Simulation plays a crucial role in various robotics applications, such as policy learning [1, 2, 3, 4, 5, 6, 7], safe and scalable robotic control evaluation [8, 9, 10, 11], and computational optimization of robot designs [12, 13, 14]. Recently, neural robotics simulators have emerged as a promising alternative to traditional analytical simulators, as neural simulators can efficiently predict robot dynamics and learn intricate physics from real-world data. For instance, neural simulators have been leveraged to capture complex interactions challenging for analytical modeling [15, 16, 17, 18], or have served as learned world models to facilitate sample-efficient policy learning [19, 20]. However, existing neural robotics simulators typically require application-specific training, often assuming fixed environments [20, 21] or simultaneous training alongside control policies [22, 23]. These limitations primarily stem from their end-to-end frameworks with inadequate representations of the global simulation state, i.e., neural models often substitute the entire classical simulator and directly map robot state and control actions ( e.g., target joint positions, target link orientations) to the robot's next state. Without encoding the environment in the state representation, the learned simulators have to implicitly memorize the task and environment details. Additionally, utilizing controller actions as input causes the simulators to overfit to particular low-level controllers used during training. Consequently, unlike classical simulators, these neural simulators often fail to generalize to novel state distributions (induced by new tasks), unseen environment setups, and customized controllers ( e.g., novel control laws or controller gains).

-- This paper presents VIMPPI, a novel control approach for underactuated double pendulum systems developed for the AI Olympics competition. We enhance the Model Predictive Path Integral framework by incorporating variational integration techniques, enabling longer planning horizons without additional computational cost. Operating at 500-700 Hz with control interpolation and disturbance detection mechanisms, VIMPPI substantially outperforms both baseline methods and alternative MPPI implementations. The AI Olympics with RealAIGym [1] competition challenges participants to develop controllers for complex robotic systems. This year's task focused on designing controllers for underactuated double pendulum systems -- the pendubot and acrobot -- with emphasis on maintaining the upper equilibrium position from various initial states [2].

A chaotic system is a highly volatile system characterized by its sensitive dependence on initial conditions and outside factors. Chaotic systems are prevalent throughout the world today: in weather patterns, disease outbreaks, and even financial markets. Chaotic systems are seen in every field of science and humanities, so being able to predict these systems is greatly beneficial to society. In this study, we evaluate 10 different machine learning models and neural networks [1] based on Root Mean Squared Error (RMSE) and R^2 values for their ability to predict one of these systems, the multi-pendulum. We begin by generating synthetic data representing the angles of the pendulum over time using the Runge Kutta Method for solving 4th Order Differential Equations (ODE-RK4) [2]. At first, we used the single-step sliding window approach, predicting the 50st step after training for steps 0-49 and so forth. However, to more accurately cover chaotic motion and behavior in these systems, we transitioned to a time-step based approach. Here, we trained the model/network on many initial angles and tested it on a completely new set of initial angles, or 'in-between' to capture chaotic motion to its fullest extent. We also evaluated the stability of the system using Lyapunov exponents. We concluded that for a double pendulum, the best model was the Long Short Term Memory Network (LSTM)[3] for the sliding window and time step approaches in both friction and frictionless scenarios. For triple pendulum, the Vanilla Recurrent Neural Network (VRNN)[4] was the best for the sliding window and Gated Recurrent Network (GRU) [5] was the best for the time step approach, but for friction, LSTM was the best.

Digital twins require computationally-efficient reduced-order models (ROMs) that can accurately describe complex dynamics of physical assets. However, constructing ROMs from noisy high-dimensional data is challenging. In this work, we propose a data-driven, non-intrusive method that utilizes stochastic variational deep kernel learning (SVDKL) to discover low-dimensional latent spaces from data and a recurrent version of SVDKL for representing and predicting the evolution of latent dynamics. The proposed method is demonstrated with two challenging examples -- a double pendulum and a reaction-diffusion system. Results show that our framework is capable of (i) denoising and reconstructing measurements, (ii) learning compact representations of system states, (iii) predicting system evolution in low-dimensional latent spaces, and (iv) quantifying modeling uncertainties.

We identify the nonlinear normal modes spawning from the stable equilibrium of a double pendulum under gravity, and we establish their connection to homoclinic orbits through the unstable upright position as energy increases. This result is exploited to devise an efficient swing-up strategy for a double pendulum with weak, saturating actuators. Our approach involves stabilizing the system onto periodic orbits associated with the nonlinear modes while gradually injecting energy. Since these modes are autonomous system evolutions, the required control effort for stabilization is minimal. Even with actuator limitations of less than 1% of the maximum gravitational torque, the proposed method accomplishes the swing-up of the double pendulum by allowing sufficient time.

Incorporating neural networks for the solution of Ordinary Differential Equations (ODEs) represents a pivotal research direction within computational mathematics. Within neural network architectures, the integration of the intrinsic structure of ODEs offers advantages such as enhanced predictive capabilities and reduced data utilization. Among these structural ODE forms, the Lagrangian representation stands out due to its significant physical underpinnings. Building upon this framework, Bhattoo introduced the concept of Lagrangian Neural Networks (LNNs). Then in this article, we introduce a groundbreaking extension (Genralized Lagrangian Neural Networks) to Lagrangian Neural Networks (LNNs), innovatively tailoring them for non-conservative systems. By leveraging the foundational importance of the Lagrangian within Lagrange's equations, we formulate the model based on the generalized Lagrange's equation. This modification not only enhances prediction accuracy but also guarantees Lagrangian representation in non-conservative systems. Furthermore, we perform various experiments, encompassing 1-dimensional and 2-dimensional examples, along with an examination of the impact of network parameters, which proved the superiority of Generalized Lagrangian Neural Networks(GLNNs).

We present a solution of the swing-up and balance task for the pendubot and acrobot for the participation in the AI Olympics competition at IJCAI 2023. Our solution is based on the Soft Actor Crtic (SAC) reinforcement learning (RL) algorithm for training a policy for the swing-up and entering the region of attraction of a linear quadratic regulator(LQR) controller for stabilizing the double pendulum at the top position. Our controller achieves competitive scores in performance and robustness for both, pendubot and acrobot, problem scenarios.

Traditionally, robots are regarded as universal motion generation machines. They are designed mainly by kinematics considerations while the desired dynamics is imposed by strong actuators and high-rate control loops. As an alternative, one can first consider the robot's intrinsic dynamics and optimize it in accordance with the desired tasks. Therefore, one needs to better understand intrinsic, uncontrolled dynamics of robotic systems. In this paper we focus on periodic orbits, as fundamental dynamic properties with many practical applications. Algebraic topology and differential geometry provide some fundamental statements about existence of periodic orbits. As an example, we present periodic orbits of the simplest multi-body system: the double-pendulum in gravity. This simple system already displays a rich variety of periodic orbits. We classify these into three classes: toroidal orbits, disk orbits and nonlinear normal modes. Some of these we found by geometrical insights and some by numerical simulation and sampling.

Imagine a world where machines can generate code to solve complex problems in the physical world around us. ChatGPT, a type of Natural Language Processor (NLP) which writes human-like responses from user input prompts can do just that. In this article, I am going to show you how. Right now, anyone can use the research release of ChatGPT -- you just need to head over to OpenAIs website and sign up for an account to try it. A lot is going on under the hood of ChatGPT and I am not going to attempt to explain it here (OpenAI gives a detailed overview of how the technology works on its website).

Numerical integrators could be used to form interpolation conditions when training neural networks to approximate the vector field of an ordinary differential equation (ODE) from data. When numerical one-step schemes such as the Runge-Kutta methods are used to approximate the temporal discretization of an ODE with a known vector field, properties such as symmetry and stability are much studied. Here, we show that using mono-implicit Runge-Kutta methods of high order allows for accurate training of Hamiltonian neural networks on small datasets. This is demonstrated by numerical experiments where the Hamiltonian of the chaotic double pendulum in addition to the Fermi-Pasta-Ulam-Tsingou system is learned from data.