Using an Extended Kalman Filter for Rigid Body Pose Estimation

[+] Author and Article Information
Kjartan Halvorsen

Biomechanics and Motor Control, Stockholm University College of Physical Education and Sports, Stockholm, Swedenkjartanh@ihs.se

Torsten Söderström, Virgil Stokes, Håkan Lanshammar

Department of Information Technology Uppsala University, Uppsala, Sweden

J Biomech Eng 127(3), 475-483 (Dec 06, 2004) (9 pages) doi:10.1115/1.1894371 History: Received February 24, 2004; Revised November 16, 2004; Accepted December 06, 2004

Rigid body pose is commonly represented as the rigid body transformation from one (often reference) pose to another. This is usually computed for each frame of data without any assumptions or restrictions on the temporal change of the pose. The most common algorithm was proposed by Söderkvist and Wedin (1993, “Determining the Movements of the Skeleton Using Well-configured Markers  ,” J. Biomech., 26, pp. 1473–1477), and implies the assumption that measurement errors are isotropic and homogenous. This paper describes an alternative method based on a state space formulation and the application of an extended Kalman filter (EKF). State space models are formulated, which describe the kinematics of the rigid body. The state vector consists of six generalized coordinates (corresponding to the 6 degrees of freedom), and their first time derivatives. The state space models have linear dynamics, while the measurement function is a nonlinear relation between the state vector and the observations (marker positions). An analytical expression for the linearized measurement function is derived. Tracking the rigid body motion using an EKF enables the use of a priori information on the measurement noise and type of motion to tune the filter. The EKF is time variant, which allows for a natural way of handling temporarily missing marker data. State updates are based on all the information available at each time step, even when data from fewer than three markers are available. Comparison with the method of Söderkvist and Wedin on simulated data showed a considerable improvement in accuracy with the proposed EKF method when marker data was temporarily missing. The proposed method offers an improvement in accuracy of rigid body pose estimation by incorporating knowledge of the characteristics of the movement and the measurement errors. Analytical expressions for the linearized system equations are provided, which eliminate the need for approximate discrete differentiation and which facilitate a fast implementation.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

Definition of rotations. Position and orientation of the axes of rotation in reference position. Axis ξ4 is fixed in the lab frame. Note that only the last axis ξ6, is fixed in the moving body.

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Figure 2

MSE in marker position. The vertical, dotted lines indicate the beginning and end of intervals with missing data. In the first and third intervals one marker was missing, and in the second and fourth intervals two markers were missing. The solid line corresponds to the EKF with the cardanic linkage description, the dashed line to the EKF with helical axis description, and the dotted line to the SVD method. The measurement noise was white and isotropic with variance 0.25. The results are based on 100 repetitions.

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Figure 3

Mean square of error in marker position. Isotropic measurement noise of variance 0.1. Intervals with missing data are indicated by the vertical dotted lines. Results based on 100 repetitions.

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Figure 4

MSE over motion sequence plotted against noise variance. Simulation results based on 200 repetitions. At a certain low level of the variance, the errors in the interpolation over the gaps in the marker data dominate, and no further improvement is seen in the MSE. The solid line corresponds to the EKF, the dashed line to the SVD/GCVSPL method, and the dash-dot line to the SVD/LP method.

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Figure 5

MSE over motion sequence plotted against degree of anisotropy. The variable γ denotes the ratio of the largest to smallest eigenvalue of the noise covariance matrix. Simulation results based on 100 repetitions. The solid line corresponds to the EKF, the dashed line to the SVD/GCVSPL method, and the dash-dot line to the SVD/LP method.

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Figure 6

MSE in marker position (distance squared) over the motion sequence. Result based on 100 repetitions. The dashed line corresponds to the SVD method and the dash-dot line to the EKF with known structure of the acceleration covariance matrix R¯a. The solid line is the result when the acceleration covariance matrix was not known.



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