8.1 Inverse dynamics calculations

A common problem encountered by the elderly is loss of balance, which sometimes results in falling. One biomechanical hypothesis suggests that falling in the elderly results from a lack of sufficient muscle strength. To study the issue of strength, suppose we asked a number of elderly women to rise from a chair in our motion lab. Our aim is to estimate the net joint torques for the major joints of the lower extremity during this task. Once we calculate the joint torques used to stand up from a chair, we can compare them to the maximum joint torques that our subjects can generate (maximum joint torques are a measure of muscle strength). This comparison will show what percentage of our subjects’ maximum strength must be used to rise from a chair.

Suppose we used high-speed video cameras, together with markers positioned judiciously over various body landmarks to estimate the time history of the absolute angular displacements of the shank, thigh, and trunk (i.e.

, , , in Figure 2). These data were subsequently filtered and numerically differentiated to obtain the absolute angular velocities and accelerations of each of these body segments over time. Assuming that the sit-to-stand activity can be characterized by predominantly planar (2-D) motion of all the body segments, a simple three-segment, planar linkage is used to model the human skeleton (Figure 2). Further assume that the body segments are connected by frictionless single degree-of-freedom revolute (i.e., hinge) joints.

In terms of kinematic data only:

(a) Please derive complete analytical expressions for the net joint torques exerted at the ankle (

), knee (), and hip () during sit-to-stand motion. Assume that the foot remains flat on the floor.

(b) Please derive complete analytical expressions for the ground reaction forces exerted in the horizontal (fore-aft) and vertical directions during the sit-to- stand motion. Again assume that the foot remains flat on the floor and that it is massless. Note that if you could measure

 and  that you could calculate 

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without any kinematic data.

(c) Would you consider this problem to be inverse or direct dynamics? Why?

(d) What is the physical significance of the intersegmental forces (e.g.,

)? Are the calculated intersegmental forces equal to the true bone-on-bone forces? Why or why not?

Figure 1. Planar 3 segment model