 function calls. The probability for VNS is set to 0.65. The optimization is deterministic, so all results can be reproduced.
This section shows the results from the eight case studies mentioned above. The actuator layout is first determined using the simplified linear-combination model. The combined effect of the resulting layout is then calculated in a FEM simulation, and, for the one-level case studies, it is finally verified using the prototype. The results are shown in in figure 6 and in table 3 for the one-level case studies and in figure 7 and in table 3 for the two-level case studies. This section compares the target, optimized, and simulated deformations in terms of the accuracy normalized with respect to the target shape and in terms of actuator number and layout. These results verify the validity of the proposed linear combination model. Also, it compares the target and measured deformations to verify the applicability of the proposed pin-joined lattice design and as a proof of concept for the model.

Comparison of the linear-combination model to the FEM simulation
As shown in table 3, the model assuming linear-combination of actuators has a minimum accuracy of 90.2% observed for the sinusoidal target shape for the two-level lattice. The average accuracy of all linear-combination optimization results is 96.6%. The minimum accuracy of the FEM simulation model results is lower, at 83.7% for the sinusoidal deformation of the two-level lattice, and on average 94.8% across all cases. A lower accuracy of the deformed shape using the linear combination model leads to a similar order of magnitude loss in the FEM simulation verification. An exception is the FEM simulation accuracy for the negative parabola along the x-axis for the one-level lattice that is higher than the linear combination model.
A high number of actuators replicate target shapes the best, as expected. For both types of lattices, the target shapes along the y-axis, where the control nodes are on the free edge, are replicated most accurately. The target shapes along the x-axis, where the control nodes are in the center of the lattice, achieve lower accuracy, except for the parabolic target shape for the one-level lattice. The deviation to the target shape of the control nodes closest to the constraints make up most of this accuracy difference. For example, for the linear target shape along the x-axis for the one-level lattice, if the error of those controls nodes is not counted in the accuracy calculation, the accuracy of the linear-combination model and FEM simulation increase from 88% to 97%.
The number of control nodes and their layout (free edge or along the center) have a much larger impact on the error than the number of variables or amplitude of the shape change. In three out of four cases for both one-and two-level lattices, the linear-combination model and the FEM simulation achieve higher levels of accuracy when the target shape is along the y-axis. However, the complexity of the target shape also impacts the error: the curvature change in the sinusoidal deformation of the two-level lattice is particularly hard to achieve, and achieves the lowest accuracy value by 5%. Also, a deformation in the negative z-direction requires at least two levels of cells, which in turn divide the maximum amplitude of the possible shape change by a factor of 1.6 for hinging and a factor of 10 for parabolic displacements.
The optimization for all target shapes converges on average with 3214 objective function calls. The number of actuators at ±30 • to the x-and y-axes of the structure and layout of the actuators correlate strongly with the target shapes. The number of actuators in either direction is even for symmetric target shapes such as hinging and deformations along the x-axis for both one-and two-level lattices. For an asymmetric deformation along the y-axis, the number of actuators in CONTROL ALONG Y-AXIS CONTROL ALONG X-AXIS either direction varies. In the twist target shape for the twolevel lattice for example, there are two more actuators in the +30 • direction than the −30 • direction. The layout on the top or bottom level of the two-level lattices also plays a role. In the sinusoidal target shape for the two-level lattice for example, the top actuators are placed when the lattice deforms concave down, and the bottom actuators are placed when the lattice deforms concave up.

Prototype
The prototype results are only given for the one-level lattice. The measured accuracy shown in table 3 is roughly 15% lower than the FEM simulation; it attains a minimum of 73.8% for the linear edge deformation (along the y-axis). On average, the measurements approximate the target shape with 80% accuracy, generally overshooting the target deformation at the free edge especially. Except for hinging, the overshooting seems to be constant among the different target shapes, which points to a general issue with the prototype. Control and accuracy decline for deformations larger than 20 mm. Although the quality of control and thus accuracy is compromised for control nodes further away from the fixed edge and the error seems to aggregate, all prototypes successfully show a strong correlation between the target behavior and their final shape.

Discussion
This section first analyses the validity and flexibility of the simplified active lattice model based on the optimized linear combination model, the FEM simulation, and the empirical results. Then, contributions of the model and the prototyping method to the field of shape morphing are discussed.
Three components of this work are analyzed: the linearcombination model, the FEM simulation of the resulting actuator layout, and the prototype. As described in the previous section, the optimization converges with an average of 96.4% accuracy for all target shapes. A smaller number of control points achieve higher accuracy, especially in the larger lattices. There is thus a trade-off between having accurate control over a few localized control points versus less accurate control over more control points across the whole shape. Overall, the problem formulation is valid and adapted to solve an actuator layout problem of this size and possibly larger efficiently and robustly, depending on the type of control accuracy needed. A higher degree of accuracy could be achieved using either a continuous actuator elongation, or several discrete elongation modes rather than a single one.
The FEM results vary more with respect to the mean than the optimized linear-combination model, although only slightly. This variation increases with a higher number of actuators in the structure, which points towards some imperfections in the modeling of actuator elongation, and an aggregation of that error with each additional actuator. As mentioned above, proximity to constrained nodes also affects accuracy, which indicates modeling imperfections of actuator elongations. However, considering the relatively large range of deformations, the FEM simulation results remain under 6.2% variation from the target shape irrespective of the size of the lattice with the exception of the sinusoidal deformation. The resulting deformations for all four target shapes are large relative to the member length in each cell. However, the assumption of small displacements that supports the use of the linear combination approach described in the background remains valid because members themselves elongate only 10%. The linear combination approach and the FEM simulation model are thus valid.
The prototype has larger deviations and uneven performances that can be explained by the margins of error introduced in fabrication and by the type of target shapes. Accuracy of control is higher for symmetric deformation modes. This can be explained by the testing conditions: since the prototype is tested with the y-axis perpendicular to the ground, gravity affects the results more for an asymmetric deformation along the y-axis. The tendency of the prototype to overshoot the target deformation indicates that the actuators elongate slightly more than expected. This error is multiplied by the number of actuators and aggregates at the free edge especially. In the optimization, a higher degree of accuracy could be achieved either with a continuous elongation range or several discrete elongation modes. However, achieving this using 3D printed soft pneumatic actuators poses considerable fabrication issues. For an industrial application, further advances are required in the field of 3D printed soft materials.
The weight and stiffness can be assessed with the ratio of actuators to beams given the assumption that actuators add mass and are weak points in the structure. As shown in table 4, the current results have the lowest ratio. The energy consumption of the structures is difficult to compare, given that existing prototypes use different actuation modes, such as electricity or heat. However, as mentioned previously, statically and kinematically determinate structures are shown to need the least activation energy [12]. The accuracy of control can be compared only with Baker and Friswell [2] and previous work by the authors, du Pasquier et al [16], since only they optimize for a precise target shape change. They offer similar shape morphing accuracy but have not verified their results with a physical prototype. The maximal range of deformation of this work is within the top three of the publications in table 4, considering that du Pasquier et al [16] shows only numerical results and Sofla et al [15] manually replaces nearly half its structure with actuators. Both of these authors achieve three target deformations, whereas this work achieves four.
To discuss the contribution of this work, its results are compared to similar publications according to the five criteria of shape-morphing defined earlier in this work: weight, stiffness, energy consumption, accuracy of control, and range and number of target shapes. As established in the introduction, the goal of this work is to solve the actuator layout problem in three steps. The simplified linear-combination model built on assumptions of static and kinematic determinacy and pin-joined lattice structures is shown to hold for several lattice sizes and target deformations. The optimization converges consistently within 4.6% of a range of target shapes relevant to applications in shapemorphing. The results are validated both numerically and with the modular prototype. This method thus solves the actuator layout problem successfully and is not limited to 3D printing. Prototypes can also be assembled from off-the-shelf mechanical actuators and bars if, for example, a higher load carrying capacity is needed.
However, the approach presented in this work is limited to statically and kinematically determinate structures. There currently is no known tool to design them automatically in 3D, so the process remains manual and lengthy. Previous work from the authors does however show an automated method for 2D and 2.5D structures [32,33]. Further, although appropriate to the problem sizes presented here, GPS performs less efficiently upwards of 400 variables, so its application to largescale structures is limited. The prototype can be further tuned to avoid the deformation overshoot problems at the free edge of the structure. Finally, the control strategy for the prototype can be improved so as to support several deformation schemes without changing the actuator layout. This will be the focus of future work.

Conclusion
This work approaches the complex problem of actuator layout in an active lattice structure in three steps. It first reduces the search space to kinematically and statically determinate lattices that deform linearly, using the example of a Kagome lattice. This assumption means that all FEM calculations can be pre-processed before the layout optimization and linear combination of actuator elongation employed. Next, the layout optimization problem is solved efficiently using GPS for two lattice sizes and four target shapes that are relevant to the field of shape-morphing lattices: hinging, twisting, convex and concave curvature, and the transition between the two. The actuator combination layout resulting from the optimization achieves the target shapes with on average 94.8% accuracy. Finally, the solutions are successfully verified both numerically through FEM simulation and experimentally with a 3D printed, modular lattice design. The modular prototype replicates the target shapes with on average 79.9% accuracy. Further work will focus on expanding the search space to overdeterminate structures and on control strategies that allow a lattice to deform into several target shapes without changing the underlying actuator layout.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.