2021 International Conference on Computer, Control and Robotics

2021 International Conference on Computer, Control and Robotics (ICCCR) | 978-1-7281-9035-8/20/$31.00 ©2021 IEEE | DOI: 10.1109/ICCCR49711.2021.9349379

Updating Potential Runaway Motion Volume in Task Space and Introducing
Minimum Volume Enclosing Ellipsoid

Hiroki Kito, Yoji Yamada, Jian Liu, Shogo Okamoto
Graduate School of Engineering
Nagoya University
Nagoya, Japane-mail: kito.hiroki@d.mbox.nagoya-u.ac.jp
Abstract—Human-robot collaboration (HRC) has gained
considerable interest among researchers, particularly in the
manufacturing field. In such systems, new risks that have notpreviously been considered may now need to be assessed. One ofthese risks is that of a robot and worker colliding with one
another. To avoid such collisions, a definition of the separation
distance between a robot and a worker is needed. According to
the ISO10218-1 standard, the safety of a worker needs to be
maintained even in the case of a single safety system fault.
Therefore, in defining the separation distance, a single fault
needs to be considered, so that a robotic manipulator may safely
run away when it occurs. As a method to quantify running away,
a potential runaway motion in the task space (PRAM-t) was
proposed. In the PRAM-t, a dynamic manipulability ellipsoidwas used to calculate the acceleration as the robot runs away.
However, when we consider running away originating from asingle fault, it can be exaggerated. Therefore, this paper
proposes a new method to compute acceleration during therunning away of a robot using an algorithm to compute
minimum volume enclosing ellipsoids. The proposed method
and the dynamic manipulability ellipsoid (DME) method arecompared using a case study.
Keywords-humanrobot
collaboration;
dynamic
manipulability; potential runaway motion in task space; single
fault

I.

INTRODUCTION

With the introduction of human-robot collaboration (HRC)
which has been intensively discussed [1], hazardous impacts
caused by collisions between industrial robots and workers are
considered to be typical of the risks due to foreseeable misuseon the worker’s side as well as to possible faults on the robot
controller’s side. According to the ISO10218-1 standard [2][3], it is required that the safety of a worker is secured even in
the case of a single fault within the safety related system, andthat the same principle of single-fault detection and protective
measures such as safety stops must be taken on the robot side.
When risks in the output reliability block [3] are taken into
account, it is necessary to define an environment where safety
can be maintained in consideration of runaway due to a single
fault of the robot. Protective separation distance (PSD),
quantitatively indicating the minimum protection separation
distance [2], is the target of safety requirements at
manufacturing sites. The appendix of [5] shows the safety
distance considering the relative motion vectors of both a

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robot and a human approaching it. in the case of personal carerobots coexisting with humans as well.
Szabo et al. used a laser scanner-type of safety-related
sensors in the SSM prototype for the first time [6]. The safety
and productivity of the SSM functionality was evaluated by
Marvel et al. [7] And they also conduct several curious
research about PSD function [8]-[10].
On the other hand, Byner et al. maximized the efficiency
of SSM by geometrically computing the distance of the robot
to the source of danger [9]. Lacevic et al expands the SSM
function to entire joints of the robot using a danger fieldalgorithm for calculating avoidance motion [12], [13]. A
number of studies have investigated the parameters that
supplement the PSD, such as basic parameters [14] and thelikelihood of intrusion [15]. Some researchers tried to
introduce psychological parameters to PSD. If a human can
avoid the robot itself, they do not collide. So avoidance actions
were investigated from the view point of safety [16]-[18].
Let us consider a relationship between a safety of HRC
and a single fault. A fault detection algorithm also being
researched actively. Brambilla et al. proposed fault detection
for robot manipulators via second-order sliding modes [19].
And same one via high-order sliding mode also be
investigated [20]. And a robust fault diagnosis and faulttolerant control system using high-order sliding mode ispresented [21].
This time, we will define the runaway action of amanipulator. The runaway action assumed in this study is
attributed to a single fault of the output robot. In addition, themost hazardous events shall be assumed when discussing
safety. Considering these two points, it can be said that theevent that we must assume is that of the robot joint causing asingle fault and the joint torque being maximized.
Furthermore, because industrial robots have a self-diagnosis
function, if a single fault such as one described above occurs,
the robot itself will detect it, and apply a brake after a certain
reaction time.
In summary, phases in a PSD process in association withthe robot runaway is decomposed as follows:1.
The output torque of one of the robot's joints is
maximized due to a single fault.
2.
After a reaction time, the joint is braked,
decelerating the robot.
3.
The robot is stopped completely.

In this paper, the above three steps are defined as“runaway.” Yamada et al. defined the space in which a robot
can move before an emergency stop during a runaway as apotential runaway space in the task space expression (PRAMt), and calculated the runaway distance using the concept of a
dynamic manipulability ellipsoid (DME) [24].

a dynamic manipulability measure of robot arms as aquantitative measure of their manipulating ability inpositioning and orienting the end-effectors, which takes thearm dynamics into consideration [24]. Furthermore, new
DMEs were proposed. Chiacchio et al. [26] researched theinfluence of gravity on the manipulators. In addition, for
redundant manipulators [27], the velocity-dependent dynamic
manipulability [28] was researched. And as an extended
concept of dynamic manipulability, the dynamic property ofmulti fingered grasping systems consisting of a multi fingered
hand and a grasped object was evaluated [29].
Since this concept is used in the PRAM-t, we start from abrief introduction of the theory. When there are no
disturbances, the equation of motion for an open-chain
manipulator with n rigid links can be expressed as:τ = M ( q ) q + τ c (q, q) + τ g (q)Figure 1. An illustration of PSD and PRAM-t.

Figure 1 illustrates a sample speed pattern which includes
both PSD (green colored area) and PRAM-t (orange-coloredarea), where vh and vr denote the relative speed of theoperator and the robot, respectively, and which define theclosest distance plane; Tri , Trl , and Tro denote the reaction
time of the protective equipment, logical control device, andmotor drive respectively. Ts , and Ts represent the stopping
time to make the speed of the robot zero and an increase in the
stopping time due to the acceleration of the robot system
respectively. Finally, at( a ) and ar(b ) denote acceleration anddeceleration performance of the robot, respectively.
In the PRAM-t, the DME is introduced as a method forcomputing acceleration when a robot runs away due to a single
fault. However, when we reconsider it, the constraint of a
DME is appropriate under our assumption. In other words, weshould make the DME constraints stricter. Therefore, we
propose introducing an algorithm for the computation ofminimum volume enclosing ellipsoids (MVCE) to the DME.
In this paper, we refer to it as a DME under the assumption ofa single fault (DMESF) and suggest to update PRAM-t byintroducing DMESF as an alternative of DME. And we alsorename updated PRAM-t, as a potential runaway motion intask space under the assumption of a single fault, (PRAM-sf).
Then we compare the DME and the DMESF using a casestudy as a pre-step of proposing PRAM-sf.
II. POTENTIAL RUNAWAY MOTION IN THE TASK SPACEA.

Dynamic Manipulability Ellipsoid
Yoshikawa et al. proposed the concept of manipulability
measure of robotic mechanisms, as discussed in [23]. This isa measure of the manipulating ability of robotic mechanisms
in positioning and orienting end-effectors. They also proposed(1)
where τ is an n × 1 vector of joint actuator torques and q ,n × 1 vectors of joint angles, joint
q and q are
angular velocities, and joint angular accelerations,
respectively. M (q) is the n ×n mass matrix that comes fromthe inertial properties of the robot, τ c (q, q ) represents thevector of Coriolis and centrifugal forces, and τ g (q)represents the vector of joint torque that comes from gravity.
Let us define the vector of the position of the tip of the robot
as x (m × 1 ). In this study, we only consider translational
acceleration. Thus, we can say that m ≤ 3. Using the m × n
Jacobian matrix J , we can obtain the relationship between x
and q .
()x = J (q)qThis equation can also be obtained by differentiating (2)with respect to time.
()x = J (q)q + J (q, q)q
The mass matrix M is positively defined and invertible.
Therefore, using Eq. (1) and Eq. (3), we can obtain thefollowing equations:x = JM −1 (τ − τ c − τ g ) + Jq

()

x = JM −1τ + xvel + x grav

()xvel = −JM −1τ c + Jq()

xg rav = − JM −1τ g()where:

It is imperative to consider torque limits as well.
− imax   i   imax ,i = 1,,n()

If the torque limits are symmetric, as in (8), τ̂ which isthe normalized τ is given by:τˆ = [1 / 1max ,, n /  nmax ]()Here, let us introduce L where,
L = diag (1max ,, nmax )ˆ = M −1L
M

()
()

In both graphs, the horizontal axis represents time, and thevertical axis represents the velocity of the manipulator. v R isthe velocity of the manipulator when a single fault occurs,
vmax is the upper limit maximum velocity of the manipulator,t R is the reaction time, and the area of each graph shows the
runaway distance.

Using Eq. (11) with Eq. (5), we obtain Eq. (12) asˆ ˆ+ x + x
x = JMτ
vel
grav

()

We can interpret xvel + xgrav as a bias of the end-effectorˆˆ,acceleration with no torque. When we focus on the term JMτˆJMτˆ is represented as an ellipsoid because ˆi  1 . If the

Figure 2. Relationship between manipulator velocity and time.

singular value decomposition is JMˆ as in Eqs. (13)-(16), thespindles can be obtained as  1u1 , , m um .

III. DYNAMIC MANIPULABILITY ELLIPSOID UNDER THE
ASSUMPTION OF A SINGLE FAULT

ˆ = UΣV T
JM()U =  u1 , , um 

()

 1

Σ =
0




0
 m 
0V =  v1 , , vn 

()

()

(Note that U , Σ ,V is m ×m, m ×n, and n ×n, respectively).
Spindles of the DME can be obtained as 𝜎1 𝒖1 ,⋯, 𝜎m 𝒖m.
B. Otential Runaway Motion in the Task Space
If the manipulator breaks down and runs away, it ispossible that the manipulator will take a diﬀerent trajectoryfrom its normal trajectory. Since this research is aimed at thecollaborative operation of industrial manipulators in a
manufacturing environment, it can be assumed that the
manipulator has a self-diagnosis function. In other words, it ispossible to terminate the manipulator in a stopped state after acertain time (reaction time) from the state at which the fault is
detected by itself and causes the manipulator to run away.
When the manipulator breaks down and begins to run
away, the speed each joint will output is uncertain. Under thiscondition, the manipulator’s end effector moves during thetime to reach the stop state and it will be positioned
indeterminately within a certain range. This is called thepotential runaway motion in the task space (PRAM-t) [22].
In PRAM-t, it is necessary to check the upper limit
maximum velocity of the manipulator in advance anddetermine whether the velocity of the manipulator reaches thisupper limit velocity during the reaction time. The manipulator
velocity pattern when the it reaches the upper limit maximum
velocity due to acceleration during runaway is determined as
Speed Pattern 1, and in the same case, the manipulator
velocity pattern when it does not reach the upper limit
maximum velocity is determined as Speed Pattern 2. These
are shown in Figure 1, the left side depicting the former andthe right side depicting the latter.

A. Disadvantage of the Original PRAM-t
The potential runaway motion in a task space is a usefultechnique for visually expressing the manipulator’s behavior
during runaway.
However, in some cases, the space obtained using thismethod may overestimate the space where the manipulator
can run away. When we calculate the potential runaway spaceof the workspace representation, a driving force may begenerated at multiple joints. Because we consider a single
fault, it is impossible for multiple joints to generate a driving
force. Therefore, the latent runaway space in the workspace
expression includes a space that cannot be reached in the eventof a single fault. Deﬁning a space larger than necessary shouldbe less productive because it leads to low work efficiency andrequires a larger task space at the manufacturing site. In thesubsequent sections, we propose a method to calculate the net
latent runaway space at the time of a single fault and explain
the concept and algorithm of the minimum volume enclosing
ellipsoid used in the method. As we wrote in introduction, wesuggest to update PRAM-t by introducing DMESF as analternative of DME. And we also rename updated PRAM-t, asa potential runaway motion in task space under the assumptionof a single fault, (PRAM-sf).
B. Minimum Volume Enclosing Ellipsoid
In this section, we discuss the algorithm for the minimum
volume enclosing ellipsoid (MVCE). In [30], the authors
proposed a method for rounding polytopes in the real number
model of computation, after which it was updated by [33]. In[31], a minimum enclosing polytope in high dimensions was
proposed. In [32], the computation of the MVCE wasresearched.
It would be useful to explain its properties brieﬂy.
Consider a set of m points in an n-dimensional space,
S = {x1, , xm} R n . To guarantee that any ellipsoidcontaining S has a positive volume, we assume that theaﬃne hull of the S spans R n . An ellipsoid in center form isgiven by:ε = {x  Rn ( x − c)T E ( x − c)  1}

(17)

where c Rn denotes the center of the ellipsoid ε andn.
E  S++
Since we want points xi to be inside ε , they must satisfy( x − c )T E ( x − c )  1

i = 1,, m.

v0det( E )det( E −1 )subject to( x − c) E ( x − c)  1
E 0

(19)

τˆ  i =  0,..., 1,...,0ˆˆ +x +xx i = JMτ
i
vel
grav

i=1,...,ni = 1, , m (20)

However, Eq. (20) is not a convex optimization problem
and, by a change of variable, we can define the ellipsoid as:
ε = { x  R n Ax − b 2  1}

(21)

1

where A = E 2 and b = E 2 c . Therefore, considering Eq.
(21), the optimization problem (20) becomes:
minimize A,blog det ( A−1 )subject to

Axi − b  1

Finally, we can obtain DMESF by computing MVEE
which including all x i .

i = 1, , m

A. Condition of Comparison
To investigate the usefulness of the proposed DMESF, wecalculated the DME and the DMESF under the same
conditions and compared their volumes. This time, we notethe conditions of calculation. We chose KUKA’s LBR iiwa
700 (KUKA) as the target manipulator for the comparison, as
shown in Figure 3. where,  B is the basic KUKA coordinate.
We defined typical five configurations of KUKA, as shown in
Figure 4 for the purpose of quantitatively evaluating theperformance of proposed DMESF. In Figure 4,  B is also thebasic coordinate of KUKA.

(22)

A0

The norm constraints Axi − b  1 , which are just convex
quadratic inequalities in the variables A and b , can be
expressed as follows:I( Ax − b)Ti

Axi − b 
0
I 

(23)

In short, the optimization problem of the MVEE can besolved by changing variables A and b .

Figure 3. KUKA LBR iiwa 700.

C. Dynamic Manipulability Ellipsoid under a Single Fault
Assumption
In this section, we consider the method to calculate
dynamic manipulability ellipsoid (DMESF). In order tocalculate the DMESF, the following constraints are added tothe joint torque vector. The constraint is that when the normof normalized torque vector is less than 1 and the torque of thejoint i is not 0 and others are 0. For example, the torque vector
when a single fault occurs on the joint i is like this:τˆ i =  0,...,ˆi ,...,0
subject()IV. COMPARISON OF DME AND DMESF

T

1

()

(18)

where v0 is the volume of the unit hypersphere in ndimensional space. Considering these, a natural formulationof the minimum volume enclosing ellipsoid (MVEE) problem
is described as follows:
minimize E,c

ˆˆ +x +xxi = JMτ
i
vel
grav

And then, we only need the largest acceleration due to asingle fault of each joint in order to calculate DMESF using
MVEE algorithm. We can obtain the set of largest
accelerations due to a single fault of each joint like this:

The volume of ε is given by:Vol(ε )=

And using (15) and (24), the acceleration due to a single
fault of the i joint xi can be obtain like this:to ˆi  1()Figure 4. Different configurations defined for comparison.

B. Results of Comparison
Figures 5–9 show the DME and DMESF comparisons foreach configuration. The axes of each graph show thedirections of acceleration, and the coordinates for each graph
are the same for the basic coordinate of KUKA. The circles
depict the acceleration, xi + , xi − (i = 1,.., n) , and the brokenlines denote the DME and DMESF in their respective graphs.
From these results, it can be seen that the DMESF isenveloped by the DME in all configurations.

Figure 9. DME and DMESF for configuration 5.

Figure 10. Comparison of volumes.
Figure 5. DME and DMESF for configuration 1.

A comparison of the volumes is shown in Figure 9. The
vertical axis shows the volume of the ellipsoid and thehorizontal axis shows the configurations. The blue bar
represents the DME and the orange bar represents the DMESF.
From this result, it can be gauged that the volume of the
DMESF is smaller than that of the DME under all conditions.
If we use the DMESF instead of the DME, the calculated
space becomes smaller, the reduction of which is importantwhen we consider introducing the PRAMSF as the separation
distance between a human and manipulator.
V. DISCUSSION

Figure 6. DME and DMESF for configuration 2.

Figure 7. DME and DMESF for configuration 3.

Figure 8. DME and DMESF for configuration 4.

In this work, we defined the worst case scenario due to asingle fault on the output manipulator. It was that a torque of
one joint becomes maximize. And we considered a computing
algorithm of the DMESF under this scenario. However,strictly speaking, there is a more dangerous case example than
the worst case we defined. The case is that multiple torques
are generated and it can occur due to a common cause failure.
Although multiple torques can be generated, the possibility ofcommon cause failure is low enough not to be taken into
account. Therefore we can ignore the scenario considering theconcept of risk assessment.
The comparison (Figure 9) shows that the volume of
DMESF is smaller than that of DME in all configurations.
And the difference of volume vary greatly according toconfigurations. About configuration 2 and 3, the difference isrelatively small and the benefit of introducing DMESF
become small in such configuration.
Finally, there is room for the improvement of human robot
collaboration tasks. The relationship between the worker's
safety and the work space which depends on the separation
distance is a tradeoff. The computation time of the method we
proposed should be short enough to implement it in an on-line
system. So, it is necessary to grasp the target problem as amultiple objective optimization one considering the worker's
safety as the high priority.

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VI. CONCLUSION
In this study, the PRAM-t was proposed as a method torepresent the space in which a manipulator can move. In thepotential runaway motion volume in the task space, a DME
was used to compute the acceleration of the manipulator whena single fault occurred at joint i of the manipulator. Further, analternative to the DME was proposed. We named it as
DMESF; it is a method that combines dynamic manipulability
and the algorithm for computing the MVEE. To evaluate theproposed method, we compared the DME and DMESF under
the same conditions for different manipulator configurations.
As a result, it was found that the DMESF is enveloped by the
DME, and the volume of the DMESF is smaller than that of
the DME. These results indicate that DMESF can represent
the substantive torques which are generated due to a single
fault. Therefore introducing the DMESF as an alternative to
the DME is useful and achieves a reduction in space.
The approach to combining the dynamic manipulability
and the method for computing a minimum volume enclosing
ellipsoid can realize a reduction of a separation distance
between a human and a manipulator while ensuring safety.
This concept should contribute to the enhancement of safety
in HRC tasks if it is visualized (e.g. on a simulator). Ongoingwork following this research includes validating DMESF,
computing PRAMSF, implementing PRAMSF to an on-line
safety related system, and evaluating the performance of the
PRAMSF as an alternative to the PRAM-t.
ACKNOWLEDGMENT
The authors would like to thank their colleague Kim
Eugene for the provision of referential data.
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