A Data-efficient Neural ODE Framework for Optimal Control of Soft Manipulators
Authors: Mohammadreza Kasaei
Keyhan Kouhkiloui Babarahmati
Zhibin Li
Mohsen Khadem
Venue: CoRL 2023 Poster (2023)
Keywords: Soft robots
Non-parametric modelling
Optimal control

Abstract: This paper introduces a novel approach for modeling continuous forward kinematic models of soft continuum robots by employing Augmented Neural ODE (ANODE), a cutting-edge family of deep neural network models. To the best of our knowledge, this is the first application of ANODE in modeling soft continuum robots. This formulation introduces auxiliary dimensions, allowing the system's states to evolve in the augmented space which provides a richer set of dynamics that the model can learn, increasing the flexibility and accuracy of the model. Our methodology achieves exceptional sample efficiency, training the continuous forward kinematic model using only 25 scattered data points. Additionally, we design and implement a fully parallel Model Predictive Path Integral (MPPI)-based controller running on a GPU, which efficiently manages a non-convex objective function. Through a set of experiments, we showed that the proposed framework (ANODE+MPPI) significantly outperforms state-of-the-art learning-based methods such as FNN and RNN in unseen-before
 scenarios and marginally outperforms them in seen-before scenarios.

A Data-efficient Neural ODE Framework for Optimal Control of Soft Manipulators
Mohammadreza Kasaei School Informatics University of Edinburgh, UK m. lkaaaaeeedda... ac. uk
Keyhan Kouhkiloui Babarahmati School of Informatics University of Edinburgh, UK kkouhkil@eds ac. uk
Zhibin Li
Mohsen Khadem
Department of Computer Science University College London, UK
School of Informatics University of Edinburgh, UK mohsen. .aaaee@edadddmddd.. ac uk
alex. www.ll@@llll.. wt..uulll...u....
Abstract: This paper introduces a novel approach for modeling continuous for- ward kinematic models of soft continuum robots by employing Augmented Neural ODE (ANODE), cutting-edge family of deep neural network models. To the best of our knowledge, this is the first application of ANODE in modeling soft continuum robots. This formulation introduces auxiliary dimensions, allowing the system's states to evolve in the augmented space which provides a richer set of dynamics that the model can learn, increasing the flexibility and accuracy of the model. Our methodology achieves exceptional sample efficiency, train- ing the continuous forward kinematic model using only 25 scattered data points. Additionally, design and implement fully parallel Model Predictive Path In- tegral -based controller running GPU, which efficiently manages a non-convex objective function. Through set of experiments, we showed that the proposed framework (ANODE+MPPI) significantly outperforms state-of-the-an learning based methods such as FNN and RNN unseen-before scenarios and marginally outperforms them in seen-before scenarios.
Keywords: Soft robots, Non-parametric modelling, Optimal control
Introduction
Soft robots are composed of compliant materials such as silicone, rubber, or elastomers, which allows them to conform to surfaces and objects while maintaining a level of physical robustness unavailable to their rigid counterparts. This level of compliance makes them suitable for broad range of applications, including medical procedures, search and rescue operations, and exploration [1]. However, the design and control of soft continuum robots present significant challenges. This is due to the inherent non-linearities and high DOF required to accurately capture the structural deformations that realize these compliant behaviours, which in turn makes it challenging to control the robot's 's motion. Several methodologies have been proposed to model soft robot controllers, broadly classified into two categories: model-based and data-driven [2].
Model-based approaches rely mathematical models represent the dynamics of the robot and use this model to design controllers. Various methodologies have been proposed in this category, including polynomial curvature fitting [3]. reduced-order finite element models [4], and lumped parameter models [5, 6). comprehensive review of physics-based models for soft robots can be found in [7]. These models have limitations, as they are based on assumptions that may not hold in all conditions and may only able to accurately describe the behaviour the robot under subset of conditions. Furthermore. these methods can be computationally expensive and may not fully capture the nonlinear behaviour exhibited by the robot. Additionally, physics- physics-based models' error tends to increase together with the model inaccuracies, e.g., fabrication inaccuracies.
7th Conference Robot Learning (CoRL. 2023), Atlanta, USA.

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MPPI
Augmented Neural
uala (Str)
Batch of samples
Man- Nen-Conver Objectiva Function
Batch estimations
f(attal.
Pénonstration
Space
GPU
Figure 1: 25 scattered data points are employed within the demonstration space for acquiring knowledge about continuous forward kinematic model (left). Subsequently, this trained model serves as the foundation for a fully parallel controller capable of managing a non-convex objective function executed GPU (right).
Learning-based approaches for soft robot modelling and control utilize collected data to develop models and controllers, either with or without relying on mathematical models. The learning-based models tend to be insensitive to physics-based modelling assumptions and fabrication inaccuracies, especially training is accomplished on the physical robot arm. These approaches aim learn kinematic or dynamics models directly [8, 10, 11, 12] or employ modellbased//modllfFee Rein- forcement Learning (RL) techniques to learn control policies [13, 14). While data-driven approaches have the potential to overcome the limitations of model- methods, they also face inherent draw- backs. First, they require a substantial amount of training data, which can be challenging to acquire for real robots and may compromise their structural integrity as demonstrated in [15), [16] and [17], where 12000, 7000, and 4096 sample points were utilized. Second, the ability to generalize knowl- edge limited, leading to poor performance in new or unseen situations [18]. Lastly, data-driven approaches lack interpretability and physical insight, making it challenging understand decision- making processes and the underlying physical principles of soft robot behaviour [13].
In this paper, we introduce a novel framework for modeling and controlling soft robots. We address the limitations of existing data-driven methods and propose a solution that is extensively tested on a real soft robot different scenarios. Our key contributions are as follows:
1. We propose a novel approach to developing continuous forward kinematic models for soft con- tinuum robots by employing Augmented Neural ODE (ANODE) [19], a state-of-the-ar family of deep neural network models. To the best of our knowledge, this is the first time that ANODE has been applied to modelling soft continuum robots. Our methodology achieves exceptional sample efficiency, training the continuous forward kinematic model using 25 real robot demonstrations.
2. Utilizing the trained model as a basis, we developed a fully parallel Model Predictive Path Inte- gral (MPPI) controller rumning on a GPU, capable of efficiently managing non-convex objective function. This controller harnesses the power parallel processing to optimize trajectory planning and control for the soft continuum robot. By leveraging the capabilities of the trained model, our controller enables the robot to navigate previously unseen trajectories within the feasible kine- matic space. In comparison to state-of-the-art data-driven approaches such as feed-forward neural network (FNN) and recurrent neural network (RNN), our framework (FK model and controller combination) demonstrates superior performance in terms of trajectory tracking error while trained on significantly fewer data.
This paper is organized as follows: the methodology of the proposed approach is outlined in Section 2. To evaluate the effectiveness of the proposed framework, series of experiments are conducted in Section 3, and the results are discussed. An ablation and comparison study with existing approaches will be presented in Section 4. Finally, the conclusion and future work are summarized Section 5.
2 Methodology
This section will provide a comprehensive overview of the key steps involved in our approach. We will start by discussing the process of generating the training dataset, explaining how we collect the necessary data to train model. Next, will focus on learning the differential kinematics of the robot, highlighting the techniques and methodologies employed in this process. Finally, will

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describe the development of controller based on the learned model, explaining how it enables the robot to execute precise and controlled movements.
2.1 Training Dataset
The robot utilized in this study is a multi-backbone robot, as depicted in Figure (left). It consists of a central flexible backbone made of compression spring and four parallel flexible rods encased around it. By manipulating the rods (pulling/pushing), the shape of the robot can be altered. More details about the robot prototype are presented in the appendix. The modeling and controlling of such robots present significant challenges, including the complexity of the coupling between actuation inputs, difficulties in modeling friction between the rods and spacers, particularly when the robot is bent, and the challenge of determining model parameters and the backbone's 's mechanical characteristics.
In this paper, we intend to utilize a limited dataset to model the behavior of the robot. The dataset was generated by an experienced operator who performed series of demonstrations using the robot. The operator manipulated the lengths of the rods move the tip in various directions. The demonstrations were conducted with the rods being pulled/pushed up to 3mm. The feasible kinematic space and demonstration space are depicted in Figure/ 1. As it is shown, the demonstration space is significantly smaller compared to the feasible kinematic space. The robot inputs, 4, E RR, and the corresponding Cartesian coordinates of the robot tip, X2 E R3, were recorded at a rate of 15 Hz to generate the training dataset, i..., D (x// 24 k=='' The position of the robot was estimated using an RGB camera, as discussed in Section 3. It is noteworthy that generating this dataset was efficient and took less than 10 minutes, and our dataset contains N=9100 samples. Only 25 points randomly selected from this dataset will be used to model the robot in the next section, while the remaining data was utilized to train other state-of-the-ard machine learning algorithms for comparison.
2.2 Learning Differential Kinematics of the Robot
We assume that the robot's behavior can be modeled by series of nonlinear differential equations:
+(1) f(x(((()()))
f R: x x RR3
with the initial conditions, xxto) x()), (t)) (0). It is assumed that a closed-form expression two consecutive states and an action to train
for the function does not exist. One multilayer perceptron (MLP) ((x,, u,) - 1x+1) or Recurrent neural network (RNN) like nonlinear auto-regressive network with exogenous inputs (NARX). MLPs and NARX models work well on the range of training data but they may struggle with extrapolation, especially if the data lies outside the range of the training data. Additionally, MLPs and NARX models, which typically operate in a discrete-time fashion, may produce predictions that exhibit discontinuities between time steps, leading to less smooth or less physically plausible extrapolations [20].
To overcome these limitations, and to develop a continuous and smooth data-efficient neural net- work to approximate the robot's model, we formulated the problem as an Augmented Neural ODE (ANODE) [19] which can naturally extend predictions beyond the observed data and time span [20]. Indeed, the robot is modeled using stiff differential equations which is characterized by having solutions with rapidly changing components as well as slowly changing components. Explicit methods like Buler (ResNet) [21] will be unstable unless the step size is taken be extremely small. There some tricks to overcome these issues but they do not fundamentally change the underlying stability properties. The key features of the ANODE in comparison to the other approaches are data efficiency, capturing complex dynamics, continuous-timg formulation, generalization, and handling of trajectory intersections [19]. In this work, we used fixed-adams (Adams- mashforth-Moulton [22] method which is an implicit method that is more stable and accurate for stiff differential equations, as they take into account not only the current and previous steps, but also the future step that we are trying to compute. This allows them handle the rapidly changing components of the solution in more robust way. Additionally, to prevent trajectories from intersecting, we expand the learning and solution space of the ODE from R x R - R to 3 x 3 - R By concatenating vector of
3

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zeros (0 px1) to each data point, we solve the ODE in this augmented space. This augmentation leads to smoother learned function f. resulting in simpler flows that can be computed by the ODE solver in fewer steps. Additionally, it alloww the ODE flow to lift points into the extra dimensions, effectively preventing trajectory intersections. This continuous-timg formulation allows the model to capture the underlying dynamics and make reasonable extrapolations based on the learned ODEs [19]. With the objective in mind, we proceed discretize the robot model described in (1) and transform it into a boundary value problem:
x² = ((()()),a()),
based on the following boundary conditions:
Xo = x(()) un u(11).
xx x(x)), Mg =((2).
considering the neural network fe as an approximation of the function f. we can determine the solution of xx4)) if we have prior knowledge of fo:
x(1)) = x(te-1) + [" fo(x()),u())dr
IK-1
thus, conventional numerical ODE solvers like the Fuler, Runge-Kutta fixed-adams algorithms can be utilized to estimate the value x(tx):
R (tg) = Of (t) (TK-1.7R)).
(5)
However, cases where fe imprecise or remains unknown, it becomes possible to assess the error in estimating the boundary values:
<= (|{((t)) x(x2)l--
To update this model, fe, adopt a random selection approach, where only 25 points chosen from the generated dataset. At each training step, single point from the dataset is selected. The loss function is estimated using equations (2-6). This error is utilized in a supervised learning manner, and the model is trained using the adjoint sensitivity method [23] to ensure memory efficiency. It is important to note that, based on equation (3), the control inputs remain constant, denoted as #o = uy. during each training step. To enhance learning efficiency, we have disregarded the dynamics of the control inputs (i.e., 0) and focused solely on learning the input-output dynamics. Details of training and network architecture are presented in the appendix.
2.3 Controller Architecture
In this section, present robot control methodology that utilizes the trained model, fo, to steer the robot towards arbitrary trajectories in accordance with non-convex cost objectives. The proposed approach is a derivative-free, sampling-based Model Predictive Control (MPC) technique, known as the Model Predictive Path Integral (MPPI) controller, which is capable of handling nonlinear [24]. First, Jacobian matrix, denoted as J. is defined through the utilization of the trained model, fe, which maps the velocity of the robot's end-effector the corresponding velocities in the configuration space:
dynamics and non-convex cost objectives
JJ,.
where J is a 3x3 3 matrix, x R and u R$ represent the robot's end-effector Cartesian coordinates and the input space, respectively. In this work, we employ numerical estimate of the Jacobian using finite difference method and batch sampling from the trained model. Now, using (7), a discrete-time stochastic dynamical system can be obtained:
X = x, + J(a, + 54,)
where Sú, the control input update which represents by a zero-mean Gaussian noise vector with variance of Σn, (δû,- NO, E,)). Using (8), the control problem can be formulated as a stochastic optimal control problem. Given finite time-horizon, E (0, 1,2, T 1). the objective of the controller is to determine optimal control sequence, (uo. H1, .... -T-1) -1) " that minimizes

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the expected cost-to-go, S(T), across all possible system trajectories, T= (xo. No. Xi.... 4r-1, with respect to (8), by taking into account the state-dependent cost function, S(T) E R:
T-1 Σ
J min a
((X))
g(x,) 2 Ru,)
(9)
where (xx)) is the terminal cost, ((x,) represents rumning cost and R E R=*x denotes a positive definite control weight matrix. order to solve this optimization problem. we adopt an iterative update law, derived in reference [24], for the implementation of the MPPI algorithm. This algorithm iteratively updates the control sequence for a predefined time horizon, utilizzig successive approximation approach:
ΣK exp (-(1/)(())ST, ou,,
U, U1
(10)
Σ exp (-(1/2)
where K represents the number of samples, = ETT ((x, (,, cost-to-go of the 4t sample, and A R* hyperparameter called inverse temperature. The cost-to- go function serves critical component in guiding the robot's decision-making process, in- corporating multiple objectives such as tracking a reference trajectory, obstacle avoidance, and considering affordance. By evaluating the future costs associated with different control inputs, the cost-to-go function allows the robot to anticipate the consequences of its actions. It con- siders the desired trajectory as a reference, aiming to minimize the deviation from this path while avoiding obstacles. Furthermore, the cost-to-go function can take into account the con- cept of affordance which encodes the relationships between actions, objects, and their result- ing effects [25]. The architecture of the proposed controller is depicted in Figure I (right).
3 Experiments
Here, a series of experiments were conducted to assess the effectiveness the proposed approach across various scenarios. Figure 9 shows the ex- perimental setup employed in our study, compris- ing flexible and expandable soft manipulator, Logitech RGB camera positioned within the robot's workspace, and user interface for recording, initi- ating, and terminating the experiments. To enable the camera to detect the position of the robot's tip, an ArUco marker [26, 27) affixed to the tip of the robot, providing feedback to the control loop- Details of the robot, the hyperparameters, the controller configuration, and the network structure can be found in the appendix.
Figure 2: Experiment setup.
3.1 Experiment Design
A series of experiments have been conducted assess the effectiveness of the proposed framework:
Static Target Tracking: different targets are pre-defined within the robot's workspace. The objective is reach these targets with minimal positional error.
Trajectory Tracking: the robot is track various trajectories in both 2D and 3D space, which include: i) square on the X-Y plane with sides measuring 0.06 ii) A circular shape in the XY plane with a radius measuring 0.03 meters: iii) A triangle with all sides measuring 0.06 meters; and = 5 sin 4tx 7
iv) An eight-shaped curve defined by the equations x a cos 21 +
where
0.03, b 0.05, T 100 seconds, and ranges from 0 to 120 seconds; v) helical trajectory is executed along the Z-axis, characterized by radius of 0.03 and a pitch of 0.02 To assess the tracking's repeatability and accuracy, three trials conducted for each trajectory.
5

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Square
Circle
Triangle
Eight
Holix
0.04
0.03
0.02
0.02
0.02
0.02
0.01
[m]
y[m]
0.12
-0.03
0.1
-0.04
-0.04
-0.03
-0.04
0.02
-0.82
[m]
[m]
[m]
x [m]
y [m]
Figure 3: Trajectory tracking results: the robot is set track diverse trajectories in both two-
dimensional (2D) and three-dimensional (3D) space. The solid-red and the dashed-black lines represent the actual and desired trajectories, respectively.
Obstacle Avoidance: in this task, the robot is set to track a helix trajectory characterized by radius of 0.03 m and a pitch of 0.02 m which is near the border of its feasible kinematic space. The objective of this task is to show how safely the robot is tracking the desired trajectory while avoiding obstacles without being unstable.
Box Opening and Test Tube Manipulation: to showcase potential application of the robot and the controller's adeptness at incorporating affordance in object manipulation during environmental interactions, we have designed demanding teleoperation task consists of two sub-tasks: opening the box and picking and placing the test tubes into the rack. In this task, the operator's 's objective is to maneuver the robot to open the of a wooden box, taking into consideration that the robot lacks the strength to counteract the force of gravity and fully extend the lid. This challenging scenario serves as compelling demonstration of the controller's ability to effectively utilize affordance while navigating the complexities of object manipulation in real- world environments.
3.2 Results and Discussions
Table 1: Statie Target Tracking Results
In the context of static target tracking, the robot's per- formance is assessed based on three metrics: steady- state error (SSE), standard deviation (o)), and settling time (ST). The obtained results are showcased in Ta- ble 1. is noteworthy that the position error remains below the threshold of mm. Additionally, the consistently low standard deviation reveals stable error pattern observed throughout all experiments.
SSE (mm)
(mm)
ST (sec)
0.82
0.68
5.72
8
0.71
0.58
4.21
0.37
0.32
3.82
In trajectory tracking experiments, the evaluation of the robot's performance involves the calculation and presentation of the root mean squared error (RMSE) and standard deviation (a)) of the error across five distinct trials. These metrics, along with a visual depiction in Figure 3, provide a comparative analysis between the robot's actual tip trajectory and the desired trajectory. Notably, the robot successfully tracks various shapes, including square, circle, triangle, eight, and helix, with a maximum RMSE of 3.2 10-3 Results are summarised in Table 2.
Representative results depicted in Figure 4 showcase the effectiveness of the MPPI algo- rithm achieving obstacle avoidance while si- multaneously maintaining stable tracking of the desired trajectories. In our implementation, the Circle
Table 2: Trajectory Tracking Results
RMSE (mm)
(mm)
x
Square
2.40
2.60
2.50
1.40
1.50
1.30
2.80
3.20
3.10
0.51
1.50
1.50
running cost is the sum of the individual cost terms including term for tracking reference trajectory, a term for avoiding obstacles, and term for penalizing jerky motions (details are presented in the appendix). The running cost plays vital role in guiding the robot's behavior by influencing the control actions to minimize deviations and ensure a safe and efficient path. By incorporating the position of the obstacles into the running cost, the robot can effectively evaluate the consequences of its actions and make decisions that prioritize obstacle avoidance and trajectory tracking. This approach enables the robot dynamically adjust its control inputs based on real-tima feedback, resulting in reliable and robust performance even in complex scenarios.
Iriangle
1.10
1.10
1.60
0.90
0.90
1.50
Eight
2.00
2.10
2.50
0.50
0.50
1.50
Helix
1.10
1.10
1.80
0.52
1.50
0.61

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0.30
0.26
6.12
0.12
0.12
0.00 02
2.01
0.04-..02
0.62
0.1
9.64
.0.02
0.2
y [m))
[m]
[m]
[m]
y [m]
[m]
Figure 4: Representative results of obstacle avoidance experiments. Blue dots represent the obstacles, solid-red and dashed-black lines are the actual and reference trajectories, respectively.
Figure 5: This of snapshots illustrates the box opening and test tube manipulation task. In this particular task, the operator's objective is to guide the robot opening the lid of the box and subsequently picking and placing the tubes into the designated racks.
In the Box Opening and Test Tube Manipulation task, the robot operates in unidirectional tele- operation mode, where the designed controller enables it to track targets specified by an operator through keyboard input. The complexity of this task arises primarily from the absence of direct force feedback, the disparity in kinematics between control interfaces and the robot, the delayed response time, and the lag between the operator's commands and the robot's actions. To address these challenges, we incorporate set of affordance terms into the running cost of the controllers (details are provided the appendix). These affordance terms can be selectively activated or de- activated by the operator limit the motion of the robot along accomplish the task (i.e, y, z), depending the task phase. The versatility of MPPI, which can handle non-convex running costs, allows us to effectively utilize these affordance terms for more intuitive and context-aware interaction between the operator, the robot, and the environment, enabling more effective and efficient teleoperation. A video showing the results is available online https:/youtu..b//60 be/tuus/-tttttttttett -5tttt and Figure shows set of snapshots of this task.
direction deemed suitable to
Ablation and Comparison Study
In this section, we conduct an ablation study to thoroughly compare the performance of the proposed ANODE-based forward kinematics model with Feedforward Neural Network (FNN) and Recurrent Neural Network (RNN) based models in four distinct scenarios. We assess the performance of the models in open-loop and closed-loop trajectory tracking scenarios for both unseen-before and seen-before scenarios, providing a comprehensive analysis of their strengths and limitations. In the open-loop scenarios, opted to exclude the MPPI controller and instead utilized simplified employed the equation (ù J*)), where Js represents the aseudo-inverse of the Jacobian and & is the reference trajectory. By implementing this modified strategy, we aimed to observe the system's 's response without the influence of the MPPI controller, focusing solely on the FK models. In the seen-before scenario, the robot was tasked with tracking an eight-trajectory X-Y plane within the demonstration space. Conversely, in the unseen-before scenario, the robot was challenged track a 3D eight-trajectory spanning the entire feasible kinematic space. These distinct scenarios were designed assess the robot's ability to adapt and generalize its knowledge across different dimensions and spatial configurations. By evaluating its performance in both seen and unseen trajectories, we gain valuable insights into the framework's capacity to handle novel situations and extrapolate its learned behaviors to unfamiliar scenarios.
approach. Specifically,
Furthermore, by comparing ANODE's performance with the traditional ML. and RNN mod- els, we aim to highlight the unique advantages of the ANODE model in capturing the un- derlying dynamics and temporal dependencies inherent in soft robot forward kinematics pre- diction. The FNN is constructed with multilayer perceptrons (MLP) to map the input ", to
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x10 ANODE
:100 MLP
x10 RNN
ANODE
[m)
y [m]
_
0.12
0.14
0.1
-0.02
-0.04
0.0
02
0.04
0.02
-2
[m] x10 -2
2
0 2
-2
0.04
0.06
[m]
[m]
[m]
x10
Figure 6: Results of the ablation study: the dashed-black lines, the solid-red and solid-blue lines correspond to the reference, closed-loop and open-loop actual trajectories, respectively.
On the other hand, the RNN adopts a NARX architecture, which takes the current state (x,, U,) as input and predicts the subsequent state X,+1- Both the FNN and RNN models are trained using the dataset generated in accordance with the methodol- ogy described in Section 2, employing mean square error as their respective loss functions. To ensure a fair and equitable compari- son, we maintained consistent network sizes across all methods. We trained three distinct FK models and integrated them within the identical control loop framework depicted Figure 1 (right). To evaluate and contrast their respective performances, we computed the RMSE and standard deviation (o) of errors across five trials, summarizing the out- comes in Table 3. Representative out- comes are visually depicted in Figure 6.
the output
Table 3: Ablation and comparison results of trajectory tracking scenarios; -O/-C tokens refer to open-loop and closed-loop results, respectively.
RMSE (mm)
(maan)
y
2
y
MLPC
6331
0.348
0370
0.531
0.348
0.69
MLP.O
0.572
0.563
0,702
0.557
0.543
0.494
RNN-C
0.542
0.537
5.100
0.54
0.537
0,498
RNNCO
0.576
0.345
3,000
0.54
0.300
0.419
ANODECC
0.105
6,127
0.116
0.105
0.127
0.116
ANODE-O
0.198
0.177
0.122
0.198
0.177
0.121
un0001
(COO
ANODE-C
0.256
0.164
0.157
0.256
0.164
0.157
ANODE-O
5,600
3,100
4.900
4.700
18000
2,900
providing further insights into the trajectory-tracking capabilities of each model. As depicted in this figure, all the models demonstrated proficiency in successfully accomplishing the seen-before sce- nario. Notably, among the tested methods, the ANODE(both open/closed loop versions) showcased superior performance, outperforming the other models in accurately accomplishing the task. In the unseen-before scenario, ANODE stood out as the only model capable of effectively generalizing its knowledge and successfully accomplishing the task. On the other hand, the MLP and RNN models encountered difficulties, leading to the robot becoming uncontrollable. The results showed that the proposed method outperformed alternative approaches significantly in unseen-before scenarios and performed slightly better in seen-before scenarios.
Conclusion and Limitations
This paper introduced soft continuum robots using ANODE. The proposed method only required 25 scattered data points. Additionally, we developed parallel MPPI-based controller running on GPU, which effectively handles non-convex objective function. This design enhances the adaptability and robustness of the learned model, enabling accurate prediction and control of soft continuum robot motion in various new scenarios. Through extensive experimentation, ablation, and comparison studies, our pro- posed framework (ANODE+MPPI) exhibits superior performance over learning- based approaches like FNN and RNN in unseen-before scenarios. also slightly outperforms them in seen-before settings. While these results are promising, there some challenges and limitations. The soft robot manipulator being studied has a limited input space comprising three variables. However, when applied to more complex robotic systems characterized by higher dimensions, the Neural ODEs encounters certain limitations. Specifically, the computational costs associated with training Neural ODEs become more challenging. This is attributed to the time-consuming nature of the forward pass, which necessitates the numerical integration of an ODE. Additionally, the proposed method- ology assumes the presence of continuous dynamies within the robot, rendering it less suitable for effectively modelling soft robots that exhibit sudden changes, instabilities, discontinuities in their behaviour.
method for modelling the continuous forward kinematic models of

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Acknowledgement
This work is supported by EU H2020 project Enhancing Healthcare with Assistive Robotic Mobile Manipulation (HARMONY, 101017008) and the Medical Research Council [MR/T023252/11.
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10

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Appendix
1
Robot Prototype
The robot, depicted in Figure 9, consists of flex- ible backbone rigidly affixed to spacers, accompa- nied by four rods fixed at the end spacer and passing through the remaining spacers with sufficient clear- ance, forming the primary body the robot.
Connector
Benew
Motor
Trackin
Marker
Extensible
Continusm
Manipulator
oftor
Camera
To drive the robot, four brushless DC motors from Maxon Motors, equipped with quadratic encoders and 150:1 reduction gearheads, are utilized. Precise motor position control is achieved through four PID position controller modules (EPOS4 Compact 50/5 CAN), which receive encoder feedback and commu- nicate with a PC using the CAN protocol establish and retrieve controller set-points and configurations. Lead screws, connected braided tubes via 3D printed connectors, are attached to the motors to con- vert motor power into tube-pulling and pushing ac- tions, A schematic of the robot is shown in Figure 8.
Figure Prototype of the flexible robotic arm composed of a reinforced multi- backbone robot. The robot is connected to four brushless DC motors using lead screws. An ArUco marker [26, 27] placed on the robot tip, and camera is used track the marker's 's position.
2 Network Architecture and Training
y
Table 4 presents a summary of the hyperparameters and network structure. should be noted that we employed an early-stopping technique to prevent overfitting when training the model. With early stopping, the model's training halted before it starts to overfit the training data, even if all iterations epochs have not been completed. This allows the model to avoid memorizing the training data excessively and improves its ability to generalize to new, unseen data.
Cross Section
X
Z
Braidal Tubes
8g
P.
Main Hackbone
l1
Figure 8: Schematic of the robot.
Table 4: Hyperparameters and network structure.
Hyperparameter
value
No. hidden neuron (0)
112 (64.32,16)
Augmented vector size (p)
64
No. of hidden layers
3
Activation functions
BLU
Learning rate
0.001
Type of ode-solver
fixed-adams
Absolute tolerance for ode-solver le-9
Relative tolerance for ode-solver
le-7
Number of iteration
9000
3 Controller Configuration
This section will provide the details of the controller configurations including its hyperparameters, running cost, and terminal cost functions. The dynamics of the controlled system is captured by the trained FK model (ANODE), while the running cost and terminal state cost are defined follows:

---

Running cost: our running cost function is composed of three costs and defined as follows:
cost_tracking = Wuacking
||x Xyeferencelf 1|
cost_obstacles = Wobstacle ((d < 0.01) + (d2 < 0.01))
cost_jerk Wjerk ||u lu- uprevious ||
cost_affordance = Waffordance affordance_measun
running_com cost cost_tracking + cost_obstacle + cost_jerk + cost_affordance
where x represents the current state of the system, Xieierence the corresponding state in the reference trajectory, denotes the current control input, and Uprevious represents the previous control input. The weights Wuracking Wobstacle: and Wjerk control the importance of each term in the overall cost function. Waffordance determines suitable metric measure that quantifies the affordance for the given task goal. The first term penalizes the deviation of the reference trajectory. These deviations are weighted by factor of 200, encouraging the system to closely follow the desired trajectory. The second term is penalty term that considers the distance between the current states and two obstacle locations, denoted as dj and d2. the distance to either obstacle less than 0.01, high penalty of 100,000 is added. This incentivizes the system to avoid approaching the obstacles too closely. To discourage jerky and abrupt movements, we considered another penalty term. This term penalizes high rates of change in acceleration or control inputs. In implementation, Wjerk is set 0.1.
Terminal cost: our terminal cost is defined as: terminal _ostttcottttoeeet Wterminal ||x Xgoal !|? where Wierminal is the weighting factor that controls the importance of the terminal cost.
The A parameter was set to 1 to balance the importance between the running cost and terminal state cost. The control inputs were constrained within the range defined by umin [-0.01,-0.01.0.001]] and umax [0.01,0.01,0.00 0..0,,0.01]. Gaussian noise with a standard deviation of Σn = 0.001 torch. eye(3) was added to control samples for exploration. The MPPI optimization process involved generating 500 control samples per iteration, with prediction horizon of 10 time steps. These parameter values were chosen to achieve effective control performance and can be fine-tuned for specific application requirements.
4 Affordance
In the context of robotics, an affordance a relationship between an actor (i.e., robot), an action performed by the actor, an object on which this action performed, and the observed effect [25]. The general idea of the affordance theory can be used in robotics provide some information of mapping between objects, agents and the actions they can take each other, there is no unified formalization of it in robotics. In our implementation, we incorporate a set of affordance terms (penalties for violating the motion restrictions) into the running cost of the controllers which can be selectively activated or deactivated by the operator, depending on the task phase. Thanks to the versatility of MPPI, which can handle non-convex running costs, allows us to effectively utilize these affordance terms for a more intuitive and context-aware interaction between the operator, the robot, and the environment, enabling more effective and efficient teleoperation. By adding the affordance measure the running cost, give more weight actions that align with the desired affordance.
5 Modeling the Entire Body of the Robot
One can model the entire body of a soft robot as continuous 3D curve. To this end, the configuration
of the main backbone can be defined using a unique set of 3D centroids, r(s.t) r) [0, E] x [0, co] -
R² x I0, oo], and family orthogonal transformations, R(s,t): [0, f] x [0, oo] - so(3) x [0, oo]
where l denotes length of the robot. The shape of the main backbone defined by
r' (s.t) = R(s,t)e3, R' (s. = R(s, (u(r)
(11)
12

---

Figure 9: Simulation setup: the simulated robot is tracking a helical trajectory.
(1)) = [O,(0, My (1).0² 0j² is the curvature vector of the deformed backbone. [-]x operator is the isomorphism between vector in R and its skew-symmetric cross product matrix, and e3 [0, 0, 1)" is the unit vector aligned with the z-axis of the global coordinate frame. This can be formulated as an ANODE problem and we can calculate the robot end-effector's position as the Cartesian coordinates of the robot's tip:
x(r) = r)((())) = (z(1)
r(s, t)ds.
(12)
The current system has 14 states including a 3D position, rotation matrix, and two inputs. To validate the answer, we need dataset to train new ANODE, and test its performance. To this end, we developed a simulated version of our robot the Pybullet simulator and we assumed that the robot can be modeled using Cosserat-rod theory [28]. The code is available online and can be downloaded from here 1, Figure shows set snapshots of simulated robot while performing task. This simulation enabled the dynamic generation of shape configuration data batches, where we produced 100 random shape configurations by sampling ux My from uniform distributions U(- U(-0.01, 0.01) and setting 4z using torch. linspace (O. 0.05, 100). For testing, we expanded the imput boundaries to incorporate unseen configurations, adjusting Hx. y to sample from u(-0.015 0.015) and 4z to span torch. linspace (0., 0.07, 480). For performance assessment, we executed 50 tests, predicting a span cm, divided into 480 steps. The resulting average Mean Squared Error (MSE) was 0.00001 494 which proves the good performance of ANODE approximate the shape of soft robot. Two representative results are shown in Figure 10.
trajectory-trackin
prediction x
position (m)
0.01
0.01
prediction true
-0.02
0.00
position postiin (m)
position (m)
0.00
-0.005
prediction y true
prediction y
0.010
00.02
predictio true
[ml
prediction
true Z
0.025
0.000
0.000
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
length (m)
length (m)
Figure 10: Two representative results of shape reconstruction.
6 Neural ODE vs Augmented Neural ODE
Neural ODEs and ANODEs belong to the same family of models. They both use differential equations to model the change in system over time. The difference between these two lies in how they handle the evolution of the system's 's state. A Neural ODE allows the state to evolve in the original state space, while an ANODE introduces auxiliary dimensions, allowing the state to evolve this augmented space. To evaluate their performances in our case, we began by evaluating the NODE's
thttps://github...
13

---

capability. We observed slight decrease in performance, particularly in open-loop scenarios with previously unseen conditions. Subsequently, to underscore ANODE's potential comparison with its counterparts (i.e. MLP, RNN, ResNet), we ventured into a more intricate task - modeling the robol's entire body which has been discussed the previous section. After generating dataset and training the models, for performance assessment, we executed 50 tests, predicting span of 7 em, divided into 480 steps. These predictions were based control inputs drawn randomly from uniform distribution u(-0.015, 0.015) em, which also covered unseen configurations. The resulting average Mean Squared Error (MSE) for ANODE stood at 0.0000) 494, distinctly lower than NODE's 0,00172892. A visual comparison of two exemplary outcomes is depicted in Figure 11. Evidently, ANODE outshone in this setup, reaffirming its superior stability and generalization capabilities over NODE.
prediction
postion (m)
position
true
true
postion (m)
0.000
postioo (m)
0.04
prediction
-0.005
prodiction
-0.000
true y
position (m)
prediction
position (m]
prediction
0.050
0.025
0.000
0.00
0.00
0.03 0.04
0.05
0.00
0.02
0.04
0.06
langth (m)
langth (m)
ANODE
NODE
Figure 11: Two representative results of shape reconstruction using ANODE and NODE.
14


Reviews:
[{"review_id": "beA9tQEnaR", "reviewer": "Reviewer_p7k6", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper is a classic application of model-based RL to a soft robotic system. For dynamics learning, the paper uses ANODE (the \"dynamics\" is really kinematic displacement of the end effector subject to control inputs). For controller, it uses MPPI. The primary conclusion of the paper is the surprising data efficiency of ANODE based dynamics learning relative to MLPs and RNNs.", "summary_of_recommendation": "Nice application of ANODEs to soft robot kinematic modeling, and trajectory optimization. The ML parts of the paper are not particularly novel, but the results may be of interest to the soft robot community. ", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "Please see questions raised above in the review.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Strengths:\n\n- The paper is overall clearly presented and well written.\n- Learning-based control of soft robots is a nascent area, and this paper presents a reasonable formulation and empirical results.\n- The method is comprehensively benchmarked\n\nWeaknesses:\n\n- Not much insight is provided into why \"ANODE\" is special for this problem class. What about vanilla ResNets which may be viewed as Euler discretization of Neural ODEs?\n- The 25-sample data efficiency is great, but 9K samples were collected - did the performance saturate quickly? That would indicate that the modeling problem is too easy.\n- Some baselines for dynamics learning might be missing, e.g., many simulators support deformables - can one use any out of the box?\n\nOther points:\n\n- Figure 1 is way too squished and I never quite understood what the robot looks like and does. Please make it much larger and clearer.\n- The dynamics setup is restrictive as the state is simply the end-effector position. What if one wishes to model the entire body of the robot?\n- What are physically the control inputs?\n- How exactly is the ANODE parameterized? \n- Does choice of integrator  (Euler/RK with different stepsizes) make a difference?\n- Below Eqn 7, why random selection - with 25 samples, one can do full-batch gradient descent.\n- What is capital X-tilde in Table 1? Unless I missed it, I do not see a definition anywhere.\n- It is possible that the conclusions of this paper are heavily tied to the robot embodiment with a small state and control space. It is not clear if ANODE will remain preferrerable for high dimensions systems. \n", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}, {"review_id": "CngoTC2a4x1", "reviewer": "Reviewer_pRah", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "In this paper, the authors present a data-efficient modeling and control approach for soft manipulators. The main idea is to use augmented Neural ODE (ANODE) model to learn the differential kinematics of the robot. Then, some type of model predicted controller is used to find the optimal input sequence for the actual robot based on the ANODE model. The proposed approach is tested on a soft manipulator and shows superior result in contrast to a plain RNN as feedforward model.", "summary_of_recommendation": "The originally of the work is limited as there is no new methodology presented but just another ML model in a standard pipeline.\n\nAfter rebuttal: Change to \"weak accept\"", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Good: Minor revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Additional details required", "questions_for_rebuttal": "- What is the performance of the controller with a Neural ODE instead of ANODE as kinematic model?\n- Is there any new methodology?\n- Could you please discuss what kind of function f might be?\n- Why did you use MPPI?\n- Line 94: In which way is this data set \"efficient\"?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "quality: The quality of the paper could be improved be more details and comparison to other approaches.\n\nclarity: The paper is mostly well written and the presented method is clear\n\noriginality: The main idea is to use the ANODE approach as model for the differential kinematics. I don’t think that this approach is original as it just use another type of ML model to learn the kinematics. \n\nsignificance: As the general pipeline is not new but just with other ML models, the significants of this work might be limited\n\nStrengths:\n- The presented approach outperforms the other tested approaches\n- The idea is quite simple but effective\n\nWeaknesses:\n\n- A non-learning baseline is missing. For instance, a (simplified) first-principle model could be used.\n- It’s not clear why a stochastic MPC approach is necessary\n- In general, the paper does not introduce any new methodology but an approach where just another ML model is used.\n- Since the model shows good performance for only 25 points, the underlying function must be quite simple. However, there are no insights or discussions on this fact.", "extra_scores": {"impact": "2: Unsure, difficult to assess."}, "extra_text": {}}, {"review_id": "2l7nZ31oNzE", "reviewer": "Reviewer_e7Lr", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper proposes a model for efficiently learning a differentiable model in a data-driven manner for controlling soft robots. Specifically, the authors use ANODE to learn a differential model based on a neural network. To generate control command using the model, MPPI is employed. The proposed method demonstrates high control performance in verification experiments using soft robots and shows potential applicability to practical complex tasks.", "summary_of_recommendation": "I believe that if the significance of using NODE in the field of soft robotics is clearly explained, it would be a valuable paper worth accepting.", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Good: Minor revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "In my opinion, the approach of training soft robot models using NODE is beneficial. However, the significance of using NODE for this domain is not clearly explained, and this needs to be clarified.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "# Strengths\n- Differential models have traditionally been used for complex soft robot models, and NODE, which learns differential models in a data-driven manner, seems to have good compatibility.\n- The proposed method is applied not only to simple tracking problems but also to practical tasks such as box and test tube manipulation, demonstrating its practicality.\n- The comparison between the proposed NODE-based method and other methods like FNN or RNN confirms superior control performance.\n\n# Weaknesses\n- In this paper, the authors use Augmented Neural Ordinary Differential Equations (ANODE) for modeling soft robots. However, the reason for using ANODE as a model is not clearly explained. How does modeling soft robots with ANODE address the issues of traditional research?\n- NODE introduces many computational expressions that are different from general neural networks due to the handling of differential equations. Can these expressions be effectively utilized in controlling soft robots? If they can be used effectively, clarifying how they are utilized would enhance the value of the paper.\n  \nFurthermore, how does the augmentation part of ANODE effectively work in this domain? Additional experiments or supplementary discussions would be beneficial.", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}]

Author rebuttal:

Meta-review: Thank you to all the reviewers for their careful evaluation and insightful comments. The paper presents a method for applying Augmented Neural Ordinary Differential Equations (ANODE) to soft robot kinematic modeling and trajectory optimization.

All three reviewers agree that the paper is well-written and presented. The paper provides comprehensive benchmarking against other approaches, and its results suggest superior control performance, especially when compared to FNN or RNN-based methods. And that the application of the proposed method to real-world tasks such as box and test tube manipulation demonstrates its practical relevance.

Still, some reviewers noted that while the paper might be technically sound, it does not introduce significant methodological advancements. The primary novelty seems to lie in the application domain rather than the method itself.

Multiple reviewers also suggested that the paper could benefit from including more baselines, especially non-learning ones or those from simulators that support deformables. Although, during the rebuttal, the authors made some arguments, a more concrete quantitative comparison with physics-based simulators will strengthen the paper and increase its relevance.

Considering the aforementioned strengths and concerns, the consensus leans towards a Weak Accept. The application domain, presentation, and empirical results are of value to the community. However, for this work to make a more significant contribution to the community, it would be beneficial for the authors to provide a clearer justification for their methodological choices and possibly explore more diverse baselines.
Decision: Accept (Poster)