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Research Papers

# Optimal Estimation of Dynamically Consistent Kinematics and Kinetics for Forward Dynamic Simulation of Gait

[+] Author and Article Information
C. David Remy

Department of Mechanical Engineering, University of Wisconsin–Madison, 1513 University Avenue, Madison, WI 53706

Darryl G. Thelen

Department of Mechanical Engineering, University of Wisconsin–Madison, 1513 University Avenue, Madison, WI 53706thelen@engr.wisc.edu

J Biomech Eng 131(3), 031005 (Jan 06, 2009) (9 pages) doi:10.1115/1.3005148 History: Received August 19, 2007; Revised July 16, 2008; Published January 06, 2009

## Abstract

Forward dynamic simulation provides a powerful framework for characterizing internal loads and for predicting changes in movement due to injury, impairment or surgical intervention. However, the computational challenge of generating simulations has greatly limited the use and application of forward dynamic models for simulating human gait. In this study, we introduce an optimal estimation approach to efficiently solve for generalized accelerations that satisfy the overall equations of motion and best agree with measured kinematics and ground reaction forces. The estimated accelerations are numerically integrated to enforce dynamic consistency over time, resulting in a forward dynamic simulation. Numerical optimization is then used to determine a set of initial generalized coordinates and speeds that produce a simulation that is most consistent with the measured motion over a full cycle of gait. The proposed method was evaluated with synthetically created kinematics and force plate data in which both random noise and bias errors were introduced. We also applied the method to experimental gait data collected from five young healthy adults walking at a preferred speed. We show that the proposed residual elimination algorithm (REA) converges to an accurate solution, reduces the detrimental effects of kinematic measurement errors on joint moments, and eliminates the need for residual forces that arise in standard inverse dynamics. The greatest improvements in joint kinetics were observed proximally, with the algorithm reducing joint moment errors due to marker noise by over 20% at the hip and over 50% at the low back. Simulated joint angles were generally within $1deg$ of recorded values when REA was used to generate a simulation from experimental gait data. REA can thus be used as a basis for generating accurate simulations of subject-specific gait dynamics.

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## Figures

Figure 1

Schematic of the residual elimination algorithm. Dynamically consistent accelerations are estimated based on the measured data, and are then numerically integrated to simulate motion. The initial conditions for this integration are optimized to yield maximal agreement of simulated and measured motions.

Figure 2

The residual elimination analysis was used to determine initial states that minimized the sum of weighted squared distances between model predicted and measured marker positions. The 38 markers that contributed to this objective function were the same as those used in the inverse kinematics routine. Higher weighting factors (numbers in brackets) were put on anatomical markers placed on the pelvis and lower extremity. Smaller weights were used for torso and upper extremity markers, and the additional tracking markers were placed on the lateral side of the thigh and shank.

Figure 3

Traditional inverse dynamics analysis produces substantial residual forces and moments (solid lines) that have no physical meaning, as shown for this sample experimental data set. The proposed algorithm eliminates these residuals completely (dashed lines) by estimating accelerations that are dynamically consistent with the whole body equations of motion.

Figure 4

Shown are the pelvic coordinates and joint angles estimated using inverse kinematics (solid lines) and the residual elimination algorithm (dashed lines) for one of the experimental data sets. A very good agreement is seen with average differences being generally less than 1deg for rotations and less than 3mm for translations.

Figure 5

Average joint moment estimation error with a 9mm SD noise. Values are given for the flexion/extension axis of lower back, hip, knee, and ankle joints, and are computed with respect to noise-free synthetic data.

Figure 6

The errors in marker positions (top), pelvic translations (middle), and joint angles (bottom) of the noise polluted data are compared to values after data processing with REA. The algorithm was able to reduce marker noise by about 9% and pelvic position errors by about 19%. Joint angle errors remained approximately the same.

Figure 7

REA was used to estimate constant offsets between the optical tracking system and the force plate reference frame to reduce the adverse effect of calibration errors. In this example, a constant offset, overlaid on a 9mm SD white noise process, has almost no influence on marker estimation error after REA is applied.

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