Optimization and Fail-Safety Analysis of Antagonistic Actuation for pHRI.

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Optimization and Fail–Safety Analysis of
Antagonistic Actuation for pHRI
Gianluca Boccadamo, Riccardo Schiavi, Soumen Sen, Giovanni Tonietti, and
Antonio Bicchi
Interdepartmental Research Centre “E. Piaggio” via Diotisalvi 2, Faculty of
Engineering University of Pisa, Italy
<bicchi,g.tonietti>@ing.unipi.it
Summary. In this paper we consider some questions in the design of actuators for
physical Human-Robot Interaction (pHRI) under strict safety requirements in all
circumstances, including unexpected impacts and HW/SW failures.
We present the design and optimization of agonistic-antagonistic actuation sys-
tems realizing the concept of variable impedance actuation (VIA). With respect to
previous results in the literature, in this paper we consider a realistic physical model
of antagonistic systems, and include the analysis of the effects of cross–coupling be-
tween actuators.
We show that antagonistic systems compare well with other possible approaches
in terms of the achievable performance while guaranteeing limited risks of impacts.
Antagonistic actuation systems however are more complex in both hardware and
software than other schemes. Issues are therefore raised, as to fault tolerance and
fail safety of different actuation schemes. In this paper, we analyze these issues and
show that the antagonistic implementation of the VIA concept fares very well under
these regards also.
1 Introduction
One of the goals of contemporary robotics research is to realize systems which
operate with delicacy in environments they share with humans, ensuring their
safety despite any adverse circumstance [1]. These may include unexpected
impacts, faults of the mechanical structure, sensors, or actuators, crashes or
malfunctional behaviours of the control software [2, 3, 4, 5].
A recent trend in robotics is to design intrinsically safe robot arms by
introducing compliance at their joints. The basic idea of this approach is that
compliant elements interposed between motors and moving links help prevent
the (heavy) reflected inertia of actuators from concurring to damage in case of
impacts. Introducing compliance, on the other hand, tends to reduce perfor-
mance of the arm. Some approaches in the direction of minimizing the perfor-
mance loss while guaranteeing safety in case of impacts have been presented

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2G. Boccadamo, R. Schiavi, S. Sen, G. Tonietti, and A. Bicchi
in the recent literature (see e.g. [6]). Among these, a method was proposed
in [7] consisting in varying the compliance of the joint transmission mech-
anism while moving the arm. This so-called Variable Stiffness Transmission
(VST) technique, and its generalization in the Variable Impedance Actuation
(VIA) concept, have been shown to be capable in theory of delivering better
performance than purely passive compliance and other techniques.
In its formulation, however, the VIA concept in [7] used a rather abstract
model of actuator and transmission, whereby the impedance could be directly
controlled to desired values in negligible time. In this paper, we consider a
more realistic model of an actuation system implementing the idea, which is
based on the use of two actuators and nonlinear elastic elements in antag-
onistic arrangement. The antagonistic solution has several advantages, and
has been used in many robotic devices before (see e.g. [8, 9]), in some cases
because of biomorphic inspiration. However, to the best of our knowledge
the introduction of nonlinear springs to achieve variable stiffness was not a
motivation for earlier work.
In this paper, we consider the implementation of the VIA concept by means
of antagonistic actuation, discuss the role of cross-coupling between antago-
nist actuators, and apply optimization methods to choose parameters which
are crucial in its design. We show that antagonistic systems can implement
effectively the VIA concept, and their performance compares well with other
possible approaches.
Antagonistic actuation systems however are more complex in both hard-
ware and software than other schemes. Issues are therefore raised as to whether
safety is guaranteed under different possible failure modes. In the paper, we
also analyze these issues and show that the antagonistic implementation of
the VIA concept fares very well under these regards also.
2 Antagonistic actuation as a VIA system
In [7] it was shown that an ideal VIA mechanism (depicted in fig. 1-a) can
effectively recover performance of mechanisms designed to guarantee safety of
humans in case of impact. The basic idea is that a VIA mechanism can be
controlled according to a stiff-and-slow/fast-and-soft paradigm: namely, to be
rather stiff in the initial and final phases of motion, when accuracy is needed
and velocity is low, while choosing higher compliance in the intermediate,
high-velocity phase, where accuracy is typically not important. Low stiffness
implies that the inertia of the rotors does not immediately reflect on the link in
case of impacts, thus allowing smoother and less damaging impacts. Such ar-
guments where supported in [7] by a detailed mechanism/control co-design op-
timization analysis, based on the solution of the so-called safe brachistochrone
problem, i.e. a minimum time control problem with constraints on the max-
imum acceptable safety risk at impacts. The model considered fig. 1-a, how-
ever, uses direct variations of impedance, which is not physically realizable.

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Optimization and Fail Safety of Antagonistic Actuation for pHRI3
a) b)
Fig. 1. The concept of Variable Impedance Actuation (a) and a possible implemen-
tation by means of antagonistic actuators (b). Effective rotor inertias are coupled
to the link inertia through nonlinear springs.
a) b)
Fig. 2. An experimental implementation of an antagonistic VIA actuator (a) and
its conceptual scheme (b).
A possible implementation of the concept via an antagonistic mechanism is
depicted in fig. 1-b. Practical implementations of antagonistic VIA systems
may assume more general configurations than the one in fig. 1-b. For instance,
in the prototype of an antagonistic VIA system depicted in fig. 2-a ([10]), the
two actuators act through a nonlinear elastic element on the link, but they
are also connected to a third elastic element cross-coupling the actuators.
Questions we consider in this section are the following: is the stiff-and-
slow/fast-and-soft control paradigm still valid, and are the good safety and
performance properties of the ideal VIA device fig. 1-a retained by an antago-
nistic implementation as in fig. 1-b? What is the role of cross-coupling elastic
elements as in fig. 2-b in antagonistic VIA actuators?
To answer these questions, we use again the analysis of solutions to the safe
brachistochrone problem, which consists in finding the optimal motor torques
τ1,τ2which drive the link position xmovbetween two given configurations in
minimal time, subject to the mechanism’s dynamics, motor torque limits, and
safety constraints. This problem is formalized for the antagonistic mechanism
of fig. 1-b as

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4G. Boccadamo, R. Schiavi, S. Sen, G. Tonietti, and A. Bicchi
⎧
⎪
⎪
⎩
where Mmov, Mrot1, Mrot2are the inertias of the link and the rotors (effective,
i.e. multiplied by the squared gear ratio); Ui,max,i = 1,2 is the maximum
torque for motor i; φi, i = 1,2 represent the impedance of deformable elements
as functions of the position of the rotors and link. A polynomial nonlinear
stiffness model is used, whereby the applied force as a function of end-point
displacement is
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
minτ
Mrot1¨ xrot1+ φ1(xrot1,xmov) = τ1
Mrot2¨ xrot2− φ2(xrot2,xmov) = τ2
Mmov¨ xlink− φ2(xrot2,xmov) − φ1(xrot1,xmov) = 0
|τ1| ≤ U1,max
|τ2| ≤ U2,max
HIC(˙ xmov, ˙ xrot1, ˙ xrot2,φ1,φ2) ≤ HICmax,
?T
01 dt
(1)
φi(xj, xk) = K1(xj− xk) + K2(xj− xk)3.
This model has been found to fit well experimental data for the device in
fig. 2-a.
The safety constraint HIC(˙ xmov, ˙ xrot1, ˙ xrot2,φ1,φ2) ≤ HICmaxdescribes
the fact that the Head Injury Coefficient of an hypothetical impact at any
instant during motion, should be limited. The HIC is an empirical measure of
biological damage used in car crash analysis literature ([11]), and depends on
both the velocity of the impacting mass, its inertia and the effective inertia of
rotors reflected through the transmission stiffness. The HIC function is rather
complex, and can only be evaluated numerically for non trivial cases. How-
ever, based on simulation studies, a conservative approximation of the HIC
function for the antagonistic mechanisms was obtained which allows rewriting
the safety constraint in the simpler form |˙ xmov| ≤ vsafe(φ1,φ2,HICmax) (cf.
[7]).
Different solutions of problem (1) have been obtained numerically, set-
ting parameters to realistic values as Mmov = 0.1 Kgm2, Mrot1 = Mrot2 =
0.6 Kgm2, HICmax= 100m2.5
A first interesting set of results is reported in fig. 3. The optimal profiles
of link velocity and joint stiffness are reported for the case where both the
initial and final configurations are required to be stiff (σ0= σf= 16 Nm/rad,
plots a and b) and when both are compliant (σ0= σf= 0.2 Nm/rad, plots c
and d). Notice in fig. 3 that the stiff-and-slow/fast-and-soft paradigm applies
also to antagonistic actuation approaches. The minimum time necessary in
the two cases is 2.4 sec and 2.65 sec, respectively. This level of performance
should be compared with what can be achieved by a simpler actuation sys-
tem, consisting of a single actuator connected to the link through a linear
elastic element (this arrangement is sometimes referred to as SEA, for Se-
ries Elastic Actuation [12]). A SEA system with a motor capable of torque
Umax= 2U1,max= 15 Nm and inertia Mrot= 2Mrot1= 1.2 Kgm2, with lin-
ear elasticity coefficient matched exactly with the required stiffness in the two
(2)
s4 , U1,max= U2,max= 7.5 Nm.

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Optimization and Fail Safety of Antagonistic Actuation for pHRI5
0 0.51
Time [s]
1.52
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Link Velocity [rad/s]
0 0.51
Time [s]
1.52
0
2
4
6
8
10
12
14
16
Stiffness [Nm/rad]
a) b)
0 0.51 1.52 2.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time [s]
Link Velocity [rad/s]
0 0.511.522.5
0
0.5
1
1.5
2
2.5
Time [s]
Stiffness [Nm/rad]
c)d)
Fig. 3. Optimization results for antagonistic actuation without cross-coupling in
a pick-and-place task. A stiff-to-stiff task is shown in a, b, while c, d, refer to a
soft-to-soft task.
cases σ = 16 and σ = 0.2, would reach the desired configuration in 3.15 sec and
3.6 sec, respectively. Moreover, it should be pointed out that the SEA system
cannot change its stiffness without modifying the mechanical hardware.
Focusing again on the antagonistic VIA system’s results, it is also inter-
esting to notice that, in the likely case that the task requires the manipulator
to be stiff at the initial and final configurations (as it would happen e.g. in a
precision pick-and-place task), the actuators are required to use a significant
portion of their maximum torque just to set such stiffness, by co–contracting
the elastic elements. However, it is also in the initial and final phases that
torque should be made available for achieving fastest acceleration of the link.
Based on this observation, it can be conjectured that some level of preload-
ing of the nonlinear elastic elements in an antagonistic VIA system could be
beneficial to performance. Elastic cross-coupling between actuators (fig. 2-b)
can have a positive effect in that it can bias the link stiffness at rest, so that
more torque is available in slow phases, while torque is used for softening the
link in fast motion. On the other hand, it is intuitive that very stiff cross-
coupling elements would drastically reduce the capability of the mechanism
to vary link stiffness, thus imposing low velocities for safety and ultimately a
performance loss.
It is therefore interesting to study the effect of cross-coupling, to determine
if there is an intermediate value of stiffness which enhances performance with
respect to the limit cases of fig. 1 (no cross-coupling) and constant stiffness
(rigid cross-coupling). To this purpose, we study a modified formulation of
the safe brachistochrone problem, namely