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Using infinitesimal symmetries for determining the first Maxwell time of geometric control problem on SH(2)
X 1 , . . . , X k , this yields a system of order k, con- manifold M of the system is three-dimensional,
sisting of equations of the form Equation (8). where, for any point q = (x, y, z) ∈ M, x and y
Next, we compute the Lie derivative of g. We are position variables and z is the angular orien-
have: tation variable of the unicycle on the hyperbolic
!
n plane. A motion m(x, y, z) on the configuration
X
L v (g) = L v a i dq i ⊗ dq i manifold M, parameterized by x, y, z ∈ R, is a
i=1 transformation that maps a point a(a 1 , a 2 ) to a
n
X point b(b 1 , b 2 ), such that:
= a i L v (dq i ⊗ dq i )
i=1 b 1 = a 1 cosh z + a 2 sinh z + x,
n
X b 2 = a 1 sinh z + a 2 cosh z + y.
= 2a i dq i L v (dq i )
Composition of two motions m 1 (x 1 , y 1 , z 1 ) and
i=1
n m 2 (x 2 , y 2 , z 2 ) is another motion m 3 (x 3 , y 3 , z 3 )
X
= 2a i dq i (i(v) d(dq i ) + d(i(v) dq i )) given as:
i=1 m 3 (x 3 , y 3 , z 3 ) = m 1 (x 1 , y 1 , z 1 ).m 2 (x 2 , y 2 , z 3 )
n
X where,
= 2a i dq i d(i(v) dq i )
x 3 = x 2 cosh z 1 + y 2 sinh z 1 + x 1 ,
i=1
n
X y 3 = x 2 sinh z 1 + y 2 cosh z 1 + y 1 ,
= 2a i dq i d(dq i (v))
z 3 = z 1 + z 2
i=1
where i(v), is the interior product. Condition The identity motion m Id is given by x = y = z =
L v (g) = 0 provides a system of order n(n+1) , 0, and inverse of a motion m(x, y, z) is given by
1
1
1
2 m −1 (x , y , z ) where,
consisting of equations of the form:
1
n n n x = −x cosh z + y sinh z,
∂h 1 ∂h n
X X X
k i h i + l i + . . . + r i = 0, (9) y = x sinh z − y cosh z,
1
∂q i ∂q i
i=1 i=1 i=1
1
z = −z.
where k i , l i , and r i are some scalar functions.
Soving the system associated with Equations (8)
The motions of the pseudo-Euclidean plane
and (9), we determine the functions h i and, con-
exhibit a group structure, with composition serv-
sequently, the generators v i of our Lie algebra of
ing as the group operation. This group, known as
symmetries. As a consequence, we can state the
the special hyperbolic group SH(2), also possesses
following result:
a smooth manifold structure, which qualifies it as
Proposition 2. The infinitesimal symmetries of a Lie group. One can equivalently formulate prob-
control system Equations (2)-(4) can be identified lem Equation (10) as a sub-Riemannian problem
by calculating the flow associated with the vector on SH(2):
field v i at time s, denoted by
˙ q = u 1 X 1 (q) + u 2 X 2 (q),
γ i (s, t) = (q 1 (s, t), . . . , q n (s, t)). This flow is ob- 2
tained by solving the system of differential equa- q ∈ M = SH(2), u = (u 1 , u 2 ) ∈ R ,
tions: q(0) = q 0 = m Id , q(t 1 ) = q 1 , (11)
∂γ i (s, t) Z t 1
= v i (γ i (s, t)), γ i (0, t) = (q 1 (t), . . . , q n (t)). 1 2 2
∂s J = u + u 2 dt → min
1
2 0
4. Infinitesimal symmetries on SH(2) where,
and the first Maxwell time
X 1 = cosh z ∂ x + sinh z ∂ y , X 2 = ∂ z .
4.1. Geometric control problem on SH(2) The problem under consideration is formulated
on a sub-Riemannian manifold (M, ∆, g). Here,
The motion of a unicycle on a hyperbolic plane
can be described using the following driftless con- M represents the underlying smooth manifold,
∆ = span{X 1 , X 2 } denotes a distribution of rank
trol system
 2, and g is a Riemannian metric defined on ∆ such
 ˙x = u 1 cosh z,
that g(X i , X j ) = δ ij , explicitly g = dx −dy +dz .
 2 2 2
˙ y = u 1 sinh z, (10)
The vector fields X 1 and X 2 , which span the dis-

˙ z = u 2 , tribution ∆, are left-invariant with respect to the

where u 1 is the translational velocity and u 2 is group structure associated with M. One can see
the angular velocity. The configuration and state that the family
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