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A Prototype of a Novel Energy Efficient Variable Stiffness Actuator

L.C. Visser∗, R. Carloni∗, F. Klijnstra∗∗and S. Stramigioli∗

Abstract—In this work, we present a proof of concept of a

novel variable stiffness actuator. The actuator design is based

on the conceptual design proposed in earlier work, and is

such that the apparent output stiffness of the actuator can

be changed independently of the output position and without

any energy cost. Experimental results show that the behavior

of the prototype is in accordance with the theoretical results

of the conceptual design, and thus show that energy efficient

variable stiffness actuators can be realized.

I. INTRODUCTION

Humans can perform tasks in many different conditions

and environments since they are able to properly adjust the

stiffness of their joints. In contrast, robot actuation is usually

stiff and is not suitable in applications in which robots need

to cooperate with humans, such as in prosthetics, rehabilita-

tion or wearable devices and social robots. Therefore, robots

should have capabilities similar to humans, and be capable of

adjusting the joint stiffness to the task and environment. This

can be achieved by using actuators, of which the apparent

output stiffness, and thus the joint stiffness, can be changed.

For this kind of actuators, commonly called variable stiff-

ness actuators, the apparent output stiffness can be changed

independently of the output position by means of a number

of internal springs and actuated degrees of freedom, which

determine how the springs are sensed at the output of the

actuator. Based on this concept, a number of variable stiffness

actuators have been presented in recent years, including

AMASC [1], VSA [2], VS-Joint [3], and MACCEPA [4].

These actuators rely on the pretension of one or more springs,

in series with the actuator output, to change the output

stiffness. This means that, to change the stiffness, the amount

of energy stored in the springs is changed. The consequence

is that energy is supplied to the actuator without doing work

at the output, i.e. energy is required to change the stiffness.

In a recent work, we presented a general port-based model

for variable stiffness actuators [5]. From the analysis of

the model, design criteria were derived for energy efficient

variable stiffness actuators, and it was shown that it is

possible to design actuators in which the stiffness can be

changed without using energy. Based on this, we designed

a conceptual actuator and validated the design with sim-

ulations. In this paper, we present a prototype realization

This work has been funded by the European Commission’s Seventh

Framework Programme as part of the project VIACTORS under grant no.

231554.

∗{l.c.visser,r.carloni,s.stramigioli}@utwente.nl, Department of Electrical

Engineering, Faculty of Electrical Engineering, Mathematics and Computer

Science, University of Twente, 7500 AE Enschede, The Netherlands.

∗∗f.klijnstra@student.utwente.nl, Deptartment of Advanced Technology,

Faculty of Science and Technology, University of Twente, 7500 AE En-

schede, The Netherlands.

C

D

∂H

∂s

˙ s

τ

˙ q

−F

˙ x

Fig. 1.

Dirac structure D defines the interconnection between the different elements

and, therefore, how power is distributed among the ports. The multi-bonds

allow any number of springs, i.e. the C-element, and any number of control

inputs (τ, ˙ q). The one dimensional port (−F, ˙ x) is the output port.

Generalized representation of a variable stiffness actuator - The

of the conceptual design. We show that experiments are

in accordance with the simulation results, and thus we can

validate the concept. In particular, we show that the apparent

output stiffness of the prototype actuator can be changed in

an energy free way, and that energy supplied to the actuator

is used only to do work on the load.

II. PORT-BASED MODEL OF VARIABLE

STIFFNESS ACTUATORS

In this Section, we recall the model of variable stiff-

ness actuators, derived by using the port-based modeling

formalism [6] and extensively described in [5]. The port-

based framework captures the essential properties of variable

stiffness actuators in terms of energy and, in particular, it

provides important insights in the power flows between the

actuator, the actuated system and the controller of the internal

degrees of freedom of the actuator.

In the formulation of the model, we assume that:

• the variable stiffness actuator has internal springs;

• there are actuated degrees of freedom that determine

how the springs are sensed at the output of the actuator;

• internal friction and inertias can be neglected.

The model is graphically depicted in Fig. 1 by using bond

graphs. The multi-dimensional C-element represents the in-

ternal springs of the actuator and is characterized by a state

s, i.e. the elongation or compression of the springs, and by

an energy function H(s), describing the amount of elastic

energy stored by the springs. The power conjugate port

variables of the element are the flow ˙ s, the rate of change

of the internal state, and, dually, the forces exerted by the

springs, i.e. the effort∂H

The internal degrees of freedom of the actuator are actu-

ated via the control port (τ, ˙ q), where the effort variable τ

describes the generalized forces that actuate the degrees of

freedom and the flow variable ˙ q denotes the generalized rate

of change of the configuration variables q. The output port

of the actuator is denoted by (−F, ˙ x), where the effort F is

∂s.

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the force generated by the actuator and the flow ˙ x is the rate

of change of the output position x.

The Dirac structure D defines how the power is distributed

between the bonds. The structure is power continuous, and

thus defines a constraint relation between the power variables

of the connected bonds. In matrix form, the Dirac structure is

represented by means of a skew symmetric matrix D(q,x):

˙ s

τ

−F

=

?

0

A(q,x)

0

−C(q,x)T

??

B(q,x)

C(q,x)

0

−A(q,x)T

−B(q,x)T

?

D(q,x)

−∂H

˙ q

˙ x

∂s

(1)

Since gyration does not regularly exist in the mechanical

domain, C(q,x) = 0. Note that the Dirac structure is not

necessarily constant and is allowed to depend on both the

configuration of the internal degrees of freedom q and the

actuator output position x.

The rate of change of the energy stored in the springs is:

dH

dt

=∂H

∂s

ds

dt=∂H

∂s

?

A(q,x)˙ q + B(q,x)˙ x

?

= τT˙ q − FT˙ x

(2)

i.e. the sum of the total power supplied via the control and

via the output port. The output force and the apparent output

stiffness of the actuator are given by:

F =∂H

∂x,

K =∂2H

∂x2

(3)

The stiffness can be changed, without injecting any energy to

the spring via the control port, by varying q while satisfying

˙ q ∈ kerA(q,x)

∀q,x

(4)

Note that a guideline for the design of energy efficient vari-

able stiffness actuators is that the apparent output stiffness

is changed without changing the elongation of the springs

and, therefore, the matrix A(q,x) of the model should have a

kernel. This means that, for such actuators, the design should

decouple the output position and the output stiffness on a

mechanical level.

III. CONCEPTUAL DESIGN

A conceptual design, which follows the guidelines of

Sec. II, has been proposed in [5] and is depicted in Fig. 2.

The actuator concept consists of one internal zero free

length spring and two actuated degrees of freedom. More

specifically, the spring is characterized by elastic constant

k and is connected to the output via a lever arm, whose

length can be varied by a linear degree of freedom q1, where

0 < q1 ≤ ℓ. The second internal degree of freedom q2

determines the position x of the output. It is assumed that

the dimensions and working conditions are such that the

angle α may be neglected. Due to the kinematics of the

system, the state s of the spring is given by s = ℓsinφ,

with sinφ =x−q2

given by the energy function H(s) =1

q1. The elastic energy stored in the spring is

2ks2. Then, the force

α

−φ

q1

q2

x

−s

k

ℓ

Fig. 2. Conceptual design of an energy efficient variable stiffness actuator -

The design is based on a lever arm with variable effective length, determined

by the linear degree of freedom q1, with 0 < q1≤ ℓ. The linear degree of

freedom q2controls the equilibrium of the output position x. The stiffness

at the output only depends on q1and on the elastic constant k of the linear

zero free length spring, whose state is s.

and the stiffness felt at the output port are respectively given

by

F =∂H

∂x= k

?ℓ

q1

?2

(x − q2),K =∂2H

∂x2= k

?ℓ

q1

?2

(5)

For this concept, in Eq. (1) we have

A(q,x) :=?A1(q,x)

B(q) :=

A2(q,x)?= −ℓ

q1

?sinφ

1?

ℓ

q1

(6)

Note that, accordingly to the design guidelines, the matrix

A(q,x) has a kernel and, therefore, the output stiffness can

be changed without supplying energy via the control port.

IV. THE VARIABLE STIFFNESS ACTUATOR

PROTOTYPE

A. Realization

The conceptual design described in Sec. III has been

realized in a test setup, shown in Fig. 3, in order to provide

a proof of concept. The realization closely matches the

concept, with the exception that the zero free length linear

spring has been replaced by an antagonistic spring setup

acting on the rotation axis of the lever arm. The springs

appear as a rotational spring with an elastic constant of

k = 0.68 Nm/rad. This arrangement is easier to realize,

while it results in exactly the same system behavior. As a

result, there is no longer a need to consider a specific working

condition for which α can be neglected. Moreover, the length

ℓ of the lever arm no longer appears in the kinematics.

The two linear internal degrees of freedom are actuated by

spindle drives with Maxon A-max brushed DC motors [7].

Sliders provide the kinematic constraints depicted in Fig. 2.

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q1actuation

q2actuation

actuator output x

Fig. 3.

concept presented in Fig. 2. The main difference is in the implementation

of the zero free length spring, which is realized by using an antagonistic

spring setup acting on the rotation point of the lever arm.

Prototype design and realization - The design closely matches the

B. Model

Before proceeding with the experimental tests for the

validation of the concept, we build the model of the setup

in order to verify if the real data are comparable with

the simulation results. In particular, by detailing the Dirac

structure of Fig. 1, we derive the bond graph model of Fig. 4,

which represents both the conceptual design and the real

system since there is no substantial difference between them.

The multidimensional port (τ, ˙ q) of Fig. 1 is split into two

separate control inputs, which separately actuate the internal

degrees of freedom q1 and q2. The subsystems labeled by

M1 and M2 contain all the relevant dynamical properties

of the two motors, as specified by the data sheets, and the

spindle drives.

The two MTF-elements (modulated transformers) imple-

ment the matrix A(q,x) of the Dirac structure (1), as given

in Eq. (6). The 0-junction represents a shared effort (the

torque) on the connected bonds and a summation of the

flows (velocities). The third MTF-element implements the

matrix B(q) of the Dirac structure (1), as given in Eq. (6).

The 1-junction represents a shared flow on the connected

bonds and represents the actuator output port with velocity

˙ x and force F. Moreover, in this model, we consider that the

system is actuating a load with inertial and friction properties

modeled by the I-element and the R-element, respectively. In

the experiments, we consider a load with mass m = 0.06 kg

and a friction coefficient r = 20 Ns/m. The value of the

friction coefficient is due to the high friction in the sliders

supporting the output motion and has been experimentally

estimated.

V. SIMULATION AND EXPERIMENTAL RESULTS

In this Section, we present a comparison between the sim-

ulation results of the model presented in Sec. IV-B and the

experimental data, both obtained in two different scenarios.

In order to have commensurable data, we implement in both

simulation and real setup the same controllers, namely PID

controllers on the velocities q1and q2, with properly tuned

parameters. Using the 20-sim simulation package [8], and

its 4C toolchain, it is possible to simulate the bond graph

model of Fig. 4 and directly export the controllers used in

C

..

k

∂H

∂s

˙ s

0

MTF

..

B(q)

−F

˙ x

1

I :m

R :r

A1(q,x):MTFMTF :A2(q,x)

τ1

˙ q1

τ2

˙ q2

M1

M2

Fig. 4.

elements model the Dirac structure as given in Eq. (6). The subsystems

M1 and M2 model the actuation of the degrees of freedom q1 and q2

respectively. The I-element and the R-element model, respectively, the

inertial and friction properties of the actuated load.

Bond graph based model of the prototype design - The MTF-

this simulation to C code, which can then run on an external

controller board to actuate the real setup.

Experiment 1 - Static output force: In the first experiment,

the aim is to determine a relation between the output force

F and the configuration q1, which is directly related to the

output stiffness by Eq. (3). The force is measured while the

output position x is fixed (i.e. ˙ x = 0). The claim is that the

stiffness can be changed without changing the energy stored

in the spring. To achieve this, the spring is loaded and the

degrees of freedom are actuated while satisfying Eq. (4).

The experiment is summarized in Fig 5. The output

position x is fixed, q1 is set at a distance of 0.076 m to

the rotation point of the lever arm, q2is such that the angle

φ = 0.15 rad. Then, q1is moved towards the rotation point in

0.005 m increments towards the final value of 0.026 m, while

q2is actuated according to Eq. (4). This implies that φ and

sinφ =

q1

are kept constant. After each increment, the

output force in Eq. (5) is measured, and it is expected that it

varies such that F(q1) = γq−1

is obtained from the kinematic analysis of the design.

The results for this experiment are presented in Fig. 6,

in which the averages of the measured output force for

a number of values of q1 are shown, together with the

standard deviation σ. The mean values of the experiments

are, except for q1= 0.076 m, all within 1σ of the theoretic

curve. The deviation for q1 = 0.076 m can be explained

by stiction present in the system. For increasing values of

q1, the force generated on the output decreases by Eq. (5),

but for this particular value for q1, this force is no longer

sufficient to overcome the stiction forces in the supporting

sliders. If the measurements for this value of q1are no longer

considered to be valid, the following curve can be fitted to

the average values of the experiments using a least square

fit, i.e. F(q1) = 0.107 · q−0.99

1

These results show that, in the prototype, the output stiffness

can be changed in an energy free way, i.e. while the energy

stored in the springs is not changed.

Experiment 2 - Dynamic output displacement: In the

x−q2

1, where γ = ksinφ = 0.101

, with a residual r2= 0.97.

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0 50 100150 200 250

0

0.1

0.2

0 50100 150200 250

0

0.05

0.1

0 50 100 150200250

0

0.01

0.02

Time [s]

Time [s]

Time [s]

φ [rad]

q1[m]

q2[m]

Fig. 5.

is varied along its configuration range. q2 is varied satisfying Eq. (4), so

that φ is kept constant and no energy is injected into or removed from the

spring. The grey lines represent the set point values for φ and q1, while the

solid black lines represent the measured experimental values.

Static output force measurements - While the spring is loaded, q1

0.02 0.030.04 0.05

q1[m]

0.06 0.070.08

0.5

1

1.5

2

2.5

3

3.5

4

4.5

F [N]

Theory

Simulation

Measured

Fig. 6.

theoretical output force. The open squares and the solid dots indicate simu-

lation results and the average experimentally measured values respectively.

The vertical bars indicate the measurement standard deviation.

Theoretic and measured output force - The solid curve shows the

second scenario, we consider that the output load consists

of a mass of 0.06 kg and we want to displace it while

keeping a constant output stiffness. With this experiment,

we aim to show that, when the stiffness is kept constant,

all energy supplied via the control port is used to do work

on the output port. This requires that, when the mass is not

accelerated, there is no energy stored in the spring. Starting

from an output position x = 0, a desired set point value

x = 0.03 m is provided, and q2is actuated to achieve this

output position. Since we want to have a constant stiffness,

q1is not actuated.

The results for both simulation and experiments are pre-

sented in Fig. 7. The simulated system responds much faster

to the step set point, although the settling time for both the

model and the prototype is of the same order of magnitude.

It can also be noted that, when q2 is actuated to displace

the load at the desired position, the spring is compressed

φ ?= 0 due to the inertial properties of the load and the

friction in the system. However, since the spring returns to

05 10 1520 25

0

0.02

0.04

05 1015 2025

0

0.02

0.04

05 101520 25

0

0.05

0.1

Time [s]

Time [s]

Time [s]

x [m]

q2[m]

φ [rad]

Fig. 7.

experiments consists of displacing a mass of 0.06 kg from x = 0 to a

desired output position x = 0.03 m (grey thick line). Only q2 is actuated

to achieve the desired output position. Some energy is stored in the spring

when the load is accelerated, but after acceleration and reaching the set

point, there is no energy left in the spring.

Simulation (dashed) and experimental (continuous) data - The

the uncompressed state φ = 0, all energy supplied via the

control port is used to do work on the load. Even though the

performances of the prototype can be improved by a more

accurate realization, the results confirm the evaluation of the

conceptual design.

VI. CONCLUSIONS AND FUTURE WORK

In this work, a prototype realization of a novel variable

stiffness actuator concept has been presented. It is shown

that it is possible to realize an actuator, whose apparent

output stiffness can be changed in an energy free way. This

is shown in static output force measurements, in which the

output force, and thus the output stiffness, is changed while

keeping the energy stored in the spring constant. In dynamic

experiments, it is shown that energy supplied via the control

port is used only to do work at the output.

Future work will focus on designing and building new

prototypes in this philosophy.

REFERENCES

[1] J.W. Hurst, J.E. Chestnutt, A.A. Rizzi, “An Actuator with Physically

Variable Stiffness for Highly Dynamic Legged Locomotion”, Proc.

IEEE Int. Conf. on Robotics and Automation, 2004.

[2] G. Tonietti, R. Schiavi, A. Bicchi, “Design and Control of a Variable

Stiffness Actuator for Safe and Fast Physical Human/Robot Interac-

tion”, Proc. IEEE Int. Conf. on Robotics and Automation, 2005.

[3] S. Wolf, G. Hirzinger, “A New Variable Stiffness Design: Matching

Requirements of the Next Robot Generation”, Proc. IEEE Int. Conf.

on Robotics and Automation, 2008.

[4] B. Vanderborght, N.G. Tsagarakis, C. Semini, R. van Ham, D.G. Cald-

well, “MACCEPA 2.0: Adjustable Compliant Actuator with Stiffening

Characteristic for Energy Efficient Hopping”, Proc. IEEE Int. Conf.

on Robotics and Automation, 2009.

[5] L.C. Visser, R. Carloni, R.¨Unal, S. Stramigioli, “Modeling and Design

of Energy Efficient Variable Stiffness Actuators”, Proc. IEEE Int.

Conf. on Robotics and Automation, 2010.

[6] V. Duindam, A. Macchelli, S. Stramigioli, H. Bruyninckx, “Modeling

and Control of Complex Physical Systems”, Springer, 2009.

[7] MaxonMotor, “A-max 22

http://ww.maxonmotor.com, 2010.

[8] Controllab Products B.V., “20-sim”, http://www.20sim.com, 2010.

brushedDC motor”,available: