Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 71

principle of virtual work to the two force systems. As for the joint torques, its
associated virtual work is

T

δ W τ 5 τ δ q (4.4)

T

where δ W is the virtual work and τ denotes the transpose of τ . For the

T , comprised of a force vector f and a moment
vector m, the virtual work performed is
δ

5 ½ f T m T

end-effector forces F

W F 5 f T δ x 1 m T

ωδ t (4.5)

where t is the angular virtual displace-

ment of the end-effector, respectively. Because the difference between the virtual
work of the joint torques and the virtual work of the end-effector forces shall be
null for all joint virtual displacements, we write

δ x is the linear virtual displacement and ωδ

T

F T J ð q Þδ

τ δ q q q (4.6)

5 '

The relationship between the joint torque vector and end-effector force/

moment vector is then given by

J T F τ 5 (4.7)

T .
Now, we extend this formulation to the case where multiple external loads
(both translational and rotational) are applied to any location of any link, not nec-
essarily to the end-effector. Let's assume that a general form of external load F k
is applied to the point at

τ 5 ½τ 1 ; τ 2 ; ...; τ n

where the torque vector is

k r k location of link k, where k r k location vector is

expressed with respect to k th local coordinate frame.
This point of application of external load can be regarded as the end-effector
for the corresponding external load. The augmented Jacobian matrix J k for this
point is derived from the linear relationship between the joint velocity vector and
the Cartesian velocity vector:

2 3

0 T 1 ð q Þ
@

0 T i ð q Þ
@

0 T k ð q Þ
@

@ @ @

k r k k r k k r k

... ...

4 5 6 3 k

J k ð q Þ 5 q 1 q i q k (4.8)

Z 0 ð q Þ ... Z i 2 1 ð q Þ ... Z k 2 1 ð q Þ

where, i k is the local z-axis vector of joint i expressed in terms of the

global coordinate system.
Therefore the joint torque vector due to the external load applied at point

1 ; ...;

5

k r k

of link k is

J k F k (4.9)
From the principle of superposition, the total joint torques due to several
external loads is obtained as a sum of all joint torques:

τ k 5

X

J k F k τ 5 (4.10)

k