Page 283 - Autonomous Mobile Robots

270 Autonomous Mobile Robots

matrix, K N ∈ R r×r is a positive definite diagonal matrix which character-
izes the electromechanical conversion between current and torque, I denotes
an r-element vector of armature current, J(q) ∈ R (n−m)×n is the matrix
associated with the constraint, and λ ∈ R n−m is the vector of constraint
forces. The terms L = diag[L 1 , L 2 , L 3 , ... , L r ], R = diag[R 1 , R 2 , R 3 , ... , R r ],
T
r
K a = diag[K a1 , K a2 , K a3 , ... , K ar ], ω =[ω 1 , ω 2 , ... , ω r ] , and ν ∈ R rep-
resent the equivalent armature inductances, resistances, back emf constants,
angular velocities of the driving motors, and the control input voltage vector,
respectively. Constraint 7.1 is assumed to be completely nonholonomic for all
n
q ∈
and t ∈
. To completely actuate the nonholonomic system, B(q) is
assumed to be a full-rank matrix and r ≥ m.
Dynamic subsystem (7.2) has the following properties [31,32]:

Property 7.1 There exists a so-called inertial parameter p and vector θ with
components depending on the mechanical parameters (mass, moment of inertia,
etc.,) such that

M(q)˙v + C(q, ˙q)v + G(q) = (q, ˙q, v, ˙v)θ (7.4)

where is a matrix of known functions of q, ˙q, v, and ˙v; and θ is a vector of
inertia parameters and assumed completely unknown in this chapter.

˙
Property 7.2 M − 2C is skew-symmetric.
T
T
If matrix N ∈ R n×n is skew-symmetric, then N =−N and Y NY = 0 for
n
all Y ∈ R .
Since J(q) ∈ R (n−m)×n , it is always realizable to find an m rank matrix
S(q) ∈ R n×m formed by a set of smooth and linearly independent vector fields
spanning the null space of J(q), that is,
T
T
S (q)J (q) = 0 (7.5)
Since S(q) =[s 1 (q), ... , s m (q)] is formed by a set of smooth and linearly
independent vector fields spanning the null space of J(q), define an auxiliary
T
m
time function v =[v 1 , ... , v m ] ∈ R such that
˙ q = S(q)v(t) = s 1 (q)v 1 + ··· + s m (q)v m (7.6)

Equation (7.6) is the so-called kinematic model of nonholonomic systems in
the literature.
Differentiating Equation (7.6) yields

˙
¨ q = S(q)v + S(q)˙v (7.7)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 270 — #4