Robotic Materials Transformable Between Elasticity and Plasticity

Abstract Robotic materials, with coupled sensing, actuation, computation, and communication, have attracted increasing attention because they are able to not only tune their conventional passive mechanical property via geometrical transformation or material phase change but also become adaptive and even intelligent to suit varying environments. However, the mechanical behavior of most robotic materials is either reversible (elastic) or irreversible (plastic), but not transformable between them. Here, a robotic material whose behavior is transformable between elastic and plastic is developed, based upon an extended neutrally stable tensegrity structure. The transformation does not depend on conventional phase transition and is fast. By integrating with sensors, the elasticity‐plasticity transformable (EPT) material is able to self‐sense deformation and decides whether to undergo transformation or not. This work expands the capability of the mechanical property modulation of robotic materials.


S1. Geometry and Shape sensing of the EPT cell
To extract the deformation information of the EPT material, it is necessary to discuss its degrees of freedom (DOFs). According to Maxwell's theory, the DOF of a mechanism is given by: in which n stands for the number of rigid bodies, g is the number of joints, and i f refers to the DOF of each joint. A single EPT unit (shown in Figure S1) consists of eight ). Accordingly, its number of DOFs is five. Each EPT unit is equipped with four angle detectors marked by purple. We assume the green angle EAF  in Figure S1 can be determined by its neighboring unit cell. Therefore, the shape of the EPT unit can be determined by these five angles. Since the shapes of the four isosceles triangles are known, the quadrilateral ABCD marked in purple can be solved to obtain the shape of the EPT cell. In ABF , AF BF L  and AB can thus be obtained by the law of sine. Figure S1. The geometry of EPT unit cell.
The other sides of the quadrilateral ABCD can be obtained in a similar way. With the four side lengths of the quadrilateral ABCD determined, only one of the angles has yet to be determined. For instance, DAB  can be obtained as: Thus, BD can be calculated by the cosine theorem: The angle of the top of the quadrilateral can then be found as: Once the quadrilateral ABCD has been completely determined, the other angles of the EPT cell can be solved similarly. By doing that, all angles of the EPT cell and the coordinate positions of each point can be determined.

S2. Deformation sensing of the EPT metamaterial
As shown in Figure 4c, the internal cell of EPT metamaterial has zero DOF, indicating that the deformation of this type of EPT cell can be fully determined by the deformation of its adjacent units. Unlike the deformation sensing of a single cell, an EPT cell in a metamaterial has many constraints, which reduce the degrees of freedom of the metamaterial. By installing angular sensors in the cells on the boundary, the 3 deformation of the entire metamaterial can be determined with a small number of sensors.
Due to a large number of constraints in the whole structure, it is necessary to rationalize the solution sequence to avoid simultaneously solving complex systems of equations. One possible solution procedure is shown in Figure S2. In the first step, the shape of the first column is solved from bottom to top; in the second step, the shape of each column is solved by repeating step 1 in the order from left to right.  Figure S2, and the corresponding known and unknown conditions are listed in Figure S3. The known angle information is from the angle detectors or neighboring cells whose shape have already been determined.
Solving six different types of cell shapes can be generalized to two types of geometric 4 problems. One type is to find 3  , 5  for known 1  , 2  , 4  ; and the other is to find 2  , 3  for known 1  , 4  , 5  . The six different positions of the cells require different numbers of sensors to determine their shape, and the angles that need to be detected are marked in purple in Figure S3. Figure S3. The conditions to solve the shape of the EPT unit in different types of cells. Figure S4a shows the EPT configuration when the number of columns m is odd, 5 while b shows the case when m is even. Based on the analysis given in Figure S3, the number of sensors required to determine the shape of an EPT metamaterial of mn  can be obtained as 2 2 1, 2 2 2, n m m is odd N n m m is even This number is much smaller than 4mn as required by the method with each cell having one sensor. In the following, we introduces the metamaterial's shape-sensing algorithm used in this study and discusses how to solve the shape of the EPT cell under different constraints.

S2.1 Solution procedure for cells -Type 1 & 3 &5
For the cells of type 1, 3, and 5, 1  , 2  , and 4  are known, with 3  and 5  to be evaluated and AE CH ∥ , AF CG ∥ . Noting that AE CH ∥ , one can get According to the parallel condition AF CG ∥ , 3  can be obtained as AC 2 AB 2 3 arcsin arctan 2 ABcos 2 / 2 AB 2 4 AB cos 2 / 2 and the bottom angle can be solved. All conditions to solve the type 1, 3, and 5 cells are obtained.

S2.2 Solution procedure for cells -Type 2 & 4 & 6
For the cells of type 2, 4, and 6, 1  , 4  , 5  are known, with 2  and 3  to be determined and AE CH ∥ and AF CG ∥ . Figure S5 shows the geometry of a deformed EPT cell under the parallel constraints, i.e., AE CH ∥ , CG AF ∥ . Let the horizontal distance between A and C be a , the distance between AF and CG be h . The following relationships among a , h , 2  and 3  can be obtained cos 3 cos 2 sin 3 sin 2 where a and h are known quantities. By solving Eq. (S7) Figure S5. Geometry of EPT unit cell under parallel constraint.
The definition of ' a and ' h are shown in the left part of Figure S5. Since they are obtained under different reference systems, ' a and ' h derived from the above equation also need to be multiplied by a rotation matrix to obtain a and h . The angle of rotation is related to the bottom angle 5  .
cos 5 sin 5 ' sin 5 cos 5 ' aa hh where 5  is marked as the green angle in Figure S5.

S2.3 Algorithm to improve shape-sensing accuracy
Using the above algorithm, calculations were performed in the order shown in Figure S2 for an EPT metamaterial, whereby 11 sensors are needed to fully determine its shape. The advantage of the algorithm is that it can be solved explicitly and thus can be calculated very fast. However, when the constraint of the structure is not idealized or one of the sensors fails, it may result in large errors. Therefore, a nonlinear programming-based algorithm is proposed here that can improve the shape sensing accuracy by increasing the number of sensors.
By way of example, the shape-sensing algorithm for a 33  EPT metamaterial is discussed. Figure S6 illustrates a possible sensor arrangement. When the procedure 8 shown in Figure S2 is adopted to solve for the unknown angles, the shape of the EPT material can be explicitly determined. is marked by the red circles in Figure S6. It can also be another set of 11 independent angles, provided that it is convenient to solve () i F θ . The dimension of i  depends 9 on the number of used sensors and can be equal to or more than 11. The sum of squares of the difference between i  (obtained from sensors) and () i F θ is taken as the objective function. A comparison of the shape-sensing results obtained using the optimization algorithm and that using the direct algorithm with only 11 sensors is shown in Figure S7. Compared to the direct algorithm, the optimization algorithm with 32 sensors shows a reduction of 75% in the mean square error of the actual shape.
However, we want to emphasize that, although the optimization algorithm provides improved results, the number of required sensors increases significantly. As a result, the time of solution increases dramatically, which hinders real-time computing, sensing and actuation.

S3. Analysis of elasticity-plasticity transformation
The analysis of the LSE in the main text is limited to elastic deformation. In this section, frictional force is taken into account in characterizing the measured plastic behavior of the EPT material. The potential energy of LSE has the form of (S12) which shows that 0 L dictates the magnitude of the moment and the ratio between 01 L and 02 L determines the equilibrium position. In addition the elastic force, frictional force also contributes to the elasticity-plasticity transformation of the LSE, which allows the unit to be stable in plastic mode. Figure S8. Model for the elasticity-plasticity transformation of the Lever-strut element.
As shown in Figure S8, the hinge can be assumed to have a linear frictional behavior. According to the force analysis, expression of the frictional resistance Eq. (S14) shows that the force at the hinge is generated only by the elasticity of the spring. The effect of the other loads can be characterized by varying  . From Eq. (S14), the expression of frictional resistance moment can be obtained as where   sign dy indicates that the direction of friction force is always opposite to the direction of displacement.  Although in the plastic mode, 0 L is not exactly equal to zero, the presence of friction can make it exhibit significant plastic characteristic and does not have enough elastic potential energy to restore it to its initial state.
13 Figure S11. Force-displacement curve of elastic rope. The length of the elastic rope is 88.6mm.

S4. Assembling of EPT robotic material
An EPT robotic material system is shown in Figure S13, consisting of a material part, a microcontroller, and a computer. The material part comprises actuators and sensors that receive instruction from the microcontroller to switch between elasticity and plasticity while transferring the deformation data to the microcontroller. The actuator for the EPT robot material is a servo motor (model DS-S006L, DFRobot) with a maximum rotation angle of 300° and a maximum torque of 11.7N cm  .The angle 14 sensor uses a potentiometer (model RV24YN20S, Shenzhen Minheng Electronic Company) with a maximum resistance of 50k .
The microcontroller model used here is Arduino Mega 2560. It receives signals from the material part of the robot through 11 analog signal ports. The microcontroller is connected to the computer through a USB port and transmits deformation data to the computer via serial communication. The computer writes the transferred data from the port to a text file in ASCII format via a Python script. This file is then read by MATLAB to reproduce the deformation of the EPT material in real-time. Figure S13. A typical EPT robotic material system.
The assembling of LSE and EPT unit cells are shown in Figure S14 and Figure S15, respectively. By tessellating the EPT unit cells, an EPT robotic material can be obtained, see Figure S16. The arrows indicate the installation direction.
15 Figure S14. Procedure for assembling LSE and its key components. Figure S15. Procedure for assembling EPT unit cell.

Movie list
Movie S1.
Principle of elasticity-plasticity transformation. We use the energy method to show the basic principle of EPT material and demonstrate the elasticity-plasticity transformation of a single Lever-strut element.

Movie S2.
From single cell to metamaterial. An EPT metamaterial is built by assembling EPT unit cells. Different responses of an EPT unit in the elastic and plastic modes after stretching and releasing are exhibited.

Movie S3.
Shape sensing of EPT robotic material. We construct the deformation sensing method of the EPT cells and show how to use the isostatic property to reduce the number of adopted sensors.

Movie S4.
Deformation-dependent mechanical property modulation. The EPT material can use its deformation information to control the elastic-plastic transition. The EPT unit cell enters the plastic mode when the cell is stretched laterally into a non-auxetic configuration and staus in elastic mode when the cell is in an auxetic configuration.