Estimation of model parameters in a dynamic system can be significantly improved with the choice of experimental trajectory. For general nonlinear dynamic systems, finding globally “best” trajectories is typically not feasible; however, given an initial estimate of the model parameters and an initial trajectory, we present a continuous-time optimization method that produces a locally optimal trajectory for parameter estimation in the presence of measurement noise. The optimization algorithm is formulated to find system trajectories that improve a norm on the Fisher information matrix (FIM). A double-pendulum cart apparatus is used to numerically and experimentally validate this technique. In simulation, the optimized trajectory increases the minimum eigenvalue of the FIM by three orders of magnitude, compared with the initial trajectory. Experimental results show that this optimized trajectory translates to an order-of-magnitude improvement in the parameter estimate error in practice.
A common metric used in these areas of experimental design---also the key metric in this paper---is the Fisher information matrix computed from observations of the system trajectory @cite_20 . Metrics on the Fisher information are used as a cost function in many optimization problems including work by Swevers on exciting'' trajectories @cite_23 . This work, as well as related works @cite_33 @cite_10 , synthesize trajectories for nonlinear systems that can be recast as linear systems with respect to the parameters.
@cite_10 @cite_23 @cite_33 @cite_20
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A common way to identify the inertial parameters of robots is to use a linear model in relation to the parameters and standard least-squares (LS) techniques. This article presents a method to generate exciting identification trajectories in order to minimize the effect of noise and error modeling on the LS solution. Using nonlinear optimization techniques, the condition number of a matrix W obtained from the energy model is minimized, and the scaling of its terms is carried out. An example of a three-degree-of-freedom robot is presented. This paper discusses experimental robot identification based on a statistical framework. It presents a new approach toward the design of optimal robot excitation trajectories, and formulates the maximum-likelihood estimation of dynamic robot model parameters. The differences between the new design approach and the existing approaches lie in the parameterization of the excitation trajectory and in the optimization criterion. The excitation trajectory for each joint is a finite Fourier series. This approach guarantees periodic excitation which is advantageous because it allows: 1) time-domain data averaging; 2) estimation of the characteristics of the measurement noise, which is valuable in the case of maximum-likelihood parameter estimation. In addition, the use of finite Fourier series allows calculation of the joint velocities and acceleration in an analytic way from the measured position response, and allows specification of the bandwidth of the excitation trajectories. The optimization criterion is the uncertainty on the estimated parameters or a lower bound for it, instead of the often used condition of the parameter estimation problem. Simulations show that this criterion yields parameter estimates with smaller uncertainty bounds than trajectories optimized according to the classical criterion. Experiments on an industrial robot show that the presented trajectory design and maximum-likelihood parameter estimation approaches complement each other to make a practicable robot identification technique which yields accurate robot models. When designing an identification experiment for a system described by nonlinear functions such as those of manipulator dynamics, it is necessary to consider whether the excitation is sufficient to provide an accurate estimate of the parameters in the presence of experimental noise. It is shown that the convergence rate and noise immunity of a parameter identifi cation experiment depend directly on the condition number of the input correlation matrix, a measure of excitation. The sensitivity of an identification experiment to unmodeled dynamics is also studied; a dimensionless measure of this sensitivity—bias susceptibility—is proposed and related to excitation. The issue of how exciting a trajectory may be is addressed, and a method is presented to maximize the exci tation. Two identification experiments reported in the litera ture are studied; analysis of these experiments shows that intuitively selected trajectories may provide poor excitation, and considerable improvement results from employing the opt... Preparations. Unbiasedness. Equivariance. Global properties. Large-sample theory. Asymptotic optimality. Author index. Subject index.