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MACCEPA: the Mechanically Adjustable Compliance
and Controllable Equilibrium Position Actuator used
in the ‘Controlled Passive Walking’ biped Veronica
Ronald Van Ham1, Bram Vanderborght2, Björn Verrelst, Michaël Van Damme & Dirk Lefeber
Vrije Universiteit Brussel, Department of Mechanical Engineering, Pleinlaan 2, 1050 Brussel, Belgium
Abstract — In this paper a novel rotational actuator with adaptable compliance is presented. First the importance of
adaptable compliance for bipedal walking is explained, and then a number of comparable designs are given with their
The MACCEPA concept and design is described in detail. The formula to calculate the generated torque is derived. It
is shown, depending on the design parameters, that the torque is a quasi linear function with respect to the angle between
equilibrium position and actual position. Also the change of the pre-tension has a quasi linear effect on the torque.
Another advantage is that the actuator can be built with standard components, e.g. electrical servo motors. Experiments
show the independent control of equilibrium position and compliance.
Finally the concept of Controlled Passive Walking is explained, which is a combination of the control strategies of
active and passive walking robots. Controlled Passive Walking requires actuators with adaptable compliance, preferably
where the control of equilibrium position and compliance are independent.
Index Terms—Adjustable Compliance, Equilibrium Position, Actuators, Compliance control, Controlled Passive Walking
Humans, like most walking animals, are walking efficiently by using the kinetical energy and the potential
energy of the lower limbs [1,2]. Human joints are actuated by at least 2 muscles, giving the possibility to
change the stiffness of a joint and to control the position.
While the shape of most bipeds is inspired by the human body, stiff electrical motors are commonly used to
actuate the joints. These bipeds do not use natural dynamics at all, resulting in high energy consumption. In
contrast with these fully actuated bipeds, there are passive walkers, which exploit the natural dynamics to
move. The mechanism is designed in such a way that the natural motions generate a humanlike walking
motion. The big advantage is that they are highly energy efficient; they only need to overcome friction and
impact losses. The major drawback is that they have only one walking speed, since the natural frequency is
determined by design.
Nowadays more and more research groups working on bipeds start to believe that natural biped walking
is a midway between both approaches, requiring actuators with adaptable compliance. By controlling both
the compliance and equilibrium position a variety of natural motions is possible, requiring a minimal
energy input to the system. This means that in the beginning of a motion, e.g. swinging the leg forward, the
compliance and equilibrium position are set, so the required swing motion is achieved in a natural way,
without control during the motion. This extension of passive walking is called “Controlled Passive
Walking”, since a controller is only required to set the appropriate values at the beginning of a motion,
selecting as such the right natural motion.
2 supported by the Fund for Scientific Research Flanders (Belgium)
One of the first realizations of a compliant electric actuator was the Series Elastic Actuator , which
consists of an electric motor in series with a spring. With this actuator a joint can be positioned while
having an inherent, but fixed compliance. For shock absorbance this is useful, but in order to use natural
dynamics this approach is limited to one natural frequency since the spring constant is fixed. This is
comparable to passive walkers , which are able to walk energy efficiently and very similar to humans,
although restricted to a single walking speed.
In Waseda University, Japan , a robot was built with variable stiffness joints. The stiffness of the joint
can be adjusted by changing the position of pulleys over which the driving wire runs. This results in a fairly
complex antagonistic driven joint system using a non-linear spring mechanism. The strong non-linearity
and complexity is a drawback of this type of actuator. At Carnegie Mellon University the AMASC
(Actuator with Mechanically Adjustable Series Compliance)  is developed. This actuator has a linear
spring characteristic, but the mechanism is too complex to be integrated into a biped.
At the Vrije Universiteit Brussel the Robotics and Multibody Mechanics research group has developed
the PPAM (Pleated Pneumatic Artificial Muscle) . These pneumatic muscles can only pull and not push,
requiring an antagonistic setup, which results in 2 controllable variables—the pressures in both muscles—
determining the compliance and equilibrium position of the joint. This setup, similar to human skeleton
muscles and straightforward since no gearing mechanism is required, is used to build the biped Lucy .
The drawback of these promising actuators is the high cost of pressurized air and the energy storage
capability in case of autonomous machines.
At the University of Pisa, Italy,  the Variable Stiffness Actuator (VIA) is designed. This mechanism
uses 3 parallel axes, of which one is the axis of the joint, and the other two are actuated by two motors. The
three axes are connected by a belt which is tensioned by 3 spring mechanisms. At Georgia Institute of
Technology, USA,  a Biologically Inspired Joint Stiffness Control, is designed. The setup can be
described as two antagonistic coupled Series Elastic Actuators , where the springs are made non-linear.
The two latter setups are easier to build but to change the equilibrium position or compliance, both
actuators have to be used. At Northwestern University the MARIONET is developed . This Exotendon-
Driven Rotary Series Elastic Actuator for Exerting Joint Torque is a rotational joint with adaptable
compliance and equilibrium position. Due to the design the force in the cable has to be continuously
Actuators with adaptable compliance are used in walking bipeds, but these actuators could also be useful
in other disciplines. The ultimate (lower)leg prostheses is an actuated system, which moves naturally as a
human body would do, and therefore gives the user more comfort. The knowledge acquired by developing
walking bipeds is applicable in the field of prostheses, and so is the use of an actuator with adaptable
compliance. Other possible applications are rehabilitation robots. Such devices are imposing gait-like
motion patterns to, for instance, the legs of a patient. In the beginning of the rehabilitation process it is
preferred to have a relatively high stiffness, which could be gradually lowered when a patient has regained
a certain level of control over his/her legs. The compliance is also very useful to absorb shocks introduced
by spasms. While classical industrial robots are built to be as stiff as possible, resulting in unsafe devices
for humans, a new trend is to incorporate compliance to make it possible to have a safe human-robot
interaction. For example , describes a soft robot arm which will assist the user to carry the load, while
the operator only has to push gently on the load.
As is shown in the above examples a straightforward, easy to control actuator with adaptable compliance
has a bright future.
In a robotic arm or bipedal leg, a rotational actuator is widely used. A rotational actuator with adaptable
compliance can be modeled as a torsion spring of which both the equilibrium position and spring constant
can be controlled separately. The torque applied by the torsion spring should be symmetrical around the
equilibrium position. Preferably the spring should be as linear as possible in the working range. To be able
to recuperate energy during passive motions, natural compliance with the use of a passive element, e.g. a
spring, is required.
Besides these requirements, from the engineering point of view, the mechanism should be simple
(simplicity results mostly in robustness and low-cost designs), lightweight (especially in autonomous
machines) and easy to control (it is preferable that the control of the equilibrium position and the
compliance is completely independent).
III. WORKING PRINCIPLE
In Fig. 1 the essential parts of a Maccepa are drawn. As can be seen there are 3 bodies pivoting around a
common rotation axis. To visualize the concept, the left body in Fig. 1 can be seen as an upper leg, the right
body as the lower leg and the rotation axis, which goes through the knee joint. Around this rotation axis, a
lever arm is pivoting, depicted as a smaller body in Fig. 1. A spring is attached between a fixed point on the
lever arm and a cable running around a fixed point on the right body to a pre-tension mechanism.
Fig. 1 Working principle of the Maccepa
The angle φ between the lever arm and the left body, is set by an electrical actuator. When α, the angle
between the lever arm and the right body, differs from zero, the force due to the elongation of the spring
will generate a torque, trying to line up the right body with the lever arm. Obviously when the angle α is
zero—this is the equilibrium position—the spring will not generate any torque. The actuator, determining
the angle φ actually sets the equilibrium position. A second actuator, pulling on the cable connected to the
spring, sets the pre-tension of the spring. This pre-tension will vary the torque for a certain angle α, thus
controlling the spring constant of an equivalent torsion spring.
IV. CALCULATION OF THE TORQUE
Fig. 2 Scheme of the Maccepa
R = Rotational joint axis
T = Torque applied by Maccepa
F = Force due to elongation of the spring
Ft = Torque generating part of F
k = Spring constant, assume a linear spring
B = Lever arm, controlling the equilibrium position
C = Distance between joint and spring tension mechanism
L = Length of the cable + rest length of the spring – position
of the pre-tensioner
P = Pre-tension of the spring, a function of the position of the
α = Angle between lever arm and right body = angle of
deviation of the equilibrium position
φ = Angle between left body and lever arm, equilibrium
sin... FCFCT t== and
give ).(.sin. LPAkCT −+=
Using the sine rule and the cosine rule results in
1.sin.. 22 BCCB
Fig. 3 Torque as a function of angle α when pre-tension is altered
Fig. 3 shows the torque generated by the Maccepa when the pre-tension is altered. Comparison of the
requirements in paragraph 2 with Fig. 3 shows that the torque is symmetrical around the equilibrium
position. The torque is also independent from the angle φ, which means the compliance and equilibrium
position can be controlled independently. For the linearity one can see that around the equilibrium position
the plot is rather linear, but for larger angles the plots are not linear anymore. In the next paragraph the
influence of the design parameters B, C and K will be studied, showing that the non-linearity of the torque
can be influenced.
Before talking about linearity, one should define the working range. As useful working range we assume
-45° to 45°. Notice that the range of the joint is not limited to 90° with this choice, since this range of -45°
to 45° only means the deviation between the actual position and the equilibrium position. The equilibrium
position can vary over a range of 360° and even more. When using the joints for bipedal walking—either
robots or prostheses—the choice of -45° to 45° is perfectly justifiable. So when looking at the linearity only
a range between 0° and 45° will be studied because of the symmetrical torque characteristics.
V. INFLUENCE OF DESIGN VARIABLES
The variables k, B and C are chosen during design and are fixed during normal operation. In this chapter
the influence of these four variables will be shown in detail.
Fig. 4 Influence of C/B
In Fig. 4 one can see that this factor determines the non-linearity of the curves. One can see the bigger
the ratio C/B the more linear the curves. Note that looking at the formula of the torque one can see that B
and C can be swapped without changing the result, so B/C should be big enough or C/B should be big
enough. In Fig. 5 the correlation coefficient of a linear regression is shown as a function of the ratio B/C.
One can see from a B/C or C/B ratio of a little above 5 the correlation coefficient is 0.99. This can be used
as a guideline during design, when working with the former mentioned range of -45° to 45°. To extend the
range, a larger C/B or B/C can be used to linearize the torque curve, obviously until certain limits due to
Fig. 5 Correlation coefficient of linear regression as a function of B/C or C/B
In Fig. 6 the influence of B is depicted. If the length of the lever arm is doubled, the torque is also
approximately doubled. Since the role of B and C can be switched, the influence of C is analogous, when
the ratio B/C is 1/5.
Fig. 6 Influence of B (or C) on the torque with C/B constant
Fig. 7 Influence of the Pre-tension P (detail of Fig. 3)
In Fig. 8 the influence of k is depicted, which is obvious looking at the formula of the torque.
Fig. 8 Influence of k
In this paragraph the influence of the different parameters is presented. From an engineering point of
view we can conclude that in order to obtain the linearity the ratio B/C or C/B should be at least 5 and that
the torque is more or less linear in the parameters B,C,k and P.
VI. FIRST EXPERIMENTAL SETUP
A first prototype was built. In Fig. 9 a CAD drawing of the setup is depicted. On the left side the body
with the pre-tension mechanism is shown, on the right side the body with the actuator, which controls the
equilibrium position. The rotation axis between these bodies is equipped with 2 roller bearings. A
potentiometer is placed on this axis to measure the angle between the two bodies, the actual joint angle.
Servomotors are used for the actuation. A servomotor is a position controlled actuator, where position
measurement and position controller are integrated in one device. This is a well suited actuator since the
Maccepa requires two positions to be set, while a servomotor is a well-known and low cost device.
Fig. 9 CAD drawing of the first Maccepa prototype
The servomotor on the right, which is placed by a bracket on the right body, has the same rotation axis as
the joint and thus sets the angle between the right body and the lever arm, which is actually the equilibrium
position of the joint. A number of different mechanisms can be used to pull on the cable to pre-tension the
spring. On the left of Fig. 9 a second servomotor with a spool is shown. By rotating this motor, the cable
will wind up changing the pre-tension of the spring and as such the compliance. However, for the first
prototype a lever arm is used for the pre-tension, since this was a standard component, shipped together
with the B525 BB-MG servomotors of Protech . This makes it more difficult to set a specific value for
the compliance, but for the first experiments we just want to set to compliance ‘high’ or ‘low’ and compare
the results. In Fig. 10 a picture of the first prototype is shown. A Microchip  PIC16F877A
microcontroller is used to control both the servos and read the analog angle information. Through a serial
interface the data is exchanged with a PC.
Fig. 10 Picture of the first Maccepa prototype
In a first experiment the difference in natural movements with altered compliance was tested. The setup
is placed so that the rotation axis is vertical; as such no external torque is applied to the joint. During the
experiment the equilibrium position is set to 0 degrees (both arms are aligned). In Fig. 11 the joint, which is
made relatively stiff, is pulled manually out of the equilibrium position and released. As a result the arm
starts oscillating around the equilibrium position with a specific frequency. After the joint stops oscillating
(around 2 sec), the joint is made more compliant, and pulled again out of the equilibrium position (2.5 sec).
Releasing the joint results in a lower natural frequency.
Fig. 11 Variation of the natural frequency for different settings of the compliance.
The experiment in Fig. 12 shows the independence between the natural frequency and the equilibrium
position. The dashed line is the set value of the equilibrium position. The compliance is not changed during
the experiment. The full line is the measured angle of the joint. In the beginning of the experiment the
equilibrium position is set to -45 degrees and the joint is pulled to zero degrees and released (at 1.5 sec).
The joint oscillates on a certain frequency, which depends on the specific compliance setting. When the
motion has stopped the equilibrium position is set to 45 degrees (artificial damping is introduced around 4
sec) and the joint is then (at 5.5 sec) pulled again to an angle of 0 degrees. Releasing the arm results in an
oscillation with the same frequency. The two experiments show that the natural frequency is a function of
the compliance, and it is independent of the equilibrium position, as is predicted by the formula for the
Fig. 12 Natural frequency independent of equilibrium position
VII. BIPED PROTOTYPE
To investigate the benefits of natural dynamics, a biped robot, called Veronica, is being built. The design
of the Maccepa was optimized to be used in a humanlike robot. Fig. 13 shows the new design, called
‘Slimline Maccepa’. Both servo motors are placed inside the left and right bodies. The motor controlling
the equilibrium position cannot be placed directly on the rotation axis; therefore a small transmission is
used. In the biped Veronica a system with 2 equal gear wheels is chosen. A pin is placed on the gear wheel
on the rotation axis, which acts as lever arm.
Fig. 13 CAD drawing of the Slimline Maccepa
Fig. 14 Picture of the biped prototype Veronica
The overall structure of the biped is a planar biped, with 6 degrees of freedom. The hip, knee and ankle
joint of each leg are pin joints, actuated each with a Slimline Maccepa. The length of the limbs is chosen to
be 30 cm, resulting in an overall size of 100 cm. Since the mechanical structure is a combination of 6
Maccepa, the control of the 12 servo motors and reading of the 6 joint angles will be done in a modular
way. Since experience was gained with the PIC controller in the first prototype, each Maccepa will be
equipped with a PIC16F876 Microcontroller, controlling both servo motors and reading the analog signal
from the rotational potentiometer. These controllers have an I2C bus, to communicate with a single master.
In the development stage the master controller will be linked to a PC by a serial port. The chosen PIC
controller is fast enough since “Controlled Passive Walking” requires less bandwidth than active controlled
walking. When debugging is over, the master microcontroller will act autonomously.
VIII. CONCEPT OF CONTROLLED PASSIVE WALKING
Passive walking robots which can walk on a flat surface require some actuation to overcome friction and
impact losses. This is mostly done by using a Series Elastic Actuator or pneumatic muscles, since these are
complaint actuators. The controller, which is not a real trajectory controller, but more a 2 faze selector. The
2 fazes can be the stand faze and the swing faze, each with their predefined settings for the SEA or
pressures for the pneumatic muscles. To know when has to be switched between the two fazes; an electrical
switch is placed at the bottom of the foot. So to walk on a certain speed, the robot is switching all the time
between 2 settings, selecting 2 natural motions. In case of the Spring Flamingo, more natural motions can
be selected, but they all have the same natural frequency, since the spring constant is fixed. The robot Mike
 can switch between 2 couples of equilibrium position and compliance, resulting in 2 natural
To expand the range of natural motions, the Maccepa actuator will be used, which can generate motion
with different natural frequencies and different equilibrium positions.
Fig. 15 Maccepa in 3 positions (A: -30°, B: 0°, C: +30°)
To explain the concept, a single vertically placed Maccepa will be used as an example, as shown in Fig.
15. The settings for compliance and equilibrium position are the same for Fig. 15 A, B and C. In Fig. 15B
no external force is applied and the position is the same as the equilibrium position, being 0 degrees. When
a torque is applied to the left, the joint will become as shown in Fig. 15A. When the joint is released from
this position with zero initial angular velocity, and friction is neglected, it will swing, in a certain time, to
the position C. Once arrived in C, with angular velocity 0, the experiment ends. Note that the same can be
done from C to A. The time needed to go from A to C depends on the moment of inertia of the pendulum,
but also on the compliance setting of the Maccepa. The more compliant, the slower the pendulum will
swing, the stiffer the faster it will swing. When the situation is not symmetrical (not -30° till +30°), the
setting for the equilibrium position has to be altered.
So starting from a position, we can make the swing motion end (= angular velocity zero) in a required
position in a required time, just by setting compliance and equilibrium position once. Of course, this can
only be achieved within certain limits.
Note that this concept, although described in a different way, is implemented in some of the robots in ,
where the compliant elements are pneumatic artificial muscles. A double pendulum (a leg) also can be
controlled from a start to an end position by setting only 4 variables at the start, based on this concept.
The Maccepa (Mechanically Adjustable Compliance and Controllable Equilibrium Position Actuator) is
presented in detail. The design variables, limited in number, are explained and their influence on the torque
characteristics is shown. Compared to other compliant mechanisms the Maccepa is straightforward,
inexpensive and the compliance is adjustable in an easy way. To control one parameter—compliance or
equilibrium position—it requires only the action of one actuator, e.g. one servomotor. These advantages
make it suitable for applications where adaptable compliance is required or is useful, e.g. dynamic walking
robots, prosthesis and robotic applications interacting with humans. A bipedal robot, Veronica, is built to
test this new concept of Controlled Passive Walking.
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