Twolevel control scheme for stabilisation of periodic orbits for planar monopedal running
ABSTRACT This study presents an online motion planning algorithm for generating reference trajectories during flight phases of a planar monopedal robot to transfer the configuration of the mechanical system from a specified initial pose to a specified final one. The algorithm developed in this research is based on the reachability and optimal control formulations of a timevarying linear system with input and state constraints. A twolevel control scheme is developed for asymptotic stabilisation of a desired periodone orbit during running of the robot. Withinstride controllers, including stance and flight phase controllers, are employed at the first level. The flight phase controller is a feedback law to track the reference trajectories generated by the proposed algorithm. To reduce the dimension of the fullorder model of running, the stance phase controller is chosen to be a parameterised timeinvariant feedback law that produces a family of twodimensional finitetime attractive and invariant submanifolds. At the second level, the parameters of the stance phase controller are updated by an eventbased update law to achieve hybrid invariance and stabilisation. To illustrate the analytical results developed for the behaviour of the closedloop system, a detailed numerical example is presented.

Article: Nonholonomic motion planning based on optimal control for flight phases of planar bipedal running
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ABSTRACT: Presented is a novel approach for online trajectory modification of joint motions to transfer a free open kinematic chain, undergoing flight phase, from a specified initial configuration to a specified final configuration. Formally, it is assumed that a nominal trajectory, computed offline, can reorient the kinematic chain (reconfiguration problem) for a given angular momentum on a time interval. A modification algorithm of body joints, based on optimal control, is developed such that for different angular momentums and time intervals, the same reconfiguration problem can be solved online. This approach can be utilised for space robotics applications and online computation of planar running trajectories during flight phases.Electronics Letters 10/2011; · 1.07 Impact Factor
Page 1
Published in IET Control Theory and Applications
Received on 3rd September 2010
Revised on 9th February 2011
doi:10.1049/ietcta.2010.0512
ISSN 17518644
Twolevel control scheme for stabilisation of periodic
orbits for planar monopedal running
N. Sadati1,2
G.A. Dumont2
K.Akbari Hamed1
W.A. Gruver3
1Intelligent Systems Laboratory, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran
2Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4
3School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
Email: sadati@sharif.edu; sadati@ece.ubc.ca
Abstract: This study presents an online motion planning algorithm for generating reference trajectories during flight phases of a
planar monopedal robot to transfer the configuration of the mechanical system from a specified initial pose to a specified final one.
The algorithm developed in this research is based on the reachability and optimal control formulations of a timevarying linear
system with input and state constraints. A twolevel control scheme is developed for asymptotic stabilisation of a desired period
one orbit during running of the robot. Withinstride controllers, including stance and flight phase controllers, are employed at the
first level. The flight phase controller is a feedback law to track the reference trajectories generated by the proposed algorithm. To
reduce the dimension of the fullorder model of running, the stance phase controller is chosen to be a parameterised timeinvariant
feedback law that produces a family of twodimensional finitetime attractive and invariant submanifolds. At the second level, the
parameters of the stance phase controller are updated by an eventbased update law to achieve hybrid invariance and stabilisation.
To illustrate the analytical results developed for the behaviour of the closedloop system, a detailed numerical example is
presented.
1Introduction
This paper presents an analytical approach for designing a
twolevel control law to asymptotically stabilise a desired
periodone orbit during running by a planar monopedal
robot. The monopedal robot is a threelink, twoactuator
planar mechanism in the sagittal plane with point foot. It is
assumed that the model of monopedal running can be
expressed by a hybrid system with two continuous phases,
including stance phase (one leg on the ground) and flight
phase (no leg on the ground), and discrete transitions
between the continuous phases, including takeoff and
landing (impact) [1, 2, Chap. 9].
The configuration of the mechanical system is specified
by the absolute orientation with respect to an inertial
world frame and by the joint angles determining the
shape of the robot. During the flight phase, the angular
momentum of the mechanical system about its centre of
mass (COM) is conserved. To reduce the dimension of
the fullorder hybrid model of running, which in turn
simplifies the stabilisation problem of the desired orbit, as
proposed by Chevallereau et al. [1], we desire that the
configuration of the mechanical system can be transferred
from a specified initial pose (immediately after the
takeoff) to a specified final pose (immediately before the
landing) during flight phases. This problem is referred to
as ‘landing in a fixed configuration or configuration
determinism at landing’ [1, 2, p. 252]. However, the flight
time and angular momentum about the COM may differ
during
reconfiguration problem must be solved online. A number
of control problems for reconfiguration of a planar
multilink robot with zero angular momentum have been
considered in the literature, for example [3–6]. For the
case that the angular momentum is not necessarily zero,
Kolmanovsky et al. [7] presented a method based on the
averaging theorem [8, Theorem 2.1] such that for any
value of the angular momentum, joint motions can
reorient the multilink arbitrarily over an arbitrary time
interval. However, when the angular momentum is not
zero, this method cannot be employed online for solving
the reconfiguration problem for monopedal running. For
this reason, we present an online reconfiguration algorithm
that solves this problem for given flight times and angular
momentums. The algorithm proposed in this paper is
expressed using the methodology of reachability and
optimal control for timevarying linear systems with input
and state constraints. This algorithm can also be utilised
for online generation of C2trajectories for free open
kinematic chains, conserving angular momentum about
their COM.
Probably the most basic tool for analysing the stability of
periodicorbitsoftimeinvariant
described by ordinary differential equations is the Poincare ´
return map. Grizzle et al. [9] showed that the Poincare ´
return map can be applied to systems with impulse effects
for analysing the stability of periodic orbits. To reduce the
dimension of the Poincare ´ return map during bipedal
consecutive steps.Consequently, the
dynamicalsystems
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walking with one degree of underactuation, the strategy of
using virtual constraints has been developed by Grizzle
et al. in [1, 2, 9–17]. For coordination of robot links, a set
of holonomic output functions, referred to as virtual
constraints, are defined and imposed to be zero by a
feedback law [18]. For the case that the corresponding zero
dynamics manifold is impact invariant, Westervelt et al.
[10] introduced the concept of hybrid zero dynamics (HZD)
which in turn results in a onedimensional restricted
Poincare ´ returnmapwith
By using the virtual constraints approach, Chevallereau
et al. [1] proposed the configuration determinism at landing
to obtain a closedform expression for the onedimensional
restricted Poincare ´ return map of running by a fivelink,
fouractuator planar bipedal robot. Moreover, to ensure that
the stance phase zero dynamics manifold is hybrid invariant
[2, p. 96] under the closedloop hybrid model of running,
an additional constraint was imposed on the vector of
generalised velocities at the end of flight phases. To satisfy
the configuration determinism at landing and hybrid
invariance, [1] utilised the approach of parameterised HZD.
Specifically, on the basis of the implicit function theorem
and a numerical nonlinear optimisation problem with an
equality constraint, the parameters of the virtual constraints
of the flight phase were updated in a stepbystep fashion
during the discrete transition from stance to flight (i.e.
takeoff). However, the stance phase controller was assumed
to be fixed.
The main contribution of this paper is to present an
analytical approach for online generation of modified
reference trajectories during flight phases of running to
satisfytheconfiguration
Moreover, by relaxing the constraint of [1] on the vector
of generalised velocities at the end of the flight phases,
we present a twolevel control scheme based on the
reconfiguration algorithm to asymptotically stabilise a
desiredperiodic orbit.In
controllers, including stance and flight phase controllers,
areemployedatthefirst
controllerisanalogousto
differences. In particular, it is chosen as a timeinvariant
and parameterised feedback law to generate a family of
finitetime attractive, zero dynamics manifolds. Unlike the
approach of [1], a continuous feedback law is employed
to track the modified reference trajectories generated by
the reconfiguration algorithm during the flight phase. To
generate a family of hybrid invariant manifolds, an event
based controller updates the parameters of the stance
phase controller during the transition from flight to stance
(i.e. impact). The terminology of an eventbased controller
is taken from [2, p. 199, 11–13].
This paper is organised as follows. Section 2 summarises
the equations of motion during the stance and flight phases
and assembles them into a hybrid model of running. In
Section 3, we treat the reconfiguration problem and an
online reconfiguration algorithm is presented. Constructive
proofs are given based on the formulation of reachability
for a timevarying linear system with input and state
constraints. The control laws for the stance and flight
phases are developed in Section 4. The timeinvariant
feedback law during the stance phase is similar to those
developed in [1, 10]. In Section 5, the stabilisation
problem is studied. Finally, simulation results of the
closedloop system in Section 6 confirm the validity of the
analyticalresults and Section 7 contains concluding
remarks.
a closedformexpression.
determinismatlanding.
this scheme,withinstride
level.
that
The
of
stance
with
phase
some[1]
2 Mechanical model of a monopedal runner
2.1Monopedal runner
A planar threelink monopedal robot with two ideal
revolute joints and point foot (see Fig. 1) is considered
throughout this paper. The joints are controlled by internal
actuators. Also, it is assumed that torques cannot be applied
at the leg end.
2.2Dynamics of the flight and stance phases
A convenient choice of the configuration variables consists of
thebody angles,theabsolute
absolute position of the monopedal with respect to the
world frame. The body angles represented by w :¼ (w1, w2)′
describe the shape of the robot, where prime denotes
matrix transpose. The absolute orientation of the robot is
represented by u, and the absolute position is represented by
the Cartesian coordinates of its COM, pcm:¼ (xcm, ycm)′.
Consequently,thegeneralised
flight phase are defined as qf:¼ (w′, u, pcm′)′¼ (q′, pcm′)′,
where q :¼ (w′, u)′. Following the notations used in
[1, 2, Chap. 3], the dynamical model during the flight
phasecanbeexpressedasthefollowingsecondorderequation
orientationandthe
coordinates duringthe
A(w)¨ q +?C(w, ˙ q)˙ q = Bu
(1)
¨ xcm= 0(2)
¨ ycm+ g0= 0 (3)
in which A is a (3 × 3) massinertia matrix,?C is a (3 × 3)
matrix containing the Coriolis and centrifugal terms,
u :¼ (u1, u2)′is a vector of actuator torques, g0 is the
gravitational constant and B :¼ [I2×202×1]′. By introducing
xf:= (q′
mechanical system during the flight phase can be expressed
in the statespace form: ˙ xf= ff(xf) + gf(xf)u. Moreover, the
state manifold for the flight phase is chosen as Xf:= TQf,
where Qf denotes the configuration space of the flight
phase. Using the principle of virtual work, a reducedorder
model for describing the evolution of the mechanical system
during the stance phase can also be obtained as follows
[2, p. 74]
f, ˙ q′
f)′as the state vector, the evolution of the
D(w)¨ q + C(w, ˙ q)˙ q + G(q) = Bu
(4)
whereD is a (3 × 3) massinertia matrix, C is a (3 × 3) matrix
containing the Coriolis and centrifugal terms and G is a
(3 × 1) gravity vector. By defining the state vector of the
stance phase as xs:= (q′
(4)canberepresented
˙ xs= fs(xs) + gs(xs)u. The state manifold is chosen as
Xs:= TQs, in which Qsis the configuration space of the
stance phase.
s, ˙ q′
s)′, where qs:¼ q and ˙ qs:= ˙ q,
instatespaceformby
2.3Openloop hybrid model of running
Following the modelling method presented by Chevallereau
et al. [1, 2, p. 249], the openloop model of monopedal
running can be expressed by the following nonlinear
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doi:10.1049/ietcta.2010.0512
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hybrid system
Ss:
˙ xs= fs(xs) + gs(xs)u,
x+
x−
x−
s? Sf
s[ Sf
s
f= Df
s(x−
s),
s
?
Sf:
˙ xf= ff(xf) + gf(xf)u,
x+
f),
x−
x−
f? Ss
f[ Ss
f
s= Ds
f(x−
f
?
(5)
In (5), the superscripts ‘–’ and ‘ + ’ denote the state of the
hybrid system immediately before and after the switching
between the state manifolds, respectively. As in [1, 2,
p. 77], we assume that the takeoff switching hypersurface
can be defined as Sf
gs(qs) is the angle of the virtual leg with respect to the
world frame (see Fig. 1) and g−
Moreover, the impact switching hypersurface is defined as
Ss
f:= {xf[ Xfy1(qf) = 0}, where y1denotes the height of
the leg end with respect to the world frame. Df
and Ds
f? Xsalso represent the takeoff and impact
maps, respectively.
s:= {xs[ Xsgs(qs) − g−
s= 0}, where
s
is a threshold value.
s: Sf
s? Xf
f: Ss
3
flight phase
Reconfiguration algorithm for the
The conservation of angular momentum about the COM of the
monopedalrobotduringtheflight phaseisexpressedinthethird
line of matrix equation (1) which can be rewritten as follows
˙u =
scm
A3,3(w)− J(w)˙ w
(6)
where scmis a constant representing the angular momentum of
themechanicalsystemabout itsCOM,
J(w) :=
(1/A3,3(w))[A3,1(w) A3,2(w)] [ R1×2and Qb is the body
configuration space. Because matrix A(w) is positive definite,
A3,3(w) . 0 for any w [ Qb. As in [1], we will assume that
the takeoff and landing occur in fixed configurations.
In particular, assume that a C2
w∗: [t∗
2] ? Qb(the evolution of body angles on the
desired orbit) can transfer the configuration of the monoped
robot during the flight phase from the initial condition
q∗
when the angular momentum about its COM is identically
equal to s∗
cm. Now, let the angular momentum about the
COM be scm, where scm= s∗
section is to present an online algorithm for generating
the modified reference trajectory w: [t1, t2] ? Qbbased on
the nominal trajectory w∗such that the configuration of the
mechanical system can be transferred from the initial
condition q∗
t2= t∗
Section 5 to simplify the stabilisation problem.
Integrating of (6) over the time interval [t1, t2] results in
nominal trajectory
1, t∗
1:= [w′∗(t∗
1), u1]′to the final condition q∗
2:= [w′∗(t∗
2), u2]′
cm. The objective of this
1to the final condition q∗
2. The results of this section will be utilised in
2, where t1= t∗
1and
u(t2) = u1+
?t2
t1
scm
A3,3(w(t))dt −
?
C
J(w)dw
(7)
where C := {c [ Qbc = w(t), t1≤ t ≤ t2} , Qb is the
path of w (t) in the body configuration space. By assuming
w(t) :¼ w∗(t(t)), where t: [t1, t2] ? [t∗
time fulfilling the following constraints:
1, t∗
2] is the virtual
1. t(t1) = t∗
2. t(t2) = t∗
3.inf
t1≤t≤t2˙ t(t) . 0
1
2
Fig. 1
Block diagram of the online reconfiguration algorithm over the time interval [t1, t2] during flight phases of monopedal running
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Equation (7) can be rewritten as follows
u(t2) = u1+
?t∗
2
t∗
1
scm
A3,3(w∗(s))
ds
˙ tWt−1(s)−
?
C∗J(w∗)dw∗
where
consequently
C∗:= {c [ Qbc = w∗(t), t∗
1≤ t ≤ t∗
2} = C,and
u(t2) − u2=
?t∗
2
t∗
1
1
A3,3(w∗(s))
scm
˙ tWt−1(s)− s∗
cm
??
ds
(8)
By defining m(s) := 1/˙ tWt−1(s) . 0 and w(s): ¼ 1/A3,
3(w∗(s)) . 0 for s [ [t∗
condition u(t2) ¼ u2 can be expressed as the following
equality constraint
1, t∗
2], and assuming scm= 0, the
?t∗
2
t∗
1
w(s)m(s)ds =s∗
cm
scm
?t∗
2
t∗
1
w(s)ds
(9)
Furthermore, from the definition of m(s), ˙ t(t) = 1/m(t(t)),
t1≤ t ≤ t2, and hence
?t∗
2
t∗
1
m(s)ds = t2− t1
(10)
Problem restatement: Determination of m(t) . 0, t∗
t∗
2such that the equality constraints in (9) and (10) are met
isequivalenttodetermining
m: [t∗
2] ? R.0which transfers the state of the following
linear timevarying system in the virtual time domain
1≤ t ≤
thecontrolinput
1, t∗
S:˙ x1= w(t)m
˙ x2= m
(11)
from (x1(t∗
where ˙ xi:= (d/dt)xifor i ¼ 1, 2 and
1), x2(t∗
1))′= (0, 0)′to (x1(t∗
2), x2(t∗
2))′= (xf
1, xf
2)′,
xf
1:=s∗
cm
scm
?t∗
2
t∗
1
w(s)ds
xf
2:= t2− t1
(12)
3.1Determination of the reachable set
The purpose of this subsection is to determine the reachable
set from the origin (at t∗
w(t) ¼ w∗(t(t)), the following relations can be obtained for
the first and second time derivatives of w(t)
1) at time t∗
2for the system S. Since
˙ w(t) =∂w∗
∂t(t(t))˙ t(t)
∂t(t(t))¨ t(t) +∂2w∗
¨ w(t) =∂w∗
∂t2(t(t))˙ t2(t)
Hence, a discontinuity of m may result in an impulsive nature
of ¨ w(t). In view of the actuator limitations, this latter fact
implies that w(t) cannot be used as a reference trajectory for
the joint angles. Thus, we present the following definition.
Definition 1: The set of admissible control inputs for the
system S is denoted by Um,Mand defined to be the set of
all continuously differentiable functions t ? (t) [ [m, M]
defined on the interval [t∗
2], where 0 , m , M.
We present a design method for obtaining an admissible
control m [ C1([t∗
2], [m, M]). For this purpose, we
consider m to be the output of a double integrator and study
the following augmented system
1, t∗
1, t∗
Sa:
˙ x1= w(t)x3
˙ x2= x3
˙ x3= x4
˙ x4= v
which can be viewed as a cascade connection of two
components. [During the continuous phases (i.e. stance and
flight), we desire that the control inputs of the mechanical
system are continuous, whereas they can be discontinuous
during the discrete transitions (i.e. takeoff and impact). For
this reason, we assume that m is the output of a double
integrator.] The first component is the system S in (11)
with x3as input and the second component is the double
integrator with a piecewise continuous function v as input.
The admissibility of m can be expressed as m ≤ x3≤ M,
which is a constraint on the state of Sa.
Definition 2: The set of admissible control inputs for the
system Sais denoted by VL1,L2and defined to be the set of
all piecewise continuous functions t ? v(t) [ [L1, L2]
defined on the interval [t∗
2], where L1, 0 , L2.
1, t∗
Definition
(x0
(xf
that the state of the system Sais transferred from the initial
point (0, 0, x0
with the constraint m ≤ x3(t) ≤ M, t∗
xf
4are free.
Itisclearthatforeveryx0
To determine Am,M,L1,L2, we study two optimal control
problems. From these problems, the optimal admissible
control inputs, vmax(t), vmin(t) [ VL1,L2, t∗
determined such that the state of the system Sa is to be
transferred from the initial point x0:= (0, 0, x0
the final point (x1(t∗
property m ≤ x3(t) ≤ M, t∗
measure Ia(v) := x1(t∗
and minimised (see point C in Fig. 2). Note that in these two
3:
For any 0 , m , M, L1, 0 , L2 and
4)′[ R2, define Am,M,L1,L2(x0
1, xf
3, x0
3, x0
4) as the set of all points
2)′[ R2for which there exists a control v [ VL1,L2such
3, x0
4)′at t∗
1to the final point (xf
1, xf
2, xf
2, where xf
3, xf
4)′at t∗
3and
2
1≤ t ≤ t∗
3? [m, M], Am,M,L1,L2(x0
3, x0
4) = f.
1≤ t ≤ t∗
2, are
3, x0
4)′at t∗
2with the
1to
2), x2(t∗
2), x3(t∗
1≤ t ≤ t∗
2) is maximised (see point D in Fig. 2)
2), x4(t∗
2, while the performance
2))′at t∗
Fig. 2
minimisation and maximisation problems for a given x2
denoted by C and D, respectively
Reachable set Am,M,L1,L2(x3
0, x4
0). The solutions of the
f
are
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optimal control problems, x2(t∗
x3(t∗
The constraint m ≤ x3(t) ≤ M can be rewritten as the
following inequality constraints
2) = xf
2is specified, whereas
2) and x4(t∗
2) are free.
S1(x) := m − x3≤ 0
S2(x) := x3− M ≤ 0
Next, we take successive virtual time derivatives of S1(x) and
S2(x) until obtaining an expression that is explicitly dependent
on v [19, p. 118]. This process will result in¨S1= −v and
¨S2= v, where¨Si:= (d2/dt2)Si, i = 1, 2. Now, define the
following Hamiltonian function
H(x,p,l,v,t):= p1w(t)x3+p2x3+p3x4+p4v+l1¨S1+l2¨S2
= p1w(t)x3+p2x3+p3x4+(p4−l1+l2)v
(13)
where x :¼ (x1, x2, x3, x4)′, p :¼ (p1, p2, p3, p4)′and
l :¼ (l1,l2)′are the state, costate and multiplier vectors,
respectively. Furthermore, in (13)
¨Si= 0,
li= 0,
ontheconstraintboundary (i.e. Si= 0)
off theconstraintboundary (i.e. Si, 0)
for i ¼ 1, 2, which can also be expressed as
v = 0, ontheconstraintboundary (i.e. Si= 0)
off theconstraintboundary (i.e. Si, 0)
li= 0,
(14)
Necessary condition for multipliers li(t), i ¼ 1, 2, to
minimise the performance measures is
li(t) ≥ 0,ontheconstraintboundary (i.e.Si= 0)(15)
Note that the maximisation of the performance measure
Ia(v) := x1(t∗
−Ia(v). The costates satisfy the following differential
equations
2) can be expressed as the minimisation of
˙ p1= 0
˙ p2= 0
˙ p3= −p1w(t) − p2
˙ p4= −p3
in which ˙ pi:= (d/dt)pi, i = 1, ..., 4. From here on, the
superscripts ‘max’ and ‘min’ will denote the solutions of the
maximisation and minimisation problems, respectively. We
first study the maximisation problem. Since the final values
xmax
1
(t∗
3
(t∗
4
(t∗
p. 200], pmax
1
(t∗
boundary conditions in combination with the costate
equations yield
2), xmax
2) and xmax
2) = −1 and pmax
2) are free, from Table 51 of [20,
(t∗
432) = pmax
(t∗
2) = 0. These
pmax
3
(t; pmax
2
) = −
?t∗
?t∗
2
t
w(s)ds − pmax
2
(t − t∗
2)
pmax
4
(t; pmax
2
) = −
2
t
?t∗
2
s
w(h)dhds +pmax
2
2
(t − t∗
2)2
From Pontryagin’s minimum principle [21], vmax(t) is
given by
vmax(t) =
L1,
L2,
undetermined,
pmax
4
pmax
4
pmax
4
− lmax
− lmax
− lmax
1
+ lmax
+ lmax
+ lmax
2
. 0
, 0
= 0
12
12
(16)
Notice
through the zero, a switching of the optimal control input
vmax(t) occurs. Assume that w(t) satisfies the following
hypothesis:
H: ˙ w(t) := (d/dt)w(t) is not zero on the open set (t∗
It will be shown that by hypothesis H the optimal control
inputs vmax(t) and vmin(t) can switch at most once, and the
singular condition does not occur. For this purpose, we
present the following result.
thatif
pmax
4
(t; pmax
2
) − lmax
1
(t) + lmax
2
(t)passes
1, t∗
2).
Lemma 1: Let m , x0
hypothesis H holds and x0
trajectories xmax(t) and xmin(t) of the system Saexist. Then,
the following statements are true:
3, M and L1, 0 , L2. Assume that
4 is such that the optimal
(a) The optimal trajectories do not enter onto the boundaries
S1¼ 0 and S2¼ 0.
(b) The optimal control inputs vmax(t) and vmin(t) can switch
at most once.
(c) The singular condition does not occur. In other words, the
sets Tmax
0
:= {t [ [t∗
4
and
Tmin
0
:= {t [ [t∗
are Lebesgue negligible.
1, t∗
1, t∗
2]pmax
2]pmin
(t) − lmax
4 (t) − lmin
1
(t) + lmax
1 (t) + lmin
2
2 (t) = 0}
(t) = 0}
Proof: To prove the statements (a), (b) and (c) of Lemma 1,
themaximisationproblem
reasonings can also be presented for the minimisation
problem.
If the optimal trajectory enters onto the constraint boundary
S1¼ 0, S2will be negative and consequently from condition
(14), vmax¼ 0 and lmax
2
= 0. Since L1, 0 , L2, from the
control input vmax(t) in (16), vmax¼ 0 results in lmax
which in combination with the necessary conditions given in
(15) yields pmax
4
≥ 0. Similarly, if the optimal trajectory
enters onto the constraint boundary S2¼ 0, then vmax¼ 0
and lmax
1
= 0. Moreover, lmax
2
pmax
4
≤ 0.
Next, we study the roots of the nonlinear equation
pmax
4
(t; pmax
2
) = 0 on the interval [t∗
also be expressed as follows
willbestudied.Similar
1
= pmax
4
= −pmax
4
which implies that
1, t∗
2]. This equation can
W(t) =pmax
2
2
(t − t∗
2)2
(17)
where W(t) :=?t∗
has at most one root in the interval [t∗
? t [ [t∗
unique and can be given by
2
t
?t∗
2
sw(h)dhds. For any pmax
2
[ R, t∗
(t; pmax
2is the
) = 0
solution of (17). We claim that the equation pmax
4
2
1, t∗
2). To show this, let
) = 0. Then, pmax
1, t∗
2) exist such that pmax
4
(? t; pmax
22
is
pmax
2
=
2
(? t − t∗
2)2W(? t)(18)
We remark that pmax
2
is positive. If there exists ˜ t [ [t∗
1, t∗
2)
1532
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doi:10.1049/ietcta.2010.0512
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Page 6
such that ˜ t = ? t and pmax
4
(˜ t; pmax
2
) = 0, then (18) implies that
W(? t)
(? t − t∗
2)2=
W(˜ t)
(˜ t − t∗
2)2
Hence, it is sufficient to show that the function k:[t∗
by k(t) := W(t)/(t − t∗
derivative of k(t) can be obtained as follows
1, t∗
2) ? R
2)2is strictly monotonic. The first
˙ k(t) =
˙W(t)(t − t∗
2) − 2W(t)
(t − t∗
2)3
=:
F(t)
(t − t∗
2)3
(19)
Assume that there exists h1[ (t∗
Since F(h1) = F(t∗
thatthere is
¨W(h2)(h2− t∗
zero.Hence,from
h3[ (h2, t∗
follows that
˙ w(h3) = 0, and this contradicts hypothesis
H. Therefore
k(t)isstrictly
consequence, the equation pmax
4
root in the interval [t∗
2).
Let
? t [ (t∗
2)bethe
pmax
4
(t; pmax
2
) = 0. Substituting (18) into pmax
1, t∗
2) such that F(h1) ¼ 0.
2) = 0, the Rolle’s theorem implies
h2[ (h1, t∗
such
2) −˙W(h2) = 0. Furthermore, ˙F(t∗
the Rolle’s theorem,
2) such that¨F(h3) = ˙ w(h3)(h3− t∗
2)that
˙F(h2) =
2) is also
there
2) = 0 which
exists
monotonic
(t; pmax
2
) = 0 has at most one
and,asa
1, t∗
1, t∗
rootofthe
(t; pmax
equation
) yields
4
2
˙ pmax
4
(? t; pmax
2
) =−˙W(? t)(? t − t∗
2) + 2W(? t)
? t − t∗
2
=F(? t)
t∗
2−? t= 0
Therefore the condition ˙ pmax
expressed as
4
(? t; pmax
2
) , 0 which can be
2
t∗
2−? t
?t∗
(t; pmax
2
? t
?t∗
2
s
w(h)dhds .
?t∗
2
? t
w(s)ds
(20)
implies that pmax
pmax
4
(t; pmax
2
not satisfied, then pmax
pmax
4
(t; pmax
2
Since m , x0
S1(x0) and
lmax
1
(t∗
that pmax
4
(t∗
Since the nonlinear equation pmax
onerootinthe
(t∗
4
(t; pmax
2
as a consequence, without loss of generality, we can assume
that pmax
4
(t∗
2
) = 0. Then there are four possible cases.
4
2
) . 0 for any t [ [t∗
2]. If condition (20) is
(t; pmax
2
) , 0 for any t [ [t∗
) . 0 for any t [ (? t, t∗
2].
3, M and xmax(t) is the optimal trajectory,
S2(x0)arenegative
1) = lmax
2
(t∗
1; pmax
2
) = 0. If pmax
4
(t∗
1, ? t) and
) , 0 for any t [ (? t, t∗
4
1, ? t) and
whichresultin
1) = 0. Without loss of generality, assume
1; pmax
2
(t; pmax
interval[t∗
2),
) = 0. In addition, vmax(t∗
) = 0, then ? t = t∗
) = 0 has at most
onthe
1) is finite and
1.
4
2
1, t∗
open set
1, t∗
2), pmax
1; pmax
Case 1: Assume that there exists ? t [ (t∗
pmax
4
(? t; pmax
2
) = 0 and inequality (20) holds. In this case, on
the interval [t∗
1, t∗
2) such that
1, ? t), vmax(t) = L1and consequently
xmax
4
(t) = x0
4+ L1(t − t∗
1)
xmax
3
(t) = x0
3+ x0
4(t − t∗
1) +L1
2(t − t∗
1)2
xmax
2
(t) = x0
3(t − t∗
1) +x0
4
2(t − t∗
1)2+L1
6(t − t∗
1)3.
Because pmax
trajectory xmax(t), t∗
S2¼ 0. From [19, p. 118], since the control of S1is obtained
only by changing¨S1, no finite control can keep the optimal
4
(t; pmax
2
) . 0 on the interval [t∗
1≤ t , ? t cannot enter onto the boundary
1, ? t), the optimal
trajectory of the system Sa on the constraint boundary
S1¼ 0, unless the following tangency constraints hold.
[The terminology of a ‘tangency constraint’ is taken from
[20, p. 118].]
N1(xmax(t)) :=
S1(xmax(t))
˙S1(xmax(t))
??
=
m − xmax
−xmax
3
(t)
4
(t)
??
=
0
0
? ?
(21)
Because L1, 0, xmax
concave function with respect to t. Thus, the condition
m , x0
3, M implies that the tangency constraints (21)
cannot be satisfied on the interval [t∗
3
(t), t∗
1≤ t , ? t is a quadratic and
1, ? t). Now, define
? x4:= xmax
4
(? t) = x0
4+ L1(? t − t∗
1)
? x3:= xmax
3
(? t) = x0
3+ x0
4(? t − t∗
1) +L1
2(? t − t∗
1)2
? x2:= xmax
2
(? t) = x0
3(? t − t∗
1) +x0
4
2(? t − t∗
1)2+L1
6(? t − t∗
1)3
Also, on the interval (? t, t∗
2], vmax(t) ¼ L2and thus
xmax
4
(t) = ? x4+ L2(t −? t)
xmax
3
(t) = ? x3+? x4(t −? t) +L2
2(t −? t)2
2(t −? t)2+L2
xmax
2
(t) = ? x2+? x3(t −? t) +? x4
6(t −? t)3
Since pmax
xmax(t), ? t , t ≤ t∗
Moreover, the tangency constraints to remain on the
boundary S2¼ 0 can be expressed as
4
(t; pmax
2
) , 0 on (? t, t∗
2cannot enter onto the boundary S1¼ 0.
2], the optimal trajectory
N2(xmax(t)) :=
S2(xmax(t))
˙S2(xmax(t))
??
=
xmax
3
(t) − M
xmax
4
(t)
??
=
0
0
? ?
(22)
The fact that xmax
function with respect to t in combination with the condition
m , ? x3, M implies that the tangency constraints in (22)
cannot be satisfied on the interval (? t, t∗
S1(xmax(t)), S2(xmax(t)) , 0 for any t [ [t∗
optimal trajectory is feasible (i.e. m ≤ xmax
L1, 0, ? x3= m will result in ? x4, 0, which in turn implies
the existence of ˆ t [ (? t, t∗
t [ (? t, ˆ t). This contradicts the feasibility of the optimal
trajectory xmax(t). In a similar manner, it can be shown that
? x3= M.
The final constraint xmax
2
(t∗
the following thirddegree equation
3
(t), ? t , t ≤ t∗
2is a quadratic and convex
2]. Consequently,
1, t∗
(t) ≤ M) and
2]. Since the
3
2] such that xmax
3
(t) , m for any
2) = xf
2can also be expressed as
L1− L2
6
(? t − t∗
2)3+ x0
3lmax+x0
4
2l2
max+L1
6l3
max= xf
2
(23)
where lmax:= t∗
calculated as follows
2− t∗
1. From (23), ? t [ R is unique and can be
? t = t∗
2+
???????????????????????????????????????????? ?
L1− L2
6
xf
2− x0
3lmax−x0
4
2l2
max−L1
6l3
max
??
3
?
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doi:10.1049/ietcta.2010.0512
1533
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Page 7
Ift∗
solution and, as a consequence, the validity of Case 1 is
confirmed.
1, ? t , t∗
2andinequality(20)issatisfied,then? tisafeasible
Case 2: There exists ? t [ (t∗
and
1, t∗
2) such that pmax
4
(? t; pmax
2
) = 0
2
t∗
2−? t
?t∗
2
? t
?t∗
2
s
w(h)dhds ,
?t∗
2
? t
w(s)ds
(24)
An analysis similar to that presented for Case 1 can be
performed. However, the thirddegree equation in (23) is
given by
L2− L1
6
(? t − t∗
2)3+ x0
3lmax+x0
4
2l2
max+L2
6l3
max= xf
2
(25)
Equation (25) has the following real and unique root
? t = t∗
2+
???????????????????????????????????????????? ?
L2− L1
6
xf
2− x0
3lmax−x0
4
2l2
max−L2
6l3
max
??
3
?
which is feasible if t∗
satisfied.
1, ? t , t∗
2and inequality (24) is
Case 3: The function pmax
[t∗
vmax(t) ; L1and similar to the analysis performed for the
interval [t∗
S2(xmax(t)) , 0. Also, the final condition xmax
be satisfied only for the following specific value of xf
4
(t; pmax
2
) is positive on the interval
is not unique. In addition,
1, t∗
2]. This implies that pmax
2
1, ? t) in Case 1, it can be shown that S1(xmax(t)),
2
(t∗
2) = xf
2can
2
xf
2= xf
2:= x0
3lmax+x0
4
2l2
max+L1
6l3
max
Case 4: If the function pmax
interval[t∗
pmax
4
(t; pmax
2
) , 0
S2(xmax(t)) , 0, t∗
xf
4
(t; pmax
not
L2. 0
2. The final condition xmax
2
) is negative on the
uniqueand
implythat
1, t∗
2], pmax
2
ismoreover,
S1(xmax(t)),
(t∗
and
1≤ t ≤ t∗
2
2) =
2can be satisfied only for the following specific value of xf
2
xf
2= ? xf
2:= x0
3lmax+x0
4
2l2
max+L2
6l3
max
The proof of Lemma 1 follows from the results obtained in
Cases 1–4.
A
Remark 1: In the minimisation problem
pmin
3 (t; pmin
2 ) =
?t∗
?t∗
2
t
w(s)ds − pmin
2 (t − t∗
2)
pmin
4 (t; pmin
2 ) =
2
t
?t∗
2
s
w(h)dhds +pmin
2
2
(t − t∗
2)2
Moreover, the nonlinear equation pmin
most one root in the interval [t∗
such that pmin
2 ) = 0. The validity of Cases 1 and 2 in
the minimisation problem is confirmed by
4 (t; pmin
1, t∗
2 ) = 0 has at
2). Let t [ [t∗
1, t∗
2) be
4 (t, pmin
2
t∗
2− t
?t∗
2
t
?t∗
2
s
w(h)dhds ,
?t∗
2
t
w(s)ds
and
2
t∗
2− t
?t∗
2
t
?t∗
2
s
w(h)dhds .
?t∗
2
t
w(s)ds
respectively. Cases 3 and 4 of the minimisation problem are
similar to the ones presented in the maximisation problem.
Remark 2: As discussed previously, by hypothesis H, the
function F(t) defined in (19) is nonzero on the interval
(t∗
2). Thus, without loss of generality, we will assume
that F(t) , 0 on the interval (t∗
imposes that Case 2 of the maximisation problem and Case
1 of the minimisation problem are not feasible.
Now let m , x0
(x0
for which the optimal solutions of the maximisation and
minimisation problems starting from the initial point
(0, 0, x0
exist. Also, denote the solutions of the
maximisation and minimisation problems corresponding to
xf
2), respectively. The functions
pmax: Vmax
R are introduced by
1, t∗
1, t∗
2). This assumption
3, M and x0
m,M,L1,L2(x0
4[ R. Define Vmax
4) to be the sets of all xf
m,M,L1,L2
2[ R
3, x0
4) and Vmin
3, x0
3, x0
4)′
2by xmax(t; xf
m,M,L1,L2(x0
2) and xmin(t; xf
3, x0
4) ? R and pmin: Vmin
m,M,L1,L2(x0
3, x0
4) ?
pmax(xf
2):=
?t∗
?t∗
2
t∗
1
w(s)xmax
3
(s; xf
2)ds = xmax
1
(t∗
2; xf
2)
pmin(xf
2):=
2
t∗
1
w(s)xmin
3 (s; xf
2)ds = xmin
1 (t∗
2, xf
2)
(26)
(see Fig. 2). We claim that if the sets Vmax
Vmin
4) are nonempty, they are connected sets.
For this purpose, we present the following lemma for which
it is shown that Vmax
A similar result can also be obtained for the set Vmin
(x0
4).
Lemma 2: Let m , x0
are two scalars such that a, b [ Vmax
any g [ (a, b), g [ Vmax
The proof is given in Appendix 1. Now we are in a position
to present the main result of this section. This result is
expressed as the following theorem which determines the
C1input m transferring the state of S from the origin at t∗
to the final point (xf
m,M,L1,L2(x0
3, x0
4) and
m,M,L1,L2(x0
3, x0
m,M,L1,L2(x0
3, x0
4) is a connected set.
m,M,L1,L2
3, x0
3, M and x0
4[ R. Assume that a , b
m,M,L1,L2(x0
m,M,L1,L2(x0
3, x0
4). Then, for
3, x0
4).
1
1, xf
2)′[ Am,M,L1,L2(x0
3, x0
4) at t∗
2.
Theorem 1: Let m , x0
L1, 0 and L2. 0 are such that
3, M and x0
4[ R. Assume that
min x0
3, x0
3+ x0
4lmax+L1
2l2
max
?
?
?
?
. m
max x0
3, x0
3+ x0
4lmax+L2
2l2
max
, M
(27)
Then, the set Am,M,L1,L2(x0
3, x0
4) is given by
Am,M,L1,L2(x0
3, x0
4)
= {(xf
1, xf
2)′[ R2xf
2≤ xf
2≤ ? xf
2, pmin(xf
2) ≤ xf
1≤ pmax(xf
2)}
where xf
(x0
2:= x0
max+ (L2/6)l3
3lmax+ (x0
4/2)l2
max+ (L1/6)l3
max, ? xf
1.
2:= x0
3lmax+
4/2)l2
maxand lmax:= t∗
2− t∗
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IET Control Theory Appl., 2011, Vol. 5, Iss. 13, pp. 1528–1543
doi:10.1049/ietcta.2010.0512
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