390 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRYum
g
l = pendulum length
m = pendulum bob mass
M =cart mass
g = acceleration due to gravityFigure 4.2. The pendulum on a cart system.The configuration space is Q = JR x S^1 and the velocity phase space, TQ
has coordinates (s, B, 8, B). The velocity of the cart relative to the lab frame is 8
and the velocity of the pendulum is the vector ( 8 + l cos() e ) -l sin() B). Thus, the
system kinetic energy isK((s, B, 8, B). =^1 "2(8, B). (
and so the Lagrangia n isM+m
ml cos()ml cos()
ml^2L(s,B,8,B) = K(s, B,8,B) - V(B),
where the potential energy is V = mgl cos().
) ( ~))
The symmetry group is that of translation in the variable s , so G = R We
do not destroy this symmetry when doing stabilization in e.Controlled Cart. Write the a bove Lagrangian as
L(s, e, 8, B) = ~ ( aB^2 + 2,BcosB 3() + 182 ) + Dcos e,
where / = M + m, ,8 = ml, a = ml^2 and D = -mgl. Positive definiteness of the
mass matrix (the Riemannian metric) corresponds to a1-,8^2 > 0. The momentum
conjugate to sis Ps = 18 +,8 cos() B. The relative equilibrium()= 0, iJ = 0 is unstable
since D < 0. This upright state is what we wish to stabilize. The equations of
motion of the cart-pendulum system subject to a control force u acting on the cart
(and no direct forces on the pendulum) are the controlled cart equations:
d BL
dt B8 = u
.!!:._ B~ _ BL = O.
dt Be BeControlled Lagrangian. The controlled Lagrangian is defined by modifying
the kinetic energy only (potentials are reserved for tracking). Let er be a real scalarand introduce a one form T = k(B)dB. Define
Lr,a = ~ [ae
2
+2,BcosB(8+k(B)B)B+1(8+k(B)B)^2 ] + ~ 1 [k(B)]
2
B^2 +DcosB.