Phase Uncertainty to State Stability of Continuous Periodic Orbits
Shishir Kolathaya
Abstract—The paper shows sufficiency conditions for stability
of continuous periodic orbits under phase uncertainty. Phase
based uncertainty is a trait of bipedal walking robots, where
the desired trajectories are parameterized by a monotonous
function. This monotonous function, called the phase variable, is
often affected by intermittent perturbations due to noisy sensors.
We will mainly focus on continuous periodic orbits obtained
via parameterized trajectories, and then analyze their stability
properties under a noisy phase estimation. In other words, our
focus is on examples where phase variables are difficult to
compute, and therefore are imperfect. We will show that stable
periodic orbits subject to phase based uncertainty are input to
state stable.
I. INTRODUCTION
This article provides a proof for phase-uncertainty-to-state
stability of continuous periodic orbits. Phase variables appear
extensively in the field of locomotion pattern generators for
bipedal robots (see Fig. 1). See [5], [9] for a brief overview
on phase uncertainty. The primary purpose of phase variables
is to modulate the desired trajectories of the actuated joints
of the robot from start to finish. Choosing a state dependent
phase variable renders the trajectory tracking control law
autonomous. This state dependency also results in the injection
of perturbations into the system via noisy sensory feedback.
Therefore stability properties of walking behaviors under an
imperfect phase determination is of interest to us.
In this manuscript we will establish preliminary results on
stability of continuous periodic orbits under imperfect phase
determinations (phase uncertainty). See [8] for a detailed
analysis on stability of hybrid periodic orbits under phase
uncertainties. Also see [7], [6] for overview on parameter-
uncertainty-to-state stability of hybrid periodic orbits.
The paper is structured as follows. Section II will introduce
a brief overview on feedback control laws used to realize stable
periodic orbits. Section III will introduce the notion of input to
state stability (ISS), and then the notion of phase uncertainty
to state stability. Finally, Section IV will introduce the main
theorem of the paper demonstrating that periodic orbits under
phase based uncertainties can be rendered phase to state stable.
II. FEEDBACK CONTROL
The goal of this section is to define the set of outputs given
the state x. Virtual constraints [13] consist of a vector of actual
outputs given as ya : TQ →Rk, and a vector of desired outputs
given as yd : R≥0 →Rk. Here yd is a map from the positive
reals, and can thus be parameterized by a phase (or time)
variable τ : TQ →R≥0 (or τ : R≥0 →R≥0 for time based). By
Shishir
Kolathaya
is
with
the
School
of
Mechanical
and
Civil
Engineering,
California
Institute
of
Technology,
Pasadena,
CA,
USA
sny@caltech.edu
Fig. 1: The red arrow indicates a specific phase variable
candidate; in this case the calf angle. As the robot progresses
throughout a step, the calf angle monotonically increases. This
is scaled appropriately with an offset to modulate from 0 to 1
for each step.
adapting a feedback linearizing controller, we can drive the
relative degree one outputs (velocity outputs)
y1(q, ˙q) = ya
1(q, ˙q)−yd
1(τ,α) ∈Rk1,
(1)
and relative degree two outputs (pose outputs)
y2(q) = ya
2(q)−yd
2(τ,α) ∈Rk2,
(2)
to zero, with α denoting the parameters of the desired tra-
jectory. k1 +k2 = k. These outputs are generally called virtual
constraints [13]. Normally, the phase variable, τ, for relative
degree two outputs is a function of the configuration τ(q).
Walking gaits, viewed as a set of desired periodic trajectories,
are modulated as functions of a phase variable to eliminate the
dependence on time [12]. In this case, the velocity (relative
degree one) outputs are: y1(q, ˙q) = ya
1(q, ˙q)−yd
1(α), where the
parameterization w.r.t. τ is absent. In terms of the states, x,
we can define the outputs as follows [1]:
y1(x) = ya
1(x)−yd
1(α)
y2(x) = ya
2(x)−yd
2(τ,α).
(3)
Feedback Linearization. The feedback linearizing controller
that drives the purely state dependent outputs y1 →0, y2 →0
is given by:
u =

Lgy1
LgLf y2
−1 
−
Lf y1
L2
f y2

+ µ

,
(4)
where Lf ,Lg denote the Lie derivatives and µ denotes the
auxiliary input applied after the feedback linearization.
Note that any effective tracking controller will theoretically
suffice (the experimental implementation uses PD control [4]).
arXiv:1707.02258v1  [math.DS]  7 Jul 2017
We therefore employ a control Lyapunov function (CLF) based
controller that can drive the following outputs to zero
η =


y1
y2
˙y2

.
(5)
If the system has outputs with more relative degrees of
freedom, then η can be accordingly modified. Applying the
controller (4) results in the following output dynamics:
˙η =


0
0
0
0
0
Ik2×k2
0
0
0


|
{z
}
F
η +


Ik1×k1
0
0
0
0
Ik2×k2


|
{z
}
G
µ,
(6)
where k1 + k2 = k. k1 is the size of the velocity outputs y1,
and k2 is the size of the relative degree two outputs y2. The
dimension of the outputs k is typically equal to the number of
control inputs m.
The auxiliary control input µ is chosen via control Lya-
punov functions (CLFs) that drives η →0. More on CLFs is
explained toward the end of this section.
Zero Dynamics. When the control objective is met such that
η = 0 for all time then the system is said to be operating on
the zero dynamics surface:
Z = {(q, ˙q) ∈πx(D)|η = 0},
(7)
for the domain D. Further, by relaxing the zeroing of the
velocity output y1 : TqQ →Rk1, we can realize partial zero
dynamics [1]:
PZ = {(q, ˙q) ∈πx(D)|y2 = 0,Lf y2 = 0}.
(8)
The humanoid DURUS, has feet and employs ankle actuation
to propel the hip forward during the continuous dynamics.
The relaxation assumption is implemented on the hip velocity,
resulting in partial zero dynamics. For the running robot
DURUS-2D, since the feet are underactuated, purely relative
degree two outputs are picked that result in full zero dynamics
of the system: PZ = Z, due to the absence of velocity outputs.
The zero dynamics are characterized by the zero dynamic
coordinates z ∈R2n−k1−2k2, which when combined with the
normal coordinates η form the transformed state space for the
full order dynamics:
˙η = Fη +Gµ
˙z = Ψ(η,z),
(9)
When the transverse and the zero dynamics are combined
together, we get the full order dynamics. Based on this
construction, we have the diffeomorphism Φ : R2n →R2n that
maps from x = (q, ˙q) to (η,z). The diffeomorphism can be
divided into parts:
Φ(x) =


Φ1(x)
Φ2(x)
Φ3(x)

=


y1(q, ˙q)
y2(q)
˙y2(q, ˙q)
z(q, ˙q)

.
(10)
Similarly, the outputs can also be divided into two parts:
η =

y1
η2

, where
η2 =

y2
˙y2

.
(11)
Control Lyapunov Function. We will define the control Lya-
punov function (CLF), and the rapidly exponentially stabilizing
control Lyapunov function (RES-CLF) as follows:
Definition 1. For the system (9), a continuously differentiable
function V : Rk1+2k2 →R≥0 is an
exponentially stabilizing
control Lyapunov function (ES-CLF) if there exist positive
constants c, ¯c,c > 0 such that for all η,z.
c∥η∥2 ≤V(η) ≤¯c∥η∥2
inf
u∈U[LfV(η,z)+LgV(η,z)u+cV(η)] ≤0.
(12)
Here Lf ,Lg are the Lie derivatives. We can accordingly de-
fine a set of controllers which render exponential convergence
of the transverse dynamics:
K(η,z) = {u ∈U : LfV(η,z)+LgV(η,z)u+cV(η) ≤0},
(13)
which has the control values that result in ˙V ≤−cV.
RES-CLF. We can impose stronger bounds on convergence
by constructing a rapidly exponentially stabilizing control
lyapunov function (RES-CLF) that can be used to stabilize the
output dynamics at a rapid rate through a user defined ε > 0.
Definition 2. For the family of continuously differentiable
functions, Vε : Rk1+2k2 →R≥0 is a rapidly exponentially
stabilizing control Lyapunov function (RES-CLF), if there
exist positive constants c1,c2,c3 > 0 such that for all 0 < ε < 1
and for all η,z,
c1∥η∥2 ≤Vε(η) ≤c2
ε2 ∥η∥2,
(14)
inf
u∈U[LfVε(η,z)+LgVε(η,z)u+ c3
ε Vε(η)] ≤0.
Therefore, we can define a class of controllers Kε:
Kε(η,z) = {u ∈U : LfVε(η,z)+LgVε(η,z)u+ c3
ε Vε(η) ≤0},
(15)
which yields the set of control values that satisfies the desired
convergence rate. Note that the set depends on the domain, due
to the dependency on f,g. Therefore, the set with the domain
subscript is denoted as Kε.
Time Dependent RES-CLF. Given (2), if the desired outputs
are parameterized by the time based phase variable instead of
the state based phase variable, we have the following output
representation
yt
1(q, ˙q) = ya
1(q, ˙q)−yd
1(τ(t),α),
(16)
for velocity outputs and
yt
2(q) = ya
2(q)−yd
2(τ(t),α),
(17)
for relative degree two (pose) outputs. The outputs are derived
from (1),(2) where the phase is now dependent on time τ(t).
With the absence of parameterization for the velocity outputs,
we have : yt
1(q, ˙q) = y1(q, ˙q). The resulting output dynamics
is obtained by taking the derivative:
˙yt
1(q, ˙q) = Lf ya
1(q, ˙q)+Lgya
1(q, ˙q)u−˙yd
1(τ(t), ˙τ(t),α)
(18)
¨yt
2(q, ˙q) = L2
f ya
2(q)+LgLf ya
2(q)u−¨yd
2(τ(t), ˙τ(t), ¨τ(t),α).
In order to drive the time dependent outputs
ηt(x) =

yt
1(x)
η2,t(x)

=


yt
1(q, ˙q)
yt
2(q)
˙yt
2(q, ˙q)

→0,
we can choose u via time dependent RES-CLFs (similar to
(15)) in the following manner:
Kt
ε(ηt,zt) = {ut ∈U : LfVt
ε(ηt)+LgVt
ε(ηt)ut + γ
ε Vt
ε(ηt) ≤0}.
(19)
This set of controllers with the domain representation will
hence be denoted as Kt
ε. A particular control solution that be-
longs to the set Kt
ε can be obtained via feedback linearization:
ut =

Lgya
1
LgLf ya
2
−1 
−
Lf ya
1
L2
f ya
2

+

˙yd
1
¨yd
2

+ µt

,
(20)
where µt is the auxiliary time based control input after
feedback linearization that can be appropriately chosen. If the
desired velocity outputs have no parameterization, then ˙yd
1 = 0
(derivative of a constant). The time based output dynamics can
be written in normal form as
˙ηt = Fηt +Gµt, ˙zt = Ψt(ηt,zt).
(21)
zt are the set of zero dynamic coordinates normal to ηt and
has the invariant dynamics ˙zt = Ψt(0,zt). Also note that the
matrices F,G are the same as the matrices used for the state
based coordinates in (9). For the time based states, ηt,zt, we
have the diffeomorphism: Φt(x) = (ηt(x),zt(x)).
RES-CLFs Obtained From Feedback Linearization. Due to
the difficulty in obtaining Lyapunov functions (in particular,
control Lyapunov functions) for nonlinear systems, we use the
linear dynamics (9),(21) obtained from feedback linearization
to realize both state and time dependent RES-CLFs. Rapid
exponential convergence is obtained by appropriately choosing
µ,µt for (9),(21) respectively. This is mainly discussed in [2]
and will be explained in brief here.
Let F,G be defined as in (9),(21), and let P be the solution
to the CARE (control algebraic Riccati equation)
FTP+PF −PGGTP+Q = 0,
(22)
for some Q = QT > 0. Since γP ≤Q, where γ = λmin(Q)
λmax(P) > 0.
λmin(.),λmax(.) denote the minimum and maximum eigenval-
ues of a given symmetric matrix respectively. By choosing
ε > 0 and letting Pε :=
 1
ε I
0
0
1
ε I

P
 1
ε I
0
0
1
ε I

and Qε :=
 1
ε I
0
0
1
ε I

Q
 1
ε I
0
0
1
ε I

, the following is satisfied
FTPε +PεF −1
ε PεGGTPε + 1
ε Qε = 0.
(23)
Motivated by (23), we can construct the following Lyapunov
function:
Vε(η) = ηTPεη.
(24)
Differentiating (24) yields
˙Vε(η) = LFVε(η)+LGVε(η)µ,
LFVε(η) = ηT(FTPε +PεF)η, LGVε(η) = 2ηTPεG.
(25)
Here µ can be picked from the set given below:
Vε(η) = ηTPεη,
(26)
Kε(η) = {µ ∈Rm : LFVε(η)+LGVε(η)µ + γ
ε Vε(η) ≤0}.
The above controller drives the state dependent outputs rapidly
exponentially to zero, ˙Vε ≤−γ
εVε, which is required by the
conditions of (14). Therefore, Vε is a valid RES-CLF.
In a similar fashion, µt can be picked from the set given
below:
Vt
ε(ηt) = ηT
t Pεηt,
(27)
Kt
ε(ηt) = {µt ∈Rm : LFVt
ε(ηt)+LGVt
ε(ηt)µt + γ
ε Vt
ε(ηt) ≤0}.
The above controller drives the time dependent outputs
rapidly exponentially to zero, ˙Vt
ε ≤−γ
εVt
ε. LG,LF are the Lie
derivatives that are similar to (25) (see [2]):
LFVt
ε = ηT
t (FTPε +PεF)ηt, LGVt
ε = 2ηT
t PεG.
(28)
For limiting the use of notations, the classes of controllers
defined in (19) and (27) are both denoted by Kt
ε with the
difference being the dependency on the number of arguments.
Therefore, a standard time based RES-CLF without substi-
tuting (20) would be denoted by Kt
ε(ηt,zt), and the time
dependent RES-CLF that utilizes the auxiliary input µt after
substituting (20) would be denoted by Kt
ε(ηt). Similarly, a
state based RES-CLF without the substitution of (4) is denoted
by Kε(η,z), and with the substitution would be denoted by
Kε(η). To summarize, the main control input u is a function
of two arguments, while the auxiliary control input µ is a
function of only one, the normal coordinates η.
III. PRELIMINARIES ON INPUT TO STATE STABILITY
In this appendix we will introduce basic definitions and
results related to input-to-state stability (ISS) for a general
nonlinear system. See [11] for a detailed survey on ISS.
Consider the following differential equation:
˙x =f(x,d),
(29)
with x taking values in Euclidean space Rn, the input d ∈Rm
for some positive integers n,m. The mapping f : Rn×Rm →Rn
is considered Lipschitz continuous and f(0,0) = 0. Note that
the dimension of the state for the robots considered are of
dimension 2n, x ∈R2n.
It can be observed that the input considered here is d.
Therefore, the construction is such that a stabilizing con-
troller u = k(x) is applied. Any deviation from this stabilizing
controller can be viewed as k(x) + d, with d being the new
disturbance input. We assume that d : R≥0 →Rm is a Lebesgue
measurable function of time: ∥d∥∞= ess. supt≥0∥d(t)∥< ∞.
We can denote this space of Lebesgue measurable functions
as Lm
∞, and therefore d ∈Lm
∞.
Class K∞and K L functions. A class K∞function is a func-
tion α : R≥0 →R≥0 which is continuous, strictly increasing,
unbounded, and satisfies α(0) = 0, and a class K L function
is a function β : R≥0 ×R≥0 →R≥0 such that β(r,t) ∈K∞for
each t and β(r,t) →0 as t →∞.
We can now define input to state stability for system (29).
Definition 3. The system (29) is input to state stable (ISS) if
there exists β ∈K L , ι ∈K∞such that
|x(t,x0)| ≤β(|x0|,t)+ι(∥d∥∞),
∀x0,d,∀t ≥0,
(30)
and (29) is considered locally ISS, if the inequality (30) is
valid for an open ball of radius r, x0 ∈Br(0).
Definition 4. The system (29) is exponential input to state
stable (e-ISS) if there exists β ∈K L , ι ∈K∞and a positive
constant λ > 0 such that
|x(t,x0)| ≤β(|x0|,t)e−λt +ι(∥d∥∞),
∀x0,d,∀t ≥0,
(31)
and (29) is considered locally e-ISS, if the inequality (31) is
valid for an open ball of radius r, x0 ∈Br(0).
Definition 5. The system is said to hold the asymptotic gain
(AG) property if there exists ι ∈K∞such that
limt→∞|x(t,x0)| ≤ι(∥d∥∞),
∀x0,d.
(32)
Definition 6. The system is said to be zero stable if there
exists β ∈K L such that:
|x(t,x0)| ≤β(|x0|,t),
∀x0,d,∀t ≥0.
(33)
The AG and ZS property are both pictorially shown in
Fig. 2.
ISS-Lyapunov functions. We can develop Lyapunov func-
tions that satisfy the ISS conditions and achieve the stability
property.
Definition 7. A smooth function V : Rn →R≥0 is an ISS-
Lyapunov function for (29) if there exist functions α, ¯α, α,
ι ∈K∞such that
α(|x|) ≤V(x) ≤¯α(|x|)
˙V(x,d) ≤−α(|x|)
for|x| ≥ι(∥d∥∞).
(34)
The following lemma establishes the relationship between the
ISS-Lyapunov function and the ISS of (29).
Lemma 1. The system (29) is ISS if and only if it admits a
smooth ISS-Lyapunov function.
Fig. 2: If the system is ISS, zero stability is achieved for a zero
input, and asymptotic gain is achieved for a bounded input.
Proof of Lemma 1 is given in [11] and in [10]. In fact
the inequality condition can be made stricter by using the
exponential estimate:
˙V(x,d) ≤−cV(x)+ι(∥d∥∞),
∀x,d.
(35)
which is then called the e-ISS Lyapunov function.
We can also use the AG propery (32) to establish ISS:
Lemma 2. The system is ISS if and only if it is zero stable
and AG.
IV. PHASE TO STATE STABILITY
We can now define the notion of phase to state stability for
continuous sytems below. A preliminary on ISS is given in
Appendix III. Systems considered are of the type
˙η = Fη +Gµ(η)+Gd
˙z = Ψ(η,z),
(36)
where a suitable Lipschitz control law µ(η) ∈Kε(η) is
applied. Since, the analysis is only for continuous dynamics,
the domain notation is ignored. A suitable control law would
have been µ(η), which is state based, but instead, a time based
control law, µ(η) + d, was applied. The time dependency is
implicit in the disturbance input d.
Definition 8. Assume a ball of radius r around the origin.
The system given by (36) is locally phase to η stable , if
there exists β ∈K L , and ι ∈K∞such that
|η(t)| ≤β(|η(0)|,t)+ι(∥d∥∞),∀η(0) ∈Br(0),∀d,∀t ≥0,
(37)
and it is locally phase to state stable , if
|(η(t),z(t))| ≤β(|(η(0),z(0))|,t)+ι(∥d∥∞),
∀η(0) ∈Br(0),∀d,∀t ≥0.
We will first establish phase to η stability, and then include
the zero dynamics to show phase to state stability. Based on the
asymptotic gain and zero stability property of the system (36)
w.r.t. the phase uncertainty d, we have the following lemma.
Lemma 3. The class of all Lipschitz continuous feedback
control laws µ(η) ∈Kε(η) applied on the system (36) yields
phase to η stability in the continuous dynamics.
Proof. Proof is provided in [9] and is straightforward due to
the fact that a stabilizing controller µ(η) is sufficient to render
the linear system phase to η stable. The ultimate bound on
η can be further obtained as
4c2
γc1ε ∥d∥∞. This ultimate upper
bound is computed in [9].
We can also realize exponential phase to state stability
of the continuous dynamics by appending a state based linear
feedback law to the time based feedback linearizing control
resulting in the dynamics
˙η
=
Fη +Gµ(η)+Gd +GByus(η,z)
˙z
=
Ψ(η,z).
(38)
Lemma 3 can now be redefined to obtain exponential phase
to state stability.
Lemma 4. Given the Lipschitz continuous control laws
µ(η) ∈Kε(η), us(η,z) ∈Ks
ε,¯ε(η,z) applied on the system
(38) yields exponential phase to η stability in the continuous
dynamics.
Proof. Proof is again provided in [9] which is similar to the
proof for Lemma 3. The ultimate bounds can also be explicitly
computed by taking the derivative of the Lyapunov function,
Vε and substituting the upper bounds. The ultimate upper
bound on η is 2¯εc2
c2
1ε2 ∥d∥∞.
Continuous Periodic Orbits. Application of a Lipschitz
continuous feedback control law µ(η) ∈Kε(η), us(η,z) ∈
Ks
ε,¯ε(η,z) results in the closed loop vector field (38). Associ-
ated with this vector field is a flow that is a function of ε, ¯ε
and also the disturbance d. Denote the flow as ϕε,¯ε,d
t
(η,z). The
flow with zero disturbance (d(t) ≡0) is periodic with period
T > 0 and a fixed point (η∗,z∗) if ϕε,¯ε,0
t
(η∗,z∗) = (η∗,z∗).
Associated with this periodic flow is the periodic orbit
O = {ϕt(η∗,z∗) ∈R2n : 0 ≤t ≤T},
(39)
Similarly, we denote the flow of the partial zero dynamics by
ϕz
t (y1,z) and the associated periodic orbit by OPZ. The periodic
orbit of the partial zero dynamics can embedded into the full
order dynamics through the canonical embedding Π0(y1,z) =
(y1,0,z). Therefore, O = Π0(OPZ).
For the periodic orbit on the zero dynamics, we have the
periodic orbit of the full order dynamics via the canonical
embedding Π0(OPZ) = O. By defining the norm ∥(η,z)∥=
∥y1∥+∥η2∥+∥z∥, we can define the distance from the periodic
orbit as
∥(η,z)∥O =
inf
(η′,z′)∈O ∥(η,z)−(η′,z′)∥
(40)
=
inf
(y′
1,z′)∈OPZ
∥(z−z′)∥+∥(y1 −y′
1)∥+∥η2∥.
The continuous dynamics is exponentially stable in each
domain if there are constants r,δ1,δ2 > 0 such that if (ηv,zv) ∈
Br(O), a neighborhood of radius r around O, it follows
that ∥ϕε,¯ε,d
(
η,z)∥O ≤δ1e−δ2t∥(ηv,zv)∥O. ∥(y1,z)∥OPZ, as men-
tioned in (40), represents the distance between z and nearest
point on the periodic orbit OPZ. Given that the partial zero
dynamics has an exponentially stable periodic orbit, there
is a Lyapunov function VPZ : R2n →R≥0 such that in a
neighborhood Br(OPZ) of OPZ (by converse Lyapunov theorem
[3]) such that
c4∥(y1,z)∥2
OPZ ≤VPZ(y1,z) ≤c5∥(y1,z)∥2
OPZ,
∂VPZ
∂z Ψ(y1,0,z)+ ∂VPZ
∂y1
˙y1 ≤−c6∥(y1,z)∥2
OPZ,


∂VPZ
∂(y1,z)

 ≤c7∥(y1,z)∥OPZ.
(41)
Define
the
composite
Lyapunov
function:
Vc(η,z) =
σVPZ(y1,z) + Vε(η), we can establish boundedness of the
dynamics of the robot when ∥d∥is bounded. In other words,
we have the following theorem, which establishes phase to
state stability of periodic orbits in continuous systems.
Theorem 1. Given that the periodic orbit Oz of the partial
zero dynamics is exponentially stable, and given the controllers
µ(η) ∈Kε(η), us(η,z) ∈Ks
ε,¯ε(η,z) applied on (38), that
render the outputs η stable w.r.t. d, then the periodic orbit
O = Π0(OPZ) obtained from the canonical embedding is
exponential phase to state stable .
Proof. Upper bounds and lower bounds on Vc are given by
Vc(η,z) ≤max{σc5, c2
ε2 }(∥(y1,z)∥2
OPZ +∥η∥2),
Vc(η,z) ≥min{σc4,c1}(∥(y1,z)∥2
OPZ +∥η∥2),
(42)
Therefore, taking the derivative:
˙Vc(η,z) = σ ∂VPZ
∂z Ψ(y1,0,z)+σ ∂VPZ
∂y1
˙y1 ...
(43)
+σ ∂VPZ
∂z (Ψ(η,z)−Ψ(y1,0,z))+ ˙Vε(η),
≤−σc6∥(y1,z)∥2
OPZ +σc7Lq∥(y1,z)∥OPZ∥η2∥+ ˙Vε(η),
≤−σc6∥(y1,z)∥2
OPZ +σc7Lq∥(y1,z)∥OPZ∥η∥+ ˙Vε(η),
where Lq is the Lipschitz constant for Ψ in (36). Substituting
for ˙Vε leads to the following expression for the Lyapunov
function:
˙Vc ≤−σc6∥(y1,z)∥2
OPZ +σc7Lq∥(y1,z)∥OPZ∥η∥
−γ
ε Vε −1
¯ε
¯Vε +2∥η∥∥Pε∥∥d∥∞
(44)
˙Vc ≤−σc6∥(y1,z)∥2
OPZ +σc7Lq∥(y1,z)∥OPZ∥η∥
−γ
ε Vε −1
¯ε c2
1∥η∥2 +2∥η∥∥Pε∥∥d∥∞
With ¯ε small enough, the disturbance can be rejected by the
expression 1
¯ε c2
1∥η∥2 for ∥η∥≥2¯εc2
c2
1ε2 ∥d∥∞; giving the following
result:
˙Vc ≤−σc6∥(y1,z)∥2
OPZ +σc7Lq∥(y1,z)∥OPZ∥η∥−γ
ε Vε
for
∥η∥≥2¯εc2
c2
1ε2 ∥d∥∞, (45)
which is the standard inequality for ISS-Lyapunov functions
(34). Therefore, for exponential convergence, σ is picked such
that c6c1
γ
ε −σ
c2
7L2q
4
> 0.
REFERENCES
[1] Aaron D Ames.
Human-inspired control of bipedal walking robots.
Automatic Control, IEEE Transactions on, 59(5):1115–1130, 2014.
[2] A.D. Ames, K. Galloway, K. Sreenath, and J.W. Grizzle.
Rapidly
exponentially stabilizing control lyapunov functions and hybrid zero
dynamics. Automatic Control, IEEE Transactions on, 59(4):876–891,
4 2014.
[3] John Hauser and Chung Choo Chung. Converse lyapunov functions for
exponentially stable periodic orbits. Systems & Control Letters, 23(1):27
– 34, 1994.
[4] Ayonga Hereid, Eric A. Cousineau, Christian M. Hubicki, and Aaron D.
Ames.
3d dynamic walking with underactuated humanoid robots:
A direct collocation framework for optimizing hybrid zero dynamics.
In 2016 IEEE International Conference on Robotics and Automation
(ICRA), pages 1447–1454, 5 2016.
[5] S. Kolathaya, A. Hereid, and A. D. Ames. Time dependent control lya-
punov functions and hybrid zero dynamics for stable robotic locomotion.
In 2016 American Control Conference (ACC), pages 3916–3921, 7 2016.
[6] Shishir Kolathaya and Aaron D. Ames.
Parameter sensitivity and
boundedness of robotic hybrid periodic orbits.
IFAC-PapersOnLine,
48(27):377 – 382, 2015.
[7] Shishir Kolathaya and Aaron D. Ames.
Parameter to state stability
of control lyapunov functions for hybrid system models of robots.
Nonlinear Analysis: Hybrid Systems, 25:174 – 191, 2017.
[8] Shishir Kolathaya and A.D. Ames. Phase uncertainty to state stability
of hybrid periodic orbits: Applications to bipedal walking and running.
Submitted to IEEE Transactions on Automatic Control, 2017.
[9] W. L. Ma, Shishir Kolathaya, and Aaron D Ames. Bipedal robotic run-
ning with DURUS-2D: Bridging the gap between theory and experiment.
In Hybrid Systems Computation and Control 2017, 2017.
[10] E. D. Sontag. Smooth stabilization implies coprime factorization. IEEE
Transactions on Automatic Control, 34(4):435–443, 4 1989.
[11] Eduardo D. Sontag. Input to State Stability: Basic Concepts and Results,
pages 163–220. Springer Berlin Heidelberg, 2008.
[12] Dario J Villarreal and Robert D Gregg.
A survey of phase variable
candidates of human locomotion.
In Engineering in Medicine and
Biology Society (EMBC), 2014 36th Annual International Conference
of the IEEE, pages 4017–4021. IEEE, 2014.
[13] E.R. Westervelt, J.W. Grizzle, C. Chevallereau, J.H. Choi, and B. Morris.
Feedback Control of Dynamic Bipedal Robot Locomotion. Automation
and Control Engineering. CRC Press, 2007.