Optimization of deep neural networks (DNNs) has been driving modern advancements in artificial intelligence. With DNNs characterized by a prolonged sequence of nonlinear propagation, determining their optimal parameters given an objective naturally fits within Optimal Control Programming. Such an interpretation of DNNs as dynamical systems has proven crucial in offering principled analysis from numerical equations to physics. In parallel to these theoretical pursuits, this paper focuses on an algorithmic perspective. Our motivated observation is the striking algorithmic resemblance between the Backpropagation algorithm for computing gradients in DNNs and the optimality conditions for dynamical systems, expressed through another backward process known as dynamic programming. Consolidating this connection, where Backpropagation admits a variational structure, solving an approximate dynamic programming up to the first-order expansion, leads to a new class of optimization methods exploring higher-order expansions of the Bellman equation. The resulting optimizer, Optimal Control Theoretic Neural Optimizer (OCNOpt), enables rich algorithmic opportunities, including layer-wise feedback policies, game-theoretic applications, and higher-order training of continuous-time models such as Neural ODEs. Extensive experiments demonstrate that OCNOpt improves upon existing methods in robustness and efficiency while maintaining manageable computational complexity, paving new avenues for principled algorithmic design grounded in dynamical systems and optimal control theory.

Quadratic programming (QP) forms a crucial foundation in optimization, appearing in a broad spectrum of domains and serving as the basis for more advanced algorithms. Consequently, as the scale and complexity of modern applications continue to grow, the development of efficient and reliable QP algorithms becomes increasingly vital. In this context, this paper introduces a novel deep learning-aided distributed optimization architecture designed for tackling large-scale QP problems. First, we combine the state-of-the-art Operator Splitting QP (OSQP) method with a consensus approach to derive **DistributedQP**, a new method tailored for network-structured problems, with convergence guarantees to optimality. Subsequently, we unfold this optimizer into a deep learning framework, leading to **DeepDistributedQP**, which leverages learned policies to accelerate reaching to desired accuracy within a restricted amount of iterations. Our approach is also theoretically grounded through Probably Approximately Correct (PAC)-Bayes theory, providing generalization bounds on the expected optimality gap for unseen problems. The proposed framework, as well as its centralized version **DeepQP**, significantly outperform their standard optimization counterparts on a variety of tasks such as randomly generated problems, optimal control, linear regression, transportation networks and others. Notably, DeepDistributedQP demonstrates strong generalization by training on small problems and scaling to solve much larger ones (up to 50K variables and 150K constraints) using the same policy. Moreover, it achieves orders-of-magnitude improvements in wall-clock time compared to OSQP. The certifiable performance guarantees of our approach are also demonstrated, ensuring higher-quality solutions over traditional optimizers.

Recent advancements in diffusion bridges for distribution transport problems have heavily relied on matching frameworks, yet existing methods often face a trade-off between scalability and access to optimal pairings during training. Fully unsupervised methods make minimal assumptions but incur high computational costs, limiting their practicality. On the other hand, imposing full supervision of the matching process with optimal pairings improves scalability, however, it can be infeasible in most applications. To strike a balance between scalability and minimal supervision, we introduce Feedback Schrödinger Bridge Matching (FSBM), a novel semi-supervised matching framework that incorporates a small portion ($<8$% of the entire dataset) of pre-aligned pairs as state feedback to guide the transport map of non-coupled samples, thereby significantly improving efficiency. This is achieved by formulating a static Entropic Optimal Transport (EOT) problem with an additional term capturing the semi-supervised guidance. The generalized EOT objective is then recast into a dynamic formulation to leverage the scalability of matching frameworks. Extensive experiments demonstrate that FSBM accelerates training and enhances generalization by leveraging coupled pairs' guidance, opening new avenues for training matching frameworks with partially aligned datasets.

The generative modeling of data on manifolds is an important task, for which diffusion models in flat spaces typically need nontrivial adaptations. This article demonstrates how a technique called `trivialization' can transfer the effectiveness of diffusion models in Euclidean spaces to Lie groups. In particular, an auxiliary momentum variable was algorithmically introduced to help transport the position variable between data distribution and a fixed, easy-to-sample distribution. Normally, this would incur further difficulty for manifold data because momentum lives in a space that changes with the position. However, our trivialization technique creates a new momentum variable that stays in a simple fixed vector space. This design, together with a manifold preserving integrator, simplifies implementation and avoids inaccuracies created by approximations such as projections to tangent space and manifold, which were typically used in prior work, hence facilitating generation with high-fidelity and efficiency. The resulting method achieves state-of-the-art performance on protein and RNA torsion angle generation and sophisticated torus datasets. We also, arguably for the first time, tackle the generation of data on high-dimensional Special Orthogonal and Unitary groups, the latter essential for quantum problems. Code is available at https://github.com/yuchen-zhu-zyc/TDM.

We present a novel second-order trajectory optimization algorithm based on Stein Variational Newton's Method and Maximum Entropy Differential Dynamic Programming. The proposed algorithm, called Stein Variational Differential Dynamic Programming, is a kernel-based extension of Maximum Entropy Differential Dynamic Programming that combines the best of the two worlds of sampling-based and gradient-based optimization. The resulting algorithm avoids known drawbacks of gradient-based dynamic optimization in terms of getting stuck at local minima, while it overcomes limitations of sampling-based stochastic optimization in terms of introducing undesirable stochasticity when applied in online fashion. To test the efficacy of the proposed algorithm, experiments are conducted in Model Predictive Control mode. The experiments include comparisons with unimodal and multimodal Maximum Entropy Differential Dynamic Programming as well as Model Predictive Path Integral Control and its multimodal and Stein Variational extensions. The results demonstrate the superior performance of the proposed algorithms and confirm the hypothesis that there is a middle ground between sampling-and gradient-based optimization that is indeed beneficial for dynamic optimization.

Rapid autonomous traversal of unstructured terrain is essential for scenarios such as disaster response, search and rescue, and planetary exploration. As a vehicle navigates at the limit of its capabilities over extreme terrain, its dynamics can change suddenly and dramatically. For example, varying terrain can affect parameters such as traction, tire slip, and rolling resistance. To achieve effective planning in such environments, it is crucial to have a dynamics model that can accurately anticipate these conditions and respond before an issue can occur. In this work, we present a hybrid model that predicts the changing dynamics induced by the terrain as a function of visual inputs. We leverage a pre-trained visual foundation model (VFM) DINOv2, which provides rich features that encodes fine-grained semantic information. To use this dynamics model for planning, we propose an end-to-end training architecture for a projection distance independent feature encoder that compresses the information from the VFM, enabling the creation of a lightweight map of the environment at runtime. We validate our architecture on an extensive dataset (hundreds of kilometers of aggressive off-road driving) collected across multiple locations as part of the DARPA Robotic Autonomy in Complex Environments with Resiliency (RACER) program. https://youtu.be/aydHxLGmnx8

Understanding complex systems by inferring trajectories from sparse sample snapshots is a fundamental challenge in a wide range of domains, e.g., single-cell biology, meteorology, and economics. Despite advancements in Bridge and Flow matching frameworks, current methodologies rely on pairwise interpolation between adjacent snapshots. This hinders their ability to capture long-range temporal dependencies and potentially affects the coherence of the inferred trajectories. To address these issues, we introduce Momentum Multi-Marginal Schrödinger Bridge Matching (3MSBM), a novel matching framework that learns smooth measure-valued splines for stochastic systems that satisfy multiple positional constraints. This is achieved by lifting the dynamics to phase space and generalizing stochastic bridges to be conditioned on several points, forming a multi-marginal conditional stochastic optimal control problem. The underlying dynamics are then learned by minimizing a variational objective, having fixed the path induced by the multi-marginal conditional bridge. As a matching approach, 3MSBM learns transport maps that preserve intermediate marginals throughout training, significantly improving convergence and scalability. Extensive experimentation in a series of real-world applications validates the superior performance of 3MSBM compared to existing methods in capturing complex dynamics with temporal dependencies, opening new avenues for training matching frameworks in multi-marginal settings.

Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in \textbf{phase space dynamics}, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

Modern distribution matching algorithms for training diffusion or flow models directly prescribe the time evolution of the marginal distributions between two boundary distributions. In this work, we consider a generalized distribution matching setup, where these marginals are only implicitly described as a solution to some task-specific objective function. The problem setup, known as the Generalized Schrödinger Bridge (GSB), appears prevalently in many scientific areas both within and without machine learning. We propose Generalized Schödinger Bridge Matching (GSBM), a new matching algorithm inspired by recent advances, generalizing them beyond kinetic energy minimization and to account for nonlinear state costs. We show that such a generalization can be cast as solving conditional stochastic optimal control, for which efficient variational approximations can be used, and further debiased with the aid of path integral theory. Compared to prior methods for solving GSB problems, our GSBM algorithm always preserves a feasible transport map between the boundary distributions throughout training, thereby enabling stable convergence and significantly improved scalability. We empirically validate our claims on an extensive suite of experimental setups, including crowd navigation, opinion depolarization, LiDAR manifolds, and image domain transfer. Our work brings new algorithmic opportunities for training diffusion models enhanced with task-specific optimality structures.

Neural networks and neural ODEs tend to be vulnerable to adversarial attacks, rendering robust optimizers critical to curb the success of such attacks. In this regard, the key insight of this work is to interpret Neural ODE optimization as a min-max optimal control problem. More particularly, we present Game Theoretic Second-Order Neural Optimizer (GTSONO), a robust game theoretic optimizer based on the principles of min-max Differential Dynamic Programming. The proposed method exhibits significant computational benefits due to efficient matrix decompositions and provides convergence guarantees to local saddle points. Empirically, the robustness of the proposed optimizer is demonstrated through greater robust accuracy compared to benchmark optimizers when trained on clean images. Additionally, its ability to provide a performance increase when adapted to an already existing adversarial defense technique is also illustrated. Finally, the superiority of the proposed update law over its gradient based counterpart highlights the potential benefits of incorporating robust optimal control paradigms into adversarial training methods.

The control of granular materials, which are found in many industrial applications, is a challenging open research problem. Granular material systems are complex-behavior (as they could have solid-, fluid-, and gas-like behaviors) and high-dimensional (as they could have many grains/particles with at least 3 DOF in 3D) systems. Recently, a machine learning-based Graph Neural Network (GNN) simulator has been proposed to learn the underlying dynamics. In this paper, we perform optimal control of a rigid body-driven granular material system whose dynamics is learned by a GNN model trained by reduced data generated via a physics-based simulator and Principal Component Analysis (PCA). We use Differential Dynamic Programming (DDP) to obtain optimal control commands that can form granular particles into a target shape. The model and results are shown to be relatively fast and accurate. The control commands are also applied to the ground truth model, i.e., physics-based simulator, to further validate the approach.

This paper proposes Distributed Model Predictive Covariance Steering (DiMPCS) for multi-agent control under stochastic uncertainty. The scope of our approach is to blend covariance steering theory, distributed optimization and model predictive control (MPC) into a single framework that is safe, scalable and decentralized. Initially, we pose a problem formulation that uses the Wasserstein distance to steer the state distributions of a multi-agent system to desired targets, and probabilistic constraints to ensure safety. We then transform this problem into a finite-dimensional optimization one by utilizing a disturbance feedback policy parametrization for covariance steering and a tractable approximation of the safety constraints. To solve the latter problem, we derive a decentralized consensus-based algorithm using the Alternating Direction Method of Multipliers. This method is then extended to a receding horizon form, which yields the proposed DiMPCS algorithm. Simulation experiments on a variety of multi-robot tasks with up to hundreds of robots demonstrate the effectiveness of DiMPCS. The superior scalability and performance of the proposed method is also highlighted through a comparison against related stochastic MPC approaches. Finally, hardware results on a multi-robot platform also verify the applicability of DiMPCS on real systems.

This article studies social system inference from a single noisy trajectory of public evolving opinions, wherein observation noise leads to the statistical dependence of samples on time and coordinates. We first propose a cyber-social system that comprises individuals in a social network and a set of information sources in a cyber layer, whose opinion dynamics explicitly takes the asymmetric cognitive bias including confirmation bias and negativity bias and the process noise into account. Based on the proposed cyber-social model, we then study the sample complexity of least-square auto-regressive model estimation, which governs the length of a single observed trajectory that is sufficient for the identified model to achieve the prescribed levels of accuracy and confidence (PAC). Building on the identified social model, we then investigate social inference, with a particular focus on the weighted network topology and the model parameters of asymmetric cognitive bias. Finally, the theoretical results and the effectiveness of the proposed inference framework are validated by the U.S. Senate Member Ideology data.

We propose Image-to-Image Schrödinger Bridge (I$^2$SB), a new class of conditional diffusion models that directly learn the nonlinear diffusion processes between two given distributions. These diffusion bridges are particularly useful for image restoration, as the degraded images are structurally informative priors for reconstructing the clean images. I$^2$SB belongs to a tractable class of Schrödinger bridge, the nonlinear extension to score-based models, whose marginal distributions can be computed analytically given boundary pairs. This results in a simulation-free framework for nonlinear diffusions, where the I$^2$SB training becomes scalable by adopting practical techniques used in standard diffusion models. We validate I$^2$SB in solving various image restoration tasks, including inpainting, super-resolution, deblurring, and JPEG restoration on ImageNet 256$\times$256 and show that I$^2$SB surpasses standard conditional diffusion models with more interpretable generative processes. Moreover, I$^2$SB matches the performance of inverse methods that additionally require the knowledge of the corruption operators. Our work opens up new algorithmic opportunities for developing efficient nonlinear diffusion models on a large scale. Project page and codes: https://i2sb.github.io/

Modeling dynamics is often the first step to making a vehicle autonomous. While on-road autonomous vehicles have been extensively studied, off-road vehicles pose many challenging modeling problems. An off-road vehicle encounters highly complex and difficult-to-model terrain/vehicle interactions, as well as having complex vehicle dynamics of its own. These complexities can create challenges for effective high-speed control and planning. In this paper, we introduce a framework for multistep dynamics prediction that explicitly handles the accumulation of modeling error and remains scalable for sampling-based controllers. Our method uses a specially-initialized Long Short-Term Memory (LSTM) over a limited time horizon as the learned component in a hybrid model to predict the dynamics of a 4-person seating all-terrain vehicle (Polaris S4 1000 RZR) in two distinct environments. By only having the LSTM predict over a fixed time horizon, we negate the need for long term stability that is often a challenge when training recurrent neural networks. Our framework is flexible as it only requires odometry information for labels. Through extensive experimentation, we show that our method is able to predict millions of possible trajectories in real-time, with a time horizon of five seconds in challenging off road driving scenarios.

Game theoretic methods have become popular for planning and prediction in situations involving rich multi-agent interactions. However, these methods often assume the existence of a single local Nash equilibria and are hence unable to handle uncertainty in the intentions of different agents. While maximum entropy (MaxEnt) dynamic games try to address this issue, practical approaches solve for MaxEnt Nash equilibria using linear-quadratic approximations which are restricted to unimodal responses and unsuitable for scenarios with multiple local Nash equilibria. By reformulating the problem as a POMDP, we propose MPOGames, a method for efficiently solving MaxEnt dynamic games that captures the interactions between local Nash equilibria. We show the importance of uncertainty-aware game theoretic methods via a two-agent merge case study. Finally, we prove the real-time capabilities of our approach with hardware experiments on a 1/10th scale car platform.

It is a crucial challenge to reconstruct population dynamics using unlabeled samples from distributions at coarse time intervals. Recent approaches such as flow-based models or Schrödinger Bridge (SB) models have demonstrated appealing performance, yet the inferred sample trajectories either fail to account for the underlying stochasticity or are unnecessarily rigid. In this article, we extend SB into phase space and propose $\underline{D}$eep $\underline{M}$omentum Multi-Marginal $\underline{S}$chrödinger $\underline{B}$ridge (DMSB), a novel computational framework that learns the smooth measure-valued spline for stochastic systems that satisfy position marginal constraints across time. By tailoring the celebrated Bregman Iteration and extending the Iteration Proportional Fitting to phase space, we manage to handle high-dimensional multi-marginal trajectory inference tasks efficiently. Our algorithm outperforms baselines significantly, as evidenced by experiments for synthetic datasets and a real-world single-cell RNA sequence dataset. Additionally, the proposed approach can reasonably reconstruct the evolution of velocity distribution, from position snapshots only, when there is a ground truth velocity that is nevertheless inaccessible.

Modern successes of diffusion models in learning complex, high-dimensional data distributions are attributed, in part, to their capability to construct diffusion processes with analytic transition kernels and score functions. The tractability results in a simulation-free framework with stable regression losses, from which reversed, generative processes can be learned at scale. However, when data is confined to a constrained set as opposed to a standard Euclidean space, these desirable characteristics appear to be lost based on prior attempts. In this work, we propose Mirror Diffusion Models (MDM), a new class of diffusion models that generate data on convex constrained sets without losing any tractability. This is achieved by learning diffusion processes in a dual space constructed from a mirror map, which, crucially, is a standard Euclidean space. We derive efficient computation of mirror maps for popular constrained sets, such as simplices and $\ell_2$-balls, showing significantly improved performance of MDM over existing methods. For safety and privacy purposes, we also explore constrained sets as a new mechanism to embed invisible but quantitative information (i.e., watermarks) in generated data, for which MDM serves as a compelling approach. Our work brings new algorithmic opportunities for learning tractable diffusion on complex domains.

This article proposes two decentralized multiagent optimal control methods that combine the computational efficiency and scalability of differential dynamic programming (DDP) and the distributed nature of the alternating direction method of multipliers (ADMM). The first one, nested distributed DDP, is a three-level architecture, which employs ADMM for consensus, an augmented Lagrangian layer for local constraints and DDP as the local optimizer. The second one, merged distributed DDP, is a two-level architecture that addresses both consensus and local constraints with ADMM, further reducing computational complexity. Both frameworks arefully decentralizedsince all computations are parallelizable among the agents and only local communication is necessary. Simulation results that scale up to thousands of cars and hundreds of drones demonstrate the effectiveness of the algorithms. Superior scalability to large-scale systems against other DDP and sequential quadratic programming methods is also illustrated. Finally, hardware experiments on a multirobot platform verify the applicability of the methods. A video with all results is provided in the supplementary material.

In this paper, we present a novel maximum entropy formulation of the Differential Dynamic Programming algorithm and derive two variants using unimodal and multimodal value functions parameterizations. By combining the maximum entropy Bellman equations with a particular approximation of the cost function, we are able to obtain a new formulation of Differential Dynamic Programming which is able to escape from local minima via exploration with a multimodal policy. To demonstrate the efficacy of the proposed algorithm, we provide experimental results using four systems on tasks that are represented by cost functions with multiple local minima and compare them against vanilla Differential Dynamic Programming. Furthermore, we discuss connections with previous work on the linearly solvable stochastic control framework and its extensions in relation to compositionality. Link to Video.

We present an algorithm, based on the Differential Dynamic Programming framework, to handle trajectory optimization problems in which the horizon is determined online rather than fixed a priori. This algorithm exhibits exact one-step convergence for linear, quadratic, time-invariant problems and is fast enough for real-time nonlinear model-predictive control. We show derivations for the nonlinear algorithm in the discrete-time case, and apply this algorithm to a variety of nonlinear problems. Finally, we show the efficacy of the optimal-horizon model-predictive control scheme compared to a standard MPC controller, on an obstacle-avoidance problem with planar robots.

This paper presents a novel control approach for autonomous systems operating under uncertainty. We combine Model Predictive Path Integral (MPPI) control with Covariance Steering (CS) theory to obtain a robust controller for general nonlinear systems. The proposed Covariance-Controlled Model Predictive Path Integral (CC-MPPI) controller addresses the performance degradation observed in some MPPI implementations owing to unexpected disturbances and uncertainties. Namely, in cases where the environment changes too fast or the simulated dynamics during the MPPI rollouts do not capture the noise and uncertainty in the actual dynamics, the baseline MPPI implementation may lead to divergence. The proposed CC-MPPI controller avoids divergence by controlling the dispersion of the rollout trajectories at the end of the prediction horizon. Furthermore, the CC-MPPI has adjustable trajectory sampling distributions that can be changed according to the environment to achieve efficient sampling. Numerical examples using a ground vehicle navigating in challenging environments demonstrate the proposed approach.

In robot learning from demonstration (LfD), a visual representation of a cost function inferred from Inverse Reinforcement Learning (IRL) provides an intuitive tool for humans to quickly interpret the underlying objectives of the demonstration. The inferred cost function can be used by controllers, for example, Model Predictive Controllers (MPCs). In this work, we improve the recently developed IRL-MPC framework, by enhancing it in a risk-sensitive formulation to be more applicable for safety-critical applications like autonomous driving. Our risk-sensitive MPCs together with the distributional costmap demonstrate lower collision rates in the CARLA simulator for autonomous driving tasks compared to other learning-based baseline methods.

Mean-Field Game (MFG) serves as a crucial mathematical framework in modeling the collective behavior of individual agents interacting stochastically with a large population. In this work, we aim at solving a challenging class of MFGs in which the differentiability of these interacting preferences may not be available to the solver, and the population is urged to converge exactly to some desired distribution. These setups are, despite being well-motivated for practical purposes, complicated enough to paralyze most (deep) numerical solvers. Nevertheless, we show that Schrödinger Bridge — as an entropy-regularized optimal transport model — can be generalized to accepting mean-field structures, hence solving these MFGs. This is achieved via the application of Forward-Backward Stochastic Differential Equations theory, which, intriguingly, leads to a computational framework with a similar structure to Temporal Difference learning. As such, it opens up novel algorithmic connections to Deep Reinforcement Learning that we leverage to facilitate practical training. We show that our proposed objective function provides necessary and sufficient conditions to the mean-field problem. Our method, named Deep Generalized Schrödinger Bridge (DeepGSB), not only outperforms prior methods in solving classical population navigation MFGs, but is also capable of solving 1000-dimensional opinion depolarization, setting a new state-of-the-art numerical solver for high-dimensional MFGs. Our code will be made available at https://github.com/ghliu/DeepGSB.

In this paper we present a scalable deep learning framework for finding Markovian Nash Equilibria in multi-agent stochastic games using fictitious play. The motivation is inspired by theoretical analysis of Forward Backward Stochastic Differential Equations and their implementation in a deep learning setting, which is the source of our algorithm’s sample efficiency improvement. By taking advantage of the permutation-invariant property of agents in symmetric games, the scalability and performance is further enhanced significantly. We showcase superior performance of our framework over the state-of-the-art deep fictitious play algorithm on an inter-bank lending/borrowing problem in terms of multiple metrics. More importantly, our approach scales up to 3000 agents in simulation, a scale which, to the best of our knowledge, represents a new state-of-the-art. We also demonstrate the applicability of our framework in robotics on a belief space autonomous racing problem.

The connection between training deep neural networks (DNNs) and optimal control theory (OCT) has attracted considerable attention as a principled tool of algorithmic design. Despite few attempts being made, they have been limited to architectures where the layer propagation resembles a Markovian dynamical system. This casts doubts on their flexibility to modern networks that heavily rely on non-Markovian dependencies between layers (e.g. skip connections in residual networks). In this work, we propose a novel dynamic game perspective by viewing each layer as a player in a dynamic game characterized by the DNN itself. Through this lens, different classes of optimizers can be seen as matching different types of Nash equilibria, depending on the implicit information structure of each (p)layer. The resulting method, called Dynamic Game Theoretic Neural Optimizer (DGNOpt), not only generalizes OCT-inspired optimizers to richer network class; it also motivates a new training principle by solving a multi-player cooperative game. DGNOpt shows convergence improvements over existing methods on image classification datasets with residual and inception networks. Our work marries strengths from both OCT and game theory, paving ways to new algorithmic opportunities from robust optimal control and bandit-based optimization.

Differential Dynamic Programming (DDP) has become a well established method for unconstrained trajectory optimization. Despite its several applications in robotics and controls, however, a widely successful constrained version of the algorithm has yet to be developed. This paper builds upon penalty methods and active-set approaches towards designing a Dynamic Programming-based methodology for constrained optimal control. Regarding the former, our derivation employs a constrained version of Bellman’s principle of optimality, by introducing a set of auxiliary slack variables in the backward pass. In parallel, we show how Augmented Lagrangian methods can be naturally incorporated within DDP, by utilizing a particular set of penalty-Lagrangian functions that preserve second-order differentiability. We demonstrate experimentally that our extensions (individually and combinations thereof) enhance significantly the convergence properties of the algorithm, and outperform previous approaches on a large number of simulated scenarios.

In this work, we present a method for obtaining an implicit objective function for vision-based navigation. The proposed methodology relies on Imitation Learning, Model Predictive Control (MPC), and an interpretation technique used in Deep Neural Networks. We use Imitation Learning as a means to do Inverse Reinforcement Learning in order to create an approximate cost function generator for a visual navigation challenge. The resulting cost function, the costmap, is used in conjunction with MPC for real-time control and outperforms other state-of-the-art costmap generators in novel environments. The proposed process allows for simple training and robustness to out-of-sample data. We apply our method to the task of vision-based autonomous driving in multiple real and simulated environments and show its generalizability. Supplementary video: https://youtu.be/WyJfT5lc0aQ

We propose a novel second-order optimization framework for training the emerging deep continuous-time models, specifically the Neural Ordinary Differential Equations (Neural ODEs). Since their training already involves expensive gradient computation by solving a backward ODE, deriving efficient second-order methods becomes highly nontrivial. Nevertheless, inspired by the recent Optimal Control (OC) interpretation of training deep networks, we show that a specific continuous-time OC methodology, called Differential Programming, can be adopted to derive backward ODEs for higher-order derivatives at the same O(1) memory cost. We further explore a low-rank representation of the second-order derivatives and show that it leads to efficient preconditioned updates with the aid of Kronecker-based factorization. The resulting method – named SNOpt – converges much faster than first-order baselines in wall-clock time, and the improvement remains consistent across various applications, e.g. image classification, generative flow, and time-series prediction. Our framework also enables direct architecture optimization, such as the integration time of Neural ODEs, with second-order feedback policies, strengthening the OC perspective as a principled tool of analyzing optimization in deep learning. Our code is available at https://github.com/ghliu/snopt.

We propose a sampling-based trajectory optimization methodology for constrained problems. We extend recent works on stochastic search to deal with box control constraints, as well as nonlinear state constraints for discrete dynamical systems. Regarding the former, our strategy is to optimize over truncated parameterized distributions on control inputs. Furthermore, we show how non-smooth penalty functions can be incorporated into our framework to handle state constraints. Simulations on cartpole and quadcopter show that our approach outperforms previous methods on constrained sampling-based optimization, in terms of quality of solutions and convergence speed.

Deep learning has enjoyed much recent success, and applying state-of-the-art model learning methods to controls is an exciting prospect. However, there is a strong reluctance to use these methods on safety-critical systems, which have constraints on safety, stability, and real-time performance. We propose a framework which satisfies these constraints while allowing the use of deep neural networks for learning model uncertainties. Central to our method is the use of Bayesian model learning, which provides an avenue for maintaining appropriate degrees of caution in the face of the unknown. In the proposed approach, we develop an adaptive control framework leveraging the theory of stochastic CLFs (Control Lyapunov Functions) and stochastic CBFs (Control Barrier Functions) along with tractable Bayesian model learning via Gaussian Processes or Bayesian neural networks. Under reasonable assumptions, we guarantee stability and safety while adapting to unknown dynamics with probability 1. We demonstrate this architecture for high-speed terrestrial mobility targeting potential applications in safety-critical high-speed Mars rover missions.

This paper presents a multirotor control architecture, where Model Predictive Path Integral Control (MPPI) and ℒ1adaptive control are combined to achieve both fast model predictive trajectory planning and robust trajectory tracking. MPPI provides a framework to solve nonlinear MPC with complex cost functions in real-time. However, it often lacks robustness, especially when the simulated dynamics are different from the true dynamics. We show that the ℒ1adaptive controller robustifies the architecture, allowing the overall system to behave similar to the nominal system simulated with MPPI. The architecture is validated in a simulated multirotor racing environment.

Sample efficiency is critical in solving real-world reinforcement learning problems where agent-environment interactions can be costly. Imitation learning from expert advice has proved to be an effective strategy for reducing the number of interactions required to train a policy. Online imitation learning, which interleaves policy evaluation and policy optimization, is a particularly effective technique with provable performance guarantees. In this work, we seek to further accelerate the convergence rate of online imitation learning, thereby making it more sample efficient. We propose two model-based algorithms inspired by Follow-the-Leader (FTL) with prediction: MoBIL-VI based on solving variational inequalities and MoBIL-Prox based on stochastic first-order updates. These two methods leverage a model to predict future gradients to speed up policy learning. When the model oracle is learned online, these algorithms can provably accelerate the best known convergence rate up to an order. Our algorithms can be viewed as a generalization of stochastic Mirror-Prox (Juditsky et al., 2011), and admit a simple constructive FTL-style analysis of performance.

This work presents a novel ensemble of Bayesian Neural Networks for control of safety-critical systems. Decision making for safety-critical systems is challenging due to performance requirements with significant consequences in the event of failure. In practice, failure of such systems can be avoided by introducing redundancies of control. Neural Networks are generally not used for safety-critical systems as they can behave in unexpected ways in response to novel inputs. In addition, there may not be any indication as to when they will fail. BNNs have been recognized for their ability to produce not only viable outputs but also provide a measure of uncertainty in these outputs. This work combines the knowledge of prediction uncertainty obtained from BNNs and ensemble control for a redundant control methodology. Our technique is applied to an agile autonomous driving task. Multiple BNNs are trained to control a vehicle in an end-to-end fashion on different sensor inputs provided by the system. We show that an individual network is successful in maneuvering around the track but crashes in the presence of unforeseen input noise. Our proposed ensemble of BNNs shows successful task performance even in the event of multiple sensor failures.

We propose the use of Bayesian networks, which provide both a mean value and an uncertainty estimate as output, to enhance the safety of learned control policies under circumstances in which a test-time input differs significantly from the training set. Our algorithm combines reinforcement learning and end-to-end imitation learning to simultaneously learn a control policy as well as a threshold over the predictive uncertainty of the learned model, with no hand-tuning required. Corrective action, such as a return of control to the model predictive controller or human expert, is taken before the failure of tasks, when the uncertainty threshold is exceeded. We validate our method on fully-observable and vision-based partially-observable systems using cart-pole and autonomous driving simulations using deep convolutional Bayesian neural networks. We demonstrate that our method is robust to uncertainty resulting from varying system dynamics as well as from partial state observability.

In this paper, we present a whole-body control framework for Wheeled Inverted Pendulum (WIP) Humanoids. WIP Humanoids are redundant manipulators dynamically balancing themselves on wheels. Characterized by several degrees of freedom, they have the ability to perform several tasks simultaneously, such as balancing, maintaining a body pose, controlling the gaze, lifting a load or maintaining end-effector configuration in operation space. The problem of whole-body control is to enable simultaneous performance of these tasks with optimal participation of all degrees of freedom at specified priorities for each objective. The control also has to obey constraint of angle and torque limits on each joint. The proposed approach is hierarchical with a low level controller for body joints manipulation and a high-level controller that defines center of mass (CoM) targets for the low-level controller to control zero dynamics of the system driving the wheels. The low-level controller plans for shorter horizons while considering more complete dynamics of the system, while the high-level controller plans for longer horizon based on an approximate model of the robot for computational efficiency.

Hybrid ground and aerial vehicles can possess distinct advantages over ground-only or flight-only designs in terms of energy savings and increased mobility. In this work we outline our unified framework for controls, planning, and autonomy of hybrid ground/air vehicles. Our contribution is three-fold: 1) We develop a control scheme for the control of passive two-wheeled hybrid ground/aerial vehicles. 2) We present a unified planner for both rolling and flying by leveraging differential flatness mappings. 3) We conduct experiments leveraging mapping and global planning for hybrid mobility in unknown environments, showing that hybrid mobility uses up to five times less energy than flying only1.1Video at https://youtu.be/nlGfYehTLpg.

Recently, there is a surge of interest in using point processes to model continuous-time user activities. This framework has resulted in novel models and improved performance in diverse applications. However, most previous works focus on the ”open loop” setting where learned models are used for predictive tasks. Typically, we are interested in the ”closed loop” setting where a policy needs to be learned to incorporate user feedbacks and guide user activities to desirable states. Although point processes have good predictive performance, it is not clear how to use them for the challenging closed loop activity guiding task. In this paper, we propose a framework to reformulate point processes into stochastic differential equations, which allows us to extend methods from stochastic optimal control to address the activity guiding problem. We also design an efficient algorithm, and show that our method guides user activities to desired states more effectively than state-of-arts.

To effectively control complex dynamical systems, accurate nonlinear models are typically needed. However, these models are not always known. In this paper, we present a data-driven approach based on Gaussian processes that learns models of quadrotors operating in partially unknown environments. What makes this challenging is that if the learning process is not carefully controlled, the system will go unstable, i.e., the quadcopter will crash. To this end, barrier certificates are employed for safe learning. The barrier certificates establish a non-conservative forward invariant safe region, in which high probability safety guarantees are provided based on the statistics of the Gaussian Process. A learning controller is designed to efficiently explore those uncertain states and expand the barrier certified safe region based on an adaptive sampling scheme. Simulation results are provided to demonstrate the effectiveness of the proposed approach.

We introduce an algorithm for autonomous control of multiple fast ground vehicles operating in close proximity to each other. The algorithm is based on a combination of the game theoretic notion of iterated best response, and an information theoretic model predictive control algorithm designed for non-linear stochastic systems. We test the algorithm on two one-fifth scale AutoRally platforms traveling at speeds upwards of 8 meters per second, while maintaining a following distance of under two meters from bumper-to-bumper.

We present a trajectory optimization approach to reinforcement learning in continuous state and action spaces, called probabilistic differential dynamic programming (PDDP). Our method represents systems dynamics using Gaussian processes (GPs), and performs local dynamic programming iteratively around a nominal trajectory in Gaussian belief spaces. Different from model-based policy search methods, PDDP does not require a policy parameterization and learns a time-varying control policy via successive forward-backward sweeps. A convergence analysis of the iterative scheme is given, showing that our algorithm converges to a stationary point globally under certain conditions. We show that prior model knowledge can be incorporated into the proposed framework to speed up learning, and a generalized optimization criterion based on the predicted cost distribution can be employed to enable risk-sensitive learning. We demonstrate the effectiveness and efficiency of the proposed algorithm using nontrivial tasks. Compared with a state-of-the-art GP-based policy search method, PDDP offers a superior combination of learning speed, data efficiency, and applicability.

We present an information-theoretic approach to stochastic optimal control problems that can be used to derive general sampling-based optimization schemes. This new mathematical method is used to develop a sampling-based model predictive control algorithm. We apply this information-theoretic model predictive control scheme to the task of aggressive autonomous driving around a dirt test track, and compare its performance with a model predictive control version of the cross-entropy method.

Sparse Spectrum Gaussian Processes (SSGPs) are a powerful tool for scaling Gaussian processes (GPs) to large datasets. Existing SSGP algorithms for regression assume deterministic inputs, precluding their use in many real-world robotics and engineering applications where accounting for input uncertainty is crucial. We address this problem by proposing two analytic moment-based approaches with closed-form expressions for SSGP regression with uncertain inputs. Our methods are more general and scalable than their standard GP counterparts, and are naturally applicable to multi-step prediction or uncertainty propagation. We show that efficient algorithms for Bayesian filtering and stochastic model predictive control can use these methods, and we evaluate our algorithms with comparative analyses and both real-world and simulated experiments.

Temporal point processes have been widely applied to model event sequence data generated by online users. In this paper, we consider the problem of how to design the optimal control policy for point processes, such that the stochastic system driven by the point process is steered to a target state. In particular, we exploit the key insight to view the stochastic optimal control problem from the perspective of optimal measure and variational inference. We further propose a convex optimization framework and an efficient algorithm to update the policy adaptively to the current system state. Experiments on synthetic and real-world data show that our algorithm can steer the user activities much more accurately and efficiently than other stochastic control methods.

We introduce an information theoretic model predictive control (MPC) algorithm capable of handling complex cost criteria and general nonlinear dynamics. The generality of the approach makes it possible to use multi-layer neural networks as dynamics models, which we incorporate into our MPC algorithm in order to solve model-based reinforcement learning tasks. We test the algorithm in simulation on a cart-pole swing up and quadrotor navigation task, as well as on actual hardware in an aggressive driving task. Empirical results demonstrate that the algorithm is capable of achieving a high level of performance and does so only utilizing data collected from the system.

In this paper we present a model predictive control algorithm designed for optimizing non-linear systems subject to complex cost criteria. The algorithm is based on a stochastic optimal control framework using a fundamental relationship between the information theoretic notions of free energy and relative entropy. The optimal controls in this setting take the form of a path integral, which we approximate using an efficient importance sampling scheme. We experimentally verify the algorithm by implementing it on a Graphics Processing Unit (GPU) and apply it to the problem of controlling a fifth-scale Auto-Rally vehicle in an aggressive driving task.

We present a data-driven stochastic optimal control framework that is derived using the path integral (PI) control approach. We find iterative control laws analytically without a priori policy parameterization based on probabilistic representation of the learned dynamics model. The proposed algorithm operates in a forward-backward sweep manner which differentiate it from other PI-related methods that perform forward sampling to find open-loop optimal controls. Our method uses significantly less sampled data to find analytic control laws compared to other approaches within the PI control family that rely on extensive sampling from given dynamics models or trials on physical systems in a model-free fashion. In addition, the learned controllers can be generalized to new tasks without re-sampling based on the compositionality theory for the linearly-solvable optimal control framework.We provide experimental results on three different systems and comparisons with state-of-the-art model-based methods to demonstrate the efficiency and generalizability of the proposed framework.

This paper considers optimal control of dynamical systems which are represented by nonlinear stochastic differential equations. It is well-known that the optimal control policy for this problem can be obtained as a function of a value function that satisfies a nonlinear partial differential equation, namely, the Hamilton-Jacobi-Bellman equation. This nonlinear PDE must be solved backwards in time, and this computation is intractable for large scale systems. Under certain assumptions, and after applying a logarithmic transformation, an alternative characterization of the optimal policy can be given in terms of a path integral. Path Integral (PI) based control methods have recently been shown to provide elegant solutions to a broad class of stochastic optimal control problems. One of the implementation challenges with this formalism is the computation of the expectation of a cost functional over the trajectories of the unforced dynamics. Computing such expectation over trajectories that are sampled uniformly may induce numerical instabilities due to the exponentiation of the cost. Therefore, sampling of low-cost trajectories is essential for the practical implementation of PI-based methods. In this paper, we use incremental sampling-based algorithms to sample useful trajectories from the unforced system dynamics, and make a novel connection between Rapidly-exploring Random Trees (RRTs) and information-theoretic stochastic optimal control. We show the results from the numerical implementation of the proposed approach to several examples.

We present two nonparametric approaches to Kullback-Leibler (KL) control, or linearly-solvable Markov decision problem (LMDP) based on Gaussian processes (GP) and Nyström approximation. Compared to recently developed parametric methods, the proposed data-driven frameworks feature accurate function approximation and efficient on-line operations. Theoretically, we derive the mathematical connection of KL control based on dynamic programming with earlier work in control theory which relies on information theoretic dualities for the infinite time horizon case. Algorithmically, we give explicit optimal control policies in nonparametric forms, and propose on-line update schemes with budgeted computational costs. Numerical results demonstrate the effectiveness and usefulness of the proposed frameworks.

In this work, we revisit the continuous-time Differential Dynamic Programming (DDP) approach for solving optimal control problems with terminal state constraints. We derive two algorithms, each for different order of expansion of the system dynamics and we investigate their performance in terms of their convergence speed. Compared to previous work, we provide a set of backward differential equations for the value function expansion by relaxing the assumption that the initial nominal control must be very close to the optimal control solution. We apply the derived algorithms to two classical optimal control problems, namely, the inverted pendulum and the Dreyfus rocket problem and show the benefit of second order expansion.

Despite the plethora of reinforcement learning algorithms in machine learning and control, the majority of the work in this area relies on discrete time formulations of stochastic dynamics. In this work we present a new policy gradient algorithm for reinforcement learning in continuous state action spaces and continuous time for free energy-like cost functions. The derivation is based on successive application of Girsanov's theorem and the use of the Radon Nikodým derivative as formulated for Markov diffusion processes. The resulting policy gradient is reward weighted. The use of Radon Nikodým extends analysis and results to more general models of stochasticity in which jump diffusions processes are considered. We apply the resulting algorithm in two simple examples for learning attractor landscapes in rhythmic and discrete movements.

Redundant tendon-driven systems such as the human hand or the ACT robotic hand are high-dimensional and nonlinear systems that make traditional control strategies ineffective. The synergy hypothesis from neuroscience suggests that employing dimensionality reduction techniques can simplify the system without a major loss in function. We define a dimensionality reduction framework consisting of separate observation and activation synergies, a first-order model, and an optimal controller. The framework is implemented for two example tasks: adaptive control of thumb posture and hybrid position/force control to enable dynamic handwriting.

We apply path integral reinforcement learning to a biomechanically accurate dynamics model of the index finger and then to the Anatomically Correct Testbed (ACT) robotic hand. We illustrate the applicability of Policy Improvement with Path Integrals (PI2) to parameterized and non-parameterized control policies. This method is based on sampling variations in control, executing them in the real world, and minimizing a cost function on the resulting performance. Iteratively improving the control policy based on real-world performance requires no direct modeling of tendon network nonlinearities and contact transitions, allowing improved task performance.

Physical contact events often allow a natural decomposition of manipulation tasks into action phases and subgoals. Within the motion primitive paradigm, each action phase corresponds to a motion primitive, and the subgoals correspond to the goal parameters of these primitives. Current state-of-the-art reinforcement learning algorithms are able to efficiently and robustly optimize the parameters of motion primitives in very high-dimensional problems. These algorithms often consider only shape parameters, which determine the trajectory between the start- and end-point of the movement. In manipulation, however, it is also crucial to optimize the goal parameters, which represent the subgoals between the motion primitives. We therefore extend the policy improvement with path integrals (PI2) algorithm to simultaneously optimize shape and goal parameters. Applying simultaneous shape and goal learning to sequences of motion primitives leads to the novel algorithm PI2Seq. We use our methods to address a fundamental challenge in manipulation: improving the robustness of everyday pick-and-place tasks.

Path integral methods have recently been shown to be applicable to a very general class of optimal control problems. Here we examine the path integral formalism from a decision-theoretic point of view, since an optimal controller can always be regarded as an instance of a perfectly rational decision-maker that chooses its actions so as to maximize its expected utility. The problem with perfect rationality is, however, that finding optimal actions is often very difficult due to prohibitive computational resource costs that are not taken into account. In contrast, a bounded rational decision-maker has only limited resources and therefore needs to strike some compromise between the desired utility and the required resource costs. In particular, we suggest an information-theoretic measure of resource costs that can be derived axiomatically. As a consequence we obtain a variational principle for choice probabilities that trades off maximizing a given utility criterion and avoiding resource costs that arise due to deviating from initially given default choice probabilities. The resulting bounded rational policies are in general probabilistic. We show that the solutions found by the path integral formalism are such bounded rational policies. Furthermore, we show that the same formalism generalizes to discrete control problems, leading to linearly solvable bounded rational control policies in the case of Markov systems. Importantly, Bellman's optimality principle is not presupposed by this variational principle, but it can be derived as a limit case. This suggests that the information-theoretic formalization of bounded rationality might serve as a general principle in control design that unifies a number of recently reported approximate optimal control methods both in the continuous and discrete domain.

We present a new approach to motion planning using a stochastic trajectory optimization framework. The approach relies on generating noisy trajectories to explore the space around an initial (possibly infeasible) trajectory, which are then combined to produced an updated trajectory with lower cost. A cost function based on a combination of obstacle and smoothness cost is optimized in each iteration. No gradient information is required for the particular optimization algorithm that we use and so general costs for which derivatives may not be available (e.g. costs corresponding to constraints and motor torques) can be included in the cost function. We demonstrate the approach both in simulation and on a mobile manipulation system for unconstrained and constrained tasks. We experimentally show that the stochastic nature of STOMP allows it to overcome local minima that gradient-based methods like CHOMP can get stuck in.

Learning complex motor skills for real world tasks is a hard problem in robotic manipulation that often requires painstaking manual tuning and design by a human expert. In this work, we present a Reinforcement Learning based approach to acquiring new motor skills from demonstration. Our approach allows the robot to learn fine manipulation skills and significantly improve its success rate and skill level starting from a possibly coarse demonstration. Our approach aims to incorporate task domain knowledge, where appropriate, by working in a space consistent with the constraints of a specific task. In addition, we also present an approach to using sensor feedback to learn a predictive model of the task outcome. This allows our system to learn the proprioceptive sensor feedback needed to monitor subsequent executions of the task online and abort execution in the event of predicted failure. We illustrate our approach using two example tasks executed with the PR2 dual-arm robot: a straight and accurate pool stroke and a box flipping task using two chopsticks as tools.

We present an approach that enables robots to learn motion primitives that are robust towards state estimation uncertainties. During reaching and preshaping, the robot learns to use line manipulation strategies to maneuver the object into a pose at which closing the hand to perform the grasp is more likely to succeed. In contrast, common assumptions in grasp planning and motion planning for reaching are that these tasks can be performed independently, and that the robot has perfect knowledge of the pose of the objects in the environment. We implement our approach using Dynamic Movement Primitives and the probabilistic model-free reinforcement learning algorithm Policy Improvement with Path Integrals (PI2). The cost function that PI2optimizes is a simple boolean that penalizes failed grasps. The key to acquiring robust motion primitives is to sample the actual pose of the object from a distribution that represents the state estimation uncertainty. During learning, the robot will thus optimize the chance of grasping an object from this distribution, rather than at one specific pose. In our empirical evaluation, we demonstrate how the motion primitives become more robust when grasping simple cylindrical objects, as well as more complex, non-convex objects. We also investigate how well the learned motion primitives generalize towards new object positions and other state estimation uncertainty distributions.

Segmenting complex movements into a sequence of primitives remains a difficult problem with many applications in the robotics and vision communities. In this work, we show how the movement segmentation problem can be reduced to a sequential movement recognition problem. To this end, we reformulate the original Dynamic Movement Primitive (DMP) formulation as a linear dynamical system with control inputs. Based on this new formulation, we develop an Expectation-Maximization algorithm to estimate the duration and goal position of a partially observed trajectory. With the help of this algorithm and the assumption that a library of movement primitives is present, we present a movement segmentation framework. We illustrate the usefulness of the new DMP formulation on the two applications of online movement recognition and movement segmentation.

Applying model-free reinforcement learning to manipulation remains challenging for several reasons. First, manipulation involves physical contact, which causes discontinuous cost functions. Second, in manipulation, the end-point of the movement must be chosen carefully, as it represents a grasp which must be adapted to the pose and shape of the object. Finally, there is uncertainty in the object pose, and even the most carefully planned movement may fail if the object is not at the expected position. To address these challenges we 1) present a simplified, computationally more efficient version of our model-free reinforcement learning algorithm PI2; 2) extend PI2so that it simultaneously learns shape parameters and goal parameters of motion primitives; 3) use shape and goal learning to acquire motion primitives that are robust to object pose uncertainty. We evaluate these contributions on a manipulation platform consisting of a 7-DOF arm with a 4-DOF hand.

Reinforcement learning (RL) is one of the most general approaches to learning control. Its applicability to complex motor systems, however, has been largely impossible so far due to the computational difficulties that reinforcement learning encounters in high dimensional continuous state-action spaces. In this paper, we derive a novel approach to RL for parameterized control policies based on the framework of stochastic optimal control with path integrals. While solidly grounded in optimal control theory and estimation theory, the update equations for learning are surprisingly simple and have no danger of numerical instabilities as neither matrix inversions nor gradient learning rates are required. Empirical evaluations demonstrate significant performance improvements over gradient-based policy learning and scalability to high-dimensional control problems. Finally, a learning experiment on a robot dog illustrates the functionality of our algorithm in a real-world scenario. We believe that our new algorithm, Policy Improvement with Path Integrals (PI2), offers currently one of the most efficient, numerically robust, and easy to implement algorithms for RL in robotics.

Recent advances on path integral stochastic optimal control [1],[2] provide new insights in the optimal control of nonlinear stochastic systems which are linear in the controls, with state independent and time invariant control transition matrix. Under these assumptions, the Hamilton-Jacobi-Bellman (HJB) equation is formulated and linearized with the use of the logarithmic transformation of the optimal value function. The resulting HJB is a linear second order partial differential equation which is solved by an approximation based on the Feynman-Kac formula [3]. In this work we review the theory of path integral control and derive the linearized HJB equation for systems with state dependent control transition matrix. In addition we derive the path integral formulation for the general class of systems with state dimensionality that is higher than the dimensionality of the controls. Furthermore, by means of a modified inverse dynamics controller, we apply path integral stochastic optimal control over the new control space. Simulations illustrate the theoretical results. Future developments and extensions are discussed.

In this paper, we introduce a modified Kalman filter that can perform robust, real-time outlier detection in the observations, without the need for manual parameter tuning by the user. Robotic systems that rely on high quality sensory data can be sensitive to data containing outliers. Since the standard Kalman filter is not robust to outliers, other variations of the Kalman filter have been proposed to overcome this issue, but these methods may require manual parameter tuning, use of heuristics or complicated parameter estimation. Our Kalman filter uses a weighted least squares-like approach by introducing weights for each data sample. A data sample with a smaller weight has a weaker contribution when estimating the current time step's state. We learn the weights and system dynamics using a variational Expectation-Maximization framework. We evaluate our Kalman filter algorithm on data from a robotic dog.