EU FP7
AMARSi
Adaptive Modular Architectures for Rich Motor Skills
ICT-248311
D 2.1
Marh 2011 (1 year)
Compliant Actuators
Authors: Matteo Laffranchi (IIT),Hide SUMIOKA (UniZu),Alexander
Sproewitz (EPFL),Dongming Gan (IIT),Nikos G. Tsagarakis (IIT)
Due date of deliverable
Actual submission date
Lead Partner
Revision
Dissemination level
15st March 2011
6th April 2011
IIT
Final
Public
This deliverable focuses on the design of compliant actuators. The state of
the art of compliant actuation systems is firstly introduced by presenting some
implementations of the different functional principles. A comparison analysis
of the features/capabilities of the presented designs motivate the introduction of
the CompAct compliant actuation module. The CompAct is the main actuation
unit developed for the compliant humanoid platform. After the introduction of
the CompAct unit the design of the lower body of the compliant humanoid is
presented followed by the description of the two compliant quadruped robots currenlty under development within the AMARSi project.
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Contents
1 Design principles for compliant actuation systems
1.1 Fixed compliance actuators . . . . . . . . . . . . . .
1.2 Variable compliance actuators . . . . . . . . . . . .
1.2.1 The series configuration . . . . . . . . . . .
1.2.2 The antagonistic configuration . . . . . . . .
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2 The compliant humanoid platform
2.1 The mechanical design of the CompAct . . . . . . . . . . . . . . . . . . . .
2.2 Compliant module stiffness model . . . . . . . . . . . . . . . . . . . . . . .
2.3 Dynamics of the CompAct actuator . . . . . . . . . . . . . . . . . . . . . .
2.4 Selection of an appropriate stiffness value based on impact studies . . . . . .
2.4.1 Impact model and dynamics analysis . . . . . . . . . . . . . . . . .
2.4.2 Worst configuration analysis . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Integrated dynamic model and the worst configurations of a 4-DOF
humanoid arm/leg robot . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Stiffness design for the joints of the 4-DOF arm/leg robot based on the
four worst case configurations . . . . . . . . . . . . . . . . . . . . .
2.5 The design of the AMARSI Compliant Humanoid . . . . . . . . . . . . . . .
2.5.1 The Mechanics of the Lower Body . . . . . . . . . . . . . . . . . . .
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3 The Oncilla robot
3.1 Summary of achievements for the quadruped robot . . . . . . . . . . . . . .
3.2 Functional principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Mechanical implementation and integration into the robotic hardware . . . .
3.4 Actuator dimensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Joint stiffness range and energy storage capacity . . . . . . . . . . . . . . . .
3.6 Actuation power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Simulation/experimental results demonstrating the performance of the compliant platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 The quadruped robot Kitty
4.1 Spinal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Biological spinal structure . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Artificial spinal structure . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Design of the quadruped robot . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Leg Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Experiments and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Experiments and results based on robot with fully actuated spine . . .
4.3.2 Experiments and results based on robot with partial actuated and passive spine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Design principles for compliant actuation systems
1 Design principles for compliant actuation systems
The creation of mechatronic systems with bio-natural properties from the actuation point of
view has been the inspiration of the robotic community for many years. This replication
though has failed so far mainly due to the complexity and inherent properties of the organic
actuation/transmission system (muscle-tendon-joint-bone) which have prevented its emulation from the traditional engineered actuation paradigms. Looking on the diverse range of
actuation techniques it can be seen that the natural world appears to have developed a much
more ubiquitous design, with the organic muscle providing power for animals ranging from
the largest whales to microbes with adaptation to cope with environmental extremes. Yet, the
potential benefits, which can be gained in any mechanism with the incorporation of biological
actuation concepts, are well known, the majority of today’s robots lack these characteristics. In these systems, key actuation technologies such as electric, pneumatic and hydraulic
have been employed and effectively used in a variety of application domains, traditionally
in the form of a stiff position or torque controlled source units to provide accurate displacement/forces for precise trajectory tracking and task execution. Recently, with the introduction
of new applications domains such as virtual/tele-presence, robot aided therapy/assistance and
personal/entertainment robotics it has become increasingly clear that the traditional actuation
approach is not suitable for addressing the high requirements of these new application domains. The requirement for closer human-robot interaction have highlighted the need for inherently safe robotic systems which can match the performance of biological systems in terms
of ability to cope with unpredicted interaction with the human and/or environment enabling
safe interaction and efficient task execution. These are key developmental features of all new
generation systems. In fact, these requirements are directly linked to the actuation system.
The lack of such an actuator unit that can mimic some of the properties of the natural muscle, e.g. compliance is probably one of the most significant barriers that prevented so far the
development of robotic systems exhibiting bio-natural functional behaviour and performance.
The introduction of these new properties within robots has the potential of solving many of the
problems of conventional stiff robots but on the other hand places new challenging demands
as the development of novel actuation prototypes with high integration density and issues as
the control of the resulting actuation systems which inevitably results more complex due to
the introduced compliance - related dynamics.
In the growing fields of wearable robotics, rehabilitation robotics, prosthetics, and walking
robots, fixed or variable compliance actuators are being designed and implemented because
of their ability to minimize large forces due to shocks, to safely interact with the user, and
their ability to store and release energy in passive elastic elements. Hence a device that can
provide compliance is needed to achieve these characteristics. From the engineering perspective, the methods for the implementation of this concept in robots are essentially three: active
control, passive compliance and hybrid active/passive solutions. The first approach consists in
the implementation of control strategies e.g. impedance/admittance control which artificially
replicate a desired level of compliance. Compliance can be incorporated into the robot by
means of a passive approach as well: in this case passive elastic elements e.g. springs are
embodied into the structure. The first part of this deliverable will report on the design principles adopted in the implementation of compliant actuators. This part is followed by a section
2
1 Design principles for compliant actuation systems
which compares the different designs to make the selection of a suitable approach to be used
in the development of the cCub humanoid which will be used within the AMARSi project.
1.1 Fixed compliance actuators
The ability of the stiff actuation units to interact with their surroundings can be increased
by means of the active approach explained previously, however the existence of delays at all
stages of the software system make these conventionally stiff actuated systems incapable of
managing high-speed contact transients because of their limited bandwidth. To address the
latest a wide range of experimental novel compliant actuation systems have been developed
during the past fifteen years. Fixed compliance actuators represent the first attempts towards
the development of compliant actuation systems. These actuators incorporate a fixed stiffness
passive element within their stucture, usually placed between the rigid actuator(s) and the
load. Fixed compliance actuators can be implemented following two main approaches, i.e.
antaongistic or series design. A conceptual schematic of the latter implementation is shown in
Fig. 1.1
Figure 1.1: Conceptual schematic of a Series Elastic Actuator
One particular type of system with inherent fixed passive compliance is the Series Elastic Actuator family, [6]. These actuators are made by the series combination ”motor-gearbox-elastic
element-link” and hence employ only one actuator and one elastic element per degree of freedom. A conceptual schematic of a SEA is presented in Fig. 1.1. In the configurations shown
in 1.1, the effort needed to drive the link is delivered by the motor-gearbox group, while θin
is the angle of the input pulley of the compliant element and the output angle is θout . The
fixed stiffness implementation has two control variables which are the input-output angular
positions of the compliant element θin and θout , Fig. 1.1. The solution proposed in [6] included a DC motor coupled to a planetary gearbox and a fixed stiffness torsion spring. This
solution has been implemented also for the linear case and has been powered by hydraulic
actuators as well, [39]. Given the low impedance (attenuated by means of the series compliance) and low friction, SEAs can achieve high fidelity force control and are hence suited
for robots operating in unstructured environment. The delicate and expensive loadcells/torque
sensors typical of force/torque controlled robots (see e.g. [23]) may ultimately induce chatter
in force control due to their high stiffness. These sensors are replaced with compliant elastic
3
1 Design principles for compliant actuation systems
elements (e.g. springs, which are robust, stable and inexpensive) in Series Elastic Actuators.
The force/torque delivered by the actuator can be estimated by means of the Hooke’s law, 17,
18 and the measurement of the compression of the compliant element:
Fa = kl x
(1)
in case of linear SEAs, where F is the force applied by the actuator, kl and x are the stiffness
and compression of the linear compliant element, respectively. Similarly as for the linear
design, in case of rotary implementations:
Ta = kt θS
(2)
where T is the torque delivered by the actuator while kt and θS are the stiffness and compression of the rotary compliant element, respectively. From the mechanical perspective linear or
torsion springs can implement the series elasticity (see e.g. the electric or hydraulic SEAs,
[39], [6]) however rubber elements held between alternating teeth have been used as well, 1.2,
[41]. The compliant element deflection state can be used to obtain precision in the estimation
Figure 1.2: Left, conceptual schematic and right picture of the elastic element which uses rubber balls
[41]
of the torque or force in case of rotary or linear SEAs. The estimated effort variable can be
then feed back to implement force/torque control, Fig. 1.3. The signal F desired is used as
reference for the feedforward and the feedback controllers F F and F B. The force F load
is estimated from the deflection of the spring element and is compared with the reference in
order to return an error F error. This latter signal, together with the measured position xload
and the reference F desired is used to generate an appropriate control signal for the electrical
motor. The main disadvantage of the preset stiffness of SEAs can be overcome by means of
the application of admittance/impedance control strategies, [17, 38, 28]. Fixed compliance
actuators can be used also in rehabilitation robotics as in [9]. In this work a prosthetic ankle
device has been developed using a SEA which employs also parallel compliance. The series
compliance is used to protect the motor-gearbox group from shock loads generated during the
foot strike. The parallel elasticity is instead engaged only when the ankle is dorsiflexed, increasing the torque control bandwidth in order to allow the prosthesis controller to capture the
torque-velocity behaviour of the human ankle in walking. The same performance improvements brought by Series Elastic Actuators (e.g. improved safety in human-robot interaction,
4
1 Design principles for compliant actuation systems
Figure 1.3: Force/Torque control of a SEA, [40]
Figure 1.4: (a) Conceptual schematic and (b) model of the ankle prosthesis [9]
ability to absorb the shocks, enhancement of the force/torque control performance) are valid
also for fixed stiffness antagonistic setups. In fact, an antagonistic joint is made of two series
elastic actuators placed antagonistically which deliver the agonist/antagonist forces F1 , F2 to
the joint as shown in Fig. 1.5. The two spring elements have fixed stiffness k. The stiffness of
the output pulley can be computed to be, [30]:
∂T
= 2r2 k
∂θ
(3)
where r is the constant radius of the pulley. Equation 12 shows that the equivalent joint stiff-
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1 Design principles for compliant actuation systems
Figure 1.5: Conceptual schematic of the antagonistic actuator setup
Figure 1.6: Legged biologically-inspired compliant robot
ness is constant for a constant stiffness of the spring components. This solution has been used
in legged robots as in [41] where series elastic actuators were used in series with tendons actuating each joint, Fig. 1.6. This bio-inspired design however has some problems as calibration
issues due to the pulleys eccentricity as well as too much distal mass, [41].
1.2 Variable compliance actuators
Variable compliance actuators are actuators which are capable of passively regulating their
physical compliance. Obvious advantages that a variable stiffness implementations offer when
compared with the fixed passive compliance units are the ability to regulate both stiffness
and position and the wide range of stiffness and energy storage capability. The advantages
gained by this capability are clearly shown in mammals: muscles and tendons change their
stiffness as a function of the motion/task they have to perform. Arm muscles assume a stiff
configuration when the arm has to perform an accurate task, while they are compliant when
they are performing the ”loading” phase of a throw. Similarly, if we analyze jumping we see
that leg muscles are compliant during the ”loading” phase of the jump or during the landing
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1 Design principles for compliant actuation systems
phase where they absorb the shock [31], while during the ”pushing” phase, they are stiff.
There are several reasons for this variation in stiffness but among others the most pressing is
the exploitation of the elastic energy stored within the muscles and tendons [32]. This enables
variable compliance actuators to achieve performance that is not possible with a conventional
stiff robotic system as Wolf et al showed when comparing a rigid joint and a compliant joint
in a SDOF setup during a throw task [33]. There work showed that there is a clear difference
between the velocities of the links and the throw distances obtained in the two cases. This
motivates researchers to implement variable compliance into the actuation systems using a
variety of different designs, however variable compliance actuators can be categorized into
two main groups. One is formed by actuators which present variable compliance placed in
series between the actuator and the load while the antagonistically-actuated joints form the
second group.
1.2.1 The series configuration
The main advantage of this type of approach is that two different actuators are used to set the
equilibrium position of the joint and the stiffness independently. Some examples of variable
stiffness actuators implemented with the series approach are presented next.
The Actuators with Adjustable Stiffness (AwAS and AwAS-II)
AwAS and AwAS-II are series type of variable stiffness actuators which can change the position and stiffness independently. In both actuators one big motor changes the position and a
small motor tunes the stiffness. Adjusting the stiffness in both actuators is done through a lever
Figure 1.7: Physical model of AwAS and AwAS-II
mechanism (Fig.1.7). A lever has three principal points; the pivot: the point around which the
lever can rotates, the spring point: the point at which springs are located and the force point:
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1 Design principles for compliant actuation systems
the point at which the force is applying to the lever [13]. In AwAS, [10, 14, 11] the force and
pivot point are kept fixed and to change the stiffness the spring point is changing.The stiffness
in this case can be defined as:
K = 2Ks r2 (2 cos2 φ − 1)
(4)
where Ks , r and φ represent stiffness of the springs, the distance between the spring point
and the pivot (which is adjustable) and the angular deflection, respectively. Using this concept
stiffness can be achieved in a good range since it depends on the square of the arm r. The
range of stiffness depends on the stiffness of the springs and length of the lever. However in
AwAS-II,[13, 12] force and spring points are kept fixed but instead the pivot point is changing.
Using the lever allows adjusting the stiffness energitically efficient since in the lever to change
each of principal points, the displacement needed to change the stiffness is perpendicular to
the force generated by the springs. Therefore the stiffness motor doesn’t need to directly
counteract against spring’s forces. The stiffness in this case can be defined as:
K = 2Ks α2 (L1 + L2 )2 cos φ
(5)
where L1 represents the distance between the pivot and the springs and L2 is the distance
L1
.Using this mechanism
between the pivot and the force. α is the ratio (adjustable) which is L
2
the stiffness can be achieved in the largest possible range from zero to infinite sice it depends
on the ratio. The ratio becomes zero when the pivot reaches the spring point and it becomes
infinitive when the pivot reaches the force point. This range does not depend on the stiffness
of the springs and lever’s length. Therefore shorter lever and softer springs can be used in this
mechansim which leads to have a lighter and more compact setup compare to the mechansim
applied to AwAS.
The Jack Spring actuator
A linear series-type variable stiffness actuator is the Jack Spring, [15]. In this solution, stiffness is varied by changing the number of active coils of the series spring. The adjustment of
the stiffness is done by rotating the spring about the coil axis. This rotation is transformed in
a linear motion of the spring along its axis thanks to the geometry of the shaft on which the
spring is mounted. In fact, this shaft is machined with the same coil geometry of the spring
and the inside/outside motion of the spring (achieved by screwing/unscrewing the spring on
its shaft) changes the effective stiffness of the actuator by varying the active spring coils, 1.8,
accordingly with the following formula which describes a coil spring stiffness:
Gd4
K=
(6)
8D3 na
In 17 G is the material shear modulus, d is the wire diameter, D is the coil diameter, and
finally na is the number of active coils. The Jack Spring actuator principle works on this last
parameter to vary the stiffness of the overall actuator. In this concept, one motor is used to
adjust the equilibrium position of the actuator and a second motor is used to adjust the stiffness.
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1 Design principles for compliant actuation systems
Figure 1.8: Active and inactive coil region in the Jack Spring Actuator
MACCEPA
Van Ham et al, [16], developed the MACCEPA actuator. Similarly as in the designs presented
previously two independent actuators, M and m are used to set the equilibrium position and
the apparent stiffness of the joint, respectively. Fig. 1.9 presents a kinematics scheme of the
MACCEPA mechanism, showing three different bodies pivoting around a central axis a. The
part of length B can rotate about the axis a and has a linear tension spring attached to one
extreme. The other end of the spring is attached to a point b which is fixed on the body shown
on the right by means of a cable. When the angle between the lever arm and the right body α is
Figure 1.9: Working principle of the MACCEPA
different from zero the force generated by the elongated spring generates a torque between the
left and the right bodies which tends to align the body shown on the right with the lever arm
of length B. When the angle α is null, the lever arm is aligned with the spring and no torque
will be generated. The smaller motor m is used to pretension the tension spring at point b by
pulling the cable connected to the spring. The length of the tensioned spring when the angle
α is null is defined as P . The relationship between the compression angle α and the torque T
can be calculated to be:
!
P −L
T = kBC sin(α) 1 + p
(7)
B 2 + C 2 − 2BC cos(α)
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1 Design principles for compliant actuation systems
where k is the spring stiffness and L is the natural length of the spring. Equation 18 shows
that the relationship between the compression angle and the delivered torque depends on the
spring preload P − L meaning that the overall joint stiffness can be regulated acting on the
preload of the spring using the motor m. The angle ϕ imposed by the main actuator sets the
equilibrium position of the joint.
1.2.2 The antagonistic configuration
Antagonistically-actuated joints employ two compliant elements to provide power to the joint.
This design is biologically-inspired, since mammalian anatomy follows the same concept,
i.e. a joint actuated by two muscles arranged in an antagonistic manner. The muscle-tendon
cooperation gives the driven link (arm, leg etc) a controllable and variable compliance. In
addition to biological muscle this type of antagonistic compliance controlled can be achieved
using both conventional two motor electric drive designs and other more biologically inspired
forms such pneumatic Muscle Actuators (pMA). In the latter case compliance is an inherent
characteristic of the actuator, while for an electrical design compliant elements (generally
springs) have to be embodied into the system. From the engineering perspective, this concept
can be implemented using three different layouts which are shown in Fig. 1.10. In literature,
these three basic setups are referres as ’simple’, ’Cross coupled’, and ’Bidirectional’, see
1.10. A simple antagonistically-actuated joint uses two driving elements. The stiffness of the
Figure 1.10: The Simple, Cross-coupled and Bi-directional antagonist setups
joint and the angular displacement of the driven link are set by means of a combination of
the actuation inputs q1 , q2 (see Fig. 1.11). By co-contraction of these actuators preloading
and, thus, tuning of the stiffness is achieved, while the rotation of the joint is obtained by
the antagonistic motion of the drives. An example of this simple antagonistic layout is
represented by a joint actuated by pneumatic muscles, Fig. 1.12. In this layout, the nonlinear
force/elongation relationship can be exploited to implement the variability of the joint stiffness
level by varying the pneumatic pressure in the bladders. When McKibben artificial muscles are
employed, the following equation can be used to model the relationship between the generated
force F , the actuator elongation L and the pressure in the tube P .
F = k L2 − L2min P
(8)
where k and Lmin are constant parameters depending on constructive details.
The same ’simple’ antagonistic actuation approach could be implemented using electrical actuation systems. Differently from the system presented previously, where pneumatic muscles
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1 Design principles for compliant actuation systems
Figure 1.11: The Simple, Cross-coupled and Bi-directional antagonist setups
Figure 1.12: Simple antagonistic setup using pneumatic muscles
are employed, in this case compliance is not an inherent property of the actuator and therefore
compliant passive elements (usually springs) have to be incorporated between the actuators
and the load. This is the case of the actuator presented in [34].
Migliore’s Actuator
This actuator uses two conventional electrical drives connected in an antagonistic manner to
the output joint through nonlinear springs. A positive angle α of the agonist servo motor
provides positive rotation of the angle θ of the output drive, while the antagonist actuator sets
an angle β in opposition to the output joint angle, Fig. 1.13. Using linear springs in the
configuration shown in 1.13, the output joint stiffness will be:
Slinear = RJ2 (k1 + k2 )
11
(9)
1 Design principles for compliant actuation systems
Figure 1.13: Conceptual schematic of the actuator presented in [34]
Figure 1.14: Mechanism to implement the stiffness nonlinearity, [34]
where RJ is the radius of the output pulley, k1 and k2 are the stiffness of the springs of the
agonist and antagonist servos. Equation 20 shows that the output joint stiffness depends only
on the constructive parameters of the joint and on the springs stiffness (i.e. constant). The
variation of stiffness of the spring components can be set to be linearly related with the cocontraction by using elastic elements with quadratic force-elongation relationship as follows:
F (x) = a(x − x0 )2 + b(x − x0 ) + c
(10)
where F is the force applied by the motors, x is the elongation of the spring component, x0 is
the natural length of the spring and a, b, c are constants. Using such spring components, the
stiffness of the output joint, S will be:
S = 2aRS RJ2 (α + β) + 2bRJ2
(11)
where RS is the radius of the driving pulleys. In this case stiffness can be varied by means of
the co-contraction of the drives (given by the sum of α and β), with constant offset 2bRJ2 . On
the other hand, the displacement of output joint is achieved by means of the agonistic motion
of the motors. Equation 21 is implemented mechanically by means of a quadratic profile of
the part named ’Frame’ of the nonlinear spring component schematically shown in Fig. 1.14.
The assembly shown in Fig. 1.14 replaces the conventional extension springs shown in the
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1 Design principles for compliant actuation systems
antagonistic setup of Fig. 1.13 therefore permitting the regulation of the joint stiffness.
Variable Stiffness Actuator (VSA)
The VSA is a Variable Stiffness Actuator which implements the so-called ’cross-coupled’
arrangement, 1.10. Referring to Fig. 1.15 the VSA consists of three pulleys (2, 3, 4) connected
by means of a timing belt (1). Pulleys 2 and 3 are actuated by means of position-controlled
backdrivable DC motors, while the output pulley (4) is connected to the link. The output joint
stiffness can be computed to be:
!
hm,1 − hm,1
h2,m − h2,m
hm,1 Lm,1
h2,m L2,m
(12)
+
σ = 2KR
+
− 2KR
hm,1
h2,m
4h3m,1
4h32,m
where K is the linear spring stiffness, R is the radius of the pulleys 2, 3 and 4, hm,1 , hm,1
and h2,m , h2,m are the natural and active lengths of springs 5 and 7. Lm,1 and L2,m are the
lengths of the belt between the pulley pairs 2-4 and 3-4. This means that stiffness can be
changed by acting on the active length of the springs and on the belt length. In detail, by
means of high/low co-contraction, high/low compression of the springs 2 and 4 (Fig. 1.15)
will generate high/low apparent output joint stiffness. In practice, the antagonist motion
Figure 1.15: CAD view and nomenclature of the VSA [19]
of the drives varies changes the apparent angle between the spring axis and the belt and this
creates the nonlinear stiffness/compression relationship whic permits the stiffness adjustment.
Agonist motion of the drives only generate displacements of the output shaft.
Actuator with Mechanically Adjustable Series Compliance (AMASC)
Another antagonistic-based design is the Actuator with Mechanically Adjustable Series Compliance (AMASC) presented in [20]. The main advantage proposed by this design is that,
similarly as for serial setups and differntly from conventional antagonistic setups, it enables
an independent control of position and stiffness. Figure 1.17 shows a simplified conceptual
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1 Design principles for compliant actuation systems
Figure 1.16: Conceptual schematic showing one of the antagonistic pairs [19]
Figure 1.17: Simplified conceptual schematic of the AMASC [20]
schematic of the AMASC. The rotation of the driving pulley on the left, J1 , sets the equilibrium position of the output link J2 , while the distance between the two pulleys x3 sets
the pretension of the nonlinear spring elements and is imposed by a second actuator. The
nonlinear function between the elongation x3 and the generated force is in this case quadratic:
Fz (z) = kz 2
(13)
where Fz (z) is the generated force and is function of the spring component elongation z. z
can be set by appropriate tuning of the degree of freedom x3 . The behaviour of this actuator is
therefore similar to that of [34] presented previously: two nonlinear quadratic spring components arranged antagonistically are employed to create the variability of stiffness of the output
joint. The nonlinear relationship between elongation and generated force is implemented in the
AMASC by means of the series of the springs with a spiral pulley - based mechanism (1.18).
The gearing of the spiral pulley-based mechanism varies proportionally with the springs deflection to create the desired quadratic relationship, 24. The two spring function when placed
in direct opposition, the single effective spring force Fef f can be computed by substituting
the (x3 − ∆x ) and (x3 + ∆x ) for z.
Fef f (x3 , ∆x ) = Fz ((x3 + ∆x )) − Fz ((x3 − ∆x ))
14
(14)
1 Design principles for compliant actuation systems
Figure 1.18: Overall conceptual schematic of the AMASC [20]
where x3 is the pretension of the two nonlinear springs and ∆x is the deflection from their rest
position. By manipulating 24 and 9 the effective force Fef f can be computed to be:
Fef f = 4Kx3 (∆x )
(15)
Equation 10 shows that the stiffness of the resulting system can be changed by adjusting the
pretension x3 .
Variable Stiffness Actuator - II (VSA-II)
The VSA - II, [21], is an antagonistic actuator which uses the bi-directional actuation principle (refer to Fig. 1.10) and can be represented in the schematic of Fig. 1.19. The VSA-II
Figure 1.19: Conceptual schematic of the VSA-II, [21]
shows several improvements with respect to the previous VSA-I, [19] as for instance a 4-bar
mechanism which shows higher load capacity and robustness. This mechanism implements a
variable transmission system used to obtain a nonlinear relationship between input and output
15
1 Design principles for compliant actuation systems
torque/displacements. Using a linear spring on the input, the relationship between deflection
and torque on the output shaft can be made nonlinear, 1.20. The link OA is actuated by a
Figure 1.20: The four bar transmission mechanism of the VSA-II, [21]
motor at O. The torsion spring of stiffness k is linear, however the stiffness at O is nonlinear
with angles β and θ and is described by the following relationship, [21]:


2
R R2
θ
R
θ
−
1
β
sin
cos 2
L
L
2
1 
− 1 + h
(16)
σ(θ) = k  q L
3 
i
2
4
θ
R
θ 2 2
1− R
sin
1
−
sin
L
2
L
2
16
2 The compliant humanoid platform
2 The compliant humanoid platform
The actuation systems presented previously can be classified in two main categories: fixed
and variable compliance actuation systems. However, each of these two groups can be implemented following antagonistic or series designs meaning that a total of four families can
be used to classify a compliant actuation system. Obvious advantages that variable stiffness
implementations offer when compared with the fixed passive compliance units are the ability
to regulate stiffness and position independently and the wide range of stiffness and energy
storage capability. This aspect is particularly interesting when low energy consumption levels
have to be reached or when the arm has to be safe and performant at the same time [30]. This
result can be explained by a biologically-inspired motivation which means that muscles and
tendons change their stiffness as a function of the motion/task they have to perform. Arm
muscles assume a stiff configuration when the arm has to perform an accurate task, while
they are compliant when they are performing the ”loading” phase of a throw. Similarly, if
we analyze jumping we see that leg muscles are compliant during the ”loading” phase of the
jump or during the landing phase where they absorb the shock [31], while during the ”pushing” phase, they are stiff. There are several reasons for this variation in stiffness but among
the most pressing is the exploitation of the elastic energy stored within the muscles and tendons [32]. This enables compliant actuators to achieve performance that is not possible with
a conventional stiff robotic system as Wolf et al showed when comparing a rigid joint and
a compliant joint in a SDOF setup during a throw task [33]. These work showed that there
is a clear difference between the velocities of the links and the throw distances obtained in
the two cases. Clearly the introduction of compliance can have very significant effects on the
performance of an actuation system relative to the classical stiff design. On the other hand the
mechanical complexity, size, weight, cost and integration are still open issues in the variable
passive compliance realizations. As a result their application to multi degree of freedom or
small scale robotic machines still remains a challenging task. At these matters the fixed compliance actuator family clearly demonstrates an advantage when compared with the variable
stiffness implementations. The compactness and simplicity of the design can be further increased by means of the use of a series approach. In fact, antagonistic designs require at least
two actuators and and two compliant elements per degree of freedom, differently from series
elastic actuator which requires one actuator and one compliant joint.
In fixed stiffness designs (as e.g. SEA) the degree of exploitation of the benefits of compliance
(especially energy efficiency and performance as the achievement of high mechanical power
peaks) is limited when compared to variable compliance implementations. However, in case
of fixed compliance actuation human and robot safety requirements can still be guaranteed if
the joint stiffness is set to an appropriate value. The selection of optimum stiffness values can
be based on the results from analyses of certain critical robot collision scenarios. In addition,
its main disadvantage of the preset passive mechanical compliance can be at some degree
minimized by combining the unit with an active stiffness control. This motivates the use of a
compact rotary SEA module for the development of the humanoid robot cCub which will be
used in this project.
In the next section the design of a new modular and compact rotary series elastic actuator
17
2 The compliant humanoid platform
(CompAct) which is used in the cCub humanoid is presented. The actuator shows particular improvements over the existing implementations based on ball-screw/die spring, torsion
springs or extension springs/cable assemblies. The compact design of the actuator is due to
the novel compliant module mechanical implementation. Despite its small size, the actuator
still retains high level of performance and it is a potential solution for small scale mutli-degree
of freedom systems when compliance is also desired.
2.1 The mechanical design of the CompAct
The mechanical realization of soft actuation unit is based on the concept of the serial elastic
actuator, but particular attention has been paid to satisfy the dimensional and weight requirements of the cCub robot. The high density of the integration is due to the novel mechanical
compliant module. To minimize dimensions while achieving high levels of rotary stiffness a
mechanical arrangement, involving a three spoke output component, an input circular pulley
and six linear springs, has been designed and fabricated. The circular component is the input
of the compliant module and is fixed to the output of the reduction drive. The three spoke
element rotates on bearings with respect to the circular base and it is coupled to it by means of
the six springs which are arranged as shown in Fig. 2.21. The three spoke component finally
forms the output of the compliant module and the mounting base of the output link.
Figure 2.21: The prototype of the CompAct series elastic actuator
The six linear springs when inserted in the arrangement shown in Figure 1 experience a precontraction equal to half of the maximum acceptable deflection. Deflections larger than the
maximum allowable are not permitted by mean of mechanical pin based locks. Two 12bit
18
2 The compliant humanoid platform
absolute position sensors have been integrated within the actuation group measuring respectively the mechanical angle of the motor after the reduction drive and the deflection angle of
the compliant module. The two sensors not only allow the monitoring of the link position
but also allow the evaluation of the joint torque. Because of the compliance introduced it is
possible to use the sensor measuring the compliant module deflection to estimate the torque.
2.2 Compliant module stiffness model
In this section the stiffness model of the three spoke spring arrangement is presented. The
deflection of the complaint module results in the generation of torques due to the compression
of the spring elements along their main axis, Figure 2. Considering one of the antagonist linear
spring pairs in Figure 3, the axial forces generated by each of the springs when the compliant
three spoke module is deflected from the equilibrium position by an angle of θS is given by:
F1 = KA (xp + x(θS )), F2 = KA (xp − x(θS ))
(17)
where xp is the spring pre-contraction and x(θS ) = Rsin(θS ) is the resulted deflection of
the two springs along their main axis, KA is the spring axial stiffness and R the length of the
spoke arm. The combined axial force applied in any of the three spokes is therefore:
Figure 2.22: Compression of the spring as a result of the module deflection
F = F1 − F2 = 2KA Rsin(θS )
(18)
The corresponding torque generated at the joint because of the axial forces of one antagonistic
pair of springs is equal to:
T = F Rcos(θS ) = 2KA R2 sin(θS )cos(θS )
(19)
So far we consider that the axial force of the spring is concentrated at one point. Considering
that the spring has an external radius of rS , it can be seen in Fig. 2.22 that the axial compression of the spring is not equal for the whole surface area being in contact with the spoke.
19
2 The compliant humanoid platform
Figure 2.23: The three spoke spring coupling arrangement
The areas that are further from the centre of rotation are subject to larger deflections creating
higher forces. As a result the torque generated by the axial deflection of the antagonistic pair
of springs can be computed by
Z R+rS
r2
1
2KA R2 sin(θS )cos(θS )dR = 2KA R2 + S sin(θS )cos(θS ) (20)
T =
2rS R−rS
3
Thus, the combined torque at the joint considering the axial forces from all three pairs is
r2
(21)
Ttotal = 3T = 6KA R2 + S sin(θS )cos(θS )
3
By directly differentiating the torque equation the rotary stiffness of the three spoke module
which is due to the axial deflection of the springs can be obtained as
∂Ttotal
r2
2cos(θS2 − 1)
(22)
= 6KA R2 + S
KS =
∂θS
3
Figure 2.24 shows the theoretical stiffness of the module within the range of the deflection
angle, for the first prototype module with the following parameters: KA = 62kN/m, R =
20.5mm, rS = 6.3mm
2.3 Dynamics of the CompAct actuator
The actuator consists of three main components, a typical brushless DC motor, a harmonic
reduction drive and the rotary compliant module introduced in the previous section. These
three components can be represented by the mechanical model shown in Figure 5 below. The
model is composed of the rotary inertia and viscous damping of the motor JM , DM , the gear
drive with the reduction ratio of N , the elastic module with an equivalent spring constant of
KS , 22, the output link inertia and axial damping coefficient JL , DL . In addition, θM , θ0
are the motor mechanical angles before and after the reduction drive, θL is the angle of the
output link and θS is the rotary defection of the elastic module with respect to θ0 such that
θL = θ0 + θS . Finally, τM is the torque provided by the actuator while τ0 is the input torque
20
2 The compliant humanoid platform
Figure 2.24: The stiffness profile of the compliant module
Figure 2.25: CompAct SEA mechanical model diagram
of the elastic element and τE is the torque imposed to the system by the load and/or the
environment. The above system can be described by the following set of dynamic equations:
JM N 2 s2 + DM N 2 s + KS θ0 − KS θL = τ0
(23)
JL s2 + DL s + KS θL − KS θ0 = τE
(24)
2.4 Selection of an appropriate stiffness value based on impact
studies
The main issue in safety and performance is the stiffness level selection and location along the
kinematic chain of the robot. This section presents a study of choosing the appropriate stiffness
21
2 The compliant humanoid platform
levels for the new compliant humanoid robot by analyzing the dynamic system during an
impact scenario aiming to ensure that the responded joint torques remain under the maximum
torque capabilities of the actuators. The stiffness selection approach in this study was to find
the worst robot configuration for each joint in the robot, in which if an accidental collision
occurs at the robot end-effector the joint will experience the highest impact torque. If the
generated torque is safe for the joints in this worst configuration, it will be safe for all other
configurations when the robot joint velocities or the end-effector velocity are within a certain
value used for the searching of the worst impact configuration.
2.4.1 Impact model and dynamics analysis
The compliant actuator can be simply modeled as in Fig. 2, by introducing a torsion compliance between the actuator and the link with equivalent stiffness k. In this way the inertia of the
actuator is decoupled from the inertia of the link, leading to a reduced responding torque under
impacts protecting the actuator from potential peak torques dangers. Assuming that there are
Figure 2.26: The compliant joint
m compliant joints in a serial robot, based on the Euler-Lagrange formulation, the dynamics
of the robot can be described by the following equations:
M (q)q̈ + C(q, q̇)q̇ + G(q) = τl
¨ + τl = τm
Mm theta
λ
θ
τl = K
−q
λ
(25)
(26)
(27)
where q and θ are the vectors of links’ joint angles and motors’ joint angles respectively,
M (q) and Mm are the link and motor inertia matrices respectively, C(q, q̇) are the matrices
of Coriolis and centrifugal terms, G(q) is the vector of gravitational torques, τl is the vector
of torques acting on the links, τm is the vector of motor torques, λ is the reducer ratio and
K is the diagonal matrix of joints’ stiffness coefficients. When the serial robot collides with
another body, the magnitude of impact force can be computed as:
Fc =
−(1 + e)v T n
nT [C]n
22
(28)
2 The compliant humanoid platform
where [C] = J[M (q)]−1 J T , [M (q)]−1 is the inverse Cartesian inertia at the end effector, n is
the unit normal vector to the plane of impact of the two bodies. v = vnv is the end-effector
velocity of the robot when impacting with magnitude v and unit direction nv . 0 < e < 1 is
the restitution coefficient denoting the type of collision (0 for purely plastic collision and 1 for
pure elastic one), with e = 1 in this study. Finally, J represents the Jacobian matrix of the
serial robot. A general impact occurs in a short period in which the impact force increases first
and then reduces in the restitution. Here Fc is an equivalent force. The impact is considered
to be very fast, thus, configurations of the links are considered to be the same with that before
the impact. The impact force can be transferred into the generalized torques exerted on the
joints by:
τc = J T nFC
(29)
τc is called impact torque in this paper. Thus, the impact force is transferred to the joint torques
in the function of the robot configuration and impact force direction.
2.4.2 Worst configuration analysis
The main weak part of the actuator is the harmonic drive reducer. Thus the aim is to derive
the joint torques on the harmonic drive which will be:
τ = τl + τc
(30)
The worst configuration identification for each joint is to find a configuration in which equation 30 returns the highest torque value for the joint. This becomes an optimization problem
where the goal is to find the worst configuration for each joint (in which 30 is maximized) by
searching the configuration space q(q1, q2, ..., qm), the normal n of the impact force and the
velocity direction nv of the robot tip based on specified 30 as:

max τi (q, nv , n)



qi min ≤ qi ≤ qi max i = 1, 2, ..., m
(31)
αvj min ≤ αvj ≤ αvj max j = 1, or1, 2



βj min ≤ βj ≤ βj max
where αvj and βj are angles representing directions of the end-effector velocity and impact
force, for planar robot j = 1, for spatial robot j = 1, 2. Other parameters are known. Thus,
m worst configurations can be identified as:


τw1 (q1 , nv1 , n1 ) 





τw2 (q2 , nv2 , n2 )
(32)
... 





τwm (qm , nvm , nm )
where τ wi stands for the worst configuration qi (qi1 , qi2 , ..., qim ) torque, which is the highest
in the whole workspace for joint i with the end-effector velocity in direction nvi and impact
force in direction ni based on the same input parameters. The worst configuration for each
joint in 32 can be taken as a configuration in which the joint needs to withstand the highest
23
2 The compliant humanoid platform
torque in order to resist the external force Fc while trying also to keep the motion of the
robot. From the above analysis, it can be seen that only equation 25 is used to get equation
31 while equations 26 and 27 which represent the actuator compliant model are not affecting
the worst configuration searching procedure described above. Having identified the worst
configuration for each joint the actuator compliant joint is then incorporated and simulation
studies (using ADAMS) are performed in the overall system considering both the robot rigid
body and actuator dynamics described by equations 25, 26 and 27.
2.4.3 Integrated dynamic model and the worst configurations of a 4-DOF
humanoid arm/leg robot
In this section the procedures described above are applied for the design of a 4-DOF compliant
robot which can represent the arm or the leg of the compliant humanoid robot. The robot
consists of a 3-DOF joint (shoulder or hip) joint and 1-DOF (elbow or knee) joint as shown in
2.27. The three rotational axes(z1 , z2 , z3 ) at the shoulder (hip) are intersecting at one point in
the shoulder (hip). The shoulder (hip) is connected with upper arm (leg) followed by another
rotational joint (z4 ) which represents the elbow joint, and the forearm (calf) as the end-effector.
The wrist (ankle) is not considered in this study. A frame coordinate system o0 x0 y0 z0 is set at
the intersecting point at the shoulder (hip) with z0 collinear with the direction of joint z1 and
y0 to the opposite direction of gravity as in 2.27. v is the velocity of the end-effector and F is
the impact force applied at the end-effector. the dynamic equations integrated with the impact
Figure 2.27: Kinematic model of the 4-DOF robot
model in 31 can be specified in terms of the searching parameters and the worst configurations
can be identified by following equations:

max τi (q1 , q2 , q3 , q4 , α1 , α2 , β1 , β2 )




−7 π9 ≤ q1 ≤ 7 π9




−5 π9 ≤ q2 ≤ 7 π9



− π2 ≤ q3 ≤ π2 i = 1, 2, 3, 4
(33)
π
− π2 ≤ q4 ≤ 5 18




0 ≤ α1 , β1 ≤ π




0
≤ α2 , β2 ≤ 2π



n.nv < 0
24
2 The compliant humanoid platform
where qi (i = 1, 2, 3, 4) are the joint angles, α1 and α2 represent the direction of end-effector
velocity, and β1 and β2 describe the direction of the impact force. Here α1 and β1 are the
angles between the directions and axis z0 , α2 and β2 are the angles between the projection
of the directions on the x0 o0 y0 plane and axis x0 . The direction of the impact force should
be within the opposite hemisphere of the velocity, indicating that the impact force should not
have positive element on the end-effector velocity direction and giving a constraint as n.nv
being negative. The input parameters of the 4-DOF robot are listed in Table 4. The robot
Table 1: Input parameters of the 4-DOF robot
Item Unit
Upper limb
Lower limb
Joint 1
Joint 2
Joint 3
Joint 4
Mass [kg]
2
2
N/A
N/A
N/A
N/A
Length [m]
0.2
0.2
N/A
N/A
N/A
N/A
Acceleration [rad/s2 ]
N/A
N/A
1
2
0.2
1
masses and link lengths and the joint angle ranges are taken from the iCub robot [42]. One
more important parameter is the magnitude of the end-effector velocity which is set as 1.5 m/s
and the joint velocities are obtained by inverse calculating using the robot Jacobian. Thus,
basing on equation 33 and the input parameters in 4 with the end-effector velocity as 1.5 m/s,
the worst configurations of all four joints can be obtained as




 τw1 (0.043, 0.58, 0.72, −1.08, 1.97, 0.66, 1.17, 0.65) 


τw2 (1.65, −0.88, 1.17, −1.13, 1.95, 6.05, 2.06, 1.17)
(34)
τw3 (0.72, 2.26, 1.04, −0.34, 2.04, 0.68, 1.37, 1.25) 





τw4 (0.28, −0.011, 1.39, −0.65, 1.71, 3.37, 2.92, 1.32)
where the unit of measure of the angles is radians. The worst case configurations are shown in
Fig. 2.28 in which the red line represents the direction of the impact force and the blue one is
the direction of the end-effector velocity. Thus, in worst configuration for joint 1 in Fig. 2.28,
joint 1 needs to support the highest joint torque to support the end-effector velocity of 1.5m/s
in the direction shown by the blue line while an impact occurs in the direction of the red line.
This is the same for the other three joints in the other figures in Fig. 2.28. Similar to the 2-DOF
case, in the worst configuration searching, another constraint is added as that the maximum
joint angle velocity is no more than 10 rad/s. This is also used to avoid extreme values of joint
angle velocities and joint torques in singular configurations when inversely calculating joint
angle velocities based on the given end-effector velocity. In Tab. 4, it can be seen that some of
the obtained joint angle velocities reach 10rad/s, indicating that the obtained configurations in
Fig. 2.28 reach the maximum joint angle constraint to give the highest values for the motion
torque with high impact torques to be the worst configurations for the joints.
25
2 The compliant humanoid platform
Figure 2.28: Worst case configurations of the 4-DOF robot
2.4.4 Stiffness design for the joints of the 4-DOF arm/leg robot based on the
four worst case configurations
Basing on the results obtained previously, a simulation model of the 4-DOF compliant robot
is built in the Adams with the system parameters shown in Table 5, in which Ji (i = 1, 2, 3, 4)
corresponds to the worst configurations of joint i as in Fig. 2.28. The joint velocities are
obtained from the worst configuration searching by inverse kinematics from the end-effector
velocity, and the input joint torques are the values needed to execute the motion of the robot in
the corresponding worst configurations to give the initial end-effector velocity as 1.5m/s and
the joint angle accelerations in Table 5. In the simulation, the maximum torque the compliant
Table 2: Input parameters of the simulation
J1
J2
J3
J4
q˙1 [rad/s]
10
10
10
-10
q˙2 [rad/s]
6.6234
-4.0564
-10
-10
q˙3 [rad/s]
-10
-10
-10
-0.5938
q˙4 [rad/s]
-10
10
-10
2.7815
τ1 [Nm]
20.6698
8.5737
7.2031
-0.5879
τ2 [Nm]
-7.7789
20.821
-19.0867
5.9208
τ3 [Nm]
4.1519
-2.0029
-11.7957
-0.1545
τ4 [Nm]
5.0046
-0.3926
0.2108
10.703
actuator can withstand is 55Nm. The results obtained are shown in Fig. 3.36. The responded
torques of the four joints are demonstrated together in the four worst configurations. The
Table 3: Stiffness of the 4-DOF robot
J1
J2
J3
J4
k1 [Nm/rad]
687
3151
716
1890
k2 [Nm/rad]
1432
286
544
303
k3 [Nm/rad]
2578
2005
744
3151
k4 [Nm/rad]
1547
2864
573
544
final stiffness values as derived from the simulation are listed in Table 6. It can be seen that
joint 2 requires comparatively smaller stiffness than other joints, which means joint 2 is more
26
2 The compliant humanoid platform
Figure 2.29: Simulation results of the 4-DOF arm/leg robot
sensitive to impact forces in these four configuration cases. By comparing the rows in 6, it
is obvious that the stiffness of the four joints in the case of joint 3 worst configuration is
relatively lower than other cases. This can also be confirmed by the joint angle velocities in
Table 5 as all the four reach the maximum 10 rad/s and this configuration is sensitive to the
impact. In Table 6, the most important values are those underlined, which shows that the
maximum stiffness for joint 1 to joint 4 should be 687 Nm/rad, 286 Nm/rad, 744 Nm/rad and
544 Nm/rad respectively. By setting the values lower than these numbers, the joint actuators
will be safe when there is an impact at the tip end with the end-effector velocity as high as 1.5
m/s. These underlined values are in the diagonal positions in Table 6, indicating that Ji case
is the worst configuration for corresponding joint i(i = 1, 2, 3, 4).
2.5 The design of the AMARSI Compliant Humanoid
The passive compliant humanoid robot [43], Fig. 2.30 developed within AMARSI is derived
from the ”iCub” baby humanoid [42, 44, 45] with similar design specifications with regards
to its dimensions, degrees of freedom (D.O.F), mass and joint range of motion. The size of
the compliant humanoid will approximate the dimensions of a 4 year old child. Regarding the
lower body of robot which is the main focus of this paper, the number of degrees of freedom
has remained unchanged with respect to the original ”iCub”. Each leg consists of 6 D.O.F: 3
D.O.F at the hip, 1 D.O.F at the knee level and 2 D.O.F at the ankle. This is a kinematic layout
used in many other bipedal robots. For the waist most humanoids usually have a relatively
simple 2 D.O.F. mechanism. For the compliant humanoid developed within AMARSI a 3
D.O.F waist was considered as this implementation offers greater motion flexibility. This
extra functionality is needed as very young children typically reach for objects from a seated
27
2 The compliant humanoid platform
position and flexibility at the waist increases their workspace. Based on above, the waist
provides pitch, roll and yaw motions for the upper body in line with the performance of the
original ”iCub” humanoid. In addition, to the above generic specifications and in order to
facilitate the experimental scenarios of the AMARSI project there is a need to incorporate
compliance properties through a synergetic combination of both active and passive compliance
principles.
Figure 2.30: The mechanical assembly of the lower body of the compliant humanoidF robot
2.5.1 The Mechanics of the Lower Body
The mechanical structure of a leg of the compliant robot and an overview of its kinematics with
the location of the D.O.F is illustrated in 2.32, [45]. From the kinematic perspective the new
lower body includes the lower torso (housing the waist module) and the two leg assemblies.
The height of the lower body from the foot to the waist is 671mm, with a maximum width and
depth (at the hips) of 176mm and 110mm respectively. The total lower body weight is 17.3kg
with each leg weighing approximately 5.9kg and the waist section including the hip flexion
motors weighing 5.5kg. The components that are subject to a low stress are fabricated in
Aluminum alloy Al6082 with the medium/highly stressed components (load bearing sections
of the housing) made of Aluminum alloy 7075 (Ergal) which has an excellent strength to
28
2 The compliant humanoid platform
weight ratio. The joint shafts are fabricated from Stainless steel 17-4PH which delivers an
excellent combination of good oxidation and corrosion resistance together with high strength.
The Compliant Leg Module
The compliant leg of the robot has a modular structure allowing for easy assembly and maintenance. The leg has an anthropomorphic kinematic form composed of the hip, the thigh with
the knee joint, the calf with the ankle joint and the foot, Fig 2.32. All these sections were
radically redesigned in the compliant leg with respect to the original ”iCub” leg assembly
to include the desired passive compliance property provided by the actuation unit presented
previously. The hip joint is based on a single side supported cantilever base structure with a
pitch-roll-yaw arrangement providing a large range of motion. The hip pitch motion is implemented using the compliant actuation module previously presented while the yaw/roll motions
use conventional stiff module (Brushless DC motor combined with a Harmonic gearbox with
a peak torque of 40Nm). The hip yaw joint has integrated torque sensing to permit the compliance regulation through active control. The hip roll is directly driven by the motor placed
Figure 2.31: Kinematic configuration and the mechanical assembly of the compliant lower body
in the centre of hip joint while torque to the hip pitch is transmitted from the hip flexion motor
29
2 The compliant humanoid platform
located in the lower torso (Fig. 2.35) through a cable stage that provides additional secondary
gearing (1.5:1) allowing for torques up to 60Nm for the hip pitch motion. The knee joint is
Figure 2.32: KThe ankle and knee compliant actuation modules exposed
directly driven by a compliant actuation group (peak torque up to 40Nm) at the centre of the
knee joint, Fig. 2.32. For the purpose of assembly optimization the compliant module for the
ankle flexion motion has been separated from its motor unit and placed at the ankle side while
the motor (40Nm) actuating this module is housed inside the calf section, Fig. 2.32 and Fig
2.33 . Torque to this joint is transferred through a solid link transmission that also provide
additional variable secondary gearing, Fig. 2.34. This permits peaks torques ranging from
45Nm-70Nm depending on the ankle flexion angle with the gear ratio curve tuned to provide
higher torques during the initial stage of the push-off gait phase (Dorsiflexion), Fig. 2.34. The
last D.O.F which produces ankle inversion/eversion uses a stiff actuator (40Nm) located on
the foot plate and directly coupled to the ankle roll joint, Fig 2.33. The sole of the foot is
divided into two parts, Fig. 2.33, forming a single toe that is spring loaded by two torsional
spring elements. The functionality of the two piece foot will be explored for the purpose of
achieving larger strides than those that can be done by humanoids with conventional solid sole
plates.
Waist Mechanism
The waist of the compliant was based on the core mechanism used in the original iCub where
the torque and power of the two actuators used for body pitch and yaw is transferred using a
cable based differential mechanism, Fig. 2.35. For the waist pitch motion the two high power
actuator assemblies (40Nm each) that drive the pitch and yaw motion apply a synchronous
motion to the two directly coupled differential input wheels. For the roll motion the motors
turn in opposite directions. Yaw is achieved through a pulley shaft directly connected to the
30
2 The compliant humanoid platform
Figure 2.33: The mechanical realization of the ’cCub” leg modules.
Figure 2.34: The Ankle flexion variable gear ratio for maximizing joint torques during the push off phase
Figure 2.35: Back and front views of the waist module
upper body frame. The actuator assembly of the yaw pulley (20Nm) is located within the centre element of the differential, Fig. 2.35. The torque is conveyed through a cable transmission
system that provides additional gearing (1.5:1) allowing for peak torques of 30Nm for torso
yaw motion.
31
3 The Oncilla robot
3 The Oncilla robot
As part of WP2 (Compliant systems) EPFL-A (Biorobotics Laboratory) is developing a novel,
small sized, compliant quadruped cat-robot (oncilla robot). We will use the codename ”oncilla” robot throughout this document to separate the former cheetah cat robot from the indevelopment AMARSi oncilla robot. The oncilla animal, also called Tigercat, is a wild cat
of approximately 2kg weight, originated in South America. The final name of the AMARSi
compliant quadruped (cat-) robot has yet to be decided for. This oncilla robot is the successor
of cheetah robot. Latter was developed and published at Biorobotics Laboratory [46], and
was the basis for the AMARSi quadruped compliant robot proposal. The AMARSi oncilla
robot is a co-development between Biorobotics laboratory (EPFL-A) and its AMARSi partner
ResLab at the University of Gentt. EPFL is currently finishing the first design phase of oncilla
robot, together with the ordered actuators we expect first test runs in about 12 weeks from now
(3rd week of June). To facilitate faster testing we will also assemble a robot version based on
RC servo motors. RC servo motors are are easily available (short delivery times), however
they are sub-optimal for the final version of oncilla robot. The RC actuated robot version will
be used to test sensors, basic mechanical principles, assembly, novel compliant elements of
oncilla robot, material, and basic locomotion patterns.
3.1 Summary of achievements for the quadruped robot
The three-segmented leg design of the former cheetah robot was fully described (biarticulate
compliant joint mechanism), and based on a single leg SLIP model calculation a specific
spring selection was made for the recent version, the oncillar robot. Oncilla robot’s leg design
was extended with a second compliant unit: a ham-string-like tension spring system. We
decided for three actuators per leg: hip (leg protraction and retraction), knee (leg extension and
contraction), and leg ablation. The first two actuators were carefully selected through reverse
estimations of torque and acceleration based on foot locus trajectory assumptions for gaits
up to 3Hz, stand-up procedures of the robot, and reverse estimations and mappings of torque
and speed hardware tests with a swinging lever. We further tested the leg design in simulation
using Webots simulation environment, and a modeled version of the cheetah/oncilla robot. We
found that leg control strategies purely exploiting the passive compliance of oncilla robot’s
legs led to a number of stable trot gaits, up to 3.5Hz (central pattern generator based control,
open loop). However stability and speed can be improved by the in-parallel activation of the
knee actuator during stance phase (double peak/hump activation). We found trot gait patterns
moving the simulated robot up to 90cm/s forward. We can confirm actual flight-phases for our
modeled oncilla robot applying above control strategy.
3.2 Functional principle
Oncilla robot inherited cheetah robot’s three-segmented, low-inertia, pantographic legs. This
choice is bio-inspired and was firstly suggested by [48], and [49, 52]. A two-joint (biarticulate)
32
3 The Oncilla robot
spanning compression spring automatically extends each limb, i.e. is the only mean to erect
the robot (”gravity loaded passive compression spring system”). New for the oncilla robot
version leg design is a second spring (tension-like mechanism) replacing one mid-segment
bar of the pantograph mechanism. The placement of this spring is often used in robotics to
represent the leg’s ham-string , e.g. Puppy Robot by [51] and Reservoir Dog from [53]. A
third compliant element in the leg design will be foot element; however its characteristics are
not fully defined yet. Non-mechanical compliance will be available through the hip joints of
the quadruped robot. Knee and hip actuator of oncilla robot feature very low gear ratios (less
than 30:1) and are fully reversible. Together with accurate position sensing (12 bit resolution)
and high speed feedback loops (planned for 500Hz-1000Hz) it will be possible to implement
a virtual compliant actuator (key word: ”force control”). The knee actuator is decoupled from
the knee joint by means of a cable mechanism, only tension forces acting at the actuator.
Hence we expect to use virtual compliance mainly in oncilla robot’s hip joint.
3.3 Mechanical implementation and integration into the robotic
hardware
A snapshot from a sketched version of oncilla robot (featuring the larger, brushless motors) is
given in Fig. 3.36. The expected weight of the robot is about 2.5kg, together with a 3000mAh
battery around 3kg. We designed oncilla robot’s legs to be roughly 50g in weight (excluding
the in-development multi-axes load cell). This will provide us with a very low-inertia leg.
Oncilla robot was designed with no extra payload capabilities. Materials used are different
types of sheet metal Aluminium (frame, limbs, standard components), glass-fiber reinforced
plastic (FR4 sheet structures), POM (bearing, fixations), brass (bearing, axes), steel (axes,
springs, motor parts), ABS plastic (compliance guidance mechanism), and strings for the cable
mechanism.
3.4 Actuator dimensioning
Oncilla robot’s brushless motors were selected after an extensive literature review (brushless
versus brushed motors, hydraulic actuation, direct drives, RC servo motors). We then assembled a large motor-gearbox combination list (this was work together with AMARSi-partner
ResLab, UGentt, Michiel D’Haene). Final motor and gear sizes were then calculated and
selected through application scenarios and tests, e.g. foot trajectory assumptions, the robot
standing up, and initial motor-gearbox tests applying a high-speed swinging lever (experiment
at ResLab, UGentt). Reverse torque and acceleration profile calculations from above scenarios allow calculating expected available torque and speed, energy loss, torque peak, and power
consumption of the load-gear-motor combination. We used standard mechatronic tools, formulas, and assumptions for latter calculations [55]. Load assumptions, available motors, and
gear combinations lead to a small set of potential/candidate motors which can drive the load
in terms of torque and speed, and don’t overheat at the same time, and range of gear ratios
for e.g. an assumed robot trot gait of about 3Hz. Calculations are done both for the hip and
the knee actuator. A third actuator (RC servo motor) is used for leg ablation; this is a new
33
3 The Oncilla robot
Figure 3.36: The figure shows the main components of oncilla robot, including the compliant elements
(sketched presentation). Clearly visible are the three-segmented legs (grey, z-shaped), distal black rubber
elements depict the feet. The estimated weight of this robot is about 3kg; this includes the weight for the
battery (around 500g).
DOF for the oncilla robot (compared to cheetah robot, e.g. used in Raibert’s quadruped, [47],
Raibert’s Bigdog [?], Boston Dynamics LittleDog [50], Semini’s HyQ [54]). We will test its
exact dimensioning with the first prototype.
3.5 Joint stiffness range and energy storage capacity
We have so far fully defined oncilla robot’s biarticulate compliant joint range. The desired effect is a non-linear leg compliance, and less a single-joint compliance. The three-segmentation
of the leg design, the placement of the attachment points of the compression spring, the selection of the spring stiffness, and the choice for the pre-stress value of the compression spring
lead to an digressive (other than linear or progressive) spring-stiffness behavior (Fig. 3.37).
Early SLIP-model tests showed that it is advantageous to use pre-stressed springs for the
robot’s limbs (Fig. 3.37, starting from 4N).
Energy storage is directly linked to the applying leg force (Fig. 3.37) by the change of the leg
length; we expect energy storage values (maximum half the leg length compression) roughly
up to 2 Joule per leg and biarticulate compliant unit. Initial dimensioning of the biarticulate
34
3 The Oncilla robot
Figure 3.37: The plot shows the applying leg force over leg length, the spring behavior of the oncilla leg is
non-linear with digressive characteristics. The configuration plotted uses a 2.3kN/m linear biarticulated
spring. Only the influence of the biarticulte compliance is shown, the ham-string compliant behavior
depends not only on the leg length but also on the angle of attack.
leg spring is based on results from a SLIP model simulation predicting good stability characteristics for a single linear, prismatic leg at characteristic hopping speeds. The resulting SLIP
model stiffness value was mapped onto our non-linear leg characteristics. The latest leg design
uses linear springs of roughly 2kN/m.
3.6 Actuation power
Based on the motor-gearbox dimensioning, we have selected and ordered Maxon brushless
motors and high-duty gearboxes with low gear ratios. The following values do not include
efficiency values for motor and gearbox, but are data sheet numbers recalculated by gear ratios. For the hip actuator this leads to a maximum rotor speed (gearbox outgoing rotor) of
80rad/s, and a recommended speed of 73rad/sec. Stall torque is 13Nm (the gearbox can tolerate peak torques of not more than 3Nm), recommended working torque 1Nm. Knee actuator
values: max speed 380rad/sec, recommended speed 350rad/sec, stall torque 2.8Nm, and recommended torque 220mNm. However the knee actuator has an additional cable mechanism
which is effectively gearing down the actuator output. Note that both described compliant
units are working either in parallel (biarticulate compliance) or in series (ham-string compliant unit) to the knee motor.
3.7 Simulation/experimental results demonstrating the
performance of the compliant platform
Our oncilla robot is currently still under construction, hence we cannot provide demos with
the hardware platform. However we used the cheetah/oncilla simulation model to extensively
35
3 The Oncilla robot
test the compliant leg design, while applying different leg joint strategies (robot in trot gait,
open-loop CPG control, and open control parameter identification using particle swarm optimization). Our test indicate that the oncilla robot will be able to facilitate the passively
compliant leg design for a range of trot gaits, i.e. multiple trot gaits with different amplitude,
offsets, phase bias etc. We also found valid, fast, and more robust gaits by supporting above
passive leg compliance with an additional active, in parallel working knee joint control. Latter
strategy leads to stable trot gaits up to 90 cm/s (more than 3 times the oncilla robot’s body
length), with flight-phases up to 10percent of leg cycle length. We will re-run these experiments, which are currently still in simulation, with the real hardware as soon as possible.
36
4 The quadruped robot Kitty
4 The quadruped robot Kitty
Based on the requirements in the T 2.1 described in the proposal, UniZu has been developing
the quadruped robot called Kitty equipped with a multi-joint spine inspired by the characteristics of biological one. The study of Kitty robot involves the novel design of spine which is
highly-compliant, actuated multi-joint system, evaluation the role of spine in the locomotion,
and investigation of rich motor skill to achieve gait transition. Over past decades, it has been
widely accepted that locomotion is generally perceived as being the function of the legs and
the trunk is considered to be carried along in a more or less passive way [58, 57, 59]. This popular hypothesis appears to have been accepted by most of roboticist as well as biologists and a
considerable amount of research has been conducted on legged robots with little consideration
on its spine. However, Gracovetsky has proposed an alternative hypothesis which emphasizes
the spinal engine, that is, locomotion is first achieved by the motion of the spine; the limbs
came after, as an improvement, not as a substitute [60]. This implies that the spine is the key
structure necessary in locomotion and maneuverability as well as in gait transition. Inspired
by this finding from biology, the kitty robot equipped with tendon-driven flexible spine has
been developed to test the hypothesis of spinal engine and investigate the rich motor skill to
achieve gait transition.
4.1 Spinal Model
4.1.1 Biological spinal structure
As an essential organ of both weight bearing and locomotion, the spine is subject to the conflict
of providing maximal stability while maintaining crucial mobility. Morphological adaptation
of the spine of mammals with respect to locomotion depends on specific biomechanical demands, for instance, the locomotion mode [61]. Therefore, we first described the general characteristics of mammalian spine, and then took a cheetah as an example to illustrate its specific
spine morphology and its role in locomotion. The spine consists of discrete bony elements,
namely vertebrae, ligaments, intervertebral discs and Zygapophyseal joint. The vertebrae are
joined by passive ligaments and kept separated by intervertebral discs. The Zygapophyseal
joints are dynamically controlled by muscular activation. Vertebrae (4.38a) are the highly
specialized bones which collectively make up the spinal column. The bony elements create
attachment points for muscles and other bones, allowing for flexible movement in a range of
directions. The spinal column provides critical support to the animal. Intervertebral discs
(4.38(b)) are located between the vertebrae, and firmly joined with the endplates of the vertebrae. They are morphologically structured soft tissue cushions serving as the spine’s shock
absorbing system, which protect the vertebrae and other structures (i.e. nerves). The discs
allow some vertebral motion: extension and flexion. Individual disc movement is very limited; however considerable motion is possible when several discs combine forces. Ligaments
(4.38(c)) are the fibrous, slightly stretchy connective tissues that hold one bone to another
in the body, forming a joint. Ligaments control the range of motion of a joint, for example,
stabilizing the joint so that the bones move in the proper alignment. Mechanically, spinal
37
4 The quadruped robot Kitty
Figure 4.38: Spine components
ligaments behave as other soft tissues of the body and they are viscoelastic with nonlinear
elastic responses [62]. A zygapophysial joint(4.38(d)) is a movable joint between the superior articular process of one vertebra and the inferior articular process of the vertebra directly
above it as shown in Figure 5. The biomechanical function of each pair of zygapophysial
joints is to guide and limit movement of the spinal motion segment. In the lumbar spine,
the zygapophysial joints function to protect the motion segment from anterior shear forces,
excessive rotation and flexion [63]. The kinematic analysis of the biological spine has not
Figure 4.39: Ranges of motion throughout the normal spine. From White and Pnjabi Clinical Biomechanices
Figure 4.40: Instantaneous axes of rotation for the lumbar vertebrae. From White and Pnjabi Clinical
Biomechanics
38
4 The quadruped robot Kitty
been well known by researchers so far because of its complex biomechanics. However, the
ranges of motion throughout the normal human’s spine have been measured as shown in Fig.
4.39, [65]. The spine can achieve combined flexion/extension, one side lateral bending and
one side axial rotation as shown in Fig. 4.40 [65]. The morphology of the spine is different
in mammals with different locomotion mode. In this report, we only took cheetah, a terrestrial quadruped mammal, as an example which is a particular case to make most use of its
spine to achieve maximal stability and maintain crucial mobility. Cheetah’s vertebrae have the
following striking features compared to other mammals.
The vertebral bodies
A slight anterior wedge shape and a decrease in sagittal diameter (represented by red line in
Fig. 4.42 (a)) towards the sacrum as shown in Fig. 4.41 are recognizable in cheetah, in whom
predominant sagittal flexion is found in locomotion.
Zygapophyseal joint shape
It reflects a mammal’s capability to resist torsion, ventral shear and dorsal shear. The cheetah’s
lumbar vertebra features inclined and plane joint surfaces as shown in Fig. 4.42 (a) which are
able to resist torsion and ventral’s shear. However, it is assumed that it counter dorsal shear
through the tensed spinal musculature.
Figure 4.41: Lumbosacral juncion, from [61]
In addition to the two factors described above, there still exist some features of spine which
might affect the locomotion, for example, the stiffness and arrangement of the ligaments; the
number of vertebrae; the stiffness and arrangement of intervertebral discs and so on. However,
there are still something remaining unclear in the study of spine, for example, how these spinal
morphological features cooperate to affect locomotion, which plays a more a more important
role in the locomotion. In order to better explain these questions, a highly-compliant, multidegree system, artificial spinal structure inspired from biology has been developed by UniZu.
4.1.2 Artificial spinal structure
This spinal structure acts not only as a beam between the forelegs and the hind, but also as
an engine to generate movement essential to locomotion. It should be designed in a modular
architecture. In this case, its morphology can be easily changed by connectors to figure out
which configuration benefits best under a certain locomotion mode and help biologist to better
39
4 The quadruped robot Kitty
Figure 4.42: Zygapophyseal joint shape, from [61]
understand the characteristic and role of spine in locomotion.
Fully-actuated spine morphology
As a first step, we started with a fully actuated artificial spine inspired by biology. Figure 6
shows the 3D representation of this spine which consists of cross-shaped rigid vertebrae made
of ABS plastic, silicon blocks and cables driven by motors. The cross-shaped rigid vertebrae
are separated by the silicon blocks, which work as intervertebral discs.
Figure 4.43: 3D representation of spinal structure
Each side of artificial vertebrae is connected by a cable through the two holes and the silicon
blocks. In the center of the vertebra, there are two faces: a convex and a concave, each of
which can match and rotate around the opposite face with the nearby vertebra as shown in
4.44. The four driven cables are pulled respectively by the four RC motors, which are able
40
4 The quadruped robot Kitty
Figure 4.44: 3D representation of the match of vertebrae
Biological spine
Vertebra
Intervertebral disk
Ligament Fig. 4.45d, e
Muscle
Artificial spine
Cross rigid segment
Silicon block
Spring
Driven-cable
Ball joint (Fig. 4.44)
Zygapophysial joint
Function
1-Make up the spinal column;
2-Create attachment points for muscles and other bones
1-Serve as the spine’s shock absorbing system;
2-Allow some vertebral motion: extension and flexion
1-Connect one vertebra to another;
2-Control the range of motion of a joint
1-Connect one vertebra to another;
1-Guide and limit movement of the spinal motion segment;
2-Protect the motion segment from anterior shear
forces excessive rotation and flexion
Table 4: Similarities in morphology between biological spinal structure and the artificial counterpart
to control the stiffness and movement of the spine. In this design, the spine can be bent in
all directions within certain predefined angle. Table 4 shows the similarities in morphology
between biological spinal structure and its artificial counterpart.
Exploration of the spine morphology
The spine was designed in a modular architecture to investigate the impact of the morphological property of the spine in locomotion. The morphological properties of spinal structure
of Kitty robot can be easily changed by the connectors. For example, the artificial vertebrae
can be replaced with differently-shaped ones. Or only part of silicon blocks are replaced by
springs, which leads to partial actuated and passive (Fig.4.45(d), Fig. 4.45(e)). As a result
of this configuration, it is more biologically close to animal’s spinal model. If we install two
rigid boards to antagonistic sides of the spine, then the DOF of spine will be constrained in the
plane perpendicular to those boards. Fig. 4.45 shows an exhibition of developed spinal structures, including rigid spine, passive spine, tendon driven spine, partially actuated and passive
spine.
41
4 The quadruped robot Kitty
Figure 4.45: Exhibition of developed spinal structures. The yellow rectangles highlight the area of the
spinal structure with different morphological properties. Red lines emphasize the springs above it.
4.2 Design of the quadruped robot
The developed quadruped robot is shown in Fig. 4.46. It is 23 [cm] long, 29 [cm] wide,
20 [cm] high and weighs 1.1[kg]. The robot features a tendon-driven spine consisting of
silicon blocks and plastic segments without actuation for legs. It was designed in modular
architecture: in addition to the leg, feet, the morphology and material properties of spinal
structure of Kitty robot can also be easily changed by the connectors.
4.2.1 Leg Design
There are 3 linear springs in each stick-shaped leg to absorb shock from the ground impact.
The legs are fixed to the body and able to be easily replaced by other kinds of legs. The bottom
of foot is glued with the material which has asymmetry friction feature contributing to control
the walking direction.
4.2.2 Actuation
Four RC motors (dynamixel RX 28) driving cables through the spine are mounted at the front
and rear parts of body to control the movement of spine. The robot does not have any kind of
actuated joints, for instance, hip or knee joints, and the legs are fixed to the body by screws.
The motors are serially connected to a PC by USB2Dynamixel which is used to transform
a USB port into a serial port RS485. Position control is taken to generate the movement of
the spine. However, a goal position should be set within the valid range from -150deg to
+150deg. Fig. 4.47 shows the maximal angle the spine can bend in the degree of ±150deg.
Due to the error from manufacture and assembly, these angles are not exactly the same, around
30 degrees.
4.3 Experiments and results
In order to observe the correlation between the morphological property and the behavior of
locomotion, a series of experiments based on two different kinds of spinal morphology have
42
4 The quadruped robot Kitty
Figure 4.46: Developed quadruped robot with a tendon-driven spine
Figure 4.47: Maximal bending angle of 4 sides in the spinal structure
been conducted. The first morphology is with fully actuated spine as shown in Fig. 4.45(c),
and the second one is with partial actuated and passive one as shown in Fig. 4.45(d). During
the experiments, several control parameter sets were tested for 3 trials, and then the best one
was chosen to calculate its average speed, standard speed. The effect of property, structure,
and control of the spine on locomotion will be illustrated in the following sections.
43
4 The quadruped robot Kitty
4.3.1 Experiments and results based on robot with fully actuated spine
Experiment setup
The configuration and feature of this fully actuated spine has already been explicitly described
in 2.2.1. Sine waves with different amplitude, frequency, and phase lag are taken to control
the spine in the beginning of the study. The definition of the parameters is shown in Table 2.
Type
Amplitude
Frequency
Phase Lag
Parameter
angleUp (aU)
angleDown (aD)
angleRight (aR)
angleLeft (aL)
frequencyUD (fUD)
frequencyRL (fRL)
phiDown(φD )
phiRight(φR )
phiLeft (φL )
Illustration
The amplitude of the sine waves controlling 4 motors ranges
from 0 to 150o due to the constraint of the motors.
If angle≥ 0, then the cable is tightening the spine, otherwise,
it is relaxing the spine.
The frequency of up-down movement and right-left
movement
The phase lag of bottom side, right side and left side with
respect to up side of the spine
Distance
The predefined travel distance of the robot
Table 5: The explanation of control parameters
Results
Moving forward is a basic criterion for locomotion and an easy start to test its performance.
Table 6 shows the parameter set taken for moving forward. The side cables were kept the
natural length without tightening and relaxing. The amplitude of up-down movement was
140deg and the frequency of up-down was 2. Phase lag is a key parameter to achieve stable
and efficient forward moving. If the phase lag is not properly set, then the power of up and
down motors will be counteracted, leading to energy inefficiency.
Parameter
Value
aU
140deg
aD
140deg
aR
0deg
aL
0deg
fUD
2
fRL
0
φD
Pi
φR
0
φL
0
Table 6: Control for walking forward
We observed that the robot is able to walk forward very fast and the fastest speed is up to
11cm/s with standard deviation of 0.42 in Tab. 7. The results show that the performance of
walking forward is stable and reproducible.
Ave Speed (cm/s)
Std speed
11.06212
0.42487
Table 7: Reults of the experiments for the forward motion
Figure 11 shows the sequential movements of the spine. The up and down movements are
obviously observed. It exhibits periodical movement which contributes to forward moving.
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4 The quadruped robot Kitty
Figure 4.48: Sequential pictures of the locomotion under given parameter set. The yellow arrow represents the walking direction.
Since the spine is fully actuated in 4 sides and able to bend in any directions, then it is possible
to achieve turning performance during the locomotion. In order to achieve this, while up-down
movement is still kept, the right-left movements is added to the spine, as shown in Tab. 8. The
only difference between turning right and left is the flip between φR and φL , which are the
phase lag with respect to the up side of the spine.
Turn right
Turn left
Params
Value
Value
aU
120deg
120deg
aD
120deg
120deg
aR
120deg
120deg
aL
120deg
120deg
fUD
2
2
fRL
2
2
φD
pi
pi
φR
0
pi
φL
pi
0
Table 8: Control for turning right and turning left
Table 9 shows that the robot is able to turn right or left easily and stably. However, the speed
is not so satisfactory, even if we increase the frequency and amplitude. We contribute the slow
speed to the inappropriate cooperation of 4 motors, which results in more unnecessary torque
to each motor. The optimization of controller is crucial to eliminate this problem. It was also
observed that the robot can turn left or right to a maximal angle similar to the other, 49deg
for turning right and 46deg for turning left (Table 9). Figure 4.49 shows the trajectories of the
robot in the case of turning right and left.
Ave Angle (degree)
Std speed
Ave speed (cm/s)
Std speed
Turning right
49.32738
0.631467
3.287622
0.126438
Turning left
46.2073
4.311225
3.443385
0.227138
Table 9: Results of the experiments
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4 The quadruped robot Kitty
Figure 4.49: Trajectories of robot turning right (a) and left (b) under given parameter sets. Red arrow
represents the walking direction
4.3.2 Experiments and results based on robot with partial actuated and
passive spine
Experiment setup
In order to investigate which morphological feature plays a more important role in locomotion,
the second spine morphology as shown in (Fig. 4.45(e)) was adopted, in which only up side
cable is actuated. In this morphology, the down-side driven-cable is replaced by a linear spring
and the side parts of the spine are kept in natural condition. Based on this configuration, the
spinal structure becomes partially actuated and partially passive. Absolute value of sine wave
was taken to control the only one motor in the spine. The motors in sides of spine are still
kept constant value 0, which means it neither pulls the cable, nor pushes the cable. The better
parameter set is in the following Tab. 10.
Amplitude of motor controlling up side of spine
Frequency of the save wave
120deg
3
Table 10: Control for walking forward
Results
We observed that the robot with this configuration can walk half as fast as the one with partial
actuated and passive spine in Tab. 11.
Ave speed (cm/s)
Std speed
6.17101
0.843323
Table 11: Results of the experiment for forward motion
The upward movement of spine doesn’t exhibit as shown in Fig. 4.50, because the spring
downside plays a passive role and is unable to pull the downward cable through the spine
which contributes mostly to the upward movement. The results showed that the locomotion
46
4 The quadruped robot Kitty
can also be achieved by spine featuring one actuated cable and one passive linear spring on
the opposite side.
Figure 4.50: Sequential pictures of the locomotion in 2 cycles when Aup =120deg, Freq=3. Red lines
represent the length of springs below it over time.
4.3.3 Conclusion
A biological spinal model has been analyzed in terms of the effect of the spinal morphological
features on the locomotion. A novel highly-compliant, multi-joint artificial spine has been developed based on this biological analysis. The development of a robot called kitty embedding
this biologically inspired spine helps biological and robotics researchers better understand the
mechanism of spinal engine behind the locomotion.
A series experiments have been conducted to test the effect of the spine on locomotion under
two kinds of spinal morphology. The first one is fully actuated spine and the second is partial
actuated and partial passive. These preliminary experiment results showed that the robot with
fully actuated spine is able to move forward fast with obvious up-down movement in the spine
and the speed is up to 11cm/s, as well as turn left or right stably. It also showed that the
robot with partial actuated and passive one can achieve the movement of moving forward as
well, but the movement of spine is just oscillating from initial position to down, which might
contribute to the slow speed which is only half as fast as the fully actuated one. All the results
emphasize the concept of spinal engine and demonstrate that only by the movement of the
spine can the robot achieve locomotion.
47
References
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