Automatic Sensor Frame Identification in Industrial Robots with Joint Elasticity∗

For robots with joint elasticity, discrepancies exist between the motor side information and the load side (i.e., end-effector) information. Therefore, high tracking performance at the load side can hardly be achieved when the estimate of load side information is inaccurate. To minimize such inaccuracies, it is desired to calibrate the load side sensor (in particular, the exact sensor location). In practice, the optimal placement of the load side sensor often varies due to the task variation necessitating frequent sensor calibrations. This frequent calibration need requires significant effort and hence is not preferable for industries which have relatively short product cycles. To solve this problem, this paper presents a sensor frame identification algorithm to automate this calibration process for the load side sensor, in particular the accelerometer. We formulate the calibration problem as a nonlinear estimation problem with unknown parameters. The Expectation-Maximization algorithm is utilized to decouple the state estimation and the parameter estimation into two separated optimization problems. An overall dual-phase learning structure associated with the proposed approach is also studied. Experiments are designed to validate the effectiveness of the proposed algorithm. NOMENCLATURE ql Load side position ∈ R n q̇m Motor side position ∈ R n q̇l Load side velocity ∈ R n ∗THIS WORK WAS SUPPORTED BY FANUC CORPORATION, JAPAN. REAL-TIME CONTROLHARDWAREAND SOFTWAREWERE PROVIDED BY NATIONAL INSTRUMENTS, INC. q̇m Motor side velocity ∈ R n q̈l Load side acceleration ∈ R n q̈m Motor side acceleration ∈ R n Ml(ql) Load side inertia matrix ∈ R n×n Mm Motor side inertia matrix (diagonal) ∈ R n×n Mn Nominal Load side inertia matrix (diagonal) ∈ R n×n C(ql , q̇l) Coriolis and centrifugal force matrix ∈ R n×n N Gear ratio matrix (diagonal) ∈ Rn×n G(ql) Gravity vector ∈ R n Dl Load side damping matrix (diagonal) ∈ R n×n Dm Motor side damping matrix (diagonal) ∈ R n×n DJ Joint damping matrix (diagonal) ∈ R n×n KJ Joint stiffness matrix (diagonal) ∈ R n×n τm Motor torque input ∈ R n fm(q̇m) Friction effect on motor side ∈ R n fl(q̇l) Friction effect on load side ∈ R n fext External force acting on end-effector ∈ R 6 dm Motor side fictitious disturbance torque ∈ R n dl Load side fictitious disturbance torque ∈ R n al Translational acceleration ∈ R 3 al,s Measurement of translational acceleration ∈ R 3 qm,s Measurement of motor side position ∈ R n R i Rotation matrix describing the frame {i} relative to the frame {i−1} ∈ R3×3 J̄s(ql ;ξs) The first three rows of the Jacobian matrix mapping from the load side joint space to the sensor frame Cartesian space ∈ R3×n ̄̇ Js(ql ;ξs) Derivative of the first three rows of the Jacobian matrix Js(ql ;ξs) ∈ R 3×n J(ql) Jacobian matrix mapping from the load side joint space to the end-effector Cartesian space ∈ R6×n nm Measurement noise of motor-side encoder ∈ R n na Measurement noise of accelerometer ∈ R 3 nw Processes noise ∈ R n ql,s Measurement of load side position ∈ R n INTRODUCTION For industrial robots with indirect drive mechanisms (e.g., gear transmissions), the load side encoder is often not available due to the cost and assembly issues. Using motor side encoder signals for feedback control, however, may not guarantee satisfactory control performance at the load side due to the robot joint dynamics. To overcome this problem, it has been suggested to attach a low-cost MEMS accelerometer at the robot end-effector to measure the translational acceleration [1–3]. Then, the accelerometer measurements are incorporated with the motor encoder signals to obtain an estimate of the end-effector position and/or velocity. Although these methods have successfully improved the load side tracking performance [4], there still remain various practical issues when applying these techniques to industrial robot systems. For example, the accelerometer location must be precisely known, but manually calibrating the sensor location requires a long period of time. Thus, the sensor based estimation scheme is difficult for industrial applications which involve frequent changes of sensor location due to task variations. More specifically, the load side accelerometer is manually mounted at the location which requires the most attention for performance improvement. Such locations may change from one task to another due to task variations. The accelerometer may be mounted magnetically to the robot end-effector so that the accelerometer location can be changed easily. In such situation, the sensor frame location needs to be identified every time when the sensor moves to a new location. A standard sensor frame calibration method [5] is to command the robot to move such that the designated sensor location touches a reference point with different postures. Then, the sensor frame location can be computed using the robot posture information and the corresponding accelerometer measurements. This calibration process requires significant effort. Therefore, developing an algorithm that can automatically detect the accelerometer’s mounting position and orientation becomes an important issue. In this work, we aim to design an algorithm which deals with the load side sensor frame identification problem using only accelerometer and motor encoder measurements over a designed trajectory. The main difficulty of this problem is that the unknown system parameters (i.e., the sensor frame location) have complex dependencies on the immeasurable system states (i.e., the load side information). Such problem is usually referred to as the “parameter estimation using incomplete data” problem, which has been studied by a number of researchers [6–8]. Among the algorithms presented in this area, the ExpectationMaximization (EM) algorithm is computationally simple and compatible with the state space model. While the algorithms presented in [7] only works for linear state space model, we propose a modified algorithm that can deal with the specific non-linear problem in this paper. This paper begins with the introduction of a standard robot dynamic model and parameterizing the sensor frame location using the Denavit-Hartenberg parameters [9]. Then, a stochastic model is designed to parameterize the model uncertainties in terms of the statistics in a Gaussian distribution. Once the model is established, the EM algorithm is applied to accomplish the state estimation and parameter estimation simultaneously. The extended Kalman filter, extended Kalman smoother, and Monte Carlo integration [10] techniques are used to obtain the information we needed in the EM algorithm. This paper also introduces a dual-phase learning structure in which the unknown parameters can be estimated systematically. The effectiveness of the proposed load side sensor frame identification algorithm is validated by experimentation on a single joint robot test-bed. SYSTEM Multi-joint Indirect Drive Model Consider an n-joint manipulator with elastic joints described by [4]: Ml (ql) q̈l +C (ql , q̇l) q̇l +G(ql)+Dl q̇l + fl(q̇l) = KJ ( Nqm−ql ) +DJ ( Nq̇m− q̇l ) − J (ql) fext Mmq̈m+Dmq̇m+ fm(q̇m) = τm−N −1 (