A Dynamical System-based Approach to Modeling Stable Robot Control Policies via Imitation Learning

Despite tremendous advances in robotics, we are still amazed by the proficiency with which humans perform movements. Even new waves of robotic systems still rely heavily on hardcoded motions with a limited ability to react autonomously and robustly to a dynamically changing environment. This thesis focuses on providing possible mechanisms to push the level of adaptivity, reactivity, and robustness of robotic systems closer to human movements. Specifically, it aims at developing these mechanisms for a subclass of robot motions called “reaching movements”, i.e. movements in space stopping at a given target (also referred to as episodic motions, discrete motions, or point-to-point motions). These reaching movements can then be used as building blocks to form more advanced robot tasks. To achieve a high level of proficiency as described above, this thesis particularly seeks to derive control policies that: 1) resemble human motions, 2) guarantee the accomplishment of the task (if the target is reachable), and 3) can instantly adapt to changes in dynamic environments. To avoid manually hardcoding robot motions, this thesis exploits the power of machine learning techniques and takes an Imitation Learning (IL) approach to build a generic model of robot movements from a few examples provided by an expert. To achieve the required level of robustness and reactivity, the perspective adopted in this thesis is that a reaching movement can be described with a nonlinear Dynamical System (DS). When building an estimate of DS from demonstrations, there are two key problems that need to be addressed: the problem of generating motions that resemble at best the demonstrations (the “how-to-imitate” problem), and most importantly, the problem of ensuring the accomplishment of the task, i.e. reaching the target (the “stability” problem). Although there are numerous well-established approaches in robotics that could answer each of these problems separately, tackling both problems simultaneously is challenging and has not been extensively studied yet. This thesis first tackles the problem mentioned above by introducing an iterative method to build an estimate of autonomous nonlinear DS that are formulated as a mixture of Gaussian functions. This method minimizes the number of Gaussian functions required for achieving both local asymptotic stability at the target and accuracy in following demonstrations. We then extend this formulation and provide sufficient conditions to ensure global asymptotic stability of autonomous DS at the target. In this approach, an estimation of the underlying DS is built by solving a constraint optimization problem, where the metric of accuracy and the stability conditions are formulated as the optimization objective and constraints, respectively. In addition to ensuring convergence of all motions to the target within the local or global stability regions, these approaches offer an inherent adaptability and robustness to changes in dynamic environments. This thesis further extends the previous approaches and ensures global asymptotic stability of DS-based motions at the target independently of the choice of the regression technique. Therefore, it offers the possibility to choose the most appropriate regression technique based on the requirements of the task at hand without compromising DS stability. This approach also provides the possibility of online learning and using a combination of two or more regression methods to model more advanced robot tasks, and can be applied to estimate motions that are represented with both autonomous and non-autonomous DS. Additionally, this thesis suggests a reformulation to modeling robot motions that allows encoding of a considerably wider set of tasks ranging from reaching movements to agile robot movements that require hitting a given target with a specific speed and direction. This approach is validated in the context of playing the challenging task of minigolf. Finally, the last part of this thesis proposes a DS-based approach to realtime obstacle avoidance. The presented approach provides a modulation that instantly modifies the robot’s motion to avoid collision with multiple static and moving convex obstacles. This approach can be applied on all the techniques described above without affecting their adaptability, swiftness, or robustness. The techniques that are developed in this thesis have been validated in simulation and on different robotic platforms including the humanoid robots HOAP-3 and iCub, and the robot arms KATANA, WAM, and LWR. Throughout this thesis we show that the DS-based approach to modeling robot discrete movements can offer a high level of adaptability, reactivity, and robustness almost effortlessly when interacting with dynamic environments

The derivatives of the SEDS optimization cost function and constraints with respect to the learning parameters

This technical report provides supplementary information for the optimization problems defined for Stable Estimator of Dynamical Systems (SEDS). Reading of this report is not necessary for researchers who only want to use SEDS learning algorithm. The report is aimed at helping those persons who want to develop SEDS, or to write their own optimization program. All the formulations reported here are developed for SEDS models; however, they can also be used for general Gaussian Mixture Model (GMM) formulations. In the case of the latter, they should be slightly modified to consider the general form of GMM. Hopefully, the report should be clear enough to help readers in that

BM: An Iterative Method to Learn Stable Non-Linear Dynamical Systems with Gaussian Mixture Models

We model the dynamics of non-linear discrete (i.e. point-to- point) robot motions as a time-independent system described by an autonomous dynamical system (DS). We propose an iterative algorithm to estimate the form of the DS through a mixture of Gaussian distributions. We prove that the resulting model is asymptotically stable at the target. We validate the accuracy of the model on a library of 2D human motions and to learn a control policy through human demonstrations for two multi- degrees of freedom robots. We show the real-time adaptation to perturbations of the learned model when controlling the two kinematically-driven robots

Comparative evaluation of approaches in T.4.1-4.3 and working definition of adaptive module

The goal of this deliverable is two-fold: (1) to present and compare different approaches towards learning and encoding movements us- ing dynamical systems that have been developed by the AMARSi partners (in the past during the first 6 months of the project), and (2) to analyze their suitability to be used as adaptive modules, i.e. as building blocks for the complete architecture that will be devel- oped in the project. The document presents a total of eight approaches, in two groups: modules for discrete movements (i.e. with a clear goal where the movement stops) and for rhythmic movements (i.e. which exhibit periodicity). The basic formulation of each approach is presented together with some illustrative simulation results. Key character- istics such as the type of dynamical behavior, learning algorithm, generalization properties, stability analysis are then discussed for each approach. We then make a comparative analysis of the different approaches by comparing these characteristics and discussing their suitability for the AMARSi project

Learning control Lyapunov function to ensure stability of dynamical system-based robot reaching motions

We consider an imitation learning approach to model robot point-to-point (also known as discrete or reaching) movements with a set of autonomous Dynamical Systems (DS). Each DS model codes a behavior (such as reaching for a cup and swinging a golf club) at the kinematic level. An estimate of these DS models are usually obtained from a set of demonstrations of the task. When modeling robot discrete motions with DS, ensuring stability of the learned DS is a key requirement to provide a useful policy. In this paper we propose an imitation learning approach that exploits the power of Control Lyapunov Function (CLF) control scheme to ensure global asymptotic stability of nonlinear DS. Given a set of demonstrations of a task, our approach proceeds in three steps: (1) Learning a valid Lyapunov function from the demonstrations by solving a constrained optimization problem, (2) Using one of the-state-of-the-art regression techniques to model an (unstable) estimate of the motion from the demonstrations, and (3) Using (1) to ensure stability of (2) during the task execution via solving a constrained convex optimization problem. The proposed approach allows learning a larger set of robot motions compared to existing methods that are based on quadratic Lyapunov functions. Additionally, by using the CLF formalism, the problem of ensuring stability of DS motions becomes independent from the choice of regression method. Hence it allows the user to adopt the most appropriate technique based on the requirements of the task at hand without compromising stability. We evaluate our approach both in simulation and on the 7 degrees of freedom Barrett WAM arm. (C) 2014 Elsevier B.V. All rights reserved

Imitation learning of Globally Stable Non-Linear Point-to-Point Robot Motions using Nonlinear Programming

Abstract — This paper presents a methodology for learning arbitrary discrete motions from a set of demonstrations. We model a motion as a nonlinear autonomous (i.e. time-invariant) dynamical system, and define the sufficient conditions to make such a system globally asymptotically stable at the target. The convergence of all trajectories is ensured starting from any point in the operational space. We propose a learning method, called Stable Estimator of Dynamical Systems (SEDS), that estimates parameters of a Gaussian Mixture Model via an optimization problem under non-linear constraints. Being time-invariant and globally stable, the system is able to handle both temporal and spatial perturbations, while performing the motion as close to the demonstrations as possible. The method is evaluated through a set of robotic experiments. I

BM: An Iterative Algorithm to Learn Stable Non-Linear Dynamical Systems with Gaussian Mixture Models

Abstract — We model the dynamics of non-linear point-topoint robot motions as a time-independent system described by an autonomous dynamical system (DS). We propose an iterative algorithm to estimate the form of the DS through a mixture of Gaussian distributions. We prove that the resulting model is asymptotically stable at the target. We validate the accuracy of the model on a library of 2D human motions and to learn a control policy through human demonstrations for two multidegrees of freedom robots. We show the real-time adaptation to perturbations of the learned model when controlling the two kinematically-driven robots. I