The Method of Inverse Dynamics

The method of inverse dynamics is increasingly being used to analyze the sequential ordering of body segment movements during an athletic event. The method is also useful to compute the joint moment that is resisted by muscle action in various modes of movement. Inverse dynamics is based on the experimental determination of velocity and acceleration terms that appear in the laws of motion. These laws are then used to evaluate the unknown forces and moments acting on parts of the body. As we illustrate in the following examples, the position and velocity of a point as well as the angular velocity of a body segment can be determined with reasonable accuracy by recording human movement with the use of digital cameras. Computational errors may be markedly larger in the evaluation of acceleration from data. Researchers use various numerical algorithms to enhance the accuracy of the inverse dynamics method. The sports mechanics literature is full of interesting articles that estimate muscle forces and muscle torques involved in baseball pitching and golf swing. Reader is referred to the references at the end of the book for an introduction to this rich area of biomechanics.

Example 9.9. Determination of Cartesian Coordinates of a Spatial Point Using Two Cameras. Two synchronized 500 frames/s cameras were used to capture the position of various markers on the throwing arms and shoulder of a baseball pitcher. For marker P placed near the elbow of the pitcher, the camera located at point O (0,0,0) and facing the positive e2 direction produced digital images which at time t = 5 s gave the following coordinates along the coordinate axes: (1.34 m, u, 2.15 m). Note that this camera could not yield information on the distance between the camera and the point in the e2 direction and therefore u is not known. The other camera, positioned at point A (-3.00 m, 2.00 m, 1.00 m) and facing the positive e1 direction, produced the following coordinates for point P at time t = 5 s: P = (v, 0.89, 1.11) where v is unknown. Determine the position of P with respect to the reference frame E positioned at point O. Evaluate the accuracy of the results.

Solution: According to the parallelogram law of vectors:

This equation can also be written in terms of its components in the E reference frame xP/O = xA/O + xP/A => 1.34 = -3 + xP/A => xP/A = 1.66 m yP/O = yA/O + yP/A => yP/O = 2 + 0.89 = 2.89 m zP/O = zA/O + zP/A = 2.15 * 1 + 1.11 = 2.11

Thus, with respect to the frame E at point O:

The percentage of error e = [(2.15 - 2.11)/2.15] X 100 = 1.9%.

Example 9.10. Orientation of the Throwing Arm. The three-dimensional coordinate values for two landmarks (reflective hemispheres of 20 mm in diameter), one positioned next to the shoulder joint and the other on the elbow, were determined using three cameras. The marker next to the shoulder joint was identified by the symbol SH and the marker next to the elbow by EL. Positions of SH and EL at time t = 5 s were as follows:

These positions were determined again 0.1 s later: rSH = 2.38 e1 + 1.85 e2 + 0.20 e3 rEL = 2.65 e1 + 2.01 e2 + 0.27 e3

Determine the unit vector b along the line segment from SH to EL. What is the time rate of change of b at time t = 5 s?

Solution: We compute b at t = 5 s and at t = 5.1 s, and identify these vectors as b- and b+, respectively. The time rate of change of b will be approximated by using the following finite-difference formula:

b- = [0.07 e1 + 0.25 e2 + 0.19 e3]/(0.0049 + 0.0625 + 0.0361)05 = 0.22 e1 + 0.78 e2 + 0.59 e3

b+ = [0.27 e1 + 0.16 e2 + 0.07 e3]/(0.0729 + 0.0256 + 0.0049)05 = 0.84 e1 + 0.50 e2 + 0.22 e3

db/dt = (0.62 e1 - 0.28 e2 - 0.37 e3)/0.1 = 6.2 e1 - 2.8 e2 - 3.7e3

Example 9.11. Angular Velocity of the Throwing Arm. In an investigation of the mechanics of arm swing, reflective markers were used to construct an orthogonal unit vector axis system with its origin at the gleno-humeral joint. The first unit vector b1 was chosen along the longitudinal axis of the upper arm. The second unit vector b2 was constructed perpendicular to b1 and was taken along the direction of the rotation axis for upper arm adduction/abduction. A third unit vector, b3, was constructed perpendicular to b1 and b2. This vector is in the direction of the rotation axis for upper arm flexion and rotation. Using the method of the previous example, the following vector quantities were determined at time t = 5 s:

b1 = 0.22 e1 + 0.78 e2 + 0.59 e3 db1/dt = 6.2 e1 - 2.8 e2 + 3.7 e3

where e1 are fixed on earth.

Determine the third unit vector b3 and its time derivative db3/dt. Determine the angular velocity of B in reference frame E and express it in terms of unit vectors in B. Solution:

db3/dt = db1/dt X b2 + b1 X db2/dt = (—e1 — 3.6 e2 — e3)

+ ( 8.3e1 + 8.0 e2 — 13.5e3) = 7.3e1 + 4.4e2 — 14.5e3

From the definition of angular velocity given in Eqn. 9.5:

W1 = (Edb2/dt) • b3 = 15.2 X (—0.16) + (—7.6) X (—0.68) + 4.9 X 0.80 = 6.7 rad/s w2 = (Edb3/dt) • b1 = 7.3 X 0.22 + 4.4 X 0.78 — 14.5 X 0.59 = —3.5 rad/s

W3 = (Edb1/dt) • b2 = 6.2 X (—0.96) — 2.8 X 0.27 + 3.7 X 0 = —6.7 rad/s

The component of the angular velocity along b1 direction may be associated with the twisting moment along the longitudional axis of the upper arm. The component along the b2 direction may be altered by muscles that cause abduction/adduction in the frontal plane. The angular velocity components in the b3 direction, on the other hand, are dependent on the muscles that flex/extend the upper arm. The projections of angular velocity along the unit vectors bi are called anatomical angular velocity components.

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