∎

11institutetext: Qianqian Xia 22institutetext: Nanjing University of
Information Science & Technology, Nanjing, 210044, China
22email<EMAIL_ADDRESS>
# Approximating continuous function on orbit spaces

Qianqian Xia

(Received: date / Accepted: date)

###### Abstract

In this paper we study a subclass of subcartesian space-the orbit space of a
proper action of Lie group on smooth manifold. We show that continuous
functions on orbit space can be approximated by smooth functions.

###### Keywords:

subcartesian space orbit space function approximation

##  1 Introduction

It is well-known that a continuous function on a smooth manifold can be
approximated by a smooth function [1] as stated by the following theorem:

Theorem 1.1 Let $M$ be a smooth manifold and $f:M\rightarrow\mathbb{R}$ be a
continuous function. Then for any $\delta>0$, there exists smooth function
$g\in C^{\infty}(M,\mathbb{R})$, such that

$|g(x)-f(x)|<\delta,$

for all $x\in M$.

There has long been perceived the need for an extension of the framework of
smooth manifolds [2] in differential geometry, which is too restrictive and
does not admit certain basic geometric intuitions. Sikorski¡¯s theory of
differential spaces studies the differential geometry of a large class of
singular spaces which both contains the theory of manifolds and allows the
investigation of singularities. It is the investigation of geometry in terms
of differentiable functions: a differential structure on a topological space
$S$ is a family $C^{\infty}(S)$ of real-valued functions on $S$ satisfying
three conditions. Functions $f\in C^{\infty}(S)$ are called smooth functions
on $S$. A differential space $S$ is said to be subcartesian [3] if every point
of $S$ has a neighbourhood diffeomorphic to a subset of some Cartesian space
$\mathbb{R}^{n}$. The theory of subcartesian spaces has been developed by
Śniatycki et.al. in recent few years. See ${[3]}$ for a systematic treatment
on this topic.

Generally speaking, it is not valid that a continuous function on a
subcartesian space $S$ can be approximated by smooth functions on $S$.
However, we study a subclass of subcartesian space in this paper. It has been
shown in ${[3]}$ that the orbit space of a proper action of a connected Lie
group on a smooth manifold is a subcartesian space. In this paper we show that
a continuous function on the orbit space $R$ can be approximated by smooth
functions on it.

This paper is organized as follows. In Section 1, we recall some basic
definitions on subcartesian space. In Section 2, we recall some basic facts on
orbit space. In Section 3, we prove our main results.

##  2 Subcartesian space

Definition 2.1 A differential structure on a topological space $S$ is a family
$C^{\infty}(S)$ of real-valued functions on $S$ satisfying the following
conditions:

1\. The family

$\\{f^{-1}(I)|f\in C^{\infty}(S)\text{and}\,I\,\text{is an open interval
in}\,\mathbb{R}\\}$

is a subbasis for the topology of $S$.

2\. If $f_{1},\cdots,f_{n}\in C^{\infty}(S)$ and $F\in
C^{\infty}(\mathbb{R}^{n})$, then $F(f_{1},\cdots,f_{n})\in C^{\infty}(S)$.

3\. If $f:S\rightarrow\mathbb{R}$ is a function such that, for every $x\in S$,
there exist an open neighborhood $U$ of $x$, and a function $f_{x}\in
C^{\infty}(S)$ satisfying

$f_{x}|U=f|U,$

then $f\in C^{\infty}(S)$. Here, the subscript vertical bar $|$ denotes a
restriction.

Definition 2.2 A map $\phi:R\rightarrow S$ is $C^{\infty}$ if
$\phi^{*}f=f\circ\phi\in C^{\infty}(R)$ for every $f\in C^{\infty}(S)$. A
$C^{\infty}$ map $\phi$ between differential spaces is a $C^{\infty}$
diffeomorphism if it is invertible and its inverse is $C^{\infty}$.

An alternative way of constructing a differential structure on a set $S$, goes
as follows. Let $\mathcal{F}$ be a family of real-valued functions on $S$.
Endow $S$ with the topology generated by a subbasis

$\\{f^{-1}(I)|f\in\mathcal{F}\,\text{and}\,I\,\text{is an open interval
in}\,\mathbb{R}\\}.$

Define $C^{\infty}(S)$ by the requirement that $h\in C^{\infty}(S)$ if, for
each $x\in S$, there exist an open subset $U$ of $S$, functions
$f_{1},\cdots,f_{n}\in F$, and $F\in C^{\infty}(\mathbb{R}^{n})$ such that

$h|_{U}=F(f_{1},\cdots,f_{n})|_{U}.$

Clearly, $\mathcal{F}\in C^{\infty}(S)$. It is proved in [3] that
$C^{\infty}(S)$ defined here is a differential structure on $S$. We refer to
it as the differential structure on $S$ generated by $\mathcal{F}$.

Let $R$ be a differential space with a differential structure $C^{\infty}(R)$,
and let $S$ be an arbitrary subset of $R$ endowed with the subspace topology
(open sets in $S$ are of the form $S\cap U$, where $U$ is an open subset of
$R$). Let

$R(S)=\\{f|_{S}|f\in C^{\infty}(R)\\}.$

Definition 2.3 The space $R(S)$ of restrictions to $S\subseteq R$ of smooth
functions on $R$ generates a differential structure $C^{\infty}(S)$ on $S$
such that the differential-space topology of $S$ coincides with its subspace
topology. In this differential structure, the inclusion map $i:S\rightarrow R$
is smooth.

In other words, $R(S)$ is the space of restrictions to $S$ of smooth functions
on $R$.

Now consider an equivalence relation $\sim$ on a differential space $S$ with
differential structure $C^{\infty}(S)$. Let $R=S/{\sim}$ be the set of
equivalence classes of $\sim$, and let $\rho:S\rightarrow R$ be the map
assigning to each $x\in S$ its equivalence class $\rho(s)$.

Definition 2.4 The space of functions on $R$, given by

$C^{\infty}(R)=\\{f:R\rightarrow\mathbb{R}|\rho^{*}f\in C^{\infty}(S)\\},$

is a differential structure on $R$. In this differential structure, the
projection map $\rho:S\rightarrow R$ is smooth.

It should be emphasized that, in general, the quotient topology of
$R=S/{\sim}$ is finer than the differential-space topology defined by
$C^{\infty}(R)$.

Definition 2.5 A differential space $S$ is subcartesian if it is Hausdorff and
every point $x\in S$ has a neighbourhood $U$ diffeomorphic to a subset $V$ of
$\mathbb{R}^{n}$ by the map $\phi:U\rightarrow\mathbb{R}^{n}$.
$(U,\mathbb{R}^{n},\phi)$ is said to be a local chart of $S$. The subcartesian
space is said to be locally connected if for every point $x\in S$, there
exists a connected neighborhood $U$ such that $(U,\mathbb{R}^{n},\phi)$ is a
local chart of $S$.

##  3 Orbit space

Consider a smooth and proper action

$\displaystyle\Phi:$ $\displaystyle G\times M\rightarrow M$ (1)
$\displaystyle(g,x)\rightarrow\Phi(g,x)=\Phi_{g}(x)=gx$

of a Lie group $G$ on a manifold $M$.

We endow the orbit space $R=M/G$ with the quotient topology. In other words, a
subset $V$ of $R$ is open if $U=\rho^{-1}(V)$ is open in $M$, where
$\rho:M\rightarrow R$ is the canonical projection (the orbit map). Let

$C^{\infty}(R)=\\{f:R\rightarrow\mathbb{R}|\rho^{*}f\in C^{\infty}(M)\\}.$

$C^{\infty}(R)$ is a differential structure on $R$.

Proposition 3.1[3] The topology of $R$ induced by $C^{\infty}(R)$ coincides
with the quotient topology.

Proposition 3.2[3] A slice through $x\in M$ for an action of $G$ on $M$ is a
submanifold $S_{x}$ of $M$ containing $x$ such that:
1\. $S_{x}$ is transverse and complementary to the orbit $Gx$ of $G$ through
$x$. In other words,

$T_{x}M=T_{x}S_{x}\oplus T_{x}(Gx).$

2\. For every $x^{\prime}\in S_{x}$, the manifold $S_{x}$ is transverse to the
orbit $Gx^{\prime}$; that is,

$T_{x^{\prime}}M=T_{x^{\prime}}S_{x}+T_{x^{\prime}}(Gx^{\prime}).$

3\. $S_{x}$ is $G_{x}$-invariant.
4\. Let $x^{\prime}\in S_{x}$. If $gx^{\prime}\in S_{x}$, then $g\in G_{x}$.

The existence of a slice through $x\in M$ is ensured by the following result.

Proposition 3.3[3] There is an open ball $B$ in $\text{hor}T_{x}M$ centred at
$0$ such that $S_{x}=\exp_{x}(B)$ is a slice through $x$ for the action of $G$
on $M$, where $\exp_{x}(v)$ is the value at 1 of the geodesics of
$G$-invariant Riemannian metric originating from $x$ in the direction $v$.
Further, the set $GS_{x}=\\{gq|g\in G\,\text{and}\,q\in S_{x}\\}$ is a $G$
invariant open neighbourhood of $x$ in $M$.

By construction, $S_{x}=\exp_{x}(B)$, where $\exp_{x}$ is an $H$-equivariant
map from a neighbourhood of $0$ in $T_{x}M$ to a neighbourhood of $x$ in $M$,
and $B$ is a ball in $\text{hor}T_{x}M$ invariant under a linear action of $H$
centred at the origin. The action of $H$ on $T_{x}M$ is linear, and it leaves
$\text{hor}T_{x}M$ invariant. Hence, it gives rise to a linear action of $H$
on $\text{hor}T_{x}M$. Moreover, the restriction of $exp_{x}$ to $B$ gives a
diffeomorphism $\psi:B\rightarrow S_{x}$, which intertwines the linear action
of $H$ on $\text{hor}T_{x}M$ and the action of $H$ on $S_{x}$.

Since $B$ is an $H$-invariant open subset of $\text{hor}T_{x}M$ and the action
of $H$ on $\text{hor}T_{x}S_{x}$ is linear, by a theorem of G.W. Schwarz,
smooth $H$-invariant functions on $S_{x}$ are smooth functions of algebraic
invariants of the action of $H$ on $\text{hor}T_{x}M$. Let
$\mathbb{R}[\text{hor}T_{x}M]^{H}$ denote the algebra of $H$-invariant
polynomials on $\text{hor}T_{x}M$. Hilbert¡¯s Theorem ensures that
$\mathbb{R}[\text{hor}T_{x}M]^{H}$ is finitely generated. Let
$\sigma_{1},\cdots,\sigma_{n}$ be a Hilbert basis for
$\mathbb{R}[\text{hor}T_{x}M]^{H}$ consisting of homogeneous polynomials. The
corresponding Hilbert map

$\displaystyle\sigma:\text{hor}T_{x}M\rightarrow\mathbb{R}^{n}:v\rightarrow\sigma(v)=(\sigma_{1}(v),\cdots,\sigma_{n}(v))$
(2)

induces a monomorphism
$\tilde{\sigma}:(\text{hor}T_{x}M)/H\rightarrow\mathbb{R}^{n}:Hv\rightarrow\sigma(v)$,
where $Hv$ is the orbit of $H$ through $v\in\text{hor}T_{x}M$ treated as a
point in $(\text{hor}T_{x}M)H$. Let $Q$ be the range of $\sigma$. By the
Tarski¨CSeidenberg Theorem, $Q$ is a semi-algebraic set in $\mathbb{R}^{n}$.
Let

$\displaystyle\phi:(\text{hor}T_{x}M)/H\rightarrow Q\subseteq\mathbb{R}^{n}$
(3)

be the bijection induced by $\tilde{\sigma}$. $\phi$ is a diffeomorphism [3].

Since $B$ is an $H$-invariant open neighbourhood of $0$ in
$\text{hor}T_{x}S_{x}$, it follows that $B/H$ is open in
$(\text{hor}T_{x}M)/H$. Hence, $B/H$ is in the domain of the diffeomorphism
$\phi:(\text{hor}T_{x}M)/H\rightarrow Q$, which induces a diffeomorphism of
$B/H$ onto $\phi(B/H)\rightarrow Q\subseteq\mathbb{R}^{n}$. Thus, $B/H$ is
diffeomorphic to a subset of $\mathbb{R}^{n}$. But $B/H$ is diffeomorphic to
$S_{x}/H$, and $S_{x}/H$ is diffeomorphic to $GS_{x}/G$. Therefore, $GS_{x}/G$
is diffeomorphic to a subset of $\mathbb{R}^{n}$.

Theorem 3.4[3] The orbit space $R=M/G$ of a proper action of $G$ on $M$ with
the differential structure $C^{\infty}(R)$ is subcartesian.

##  4 Approximating continuous function on orbit spaces

Lemam 4.1[1] Let $U$ be an open subset of $\mathbb{R}^{m}$ and let $K\subseteq
U$ be a compact subset of $\mathbb{R}^{m}$. Then for any continuous function
$f\in C^{0}(U)$ and any $\delta>0$, there exists a smooth function $g\in
C^{\infty}(U)$, such that

$|g(y)-f(y)|<\delta,$

for any $y\in K$.

Lemma 4.2 For each $y_{0}\in R$, there exist a local neighborhood $V$ of
$y_{0}$ and a compact subset $K\subseteq V$ satisfying that for any continuous
function $f\in C^{0}(V)$ and any $\delta>0$, there exists a smooth function
$g\in C^{\infty}(V)$, where $(V,C^{\infty}(V))$ is a differential subspace of
$R$, such that

$|g(y)-f(y)|<\delta,$

for any $y\in K$.

Proof. Let $x_{0}\in M$ such that $\rho(x_{0})=y_{0}$. Let $H$ be the isotropy
group of $x_{0}$ and $S_{x_{0}}=\exp_{x_{0}}(B)$ be a slice through $x_{0}\in
M$, where $\exp_{x_{0}}$ is an $H$-equivariant map from a neighbourhood of $0$
in $T_{x_{0}}M$ to a neighbourhood of $x_{0}$ in $M$, and $B$ is a ball in
$\text{hor}T_{x_{0}}M$ invariant under a linear action of $H$ centred at the
origin. Let $K\subseteq B$ be a $H$ invariant compact subset of $T_{x_{0}}M$.

Then for any continuous function $f$ on $\rho(GS_{x_{0}})$, it follows that
$\exp^{*}(\rho^{*}f|_{S_{x_{0}}})$ is a continuous function on $B$. From Lemma
4.1 we know that any for $\delta>0$ there exists a smooth function $h\in
C^{\infty}(B)$, such that

$|h(x)-\exp^{*}(\rho^{*}f|_{S_{x_{0}}})(x)|<\delta,$

for any $x\in K$.

Now consider the smooth function $h\circ\exp^{-1}$ on $S_{x}$, which satisfies
that $|h\circ\exp^{-1}(x)-\rho^{*}f|_{S_{x_{0}}}(x)|<\delta,$ for any
$x\in\exp(K)$, where $\exp(K)$ is an $H$-invariant compact subset in $M$.
Since $H$ is compact, we may average $h\circ\exp^{-1}$ over $H$, obtaining a
$H$-invariant function

$\tilde{h}=\int_{H}\Phi^{*}_{g}(h\circ\exp^{-1})d\mu(g),$

where $d\mu(g)$ is the Haar measure on $H$ normalized so that $volH=1$.

The set $GS_{x_{0}}$ is a $G$-invariant open neighbourhood of $x_{0}$ in $M$.
We can define a $G$-invariant function $\tilde{f}_{1}$ on $GS_{x_{0}}$ as
follows. For each $x^{\prime\prime}\in GS_{x_{0}}$, there exists $g\in G$ such
that $x^{\prime\prime}=gx^{\prime}$ for $x^{\prime}\in S_{x}$, and we set

$\tilde{f}_{1}(x^{\prime\prime})=\tilde{h}(x^{\prime}).$

Since $\tilde{h}$ is $H$-invariant, the function $\tilde{f}_{1}$ is well
defined on $GS_{x_{0}}$ and is $G$-invariant. Hence we have a smooth function
$f_{1}\in C^{\infty}(\rho(S_{x_{0}}))$ such that
$\rho^{*}f_{1}=\tilde{f}_{1}$. Besides, for each $y\in\rho(\exp(K))$, we have

$\displaystyle|f_{1}(y)-f(y)|$ $\displaystyle=$
$\displaystyle|\tilde{h}(x)-\rho^{*}f(x)|=|\int_{H}(\Phi^{*}_{g}(h\circ\exp^{-1}))(x)d\mu(g)-\int_{H}(\Phi^{*}_{g}(\rho^{*}f))(x)d\mu(g)|$
(4) $\displaystyle=$
$\displaystyle|\int_{H}(\Phi^{*}_{g}(h\circ\exp^{-1})-\Phi^{*}_{g}(\rho^{*}f))(x)d\mu(g)|$
$\displaystyle\leq$
$\displaystyle\int_{H}|(\Phi^{*}_{g}(h\circ\exp^{-1})-\Phi^{*}_{g}(\rho^{*}f))(x)|d\mu(g)$
$\displaystyle<$ $\displaystyle\int_{H}\delta d\mu(g)$ $\displaystyle=$
$\displaystyle\delta.$

Hence for $y_{0}\in R$ and for $x\in\rho^{-1}(y)$,there exist a local
neighborhood $\rho(GS_{x})$ of $y_{0}$ and a compact subset
$\rho(K)\subseteq\rho(GS_{x})$ satisfying that for any continuous function
$f\in C^{0}(\rho(GS_{x}))$ and any $\delta>0$, there exists a smooth function
$f_{1}\in C^{\infty}(\rho(GS_{x}))$, where
$(\rho(GS_{x}),C^{\infty}(\rho(GS_{x})))$ is a differential subspace of $R$,
such that

$|f_{1}(y)-f(y)|<\delta,$

for any $y\in\rho(K)$. Hence the result follows immediately.

Lemma 4.3[2] Let $U,V$ be two open subsets of the smooth manifold $M$
satisfying that $\bar{U}$ is compact and $\bar{U}\cap\bar{V}=\emptyset$. Then
there exist smooth function $f\in C^{\infty}(M)$ such that

$\displaystyle f(x)=1,x\in\bar{U},$ $\displaystyle
0<f(x)<1,x\in{M/{\bar{V}\cup\bar{U}}}$ $\displaystyle f(x)=0,x\in\bar{V}.$ (5)

Lemma 4.4 Let $x\in M$ and let $0\in W\subseteq V\subseteq U\subseteq B$ be
$H$-invariant open subsets of $\text{hor}T_{x}M$ such that $\bar{W}\subseteq
V$ and $\bar{V}$ are compact, where $B$ is satisfies that $\exp(B)=S_{x}$. Let
$G$ be an open subset of $R$ and $(\rho(\exp(U)),\psi)$ be the local
coordinate for $R$ induced by the Hilbert map (2). Let
$f:R\rightarrow\mathbb{R}$ be a continuous map satisfying that $f|_{G}$ is
smooth. Then for any $\delta>0$, there exists a continuous map
$g:R\rightarrow\mathbb{R}$, such that

(1) $g(y)=f(y)$, for any $y\in R/\rho(\exp(V))$;

(2) $g|_{G\cup\rho(\exp(W))}$ is smooth;

(3) $|g(y)-f(y)|<\delta$, for all $y\in R$,

where $\sigma$ is given by (2).

Proof. It follows from Lemma 4.3 that there exists smooth function
$\eta:\text{hor}_{x}M\rightarrow\mathbb{R}$ such that

$\displaystyle\eta(x)=1,x\in\bar{W},$ $\displaystyle
0<\eta(x)<1,x\in{V/{\bar{W}}}$
$\displaystyle\eta(x)=0,x\in\text{hor}T_{x}M/{V},$ (6)

which yields a smooth function $\eta\circ\exp^{-1}$ on $S_{x}$. Since $V,W$
are $H$-invariant, then by averaging $\eta\circ\exp^{-1}$ over $G_{x}$ we get
a $G_{x}$ invariant smooth function $\tilde{\eta}$ on $S_{x}$ satisfying that

$\displaystyle\tilde{\eta}(x)=1,x\in\exp(\bar{W}),$ $\displaystyle
0<\tilde{\eta}(x)<1,x\in\exp({V/{\bar{W}}})$
$\displaystyle\tilde{\eta}(x)=0,x\in S_{x}/\exp(V),$ (7)

which can be extended to a smooth $G$ invariant function on $M$. Hence we get
a function $\bar{\eta}\in C^{\infty}(R)$ satisfying that

$\displaystyle\bar{\eta}(y)=1,y\in\rho(\exp(\bar{W})),$ $\displaystyle
0<\bar{\eta}(y)<1,y\in\rho(\exp({V/{\bar{W}}}))$
$\displaystyle\bar{\eta}(y)=0,y\in R/\rho(\exp(V)).$ (8)

It follows from Lemma 4.2 that the function $f|_{U}$ can be approximated by
smooth functions on $U$. That is, for any $\delta>0$, there exists smooth
function $g_{0}\in C^{\infty}(\rho(\exp(U)))$ such that
$|g_{0}(y)-f(y)|<\delta$, for $y\in\rho(\exp(\bar{V}))$.

Since $f=(1-\bar{\eta})f+\bar{\eta}f$, we define

$g=(1-\bar{\eta})f+\bar{\eta}g_{0}.$

Then the result follows immediately.

Lemma 4.5[1] Let $M$ be a second countable locally compact Hausdorff
topological space. Then there exist countable many open sets
$G_{1},G_{2},\cdots,G_{k},\cdots$ satisfying

(1) $cl(G_{j})$ is compact, $j=1,2,\cdots$;

(2) $cl(G_{j})\subseteq G_{j+1},j=1,2,\cdots$;

(3) $\cup G_{j}=\cup cl(G_{j})=M,$
where $cl(G_{j})$ denotes the closure of $G_{j},j=1,2,\cdots$.

Lemma 4.6 There exist locally finite open covers
$(U_{j})_{j\in\mathbb{Z}_{>0}},(V_{j})_{j\in\mathbb{Z}_{>0}},(W_{j})_{j\in\mathbb{Z}_{>0}}$
of $R$ such that $cl(U_{j})\subseteq V_{j},cl(V_{j})\subseteq W_{j}$, and
$cl(U_{j}),cl(V_{j}),cl(W_{j})$ are compact, for each $j>0$, where
$(W_{j},\mathbb{R}^{n_{j}},\phi_{j})$ is a local chart of $R$ induced by the
Hilbert map (2).

Proof. From Lemma 4.5 we know that there exist countable open sets
$G_{1},\cdots,G_{k},\cdots$ satisfying conditions (1), (2) and (3) in Lemma
4.5. It follows that $cl(G_{h})/{G_{h-1}}$ is compact, $G_{h+1}/{cl(G_{h-2})}$
is open and $cl(G_{h})/{G_{h-1}}\subseteq G_{h+1}/{cl(G_{h-2})}$. On the other
hand we know the local charts induced by the Hilbert map (2) of $R$ form an
open cover of $R$. Then for $y\in cl({G}_{h})/{G_{h-1}}$, there exist a local
chart $(\mathscr{V},\phi)$ of $y$ induced by the Hilbert map (2). Consider the
$H$ invariant open set $\sigma^{-1}((G_{h+1}/{cl({G}_{h-2})}\cap\mathscr{V})$
in $\text{hor}T_{x}M$, where $\rho(x)=y$. There exists an open ball
$B_{\epsilon}$ such that
$\bar{B}_{\epsilon}\subseteq\sigma^{-1}((G_{h+1}/{cl({G}_{h-2})}\cap\mathscr{V})$
centered at $0$. Let $\mathscr{W}=\sigma(B_{\epsilon})$. Hence $\mathscr{W}$
is an open subset containing $y$ such that
$cl(\mathscr{W})\subseteq(G_{h+1}/{cl({G}_{h-2})})\cap\mathscr{V}$.

It follows that

(1) $y\in W\subseteq(G_{h+1}/{cl({G}_{h-2})})\cap\mathscr{V}$;

(2) $\phi(y)=0$ and $\phi(W)=\phi(\mathscr{W})\subseteq\phi(\mathscr{V})$;
Since $cl(W)=\sigma(\bar{B}_{\epsilon})$ and $\bar{B}_{\epsilon}$ is compact,
it follows that $cl(W)$ is compact.

Let $\mathscr{W}_{1}=\sigma(B_{\epsilon_{1}})$ where
$0<\epsilon_{1}<\epsilon$. Then $\mathscr{W}_{1}$ is an open set containing
$y$ such that $cl(\mathscr{W}_{1})\subseteq W$. Denote by $V=\mathscr{W}_{1}$.
And let $\mathscr{W}_{2}=\sigma(B_{\epsilon_{2}})$ where
$0<\epsilon_{2}\textless\epsilon_{1}$. Then $\mathscr{W}_{2}$ be an open set
containing $y$ such that $cl(\mathscr{W}_{2})\subseteq V$. Denote by
$U=\mathscr{W}_{2}$. Then we have $cl(U)\subseteq V$ and $cl(V)\subseteq W$.

Since ${cl({G}_{h}})/{G_{h-1}}$ is compact, there exist finitely many points
$y_{h,1},y_{h,2},\cdots$, $y_{h,k_{h}}\in cl({G}_{h})/{G_{h-1}}$, such that
the corresponding open sets $U_{h,1},U_{h,2},\cdots,U_{h,k_{h}}$ form an open
cover of $cl(G_{h})/G_{h-1}$. We claim that the corresponding open sets

$(U_{1,1},U_{1,2},\cdots,U_{1,k_{1}};U_{2,1},U_{2,2},\cdots,U_{2,k_{2}};\cdots),$
$(V_{1,1},V_{1,2},\cdots,V_{1,k_{1}};V_{2,1},V_{2,2},\cdots,V_{2,k_{2}};\cdots),$
$(W_{1,1},W_{1,2},\cdots,W_{1,k_{1}};W_{2,1},W_{2,2},\cdots,W_{2,k_{2}};\cdots),$

satisfy the conditions in the lemma. We only need to prove the local
finiteness of $(W_{i,j})$. Given $y\in R$, assume that $y\in G_{r}$, then it
follows from the above construction that there exist finite many $W_{i,j}$
that intersect $G_{r}$. In fact,

$W_{i,j}\cap G_{r}=\emptyset,i\geq r+2,1\leq j\leq k_{i}.$

This completes the proof of the lemma.

Lemma 4.7 Let $f:R\rightarrow\mathbb{R}$ be a continuous function on $R$. Then
for any $\delta>0$, there exists $g\in C^{\infty}(R)$, such that

$|g(y)-f(y)|<\delta,$

for any $y\in R$.

Proof. From Lemma 4.6 we know that there exists locally finite open covers
$(U_{j})_{j\in\mathbb{Z}_{>0}}$, $(V_{j})_{j\in\mathbb{Z}_{>0}}$,
$(W_{j})_{j\in\mathbb{Z}_{>0}}$ of $R$ such that $cl(U_{j})\subseteq
V_{j},cl(V_{j})\subseteq W_{j}$, and $cl(U_{j}),cl(V_{j}),cl(W_{j})$ are
compact, for each $j>0$, where $(W_{j},\mathbb{R}^{n_{j}},\phi_{j})$ is a
local chart of $R$ induced by the Hilbert map (2).

Set $W_{0}=\emptyset,f_{0}=f$. Assume we have continuous function $f_{k}$ on
$R$ such that $f_{k}|_{G_{k}}$ is smooth, where

$G_{k}=\cup_{j=0}^{k}W_{j}.$

Then it follows from Lemma 4.4 that there exists continuous function $f_{k+1}$
on $R$, such that $f_{k+1}|_{G_{k+1}}$ is smooth, where

$G_{k+1}=\cup_{j=0}^{k+1}W_{j}.$

Besides, $f_{k+1}|_{M/V_{k+1}}=f_{k}$, and

$\displaystyle|f_{k+1}(y)-f_{k}(y)|<\frac{\delta}{2^{k+1}},$ (9)

for all $y\in R$.

Let

$g(y)=\lim_{k\rightarrow\infty}f_{k}(y).$

It follows from (9) that $g$ is well-defined. And

$|g(y)-f(y)|<\delta,$

for any $y\in R$.

We claim that $g\in C^{\infty}(R)$. For $y\in R$, there exists
$l\in\mathbb{Z}_{>0}$, such that $y\in W_{l}$. Now consider the function
$f_{l},f_{l+1},\cdots$ which are smooth on $W_{j}$. It follows that
$\rho^{*}f_{l},\rho^{*}f_{l+1},\cdots$ are smooth functions on the open
subsets $\rho^{-1}(W)$ of $M$, which satisfies

$\displaystyle|\rho^{*}f_{k+1}(x)-\rho^{*}f_{k}(x)|<\frac{\delta}{2^{k+1}},$
(10)

for $k\geq l$ and $x\in\rho^{-1}(W)$. It follows that
$\lim_{k\rightarrow\infty}\rho^{*}f_{k}\in C^{\infty}(\rho^{-1}(W))$. Besides,
it is also $G$ invariant which descends to $g$ on $W$. Hence $g|_{W}\in
C^{\infty}(W)$. Since $y$ is arbitrary, we have $g\in C^{\infty}(R)$. This
completes the proof.

Theorem 4.8 Let $f:R\rightarrow\mathbb{R}^{n}$ be a continuous function on
$R$. Then for any $\epsilon>0$, there exists $g\in
C^{\infty}(R;\mathbb{R}^{n})$, such that

$|g(y)-f(y)|<\epsilon,$

for any $y\in R$. Besides, $g$ is homotopic to $f$.

Proof. Let $\delta=\epsilon/{\sqrt{n}}$. It follows from Lemma 4.7 that there
exist smooth functions $g_{1},\cdots,g_{n}\in C^{\infty}(R)$ such that

$|g_{i}(y)-f_{i}(y)|<\delta,$

for any $y\in R$, where $f=(f_{1},\cdots,f_{n})$. Consider the smooth map
$g=(g_{1},\cdots,g_{n})$. We have that

$|g(y)-f(y)|<\epsilon,$

for any $y\in R$. Besides, define

$F(t.y)=(1-t)g(y)+tf(y),$

for $(t,y)\in I\times R$. It is obvious that $F$ defines a homotopy from $g$
to $f$. Hence the result follows immediately.

## References

  * (1) Zhang, Z.S., Lecture notes on differential topology. Beijing: Peking University Press, 1996 (in Chinese)
  * (2) Chen, W.H., An introduction to differential manifold. Beijing: Higher Education Press, 2002 (in Chinese)
  * (3) Śniatycki, J., Differential geometry of singular spaces and reduction of symmetry. Cambridge: Cambridge University Press, 2013.


# CAN Coach: Vehicular Control through Human Cyber-Physical Systems

Matthew Nice<EMAIL_ADDRESS>0000-0003-3609-1760 Vanderbilt
University1025 16th Ave S.NashvilleTennessee37212 ,  Safwan Elmadani
University of ArizonaTusconArizona85721 ,  Rahul Bhadani  University of
ArizonaTusconArizona85721 ,  Matt Bunting  University of
ArizonaTusconArizona85721 ,  Jonathan Sprinkle  0000-0003-4176-1212 University
of ArizonaTusconArizona85721 and  Dan Work  Vanderbilt University1025 16th Ave
S.NashvilleTennessee37212

(2021)

###### Abstract.

This work addresses whether a human-in-the-loop cyber-physical system (HCPS)
can be effective in improving the longitudinal control of an individual
vehicle in a traffic flow. We introduce the CAN Coach, which is a system that
gives feedback to the human-in-the-loop using radar data (relative speed and
position information to objects ahead) that is available on the controller
area network (CAN). Using a cohort of six human subjects driving an
instrumented vehicle, we compare the ability of the human-in-the-loop driver
to achieve a constant time-gap control policy using only human-based visual
perception to the car ahead, and by augmenting human perception with audible
feedback from CAN sensor data. The addition of CAN-based feedback reduces the
mean time-gap error by an average of 73%, and also improves the consistency of
the human by reducing the standard deviation of the time-gap error by 53%. We
remove human perception from the loop using a ghost mode in which the human-
in-the-loop is coached to track a virtual vehicle on the road, rather than a
physical one. The loss of visual perception of the vehicle ahead degrades the
performance for most drivers, but by varying amounts. We show that human
subjects can match the velocity of the lead vehicle ahead with and without
CAN-based feedback, but velocity matching does not offer regulation of vehicle
spacing. The viability of dynamic time-gap control is also demonstrated. We
conclude that (1) it is possible to coach drivers to improve performance on
driving tasks using CAN data, and (2) it is a true HCPS, since removing human
perception from the control loop reduces performance at the given control
objective.

human-in-the-loop, cyber-physical systems, controller area network, vehicles

††journalyear: 2021††copyright: acmcopyright††conference: ACM/IEEE 12th
International Conference on Cyber-Physical Systems (with CPS-IoT Week 2021);
May 19–21, 2021; Nashville, TN, USA††booktitle: ACM/IEEE 12th International
Conference on Cyber-Physical Systems (with CPS-IoT Week 2021) (ICCPS ’21), May
19–21, 2021, Nashville, TN, USA††price: 15.00††doi:
10.1145/3450267.3450541††isbn: 978-1-4503-8353-0/21/05††ccs: Applied computing
Engineering††ccs: Computer systems organization Real-time systems††ccs:
Computer systems organization Sensors and actuators††ccs: Computer systems
organization Embedded and cyber-physical systems Figure 1. Ego vehicle with
CAN Coach (White Toyota RAV4) and lead vehicle (Silver Honda Fit) in formation
in the right lane.

##  1\. Introduction

The goal of Lagrangian control is to control a few particles within a flow in
order to effect the overall flow. In the domain of transportation, this may be
carried out by control of a few vehicles in the flow of traffic in order to
achieve a control objective. In order to regulate certain kinds of traffic
phenomena, it is necessary to go beyond velocity control (which might leave
space that encourages lane-changing) in order to control the spacing between
the control (ego) vehicle and the lead vehicle.

Research has been done to evaluate the idea of Lagrangian control of traffic
with automated vehicles, demonstrating that vehicle automation can smooth
traffic waves both in simulation (Vinitsky et al., 2018) and in full-scale
experiments (Stern et al., 2017). These controllers dampen traffic waves
through prescribed time-gap following, but they depend on advanced sensing and
computation from the vehicles of tomorrow in order to close the loop with an
accuracy and dependability that is on par with human-driven vehicles, with a
requirement that 5% or more of the vehicles must be controlled vehicles.

This raises a motivating question: can a human driver drive in a manner that
dampens traffic waves, and thus accelerate the potential to adopt Lagrangian
control by having the human perform some tasks that might otherwise require
advanced sensors? If a human driver can be enabled to accurately follow
commands that would enable Lagrangian control, then it may be possible to
rapidly deploy technology in the vehicles of today that combines the advanced
perception found in human drivers with advice from a supervisory controller
that provides updated time-gap set points to carry out a control objective.

In this paper, we ask drivers to follow a vehicle (the lead vehicle) in their
vehicle (the ego vehicle) in order to determine how to positively improve
time-gap following through the use of a CAN Coach. This CAN Coach uses
information from the CAN bus such as instantaneous velocity along with
estimates of the relative position and velocity of the lead vehicle. These
data were collected using only data from the CAN in the ego vehicle.111This
work was conducted under IRB approval #200343. Drivers are asked to follow a
lead vehicle with several different objectives, as well as to follow an
imaginary vehicle (a ghost vehicle) by using the CAN Coach to follow a time-
gap (a measure of time separation between the vehicles), but without the
visual feedback to the human-in-the-loop from the lead vehicle.

Our results show that when coached by the CAN Coach, human-in-the-loop drivers
can closely control the time-gap between the lead vehicle and the ego vehicle,
i.e. space as a function of speed. The driving behavior with the CAN Coach is
significantly improved compared to when subjects are instructed to use their
own perception to achieve the control objective. When following the ghost
vehicle, performance degrades due to the lack of a human-perceivable vehicle.

The contribution of this paper is the demonstration of an HCPS that is
effective in controlling an individual vehicle which could be used for
Lagrangian control of traffic flow. The CAN-based feedback system that is used
(CAN Coach) enables drivers to achieve precise control objectives they could
not otherwise achieve by giving extrasensory feedback informing adjustments of
velocity and spacing. In other words, the CAN Coach can coach drivers to
achieve “superhuman” driving tasks, and enables a human-in-the-loop to drive
in a manner that could dampen traffic waves.222Here, we refer to superhuman in
a colloquial sense, i.e., improving on what human drivers can do without
assistance. If successful, the application could scale widely given the low
cost of the technology and the pervasive availability of the vehicle sensors;
this would demonstrate the feasibility to deploy Lagrangian control techniques
with the vehicle fleet of today.

The remainder of this article is organized as follows. In Section 2, we
discuss works informing the formation of CAN Coach. In Section 3, we provide
an overview of CAN data, the hardware platform, and the software platform. In
Section 4, we describe the feedback design and the experimental design.
Section 5 describes CAN data validation, and results of the CAN Coach system
experiments. Finally, Section 6 outlines the planned extensions from our work
presented here.

##  2\. Related Work

To effect traffic phenomena such as traffic waves via Lagrangian control, it
is necessary to be able to control the velocity and spacing of individual
vehicles. One approach to reduce the severity of traffic jams is through
vehicular-based (Forster et al., 2014) variable speed limits, which is an
adaptation of classical infrastructure based approaches (Breton et al., 2002).
The work (Delle Monache et al., 2019) considers models for autonomous vehicles
to control traffic flow composed primarily of human-piloted vehicles.

Designing human-in-the-loop vehicle control systems via modeling and
simulation (Delle Monache et al., 2019; Breton et al., 2002; Forster et al.,
2014; Sullman et al., 2015) has drawbacks due to the fidelity of the modeling.
Driver assistance systems to promote truck eco-driving at signalized
intersections have been tested on real vehicles in (Wang et al., 2019).
Similarly, field experiments demonstrating human traffic flow stabilization
using an automated vehicle are considered in (Stern et al., 2017).

The use of CAN data as a data source for analysis of driving behavior such as
lane changing, turning, and driver categorization is (Fugiglando et al., 2018;
Wakita et al., 2006) becoming increasingly recognized. In the work (Carmona et
al., 2015), a driver behavior identification tool is presented that fuses CAN
data with sensor data from an inertial measurement unit and a GPS unit to
classify normal and aggressive maneuvers in real-time. Unlike CAN Coach
introduced next, the analysis is not used to close the loop on the human
driver in real-time. However it does establish the potential of using sensor
fusion at run-time to understand driver behavior.

Using technology to improve vehicle control and properties of traffic is a
longstanding area of interest. Notable examples include the use of Cooperative
Adaptive Cruise Control (CACC) (Milanés et al., 2013; Naus et al., 2010)
systems, which provide longitudinal control of the vehicle using information
exchanged between vehicles via vehicle-to-vehicle communication to improve
stability and throughput. In this present work we take a different approach,
and instead explore how sensing can enhance human drivers in a control task,
rather than automating the task and relegating the human to a monitoring role.

Though there can be complex trade-offs in augmenting human driver sensing via
ADAS, it can improve safety and comfort (Li et al., 2011). This work does not
aim to mimic current ACC with a human-in-the-loop; instead it explores how
current vehicle technologies may be used to enhance human driving.

A closely related work to our own is (Wu et al., 2016), which implements a
real-time speed advisory system using vehicle-to-vehicle communications to
allow a driver to accurately match the speed of the vehicle ahead. As we show
in Section 5, constant time-gap based feedback can offer the ability to
regulate the vehicle speed and position, which is not possible using only
velocity-based feedback. Moreover, using CAN data, we avoid the need for a
vehicle-to-vehicle communication system, since radar-based relative speed and
space-gap data is directly available on the CAN.

##  3\. CAN Coach System Background

This section provides an overview of the CAN data, hardware platforms, and the
software platforms used in our study.

###  3.1. CAN Data

To use the CAN Coach, depending on the control objective, we need measurements
of the velocity of the ego vehicle, $v$, velocity of the leading vehicle,
$v_{\mathrm{lead}}$ and the space-gap, $s$, the distance between the front
bumper of the ego vehicle and the rear bumper of the lead vehicle (see Figure
1). By accessing the CAN data in the ego (following) vehicle, this information
can be found via radar sensors and wheel encoders.

Vehicles with ACC typically have front-facing radar, which directly measures
information we need to compute the space-gap and relative velocity
$v_{\mathrm{lead}}-v$. Note that for feedback in real-time, there are two
related challenges with radar data. First, incoming radar data tracks many
objects in a wide field of view and thus needs to be processed to extract a
high quality estimate of the location and relative velocity of the lead
vehicle. Second, any processing on the radar data must be done quickly so that
the feedback to the human-in-the-loop remains relevant. These
challenges—briefly discussed in Section 3.3.1—are otherwise outside the scope
of this paper.

###  3.2. Hardware

The base vehicle used in this work is a stock 2020 Toyota Rav4 Hybrid vehicle.
The vehicle has as standard equipment an Adaptive Cruise Control (ACC) system
and Lane Tracing Assist (LTA) system. The ACC system depends on a stock
forward looking radar unit that transmits relevant data on the CAN. To access
this data, we used a Gray Panda manufactured by Comma.ai (Santana and Hotz,
2016) as a data logging device. The Gray Panda is connected to a Raspberry Pi
4 via USB. The Pi decodes the messages, transforms them into Robotic Operating
System (ROS) messages, executes the CAN Coach to generate an auditory feedback
for the human-in-the-loop, and records all data from the car and from all ROS
nodes.

###  3.3. Software

In this section, we discuss software that was developed to implement the CAN
Coach System, execute the driving scenarios in a repeatable manner, and
perform analysis of the captured data. Figure 2 shows how the information
flows through the system.

Comma.ai Panda devices are used as interfaces to allow for reading directly
from the CAN bus. Libpanda is a C++ based library to aid in the construction
of custom software, abstracting the USB interface for easy reading and writing
to the CAN bus and for easy reading of the GPS module (if installed) (The
University of Arizona, 2020). The libpanda library was developed in order to
capture all data available from the connected CAN buses, rather than the
subset of data selected by Comma.ai for use by their open-source control
packages.

The Robotic Operating System (ROS) is an open source framework for robotics.
It provides the necessary tools and libraries to create and run peer-to-peer
processes called nodes. ROS enables modular software development by utilizing
reusable code packages (Quigley et al., 2009).

The CAN Coach System consists of these ROS nodes:

  * •

CAN Coach node: the core node in the CAN Coach System, it processes relative
vehicle velocity and distance and generates feedback for the human-in-the-
loop;

  * •

Ghost Mode: uses a simulated velocity of a lead vehicle in order to create a
virtual vehicle’s position in front of the ego vehicle;

  * •

Director: facilitates communication between the
human-in-the-loop, mode changer, and CAN Coach, and ensures consistent
execution of the experiment for each subject without the need to have an
additional researcher advancing each test mode;

  * •

Mode Changer: allows the human-in-the-loop to advance (or reverse) the mode of
the test while driving.

####  3.3.1. CAN Coach

The CAN Coach, the eponymous ROS node, is what puts the human into the loop
with the vehicle sensors, with audio feedback derived from CAN data in real
time. The CAN Coach subscribes to velocity, relative distance, and relative
velocity data obtained from the CAN bus, and when in the Ghost Mode subscribes
to that node for relative distance and velocity of the virtual lead car.

With this information, the CAN Coach uses the 16 hi-frequency radar signals
(20 Hz) that correspond to all tracked objects, along with a low-frequency
radar trace (1 Hz) that corresponds to the lead vehicle’s distance, in order
to determine the relative distance $s$ and relative velocity
$v_{\mathrm{lead}}-v$ between the ego and lead vehicles. This is achieved by
finding a match between the most recent location of the lead vehicle and a
buffer of recent raw radar data. With a match to the raw radar position, there
is a corresponding relative velocity measurement. The CAN Coach can then
communicate this information to the human-in-the-loop by producing the sound
apt for the current control objective and feedback type.

Figure 2. Diagram showing how information flows through the system.

####  3.3.2. Ghost Mode

The Ghost Mode is designed to measure how well the human-in-the-loop can track
a lead vehicle through feedback from the CAN Coach, but without the ability to
see the lead vehicle. Additional information exploring the motivation and
merit of the Ghost Mode is provided in Section 4.2.3.

In order to carry out this mode of the experiment, relative distance and
relative velocity are measured from the ego vehicle to a virtual ‘ghost’
vehicle—rather than obtained from the CAN bus. Through code generated from a
Simulink model, the ego vehicle’s velocity signal and a simulated constant
velocity of the ghost vehicle are integrated to simulate the relative distance
and relative velocity between the ego vehicle and lead (ghost) vehicle.

The CAN Coach calculates the time-gap in Ghost Mode from the ego velocity, and
the difference in the distance traveled for both vehicles during ghost mode
plus a constant of 65 m. A companion Stateflow model periodically determines
whether the virtual space gap between the ego and ghost vehicles has either
grown too large (which may result in unsafe high speed by the human-in-the-
loop to catch up) or too small (which may result in unsafe low speed to allow
the ghost vehicle to pass). If the virtual space gap $s>100$ or $s<-30$, then
the virtual distance is reset to 0 m (plus the constant 65 m) to permit safe
execution of the experiment. The constant is added because at the initialized
speed, 29 m/s, and time-gap set point, 2.25 s, the target space-gap is 65 m.
This ensures that in the first minutes of Ghost Mode, if the ghost vehicle is
reset it will return to the target state.

####  3.3.3. Director and Mode Changer Nodes

Due to ongoing health and safety practices, only one person is allowed in a
vehicle at a time—which introduces challenges in carrying out driving
experiments where a passenger might otherwise control the experiment mode. We
address this challenge through these supervisory nodes that require no
synchronous interaction from the driver: a state-based Director, and a Mode
Changer.

An abridged schematic of the Director node, simplified for readability, is
provided in Figure 3. The Director advances modes based on elapsed time,
yielding a precise, repeatable sequence with a fixed duration for each segment
of the CAN Coach experiment. The Director node publishes the set point,
feedback type, and current mode to the CAN Coach node every 0.5 s.

The Director subscribes to the Mode Changer, allowing the human-in-the-loop to
advance (or reverse) the experiment mode. For example, if the lead driver
thinks that the data would not be useful for analysis, that driver could
request the human-in-the-loop subject to reset the mode using the vehicle’s
light stalk. The experiment’s modes are the tested pairs of control objectives
and feedback types.

Figure 3. The Director node: Each experiment state lasts for a precise
duration, or can be advanced or reversed based on an input.

##  4\. Experimental Methods

This section describes the decisions made to design the feedback within the
CAN Coach, and to design the experiments.

###  4.1. Feedback Design

The CAN Coach provides simple feedback to the human-in-the-loop, because
complexity can be a burden (Bazilinskyy et al., 2018). In CAN Coach, when the
human-in-the-loop is a threshold amount away from the control target, an
audible signal is output to inform the human-in-the-loop of an action to take.
We choose audio feedback because (Scott and Gray, 2008) shows auditory driving
warnings correspond with better driver reaction times than visual warnings.
Some commercial systems use discrete audible feedback, such as Forward
Collision Warning Systems. They take into account both the physical driving
context and human interaction; the precise timing of the discrete intervention
is key to the system’s success (Jamson et al., 2008; Abe and Richardson,
2006). Literature suggests, however, that continuous feedback (Seppelt and
Lee, 2007; Fairclough et al., 1997) is preferred over discrete feedback (Risto
and Martens, 2014) for CAN Coach, because continuous feedback allows for
greater control of precision over an extended period of time.

Under time-gap control, if the current time-gap differs from the desired time-
gap (the set point) by more than $\pm$0.05 s, a sound is output to increase or
shorten the time-gap. Under velocity matching control, if the relative
velocity to the vehicle ahead differs by more than $\pm$0.4 m/s, a sound is
given as output to increase or decrease the velocity.

There are only three messages the feedback communicates: speed up, slow down,
or do nothing. A high pitched sound is used to indicate the human-in-the-loop
should accelerate; a low pitched sound is used to indicate the human-in-the-
loop should decelerate. No sound is emitted when the human-in-the-loop is near
the set point.

###  4.2. Experimental Design

####  4.2.1. Description of the Driving Environment

The
experiments conducted in this work are conducted with two vehicles on a
freeway. A lead vehicle is used during the experiment to provide consistency
in the testing environment, and to trace the route for the ego vehicle, at a
speed of 65 mph ($\approx 29.0\;\mathrm{m/s})$. The ego vehicle is
instrumented with the CAN Coach, and it follows immediately behind the lead
vehicle while executing desired control objectives. Each drive begins and ends
in the same locations for each tested driver, for a consistent roadway grade
along the route on which the drivers are tested. The tests occurred on a 55
mile freeway route (Figure 4) at off peak hours to reduce interactions between
the vehicles involved in the experiment and other road users. Each experiment
lasts about 57 minutes per driver.

Figure 4. Route used to conduct the CAN Coach experiments.

Prior to the start of each experiment, each driver is given an overview of the
experiments, the route, the technologies used in the experiments, and the
safety protocols. Drivers are informed to follow instructions given during the
experiment to the best of their ability without compromising safety.
Additional instructions are provided to the driver automatically using a
standardized script prompted from the Director (Section 3.3.3), eliminating
the need for any experimental staff to be in the vehicle with the human-in-
the-loop subject.

At the start of the experiment, each driver is given a period of time to
operate the vehicle following the lead vehicle under a Normal Driving regime.
The driver is given instructions to follow the vehicle ahead without changing
lanes.

Recall that the lead vehicle velocity is denoted by $v_{\text{lead}}$ and the
ego vehicle velocity be denoted by $v$ (Figure 1). The relative velocity is
$\Delta v:=v_{\text{lead}}-v$, where a negative relative velocity indicates
the ego vehicle is catching up, while a positive relative velocity indicates
the ego vehicle is falling behind. The space-gap, denoted by $s$, is the
distance between the front bumper of the ego vehicle and the rear bumper of
the lead vehicle. The time-gap, denoted $\tau:=s/v$, is the time required to
travel the space-gap distance $s$ when traveling at the velocity $v$ (Loulizi
et al., 2019).

In all experiments, the lead vehicle speed is set at 29.0 m/s (65 mph) using a
standard cruise control system. Variations in the lead vehicle speed occur due
to variations in the grade of the roadway as well as traffic conditions.

Six drivers consisting of university students and staff were recruited to
conduct the experiments. All drivers passed a university-required driver
safety course.

####  4.2.2. Description of Control Objectives

During each experiment, the human-in-the-loop driver of the ego vehicle is
asked to adjust the speed of the vehicle to achieve a desired control
objective. Only one objective is used at a time. The control objectives are as
follows.

  * •

Constant Time-gap: Drive the ego vehicle such that the time-gap to the vehicle
ahead is $\tau=2.25$ s.

  * •

Velocity Matching: Match the velocity of the lead vehicle (i.e. so that
$\Delta v=0$). The set speed of the lead vehicle is not announced to the
driver (i.e., they are not informed that the lead vehicle is traveling at a
fixed speed of 29.0 m/s). This meaningfully differs from Constant Time-gap
because there is no fixed target for following distance.

  * •

Dynamic Time-gap: Drive the ego vehicle such that the time-gap is $\tau=2.25$
s or $\tau=1.8$ s. The desired time-gap changes every 60 s.

####  4.2.3. Feedback Type

To meet the control objective, the driver is given varying degrees of
information:

  * •

Instructed Driving (human visual perception feedback): The human-in-the-loop
is verbally given the specific control objective (e.g., drive with a specified
time-gap). In the case of the dynamic time-gap control objective, the human-
in-the-loop is told when the time-gap changes, and the value of the new time-
gap.

  * •

Coached Driving (human visual perception and CAN feedback): The human-in-the-
loop is given all of the information provided in the Instructed Driving setup,
as well as additional feedback using CAN data.

  * •

Ghost Mode (CAN feedback but without human visual perception): Ghost mode is a
special setup in which the lead vehicle is replaced by a virtual (i.e., ghost)
vehicle. CAN Coach is used to provide feedback to help the human-in-the-loop
of the ego vehicle achieve a control objective relative to the ghost vehicle.
The point of the ghost mode is to remove the possibility of the human-in-the-
loop from visualizing the space-gap or the velocity of the lead vehicle. The
human-in-the-loop is given the same instructions and feedback as for Coached
Driving, with the major modification that the real lead vehicle is replaced by
the ghost vehicle.

##  5\. Results

This section presents the results from each human-in-the-loop subject under
different control objectives and feedback types. We first consider the
constant time-gap control objective, and compare the performance of humans-in-
the-loop using Instructed Driving to the performance when CAN Coach is used to
provide feedback. We show that CAN Coached driving improves the ability of the
human-in-the-loop to achieve the control objective. Next, we consider the
importance of the the ability of the human-in-the-loop to see the vehicle
ahead by comparing driving under Coached Driving, to driving in Ghost Mode.
Finally we consider additional control objectives including velocity matching
and dynamic time-gap matching.

###  5.1. Data Preprocessing

CAN velocity and space-gap data are validated using GPS devices following the
process described in (Wang et al., 2020). Irrelevant data from the experiments
must be removed in preprocessing. These data are collected due to experimental
conditions on an open roadway. For example, we discard data near and on the
ramp connecting the two freeways that form the driving route. For some drivers
during some tests, a vehicle not part of the experiment can have undue
influence the experiment, e.g., by cutting in between the lead vehicle and the
instrumented ego vehicle. These data are removed prior to analysis. Through
manual inspection of dash-mounted video data recorded during the tests, we
determine simple thresholds that remove data corresponding to these anomalies
outside of the intended testing environment. Specifically, we observe that we
can eliminate data corresponding to these events by discarding data below the
10th percentile velocity, and when the relative velocity is below the 5th
percentile or above the 99th percentile.

###  5.2. Constant Time-gap Control

In the first set of experiments, we compare the ability of a human-in-the-loop
to achieve the constant time-gap control objective. We first present the
results under Instructed Driving (visual feedback) compared to Coached Driving
(visual and CAN feedback) conditions. Then we remove the possibility of visual
feedback by considering a constant time-gap control to a ghost vehicle.

We consider the following performance measures. The control objective is to
achieve a desired time-gap of $\tau_{\text{desired}}=2.25$ s. The time-gap
error at each measurement time instant $t$ is computed as
$\varepsilon_{\tau}(t):=\tau_{\text{desired}}-\tau(t)$. The empirical
distribution of the time-gap error over the duration of the test can be
computed, as well as the mean error and the standard deviation of the error.
Similarly, we can compute a corresponding space-gap error as
$\varepsilon_{s}(t)=v(t)\tau_{\text{desired}}-s(t)$, and we can compute the
space-gap error statistics.

(a) Instructed Driving

(b) Coached Driving

Figure 5. Driver 1. (a) Instructed Driving and (b) Coached Driving for the
Constant Time-gap control objective. Horizontal line indicates the desired
time-gap of 2.25 s.

####  5.2.1. Instructed vs. Coached Driving: Driver 1

We
compare the results of a single driver under Instructed Driving to Coached
Driving. Under the Instructed Driving (where feedback is based only on human
visual perception), Driver 1 does not achieve the control objective. Figure
5(a) shows 6 minutes (360 seconds) of instructed driving. The driver has a
time-gap that is consistently too low. In contrast, Figure 5(b) shows the same
driver and control objective under the Coached setting in which visual and CAN
feedback are provided. It is clear from the figures that CAN Coach helps the
driver correct the time-gap to achieve the control objective. To summarize the
time-series shown in Figure 5, histograms of the time-gap error with and
without CAN feedback is provided in Figure 6. Under Instructed Driving, the
driver has a mean time-gap error of -0.3 s, and a standard deviation of 0.22
s. Under Coached driving, the mean time-gap error is 0.02 s, and the standard
deviation is 0.13 s. In summary, the addition of feedback based on the CAN
reduces the mean time-gap error by 93% and the standard deviation by 41%.

Figure 6. Driver 1 time-gap error distribution for Instructed and Coached
Driving under Constant Time-gap control.

It is also possible to explore the corresponding space-gap error and the
relative velocity to the lead vehicle under Instructed and Coached Driving.
Comparing 7(a) and 7(b), we see that though there are marginal differences in
relative velocity between Instructed and Coached Driving, there are important
differences in the space-gap error. Coached driving substantially reduces the
mean space-gap error (from -8.71 m to 0.088 m) as well as the standard
deviation of the space-gap error (from 6.15 m to 3.65 m).

(a) Space-gap Error Distribution

(b) Relative Velocity Distribution

Figure 7. Driver 1 space-gap error and relative velocity to the lead vehicle
under Constant Time-gap control.

####  5.2.2. Instructed vs. Coached Driving: All Drivers

Figure 8 shows the results comparing instructed driving to coached driving for
all six drivers. Examining Figure 8(a), it is clear that the drivers had wide
performance variation under Instructed Driving. This is true both in terms of
average error from the set point, as well as the standard deviation. While
aiming for a 2.25 s time-gap, the observed average time-gaps range from from
1.95 s (for Driver 1) to 3.61 s (for Driver 4). The standard deviation of the
time-gap range from 0.22 s (Driver 1) to 1.01 s (Driver 4).

(a) Instructed Driving

(b) Coached Driving

Figure 8. Time-gap error distributions for all drivers under (a) Instructed
Driving and (b) Coached Driving under Constant Time-gap control.

Figure 8(b) presents the time-gap errors for all six drivers under Coached
Driving. All drivers are able to obtain a low mean error on the time-gap. The
smallest time-gap error is 0.02 s (Driver 1) and the largest is 0.1 s (Driver
4). The variance of the errors are also substantially reduced. The standard
deviations range from 0.13 s (Driver 1) to 0.28 s (Driver 5). To put the
standard deviations into perspective, note that the standard deviations of all
six drivers under Coached Driving are lower than the second most consistent
driver (Driver 2; standard deviation of 0.34 s) under the Instructed mode.

Across all drivers (Table 1), the mean time-gap error reduced from 0.44 s
under Instructed Driving to 0.05 s under Coached Driving, which is a 73%
reduction in mean error. The drivers on average are more consistent, with the
error standard deviation reduced from 0.49 s to 0.2 s (a 53% reduction). Only
Driver 6 saw no reduction in mean error when operating under coached mode,
which is due to the fact that Driver 6 had a very low 0.06 s mean error in
instructed mode. All other drivers saw mean error reductions above 70% as a
result of the feedback from the CAN bus.

| Instructed | Coached | % reduction
---|---|---|---
Driver | mean | std | mean | std | mean | std
1 | 0.30 | 0.22 | 0.02 | 0.13 | 93% | 41%
2 | 0.14 | 0.34 | 0.04 | 0.18 | 71% | 47%
3 | 0.38 | 0.48 | 0.04 | 0.18 | 89% | 63%
4 | 1.36 | 1.01 | 0.10 | 0.16 | 93% | 84%
5 | 0.39 | 0.46 | 0.03 | 0.28 | 92% | 39%
6 | 0.06 | 0.42 | 0.06 | 0.24 | 0% | 43%
Avg | 0.44 | 0.49 | 0.05 | 0.2 | 73% | 53%

Table 1. Performance summary of Instructed and Coached driving under Constant
Time-gap control. Units in sec.; Percent reduction is computed using
Instructed as the baseline.

####  5.2.3. Removing Visual Feedback via Ghost Mode

In this section we consider the importance of human-based visual perception in
the control loop. The expectation is that by removing the ability of the
human-in-the-loop to also sense the vehicle ahead, the ability to follow the
vehicle will degrade. The use of Ghost Mode allows for an ablation analysis
because it removes the human perception of the lead vehicle by replacing it
with an abstract ghost vehicle.

We first highlight the results of Driver 1 and then present the summary
results for the remaining drivers. Figure 9 shows the time-gap error under
Coached and Ghost Mode driving conditions. There is a drop in performance of
the human-in-the-loop when following a ghost vehicle compared to a real
vehicle, as quantified by the mean time-gap error. The mean time-gap error of
0.02 s under Coached Driving conditions increases to 0.15 s in Ghost Mode.
Similarly, the standard deviation increases from 0.13 s in Coached Driving to
0.26 s when in Ghost Mode.

Next we consider the range of performances across the drivers. Figure 10
summarizes the mean time-gap error and standard deviation of the time-gap
error under Coached Driving and Ghost Mode driving. Figure 10(a) shows the the
mean time-gap error increases for Drivers 1, 4, 5, and 6 when following a
virtual vehicle rather than a real one. The mean time-gap error is identical
for Driver 3 under both feedback modes, while the mean error for driver 2 is
slightly reduced under Ghost Mode relative to Coached Driving. The mean time-
gap error across all drivers is 0.05 s during Coached Driving, compared to
0.13 s in Ghost Mode. The standard deviation of the time-gap error
distributions under Coached and Ghost driving is shown in Figure 10(b). The
standard deviation increases for all drivers except Driver 2 when following a
ghost vehicle compared to a real one. Combined, all drivers except Driver 2
saw either an increase in the mean error, standard deviation of the error, or
both. This indicates that most of the humans-in-the-loop contribute to
achieving the control task by seeing the vehicle ahead, even with CAN Coach
feedback.

Note that performance under Ghost Mode is still improved compared to driving
using only visual perception in the Instructed mode.

Figure 9. Driver 1. Time-gap error distribution for Coached Driving and Ghost
Mode feedback under Constant Time-gap control.

(a) Mean time-gap error for Coached Driving and Ghost Mode.

(b) Standard deviation of the time-gap for Coached Driving and Ghost Mode.

Figure 10. Comparison of mean time-gap error and standard deviation for
Coached Driving and Ghost Mode.

###  5.3. Velocity Matching Control

Next we consider a new control objective, in which the human-in-the-loop is
asked to match the velocity of the vehicle ahead. There is a moderate
reduction in standard deviation of the relative velocity for Coached Driving
over Instructed Driving for the velocity matching control objective. Coached
Driving feedback reduces mean relative velocity by 0.03 m/s, and reduces
standard deviation by 0.16 m/s compared to Instructed Driving. The improvement
is modest because the drivers under Instructed Driving were able to match the
velocity of the lead vehicle reasonably well. As a consequence, CAN feedback
resulted in only a slight reduction in error.

The most important and intuitive observation is that Velocity Matching does
not standardize the space-gap across the drivers. The space-gap distribution
under Coached Velocity Matching is shown in Figure 11(a), which can be
compared to the space-gap variation under the constant time-gap control
objective (Figure 11(b)). When under velocity matching control, mean space-
gaps ranged from 51.8 m (Driver 5) to 80.5 m (Driver 3). Under time-gap
control similar relative velocity errors are observed, but the mean space-gaps
are more tightly bounded. They range from 64.0 m (Driver 2) to 69.2 m (Driver
4). In summary, controlling the time-gap provides similar velocity control and
better space-gap regulation than velocity based control.

(a) Velocity matching

(b) Constant Time-gap

Figure 11. Space-gap distributions with Coached feedback for (a) Velocity
Matching and (b) Constant Time-gap control.

###  5.4. Dynamic Time-gap

Finally, we consider the consequences of changing the time gap between two
fixed values in a Dynamic Time-gap objective. Figure 12(a) shows the time-gap
error distribution under the Instructed Driving conditions, while Figure 12(b)
shows the time-gap error distribution under the Coached Driving condition. The
mean time-gap error decreases from 0.21 s (Instructed) to 0.08 s (Coached).
The standard deviation is decreased from 0.40 s (Instructed) to 0.30 s
(Coached). This shows that CAN-based feedback can more accurately and
precisely control the vehicle than without CAN-based feedback, even as the
control objective increases in difficulty.

When comparing Dynamic Time-gap Coached Driving to the Constant Time-gap
Coached Driving, we see that performance was slightly worse with Dynamic Time-
gap than with Constant Time-gap. From Constant Time-gap Coached Driving to
Dynamic Time-gap Coached Driving, the mean time-gap error increased from 0.05
s to 0.08 s; the mean standard deviation increased from 0.19 s to 0.30 s.

(a) Instructed Driving

(b) Coached Driving

Figure 12. Time-gap error distributions for all drivers under (a) Instructed
Driving and (b) Coached Driving under Dynamic Time-gap control.

##  6\. Conclusions and Future Work

With an eye towards scalable Lagrangian traffic control, this work considered
the possibility of vehicular control through a human-in-the loop system. By
providing the driver feedback using sensor data reported on the CAN, drivers
of varying skill levels are able to consistently achieve a desired time-gap
control objective which effectively regulates both the velocity and the space-
gap of the ego vehicle. The CAN Coach can coach drivers to achieve driving
tasks that were not possible without feedback, opening a door to modify driver
behavior for the benefit of the overall traffic flow. Though changing time-gap
affects safety and efficiency, this work focuses on the successful
implementation of a control objective, not the merits of the control objective
itself.

In our future work we are interested in understanding the limits on the
complexity of the control objective. For example it may be possible for
drivers to explicitly follow a complex desired trajectory using time-gap based
feedback to the desired trajectory, or for drivers to be coached to explicitly
achieve a traffic wave stabilizing control policy. Enhanced driver assistance
interfaces and experiments with a larger cohort of drivers will be important
to generalize the findings presented here. Personalized feedback to drivers
could improve performance and minimize intervention.

###### Acknowledgements.

This material is based upon work supported by the National Science Foundation
under Grant No. CNS-1837652. This material is based upon work supported by the
U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy
(EERE) under the Vehicle Technologies Office award number CID DE-EE0008872.
The views expressed herein do not necessarily represent the views of the U.S.
Department of Energy or the United States Government.

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##  7\. Appendix

This section contains an analysis of a stock ACC system tested under the same
protocol as the Dynamic Time-gap, and comments on cut-ins that occur during
data collection.

###  7.1. ACC Dynamic Time-gap Test

Figure 13. Time-gap error distributions for the stock ACC system under Dynamic
Time-gap control. This distribution includes cut-in events.

Though the aim of this work is not to recreate ACC with human drivers, it is
interesting to compare between ACC driving and human driving with CAN Coach.
In Figure 13 we can see how the stock ACC system compares to the human drivers
with CAN Coach in Figure 12(b).

The same dynamic time-gap protocol was run, but this time with the stock ACC
active. The ACC has a mean time gap error of -0.03 s and a standard deviation
of 0.40 s. As a side note, the ACC error calculation is based on an assumed
constant time-gap strategy of the stock ACC system and the stated time-gap
settings in the owner’s manual of the vehicle. The ACC driving shows a smaller
mean error (0.03 s ¡ 0.08 s), and a higher standard deviation (0.40 s ¿ 0.30
s) in comparison to the average human driver under Dynamic Time-gap Coached
Driving.

###  7.2. Discussion on Cut-Ins

The planned experiments were not structured to account for the semi-random
nature of vehicle cut-ins. If, for example, during one driver’s test there
were two cut-in events and during another driver’s test there were none, the
results could be skewed. Broadly, the number and nature of cut-ins across
drivers could be very different and bias the results. It would be interesting
to do an analysis testing the velocity and spacing thresholds for cut-ins, and
even make an online predictive model based on the state-space sensed by a
vehicle in real-time.

In practice, the number of cut-ins were minimal. Each driver had 2.6 cut-in
events on average over the course of the nearly hour-long drives. Some of
these cut-ins are during the Ghost Mode, where the driver is following a Ghost
Car. In this setting the cut-in is even more abstract: there is a real vehicle
cutting in front of the CAN Coach equipped vehicle which is following a
virtual vehicle. Drivers were not given any specific guidance on how to
respond to cut in events, and the sample size is small by construction (i.e.,
the selection of a remote freeway). Consequently we do not analyze the cut in
data rigorously in this article, rather we discard the data surrounding the
events.