arXiv:1605.02577v1  [math.OC]  9 May 2016
Proofs of the Technical Results Justifying an Algorithm of Reactive
3D Navigation for a Surface Scan by a Nonholonomic Mobile Robot∗
Alexey S. Matveev†, Kirill S. Ovchinnikov†, and Andrey V.Savkin‡
1
Introduction
The paper addresses autonomous navigation of a mobile robot in missions such as aerial and underwater
photography, environmental monitoring, remote sensing, border patrol, environmental monitoring, imaging and
inspecting various 3D structures. For many of these and other missions, it is needed to drive a robot in a 3D space
so that it advances to a certain 2D surface, then moves close to it and performs its scanning, i.e., sequentially
visits tight vicinities of all its points. The seminal paper [10] classiﬁes such behavior as sweep coverage of the
surface. Apart from the afore-mentioned examples, navigation of mobile robots for sweep coverage may be of
interest for environmental studies, geological exploration, underwater archeology [4], seabed exploration [12],
rescue and search operations, ﬁne-scale imaging and inspection of underwater, air, or space structures, e.g.,
coral reefs or ship hulls [13, 16], spray painting [2], oil and gas pipeline inspection, infrastructure surveillance,
damage assessment, recovery, and protection of 3D objects, localization, reduction, and stopping of underwater
oil spills [32], 3D building reconstruction, and aerial city surveillance, to name just a few.
Sweep coverage of planar objects in a two-dimensional workspace, also known as area coverage, enjoys a
substantial literature; we refer the reader to [6, 11, 14, 32] for comprehensive surveys.
Most of these works
assume that both the area to be covered and robot reside in a common plane. However, this fails to be true
for many applications, e.g., for inspecting the surfaces of chimney-stalks or gas storage tanks. Meanwhile, only
a few works address special challenges arisen in coverage of essentially non-planar surfaces when moving in a
three-dimensional workspace.
Vertically projectively planar surfaces [12] can be handled via direct application of planar terrain-covering
algorithms (e.g., lawnmover-like patterns) to the robot’s vertical projection onto a horizontal plane with either
following the elevation proﬁle of the terrain within a certain range of depths [12] or successively moving in evenly
spaced parallel planes [20]. In [3], it is advocated to follow the horizontal slice of a closed orientable surface with a
pre-speciﬁed margin and to vertically lift, from time to time, the slicing plane by a given step. Critical points are
detected to partition a topologically complex surface into simple parts to be individually scanned. The problem
of coverage path planning directly on the surface is addressed in [1, 2] based on a cellular decomposition of
the surface into geometrically simple patches (extruded surfaces homeomorphic to a disk), which are subjected
to relatively independent scans. This opens the door for primarily “planar” treatment of any patch, whereas
performance issues may cause the need to allow for surface contortion.
A relevant problem of ﬁnding an
optimal family of curve segments on a surface is considered in [17]. Paper [11] recommends segmentation of
the seabed into low-relief and high-slope subregions based on a bathymetric map; the former are handled via
classical lawnmower survey, the latter are handled by following constant-depth horizontal bathymetric contours
at iteratively increasing depths. A somewhat similar approach to coverage path planning for arable farming is
employed in [15], where the terrain is partitioned into subregions based mainly on the slope steepness.
Cluttered workspaces and surfaces with occluded parts visible only from special viewpoints are typically
treated via consecutive solution of two NP-hard problems [7, 8, 30]. First, a kind of the standard art gallery
problem is solved in order to determine a set of viewpoints that establish full visual coverage of the surface.
Second, a variant of the traveling salesman problem is solved to acquire a shortest path visiting all these
points. The accompanying hard computational burden can be somewhat eased by using a random sampling
machinery [7,8].
Thus, the entire previous research is focused on geometrical path planning, implementation of following
the planned path was out of discussion. However, this implementation may be a real problem in the face of
practical limitations of the robot; examples are concerned with, but are not limited to, speedy, yet safe, following
along a highly contorted surface or reliable estimation of the surface’s slope based on noisy data. Moreover,
∗This work was supported by the Australian Research Council, and by the Russian Science Foundation 14-21-00041 and the
Saint Petersburg State University.
†Department of Mathematics and Mechanics, Saint Petersburg State University, St. Petersburg, Russia
‡School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia
1
kinematic, dynamic, or sensing constraints may cause infeasibility of the planned maneuvers or operations, like
a sharp vertical turn to escape to a higher plane or detecting the end of a full turn around a horizontal slice of
the surface. When solving purely navigation tasks, most of the previous works either hinge on a known map
of the environment, which may be unavailable in practice, or involve computationally expensive sensory data
processing, close to building such a map.
On the other hand, there is a growing demand to perform coverage operations faster, yet with improved
safety and eﬃciency, e.g., closer to the surface, and by means of simpler and cheaper hardware. This calls
for computationally inexpensive algorithms that allow a mobile robot to quickly make and implement naviga-
tion decisions and ensure safety, while respecting all perceptional and computational limitations and motion
constraints that may apply.
This paper is aimed at demonstration that even in the face of challenges from nonholonomic constrains,
under-actuation, and ﬁnite control range, safe sweep coverage of a 2D surface embedded in a 3D workspace can
be achieved at the lowest control level via a computationally primitive reﬂex-like rules. They straightforwardly
transform the current sensor readings into a current control input and do not use complex or dubious operations
like even short-term motion planning, building a map of even a tight vicinity of the scene or surface, or depositing
marks in the environment. Moreover, the above challenges are enhanced by poor perceptual capacity of the
robot that is conﬁned to access only to a certain direction in 3D, to the robot’s coordinate along it, and to the
distance from the scanned surface in the perpendicular plane. Finally, no a priori knowledge about the surface
is available and the memory size of the robot is drastically small so that it is impossible to memorize, in any
form, online data about the surface in even “next to nothing” portions.
To the above end, we address control at the kinematic level of a robot described by the standard model of the
3D nonholonomic unicycle: The robot travels at a constant speed in 3D and is steered by the acceleration vector,
which is bounded in magnitude and is constantly normal to the velocity. This classic model applies to many
mechanical systems, e.g., constant-speed vehicles moving in the surge direction and controlled by bounded yaw
and pitch rates, like ﬁxed-wing aircrafts or torpedo-like underwater vehicles. Moreover, this model is relevant
for any systems whose acceleration can be freely manipulated within a disk normal to the velocity vector, like
for many rotorcraft systems [5]. The objective is to perform sweep coverage of the surface via spiraling around
it with a drift in the assigned direction.
In this paper, we deal with compact surfaces that are not normal to this direction within the scan range.
At the same time, the respective ﬁndings may be used as primitives when handling more complex surfaces
via their segmentation. To start with, we disclose a trade-oﬀbetween the surface’s contortion and the limited
turning capacity of the robot necessary for the requested sweep coverage to be feasible. After this, we assume
that these conditions are satisﬁed in a slightly enhanced form, propose a novel navigation law, and show,
via a mathematically rigorous nonlocal convergence result, that the control objective is achieved by this law
provided that it is properly tuned. Thus this law solves the problem almost whenever it is solvable. Explicit
recommendations on tuning the controller are also oﬀered.
This paper develops some ideas from [21–29]. The form of the 3D kinematic model is borrowed from [24],
where its detailed discussion, including relations with a more conventional form, is oﬀered. The other papers
handle 2D environments and are threaded by a general regulation paradigm stemming from the biologically
inspired equiangular navigation law [33]. However, the 3D context causes serious extra challenges and makes
the ﬁndings of [21–23,25–28] mostly inapplicable. For example, though the targeted “spiralling” can be viewed
as 2D motion about the surface combined with a perpendicular drift, the properties of 2D motion are far from
key those of a Dubins car treated in the above papers, to say nothing that the respective loops of regulation
of these 2D motion and drift are not decoupled, whereas the inﬂuence of the latter on the former carries a
potential of failure and calls for thorough extra research.
The body of the paper is organized as follows. Section 2 describes the problem setup, its technical speciﬁca-
tion is give in Section 4. Section 3 is devoted to necessary conditions and associated assumptions of theoretical
analysis. The control law and the main results are presented in Sections 5. These results are illustrated in
Section 6 for special scenarios.
The extended introduction and discussion of the proposed control law are given in the paper submitted
by the authors to the IFAC journal Automatica. This text basically contains the proofs of the technical facts
underlying justiﬁcation of the convergence at performance of the proposed algorithm in that paper, which were
not included into it due to the length limitations. To make the current text logically consistent, we reproduce
the problem statement and notations.
2
2
Description of the System and General Setup of the Problem
A mobile robot travels in the three-dimensional space R3 with a constant surge speed v and is controlled by
bounded pitching q and yawing r rates. ∗There is an unknown domain D ⊂R3 in the space. The objective is
to drive the robot to the desired distance d0 to D and to scan and patrol its boundary surface ∂D afterwards,
with maintaining this distance d0.
By following [24], we do not come into details of the roll motion (which is of minor importance for the
problem at hand) and employ the abridged kinematic model of 3D unicycle:
˙r = v⃗ı,
˙⃗ı = ⃗u,
⟨⃗u;⃗ı⟩= 0,
∥⃗u∥≤u.
(2.1)
Here r = col(x, y, z) is the robot’s position, x, y, z are its absolute Cartesian coordinates, ⃗ı is the unit vector
along its centerline, ⟨·; ·⟩and ∥· ∥are the standard Euclidian inner product and norm, respectively, ⃗u is the
two-dimensional control, and the upper bound u on its magnitude is given. In (2.1), the equation ⟨⃗u;⃗ı⟩= 0
keeps the length of ⃗ı constant, as is required. In eﬀect, (2.1) comes to setting up the speed v and minimal
turning radius R := v/u of the robot: It travels with the speed v over space curves whose curvature radii are
no less than R.
Remark 2.1 in [24] discusses replacement of the “abstract” control ⃗u by the conventional pitching q and
yawing r rates in various contexts, based on a one-to-one correspondence ⃗u ↔(q, r). That remark also shows
that the model (2.1) is applicable whenever the speed v can be kept constant by a proper control, whereas
the acceleration can be simultaneously manipulated within a disk perpendicular to the velocity ⃗v and centered
at the origin; then ⃗ı := ⃗v/v. This holds for helicopters, submarine-like vehicles, and many other mechanical
systems that move not necessarily in the surge direction [5,31]. Since our main results are expressed in terms
of ⃗u, they cover this case as well.
The robot has access to a certain unit vector ⃗h ∈R3 in the local frame attached to its body.
It also
measures its own coordinate h(r) = ⟨r;⃗h⟩+ const along the direction given by ⃗h. For the convenience’s sake
and to indicate a typical, though not mandatory, scenario, ⃗h and h are termed the vertical vector and altitude,
respectively. The “horizontal” distance d(r) := minr′∈D∩Hr ∥r −r′∥to the domain D is also accessible. Here
Hr := {r′ : ⟨r′ −r;⃗h⟩= 0} is the horizontal plane passing through the current location r of the robot, and
infr′∈Ø d := ∞. No other measurements are available. A safety margin d ≥dsafe > 0 should be always respected.
To specify the problem, we note that in our theoretical analysis, D is viewed as a compact 3-dimensional
smooth manifold with boundary. Its boundary surface ∂D is a 2-dimensional smooth manifold in R3 with
ﬁnitely many connected components. One of them is the “outer” boundary of D, the others are attached to
the “holes” in D. The objective is to scan a certain component B†, e.g., the “outer” one. The scan should be
done within a given range of altitudes h ∈[h−, h+], where h−< h+. In the horizontal ﬂat layer L deﬁned by
this range, the surface B† may in turns, look like a set of several isolated pieces; the scan objective is focused
on only one of them B. So the directive to maintain a distance of d0 in fact addresses the distance to B.
The scan itself is meant as a robot’s motion for which the point br of B horizontally nearest to the robot
sweeps over the entire B so that any point b of B appears to be in a close proximity of the path traced by br.
Then b is at approximately a distance of d0 = ∥r −br∥from the robot at some time, which distance is viewed
as favorable for achieving a certain “higher level” objective, like monitoring, recording, processing or inspection
of the boundary, or prevention of intrusions through B. Finally, repeated scans should be arranged: a complete
surface scan of B should be rerun over and over again.
Now we pass to formal details of the problem setup.
3
Assumptions Underlying Theoretical Analysis, Necessary Condi-
tions, and Notations
This speciﬁcation proceeds from the following.
Assumption 3.1. The domain D is a compact 3-dimensional C3-smooth manifold with boundary ∂D. Within
the horizontal layer L := {r : h−≤h ≤h+}, the unit inner normal Nb to ∂D is not vertical (co-linear to ⃗h) at
any point b from the boundary piece B† of interest.
Here “inner”=“directed inwards D”. Due to the second claim, B† ∩L has ﬁnitely many connected compo-
nents, each vertically spreading from h−to h+. As was noted, one of them B is to be scanned. Let Dout be the
part of the free space in the layer L whose boundary includes B.
We would like to organize scan as circumnavigation of B at a distance of d = d0 with a small increment ∆h in
altitude from one round of circumvention to the next one so that the robot spirals around B. Circumnavigation
∗The projections of the angular velocity on the pitch and yaw axes of the standard robot-ﬁxed Cartesian frame
3
may be either outer or inner, depending on the side of B that hosts Dout. More precise idea about the desired
maneuver will be given and justiﬁed in the next section based on our main assumptions about the surface. In
turns, they are slightly enhanced conditions necessary for a robot with a limited turning radius to be capable
of coping with contortion of the scanned surface B. To come into details of these conditions and assumptions,
we need the following notations.
• Tb — the plane tangent to B at point b ∈B;
• θb ∈[0, π] — the angle between the normal N(b) and ⃗h;
• Sb(V ) = −DV N ∈Tr — the shape operator, i.e., minus the derivative of N in the tangent direction V ;
• IIb[V ; W] = ⟨Sb(V ); W⟩— the second fundamental form;
• γ(h∗) = {r ∈B : h(r) = h∗} – the horizontal slice of B at the altitude h∗; γb := γ[h(b)];
• ⃗a1 × ⃗a2 — the cross-product of two vectors ⃗ai ∈R3;
• T0(b) = Nb×⃗h
sin θb — the unit vector tangential to the slice γb at b ∈B and directed so that D is to the left;
• T⊥(b) := Nb × T0(b) ∈Tb — the second component of orthogonal basis (T0, T⊥) of the tangent plane Tb;
• ⃗nb := Nb−cos θb⃗h
sin θb
— the inner unit normal to the slice γb in the host horizontal plane;
• br — the point of B horizontally nearest to r.
Within the horizontal layer h(b) ∈[h−, h+], Assumption 3.1 ensures that θb ∈(0, π) and T0(b) is well deﬁned.
As was remarked, we are interested in circumnavigation of B with a small increment ∆h in altitude between
the rounds of circumnavigation. Looking for necessary conditions, we focus on the idealized limit case where
∆h = 0 and so the ideal maneuver comes to planar horizontal circumnavigation of a horizontal slice of B with
a constant margin of d0. By Lemma 1 in [26] and the associated discussion, the following property is nearly
necessary for this maneuver to be feasible at any altitude h ∈[h−, h+].
Assumption 3.2. The “minimum distance” point br ∈B is unique for any location r ∈Dout at a horizontal
distance of d(r) = d0 from B, and such locations do exist.
This assumption is necessarily fulﬁlled if any horizontal slice of the domain D is convex.
The next condition is that the robot’s minimum turning radius R should be small enough as compared with
contortion of B evaluated by the second fundamental form.
Lemma 3.1. Let the robot can circumnavigate the horizontal slice γ of B at any altitude h ∈[h−, h+], moving
in Dout and in the associated horizontal plane, with always maintaining the distance d0 to γ. Then
R = v
u ≤
sin θb
|IIb[T0; T0] | + d0sgn IIb[T0; T0]
(3.2)
for any b ∈B, where T0 := T0(b), a/0 := ∞∀a > 0, and sgn 0 := 0.
The proof of this claim employs several technical facts concentrated in the next lemma. To state it, we recall
that DV is the derivative in direction V and [⃗a,⃗b,⃗c ] = ⟨⃗a;⃗b × ⃗c⟩is the scalar triple product.
Lemma 3.2. At any point b ∈B and for any V ∈Tb,
⟨N;⃗h⟩= cos θ, ⟨N; T0⟩= ⟨N; T⊥⟩= 0,
(3.3)
⟨⃗h; T0⟩= ⟨⃗h;⃗n⟩= 0, ⟨⃗h; T⊥⟩= −⟨N;⃗n⟩= −sin θ,
(3.4)
⟨⃗n; T0⟩= ⟨T⊥; T0⟩= 0,
⟨⃗n; T⊥⟩= cos θ,
(3.5)
DV θ = −II[V ; T⊥] ,
(3.6)
DV ⃗n = −II[V ; T0]
sin θ
T0,
DV T0 = II[V ; T0]
sin θ
⃗n
(3.7)
DvT⊥= II[V ; T⊥] N −cot θ · II[V ; T0] T0.
(3.8)
4
Proof: Formulas (3.3)—(3.5) are immediate from the deﬁnitions of θ, T0, T⊥, and ⃗n since
⟨N;⃗n⟩=
*
N; N −cos θ⃗h
sin θ
+
= 1 −cos θ⟨N;⃗h⟩
sin θ
= 1 −cos2 θ
sin θ
= sin θ,
⟨⃗h;⃗n⟩=
*
⃗h; N −cos θ⃗h
sin θ
+
= ⟨⃗h; N⟩−cos θ
sin θ
= 0,
⟨⃗h; T⊥⟩=
D
⃗h; N × T0
E
=
h
⃗h, N, T0
i (a)
= −
h
T0, N,⃗h
i
= −

T0; N × ⃗h
| {z }
=sin θT0

= −sin θ;
(3.9)
⟨⃗n; T⊥⟩=
*
N −cos θ⃗h
sin θ
; N × T0
+
= −cos θ
sin θ
D
⃗h; N × T0
E (3.9)
== cos θ.
Here (a) holds since the triple scalar product is anti-symmetric. By diﬀerentiating cos θ = ⟨N;⃗h⟩, we arrive
at −DV θ sin θ = −
D
S(V );⃗h
E (b)
= −⟨S(V );⃗π⟩, where ⃗π := ⃗h −cos θN and (b) holds since S(V ) ∈T and
N is normal to the tangent plane T. Since ⟨⃗π; N⟩= 0 ⇒⃗π ∈T, we have ⃗π = cT0 + eT⊥, c, e ∈R. Here
c = ⟨⃗h −cos θN; T0⟩= 0 and e = ⟨⃗h −cos θN; T⊥⟩= −sin θ by (3.3) and (3.4). Thus we arrive at (3.6). The
ﬁrst part of (3.7) holds since ⟨⃗n;⃗h⟩= 0 ∀b ⇒
D
DV ⃗n;⃗h
E
= 0 and ∥⃗n∥= 1 ∀b ⇒⟨DV ⃗n;⃗n⟩= 0 imply that DV ⃗n
is co-linear with ⃗n × ⃗h = N−cos θ⃗h
sin θ
× ⃗h = N×⃗h
sin θ = T0, i.e., Dv⃗n = ζT0, where
ζ = ⟨DV ⃗n; T0⟩=
*
DV
N −⃗h cos θ
sin θ
; T0
+
=
*
−S(V ) + ⃗hDV θ sin θ
sin θ
; T0
+
−⟨⃗n; T0⟩DV θ cot θ
(3.4), (3.5)
======
−S(V )
sin θ ; T0

.
Similar arguments assure that DV T0 = ξ⃗n. Here
ξ = ⟨DV T0;⃗n⟩=
*
DV
N × ⃗h
sin θ ;⃗n
+
= −
*
S(V ) × ⃗h
sin θ
;⃗n
+
−⟨T0;⃗n⟩DV θ cos θ
sin θ
(3.5)
= −[⃗n, S(V ),⃗h]
sin θ
(a)
= [S(V ),⃗n,⃗h]
sin θ
=
*
S(V )
sin θ ; N −cos θ⃗h
sin θ
× ⃗h
+
=
*
S(V )
sin θ ; N × ⃗h
sin θ
+
,
where N×⃗h
sin θ = T0, which completes the proof of (3.7). Finally, DV T⊥lies in the plane normal to T⊥since
∥T⊥∥= 1 ∀b. By (3.3) and (3.5), T0 and N constitute a basis of this plane, and so DV T⊥= ςT0 + βN. It
remains to note that
DvT⊥= DV (N × T0) = (DV N) × T0 + N × DV T0,
ς = ⟨DV T⊥; T0⟩= II[V ; T0]
sin θ
*
T0; N ×
 
N −cos θ⃗h
sin θ
!+
= −II[V ; T0] cot θ
*
T0; N × ⃗h
sin θ
+
= −II[V ; T0] cot θ,
β = ⟨DvT⊥; N⟩= ⟨(DV N) × T0; N⟩= −⟨S(V ) × T0; N⟩
= −[N, S(V ), T0]
(a)
= [S(V ), N, T0] = ⟨S(V ); N × T0⟩= ⟨S(V ); T⊥⟩= II[V ; T⊥] .
□
Proof of Lemma 3.1.
Let ̺(σ) be a regular parametric representation of γ near br.
Here σ is the
arclength, σ = 0 represents br, and σ ascends in the direction of T0. Then f(σ) := 1/2∥̺(σ) −r∥2 ≥f(0).
Hence df(σ)
dσ (0) = ⟨T0(br); br −r⟩= 0 ⇒br −r = d0⃗nbr. Furthermore,
0 ≤d2f(σ)
dσ2
(0) = d
dσ ⟨T0[̺(σ)]; ̺(σ) −r⟩

σ=0
(3.7)
==
IIbr[T0(br); T0(br)]
sin θ
⃗nbr; d0⃗nbr

+ ⟨T0(br); T0(br)⟩
= 1 + d0
IIbr[T0(br); T0(br)]
sin θ
=: C(r).
Meanwhile, the robot’s motion can be parametrized as follows r(t) = ̺[σ(t)] −d0⃗nρ[σ(t)], where σ(t) is the
parameter of br(t). With regard to (2.1) and (3.7), we have
v⃗ı = ˙r = ˙σT0 + d0 ˙σ IIbr[T0(br); T0(br)]
sin θ
T0 = C ˙σT0.
5
So ˙σ = v/C and the robot’s acceleration ⃗u is normal to T0 by (2.1); it also horizontal since the robot moves in
the horizontal plane. It follows that ⃗u = u⃗n, where
u = ⟨⃗u;⃗n⟩=
D
˙⃗ı;⃗n
E (3.7)
== 1
v
d(C ˙σ)
dt
T0;⃗n

+ C ˙σ2
v
II[T0; T0]
sin θ
⃗n;⃗n

(3.5)
== v
C
II[T0; T0]
sin θ
.
By putting this into the last relation |u| = ∥⃗u∥≤u from (2.1) and invoking that C ≥0 by the foregoing, we
arrive at (3.2) via elementary transformations.
□
Thus (3.2) is needed for capacity to maintain d ≡d0, which at least means that whenever d = d0 and ˙d = 0,
a feasible control exists such that ¨d = 0. It does not come as a surprise that putting < in place of ≤in (3.2)
implies the possibility to freely manipulate the sign of ¨d by picking proper feasible controls, thus making the
output d locally controllable. In problems of regulating an output to a desired value, the output controllability
is conventionally needed at all points on the transient portion of the trajectory. In view of this, we enhance
(3.2) by the replacement “≤” 7→“<” and extend the resultant inequality on the entire operational zone Dop.
For convenience’s sake, we deﬁne this zone in terms of h and d. Since upward and downward scans should be
terminated and commenced, respectively, at the altitude h = h+ and the vertical velocity cannot be instantly
reversed, transfer from the former to the latter inevitably involves a maneuver at altitudes h > h+. Similarly,
the opposite transfer is through altitudes h < h−. So we expand the altitude range from [h−, h+] to [h−−
∆h, h+ + ∆h] (with a certain ∆h > 0) in the deﬁnition of the operational zone. It is also characterized by the
extreme values d−< d+ taken by d in it, where dsafe ≤d−< d0 < d+. Overall,
Dop := {r ∈Dout : h−−∆h ≤h(r) ≤h+ + ∆h, d−≤d(r) ≤d+},
and we arrive at the following assumption, which covers Assumption 3.2 and the second part of Assumptions 3.1.
Assumption 3.3. Assumption 3.2 is valid with h+ := h+ + ∆h and h−:= h−−∆h. For any point r ∈Dop,
(3.2) holds with < put in place of ≤and d0 := d(r).
From now on, B denotes the extension of the original B to the lateral piece of the boundary of Dop.
Our last assumption addresses the initial location rin and orientation ⃗ıin of the robot. To simplify the
formulations, we assume that the latter is horizontal ⟨⃗ıin;⃗h⟩= 0, which can be always achieved by a preliminary
maneuver. Then ⟨⃗u;⃗h⟩≡0 guarantees that the robot remains in the initial horizontal plane. There are two
options of such a motion with a maximum and constant actuation
∥⃗u∥≡u,
(3.10)
i.e., the left and right turn, respectively. The respective circular paths C±
in are called the initial circles; the
disks D±
in bounded by them are called the initial disks. Under the control law proposed in Section 5, the robot
initially traces a part of C±
in. The last assumption guarantees that the robot does not leave the operation zone
when doing so.
Assumption 3.4. The both initial disks lie in the interior of the operational zone.
4
Speciﬁcation of the Problem Setup
Deﬁnition 4.1. A maneuver of the robot in Dop is called a full scan of B with preciseness ε∗> 0 if for any
point b ∈B, there exists a point r on the robot’s path such that ∥b −br∥< ε∗.
In fact, the core objective is to perform such a scan with small enough ε∗. This leaves much freedom with
respect to the paths of both the robot and its projection br. The next deﬁnition speciﬁes them by choosing a
spiral-like pattern of motion along the surface of interest.
Deﬁnition 4.2. A maneuver in Dout is called the pure spiralling scan of B with vertical rate η ∈(−v, v) and
within the altitude range [h−, h+] if the altitude h runs the entire length of this range with the constant rate
˙h = η, and the horizontal distance d from the robot to B is kept equal to d0.
For ε, εd > 0, a maneuver is called an (ε, εd)-approximate spiralling scan of B with vertical rate η and
within the altitude range [h−, h+] if the above claims hold with ˙h = η and d ≡d0 replaced by |˙h −η| ≤ε and
|d −d0| ≤εd, respectively, and | ˙d| ≤ε.
This scan may be upward (η > 0) or downward (η < 0). The forthcoming Lemma 4.1 shows that if |η|, ε, and
εa are small enough, approximate spiralling scan of B with vertical rate η and within the altitude range [h−, h+]
guarantees full scan of B in the sense of Deﬁnition 4.1 with high preciseness. Hence such a scan is a fascinating
option. We are interested in repeatedly performing such scans in the sense of the following deﬁnition.
6
Deﬁnition 4.3. We say that the surface B is repeatedly scanned by the robot with the vertical speed η∗> 0 if
d(t) ≥dsafe ∀t and the time interval [0, ∞) can be partitioned into subintervals 0 < t−
0 < t+
0 < t−
1 < t+
1 < . . . so
that t±
k →∞as k →∞, and the following claims hold:
i. During any time interval [t−
k , t+
k ], the robot performs an (εk, εd
k)-approximately pure scan with the vertical
rate ηk = ±η∗, where εk, εd
k →0 as k →∞;
ii. During any time interval Ik := (t+
k , t−
k+1), the altitude h is outside the range [h−, h+];
iii. The intervals Ik are bounded supk(t−
k+1 −t+
k ) < ∞.
It is easy to see that ηk+1 = −ηk ∀k.
The intervals Ik accommodate transients required to pass from
an upward scan to downward one, and vice versa; the interval [0, t−
0 ] accommodates the transient from the
initial state. The convergence εk, εd
k →0 means that the robot’s maneuver becomes a nearly pure scan as time
progresses.
It is required to design a controller that ensures repeated scans in the sense of Deﬁnition 4.3.
In conclusion of the section, we ﬂesh out the above remark about the relationship between scans in the sense
of Deﬁnitions 4.1 and 4.2, respectively.
To this end, we suppose that Assumptions 3.2 and 3.3 hold, and start with introduction of “pseudo-
cylindrical” coordinates on B. The horizontal slice γ(h−) is a connected compact closed smooth curve. So
there is an isometrical isomorphism ς ∈Sl
0
j
←→γ(h−) between the circle Sl
0 of a proper radius l and γ(h−). The
integral curves of the diﬀerential equation ˙b = −T⊥(b)/ sin θb, b ∈B on the compact manifold B are extensible
at least while remaining in the range of altitudes [h−, h+] [18, Ch. IV], whereas ˙h = 1 on them. Hence these
curves span the range of altitudes [h−, h+] and can be parametrized by h ∈[h−, h+]; let b = b(h, ς), h ∈[h−, h+]
be the integral curve starting from b(h−, ς) = j(ς). Smooth dependence of the integral curve on initial data and
uniqueness of the solution of the associated Cauchy problem imply that (h, l) ∈[h−, h+] × Sl
0 →b(h, ς) ∈B
is the diﬀeomorphism between the indicated manifolds. Thus h and ς can be viewed as “pseudo-cylindrical”
coordinates on B.
Lemma 4.1. Suppose that Assumptions 3.2 and 3.3 are valid. There exist η0, εd
0, K, M > 0 such that if
0 < |η| < η0,
ε < |η|/2,
εd < εd
0,
the following claims hold for any (ε, εd)-approximate spiralling scan (in Dop) of B with vertical rate η and
within the altitude range [h−, h+]:
i) This maneuver is a full scan of B with preciseness K|η|;
ii) The coordinate ς of br completes no less than N ≥M h+−h−
|η|
−2 full runs over the circle Sl
0.
The proof of this lemma is prefaced by several technical facts.
Lemma 4.2. While the robot moves in the operational zone,
˙h = v⟨⃗ı;⃗h⟩,
¨h = v⟨⃗u;⃗h⟩,
˙d = −v⟨⃗ı;⃗n⟩−˙h cot θ,
(4.11)
¨d = −v⟨⃗u;⃗n⟩−¨h cot θ + λ2
II[T0; T0]
sin θ + dII[T0; T0] −2
˙hλ
sin θ
II[T⊥; T0]
sin θ + dII[T0; T0]
+
˙h2
sin3 θ
"
II[T⊥; T⊥] −d
II[T0; T⊥]2
sin θ + dII[T0; T0]
#
,
(4.12)
where
λ := sgn ⟨⃗ı; T0⟩
s
v2 −
˙h2
sin2 θ −2˙h ˙d cot θ −˙d2.
(4.13)
Proof: The ﬁrst two formulas in (4.11) are immediate from (2.1) since h = ⟨r;⃗ı⟩+ const. The velocity
⃗τ(t) =
dbr(t)
dt
of the point br(t) ∈B nearest to r(t) is tangential to B and so ⃗τ = ζT0 + ξT⊥. By diﬀerentiating
7
the equation r = b −d⃗n with regard to (2.1) and (3.7), we see that
v⃗ı = ξT⊥−˙d⃗n +

ζ + dII[⃗τ; T0]
sin θ

|
{z
}
λ
T0;
(4.14)
˙h
(4.11)
== v⟨⃗ı;⃗h⟩
(3.4)
== −ξ sin θ ⇒ξ = −
˙h
sin θ;
(4.15)
˙d
(3.5)
= −v⟨⃗ı;⃗n⟩+ ξ cos θ = −v⟨⃗ı;⃗n⟩−˙h cot θ ⇒(4.11),
v2 = ⟨v⃗ı; v⃗ı⟩
(3.5)
==
˙h2
sin2 θ + 2˙h ˙d cot θ + ˙d2 + λ2 ⇒(4.13),
= ζ + dζ II[T0; T0]
sin θ
+ dξ II[T⊥; T0]
sin θ
⇒ζ = λ + d
˙h
sin2 θII[T⊥; T0]
1 + d II[T0;T0]
sin θ
.
By diﬀerentiating (4.14), we get due to (2.1), (3.7), (3.8),
v⃗u = ˙ξT⊥+ ξ {II[⃗τ; T⊥] N −cot θ · II[⃗τ; T0] T0} −¨d⃗n + dII[⃗τ; T0]
sin θ
T0 + ˙λT0 + λII[⃗τ; T0]
sin θ
⃗n;
v⟨⃗u;⃗n⟩
(3.4), (3.5)
====== −
" ¨h
sin θ −ξII[⃗τ; T⊥] cot θ
#
cos θ −¨d + ξ sin θII[⃗τ; T⊥] + λII[⃗τ; T0]
sin θ
= −¨h cot θ −¨d + ξII[⃗τ; T⊥] + λII[⃗τ; T0]
sin θ
.
Here with regard to (4),
ξII[⃗τ; T⊥] + λII[⃗τ; T0] = ξ2II[T⊥; T⊥] + ξζII[T0; T⊥] + λζII[T0; T0] + λξII[T⊥; T0]
= ξ2II[T⊥; T⊥] + λξII[T⊥; T0] + ξII[T0; T⊥]
λ sin θ −dξII[T⊥; T0]
sin θ + dII[T0; T0]

+λII[T0; T0]
λ sin θ −dξII[T⊥; T0]
sin θ + dII[T0; T0]

= ξ2
"
II[T⊥; T⊥] −d
II[T0; T⊥]2
sin θ + dII[T0; T0]
#
+ λ2 sin θ
II[T0; T0]
sin θ + dII[T0; T0] + 2ξλ
sin θII[T⊥; T0]
sin θ + dII[T0; T0]
Summarizing, we arrive at (4.12).
□
Lemma 4.3. If the robot is not vertically oriented and lies in Dop, the following relations hold:
⟨⃗ı; N⟩= −˙dsin θ
v
, ⟨⃗ı⊥
h ;⃗n⟩= cot α
v
h
˙d + cot θ˙h
i
,⃗ı⊥
⊥=
⃗h ×⃗ı
sin α ,
⟨⃗ı⊥
⊥;⃗n⟩=
λ
v sin α, cos α =
˙h
v , sin α =
p
v2 −˙h2
v
.
(4.16)
Proof: The ﬁrst formula in (4.16) follows from (4.14) and (3.3), (3.4). The second formula holds since
⟨⃗ı⊥
h ;⃗n⟩=
*⃗h −cos α⃗ı
sin α
; N −cos θ⃗h
sin θ
+
= ⟨⃗h; N⟩−cos α⟨⃗ı; N⟩−cos θ + cos α cos θ⟨⃗ı;⃗h⟩
sin α sin θ
The third relation holds since ⃗ı⊥
⊥= ⃗ı⊥
h ×⃗ı =
⃗h−cos α·⃗ı
sin α
×⃗ı =
⃗h×⃗ı
sin α. The the fourth formula is true thanks to
(4.14) since
⟨⃗ı⊥
⊥;⃗n⟩=
*⃗h ×⃗ı
sin α ;⃗n
+
= [ N−⃗h cos θ
sin θ
, h,⃗ı]
sin α
=
[N, h,⃗ı]
sin α sin θ;
[⃗ı, N, h]
sin α sin θ =

⃗ı; N×h
sin θ

sin α
= ⟨⃗ı; T0⟩
sin α .
The last two formulas hold since ˙h = v⟨⃗ı;⃗h⟩= v cos α.
□
Proof of Lemma 4.1. As was shown in the proof of Lemma 4.2, dbr
dt = ζT0 −
˙h
sin θT⊥, where ζ is given by
(4.13) and (4). By Assumption 3.3, sin θ + dII[T0; T0] > 0. Since Dop is compact, the above strict inequality
8
and sin θ > 0 imply that the concerned quantities are uniformly (over Dop) bounded from below by positive
constants.
Since |˙h| ≤|η| + |˙h −η| ≤3|η|/2, | ˙d| ≤ε < |η|/2 by Deﬁnition 4.3, picking η0 and εd
0 small
enough ensures that in (4.13) and (4), |λ| ≥v/2 and |ζ| ≥ζ∗> 0, where the constant ζ∗depends only on
B, d0 and v. It follows that the coordinate ς(t) of br(t) obeys the inequality | ˙ς| ≥kζ∗, where k > 0 is the
coeﬃcient of distortion due to the change of the variables. Due to continuity, ς does not change the direction
of motion over Sl
0. Thus it completes a full run over Sl
0 for no less than l/(kζ∗) time units. Meanwhile h runs
the entire length of [h−, h+] with the speed |˙h| ∈[|η|/2, 3/2|η|], which takes a time T ≥2(h+ −h−)/(3|η|).
Hence any point on the cylinder [h−, h+] × Sl
0 lies at a distance of no more than 3/2|η|/(kζ∗) from some
point of the path traced by the “pseudo-cylindrical” coordinates of br(t). Since the map b(h, ς) is Lipschitz
continuous [18, Ch. IV], this implies i).
To prove ii), it suﬃces to note that for time T , the coordinate ς
makes no less than ⌊T kζ∗/l⌋−1 ≥⌊2(h+ −h−)kζ∗/(3|η|l)⌋−1 ≥M h+−h−
|η|
−2 complete runs over Sl
0, where
M := 2kζ∗/(3l).
□
5
The Proposed Control Law and its Convergence
The proposed control law assumes that the robot is not vertically oriented, i.e., the angle α ∈[0, π] between⃗ı and
⃗h diﬀers from 0 and π. Then the orthogonal projection ⃗h−cosα⃗ı of ⃗h onto the plane ⃗ı⊥normal to ⃗ı is nonzero.
Let ⃗ı⊥
h :=
⃗h−cos α·⃗ı
sin α
stand for the normalization of this projection to the unit length, and let ⃗ı⊥
⊥:= ⃗ı⊥
h ×⃗ı. The
pair (⃗ı⊥
h ,⃗ı⊥
⊥) forms an orthonormal basis in ⃗ı⊥and is computable by the robot in its local reference frame. To
achieve the control objective, we employ a hybrid controller with three discrete states (modes) IN, S+, and
S−, which mean “initial maneuver”, “upward scan” and “downward scan”, respectively. The duration Tin > 0
of the initial maneuver and the scan vertical speed η∗> 0 are controller parameters. They are converted into
the current vertical rate η depending on the mode:
η := 0 in IN, η := η∗in S+, η := −η∗in S−.
(5.1)
The logic of switching the modes is as follows (see Fig. 1):
S+ 7→S−whenever h > h+,
S−7→S+ whenever h < h−.
(5.2)
IN
at time t=Tin
7−−−−−−−−→





S+
if h ≤h−,
S±
if h ∈(h−, h+),
S−
if h ≥h+.
(5.3)
The initial mode is IN; in the middle row from (5.3), an arbitrary choice between S+ and S−is performed.
(a)
(b)
Figure 1: Switching logic
The control law is given by the simple formula
⃗u = −uhsgn
h
˙h −η
i
⃗ı⊥
h +
q
u2 −u2
hsgn
h
˙d + χ(d −d0)
i
⃗ı⊥
⊥.
(5.4)
Here χ(·) is a linear function with saturation
χ(p) :=
(
γp
if |p| ≤δ,
sgn (p)µ
otherwise,
µ := γδ,
(5.5)
and uh ∈(0, u), γ > 0, δ > 0 are controller parameters. Since the unit vectors ⃗ı⊥
h ,⃗ı⊥
⊥, and ⃗ı are mutually
perpendicular, the control (5.4) is feasible, i.e., meets the last two requirements from (2.1). The time derivatives
˙h and ˙d can be accessed via, e.g., numerical diﬀerentiation.
Solutions of the system closed by the discontinuous controller (5.4) are meant in the Filippov sense [9].
Now we show that the control objective can always be achieved by means of the proposed control law under
the above nearly unavoidable assumptions.
9
Theorem 5.1. Suppose that Assumptions 3.3 and 3.4 hold. Then there exist controller parameters η∗, uh, µ, δ,
and Tin such that the following claim holds:
(i) The proposed controller ensures a repeated scan of the surface B with the requested vertical speed and
constantly maintaining a non-vertical orientation ⃗ı × ⃗h ̸= 0 ∀t.
Moreover, for any compact set Qin = {(r,⃗ı)} of initial states each element of which satisﬁes Assumption 3.4
and any η∗> 0, there exist common values of the controller parameters for which (i) holds with the vertical
speed of scan η∗≤η∗whenever the initial state lies in Qin.
The proof of the results stated in this section are given in the journal version of the paper submitted to
Automatica. The ﬁrst claim of Theorem 5.1 is a particular case of the second one (Qin contains only one state).
Now we detail controller tuning under which the conclusion of Theorem 5.1 holds.
As a preliminary step, we note that by Assumption 3.4,
u
v >
|II[T0; T0] |
sin θ + dII[T0; T0],
sin θ + dII[T0; T0] > 0
(5.6)
in the operational zone. Since this zone is compact, there exist k > 1 and ∆> 0 such that in this zone,
u
v > k
|II[T0; T0] |
sin θ + dII[T0; T0] + 2∆.
(5.7)
After ﬁnding such k and ∆, we pick a free design parameter κ ∈(0, 1).
Tuning uh ∈(0, u) from (5.4). To this end, we ﬁrst modify the deﬁnition of the initial disk via putting
∥⃗u∥≡
p
u2 −u2
h in place of (3.10). This results in two larger horizontal disks D±
in(uh) whose boundaries are
tangential to C±
in at the initial location of the robot and whose radius
v
√
u2−u2
h is close to the radius v
u of C±
in
for uh ≈0. Thanks to Assumption 3.4, there is u0
h ∈(0, u) such that the disks D±
in(uh) lie in the operational
zone for all uh ∈(0, u0
h]. The parameter uh is chosen so that
0 < uh ≤min
n
u
p
1 −k−1; u0
h
o
,
(5.8)
uh ≤
v√1 −κ
√
k
h
κ
2√1−κ + | cot θ|
i∆in Dop.
(5.9)
Tuning η∗from (5.1) and µ, γ from (5.5). These parameters are chosen so that in the operational zone,
η2
∗
sin2 θ + 2η∗µ| cot θ| + µ2 < κv2,
(5.10)
2η∗µ ≤(k −1)v2| tan θ|,
(5.11)
η2
∗
sin3 θ
II[T⊥; T⊥] −d
II[T0; T⊥]2
sin θ + dII[T0; T0]
 + γµ + 2v
√
k
sin θ
|II[T⊥; T0] |
sin θ + dII[T0; T0]η∗≤v2√1 −κ
√
k
∆,
(5.12)
η∗< v
s
1 −

1 −∆huh
v
2
+
, [a]+ := max{0, a}.
(5.13)
Finally, the duration of the initial mode should be chosen so that
Tin >
2π
p
u2 −u2
h
.
(5.14)
The recommended choice is feasible since (5.8)—(5.12) are satisﬁed by all small enough uh, η∗, µ, and γ. This
can be viewed as a guideline for experimentally tuning the controller. If an a priory knowledge about the domain
is available, (5.8)—(5.12) can be used for analytically tuning the controller, which is based on estimation of the
involved parameters θ and II[·; ·] of the surface. In the next section, we illustrate this with simple examples.
6
Illustrative Examples:
Inner and Outer Scans of an Unknown
Surface of Revolution
Let the surface of interest B be generated by rotating a C3-smooth planar regular curve C around a vertical
axis A. Also, let the generatrix C lie in a plane containing A, do not intersect A, and let its tangent TC be
10
never normal to A. It follows that C can be viewed as the graph of the function ρ(h) that gives the distance
between C and A at the altitude h ∈Ih := [h−−∆h, h+ + ∆h].
Assuming that the z-axis of the world frame is aligned with A, we parametrize x = ρ(h) cos ϕ, y =
ρ(h) sin ϕ, z = h the surface B by ϕ and h ∈Ih. By using this, it is easy to show (see [19, Sec. 5.10] for
details) that
N = −
(−1)σ
p
1 + (ρ′)2


cos ϕ
sin ϕ
−ρ′

, θ = π
2 −(−1)σβ,
II[T0; T0] =
(−1)σ
ρ
p
1 + (ρ′)2 = (−1)σ cos β
ρ
, II[T0; T⊥] = 0,
II[T⊥; T⊥] =
(−1)σ+1ρ′′
(1 + (ρ′)2)3/2 = (−1)σ+1c.
Here β(h) = arctanρ′(h) is the angle from ⃗h to TC, σ = 0 and σ = 1 in the case of outer and inner scan,
respectively, and c = c(h) is the signed curvature of C at the altitude h (c > 0 at concavities of the graph).
After simple computations, the necessary conditions from Section 3 take the form R ≤ρ(h) + (−1)σd0 with an
apparent sense: The minimal turning radius R is so small that the robot can horizontally rotate around A at a
distance of ρ + (−1)σd0, which is equivalent to horizontally following the surface B with the margin d = d0.
Now we comment on analytically tuning the controller, assuming that the available knowledge about the
unknown curve C comes to the following estimates:
• β ∈[0, π/2) — an upper bound on the tangent angle |β|;
• 0 < ρ−< ρ+ — a lower and upper estimate of the distance ρ(h), h ∈Ih from A to the curve C, respectively;
• c+ — an upper estimate of the curvature |c| of C.
Based on these data, the controller should be tuned so that from any initial location rin with h(rin) ∈Ih and
ρ(rin) ∈[ρ−
in, ρ+
in], the robot is driven to a horizontal distance of d0 to B and then repeatedly scan B with
d ≈d0. Here and in the sequel, ρ(r) is the distance from r to A, and 0 < ρ−
in < ρ+
in are given.
6.1
Outer Scan
We assume that
ρ−
in > ρ+ + dsafe + 2R,
ρ−
in > 3R + ρ+ −ρ−,
ρ−+ d0 > R.
(6.1)
The ﬁrst two inequalities mean that the initial location rin is far enough from A; the ﬁrst of them guarantees
that rin is on the correct side of B. The last inequality enhances the controllability condition R ≤ρ(h)+ d0 ∀h.
Now we put
d∗:= 1
2

max{dsafe; R −ρ−} + min{d0; ρ−
in −ρ+ −2R}

,
∆ρ := ρ−
in −ρ+ −2R −d∗
(6.1)
> 0,
u0
h = u
s
1 −
1
[1 + ∆ρ/(2R)]2 .
(6.2)
Then we pick free design parameters: κ ∈(0, 1) and (k, ∆) from the following triangle in the plane of (k, ∆)’s
R−1 >
k
ρ−+ d∗
+ 2∆,
k > 1.
(6.3)
Proposition 6.1. Let the parameters uh, η∗, µ, γ > 0 and Tin of the controller be chosen so that (5.14) holds
and
0 < uh ≤min


u
p
1 −k−1; u0
h;
v√1 −κ
√
k
h
κ
2√1−κ + tan β
i∆


,
η2
∗
cos2 β + 2η∗µ tan β + µ2 < κv2,
(6.4)
2η∗µ ≤(k −1)v2 cot β,
(6.5)
η∗< v
s
1 −

1 −∆huh
v
2
+
,
η2
∗c+
cos3 β < v2√1 −κ
√
k
∆,
(6.6)
γ ≤µ−1
v2√1 −κ
√
k
∆−η2
∗c+
cos3 β

.
(6.7)
11
Then claim (i) of Theorem 5.1 is true whenever h(rin) ∈Ih, ρ(rin) ∈[ρ−
in, ρ+
in], and an outer scan is performed.
Here the range of uh’s is given in a closed form, whereas (6.4), (6.5), and (6.6) explicitly describe a domain
of (η∗, µ)’s bounded by an ellipse, hyperbola, and a straight line. After µ is chosen, (6.7) bounds γ in an explicit
form.
Proof of Proposition 6.1. By putting d−:= d∗, d+ := ∞, it is easy to see that Assumptions 3.3 and 3.4
hold and (5.7) takes the form (6.3). It is straightforward to check that the quantity (6.2) meets the requirements
preceding formula (5.8). By maximizing over θ = π/2 −β(h), h ∈Ih, the requirements (5.8)—(5.11), (5.13) are
reduced to those from the proposition’s body, except for (6.7) and the second part of (6.6). The last inequalities
are similarly obtained from (5.12).
□
6.2
Inner Scan
We assume that
ρ−> ρ+
in + dsafe + 2R,
ρ−
in > 3R + ρ+ −ρ−,
ρ−> d0 + R.
(6.8)
By the last inequality, the space inside B is “wide enough”; the ﬁrst two ones mean that the initial location is
far enough from both B and the axis A. We put
d† := 1
2

max{d0; ρ+ −ρ−
in + 2R} + ρ−−R

,
∆ρ = min{d† + ρ−
in −ρ+ −2R; ρ−−ρ+
in −dsafe −2R}
and deﬁne u0
h by (6.2). The free design parameters: κ ∈(0, 1) and (k, ∆) are chosen subject to
R−1 >
k
ρ−−d†
+ 2∆,
k > 1.
(6.9)
Proposition 6.2. Let the parameters uh, η∗, µ, γ > 0 and Tin be chosen like in Proposition 6.1. Then claim (i)
of Theorem 5.1 is true whenever h(rin) ∈Ih, ρ(rin) ∈[ρ−
in, ρ+
in], and an inner scan is performed.
The proof is similar to that of Proposition 6.1.
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