Rapid Convex Optimization of Centroidal Dynamics using Block Coordinate Descent

In this paper we explore the use of block coordinate descent (BCD) to optimize the centroidal momentum dynamics for dynamically consistent multi-contact behaviors. The centroidal dynamics have recently received a large amount of attention in order to create physically realizable motions for robots with hands and feet while being computationally more tractable than full rigid body dynamics models. Our contribution lies in exploiting the structure of the dynamics in order to simplify the original non-convex problem into two convex subproblems. We iterate between these two subproblems for a set number of iterations or until a consensus is reached. We explore the properties of the proposed optimization method for the centroidal dynamics and verify in simulation that motions generated by our approach can be tracked by the quadruped Solo12. In addition, we compare our method to a recently proposed convexification using a sequence of convex relaxations as well as a more standard interior point method used in the off- the-shelf solver IPOPT to show that our approach finds similar, if not better, trajectories (in terms of cost), and is more than four times faster than both approaches. Finally, compared to previous approaches, we note its practicality due to the convex nature of each subproblem which allows our method to be used with any off-the-shelf quadratic programming solver.

Model-free Reinforcement Learning for Robust Locomotion Using Trajectory Optimization for Exploration

In this work we present a general, two-stage reinforcement learning approach for going from a single demonstration trajectory to a robust policy that can be deployed on hardware without any additional training. The demonstration is used in the first stage as a starting point to facilitate initial exploration. In the second stage, the relevant task reward is optimized directly and a policy robust to environment uncertainties is computed. We demonstrate and examine in detail performance and robustness of our approach on highly dynamic hopping and bounding tasks on a real quadruped robot.

$\mathcal{N}$IPM-HLSP: An Efficient Interior-Point Method for Hierarchical Least-Squares Programs

Hierarchical least-squares programs with linear constraints (HLSP) are a type of optimization problem very common in robotics. Each priority level contains an objective in least-squares form which is subject to the linear constraints of the higher priority hierarchy levels. Active-set methods (ASM) are a popular choice for solving them. However, they can perform poorly in terms of computational time if there are large changes of the active set. We therefore propose a computationally efficient primal-dual interior-point method (IPM) for HLSP's which is able to maintain constant numbers of solver iterations in these situations. We base our IPM on the null-space method which requires only a single decomposition per Newton iteration instead of two as it is the case for other IPM solvers. After a priority level has converged we compose a set of active constraints judging upon the dual and project lower priority levels into their null-space. We show that the IPM-HLSP can be expressed in least-squares form which avoids the formation of the quadratic Karush-Kuhn-Tucker (KKT) Hessian. Due to our choice of the null-space basis the IPM-HLSP is as fast as the state-of-the-art ASM-HLSP solver for equality only problems.

Learning Dynamical Systems from Noisy Sensor Measurements using Multiple Shooting

Modeling dynamical systems plays a crucial role in capturing and understanding complex physical phenomena. When physical models are not sufficiently accurate or hardly describable by analytical formulas, one can use generic function approximators such as neural networks to capture the system dynamics directly from sensor measurements. As for now, current methods to learn the parameters of these neural networks are highly sensitive to the inherent instability of most dynamical systems of interest, which in turn prevents the study of very long sequences. In this work, we introduce a generic and scalable method based on multiple shooting to learn latent representations of indirectly observed dynamical systems. We achieve state-of-the-art performances on systems observed directly from raw images. Further, we demonstrate that our method is robust to noisy measurements and can handle complex dynamical systems, such as chaotic ones.

A Robustness Analysis of Inverse Optimal Control of Bipedal Walking

Cost functions have the potential to provide compact and understandable generalizations of motion. The goal of Inverse Optimal Control (IOC) is to analyze an observed behavior which is assumed to be optimal with respect to an unknown cost function, and infer this cost function. Here we develop a method for characterizing cost functions of legged locomotion, with the goal of representing complex humanoid behavior with simple models. To test this methodology we simulate walking gaits of a simple 5 link planar walking model which optimize known cost functions, and assess the ability of our IOC method to recover them. In particular, the IOC method uses an iterative trajectory optimization process to infer cost function weightings consistent with those used to generate a single demonstrated optimal trial. We also explore sensitivity of the IOC to sensor noise in the observed trajectory, imperfect knowledge of the model or task, as well as uncertainty in the components of the cost function used. With appropriate modeling, these methods may help infer cost functions from human data, yielding a compact and generalizable representation of human-like motion for use in humanoid robot controllers, as well as providing a new tool for experimentally exploring human preferences.

Simultaneous Navigation and Construction Benchmarking Environments

We need intelligent robots for mobile construction, the process of navigating in an environment and modifying its structure according to a geometric design. In this task, a major robot vision and learning challenge is how to exactly achieve the design without GPS, due to the difficulty caused by the bi-directional coupling of accurate robot localization and navigation together with strategic environment manipulation. However, many existing robot vision and learning tasks such as visual navigation and robot manipulation address only one of these two coupled aspects. To stimulate the pursuit of a generic and adaptive solution, we reasonably simplify mobile construction as a partially observable Markov decision process (POMDP) in 1/2/3D grid worlds and benchmark the performance of a handcrafted policy with basic localization and planning, and state-of-the-art deep reinforcement learning (RL) methods. Our extensive experiments show that the coupling makes this problem very challenging for those methods, and emphasize the need for novel task-specific solutions.

Fast and Accurate Multi-Body Simulation with Stiff Viscoelastic Contacts

The simulation of multi-body systems with frictional contacts is a fundamental tool for many fields, such as robotics, computer graphics, and mechanics. Hard frictional contacts are particularly troublesome to simulate because they make the differential equations stiff, calling for computationally demanding implicit integration schemes. We suggest to tackle this issue by using exponential integrators, a long-standing class of integration schemes (first introduced in the 60's) that in recent years has enjoyed a resurgence of interest. We show that this scheme can be easily applied to multi-body systems subject to stiff viscoelastic contacts, producing accurate results at lower computational cost than classic explicit schemes. In our tests with quadruped and biped robots, our method demonstrated stable behaviors with large time steps (10 ms) and stiff contacts ($10^5$ N/m). Its excellent properties, especially for fast and coarse simulations, make it a valuable candidate for many applications in robotics, such as simulation, Model Predictive Control, Reinforcement Learning, and controller design.

Impedance Optimization for Uncertain Contact Interactions Through Risk Sensitive Optimal Control

This paper addresses the problem of computing optimal impedance schedules for legged locomotion tasks involving complex contact interactions. We formulate the problem of impedance regulation as a trade-off between disturbance rejection and measurement uncertainty. We extend a stochastic optimal control algorithm known as Risk Sensitive Control to take into account measurement uncertainty and propose a formal way to include such uncertainty for unknown contact locations. The approach can efficiently generate optimal state and control trajectories along with local feedback control gains, i.e. impedance schedules. Extensive simulations demonstrate the capabilities of the approach in generating meaningful stiffness and damping modulation patterns before and after contact interaction. For example, contact forces are reduced during early contacts, damping increases to anticipate a high impact event and tracking is automatically traded-off for increased stability. In particular, we show a significant improvement in performance during jumping and trotting tasks with a simulated quadruped robot.

Leveraging Forward Model Prediction Error for Learning Control

Learning for model based control can be sample-efficient and generalize well, however successfully learning models and controllers that represent the problem at hand can be challenging for complex tasks. Using inaccurate models for learning can lead to sub-optimal solutions, that are unlikely to perform well in practice. In this work, we present a learning approach which iterates between model learning and data collection and leverages forward model prediction error for learning control. We show how using the controller's prediction as input to a forward model can create a differentiable connection between the controller and the model, allowing us to formulate a loss in the state space. This lets us include forward model prediction error during controller learning and we show that this creates a loss objective that significantly improves learning on different motor control tasks. We provide empirical and theoretical results that show the benefits of our method and present evaluations in simulation for learning control on a 7 DoF manipulator and an underactuated 12 DoF quadruped. We show that our approach successfully learns controllers for challenging motor control tasks involving contact switching.

Learning a Centroidal Motion Planner for Legged Locomotion

Whole-body optimizers have been successful at automatically computing complex dynamic locomotion behaviors. However they are often limited to offline planning as they are computationally too expensive to replan with a high frequency. Simpler models are then typically used for online replanning. In this paper we present a method to generate whole body movements in real-time for locomotion tasks. Our approach consists in learning a centroidal neural network that predicts the desired centroidal motion given the current state of the robot and a desired contact plan. The network is trained using an existing whole body motion optimizer. Our approach enables to learn with few training samples dynamic motions that can be used in a complete whole-body control framework at high frequency, which is usually not attainable with typical full-body optimizers. We demonstrate our method to generate a rich set of walking and jumping motions on a real quadruped robot.