An analytical model to quantify the impact of the propagation of uncertainty in knee joint angle computation

Abstract

Joint kinematics are typically described using Cardan angles or the attitude vector and its projection on the joint axes. Whichever the notation used, the uncertainties present in gait measurements affect the computed kinematics, especially for the knee joint. One notation - the attitude vector - enables the derivation of an analytical model of the propagation of uncertainty. Thus, the objective of this study was to derive this analytical model and assess the propagation of uncertainty in knee joint angle computation. Multi-session gait data acquired from one asymptomatic adult participant was used as reference data (experimental mean curve and standard deviations). Findings showed that an input uncertainty of 5° in the attitude vector and joint axes parameters matched experimental standard deviations. Taking each uncertainty independently, the cross-talk effect could result from uncertainty in the orientation of either the attitude vector (intrinsic variability) or the first joint axis (extrinsic variability). We concluded that the model successfully estimated the propagation of input uncertainties on joint angles and enabled an investigation of how that propagation occurred. The analytical model could be used to a priori estimate the standard deviations of experimental kinematics curves based on expected intrinsic and extrinsic uncertainties.

Keywords: Euler and cardan angles; Joint coordinate system; attitude vector; cross-talk; knee kinematics; reproducibility.

Figure 1.

Representation of definition of the cardan angles and axis (A) and attitude vector (B) for the knee joint.

Figure 2.

Comparison between the experimental variabilities and the theoretical standard uncertainties (u) corresponding to 2°, 5° and 10° of input uncertainty in rotation angle θ, the orientation of the rotation axis k and in the orientation of joint axes e₁ and e₃. The dotted blue line separates the stance and swing phases. The presented output uncertainties represent the movement of flexion-extension (u(θ₁)), varus-valgus (u(θ₂)) and internal-external rotation (u(θ₃)).

Figure 3.

Impact of the standard input uncertainty (u) of 5° in the rotation angle θ, the in the orientation of the rotation axis k and in the orientation of joint axes e₁ and e₃ on knee joint angles. The solid blue line and the blue corridor represent the mean and standard deviation of inter-session experimental variability, respectively, and the red corridor represents the theoretical standard uncertainty (u). The dotted blue line separates the stance and swing phases. The presented output uncertainties represent the movement of flexion-extension (u(θ₁)), varus-valgus (u(θ₂)) and internal-external rotation (u(θ₃)).