# Learning Joint Space Reference Manifold
for Reliable Physical Assistance

Amirreza Razmjoo1,2, Tilen Brecelj3, Kristina Savevska3, Aleš Ude3, Tadej
Petrič3, and Sylvain Calinon1,2 *This work was supported by the SWITCH project
(https://switch-project.github.io/), funded by the Swiss National Science
Foundation and Slovenian Research Agency with grant number N2-0153.1 Idiap
Research Institute, CH<EMAIL_ADDRESS>École Polytechnique Fédérale
de Lausanne (EPFL), CH3 Jožef Stefan Institute, SI<EMAIL_ADDRESS>
###### Abstract

This paper presents a study on the use of the Talos humanoid robot for
performing assistive sit-to-stand or stand-to-sit tasks. In such tasks, the
human exerts a large amount of force (100–200 N) within a very short time (2–8
s), posing significant challenges in terms of human unpredictability and robot
stability control. To address these challenges, we propose an approach for
finding a spatial reference for the robot, which allows the robot to move
according to the force exerted by the human and control its stability during
the task. Specifically, we focus on the problem of finding a 1D manifold for
the robot, while assuming a simple controller to guide its movement on this
manifold. To achieve this, we use a functional representation to parameterize
the manifold and solve an optimization problem that takes into account the
robot’s stability and the unpredictability of human behavior. We demonstrate
the effectiveness of our approach through simulations and experiments with the
Talos robot, showing robustness and adaptability.

##  I INTRODUCTION

The integration of robots into our daily lives is becoming increasingly
feasible, and their presence in society is no longer a distant possibility.
This possesses the capability to provide different forms of assistance to
individuals in their daily lives, including social [1], collaborative [2], and
physical support [3]. The current study specifically focuses on physical
assistance, in which a humanoid robot, Talos, aids users in performing sit-to-
stand or stand-to-sit (STS) tasks.

Previous research in STS tasks has mainly concentrated on utilizing
optimization techniques to estimate optimal assistance, with the aim of
solving the problem offline [4, 5, 6]. The experimental setup utilized in
these studies involved low-dimensional, fully actuated systems, allowing for
the admissible implementation of pre-planned assistance in the reproduction
phase. In this scenario, the human acts as a follower system, adapting his/her
behavior to the pre-planned actions of the robot. However, this paper aims to
achieve a more human-like STS task, where the robot plays the role of a
follower and reacts to the human’s actions while considering different
constraints. A similar study to our work is the one conducted by Shimon et al.
[7], where a ballbot was used to assist a human in an STS task by controlling
its leaning angle based on the force received from the human.

|
---|---
a) Human-human interaction | b) Dynamic model
|
c) Simulation | d) Real robot

Figure 1: Experimental setup. (a) Force and human articulation position data
collection during human-human interaction (only force data is utilized in this
paper), (b) Robot’s dynamic model, (c) Simulation setup where the human is
modeled as an external force, and (d) human-robot interaction scenario.

Humanoid robots hold a distinct position among various types of robots, as
their physical resemblance to humans has fueled aspirations that humanoid
robots may one day perform tasks at a level comparable to humans. However,
these robots currently face numerous challenges, including underactuated
dynamics, high dimensionality, and unilateral contact with the environment.
Overcoming these research challenges has led to the development of reduced-
order models such as the linear inverted pendulum mode (LIPM) [8] or
centroidal dynamics [9], to reduce the dimensionality of the system.
Additionally, different strategies such as hip or ankle strategies have been
developed to recover the robot from unstable states [10]. Optimization-based
methods have been employed to consider the locomotion of the robot while
planning for the unactuated states and constraints [9, 11]. Another approach
is to partition the robot into multiple kinematic chains, with some of them
focusing on task performance and others being responsible for balance
maintenance [12].

In the context of physical interactions between humanoid robots and humans,
such as STS tasks, the aforementioned issues to control a humanoid become
significantly more critical. STS tasks are challenging as they require humans
to exert substantial force in a short period. According to data published in
[7] and as demonstrated in this paper, the applied force can range between
100–200 N within 2–8 seconds. Consequently, the robot must respond promptly to
these forces while maintaining its stability and adhering to all constraints.
Optimization-based approaches utilized in other studies are mostly impractical
for this problem as they can be time-consuming and they usually rely on
accurate environmental models that are unfeasible here due to the human
unpredictability.

One approach to limit these issues can be to restrict the robot’s motion to a
manifold, enabling the robot to avoid solving a high-dimensional problem, as
its motion is now constrained to exist solely on the manifold. It also enables
the robot to exclude certain constraints, such as joint limits, from its
consideration, given that these constraints have already been accounted for in
the manifold.

Reduced models, such as centroidal dynamics, can also offer a promising
solution. However, a critical question remains challenging: how to convert
center-of-mass (CoM) motion into joint configurations while maintaining system
stability. Considering the robot and task redundancy, one potential solution
is to utilize methods such as the centroidal momentum matrix presented in [13]
to map the robot’s CoM acceleration back to the joint acceleration by solving
a regression problem. Constraints or other tasks can be considered in the null
space of this problem. Unfortunately, this method cannot guarantee that all
constraints will be met. Furthermore, the regression approach used in these
methods can lead to unintended behaviors if the user behaves unexpectedly.
Restricting the inverse kinematic problem to a manifold can alleviate certain
challenges. Specifically, kinematic constraints become redundant, as the
relevant information is already incorporated within the manifold.
Additionally, this reference can be computed and pre-evaluated offline,
thereby it ensures that the robot’s trajectory remains constrained within the
manifold, regardless of the user’s interactions with the robot.

Motivated by the objective of employing a manifold, this paper tackles the
problem of robot control by addressing a fundamental question of how to find
this manifold for physical STS assistance. Some simplifying assumptions are
made to enable the robot to control its stability by regulating its CoM
exclusively in the anterior/posterior direction, similar to the approach taken
in [7]. This leads to a one-degree-of-freedom problem which motivates us to
find a robust correlation among the redundant robot joint configurations as a
manifold to enable the robot to exhibit robust behavior in the face of
unpredictable human actions. The paper demonstrates the effectiveness of this
approach through simulation experiments involving various human behaviors,
where a real-world robotic experiment is conducted to verify the feasibility
of the proposed method.

##  II BACKGROUND

###  II-A Robot Stability

Different approaches have been proposed to measure the robot stability, such
as the Zero Moment Point (ZMP) [8], which is also known as the Center of
Pressure (CoP), and the Contact Wrench Cone (CWC) [14]. In this work, we focus
on the ZMP, which is the point on the ground where the horizontal moment
applied on the robot is zero. For the 2D robot depicted in Fig. 1, this point
can be calculated as

$x_{\text{ZMP}}=\frac{{\tau}_{\text{f}}}{f_{\text{f2}}}\\!\underset{}{\overset{\text{static
cond.}}{\rightarrow}}\\!x_{\text{ZMP}}=\frac{f_{\text{h1}}x_{\text{h2}}\\!-\\!f_{\text{h2}}x_{\text{h1}}\\!-\\!mgx_{\text{CoM1}}}{-mg-
f_{\text{h2}}},$ (1)

where $g=-9.81\,\frac{\text{m}}{\text{s}^{2}}$ is the gravity acceleration.
According to the data we have received from the human side (Fig. 3),
$f_{\text{h2}}x_{\text{h1}}\ll f_{\text{h1}}x_{\text{h2}}$ and
$f_{\text{h2}}\ll mg$, so (1) can be simplified to

$x_{\text{ZMP}}\approx\frac{f_{\text{h1}}x_{\text{h2}}-mgx_{\text{CoM1}}}{-mg}.$
(2)

For the robot to be stable, $x_{\text{zmp}}$ should be located inside the
support polygon. For the 2D case, this support polygon can be denoted as
$\delta^{-}\leq x_{\text{ZMP}}\leq\delta^{+},$ where $\delta^{\pm}$ correspond
to the two ends of the foot location on the ground.

###  II-B Functional Representation

Parameterizing a trajectory as a functional representation, such as motion
primitives, has become a widely used approach in the field of Learning from
Demonstration (LfD), as evidenced by the popularity of methods such as
Dynamical Movement Primitives (DMP) [15] or Probabilistic Movement Primitives
(ProMP) [16], see [17] for an overview of using basis functions for motion
parameterization. This technique can reduce the number of parameters to
describe a movement while ensuring its smoothness, which is often difficult to
achieve otherwise. Due to these advantages, they have also proven to be useful
in optimal control problem formulation [5].

Consider a $d$-dimensional variable $\bm{q}(s)$ on a 1D manifold, where
$s\in[0,1]$ is a mapping variable that indicates the relative position of the
point on the manifold. It is assumed that each element $q_{i}(s)$ of
$\bm{q}(s)$ where $i\in\\{1,2,\ldots,d\\}$ can be expressed as a weighted
combination of $n$ basis functions $q_{i}(s)=\bm{\phi}(s)\bm{w}_{i}$, where
$\bm{\phi}(s)\in\mathbb{R}^{1\times n}$ is the vector of basis functions at
the location $s$, and $\bm{w}_{i}$ is the corresponding weighting vector.

By concatenating all dimensions, we can obtain a similar formulation
$\bm{q}(s)=\bm{\Psi}(s)\bm{w}$ with $\bm{\Psi}(s)=\bm{\phi}(s)\otimes\bm{C}$,
where $\otimes$ is the Kronecker product, $\bm{C}\in\mathbb{R}^{d\times r}$ is
a coordination matrix, and $\bm{w}$ is the concatenated weighting vector.
Often, $\bm{C}=\bm{I}_{d}$ is used as coordination matrix to decouple the $d$
dimensions. However, alternative selections can be made to establish
correlations among the variables.

Another approach to describe a manifold is to represent this with $K$ discrete
points uniformly spread on the manifold. Compared to this method, the
functional representation approach not only reduces the dimensionality of the
path from $d\times K$ to $r\times n$ (where $n\ll K$ and $r\leq d$) but also
ensures smoothness due to the continuity of the basis functions. In this
paper, Bernstein polynomials [18] were chosen as basis functions (also used
for Bézier curves), although other alternatives such as Radial Basis Functions
(RBFs) can also be used.

##  III Methodology

In this section, we introduce a set of assumptions to simplify the problem,
followed by our objective to construct a manifold. The manifold can be
regarded as a lookup table whose smoothness implementation on the actual robot
is ensured by exploiting the functional representation idea.

###  III-A Assumptions

To simplify the problem, several assumptions are made:

  1. 1.

The robot will not move its feet during the task, meaning that no stepping
action is required.

  2. 2.

The two feet of the robot are located in the same position when viewed from
the sagittal plane and the forces applied to the robot’s hands are identical,
resulting in symmetrical behavior on the left and right sides of the robot.
This correlation can be formulated through the coordination matrix $\bm{C}$
presented in Sec. II-B.

  3. 3.

We control only the ankle, knee, hip, shoulder (3 DoF), and elbow joints,
which are referred to as active joints
$\bm{q}^{\text{active}}\in\mathcal{R}^{14}$ hereafter.

  4. 4.

The robot can apply sufficient effort to achieve a desired behavior.

The first two assumptions allow the problem to be simplified by controlling
only the robot CoM in the anterior/posterior direction, similarly as in [7].
The fourth assumption allows the problem to not consider the actuator limits
during the planning phase which is a common assumption when working with
humanoids [19, 9].

###  III-B Deriving a Spatial Reference

The problem addressed in this paper can be formulated as finding a suitable
configuration for the humanoid robot to maintain its balance while being
subjected to external forces from a human in a collaborative task, namely

$\begin{gathered}\ddot{\bm{Q}}^{*}=[\ddot{\bm{q}}^{*}_{0},\ddot{\bm{q}}^{*}_{1},\ldots,\ddot{\bm{q}}^{*}_{T}]=\underset{\ddot{\bm{Q}}}{\arg\min}\sum_{t=0}^{t=T}c_{t}(\bm{q}_{t},\bm{f}_{t},\bm{u}_{t}),\\\
\text{s.t.}\ \bm{q}(0)=\bm{q}_{0},\ \bm{q}_{t}\in\mathcal{Q}^{\text{active}},\
\bm{u}_{t}\in\mathcal{U}^{\text{active}},\\\ \
\text{DY}(\bm{q}_{t},\dot{\bm{q}}_{t},\ddot{\bm{q}},\bm{f}_{t},\bm{u}_{t})=0,\
g(\bm{q}_{t},\dot{\bm{q}}_{t},\ddot{\bm{q}},\bm{f}_{t},\bm{u}_{t})\leq 0\\\
\text{ZMP}(\bm{q}_{t},\dot{\bm{q}}_{t},\ddot{\bm{q}},\bm{f}_{t},\bm{u}_{t})\in\delta^{\pm},\
\ \ \forall t\in\\{0,\ldots,T\\},\end{gathered}$ (3)

where for each time step $t$, the cost function is defined as $c_{t}$, the
force applied at the robot’s hands is denoted as $\bm{f}_{t}$, and the control
command as $\bm{u}_{t}$. The dynamic equation of the system is represented by
the function $\text{DY}(\cdot)$, while the ZMP point of the system is
calculated using the function $\text{ZMP}(\cdot)$. Other constraints that may
be necessary for the task are incorporated through the function $g(\cdot)$.
The feasible set for active joint angles and their control commands are
denoted by $\mathcal{Q}^{\text{active}}$ and $\mathcal{U}^{\text{active}}$,
respectively.

To incorporate feedback from the environment or to adapt to different humans,
the problem represented by (3) must be solved recurrently using a method such
as model predictive controller (MPC). However, this approach does not seem
suitable for pHRI tasks as the environmental model can vary significantly at
each time step. Instead of solving (3) for the whole trajectory, our focus is
to find a desired terminal configuration $\bm{q}_{T}$ at each time, which is
expected to be static at the end of the task, i.e.,
$\dot{\bm{q}}_{T}=\ddot{\bm{q}}_{T}=\bm{0}$. Therefore, we can assume static
motion and use the formula derived for the ZMP (2), which allows us to define
the ZMP as a function of the current configuration $\bm{q}$ and the horizontal
external force applied by the user $f_{\text{h1}}$, namely

$\begin{gathered}\bm{q}^{*}_{T}=\underset{\bm{q}_{T}}{\arg\min}\ \
c(\bm{q}_{T},f_{\text{h1}}),\\\ \text{s.t.}\ \
\bm{q}_{T}\in\mathcal{Q}^{\text{active}},\
\text{ZMP}(\bm{q}_{T},f_{\text{h1}})\in\delta^{\pm},\
g(\bm{q}_{T},f_{\text{h1}},f_{\text{h2}})\leq 0.\end{gathered}$ (4)

For the sake of enhancing readability, the notation of $T$ is omitted
hereafter. The cost function $c$ should be defined to consider the goal of the
task which is the stability of the robot. One potential approach for realizing
this objective involves defining the cost function as

$c_{\text{std}}(\bm{q},f_{\text{h1}})=\|\text{ZMP}(\bm{q},f_{\text{h1}})\|^{2}.$
(5)

However, due to the unpredictable nature of human behavior, the desired
configuration should be suitable for a wide range of possible force inputs.
Considering this unpredictability seems to be a very crucial point that is
usually overlooked in previous research [4, 7] due to their low-dimensional or
fully actuated robotic systems. Therefore, an alternative approach to defining
the cost function is

$c_{\text{rob}}(\bm{q})=\sum_{p=0}^{p=f_{\text{max}}}{\Big{\|}\text{ReLU}\big{(}\text{ZMP}(\bm{q},p)-\delta^{+}\big{)}\Big{\|}}^{2}\\\
+{\Big{\|}\text{ReLU}\big{(}\delta^{-}-\text{ZMP}(\bm{q},p)\big{)}\Big{\|}}^{2},$
(6)

where $f_{\text{max}}$ is the maximum force that we expect to receive from the
user. This cost penalizes the system quadratically according to how much it
has violated the boundaries.

###  III-C Functional Representations

Combining (6) and (3) will help us obtain a robust configuration of the robot
that specifies where the robot should be according to different forces (e.g.,
using a lookup table). However, solving this problem in its current form can
have some drawbacks. For example, we need to solve (3) for each value of
$f_{\text{h1}}$, which can result in a jump between different solutions, see
Fig. 2.a. The aforementioned behavior can potentially influence the quality of
human-robot interaction, in addition to affecting the sensory information,
particularly the force values, if the robot experiences shaking during its
operation. To overcome this limitation, we deploy the idea of a functional
representation as it can correlate the solution for different force values.
Specifically, we divide the problem into two parts by introducing a new
intermediate variable $s\in[0,1]$ and assuming that the desired configuration
is composed of some basis functions that are functions of $s$ such that

|
---|---
a) Elbow joint angles | b) ZMP point

Figure 2: Exploring the impact of diverse steps on varying force magnitudes.

$\bm{q}(s)=\bm{\Psi}(s)\bm{w},\ \text{with}\ s=\ell(f_{\text{h1}}),$ (7)

where a function $\ell(\cdot)$ maps the force to the value of $s$. In this
formulation, $\bm{q}$ can be regarded as a spatial reference, while the
mapping variable $s$ can be regarded as a latent variable that determines the
position along the reference. In this work, we assume a simple linear
relationship between $s$ and the applied force, i.e.,
$s=\frac{f_{\text{h1}}}{f_{\text{max}}}$, and leave it for future studies to
develop an enhanced controller that operates effectively on this one-
dimensional manifold. Now, the problem consists of finding the desired
coefficients $\bm{w}$ for the entire task, which can be solved by minimizing
the optimization problem

$\begin{gathered}\bm{w}^{*}=\underset{\bm{w}}{\text{argmin}}\sum_{i=0}^{i=D}c_{\text{rob}}(\bm{q}(s_{i})),\\\
\text{s.t.}\ \ \bm{q}(s_{i})\in\mathcal{Q}^{\text{active}},\ \
\text{ZMP}(\bm{q}(s_{i}),f_{\text{h1}}(s_{i}))\in\delta^{\pm},\\\
f_{\text{h1}}(s_{i})=sf_{\text{max}},\ \ \ g(\bm{q}(s),f_{\text{h1}}(s))\leq
0,\end{gathered}$ (8)

where $s_{i}=\frac{i}{D}$ for $i\in\\{0,1,2,\ldots,D\\}$. The summation in (8)
is going to be solved for $D+1$ discrete points on the manifold.

##  IV Experiment

This section analyses the efficacy of the proposed method through both
simulation and real-world experimentation by conducting two experiments. The
first experiment, which we refer to as the _robust case_ , has utilized
$c_{\text{rob}}$ to find a robust solution for a range of force values, while
the second experiment considers only the current force values, i.e.,
$c=c_{\text{std}}$, and is referred to as the _standard case_. The videos
presenting all the experiments are accessible through the supplementary
materials or via https://www.youtube.com/watch?v=GQAad6GFPlE

###  IV-A Finding the Spatial Reference

The initial step in this study is to solve (8) for the robotic system. Based
on the data gathered from human-human interaction (Fig. 3), it has been
determined that a force of approximately $90\,\text{N}$ is required at each
hand to enable standing up. Thus, the robot must be capable of withstanding
forces up to $180\,\text{N}$ to replicate the motion. In consideration of
safety concerns, a maximum value of $f_{\text{max}}=200\,\text{N}$ has been
selected. Although the ZMP for Talos can fall within a range of $\pm
10\,\text{cm}$ around the foot, a conservative margin of $\delta=5\,\text{cm}$
has been incorporated to account for unmodeled factors such as friction and to
accommodate the simplifying assumptions utilized in this study. Additionally,
to ensure safety, the system has been constrained to not exceed a hand
position displacement of $10\,\text{cm}$ from its default configuration.

Bézier curves with 11 control points are utilized in this study (number
selected empirically). Moreover, to exploit the symmetrical characteristics of
the 14 active joints, the coordination matrix $\bm{C}$ has been defined as

$\bm{C}_{14\times 7}=\begin{bmatrix}1&1&0&0&\ldots&0&0\\\
0&0&1&1&\ldots&0&0\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&\ldots&1&1\end{bmatrix}^{\scriptscriptstyle\top}.$ (9)

We have solved (8) offline by discretizing $s$ into 20 equally spaced points.
The resulting elbow joint and ZMP values are illustrated in Fig. 2. We have
used the SLSQP optimization method, available in the SciPy package [20] for
solving the optimization problem.

Figure 3: The force that the user applied at one of his/her hands when
interacting with another user.

###  IV-B Results

####  IV-B1 Simulation

The simulation was executed using a ROS interface on the Docker file provided
by PAL Robotics. Gazebo was utilized for physical simulation, with a PD
position controller applied to the robot’s motor current. It should be noted
that the method described here is not restricted to a position controller and
that alternative controllers could also be utilized. The control system reads
the force values applied to the robot, calculates the desired latent position
on the manifold and its corresponding active joint configurations, and
publishes the desired values at 20 Hz.

To replicate the human behaviors in the simulation, a sinusoidal force was
applied for $2h$ seconds, reaching its maximum value of
$M\in[100,200]\,\text{N}$ within $h\in[1,4]$ seconds. The results of this
simulation are presented in Fig. 4. The findings suggest that the robust
version of the system demonstrates a significantly higher success rate in
handling the challenges inherent to the STS task. It is worth noting that the
spatial reference generated for this experiment assumed static motion, which
may not hold true during the experiment. This underlying assumption may have
contributed to the system failure in certain instances. However, the system
developed by taking into account the unpredictability of human behavior
demonstrates greater resilience and produces more robust spatial references.

|
---|---
a) Robust | b) Standard

Figure 4: Examining the performance of the two systems operating under
different force magnitudes $M\ [N]$ (columns), and its rising time $h\ [s]$
(rows). The force is applied for $2h$ seconds. Green and gray blocks represent
successful and failure motions, respectively.

####  IV-B2 Real Robot

Following the verification of results in the simulation, a real-world
experiment was conducted to validate the system performance in a practical
setting. Participants were asked to perform the STS task on the robot, and the
same setup was employed as in the simulation, with the only difference being
the replacement of the Gazebo simulator with the real robot.

The force values applied by the participants on the robot are depicted in Fig.
5, indicating that the maximum assumed force was not exceeded. Figure 5 also
illustrates how the ZMP point on the robot varied during the real-world
experiment. Although the controller was derived under the static assumption,
the system attempted to remain within the specified boundary. Nevertheless, in
some instances, the ZMP deviated from the marginal boundary $\delta$ while
remaining within the stable region. Although the control mechanism itself was
not capable of precisely controlling the ZMP point, the manifold derived by
the robust case displayed appropriate resilience in preventing the system from
going beyond the marginal boundary and stable region.

|  |  |
---|---|---|---
|  |  |

Figure 5: The forces users applied at each robot’s hand (the first row), and
their corresponding ZMP change (the second row). Different columns correspond
to different users.

##  V Discussion

This section discusses the effects of the steps taken to solve the STS task
using the proposed method. Without considering the use of a functional
representation, there could be big discontinuities in the joint space, as
shown in Fig. 2.a with jumps up to $0.2$ rad ($\approx 11.46$ degrees) for the
elbow joint. We observed that a functional representation eliminates the jump
in the spatial reference by correlating the optimization solution for all
force values using the location-independent variables $\bm{w}$.

We incorporated human unpredictability into the cost function by optimizing it
over the range of the expected human behavior. The results, as illustrated in
Fig. 2.b, demonstrate that our approach allows the robot to utilize the
marginal boundary more effectively. Specifically, at the beginning of the
task, the robot anticipates the possibility of a force being applied by the
human and adjusts its ZMP point accordingly to provide a larger safety margin.
Conversely, the standard method focuses solely on regulating the ZMP point of
each force around zero. The advantages of this robustness are demonstrated in
Fig. 4. To further enhance this approach, future studies may consider
weighting each potential human behavior by its occurrence probability. This
would enable the system to better adapt to more likely scenarios, and
potentially lead to more personalized and efficient task performance.

The method that we propose has some remaining limitations. Fig. 5 shows that
the ZMP points slightly violate the marginal boundaries $\delta$. This is
because the method does not have direct control over the ZMP point and relies
on the robustness of the manifold. This error can also be related to unmodeled
dynamics such as the friction effect, especially when there is no external
force, where ZMP stays around $-0.07\,\text{cm}$ instead of
$-0.05\,\text{cm}$. In future work, we will investigate adding feedback on the
ZMP point to control it more directly on the manifold. Another idea could be
to solve (3) on the manifold, removing the need to consider some constraints
such as joint limits, and optimizing a one-dimensional variable $\ddot{s}$
over a horizon. The efficiency of these methods should be investigated in
future studies.

One observation from the experimental results is that for the user shown in
the rightmost column in Fig. 5, the estimated ZMP point using a static
assumption differs significantly from the actual one, despite the successful
completion of the task. This phenomenon is also evident in other experiments
(not depicted here), particularly when the user applies forces below 40 N on
each hand (80 N on the robot). Our preliminary hypothesis is that, as shown in
Fig. 2.b when the force is below 40 N (20 N at each hand), the robot’s
movement is concentrated on maintaining the ZMP on the boundary, which
requires agile motion of CoM to compensate the external force. Consequently,
the static assumption is more likely to be violated in this phase. This can
also be seen by comparing the amount of motion in Fig. 2.a at the beginning of
the task versus the remainder of the task. Additionally, since the robot’s
motion is controlled through a position controller, the accuracy of the
calculated ZMP depends on the controller’s precision in tracking the reference
motion, which is expected to be less accurate for agile motions. To address
this issue, we can either modify the underlying controller or incorporate an
online controller. Furthermore, unlike the others, this user has applied force
of comparable or greater magnitude in the vertical direction, as opposed to
the horizontal direction. We initially assumed that the impact of this force
would be negligible compared to the effect of the horizontal force. However,
for this particular user, this assumption may not hold true. This outcome may
be attributed to the non-adaptive initial pose of the robot’s hand, which
might not have been comfortable for certain users, resulting in non-intuitive
force patterns. While this force was insufficient to cause the robot to fall,
we are motivated to consider both forces in our future formulation, which will
result in higher dimensional manifolds.

##  VI CONCLUSIONS

We demonstrated how the concept of the manifold can be leveraged to
effectively utilize the redundancy of a problem to construct an interpretable
low-dimensional latent space for the robot. This manifold can be interpreted
as local coordination among the system, which correlates all of the joints in
accordance with the desired position of the robot on the manifold. We analyzed
the impact of different stages of the proposed method on the final results.
While incorporating the functional representation idea leads to a smooth
manifold, considering the uncertainty associated with human behavior enables
the system to exploit the redundancy in a broader way.

In this study, we assumed a simple controller that maps the force value to the
position of the robot on the manifold and does not address online controller
design. Future work could involve the use of more sophisticated controllers
that would update the desired location based on various parameters such as the
zero moment point. It may be possible to utilize a fast model predictive
controller (MPC) control strategy acting on this 1D manifold. Furthermore, an
area for improvement is optimization under uncertain human behaviors. This can
be achieved through various methods of predicting human behavior and weighting
different behaviors in the cost function according to their likelihood of
occurrence. Moreover, extending the concept of manifold to multidimensional
manifolds may lead to interesting results. Lastly, an intriguing avenue to
pursue would involve jointly optimizing over both the predefined coordination
matrix $\bm{C}$ and the superposition weights $\bm{w}$. Such an approach could
potentially generate even more significant synergies between the joints,
thereby offering additional opportunities for the controller to exploit.

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