Joint Energetics
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Joint Power
Muscle Power
Work Done by the Muscles
Segment Energetics
References & Related Literature

Joint Power

Power is defined as the rate of work or the rate of energy flow. Two power measures can be obtained from the joint kinetics: the joint power and the muscle power. The joint power is the scalar product of the net joint force and the joint velocity:

    [1]

where P(j) = the joint power, F = the net joint force, and v = the velocity of the joint. Note here that power is a scalar: it does not have direction. As shown in Figure 1, the joint power of the foot and shank at the ankle are

    [2]

   [3]

where FAK/FT = the net joint force acting on the foot at the ankle, FAK/SH = the net joint force acting on the shank at the ankle, and vAK = the ankle velocity. [3] shows that the foot and the shank show joint powers of the same magnitude with the opposite sign at the ankle. This suggests that the ankle joint only transfers energy from foot to the shank, vice versa. Between the two segments forming a joint, one segment always gains energy at the rate the other loses it, vice versa.

Precisely speaking, the joint power is the rate of energy transfer through the joint caused by the linear motion of the joint.

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Muscle Power

The muscle power is defined as the scalar product of the joint torque and the segment's angular velocity:

   [4]

where P(M) = the muscle power, T = the joint torque, and w = the angular velocity. For example, the muscle powers of the foot and the shank at the ankle (Figure 2) are

   [5]

   [6]

where TAK/FT = the ankle joint torque acting on the foot, TAK/SH = the ankle joint torque acting on the shank, wFT = the angular velocity of the foot, and wSH = the angular velocity of the shank. It appears that there is no apparent relationship between the muscle powers of the foot and shank since the angular velocities of the foot and shank can be very different.

At the muscle, two things happen: (1) the muscle transfers energy from one segment it attaches to to the other, and (2) the muscle does a work through contraction. The energy transfer happens from one segment to the other and the net change in the energy in the two segments due to the energy transfer must be 0. On the other hand, the muscle can either add (positive work) or drain (negative work) energy to or from both segments at the same time by doing a work. At any rate, sum of the two muscle powers shown in [5] and [6] must yield

   [7]

where P(M)AK = the rate of work done by the muscles at the ankle.

If the ankle joint torque and the difference in the angular velocity are in the same direction, P(M)AK becomes positive, suggesting concentric contraction of the muscle while a negative power means eccentric activation of the muscle.

On thing needs to be understood clearly here is the fact that the works done by the muscles to both segments involved in a joint are not necessarily equal. The actual rates of work done by the muscles to the individual segments depend on the moments of inertia of the segments. The only thing guaranteed here is that the torques exerted by the muscles to the segments are equal as the Third Law of Motion clearly states.

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Work Done by the Muscles

The amount of work done by muscles at a joint can be computed from the rate of work at the joint through time-integration. For example, the work done by the muscles at the ankle (Figure 2) is

    [8]

where to = the initial time of the movement, t1 = the final time, WAK = the amount of work done by the muscles at the ankle. [8] is in fact equal to the area under the muscle power-time curve. Throughout a movement, the muscle power can show both positive and negative phases. Computing the amount of work done in each phase is very important in the sense that it really gives the amount of efforts put by the muscles either concentrically and eccentrically. Integrating the muscle power-time curve for the entire phase is somewhat meaningless since the positive and negative phases cancel each other and one can only get the net work done. The muscles consume energy regardless of the mode of contraction (concentric vs. eccentric) and it is only reasonable to add all the positive and negative works done.

This can be generalized as

    [9]

where i = phase number, to(i) = the initial time of the i-th phase, t1(i) = the final time, and W(+/-) = the absolute amount of work done by muscles regardless of the contraction mode. [9] is commonly used to compute the amount of internal work done by muscles and the efficiency of the human machine.

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Segment Energetics

From [5] and [6], the overall rate of change in the mechanical energy of the foot and shank are

   [10]

where PSEGMENT = the overall power (rate of change in the mechanical energy) of the segment, PJOINT/SEGMENT = the energy flow rate into the segment at a joint, P(E) = the power (rate of work) caused by the external forces such as the ground reaction force, PE = the potential energy of the segment, KE = the kinetic energy of the segment, and ME = the mechanical energy of the segment. [10] basically says that the energy flow through the joints and muscles into a segment at both ends (proximal & distal) and the work done by the external forces cause change in the mechanical energy (kinetic energy + potential energy) of the segment.

In the foot, the power caused by the external forces is likely to be zero in most cases since either the foot is stationary on the ground (zero velocity), or the foot is off the ground (zero ground reaction force). If the foot moves while maintaining a contact with the ground such as in slipping or skating, there will be a negative work done by the ground reaction force to the foot. If the main focus is on the work done by the muscles, this power or work is of course not that important.

From [10], the change in the mechanical energy of the foot and shank are

    [11]

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References and Related Literature

Winter, D.A. (1990). Biomechanics and motor control of human movement (2nd Ed.), New York, NY: John Wiley & Sons, Inc.

Winter, D.A. (1983). Moments of force and mechanical power in jogging. J. Biomechanics 16, 91-97.

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Young-Hoo Kwon, 1998-