Page 1

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 0.51 1.52 2.5

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

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6G. Boccadamo, R. Schiavi, S. Sen, G. Tonietti, and A. Bicchi

10

?2

10

0

10

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

K* [Nm/rad]

a)

Minimum Time [s]

0 0.51 1.52 2.5

0

1

2

3

4

5

Time [s]

b)

Link Velocity [rad/s]

00.511.52 2.5

0

2

4

6

8

10

12

14

16

Time [s]

c)

Stiffness [Nm/rad]

Fig. 4. a) Minimum time to reach the target configuration vs. cross-coupling stiffness

K3 (L0 = 1). Solid: σ0 = σf = 16 Nm/rad; dashed: σ0 = σf = 0.2 Nm/rad.

Optimization results for antagonistic actuation with cross-coupling in a pick-and-

place, stiff-to-stiff task (solid) and soft-to-soft task (dashed): velocity (b) and joint

stiffness (c).

⎧

⎪

⎪

⎩

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

minτ

Mrot1¨ xrot1+ φ1(xrot1,xmov) − φ3(xrot1,xrot2) = τ1

Mrot2¨ xrot2+ φ2(xrot2,xmov) + φ3(xrot1,xrot2) = τ2

Mmov¨ xlink+ φ2(xrot2,xmov) − φ1(xrot1,xmov) = 0

|τ1| ≤ U1,max

|τ2| ≤ U2,max

HIC(˙ xmov, ˙ xrot1, ˙ xrot2,φ1,φ2) ≤ HICmax,

where φ3(xrot1,xrot2) indicates the cross-coupling elasticity. In particular, we

study how optimal solutions vary in different instances of the problem with in-

creasing cross-coupling stiffness. A linear stiffness model is assumed for cross-

coupling, with φ3(xrot1,xrot2) = K3(L0+xrot2−xrot1), where L0denotes the

preload offset.

In fig. 4 numerical solutions obtained for problem (3) are reported, for the

two stiff-to-stiff and soft-to-soft tasks already considered in fig. 3). It can be

observed from fig. 4-a that an optimal value of cross-coupling K3 exists in

both cases, but they do not coincide. Setting for instance K3= 0.04 Nm/rad,

the optimal value for the stiff-to-stiff task, we obtain the optimal link velocity

and stiffness plots reported in fig. 4-b and -c, and the optimal times 2.35

sec and 2.65 sec, respectively. Optimizing the design for one task can hence

decrease performance in others.

?T

01 dt

(3)

3 Fail Safety

In this section we analyze how antagonistic VIA mechanisms behave under

failure of some of their components, and compare their ability to remain safe

in spite of failures, with those of other possible actuation structures.

A large variety of possible failure modes should be considered in order to

assess fail safety of a robot system. In our case, we focus on the following

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Optimization and Fail Safety of Antagonistic Actuation for pHRI7

possible events: mechanical failures, consisting in breakage of one or more of

the elastic elements, leading to K → 0, and HW/SW control failures, whereby

one or more of the actuators delivers an uncontrolled torque. In the latter case,

we consider that the torque can either remain “stuck” at some value, or default

to zero, or to ±Umax.

An extensive simulation campaign has been undertaken to study the effec-

tive HIC at impacts of an antagonistic VIA mechanism (fig. 1-b), evaluating

all possible failure mode combinations. To be conservative, the HIC is eval-

uated assuming an impact occurs after some time from the instant of the

fault event, during which time the mechanism could increase its momentum.

The diagram in fig. 5 reports the worst-case HIC during the execution of a

typical pick-and-place task, under a nominal optimal control. Labels in dif-

ferent regions indicate which failure mode determines the worst-case HIC of

impacts. It can be noticed from fig. 5 that the worst case impact has maxi-

1 1.52 2.53 3.5

0

10

20

30

40

50

60

70

80

90

100

Time [S]

HIC [m2.5/s4]

τ1 = stuck

τ2 = stuck

τ1 = U1,max

τ2 = U2,max

K1 = 0

τ2 = stuck

K1 = 0

τ2 = U2,max

τ1 = U1,max

τ2 = U2,max

Fig. 5. HIC values under different failures mode for VIA.

mum HIC value around 100. This has been obtained by suitably choosing the

mechanism parameters, in particular by setting U1,max= U,2max= 1.5Nm.

The performance under these design choice is around 3 sec. Should different

values of worst-case HIC be specified, the antagonistic VIA mechanism could

be designed with different parameters to accommodate for the specifications.

It is interesting to compare these results with other existing actuation

systems for pHRI. Namely, we will make reference to SEA and to the so-called

Distributed Macro-Mini (DM2) actuation scheme ([13]). The latter basically

consists of a SEA with a small inertia, low torque, high bandwidth actuator

connected directly to the link inertia to actively damp oscillations.

In order to obtain a meaningful comparison, we consider optimized SEA

and DM2implementations, with equal rotor and link inertia, which obtain

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8 G. Boccadamo, R. Schiavi, S. Sen, G. Tonietti, and A. Bicchi

a similar performance of 3 sec in the same task. Fig. 6 describes the worst-

case HIC values obtained for SEA and DM2, respectively. Failure modes have

been considered which are analogous to those described above, as are all other

parameters in the simulations. It can be observed that, in some modes (notably

for control torques defaulting to their maximum value, τ → Umax), both SEA

and DM2can be unsafe (HIC ? 100). The DM2scheme exhibits slightly

better fail safety characteristics than SEA. An explanation of the apparently

superior fail-safety characteristics of antagonistic VIA actuation is that such

scheme achieves comparable nominal performance by employing two motors

each of much smaller size than what necessary in the SEA and DM2.

1 1.52 2.5

Time [s]

3 3.54

0

100

200

300

HIC (DM2) [m2.5/s4]

1 1.52 2.5

Time [s]

3 3.54

0

50

100

150

200

HIC (SEA) [m2.5/s4]

τ1 = U1,MAX

τ2 = U2, MAX

τ1 = ?U1,MAX

τ2 = τ2, opt

τ1 = U1,MAX

τ2 = U2, MAX

τ1 = ?U1,MAX

τ2 = τ2, opt

τ1 = U1,MAX

τ2 = U2, MAX

τ = UMAX

τ1 = ?UMAX

τ1 = ?UMAX

τ = UMAX

τ = UMAX

Fig. 6. Maximum of the HIC value in various failure cases for SEA (top) and DM2

(bottom).

4 Conclusion

We discussed the design of variable stiffness actuation systems based on an-

tagonistic arrangements of nonlinear elastic elements and motors. The basic

stiff-and-slow/fast-and-soft idea of the VIA approach was shown to apply and

to be effective for realistic models of antagonistic actuation.

In order for the system to guarantee safety of operations in the proximity

of humans, its behaviour must remain safe in conditions where functionality

is degraded by possible failures. A thorough examination of possible HW and

SW failures of the system has been considered, and results have been compared

with those of other possible solutions, with favourable result.

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Optimization and Fail Safety of Antagonistic Actuation for pHRI9

5 Acknowledgments

Work partially done with the support of the EU Network of Excellence EU-

RON - European Robotic Network, under the Prospective Research Project

PHRIDOM

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