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Learning physics-informed simulation models for soft robotic manipulation: A case study with dielectric elastomer actuators

Soft actuators offer a safe, adaptable approach to tasks like gentle grasping and dexterous manipulation. Creating accurate models to control such systems however is challenging due to the complex physics of deformable materials. Accurate Finite Element Method (FEM) models incur prohibitive computational complexity for closed-loop use. Using a differentiable simulator is an attractive alternative, but their applicability to soft actuators and deformable materials remains underexplored. This paper presents a framework that combines the advantages of both. We learn a differentiable model consisting of a material properties neural network and an analytical dynamics model of the remainder of the manipulation task. This physics-informed model is trained using data generated from FEM, and can be used for closed-loop control and inference. We evaluate our framework on a dielectric elastomer actuator (DEA) coin-pulling task. We simulate the task of using DEA to pull a coin along a surface with frictional contact, using FEM, and evaluate the physics-informed model for simulation, control, and inference. Our model attains<5% simulation error compared to FEM, and we use it as the basis for an MPC controller that requires fewer iterations to converge than model-free actor-critic, PD, and heuristic policies.


I. INTRODUCTION
Soft robotic actuators provide a safe, adaptive, low-cost solution for movement tasks such as grasping and dexterous manipulation [1]. Precision manipulation using soft actuators however is a major challenge, as it requires modelling the deformable actuators' dynamic within the context of the manipulation task [2]. Such models are then used to learn accurate control strategies via simulation [3]. Recently differentiable simulators have been used to learn controllers in closed-loop scenarios by allowing the use of gradientbased optimization methods (e.g., Model Predictive Control, MPC) [4]. They have also been used for inference and data generation tasks.
Simulating deformable robots and contact rich manipulation is expensive [3]. Traditional methods model such dynamics by decomposing their geometry. For example, Position Based Dynamics approximates multi-body physics by deconstructing the system into particles [5]. However, these methods fail to accurately capture the true underlying physics, making it difficult to meaningfully interpret or constrain the particles. The continuum mechanics and contact dynamics of deformable materials are difficult to model with such approximate methods, leading to physically unrealistic results, which hinders model-based control. Physically accurate simulation of soft robotic manipulation requires modelling the underlying equations, defined by complex Ordinary/Partial Differential Equations (ODEs/PDEs).
Finite Element Methods (FEMs) provide a numerical method for solving such equations. Yet despite the ability of FEMs to accurately model such phenomena, integrating FEM simulation with closed-loop control is challenging due to their computationally expensive meshing: unless the meshes are dense and cover the domain, fidelity is poor. This paper's key idea is to generate data from an accurate (but slow) FEM model to learn an approximate (but fast) physics-informed model f for soft robotic manipulation.
Our framework uses f as a differentiable simulator for simultaneous closed-loop control and inference. Our model f is composed of two parts: a material network mwhich approximates deformable material behaviour (e.g., hyperelastic) -and dynamics d -equations representing the physical context of manipulation task (sliding motion under frictional contact with the surface).
Our framework's objective is to learn fast physicsinformed models that can be evaluated in real-time without significant loss in accuracy, and to use these models for control. As an illustrative use case, we focus on soft robotic pulling task using Dielectric Elastomer Actuators (DEAs). DEAs are soft actuators made using electroactive polymers that convert electrical work to mechanical work via expanding or bending motion. In our task, the goal is to pull a stationary coin by deforming the free end of the DEA (Fig. 1). We learn the physics-informed model for this pulling motion f , and evaluate its accuracy as a simulator against FEM simulations. For control, we use the differentiability of f to learn a model-based control policy (with the solver from [6]) and infer the parameters of the system's dynamics. 1 Our main contributions are: (i) a closed-loop control framework for soft robotic manipulation, that uses a differentiable physics-informed model f trained using FEM (Section II), (ii) the design of an exemplary DEA pulling task (Section III), that is simulated in FEM (Section IV), and (iii) performance evaluation of model f and its use in closed-loop control. For the latter, we compare simulation accuracy of f with both a FEM model and a baseline neural network. Additionally, we compare the model-based control policy (MPC with f ), with (i) a model-free control policy (learnt using the Soft Actor-Critic SAC algorithm [7]), (ii) a PD control policy (evaluated previously for DEA control [8]), and (iii) a heuristic control policy 2 (inspired by typical soft-robotic control policies [3]). We design experiments (Section VI) across 8 DEA pulling setups to evaluate our framework and answer the following questions: (Q1) How to define f using the physical laws of the system?
What is the simulation accuracy of f in a system with new unknown parameters (e.g., frictional coefficient)? (Q2) What is the performance of model-based control policy (based on f ), compared to other control policies? (Q3) What is the accuracy of the inferred model parameters?
Our results show f provides ≤ 5% simulation error compared to FEM. Further, in closed-loop control, an MPC using f outperforms all other policies, while simultaneously inferring system properties with ≤ 10% error. Vidoes of the manipulation task and other supplementary materials are available at https://sites.google.com/view/phy-informed-simsoft-robot/home.

A. Related Work
Soft robots are inspired by biological systems, e.g., where animals use muscles to achieve safe actuation and control [1]. Engineers use soft actuators to develop similarly safe, quick, adaptable, and precise robotic manipulation [2]. These soft actuators generate mechanical work under a specific actuation, e.g., shape memory alloys respond to thermal actuation, hydraulic actuators respond to pressure, etc. Learning control for soft actuators requires accurate simulation models that are used inside the control loop [4]. Designing simulator models for soft actuators is a challenging task, traditionally using particle based models (e.g., liquids [9]). In recent years, researchers are using FEM modelling that allows highly accurate modelling of deformable materials (e.g., fabric [10], composite materials [11]).
Dielectric elastomers (DE) are electroactive polymers that produce deformation under the influence of an external electric field. DEA are soft actuators that use thin layers of DE materials to achieve actuation under the stimulus of electric activation. DEAs provide fast and large deformation, are lightweight, and have a high energy density, which makes them promising candidates for soft robotic applications [12]. 1 For the case study of DEA pulling, coin mass mc and kinetic friction coefficient µ b are inferred. For details, refer to Section III-B. 2 The heuristic policy linearly ramps up actuation voltage until the 'episodic' task terminates. For details, refer to Section VI Hence, DEA has been explored to design soft robotic grippers [13], underwater robots [14], crawling robots [15], etc.
There have been several previous approaches to modelling DE behaviour. A mathematical fractional Kelvin-Voigt model for DEA is presented in [8], where they eliminate overshoot in PID control. In [16], a spring-dashpots model describes the DEA's dynamics, with model parameters determined from the response of the robot. FEM models for DE material have been explored, where deformation in unimoph DEA with inhomogeneous geometry modeled in [17] uses piezoelectric elements. A simplified finite element analysis of a dielectric bending actuator is performed in [13]. The FEM model for a gripping actuator in [18] is based on a custom defined material. These methods focus on modelling the DEA behaviour in isolation (separate from the manipulation task). Thus, they lack an understanding of the task context in which the DEA manipulator is being used. In contrast, our method simulates the complete manipulation task, allowing us to learn both DEA behaviour and the task specific context.
An accurate simulator of the manipulation task can assist in learning a controller. For example, a position based dynamics simulator (defined using particle interactions) is used in [9] to develop control strategies for pouring liquid. A FEM based differentiable simulator in [19] is used to learn control strategies for cutting. The above methods are designed to control one specific manipulation task (e.g., in [20], the model is explicitly engineered for cutting). In contrast, our control framework can be used for any robotic manipulation task that can be simulated in FEM.
Learning based modelling and control can be used for soft robot manipulation. For example, a differentiable model of a soft robot's quasi-static physics is learned in [21], and then used to perform gradient-based optimization to perform open-loop control. In [22], a recurrent neural network is used as a fixed forward model within a policy learning algorithm for closed-loop soft robot control. In contrast, our trained physics-informed model f can infer properties of new unknown setups, adapting during the closed-loop control process.

II. CONTROL FRAMEWORK
The objective of our control framework is to learn a control policy π for a manipulation task. Figure 2 shows the closedloop control design using the trained physics-informed model f of the manipulation task along with policy π.
A model-based control approach utilizes a forward model of the system: f : S × A → S, where S is the state space, and A is the control action space. For each timestep t, the state is s t ∈ S, and the control action is a t ∈ A. For MPC, the optimal control actions at each timestep is estimated by solving the optimization problem defined in Eq. (1), where a init is the initial action. We particularly choose to use a physics-informed f , which is differentiable, and allows us to use gradient-based methods to solve this optimization problem (such as finite-horizon Linear Quadratic Regulator, LQR [6]). The objective of the control (e.g., get to a target location) is used to define the cost function C : S × A → R (e.g., distance from the target location). For example, in DEA pulling, cost function C is defined for the objective of achieving target state (i.e., the target location of the coin) with penalty on control actions to minimize actuation voltage of the DEA. Eq. (2) shows the cost function, where s * t is target state, and w s , w a the state and action penalties.
arg min The physics of the robotic manipulation task includes, (i) the physical laws of the deformable material behaviour, e.g., electromechanical/hyperelastic behaviour, characterized by high order ODEs/PDEs that are computationally complex, and, (ii) the physical laws built according to the context of the manipulation task, e.g., sliding motion laws, or gravity. In modeling these physics, interaction variables z are introduced to describe the contact properties (e.g., force, stress, pressure) between the deformable material and its surroundings. In particular, for DEA pulling, z are the forces exerted by the DEA actuator on the contact surface. We simulate the manipulation task using a FEM model to numerically solve the associated physics equations. In this model, state s t+1 and interaction variables z t are simulated, given s t and a t . The FEM model for DEA pulling is described in Section IV. How the simulated data is then used to train a physics-informed model f is described below.

A. Physics-informed model (f )
The physics-informed model f has two parts: (i) The material network (m): A function approximator with weights θ estimating interaction variablesẑ (Eq. (5)). These interaction variables (z) characterize deformable material behaviour in the manipulation task (e.g., forces by DEA on contact surface). (ii) The dynamics (d): Physical laws characterizing the motion/dynamics of the system in the form of mathematical a t ← select action at t using a fixed policy; 4: s t+1 , z t = FEM (s t , a t );
where θ are material neural network weights, and φ the parameters (e.g., mass) for system dynamics. The material model m of a deformable material can describe an actuator (e.g., DEA), or the manipulated object (e.g., cloth) depending on the manipulation task. The interaction variables z depend only on the deformable material (i.e., the material network m), and are not impacted by the dynamics d of the manipulation. For example, in DEA pulling, m describes the actuator behaviour of a unimorph DEA (Section IV). Our model f is a physics informed neural network [23], where physical rules can be imposed on interaction variables z in the form of ODEs/PDEs. However, for the case study of DEA pulling, a system of linear equations sufficiently defines dynamics d. We next show how to define f using physical laws. Model f is differentiable, and captures the manipulation task physics through dynamics d. In addition to its use as a simulator to generate data, we also use it for inference, and for gradient-based control learning.

B. Training and policy synthesis
The physics-informed model f and policy π are learnt in two steps. First, a Learning step optimizes weights θ of m, using data generated by the FEM model of the task. Second, a Control step, where policy π is learnt via environment interactions. These interactions are used to infer parameters φ (e.g., coin mass) of dynamics d, informing f . We then use f to learn π. a) Learning The weights θ are optimized by minimizing the loss function based on the error in estimating material model, Algorithm 2 Control Input: Trained weights θ 1: Randomly initialize parameters φ, and ψ, and fix θ; 2: s 1 ← env .reset(); 3: while not stoping condition do 4: a t = π(s t ; ψ); 5: s t+1 , r t ← env .step(a t ); 6: Replay buffer R ← R ∪ (s t , a t , r t , s t+1 );

7:
if it's time to update then 8: Randomly sample B transitions from ∼ R; 9: ; // Update φ usingŝ, Eq. (7) 12: // Policy Update Update ψ by policy defined updates, e.g., SAC [7]; 14: end if 15: end while i.e., m: (z t −ẑ t ), and the error in enforcing dynamics, i.e., d: (s t −ŝ t ). Incorporating the loss encountered inŝ ensures that our model adheres to dynamics d (as shown by optimization of physics informed neural networks [23]). Algorithm 1 shows how θ is optimized by fixing parameters φ and minimizing the learning loss L l (Eq. (6)). A fixed policy is used to select actions a t (e.g., a random or uniform policy). The learning rate for weights θ is α θ and number of data samples is N .
b) Control Algorithm 2 shows our steps for closed-loop control. First, in inference, parameters φ are optimized by minimizing the dynamics estimation error, i.e., d: (s t −ŝ t ). Eq. (7) gives the loss function, with learning rate γ φ and batch size B.
Second, we learn policy π with weights ψ. Updates in ψ are defined using the underlying policy π and its training objective. For the model-free policy trained using the SAC algorithm, ψ is updated based on the loss function from [7]; the weights ψ represent weights of the neural network. For model-based MPC, we use our trained model f and gradientbased optimization to estimate the best action. Thus, with MPC, updating parameters ψ (line 13) is unnecessary.

III. SOFT ROBOTIC DEA PULLING
We design the manipulation task of coin pulling using a unimorph Dielectric Elastomers Actuators (DEAs) to evaluate the framework proposed in Section II. The deformable DEA actuator is made of Dielectric Elastomers (DEs), which are a type of electroactive polymers that produce mechanical strain under the influence of electric voltage. Thus, a DE  Force along y-axis by DEA on coin c membrane expands its area when a voltage is applied across its thickness [24]. Figure 3(a) shows a unimorph DEA, with one active and one constraining layer. The active layer expands under externally applied voltage causing the bending motion. The DEA is fixed at one end, and the other end rests freely on a circular coin c. On actuation, the DEA acts as a soft robotic finger, pulling the coin. A controller policy π can be learnt to achieve a certain displacement in the coin. Figure 3(b) shows the 2D view of the setup, where the mass of the coin is m c , the kinetic friction coefficient between the coin and DEA is µ t , and the kinetic friction coefficient between the coin and bottom surface is µ b . The displacement of the coin depends on such parameters of the system. A pulling coin setup C c is characterized by fixed values of {m c , µ t , µ b }. Setups C 1 , C 2 , . . . represent pulling different coins, based on different parameter values. 3 The physics-informed model f of the system is defined by the variables shown in Table I. The state of the system at time-step t is characterized by the location x t and velocity u t of the coin along the x-axis. The action comprises the voltage (V t ) applied on the DEA and ∆t, 4 and the hidden variables are the forces (F x and F y ) applied by the DEA on the top surface of the coin.

A. Material network (m)
Modeling non-linear properties of DEs require modeling the effects of hyperelasticity and Maxwell stress [24]. On application of voltage V, maxwell stress causes the bending actuation in DEA. The actuated DEA exerts forces F x and F y on the top surface of the coin, which results in its motion. The material network used to estimate these forces is defined in Eq. (8). We simulate DEA pulling using FEM, to generate 3 We consider the coins to be of fixed dimensions (i.e., fixed volume), and thus change the mass mc by changing the density ρ of the coin material. For further details, please refer to Section VI-A. 4 Note that in FEM, the time between successive simulation datapoints may vary. Thus, ∆t is variable during training the model f . data and optimize weights θ (Section IV).

B. Dynamics (d)
Physical laws of the pulling setup define the system dynamics d (Section II-A). There are two stages during pulling: static friction (forces are applied but there is no motion), and kinetic friction (applied forces cause motion in coin). An actuation threshold voltage V T is required to achieve a minimum coin displacement (i.e., to get to the stage of kinetic friction). Coin acceleration A t is due to the net force in the x-axis (Eq. (9)), where F µ is the opposing frictional force. We calculate F µ using Eq. (10), assuming linear growth in frictional force during the stage of static friction, and a no-slip condition on the top surface.
where g is gravitational acceleration (9.8 m/s 2 ). We assume a frictional velocity decay for a moving coin if DEA actuation is stopped (i.e., V t = 0). These dynamics disregard non-linear motion in coin (with high DEA actuation voltages, where DEA loses contact with the coin surface). The dynamics d of this pulling setup is characterized by parameters φ (Section II), which are: coin mass m c and frictional coefficient µ b . While training m, these values are fixed. We infer φ during closed-loop environment interactions.
The next location and velocity of the coin, i.e.,x t+1 and u t+1 , are calculated using A t , x t , u t , and ∆t and equations of motion. Thus, the dynamics d is a set of linear equations based on the laws of motion.

IV. FEM OF DEA PULLING
FEM is a numerical method for solving differential equations of a physical system. Its complexity depends on both the task and available computational power. The physical system of DEA pulling consists of a unimorph DEA on the fixed surface and a solid coin (Section III). We use commercially available software ABAQUS [25] to build a 3D model of DEA pulling. Figure 4 shows the simulated setup, during the starting stage and the stages of static and kinetic friction. In the stage of static friction, we clearly see no motion in the coin even when actuating the DEA.
a) DE Material To model the non-linear elastic behaviour in DE membranes, recent works use hyperelastic material models, like the Gent and the Neo-Hookean models [26]. For the small strains present in the thin membrane of our DEA (≤ 10%), such hyperelastic behaviour reduces to linear elasticity. Furthermore, since no commercial FEM packages provide DE elements out of the box, we approximate the behaviour of our DE material in FEM using piezoelectric materials elements [27]. We modify the piezoelectric finite elements material properties to model the Maxwell stress effect observed in dielectric materials. The Maxwell stress p on the DE membrane is given by Eq. (11) [24]. Similar to DE, piezoelectric materials exhibit strain when in the presence of electric fields. The piezoelectric stress is given by Eq. (12).
where V is the applied voltage across thickness z of the DE membrane, the relative permittivity is e r , and the permittivity of free space is e 0 . The piezoelectric elastic stiffness matrix is D E ijkl , the strain tensor is kl , the stress coefficient is e mij and the electric potential gradient is E m . As detailed in [17] the piezoelectric stress becomes approximately equal to the Maxwell stress for a thin membrane, such that the strain e zz in the direction of thickness (z-axis) is given by: A. FEM simulation settings Figure 4(a) shows our FEM setup when the DEA is inactive (zero voltage). The mesh consists of an 8-node linear brick (ABAQUS element type C3D8E). Each DE membrane has 10 elements. 5 For meshing the coin, ABAQUS's internal meshing strategy is used to generate 20 elements.
For both the active and constraining layers of the DE material, the Poisson's ratio is 0.5 and Young's modulus is 0.56 MPa [28]. We use Eq. (13) to calculate e zz = 3.68. All other piezoelectric coefficients are zero. We assume elastic behaviour for the coin. The bottom surface and the fixed end of the DEA are constrained using encastre boundary condition. The top surface assumes a no-slip condition.
The coin rests on the frictional surface with coefficient µ b ∈ {0.2, 0.25} (defined as tangential behaviour in the contact interactions in ABAQUS). Similarly, the free end of the DEA rests atop the coin with µ t ∈ {0.5, 0.55}. To simulate a real scenario, we include gravitational load (g = 9.8 m/s 2 ). We also do not assume a no-slip condition between the top of the coin and DEA, in contrast to the dynamics in Section III-B. Coin mass (m c ) is calculated using its volume and density ρ ∈ {7.7, 7.8} g/cm 3 , thus, making m c ∈ {1.36, 1.38} g. The total time simulated in FEM is 1 s, with ∆t between points determined by the internal solver.

V. PARAMETERS SETTINGS
This section details the architectures, parameters, and training procedures used to evaluate our framework. An experimental setup C is defined using differing values of µ t , µ b , m c (Section III). We use 8 setups, denoted {C 1 , C 2 , . . . , C 8 }. An FEM model is developed for each setup. To collect the dataset for each model, we apply a linearly increasing electric potential load (V t ∈ {0.0, 400.0}V ) to the top surface of the active DE layer. Table I describes the values collected at each timestep t. Each dataset contains 1000-2000 data points. Across setups, x 0 , u 0 = 0.0, 0.0 .
During learning, we initialize parameters φ (m c and µ b , Section III-B) of dynamics d using the true values from the FEM model. Note that µ t is not used as a parameter in the dynamics of f , but, is required for the FEM modeling. For each setup, the threshold voltage (V T ) is the voltage required to achieve a displacement of −10 −5 m. We assume the coin loses contact with the DEA for V t ≥ 300 V.
Physics-informed model f is developed using Pytorch [29] and contains the material network m and the dynamics d (Section II). The material network m is a fully connected neural network with four input nodes, two output nodes, three hidden layers with 128 neurons each, and rectified linear (ReLU) activation functions. An ADAM optimizer with learning rate 0.001 is used to minimize Eq. (6) and Eq. (7) for 1,000 iterations. An early stopping criterion is used based on validation loss with 0.0 minimum change. The baseline neural network uses RelU activations, while the recurrent neural network uses LSTM cells. The LSTM uses an 8-step recurrence.
The target state for the controller is x T = −1.0 mm, i.e., goal is to achieve a 1 mm displacement. The manipulation 'episode' is simulated, and terminates when x t − x T < 0.01 mm. To simulate real-world discrepancies, we add gaussian noise to simulated state x t with zero mean and standard deviation of 0.001 m. We average results for 10 'episodes' for all controllers. Batch size (Algorithm 2) is 256. During control, the time increment ∆t is fixed to 0.001 s-the policy does not provide it as an output. For inference, m c is initialized to 0.001 g and µ b is initialized to 0.2. We use the model-based control policy from [6]. For MPC, we set the number of timesteps 20, LQR iterations to 20, and action penalty to 0.001. The model-free policy is trained using Soft Actor-Critic (SAC) [7]. For SAC, fully connected neural networks are used for actor and critic with two hidden layers of 256 neurons each. The value of τ (soft updates) is set to 0.005, and networks are optimized using MSE loss and the ADAM optimizer. For the PD controller, the value of K p is set to −0.5 and K d is set to 5.

VI. EXPERIMENTAL RESULTS
This section presents experiments describing the high simulation and inference accuracy of f , while also developing an effective closed-loop soft robotic controller. For the case of DEA pulling, we evaluate, (i) the accuracy of the f as a simulator, and (ii) the accuracy of f in inference, and (iii) the closed-loop MPC controller that utilizes f .

A. Simulation (Q1)
We design an experiment to answer Q1, i.e., to show that f can simulate data for new parameter settings, we train and simulate on different setups (defined in Section III). Note that different setups represent different coins, with different mass m c and frictional coefficients µ b and µ t .
A simulation experiment set has coin setups given by {C train , C val , C test }. FEM data from {C train , C val } is used to optimize weights θ (material network). The objective of learning step (Algorithm 1) is to optimize weights θ of the material network m. During training and validation, the material network m is same across coin setups (i.e., same θ for C train and C val ) and dynamics d are specific to coin setups (i.e., φ based on C train and C val ). During testing, data is simulated recursively (for T = 1 s) in a test setup C test , using simulator f . This f consists of (i) a material network with previously optimized θ, and (ii) the dynamics with parameters φ test (based on C test ). We present averaged results for 6 experiment sets (E.g., for first set, {C train , C val , C test } are {C 1 , C 2 , C 3 }, the second are {C 2 , C 3 , C 4 }, etc.).
We evaluate the absolute errors encountered in simulatinĝ s t+1 (x t and u t ) at each timestep t. For example, error in x t is given by e x t = |x t − x t |, wherex t is the location simulated using f , and x t is the true value (from the FEM dataset). Absolute errors in data simulated using a black-box baseline Neural Network (NN ) and long short term memory recurrent neural network (LSTM ) trained using data from C train and C val are also included (e.g., a NN that simply approximatesŝ t+1 = NN (s t , a t ; w)). Table II provides the mean absolute errors in simulating location (x t ) and forces using physics-informed model f and the baseline models across all test setups. Figure 5 shows the absolute error in x for all test setups for f , NN , and LSTM . In all cases, f outperforms the baseline models. Additionally, the average absolute error in x simulated using f is less than 0.05 times the magnitude of the actual values, i.e., we note approximately ≤ 5% error compared the FEM simulation.
In the first half of the simulation time during static friction (Section III-B), we see negligible displacements and increasing F y . In the latter half, the kinetic frictional force becomes stable, and we see a change in coin location (x t ). Our model  f accurately estimates x in both stages. We see a similar accuracy for f compared to the FEM, NN , and LSTM in simulating velocity u x for the coin. While decreasing the neurons per layer in f from 128 to 64 (Section V) results in no statistically significant increase in average error, decreasing the capacity of baseline NN by the same amount results in an increase from 15% average error to 50%. Real-time control in manipulation tasks requires fast simulations. For 1 second of simulation with 700 points, the FEM model takes 130 seconds (average for all test coins)a prohibitive duration for real-time control. In comparison, our physics-informed model f takes ≤ 0.7 seconds. While both baseline models are also faster than FEM, (0.3 and 1.2 seconds for NN and LSTM respectively), they provide poor accuracy compared to f . Our proposed model f is fast as opposed to FEM but provides high fidelity to FEM as a simulation model of soft robotic manipulation.
The absolute error in forces F x and F y in the region of static friction is higher compared to the region of kinetic friction. This happens because we optimize material network m using a physics informed loss function, i.e., a loss function that is based on the error in the next state s and the error in the interaction variables z (Eq. (6)). Optimizing m using this loss function assists in learning the overall manipulation task behaviour, as opposed to only learning the outputs of m (F x and F y ). This behaviour is non-restrictive, as the objective of our model is to learn the next state of the motion, which is simulated accurately.

B. Control (Q2)
In this experiment, we evaluate f to answer Q2 (performance of model-based MPC compared to other control policies?). In the control step (Algorithm 2), we learn closedloop control for test setups C 1 and C 2 . Prior to this, in the learning step (Algorithm 1), we train the model f a using the FEM data from setups {C 3 , . . . , C 8 }. We do not use the data from C 1 and C 2 during learning to avoid information leakage. For C 1 and C 2 , we learn the following policies: The Heur policy is inspired by typical soft-robotic control policies [3]. The environment is simulated by trained physics-informed models f s,1 and f s,2 . This is due to the lack of a real-world DEA setup (However, Section VI-A shows our physics-informed models are accurate simulators). Figure 6 shows the average coin location x t during closedloop control of test setup C 2 . Average is calculated across 10 episodes to compare the MPC policy (model-based), with SAC policy (model-free), a PD policy and a Heur policy. We notice similar results for both test setups C 1 and C 2 . The variance in coin position is due to the noise added in the simulated observations to estimate a realistic scenario.
The coin reaches the target in ≤ 200 iterations under the MPC policy, i.e., ≤ 200 observations are captured in the control loop. In contrast, SAC policy, PD policy, and Heur capture approximately 10,000, 1500, and 500 iterations. Thus, during control, the MPC policy reaches the target in the least number of steps, where the computation time per iteration (average 1.2 s) is not computationally prohibitive for a real world manipulation. We note similar computation times for the model-free SAC policy (average 0.005 s) and PD policy (average 0.001 s).
During control using Heur policy we notice sudden motion towards x T after approximately 280 iterations. This represents the transition from stage of static friction to stage of kinetic friction. The Heur policy linearly ramps up actuation voltage every iteration, and thus, does not depend on location feedback. In contrast, both PD policy and SAC policy rely on observed location, and take longer to reach and manipulate the coin in the stage of kinetic friction.

C. Inference (Q3)
This experiment evaluates how accurately f infers the model parameters. Algorithm 2 infers m c and µ b for test setups C 1 and C 2 . Similar to the control experiment, in the learning step, we train f a using FEM data from setups {C 3 , . . . , C 8 }. The mass m c and frictional coefficient µ b inferred by physics-informed model f are compared against their real values (used in C test during control with MPC policy). Inferred m c converges to ≤ 10% error compared to the real value within 2,000 iterations, while µ b , the converges within 300 iterations. Similar results hold across both test setups and all episodes.

VII. CONCLUSIONS
This paper presents a framework to learn a differentiable simulator to develop a corresponding controller for soft robotic manipulation. We defined a physics-informed model f consisting of a material network m, and dynamics d. This model f can be used as a simulator for data-generation, inference, and control policy optimization. We designed a soft-robotics case study where a coin is pulled using unimorph DEA. FEM simulation of the DEA generated data to train f . Our experiments used multiple setups to evaluate the framework in learning f and model-based control.
From our analyses, we conclude, (i) the physics-informed model f trained using the proposed framework can simulate new setups (characterized by parameters φ) with ≤ 5% error compared to FEM (Fig. 5); (ii) a closed-loop MPC policy based on differentiable model f outperformed all other policies in orders of hundreds of iterations (Section VI-B); (iii) f can be used for accurate inference of the parameters φ: m c and µ b (Section VI-C). Future research directions include: Further evaluating this control framework with physical soft robotic actuators; Evaluating the model f in tasks with large degrees of freedom; and exploring model-based policies with lower computational requirements compared to MPC. ACKNOWLEDGMENT S. Ramamoorthy would like to acknowledge financial support from the Alan Turing Institute, for the project 'Enabling advanced autonomy through human-AI collaboration'. We thank Mukul Sahu for his comments on FEM modeling.