Research Article
Model-based sensorless robot collision
detection under model uncertainties
with a fast dynamics identification
International Journal of Advanced
Robotic Systems
May-June 2019: 1–15
ª The Author(s) 2019
DOI: 10.1177/1729881419853713
journals.sagepub.com/home/arx
Pengfei Cao1,2 , Yahui Gan1,2 and Xianzhong Dai1,2
Abstract
This article presents a novel model-based sensorless collision detection scheme for human–robot interaction. In order to
recognize external impacts exerted on the manipulator with sensitivity and robustness without additional exteroceptive
sensors, the method based on torque residual, which is the difference between nominal and actual torque, is adopted using
only motor-side information. In contrast to classic dynamics identification procedure which requires complicated symbolic derivation, a sequential dynamics identification was proposed by decomposing robot dynamics into gravity and
friction item, which is simple in symbolic expression and easy to identify with least squares method, and the remaining
structure-complex torque effect. Subsequently, the remaining torque effect was reformulated to overcome the structural
complexity of original expression and experimentally recovered using a machine learning approach named Lasso while
keeping the involving candidates number reduced to a certain degree. Moreover, a state-dependent dynamic threshold
was developed to handle the abnormal peaks in residual due to model uncertainties. The effectiveness of the proposed
method was experimentally validated on a conventional industrial manipulator, which illustrates the feasibility and simplicity of the collision detection method.
Keywords
Sensorless collision detection, human–robot interaction, dynamics identification, model uncertainty, Lasso
Date received: 9 November 2017; accepted: 23 April 2019
Topic: AI in Robotics; Human Robot/Machine Interaction
Topic Editor: Chrystopher L Nehaniv
Associate Editor: Lajos Nagy
Introduction
Robot collaborating with humans in both domestic and
industrial area is increasingly becoming a research focus,
and much effort has been devoted to realize human–robot
collaboration in which both counterparts share the same
workspace to accomplish a goal via different interaction
modes.1,2 Compared to conventional robots strictly confined within safe fence to avoid any possibility of physical
interaction when robot is performing a predefined task,
collaborative robots, also known as co-bots, have been
regarded as a new generation of robots with the feasibility
to complete the complementary task with close involvement of operator or direct physical contact with human for
cooperation through high-level autonomy. The arising concern of safety is a critical challenge when a close distance
between humans and robotic apparatus is typically
demanded for collaboration scenario since the chance of
1
School of Automation, Southeast University, Nanjing, People’s Republic
of China
2
Key Laboratory of Measurement and Control of Complex Systems of
Engineering, Ministry of Education, Nanjing, People’s Republic of China
Corresponding author:
Yahui Gan, School of Automation, Southeast University, No. 2 Sipailou,
Xuanwu, Nanjing 210096, People’s Republic of China.
Email: y.h.gan@seu.edu.cn
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2
unexpected collision significantly increases due to one’s
frequent intervention into the robot’s operation space.3–5
To address the issues of safety, several related technical
standards have been proposed to provide guidance for collaborative robots in order to minimize the potential injury
risks caused by collisions between the robot and human
operator or its surrounding environment.6–8
Generally speaking, effective schemes dealing with collision can be divided into pre-collision and post-collision
strategies based on when the active actions are given with
respect to time of impact. On the one hand, the former
strategies can usually achieve desirable collision-free performance by actively providing appropriate collision
avoidance path using additional sensors to perceive
unstructured surrounding environment, which is sometimes
sophisticated and probably computational heavy.9–12 For
example, human–robot coexistence can be guaranteed by
introducing preventive approaches aiming at monitoring
the operator’s distance to robot body with stereo vision10
and proximity sensors.12 On the other hand, post-collision
strategies give a reactive response to the impact without the
requirements of real-time locating operator’s position and a
complex path planning algorithm to avoid obstacles.2,13–18
Meanwhile, the algorithmic simplicity of post-collision
scheme is always at the cost of the potential collision
impact due to its intrinsic property that collision could not
be detected until the impact between the robot and its surroundings occurs, which highlights the significance of how
to accurately diagnose the abnormal signals resulting from
the unexpected collision. However, the post-collision
scheme, which succeeds in minimizing the unintended collision threats, is attractive to reduce the costs of external
sensing and potential failures due to the integration of additional perceptive components, making the approach preferable for cost-sensitive industrial applications.2,19,20 In
human–robot collaboration context, a comprehensive risk
assessment is particularly required for robotic system
adopting the post-collision strategy to embed safety scheme
such as collision detection and reaction into the control
system. Crash testing in robotics by Haddadin et al. experimentally manifested how the factors like robot mass and
velocity affect the injury severity index from both biomechanics and forensics aspects through human–robot impact
voluntary testing.4,21 It was demonstrated that a lightweight
manipulator with compliant actuation is less likely to cause
fatal injuries.22–25 Lightweight designed robots are inherently suitable for close human–robot interaction with the
desirable features of low inertia and reduced weight which
allows driving torque output to decrease. Thus, the injury
caused by collision is also considerably minimized in contrast to conventional heavy and stiff manipulators.22,23 In
terms of compliant actuation, flexibility is introduced into
the actuator,25 even with adjustable impedance property.26
Apart from the human-friendly design with compliant and
lightweight property which does not necessarily ensure the
operator’s safety, control for safe physical interaction
International Journal of Advanced Robotic Systems
allows the realization of collision handling with the primary motivation of detecting the impact-deduced signal
as fast as possible before the robot has imposed excessive
force on the operator. A simple but feasible approach is to
monitor non-model-based signals such as the instantaneous
transients in current or the variation of position, which to
some extent characterizes the system fault in control system caused by external impact.27 These approaches suffered from the drawbacks of low sensitivity and system
dependence as moderate impact may produce too sight
influence to observe and the detection threshold is heavily
dependent on control performance. Other accessible methods principally rely on the model-based signal, which
requires the complex nonlinear system dynamic description
and accurate dynamic parameters provided by robot manufactures or experimentally calibrated.13,19,20,28 A straightforward idea is to compare the actual torques with the
estimated torques, which is easily calculated using the
inverse dynamics providing real-time robot motion information, to generate the residual signal as the difference of
these two torques. Thus, the residual signal is attributed to a
collision under the assumption of precise dynamic parameters and modeling. This method has been refined by
employing an observer of the generalized momentum to
avoid double numerical differentiation of position to calculate the joint acceleration, which inherently reduces the
measurement noise and improves the accuracy of the residual signal.13,15 In a similar perspective, several disturbance observers were also discussed to estimate the
exogenous disturbance imposed by the environment.29–31
In summary, the effectiveness of these model-based methods critically relies on the accuracy of relevant dynamic
parameters which always keep confidential and unavailable
to users and need parametric calibration by a meticulous
identification experiment, since the parametric errors structurally affect the residual signal as external disturbances.
The dynamic parameter identification has been extensively
researched to explore the issues including symbolic calculation of base parameter, excitation trajectory design, physical feasibility, and estimation method, which requires
much effort and attention to derive the sophisticated symbolic expression and enhance the accuracy with a wellconditioned excitation trajectories. 32–35 One practical
approach which combines machine learning method with
model identification procedure enables the simplification
of symbolic computation and a satisfying prediction performance.36–39 For instance, a data-driven machine learning approach was presented in literature39 to learn the
dynamics of a planar two-link robot using a regressor selection technique while this method is limited to a class of
planar robot whose fiction and gravity effect has little influence on the overall dynamics. Regardless of the parameter
identification accuracy, the collision is eventually judged
with the comparison of the residual signal and a certain
threshold. A primary concern in designing threshold is how
to achieve a trade-off between prevention of false alarm
Cao et al.
and detection sensitivity at the same time. It is worthwhile
to tune the collision detection threshold rather than select a
static threshold to achieve better detection performance.21,17 The generation of a dynamic threshold was
elaborated in literature17 through an adaptive residual evaluation to tackle the uncertainties in robot dynamic parameters for a two-link robot, while the online computational
burden would significantly increase for robot with more
degrees of freedom (DOFs), and estimation errors would
accumulate as a recursive scheme was adopted in the
threshold formulation.
In this article, a novel sensorless collision detection
algorithm for human–robot collaboration scenario is presented on a conventional industrial robot without prior
knowledge of dynamic parameters. Based on the modelbased collision handing framework, the unknown dynamic
parameters of the manipulator, which is indispensable to
reconstruct the inverse dynamics and consequently generate the residual signal, are estimated with a sequential identification procedure. The first step of the identification
experiment is to predict the gravity and friction accounting
for the major components of torque when robot is running
in collaboration mode. Subsequently, a machine learning
approach named Lasso is introduced to identify the remaining torque, avoiding the necessity of symbolic derivation of
the sophisticated expression. Furthermore, by synthesizing
state-dependent dynamic threshold, the residual signal is
evaluated in case of model uncertainties.
The rest of this article is organized as follows. Firstly,
the dynamics modeling and model-based collision detection scheme are illustrated. Then, a fast dynamics identification is described with a sequential identification method.
The gravity and friction effects are estimated with a classic
method, and details are provided on the formulation of the
remaining torque prediction using a machine learning
approach. The residual signal is analyzed with a dynamic
threshold to detect external disturbances. Subsequently, the
proposed collision detection scheme is experimentally validated on a robot performing a predefined task. Finally,
collusions and future directions are provided.
Model-based collision detection
Robot dynamic model
For n-DOFs serial manipulator, let q m 2 Rn and t m 2 Rn ,
respectively, denote the motor positions and torques, and
q 2 Rn and t 2 Rn , respectively, denote the positions and
torques after the gear transmission. These according variables are connected with the transmission matrix
R red 2 Rnn with q ¼ R1
red q m and t ¼ R red t m . However,
in the presence of joint elasticity, the link positions q 2 Rn
are not identical to the transmitted motor positions q . Stiff
effects can be adopted to represent the flexible transmission
model, along with additional damping effects. Without loss
of generality, only the stiff model will be considered in this
3
study to simplify the analysis. Indeed, the above positions
are involved in the dynamic model for robot with elastic
joints as follows
M l ðq Þ€
q þ C ðq ; q_ Þq_ þ G ðq Þ þ t f l ðq_ Þ þ K ðq q Þ ¼ t ext
ð1Þ
€ þ t f m ðq_ Þ K ðq q Þ ¼ t
Bq
ð2Þ
where M l ðq Þ 2 Rnn is the positive definite inertia matrix
of robot link; C ðq ; q_ Þq_ 2 Rn is the centrifugal and Coriolis
torque; G ðq Þ 2 Rn is the gravity item; B 2 R nn is the
constant, diagonal matrix of the motor inertia moments;
K 2 Rnn is the constant, diagonal joint stiffness matrix;
t ext 2 Rn is the effective torque at joint level arising from
the external impact force exerted on the robot link; and
t f l ðq_ Þ 2 Rn and t f m ðq_ Þ 2 Rn are, respectively, the joint
and motor friction torques. The Coulomb–viscous model is
the most common model to characterize friction effect and
thus the joint and motor friction torques could be depicted
as t f l ¼ F vl q_ þ F cl signðq_ Þ at joint level and t f m ¼
F vm q_ þ F cm signðq_ Þ at motor level where ðF vl ; F vm Þ are
viscous friction coefficients and ðF cl ; F cm Þ are Coulomb
friction coefficients. Equations (1) and (2) are referred to
as link equation and motor equation, respectively. By summarizing equations (1) and (2), we obtain
q þ C ðq ; q_ Þq_ þ G ðq Þ þ t f l ðq_ Þ þ B q€ þ t f m ðq_ Þ
M l ðq Þ€
¼ t þ t ext
ð3Þ
Note that a completely rigid joint is always taken into
consideration for simplified analysis (K ! 1), then
q ¼ q , and we obtain the dynamic characteristic of nDOFs serial robot with rigid links and joints as follows
M ðq Þ€
q þ C ðq ; q_ Þq_ þ G ðq Þ þ t f ðq_ Þ ¼ t þ t ext
ð4Þ
where M ðq Þ ¼ M l ðq Þ þ B 2 Rnn represents the lumped
inertia and t f ðq_ Þ 2 Rn captures the combined friction
effects at the link level as t f ðq_ Þ ¼ F v q_ þ F c signðq_ Þ with
viscous friction coefficients F v ¼ F vl þ F vm and Coulomb
friction coefficients F c ¼ F cl þ F cm . Although equation
(4) is the ordinary form of the rigid robot and has been
thoroughly researched theoretically and experimentally,
we derive this equation from engineering aspect for further
implementation on a manipulator.
The symbolic dynamic model equation in equation (4)
can be firstly derived using Lagrange equation or Newton–
Euler method, which can be decomposed such that the
dynamic model can be reformulated as a linear regression
form. The dynamics equation can be written in linear
expression with respect to its dynamic parameters as
t ¼ H ðq ; q_ ; q€ Þp
ð5Þ
where vector p represents the dynamic parameter set, and
the matrix H ðq ; q_ ; q€ Þ is always referred to as regression
4
International Journal of Advanced Robotic Systems
matrix or regressor. For each link, its dynamic parameters
consist of six components of the inertia tensor, three components of the first moment, mass, and the viscous and
Coulomb friction coefficients. Instead of including every
individual parameter, a base parameter set is preferable in
model identification since some parameters have no influence on system dynamics and some can be regrouped in
linear combinations of its original set. The base parameter
set has a lower dimension than the original one. After
analyzing the symbolic expression of robot’s dynamics, a
well-conditioned trajectory is delicately designed for the
actual robot to execute in order to excite robot’s dynamic
characteristic. Given an observation of robot joint motion
q and generalized joint forces/torques t for N samples at
sampling time tk , the dynamic parameter vector p may be
determined by least square regression as
^ ¼ arg min kH s p t s k
p
ð6Þ
p
3
H ðq ðt 1 Þ; q_ ðt 1 Þ; q€ ðt 1 ÞÞÞ
7
6
..
7 and t s ¼
where H s ¼ 6
.
5
4
H ðq ðtN Þ; q_ ðtN Þ; q€ ðtN ÞÞÞ
2
2
3
t ðt 1 Þ
6 . 7
6 . 7.
4 . 5
t ðtN Þ
Since the stacked regression matrix H s is typically not
full rank, the Moore–Penrose pseudoinverse is used to find
^ ¼ H þ t s , where superscript þ
the estimation solution as q
s
denotes the pseudoinverse operation.
Residual generation under model uncertainties
In human–robot collaboration scenario where humans are
close to the robot to perform cooperative tasks by sharing
workspace, safety is the fundamental issue to be considered
such that effective safe regulations must be utilized to
ensure human safety in case of unexpected collisions. To
address the collision detection issues for safe pHRI, several
model-based approaches have been proposed based on
residual signal using the difference between nominal and
actual values such as torque, energy, and generalized
momentum.2,13,15 These approaches have been devoted
much attention to detect the unexpected physical collision
between the manipulator and its environment, since only
the proprioceptive sensors were used in the detection procedure. It is not only economical to keep the expenses low
for external sensing, but also easy to implement by reducing the integration elements, leading to lower maintenance
costs and potential failures. For this type of collision detection approach, a straightforward residual method can be
computed as the difference between the estimated torque
^ and the actual torque t
t
^ t
r ¼t
ð7Þ
^ results from the inverse dynamic model (4) with
where t
the estimated parameters and measured robot motion information. The estimated torque is stated as
^ ð^q ; ^q_Þ^q_ þ G
^ ð^q Þ þ t
^ ð^q Þ^q€ þ C
^ ¼M
^ f ð^q_Þ
t
ð8Þ
where ð^Þ denotes the measured or estimated value. For
rigid robot with perfect system modeling, accurate dynamic
parameters, and well-measured robot motion variables, the
residual in equation (7) equals the external torque t ext
r ideal ¼ t ext
ð9Þ
However, the estimated torque value (8) contains modeling and measurement errors, and it yields
r ¼ t ext þ dt
ð10Þ
where dt denotes the torque errors induced by uncertainties including model uncertainties dt mdl and measurement
errors dt meas . Generally, model uncertainties dt mdl contain
modeling errors due to incomplete or simplified expression
of robot dynamics and dynamic parametric errors accounting for the inaccuracy of dynamic parameters such as link
inertia and mass. Unfortunately, robot dynamic parameters
are always confidential to be available from the robot manufactures even though they can be computed based on computer-aided design (CAD) model or by a well-designed
parameter identification experiment, while such identification procedure is always complicated for implementation
and the parametric accuracy remains to be verified. Moreover, the inverse dynamic model equation containing such
dynamic parameters is too complex to be wildly used in the
industrial applications as the computation complexity of
inverse dynamic model challenges the limited robot controller capacity. Hence, a simplified model is preferred in
the robot dynamic modeling with reduced expression by
omitting the items having the irrelevant influence on the
overall dynamic property while keeping fairly satisfied
performance. Another source of the torque errors comes
from the measurement. As only the position encoder is
available in acquiring robot motion data for most industrial manipulators, thus joint velocity and its derivatives
are often approximated using numerical differentiation,
consequently amplifying the existing noise at position
level. However, for a well-parameterized positioncontrolled robot running a commanded motor trajectory
with smooth derivatives of higher order, q and its derivatives can be replaced with q d and its derivatives or
numerically differentiated after a well-tuned low-pass filter to remove noise. It is clear that the torque errors dt
specify the dependence on the acceleration (inertia uncertainty), speed (friction, centrifugal, and Coriolis uncertainty), and position (gravity uncertainty).
Proposed collision detection scheme
In order to formulate an effective model-based collision
detection scheme without additional sensors, identification
of dynamic parameter is the fundamental basis for the sensorless collision detection. A major difficulty of the classic
dynamic parameter identification lies in the derivation of
Cao et al.
5
the symbolic expression of robot dynamics and regrouping
the linear form, which requires much effort and attention
even with the assistance of commercial software for symbolic computation. In contrast to traditional method, an
alternative to handle model identification is machine learning approach.36,37,39 This method offers a framework for
automatically learning the complex dynamics without
incorporating knowledge about the symbolic expression
while still keeping a satisfactory performance for robot
control and planning. In literature,39 a fast modeling and
identification of robot dynamics was proposed based on
Lasso40 and a simple example of a two-link direct-drive
robot was demonstrated to show the effectiveness of this
promising approach. Although this method can skip the
analytical derivation and identify the candidate terms that
act on robot dynamics by a regressor selection technique,
the condition of this experiment is limited to a planar
manipulator that has no gravity and friction effects, which
is obviously not suitable for most robotic apparatus. In this
section, an approach based on Lasso is presented to remedy
the defect in39 to realize dynamic modeling and identification for a generic articulated manipulator so as for further
collision detection scheme.
Gravity identification
Robot dynamic model identification, which has been intensively researched, is absolutely complex compared to the
gravity and friction identification. Here, we will first examine the problem of gravity identification and design a
method for accurate identification. The identification of
gravity is only position-dependent and the gravity item can
be expressed in the following linear parameterized form
G ðq Þ ¼ H G p G
ð11Þ
where H G is the regression matrix of gravity after regrouping the gravity symbolic expression with respect to the base
gravity parameters set p G . At the same time, it is not easy
to extract the pure contribution of gravity from joint torque.
Even if the robot is moving in a quasi-static state, the friction effect still plays an unignored role in robot dynamics.
In order to obtain the contribution due to gravity effect, the
following analysis is conducted by an in-depth study of
robot dynamics. Assume two robot configurations q 1 , q 2
satisfy
q1 ¼ q2 ¼ φ
q_ 1 ¼ q_ 2 ¼ φ_
q€ 1 ¼ q€ 2 ¼ φ€
ð12Þ
Then adding the resulting torques according to these two
configurations yields
_ φ_ þ G ðφÞ
t 1 þ t 2 ¼ 2½M ðφÞφ€ þ C ðφ; φÞ
ð13Þ
_ < e (e is a small positive value), then
If φ€ ¼ 0 and jφj
the previous equation in equation (13) becomes
t 1 þ t 2 2G ðφÞ
ð14Þ
Hence, the contrition due to gravity in joint torque
can be derived if the above motion assumptions are satisfied by moving the robot to a certain position from the
opposite direction at a low velocity with a constant value
so as to eliminate other torque effects. To estimate the
gravity parameters, we extracted the gravity effect from
joint torque by gathering data from 100 different robot
configurations and applied least squares method on equation (11).
Friction identification
For electromechanical systems including robot, friction
effect plays a significant role in system modeling and
control, which is usually difficult to predict especially
when robot changes its motion direction. Many
researchers have proposed different friction models and
identification methods for robot control and simulation. 41–44 In this article, we only employ the basic
Coulomb and viscous friction model due to its simple
description which captures most characteristic of the
friction phenomenon yet. Similar to the extraction of
gravity effect from joint torque, the friction effect must
be processed in order to identify the friction parameters. Suppose two robot motion states satisfy the
condition in equation (12), then the difference of these
according torques is
_
t 1 t 2 ¼ 2t f ðφÞ
ð15Þ
It is intuitive to design a sinusoidal motion for joint i in
the form of qi ðtÞ ¼ sinð 2p
T tÞ (T is the cycle time) at joint
space to meet the requirements of the above-mentioned
motion states if two sampling time are chosen as
t 1 ¼ kT =4 DT for q1 and t 2 ¼ kT =4 þ DT for q 2 with
k ¼ 1; 3 and 0 < DT < T =4. By applying the designed
trajectory for the robot to execute and subtracting the joint
torques in the according motion states, the equivalent effect
due to friction can be acquired based on the analysis in
equation (15). The friction torque is characterized in Coulomb–viscous model and is identified in an axis-by-axis
manner. The friction of the i th axis could be parameterized
in a linear form
F ci
t f i ðq_ i Þ ¼ ½signðq_ i Þ q_ i F vi
where F ci and F vi are the Coulomb and viscous parameters
of the i th joint, respectively. Gathering several groups of
the joint velocity and friction torque and applying the least
squares method, the friction parameters could be experimentally calibrated. We gathered 50 pairs of torques at the
corresponding motion states in which the absolute values of
each component in equation (4) are equal despite of the
sign of the friction items.
6
International Journal of Advanced Robotic Systems
Remaining torque identification
Since the gravity and friction items have been identified,
thus these two components can be removed from robot joint
torque for the remaining complex, nonlinear components
model learning. Here, the sum of inertial torque M ðq Þ€
q
_
_
and centrifugal and Coriolis torque C ðq ; q Þq is designated
as t r . Using the linearity of the dynamic model, the
remaining torque t r also keeps this linear property with
respect to relevant dynamic parameter vector
p r 2 Rk ðk 10nÞ in the following form
t r ¼ H r ðq ; q_ ; q€ Þp r
ð16Þ
It is reasonable to consider the elements in regressor
H r ðq ; q_ ; q€ Þ should be linear combinations of certain candidate terms defined by functions of joint position, velocity, and acceleration multiplied by sine and/or cosine
value of q . Indeed, each element in H r can be calculated as
H ijr ¼
r
X
ð17Þ
dijk g k
k¼1
where H ijr is element in the ith row and jth column of matrix
H r . Candidate g k in equation (17) is chosen from A B,
where denotes the Kronecker product, A ¼ ½€
q1 ; €q2 ; ;
T
€
qn ; q_ 1 q_ 1 ; q_ 1 q_ 2 ; ; q_ n1 q_ n ; q_ n q_ n 2 R and B ¼ w ðq 1 Þ
w ðq2 Þ w ðqn Þ 2 Rr ; w ðqi Þ ¼ ½1; sinqi ; cosqi ; sinð2qi Þ;
cosð2qi Þ; ; sinðmqi Þ; cosðmqi ÞT ði ¼ 1; 2; ; nÞ, m is
an positive integer and is set to be 2 (see Appendix 1). The
dimension of A is ¼ ðn2 þ 3nÞ=2 and r ¼ ð2m þ 1Þn for
B. dijk can be set to zero when the corresponding g k does
not show in H ijr or alternatively an unknown nonzero value
decided by robot’s structure. Therefore, according to the
expression in equation (17), the remaining torque for joint i
can be transformed into
t ir ¼
k
X
j¼1
H ijr pjr ¼
r
X
gk p
~ ik
r
ð18Þ
k¼1
Pk
j
~ ik
where p
~ ik
r ¼
r
j¼1 d ijk pr . Furthermore, by designating p
i
~ r , which is the converted
as the kth component of vector p
dynamic parameter set for ith joint, g k as the kth compo~ r , the expression in equation (18) can be
nent of vector H
restated in a compact form as
~ rp
~r i
t ir ¼ H
ð19Þ
~ r no
In the transformed expression (19), the regressor H
longer contains the information specified by system
mechanical structure, which otherwise has to be provided
in equation (16) with cautious and tedious symbolic computation, making the dynamic parameter identification procedure more convenient. Compared to the original
parameter set defined in equation (16), the converted para~ r mapping from p r is identified separately for
meter set p
each joint and has sparse feature due to the conversion,
while the burden of symbolic computation is significantly
~ r in equareduced. It is worth noting that the regressor H
tion (19) is equal for all joints and the size of parameter
~ ri is significantly larger than 10n, leading to more
vector p
computation time for the regression. However, the regression is done offline to select an adequate size of the parameters. Thus, the computation time for regression analysis
is not a concern here.
In order to identify the dynamic parameter while reducing the feature dimension to a certain degree by shrinking
some coefficient estimates toward zero, a regressor selection method named Lasso is a privilege to employ.39 Lasso
is a regression analysis method that incorporates the least
squares method to minimize regression error and an l 1
penalty term to yield sparse model by forcing some coefficients to zero. Generally, Lasso has the advantage of effectively producing simpler and more interpretable models
that involve only a subset of the predicator by introducing
an l 1 regulation term to penalize large coefficients.
Observing N groups of data and utilizing the Lasso
method, the parameter identification problem for ith joint
is formulated as
i
2
H
~ r sp
~ r 1
~ ri t ir s 2 þ lp
ð20Þ
min
i
~r
p
2 i
3
3
t r ðt 1 Þ
~ r ðq ðt 1 Þ; q_ ðt 1 Þ; q€ ðt 1 ÞÞ
H
6 . 7
7 i
6
..
7
7, t ¼ 6
¼6
5 rs
4
4 .. 5,
.
~ r ðq ðtN Þ; q_ ðtN Þ; q€ ðtN ÞÞ
H
t ir ðtN Þ
2
~ rs
where H
and l > 0 is an adjustable tunning parameter.
~ ri has many
It is sufficient to predict that the vector p
zero elements which have to be forced into zero in the
regression in equation (20). To this end, the choice of the
tuning parameter l critically trades off the regression variance and sparsity which indicates how many variables are
involved in the regression. For instance, the solution of
equation (20) is exactly equal to the least squares regression when l ¼ 0, and when l increases to some extent, all
coefficients estimated are substantially forced to approximate zero in order to regulate the penalty term. A simple
way to tackle this problem is cross-validation which can
properly maintain the balance of estimation variance and
variable selection, which has been implemented in
MATLAB Lasso function.
For classic robot dynamic parameter identification, it
is essentially significant to design a well-conditioned
excitation trajectory in order to reduce estimation
error.33 When it comes to machine learning method, it
also requires a sufficiently large amount of data that can
contain the necessary information for modeling as much
as possible to improve estimation quality. However,
designing a well-conditioned trajectory is not the focus
of this article and a practical trajectory generation
approach inspired by finite Fourier series trajectory33
is briefly introduced here.
Cao et al.
7
The basic form of the excitation trajectory for the ith
joint is as follows
W k
X
ai
bki
sinð!b ktÞ cosð!b ktÞ þ qi0
qi ðtÞ ¼
!b k
!b k
k¼1
Residual/Nm
k¼1
€
qi ðtÞ ¼
aki ksinð!b ktÞ þ bki kcosð!b ktÞ
10
w/o collision
with collision
upper threshold
5
W X
q_ i ðtÞ ¼
aki cosð!b ktÞ þ bki sinð!b ktÞ
W X
15
0
-5
k¼1
ð21Þ
where !b represents the fundamental frequency of the
motion for all joints and W is the order of the harmonic
of the sine and cosine functions. aki and bki are the parameters deciding the patterns of the trajectory which are
determined through optimization. qi0 is an auxiliary parameter which guarantees the robot to recover to its initial
configuration at the end of each period. In order to avoid a
sudden change at the start and end of each period when
calculating a continuous trajectory, boundary constraints
are imposed on the start and end motion states of each
period. Additionally, the exciting trajectory for each joint
is bounded for demanded specifications. Given the above
requirements, the problem of the excitation trajectory
design for each period is formulated as
min J
aki ;bki
s:t:
qð0Þ ¼ qðtf Þ ¼ 0
_
_ fÞ ¼ 0
qð0Þ
¼ qðt
€
qð0Þ ¼ €
qðtf Þ ¼ 0
qmin qðtÞ qmax
_ q_ max
q_ min qðtÞ
€ðtÞ q
€max
€min q
q
ð22Þ
where J is an optimization objective to be determined afterwards; qmax ; q_ max , and €
qmax denote the upper physical limits
of joint position, velocity, and acceleration, respectively,
qmin as the corresponding lower limits
with qmin ; q_ min , and €
of relevant joint variables. Usually, the optimization objective can be selected as the condition number of the regression matrix in order to plan a well-conditioned trajectory
for accurate parameter estimation. Instead of optimizing a
complex nonlinear objective function, a practical optimization function which can generate the necessary information
required in the regression is adopted as
J¼
n X
M X
z k a k þ h b k 2
k i 2
i
ð23Þ
i¼1 k¼1
where both z k and hk are randomly generated between 1
and 1. This selection of optimization objective can not
only simplify the symbolic derivation of the nonlinear
function used in the study by Swevers et al.,33 but also
-10
-15
120
lower threshold
121
122
123
124
125
126
127
128
129
Time/sec
Figure 1. Residual signals for free motion (blue, solid line) and
collision (red, dashed line) for joint 1.
can include a variety of motion patterns for a good data
fitting hereafter.
Residual threshold design
As the elaboration in previous section, the residual signal
consists of two parts including the external torque effects
indicating impacts and the unfavorable disturbances due to
parametric uncertainties and model inaccuracies which corrupt the collision detection. It is observed that even under
the free motion circumstance, the residual signal has the
trajectory-dependent variation in which the peaks due to
model uncertainties surprisingly larger than the peak produced by the collision, revealing a static threshold above
the maximal residual amplitude is not suitable to obtain a
good collision detection performance (see Figure 1). In
order to overtake the inappropriate static threshold, the
peak caused by collision may have to rise so excessively
that collision may be critically harmful to some extent.
A reliable and achievable alternative to distinguish the
meaningful peaks caused by collision from false alarm
peaks provoked by model uncertainties is to adopt a
time-varying dynamic threshold. Instead of the method
proposed in literature,17 the residual signal is considered
as moderate when there is no collision; otherwise, further
model refinement and identification has to be implemented
to modify the residual. Thus, for a relatively accurate
model identification, the residual can be bounded with a
dynamic threshold dependent on trajectory. The inner
structure of the residual is not modeled in detail but deemed
to be a black box. When there is no collision, the dynamic
threshold for each joint holds
jri j < ridet ¼ ⌿ i u i þ ui ; i ¼ 1; ; n
ð24Þ
The threshold ridet is built on input u i composed of different functions of robot states. Taking the residual and
8
International Journal of Advanced Robotic Systems
motion states into consideration, it is intuitive to choose the
dependence of the residual on acceleration and speed
by setting
h
i
2 T
u i ¼ j€
qi j jq_ i j eri q_ i
ð25Þ
The last item in u i is to compensate for the uncertainty
when robot changes its motion direction, and the corresponding parameter ri determines the slope of the peaks
which is manually tuned for each link. Moreover,
the threshold is characterized by a diagonal matrix
⌿ i ¼ diagð i1 ; i2 ; i3 Þ 2 R 33 which is subsequently
determined by optimization. Additional parameter ui is
added to ensure a certain safety margin, which guarantees
the robust detection. In order to ascertain the related
unknown parameters to construct a functional threshold,
an optimization approach is proposed to compute these
parameters. For compactness, we only present how the
ith threshold for joint i is practicably designed by solving
the following problem
min a
⌿ i ;ui
s:t:
ð26Þ
a ¼ ridet jri j
a xi
Figure 2. Experimental platform of Efort ER3A robot.
CD ¼
0
1
if
else
8i : jri j <
ridet
Z4
Y3
Experiment setup
The proposed collision detection scheme has been performed on Efort ER3A industrial robot in our lab (see
Figure 2). The 6-DOF manipulator has a spherical wrist,
3 kg of payload capability, and maximal stretch of 0.63 m
from the base. The link frames of the Efort ER3A robot are
illustrated in Figure 3 and geometry information of the
manipulator in modified Denavit and Hartenberg (MDH)
representation is given in Table 1. The robot is powered
with AC servo motors, which are connected with harmonic
X3
TOOL
Y4
a3
X6
X5
Z6
Y5
Z3
Z5
Y6
a2
Z1
Y1
X1
a1
X2
Y2
Z0
ð27Þ
Experiment and validation
X4
d4
where xi > 0. The definition in equation (26) can not only
keep the threshold from the residual with a secure range
larger than xi to insure detection robustness but also force
the threshold to approximate the residual as close as possible to realize accurate detection of tiny external disturbances. The problem in equation (26) is convex and thus
can be solved rapidly with an optimal solution. Here, we
use the information of robot motion and residual value
acquired from the previous training trajectory. Eventually,
the binary detection output CD which denotes the occurrence of an external collision is computed according to the
comparison between absolute value of residual signal and
the proposed threshold
Z2
Y0
X0
Figure 3. Link frames of Efort ER3A robot.
Table 1. The MDH parameters of Efort ER3A.
Link i
1
2
3
4
5
6
ai1 (rad)
di (m)
ai1 (rad)
qi (rad)
0
p=2
0
p=2
p=2
p=2
0
0
0
0.299
0
0
0
0.050
0.270
0.08
0
0
0
p=2
0
0
0
0
MDH: modified Denavit and Hartenberg.
Cao et al.
9
Controller system
Target PC
External
sensors
Teach
pendant
UDP
TCP /IP
N etwork
switch
EtherCAT master
Industrial PC
CPU : Intel Celeron M @
sdafdad 1.3GHz
RTOS : VxWorks
RAM : 512 M
Hilscher
CIFX 104CRE PC cards
Digital servo system
EtherCAT slave
Servo driver 1
Motor 1
Servo driver 1
Motor 2
N etwork
adapter
Servo driver n
TCP /IP
Motor n
Host PC
Figure 4. Control framework of Efort ER3A robot.
gears to increase the effective torque. The robot system
uses an industrial computer equipped with 2G RAM and
a Hilscher cifX bus communication card, running VxWorks
RTOS to realize real-time control of the robot servo drivers. The communication card enables the EtherCAT communication between the servo drivers as the slave devices
and the industrial computer as the master device, which
allows sending motion or torque commands to control
robot and receiving real-time robot’s information such
as joint positions and currents at every 4 ms. Since the
control architecture is open, the users can program the
robot through a teach pendant or use the Cþþ language
with a graphical user interface on a personal computer.
Hence, with appropriate programming, the robot control
system can implement motion control on both joint and
Cartesian space and collect data from both proprioceptive sensors (e.g. motor position encoder and motor current sensor) and exteroceptive sensors (e.g. force/torque
sensor, vision sensor). The overall overview of robotic
control system is shown in Figure 4.
The robot is position-controlled with a well-tuned
proportional-derivative control law embedded in the servo
drivers and only the first three joints are activated for simplicity throughout the article. The input provided to the
controller is typically reference position q d at the joint
level and the available output from the servo drivers including the joint positions q measured by motor-side encoders
and the motor currents I. Meanwhile, the useful joint toque
t can be obtained using the motor currents with the following transformation
t ¼ K I ¼ diagðk 1 ; :::; k n ÞI
ð28Þ
where k i is the current-to-torque gain of the ith motor provided by the manufacturer as a constant. The acquired
motor torque is, however, too noisy to be directly used in
distinguishing the external torque as the random noise signal will annihilate the torque transient caused by collision.
On the other hand, the high-frequency noise can be greatly
reduced using an appropriate low-pass filter, and a smooth
torque is retained while an effective filtering can prevent
the effect of hard collisions from overly filtering and avoid
corrupting the sensitivity of the collision detection scheme
in principle. It is worth noting that the implementation of
suitable filtering of the torque requires that the main frequency distribution of external impact torque is within the
filter cutoff frequency and the filter has a stable structure
and no accumulated filtering errors in digital implementation. In order to solve this problem in filer design, a scheme
of distinguishing between intended contact and unexpected
collision is introduced in the study by Cho et al.18 The lowpass filter HðzÞ was finally chosen as
HðzÞ ¼
5
X
hðiÞzi
ð29Þ
i¼0
with h0 ¼ h5 ¼ 0:1118, h1 ¼ h4 ¼ 0:1677, and h2 ¼
h3 ¼ 0:2205, which represent a fifth-order finite impulse
response low-pass using a cutoff frequency of 10 Hz and a
Kaiser window. The cutoff frequency was selected to
remove the noise signal as much as possible while preserve
the meaningful torque data. At the same time, the order of
the filter was limited to 5 to reduce the resulting time delay.
Dynamics identification and residual threshold design
In the case of the Efort Ertext 3A robot, gravity effect only
presents at the second and third joints and the base gravity
parameters can be parameterized with four grouped expressions of mass and center mass. Based on the analysis in the
gravity and friction identification, the identification procedure was conducted on the industrial robot sequentially.
The corresponding parameters of the first three joints are
summarized in Table 2.
In our experiment of training trajectory generation,
some basic parameters were designed with W ¼ 5, fundamental frequency !b ¼ 2p=5. The physical bounds on joint
position, velocity, and acceleration are given in Table 3.
According to the training trajectory generation scheme
based on the optimization problem in equation (22), we
finally produced a trajectory composed of 50 periods of
motion data which is continuous at the beginning and
end of the individual trajectory for training the remaining
torque. Figure 5 shows part of joint position, velocity, and
10
International Journal of Advanced Robotic Systems
Table 2. Dynamic parameters of gravity and friction effects.
Value
kg m
kg m
kg m
kg m
Nm s/rad
Nm
Nm s/rad
Nm
Nm s/rad
Nm
0.6595
0.2278
0.2139
0.8865
6.2073
3.1127
7.0136
5.2570
4.0979
4.5163
(a)
5
0
-5
-10
-15
120
(b)
122
124
126
128
130
132
134
136
138
140
122
124
126
128
130
132
134
136
138
140
122
124
126
128
130
132
134
136
138
140
30
20
Table 3. Motion constraints for training trajectories.
Joint
measured torque
estimated torque
15
10
τ1 /Nm
m 3 l 3y
m 3 l 3x
m 2 l 2y
m 2 l 2x þ m 3 a 2
F v1
F c1
F v2
F c2
F v3
F c3
Unit
τ2 /Nm
Parameters
1
2
3
+90
60
90
+60
60
90
+60
60
90
10
0
-10
-20
Position range ( )
Maximum velocity ( /s)
Maximum acceleration ( /s2)
120
(c)
20
15
Joint position/deg
(a)
Joint 1
Joint 2
Joint 3
40
0
-10
120
0
Time/sec
-20
122
124
126
128
130
132
134
136
138
140
(b)
60
Joint velocity/(deg/s)
5
-5
20
-40
120
40
20
0
-20
-40
120
122
124
126
128
122
124
126
128
130
132
134
136
138
140
130
132
134
136
138
140
(c)
Joint acceleration/(deg/s2 )
τ3 /Nm
10
50
0
-50
120
Time/sec
Figure 5. Reference trajectories for training. (a) Joint position.
(b) Joint velocity. (c) Joint acceleration.
acceleration of the training trajectory. Consequently, the
demanded motion was executed by the robot to acquire the
relevant joint information including actual joint position
Figure 6. Actual joint torque and estimated torque based on the
proposed method for trajectories in Figure 5.
and torque. For joint actual position, a low-pass, zero phase
filter and a central difference algorithm were used to calculate joint velocity and acceleration. Meanwhile, the joint
torque effect due to gravity and friction was removed to
obtain the remaining torque t r for further data process.
Once the income and outcome given in equation (20) are
available, the Lasso algorithm automatically computes the
parameter estimation with variable selection using the
package provided by MATLAB.
In the context of a three joint articulated manipulator,
because the robot’s motion is centrosymmetric with respect
to joint 1, w ðq 1 Þ can be excluded from set B. Hence, B has
a reduced number of candidates to 25, while A has the
~ r is
dimension of 9; the total number of elements in H
calculated as 225, leading to the dimension of the converted
~ ir shares the same dimension value
dynamic parameter set p
which is larger than the dimension of p r .
The regularization progressively strengthens as the
value of l increases, resulting in fewer nonzero regression
coefficients and generally larger cross-validated mean
square error suggesting inadequate fit of the responses.
However, the parameter l was automatically selected with
the optimal value provided by Lasso function. Practically,
for coefficient p
~ ik
r which is less than 0.2, this coefficient is
Cao et al.
11
(a)
with collision
w/o collision
threshold
12
|r1 |/Nm
10
8
6
4
2
0
120
|r2 |/Nm
(b)
|r3 |/Nm
124
126
128
130
132
134
136
138
140
122
124
126
128
130
132
134
136
138
140
122
124
126
128
130
132
134
136
138
140
15
10
5
0
120
(c)
122
15
10
5
0
120
Time/sec
Figure 7. Residual signals for free motion (blue, solid line), collision (red, dashed line), and threshold (green, dashed line) for training
trajectory in Figure 5.
regarded as having negligible influence on the entire
dynamics and always discarded in recovering the remaining torque. For a group of l value, the cross-validation error
for each value of l is automatically computed. At the same
time, the relevant coefficient estimates could also be
obtained for corresponding l value. With the trace plot of
coefficients and cross-validated mean square error plot of
Lasso fit, one could visually observe the difference of l
selection making on the coefficient dimension reduction
~ r i are
and estimation error. If the zero components in p
selected more, that is, higher value of l parameter in the
Lasso algorithm is preferred, then the threshold margin x i
may be manually set more conservative. In our implementation, we chose the l which reduces the dimension of the
coefficients larger than 0.2 to 10 and keeps the estimation
error satisfactory as well. Eventually, the recovered robot
inverse dynamics composed of the abovementioned three
parts and the actual toque is illustrated in Figure 6 corresponding to the motion in Figure 5.
Figure 6 compares the measured and predicted torques,
revealing the inverse dynamics can be reconstructed given
the demanded motion. It is worth noting that the predicted
torque roughly fits the actual torque well expect some derivation and there exists slight torque variation during the
motion because a simplified dynamics identification process is adopted in this article. Meanwhile, the absolute
Table 4. Parameters of the proposed dynamic threshold.
Joint i
i1
i2
i3
ui
xi
ri
1
2
3
0.1433
0.2295
5.1588
3.4021
2.0000
150
0.335
0.7350
5.5866
5.6258
2.0000
200
0.7733
1.3615
9.1100
3.6457
2.0000
150
residual value under free motion in Figure 5 can be calculated once the inverse dynamics is available, and the torque
residual for each joint is depicted in Figure 7.
It is observed that the residual signal fluctuates near zero
except an abrupt peak arises when velocity changes its sign,
which implies the reliability and practicality of the proposed dynamics identification. The abrupt peak arises
when velocity changes its direction since it is difficult to
accurately model the friction in low-velocity range. When
velocity traverses the low-velocity regime, several phenomena like Stribeck effect, predisplacement, rate dependence, and hysteresis will significantly dominate the
dynamics of friction. To address these dynamic behaviors
in low-velocity regime, several friction models incorporating more coefficients and complicated structure were
proposed such as LuGre model, Maxwell-slip model, and
12
International Journal of Advanced Robotic Systems
(a)
with collision
w/o collision
threshold
|r1 |/Nm
10
5
0
10
11
12
13
14
15
16
17
18
19
20
11
12
13
14
15
16
17
18
19
20
11
12
13
14
15
16
17
18
19
20
(b)
20
|r2 |/Nm
15
10
5
0
10
(c)
20
|r3 |/Nm
15
10
5
0
10
Time/sec
Figure 8. Residual signals for free motion (blue, solid line), collision (red, dashed line), and threshold (green, dashed) for validation
trajectory.
Leuven model. These models could to some extent illustrate the property of friction dynamics in low velocity, but
the identification procedure is not trivial and the accuracy
cannot be guaranteed. Indeed, the Coulomb–viscous model
is still the most popular one in constructing the friction in
robotics due to its linearity in parameters and satisfactory
description of friction dynamics in most velocity regime.
In addition, collisions were imposed by operator on the
manipulator when it executed the same trajectory and corresponding residual signals were computed as presented in
Figure 7. In contrast to collision-free scenario, the peaks
due to collisions have the influence on the residual as the
model uncertainties do. Hence, the torque errors induced by
the dynamics uncertainties including unmodeled dynamics
and estimation bias will cause adverse effects on accurate
collision detection, and this is tackled by employing the
dynamic threshold.
The parameters for threshold is given in Table 4, and the
corresponding reconstructed threshold is compared with
the torque residual, as shown in Figure 7, illustrating the
acquired threshold can bound the residual signal to ensure
the detection mechanism not be activated by false alarms
due to measurement errors and dynamic model uncertainties as well as small enough to allow fast detection of
collision. The effectiveness of the proposed threshold was
validated for a different trajectory as detailed in the following section.
Validation
The proposed scheme for sensorless robot collision detection was experimentally validated on the industrial robot
ER3A performing a continuous cyclic motion task with a
triangle curve in Cartesian space. The triangle motion in
Cartesian space is with the maximal velocity of 0.2 m/s and
trajectories in the joint space are calculated using the
inverse kinematics.
Figure 8 shows the residual signal for each joint under
different collision conditions. For example, a collision near
10 and 14 s is relatively moderate and these joint peaks may
not be detected if static threshold was adopted. Thanks to
the proposed dynamic threshold, the tiny torque peaks
resulting from collision could be sensitively identified. For
example, a collision occurred near 14 s and the effective
torque caused by collision at joint 1 is smaller than the
maximal dynamics error induced by model uncertainties.
The dynamic residual threshold can also effectively recognize the source of the torque peaks. For the collision at t ¼
Cao et al.
18 s, the peaks in residual signals were recognized at all
joints, suggesting the unexpected collision with slight
impact occurred at the body before the third joint. The
effective external torque at joint level satisfies
t ¼ J Tc F ext , where F ext is the Cartesian contact force and
J c ðq Þ is the Jacobian matrix at the collision point. When
the external force F ext belongs to the kernel of J Tc ðq Þ, the
external force will be badly reflected at the joint level. The
collision peak at t ¼ 14 s is only detected at the first joint,
while the slight peak at the second joint could not be sensed.
This may be owing to that the matrix J Tc ðq Þ is illconditioned at that configuration. Generally speaking, the
collision detection scheme could locate the probable collision position, which may be utilized for further collision
reaction scheme. In our article, a simple collision reaction
was adopted by stopping the manipulator for 1 s to prevent
harm to operators as soon as the collision was detected.
Conclusion
In this article, a current-based whole body collision detection is presented on a conventional industrial manipulator
based on a quick dynamic model identification using Lasso
and a residual evaluation method to overcome the dynamic
model uncertainties. The presented method uses only proprioceptive sensors providing motor-side information without additional exteroceptive sensors. The collision effect is
diagnosed as system fault through comparison between the
actual torques related to the internal motor currents and the
reconstructed torque from the joint position and its derivatives. In this framework, a sequential identification of
dynamic parameters is introduced where the major components of the dynamics (i.e. gravity and friction) in pHRI
scenario are identified in the first step and the remaining
torque is formulated with a simplified model. On the basis
of this model, Lasso algorithm is adopted to predict the
relevant parameters along with parameter selection in order
to capture the characteristic of the remaining torque. The
residual signal which is defined as the difference of the
actual and predicted torque is used to recognize whether
the external impact is imposed on the robot. To tackle the
irregular peaks due to dynamic uncertainties, dynamic
threshold is imposed on the residual which ensures the
sensitivity of the detection scheme as well as the robustness
to false alarm caused by system errors.
The method has been implemented on a conventional
industrial robot ER3A with EtherCAT protocol for realtime data communication. Instead of the classic parameter
identification with complicated symbolic derivation and
heavy computation burden for real-time control, the proposed method can fast and effectively reconstruct the
inverse dynamic model. Consequently, the residual signal
for collision detection is generated through the simplified
model, and threshold is conceived to remedy the peaks due
to system uncertainties. Compared with a static threshold,
13
the dynamic threshold reduces the collision torque sensitivity from 10 Nm to 5 Nm while maintaining the robustness. Although the joint sensitivity does not share linear
relationship with respect to the impact sensitivity in the
Cartesian space, the impact can even be impossibly sensed
if the impact force belongs to transposed Jacobian matrix
kernel space. However, the robot always holds favorable
configurations when robot shares workspace when collaborating with humans; external impact is reflected on the
joint space with high fidelity which guarantees the detection of collisions occurs on the whole body of the robot. In
our experiment, the collisions can be detected with the
sensitivity less than 5 Nm in joint space with a short time
delay less than 50 ms.
Despite a fast and simplified model identification, a
dynamic threshold for residual signal is proposed to
achieve a satisfactory whole body collision detection on
an industrial manipulator; our future plan is to evaluate
the sensitivity of forces in Cartesian space of the collision
classification using a 6D F/T sensor. Additionally, the
presented idea in recovering dynamics and residual
threshold generation can be brought into the external force
estimation at the end effector so as to realize a low-cost
virtual force sensor which benefits the force-control scenario like assembly and human–robot cooperation. Further investigation directions in the collision detection also
include collision detection scheme of a robot with payload
fixed at the end effector and how the factors like speed and
stiffness affect the accuracy of the collision and corresponding strategies.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial support
for the research, authorship, and/or publication of this article: This
work was supported in part by Natural Science Foundation of
China under Grant Numbers 61873308, 61503076, and
61175113 and Natural Science Foundation of Jiangsu Province
under Grant Number BK20150624.
ORCID iD
Pengfei Cao
https://orcid.org/0000-0003-4642-1102
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Appendix 1
Derive the robot dynamics equation using Lagrangian
method, the kinetic energy T i and potential energy U i for
robot i th link could be expressed as
1
1
T i ¼ mi v Tci v ci þ v Ti 0 R i I ci 0 R Ti vi
2
2
U i ¼ mi gT 0 rci
ð1AÞ
ð1BÞ
where mi is the i th link mass, I ci is the link inertia tensor
expressed in i th link frame, vci is linear velocity of the link
mass center, and v i is angular velocity of the link; both
velocities are expressed in the base frame f0g. 0 R i is the
rotation matrix from link i frame to the base frame and 0 r ci
is the position of the center of mass of link i with respect to
base frame. g is the gravity acceleration vector in the base
frame. Based on homogeneous transformation, the position
of the center of mass of i th link 0 r ci could be written as
function of q
i 0 r ci
r ci
1
0
i1
ð1CÞ
¼ A 1 ðq1 Þ A 2 ðq 2 Þ A i ðqi Þ
1
1
where s1 A s is the transformation matrix between frame fsg
and fs 1g (1 s i) and i r ci is the center of mass with
respect to the link frame {i} and a constant value. Thus, there
will be cosðqs Þ and sinðqs Þ in the symbolic expression of 0 r ci
rather than cosð2qs Þ and sinð2qs Þ, as well as the Jacobian of
0
r ci . It is necessary to express the kinetic energy as a function of joint variables with the Jacobian computation
1
1
T i ¼ mi q_ T J TLi J Li q_ þ q_ T J TAi 0 R i I ci 0 R Ti J Ai q_
2
2
ð1DÞ
where JLi and JAi are linear and angular velocity Jacobian
matrix, respectively. Obviously, it is rather cosðqs Þ, sinðqs Þ
than cosð2qs Þ, sinð2qs Þ (1 s i) that are included in
these two Jacobian matrix. The total kinetic energy of the
robot is
T ¼
n
X
i¼1
Ti¼
n X
n
1X
1
bij ðqÞq_ i q_ j ¼ q_ T B ðq Þ
2 i¼1 j¼1
2
ð1EÞ
P
where B ðq Þ ¼ ni¼1 ðmi J TLi J Li þ J TAi 0 R i I ci 0 R Ti J Ai Þ. It
should be noted that the total kinetic energy T is quadratic
to JLi and JAi ; hence, it will include cosð2qi Þ and sinð2qi Þ.
To this point, the Lagrangian Lðq ; q_ Þ for the manipulator
and generalized forces r performing work on joint coordinates are
Lðq ; q_ Þ ¼ T ðq ; q_ Þ Uðq Þ
ð1FÞ
d @L @L
¼r
dt @ q_
@q
ð1GÞ
Combining equations (1E) to (1G) yields
n
n X
n X
X
@bij 1 @bjk
@U
bij ðqÞ€qi þ
¼ ri :
q_ k q_ j þ
2
@q
@q
@q
k
i
i
j¼1
j¼1 k¼1
ð1HÞ
Usually, the generalized force r is written in the following form
M ðq Þq_ þ C ðq ; q_ Þq_ þ G ðq Þ ¼ r
ð1IÞ
Finally, it could be concluded that the elements in M ðq Þ
and C ðq ; q_ Þ contain cosðqi Þ; sinðqi Þ; cosð2qi Þ and sinð2qi Þ;
no more high order of cosðmqi Þ; sinðmqi Þ (m is a positive
integer) will be included. Thus, m ¼ 2.