The final step is to convert the equations into phase space form. A -dimensional phase vector is introduced as . The task is to obtain , which represents scalar equations. The first equations are for from to . The final equations are obtained by solving for .

Suppose that the general form in (13.142) is used. Solving for yields

The phase variables are then substituted in a straightforward manner. Each for from to becomes , and , , and are expressed in terms of . This completes the specification of the phase transition equation.

and the potential energy of is

The kinetic energy of is

in which denotes the position of the center of mass of and is given from (3.53) as

The potential energy of is

At this point, the Lagrangian function can be formed as

and inserted into (13.118) to obtain the differential constraints in implicit form, expressed in terms of , , and . Conversion to the phase space is performed by solving the implicit constraints for and assigning , in which is a four-dimensional phase vector.

Rather than performing the computations directly using (13.118), the constraints can be directly determined using (13.140). The terms are

(13.155) |

in which

(13.156) |

and

(13.157) |

The final term is defined as

(13.158) |

The dynamics can alternatively be expressed using , , and in (13.142). The Coriolis matrix is defined using (13.143) to obtain

in which is defined in (13.152) and is a function of . For convenience, let

The resulting expression, which is now a special form of (13.142), is

The phase transition equation is obtained by letting
and substituting the
state variables into (13.161). The variables
and
become
and
,
respectively. The equations must be solved for
and
. An extension of this model to motors that have gear ratios and
nonnegligible mass appears in [856].

The example provided here barely scratches the surface on the possible systems that can be elegantly modeled. Many robotics texts cover cases in which there are more links, different kinds of joints, and frictional forces [366,725,856,907,994].

The phase transition equation for chains of bodies could alternatively be derived using the Newton-Euler formulation of mechanics. Even though the Lagrangian form is more elegant, the Newton-Euler equations, when expressed recursively, are far more efficient for simulations of multibody dynamical systems [366,863,994].

Steven M LaValle 2020-08-14