Biomedical Engineering Reference
In-Depth Information
Bidirectional forces (force motors and actuators) are treated as two forces
equal in magnitude and opposite in direction, acting on two segments.
Equation (8.32) is then modified to include the other end point. Similarly,
an external torque or torque motor i acting between frames j and k is given
by the list:
trq(i )
=
[ j , k , τ x , τ y , τ z ]
(8.33)
where x , τ y , τ z ) are the components of the externally applied torque. The
contribution of the externally applied forces and torques to the generalized
forces of the system is obtained by:
δW
δq i
Q i
=
(8.34)
where W is the total work done by the external forces and torques [similar
to Equation (8.6)].
8.5
DESIGNATION OF JOINTS
Under Lagrangian dynamics, there are two ways of treating joints in a given
model because of the fact that action and reaction at joints do not perform
work. If joint reaction forces are not required, then a joint can be viewed as
a common point between two adjacent segments. To obtain joint forces, a
joint is set apart into two points, each on the appropriate segment LRS. An
equation, known as loop closure (or constraint ) equation , is then written to
state the relation between these points. The reaction forces are those forces
λ that maintain the constraint. In the case of pin joints or ball and socket
joints, the relation is that the vector between the two points is zero. Other
joint types are beyond the limitations of this introductory topic.
8.6
ILLUSTRATIVE EXAMPLE
A standing human is modeled as an inverted pendulum. The three elements
shown in Figure 8.5 a represent the leg, thigh, and trunk segments. The seg-
ment lengths are represented by l . The locations of the segment centers of
mass are represented by r , all measured from the distal end. A horizontal
disturbance force F is applied at the hip. Horizontal feet disturbances were
simulated as a fixed acceleration
x of the ground contact point. The joint
torques T 1 , T 2 , and T 3 represent the ankle, knee, and hip torques, respectively.
The system has four variables, q 1 (
¨
θ 3 ) .
Therefore, four differential equations are expected. As a first step, the model is
replaced by the appropriate references LRS and points pt . Model parameters
=
x ) , q 2 (
=
θ 1 ) , q 3 (
=
θ 2 ) , and q 4 (
=
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