A key component of our dynamical modeling of the biped is the use
of the Reduced Dynamics Algorithm presented in Section 
of the
Appendix. We have already mentioned that because of the contact
constraints in phase one and in phase two of walking, we are faced with a
differential-algebraic system.
Two courses of actions are possible when it is necessary to integrate
the dynamics, one being the use of specially
tailored integration routines which often require the partial
derivatives of the various contact constraints.
The preferable approach, however, is to use a reduced unconstrained
set of dynamics which evolve on the constraint manifold.
Then it is possible to use standard integration procedures.
Recall that one of the primary difficulties of the Reduced Dynamics Algorithms is that the inverse kinematics must be used to solve for the dependent states. For the biped, the first task is to solve for the angle which determines the position of each leg. This is easy since our problem is equivalent to solving for the joint angles of a 2-link manipulator when its endpoints are known.
The following well-known solution comes from
Spong and Vidyasagar [13].
In
Figure 9
is displayed the inverse kinematics problem.
Let 
and 
be the desired joint angles,
and 
the lengths of the upper and lower legs
respectively, and 
and 
two intermediate
angles.
We assume that one end of the 2-link arm has been transferred to
the origin while the other end
has coordinates (x,y). From the Law of Cosines,
Also, 
. In our case,
since the knees only bend in one direction, the minus sign is
always used so that
Now may be obtained easily from two intermediate
angles and .
Let the angle 
while
.
The final expression for is
The dependent position states then are
and 
.
Now we turn to computing the velocities
along the manipulator.
The velocities are known at both ends of the 2-link
manipulator, so
the Collision Algorithm can be used to give the
joint velocities uniquely.
This is done by initializing the algorithm for a 2-link manipulator
with the known spatial
velocity at the joint connecting the upper leg and the torso.
The joint angles have been previously calculated and the joint
velocities are arbitrary as there will only be one solution.
The updated velocities will then correspond to the values of
the states 
and 
.
