11institutetext: Institute of Digitalization in Medicine, Faculty of Medicine
and Medical Center - University of Freiburg, Freiburg, Germany
22institutetext: Clinic of Neurology and Neurophysiology, Medical Centre-
University of Freiburg, Faculty of Medicine, University of Freiburg,
Breisacher Straße 64, 79106, Freiburg im Breisgau
22email: {vittorio.lippi, christoph.maurer, thomas.mergner}@uniklinik-
freiburg.de

# Human-Likeness Indicator for Robot Posture Control and Balance

Vittorio Lippi 1122 0000-0001-5520-8974 Christoph Maurer 22
0000-0001-9050-279X Thomas Mergner 22 0000-0001-7231-164X

###### Abstract

Similarly to humans, humanoid robots require posture control and balance to
walk and interact with the environment. In this work posture control in
perturbed conditions is evaluated as a performance test for humanoid control.
A specific performance indicator is proposed: the score is based on the
comparison between the body sway of the tested humanoid standing on a moving
surface and the sway produced by healthy subjects performing the same
experiment. This approach is here oriented to the evaluation of a human-
likeness. The measure is tested using a humanoid robot in order to demonstrate
a typical usage of the proposed evaluation scheme and an example of how to
improve robot control on the basis of such a performance indicator score.

###### Keywords:

Humanoids Benchmarking Human likeness Posture Control Balance

##  1 INTRODUCTION

###  1.1 Overview

The benchmarking of humanoid performance is gaining interest in the research
community [40, 6, 9, 46, 47, 27, 28]. The performance of a humanoid is a
complex issue that covers several aspects, e.g. sensor fusion, cognitive and
motor functions, mechanics, and energy efficiency. In particular, a recent
European project, EUROBENCH [42, 45, 44] is proposing to implement standard
and repeatable experimental procedures to evaluate and compare the performance
of different robots. This work describes one specific posture control
performance indicator to be implemented within the project, evaluating human-
likeness based on the similarity between human and robot responses to external
disturbances [28, 27, 24].

Posture control is here particularly relevant because falling is one of the
typical reasons for failures for humanoids [4, 5, 13]. This work focuses on
posture control from the point of view of human-likeliness. A formal and
unanimous definition of human-likeness is still missing, although the concept
is relevant both for robotics and neuroscience [46].

Figure 1: Posture control experiment with a moving support surface, a 6-Dof
Stuart Platform. On the left, a human subject with active markers visible on
the back (blue triangular plates) is shown. On the right, the Lucy robot
standing on a different platform that implements the same support surface tilt
profile.

Notwithstanding the difficulty in defining human-like behavior, it is expected
to be superior to the one of state-of-the-art robots [36]. Human-likeness is
envisaged to be associated with some advantages such as mechanical compliance
(also important for safety) and low energy consumption. Both of these features
are typically associated with the low feedback loop gain typical for
biological systems in face of the presence of neural delay. In robotics, the
delay associated with the transport of signals is usually negligible, but the
robustness in face of delays may be important in limiting the required
computational resources needed to implement the real-time control [37]. Often
human-likeness is defined (1) in terms of perception from the point of view of
human observers [49, 38, 1], (2) identified with the presence of a specific
feature, e.g. reproducing human trajectories [21] or exhibiting mechanical
compliance [8], or defined in relation to a set of tasks, e.g. in challenges
as used in the DARPA challenge [31, 10]. The measure proposed here is based on
body kinematics with the aim to provide an indicator of human-likeness that is
repeatable and objective.

The idea of testing robots in the experimental setup used for human posture
control dates was inspired by the necessity of implementing and testing human
posture control models in a “real world” set-up, including factors that are
not usually well accounted for by simulations such as sensor and actuation
noise [14, 25, 32, 34]. Once this approach was defined, posture control
experiments provided a way to compare alternative control systems (bio-
inspired or not) on the same robotic platform [2, 37]. The general
benchmarking issue represents a generalization and a formalization of such
studies. In general, the mutual inspiration between humanoid robotics and
neuroscience is well recognized and considered promising in both the robotics
and neuroscience fields [7].

This article represents an extended version of [24], a conference paper where
the benchmarking principle was tested with simulated experiments. In the
present work, the system is tested with a real humanoid robot. In the
remainder of the Introduction, we describe the proposed benchmarking approach
summarizing the method previously detailed in [24]. In the Method section, the
robotic platform, the control system, and the experimental testbed are
described as well as the set-up used to produce the human data-set. The
Results section will show the scores obtained with different settings in the
control system. The implications of the proposed performance indicator and the
limitations associated with human likeliness in humanoid posture control will
be discussed in the conclusions.

###  1.2 Description of the Approach

The proposed benchmarking measure is based on a data-set of results from human
experiments. The experimental data consist of the body sway of human body
segments induced by an external stimulus, here specifically the tilt of the
support surface in the body sagittal plane. The measured body sway is
characterized as a Frequency Response Function, FRF, i.e. an empirical
transfer function between the stimulus characterized by a Pseudo Random
Ternary Sequence (PRTS) profile [39] and the response computed on specific
frequency points. The data-set has been produced with healthy subjects. The
comparison aims to assess the similarity in the balancing behavior between the
robot and a set of average healthy subjects. Such measure does not make
explicit assumptions about specific advantages of the human behavior, in
contrast to the performance indexes proposed in other studies that identify
some goal like minimum torque or energy consumption in robots [20], or a
specific problem in human subjects [43].

Body sway profiles FRFs have been chosen as a basis for the benchmarking
because such analysis has already been studied in several previous
publications with human subjects. These provide a reference for comparison and
tools for analysis. There are several reasons why the body sway induced by
support surface tilt has been repeatedly used to study human posture control.
As a repeatable stimulus, it can be used to formally characterize the behavior
in terms of responses to a well-defined input. Furthermore, the tilting
support surface requires the balancing mechanism to integrate vestibular input
and proprioception (and vision, with eyes open), and hence it is well apt to
study human sensor fusion mechanisms.

###  1.3 Human Data-Set

A group of 38 young healthy subjects serves as a reference for the human-
likeness criterion. The subjects were presented with a stimulus consisting of
a tilt of the support surface in the body sagittal plane, while the recorded
output was the body sway. The typical set-up is shown in Fig. 1, the body sway
tracking was performed using active markers (Optotrak 3020; Waterloo, ON,
Canada), attached to subjects’ hips and shoulders and the platform. A PC with
custom software was used to generate the support surface tilt in the body-
sagittal plane. The marker positions were recorded at 100 Hz using software
written in LabView (National Instruments; Austin, TX, United States). The
profile used for the stimulus is a pseudorandom ternary signal, PRTS [39]. The
peak-to-peak amplitude was set to $1^{\circ}$. The amplitude is rather small
compared to what a healthy subject can withstand. Usually, in similar studies,
the tilt may go up to 8∘ and more [39, 3]. This was motivated by the aim to
provide a data-set that could be compared safely with elderly subjects and
patients and, in the specific case of robotic benchmarking, can be used to
characterize the behavior of the robot without the risk of making it fall (and
potentially break).

###  1.4 Performance indicator

The performance indicator is here a measure of similarity with human behavior.
The body sway profiles, i.e. the angular sway of the body COM with respect to
the ankle joint, are used to characterize and compare the responses. The
comparison is defined in terms of the norm of the difference between frequency
response functions (FRFs). The PRTS power-spectrum has a profile with peaks at
$f_{peak}$ separated by ranges of frequencies with no power [19, 39, 23]. The
response is evaluated for a specific set of frequencies where the PRTS
spectrum has peaks: $\mathbf{f_{peak}}$=[0.05, 0.15, 0.25, 0.35, 0.45, 0.55,
0.65, 0.75, 0.85, 0.95, 1.05, 1.15, 1.25, 1.35, 1.45, 1.55, 1.65, 1.75, 1.85,
1.95, 2.05, 2.15, 2.25, 2.35, 2.45] Hz. Such a discrete spectrum is then
transformed into a vector of 11 components by averaging the FRF over
neighboring frequencies as illustrated in Fig. 2:

$f_{x(k)}=\frac{\sum_{i\in B_{k}}f_{peak(i)}}{N_{k}}$ (1)

where $k$ and $i$ are the indexes of the components of the frequency vectors,
and $B_{k}$ is a set of $N_{k}$ frequencies averaged to obtain the $k^{th}$
sample. The $B_{k}$ are shown in Fig. 2 as white and pink bands (notice that
the bands are overlapping, light-pink has been used to indicate a zone
belonging to two adjacent bands). Similarly the Fourier transform of the PRTS
$P(f_{x})$ and the Fourier transform of the responses are averaged over the
bands $B_{k}$ before computing the FRFs. The final representation of the FRF
is a function of the 11 frequencies $f_{x}$ =[0.1, 0.3, 0.6, 0.8, 1.1, 1.4,
1.8, 2.2, 2.7, 3.5, 4.4]

In a previous description of the system in [27] the frequency vector had 16
components as proposed in other works using the PRTS, e.g. [39, 12, 23, 22].
In this work, a shorter version of the signal is used, which is considered
safer and less fatiguing for human subjects, sometimes elderly patients, and
convenient for robotics experimenters. Consequently, the discrete Fourier
transform of the signal and the resulting FRFs have fewer components.

The choice of the frequencies in $B_{k}$ and their overlapping follows the
method described in [39], but here is adapted to the 11 frequencies
considered. The rationale behind such a choice was to get a representation
with the frequencies equally spaced on a logarithmic scale, which is often
used in posturography papers to present the FRF, with the overlapping
providing a smoothing effect. The FRF is computed from the 11 components of
the Fourier transform of the input $U$, and the output $Y$ as

$H=\frac{G_{UY}}{G_{U}}$ (2)

where $G_{UY}=U^{*}\odot Y$ and $G_{U}=U^{*}\odot U$ are empirical estimations
of the cross power spectrum and the input power spectrum (“$\odot$” is the
Hadamard, element-wise product). The peaks of the PRTS power-spectrum have
larger values at lower frequencies [19]. This implies a better signal-to-noise
ratio for the first components. A weighting vector $\mathbf{w}$ based on
$P(f_{x})$ is then defined in a similar way to eq. 1, but considering the
power

$w_{k}=\sqrt{\sum_{i\in B_{k}}||P(f_{peak(i)})||^{2}}$ (3)

. The distance between two FRFs is defined and the norm of the difference
weighted by the precision matrix, i.e. the inverse of the covariance matrix
$\Sigma$, computed for the data-set of normal subjects. Before doing this, the
FRF is expanded into a vector with the real and imaginary components as
separated elements, i.e. 22 components. This together with the foretold
weighting leads to the definition of the norm:

$D=\sqrt{S\Delta^{T}\Sigma^{-1}\Delta S}$ (4)

where $S=diag([\mathbf{w},\mathbf{w}])$ is the diagonal matrix representing
the re-weighting due to the power-spectrum, repeated twice to cover the 22
elements, and $\Delta$ is the difference between the two FRFs expanded to 22
components. This approach does not require model identification because it is
performed on the basis of the data. The comparison can be performed between
the tested robot and the average of the groups of humans (healthy or with
defined deficient conditions) or between two single samples to quantify how
much two robots differ from each other. The score of human-likeness is
obtained comparing the sample with the mean of the human sample set
$\mathbf{\mu}$ so that

$\Delta=H-\mathbf{\mu}$ (5)

. The parameters $\mathbf{\mu}$ and $\Sigma$ defining the score are given in
Fig. 2. the score in eq. 4 resembles a Mahalanobis distance

$D=\sqrt{\Delta^{T}\Sigma^{-1}\Delta}$ (6)

, but with the addition of weights. Assuming a joint normal distribution for
the weighted FRFs, $\mathcal{N}(\mu,\Sigma)$, the Mahalanobis distance defines
probability density, where a smaller distance associated with higher
probability density.

Figure 2: Input and parameters used in the definition of the performance
indicator. A Magnitude of the spectrum of the input PRTS, with the typical
“comb” profile alternating peaks to zeros. B Example of the empirical transfer
function from a trial. C 11-component FRF associated with the transfer
function in B. Note that, as the FRF is averaged in the complex domain the
magnitude of the average shown in the plot is not the average of the
magnitudes. The bands on the background show the frequency ranges over which
the spectrum is averaged: white and pink represent ranges associated with
groups of frequencies. The sets of frequencies are overlapping, with light-
pink bands belonging to both the two contiguous groups, and a sample on the
transition between two bands belongs to both the groups. D Weights
representing the values of $P(f_{x})$ in eq. 4. E Inverse of $\Sigma$ from eq.
4. Notice how the values on the diagonal corresponding to the highest
frequencies are associated with high values (less variance in the dataset).
This would make the metrics particularly sensitive to accidental changes in
such components (e.g. due to noise). This is compensated by the weighting
profiles in D, which associate almost zero weight to high-frequency
components. D $\mathbf{\mu}$ from eq. 5.

##  2 MATERIALS AND METHODS

###  2.1 Humanoid Robot

The test is performed with the humanoid robot Lucy, a custom platform designed
to simulate human posture control[41]. Lucy Posturob has 14 degrees of freedom
controlling posture in the frontal and the sagittal plane. The anthropometric
parameters of the robot are shown in Table 1

Body mass | 15.30 | kg
---|---|---
Upper body mass (including pelvis) | 9.50 | kg
Thighs mass | 2.80 | kg
Shanks mass | 3.00 | kg
Total height (from ankle joints) | 1.52 | m
Body COM height (from ankle joints) | 0.68 | m
Shank COM height | 0.32 | m
Thigh COM height | 0.31 | m
Table 1: Robot anthropometrics.

###  2.2 Control System

The robot is controlled using the Disturbance Estimation and Compensation
model, DEC[29, 25]. The DEC model was proposed as a model of human posture
control in steady-state [33]. The DEC exploits human-like sensor-fusion [15,
16, 14] to reconstruct external disturbances having an impact on body posture
and balance, i.e. gravity, support surface tilt, support surface acceleration,
external push. The reconstructed disturbances are compensated using a servo
controller implemented as a PD. The resulting control law for a joint can be
expressed in Laplace domain as

$T_{joint}=(K_{p}+sK_{d})e^{\Delta_{t}s}\left(-\epsilon+\alpha_{grav}+\alpha_{trans}+\alpha_{push}\right)(K_{p}^{pass}+sK_{d}^{pass})\alpha_{joint}$
(7)

Where $T_{joint}$ is the torque applied at a joint, $\alpha_{grav}$,
$\alpha_{trans}$, and$\alpha_{push}$ are angle equivalent of the external
disturbances, i.e. the required torque divided by the proportional gain
$K_{p}$111In the ideal formulation proposed in [32] the controller gain
$K_{p}\cong m\cdot g\cdot h$, where $m$ is the body (or segment) mass, $g$ the
gravity acceleration, and $h$ the height of the COM, is equal or slightly
smaller than the one required to compensate the gravity torque (that is, with
small-angle approximation, $T_{g}=m\cdot g\cdot h\cdot\alpha_{bs}$).Because of
this, the angle equivalent of the gravity torque is the body sway
$\alpha_{bs}$. In general additional gains are imposed on the disturbance
compensations in order to reproduce human behavior (that does not compensate
disturbances perfectly) or to achieve the desired dynamic response on a robot.
Notice that the compensating torque is not exactly equal to the estimated
disturbance because of the derivative component of the servo loop $sK_{d}$..
The support surface tilt is compensated by identifying the support surface
tilt and using it in the definition of the controlled error $\epsilon$. The
specific definition of $\epsilon$ may change depending on which variable is
controlled (e.g. joint angle or COM sway of the above segments). The model is
nonlinear because it includes sensory thresholds [30] and the disturbance
estimators are non-linear functions. The control loop includes a lumped delay
(one for each joint). Specifically the threshold non-linearity is formulated
as:

$\hat{\alpha}_{FS}=\int_{0}^{t}f_{\theta}\left(\dot{\bar{\alpha}}_{FS}\right)$
(8)

where $\dot{\bar{\alpha}}_{FS}$ is the support surface rotation velocity (FS =
foot-in-space)reconstructed from the vestibular and the proprioceptive input
$\hat{\alpha}_{FS}$ is the reconstructed signal used in the servo loop (Fig. 3
A). The function $f_{\theta}$ is the dead-band threshold:

$f_{\theta}(\alpha)=\left\\{\begin{array}[]{llc}\alpha+\theta&if&\alpha<-\theta\\\
0&if&-\theta<\alpha<\theta\\\ \alpha-\theta&if&\alpha>\theta\\\
\end{array}\right.$ (9)

The formulation of the DEC model leads to a modular control system with one
control module for each degree of freedom, based on a servo loop controlled
with a PD. The servo loop concept is established in neurological models [48,
35]. The modularity, in this specific case, was proposed as a generalization
of the two DOF model presented in [14]. Interestingly the resulting
architecture with low-level servo loops resembles state-of-the-art humanoid
control systems, e.g. [18, 17]. The modular control scheme is shown in Fig. 3
B. The control parameters are reported in Table 2

Symbol | Quantity | Value
---|---|---
ANKLE |  |  |
P | proportional gain | 119.57 | Nm/rad
D | derivative gain | 11.95 | Nms/rad
$G_{ext}$ | external torque gain | 0.5 |
$\omega_{ext}$ | External torque filter cutoff frequency | 5 | rad/s
G | loop gain (multiplies everything) | 1.2 |
KNEE |  |  |
P | proportional gain | 55.72 | Nm/rad
D | derivative gain | 0.4458 | Nms/rad
$G_{ext}$ | external torque gain | 0.5 |
$\omega_{ext}$ | External torque filter cutoff frequency | 5 | rad/s
HIP |  |  |
P | proportional gain | 22.71 | Nm/rad
D | derivative gain | 5.67 | Nms/rad
$G_{ext}$ | external torque gain | 0.5 |
$\omega_{ext}$ | External torque filter cutoff frequency | 5 | rad/s
PELVIS |  |  |
P | proportional gain | 10.59 | Nm/rad
D | derivative gain | 0.07 | Nms/rad
$G_{ext}$ | external torque gain | 0 |
$\omega_{ext}$ | External torque filter cutoff frequency | N.A. | rad/s
Table 2: Control parameters for the sagittal plane from[26]. The
$\omega_{ext}$ is the cutoff frequency of a Butterworth filter applied to the
external contact torque (push) feedback. A gain G of 1.2 was applied to the
output of the PD controller, leading to a slightly overcompensation of the
external disturbances. in the here presented experiments this will be modified
in order to obtain different responses. Figure 3: The DEC control system. (A)
The servo loop concept showing the compensation of external disturbances. The
control scheme is presented for the single inverted pendulum (SIP) case, the
generalization to more DOFs is produced with modules controlling each joint as
a SIP. (B) The modular contol system in the sagittal plane. (C) The modelling
in the frontal plane. With fixed feet position and neglecting knee bend, legs,
pelvis and the support surface represent a four bar linkage with one DOF, an
additional DOF is represented by the link between the hip and the trunk. This
is mapped to a DIP model similar to the one used in the description of human
balance from [11]. The torque feedback is computed on the DIP model and then
mapped back to the actuators. (D) Two example positions with the associated
DIP kinematics. (E) The control scheme for the frontal plane with two modules.
The two degrees of freedom represented by the rotation of the legs around the
vertical axis are controlled with a PD feedback (simulating a passive DOF).

###  2.3 Test Protocol

The robot Lucy has been tested five time with different control system
configurations. Specifically the implementation of the DEC from [26] was used
as base and tested, modified version consisted in one without derivative
control of the ankle joints and one with twice the derivative gain. Two
further trials have been performed with a version of the controller with a
lower loop gain (multiplying the output of the PD controller). In one case the
gain was 1 and in the other 0.8, versus the 1.2 from [26]. The body sway used
to compute the human likeness measure was computed using the internal sensors
of the robot. The signal were synchronized alligning the ideal PRTS samples
with the profile of the foot-in-space tilt $\alpha_{FS}$ computed with
internal sensors. The alignment was performed on the basis of maximum auto-
correlation.

##  3 RESULTS

The results obtained, ordered by score are:

Control |  Score |  CDF
---|---|---
No D |  2.9798 |  92.1053 %
Lower gain (1) |  3.1297 |  92.9825 %
Lowest gain (0.8) |  3.3708 |  95.6140 %
Standard DEC[26] |  3.3823 |  95.6140 %
Double D |  3.5646 |  95.6140 %
Table 3: Scores obtained with different control configurations.

The values in the third column give estimates of the cumulative distribution
function (CDF) of the data-set at the given score. It is computed by counting
the fraction of samples with a score smaller than the one produced in the
specific simulation (due to the discrete nature of such measure, two results
with a slightly similar score may have the same CDF). Fig. 4 shows the
obtained scores together with human samples for reference. The Mahalanobis
distance is also plotted as function of the score.

Figure 4: Mahalanobis distance for the samples in the data-set (red dots) as a
function of the metric from eq. 4 and the bootstrap estimated cumulative
distribution of the metric (blue line) and its variance (light blue bands) .
The human-likeness score is smaller than the Mahalanobis distance because if
the coefficients in Fig. 2 D that are $\leq 1$, and in general there is a
spread of the Mahalanobis distances of samples associated with the same D
because the weighting almost removes the variation due to high-frequency
components. The stars represent the results obtained with different
configurations. Figure 5: Energy produced by the actuators of the ankles,
computed indirectly on the basis of torque and joint angle profiles.

##  4 Discussion

The results in Table 3 show how the best score was obtained without derivative
control on the ankle while the largest score was the one with twice the
derivative gain. Loop gains smaller than the base model from [26] led to
better scores although the lowest gain tested (0.8) led to a score that was
similar to the one of the original control. Interestingly the result confirms
the idea that human like behavior is in general relaxed (i.e. slightly under-
compensation of disturbances). The larger Mahlanobis distance exhibited by the
controller with the lowest gain (0.8) and with no D is due to oscillations
produced at frequencies that are cut by the weights in eq. 3. Such
oscillations are induced by the lack of sufficient damping. One may ask, for
example, if the human like behavior is advantageous in terms of energy. An
indirect estimation of the energy used by the ankle actuators during the
trials is shown in Fig. 5. The picture suggests that the most efficient is the
original control system (that was carefully tuned to reduce the sway, in
contrast to the other that were produced modifying D and G with the sole
purpose of creating examples). The lowest gain $G=0.8$ produces less energy
consumption than $G=1$ but more than the original $G=1.2$. This nonlinear
relationship is the result of a trade off between the smaller oscillations
produced by higher gains and the smaller torques associated with smaller
gains. The Example with double derivative gain is the less human like and the
less efficient.

##  5 Conclusions

This work proposed a score for the evaluation of a human-likeness in humanoid
posture control experiments. The experimental set up and the analysis pipeline
has been described. The examples shown how different combinations of
parameters lead to different scores, suggesting that low loop gain results in
more human like FRF profiles. A final consideration on energy consumption by
ankle actuators shown that the most human like behavior is not always the most
efficient. Hence human-likeness should be associated with other measures to
assess the performance of the robot completely. In general human like
movements can be an advantage by themselves in human-robot interaction and for
assistive devices, while in other contexts it can be sacrificed for efficiency
and precision.

##  6 Testbed and code availability

The presented benchmarking performance indicator is par of the project
EUROBENCH https://eurobench2020.eu/.The EUROBENCH framework includes several
tests and performance indicators provided with the aim of testing the
performance of robots at any stage of development.

The code will be updated in the docker hub:
https://hub.docker.com/repository/docker/eurobenchtest/pi_comtest.

In Linux, code can be run without being installed locally, as indicated on the
readme page: https://github.com/eurobench/pi_comtest#docker-based-code-access

##  7 Acknowledgment

|  This work is supported by the project COMTEST, a sub-project of EUROBENCH
(European Robotic Framework for Bipedal Locomotion Benchmarking,
www.eurobench2020.eu) funded by H2020 Topic ICT 27-2017 under grant agreement
number 779963.
---|---

## References

  * [1] Abel, M., Kuz, S., Patel, H.J., Petruck, H., Schlick, C.M., Pellicano, A., Binkofski, F.C.: Gender effects in observation of robotic and humanoid actions. Frontiers in psychology 11, 797 (2020)
  * [2] Alexandrov, A.V., Lippi, V., Mergner, T., Frolov, A.A., Hettich, G., Husek, D.: Human-inspired eigenmovement concept provides coupling-free sensorimotor control in humanoid robot. Front. Neurorobot. 11, 22 (2017)
  * [3] Assländer, L., Hettich, G., Mergner, T.: Visual contribution to human standing balance during support surface tilts. Hum. Mov. Sci. 41, 147–164 (2015)
  * [4] Atkeson, C.G., Babu, B.P.W., Banerjee, N., Berenson, D., Bove, C.P., Cui, X., DeDonato, M., Du, R., Feng, S., Franklin, P., et al.: No falls, no resets: Reliable humanoid behavior in the darpa robotics challenge. In: 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids). pp. 623–630. IEEE (2015)
  * [5] Atkeson, C.G., Benzun, P.B., Banerjee, N., Berenson, D., Bove, C.P., Cui, X., DeDonato, M., Du, R., Feng, S., Franklin, P., et al.: What happened at the DARPA Robotics Challenge finals. In: The DARPA Robotics Challenge Finals: Humanoid Robots to the Rescue, pp. 667–684. Springer (2018)
  * [6] Behnke, S.: Robot competitions-ideal benchmarks for robotics research. In: Proc. of IROS-2006 Workshop on Benchmarks in Robotics Research. Institute of Electrical and Electronics Engineers (IEEE) (2006)
  * [7] Cheng, G., Hyon, S.H., Morimoto, J., Ude, A., Hale, J.G., Colvin, G., Scroggin, W., Jacobsen, S.C.: Cb: A humanoid research platform for exploring neuroscience. Advanced Robotics 21(10), 1097–1114 (2007)
  * [8] Colasanto, L., Van der Noot, N., Ijspeert, A.J.: Bio-inspired walking for humanoid robots using feet with human-like compliance and neuromuscular control. In: 2015 IEEE-RAS 15th international conference on humanoid robots (humanoids). pp. 26–32. IEEE (2015)
  * [9] Conti, R., Giovacchini, F., Saccares, L., Vitiello, N., Pons, J.L., Torricelli, D.: What do people expect from benchmarking of bipedal robots? Preliminary results of the EUROBENCH survey. In: International Symposium on Wearable Robotics. pp. 132–136. Springer (2018)
  * [10] DeDonato, M., Dimitrov, V., Du, R., Giovacchini, R., Knoedler, K., Long, X., Polido, F., Gennert, M.A., Padır, T., Feng, S., et al.: Human-in-the-loop control of a humanoid robot for disaster response: a report from the darpa robotics challenge trials. Journal of Field Robotics 32(2), 275–292 (2015)
  * [11] Goodworth, A.D., Peterka, R.J.: Influence of stance width on frontal plane postural dynamics and coordination in human balance control. Journal of Neurophysiology 104(2), 1103–1118 (2010)
  * [12] Goodworth, A.D., Peterka, R.J.: Identifying mechanisms of stance control: a single stimulus multiple output model-fit approach. Journal of Neuroscience Methods 296, 44–56 (2018)
  * [13] Guizzo, E., Ackerman, E.: The hard lessons of DARPA’s robotics challenge [News]. IEEE Spectrum 52(8), 11–13 (2015)
  * [14] Hettich, G., Assländer, L., Gollhofer, A., Mergner, T.: Human hip—ankle coordination emerging from multisensory feedback control. Hum Mov Sci 37, 123–146 (2014)
  * [15] Hettich, G., Lippi, V., Mergner, T.: Human-like sensor fusion mechanisms in a postural control robot. In: Londral, A.E., Encarnacao, P., Pons, J.L. (eds.) Proceedings of the International Congress on Neurotechnology, Electronics and Informatics. Vilamoura, Portugal. pp. 152–160. (2013)
  * [16] Hettich, G., Lippi, V., Mergner, T.: Human-like sensor fusion implemented in the posture control of a bipedal robot. In: Neurotechnology, Electronics, and Informatics, pp. 29–45. Springer (2015)
  * [17] Hyon, S.H., Osu, R., Otaka, Y.: Integration of multi-level postural balancing on humanoid robots. In: 2009 IEEE international conference on robotics and automation. pp. 1549–1556. IEEE (2009)
  * [18] Ishihara, K., Morimoto, J.: Computationally Affordable Hierarchical Framework for Humanoid Robot Control. In: 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). pp. 7349–7356. IEEE (2021)
  * [19] Jilk, D.J., Safavynia, S.A., Ting, L.H.: Contribution of vision to postural behaviors during continuous support-surface translations. Experimental Brain Research 232(1), 169–180 (2014)
  * [20] Kashiri, N., Ajoudani, A., Caldwell, D.G., Tsagarakis, N.G.: Evaluation of hip kinematics influence on the performance of a quadrupedal robot leg. In: Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,. pp. 205–212. INSTICC, SciTePress (2016). https://doi.org/10.5220/0005986502050212
  * [21] Kim, S., Kim, C., You, B., Oh, S.: Stable whole-body motion generation for humanoid robots to imitate human motions. In: 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems. pp. 2518–2524. IEEE (2009)
  * [22] Lippi, V., Assländer, L., Akcay, E., Mergner, T.: Body sway responses to pseudorandom support surface translations of vestibular loss subjects resemble those of vestibular able subjects. Neuroscience Letters 736, 135271 (2020)
  * [23] Lippi, V., Brands, K.G., Seel, T.: Real-time implementation and evaluation of magnetometerless tracking system for human and humanoid posture control benchmarking based on inertial sensors. In: Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO. pp. 675–680. INSTICC, SciTePress (2020). https://doi.org/10.5220/0009869106750680
  * [24] Lippi, V., Maurer, C., Mergner, T.: Evaluating Robot Posture Control and Balance by Comparison to Human Subjects using Human Likeness Measures. In: Proceedings of the 2nd International Conference on Robotics, Computer Vision and Intelligent Systems - ROBOVIS,. pp. 77–85. INSTICC, SciTePress (2021). https://doi.org/10.5220/0010646900003061
  * [25] Lippi, V., Mergner, T.: Human-derived disturbance estimation and compensation (DEC) method lends itself to a modular sensorimotor control in a humanoid robot. Front. Neurorobot. 11, 49 (2017)
  * [26] Lippi, V., Mergner, T.: A challenge: Support of standing balance in assistive robotic devices. Applied Sciences 10(15), 5240 (2020)
  * [27] Lippi, V., Mergner, T., Maurer, C., Seel, T.: Performance Indicators of Humanoid Posture Control and Balance Inspired by Human Experiments. In: 2020 International Symposium on Wearable Robotics and Rehabilitation (WeRob) (2020)
  * [28] Lippi, V., Mergner, T., Seel, T., Maurer, C.: COMTEST project: A complete modular test stand for human and humanoid posture control and balance. In: 2019 IEEE-RAS 19th International Conference on Humanoid Robots (Humanoids) Toronto, Canada. October 15-17 (2019)
  * [29] Lippi, V., Mergner, T., Szumowski, M., Zurawska, M.S., Zielińska, T.: Human-inspired humanoid balancing and posture control in frontal plane. In: ROMANSY 21-Robot Design, Dynamics and Control: Proceedings of the 21st CISM-IFToMM Symposium, June 20-23, Udine, Italy. vol. 569, pp. 285–292. Springer (2016)
  * [30] Lippi, V., Molinari, F.: Lyapunov Stability of a Nonlinear Bio-inspired System for the Control of Humanoid Balance. In: Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics - ICINCO. pp. 726–733. INSTICC, SciTePress (2020). https://doi.org/10.5220/0009970307260733
  * [31] Luo, J., Zhang, Y., Hauser, K., Park, H.A., Paldhe, M., Lee, C.G., Grey, M., Stilman, M., Oh, J.H., Lee, J., et al.: Robust ladder-climbing with a humanoid robot with application to the darpa robotics challenge. In: 2014 IEEE International Conference on Robotics and Automation (ICRA). pp. 2792–2798. IEEE (2014)
  * [32] Mergner, T.: A neurological view on reactive human stance control. Annual Reviews in Control 34(2), 77–198 (2010)
  * [33] Mergner, T., Maurer, C., Peterka, R.J.: A multisensory posture control model of human upright stance. Progress in Brain Research 142, 189–201 (2003)
  * [34] Mergner, T., Schweigart, G., Fennell, L.: Vestibular humanoid postural control. Journal of Physiology - Paris 103, 178–194 (2009)
  * [35] Merton, P.: Speculations on the servo-control of movement. In: Ciba Foundation Symposium-The Spinal Cord. pp. 247–260. Wiley Online Library (1953)
  * [36] Nori, F., Peters, J., Padois, V., Babic, J., Mistry, M., Ivaldi, S.: Whole-body motion in humans and humanoids. In: Workshop on New Research Frontiers for Intelligent Autonomous Systems (2014)
  * [37] Ott, C., Henze, B., Hettich, G., Seyde, T.N., Roa, M.A., Lippi, V., Mergner, T.: Good posture, good balance: comparison of bioinspired and model-based approaches for posture control of humanoid robots. IEEE Robotics & Automation Magazine 23(1), 22–33 (2016)
  * [38] Oztop, E., Franklin, D.W., Chaminade, T., Cheng, G.: Human–humanoid interaction: is a humanoid robot perceived as a human? International Journal of Humanoid Robotics 2(04), 537–559 (2005)
  * [39] Peterka, R.: Sensorimotor integration in human postural control. Journal of neurophysiology 88(3), 1097–1118 (2002)
  * [40] del Pobil, A.P., Madhavan, R., Messina, E.: Benchmarks in robotics research. In: Proc. of IROS-2006 Workshop on Benchmarks in Robotics Research. Citeseer (2006)
  * [41] Posture Control Laboratory, Uniklinik Freiburg - Neurozentrum.: Lucy posturob, https://www.uniklinik-freiburg.de/neurologie/forschung/neurologische-arbeitsgruppen/postural-control.html
  * [42] Remazeilles, A., Dominguez, A., Barralon, P., Torricelli, D.: Towards a Unified Terminology for Benchmarking Bipedal Systems. In: Moreno, J.C., Masood, J., Schneider, U., Maufroy, C., Pons, J.L. (eds.) Wearable Robotics: Challenges and Trends. pp. 609–613. Springer International Publishing, Cham (2022)
  * [43] Singh, N.K.: Detection of hesitant dynamic postural control. In: 2016 IEEE 12th International Colloquium on Signal Processing & Its Applications (CSPA). pp. 36–40. IEEE (2016)
  * [44] The EUROBENCH consortium: Eurobench Github Repository (2022), https://github.com/eurobench
  * [45] The EUROBENCH consortium: Eurobench Software documentation (2022), https://eurobench.github.io/sofware_documentation/latest/index.html
  * [46] Torricelli, D., Mizanoor, R.S., Gonzalez, J., Lippi, V., Hettich, G., Asslaender, L., Weckx, M., Vanderborght, B., Dosen, S., Sartori, M., et al.: Benchmarking human-like posture and locomotion of humanoid robots: A preliminary scheme. In: Conference on Biomimetic and Biohybrid Systems. pp. 320–331. Springer (2014)
  * [47] Torricelli, D., Mizanoor, R.S., Lippi, V., Weckx, M., Mathijssen, G., Vanderborght, B., Mergner, T., Lefeber, D., Pons, J.L.: Benchmarking human likeness of bipedal robot locomotion: State of the art and future trends. In: Metrics of Sensory Motor Coordination and Integration in Robots and Animals, pp. 147–166. Springer (2020)
  * [48] Wiener, N.: Cybernetics or Control and Communication in the Animal and the Machine. Technology Press (1948)
  * [49] von Zitzewitz, J., Boesch, P.M., Wolf, P., Riener, R.: Quantifying the human likeness of a humanoid robot. International Journal of Social Robotics 5(2), 263–276 (2013)


# Reply to Comment on “Multitime Quantum Communication: Interesting But Not
Counterfactual” by L. Vaidman

Robert B. Griffiths
Department of Physics
Carnegie Mellon University
Pittsburgh, PA 15213 Electronic address<EMAIL_ADDRESS>
(Version of 22 September 2023)

###### Abstract

This is a response to comments and criticisms found in the preceding Comment
[1] by Vaidman on the paper [2].

A significant part of Vaidman’s Comment [1] is devoted to a discussion of
_counterfactuals_ , starting with a quotation from Penrose. The use of
counterfactuals in discussions of quantum foundations in fact goes back much
earlier, see e.g. [3]. The basic idea involved in a counterfactual is a
comparison between two (or more) situations: one the “actual” world and the
other the “counterfactual” world which differs from the former in certain
specified ways. One then considers various consequences of these differences.
In quantum theory this can lead to difficulties and paradoxes when the
physical properties of interest in the two worlds are represented by
incompatible observables or noncommuting projectors. The consistent histories
(CH) approach to quantum theory avoids such paradoxes by refusing to compare
incompatibles; see [4] and Ch. 19 of [5]. For the way in which CH resolves the
(supposedly) interaction-free measurement paradox mentioned by Vaidman, see
Ch. 21 of [5].

In [2], the paper addressed by Vaidman’s Comment, it is argued that the claim
of counterfactual communication by Salih et al. in [6] fails in that it
incorrectly assigns a probability for a photon to be in the communication
channel connecting Alice and Bob at intermediate times when quantum
interference effects are important, as well as incorrectly counting the number
of times it passes through the channel. While both errors are significant, the
first is more interesting in that it raises the question of what can properly
be said about a quantum particle’s location at an intermediate time given a
wavefunction evolving unitarily from an initial state on its way to a later
measurement.

In Hilbert space quantum theory—which is to say the basic principles set forth
by von Neumann [7], see in particular Sec. III.5—a physical property is
represented by a projector $P$ on a Hilbert subspace. In particular if the
property is that the particle is in some spatial region $R$, the projector $P$
applied to the position wavefunction $\psi(\textbf{r})$ leaves it unchanged
for all $\textbf{r}\in R$, but sets it equal to zero elsewhere. Consequently,
if $\psi(\textbf{r})$ is nonzero both for r in some region inside $R$ and also
some other region outside $R$, the projector $|\psi\rangle\langle\psi|$
corresponding to the particle’s wavefunction (assumed normalized) does not
commute with $P$, and when projectors do not commute—this is the essence of
quantum uncertainty principles—there is no meaningful way to discuss whether
or not the particle is in $R$. A well-known example is the double slit paradox
where, in the presence of interference, one cannot meaningfully say which slit
the photon passed through. For this reason the CH interpretation of quantum
mechanics considers the conjunction of two properties represented by
noncommuting projectors to be meaningless: To say the particle is in or
outside $R$ makes no more sense than to discuss whether $S_{x}$ is $+1/2$ of
$-1/2\hbar$ for a spin half particle when $S_{z}$ is $+1/2\hbar$.

Vaidman tries to get around this difficulty by asking whether a quantum
particle leaves a trace of its presence at a particular location via a weak
interaction with some other physical system at this location. That such a weak
measurement does not resolve the problem but simply generates more paradoxes
was known to Feynman; see his discussion in [8] of a weak light source
following the double slit—in his case the two holes with a coherent electron
wave passing through them. For an analysis of this situation based on
consistent quantum principles, see Sec. 13.5 of [5]. Vaidman’s nested Mach-
Zehnder paradox [9] has the same general character. Its resolution when weak
measurements are analyzed using consistent quantum principles, [10], was not
discussed in Vaidman’s Comment [11] on that paper. The objections to the
nested Mach-Zehnder paradox by Englert et al. in [12] are similar: they argue
that one cannot assign a probability to a particle’s following a particular
path when it is in a coherent superposition of amplitudes on different paths.

Towards the end of [1] Vaidman discusses the use of a quantity called _Cost_ ,
used in [2] as a measure of channel usage. In response it may be noted that
Cost was introduced as a replacement for the misleading use of “probability”
in [6], as in much of the succeeding literature. In its favor is the fact that
Cost is a well-defined mathematical quantity in situations where probabilities
cannot be consistently assigned, and its use leads to the rigorous bound in
Sec. III D of [2], probably the most interesting technical result in that
paper. However, as with any novel idea, only the future will show whether Cost
is really useful or needs to be replaced by something else. Vaidman’s concern
that the analysis using Cost includes both cases in which a communication
protocol succeeds as well as when it fails is not relevant to the situation
considered in [6], where the protocol always succeeds with high probability—an
instance of what is called a _full_ protocol in Sec. III C of [2]. The
contrasting case of _partial_ protocols—as, for example, when by convention
the non-arrival of a photon in Alice’s apparatus at a particular time signals
that Bob has transmitted the bit 1—requires a separate discussion, which might
be a useful subject for some future paper.

I am grateful to Carnegie-Mellon University and its Physics Department for
continuing support of my activities as an emeritus faculty member.

## References

  * [1] Lev Vaidman.  Comment on “Multitime quantum communication: Interesting but not counterfactual”.  arXiv:2306.16756, 2023.
  * [2] Robert B. Griffiths.  Multitime quantum communication: Interesting but not counterfactual.  Phys. Rev. A, 107:062219, 2023.  arXiv:2301.01730 v3.
  * [3] Henry Pierce Stapp.  S-matrix interpretation of quantum theory.  Phys. Rev. D, 3:1303–1320, 1971.
  * [4] Robert B. Griffiths.  Consistent quantum counterfactuals.  Phys. Rev. A, 60:5–9, 1999.
  * [5] Robert B. Griffiths.  Consistent Quantum Theory.  Cambridge University Press, Cambridge, U.K., 2002.  http://quantum.phys.cmu.edu/CQT/.
  * [6] Hatim Salih, Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy.  Protocol for direct counterfactual quantum communication.  Phys. Rev. Lett., 110:170502, 2013.  arXiv:1206.2042.
  * [7] John von Neumann.  Mathematical Foundations of Quantum Mechanics.  Princeton University Press, Princeton, 1955,2018.  Translation by R. T. Beyer of Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, Berlin, 1932.
  * [8] R. P. Feynman, R. B. Leighton, and M. Sands.  The Feynman Lectures on Physics, volume III: Quantum Mechanics.  Addison-Wesley, Reading, Mass., 1965.  Ch. 1.
  * [9] L. Vaidman.  Past of a quantum particle.  Phys. Rev. A, 87:052104, 2013.
  * [10] Robert B. Griffiths.  Particle path through a nested Mach-Zehnder interferometer.  Phys. Rev. A, 94:032115, 2016.  arXiv:1604.04596.
  * [11] L. Vaidman.  Comment on ‘particle path through a nested mach-zehnder interferometer’.  Phys. Rev. A, 95:066101, 2017.  arXiv:1610.07734.
  * [12] Berthold-Georg Englert, Kelvin Horia, Jibo Dai, Yink Loong Len, and Hui Khoon Ng.  Past of a quantum particle revisited.  Phys. Rev. A, 96:022126, 2017.  arXiv:1704.03722 v2.