A Family of Iterative Gauss-Newton Shooting Methods for Nonlinear Optimal Control

This paper introduces a family of iterative algorithms for unconstrained nonlinear optimal control. We generalize the well-known iLQR algorithm to different multiple-shooting variants, combining advantages like straight-forward initialization and a closed-loop forward integration. All algorithms have similar computational complexity, i.e. linear complexity in the time horizon, and can be derived in the same computational framework. We compare the full-step variants of our algorithms and present several simulation examples, including a high-dimensional underactuated robot subject to contact switches. Simulation results show that our multiple-shooting algorithms can achieve faster convergence, better local contraction rates and much shorter runtimes than classical iLQR, which makes them a superior choice for nonlinear model predictive control applications.Comment: 8 page

Momentum Control with Hierarchical Inverse Dynamics on a Torque-Controlled Humanoid

Hierarchical inverse dynamics based on cascades of quadratic programs have been proposed for the control of legged robots. They have important benefits but to the best of our knowledge have never been implemented on a torque controlled humanoid where model inaccuracies, sensor noise and real-time computation requirements can be problematic. Using a reformulation of existing algorithms, we propose a simplification of the problem that allows to achieve real-time control. Momentum-based control is integrated in the task hierarchy and a LQR design approach is used to compute the desired associated closed-loop behavior and improve performance. Extensive experiments on various balancing and tracking tasks show very robust performance in the face of unknown disturbances, even when the humanoid is standing on one foot. Our results demonstrate that hierarchical inverse dynamics together with momentum control can be efficiently used for feedback control under real robot conditions.Comment: 21 pages, 11 figures, 4 tables in Autonomous Robots (2015

Balancing experiments on a torque-controlled humanoid with hierarchical inverse dynamics

Recently several hierarchical inverse dynamics controllers based on cascades of quadratic programs have been proposed for application on torque controlled robots. They have important theoretical benefits but have never been implemented on a torque controlled robot where model inaccuracies and real-time computation requirements can be problematic. In this contribution we present an experimental evaluation of these algorithms in the context of balance control for a humanoid robot. The presented experiments demonstrate the applicability of the approach under real robot conditions (i.e. model uncertainty, estimation errors, etc). We propose a simplification of the optimization problem that allows us to decrease computation time enough to implement it in a fast torque control loop. We implement a momentum-based balance controller which shows robust performance in face of unknown disturbances, even when the robot is standing on only one foot. In a second experiment, a tracking task is evaluated to demonstrate the performance of the controller with more complicated hierarchies. Our results show that hierarchical inverse dynamics controllers can be used for feedback control of humanoid robots and that momentum-based balance control can be efficiently implemented on a real robot.Comment: appears in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 201

Dynamic Estimation of Rigid Motion from Perspective Views via Recursive Identification of Exterior Differential Systems with Parameters on a Topological Manifold

We formulate the problem of estimating the motion of a rigid object viewed under perspective projection as the identification of a dynamic model in Exterior Differential form with parameters on a topological manifold. We first describe a general method for recursive identification of nonlinear implicit systems using prediction error criteria. The parameters are allowed to move slowly on some topological (not necessarily smooth) manifold. The basic recursion is solved in two different ways: one is based on a simple extension of the traditional Kalman Filter to nonlinear and implicit measurement constraints, the other may be regarded as a generalized "Gauss-Newton" iteration, akin to traditional Recursive Prediction Error Method techniques in linear identification. A derivation of the "Implicit Extended Kalman Filter" (IEKF) is reported in the appendix. The ID framework is then applied to solving the visual motion problem: it indeed is possible to characterize it in terms of identification of an Exterior Differential System with parameters living on a C0 topological manifold, called the "essential manifold". We consider two alternative estimation paradigms. The first is in the local coordinates of the essential manifold: we estimate the state of a nonlinear implicit model on a linear space. The second is obtained by a linear update on the (linear) embedding space followed by a projection onto the essential manifold. These schemes proved successful in performing the motion estimation task, as we show in experiments on real and noisy synthetic image sequences

On Time Optimization of Centroidal Momentum Dynamics

Recently, the centroidal momentum dynamics has received substantial attention to plan dynamically consistent motions for robots with arms and legs in multi-contact scenarios. However, it is also non convex which renders any optimization approach difficult and timing is usually kept fixed in most trajectory optimization techniques to not introduce additional non convexities to the problem. But this can limit the versatility of the algorithms. In our previous work, we proposed a convex relaxation of the problem that allowed to efficiently compute momentum trajectories and contact forces. However, our approach could not minimize a desired angular momentum objective which seriously limited its applicability. Noticing that the non-convexity introduced by the time variables is of similar nature as the centroidal dynamics one, we propose two convex relaxations to the problem based on trust regions and soft constraints. The resulting approaches can compute time-optimized dynamically consistent trajectories sufficiently fast to make the approach realtime capable. The performance of the algorithm is demonstrated in several multi-contact scenarios for a humanoid robot. In particular, we show that the proposed convex relaxation of the original problem finds solutions that are consistent with the original non-convex problem and illustrate how timing optimization allows to find motion plans that would be difficult to plan with fixed timing.Comment: 7 pages, 4 figures, ICRA 201

Recent Advances in Computational Methods for the Power Flow Equations

The power flow equations are at the core of most of the computations for designing and operating electric power systems. The power flow equations are a system of multivariate nonlinear equations which relate the power injections and voltages in a power system. A plethora of methods have been devised to solve these equations, starting from Newton-based methods to homotopy continuation and other optimization-based methods. While many of these methods often efficiently find a high-voltage, stable solution due to its large basin of attraction, most of the methods struggle to find low-voltage solutions which play significant role in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in methods for solving power flow equations to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to the power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Groebner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction.Comment: 13 pages, 2 figures. Submitted to the Tutorial Session at IEEE 2016 American Control Conferenc

Differentiable world programs

L'intelligence artificielle (IA) moderne a ouvert de nouvelles perspectives prometteuses pour la création de robots intelligents. En particulier, les architectures d'apprentissage basées sur le gradient (réseaux neuronaux profonds) ont considérablement amélioré la compréhension des scènes 3D en termes de perception, de raisonnement et d'action. Cependant, ces progrès ont affaibli l'attrait de nombreuses techniques ``classiques'' développées au cours des dernières décennies. Nous postulons qu'un mélange de méthodes ``classiques'' et ``apprises'' est la voie la plus prometteuse pour développer des modèles du monde flexibles, interprétables et exploitables : une nécessité pour les agents intelligents incorporés. La question centrale de cette thèse est : ``Quelle est la manière idéale de combiner les techniques classiques avec des architectures d'apprentissage basées sur le gradient pour une compréhension riche du monde 3D ?''. Cette vision ouvre la voie à une multitude d'applications qui ont un impact fondamental sur la façon dont les agents physiques perçoivent et interagissent avec leur environnement. Cette thèse, appelée ``programmes différentiables pour modèler l'environnement'', unifie les efforts de plusieurs domaines étroitement liés mais actuellement disjoints, notamment la robotique, la vision par ordinateur, l'infographie et l'IA. Ma première contribution---gradSLAM--- est un système de localisation et de cartographie simultanées (SLAM) dense et entièrement différentiable. En permettant le calcul du gradient à travers des composants autrement non différentiables tels que l'optimisation non linéaire par moindres carrés, le raycasting, l'odométrie visuelle et la cartographie dense, gradSLAM ouvre de nouvelles voies pour intégrer la reconstruction 3D classique et l'apprentissage profond. Ma deuxième contribution - taskography - propose une sparsification conditionnée par la tâche de grandes scènes 3D encodées sous forme de graphes de scènes 3D. Cela permet aux planificateurs classiques d'égaler (et de surpasser) les planificateurs de pointe basés sur l'apprentissage en concentrant le calcul sur les attributs de la scène pertinents pour la tâche. Ma troisième et dernière contribution---gradSim--- est un simulateur entièrement différentiable qui combine des moteurs physiques et graphiques différentiables pour permettre l'estimation des paramètres physiques et le contrôle visuomoteur, uniquement à partir de vidéos ou d'une image fixe.Modern artificial intelligence (AI) has created exciting new opportunities for building intelligent robots. In particular, gradient-based learning architectures (deep neural networks) have tremendously improved 3D scene understanding in terms of perception, reasoning, and action. However, these advancements have undermined many ``classical'' techniques developed over the last few decades. We postulate that a blend of ``classical'' and ``learned'' methods is the most promising path to developing flexible, interpretable, and actionable models of the world: a necessity for intelligent embodied agents. ``What is the ideal way to combine classical techniques with gradient-based learning architectures for a rich understanding of the 3D world?'' is the central question in this dissertation. This understanding enables a multitude of applications that fundamentally impact how embodied agents perceive and interact with their environment. This dissertation, dubbed ``differentiable world programs'', unifies efforts from multiple closely-related but currently-disjoint fields including robotics, computer vision, computer graphics, and AI. Our first contribution---gradSLAM---is a fully differentiable dense simultaneous localization and mapping (SLAM) system. By enabling gradient computation through otherwise non-differentiable components such as nonlinear least squares optimization, ray casting, visual odometry, and dense mapping, gradSLAM opens up new avenues for integrating classical 3D reconstruction and deep learning. Our second contribution---taskography---proposes a task-conditioned sparsification of large 3D scenes encoded as 3D scene graphs. This enables classical planners to match (and surpass) state-of-the-art learning-based planners by focusing computation on task-relevant scene attributes. Our third and final contribution---gradSim---is a fully differentiable simulator that composes differentiable physics and graphics engines to enable physical parameter estimation and visuomotor control, solely from videos or a still image

A Global Steering Method for Nonholonomic Systems

In this paper, we present an iterative steering algorithm for nonholonomic systems (also called driftless control-affine systems) and we prove its global convergence under the sole assumption that the Lie Algebraic Rank Condition (LARC) holds true everywhere. That algorithm is an extension of the one introduced in [21] for regular systems. The first novelty here consists in the explicit algebraic construction, starting from the original control system, of a lifted control system which is regular. The second contribution of the paper is an exact motion planning method for nilpotent systems, which makes use of sinusoidal control laws and which is a generalization of the algorithm described in [29] for chained-form systems