# Representing polynomials by degenerate Bernoulli polynomials

Dae San Kim  Department of Mathematics, Sogang University, Seoul 121-742,
Republic of Korea<EMAIL_ADDRESS>and  Taekyun Kim  Department of
Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the problem of representing any polynomial in terms
of the degenerate Bernoulli polynomials and more generally of the higher-order
degenerate Bernoulli polynomials. We derive explicit formulas with the help of
umbral calculus and illustrate our results with some examples.

###### Key words and phrases:

Degenerate Bernoulli polynomial; Higher-order degenerate Bernoulli polynomial;
Umbral calculus

###### 2000 Mathematics Subject Classification:

05A19; 05A40; 11B68; 11B83

##  1\. Introduction and preliminaries

In [11], the following identity is obtained from formulas (see Remark 3.2)
expressing any polynomial in terms of Bernoulli polynomials $B_{n}(x)$ after a
slight modification:

(1.1)
$\displaystyle\sum_{k=1}^{n-1}\frac{1}{2k\left(2n-2k\right)}B_{2k}\left(x\right)B_{2n-2k}\left(x\right)+\frac{2}{2n-1}B_{1}\left(x\right)B_{2n-1}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{k=1}^{n}\frac{1}{2k}\dbinom{2n}{2k}B_{2k}B_{2n-2k}\left(x\right)+\frac{1}{n}H_{2n-1}B_{2n}\left(x\right)+\frac{2}{2n-1}B_{1}\left(x\right)B_{2n-1},$

where $n\geq 2$ and $H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$.

Letting $x=0$ in (1.1), we obtain a slight variant of the well-known Miki’s
identity:

(1.2)
$\sum_{k=1}^{n-1}\frac{1}{2k\left(2n-2k\right)}B_{2k}B_{2n-2k}=\frac{1}{n}\sum_{k=1}^{n}\frac{1}{2k}\dbinom{2n}{2k}B_{2k}B_{2n-2k}+\frac{1}{n}H_{2n-1}B_{2n}.$

Letting $x=\frac{1}{2}$ in (1.1), we get the Faber-Pandharipande-Zagier (FPZ)
identity:

(1.3)
$\sum_{k=1}^{n-1}\frac{1}{2k\left(2n-2k\right)}\overline{B}_{2k}\overline{B}_{2n-2k}=\frac{1}{n}\sum_{k=1}^{n}\frac{1}{2k}\dbinom{2n}{2k}B_{2k}\overline{B}_{2n-2k}+\frac{1}{n}H_{2n-1}\overline{B}_{2n},$

where
$\overline{B}_{n}=\left(2^{1-n}-1\right)B_{n}=B_{n}\left(\frac{1}{2}\right)$,
for all $n=0,1,2,\dots.$

It should be emphasized that our proof of Miki’s and Faber-Pandharipande-
Zagier identities follows from the simple formulas in Remark 3.2, involving
only derivatives and integrals of the given polynomials. However, the other
proofs of Miki’s and FPZ identities are quite involved. Indeed, for the Miki’s
identity, Miki [16] uses a formula for the Fermat quotient $\frac{a^{p}-a}{p}$
modulo $p^{2}$, Shiratani-Yokoyama [20] utilizes $p$-adic analysis and Gessel
[6] exploits two different expressions for Stirling numbers of the second kind
$S_{2}\left(n,k\right)$.
In 1998, Faber and Pandharipande found that certain conjectural relations
between Hodge integrals in Gromov-Witten theory require the FPZ identity. A
proof of the FPZ identity was given by Zagier in the appendix of [5]. Dunne-
Schubert [4] also proves the FPZ identity by making use of the asymptotic
expansion of some special polynomials coming from the quantum field theory
computations.

Analogous formulas to Remark 3.2 can be obtained for the representations by
Euler and Genocchi polynomials. Many interesting identities have been derived
by using these formulas and the one in Remark 3.2 (see [7-12]). The list in
the References are far from being exhaustive. The interested reader can easily
find more related papers in the literature. Also, we should mention here that
there are other ways of obtaining the same result as the one in (1.1). One of
them is to use Fourier series expansion of the function obtained by extending
by periodicity 1 of the polynomial function restricted to the interval $[0,1)$
(see [13-15]).

The aim of this paper is to derive formulas (see Theorem 3.1) expressing any
polynomial in terms of the degenerate Bernoulli polynomials (see [1]) with the
help of umbral calculus (see [3,18,19,21]) and to illustrate our results with
some examples. This can be generalized to the higher-order degenerate
Bernoulli polynomials. Indeed, we deduce formulas of representing any
polynomial in terms of the higher-order degenerate Bernoulli polynomials (see
Theorems 4.1 and 4.2) again by using umbral calculus. The contribution of this
paper is the derivation of such formulas which, we think, have many potential
applications. We have obtained formulas for representing any polynomial by the
higher-order degenerate Euler polynomials and the higher-order degenerate
Genocchi polynomials which will appear elsewhere.

The outline of this paper is as follows. In Section 1, we recall some
necessary facts that are needed throughout this paper. In Section 2, we go
over umbral calculus briefly. In Section 3, we derive formulas expressing any
polynomial in terms of the degenerate Bernoulli polynomials. In Section 4, we
derive formulas representing any polynomial in terms of the higher-order
degenerate Bernoulli polynomials. In Section 5, we illustrate our results with
some examples. Finally, we conclude our paper in Section 6.

The Bernoulli polynomials $B_{n}(x)$ are defined by

(1.4) $\frac{t}{e^{t}-1}e^{xt}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!}.$

When $x=0$, $B_{n}=B_{n}(0)$ are called the Bernoulli numbers. We observe that
$B_{n}(x)=\sum_{j=0}^{n}\binom{n}{j}B_{n-j}x^{j},\,\frac{d}{dx}B_{n}(x)=nB_{n-1}(x)$.
The first few terms of $B_{n}$ are given by:

$\displaystyle
B_{0}=1,\,B_{1}=-\frac{1}{2},\,B_{2}=\frac{1}{6},\,B_{4}=-\frac{1}{30},\,B_{6}=\frac{1}{42},\,B_{8}=-\frac{1}{30},\,B_{10}=\frac{5}{66},$
$\displaystyle\,B_{12}=-\frac{691}{2730},\dots;B_{2k+1}=0,\,\,(k\geq 1).$

More generally, for any nonnegative integer $r$, the Bernoulli polynomials
$B_{n}^{(r)}(x)$ of order $r$ are given by

(1.5)
$\bigg{(}\frac{t}{e^{t}-1}\bigg{)}^{r}e^{xt}=\sum_{n=0}^{\infty}B_{n}^{(r)}(x)\frac{t^{n}}{n!}.$

The Euler polynomials $E_{n}(x)$ are defined by

(1.6) $\frac{2}{e^{t}+1}e^{xt}=\sum_{n=0}^{\infty}E_{n}(x)\frac{t^{n}}{n!}.$

When $x=0$, $E_{n}=E_{n}(0)$ are called the Euler numbers. We observe that
$E_{n}(x)=\sum_{j=0}^{n}\binom{n}{j}E_{n-j}x^{j},\,\frac{d}{dx}E_{n}(x)=nE_{n-1}(x)$.
The first few terms of $E_{n}$ are given by:

$\displaystyle
E_{0}=1,\,E_{1}=-\frac{1}{2},\,E_{3}=\frac{1}{4},\,E_{5}=-\frac{1}{2},\,E_{7}=\frac{17}{8},\,B_{9}=-\frac{31}{2},\dots;$
$\displaystyle E_{2k}=0,\,\,(k\geq 1).$

The Genocchi polynomials $G_{n}(x)$ are defined by

(1.7) $\frac{2t}{e^{t}+1}e^{xt}=\sum_{n=0}^{\infty}G_{n}(x)\frac{t^{n}}{n!}.$

When $x=0$, $G_{n}=G_{n}(0)$ are called the Genocchi numbers. We observe that
$G_{n}(x)=\sum_{j=0}^{n}\binom{n}{j}G_{n-j}x^{j}$,
$\frac{d}{dx}G_{n}(x)=nG_{n-1}(x)$, and $\mathrm{deg}\,G_{n}(x)=n-1$, for
$n\geq 1$. The first few terms of $G_{n}$ are given by:

$\displaystyle
G_{0}=0,\,G_{1}=1,\,G_{2}=-1,\,G_{4}=1,\,G_{6}=-3,\,G_{8}=17,\,G_{10}=-155$
$\displaystyle G_{12}=2073,\dots;G_{2k+1}=0,\,\,(k\geq 1).$

For any nonzero real number $\lambda$, the degenerate exponentials are given
by

(1.8) $\displaystyle e_{\lambda}^{x}(t)=(1+\lambda
t)^{\frac{x}{\lambda}}=\sum_{n=0}^{\infty}(x)_{n,\lambda}\frac{t^{n}}{n!},$
$\displaystyle e_{\lambda}(t)=e_{\lambda}^{1}(t)=(1+\lambda
t)^{\frac{1}{\lambda}}=\sum_{n=0}^{\infty}(1)_{n,\lambda}\frac{t^{n}}{n!}.$

Carlitz [1] introduced a degenerate version of the Bernoulli polynomials
$B_{n}(x)$, called the degenerate Bernoulli polynomials and denoted by
$\beta_{n,\lambda}(x)$, which is given by

(1.9)
$\frac{t}{e_{\lambda}(t)-1}e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}\beta_{n,\lambda}(x)\frac{t^{n}}{n!}.$

For $x=0$, $\beta_{n,\lambda}=\beta_{n,\lambda}(0)$ are called the degenerate
Bernoulli numbers.

More generally, for any nonnegative integer $r$, the degenerate Bernoulli
polynomials $\beta_{n,\lambda}^{(r)}(x)$ of order $r$ are given by

(1.10)
$\bigg{(}\frac{t}{e_{\lambda}(t)-1}\bigg{)}^{r}e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}\beta_{n,\lambda}^{(r)}(x)\frac{t^{n}}{n!}.$

We remark that $\beta_{n,\lambda}(x)\rightarrow B_{n}(x)$, and
$\beta_{n,\lambda}^{(r)}(x)\rightarrow B_{n}^{(r)}(x)$, as $\lambda$ tends to
$0$.

We recall some notations and facts about forward differences. Let $f$ be any
complex-valued function of the real variable $x$. Then, for any real number
$a$, the forward difference $\Delta_{a}$ is given by

(1.11) $\Delta_{a}f(x)=f(x+a)-f(x).$

If $a=1$, then we let

(1.12) $\Delta f(x)=\Delta_{1}f(x)=f(x+1)-f(x).$

In general, the $n$th oder forward differences are given by

(1.13) $\Delta_{a}^{n}f(x)=\sum_{i=0}^{n}\binom{n}{i}(-1)^{n-i}f(x+ia).$

For $a=1$, we have

(1.14) $\Delta^{n}f(x)=\sum_{i=0}^{n}\binom{n}{i}(-1)^{n-i}f(x+i).$

Finally, we recall that the Stirling numbers of the second kind $S_{2}(n,k)$
can be given by means of

(1.15)
$\frac{1}{k!}(e^{t}-1)^{k}=\sum_{n=k}^{\infty}S_{2}(n,k)\frac{t^{n}}{n!}.$

##  2\. Review of umbral calculus

Here we will briefly go over very basic facts about umbral calculus. For more
details on this, we recommend the reader to refer to [3,18,19,21]. Let
$\mathbb{C}$ be the field of complex numbers. Then $\mathcal{F}$ denotes the
algebra of formal power series in $t$ over $\mathbb{C}$, given by

$\mathcal{F}=\bigg{\\{}f(t)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}~{}\bigg{|}~{}a_{k}\in\mathbb{C}\bigg{\\}},$

and $\mathbb{P}=\mathbb{C}[x]$ indicates the algebra of polynomials in $x$
with coefficients in $\mathbb{C}$.

Let $\mathbb{P}^{*}$ be the vector space of all linear functionals on
$\mathbb{P}$. If $\langle L|p(x)\rangle$ denotes the action of the linear
functional $L$ on the polynomial $p(x)$, then the vector space operations on
$\mathbb{P}^{*}$ are defined by

$\langle L+M|p(x)\rangle=\langle L|p(x)\rangle+\langle
M|p(x)\rangle,\quad\langle cL|p(x)\rangle=c\langle L|p(x)\rangle,$

where $c$ is a complex number.

For $f(t)\in\mathcal{F}$ with $\displaystyle
f(t)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}\displaystyle$, we define the
linear functional on $\mathbb{P}$ by

(2.1) $\langle f(t)|x^{k}\rangle=a_{k}.$

From (2.1), we note that

$\langle t^{k}|x^{n}\rangle=n!\delta_{n,k},\quad(n,k\geq 0),$

where $\delta_{n,k}$ is the Kronecker’s symbol.

Some remarkable linear functionals are as follows:

$\displaystyle\langle e^{yt}|p(x)\rangle=p(y),$ (2.2) $\displaystyle\langle
e^{yt}-1|p(x)\rangle=p(y)-p(0),$
$\displaystyle\bigg{\langle}\frac{e^{yt}-1}{t}\bigg{|}p(x)\bigg{\rangle}=\int_{0}^{y}p(u)du.$

Let

(2.3) $f_{L}(t)=\sum_{k=0}^{\infty}\langle L|x^{k}\rangle\frac{t^{k}}{k!}.$

Then, by (2.1) and (2.3), we get

$\langle f_{L}(t)|x^{n}\rangle=\langle L|x^{n}\rangle.$

That is, $f_{L}(t)=L$. Additionally, the map $L\longmapsto f_{L}(t)$ is a
vector space isomorphism from $\mathbb{P}^{*}$ onto $\mathcal{F}$.

Henceforth, $\mathcal{F}$ denotes both the algebra of formal power series in
$t$ and the vector space of all linear functionals on $\mathbb{P}$.
$\mathcal{F}$ is called the umbral algebra and the umbral calculus is the
study of umbral algebra. For each nonnegative integer $k$, the differential
operator $t^{k}$ on $\mathbb{P}$ is defined by

(2.4) $t^{k}x^{n}=\left\\{\begin{array}[]{cc}(n)_{k}x^{n-k},&\textrm{if $k\leq
n$,}\\\ 0,&\textrm{if $k>n$.}\end{array}\right.$

Extending (2.4) linearly, any power series

$f(t)=\sum_{k=0}^{\infty}\frac{a_{k}}{k!}t^{k}\in\mathcal{F}$

gives the differential operator on $\mathbb{P}$ defined by

(2.5) $f(t)x^{n}=\sum_{k=0}^{n}\binom{n}{k}a_{k}x^{n-k},\quad(n\geq 0).$

It should be observed that, for any formal power series $f(t)$ and any
polynomial $p(x)$, we have

(2.6) $\langle f(t)|p(x)\rangle=\langle 1|f(t)p(x)\rangle=f(t)p(x)|_{x=0}.$

Here we note that an element $f(t)$ of $\mathcal{F}$ is a formal power series,
a linear functional and a differential operator. Some notable differential
operators are as follows:

$\displaystyle e^{yt}p(x)=p(x+y),$ (2.7)
$\displaystyle(e^{yt}-1)p(x)=p(x+y)-p(x),$
$\displaystyle\frac{e^{yt}-1}{t}p(x)=\int_{x}^{x+y}p(u)du.$

The order $o(f(t))$ of the power series $f(t)(\neq 0)$ is the smallest integer
for which $a_{k}$ does not vanish. If $o(f(t))=0$, then $f(t)$ is called an
invertible series. If $o(f(t))=1$, then $f(t)$ is called a delta series.

For $f(t),g(t)\in\mathcal{F}$ with $o(f(t))=1$ and $o(g(t))=0$, there exists a
unique sequence $s_{n}(x)$ (deg $s_{n}(x)=n$) of polynomials such that

(2.8)
$\big{\langle}g(t)f(t)^{k}|s_{n}(x)\big{\rangle}=n!\delta_{n,k},\quad(n,k\geq
0).$

The sequence $s_{n}(x)$ is said to be the Sheffer sequence for $(g(t),f(t))$,
which is denoted by $s_{n}(x)\sim(g(t),f(t))$. We observe from (2.8) that

(2.9) $s_{n}(x)=\frac{1}{g(t)}p_{n}(x),$

where $p_{n}(x)=g(t)s_{n}(x)\sim(1,f(t))$.

In particular, if $s_{n}(x)\sim(g(t),t)$, then $p_{n}(x)=x^{n}$, and hence

(2.10) $s_{n}(x)=\frac{1}{g(t)}x^{n}.$

It is well known that $s_{n}(x)\sim(g(t),f(t))$ if and only if

(2.11)
$\frac{1}{g\big{(}\overline{f}(t)\big{)}}e^{x\overline{f}(t)}=\sum_{k=0}^{\infty}\frac{s_{k}(x)}{k!}t^{k},$

for all $x\in\mathbb{C}$, where $\overline{f}(t)$ is the compositional inverse
of $f(t)$ such that $\overline{f}(f(t))=f(\overline{f}(t))=t$.

The following equations (2.12), (2.13), and (2.14) are equivalent to the fact
that $s_{n}\left(x\right)$ is Sheffer for
$\left(g\left(t\right),f\left(t\right)\right)$, for some invertible $g(t)$:

(2.12) $\displaystyle f\left(t\right)s_{n}\left(x\right)$
$\displaystyle=ns_{n-1}\left(x\right),\quad\left(n\geq 0\right),$ (2.13)
$\displaystyle s_{n}\left(x+y\right)$
$\displaystyle=\sum_{j=0}^{n}\binom{n}{j}s_{j}\left(x\right)p_{n-j}\left(y\right),$

with $p_{n}\left(x\right)=g\left(t\right)s_{n}\left(x\right),$

(2.14)
$s_{n}\left(x\right)=\sum_{j=0}^{n}\frac{1}{j!}\big{\langle}{g\left(\overline{f}\left(t\right)\right)^{-1}\overline{f}\left(t\right)^{j}}\big{|}{x^{n}\big{\rangle}}x^{j}.$

If $s_{n}(x)\sim(g(t),f(t))$, then the following recurrence relation holds:

(2.15)
$s_{n+1}(x)=\bigg{(}x-\frac{g^{\prime}(t)}{g(t)}\bigg{)}\frac{1}{f^{\prime}(t)}s_{n}(x).$

For $s_{n}(x)\sim(g(t),f(t))$, and $r_{n}(x)\sim(h(t),l(t))$, we have

(2.16)
$s_{n}\left(x\right)=\sum_{k=0}^{n}C_{n,k}r_{k}\left(x\right),\quad\left(n\geq
0\right),$

where

(2.17)
$C_{n,k}=\frac{1}{k!}\bigg{\langle}{\frac{h\left(\overline{f}\left(t\right)\right)}{g\left(\overline{f}\left(t\right)\right)}l\left(\overline{f}\left(t\right)\right)^{k}}\Big{|}{x^{n}}\bigg{\rangle}.$

Let $p_{n}(x),\,q_{n}(x)=\sum_{k=0}^{n}q_{n,k}x^{k}$ be sequences of
polynomials. Then the umbral composition of $q_{n}(x)$ with $p_{n}(x)$ is
defined to be the sequence

(2.18) $q_{n}({\bf{p}}(x))=\sum_{k=0}^{n}q_{n,k}p_{k}(x).$

##  3\. Representation by degenerate Bernoulli polynomials

Our interest here is to derive formulas expressing any polynomial in terms of
the degenerate Bernoulli polynomials.

From (1.9) and (1.8), we first observe that

(3.1) $\displaystyle\beta_{n,\lambda}(x)\sim\big{(}g(t)$
$\displaystyle=\frac{e^{t}-1}{f(t)}=\frac{\lambda(e^{t}-1)}{e^{\lambda
t}-1},f(t)=\frac{1}{\lambda}(e^{\lambda t}-1)\big{)},$ (3.2)
$\displaystyle(x)_{n,\lambda}\sim(1,f(t)=\frac{1}{\lambda}(e^{\lambda t}-1)).$

From (1.11), (2.2), (2.7), (2.12), (3.1) and (3.2), we note that

(3.3) $\displaystyle f(t)\beta_{n,\lambda}(x)$
$\displaystyle=n\beta_{n-1}(x)=\frac{1}{\lambda}(e^{\lambda
t}-1)\beta_{n}(x)=\frac{1}{\lambda}\Delta_{\lambda}\beta_{n}(x),$ (3.4)
$\displaystyle f(t)(x)_{n,\lambda}=n(x)_{n-1,\lambda}.$

It is immediate to see from (1.9) that

(3.5) $\displaystyle\Delta\beta_{n,\lambda}(x)$
$\displaystyle=\beta_{n,\lambda}(x+1)-\beta_{n,\lambda}(x)=n(x)_{n-1,\lambda},$
(3.6) $\displaystyle\beta_{n,\lambda}(1)-\beta_{n,\lambda}=\delta_{n,1},$

where $\delta_{n,1}$ is the Kronecker’s delta.

Now, we assume that $p(x)\in\mathbb{C}[x]$ has degree $n$, and write
$p(x)=\sum_{k=0}^{n}a_{k}\beta_{k,\lambda}(x)$. Let $h(x)=p(x+1)-p(x)=\Delta
p(x)$. Then, from (3.4) and (3.5), we have

$\displaystyle h(x)$
$\displaystyle=\sum_{k=0}^{n}a_{k}(\beta_{k,\lambda}(x+1)-\beta_{k,\lambda}(x))$
(3.7) $\displaystyle=\sum_{k=0}^{n}a_{k}k(x)_{k-1,\lambda}$
$\displaystyle=f(t)\sum_{k=0}^{n}a_{k}(x)_{k,\lambda}$

For $r\geq 1$, from (3.7) and (3.4) we obtain

(3.8) $\displaystyle(f(t))^{r-1}h(x)$
$\displaystyle=(f(t))^{r}\sum_{k=0}^{n}a_{k}(x)_{k,\lambda}$
$\displaystyle=\sum_{k=r}^{n}k(k-1)\cdots(k-r+1)a_{k}(x)_{k-r,\lambda}.$

Letting $x=0$ in (3.8), we finally get

(3.9)
$a_{r}=\frac{1}{r!}(f(t))^{r-1}h(x)|_{x=0}=\frac{1}{r!}\langle(f(t))^{r-1}|h(x)\rangle,\,\,(r\geq
1),$

An alternative expression of (3.9) is given by
As $f(t)h(x)=\frac{1}{\lambda}(e^{\lambda
t}-1)h(x)=\frac{1}{\lambda}\Delta_{\lambda}h(x)$,

$\displaystyle a_{r}$
$\displaystyle=\frac{1}{r!\lambda^{r-1}}\Delta_{\lambda}^{r-1}h(x)|_{x=0}$
(3.10)
$\displaystyle=\frac{1}{r!\lambda^{r-1}}(\Delta_{\lambda}^{r-1}p(1)-\Delta_{\lambda}^{r-1}p(0))$
$\displaystyle=\frac{1}{r!\lambda^{r-1}}\Delta_{\lambda}^{r-1}\Delta p(0).$

From (1.13), we have another alternative expression of (3.9) which is given by

$\displaystyle a_{r}$
$\displaystyle=\frac{1}{r!\lambda^{r-1}}\Delta_{\lambda}^{r-1}h(x)|_{x=0}$
(3.11)
$\displaystyle=\frac{1}{r!\lambda^{r-1}}\sum_{k=0}^{r-1}\binom{r-1}{k}(-1)^{r-1-k}h(x+k\lambda)|_{x=0}$
$\displaystyle=\frac{1}{r!\lambda^{r-1}}\sum_{k=0}^{r-1}\binom{r-1}{k}(-1)^{r-1-k}(p(1+k\lambda)-p(k\lambda)).$

By using (1.15), we obtain yet another expression of (3.9) that is given by

$\displaystyle a_{r}$
$\displaystyle=\frac{1}{r!}\langle(f(t))^{r-1}|h(x)\rangle$ (3.12)
$\displaystyle=\frac{1}{r\lambda^{r-1}}\bigg{\langle}\frac{1}{(r-1)!}(e^{\lambda
t}-1)^{r-1}\bigg{|}h(x)\bigg{\rangle}$
$\displaystyle=\frac{1}{r\lambda^{r-1}}\bigg{\langle}\sum_{l=r-1}^{\infty}S_{2}(l,r-1)\frac{\lambda^{l}t^{l}}{l!}\bigg{|}h(x)\bigg{\rangle}$
$\displaystyle=\frac{1}{r}\sum_{l=r-1}^{n}S_{2}(l,r-1)\frac{\lambda^{l-r+1}}{l!}(p^{(l)}(1)-p^{(l)}(0)),$

where $p^{(l)}(x)=(\frac{d}{dx})^{l}p(x)$.

Now, it remains to determine $a_{0}$. We first note from (2.2), (2.6), (3.3)
and (3.6) that

$\displaystyle g(t)p(x)|_{x=0}$ $\displaystyle=\langle g(t)|p(x)\rangle$
$\displaystyle=\bigg{\langle}\frac{e^{t}-1}{f(t)}\bigg{|}p(x)\bigg{\rangle}$
(3.13)
$\displaystyle=\sum_{k=0}^{n}a_{k}\bigg{\langle}\frac{e^{t}-1}{f(t)}\bigg{|}\beta_{k,\lambda}(x)\bigg{\rangle}$
$\displaystyle=\sum_{k=0}^{n}a_{k}\bigg{\langle}\frac{e^{t}-1}{f(t)}\bigg{|}f(t)\frac{1}{k+1}\beta_{k+1,\lambda}(x)\bigg{\rangle}$
$\displaystyle=\sum_{k=0}^{n}\frac{a_{k}}{k+1}\bigg{\langle}e^{t}-1\bigg{|}\beta_{k+1,\lambda}(x)\bigg{\rangle}$
$\displaystyle=\sum_{k=0}^{n}\frac{a_{k}}{k+1}\delta_{k+1,1}=a_{0}.$

We want to find more explicit expression for (3.13). As $\frac{\lambda
t}{e^{\lambda
t}-1}e^{xt}=\sum_{n=0}^{\infty}\lambda^{n}B_{n}(\frac{x}{\lambda})\frac{t^{n}}{n!}$,
we note from (2.10) that $\lambda^{n}B_{n}(\frac{x}{\lambda})=\frac{\lambda
t}{e^{\lambda t}-1}x^{n}$. To proceed further, we let
$p(x)=\sum_{i=0}^{n}b_{i}x^{i}$.

From (2.2), (3.1) and (3.13), noting that $g(t)=\frac{e^{t}-1}{t}\frac{\lambda
t}{e^{\lambda t}-1}$, we have

$\displaystyle a_{0}$
$\displaystyle=\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\frac{\lambda
t}{e^{\lambda t}-1}p(x)\bigg{\rangle}$
$\displaystyle=\sum_{i=0}^{n}b_{i}\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\lambda^{i}B_{i}\big{(}\frac{x}{\lambda}\big{)}\bigg{\rangle}$
(3.14)
$\displaystyle=\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\sum_{i=0}^{n}b_{i}\lambda^{i}B_{i}\big{(}\frac{x}{\lambda}\big{)}\bigg{\rangle}$
$\displaystyle=\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}p\big{(}\lambda{\bf{B}}\big{(}\frac{x}{\lambda}\big{)}\big{)}\bigg{\rangle}$
$\displaystyle=\int_{0}^{1}p\big{(}\lambda{\bf{B}}\big{(}\frac{u}{\lambda}\big{)}\big{)}\,du,$

where $p\big{(}\lambda{\bf{B}}\big{(}\frac{x}{\lambda}\big{)}\big{)}$ denotes
the umbral composition of $p(x)$ with
$\lambda^{i}B_{i}\big{(}\frac{x}{\lambda}\big{)}$, that is, it is given by
$p\big{(}\lambda{\bf{B}}\big{(}\frac{x}{\lambda}\big{)}\big{)}=\sum_{i=0}^{n}b_{i}\lambda^{i}B_{i}\big{(}\frac{x}{\lambda}\big{)}$.
Finally, from (3.9)–(3.11) and (3.14) we get the following theorem.

######  Theorem 3.1.

Let $p(x)\in\mathbb{C}[x],\mathrm{deg}\,p(x)=n$. Then we have
$p(x)=\sum_{k=0}^{n}a_{k}\beta_{k,\lambda}(x)$,
where

$\displaystyle
a_{0}=g(t)p(x)|_{x=0}=\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\frac{\lambda
t}{e^{\lambda
t}-1}p(x)\bigg{\rangle}=\int_{0}^{1}p\big{(}\lambda{\bf{B}}\big{(}\frac{u}{\lambda}\big{)}\big{)}du,$
$\displaystyle a_{k}=\frac{1}{k!}(f(t))^{k-1}(p(x+1)-p(x))|_{x=0}$
$\displaystyle=\frac{1}{k!\lambda^{k-1}}\big{\langle}\big{(}e^{\lambda
t}-1\big{)}^{k-1}\big{|}p(x+1)-p(x)\big{\rangle}$
$\displaystyle=\frac{1}{k!\lambda^{k-1}}\Delta_{\lambda}^{k-1}\Delta p(0)$
$\displaystyle=\frac{1}{k!\lambda^{k-1}}\sum_{j=0}^{k-1}\binom{k-1}{j}(-1)^{k-1-j}(p(1+j\lambda)-p(j\lambda))$
$\displaystyle=\frac{1}{k}\sum_{l=k-1}^{n}S_{2}(l,k-1)\frac{\lambda^{l-k+1}}{l!}(p^{(l)}(1)-p^{(l)}(0)),\,\,\mathrm{for}\,\,k=1,2,\dots,n,$

where $g(t)=\frac{\lambda(e^{t}-1)}{e^{\lambda
t}-1},\,\,f(t)=\frac{1}{\lambda}(e^{\lambda t}-1)$, and
$p\big{(}\lambda{\bf{B}}\big{(}\frac{x}{\lambda}\big{)}\big{)}$ denotes the
umbral composition of $p(x)$ with
$\lambda^{i}B_{i}\big{(}\frac{x}{\lambda}\big{)}$.

######  Remark 3.2.

Let $p(x)\in\mathbb{C}[x],with\,\,\mathrm{deg}\,p(x)=n$. Write
$p(x)=\sum_{k=0}^{n}a_{k}B_{k}(x)$. As $\lambda$ tends to $0$,
$g(t)\rightarrow\frac{e^{t}-1}{t},\,\,f(t)\rightarrow
t,\,\,p\big{(}\lambda{\bf{B}}\big{(}\frac{x}{\lambda}\big{)}\big{)}\rightarrow
p(x)$. Thus we recover from Theorem 3.1 the result obtained in [11]. Namely,
we have

$\displaystyle a_{0}=\int_{0}^{1}p(t)dt,\quad
a_{k}=\frac{1}{k!}(p^{(k-1)}(1)-p^{(k-1)}(0)),\,\,\mathrm{for}\,\,k=1,2,\dots,n.$

######  Remark 3.3.

We have two other expressions for $a_{0}$.

(a) With $g(t)=\frac{\lambda(e^{t}-1)}{e^{\lambda
t}-1},f(t)=\frac{1}{\lambda}(e^{\lambda t}-1)$, from (2.15) we have

(3.15)
$\beta_{n+1,\lambda}(x)=\bigg{(}x-\frac{g^{\prime}(t)}{g(t)}\bigg{)}\frac{1}{f^{\prime}(t)}\beta_{n,\lambda}(x).$

By making use of (3.6) and (3.15), we have

$\displaystyle\bigg{\langle}e^{t}-1\bigg{|}\bigg{(}x-\frac{g^{\prime}(t)}{g(t)}\bigg{)}\frac{1}{f^{\prime}(t)}p(x)\bigg{\rangle}$
$\displaystyle=\sum_{k=0}^{n}a_{k}\langle
e^{t}-1|\beta_{k+1,\lambda}(x)\rangle$
$\displaystyle=\sum_{k=0}^{n}a_{k}(\beta_{k+1,\lambda}(1)-\beta_{k+1,\lambda})$
$\displaystyle=a_{0}.$

(b) By substituting $x=0$ into $p(x)=\sum_{k=0}^{n}a_{k}\beta_{k,\lambda}(x)$,
we get

$a_{0}=p(0)-a_{1}\beta_{1,\lambda}-\cdots-a_{n}\beta_{n,\lambda}.$

######  Remark 3.4.

Theorems 3.1 has been applied to many polynomials in oder to obtain
interesting identities for certain special polynomials and numbers. Some of
the polynomials that have been considered are as follows:

(a)

$\sum B_{i_{1}}(x)\cdots B_{i_{r}}(x)E_{j_{1}}(x)\cdots
E_{j_{s}}(x)G_{k_{1}+1}(x)\cdots G_{k_{t}+1}(x)x^{l},$

where the sum is over all nonnegative integers
$i_{1},\cdots,i_{r},j_{1},\cdots,j_{s},k_{1},\cdots,k_{t},l$ such that
$i_{1}+\cdots+i_{r}+j_{1}+\cdots+j_{s}+k_{1}+\cdots+k_{t}+l=n$, and $r,s,t,l$
are nonnegative integers with $r+s+t\geq 1$.
(b)

$\sum\frac{B_{i_{1}}(x)\cdots B_{i_{r}}(x)E_{j_{1}}(x)\cdots
E_{j_{s}}(x)G_{k_{1}+1}(x)\cdots G_{k_{t}+1}(x)x^{l}}{i_{1}!\cdots
i_{r}!j_{1}!\cdots j_{s}!(k_{1}+1)!\cdots(k_{t}+1)!l!},\\\ $

where the sum is over all nonnegative integers
$i_{1},\cdots,i_{r},j_{1},\cdots,j_{s},k_{1},\cdots,k_{t},l$ such that
$i_{1}+\cdots+i_{r}+j_{1}+\cdots+j_{s}+k_{1}+\cdots+k_{t}+l=n$, and $r,s,t,l$
are nonnegative integers with $r+s+t\geq 1$.
(c)

$\sum\frac{B_{i_{1}}(x)\cdots B_{i_{r}}(x)E_{j_{1}}(x)\cdots
E_{j_{s}}(x)G_{k_{1}+1}(x)\cdots G_{k_{t}+1}(x)x^{l}}{i_{1}\cdots
i_{r}j_{1}\cdots j_{s}(k_{1}+1)\cdots(k_{t}+1)l},\\\ $

where the sum is over all positive integers
$i_{1},\cdots,i_{r},j_{1},\cdots,j_{s},l$ and nonnegative integers
$k_{1},\cdots,k_{t}$ such that
$i_{1}+\cdots+i_{r}+j_{1}+\cdots+j_{s}+k_{1}+\cdots+k_{t}+l=n$, and $r,s,t,l$
are nonnegative integers with $r+s+t\geq 1$.

##  4\. Representation by higher-order degenerate Bernoulli polynomials

Our interest here is to derive formulas expressing any polynomial in terms of
the higher-order degenerate Bernoulli polynomials.

With $g(t)=\frac{e^{t}-1}{f(t)}=\frac{\lambda(e^{t}-1)}{e^{\lambda
t}-1},f(t)=\frac{1}{\lambda}(e^{\lambda t}-1)$, from (1.10) we note that

$\beta_{n,\lambda}^{(r)}(x)\sim(g(t)^{r},f(t)).$

Also, from (2.12), we have

(4.1) $f(t)\beta_{n,\lambda}^{(r)}(x)=n\beta_{n-1,\lambda}^{(r)}(x),$

and from the generating function of the higher-order degenerate Bernoulli
polynomials, it is immediate to see that

(4.2)
$\Delta\beta_{n,\lambda}^{(r)}(x)=\beta_{n,\lambda}^{(r)}(x+1)-\beta_{n,\lambda}^{(r)}(x)=n\beta_{n-1,\lambda}^{(r-1)}(x).$

Now, we assume that $p(x)\in\mathbb{C}[x]$ has degree $n$, and write
$p(x)=\sum_{k=0}^{n}a_{k}\beta_{k,\lambda}^{(r)}(x)$. It is important to
observe from (4.1) and (4.2) that

$\displaystyle g(t)\beta_{n,\lambda}^{(r)}(x)$
$\displaystyle=\frac{e^{t}-1}{f(t)}\beta_{n,\lambda}^{(r)}(x)$ (4.3)
$\displaystyle=\frac{e^{t}-1}{f(t)}f(t)\frac{\beta_{n+1,\lambda}^{(r)}(x)}{n+1}$
$\displaystyle=\frac{1}{n+1}(\beta_{n+1,\lambda}^{(r)}(x+1)-\beta_{n+1,\lambda}^{(r)}(x))$
$\displaystyle=\beta_{n,\lambda}^{(r-1)}(x).$

Thus, from (4.3) we have
$g(t)^{r}\beta_{n,\lambda}^{(r)}(x)=\beta_{n,\lambda}^{(0)}(x)=(x)_{n,\lambda}$,
and hence

(4.4)
$g(t)^{r}p(x)=\sum_{l=0}^{n}a_{l}\,g(t)^{r}\beta_{l,\lambda}^{(r)}(x)=\sum_{l=0}^{n}a_{l}(x)_{l,\lambda}.$

By using (4.4) and (3.4), we observe that

(4.5) $\displaystyle f(t)^{k}g(t)^{r}p(x)$
$\displaystyle=\sum_{l=0}^{n}a_{l}f(t)^{k}(x)_{l,\lambda}$
$\displaystyle=\sum_{l=k}^{n}a_{l}\,l(l-1)\cdots(l-k+1)(x)_{l-k,\lambda}.$

By evaluating (4.5) at $x=0$, we obtain

(4.6) $a_{k}=\frac{1}{k!}f(t)^{k}g(t)^{r}p(x)|_{x=0}=\frac{1}{k!}\langle
f(t)^{k}g(t)^{r}|p(x)\rangle.$

This also follows from the observation $\langle
g(t)^{r}f(t)^{k}|\beta_{l,\lambda}^{(r)}(x)\rangle=l!\,\delta_{l,k}.$

To proceed further, we note that $f(t)g(t)=f(t)\frac{e^{t}-1}{f(t)}=e^{t}-1$.

Assume first that $r>n$. Then $r>k$, for all $k=0,1,\dots,n$.

$\displaystyle a_{k}$ $\displaystyle=\frac{1}{k!}\langle
f(t)^{k}g(t)^{r}|p(x)\rangle$
$\displaystyle=\frac{1}{k!}\langle(e^{t}-1)^{k}g(t)^{r-k}|p(x)\rangle$ (4.7)
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}\langle
e^{jt}|g(t)^{r-k}p(x)\rangle$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}g(t)^{r-k}p(x)|_{x=j}$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}g(t)^{r-k}p(j).$

Next, we assume that $r\leq n$. If further $0\leq k<r$, then $a_{k}$ is the
same as the expression in (4.7). Let $r\leq k\leq n$. Then we have

$\displaystyle a_{k}$ $\displaystyle=\frac{1}{k!}\langle
f(t)^{k}g(t)^{r}|p(x)\rangle$ (4.8) $\displaystyle=\frac{1}{k!}\langle
f(t)^{k-r}(e^{t}-1)^{r}|p(x)\rangle$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{r}(-1)^{r-j}\binom{r}{j}\langle
e^{jt}|f(t)^{k-r}p(x)\rangle$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{r}(-1)^{r-j}\binom{r}{j}f(t)^{k-r}p(j).$

Summarizing the results so far, from (4.7) and (4.8) we obtain the following
theorem.

######  Theorem 4.1.

Let $p(x)\in\mathbb{C}[x],\mathrm{deg}\,p(x)=n$. Let
$g(t)=\frac{\lambda(e^{t}-1)}{e^{\lambda
t}-1},f(t)=\frac{1}{\lambda}(e^{\lambda t}-1)$. Then we have
(a) For $r>n$, we have

$p(x)=\sum_{k=0}^{n}\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}g(t)^{r-k}p(j)\beta_{k,\lambda}^{(r)}(x).$

(b) For $r\leq n$, we have

$\displaystyle p(x)$
$\displaystyle=\sum_{k=0}^{r-1}\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}g(t)^{r-k}p(j)\beta_{k,\lambda}^{(r)}(x)$
$\displaystyle+\sum_{k=r}^{n}\frac{1}{k!}\sum_{j=0}^{r}(-1)^{r-j}\binom{r}{j}f(t)^{k-r}p(j)\beta_{k,\lambda}^{(r)}(x).$

We would like to find more explicit expressions for the results in Theorem
4.1.

First, we note from (1.13) and (1.15) that

$\displaystyle f(t)^{k-r}p(j)$ $\displaystyle=f(t)^{k-r}p(x)|_{x=j}$ (4.9)
$\displaystyle=\frac{1}{\lambda^{k-r}}\Delta_{\lambda}^{k-r}p(x)|_{x=j}$
$\displaystyle=\frac{1}{\lambda^{k-r}}\sum_{l=0}^{k-r}\binom{k-r}{l}(-1)^{k-r-l}p(j+l\lambda).$
$\displaystyle=(k-r)!\sum_{l=k-r}^{n}S_{2}(l,k-r)\frac{\lambda^{l-k+r}}{l!}p^{(l)}(j).$

Let $I$ be the linear integral operator defined on $\mathbb{P}$, which is
given by

(4.10) $Iq(x)=\frac{e^{t}-1}{t}q(x)=\int_{x}^{x+1}q(u)\,du.$

From $\big{(}\frac{\lambda t}{e^{\lambda
t}-1}\big{)}^{a}e^{xt}=\sum_{n=0}^{\infty}\lambda^{n}B_{n}^{(a)}(\frac{x}{\lambda})\frac{t^{n}}{n!}$,
we get

(4.11)
$\lambda^{n}B_{n}^{(a)}\big{(}\frac{x}{\lambda}\big{)}=\bigg{(}\frac{\lambda
t}{e^{\lambda t}-1}\bigg{)}^{a}x^{n},$

and hence, from (4.11), we see that

(4.12) $\bigg{(}\frac{\lambda t}{e^{\lambda
t}-1}\bigg{)}^{r-k}p(x)=p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)},$

where $p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}$
denotes the umbral composition of $p(x)$ with
$\lambda^{i}B_{i}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}$.

Now, noting that $g(t)=\frac{e^{t}-1}{t}\frac{\lambda t}{e^{\lambda t}-1}$,
from (4.10) and (4.12), we have

$\displaystyle g(t)^{r-k}p(j)$ $\displaystyle=g(t)^{r-k}p(x)|_{x=j}$
$\displaystyle=\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{r-k}\bigg{(}\frac{\lambda
t}{e^{\lambda t}-1}\bigg{)}^{r-k}p(x)\bigg{|}_{x=j}$ (4.13)
$\displaystyle=\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{r-k}p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\bigg{|}_{x=j}$
$\displaystyle=I^{r-k}p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\bigg{|}_{x=j}$
$\displaystyle=\bigg{\langle}e^{jt}\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{r-k}\bigg{|}\bigg{(}\frac{\lambda
t}{e^{\lambda t}-1}\bigg{)}^{r-k}p(x)\bigg{\rangle}.$

From (1.15), we see that
$\big{(}\frac{e^{t}-1}{t}\big{)}^{m}=\sum_{l=0}^{\infty}S_{2}(l+m,m)\frac{m!}{(l+m)!}t^{l}$,
and hence from (4.12) and (4.13) we get another expression for
$g(t)^{r-k}p(j)$ in the following:

$\displaystyle g(t)^{r-k}p(j)$ $\displaystyle=\bigg{(}\frac{\lambda
t}{e^{\lambda
t}-1}\bigg{)}^{r-k}\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{r-k}p(x)\bigg{|}_{x=j}$
$\displaystyle=\sum_{l=0}^{n}S_{2}(l+r-k,r-k)\frac{(r-k)!}{(l+r-k)!}p^{(l)}\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\big{|}_{x=j}.$

Now, from Theorem 4.1, (4.9) and (4.12), we finally arrive at the following
theorem.

######  Theorem 4.2.

Let $p(x)\in\mathbb{C}[x],\mathrm{deg}\,p(x)=n$. Then we have
(a) For $r>n$, we have

$\displaystyle p(x)$
$\displaystyle=\sum_{k=0}^{n}\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}I^{r-k}p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\bigg{|}_{x=j}\beta_{k,\lambda}^{(r)}(x)$
$\displaystyle=\sum_{k=0}^{n}\frac{1}{k!}\sum_{j=0}^{k}\sum_{l=0}^{n}(-1)^{k-j}\binom{k}{j}\frac{(r-k)!}{(l+r-k)!}S_{2}(l+r-k,r-k)$
$\displaystyle\quad\quad\quad\times
p^{(l)}\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\big{|}_{x=j}\beta_{k,\lambda}^{(r)}(x).$

(b) For $r\leq n$, we have

$\displaystyle p(x)$
$\displaystyle=\sum_{k=0}^{r-1}\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}I^{r-k}p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\bigg{|}_{x=j}\beta_{k,\lambda}^{(r)}(x)$
$\displaystyle+\sum_{k=r}^{n}\frac{1}{k!\lambda^{k-r}}\sum_{j=0}^{r}\sum_{l=0}^{k-r}(-1)^{k-j-l}\binom{r}{j}\binom{k-r}{l}p(j+l\lambda)\beta_{k,\lambda}^{(r)}(x)$
$\displaystyle=\sum_{k=0}^{r-1}\frac{1}{k!}\sum_{j=0}^{k}\sum_{l=0}^{n}(-1)^{k-j}\binom{k}{j}\frac{(r-k)!}{(l+r-k)!}S_{2}(l+r-k,r-k)$
$\displaystyle\quad\quad\quad\times
p^{(l)}\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\big{|}_{x=j}\beta_{k,\lambda}^{(r)}(x)$
$\displaystyle+\sum_{k=r}^{n}\frac{1}{k!}\sum_{j=0}^{r}\sum_{l=k-r}^{n}(-1)^{r-j}\binom{r}{j}(k-r)!\frac{\lambda^{l-k+r}}{l!}S_{2}(l,k-r)p^{(l)}(j)\beta_{k,\lambda}^{(r)}(x)$

Here $I$ is the linear operator given by $Iq(x)=\int_{x}^{x+1}q(u)\,du,$ and
$p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}$
denotes the umbral composition of $p(x)$ with
$\lambda^{i}B_{i}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}$.

######  Remark 4.3.

Let $p(x)\in\mathbb{C}[x],with\,\,\mathrm{deg}\,p(x)=n$. Write
$p(x)=\sum_{k=0}^{n}a_{k}B_{k}^{(r)}(x)$. As $\lambda$ tends to $0$,
$g(t)\rightarrow\frac{e^{t}-1}{t},\,\,f(t)\rightarrow
t,\,\,p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\rightarrow
p(x)$. Thus, from Theorem 4.1, we recover the following result obtained in
[8]:

(a) For $r>n$, we have

$p(x)=\sum_{k=0}^{n}\bigg{(}\sum_{j=0}^{k}\frac{1}{k!}(-1)^{k-j}\binom{k}{j}I^{r-k}p(j)\bigg{)}B_{k}^{(r)}(x).$

(b) For $r\leq n$, we have

$\displaystyle p(x)=$
$\displaystyle\sum_{k=0}^{r-1}\bigg{(}\sum_{j=0}^{k}\frac{1}{k!}(-1)^{k-j}\binom{k}{j}I^{r-k}p(j)\bigg{)}B_{k}^{(r)}(x)$
$\displaystyle+\sum_{k=r}^{n}\bigg{(}\sum_{j=0}^{r}\frac{1}{k!}(-1)^{r-j}p^{(k-r)}(j)\bigg{)}B_{k}^{(r)}(x).$

Here $I$ is the integral operator $Iq(x)=\int_{x}^{x+1}q(u)\,du$.

##  5\. Examples

(a) Here we illustrate Theorem 3.1, with $p(x)=B_{n}(x)$. Let
$B_{n}(x)=\sum_{k=0}^{n}a_{k}\beta_{k,\lambda}(x)$.

Then, as $B_{n}(x)=\sum_{j=0}^{n}\binom{n}{j}B_{n-j}x^{j}$ and
$\frac{d}{dx}\frac{1}{j+1}B_{j+1}(x)=B_{j}(x)$, we have

$\displaystyle a_{0}$
$\displaystyle=\int_{0}^{1}B_{n}\big{(}\lambda{\bf{B}}\big{(}\frac{u}{\lambda}\big{)}\big{)}du$
(5.1)
$\displaystyle=\sum_{j=0}^{n}\binom{n}{j}B_{n-j}\lambda^{j}\int_{0}^{1}B_{j}\big{(}\frac{u}{\lambda}\big{)}du$
$\displaystyle=\sum_{j=0}^{n}\binom{n}{j}B_{n-j}\frac{\lambda^{j+1}}{j+1}\big{(}B_{j+1}\big{(}\frac{1}{\lambda}\big{)}-B_{j+1}\big{)},$

which is, recalling that $\lambda^{n}B_{n}(\frac{x}{\lambda})=\frac{\lambda
t}{e^{\lambda t}-1}x^{n}$ and from Theorem 3.1, also equal to

$\displaystyle a_{0}$
$\displaystyle=\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\frac{\lambda
t}{e^{\lambda t}-1}B_{n}(x)\bigg{\rangle}$
$\displaystyle=\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\frac{\lambda
t}{e^{\lambda t}-1}\frac{t}{e^{t}-1}x^{n}\bigg{\rangle}$ (5.2)
$\displaystyle=\bigg{\langle}1\bigg{|}\frac{\lambda t}{e^{\lambda
t}-1}x^{n}\bigg{\rangle}$
$\displaystyle=\bigg{\langle}1\bigg{|}\lambda^{n}B_{n}\big{(}\frac{x}{\lambda}\big{)}\bigg{\rangle}$
$\displaystyle=\lambda^{n}B_{n}.$

Incidentally, (5.1) and (5.2) together give us the following polynomial
identity:

$\sum_{j=0}^{n}\binom{n}{j}B_{n-j}\frac{x^{j+1}}{j+1}\big{(}B_{j+1}\big{(}\frac{1}{x}\big{)}-B_{j+1}\big{)}=x^{n}B_{n}.$

Further, for $k$ with $1\leq k\leq n$, we have

$\displaystyle a_{k}$
$\displaystyle=\frac{1}{k!\lambda^{k-1}}\big{\langle}\big{(}e^{\lambda
t}-1\big{)}^{k-1}\big{|}B_{n}(x+1)-B_{n}(x)\big{\rangle}$ (5.3)
$\displaystyle=\frac{1}{k!\lambda^{k-1}}\big{\langle}\big{(}e^{\lambda
t}-1\big{)}^{k-1}\big{|}nx^{n-1}\big{\rangle}$
$\displaystyle=\frac{1}{k!\lambda^{k-1}}\Delta_{\lambda}^{k-1}nx^{n-1}|_{x=0}$
$\displaystyle=\frac{n}{k!\lambda^{k-1}}\Delta_{\lambda}^{k-1}0^{n-1}.$

Noting that
$B_{n}(x)\sim\big{(}\frac{e^{t}-1}{t},t\big{)},\,\,\beta_{n,\lambda}(x)\sim\big{(}\frac{\lambda(e^{t}-1)}{e^{\lambda
t}-1},\frac{1}{\lambda}(e^{\lambda t}-1)\big{)}$, one can determine the
coefficients also by using (2.17). Thus, from (5.2) and (5.3) we have

$B_{n}(x)=\lambda^{n}B_{n}\beta_{0,\lambda}(x)+\sum_{k=1}^{n}\frac{n}{k!\lambda^{k-1}}\Delta_{\lambda}^{k-1}0^{n-1}\beta_{k,\lambda}(x).$

(b) Here we illustrate Theorem 3.1, for
$p(x)=\sum_{k=1}^{n-1}\frac{1}{k(n-k)}B_{k}(x)B_{n-k}(x),\,\,(n\geq 2)$. For
this, we first recall from [11] that

(5.4)
$p(x)=\frac{2}{n}\sum_{l=0}^{n-2}\frac{1}{n-l}\binom{n}{l}B_{n-l}B_{l}(x)+\frac{2}{n}H_{n-1}B_{n}(x),$

where $H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$ is the harmonic number and a
slight modification of (5.4) gives the identity in (1.1). Let
$p(x)=\sum_{k=0}^{n}a_{k}\beta_{k,\lambda}(x)$. Then we have

$\displaystyle a_{0}$
$\displaystyle=\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\frac{\lambda
t}{e^{\lambda t}-1}p(x)\bigg{\rangle}$ (5.5)
$\displaystyle=\frac{2}{n}\sum_{l=0}^{n-2}\frac{1}{n-l}\binom{n}{l}B_{n-l}\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\frac{\lambda
t}{e^{\lambda t}-1}B_{l}(x)\bigg{\rangle}$
$\displaystyle\quad\quad+\frac{2}{n}H_{n-1}\bigg{\langle}\frac{e^{t}-1}{t}\bigg{|}\frac{\lambda
t}{e^{\lambda t}-1}B_{n}(x)\bigg{\rangle}$
$\displaystyle=\frac{2}{n}\sum_{l=0}^{n-2}\frac{1}{n-l}\binom{n}{l}B_{n-l}\lambda^{l}B_{l}+\frac{2}{n}H_{n-1}\lambda^{n}B_{n}.$

For $k$, with $1\leq k\leq n$, we obtain

$\displaystyle k!\lambda^{k-1}a_{k}$
$\displaystyle=\big{\langle}\big{(}e^{\lambda
t}-1\big{)}^{k-1}\big{|}p(x+1)-p(x)\big{\rangle}$ (5.6)
$\displaystyle=\frac{2}{n}\sum_{l=1}^{n-2}\frac{1}{n-l}\binom{n}{l}B_{n-l}\big{\langle}\big{(}e^{\lambda
t}-1\big{)}^{k-1}\big{|}B_{l}(x+1)-B_{l}(x)\big{\rangle}$
$\displaystyle\quad\quad+\frac{2}{n}H_{n-1}\big{\langle}\big{(}e^{\lambda
t}-1\big{)}^{k-1}\big{|}B_{n}(x+1)-B_{n}(x)\big{\rangle}$
$\displaystyle=\frac{2}{n}\sum_{l=1}^{n-2}\frac{l}{n-l}\binom{n}{l}B_{n-l}\Delta_{\lambda}^{k-1}0^{l-1}+2H_{n-1}\Delta_{\lambda}^{k-1}0^{n-1}.$

Thus, from (5.5) and (5.6) and for $n\geq 2$, we have

$\displaystyle\sum_{k=1}^{n-1}\frac{1}{k(n-k)}B_{k}(x)B_{n-k}(x)$
$\displaystyle=\frac{2}{n}\bigg{(}\sum_{l=0}^{n-2}\frac{1}{n-l}\binom{n}{l}B_{n-l}\lambda^{l}B_{l}+H_{n-1}\lambda^{n}B_{n}\bigg{)}\beta_{0,\lambda}(x)$
$\displaystyle\quad\quad+\frac{2}{n}\sum_{k=1}^{n}\frac{1}{k!\lambda^{k-1}}\bigg{(}\sum_{l=1}^{n-2}\frac{l}{n-l}\binom{n}{l}B_{n-l}\Delta_{\lambda}^{k-1}0^{l-1}+nH_{n-1}\Delta_{\lambda}^{k-1}0^{n-1}\bigg{)}\beta_{k,\lambda}(x).$

(c) In [11], it is shown that the following identity holds for $n\geq 2$:

$\sum_{k=1}^{n-1}\frac{1}{k(n-k)}E_{k}(x)E_{n-k}(x)=\frac{4E_{n+1}}{n^{2}(n+1)}-\frac{4}{n}\sum_{l=1}^{n}\frac{\binom{n}{l}(H_{n-1}-H_{n-l})}{n-l+1}E_{n-l+1}B_{l}(x),$

where $H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$ is the harmonic number.

Write
$\sum_{k=1}^{n-1}\frac{1}{k(n-k)}E_{k}(x)E_{n-k}(x)=\sum_{k=0}^{n}a_{k}\beta_{k,\lambda}(x)$.

By proceeding similarly to (b), we obtain the following identity:

$\displaystyle\sum_{k=1}^{n-1}\frac{1}{k(n-k)}E_{k}(x)E_{n-k}(x)$
$\displaystyle=\frac{4}{n}\bigg{(}\frac{E_{n+1}}{n(n+1)}-\sum_{l=1}^{n}\frac{\binom{n}{l}(H_{n-1}-H_{n-l})}{n-l+1}E_{n-l+1}\lambda^{l}B_{l}\bigg{)}\beta_{0,\lambda}(x)$
$\displaystyle\quad\quad-\frac{4}{n}\sum_{k=1}^{n}\frac{1}{k!\lambda^{k-1}}\bigg{(}\sum_{l=1}^{n}\frac{l\,\binom{n}{l}(H_{n-1}-H_{n-l})}{n-l+1}E_{n-l+1}\Delta_{\lambda}^{k-1}0^{l-1}\bigg{)}\beta_{k,\lambda}(x).$

(d) In [15], it is proved that the following identity is valid for $n\geq 3$:

(5.7)
$\sum_{k=1}^{n-1}\frac{1}{k(n-k)}G_{k}(x)G_{n-k}(x)=-\frac{4}{n}\sum_{k=0}^{n-2}\binom{n}{k}\frac{G_{n-k}}{n-k}B_{k}(x).$

Again, by proceeding analogously to (b), we get the following identity:

$\displaystyle\sum_{k=1}^{n-1}\frac{1}{k(n-k)}G_{k}(x)G_{n-k}(x)$
$\displaystyle=-\frac{4}{n}\bigg{(}\sum_{l=0}^{n-2}\binom{n}{l}\frac{G_{n-l}}{n-l}\lambda^{l}B_{l}\bigg{)}\beta_{0,\lambda}(x)$
$\displaystyle\quad-\frac{4}{n}\sum_{k=1}^{n-2}\frac{1}{k!\lambda^{k-1}}\bigg{(}\sum_{l=1}^{n-2}\binom{n}{l}\frac{l\,G_{n-l}}{n-l}\Delta_{\lambda}^{k-1}0^{l-1}\bigg{)}\beta_{k,\lambda}(x).$

(e) Nielsen [17,2] expressed products of two Bernoulli polynomials in terms of
Bernoulli polynomials. Namely, for positive integers $m$ and $n$, with
$m+n\geq 2$,

$B_{m}(x)B_{n}(x)=\sum_{r}\left\\{\binom{m}{2r}n+\binom{n}{2r}m\right\\}\frac{B_{2r}B_{m+n-2r}(x)}{m+n-2r}+(-1)^{m+1}\frac{B_{m+n}}{\binom{m+n}{m}}.$

Again, in a similar way to (b), we can show that

$\displaystyle B_{m}(x)B_{n}(x)$
$\displaystyle=\bigg{(}\sum_{r}\left\\{\binom{m}{2r}n+\binom{n}{2r}m\right\\}\frac{B_{2r}B_{m+n-2r}}{m+n-2r}\lambda^{m+n-2r}+(-1)^{m+1}\frac{B_{m+n}}{\binom{m+n}{m}}\bigg{)}\beta_{0,\lambda}(x)$
$\displaystyle\quad\quad+\sum_{k=1}^{m+n}\frac{1}{k!\lambda^{k-1}}\sum_{r}\left\\{\binom{m}{2r}n+\binom{n}{2r}m\right\\}B_{2r}\Delta_{\lambda}^{k-1}0^{m+n-2r-1}\beta_{k,\lambda}(x).$

(f) Nielsen [17,2] also represented products of two Euler polynomials in terms
of Bernoulli polynomials as follows:

$\displaystyle E_{m}(x)E_{n}(x)$
$\displaystyle=-2\sum_{r=1}^{m}\binom{m}{r}E_{r}\frac{B_{m+n-r+1}(x)}{m+n-r+1}$
$\displaystyle\quad-2\sum_{s=1}^{n}\binom{n}{s}E_{s}\frac{B_{m+n-s+1}(x)}{m+n-s+1}$
$\displaystyle\quad+2(-1)^{n+1}\frac{m!\,n!}{(m+n+1)!}E_{m+n+1}.$

In the same way as (b), we can show that

$\displaystyle E_{m}(x)E_{n}(x)$
$\displaystyle=-2\bigg{(}\sum_{r=1}^{m}\binom{m}{r}E_{r}\frac{B_{m+n-r+1}\lambda^{m+n-r+1}}{m+n-r+1}+\sum_{s=1}^{n}\binom{n}{s}E_{s}\frac{B_{m+n-s+1}\lambda^{m+n-s+1}}{m+n-s+1}$
$\displaystyle\quad\quad+(-1)^{n}\frac{m!\,n!}{(m+n+1)!}E_{m+n+1}\bigg{)}\beta_{0,\lambda}(x)$
$\displaystyle\quad\quad-\sum_{k=1}^{m+n}\frac{2}{k!\lambda^{k-1}}\bigg{(}\sum_{r=1}^{m}\binom{m}{r}E_{r}\Delta_{\lambda}^{k-1}0^{m+n-r}+\sum_{s=1}^{n}\binom{n}{s}E_{s}\Delta_{\lambda}^{k-1}0^{m+n-s}\bigg{)}\beta_{k,\lambda}(x).$

(g) As the last example, we would like to express the sum on the left hand
side of (5.7) in terms of the degenerate Bernoulli polynomials of order $r$.
For this, we first observe that, for any positive integer $a$,

(5.8)
$\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{a}B_{n}^{(r)}\big{(}\frac{x}{\lambda}\big{)}=I^{a}B_{n}^{(r)}\big{(}\frac{x}{\lambda}\big{)}=\frac{\lambda^{a}}{\langle
n+1\rangle_{a}}\sum_{m=0}^{a}(-1)^{a-m}\binom{a}{m}B_{n+a}^{(r)}\big{(}\frac{x+m}{\lambda}\big{)},$

where $I$ is the integral operator $Iq(x)=\int_{x}^{x+1}q(u)\,du,$ and
$\langle n+1\rangle_{a}=(n+1)\cdots(n+a)$. Let
$p(x)=\sum_{k=1}^{n-1}\frac{1}{k(n-k)}G_{k}(x)G_{n-k}(x),\,\,(n\geq 3)$. Then
$p(x)=-\frac{4}{n}\sum_{l=0}^{n-2}\binom{n}{l}\frac{G_{n-l}}{n-l}B_{l}(x).$ In
order to apply Theorem 4.2, we need to compute (see (4.13))

$I^{r-k}p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\bigg{|}_{x=j}=\bigg{\langle}e^{jt}\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{r-k}\bigg{|}\bigg{(}\frac{\lambda
t}{e^{\lambda t}-1}\bigg{)}^{r-k}p(x)\bigg{\rangle},$

which is equal to

$\displaystyle
I^{r-k}p\big{(}\lambda{\bf{B}}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\big{)}\bigg{|}_{x=j}$
(5.9)
$\displaystyle=-\frac{4}{n}\sum_{l=0}^{n-2}\binom{n}{l}\frac{G_{n-l}}{n-l}\bigg{\langle}e^{jt}\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{r-k}\bigg{|}\bigg{(}\frac{\lambda
t}{e^{\lambda t}-1}\bigg{)}^{r-k}B_{l}(x)\bigg{\rangle}$
$\displaystyle=-\frac{4}{n}\sum_{l=0}^{n-2}\binom{n}{l}\frac{G_{n-l}}{n-l}\lambda^{l}\bigg{\langle}e^{jt}\bigg{|}\bigg{(}\frac{e^{t}-1}{t}\bigg{)}^{r-k-1}B_{l}^{(r-k)}\big{(}\frac{x}{\lambda}\big{)}\bigg{\rangle}$
$\displaystyle=-\frac{4}{n}\sum_{l=0}^{n-2}\binom{n}{l}\frac{G_{n-l}}{n-l}\frac{\lambda^{l+r-k-1}}{\langle
l+1\rangle_{r-k-1}}\sum_{m=0}^{r-k-1}(-1)^{r-k-1-m}\binom{r-k-1}{m}B_{l+r-k-1}^{(r-k)}\big{(}\frac{j+m}{\lambda}\big{)},$

where we used (5.8). Assume that $n\geq r$ and $n\geq 3$. Then, from Theorem
4.2 and (5.9), we obtain

$\displaystyle\sum_{k=1}^{n-1}\frac{1}{k(n-k)}$ $\displaystyle
G_{k}(x)G_{n-k}(x)=-\frac{4}{n}\sum_{k=0}^{r-1}\bigg{(}\frac{1}{k!}\sum_{j=0}^{k}\sum_{l=0}^{n-2}\sum_{m=0}^{r-k-1}(-1)^{r-j-m-1}\binom{k}{j}\binom{n}{l}$
$\displaystyle\times\frac{G_{n-l}}{n-l}\frac{\lambda^{l+r-k-1}}{\langle
l+1\rangle_{r-k-1}}\binom{r-k-1}{m}B_{l+r-k-1}^{(r-k)}\big{(}\frac{j+m}{\lambda}\big{)}\bigg{)}\beta_{k,\lambda}^{(r)}(x)$
$\displaystyle+\sum_{k=r}^{n-2}\bigg{(}\frac{1}{k!\lambda^{k-r}}\sum_{j=0}^{r}\sum_{l=0}^{k-r}\sum_{m=1}^{n-1}\frac{(-1)^{k-j-l}}{{m(n-m)}}\binom{r}{j}\binom{k-r}{l}G_{m}(j+l\lambda)$
$\displaystyle\times G_{n-m}(j+l\lambda)\bigg{)}\beta_{k,\lambda}^{(r)}(x).$

##  6\. Conclusion

In this paper, we were interested in representing any polynomial in terms of
the degenerate Bernoulli polynomials and of the higher-order degenerate
Bernoulli polynomials. We were able to derive formulas for such
representations with the help of umbral calculus. We showed that, by letting
$\lambda$ tends to zero, they agree with the previously found formulas for
representations by the Bernoulli polynomials and by the higher-order Bernoulli
polynomials. Further, we illustrated the formulas with some examples.

Even though the method adopted in this paper is elementary, they are very
useful and powerful. Indeed, as we mentioned in the Section 1, both a variant
of Miki’s identity and Faber-Pandharipande-Zagier (FPZ) identity follow from
the one identity (see (1.1)) that can be derived from a formula (see Remark
3.2) involving only derivatives and integrals of the given polynomial, while
all the other proofs are quite involved. We recall here that the FPZ identity
was a conjectural relations between Hodge integrals in Gromov-Witten theory.
In addition, this paper demonstrates that umbral calculus is a very convenient
tool.

It is one of our future research projects to continue to find formulas
representing polynomials in terms of some specific special polynomials and to
apply those in discovering some interesting identities.

## References

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# Workspace Aware Online Grasp Planning

Iretiayo Akinola, Jacob Varley, Boyuan Chen, and Peter K. Allen  This work was
supported in part by National Science Foundation grants IIS-1208153 and
IIS-1527747. Authors are with the Department of Computer Science, Columbia
University, New York, NY 10027, USA. (iakinola, jvarley, bchen,
<EMAIL_ADDRESS>
###### Abstract

This work provides a framework for a workspace aware online grasp planner.
This framework greatly improves the performance of standard online grasp
planning algorithms by incorporating a notion of reachability into the online
grasp planning process. Offline, a database of hundreds of thousands of unique
end-effector poses were queried for feasability. At runtime, our grasp planner
uses this database to bias the hand towards reachable end-effector
configurations. The bias keeps the grasp planner in accessible regions of the
planning scene so that the resulting grasps are tailored to the situation at
hand. This results in a higher percentage of reachable grasps, a higher
percentage of successful grasp executions, and a reduced planning time. We
also present experimental results using simulated and real environments.

##  I Introduction

Grasp planning and motion planning are two fundamental problems in the
research of intelligent robotic manipulation systems. These problems are not
new, and many reasonable solutions have been developed to approach them
individually. Most of the research has treated these problems as distinct
research areas focusing either on: 1) how to get a set of high quality
candidate grasps [1][2] or 2) how to generate viable trajectories to bring the
gripper to the desired hand configuration [3][4][5]. While it is often
convenient to treat grasp and motion planning as distinct problems, we
demonstrate that solving them jointly can improve performance and reduce
computation time. Grasp planners unaware of the robot workspace and without a
notion of reachability often spend significant time and resources evaluating
grasps that may simply be unreachable. For example, a reachability unaware
planner is equally likely to return impossible grasps that assume the robot
will approach the object from the object’s side furthest from the robot. Our
planner avoids unreachable areas of the workspace dramatically reducing the
size of the search space. This lowers online planning time, and improves the
quality of the planned grasps as more time is spent refining quality grasps in
reachable portions of the workspace, rather than planning grasps that may be
stable but are unreachable.

Figure 1: Top Row: Visualization of cross sections of the precomputed
reachable space for a Fetch Robot and Staubli Arm with Barrett Hand. Green
arrows represent reachable poses, red arrows unreachable. This space is
computed offline, once for a given robot. Bottom Row: Signed Distance Field
generated from the above reachability spaces. Figure 2: Workspace Aware Online
Grasping Framework - Offline: 1) the robot’s reachability space is queried for
IK solutions that are free of collisions with the robot itself and static
objects such as walls and tables. 2) An SDF is created from the reachability
space. Online: 3) Grasp planning is quickly accomplished utilizing the
reachability space SDF. 4) A motion plan for one of the planned grasps. 5)
Trajectory executed by the robot for a stable grasp.

In order to generate grasps which are stable and reachable, we propose a new
energy function for use with simulated annealing grasp planners [1][2]. This
energy function is a weighted combination of a grasp stability term, and a
grasp reachability term. Our grasp reachability term is inspired by the idea
of a precomputed reachability space which has demonstrated utility in other
robotics tasks [6][7]. In our work, we generate a densely sampled reachability
space for our robot as shown in Fig. 1(Top Row). This offline computation
checks whether an Inverse Kinematic (IK) solution exists for a given pose.
This database in and of itself has utility for filtering lists of precomputed-
grasps and quickly removing unreachable grasps, leaving only reasonable grasps
as possible results. We further postprocess our reachability space and compute
a Signed Distance Field (SDF) representation in Fig. 1(Bottom Row). The SDF is
used in our online grasp planning energy function to guide the hand towards
reachable regions of the robot’s workspace. Obstacles present in the grasping
scene can be embedded into the space prior to SDF generation which results in
a field that repels the grasp search away from poses that collide with the
obstacles.

Experiments in both simulation and physical environments have been performed
to assess the performance of our method on multiple objects and under various
poses. We demonstrate that our reachability aware grasp planner results in a
larger number of reachable grasps, a larger number of successful grasp
executions, and a reduced planning time compared to other online grasp
planning methods.

Our method for workspace aware online grasp planning is overiewed in Fig. 2.
This framework has a number of advantages. First, it combines two closely
coupled processes into one; rather than going back and forth between grasp
planning and trajectory planning until a feasible grasp is found, we solve for
a fully reachable grasp at once. It increases the probability of finding
feasible grasps quickly since the optimization process is guided by our energy
function within reachable spaces. Also, our method is well suited for online
planning which in general is more adaptable and robust than the traditional
offline grasp planning approach and works in the case of novel objects for
which precomputed grasps do not exist.

In summary, we take a closer look at the problem of grasp quality and
reachability analysis of grasps in an integrated and structured manner. The
contributions of this work include: 1) A novel representation of reachability
space that uses signed-distance field (SDF) to provide a gradient field during
grasp planning; 2) An online grasp planning system with an integrated notion
of reachability; 3) A formulation that ranks a list of grasps (e.g. from a
database) based on their combined reachability and grasp quality; 4) A method
of embedding of obstacles into the reachability space during grasp planning;
5) experimental results using our framework compared to other grasp planning
solutions evaluated in multiple grasping scenarios.

##  II Related Work

Different works have highlighted the problem of generating non-reachable
planned grasps from grasp planning systems. For example, a 2016 review of the
Amazon Picking Challenge (APC) [8] explicitly highlighted the fact that many
teams opted against the planning paradigm (choosing an online feedback
paradigm instead). Since grasp planning metrics typically ignore the notion of
reachability, a large percentage of the planned grasps end up being
unreachable and not useful for the robot as illustrated in Fig. 3 . Hence, an
extra computational step is required to check each of the planned grasps for
reachability; a top performing APC team [9] uses this post-check solution
which shows that this limited partial solution is built into even competitive
systems. This post-check solution has a number of drawbacks: first, the post-
processing check for reachability might not contain any reachable grasp which
implies that the entire grasp planning process has to be repeated. Another
closely related problem with post-reachability checking is that more time is
incurred when replanning grasps that fail the reachability test stage. Also,
many systems, such as those in APC, are highly tuned to specific use-cases
which do not generalize well especially when the grasping environment changes
in small ways. Our system presents a way to automatically incorporate the
environment into grasp planning by augmenting the reachability space.

Many grasp planning platforms such as GraspIt![10], OpenRave[5], and
OpenGrasp[11] only consider properties specific to the end-effector during
grasp planning. Given the hand description and the object, these platforms
compute grasp quality for different hand-object configurations using metrics
such as force/form closure and return the best quality grasps found. This
approach ignores reachability of the resulting grasp as often the arm is not
even modeled in the simulator, and leaves the burden of verifying the grasp
outputs to the subsequent stages of robotics applications. Our work instead
incorporates the notion of reachability during planning thereby increasing the
quality of the output of the grasp planning processing, from the viewpoint of
the subsequent stages. The reachability space aptly captures the information
about the robot geometry and kinematics; our approach incorporates this
information into the grasp planning process in a way that produces feasible
reachable results as input for trajectory planning. Since this involves a
constant time lookup, incorporating it gives extra advantage at slight
computational cost.

The reachability space presents an intuitive way to introduce soft constraints
to the grasp planning process. One can restrict the allowable grasp
configuration space by adjusting the reachability space to reflect such
preferences. Though we have not shown it in this work, this idea has been
demonstrated in a previous work [12]. In their work, they use an ”Environment
Clearance Score” to guide grasps away from obstacles in the robot workspace.
The same work introduced the idea of ”Grasp-scoring function” which uses an
approximate kinematics of the robot to score a grasp. This is different from
our work in the sense that we build the actual reachable space using the exact
kinematics of the robot. Also, while they use the scoring function on pre-
computed grasps, we use the reachability value directly for online grasp
planning.

The concept of offline reachability analysis exists in the literature [7] [6].
However, many of these works use offline precomputed grasps. While our
approach shares the concept of offline reachability analysis with these works,
we differ in that we incorporate the reachability information directly into
online grasp planning. For example, a method [13] uses capability maps and an
additional inverse kinematics check to filter out sampled grasp poses. Since
their grasp sampling is uniform, the chances of sampling reachable poses do
not improve over time. Conversely, our method uses a gradient field to guide
the grasp planning process to reachable portions. Note that our work treats
grasping as a self-contained task that comes up with useful outputs for
subsequent stages of robotic operations while these works give more
consideration to the manipulation tasks.

Haustein etal.[14] has recently looked at combining grasp planning and motion
planning. Similar to [15], their method uses a bidirectional search approach
to randomly sample grasp candidates in one direction and sample arm
trajectories in the opposite direction until a connection is made to form a
hand grasp/arm trajectory pair. Our method is complementary to theirs in that
we guide the very high-dimensional grasp goal sampling process to regions that
are reachable leading to faster connection of sample grasps to the sampled arm
trajectories. Our work provides a simple and effective upgrade to common
standalone grasp planning approaches as well as to the inner grasp search
modules of integrated grasp and motion planning systems. While our work is
similar to [12] [7] in being workspace aware, ours differs in a number of
ways: (1) we focus on solving online grasp planning, (2) we compute the actual
reachability map, not an approximation as in [12], (3) we use a novel
representation of the reachability space, and (4) we use a different
optimization technique, simulated annealing.

##  III Offline Reachability Space Generation

A grasp is reachable if a motion plan can be found to move the arm from its
current configuration to a goal configuration that places the hand at desired
grasp location. This is not always possible for a number of reasons typically
because no inverse kinematics(IK) solution exists to place the hand at the
desired grasp pose, self collision with other parts of the robot, or collision
with obstacles in the planning scene.  Algorithm 1 Reachability Space
Generation 1:procedure GenerateReachabilitySpace 2: poses =
uniformSampleWorkspace()  3: pose2Reachable = $\\{\\}$ 4: for pose in poses do
5: reachable = hasCollisionFreeIK(pose)  6: pose2Reachable[pose] = reachable
7: SDF = computeSDF(pose2Reachable)  8: return SDF

###  III-A Reachability Space Representation

We observe that this definition of the original reachability space is binary,
either 1 (reachable), or 0 (not reachable), and has no gradient to indicating
the direction from non-reachable regions to reachable portions of the
workspace. In the context of Simulated Annealing for grasp planning, it is
beneficial to provide an energy function that has a landscape that can guide
solutions to more desired regions of the annealing space. This observation
informed a novel signed-distance-field (SDF) representation of the
reachability space that is amenable to optimization formulations. Our SDF maps
a pose to a value representing the distance to the manifold or boundary
between the unreachable poses and reachable poses and can be interpreted as
follows $d_{sdf}=SDF(pose)$:

  * •

$d_{sdf}=0$: pose lies exactly on boundary between reachable and unreachable
grasp poses.

  * •

$d_{sdf}>0$: pose lies within the reachable region of the workspace, and is
distance $d_{sdf}$ away from the boundary.

  * •

$d_{sdf}<0$: pose lies outside the reachable region of the workspace, and is
distance $d_{sdf}$ away from the boundary.

This field can then serve as criteria for classifying a grasp as either lying
in a reachable space or not. And nearby hand configurations can be ordered
based on $d_{sdf}$, their distance from this boundary.

Grid resolution and metric choices are two important parameters for the
reachability space and SDF generation. The 6D hand pose of a given grasp has
the 3D translational (linear) component and the 3D rotation (angular)
component, two physically different quantities with different units. We had to
systematically determine good resolution levels and a good ratio between unit
linear measurements and unit angular measurements, to obtain a unified 6D
metric space suitable for our purpose. Effectively, our metric can be defined
as:

$d_{sdf}=\sqrt{||\Delta xyz/res_{lin}||^{2}+r||\Delta rpy/res_{rot}||^{2}}$

where $\Delta xyz$ and $res_{lin}$ (in centimeters) are the translational
distance and resolution respectively, $\Delta xyz$ and $res_{lin}$ (in
radians) are the rotational distance and resolution, and $r$ is the relative
metric ratio i.e. a distance of $res_{lin}$cm $\equiv r*res_{rot}$rad.

We varied translational resolution $res_{lin}=5,10,20$cm, angular resolution
$res_{rot}=\pi/8,\pi/4,\pi/2$rad, and translational to rotational metric ratio
$r=0.1,1,2,5,10$; and checked the performance of the SDF generated for each
combination. First, we randomly generated 10,000 grasp poses and checked for
the existence of an IK solution i.e. the grasp’s reachability. Then, for each
of the resolution levels, we evaluated the quality of the resulting SDF by
checking what percentage of the random grasp poses were correctly classified
as reachable (or not) by the SDF manifold. Reachable grasps evaluate to
positive sdf values, unreachable grasps evaluate to negative sdf values. The
time to generate SDF from binary reachability space was recorded in each case.
From this parameter sweep, the densest resolution level $res_{lin}=5$cm,
$res_{rot}=\pi/8$rad gave 0.992 & 0.973 as accuracy and precision respectively
but took 177secs to generate the SDF. We observed that 10cm translational,
$\pi/4$ rotational resolution levels and metric ratio $r=1$ still gave a high
accuracy (0.979) and precision (0.92) with the sdf generation time under
3secs. These choices defined the metric space for the 6D hand pose
reachability space: a 10cm translation and a $\pi/4$rad rotation are
equidistance in the reachability space. Note that we take into consideration
the cyclical nature of the rotational degrees of freedom during SDF generation
as angles wrap around at $2\pi$.

###  III-B Reachability Space Generation

Reachable-space generation is a one-time process for a given robot. Algorithm
1 details the process for generating the Reachable Space. We search a
discretization of the 6D pose space of the robot to determine which
configurations will be reachable by the hand. For the Fetch Robot, we sampled
the workspace using the values in meters centered at the robot’s base shown in
Table I. In total 675,840 unique poses where queried for reachability. 102,692
were reachable and 573,148 were unreachable. This was computed using Moveit!’s
IK solver which checks for a valid collision free IK solution for a given
pose. The 6D pose space containing the binary IK query results was converted
into an SDF using the fast marching method from scikit-
fmm111https://github.com/scikit-fmm/scikit-fmm. Once this computation is done,
the SDF can be queried with any pose inside the space to determine both its
binary reachability and distance to the boundary separating reachable and non
reachable poses. Reachability space generation codes and data will be made
available on the project website:
http://crlab.cs.columbia.edu/reachabilityawaregrasping/

TABLE I: Discretization of robot workspace for SDF generation for the Fetch robot. Values are in meters (x, y, z) and radians (roll, pitch, yaw). The origin of the space is centered at the robot base. The roll was sampled less because the Fetch robot can twist its wrist to achieve any roll for a give hand pose. | X | Y | Z | Roll | Pitch | Yaw
---|---|---|---|---|---|---
min | 0.0 | -1.1 | 0.0 | -PI | -PI | -PI
max | 1.2 | 1.1 | 2.0 | PI | PI | PI
step | 0.1 | 0.1 | 0.1 | PI | PI/4.0 | PI/4.0

##  IV Online Reachability-Aware Grasp Planning

Grasp planning is the process of finding ”good grasps” for an object that can
be executed using an articulated robotic end effector. A grasp is typically
classified as ”good” based on how well it can withstand external disturbances
without dropping the object. The Ferrari-Canny method [16] is a common way to
evaluate robot grasps; it defines how to quantitatively measure the space of
disturbance wrenches that can be resisted by a given grasp. Other techniques
are typically built around this metric as shown in this review [17]. Grasp
planning frameworks such as GraspIt! use these grasp metrics of force and form
closure as objective functions for optimization. Since this formulation has no
notion of reachability, the grasp results, while stable, might require the end
effector to be placed in a pose that is impossible to reach given the robot’s
current location.

###  IV-A Simulated Annealing for Grasp Planning

Grasp planning can be thought of as finding low energy configurations within
the hand object space. This search is done in a multidimensional space where
grasps are sampled and evaluated. The 6 dimensional (x, y, z, roll, pitch,
yaw) space we generated with our SDF is a subspace of the 6 + N dimensional
grasp search space. The additional N dimensions represent the EigenGrasps or
eigen vectors of the end effector Degree of Freedom (DOF) space as described
in [1]. For the Fetch (1 DOF), N=1, and for the Barrett Hand (4 DOF), N=2.
These 6 + N dimensions describe both the pose and DOF values of the end
effector. The goal of grasp planning is to find low energy points in this 6 +
N dimensional space. These points represent the pose and hand configuration of
”good grasps”.

Simulated annealing (SA) [18] – a probabilistic technique for approximating
global optimums for functions lends itself nicely for our formulation. SA
presents a number of advantages in grasp planning[19]. For example, it does
not need an analytical gradient which can be computationally infeasible; it
handles the high nonlinearity of grasp quality functions; and it is highly
adaptable to constraints. Our approach implicitly guides the annealing process
ensuring that sampling is done in regions of reachable space which increases
the probability of getting useful grasp solutions. While online grasping is
typically avoided due to time cost of exploring a larger search space, this
work makes online grasping more feasible since the majority of the space which
is unreachable need not be searched. With our new energy formulation, the
annealing process will quickly drive the hand towards reachable grasp
locations. Section V below shows that this new energy function increases the
success probability of online grasping significantly.

Algorithm 2 Contact And Potential Energy (SA-CP)

1:procedure ContactAndPotentialEnergy

2: $e_{p}$ = potentialEnergy()

3: stable = $e_{p}$ $<$ 0

4: if stable then

5: return $e_{p}$

6: else

7: $e_{contact}$ = contactEnergy()

8: return $e_{contact}$

Algorithm 3 Reachability-Aware Energy (SA-OURS)

1:procedure ReachabilityAwareEnergy

2: $e_{p}$ = potentialEnergy()

3: $e_{reach}$ = reachabilityEnergy()

4: stable = $e_{p}$ $>$ 0

5: reachable = $e_{reach}$ $<$ 0

6: if reachable && stable then

7: return $e_{p}$ \+ $\alpha_{1}$ $\cdot$ $e_{reach}$

8: else if reachable && !stable then

9: $e_{contact}$ = contactEnergy()

10: return $e_{contact}$ \+ $\alpha_{2}$ $\cdot$ $e_{reach}$

11: else //!reachable

12: $e_{contact}$ = contactEnergy()

13: return $e_{contact}$ \+ $\alpha_{3}$ $\cdot$ $e_{reach}$

###  IV-B Novel Grasp Energy Formulation

In our work, we augment the Contact and Potential grasp energy function from
[1]. The Contact and Potential energy function is described in Algorithm 2 and
consists of a potential energy term ($e_{p}$) measuring the grasps potential
to resist external forces and torques, and a contact energy term
($e_{contact}$) that defines the proximity of the hand to the object being
grasped. These two terms define the grasp fitness function used by the
optimization algorithm (simulated annealing) to search for good grasp
configurations. Our Reachability Aware method augments the Contact and
Potential energy function with a reachability energy term ($e_{reach}$) that
encodes the kinematic constraints of the robot. We use the reachability SDF
($e_{reach}:=d_{sdf}$) described previously as a regularizing term to obtain a
new cost functions that drives the grasp planning optimization towards
reachable hand configurations. As shown in [20] discretized SDFs still model
and behave like a continuous function so it can be used during the simulated
annealing optimization process to drive the search towards reachable hand
configurations. Our reachability-aware energy function $E$ is given as:

$E=G+\alpha R$ (1)

where $G$ is a metric of grasp quality (e.g. force closure), $R$ is a measure
of reachability of a given grasp pose and $\alpha$ defines the tradeoff
between the conventional grasp metric and the reachability value. To optimize
the overall grasp energy, we use the simulated annealing search according to
[21].

In terms of implementation, we build on the GraspIt! simulator and combine the
existing energy metrics with the new reachable energy term. The energy
components have forms that we exploit. The grasp energy term (as described in
[1] and Algorithm 2) is negative when the grasp has force closure and positive
otherwise. The reachability energy has similar meaning which gives four
quadrants of possibilities. We chose the weights of each quadrants so the
range of values for the reachability term and grasp term are of the same order
of magnitude. This provides a gradient from bad to good quality (reachable and
force-closure) grasps. Algorithm. 3 shows specifically how the energy terms
are combined. $\alpha$ values were set to $\alpha_{1}=-0.1$, $\alpha_{2}=-10$,
$\alpha_{3}=-10$ to obtain a function where more negative energy values
correspond to more reachable stable grasps, and positive high energy values
correspond to unreachable and unstable grasps.

Since our reachability space is represented as a grid of discretized SDFs, we
use multi-linear interpolation to get ($e_{reach}:=d_{sdf}$) for a given grasp
pose from the reachability SDF grid. Each query pose during the optimization
is situated in the robot’s reachable space to get the cell location and the
reachability value is given as the weighted sum of the values for the $2^{N}$
($N=6$) corners of the hypervoxel where the query point falls in the pre-
computed SO(6) reachable space. The $2^{N}$ corner values are looked up in the
pre-computed reachability space.

$R[p_{query}]=\sum_{i=1}^{2^{N}}w_{i}R[p_{i}]$ (2)

where $p_{query}=[x_{q},y_{q},z_{q},r_{q},p_{q},y_{q}]$ is the query pose,
$p_{i}=[x_{i},y_{i},z_{i},r_{i},p_{i},y_{i}]$ for $i=1...2^{N}$ correspond to
the neighbouring corners in the reachability space grid and $w_{i}$ is the
proportion of the grid occupied by creating a volume from connecting the query
point and the corner $p_{i}$. See [22] for more details.

###  IV-C Embedding Obstacles in the Reachability Space

Obstacles whose poses are not expected to change in relation to the robot can
be placed in the scene while generating the initial reachability space. This
approach is applicable to modeling tables, walls and fixtures around fixed
base robotic arms, or static room environments for mobile manipulators. This
is useful for the Staubli Arm and attached Barrett Hand as shown in Fig. 1.
This robot is fixed to the table, next to the walls, making it reasonable to
incorporate these obstacles during the initial reachability space creation.

In order to incorporate non-static obstacles into our precomputed reachability
space, we mask out regions in the robot’s binary reachability space that
overlap with an obstacle prior to generating the SDF representation; this
corresponds to modifying the reachable space in the first block of Fig. 2.
Given the obstacles’ geometries and poses, we first collate grid locations of
the reachability space that overlap with obstacle geometries. All grasp poses
at these locations (including those that were otherwise reachable) are marked
as unreachable. Then, we regenerate an SDF representation of the resulting
binary reachable space. The SDF generation is fast, taking 2-3 seconds on
average, and this step is only required when the workspace changes. While this
fast obstacle-masking-procedure considers only the hand (not full arm
configuration) and might miss out on some other grasp poses that become
infeasible due to obstacles, we’ve found that it is a reasonable approximation
that works well in practice. The masked out obstacle locations imposes a
negative field that pushes the SDF manifold further away from the obstacles
and our experiments show that this effect results in an improved grasp
planner.

##  V Experiments

We describe different experiments in both simulation and physical environments
to assess the performance of our method on multiple objects and under various
poses. All experiments (simulated and real) were run on two robot platforms:
the Fetch mobile manipulator and the Barrett hand mounted on a Staubli Arm
(Staubli-Barrett). The Staubli-Barrett set-up (see Fig 1) has additional walls
to limit the workspace of the robot as a safety precaution to ensure the robot
does not elbow objects outside of it’s workspace during grasping. This safety
box heavily constrains the range of grasping directions as many grasp poses
will be invalidated because of collision with the walls. Static objects such
as the walls were included in the scene when constructing the reachability
space offline.

For each experiment, we compared a typical grasp planning method with our
reachability-aware method:
Sim. Ann. Contact and Potential (SA-C&P): Simulated Annealing in GraspIt!
using the Contact and Potential energy function from Algorithm 2.
Sim. Ann. Reachability-Aware (SA-Ours): Simulated Annealing in GraspIt! using
the newly proposed energy function from Algorithm 3.

###  V-A Evaluation Metrics

The metrics used for evaluating our methods include:
Percent Reachable Grasps: The number of reachable grasps divided by the total
number of grasps generated from a run of a given grasp planner.
Number of Required Plan Attempts: The grasp planner returns a list of grasps
ordered according to quality. We record how many of these grasps we have to go
through until a valid path can be planned.
Lift Success: Here we execute the first valid grasp from the set returned by
the planner. A grasp is successful if the gripper placed at the grasp pose can
successful lift the object off the ground. We use the Klamp’t simulator [23]
to test this in simulation. For the real world experiments, we executed the
grasps on the real robot and checked if the object was picked up.

Figure 3: Reachability results for uniformly sampled grasps (Red arrows
reachable, blue unreachable). Even for objects well within the bounds of the
robot workspace, there are many invalid approach directions which are not
easily modeled by simple heuristics.

###  V-B Grasp Planning with Dynamic Obstacles Experiment

Here we set-up a typical grasp planning scenario: in a virtual scene, three
objects were placed on a table. One is the target object to be grasped while
the other two are obstacles that the robot must not collide with during
grasping (Fig 5). We used 4 different meshes as the target object. We placed
the target object in 9 difficult-to-reach poses that were either at the extent
of the robot’s workspace, or extremely close to the robot. Both of these
situations drastically limit the number of valid approach directions that the
robot can use to grasp the object. For each pose, GraspIt!’s Simulated
Annealing planner was run for varying number of planning steps using both the
SA-C&P and SA-Ours energies. We used two versions of our reachability-aware
planner (SA-Ours), one with unmodified SDF (Section IV-B) and the other with
obstacles-embedded SDF (Section IV-C). For each planner, we checked the
reachability of all planned grasps and evaluate the fraction of planned grasps
that are reachable after running the planner for a given number of steps.

(a) Fetch Robot Results

(b) Staubli-Barrett Results

Figure 4: Mean fraction of reachable grasps using different energy functions
for varying planning duration. The red plot shows that using pure grasp
planning with no notion of reachability gives grasp results that have a low
chance of being reachable (48.2% for Fetch and 18.8% for the Staubli-Barrett).
The blue plot shows that our reachability aware grasp planner results in a
higher fraction ($>25\%$ increase) of reachable grasps for both robots. The
green plot shows the additional gain obtained when we embed obstacles into the
SDF reachability space. This results in reachable grasps that have feasible IK
results and do not collide with the obstacles hence an increase in the overall
fraction of reachable grasps.

The results (Fig 4) show the significance of our reachability-aware grasp
planning approach compared with a typical grasp planner on the same setup.
Given the same amount of time, the planners each return twenty grasp solutions
and we check what fraction of them are achievable by the robot and report the
average over all 36 simulated object poses (4 objects, 9 poses). As expected,
a typical grasp planner that has no notion of reachability (shown in red) will
yield a large percentage of grasps that are not feasible for the robot. Even
with additional planning time, there is no improvement in the odds of
returning reachable grasps. On the other hand, our method (shown in blue)
yields significantly higher fraction of reachable grasps ($>25\%$
improvement). In addition, the fraction of reachable grasps increases quickly
with planning time as our method optimizes for both stability and reachability
of grasps. While the blue plot uses the original offline SDF, the green plot
shows that by embedding the new obstacles into the precomputed SDF, we are
able to increase the percentage of reachable grasps (Fetch: 95.7%, Staubli-
Barrett: 86.1%) compared to (84.3%, 50.2%) respectively when the precomputed
SDF only avoids self collision and static objects such as the walls.

After demonstrating that our method increases the feasibility of grasp
planning results, we also observe that this method also leads to a significant
speedup of the grasp planning process since the reachability energy term
guides the annealing process quickly to reachable regions hence reducing the
search space. Once GraspIt! returns a list of grasps, we iterate through the
list in order of grasp quality and attempt to plan a valid path for each
grasps. When a valid grasp is found, we use the Klampt! simulator to get the
lift success. For a simple parallel jaw gripper like the Fetch gripper, there
was no significant difference in the percentage of lift success (SA-C&P: 0.95
and SA-Ours: 0.93) but our method requires less number of motion planning
attempts (SA-C&P: 2.75 and SA-Ours: 1.10) to find a valid grasp in the list
and overall returns a higher percentage of useful grasps (reachable and lift
success). Our grasp energy formulation ensures that a top ranked grasp has a
high chance of being reachable. The difference in grasp quality was more
pronounced with the 4 DOF Barrett hand. Our method not only requires less
number of motion planning attempts (SA-C&P: 6.2 and SA-Ours: 1.27), it also
achieves 20% higher lift success rate (SA-C&P: 0.75 and SA-Ours: 0.95). This
is because our planner spends most of it’s time refining grasps in the much-
reduced reachable regions.

###  V-C Real Robot Crowded Scene Experiment

Figure 5: Crowded scene for real world experiments. Top (Fetch Robot): Our
Reachability-Aware planner (SA-Ours) was able to successfully grasp the
shaving cream bottle 3/3 times, while annealing without the reachability space
(SA-C&P) failed 2/3 times. Bottom (Staubli-Barrett Robot): SA-Ours
successfully grasped the pringles bottle 5/5 times, while SA-C&P failed 3/5
times.

To verify our planner and demonstrate that it works outside of simulation, the
simulated annealing based grasp planner was run with a variable number of
annealing steps and the success rate was reported in Table II and III. This
experiment compares two grasp energy formulations: SA-Ours and SA-C&P.

TABLE II: Grasp success results on real robot (Fetch) with a crowded scene.
Each method was given 3 attempts to plan and execute a grasp on the shaving
cream bottle. Crowded Scene Grasp Planning (Fetch)
---
| Search
---
Energy
# steps |  | Success
---
Rate
| Mean Grasp
---
Planning Time (s)
SA-Ours | 10K | 100.0% | 8.706
SA-C&P | 10K | 33.3% | 8.360
SA-C&P | 40K | 66.6% | 32.31
TABLE III: Grasp success results on real Staubli-Barrett robot with a crowded
scene (Fig 5). Each method was given 5 attempts to plan and execute a grasp on
the pringles bottle. Crowded Scene Grasp Planning (Barrett)
---
| Search
---
Energy
# steps |  | Success
---
Rate
| Mean Grasp
---
Planning Time (s)
SA-Ours | 10K | 100.0% | 11.36
SA-C&P | 10K | 40% | 9.53
SA-C&P | 40K | 60% | 31.98

Multiple objects were placed in the planning scene as shown in Fig 5. The
additional objects reduce the range of possible grasps since the obstacles
make many grasp poses infeasible. Note that both methods are aware of the
obstacles during grasp planning and avoid grasps that put the hand in
collision with obstacles. Table II shows that our method, SA-Ours, is able to
achieve grasp success – with the Fetch robot picking the object three times
out of three within the limited planning time. Conversely, SA-C&P fails to
produce a reachable grasp 2/3 times when run for 10,000 steps, and fails once
even when it is allowed to run for 40,000 steps. Despite being allowed to run
for a long duration, the naive planner spends much of its time exploring the
back half of the bottle which is completely unreachable to the Fetch robot.

We repeated the same experiment with Staubli-Barrett robot (bottom of Fig 5).
Though the objects were placed roughly at the centre of the workspace, the
presence of walls and distractor objects significantly limit the range of
feasible grasps for the target object. Each method had 5 trials and the
results followed the same pattern: SA-Ours achieves grasp success all five
times within the limited planning time (10,000 steps) while SA-C&P fails 2/5
times even when run for 40,000 steps.

##  VI Conclusion and Future Work

This work provides a framework for a workspace aware grasp planner. This
planner has greatly improved performance over standard online grasp planning
algorithms because of its ability to incorporate a notion of reachability into
the online grasp planning process. This improvement is accomplished by
leveraging a large precomputed database of over 675,840 unique end-effector
poses which have been tested for reachability. At runtime, our grasp planner
uses this database to bias the hand towards reachable end effector
configurations. This bias allows the grasp planner to generate grasps where a
significantly higher percentage of grasps are reachable, a higher percentage
result in successful grasp executions, and the planning time required is
reduced. It has been experimentally tested in both simulated and physical
environments with different arm/gripper combinations.

Several future research directions include: utilizing our computed
reachability workspace to help mobile robots navigate to optimal locations for
manipulating objects and improvements for speeding up the process of
incorporating dynamic objects into our notion of reachability to further
assist the grasp planner.

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