Incremental Semiparametric Inverse Dynamics Learning Raffaello Camoriano ∗† , Silvio Traversaro ‡ , Lorenzo Rosasco ⋄ , Giorgio Metta △ , and Francesco Nori ‡ Abstract— This paper presents a novel approach for incre- mental semiparametric inverse dynamics learning. In partic- ular, we consider the mixture of two approaches: Parametric modeling based on rigid body dynamics equations and non- parametric modeling based on incremental kernel methods, with no prior information on the mechanical properties of the system. This yields to an incremental semiparametric approach, leveraging the advantages of both the parametric and nonparametric models. We validate the proposed technique learning the dynamics of one arm of the iCub humanoid robot. I. INTRODUCTION In order to control a robot a model describing the relation between the actuator inputs, the interactions with the world and bodies accelerations is required. This model is called the dynamics model of the robot. A dynamics model can be obtained from first principles in mechanics, using the techniques of rigid body dynamics (RBD) [1], resulting in a parametric model in which the values of physically mean- ingful parameters must be provided to complete the fixed structure of the model. Alternatively, the dynamical model can be obtained from experimental data using Machine Learning techniques, resulting in a nonparametric model. Traditional dynamics parametric methods are based on several assumptions, such as rigidity of links or that friction has a simple analytical form, which may not be accurate in real systems. On the other hand, nonparametric methods based on algorithms such as Kernel Ridge Regression (KRR) [2], [3], [4], Kernel Regularized Least Squares (KRLS) [5] or Gaussian Processes [6] can model dynamics by extrapolat- ing the input-output relationship directly from the available data 1 . If a suitable kernel function is chosen, then the nonparametric model is a universal approximator which can account for the dynamics effects which are not considered by the parametric model. Still, nonparametric models have ∗ Corresponding author. † Raffaello Camoriano is with iCub Facility, Istituto Italiano di Tec- nologia, Via Morego 30, Genoa 16163, Italy, and DIBRIS, Universit` a degli Studi di Genova, Via All’Opera Pia, 13, Genoa 16145, Italy. Email: raffaello.camoriano@iit.it ‡ Silvio Traversaro and Francesco Nori are with RBCS Department, Istituto Italiano di Tecnologia, Via Morego 30, Genoa 16163, Italy. Email: name.surname@iit.it ⋄ Lorenzo Rosasco is with LCSL, Istituto Italiano di Tecnologia and Massachusetts Institute of Technology, Cambridge, MA 02139, USA, and DIBRIS, Universit` a degli Studi di Genova, Via All’Opera Pia, 13, Genoa 16145, Italy. Email: lrosasco@mit.edu △ Giorgio Metta is with iCub Facility, Istituto Italiano di Tecnologia, Via Morego 30, Genoa 16163, Italy. Email: giorgio.metta@iit.it 1 Note that KRR and KRLS have a very similar formulation, and that these are also equivalent to the techniques derived from Gaussian Processes, as explained for instance in Chapter 6 of [4]. TABLE I: Summary of related works on semiparametric or incremental robot dynamics learning. Author, Year Parametric Nonparametric Nguyen-Tuong, 2010 [7] Batch Batch Gijsberts, 2011 [8] - Incremental Tingfan Wu, 2012 [9] Batch Batch De La Cruz, 2012 [10] CAD ∗ Incremental Camoriano, 2015 Incremental Incremental ∗ In [10] the parametric part is used only for initializing the nonparametric model. no prior knowledge about the target function to be approxi- mated. Therefore, they need a sufficient amount of training examples in order to produce accurate predictions on the entire input space. If the learning phase has been performed offline, both approaches are sensitive to the variation of the mechanical properties over long time spans, which are mainly caused by temperature shifts and wear. Even the inertial parameters can change over time. For example if the robot grasps a heavy object, the resulting change in dynamics can be described by a change of the inertial parameters of the hand. A solution to this problem is to address the variations of the identified system properties by learning incrementally, continuously updating the model as long as new data becomes available. In this paper we propose a novel technique that joins parametric and nonparametric model learning in an incremental fashion. Classical methods for physics-based dynamics modeling can be found in [1]. These methods require to identify the mechanical parameters of the rigid bodies composing the robot [11], [12], [13], [14], which can then be employed in model-based control and state estimation schemes. In [7] the authors present a learning technique which combines prior knowledge about the physical structure of the mechanical system and learning from available data with Gaussian Process Regression (GPR) [6]. A similar approach is presented in [9]. Both techniques require an offline training phase and are not incremental, limiting them to scenarios in which the properties of the system do not change significantly over time. In [10] an incremental semiparametric robot dynamics learning scheme based on Locally Weighted Projection Re- gression (LWPR) [15] is presented, that is initialized using a linearized parametric model. However, this approach uses a fixed parametric model, that is not updated as new data becomes available. Moreover, LWPR has been shown to underperform with respect to other methods (e.g. [8]). arXiv:1601.04549v1 [stat.ML] 18 Jan 2016