analysis for completeness. We consider $1/N$-expansion instead of
$1/c$-expansion as in [46] since the expressions become simpler. However, we
can easily relate them by consistently make the replacement
$1/N\to(1-\lambda^{2})/c$ at the order we are interested in as before.

Since the conformal dimension $\Delta_{+}$ is given by

$\displaystyle\Delta_{+}=\frac{(N-1)(2N+1+k)}{N(N+k)}=1+\lambda-\frac{1}{N}+\mathcal{O}(N^{-2})\,,$
(A.9)

we have

$\displaystyle(1-z)^{-\Delta_{+}+1+\lambda}=1-\frac{1}{N}\sum_{l=1}^{\infty}\,\frac{1}{l}\,z^{l}+\mathcal{O}(N^{-2})\,.$
(A.10)

Moreover, we can expand Thus the first line in the right hand side of (4.17)
becomes

$\displaystyle|z|^{2\Delta_{+}}G_{++(1)}(z)=1+\left[\frac{1}{N}\sum_{n=1}^{\infty}\left(-\frac{1}{n}+\frac{\Gamma(1-\lambda)\,\Gamma(n)}{\Gamma(n-\lambda)}\right)z^{n}+\text{c.c}\right]+\mathcal{O}(N^{-2})\,.$
(A.11)

The vacuum conformal block is

$\displaystyle\mathcal{V}_{0}=1+\frac{1}{2}\left(\frac{1+\lambda}{1-\lambda}\right)z^{2}\,{}_{2}F_{1}(2,2;4;z)+\mathcal{O}(N^{-2})\,,$
(A.12)

while the general block with an intermediate state with conformal weight
$h_{p}$ is

$\displaystyle\mathcal{V}_{p}=z^{h_{p}}\,{}_{2}F_{1}(h_{p},h_{p};2h_{p};z)+\mathcal{O}(N^{-1})\,.$
(A.13)

Let us write down

$\displaystyle|z|^{2\Delta_{+}}G_{++(1)}(z)=1+\left[\frac{1}{N}\sum_{s=2}^{\infty}\left(D^{(s)}_{+}\right)^{2}\,z^{s}\,{}_{2}F_{1}(s,s;2s;z)+\text{c.c}\right]+\mathcal{O}(N^{-2})\,.$
(A.14)

Comparing the $z$-expansions in the both sides, we have the equations

$\displaystyle-\frac{1}{n}+\frac{\Gamma(1-\lambda)\,\Gamma(n)}{\Gamma(n-\lambda)}=\Gamma(n)^{2}\,\sum_{s=2}^{n}\frac{\Gamma(2s)\,\left(D^{(s)}_{+}\right)^{2}}{\Gamma(s)^{2}\,\Gamma(s+n)\,(n-s)!}\,.$
(A.15)

Here we have used

$\displaystyle
z^{s}\,{}_{2}F_{1}(s,s;2s;z)=\frac{\Gamma(2s)}{\Gamma(s)^{2}}\,\sum_{n=0}^{\infty}\frac{\Gamma(s+n)^{2}}{\Gamma(2s+n)}\,\frac{z^{n+s}}{n!}\,.$
(A.16)

The non-trivial solutions to (A.15) were found to be [46]

$\displaystyle
D^{(s)}=\eta^{s}_{+}\,\sqrt{\frac{\Gamma(s)^{2}\,\Gamma(1-\lambda)\,\Gamma(s+\lambda)}{\Gamma(2s-1)\,\Gamma(1+\lambda)\,\Gamma(s-\lambda)}}={\sqrt{\frac{N}{cB^{(s)}}}}\,C_{+}^{(s)}\,,$
(A.17)

where $\eta^{s}_{+}=\pm 1$.

Next, we consider the second term in (4.17). The coefficient $\mathcal{N}_{+}$
can be expanded as

$\displaystyle\mathcal{N}_{+}=1+\frac{1}{N}\frac{2\lambda}{1+\lambda}+\mathcal{O}(N^{-2})\,.$
(A.18)

Applying a well-known formula for hypergeometric function:

$\displaystyle\begin{aligned}
&{}_{2}F_{1}(\alpha,\beta;\gamma;z)=\frac{\Gamma(\gamma)\,\Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha)\,\Gamma(\gamma-\beta)}\,{}_{2}F_{1}(\alpha,\beta;\alpha+\beta-\gamma+1;1-z)\\\
&\quad+\frac{\Gamma(\gamma)\,\Gamma(\alpha+\beta-\gamma)}{\Gamma(\alpha)\,\Gamma(\beta)}\,(1-z)^{\gamma-\alpha-\beta}\,{}_{2}F_{1}(\gamma-\alpha,\gamma-\beta;\gamma-\alpha-\beta+1;1-z)\,,\end{aligned}$
(A.19)

we can rewrite

$\displaystyle
z^{-\Delta_{+}+1+\lambda}\,{}_{2}F_{1}\left(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;z\right)$
$\displaystyle\quad=z^{-\Delta_{+}+1+\lambda}\,\frac{\Gamma(2+\lambda)\,\Gamma(1+\lambda)}{\Gamma\left(1+\lambda-\frac{\lambda}{N}\right)\,\Gamma\left(2+\lambda+\frac{\lambda}{N}\right)}\,{}_{2}F_{1}\left(1+\frac{\lambda}{N},-\frac{\lambda}{N};-\lambda;1-z\right)$
(A.20)
$\displaystyle\qquad+z^{-\Delta_{+}+1+\lambda}\,(1-z)^{1+\lambda}\,\frac{\Gamma(2+\lambda)\,\Gamma(-1-\lambda)}{\Gamma\left(1+\frac{\lambda}{N}\right)\,\Gamma\left(-\frac{\lambda}{N}\right)}\,{}_{2}F_{1}\left(1+\lambda-\frac{\lambda}{N},2+\lambda+\frac{\lambda}{N};2+\lambda;1-z\right)\,.$

Since we can expand

$\displaystyle\frac{\Gamma(2+\lambda)\,\Gamma(1+\lambda)}{\Gamma\left(1+\lambda-\frac{\lambda}{N}\right)\,\Gamma\left(2+\lambda+\frac{\lambda}{N}\right)}=1-\frac{1}{N}\frac{\lambda}{1+\lambda}+\mathcal{O}(N^{-2})\,,$
(A.21)

the first term in the right hand side of (A.20) can be expressed as

$\displaystyle
1+\sum_{s=2}^{\infty}\frac{(C^{s}_{\pm})^{2}}{B^{s}}\,(1-z)^{s}\,{}_{2}F_{1}(s,s;2s;1-z)-\frac{1}{N}\frac{\lambda}{\lambda+1}+\mathcal{O}(c^{-2})\,,$

where we have used (A.14) but replacing $z$ by $1-z$. Further applying

$\displaystyle{}_{2}F_{1}(\alpha,\beta;\gamma;z)=(1-z)^{\gamma-\alpha-\beta}\,{}_{2}F_{1}(\gamma-\alpha,\gamma-\beta;\gamma;z)\,,$
(A.22)

the second term can be written as

$\displaystyle\begin{aligned}
&(1-z)^{1+\lambda}\,\frac{\Gamma(2+\lambda)\,\Gamma(-1-\lambda)}{\Gamma\left(1+\frac{\lambda}{N}\right)\,\Gamma\left(-\frac{\lambda}{N}\right)}\,{}_{2}F_{1}\left(1+\lambda-\frac{\lambda}{N},2+\lambda+\frac{\lambda}{N};2+\lambda;z\right)\\\
&\quad=z^{-1-\lambda}\,(1-z)^{1+\lambda}\,\frac{\Gamma(2+\lambda)\,\Gamma(-1-\lambda)}{\Gamma\left(1+\frac{\lambda}{N}\right)\,\Gamma\left(-\frac{\lambda}{N}\right)}\,{}_{2}F_{1}\left(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;z\right)\\\
&\quad=-\frac{\lambda\pi\csc(\lambda\pi)}{N}\,z^{-1-\lambda}\,(1-z)^{1+\lambda}+\mathcal{O}(N^{-2})\,.\end{aligned}$
(A.23)

###  A.3 Shadow blocks for conserved currents

In this appendix, we consider the conformal wave expansions of
$\langle\mathcal{O}_{h_{1}}(\infty)\mathcal{O}_{h_{2}}(1)\mathcal{O}_{h_{3}}(z)\mathcal{O}_{h_{4}}(0)\rangle$
in terms of the exchange operator with conformal weights $(h,\bar{h})$. For
the application to our examples, we set $h_{1}=h_{2}$ and $h_{3}=h_{4}$. We
use the following formula given in [58, 49] as

$\displaystyle\frac{\Gamma(\bar{h})\Gamma(1-\bar{h})}{\Gamma(1-h)\Gamma(h)}\frac{1}{\pi}\int
d^{2}z\mathcal{F}_{12}^{h,\bar{h}}(z,\bar{z})\mathcal{F}_{34}^{1-h,1-\bar{h}}(z,\bar{z})=\frac{1}{z_{12}^{2h_{1}}z_{34}^{2h_{3}}}\frac{1}{\bar{z}_{12}^{2\bar{h}_{1}}\bar{z}_{34}^{2\bar{h}_{3}}}\mathcal{I}(z,\bar{z})$
(A.24)

with cross ratios

$\displaystyle
z=\frac{z_{12}z_{34}}{z_{13}z_{24}}\,,\quad\bar{z}=\frac{\bar{z}_{12}\bar{z}_{34}}{\bar{z}_{13}\bar{z}_{24}}\,.$
(A.25)

The conformal partial wave is

$\displaystyle\frac{(-1)^{2(h-\bar{h})}}{k_{1-h,1-\bar{h}}}\mathcal{I}_{h,\bar{h}}(z,\bar{z})=\frac{(-1)^{2(h-\bar{h})}}{k_{1-h,1-\bar{h}}}\frac{\Gamma(\bar{h})^{2}}{\Gamma(1-h)^{2}}F_{h,\bar{h}}(z,\bar{z})+\frac{1}{k_{h,\bar{h}}}\frac{\Gamma(1-\bar{h})^{2}}{\Gamma(h)^{2}}F_{1-h,1-\bar{h}}(z,\bar{z})\,,$
(A.26)

where we have defined

$\displaystyle
F_{h,\bar{h}}(z,\bar{z})=z^{h}{}_{2}F_{1}(h,h;2h;z)\bar{z}^{\bar{h}}{}_{2}F_{1}(\bar{h},\bar{h};2\bar{h};\bar{z})\,.$
(A.27)

We then examine a four-point conformal partial wave of exchange of conserved
spin-$s$ current for
$\langle\mathcal{O}_{h_{1}}(\infty)\mathcal{O}_{h_{1}}(1)\mathcal{O}_{h_{3}}(z)\mathcal{O}_{h_{3}}(0)\rangle$.
It is given by a sum of conformal block of conserved spin-$s$ current and its
shadow block as in (4.14), see [61] for some properties. In general, a
conformal partial wave is a sum of conformal block and its shadow block such
that there are no monodromy around, say, $z=0$ and $z=1$. In the current case,
it is obvious that there are no monodromy around $z=0$, thus we focus on the
behavior near $z=1$. As in (4.6) of [62], the first term of (4.14) (after
exchanging $1-z$ by $z$) can be written as

$\displaystyle
z^{s}{}_{2}F_{1}(s,s;2s;z)=-\log(1-z)z^{s}\frac{\Gamma(2s)}{\Gamma(s)^{2}}{}_{2}F_{1}(s,s;1;1-z)+\text{holomorphic
at $z=1$}\,.$ (A.28)

On the other hand, because of

$\displaystyle\bar{z}{}_{2}F_{1}(1,1;2;\bar{z})=-\log(1-\bar{z})\,,$ (A.29)

the second term is equal to

$\displaystyle-\log(1-\bar{z})\frac{\Gamma(2s)}{\Gamma(s)^{2}}{}_{2}z^{s}F_{1}(s,s;1;1-z)\,.$
(A.30)

Therefore, the sum of two terms does not have any monodromy at $z=1$. This is
actually an important property of conformal partial wave, which might be seen
from the integral representation (A.24). It might be also useful to rewrite
the shadow block as

$\displaystyle\begin{aligned}
\frac{\Gamma(2s)\Gamma(2s-1)}{\Gamma(s)^{4}}z^{1-s}{}_{2}F_{1}(1-s,1-s;2-2s;z)=\frac{\Gamma(2s)}{\Gamma(s)^{2}}\sum_{n=0}^{s-1}(-1)^{n}\frac{\Gamma(2s-n-1)}{\Gamma(s-n)^{2}}\frac{z^{n+1-s}}{n!}\,.\end{aligned}$
(A.31)

We begin by studying $G_{-+}(z)$ in (4.2). The coefficients in front of
conformal blocks of spin-$s$ current exchange are obtained by solving the
constraint equations

$\displaystyle\frac{n-1}{n}=\frac{\Gamma(s)^{2}}{\Gamma(2s)}\sum_{s=2}^{n}\frac{(D^{(s)}_{-+})^{2}\Gamma(n)^{2}}{\Gamma(s+n)(n-s)!}$
(A.32)

for $n=2,3,\ldots$, see (4.14) of [46]. Here we have

$\displaystyle(D^{(s)}_{-+})^{2}=\frac{\Gamma(s)^{2}}{\Gamma(2s-1)}=\frac{(-1)^{s}C_{+}^{(s)}C_{-}^{(s)}N}{cB^{(s)}}$
(A.33)

for $s=2,3,\ldots$. From the sum of shadow blocks over $s=2,3,\ldots$, we find

$\displaystyle\left[\sum_{n=0}^{\infty}\sum_{s=n+1}^{\infty}(-1)^{s-n-1}\frac{(2s-1)\Gamma(s+n)}{\Gamma(n+1)^{2}}\frac{(1-z)^{-n}}{(s-n-1)!}\right]-1=-1\,.$
(A.34)

Multiplying $-\log\bar{z}$, we obtain the corresponding term in (4.15). The
result may be the expected one. The sum of conformal blocks over
$s=2,3,\ldots$ has the term $\frac{1}{N}\log z$ for the holomorphic part as in
(4.2). Thus the sum of shadow blocks over $s=2,3,\ldots$ should be
$\tfrac{1}{N}\log\bar{z}$, which makes the sum of the two parts to have no
monodromy at $z=1$.

We then move to $G_{++}(z)$ in (4.17). Since the computation of (2.23) leads
to a sum of conformal partial waves, we would like to compute

$\displaystyle\sum_{s=2}^{\infty}\frac{(C_{+}^{(s)})^{2}}{cB^{(s)}}\mathcal{I}_{s,0}\,.$
(A.35)

As explained in appendix A.2, the sum over conformal blocks of spin-$s$
exchange is given by

$\displaystyle(1-z)^{\frac{1}{N}}{}_{2}F_{1}(1+\tfrac{\lambda}{N},-\tfrac{\lambda}{N};-\lambda;z)-1+\mathcal{O}(N^{-2})\,.$
(A.36)

From the arguments of monodromy around $z=1$, we can deduce the sum of shadow
blocks as in the previous example. Namely, the sum of shadow blocks should be

$\displaystyle\frac{1}{N}\log(1-\bar{z})\,.$ (A.37)

##  Appendix B Bulk interpretations

In the main context, we mainly work on the dual CFT and did not deal with the
bulk theory in explicit details. However, things sometimes become intuitively
clearer from the bulk viewpoints. In this appendix, for completeness, we will
summarize the essential details about embedding formalism for Euclidean AdSd+1
following [63]. We have also utilized specific features in 2d CFT, such as,
holomorphic factorization. We show that the property can be seen from the bulk
viewpoints as well.

###  B.1 Embedding formalism for Euclidean AdSd+1

In this work, we compute bulk correlators in terms of dual CFT. Even so, it is
illustrative to see which bulk computations correspond to boundary ones. In
order to explain the bulk computations, it is useful to introduce embedding
formulation, where Euclidean AdSd+1 space is expressed by a hypersurface,

$\displaystyle X^{2}=-1\,,\quad X^{0}>0\,,$ (B.1)

in $(d+2)$ dimensional Minkowski space $\mathbb{M}^{d+2}$. Here we set the AdS
radius to be one as $\ell_{\text{AdS}}=1$. We may use the light-cone
coordinates $X^{A}=(X^{+},X^{-},\vec{X})$ with the metric

$\displaystyle X^{2}=\eta_{AB}X^{A}X^{B}=-X^{+}X^{-}+\vec{X}\cdot\vec{X}\,.$
(B.2)

The Poincaré coordinates (2.2) are given by
$X=\frac{1}{y}(1,y^{2}+\vec{x}^{2},\vec{x})$. The AdS boundary is located at
$y=0$, which may be expressed by light rays

$\displaystyle P^{2}=0,\quad P=\lambda P$ (B.3)

with $\lambda\in\mathbb{R}$.

We consider a symmetric traceless tensor field $\sigma_{i_{1}\cdots i_{s}}$ on
AdSd+1, which is related to a symmetric traceless tensor field
$\Sigma_{A_{1}\cdots A_{s}}$ on $\mathbb{M}^{d+2}$ as

$\displaystyle\sigma_{i_{1}\cdots i_{s}}=\frac{\partial X^{A_{1}}}{\partial
x^{i_{s}}}\cdots\frac{\partial X^{A_{s}}}{\partial
x^{i_{s}}}\Sigma_{A_{1}\cdots A_{s}}(X)\,.$ (B.4)

We define $\Sigma_{A_{1}\cdots A_{s}}$ on the surface $X^{2}=-1$ and assign
the transverse condition

$\displaystyle X^{A_{1}}\Sigma_{A_{1}\cdots A_{s}}(X)=0\,.$ (B.5)

Introducing bulk polarization vectors $W^{A}$, we define

$\displaystyle\Sigma(X,W)=W^{A_{1}}\cdots W^{A_{s}}\Sigma_{A_{1}\cdots
A_{s}}(X)\,,$ (B.6)

where we assign $W^{2}=0$, $X\cdot W=0$ corresponding to the traceless and
transverse conditions, respectively. On the AdS boundary described by
$P^{2}=0$, we also define a symmetric traceless operator $\Xi_{A_{1}\cdots
A_{s}}(P)$. We require $\Xi(\lambda P)=\lambda^{-\Delta}\Xi(P)$ with conformal
dimension $\Delta$. Introducing polarization vectors $Z^{A}$, we define

$\displaystyle\Xi(P,Z)=Z^{A_{1}}\cdots Z^{A_{J}}\Xi_{A_{1}\cdots A_{J}}(P)$
(B.7)

with $Z^{2}=0$, $P\cdot Z=0$. We also assign $P^{A_{1}}\Xi_{A_{1}\cdots
A_{J}}=0$, which can be encoded by requiring $\Xi(P,Z+\alpha Z)=\Xi(P,Z)$.

We then consider a bulk to bulk propagator of spin-$s$ symmetric traceless
field with dual scaling dimension $\Delta$. The propagator between two AdS
bulk points $X_{1},X_{2}$ and polarization vectors $W_{1},W_{2}$ is denoted by
$G_{\Delta,s}(X_{1},X_{2};W_{1},W_{2})$. Similarly, a bulk-to-boundary
propagator is denoted by $K_{\Delta,s}(X,P;W,Z)$. The AdS harmonic function is
defined by

$\displaystyle\begin{aligned} &\Omega_{\nu,s}(X_{1},X_{2};W_{1},W_{2})\\\
&\quad=\frac{\nu^{2}}{\pi s!(\frac{d}{2}-1)_{s}}\int
dPK_{\frac{d}{2}+i\nu,s}(X_{1},P;W_{1},D_{Z})K_{\frac{d}{2}-i\nu,s}(X_{2},P;W_{2},Z)\,.\end{aligned}$
(B.8)

Here $(a)_{n}=a(a+1)\cdots(a+n)$ and

$\displaystyle D_{Z}=\left(\frac{d}{2}-1+Z\cdot\frac{\partial}{\partial
Z}\right)\frac{\partial}{\partial
Z_{A}}-\frac{1}{2}Z^{A}\frac{\partial^{2}}{\partial Z\cdot\partial Z}\,.$
(B.9)

The bulk to bulk propagator is now expressed in terms of the AdS harmonic
function as

$\displaystyle G_{\Delta,s}(X_{1},X_{2};W_{1},W_{2})=\sum_{l=0}^{s}\int d\nu
a_{l}(\nu)((W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2}))^{s-l}\Omega_{\nu,l}(X_{1},X_{2};W_{1},W_{2})\,,$
(B.10)

where

$\displaystyle a_{s}(\nu)=\frac{1}{\nu^{2}+(\Delta-\frac{d}{2})^{2}}$ (B.11)

for the highest spin contribution. See [63] for more detailed analysis.

We will consider three-point function of scalar-scalar-higher spin current in
a CFT. Let us denote dual fields by $\phi^{\text{AdS}}_{1}$,
$\phi^{\text{AdS}}_{2}$, $\sigma^{\text{AdS}}_{i_{1}\cdots i_{s}}$ and
introduce a type of interaction

$\displaystyle
g\int_{\text{AdS}}dx\sqrt{g}(\phi^{\text{AdS}}_{1}\nabla_{i_{1}}\cdots\nabla_{i_{s}}\phi^{\text{AdS}}_{2})\sigma_{\text{AdS}}^{i_{1}\cdots
i_{s}}\,.$ (B.12)

Then the three-point correlation function in the dual CFT can be computed via
Witten diagram as [63]

Figure 5: Witten diagram dual to three-point function of scalar-scalar-higher
spin current.

$\displaystyle\begin{aligned}
&\left\langle\mathcal{O}_{1}^{\text{AdS}}(P_{1})\mathcal{O}_{2}^{\text{AdS}}(P_{2})J^{\text{AdS}}(P_{3},Z)\right\rangle\\\
&=\frac{g}{\sqrt{\mathcal{C}_{\Delta_{1},0}\mathcal{C}_{\Delta_{2},0}\mathcal{C}_{\Delta,s}}}\int
dXK_{\Delta_{2},0}(X,P_{2})\frac{K_{\Delta,s}(X,P_{3};K,Z)(W\cdot\nabla)^{s}K_{\Delta_{1},0}(X,P_{1})}{s!(\frac{d-1}{2})_{s}}\\\
&=\frac{g}{\sqrt{\mathcal{C}_{\Delta_{1},0}\mathcal{C}_{\Delta_{2},0}\mathcal{C}_{\Delta,s}}}b(\Delta_{1},\Delta_{2},\Delta,s)\frac{((Z\cdot
P_{1})P_{23}-(Z\cdot
P_{2})P_{13})^{s}}{P_{12}^{\frac{\Delta_{1}+\Delta_{2}-\Delta+s}{2}}P_{13}^{\frac{\Delta_{1}+\Delta-\Delta_{2}+s}{2}}P_{23}^{\frac{\Delta+\Delta_{2}-\Delta_{1}+s}{2}}}\,.\end{aligned}$
(B.13)

Here $\mathcal{C}_{\Delta,s}$ is the coefficient of two-point function and
$P_{ij}=-2P_{i}\cdot P_{j}$. Moreover, $K$ is a projection operator defined by

$\displaystyle\begin{aligned} K&=\frac{d-1}{2}\left[\frac{\partial}{\partial
W^{A}}+X_{A}\left(X\cdot\frac{\partial}{W}\right)\right]+\left(W\cdot\frac{\partial}{\partial
W}\right)\frac{\partial}{\partial W^{A}}\\\
&\quad+X_{A}\left(W\cdot\frac{\partial}{\partial
W}\right)\left(X\cdot\frac{\partial}{\partial
W}\right)-\frac{1}{2}W_{A}\left[\frac{\partial^{2}}{\partial W\cdot\partial
W}+\left(X\cdot\frac{\partial}{\partial W}\right)^{2}\right]\,.\end{aligned}$
(B.14)

The coefficient function was computed in [63] as

$\displaystyle\begin{aligned} &b(\Delta_{1},\Delta_{2},\Delta,s)\\\
&\quad=\mathcal{C}_{\Delta_{1},0}\mathcal{C}_{\Delta_{2},0}\mathcal{C}_{\Delta,s}\frac{\pi^{\frac{d}{2}}\Gamma(\frac{\Delta_{1}+\Delta_{2}+\Delta-d+s}{2})\Gamma(\frac{\Delta_{1}+\Delta_{2}-\Delta+s}{2})\Gamma(\frac{\Delta+\Delta_{1}-\Delta_{2}+s}{2})\Gamma(\frac{\Delta+\Delta_{2}-\Delta_{1}+s}{2})}{2^{1-s}\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\Delta+s)}\,.\end{aligned}$
(B.15)

Finally, we consider four-point correlation function of scalar operators

$\displaystyle\left\langle\mathcal{O}_{1}(P_{1})\mathcal{O}_{2}(P_{2})\mathcal{O}_{3}(P_{3})\mathcal{O}_{4}(P_{4})\right\rangle\,.$
(B.16)

We focus on the contribution coming from the Witten diagram with the exchange
of a spin-$s$ field. Then the four-point correlation functions can be computed
from the bulk by

Figure 6: Witten diagram dual to a contribution to four-point function of
scalar operators. The diagram can be written as a product of two three-point
functions by applying the split representation of bulk to bulk propagator.

$\displaystyle\begin{aligned} &\frac{g^{2}}{(s!(\frac{d-1}{2})_{s})^{2}}\int
dX_{1}dX_{2}K_{\Delta_{1},0}(P_{1},X_{1})\left[(K_{1}\cdot\nabla_{1})^{s}K_{\Delta_{2},0}(P_{2},X_{1})\right]\\\
&\quad\times
K_{\Delta_{3},0}(P_{3},X_{2})\left[(K_{2}\cdot\nabla_{2})^{s}K_{\Delta_{4},0}(P_{4},X_{2})\right]\sum_{l=0}^{s}\int
d\nu a_{l}(\nu)\Omega_{\nu,s}(X_{1},X_{2},W_{1},W_{2})\,.\end{aligned}$ (B.17)

For the bulk to bulk propagator, we have used (B.10). If we apply the split
representation of AdS harmonics (B.8), then the above expression is reduced to

$\displaystyle\left\langle\mathcal{O}_{1}(P_{1})\mathcal{O}_{2}(P_{2})\mathcal{O}_{3}(P_{3})\mathcal{O}_{4}(P_{4})\right\rangle=\sum_{l=0}^{\infty}\int
d\nu c_{l}(\nu)F_{\nu,l}(P_{i})$ (B.18)

with

$\displaystyle F_{\nu,l}(P_{i})=\int
dP_{5}\left\langle\mathcal{O}_{1}^{\text{AdS}}(P_{1})\mathcal{O}_{2}^{\text{AdS}}(P_{2})J^{\text{AdS}}_{\frac{d}{2}+i\nu,l}(P_{5},D_{Z})\right\rangle\left\langle
J^{\text{AdS}}_{\frac{d}{2}-i\nu,l}(P_{5},Z)\mathcal{O}_{3}^{\text{AdS}}(P_{3})\mathcal{O}_{4}^{\text{AdS}}(P_{4})\right\rangle\,.$
(B.19)

Here we have used the bulk expression of three-point function in (B.13). The
expressions (B.19) and (B.18) are definitions of conformal partial waves and
their expansions of four-point function, respectively, see fig. 6.

###  B.2 Three-point function in the holographic gauge

In this appendix, we obtain the coordinate dependence of three-point function
(3.13) from the bulk theory. We will see that the holographic gauge for
higher-spin gauge fields is suitable to make holomorphic factorization
transparency. For a scalar field, we use the bulk to boundary propagator in
the global coordinates:

$\displaystyle
K_{\Delta}(y,z;z^{\prime})=C_{\Delta}^{\text{AdS}}\left(\frac{y}{y^{2}+|z-z^{\prime}|^{2}}\right)^{\Delta}\,.$
(B.20)

For a spin-$s$ field $\varphi_{\mu_{1}\cdots\mu_{s}}$, we use a holographic
gauge with $\varphi_{\mu_{1}\cdots\mu_{s}}=0$ if at least one of $\mu_{l}=y$.
For two dimensions, only physical fields are $\varphi_{z\cdots z}$ and
$\varphi_{\bar{z}\cdots\bar{z}}$. Then the bulk to boundary propagator may be
obtained as [64]

$\displaystyle
K_{l,l}(y,z;z^{\prime})=C_{l}^{\text{AdS}}y^{-l}K_{l}(y,z;z^{\prime})\,.$
(B.21)

Let us assume the bulk coupling
$g\bar{\phi}\nabla^{\mu_{1}}\cdots\nabla^{\mu_{s}}\phi\varphi_{\mu_{1}\cdots\mu_{s}}$.
In the holographic gauge, only the non-trivial contribution is
$g(\bar{\phi}(\nabla^{z})^{s}\phi\varphi_{z\cdots
z}+\phi(\nabla^{\bar{z}})^{s}\phi\varphi_{\bar{z}\cdots\bar{z}})$. Let us
focus only on one term, which leads to

$\displaystyle
A(z_{1},z_{2},z_{3})=-g\int\frac{dwd\bar{w}dy}{y^{3}}y^{-l}K_{l,l}(y,w;z_{1})y^{2l}\partial_{\bar{w}}^{l}K_{\Delta}(y,w;z_{2})K_{\Delta}(y,w;z_{3})\,.$
(B.22)

As in [65], we utilize the conformal symmetry of CFT under $z\to
z^{\prime}=-1/z$ and $\bar{z}\to\bar{z}^{\prime}=-1/\bar{z}$. The symmetry
corresponds to the AdS isometry associated with $y\to y^{\prime}=y|z|^{-2}$.
The computation becomes simplified as

$\displaystyle\begin{aligned}
A(z_{1},z_{2},0)&=-gC_{\Delta}^{\text{AdS}}\frac{1}{{\bar{z}}_{1}^{2l}|z_{2}|^{2\Delta}}\int\frac{dw^{\prime}d\bar{w}^{\prime}dy^{\prime}}{(y^{\prime})^{3}}K_{l,l}(y^{\prime},w^{\prime};z_{1}^{\prime})\partial_{\bar{w}^{\prime}}^{l}K_{\Delta}(y^{\prime},w^{\prime};z_{2}^{\prime})(y^{\prime})^{\Delta+l}\,.\end{aligned}$
(B.23)

Here notice that

$\displaystyle\varphi_{z\cdots z}=\left(\frac{\partial z^{\prime}}{\partial
z}\right)^{l}\varphi_{z^{\prime}\cdots
z^{\prime}}=\frac{1}{z^{2l}}\varphi_{z^{\prime}\cdots z^{\prime}}\,.$ (B.24)

According to (22) and (23) of [65], we may conclude that the integral is
proportional to $(\bar{z}_{1}^{\prime}-\bar{z}_{2}^{\prime})^{-l}$. Thus, we
have

$\displaystyle
A(z_{1},z_{2},0)\propto\frac{1}{{\bar{z}_{1}}^{4l}|z_{2}|^{2\Delta}}\frac{1}{(\bar{z}_{1}^{\prime}-\bar{z}_{2}^{\prime})^{l}}=\frac{\bar{z}_{2}^{l}}{\bar{z}_{1}^{l}(\bar{z}_{2}-\bar{z}_{1})^{l}}\frac{1}{|z_{2}|^{2\Delta}}\,.$
(B.25)

In order to compute the coefficient, we need to evaluate the integral. For
this, we rewrite it as

$\displaystyle\begin{aligned} &\lim_{\epsilon\to
0}(-1)^{l}\partial_{\bar{z}_{2}^{\prime}}^{l}\int\frac{dw^{\prime}d\bar{w}^{\prime}dy^{\prime}}{(y^{\prime})^{3}}K_{l,l}(y^{\prime},w^{\prime};z_{1}^{\prime})K_{\Delta}(y^{\prime},w^{\prime};z_{2}^{\prime})(y^{\prime})^{\Delta+l+2\epsilon}\\\
&\quad=\lim_{\epsilon\to
0}(-1)^{l}\partial_{\bar{z}_{2}^{\prime}}^{l}C_{l}^{\text{AdS}}C_{\Delta}^{\text{AdS}}I[2\Delta+2l+2\epsilon-3,l,\Delta,2]|\bar{z}_{1}^{\prime}-\bar{z}_{2}^{\prime}|^{2\epsilon}\,.\end{aligned}$
(B.26)

Here $I(a,b,c,d)$ is given by (23) of [65] as

$\displaystyle\begin{aligned}
I[2\Delta+2l+2\epsilon-3,l,\Delta,2]&=\frac{\pi}{2}\frac{\Gamma(\Delta+l+\epsilon-1)\Gamma(-\epsilon)}{\Gamma(l)\Gamma(\Delta)}\frac{\Gamma(\Delta+\epsilon)\Gamma(l+\epsilon)}{\Gamma(\Delta+l+2\epsilon)}\\\
&=-\frac{1}{\epsilon}\frac{\pi}{2(\Delta+l-1)}+\mathcal{O}(\epsilon^{0})\,.\end{aligned}$
(B.27)

In total, we find

$\displaystyle\begin{aligned}
A(z_{1},z_{2},0)&=g(C_{\Delta}^{\text{AdS}})^{2}C_{l}^{\text{AdS}}\frac{(-1)^{l+1}\Gamma(l)\pi}{2(\Delta+l-1)}\frac{\bar{z}_{2}^{l}}{\bar{z}_{1}^{l}(\bar{z}_{2}-\bar{z}_{1})^{l}}\frac{1}{|z_{2}|^{2\Delta}}\,.\end{aligned}$
(B.28)

###  B.3 Geodesic Witten diagrams and two-dimensional global conformal blocks

#####

In [49], an expression for two-dimensional global conformal partial wave with
arbitrary external twists was derived, which can be written as:

$W^{(s)}_{h,\bar{h}}(z_{i})=\frac{1}{\pi}\int dz_{0}d\bar{z}_{0}{\cal
F}_{12}^{h\bar{h}}(z_{0},\bar{z}_{0}){\cal
F}_{34}^{1-h1-\bar{h}}(z_{0},\bar{z}_{0})=K_{h_{i}}^{(s)}(z_{i})K_{\bar{h}_{i}}^{(s)}(\bar{z}_{i})\hat{{\cal
I}}_{h,\bar{h}}(\eta,\bar{\eta})\,,$ (B.29)

with

${\cal
F}_{12}^{h\bar{h}}(z_{0},\bar{z}_{0})=\frac{1}{(z_{12}^{2})^{\frac{h_{1}+h_{2}-h_{0}}{2}}(z_{10}^{2})^{\frac{h_{0}+h_{1}-h_{2}}{2}}(z_{20}^{2})^{\frac{h_{0}+h_{2}-h_{1}}{2}}}\frac{1}{(\bar{z}_{12}^{2})^{\frac{\bar{h}_{1}+\bar{h}_{2}-\bar{h}_{0}}{2}}(\bar{z}_{10}^{2})^{\frac{\bar{h}_{0}+\bar{h}_{1}-\bar{h}_{2}}{2}}(\bar{z}_{20}^{2})^{\frac{\bar{h}_{0}+\bar{h}_{2}-\bar{h}_{1}}{2}}}$
(B.30)

where the holomorphic and anti-holomorphic twists are related to the scaling
dimensions $\\{\Delta_{i}\\}$ and spins $\\{s_{i}\\}$ via
$h_{i}+\bar{h}_{i}=\Delta_{i},h_{i}-\bar{h}_{i}=s_{i}$ and
$s_{0}+s_{1}+s_{2}\in\mathbb{Z}$, while the overall kinematic factor
$K_{h_{i}}^{(s)}(z)$ is given by:

$K_{h_{i}}^{(s)}(z_{i})=\frac{1}{(z_{12}^{2})^{\frac{h_{12}^{+}}{2}}(z_{34}^{2})^{\frac{h_{34}^{+}}{2}}}\left(\frac{z_{24}^{2}}{z_{14}^{2}}\right)^{h_{12}^{-}}\left(\frac{z_{14}^{2}}{z_{13}^{2}}\right)^{h_{34}^{-}},\quad
h_{ij}^{\pm}=h_{i}\pm h_{j}\,,$ (B.31)

similarly for $K_{\bar{h}_{i}}(\bar{z}_{i})$. The kinematic integral over
product of three-point functions can be performed and yields:

$\displaystyle\begin{aligned} \hat{{\cal
I}}^{(s)}_{h,\bar{h}}(\eta,\bar{\eta})&=\frac{\Gamma(1-2h)}{\Gamma(2\bar{h})}\frac{\Gamma(h-h_{12}^{-})\Gamma(\bar{h}+\bar{h}_{12}^{-})}{\Gamma(1-h-h_{12}^{-})\Gamma(1-\bar{h}+\bar{h}_{12}^{-})}F_{h,\bar{h}}^{(s)}(\eta,\bar{\eta})\\\
&\quad+\frac{\Gamma(2h-1)}{\Gamma(2-2\bar{h})}\frac{\Gamma(1-h-h_{34}^{-})\Gamma(1-\bar{h}+\bar{h}_{34}^{-})}{\Gamma(h-h_{34}^{-})\Gamma(\bar{h}+\bar{h}_{34}^{-})}F_{1-h,1-\bar{h}}^{(s)}(\eta,\bar{\eta})\end{aligned}$
(B.32)

and

$F_{h,\bar{h}}^{(s)}(\eta,\bar{\eta})=\eta^{h}\,{}_{2}F_{1}\left(h-h_{12}^{-},h+h_{34}^{+};2h;\eta\right)\bar{\eta}^{\bar{h}}\,{}_{2}F_{1}\left(\bar{h}-\bar{h}_{12}^{-},\bar{h}+\bar{h}_{34}^{+};2\bar{h};\bar{\eta}\right)\,.$
(B.33)

We can then expand the four-point correlation function of primary operators
$\varphi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})$ in terms of resultant global
conformal block:

$\left\langle\prod_{i=1}^{4}\varphi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\right\rangle=K_{h_{i}}^{(s)}(z_{i})K_{\bar{h}_{i}}^{(s)}(\bar{z}_{i})\sum_{h,\bar{h}}a_{h,\bar{h}}F_{h,\bar{h}}^{(s)}(z,\bar{z})\,.$
(B.34)

If we now focus on the factorizable three-point function (B.30) of scalar-
scalar-higher spin current in two-dimensional CFTs such that
$2h_{1,2}=2\bar{h}_{1,2}=\Delta_{1,2}$, it can also be expressed in terms of
the embedding formalism by choosing the following embedding coordinates in
$\mathbb{M}^{1+3}$:

$P_{i}^{A}=(P_{i}^{+},P_{i}^{-},P_{i},\bar{P}_{i})=(1,z_{i}\bar{z}_{i},z_{i},\bar{z}_{i}),\quad
i=0,1,2$ (B.35)

where we have chosen complex coordinates $(z_{i},\bar{z}_{i})$ in the last two
spatial embedding coordinates. For such coordinate choice, the polarization
vector $Z_{i}$ satisfying transverse condition $P_{i}\cdot Z_{i}=0$ in (B.13)
can be parameterized as:

$Z^{A}_{i}=(0,w_{i}\bar{z}_{i}+\bar{w}_{i}z_{i},w_{i},\bar{w}_{i})\,,$ (B.36)

where $(w_{i},\bar{w}_{i})$ parameterize the otherwise arbitrary two
dimensional polarization vector. Since it is only used to perform tensor index
contraction, we can treat $w_{i}$ and $\bar{w}_{i}$ as independent complex
variables, and for our case if we can make the following choice:

$Z^{A}_{0}=\mathcal{Z}^{A}_{0}=\left(1,-2\frac{\hat{z}_{12}z_{0}}{\hat{z}_{10}\hat{z}_{20}},,-2\frac{\hat{z}_{12}}{\hat{z}_{10}\hat{z}_{20}},0\right)\,,\quad\hat{z}_{ij}=\frac{z_{ij}}{|z_{ij}|}\,,\quad
i,j=0,1,2\,.$ (B.37)

The direct substitutions and simple calculations then shows that (B.30) is
indeed proportional to (B.13) for $d=2$.

This choice of polarization vector also leads us to directly rewrite (B.30) in
terms of a single integral giving rise to geodesic Witten diagram. Starting
with the scalar-scalar-scalar case, we have the following identity:

$\displaystyle\begin{aligned}
&\frac{1}{|z_{12}|^{{\Delta_{1}+\Delta_{2}-\Delta_{0}}}|z_{10}|^{{\Delta_{0}+\Delta_{1}-\Delta_{2}}}|z_{20}|^{{\Delta_{0}+\Delta_{2}-\Delta_{1}}}}\\\
&\quad=\frac{2}{{\rm
B}(\frac{\Delta_{0}+\Delta_{1}-\Delta_{2}}{2},\frac{\Delta_{0}-\Delta_{1}+\Delta_{2}}{2})}\int^{\infty}_{-\infty}d\lambda\prod_{i=0}^{2}\frac{1}{(-2P_{i}\cdot
X(\lambda))^{\Delta_{i}}}\end{aligned}$ (B.38)

where $B(x,y)$ is the usual beta function and

$X^{A}(\lambda)=\frac{P_{1}^{A}e^{\lambda}+P_{2}^{A}e^{\lambda}}{(-2P_{1}\cdot
P_{2})^{\frac{1}{2}}}\,,\quad-\infty<\lambda<+\infty$ (B.39)

can be regarded geometrically as a AdS3 bulk coordinate restricted to move
along the geodesic connecting the boundary points $P_{1}$ and $P_{2}$. We can
next apply the differential identity given in equation (130) of [63]:

$\frac{1}{\left(\frac{\Delta_{0}+\Delta_{1}^{\prime}-\Delta_{2}}{2}\right)_{s_{0}}}\frac{2}{{\rm
B}(\frac{\Delta_{0}+\Delta_{1}^{\prime}-\Delta_{2}}{2},\frac{\Delta_{0}-\Delta_{1}^{\prime}+\Delta_{2}}{2})}\int^{\infty}_{-\infty}d\lambda(D_{01}|_{Z_{0}=\mathcal{Z}_{0}})^{s_{0}}\prod_{i=0}^{2}\frac{1}{(-2P_{i}\cdot
X(\lambda))^{\Delta_{i}}}|_{\Delta_{1}\to\Delta_{1}^{\prime}}$ (B.40)

where the differential operator is given by:

$D_{01}=(P_{1}\cdot Z_{0})\left(Z_{0}\frac{\partial}{\partial
Z_{0}}-P_{0}\frac{\partial}{\partial P_{0}}\right)+(P_{1}\cdot
P_{0})\left(Z_{0}\frac{\partial}{\partial P_{0}}\right)$ (B.41)

and we have made the replacements of the scaling dimension
$\Delta_{1}\to\Delta_{1}^{\prime}=\Delta_{1}-s_{0}$ and the polarization
vector $Z_{0}\to\mathcal{Z}_{0}$ in $D_{01}$ after the differentiation. We can
then identify the resultant geodesic integral with the three point scalar-
scalar-spin interaction along the geodesic. We can next apply similar
construction in [66] which utilizing the split representation of the AdS
harmonic function reviewed earlier to construct the holographic dual
configuration to global conformal partial wave (B.29).

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# ExoRecovery: Push Recovery with a Lower-Limb Exoskeleton
based on Stepping Strategy

Zeynep Özge Orhan1,2, Milad Shafiee2, Vincent Juillard1,
Joel Coelho Oliveira1, Auke Ijspeert2, and Mohamed Bouri1 1 Zeynep Özge Orhan,
Vincent Juillard, Joel Coelho Oliveira, Mohamed Bouri are with the REHAssist
group, Ecole Polytechnique Federale de Lausanne (EPFL). (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>mohamed.bouri@epfl.ch)2 Zeynep Özge Orhan, Milad Shafiee and Auke Ijspeert are
with the BioRobotics Laboratory, Ecole Polytechnique Federale de Lausanne
(EPFL). (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Balance loss is a significant challenge in lower-limb exoskeleton
applications, as it can lead to potential falls, thereby impacting user safety
and confidence. We introduce a control framework for omnidirectional recovery
step planning by online optimization of step duration and position in response
to external forces. We map the step duration and position to a human-like foot
trajectory, which is then translated into joint trajectories using inverse
kinematics. These trajectories are executed via an impedance controller,
promoting cooperation between the exoskeleton and the user. Moreover, our
framework is based on the concept of the divergent component of motion, also
known as the Extrapolated Center of Mass, which has been established as a
consistent dynamic for describing human movement. This real-time online
optimization framework enhances the adaptability of exoskeleton users under
unforeseen forces thereby improving the overall user stability and safety. To
validate the effectiveness of our approach, simulations, and experiments were
conducted. Our push recovery experiments employing the exoskeleton in zero-
torque mode (without assistance) exhibit an alignment with the exoskeleton’s
recovery assistance mode, that shows the consistency of the control framework
with human intention. To the best of our knowledge, this is the first
cooperative push recovery framework for the lower-limb human exoskeleton that
relies on the simultaneous adaptation of intra-stride parameters in both
frontal and sagittal directions. The proposed control scheme has been
validated with human subject experiments.

##  I Introduction

Lower-limb exoskeletons (LLEs), while promising for assisting those with
walking impairments, face significant challenges in maintaining balance and
stability, particularly in real-world scenarios with various external
disturbances [1, 2, 3]. Balance is a generic term describing the body posture
dynamics to prevent falling and it is a crucial ability to ensure upright
standing [4, 5]. Researchers revealed three elementary strategies of human
balance recovery such as ankle and hip strategies, and stepping strategy with
variable step duration [6, 7, 8, 4]. When the disturbances are too large to
handle with the aforementioned strategies, they are tackled with a stepping
strategy.

When individuals experience balance perturbations, executing a well-
coordinated step can help restore equilibrium and prevent falls. By adjusting
the position of the feet, the stepping strategy allows individuals to shift
their center of mass (CoM), stabilize their posture, and counteract external
disturbances. The implementation of an effective stepping strategy with a
suitable step length and step duration is particularly important in LLEs, as
it enables users to regain balance and navigate safely in challenging
environments.

Figure 1: The 2nd prototype of autonomyo lower-limb exoskeleton. A simplified
kinematics model of the autonomyo exoskeleton is illustrated.

Although the majority of the studies focused on gait assistance with LLEs,
balance control in standing and walking is crucial for the control of an LLE
for posture and gait assistance [9, 10]. Recently, the balance strategies of
humans also inspired researchers in the field of LLEs to assist the balance of
users [11, 10].

Most of the effort in the standing balance assistance is taking advantage of
actuated ankle joints [12, 13, 14, 15, 16, 17]. There are other studies that
investigated ankle, hip, and combined strategy in terms of subject-specific
stability limits, slip prevention, importance of hip abd./add. for weight
shift and lateral foot placement [18, 1, 19].

Duburcq et al. [20] pioneered a push recovery controller based on
reinforcement learning, marking a substantial advancement. They successfully
demonstrated reactive push recovery with a humanoid robot, utilizing deep
reinforcement learning. Notably, this study did not involve direct interaction
between the exoskeleton and the user, and the exoskeleton functioned primarily
as a humanoid robot.

In the study by Zhang et al. [21], they proposed a method based on the capture
point (CP) concept to enhance the balance restoration capabilities of LLEs
under significant interference conditions. However, this study did not address
step duration optimization, and its focus was exclusively on the sagittal
plane. Similarly, in [22], an xCoM-based balance assistance strategy in the
sagittal plane is proposed for disturbances under forward and backward
directions. Vallery et al. [23], proposed another xCoM-based balance
controller that provides calculated feed-forward trajectories. The controller
is triggered in case of balance loss and is used to keep users in an upright
standing position for a stationary exoskeleton. In [24], a balance assistance
strategy in the sagittal plane is suggested based on the zero-moment point
(ZMP) model-based method. The ZMP has been used to generate the trajectory of
movement during the stance phase and assistive torques are designed based on
the minimization of modulating of virtual potential energy.

It’s worth noting that the CP, Divergent Component of Motion (DCM) and the
xCoM share the same definition. The terms CP and DCM are commonly used within
the robotics community, while xCoM is more frequently employed in the
biomechanics community. Although CP and ZMP-based methods are widely used in
the literature on humanoids [25, 26, 27, 28, 29] and also for walking pattern
generation for exoskeletons [19, 30, 31], it has not been investigated in
detail for step position and duration adaptation of exoskeletons in sagittal
and frontal planes. Furthermore, these methods were tested in simulations,
leaving room for further hardware validation [32, 33, 34, 35, 36].
Farkhatdinov et al. [37] introduced a push recovery mechanism for the human-
exoskeleton. However, this approach primarily focused on applying assistive
torque without model-based optimization or consideration of the system’s
dynamics.

As the exoskeleton user actively participates in the recovery step, it is
crucial to override the user’s behaviors to provide assistance. Instead, the
exoskeleton and the user symbiotically move toward the human desired recovery
pose. To the best of our knowledge, currently, there is no robotic exoskeleton
that can support adaptive step position and duration for balance recovery in
sagittal and frontal planes that are collaboratively interacting with
exoskeleton users.

The present work describes an experimental study on exoskeleton-assisted
recovery stepping together with its theoretical background for balance
recovery in case of perturbation in healthy individuals. The introduced
framework offers targeted joint trajectories to facilitate adjustments in
stepping location, encompassing hip abd./add. for step-width adaptation, as
well as knee and hip flex./ext. for step-length adaptation. In particular, the
contributions of this paper are the following:

  * •

We have developed a user-cooperative omnidirectional recovery stepping control
strategy in case of a balance loss. The suggested strategy is implemented and
verified through simulations and conducted in-lab experimental evaluations
with human subjects.

  * •

We propose a framework to detect a possible balance loss under severe
perturbations based on divergent component of motion dynamics.

  * •

We have implemented a bio-inspired recovery step trajectory based on human
foot position during gait.

##  II Methods

###  II-A Linear Inverted Pendulum Model (LIPM)

Figure 2: DCM, CoM and CoP Points correlations for Centroidal Dynamics

The LIPM has been widely utilized to describe the dynamics of the CoM for
bipedal locomotion [38]. The LIPM assumes a constant rate of change of
centroidal angular momentum and movement of the CoM height within a plane. In
[17], it has been suggested that for symbiotic human-exoskeleton balance
control, CoM kinematics-based feedback could be beneficial to precede
physiological responses. Also, these assumptions are suitable for planning
humanoid robot locomotion, as research on human walking indicates minimal
variations in centroidal angular momentum and CoM height[39]. Based on the
assumptions, the equations of motion for the LIPM can be derived as follows:

$\vspace{-0.25cm}\mathbf{\ddot{x}}=\omega^{2}(\mathbf{x}-\mathbf{cop})\vspace{-0.02cm}$
(1)

in which $\mathbf{x}=[x_{com},y_{com}]^{T}$ is the horizontal position of the
CoM, $\omega_{0}=\sqrt{\frac{g}{\Delta z}}$ is the natural frequency of the
LIPM, and $\mathbf{cop}=[cop_{x},cop_{y}]^{T}$ is the horizontal position of
the center of pressure (CoP).

###  II-B Divergent Component of Motion

In this section, we provide an overview of the DCM concept’s background. The
dynamics of the CoM, as modeled by the LIPM, can be split into stable and
unstable components. The unstable component is referred to as the DCM and is
defined as follows:

$\bm{\xi}=\mathbf{x}+\frac{\mathbf{\dot{x}}}{{\omega}}\vspace{-0.1cm}$ (2)

Throughout this study, DCM is represented as $\xi$ as the notation in [40].
From (2), the CoM dynamics is given by:

$\mathbf{\dot{x}}={{\omega}}(\bm{\xi}-\mathbf{x})\vspace{-0.1cm}$ (3)

By differentiating (2) and substituting (1), the DCM dynamics is expressed as
:

$\dot{\bm{\xi}}={{\omega}}(\bm{\xi}-\mathbf{cop})\vspace{-0.2cm}$ (4)

Fig. (2) illustrates the relationship between DCM dynamics, CoM, and the CoP.
By re-arranging DCM dynamics (4), the following ordinary differential equation
(ODE) holds:

$\dot{\bm{\xi}}-\omega{\bm{\xi}}=-{\omega}\hskip
2.04877pt\mathbf{cop}_{0}\vspace{-0.2cm}$ (5)

The solution to (5) writes:

$\bm{\xi}(t)=e^{\int{\omega}dt}\left[\int\left(-{\mathbf{cop}_{0}}\hskip
2.04877pt\omega\right)e^{\int-{\omega}dt}dt+\mathbf{C}\right],\vspace{-0.2cm}$
(6)

where $C\in\mathbb{R}^{2}$ is the vector of unknown coefficients that can be
found by imposing the boundary conditions. Therefore, we can find these
coefficients by solving the problem (6) either as an initial value problem,
namely

$\bm{\xi}(0)=\bm{\xi}_{0}=\mathbf{cop}_{0}+\mathbf{C}_{0},\vspace{-0.1cm}$ (7)

or as a final value problem:

$\bm{\xi}(T)=\bm{\xi}_{T}=\mathbf{cop_{0}}+\mathbf{C_{f}}\hskip
0.91042pte^{\omega{T}}.\vspace{-0.1cm}$ (8)

Therefore, by solving the equation (4) as an initial value problem, we arrive
at the following equation that represents the time evolution of the DCM:

$\bm{\xi}=(\bm{\xi}_{0}-{\mathbf{{cop_{0}}}})\hskip
1.42271pt{{\exp}}({{\omega}}{{t}})+\mathbf{{cop_{0}}}\vspace{-0.1cm}$ (9)

We also can solve the CoM dynamics (3) by treating it as an initial value
problem:

$\mathbf{x}=(\mathbf{{x}_{0}}-\bm{\xi}_{0})\hskip
1.42271pt{{\exp}}({{-\omega}}{{t}})+\bm{\xi}_{0}\vspace{-0.1cm}$ (10)

As evident from the above equation, the CoM exhibits stable dynamics, with the
exponential term being negative. However, the DCM exhibits unstable dynamics,
characterized by a positive exponential term. This indicates that the
difference between $\xi_{0}$ and ${cop_{0}}$ increases exponentially over
time. Therefore, a prerequisite for ensuring the stability of the CoM
trajectory is that the DCM trajectory remains stable. As stated in [41], the
concept of DCM provides a relatively straightforward approach for formulating
stability requirements in walking. A simple rule proves effective in ensuring
walking stability: when placing the foot, position the Center of Pressure
(CoP) at a certain distance behind and outward from the DCM at the moment of
foot contact. This distance between the CoP and the DCM during foot placement
is referred to as the DCM offset, and minimizing this distance is crucial for
maintaining viable states.

###  II-C Step Adaptation Controller

This section presents the step adaption mechanism based on DCM dynamics [29,
28]. More precisely, we present below a step adjustment strategy that
optimizes the next step position and timing based on the measured DCM.

To find a DCM trajectory that satisfies both the initial and the final
condition problems, the coefficient $C_{0}$ must equal $C_{f}$. Thus, by
combining (7) and (8), one has:

$\bm{\xi}_{0}-\mathbf{cop}_{0}=\left(\bm{\xi}_{T}-\mathbf{cop}_{0}\right)e^{-\omega{T}}.$
(11)

Now by defining $\sigma=e^{\omega{T}}$ we obtain :

$\bm{\xi}_{T}+\mathbf{cop}_{0}(-1+\sigma)-\bm{\xi}_{0}\sigma=0.\vspace{-0.1cm}$
(12)

Let $\mathbf{cop}_{T}$ represent the CoP position at the start of the next
step, and $\bm{\gamma_{T}}=\bm{\xi_{T}}-\mathbf{cop_{T}}$ denote the DCM
offset for the next step (i.e, the end of this step). Therefore,
straightforward calculations lead to:

$\bm{\gamma_{T}}+\mathbf{cop}_{T}+\left(\mathbf{cop}_{0}-\bm{\xi}_{0}\right)\sigma=\mathbf{cop}_{0}.\vspace{-0.1cm}$
(13)

The step adjustment problem can be formalized as a constrained optimization
problem, wherein the search variables consist of $\gamma_{T}$,
$\mathbf{cop}_{T}$, and $\sigma$, and the cost function is appropriately
defined in a quadratic manner. It’s worth noting that the desired final DCM
position and step timing are dependent on $\gamma_{T}$ and $\sigma$,
respectively. Additionally, $\mathbf{cop}_{T}$ is assumed to be located at the
center of the foot at the start of the next step. Therefore, we can treat this
position as the target for the upcoming footstep placement. The selected cost
function aims to minimize the deviation of the desired gait values from the
nominal ones:

$\begin{split}J&=\alpha_{1}\mathinner{\\!\left\lVert\mathbf{cop}_{T}-{\mathbf{cop}}_{T,nom}\right\rVert}^{2}+\alpha_{2}\mathinner{\\!\left\lVert\bm{\gamma}_{T}-\bm{\gamma}_{nom}\right\rVert}^{2}\\\
&+\alpha_{3}|\sigma-e^{\omega{T_{nom}}}|^{2},\end{split}\vspace{-0.5cm}$ (14)

where $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ are positive numbers and the
next ZMP position $\mathbf{cop}_{T,nom}$, step duration $T_{nom}$ and next DCM
offset $\gamma_{nom}$ are the desired values.

We also present the following set of inequality constraints:

$\begin{bmatrix}I_{2}&0_{2\times 1}&0_{2}\\\ -I_{2}&0_{2\times 1}&0_{2}\\\
0_{1\times 2}&I_{1}&0_{1\times 2}\\\ 0_{1\times 2}&-I_{1}&0_{1\times
2}\end{bmatrix}\begin{bmatrix}\mathbf{cop}_{T}\\\ \sigma\\\
\bm{\gamma}_{T}\end{bmatrix}\leq\begin{bmatrix}\mathbf{cop}_{T,max}\\\
-\mathbf{cop}_{T,min}\\\ \sigma_{max}\\\ -\sigma_{min}\\\
\end{bmatrix},\vspace{-0.3cm}$ (15)

Here, $\mathbf{cop}_{T,max}$ and $\mathbf{cop}_{T,min}$ are in
$\mathbb{R}^{2}$, while $\sigma_{max}$ and $\sigma_{min}$ belong to
$\mathbb{R}$. These inequality constraints are established considering the
constraints imposed by leg kinematics on the maximum step length and by the
maximum achievable velocity on the minimum step duration. Lastly, the
relationship described in (13) is considered as an equality constraint. Due to
the quadratic and linear dependence of the cost function and constraints on
the unknown variables, the entire framework can be formulated as a Quadratic
Programming (QP) problem. At each control cycle, the QP problem is solved by
substituting $\xi_{0}$ with the current DCM position and dynamically shrinking
the single support duration as the robot executes the step. It is worth noting
that during push recovery with a human-exoskeleton, the controller attempts to
minimize the DCM offset. However, at the end of the recovery step, the
capturability constraint for stopping movement, which mandates a DCM offset of
zero, is managed with the use of the exoskeleton, employing an ankle strategy.
This is consistent with the concept of an assisting mode where humans and
robots collaborate to maintain balance.

###  II-D Control Strategy Implementation

####  II-D1 autonomyo exoskeleton

The LLE autonomyo is developed to partially assist people who have walking
impairments due to neuromuscular deficits [42]. The second prototype of the
autonomyo exoskeleton (autonomyo v2) has 6 degrees of freedom (DoF), 3 active
and 3 passive per leg, as in the first design. The 3 active DoF are on the hip
(abd./add. and flex./ext.) and knee (flex./ext.) joints as shown in Fig. 1.
The remaining passive DoF are at the ankle joint (eversion/inversion,
dorsiflex./plantar flex., and abd./add.).

For hip and knee flex./ext. actuation, each unit consists of a brushless motor
(BP4, Faulhaber AG, Switzerland) and a corresponding gearbox (42GPT, Faulhaber
AG, Switzerland) with a 108:1 transmission ratio together with an integrated
torque sensor at the actuator side. An additional cable transmission (2.6:1)
is utilized for hip and knee flex./ext. actuation, while a ball screw
transmission is used for hip abd./add..

The actuators, batteries, and electronics of the system are mainly placed in
the back modules. The exoskeleton has three interfaces at the foot, the shank,
and the trunk for the physical connection to the user. The weight of the
device is about 20 kg including the batteries. The size of the lower body
segments is adjustable according to user-specific measurements. The
controllers are implemented on the embedded computer of autonomyo v2
(BeagleBone Black, Texas Instruments, USA). Wireless communication with
autonomyo v2 is established through a Wi-Fi module.

####  II-D2 Center of Mass Estimation

The CoM position is estimated based on the trunk roll and pitch angles that
are reconstructed through the accelerometer and gyroscope data collected
through the MPU6050 IMU module on the back module of the exoskeleton. The CoM
is assumed at a constant height that has been in the same place as the sensor
placement. The exoskeleton is considered as a rigid leg with a fixed length
during the stance pose and for small angles.

####  II-D3 Balance Loss Detection

For effective balance assistance, real-time assessment of balance and timely
detection of upcoming balance loss are paramount. Posturography, a reliable
method for objectively quantifying postural sway and balance control,
traditionally relies on force plate measurements to assess ground reaction
forces [43]. A key metric in this assessment is the sway area, the $95\%$
confidence ellipse around the mean postural sway in both anteroposterior and
mediolateral directions [44].

Since the postural sway is based on the movement of the body’s CoM [45], to
set the limits for detecting these perturbations, we were inspired by the
elliptical shape often used to represent the sway area. This choice allows us
to effectively define boundaries within which DCM dynamics can be considered
normal or indicative of a perturbation event.

Figure 3: Comparison of foot trajectory data collected during walking to the
designed foot trajectory Figure 4: The overview of the controller scheme. The
step planner is activated when a recovery step is required based on the
balance loss detector.

####  II-D4 Inverse Kinematics of autonomyo

Each side of the exoskeleton is modeled as a three-degree-of-freedom robotic
arm considering the foot is an end-effector and the trunk is the ground as
shown in Fig. 1. The desired foot trajectory is denoted as ${x,y,z}$, where
$l_{0}$ and $l_{1}$ are the distances between the middle of the trunk
connection to the center of rotation of the hip abd./add. joint, between the
center of rotations of the hip abd./add. and hip flex./ext. joints. The thigh
length is shown as $l_{2}$, and $l_{3}$ is the shank length.

A geometric approach is followed to obtain an analytical equation for the
inverse kinematics. Since the desired trajectory for the foot is designed
${x,y,z}$, through the inverse kinematics hip abd./add., hip flex./ext., and
knee flex./ext. joint trajectories are generated by using Eqn. 16.

$\begin{split}\theta_{1}&=\frac{\pi}{2}+\text{arctan}(\frac{y}{z})-\text{arctan}(\frac{\sqrt{r^{2}-d^{2}}}{d})\\\
\theta_{3}&=\text{arctan}(\frac{-D}{\sqrt{1-D^{2}}})\\\
\theta_{2}&=\text{arctan}(\frac{\sqrt{r^{2}-d^{2}}}{x})-\text{arctan}(\frac{{l_{2}}+{l_{3}}\cos\theta_{3}}{{l_{3}}\sin\theta_{3}})\end{split}$
(16)

where $r^{2}=y^{2}+z^{2}$ and
$D=\frac{r^{2}-{l_{1}}^{2}+x^{2}-{l_{2}}^{2}-{l_{3}}^{2}}{2{l_{2}}{l_{3}}}$.The
$\theta_{1}$, $\theta_{2}$, and $\theta_{3}$ are the hip abd./add., hip
flex./ext., and knee flex./ext. joint angles, respectively.

####  II-D5 Foot Trajectory Design

Once the optimization problem described in Sec.II-C is solved, the desired
foot placement is obtained. From the initial point to the final point, a
trajectory is required to perform the desired action. The foot trajectory is
designed based on collected data on the human foot position on the vertical
axis. The data is collected while walking on a treadmill with an IMU-based
motion capture system, Xsens [46]. Based on this data fifth-order splines are
generated to follow these trajectories where the peak occurs at the $40\%$
percent of the total duration with a peak foot height of $0.07m$. The
comparison with collected data to the designed trajectory for a motion of $1s$
is illustrated in Fig. 3.

The speed and acceleration of the start and end of each joint are taken as
zero. For the in-plane motion, the initial position is taken as zero where the
final step position is given by the CoP position as the result of the
optimization problem.

Figure 5: The tested optimization weight vectors ($w_{1}$, $w_{2}$, $w_{3}$)
resulted in step durations and next step positions.
Figure 6: (A) An illustration of the conducted experiment where on the left
subject is standing straight and on the right the subject is landed on the
recovery step position. (B) Example of the angle difference between measured
CoP position and the computed step positions during zero-torque control mode
experiments. (C) The experimental results of three subjects show the computed
angle differences between the actual recovery center of pressure positions and
the computed center of pressure positions next during push recovery in zero-
torque control mode when the exoskeleton was not applying assistive torques in
the desired trajectory.

####  II-D6 Impedance Controller

An impedance controller strategy is selected for the implementation of
recovery stepping since it provides safe and intuitive assistance during
balance recovery in cooperation with the user. As suggested in [23], open-loop
assistance is provided, where the exoskeleton only assists when the balance
loss is detected. To perform the recovery step, a virtual spring is rendered
around the desired trajectory of the joint angles.

The controller architecture for the hip flex./ext. and knee/flex./ext. is
shown in Fig.4 where the impedance control loop is followed by the inner
P-torque control loop. Spring stiffness of impedance controller is selected as
[1.5-0.4-0.4] Nm/deg for hip abd./add., and hip and knee flex./ext.,
respectively. Since there is no torque sensor placed at the hip abd./add.
joint, torque is controlled over the motor current.

When there is no need to take a recovery step or once the stepping is
finished, the exoskeleton is in a zero-torque control mode where the spring
stiffness of the impedance controller is set to zero to interfere with the
movement of the subject minimally.

##  III Experimental Results

###  III-A Effect of Optimization Parameters on Recovery Step

The constrained optimization problem from Sec. II-C is solved using the open-
source software qpOASES [47]. To be able to generate more natural recovery
steps in terms of step duration and next CoP position, several distinct weight
combinations are tested as $w_{1}$, $w_{2}$, and $w_{3}$ as shown in Fig. 5.
While trying to minimize the DCM offset, to be able to cooperate with the
exoskeleton user, a set that resulted in shorter step lengths with longer
durations, $w_{3}$ is selected.

###  III-B Real-World Experiments

Three healthy subjects with the age of 27 $\pm$ 2.65 years, a height of 168.3
$\pm$ 3.78 cm, and a body mass of 67.67 $\pm$ 5.51 kg were asked to perform
the trial. The experimental scenario included two different tests one with
zero-torque mode and one with assistive mode. In the zero-torque mode, the
exoskeleton can be perceived as in transparent mode where the users should
ideally limited by the passive dynamics of the system. In the assistive mode,
the calculated torques are provided based on the controller scheme explained
in Sec. II-D.

During the experiments, participants stood upright on a measurement grid,
facing towards the large surround. We instructed subjects to maintain a
standing balance throughout each trial. The subjects are pushed from various
directions from their back as depicted in Fig.6-(A). If the subject was taking
a step to recover from the external push, the step is logged and recorded on
the grid manually in a simultaneous manner. If the subject was using another
recovery strategy the step is not recorded. Also, if the found solution was
not in the direction of the external perturbation due to incoherent CoM
movements by the resistance of the user, the steps are not considered.

Although the developed recovery stepping strategy is implemented
omnidirectional, it has been tested in the forward direction since fear of
falling backward could potentially lead to inaccurate CoM changes [48]. Also,
with our push strategy since the subjects were more prone to use the loading-
unloading balance strategy side steps were not tested.

####  III-B1 Experiments in Zero-torque Mode

Our push recovery experiments were conducted under zero-torque control mode,
with the aim of evaluating the intuitiveness of the derived trajectories and
analyzing the extent of symbiotic interaction between the exoskeleton and the
user’s intentions. For each subject 5 recovery steps are taken into account
per each side as left and right. During these recovery steps in response to
perturbations, we compared the measured step positions with the computed step
positions, employing angle differences as a primary metric. An illustrative
example of this comparison is presented in Fig. 6-(B), with the swing leg
serving as the reference point for the drawn lines. The results of this
analysis are presented in Fig. 6-(C).

Figure 7: The desired torques are depicted for right hip flex./ext. and knee
flex./ext joints after the impedance controller during the experiments

####  III-B2 Experiments with the Controller

We present the joint trajectories, in both the zero-torque control mode and
the balance recovery assistance achieved through the impedance controller in
Fig. 8. The desired torques for hip flex./ext. and knee flex./ext joints are
depicted for the right side as an example of the contribution of the
exoskeleton in Fig. 7.

Figure 8: The experimental results are presented with joint trajectories where
the desired positions are computed via inverse kinematics of the found step
position and measured trajectories are from exoskeleton joints. Joint
trajectories for zero-torque control mode and impedance controller are given
for the right side.

##  IV Discussion

In Fig. 5, we present the outcomes of our systematic exploration of weight
parameter combinations within the optimization problem. These experiments
highlight our capacity to modulate step durations and step lengths
effectively. For the real-world experiments, we selected a specific weight
configuration, denoted as $w_{3}$, which yielded a stepping pattern
characterized by shorter step lengths and longer durations compared to other
weight settings. It’s worth emphasizing that these weight values are not
rigidly predetermined. Instead, they should be tuned to individual users’
needs, physical conditions, and personal preferences.

The results of the initial set of experiments conducted under zero-torque
control are presented in Fig. 6-(C), showing that the mean difference in the
angle of landed position to computed recovery position is in between
[$-5\degree-10\degree$]. It’s important to acknowledge that that the computed
position is based on the simplified kinematic model of the exoskeleton and
simple CoM position estimation while the measured position is based on real-
world experimental data. Given the inherent inaccuracies in the modeling and
CoM estimation, this comparison may not provide a fair assessment of the
computed position’s accuracy. Nevertheless, these results provide valuable
insights into the effectiveness of our approach in aligning the exoskeleton’s
contribution with user intentions.

For the same experiment the desired and measured joint trajectories as
presented in Fig. 8, reveal our success in achieving this objective, as
evidenced by the shape of the trajectories that closely resemble those of
human trajectories. These findings underscore the potential of our approach to
facilitate intuitive and natural interactions between users and the
exoskeleton. Note that certain deviations exist in the maximum values and peak
timing of hip and knee joint flex./ext. when compared to the computed
trajectories. Consequently, these results suggest improvement in the design of
foot trajectories, ensuring their alignment with the specific dynamics of
recovery steps, rather than relying on walking trajectories as employed in
this study.

For the experiments, in assistance mode our results show that with the
exoskeleton’s assistance, the trajectories become notably smoother and exhibit
reduced oscillations with a reduction in the standard deviation of the
trajectory as shown in Fig. 8. The application of assistive torques depicted
in Fig. 7 highlights the exoskeleton’s active contribution to trajectory
control. These improvements show the exoskeleton’s ability to enhance a more
seamless recovery stepping.

Meanwhile, it should also be noted that since in the suggested approach
assistance is only applied during stepping, there is a steady-state error in
the final pose. Also, due to the nature of the impedance controller, the user
is allowed to deviate from the desired trajectory. The controller gain should
be adaptive based on the need for the assistance of the user to be able to
supplement the required assistance.

In summary, real-world experiments present the potential of our approach to
align exoskeleton behavior with user intentions, enhance trajectory precision,
and contribute to more synergistic recovery stepping. Nevertheless, ongoing
research and refinement are essential to address identified limitations and
further optimize the integration of exoskeleton assistance for various users
and conditions.

##  V Conclusion

In this study, we introduce a push recovery framework for collaborative human-
exoskeleton systems, which relies on a stepping strategy. We formulate the
stepping strategy as an online optimization problem aimed at determining the
optimal step position and duration, allowing for the recovery of balance under
severe external disturbances. Our results demonstrate that the proposed
framework consistently generates omnidirectional stride parameters that align
closely with those of a human subject. We believe that this paper represents a
useful starting point for implementing collaborative push recovery of
synergistic cooperation between exoskeletons and humans.

Future endeavors should focus on the assessment of user effort and explore
conditions conducive to the application of reactive stepping during various
daily living activities. Moreover, systematic evaluations involving healthy
control groups and individuals with balance deficiencies are warranted to
refine our approach. Incorporating hip and ankle strategies in subsequent
studies holds the potential to improve the versatility of the approach. Apart
from its potential for standing balance, the suggested strategy could be
combined with balance assistance during gait. Furthermore, Incorporating deep
reinforcement learning and central pattern generators into the current
framework can enhance push recovery during locomotion and promote consistent
behavior across various scenarios and individuals [49, 50, 51].

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