Maximum voluntary joint torque as a function of joint angle and angular velocity: model development and application to the lower limb

Abstract

Measurements of human strength can be important during analyses of physical activities. Such measurements have often taken the form of the maximum voluntary torque at a single joint angle and angular velocity. However, the available strength varies substantially with joint position and velocity. When examining dynamic activities, strength measurements should account for these variations. A model is presented of maximum voluntary joint torque as a function of joint angle and angular velocity. The model is based on well-known physiological relationships between muscle force and length and between muscle force and velocity and was tested by fitting it to maximum voluntary joint torque data from six different exertions in the lower limb. Isometric, concentric and eccentric maximum voluntary contractions were collected during hip extension, hip flexion, knee extension, knee flexion, ankle plantar flexion and dorsiflexion. Model parameters are reported for each of these exertion directions by gender and age group. This model provides an efficient method by which strength variations with joint angle and angular velocity may be incorporated into comparisons between joint torques calculated by inverse dynamics and the maximum available joint torques.

Figure 1.

Maximum knee extension torque measurements from a single subject (left), and the corresponding maximum torque surface estimated by the model (right). Positive joint angle indicates flexion. Positive angular velocity indicates concentric motion, while negative indicates eccentric motion.

Figure 2.

To quantify the overall predictive abilities of the model, linear regression was performed between the predicted and measured joint torques for all subjects. The data and regression line for knee extension torques are shown. R² values are given for all six exertion directions.

Figure 3.

Torque surfaces created using means of parameters reported in Table 3. These surfaces illustrate the variation of maximum joint torque (N-m) with joint angle (deg) and angular velocity (deg/s) as well as the marked differences between joints and exertion directions. Positive joint angle indicates flexion (dorsiflexion). Positive angular velocity indicates concentric motion, while negative indicates eccentric motion.