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A Bayesian Approach to Nonlinear Parameter Identification for Rigid A Bayesian Approach to Nonlinear Parameter Identification for Rigid Body Dynamics Jo-Anne Ting∗, Michael Mistry∗, Jan Peters∗, Stefan Schaal∗† and Jun Nakanishi†‡ ∗Computer Science, University of Southern California, Los Angeles, CA, 90089-2520 †ATR Computational Neuroscience Labs, Kyoto 619-0288, Japan ‡ICORP, Japan Science and Technology Agency Kyoto 619-0288, Japan Email: {joanneti, mmistry, jrpeters, sschaal}@usc.edu, [email protected] Abstract— For robots of increasing complexity such as hu- robots such as humanoid robots have significant additional manoid robots, conventional identification of rigid body dynamics nonlinear dynamics beyond the rigid body dynamics model, models based on CAD data and actuator models becomes due to actuator dynamics, routing of cables, use of protective difficult and inaccurate due to the large number of additional nonlinear effects in these systems, e.g., stemming from stiff shells and other sources. In such cases, instead of trying to wires, hydraulic hoses, protective shells, skin, etc. Data driven explicitly model all possible nonlinear effects in the robot, parameter estimation offers an alternative model identification empirical system identification methods appear to be more method, but it is often burdened by various other problems, useful. Under the assumption that a rigid body dynamics such as significant noise in all measured or inferred variables (RBD) model is sufficient to capture the entire robot dynamics, of the robot. The danger of physically inconsistent results also exists due to unmodeled nonlinearities or insufficiently rich data. this problem is theoretically straightforward as all unknown In this paper, we address all these problems by developing a parameters of the robot such as mass, center of mass and Bayesian parameter identification method that can automatically inertial parameters appear linearly in the rigid body dynamics detect noise in both input and output data for the regression equations [3]. Hence, after an appropriate re-arrangement of algorithm that performs system identification. A post-processing the RBD equations of motion, parameter identification can be step ensures physically consistent rigid body parameters by nonlinearly projecting the result of the Bayesian estimation onto performed with linear regression techniques. constraints given by positive definite inertia matrices and the Several problems, however, make this seemingly straight- parallel axis theorem. We demonstrate on synthetic and actual forward empirical RBD system identification approach more robot data that our technique performs parameter identification challenging. Firstly, for high dimensional robotic systems, it is with 5 to 20% higher accuracy than traditional methods. Due not easy to generate sufficiently rich data so that all parameters to the resulting physically consistent parameters, our algorithm enables us to apply advanced control methods that algebraically will be properly identifiable. Secondly, sensory data collected require physical consistency on robotic platforms. from a robot is noisy. Noise sources exist in both input and output data, and this effect is additionally amplified by I. INTRODUCTION numerical differentiation done to obtain derivative data from Advanced robot control algorithms usually rely on model- the sensors. Traditional linear regression techniques are only based control techniques in order to accomplish a desired level capable of dealing with output noise, and the presence of input of accuracy and compliance. Typical examples include com- noise introduces a persistent bias to the regression solution. puted torque control, inverse dynamics control and operational Digital filtering of the data can eliminate some noise, but it space control [1], [2]. Depending on their sophistication, these often eliminates important structure as well. Techniques exist model-based controllers also have different levels of demands such as Total Least Squares (TLS) [4], [5], otherwise known as on the quality of the identified robot model. For instance, orthogonal least-squares regression [6], to perform parameter computed torque control is generally rather insensitive to estimation, but it assumes that the variances of input noise and modeling errors, while operational space control, with its output noise are the same [7]. In real-world systems where this explicit use of both the rigid body dynamics inertia matrix and assumption does not hold, the resulting parameter estimates the centripetal/Coriolis force vector, degrades significantly in will be biased [8]. Thirdly, there is no mechanism in the face of modeling errors. Thus, accurate model identification regression problem for RBD model identification that ensures is a highly important topic for advanced robot control, and the identified parameters are physically plausible. Particularly many modern robotics applications rely on it (e.g., as in haptic in the light of insufficiently rich data and nonlinearities beyond robotic devices, robotic surgery and the safe application of the RBD model, one often encounters physically incorrectly compliant assistive robots in human environments). identified parameters such as negative values on the diagonal Ideally, system identification can be performed based on the of an inertial matrix. Using physically incorrect data in model- CAD data of a robot provided by the manufacturer, at least based control leads to dangerously unstable controllers. The in the context of rigid body dynamic systems—which will be final problem with empirical RBD system identification is that the scope of this paper. However, many modern light-weight some RBD parameters of a robot are not identifiable at all [3]. As a result, the regression problem for RBD parameter estima- to be suitable for this dimensionality. The re-arrangement of tion is almost always numerically ill-conditioned and bears the the RBD equations for parameter estimation creates a matrix danger of generating parameter estimates that strongly deviate where the input vectors x are arranged in the rows of the from the true values, despite a seemingly low error fit of the matrix X and the corresponding scalar outputs y are the data. coefficients of the vector y. A general model for such linear Various methods exist to deal with some of the problems regression with noise-contaminated input and output data is: mentioned above such as regression based on singular-value d decomposition (SVD), ridge regression to cope with the ill- y = wzmtm + y conditioned data [9], or TLS and orthogonal-least squares (1) mX=1 to address input noise [6]. Nevertheless, a comprehensive x = w t + approach to address the entire set of issues has not been m xm m xm suggested so far. Recent work such as [10] has addressed where d is the number of input dimensions, t is noiseless input the problem of input noise, but in the context of system data composed of tm components, wz and wx are regression identification of a time-series, while ignoring the problems as- vectors composed of wzm and wxm components respectively, sociated with ill-conditioned data in high dimensional spaces. and y and x are additive mean-zero noise. Only X and y In this paper, we suggest a Bayesian estimation approach to are observable. Note that if the input data is noiseless (that the RBD parameter estimation problem that has all the desired is, xm = wxmtm) and if we set wxm = 1, then we obtain T properties: the familiar linear regression equation of y = wz x + y for • Explicitly identifies input and output noise in the data noiseless input data and noisy output data. The slightly more • Is robust in face of ill-conditioned data general formulation above will be useful in preparing our novel • Detects non-identifiable parameters estimation algorithm. • Produces physically correct parameter estimates The OLS estimate of the regression vector βOLS is T −1 T A key component of our technique is a recently developed (X X) X y, where βOLS is composed of the parameters Bayesian machine learning framework that enables us to recast wz and wx, as discussed in the following paragraph. The ordinary least squares (OLS) regression in a more advanced first major issue with OLS regression in high dimensional T −1 algorithm for input noise clean-up and numerical robustness, spaces is that the full rank assumption of (X X) is often especially for very high dimensional estimation problems. A violated due to underconstrained datasets. For more than 500 post-processing step ensures that the rigid body parameters are input dimensions, the matrix inversion required in OLS also physically consistent by nonlinearly projecting the results of becomes rather expensive. Ridge regression can fix the prob- the Bayesian estimate onto the constraints. We will sketch the lem of ill-conditioned matrices by introducing an uncontrolled derivation of this algorithm and compare its results with other amount of bias. There exist also alternative methods to invert approaches in the context of identification of RBD parameters the matrix more efficiently [11], [12], as for instance through on synthetic data and on a robotic vision head. On average, our singular value decomposition factorization (e.g., using the algorithm achieves a 5 to 20% improvement when compared Matlab pinv() function). Nevertheless, all these methods are to other standard techniques for the identification of RBD unable to model noise in input data and require the manual parameters. tuning of meta parameters, which can strongly influence the The remaining paper is structured as follows. First, we quality
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