# Measuring “Why” in Recommender Systems: a Comprehensive Survey on the
Evaluation of Explainable Recommendation

Xu Chen1 Yongfeng Zhang2 Ji-Rong Wen1 1Beijing Key Laboratory of Big Data
Management and Analysis Methods,
Gaoling School of Artificial Intelligence, Renmin University of China
2Department of Computer Science, Rutgers University<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Explainable recommendation has shown its great advantages for improving
recommendation persuasiveness, user satisfaction, system transparency and
among others. A fundamental problem of explainable recommendation is how to
evaluate the explanations. In the past few years, various evaluation
strategies have been proposed. However, they are scattered in different
papers, and there lacks a systematic and detailed comparison between them. To
bridge this gap, in this paper, we comprehensively review the previous work,
and provide different taxonomies according to the evaluation perspectives and
evaluation methods. Beyond summarization, we also analyze the (dis)advantages
of existing evaluation methods, and provide a series of guidelines on how to
select them. The contents of this survey are concluded from more than 100
papers from top-tier conferences like IJCAI, AAAI, TheWebConf, Recsys, UMAP
and IUI, and the complete comparisons are presented at
https://shimo.im/sheets/VKrpYTcwVH6KXgdy/MO
DOC/. With this survey, we finally aim to provide a clear and comprehensive
review on the evaluation of explainable recommendation.

##  1 Introduction

Artificial intelligence (AI) has deeply revolutionized people’s daily life and
production, bringing effectiveness, efficiency, reliability, automation and
etc. Among different AI tasks, many are purely objective, where the ground
truths are irrelevant with subjective factors. Computer vision (CV) is a
typical domain falling into this category. For example, in the task of image
classification, the models are built to accurately predict the labels like a
cat or a dog. These labels are objective facts, which do not change with human
feelings. In many other domains, the tasks are more subjective, where there is
no rigid ground truth, and the target is to improve the utilities of certain
stakeholders. Recommender system is a typical subjective AI task, where the
stakeholder utilities not only include the recommendation accuracy, but also
contain a lot of beyond-accuracy aspects, among which explainability is a
significant and widely studied one.

Basically, explainable recommendation solves the problem of “why”, that is,
why the items are recommended. By providing explanations, people can make more
informed decisions, and the recommendation persuasiveness and algorithm
transparency can be improved. In the field of explainable recommendation,
evaluation is a fundamental problem, which judges the model availability and
helps to make model selections. Comparing with the other evaluation tasks,
evaluating recommendation explanations is much more difficult because the
ground truth is hard to obtain and human feelings are not easy to approximate.
To alleviate these difficulties, the past few decades have witnessed a lot of
promising evaluation strategies. However, these strategies are independently
proposed in different papers, and has various prerequisites and implementation
characters. There lacks a systematic comparison between them, which may hinder
the evaluation standardization and be not friendly to the newcomers to this
domain.

To bridge the above gap, in this paper, we thoroughly conclude the previous
work, and present a clear summarization on existing evaluation strategies. In
specific, recommendation explanations may serve for different targets, for
example, the users, providers and model designers. For different targets, the
evaluation perspectives are also various, and we summarize four major
evaluation perspectives from the previous work, that is, the explanation
effectiveness, transparency, persuasiveness and scrutability. For evaluating
the explanations from the above perspectives, there exists a lot of evaluation
methods. We conclude these methods into four categories including the case
studies, quantitative metrics, crowdsourcing and online experiments. For each
method, we detail its main characters, and analysis its advantages and
shortcomings, At last, we propose a series of guidelines on how to select
these evaluation methods in real-world problems. The main contributions of
this paper are summarized as follows:

$\bullet$ We provide a systematic and comprehensive survey on the evaluation
of explainable recommendation, which to the best of our knowledge, is the
first survey on this topic.

$\bullet$ We propose different taxonomies for existing papers according to the
evaluation perspectives and evaluation methods. The references cover more than
100 papers from top tier conferences like IJCAI, AAAI, TheWebConf, Recsys,
UMAP and IUI, and their complete summarization is presented at
https://shimo.im/sheets/VKrpYTcwVH6KXgdy/MODOC/.

$\bullet$ We analysis the advantages and shortcomings of existing evaluation
methods and propose several guidelines on how to choose them in different
scenarios.

##  2 Related Survey

Recommender system is becoming increasingly important, with dozens of papers
published each years on top tier conferences/journals. For better summarizing
and comparing these work, people have conducted many promising survey studies.
For general recommendation, early surveys mainly focus on shallow
architectures like matrix factorization or heuristic methods Adomavicius
(2004). Later on, with the ever prospering of deep neural networks, people
have designed a large amount of neural recommender models. Correspondingly,
many comprehensive surveys on deep recommender algorithms are proposed Zhang
(2019); Wu (2021); Fang (2020). Beyond concluding recommender models from the
architecture perspective, many surveys focus on how to leverage the side
information. For example, Bouraga (2014) summarizes the models on
incorporating knowledge graph into recommender systems. Chen (2013) reviews
the algorithms on social recommendation. Kanwal (2021) focuses on the
combination between the text information and recommender models. Ji (2017)
summarizes the models on image based recommendation. In addition, people also
conduct surveys according to different recommendation domains. For example,
Song (2012) summarizes the algorithms in music recommendation. Goyani (2020)
focus more on the movie recommender models. Wei (2007) is tailored for the
algorithms in e-commerce. Yu (2015) is a comprehensive survey on point-of-
interest (POI) recommendation. We have also noticed that there are some
surveys on recommendation evaluations Gunawardana (2009); Silveira (2019).
However, they focus more on the metrics like accuracy, novelty, serendipity
and coverage, which differs from our target on explanation evaluation. In the
past, we have published a survey on explainable recommender models Zhang
(2018), but it aims to summarize this field in a general sense. However, in
this paper, we focus on the evaluation part, and include many latest studies.

Table 1: Summarization of the evaluation perspectives.

Evaluation perspective |  Evaluation problem |  Representative papers |  Serving target
---|---|---|---
Effectiveness |  Whether the explanations are useful for the users to make more accurate/faster decisions? |  Hada (2021); Guesmi (2021); Wang (2020); Gao et al. (2019); Chen et al. (2013) |  Users
Transparency |  Whether the explanations can reveal the internal working principles of the recommender models? |  Chen (2021a); Sonboli (2021); Li (2021c, 2020c); Fu (2020) |  Model designers
Persuasiveness |  Whether the explanations can increase the click/purchase rate of the users on the items? |  Musto et al. (2019); Tsai and Brusilovsky (2019); Balog and Radlinski (2020); Tsukuda (2020); Chen and Wang (2014) |  Providers
Scrutability |  Whether the explanations can exactly correspond to the recommendation results? |  He et al. (2015); Tsai and Brusilovsky (2019); Cheng et al. (2019); Balog et al. (2019); Liu (2020) |  Model designers

##  3 Evaluation Perspectives

Recommendation explanations may serve for different purposes, _e.g._ ,
providing more recommendation details, so that people can make more informed
decisions, or improving recommendation persuasiveness, so that the providers
can obtain more profit. For different purposes, the explanations should be
evaluated from different perspectives. In the literature, the following
evaluation perspectives are usually considered.

###  3.1 Effectiveness

Effectiveness aims to evaluate “whether the explanations are useful for the
users to make more accurate/faster decisions Balog and Radlinski (2020)?” or
sometimes “whether the users are more satisfied with the explanations Xian
(2021)?” It is targeted at improving the utilities of the users, that is,
better serving the users with the explanations. By enhancing the
effectiveness, the user experience can be improved, which may consequently
increase the user conversion rate and stickiness. Here, we use the term
“effectiveness” in a broad sense, which covers the concepts of efficiency,
satisfaction and trust in many literature Balog and Radlinski (2020).

###  3.2 Transparency

Transparency aims to evaluate “whether the explanations can reveal the
internal working principles of the recommender models Tai (2021); Sonboli
(2021); Li (2021b)?” It encourages the algorithms to produce explanations
which can open the “black box” of the recommender models. With better
transparency, the model designers can know more about how the recommendations
are generated, and better debug the system.

###  3.3 Persuasiveness

Persuasiveness aims to evaluate “whether the explanations can increase the
interaction probability of the users on the items Tsukuda (2020); Balog and
Radlinski (2020); Musto et al. (2019)?” In general, persuasiveness aims to
enhance the utilities of the providers (_e.g._ , sellers on the e-commerce
websites, content creators on the micro video applications). Comparing with
transparency, persuasiveness does not care about whether the explanations can
honestly reflect the model principles. It focuses more on whether the
explanations can persuade users to interact with the recommendations, so that
the business goals can be achieved.

###  3.4 Scrutability

Scrutability aims to evaluate “whether the explanations can exactly correspond
to the recommendation results?” Cheng et al. (2019); Tsai and Brusilovsky
(2019); He et al. (2015). In other words, if the explanations are different,
then whether and how the recommendation list is altered. This perspective is
closely related with transparency yet one does not imply the other.
Transparency pays more attention to the details inside the model, aiming to
whiten the working mechanisms. Scrutability cares more about the relations
between the explanations and outputs, which can be model agnostic.

We summarize the above evaluation perspectives in Table 1. We can see,
recommendation explanations mainly serve for three types of stakeholders, that
is, the users, providers and model designers. For different stakeholders, the
evaluation perspectives are different: effectiveness cares more about the
utilities of the users, persuasiveness is more tailored for the providers, and
transparency and scrutability serve more for the model designers. If one
compares the explanations in recommender systems and general machine learning,
she may find that the latter mainly focus on the transparency and
scrutability, since it is not related with the users and provides. In this
sense, explainable recommendation can be more complex, since it needs to
trade-off different stakeholders.

######  Remark.

(1) The above four perspectives are not always independent, they may have both
overlaps and contradictions. For example, the explanations for improving
transparency may satisfy the curiosities of some users on how the recommender
models work, and therefore the explanation effectiveness can be enhanced. In
another example, to persuade users to purchase more items (_i.e._ ,
persuasiveness), the explanations may not accurately reveal the model working
mechanism (_i.e._ , transparency), but have to highlight the features which
can promote sales. When designing and evaluating explainable recommender
models, it is necessary to predetermine which perspectives to focus on, and
how much we can sacrifice on the other perspectives. (2) We have noticed there
are many other evaluation perspectives, such as the fluency of the explanation
Chen (2021b) or whether a user-item pair is explainable Liu et al. (2019).
However, these perspectives are not common, only appearing in a small amount
of papers. Thus we do not elaborate them in this survey.

##  4 Evaluation Methods

To evaluate recommendation explanations from the above perspectives, people
have designed quite a lot of evaluation methods. By going over the previous
papers, we find that there are mainly four categories, that is: case studies,
quantitative metrics, crowdsourcing and online experiments. In the following,
we elaborate these methods more in detail.

###  4.1 Evaluation with Case Studies

In this category, people usually present some examples to illustrate how the
recommender model works and whether the generated explanations are aligned
with human intuitions. Usually, the recommender models are firstly learned
based on the training set, and then the examples are generated based on the
intermediate or final outputs by feeding the testing samples into the
optimized model. More specifically, the examples can be both positive and
negative Chen et al. (2019). By the positive examples, the effectiveness of
the model can be demonstrated, while by the negative ones, the readers can
understand when and why the model fails.

The most popular case studies are conducted by visualizing the attention
weights Li (2021b, c); Chen et al. (2019). For example, in fashion
recommendation, Li (2021b) presents the matching degree between the tops and
bottoms across different attributes. In image recommendation, Chen (2017)
visualizes the importances of different user previously interacted images as
well as the regions in each image. In graph based recommendation, Li (2021c)
illustrates which neighbors, attributes and structure contexts are more
important for the current user-item interaction prediction.

In many studies, people present template-based or fully generated natural
language explanations as case studies. For example, Chen (2020); Zhang et al.
(2014) show the explanations by assembling the model predicted attributes and
the pre-defined templates. Li (2020b, 2021a) present the explanations which
are completely generated from the models.

When evaluating knowledge-based recommender models, people usually conduct
case studies by showing the reasoning paths between the users and items. For
example, in Wang et al. (2019), the authors present the importances of
different paths for a given user-item pair. In Xian (2020); Fu (2020), the
authors present examples to reveal the model routing process from the users to
the items.

In many papers, the explanation items/features are presented as intuitive case
studies. For example, Liu (2020); Li (2020c) highlights the items which are
significant for the current model prediction. Yang et al. (2019) shows
important feature combinations to reveal the model working principles.

In aspect-based recommender models, the case studies are usually conducted by
presenting the learned aspects for the users/items. For example, in Tan et al.
(2016), the authors show the top words in the learned aspects to demonstrate
the effectiveness of the model in capturing user preference.

The major advantage of case studies lies in its intuitiveness, which makes the
readers easily understand how the explanations look like. However, there are
many significant weaknesses. To begin with, it can be biased, since one cannot
evaluate and present all the samples. And then, due to the lack of
quantitative scores, it is difficult to compare different models and make
accurate model selections.

###  4.2 Evaluation with Quantitative Metrics

In order to more precisely evaluate the explanations, people have designed
many quantitative metrics. Basically, this category of methods assumes that
the stakeholder utilities can be approximated and evaluated via parameterized
functions. In the following, we elaborate the representative metrics.

Intuitively, if the explanations can accurately reflect the user preferences
on the items, then the user satisfaction can be improved. As a result, many
papers Li (2021a); Hada (2021); Li (2020a) leverage the user reviews, which
pool comprehensive user preferences, as the explanation ground truth. In these
studies, the explainable recommendation problem is regarded as a natural
language generation (NLG) task, where the goal is to accurately predict the
user reviews. Under this formulation, the following metrics are usually
leveraged to evaluate the explanation qualities.

$\bullet$ BLEU and ROUGE scores. BLEU Papineni (2002) and ROUGE Lin (2004)
scores are commonly used metrics for natural language generation, where the
key insight is to count the world (or n-gram) level overlaps between the model
predicted and real user reviews.

$\bullet$ Unique Sentence Ratio (USR), Feature Coverage Ratio (FCR) and
Feature Diversity (FD). These metrics aim to evaluate the diversity of the
generated reviews Li (2020a). USR evaluates the diversity on the sentence
level. In specific, suppose the set of unique reviews is $\mathcal{S}$, and
the total number of reviews is N, then this metric is computed as
$\text{USR}=\frac{|\mathcal{S}|}{N}$. FCR and FD compute the diversity on the
feature level. In the former metric, suppose the number of distinct features
in all the generated reviews is $M$, and the total number of features is
$N_{f}$, than $\text{FCR}=\frac{M}{N_{f}}$. For the latter one, suppose the
feature set in the generated review for user-item pair $(u,i)$ is $F_{u,i}$,
then
$\text{FD}=\frac{1}{N(N-1)}\sum_{(u,i)\neq(u^{\prime},i^{\prime})}|F_{u,i}\cap
F_{u^{\prime},i^{\prime}}|$, where $N$ is the total number of different user-
item pairs.

$\bullet$ Feature-level Precision (FP), Recall (FR) and $F_{1}$ (FF) In many
papers Tan (2021); Tai (2021), the explanation quality is evaluated by
comparing the features in the predicted and real user reviews. For each user-
item pair $(u,i)$, suppose the predicted and real feature sets are $S_{u,i}$
and $T_{u,i}$, respectively. Then the feature-level precision, recall and
$F_{1}$ for this user-item pair are computed as
$\text{FP}_{u,i}=\frac{|S_{u,i}\cap T_{u,i}|}{|S_{u,i}|}$,
$\text{FR}_{u,i}=\frac{|S_{u,i}\cap T_{u,i}|}{|T_{u,i}|}$ and
$\text{FF}_{u,i}=\frac{2\cdot\text{FP}\cdot\text{FR}}{\text{FP}+\text{FR}}$,
respectively. The final results are averaged across all the user-item pairs,
that is, $\text{FP}=\frac{1}{N}\sum_{u,i}\text{FP}_{u,i}$,
$\text{FR}=\frac{1}{N}\sum_{u,i}\text{FR}_{u,i}$ and
$\text{FF}=\frac{1}{N}\sum_{u,i}\text{FF}_{u,i}$, where $N$ is the number of
user-item pairs.

$\bullet$ Feature Matching Ratio (FMR). In Li (2020a), the reviews are
generated under the control of a feature. FMR evaluates whether this feature
can successfully appear in the predicted results. It is formally computed as
$\text{FMR}=\frac{1}{N}\sum_{u,i}\bm{1}(f_{u,i}\in S_{u,i})$, where $f_{u,i}$
is the input feature, $S_{u,i}$ is the predicted review, $N$ is the number of
user-item pairs, and $\bm{1}$ is the indicator function.

In counterfactual explainable recommendation Tan (2021), the following two
metrics are usually leveraged to evaluate how the model prediction changes
when the key features/items (which are used as the explanations) are altered.

$\bullet$ Probability of Necessity (PN). This metric aims to evaluate “if the
explanation features of an item had been ignored, whether this item will be
removed from the recommendation list?” Formally, Suppose $A_{ij}$ is the
explanation feature set of item $j$ when being recommended to user $i$, let
$\text{PN}_{ij}=1$, if item $j$ no longer exists in the recommendation list
when one ignores the features in $A_{ij}$, otherwise $\text{PN}_{ij}=0$. The
final PN score is computed as
$\sum_{i,j}\frac{\text{PN}_{ij}}{\bm{1}({|A_{ij}|>0})}$.

$\bullet$ Probability of Sufficiency (PS). This metric aims to evaluate “if
only the explanation features of an item are remained, whether this item will
be still in the recommendation list?” Let $\text{PS}_{ij}=1$, if item $j$
exists in the recommendation list when one only remains the features in
$A_{ij}$, otherwise $\text{PS}_{ij}=0$. The final PS score is computed as
$\sum_{i,j}\frac{\text{PS}_{ij}}{\bm{1}({|A_{ij}|>0})}$.

Similar evaluation metric is also leveraged in Liu (2020), which measures the
explanation quality based on the performance change before and after the key
history items/entities are removed from the model input. The key intuition is
that if the explanations are the key history items/entities underlying the
recommendations, then removing them should greatly change the recommendation
performance. The formal definition is as follows:

$\bullet$ Performance Shift (PS). Suppose the original recommendation
performance is $r$, after removing the key history items/entities, the
performance is changed to $r^{\prime}$, then
$\text{RS}=\frac{r-r^{\prime}}{r}$. Here the performance can be evaluated with
different metrics, such as precision, recall, NDCG and so on.

In Abdollahi and Nasraoui (2017), the authors firstly define which items can
be explained based on the user-item interaction graph, and then the following
two metrics are proposed to evaluate the explainability of the
recommendations.

$\bullet$ Mean Explainability Precision (MEP) and Mean Explainability Recall
(MER). For a user $u$, suppose the set of items which can be explained is
$N_{u}$, the recommended item set is $M_{u}$, then
$\text{MEP}=\frac{|N_{u}\cap M_{u}|}{|M_{u}|}$ and
$\text{MEP}=\frac{|N_{u}\cap M_{u}|}{|N_{u}|}$.

Based on the above quantitative metrics, different models can be compared,
which is important for model selection and benchmarking this field. In
addition, quantitative metrics are usually very efficient, since most of them
can be computed in a short time. As for the shortcomings, quantitative metrics
are sometimes not fully aligned with the evaluation goals. For example, BLUE
score only evaluates the word-level overlaps between the generated and real
explanations, but it does not directly tell whether the explanations are
reasonable or not. What’s more, when the stakeholder utilities are too
complicated, existing quantitative metrics, which are heuristically designed
by a small amount of people, may have the risk of misleading the evaluation
results.

###  4.3 Evaluation with Crowdsourcing

As mentioned before, recommender system is a subjective AI task, thus
involving human feelings into the evaluation process is a direct and natural
idea. Such evaluation methods are called crowdsourcing, and there are mainly
three strategies, which are elaborated as follows.

$\bullet$ Crowdsourcing with public datasets. In this method, the recommender
models are trained based on the public datasets. And then, many annotators are
recruited to evaluate the model generated explanations based on a series of
questions. There are two key points in the evaluation process.

(i) Annotation quality control. In order to accurately evaluate the
explanations, controlling the annotation quality is a necessary step. In the
previous work, there are two major strategies: one is based the voting
mechanism, for example, in Chen et al. (2019), there are three annotators for
each labeling task, and the final result is available only when more than two
annotators have the same judgment. The other strategy is based on computing
certain statistical quantities. For example, in Chen et al. (2018), Cohen’s
Weighted $\kappa$ Cohen (1968) is leveraged to assess the inter-annotator
agreements and remove the outlier annotations.

(ii) Annotation question designs. We find that the annotation questions in
different papers are quite diverse. In general, they can be classified into
three categories. The first category are “model-agnostic” questions, where the
problems do not dependent on any specific model. In Chen et al. (2019), the
annotators are required to simulate themselves as the real users, and then
label out the ideal explanations from all the possible ones. Then the labeled
explanations are regarded as the ground truth, and the proposed model and
baselines are evaluated by comparing their predicted explanations with the
ground truth. The second category are “single-model” questions, where the
problems focus on only one explainable model. In Li (2021b), the annotators
are asked to up-vote or down-vote the explanations generated from the designed
model and the baseline, respectively. In Xian (2021); Li (2020b), the
annotators have to evaluate whether the proposed model can help users make
better purchase decision. In Chen (2021b), the authors ask the annotators to
label whether the explanations produced from the designed model are fluent and
useful. In Tao et al. (2019); Wang et al. (2018), the annotators are required
to answer whether they are satisfied with the generated explanations, whether
the explanations can really match their preferences and whether the
explanations can provide more information for users to make a decision. The
third category are “pair-model” questions, where the problems focus on the
comparison between a pair of explainable models. In Li (2020b), the annotators
need to answer which explanations produced from the designed model and
baseline are more closer to the ground truth. In Tao et al. (2019), the
annotation questions are: between A and B, whose explanations do you think can
better help you understand the recommendations? and whose explanations can
better help you make a more informed decision? In Chen et al. (2018), the
annotators are asked to compare the usefulness of the explanations generated
from the proposed model and the baselines.

In the above questions, the first category is the most difficult, since the
annotators have to make decisions from a large amount of candidates. However,
the labeled datasets are model-agnostic, which can be reused for different
models and experiments. In the second and third categories, the annotators
only have to answer yes or no, or select from a small amount of candidate
answers, but the annotations cannot be reused, which is costly and not
scalable.

$\bullet$ Crowdsourcing by injecting annotator data into public dataset. A
major problem in the above method is that the annotator preferences may
deviate from the real users’, which may introduce noises into the evaluation.
As a remedy for this problem, Gao et al. (2019) designs a novel crowdsourcing
method by combining the annotator generated data with the public dataset. In
specific, the evaluations are conducted based on the Yelp
dataset111https://www.yelp.com/dataset, which is collected from yelp.com. The
authors recruit 20 annotators, and require each annotator to write at least 15
reviews on yelp.com. Then, the reviews written by the annotators are infused
into the original Yelp dataset, and the explainable models are trained based
on the augmented dataset. In the evaluation process, the annotators are
required to score from 1 to 5 on the explanations according to their
usefulness. Since the data generated by the annotators is also incorporated
into the model training process, the feedback from the annotators are exactly
the real user feedback.

Table 2: Summarization of the evaluation methods. In the second and third
column, we present the most significant strengths and shortcomings of
different evaluation methods. In the last column, we present the perspectives
that a method is usually leveraged to evaluate.

Evaluation methods |  Strengths |  Shortcomings |  Representative papers |  Evaluation perspectives
---|---|---|---|---
Case studies |  Better intuitiveness |  Bias; Cannot make comparisons |  Xian (2020, 2020); Fu (2020); Liu (2020); Barkan (2020); Li (2020c) |  Effectiveness; Transparency
Quantitative metrics |  Quantitative evaluation; Easy to benchmark; High efficiency |  Deviating from the explanation goals; Less effective approximation |  Li (2021a); Hada (2021); Li (2020a); Tan (2021); Tai (2021) |  Effectiveness; Scrutability
Crowdsourcing |  Based on real human feelings |  High cost |  Li (2021b); Xian (2021); Li (2020b); Chen (2021b); Wang et al. (2018) |  Effectiveness; Scrutability; Transparency; Persuasiveness
Online experiments |  High reliable |  High cost |  Zhang et al. (2014); Xian (2021) |  Effectiveness; Persuasiveness

$\bullet$ Crowdsourcing with fully constructed datasets. In this category, the
datasets are fully constructed by the annotators. In general, there are four
steps in the evaluation process, that is, (i) recruiting annotators with
different background (_e.g._ , age, sex, nationality and etc.), (ii)
collecting annotator preferences, (iii) generating recommendations and
explanations for the annotators, and (iv) evaluating the explanations by
asking questions on different evaluation perspectives. Usually, The number of
annotators in this category is much larger than above two methods, and the
experiment settings and studied problems are quite diverse. For example, in
Musto et al. (2019), there are 286 annotators (with 76.3% males, and 48.4%
PhDs), and they are asked to evaluate the explanation transparency and
effectiveness. In Naveed (2020), the authors aim to study whether feature-
based collaborative filtering (CF) models can lead to better explanations than
the conventional CF’s. There are 20 annotators, among which 14 are females
with the age ranges from 21 to 40. In Hernandez-Bocanegra (2020), the authors
investigate whether the type and justification level of the explanations
influence the user perceptions. To achieve this goal, 152 annotators are
recruited, where there are 87 females, and the average age is 39.84. In Balog
and Radlinski (2020), the authors recruit 240 annotators to study the
relations between the explanation effectiveness, transparency, persuasiveness
and scrutability. In Tsukuda (2020), the authors recruit 622 annotators to
evaluate the influence of the explanation styles on the explanation
persuasiveness.

In the above three types of crowdsourcing methods, the first one is fully
based on the public datasets, the second one is based on the combination
between the annotator generated data and the public dataset, and the last one
is completely based on the annotator generated data. Comparing with the case
studies and quantitative metrics, crowdsourcing is directly based on real
human feelings. However, it can be much more expensive due to the cost of
recruiting annotators.

###  4.4 Evaluation with Online Experiments

In the previous work, we also find a few papers, which evaluate recommendation
explanations with online experiments. In specific, in Zhang et al. (2014), the
online users are split into three groups. For the users in the first group,
the explanations are generated from the proposed model. For the second group
users, the explanations are produced from the baseline. For the last group,
there is no explanations. After running the system for a short time, the
authors compute the click through rate (CTR) in each group to evaluate whether
the explanations are useful. Similar method are also leveraged in Xian (2021),
where the conversion and revenue are compared before and after presenting the
explanations to the users. We believe online experiments can be more reliable.
However, its cost is very high, since it may disturb the real production
environments, and impact user experiences.

We summarize the above evaluation methods in Table 2, where we can see: for
different evaluation methods, they are usually leveraged to evaluate different
perspectives. For example, case studies are mainly used to evaluate
explanation effectiveness and transparency, while crowdsourcing can be
leveraged to evaluate all the perspectives defined in section 3, since it can
flexibly design questions for the annotators. From the second and third
columns, we can see different evaluation methods have their own strengths and
shortcomings, and no one method can take all the advantages.

Guidelines for selecting the evaluation methods. Based on different characters
of the evaluation methods, we propose several guidelines on using them in
real-world applications. (i) If the recommender models serve for high-stake
tasks such as health caring and finance, then more reliable evaluation methods
should be selected, since the cost of mis-explaining a recommendation, so that
the users fail to make correct decisions can be much larger than that of
conducting the evaluations. (ii) For general recommendation tasks, if the
evaluation budget is limited, one has to trade-off the strengths and
shortcomings of different evaluation methods. (iii) From Table 2, we can see
the same perspective can be evaluated using different methods, thus one can
simultaneously using these methods to take their different advantages. For
example, when the model is not well optimized, we can use case studies to
roughly evaluate the explanation effectiveness. Once the model has been tuned
better, quantitative metrics can be leveraged to demonstrate whether the
explanations are indeed effective.

##  5 Conclusion and Outlooks

In this paper, we summarize existing explainable recommendation papers with a
focus on the explanation evaluation. In specific, we introduce the main
evaluation perspectives and methods in the previous work. For each evaluation
method, we detail its characters, representative papers, and also highlight
its strengths and shortcomings. At last, we propose several guidelines for
better selecting the evaluation methods in real-world applications.

By this survey, we would like to provide a clear summarization on the
evaluation strategies of explainable recommendation. We believe there is still
much room left for improvements. To begin with, since the heuristically
designed quantitative metrics can be incompetent for evaluating the complex
stakeholder utilities, one can automatically learn the metric functions, where
one can incorporate a small amount of human labeled data as the ground truth.
And then, for benchmarking this field, one can build a large scale dataset,
which incorporates the ground truth on the user effectiveness, recommendation
persuasiveness and so on. At last, existing methods mostly evaluate the
explanations within a specific task. However, we believe human intrinsic
preferences should be stable and robust across different tasks. Thus, in order
to evaluate whether the derived explanations can indeed reveal such human
intrinsic preferences, developing task-agnostic evaluation methods should be
an interesting research direction.

## References

  * Abdollahi and Nasraoui [2017] Behnoush Abdollahi and Olfa Nasraoui.  Using explainability for constrained matrix factorization.  In Proceedings of the Eleventh ACM Conference on Recommender Systems, pages 79–83, 2017.
  * Adomavicius [2004] Gediminas et al. Adomavicius.  Recommendation technologies: Survey of current methods and possible extensions.  2004\.
  * Balog and Radlinski [2020] Krisztian Balog and Filip Radlinski.  Measuring recommendation explanation quality: The conflicting goals of explanations.  In SIGIR, 2020.
  * Balog et al. [2019] Krisztian Balog, Filip Radlinski, and Shushan Arakelyan.  Transparent, scrutable and explainable user models for personalized recommendation.  In Proceedings of the 42nd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 265–274, 2019.
  * Barkan [2020] Oren et al. Barkan.  Explainable recommendations via attentive multi-persona collaborative filtering.  In Recsys, 2020.
  * Bouraga [2014] Sarah et al. Bouraga.  Knowledge-based recommendation systems: A survey.  IJIIT, 2014.
  * Chen and Wang [2014] Li Chen and Feng Wang.  Sentiment-enhanced explanation of product recommendations.  In Proceedings of the 23rd international conference on World Wide Web, pages 239–240, 2014.
  * Chen et al. [2013] Wei Chen, Wynne Hsu, and Mong Li Lee.  Tagcloud-based explanation with feedback for recommender systems.  In Proceedings of the 36th international ACM SIGIR conference on Research and development in information retrieval, pages 945–948, 2013.
  * Chen et al. [2018] Chong Chen, Min Zhang, Yiqun Liu, and Shaoping Ma.  Neural a entional rating regression with review-level explanations.(2018).  2018\.
  * Chen et al. [2019] Xu Chen, Hanxiong Chen, Hongteng Xu, Yongfeng Zhang, Yixin Cao, Zheng Qin, and Hongyuan Zha.  Personalized fashion recommendation with visual explanations based on multimodal attention network: Towards visually explainable recommendation.  In Proceedings of the 42nd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 765–774, 2019.
  * Chen [2013] Song et al. Chen.  Social network based recommendation systems: A short survey.  In ICSC. IEEE, 2013.
  * Chen [2017] Jingyuan et al. Chen.  Attentive collaborative filtering: Multimedia recommendation with item-and component-level attention.  In SIGIR, 2017.
  * Chen [2020] Tong et al. Chen.  Try this instead: Personalized and interpretable substitute recommendation.  In SIGIR, 2020.
  * Chen [2021a] Hongxu et al. Chen.  Temporal meta-path guided explainable recommendation.  In WSDM, 2021.
  * Chen [2021b] Zhongxia et al. Chen.  Towards explainable conversational recommendation.  In IJCAI, 2021.
  * Cheng et al. [2019] Weiyu Cheng, Yanyan Shen, Linpeng Huang, and Yanmin Zhu.  Incorporating interpretability into latent factor models via fast influence analysis.  In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 885–893, 2019.
  * Cohen [1968] Jacob Cohen.  Weighted kappa: nominal scale agreement provision for scaled disagreement or partial credit.  Psychological bulletin, 1968.
  * Fang [2020] Hui et al. Fang.  Deep learning for sequential recommendation: Algorithms, influential factors, and evaluations.  TOIS, 2020.
  * Fu [2020] Zuohui et al. Fu.  Fairness-aware explainable recommendation over knowledge graphs.  In SIGIR, 2020.
  * Gao et al. [2019] Jingyue Gao, Xiting Wang, Yasha Wang, and Xing Xie.  Explainable recommendation through attentive multi-view learning.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 3622–3629, 2019.
  * Goyani [2020] Mahesh et al. Goyani.  A review of movie recommendation system.  ELCVIA: electronic letters on computer vision and image analysis, 2020.
  * Guesmi [2021] Mouadh et al. Guesmi.  On-demand personalized explanation for transparent recommendation.  In UMAP, 2021.
  * Gunawardana [2009] Asela et al. Gunawardana.  A survey of accuracy evaluation metrics of recommendation tasks.  JMLR, 2009.
  * Hada [2021] Deepesh V aet al. Hada.  Rexplug: Explainable recommendation using plug-and-play language model.  In SIGIR, 2021.
  * He et al. [2015] Xiangnan He, Tao Chen, Min-Yen Kan, and Xiao Chen.  Trirank: Review-aware explainable recommendation by modeling aspects.  In Proceedings of the 24th ACM International on Conference on Information and Knowledge Management, pages 1661–1670, 2015.
  * Hernandez-Bocanegra [2020] Diana C et al. Hernandez-Bocanegra.  Effects of argumentative explanation types on the perception of review-based recommendations.  In UMAP, 2020.
  * Ji [2017] Zhenyan et al. Ji.  A survey of personalised image retrieval and recommendation.  In National Conference of Theoretical Computer Science, 2017.
  * Kanwal [2021] Safia et al. Kanwal.  A review of text-based recommendation systems.  2021\.
  * Li [2020a] Lei et al. Li.  Generate neural template explanations for recommendation.  In CIKM, 2020.
  * Li [2020b] Lei et al. Li.  Towards controllable explanation generation for recommender systems via neural template.  In TheWebConf, 2020.
  * Li [2020c] Xueqi et al. Li.  Directional and explainable serendipity recommendation.  In TheWebConf, 2020.
  * Li [2021a] Lei et al. Li.  Personalized transformer for explainable recommendation.  arXiv preprint arXiv:2105.11601, 2021.
  * Li [2021b] Yang et al. Li.  Attribute-aware explainable complementary clothing recommendation.  World Wide Web, 2021.
  * Li [2021c] Zeyu et al. Li.  You are what and where you are: Graph enhanced attention network for explainable poi recommendation.  In CIKM, 2021.
  * Lin [2004] Chin-Yew Lin.  Rouge: A package for automatic evaluation of summaries.  In Text summarization branches out, pages 74–81, 2004.
  * Liu et al. [2019] Huafeng Liu, Jingxuan Wen, Liping Jing, Jian Yu, Xiangliang Zhang, and Min Zhang.  In2rec: Influence-based interpretable recommendation.  In Proceedings of the 28th ACM International Conference on Information and Knowledge Management, pages 1803–1812, 2019.
  * Liu [2020] Ninghao et al. Liu.  Explainable recommender systems via resolving learning representations.  In CIKM, 2020.
  * Musto et al. [2019] Cataldo Musto, Pasquale Lops, Marco de Gemmis, and Giovanni Semeraro.  Justifying recommendations through aspect-based sentiment analysis of users reviews.  In Proceedings of the 27th ACM Conference on User Modeling, Adaptation and Personalization, pages 4–12, 2019.
  * Naveed [2020] Sidra et al. Naveed.  On the use of feature-based collaborative explanations: An empirical comparison of explanation styles.  In UMAP, 2020.
  * Papineni [2002] et al. Papineni, Kishore.  Bleu: a method for automatic evaluation of machine translation.  In ACL, pages 311–318, 2002.
  * Silveira [2019] Thiago et al. Silveira.  How good your recommender system is? a survey on evaluations in recommendation.  IJMRC, 2019.
  * Sonboli [2021] Nasim et al. Sonboli.  Fairness and transparency in recommendation: The users’ perspective.  In UMAP, 2021.
  * Song [2012] Yading et al. Song.  A survey of music recommendation systems and future perspectives.  In CMMR, 2012.
  * Tai [2021] Chang-You et al. Tai.  User-centric path reasoning towards explainable recommendation.  In SIGIR, 2021.
  * Tan et al. [2016] Yunzhi Tan, Min Zhang, Yiqun Liu, and Shaoping Ma.  Rating-boosted latent topics: Understanding users and items with ratings and reviews.  In IJCAI, volume 16, pages 2640–2646, 2016.
  * Tan [2021] Juntao et al. Tan.  Counterfactual explainable recommendation.  In CIKM, 2021.
  * Tao et al. [2019] Yiyi Tao, Yiling Jia, Nan Wang, and Hongning Wang.  The fact: Taming latent factor models for explainability with factorization trees.  In Proceedings of the 42nd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 295–304, 2019.
  * Tsai and Brusilovsky [2019] Chun-Hua Tsai and Peter Brusilovsky.  Evaluating visual explanations for similarity-based recommendations: User perception and performance.  In Proceedings of the 27th ACM Conference on User Modeling, Adaptation and Personalization, pages 22–30, 2019.
  * Tsukuda [2020] Kosetsu et al. Tsukuda.  Explainable recommendation for repeat consumption.  In Recsys, 2020.
  * Wang et al. [2018] Nan Wang, Hongning Wang, Yiling Jia, and Yue Yin.  Explainable recommendation via multi-task learning in opinionated text data.  In The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval, pages 165–174, 2018.
  * Wang et al. [2019] Xiang Wang, Dingxian Wang, Canran Xu, Xiangnan He, Yixin Cao, and Tat-Seng Chua.  Explainable reasoning over knowledge graphs for recommendation.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 5329–5336, 2019.
  * Wang [2020] Chao et al. Wang.  Personalized employee training course recommendation with career development awareness.  In TheWebConf, 2020.
  * Wei [2007] Kangning et al. Wei.  A survey of e-commerce recommender systems.  In ICSSM, 2007.
  * Wu [2021] et al. Wu, Le.  A survey on neural recommendation: From collaborative filtering to content and context enriched recommendation.  arXiv preprint arXiv:2104.13030, 2021.
  * Xian [2020] Yikun et al. Xian.  Cafe: Coarse-to-fine neural symbolic reasoning for explainable recommendation.  In CIKM, 2020.
  * Xian [2021] Yikun et al. Xian.  Ex3: Explainable attribute-aware item-set recommendations.  In Recsys, 2021.
  * Yang et al. [2019] Xun Yang, Xiangnan He, Xiang Wang, Yunshan Ma, Fuli Feng, Meng Wang, and Tat-Seng Chua.  Interpretable fashion matching with rich attributes.  In Proceedings of the 42nd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 775–784, 2019.
  * Yu [2015] Yonghong et al. Yu.  A survey of point-of-interest recommendation in location-based social networks.  In AAAIW, 2015.
  * Zhang et al. [2014] Yongfeng Zhang, Guokun Lai, Min Zhang, Yi Zhang, Yiqun Liu, and Shaoping Ma.  Explicit factor models for explainable recommendation based on phrase-level sentiment analysis.  In Proceedings of the 37th international ACM SIGIR conference on Research & development in information retrieval, pages 83–92, 2014.
  * Zhang [2018] Yongfeng et al. Zhang.  Explainable recommendation: A survey and new perspectives.  arXiv preprint arXiv:1804.11192, 2018.
  * Zhang [2019] Shuai et al. Zhang.  Deep learning based recommender system: A survey and new perspectives.  ACM Computing Surveys (CSUR), 2019.


# Human-robot collaborative object transfer using human motion prediction
based on Cartesian pose Dynamic Movement Primitives

Antonis Sidiropoulos1,a, Yiannis Karayiannidis2,b and Zoe Doulgeri1,a
1Aristotle University of Thessaloniki, Department of Electrical and Computer
Engineering, Thessaloniki 54124, Greece<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Technology, Department of
Electrical Engineering, Göteborg, Sweeden<EMAIL_ADDRESS>The research
leading to these results has received funding by the EU Horizon 2020 Research
and Innovation Programme under grant agreement No 820767, project
CoLLaboratE.b This research is co-financed by the Swedish Research Council
(VR).

###### Abstract

In this work, the problem of human-robot collaborative object transfer to
unknown target poses is addressed. The desired pattern of the end-effector
pose trajectory to a known target pose is encoded using DMPs (Dynamic Movement
Primitives). During transportation of the object to new unknown targets, a
DMP-based reference model and an EKF (Extended Kalman Filter) for estimating
the target pose and time duration of the human’s intended motion is proposed.
A stability analysis of the overall scheme is provided. Experiments using a
Kuka LWR4+ robot equipped with an ATI sensor at its end-effector validate its
efficacy with respect to the required human effort and compare it with an
admittance control scheme.

This paper is a Post-Print version (i.e. final draft post-refereeing). The
publisher’s version can be accessed through DOI:
10.1109/ICRA48506.2021.9562035

##  I Introduction

The technological advancements during the past few decades have given a lot of
momentum to various research fields in robotics. Undoubtedly, physical human-
robot interaction constitutes a prolific research area, with numerous
applications, both in industrial and household environments [1]. The prospect
of having robots work collaboratively with humans has attracted a lot of
attention and can greatly enhance everyday life. For instance, tasks like
lifting or transferring objects occur quite frequently. Robots could actively
assist humans execute such tasks, mitigating their fatigue. To this end, a
consensus of a common trajectory and synchronization is required to execute
the task in synergy. Discord between the partners’ intention will result in
high interaction forces [2]. Hence, there is a need for intelligent proactive
robot control strategies with human intention prediction in order to minimize
human effort, promoting productivity and efficiency.

For the improvement of human robot collaboration, a lot of emphasis is placed
upon the prediction of the human’s motion/intention. Human motion prediction
algorithms are typically based on a minimum jerk motion model [3, 4, 5]. An
impedance model whose reference trajectory is produced by a radial basis
function (RBF) network adapted online by the human exerted force, was
suggested in [6]. Compared to admittance, the forces required by the human
were reduced, however boundedness of the reference trajectory is not proved
and their results show that the estimated trajectory lagged and did not
converge to the actual one. In [7], Iterative Learning Control is employed to
adjust the DMP’s trajectory for transferring an object to a new position.
However, this requires several repetitions for each new target until the task
is learned. The prediction of the goal in a high-five application with DMPs
using a KF was proposed in [8]. In [9] the idea of using an EKF was introduced
to predict on-line the intended handover position and time based on the
current position of the human hand and a DMP that parameterizes its motion.
Human-robot handover without involving any prediction of the handover pose and
time is considered in [10]. However, in [8, 9, 10], the human and the robot
are not physically coupled except at the end of the motion.

In our previous work, human-robot collaborative object transfer is addressed,
focusing only on Cartesian position [11]. In this paper, we extend our method
to include not only the position but also the target orientation. The robot is
only aware of the pattern of motion, but is agnostic to the target pose and
how fast the movement should be executed (time scaling). The objective is to
render the robot proactive in the execution of the learned motion pattern
scaled to unknown targets and time scales by anticipating humans’ intention,
thus minimizing their effort. To the best of our knowledge all previous works
on predicting human motion focus solely on the Cartesian position. A possible
reason is that any parameterization of orientation is non-linear which
complicates the formulation and design of the controller and observer. Hence,
any extension including the prediction of the target orientation and its real-
time use in the control loop is far from trivial and it can prove to be quite
challenging. In this work, a modification of the typical DMP orientation
formulation was required in order to solve the control problem. The main
contribution of this work is therefore a combined control and prediction
scheme that includes both target orientation and position and its stability
proof. The proposed scheme’s practical efficiency is demonstrated in a number
of experiments.

##  II Proposed approach

Our focus is on tasks, where the motion pattern is essentially the same, but
the initial/target poses and time duration of the movement can be different
leading to the spatio-temporal scaling of the motion pattern. This occurs for
instance in box stacking tasks. We make use of DMPs to encode the Cartesian
pose of the robot’s end-effector during a point to point motion, that is
recorded from a kinesthetic demonstration. This recorded motion pattern is
employed by the robot for collaborative transportation of an object to
different targets and at different time scalings unknown to the robot. Human-
Human collaborative object transfer studies reveal that object motion is
highly correlated with a motion pattern [12, 2].

For position, let the motion to a new target $\bm{p}_{g}\in\mathbb{R}^{3}$
with time scaling $\tau_{p}$, be generated by the position DMP model:

$\ddot{\bm{p}}=\frac{1}{\tau_{p}^{2}}(\alpha_{z}\beta_{z}(\bm{p}_{g}-\bm{p})-\alpha_{z}\tau_{p}\dot{\bm{p}}+g_{f}(t/\tau_{p})\bm{K}_{p_{g}}\bm{f}_{p}(t/\tau_{p}))$
(1)

where $\bm{p},\dot{\bm{p}},\ddot{\bm{p}}\in\mathbb{R}^{3}$ are the position,
velocity and acceleration of the robot’s end-effector and the rest terms are
detailed in Appendix B. Accordingly for the orientation, the motion to a new
target orientation $\bm{Q}_{g}\in\mathbb{S}^{3}$ with time scaling $\tau_{o}$
is generated by the following orientation DMP formulation:

$\ddot{\bm{q}}^{\prime}=\frac{1}{\tau_{o}^{2}}(\alpha_{z}\beta_{z}(\bm{q}^{\prime}_{g}-\bm{q}^{\prime})-\alpha_{z}\tau_{o}\dot{\bm{q}}^{\prime}+g_{f}(t/\tau_{o})\bm{K}_{q_{g}}\bm{f}_{o}(t/\tau_{o}))$
(2)

where $\bm{q}^{\prime}=\log(\bm{Q}*\bar{\bm{Q}}_{0})\in\mathbb{R}^{3}$ with
$\bm{Q}$, $\bm{Q}_{0}\in\mathbb{S}^{3}$ being the current and initial
orientation and $\bar{\bm{Q}}_{0}$ the conjugate of $\bm{Q}_{0}$, $\log(.)$
the quaternion logarithm and
$\bm{q}^{\prime}_{g}=\log(\bm{Q}_{g}*\bar{\bm{Q}}_{0})$, Unit quaternion
preliminaries can be found in Appendix A, and more details on (2) in Appendix
B.

In the above DMP models, the forcing terms $\bm{f}_{p},\bm{f}_{o}$ comprise of
a weighted sum of Gaussians which encode the motion pattern. The matrices
$\bm{K}_{p_{g}}$, $\bm{K}_{q_{g}}$ provide the spatial scaling of the motion
based on the target and the gating function $g_{f}(.)$ ensures that the
forcing term vanishes at the end of the motion (Appendix B).

###### Remark 1

Writing the DMP model with the orientation anchored to the initial orientation
$\bm{Q}_{0}$ is important as it decouples $\bm{q}^{\prime}$ from $\bm{Q}_{g}$
which greatly simplifies the orientation DMP formulation in case time varying
target estimates are to be utilized. This is in contrast to
$\bm{q}^{\prime}=\log(\bm{Q}_{g}*\bar{\bm{Q}})$ from [13], where the
orientation is anchored to the target orientation $\bm{Q}_{g}$. This
formulation is exploited in the design of the reference model detailed in the
next Section.

###### Remark 2

The use of separate time scaling variables for position and orientation,
$\tau_{p},\tau_{o}$, is adopted. This provides greater versatility and
generalization, since it accommodates also the scenario of having position and
orientation patterns with different temporal duration during execution.

During the collaborative object transfer we assume the robot grasps the object
rigidly and compensates its weight, while a F/T sensor is mounted at its end-
effector. The external force/torque exerted by the human is calculated by
subtracting from the F/T sensor’s measurement the object’s wrench. It is
assumed that no other contacts with the environment occur, as this would
require additional sensors to discriminate contact forces from the human
intended forces. A velocity controlled robot is considered, implying that any
reference velocity can be accurately tracked. The core idea is to design a
control scheme that will: 1) render the robot compliant to the forces/torques
exerted by the human so that he can move the object along the trajectory
intended by him, 2) make the robot proactive in the tracking of the intended
trajectory, by estimating online the target and time-scaling, in order to
minimize the effort required by the human. To achieve this objective we
propose the control scheme depicted in Fig. 1 where:

  * •

The DMP based reference model takes as input the external force/torque exerted
by the human, and using the current state
$\bm{p},\dot{\bm{p}},\bm{Q},\bm{\omega}$ (pose, velocity) and the estimates of
the target pose and time scalings,
$\hat{\bm{p}}_{g},\hat{\bm{Q}}_{g},\hat{\tau}_{p},\hat{\tau}_{o}$ generates a
DMP-model based estimated trajectory, that is shaped by the external
force/torque.

  * •

The DMP based EKF takes as input the force/torque exerted by the human and the
current state to produce estimates of the target pose and time-scaling.

In the subsequent sections we dwell on the formulation and design of the
reference model and the observer.

Figure 1: Proposed approach.

##  III Reference model

The reference model for position is formulated as:

$\bm{M}_{p}\ddot{\bm{p}}=\bm{M}_{p}\hat{\ddot{\bm{p}}}+\bm{f}_{ext}$ (3)
$\hat{\ddot{\bm{p}}}=\frac{1}{\hat{\tau}_{p}^{2}}(\alpha_{z}\beta_{z}(\hat{\bm{p}}_{g}-\bm{p})-\alpha_{z}\hat{\tau}_{p}\dot{\bm{p}}+g_{f}(t/\hat{\tau}_{p})\hat{\bm{K}}_{p_{g}}\bm{f}_{p}(t/\hat{\tau}_{p}))$

where $\bm{f}_{ext}$ is the force exerted by the human,
$\bm{M}_{p}=diag(m_{x},m_{y},m_{z})>0$ is the reference model’s inertia which
weights the effect of $\bm{f}_{ext}$ on the reference model’s trajectory and
$\hat{\ddot{\bm{p}}}$ is the estimated acceleration based on the position DMP
model (1), using the estimated target $\hat{\bm{p}}_{g}$ and time scaling
$\hat{\tau}_{p}$.

Similar to position, the reference model for orientation is formulated as:

$\bm{M}_{o}\dot{\bm{\omega}}=\bm{M}_{o}\hat{\dot{\bm{\omega}}}+\bm{\tau}_{ext}$
(4)
$\hat{\dot{\bm{\omega}}}=vec\\{2(\bm{J}_{q^{\prime}}\dot{\bm{q}}^{\prime}+\bm{J}_{q^{\prime}}\hat{\ddot{\bm{q}}}^{\prime})*\bar{\bm{Q}}^{\prime}-\frac{1}{2}\begin{bmatrix}||\bm{\omega}||^{2}\\\
\bm{0}_{3\times 1}\end{bmatrix}\\}$ (5)

with

$\hat{\ddot{\bm{q}}}^{\prime}=\frac{1}{\hat{\tau}_{o}^{2}}(\alpha_{z}\beta_{z}(\hat{\bm{q}}^{\prime}_{g}-\bm{q}^{\prime})-\alpha_{z}\hat{\tau_{o}}\dot{\bm{q}}^{\prime}+g_{f}(t/\hat{\tau}_{o})\hat{\bm{K}}_{q_{g}}\bm{f}_{o}(t/\hat{\tau}_{o}))$
(6)

where $\bm{M}_{o}=diag(I_{x},I_{y},I_{z})>0$ is the reference model’s inertia,
$\bm{\tau}_{ext}$ is the torque exerted by the human and
$\hat{\dot{\bm{\omega}}}$ is the estimated rotational acceleration
corresponding to the estimated acceleration $\hat{\ddot{\bm{q}}}^{\prime}$
(see (20) from Appendix A). Finally $vec\\{.\\}$ denotes the vector part of a
quaternion. Equation (6) is derived from (2) using the estimated target
$\hat{\bm{q}}^{\prime}_{g}=\log(\hat{\bm{Q}}_{g}*\bar{\bm{Q}}_{0})$ and time
scaling $\hat{\tau}_{o}$.

###### Remark 3

Further to remark 1, notice that in (4), (5), if
$\bm{q}^{\prime}=\log(\bm{Q}_{g}*\bar{\bm{Q}})$ was used instead of
$\bm{q}^{\prime}=\log(\bm{Q}*\bar{\bm{Q}}_{0})$ then apart from
$\hat{\bm{q}}^{\prime}_{g}$, all other terms would also be dependent on
$\hat{\bm{Q}}_{g}$ (and some of them on its derivative too). This would
complicate the design of the reference model and the results of the stability
analysis in Section V would most likely be inconclusive.

##  IV DMP-based EKF observer for target and time scale prediction

We assume that the trajectory generated from the reference model (3), (4)
corresponds to the trajectory produced by the DMP models (1), (2) for constant
target $\bm{p}_{g},\bm{Q}_{g}$ and time-scalings $\tau_{p},\tau_{o}$, which we
want to estimate. This assumption is reasonable because, as highlighted in
Section II, we focus on tasks where the motion pattern is essentially the
same, but the initial/target poses and time duration of the movement can be
different leading to the spatio-temporal scaling of the motion pattern. To
this end, we construct a DMP-based EKF with states
$\bm{\theta}_{p}=\left[\bm{p}_{g}^{T}\ \tau_{p}\right]^{T}$,
$\bm{\theta}_{o}=\left[{\bm{q}^{\prime}_{g}}^{T}\ \tau_{o}\right]^{T}$ with
the following state and measurement equations for $i\in\\{p,o\\}$,:

$\dot{\bm{\theta}}_{i}=\bm{0}_{4\times 1}$ (7)
$\bm{z}_{i}=\bm{h}_{i}(\bm{\theta}_{i},\bm{s}_{i},t)$ (8)

where $\bm{z}_{p}=\ddot{\bm{p}}$, $\bm{z}_{o}=\dot{\bm{\omega}}$,
$\bm{s}_{p}=[\bm{p}^{T}\ \dot{\bm{p}}^{T}]^{T}$,
$\bm{s}_{o}=[\bm{Q}^{{}^{\prime}T}\ \bm{\omega}^{T}]^{T}$ and:

$\bm{h}_{p}(\bm{\theta}_{p},\bm{s}_{p},t)\triangleq\frac{1}{\tau_{p}^{2}}(\alpha_{z}\beta_{z}(\bm{p}_{g}-\bm{p})-\alpha_{z}\tau_{p}\dot{\bm{p}}+g_{f}(\frac{t}{\tau_{p}})\bm{K}_{p_{g}}\bm{f}_{p}(\frac{t}{\tau_{p}}))$
$\bm{h}_{o}(\bm{\theta}_{o},\bm{s}_{o},t)\triangleq
vec\\{2(\bm{J}_{q^{\prime}}\dot{\bm{q}}^{\prime}+\bm{J}_{q^{\prime}}\ddot{\bm{q}}^{\prime})*\bar{\bm{Q}}^{\prime}\\}$

with $\ddot{\bm{q}}^{\prime}$ given by (2).

For each system $i\in\\{p,o\\}$ in (7)-(8) we construct a fading memory EKF
observer [14, 15] with projection [15] and normalization [16]:

$\dot{\hat{\bm{\theta}}}_{i}=\bm{K}_{i}(\bm{z}_{i}-\hat{\bm{z}}_{i})$ (9)
$\hat{\bm{z}}_{i}=\bm{h}_{i}(\hat{\bm{\theta}}_{i},\bm{s}_{i},t)$ (10)

where $\hat{\bm{\theta}}_{i}$ is the state estimate of the $i_{th}$ observer
and $\bm{K}_{i}\in\mathbb{R}^{4\times 3}$ is a time varying gain matrix given
by:

$\bm{K}_{i}=\bm{N}_{i}(t)\bm{P}_{i}(t)\bar{\bm{C}}_{i}^{T}(t)\bm{R}_{i}^{-1}$
(11)

where $\bm{N}_{i}(t)$ is a projection matrix which ensures the estimates
respect certain bounds, $\bm{P}_{i}(t)$ is given by the solution of the the
following equation [14], [15]:

$\dot{\bm{P}}_{i}=\begin{cases}2a_{p}\bm{P}_{i}+(\bm{K}_{i}\bm{R}_{i}-\bm{P}_{i}\bar{\bm{C}}_{i}^{T})\bm{R}_{i}^{-1}(\bm{K}_{i}\bm{R}_{i}-\bm{P}_{i}\bar{\bm{C}}_{i}^{T})^{T}\\\
+\bm{Q}_{i}-\bm{P}_{i}\bar{\bm{C}}_{i}^{T}(t)\bm{R}_{i}^{-1}\bar{\bm{C}}_{i}\bm{P}_{i}\quad,\
||\bm{P}_{i}||\leq\rho_{2}\\\ \bm{0}\ ,\ otherwise\end{cases}$ (12)

$a_{p}>0$, $\bm{R}_{i}$, $\bm{Q}_{i}$ are the measurement and process noise
covariance matrices and $\bar{\bm{C}}_{i}(t)$ the linearized measurement
equation matrix with normalization found by
$\bar{\bm{C}}_{i}(t)=\bm{C}_{i}(t)/c_{n,i}$, where
$\bm{C}_{i}(t)=\left.\frac{\partial\bm{h}_{i}(\bm{\theta}_{i},\bm{u}_{i},t)}{\partial\bm{\theta}_{i}}\right|_{\bm{\theta}_{i}=\hat{\bm{\theta}}_{i}}$
and $c_{n,i}=\sqrt{1+\lambda_{max}(\bm{C}_{i}(t)\bm{C}_{i}^{T}(t))}$.
Normalization is used to ensure the boundedness of the observer’s update law,
irrespective of the boundedness of $\bm{s}_{i}$ [16]. Notice that
$\bm{C}_{i}(t)=\left.\frac{\partial\bm{h}_{i}(\bm{\theta}_{i},\bm{u}_{i},t)}{\partial\bm{\theta}_{i}}\right|_{\bm{\theta}_{i}=\hat{\bm{\theta}}_{i}}$
can be computed analytically, similarly to [9], but its analytical expression
is omitted here due to space restrictions. It is worth mentioning that if
$\log(\bm{Q}_{g}*\bar{\bm{Q}})$ was used instead of $\bm{q}^{\prime}_{g}$,
then the analytic derivation becomes practically intractable and one would
have to resort to numerical differentiation, increasing the observer’s
computational complexity and possibly compromising its performance. The design
constant $\rho_{2}>0$ is used to ensure that the covariance matrix remains
bounded. The projection matrix $\bm{N}_{i}$ is derived from the constraints
$\bm{D}_{i}\bm{\theta}_{i}\leq\bm{d}_{i}$ based on a least squares approach
[15], where $\bm{d}_{i}=[\bar{\bm{\theta}}_{i}^{T}\
-\underline{\bm{\theta}}_{i}^{T}]^{T}$ and $\bm{D}_{i}=\begin{bmatrix}\
\bm{I}_{4}\ -\bm{I}_{4}\end{bmatrix}^{T}$. Bounds $\bar{\bm{\theta}}_{i},\
\underline{\bm{\theta}}_{i}$ stem from the physical interpretation of the
estimated parameters, i.e. the target position is constrained by the robot’s
workspace. Moreover, the time scalings are positive and finite. The
projections matrix $\bm{N}_{i}$ is then given by [15]:

$\bm{N}_{i}=\left(\bm{I}_{4}-\bar{\bm{D}}_{i}^{T}(\bar{\bm{D}_{i}}\bar{\bm{D}}_{i}^{T})^{-1}\bar{\bm{D}}_{i}\right)$
(13)

with the active constraints satisfying
$\bar{\bm{D}}_{i}\hat{\bm{\theta}}_{i}=\bar{\bm{d}}_{i}$ and
$\bar{\bm{D}}_{i,j}\dot{\hat{\bm{\theta}}}_{i}>\bm{0}$, where
$\bar{\bm{D}}_{i,j}$ is the $j_{th}$ row of $\bar{\bm{D}}_{i}$ and
$\bar{\bm{D}}_{i},\bar{\bm{d}}_{i}$ are subset of the rows of
$\bm{D}_{i},\bm{d}_{i}$ respectively.

Finally, the measurement error can be obtained from (3), (4), i.e.
$\bm{z}_{p}-\hat{\bm{z}}_{p}=\ddot{\bm{p}}-\hat{\ddot{\bm{p}}}=\bm{M}_{p}^{-1}\bm{f}_{ext}$
and
$\bm{z}_{o}-\hat{\bm{z}}_{o}=\dot{\bm{\omega}}-\hat{\dot{\bm{\omega}}}=\bm{M}_{o}^{-1}\bm{\tau}_{ext}$.
Therefore, for the update law (9) we can utilize the following expression:

$\dot{\hat{\bm{\theta}}}_{i}=\bm{K}_{i}\bm{M}_{i}^{-1}\frac{\bm{\nu}_{i}}{c_{n,i}}$
(14)

where $\bm{\nu}_{p}=\bm{f}_{ext}$ and $\bm{\nu}_{o}=\bm{\tau}_{ext}$.

###### Remark 4

For an observer of the form (9), the covariance matrix $\bm{P}$ satisfies [17]
$\dot{\bm{P}}=(\bm{K}\bm{R}-\bm{P}\bm{C}^{T})\bm{R}^{-1}(\bm{K}\bm{R}-\bm{P}\bm{C}^{T})^{T}+\bm{Q}-\bm{P}\bm{C}^{T}\bm{R}\bm{C}\bm{P}$
for any matrix $\bm{K}$. In the case of the fading memory filter [15], [14]
the term $a_{p}\bm{P}$, $a_{p}>0$ is added to $\dot{\bm{P}}$. The Kalman gain
is derived by minimizing $\dot{\bm{P}}$ w.r.t. $\bm{K}$ which yields
$\bm{K}=\bm{P}\bm{C}^{T}\bm{R}^{-1}$. When projection is employed, the Kalman
gain is further modified by the projection matrix $\bm{N}$ [15], resulting in
(11) which in turn yields the covariance matrix update (12).

###### Remark 5

We chose specifically the EKF with fading memory since is has been shown to be
robust against non-linearities (see [14] and Chapter $7.4$ from [15]).
Moreover, we compared it to the UKF (Unscented Kalman Filter) and we found the
EKF to perform slightly better in our case, which has also been observed in
other applications [18, 19, 20].

##  V Stability Analysis

Due to the nonlinearity of the motion pattern with respect to the target and
time scaling, the estimates of the EKF are guaranteed to converge to the
actual ones locally, as studied in [9]. Thanks to the projection however the
estimates remain bounded. However, this does not imply that the reference
model’s state (hence the robot’s state) will remain bounded, which is crucial.
Concerning this, the following theorem can be proven regarding the reference
model:

###### Theorem 1

The model reference given by (3), (4) along with the observer based on the EKF
given by (14) ensures that
$\bm{p},\dot{\bm{p}},\bm{Q},\bm{\omega}\in\bm{L}_{\infty}$ if
$\bm{f}_{ext},\bm{\tau}_{ext}\in\bm{L}_{\infty}\cap\bm{L}_{2}$.

###### Proof:

The proof for the boundedness of $\bm{p},\dot{\bm{p}}$ given that
$\bm{f}_{ext}\in L_{2}\cap L_{\infty}$ is identical to the proof in [11]. The
orientation is also bounded since $\bm{Q}\in\mathbb{S}^{3}$. For the
rotational velocity from (4), (5) we get:

$\dot{\bm{\omega}}=vec\\{2(\bm{J}_{q^{\prime}}\dot{\bm{q}}^{\prime}+\bm{J}_{q^{\prime}}\hat{\ddot{\bm{q}}}^{\prime})*\bar{\bm{Q}}^{\prime}-\frac{1}{2}\begin{bmatrix}||\bm{\omega}||^{2}\\\
\bm{0}_{3\times 1}\end{bmatrix}\\}+\bm{M}_{o}^{-1}\bm{\tau}_{ext}$

Substituting $\dot{\bm{\omega}}$ from (20), replacing $\bm{Q}$ by
$\bm{Q}^{\prime}$ and $\bm{q}$ by $\bm{q}^{\prime}$ we can arrive after some
mathematical calculations at:

$\displaystyle\bm{J}_{q^{\prime}}(\ddot{\bm{q}}^{\prime}-\hat{\ddot{\bm{q}}}^{\prime})=\frac{1}{2}\bm{T}_{ext}*\bm{Q}^{\prime}$

where $\bm{T}_{ext}=[0\ (\bm{M}_{o}^{-1}\bm{\tau}_{ext})^{T}]^{T}$.
Multiplying both sides by $\bm{J}_{Q^{\prime}}$ and using the identity
$\bm{J}_{Q^{\prime}}\bm{J}_{q^{\prime}}=\bm{I}_{3}$ [13], we arrive at:

$\ddot{\bm{q}}^{\prime}-\hat{\ddot{\bm{q}}}^{\prime}=\frac{1}{2}\bm{J}_{Q^{\prime}}(\bm{T}_{ext}*\bm{Q}^{\prime})$

Substituting $\hat{\ddot{\bm{q}}}^{\prime}$ from (6) we obtain:

$\hat{\tau}_{o}^{2}\ddot{\bm{q}}^{\prime}=-\alpha_{z}\beta_{z}\bm{q}^{\prime}-\alpha_{z}\hat{\tau}_{o}\dot{\bm{q}}^{\prime}+\hat{\tau}_{o}^{2}\bm{d}$
(15)

where
$\bm{d}=\frac{1}{\hat{\tau}_{o}^{2}}(\alpha_{z}\beta_{z}\hat{\bm{q}}^{\prime}_{g}+g_{f}(\hat{x})\hat{\bm{K}}_{q_{g}}\bm{f}_{o}(\hat{x})+\frac{1}{2}\bm{J}_{Q^{\prime}}(\bm{T}_{ext}*\bm{Q}^{\prime}))$
can be viewed as a time varying bounded disturbance since
$\hat{\tau}_{o},\hat{\bm{q}}^{\prime}_{g},\bm{T}_{ext},\bm{Q}^{\prime},\bm{J}_{Q^{\prime}},\bm{f}_{o}(\hat{x})\in\bm{L}_{\infty}$.
It is important to highlight that the derivation of (15) was made possible by
the modified orientation DMP introduced in this work (Appendix B). Thus, for
the rest of the proof we can follow [11], to show that
$\bm{q}^{\prime},\dot{\bm{q}}^{\prime}\in\bm{L}_{\infty}$ hence
$\bm{q},\dot{\bm{q}}\in\bm{L}_{\infty}$, therefore from (19) we conclude that
$\bm{\omega}\in\bm{L}_{\infty}$.

The proof for the boundedness of $\bm{q},\dot{\bm{q}}$ is provided for
completeness in Appendix C. ∎

Notice that $\bm{f}_{ext},\bm{\tau}_{ext}\rightarrow\bm{0}$, (14) implies
$\dot{\hat{\bm{\theta}}}_{i}\rightarrow 0$, and from (3), (4)
$\bm{p}\rightarrow\hat{\bm{p}}_{g}$, $\dot{\bm{p}}\rightarrow\bm{0}$,
$\bm{Q}\rightarrow\hat{\bm{Q}}_{g}$, $\bm{\omega}\rightarrow\bm{0}$.
Considering that the condition $\bm{f}_{ext},\bm{\tau}_{ext}\rightarrow\bm{0}$
is met iff the human has reached the desired target pose, this further implies
that $\hat{\bm{p}}_{g}\rightarrow\bm{p}_{g}$,
$\hat{\bm{Q}}_{g}\rightarrow\bm{Q}_{g}$.

###### Remark 6

Theorem 1 holds for any observer combined with the reference model (3), (4) as
long as it produces bounded estimates and the estimation update law is bounded
and has bounded energy, i.e. $\hat{\bm{\theta}}\in L_{\infty}$ and
$\dot{\hat{\bm{\theta}}}\in L_{\infty}\cap L_{2}$.

##  VI Experimental results

The experimental setup consists of a Kuka LWR4+ robot equipped with an ATI F/T
sensor at its wrist. A rectangular long box is mounted at the robot’s wrist as
shown in Fig. 2. The object’s dynamics were identified offline and were
compensated during the experiments. The training phase involved a single
demonstration by kinesthetically guiding the robot holding the box from the
other side, and training a DMP for position and orientation. During the
collaborative object transfer to new targets from different initial poses the
robot is under velocity control and is driven by the reference velocity
produced by the proposed approach DMP+EKF (3), (4). We have further
implemented an admittance controller for comparison. The control cycle was set
to $2ms$. The parameters chosen for the DMP are $N=30$, $\alpha_{z}=40$,
$\beta_{z}=10$, for the reference model $\bm{M}_{p}=2\bm{I}_{3}$,
$\bm{M}_{o}=0.1\bm{I}_{3}$, and for the observer
$\bm{P}_{p}(0)=\bm{P}_{o}(0)=diag(1,1,1,10)$,
$\bm{R}_{p}=\bm{R}_{o}=2000\bm{I}_{3}$,
$\bm{Q}_{p}=\bm{Q}_{o}=0.001\bm{I}_{4}$, $a_{p}=1.001$, $\rho_{2}=10000$,
$\bar{\bm{\theta}}_{p}=[0.75\ 0.7\ 0.95\ 60.0]^{T}$,
$\underline{\bm{\theta}}_{p}=[-0.75\ -0.7\ -0.2\ 1.0]^{T}$,
$\bar{\bm{\theta}}_{o}=[2\pi\ 2\pi\ 2\pi\ 60.0]^{T}$,
$\underline{\bm{\theta}}_{o}=[-2\pi\ -2\pi\ -2\pi\ 1.0]^{T}$. For the EKF the
discrete implementation was employed [15]. The admittance model
$\bm{M}\dot{\bm{V}}+\bm{D}\bm{V}=\bm{F}_{ext}$, with
$\bm{V}=[\dot{\bm{p}}^{T}\ \bm{\omega}^{T}]^{T}$ and
$\bm{F}_{ext}=[\bm{f}_{ext}^{T}\ \bm{\tau}_{ext}^{T}]^{T}$ was utilized with
the following parameters, tuned manually for best performance:
$\bm{M}=diag(1.3\bm{I}_{3},0.08\bm{I}_{3})$ and
$\bm{D}=diag(25\bm{I}_{3},0.6\bm{I}_{3})$. In all scenarios, the desired
target pose is about $1$ cm from the placing surface, so that no contact
forces emerge.

Results for three different collaborative object transfers are shown in Fig.
3. In all cases the initial target estimate was set equal to the robot’s
initial pose. The initial time scaling estimate was set to $6$, while the
demonstrations duration was $4.7$ sec. Thus we make a more conservative
initial estimate of the time scaling by assuming that the motion will be
executed slower than in the demonstration. The DMP clock and the estimation
process commence when $||\bm{F}_{ext}||>1$ N, to synchronize the robot with
the human interaction. Notice that the estimates converge to a small region
around the actual target either before the end of the motion (e.g.
$\hat{\bm{p}}_{g,y}$ and $\hat{\bm{q}}^{\prime}_{x}$ in experiment $1$) or by
evolving towards the right direction (e.g. $\hat{\bm{p}}_{g,y}$ and
$\hat{\bm{q}}^{\prime}_{z}$ in experiment $2$). This is crucial as it
contributes considerably to the reduction of the human’s effort, as can be
seen in Fig. 4, where the absolute power required by the user using DMP+EKF
(blue graphs) is compared to using admittance (red graphs). Concerning the
times scaling, increase in the estimates means that the estimator infers that
the motion is to be executed slower, while decrease implies the converse.
Moreover, all time scaling estimates converge ultimately to a steady state
value.

We have further conducted experiments of a collaborative object transfer to a
new target with 5 users. Each user repeated the experiment 5 times to extract
some statistical measures. In all cases the initial target estimate was set
equal to the robot’s initial pose. In Fig. 5 the average power $\pm$ the
standard deviation over five repetitions are depicted for each user. Using
DMP+EKF the total absolute work for all users varied within $[1.28,\ 2.11]\
J$, the mean $L_{2}$ norm of the force $[0.77,\ 1.17]\ N$ and the mean $L_{2}$
norm of the torque $[0.13,\ 0.17]\ Nm$. In contrast, with admittance, the work
varied within $[7.1,\ 11.9]\ J$, the mean $L_{2}$ norm of the force $[4.26,\
6.36]\ N$ and the mean $L_{2}$ norm of the torque $[0.42,\ 0.71]\ Nm$.

Figure 2: Experimental setup.

(a) experiment 1

(b) experiment 2

(c) experiment 3

Figure 3: Estimation results for the three first object transfer experiments.
The estimates of the target position and orientation are plotted with solid
blue lines and the corresponding robot’s position/orientation with dash-dotted
green line. The time scaling estimates are plotted with light brown lines. For
orientation all quantities are expressed as the quaternion logarithm w.r.t the
initial orientation $\bm{Q}_{0}$

(a) experiment 1

(b) experiment 2

(c) experiment 3

Figure 4: Absolute power for the three first object transfer experiments. The
blue graphs correspond to DMP+EKF and the red graphs to admittance. Figure 5:
Power $\pm$ the standard deviation over five object transfer repetitions for
each user. The blue graphs correspond to using DMP+EKF and the red graphs to
Admittance.

##  VII Conclusions

In this work a DMP-based reference model and an EKF observer for predicting
the target pose and time scaling for assisting the human proactively in the
transportation of an object was proposed. The stability analysis that was
carried out proves that the proposed scheme guarantees the boundedness of the
reference model and observer. Experimental results validate the proposed
approach, highlighting its practical benefits and efficiency with respect to
human effort minimization. In this work the object’s weight was assumed known
a priori and only human exerted forces were considered. Our future work is
oriented towards the extension of the proposed method to handle unknown object
dynamics and the discrimination of human wrenches from the wrenches emerging
due to contact with the environment.

## Appendix A - Unit Quaternion Preliminaries

Given a rotation matrix $\bm{R}\in SO(3)$, an orientation can be expressed in
terms of the unit quaternion $\bm{Q}\in\mathbb{S}^{3}$ as $\bm{Q}=[w\
\bm{v}^{T}]^{T}=[\cos(\theta)\ \sin(\theta)\bm{k}^{T}]$, where
$\bm{k}\in\mathbb{R}^{3}$, $2\theta\in[0\ 2\pi)$ are the equivalent unit axis
- angle representation. The quaternion product between the unit quaternions
$\bm{Q}_{1}$, $\bm{Q}_{2}$ is
$\bm{Q}_{1}*\bm{Q}_{2}=\begin{bmatrix}w_{1}w_{2}-\bm{v}_{1}^{T}\bm{v}_{2}\\\
w_{1}\bm{v}_{2}+w_{2}\bm{v}_{1}+\bm{v}_{1}\times\bm{v}_{2}\end{bmatrix}$. The
inverse of a unit quaternion is equal to its conjugate which is
$\bm{Q}^{-1}=\bar{\bm{Q}}=[w\ -\bm{v}^{T}]^{T}$. The quaternion logarithm is
$\bm{q}=\log(\bm{Q})$, where $\log:\ \mathbb{S}^{3}\rightarrow\mathbb{R}^{3}$
is defined as:

$\log(\bm{Q})\triangleq\left\\{\begin{matrix}2\cos^{-1}(w)\frac{\bm{v}}{||\bm{v}||},\
|w|\neq 1\\\ [0,0,0]^{T},\ \text{otherwise}\end{matrix}\right.$ (16)

The quaternion exponential is $\bm{Q}=\exp(\bm{q})$, where $\exp:\
\mathbb{R}^{3}\rightarrow\mathbb{S}^{3}$ is defined as:

$\exp(\bm{q})\triangleq\left\\{\begin{matrix}[\cos(||\bm{q}/2||),\sin(||\bm{q}/2||)\frac{\bm{q}^{T}}{||\bm{q}||}]^{T},\
||\bm{q}||\neq 0\\\ [1,0,0,0]^{T},\ \text{otherwise}\end{matrix}\right.$ (17)

If we limit the domain of the exponential map $\exp:\
\mathbb{R}^{3}\rightarrow\mathbb{S}^{3}$ to $||\bm{v}||<\pi$ and the domain of
the logarithmic map to $\mathbb{S}^{3}/([-1,0,0,0]^{T})$, then both mappings
become one-to-one, continuously differentiable and inverse to each other. The
equations that relate the time derivative of $\bm{q}$ and the rotational
velocity $\bm{\omega}$ and acceleration $\dot{\bm{\omega}}$ of $\bm{Q}$ are
[13]:

$\dot{\bm{q}}=\bm{J}_{Q}(\bm{\Omega}*\bm{Q})$ (18)
$\bm{\Omega}=2(\bm{J}_{q}\dot{\bm{q}})*\bar{\bm{Q}}$ (19)
$\dot{\bm{\Omega}}=2(\dot{\bm{J}}_{q}\dot{\bm{q}}+\bm{J}_{q}\ddot{\bm{q}})*\bar{\bm{Q}}-\frac{1}{2}\begin{bmatrix}||\bm{\omega}||^{2}\\\
\bm{0}_{3\times 1}\end{bmatrix}$ (20)

where $\bm{\Omega}=[0\ \bm{\omega}^{T}]^{T}$ and

$\bm{J}_{Q}=2\begin{bmatrix}\frac{\theta\cos(\theta)-\sin(\theta)}{\sin^{2}(\theta)}\bm{k}&\frac{\theta}{\sin(\theta)}\bm{I}_{3}\end{bmatrix}$
(21) $\bm{J}_{q}=\frac{1}{2}\begin{bmatrix}-\sin(\theta)\bm{k}^{T}\\\
\frac{\sin(\theta)}{\theta}(\bm{I}_{3}-\bm{k}\bm{k}^{T})+\cos(\theta)\bm{k}\bm{k}^{T}\end{bmatrix}$
(22)

## Appendix B - DMP Preliminaries

###  VII-A Cartesian position encoding

A DMP for encoding a Cartesian position point-to-point motion can be expressed
as [21]:

$\displaystyle\ddot{\bm{p}}=\frac{1}{\tau^{2}}(\alpha_{z}\beta_{z}(\bm{p}_{g}-\bm{p})-\alpha_{z}\tau\dot{\bm{p}}+g_{f}(x)\bm{K}_{p_{g}}\bm{f}(x))$
(23) $\displaystyle\tau\dot{x}=1\ ,\ x(0)=0$ (24)

The desired motion is encoded by the forcing term,
${\bm{f}}(x)=\frac{\sum_{i=1}^{N}\bm{W}_{i}\psi_{i}(x)}{\sum_{i=1}^{N}\psi_{i}(x)}$
which is the weighted sum of $N$ Gaussian kernels, with
$\psi_{i}(x)=\exp(-h_{i}(x-c_{i})^{2})$. The matrix
$\bm{W}\in\mathbf{R}^{3\times N}$, with $\bm{W}_{i}$ denoting the $i_{th}$
column, contains in each row the weights for each Cartesian coordinate, which
can be learned using Least Squares or Locally Weighted Regression (LWR) [22]
based on the demonstrated data. The DMP evolves based on the phase variable
$x$, used to avoid direct time dependency. We use a linear canonical system
(24) as in [11]. The variable $\tau>0$ provides temporal scaling of the
encoded motion pattern and the matrix
$\bm{K}_{p_{g}}=diag(\bm{p}_{g}-\bm{p}_{0})$ provides spatial scaling. The
sigmoid gating $g_{f}(x)=\frac{1}{1+e^{a_{g}(x-c_{g})}}$ ensures that the
forcing term fades to zero at $x=1$ as in [23], thus for $x\geq 1$ (23) acts
as a pure spring-damper and converges asymptotically to the goal $\bm{p}_{g}$.
Integrating (24) and substituting $x$ in (23) we get (1).

###  VII-B Cartesian orientation encoding

A formulation for orientation DMP that avoids undesired oscillations during
reproduction as opposed to [7] is proposed in [13] and can be written as
follows :

$\displaystyle\ddot{\bm{q}}^{\prime}=\frac{1}{\tau^{2}}(\alpha_{z}\beta_{z}(\bm{q}^{\prime}_{g}-\bm{q}^{\prime})-\alpha_{z}\tau\dot{\bm{q}}^{\prime}+g_{f}(x)\bm{K}_{q_{g}}\bm{f}(x))$
(25) $\displaystyle\tau\dot{x}=1\ ,\ x(0)=0$ (26)

where, in [13], $\bm{q}^{\prime}=\log(\bm{Q}_{g}*\bar{\bm{Q}})$ with $\bm{Q}$,
$\bm{Q}_{g}$ the current and target orientation in unit quaternions
respectively and $\bm{K}_{q_{g}}=diag(\log(\bm{Q}_{g}*\bar{\bm{Q}}_{0}))$.
However, $\bm{q}^{\prime}$ couples the DMP’s state to $\bm{Q}_{g}$, which in
our case is estimated online. Adopting this would incur extra dynamics to the
DMP, affecting the execution and making the derivation of the proof in section
V not possible. Instead, we retain the formulation given by (25), which
exploits the use of the logarithmic map and define
$\bm{q}^{\prime}=\log(\bm{Q}*\bar{\bm{Q}}_{0})$, where $\bm{Q}_{0}$ is the
initial orientation. This is also in line with the position DMP, which is also
anchored to the initial position, i.e. (23) can be also written as
$\ddot{\bm{p}}^{\prime}=\frac{1}{\tau^{2}}(\alpha_{z}\beta_{z}(\bm{p}^{\prime}_{g}-\bm{p}^{\prime})-\alpha_{z}\tau\dot{\bm{p}}^{\prime}+g_{f}(x)\bm{K}_{p_{g}}\bm{f}(x))$,
where $\bm{p}^{\prime}=\bm{p}-\bm{p}_{0}$ and
$\bm{p}^{\prime}_{g}=\bm{p}_{g}-\bm{p}_{0}$. Integrating (26) and substituting
$x$ in (25) we get (2).

## Appendix C - Proof of Theorem 1

In the following we complete the proof of Theorem 1 from Section V.

###### Proof:

The system given by (15), with $\bm{d}$ being a time varying bounded
disturbance, can be decoupled in each dimension. Therefore we will conduct the
analysis for one element of $\bm{q}^{\prime}$ with dynamics:

$\hat{\tau}_{o}^{2}\ddot{y}=-\alpha_{z}\beta_{z}y-\alpha_{z}\hat{\tau}_{o}\dot{y}+\hat{\tau}_{o}^{2}d$
(27)

Introducing the state variable $\bm{\zeta}=[y\ \dot{y}]^{T}$ (27) can be
written in matrix form: $\dot{\bm{\zeta}}=\bm{A}(t)\bm{\zeta}+\bm{B}d$ where
$\bm{A}(t)=\frac{1}{\hat{\tau}_{o}^{2}}\begin{bmatrix}0&1\\\
-\alpha_{z}\beta_{z}&-\alpha_{z}\hat{\tau}_{o}\end{bmatrix}$ and
$\bm{B}=\begin{bmatrix}0\\\ 1\end{bmatrix}$. Since $\alpha_{z},\beta_{z}>0$
and $\hat{\tau}_{o}$ is positive and bounded,
$Re\\{\lambda_{i}(\bm{A}(t))\\}\leq-\sigma_{s}$ $\forall t\geq 0$, $i=1,2$,
where $\sigma_{s}>0$ is constant. Taking into account that
$\bar{\bm{C}}(t),\bm{P}(t)\in\bm{L}_{\infty}$ and
$\bm{\tau}_{ext}\in\bm{L}_{\infty}\cap\bm{L}_{2}$ it follows from (14) that
$\dot{\hat{\tau}}_{o}\in\bm{L}_{\infty}\cap\bm{L}_{2}$. Moreover,
$\dot{\bm{A}}(t)=\bm{A}_{1}(t)\dot{\hat{\tau}}_{o}$, where
$\bm{A}_{1}(t)=\begin{bmatrix}0&-2/\hat{\tau}_{o}^{3}\\\
2\alpha_{z}\beta_{z}/\hat{\tau}_{o}^{3}&\alpha_{z}/\hat{\tau}_{o}^{2}\end{bmatrix}$
and since $||\bm{A}_{1}(t)||\in\bm{L}_{\infty}$ and
$\dot{\hat{\tau}}_{o}\in\bm{L}_{\infty}\cap\bm{L}_{2}$ we will also have that
$||\dot{\bm{A}}(t)||\leq||\bm{A}_{1}(t)|||\dot{\hat{\tau}}_{o}|\in\bm{L}_{\infty}\cap\bm{L}_{2}$.
Finally, since $\bm{A}(t)$ is differentiable and bounded, based on Theorem
$3.4.11$ from [16], the origin is uniformly globally asymptotically stable
equilibrium for the system $\dot{\bm{\zeta}}=\bm{A}(t)\bm{\zeta}$. Therefore
there exist matrices $\bm{\Pi}(t)=\bm{\Pi}^{T}(t)>0$ and
$\bm{Q}(t)=\bm{Q}^{T}(t)>0$ with
$\dot{\bm{\Pi}}=-\bm{A}^{T}(t)\bm{\Pi}(t)-\bm{\Pi}(t)\bm{A}(t)-\bm{Q}(t)$
satisfying $0<\pi_{1}<||\bm{\Pi}(t)||<\pi_{2}$ and
$0<q_{1}<||\bm{Q}(t)||<q_{2}$ [24]. Consequently the system
$\dot{\bm{\zeta}}=\bm{A}(t)\bm{\zeta}+\bm{B}d$ is uniformly ultimately
bounded, which can be shown easily using the Lyapunov function
$V=\bm{\zeta}^{T}\bm{\Pi}(t)\bm{\zeta}$. Hence, $y,\dot{y}\in\bm{L}_{\infty}$.
The same analysis holds for each dimension, so
$\dot{\bm{q}}\in\bm{L}_{\infty}$, therefore from (19) we conclude that
$\bm{\omega}\in\bm{L}_{\infty}$. ∎

## References

  * [1] B. Chandrasekaran and J. M. Conrad, “Human-robot collaboration: A survey,” in _SoutheastCon 2015_ , April 2015, pp. 1–8.
  * [2] S. Miossec and A. Kheddar, “Human motion in cooperative tasks: Moving object case study,” in _2008 IEEE International Conference on Robotics and Biomimetics_ , Feb 2009, pp. 1509–1514.
  * [3] Y. Maeda, T. Hara, and T. Arai, “Human-robot cooperative manipulation with motion estimation,” in _Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180)_ , vol. 4, 2001, pp. 2240–2245 vol.4.
  * [4] B. Corteville, E. Aertbelien, H. Bruyninckx, J. D. Schutter, and H. V. Brussel, “Human-inspired robot assistant for fast point-to-point movements,” in _Proceedings 2007 IEEE International Conference on Robotics and Automation_ , April 2007, pp. 3639–3644.
  * [5] T. Flash and N. Hogans, “The coordination of arm movements: An experimentally confirmed mathematical model,” _Journal of neuroscience_ , vol. 5, pp. 1688–1703, 1985.
  * [6] Y. Li and S. S. Ge, “Human–robot collaboration based on motion intention estimation,” _IEEE/ASME Transactions on Mechatronics_ , vol. 19, no. 3, pp. 1007–1014, June 2014.
  * [7] A. Gams, B. Nemec, A. J. Ijspeert, and A. Ude, “Coupling movement primitives: Interaction with the environment and bimanual tasks,” _IEEE Transactions on Robotics_ , vol. 30, no. 4, pp. 816–830, Aug 2014.
  * [8] G. Maeda, M. Ewerton, R. Lioutikov, H. B. Amor, J. Peters, and G. Neumann, “Learning interaction for collaborative tasks with probabilistic movement primitives,” in _2014 IEEE-RAS International Conference on Humanoid Robots_ , Nov 2014, pp. 527–534.
  * [9] D. Widmann and Y. Karayiannidis, “Human motion prediction in human-robot handovers based on dynamic movement primitives,” in _European Control Conference_ , June 2018.
  * [10] A. Sidiropoulos, E. Psomopoulou, and Z. Doulgeri, “A human inspired handover policy using gaussian mixture models and haptic cues,” _Autonomous Robots_ , vol. 43, no. 6, pp. 1327–1342, Aug 2019. [Online]. Available: https://doi.org/10.1007/s10514-018-9705-x
  * [11] A. Sidiropoulos, Y. Karayiannidis, and Z. Doulgeri, “Human-robot collaborative object transfer using human motion prediction based on dynamic movement primitives,” in _2019 18th European Control Conference (ECC)_ , June 2019, pp. 2583–2588.
  * [12] E. Noohi, M. Žefran, and J. L. Patton, “A model for human-human collaborative object manipulation and its application to human-robot interaction,” _IEEE Transactions on Robotics_ , vol. 32, no. 4, pp. 880–896, 2016.
  * [13] L. Koutras and Z. Doulgeri, “A correct formulation for the orientation dynamic movement primitives for robot control in the cartesian space,” in _Proceedings of The 3rd Conference on Robot Learning_ , 30 Oct –1 Nov 2019\.
  * [14] K. Reif, F. Sonnemann, and R. Unbehauen, “An ekf-based nonlinear observer with a prescribed degree of stability,” _Automatica_ , vol. 34, no. 9, pp. 1119 – 1123, 1998.
  * [15] D. Simon, _Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches_. New York, NY, USA: Wiley-Interscience, 2006.
  * [16] P. A. Ioannou and J. Sun, _Robust Adaptive Control_. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 1995.
  * [17] R. M. Murray, “Optimization-based control,” 2010.
  * [18] J. J. LaViola, “A comparison of unscented and extended kalman filtering for estimating quaternion motion,” in _Proceedings of the 2003 American Control Conference, 2003._ , vol. 3, June 2003, pp. 2435–2440 vol.3.
  * [19] M. St-Pierre and D. Gingras, “Comparison between the unscented kalman filter and the extended kalman filter for the position estimation module of an integrated navigation information system,” in _IEEE Intelligent Vehicles Symposium, 2004_ , June 2004, pp. 831–835.
  * [20] X. Wu and Y. Wang, “Extended and unscented kalman filtering based feedforward neural networks for time series prediction,” _Applied Mathematical Modelling_ , vol. 36, no. 3, pp. 1123 – 1131, 2012. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0307904X11004501
  * [21] A. J. Ijspeert, J. Nakanishi, H. Hoffmann, P. Pastor, and S. Schaal, “Dynamical movement primitives: Learning attractor models for motor behaviors,” _Neural Comput._ , vol. 25, no. 2, pp. 328–373, Feb 2013.
  * [22] S. Schaal and C. G. Atkeson, “Constructive incremental learning from only local information,” _Neural Comput._ , vol. 10, no. 8, pp. 2047–2084, Nov. 1998.
  * [23] T. Kulvicius, K. Ning, M. Tamosiunaite, and F. Worgötter, “Joining movement sequences: Modified dynamic movement primitives for robotics applications exemplified on handwriting,” _IEEE Transactions on Robotics_ , vol. 28, no. 1, pp. 145–157, Feb 2012.
  * [24] H. J. Marquez, “Nonlinear control systems: Analysis and design,” _IEEE Transactions on Automatic Control_ , vol. 49, pp. 1225–1226, 2003.