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q ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
u_1
e1ðrÞdrds
x €
1 x € 2
Re-ordering the last equation leads to the following expression:
The kinematic model of the system is stated as follows:
2ðsÞds
u_1
h ¼ u2
k2s2 þ k1s þ k0 ðyðsÞ
y ðsÞÞ sðs þ k3Þ
4
(26)
euðsÞ¼
¼
forces the system
(30)
to asymptotically track the given reference trajectories ðx 1ðtÞ; x
e2ðtÞ ¼ x2ðtÞ x 2ðtÞ
(25)
5
x_2x € 1
x_1x € 1 þ x_2x €
2 u_1 ¼
(31)
x_ 2ðtÞ
The idealized model of a single axis two wheeled vehicle is depicted in Fig. 1. The axis is of
length L and each wheel of radius R is powered by a direct current motor yielding variable angular
speeds x1, x2, respectively. The position variables in X1 and X2 axes are (x1, x2), and h is the
orientation angle of the robot. The linear velocities of the points of contact of the wheels respect to
the ground are given by v1 ¼ x1R and v2 ¼ x2R. In this case, the only measurable variables are x1,
x2. This system is subject to non holonomic restrictions.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
(22)
(23)
The relationship between the inputs and the flat outputs highest derivatives is not invertible due
to an ill defined relative degree.
ðtÞÞ.
x_
1ðtÞ ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
(32)
7
7
7
8 x_1 ¼ u1 cos h
051001-4 / Vol. 137, MAY 2015
þ x_
x_ 1ðtÞÞ k1; 2e1ðtÞ
1ðsÞds
The control objective is stated as follows: given a set of desired output trajectories: ðx 1ðtÞ; x
2ðtÞÞ, devise output feedback control laws u1, u2, such that the output coordinates (x1, x2) perform
an asymptotic tracking of the given trajectories which is also capable of rejecting some additive
constant disturbances affecting the non linear output behavior.
This simple example suggests that GPI control is equivalent to classical robust compensator design,
with no need for asymptotic state estimators.
x_1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
>:
To overcome this fact, let us introduce, as an extended auxiliary control input, the time derivative of
u1. We have
(36)
2ðtÞ ¼ x € 2ðtÞ k2; 3ð ^ x_2ðtÞ
uðsÞ ¼ s
x €
2x_1 u2 ¼
with
(35)
Transactions of the ASME
2 q ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2 ðtÞ
4 Controller Design This system is
differentially flat, with flat outputs given by the pair of coordinates (x1, x2) which describe the
position of the mid dle point of the axis connecting the wheels. These two variables differentially
parameterize all system variables, including the inputs, as follows:
4
(30)
e2ðrÞdrds
s
4.1 Feedforward Linearization Control Scheme. In this case, the exact feedforward linearization
scheme will be used to solve the problem of controlling the nonholonomic robot. The following
theorem details the proposed control laws and their effects.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
k2s2 þ k1s þ k0 eyðsÞ
sðs þ k3Þ
(29)
þ x_
This control input extension yields now an invertible relation between the control inputs and the
flat outputs highest derivatives.
(27)
(34)
3
k1; 3ð ^ x_1ðtÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
_
Finally,
6
6
6
5
> <
y ðsÞ
6
6
6
6
u1 ¼
This relation is of the form
ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
(33)
x_2
¼
x_ 1ðtÞ
(28)
h ¼ arctanð Þ x_2 = x_1
x_ 2ðtÞÞ k2; 2e2ðtÞ
THEOREM 8. Given a set of initial conditions, (x1 (0), x2 (0)), suffi ciently close to the initial
value of the given nominal trajectories, ðx 1ðtÞ; x 2ðtÞÞ, the following multivariable dynamical
feedback with trol law
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
3
x_1
(37)
euðsÞ
euðsÞ¼ k3
þ x_
(24)
^ x_1 ¼ ðt
^ x_2 ¼ ðt
7
7
7
7
1ðtÞ ¼ x € 1ðtÞ
2 p x_2 þ x_
s
e1ðtÞ ¼ x1ðtÞ x 1ðtÞ
x_2 ¼ u1 sin h