Maximum High Jump with a Robotic Leg - Student Projects

Master Thesis
Maximum High Jump with a
Robotic Leg
April 2008
Supervised by: Author:
C. David Remy Jonas Fisler
Prof. Keith Buffinton
Autonomous Systems Lab
Prof. Roland Siegwart

Abstract i
Abstract
Jumping is one of the most dynamic movements that animals and humans can
perform. It is therefore no surprise, that jumping poses many challenges to
a robotic system. A leg that is able to jump requires a durable mechanical
construction, powerful actuation, and sophisticated control algorithms.
In this thesis a three degree of freedom robotic leg was developed that is
capable of such a highly dynamic movement. This included a sophisticated
mechanical construction that minimized the overall mass and especially the
inertia of the leg, series elastic actuation to recover energy, increase the peak
power of the motors and enhance the shock tolerance of the actuators, and
control inputs that optimally exploited the stretch-shortening-cycle of these
actuators.
Within the project, several leg design variations were evaluated. The most
promising ones were then modeled and an optimal combination of motor, gear,
and elasticity was determined in simulation. The final mechanical design was
performed in CAD, resulting in production ready drawings of all parts. After
successful assembly, the capability of the leg to jump was shown in experiments.
Furthermore, an optimal feedforward control algorithm was proposed which can
significantly improve the jumping performance of the leg.
The results of this thesis are intended to be used in the develop of a
quadruped robot that is capable of dynamic walking and running.

Contents ii
Contents
Abstract i
List of Tables iii
List of Figures iii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Objectives of this thesis . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . 4
2 Design Evaluation and Simulation 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 First design evaluation and kinematic models . . . . . . . . . . . 5
2.3 Dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Mechanical Design 20
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Spring selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Gear selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Hip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Thigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.7 Knee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.8 Shank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.9 Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.10 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Control 30
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Improved model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Optimized feedforward control . . . . . . . . . . . . . . . . . . . 33
4.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Assembly and Test 39
5.1 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

List of Tables iii
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 Summary and Future Work 47
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A Appendix 48
A.1 Derivation of kinematic equations for the pantographic design . . 48
A.2 Morphological box . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.3 Data sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.4 Parts list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
References 63
List of Tables
1 Design considerations . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Some values comparing final velocity to jump height . . . . . . . 14
3 Overview motors and gearboxes . . . . . . . . . . . . . . . . . . . 15
4 Optimized jump height . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Comparison of three compression springs . . . . . . . . . . . . . . 21
6 Mass properties of the leg identified from the 3D-CAD model . . 32
List of Figures
1 Series elastic actuation . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Previous leg designs . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Three designs by Toepfer . . . . . . . . . . . . . . . . . . . . . . 4
4 Mechanical design variants . . . . . . . . . . . . . . . . . . . . . . 5
5 Consideration to motor torque . . . . . . . . . . . . . . . . . . . 6
6 Pantographic leg design . . . . . . . . . . . . . . . . . . . . . . . 7
7 Ballpoint leg design . . . . . . . . . . . . . . . . . . . . . . . . . 8
8 Motor knee design . . . . . . . . . . . . . . . . . . . . . . . . . . 9
9 Cable design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
10 Comparison of ∂y
∂φ . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
11 Block diagram of dynamic model . . . . . . . . . . . . . . . . . . 10
12 Simplified leg in SimMechanics . . . . . . . . . . . . . . . . . . . 13
13 Comparison of motor optimization . . . . . . . . . . . . . . . . . 16
14 Optimized launch for a jump . . . . . . . . . . . . . . . . . . . . 17

List of Figures iv
15 Force on the ground before take-off . . . . . . . . . . . . . . . . . 18
16 Differential drive . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
17 Load estimation hip . . . . . . . . . . . . . . . . . . . . . . . . . 22
18 Hip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
19 Hip detail 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
20 Hip detail 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
21 Hip detail 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
22 Thigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
23 Knee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
24 Cable winding knee . . . . . . . . . . . . . . . . . . . . . . . . . . 29
25 Shank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
26 Dyneema cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
27 Leg geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
28 SimMechanics model . . . . . . . . . . . . . . . . . . . . . . . . . 32
29 Power contraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
30 Trajectories 1 of optimized jump . . . . . . . . . . . . . . . . . . 37
31 Trajectories 2 of optimized jump . . . . . . . . . . . . . . . . . . 38
32 Circlip of the knee shaft . . . . . . . . . . . . . . . . . . . . . . . 39
33 Mounting torsion spring . . . . . . . . . . . . . . . . . . . . . . . 40
34 Support of RE 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
35 Cable on pulley . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
36 Test setup: mounting . . . . . . . . . . . . . . . . . . . . . . . . . 42
37 Trajectory control . . . . . . . . . . . . . . . . . . . . . . . . . . 45
38 Picture series of a jump . . . . . . . . . . . . . . . . . . . . . . . 46

1 Introduction 1
1 Introduction
1.1 Motivation
Most commonly, machines move around on wheels because they can achieve very
good efficiencies with relatively simple mechanical implementations. Balance
is usually not an issue for wheeled systems, and they can move at high speed.
But these amenities are limited to flat surfaces.
On rough terrain legged systems have clearly advantages. They can climb
up mountains and stairs, wade in mud or snow, or balance over rocks. When
walking, small obstacles can be overstepped, and the direction of travel imme-
diately changed. If the system stumbles and falls it can rise again with the
help of legs. Another advantage of legged systems is their ability to manipulate
objects like for example a ball.
Engineers gained interest to build artificial legs to overcome the drawbacks
of wheels. A recently launched project at the Autonomous Systems Lab at
ETH Zurich deals with the design and construction of an energy efficient robotic
quadruped that is capable of various dynamic motions including fast walking,
trotting, bounding, and galloping. The robot will exploit natural passive dy-
namics to be energetically efficient while maximizing dynamic performance.
Within this context, there are two main difficulties. Firstly, the various gaits
show complex patterns, which pose difficult challenges on the computational
algorithms and the computational power. Secondly, rapid leg movements or
jumps need explosive leg power. Power is usually heavy to carry around and
poses a major limitation.
Keeping a dynamically moving robot stable requires a sophisticated control
algorithm, which necessitates quick and accurate actuators. To allow a robot
to run fast and efficient it must be able to place its legs in the short instance
between two steps and -in about the same time- deliver the necessary impulses
to keep its center of gravity in the air.
1.2 Previous work
1.2.1 Legged robotics
The first four-legged autonomous robot, the “phoney pony“, walked in 1966 [1,
2]. Matsuoka [3] realized a monopod able to move dynamically, and Raibert et
al. [4] developed the first dynamically walking quadruped. In recent years,
efforts increased to develop legged robots capable of dynamic maneuvers like
jumping, bounding, trotting, and galloping [5]-[14].

1.2 Previous work 2
To build legged systems, Robinson et al.recommend to use series elastic
actuation, because this introduces serious advantages for electric actuators.
Figure 1: Series elastic actuation
Figure 1 demonstrates the basic principle of series elastic actuation. The torque
or force of the motor is transmitted to the load through a spring. This decouples
the actuators position from the load position, and low-pass filters the actuator’s
output, leading to the following advantages.
Shock tolerance of the actuator is increased. It has a lower reflected inertia,
and causes less damage to the environment. More stable and accurate force
control is possible. Force bandwith (especially for spring- or damper-like out-
put) is increased except for zero motion. The elasticity additionally implements
a means to store energy, particularly for systems that undergo periodic motion.
Series elastic actuation should be used when high constraints are posed onto
the power to mass ratio, or impact shocks might damage actuator and gears.
For a careful selection process of the components for series elastic actuation,
Robinson et al. [15] show some design guidelines. In 2006, Paluska et al. [16]
investigate actuator power and work output of electric motors and state that
series elastic actuation can temporarily increase the motor energy delivered to
a mass by a factor of four.
1.2.2 Leg design for dynamic motion of a quadruped
In 1995, Mennitto [17] (Fig. 2(a)) developed a 8 kg three limb leg prototype
for a quadruped. The actuation is built on a concept called LADD (linear
to angular displacement device). The LADD is very efficient and light-weight
but there is no commercially available solution, and the manufacturing of the
LADD is complex and takes a long time to test.
Remic [18] built a protoype leg, in 2005, consisting of two limbs (Fig. 2(b)).
Weaknesses of this design were corrected and a theoretical model built by
Knox [19] and Curran [20]. The leg is able to successfully perform a series
of vertical jumps. Their work has a similar goal as the project at ETH, but
does not show the evaluation and selection process of the leg construction.
In a thesis on legged robotics at the Autonomous Systems Lab, Toepfer [21]
evaluated three leg designs: ’Linear leg’, ’2-motor-leg’ with linear elasticity or

1.3 Objectives of this thesis 3
(a) Mennitto (b) Remic
Figure 2: Previous leg designs by Mennitto and Remic
with rotational elasticity, and ’pantographic leg’ (Fig. 3). After simulation, he
concluded that the design ’linear leg’ is inferior to the other designs, and rec-
ommended the ’pantographic leg’, of which a first CAD-model was developed.
1.3 Objectives of this thesis
In this thesis a single prototype leg is developed meeting the requirements for
dynamic movements. It is tested in a scenario consisting of a single jump with
the overall goal to maximize a high jump. Experience shall be gained for further
improvements and construction of robotic legs.
The challenge in the development of the mechanical system lies in the nec-
essary dynamics. The moving parts are required to have small masses and
moments of inertia, and must yet be able to deliver large forces and moments.
The design has to withstand high loads during landing and take-off. Moreover,
it is desired that the structure is able to passively store energy and recover it at
a later instance, which significantly improves the efficiency of the locomotion.
Previous studies at the lab estimated a mass of 5 kg for the leg including
a fourth of payload like cameras or sensors. The leg length should be ap-
proximately 40 cm long consisting of two limbs. Although for a jump the full
movement range is not necessary, the leg is supposed to be able to fold and to
stretch as far as possible. This will allow future experiments like for example
standing up from the ground.
Electrical energy supply is limited by a maximum voltage of 36 V, providing

1.4 Organization of this thesis 4
(a) Linear leg (b) 2-motor-leg (c) Pantographic
leg
Figure 3: Three designs that were evaluated by Toepfer [21].
a maximum current of 10 A.
1.4 Organization of this thesis
In Chapter 2, studies on several designs are modeled and simulated to analyze
the occuring forces and moments before take-off. An optimization algorithm
is run to determine a good selection of motor, gearbox and elasticity to reach
maximal height during a jump
The detailed design of the leg, and the evaluation process of the mechanical
parts are explained in Chapter 3.
In Chapter 4, the modeling is revised and adapted to the actual leg construc-
tion. An optimal feedforward control trajectory for the motors is developed and
implemented into software.
Section 5 finally shows the manufacturing and assembly of the prototype
and the results and experiences that were made in a first series of tests.

2 Design Evaluation and Simulation 5
2 Design Evaluation and Simulation
2.1 Introduction
In this section, four mechanical designs are analyzed and compared. The models
for actuation (motor), elasticity (spring), and the mechanical dynamics of the
selected design will be derived and simulated. Motor, spring, and leg are form-
ing a series elastic actuator as described in section 1.2.1. With these models,
optimization algorithms are run to select an optimal motor, spring and initial
leg position to perform a maximal jump for each design. These components
and the estimated forces and torques will form the basis for the mechanical
construction in Chapter 3.
2.2 First design evaluation and kinematic models
Toepfer [21] compared several designs (Section 1.2.2), and proposed the ’pan-
tographic leg design’ to achieve a maximum jump. Also, he pointed out that a
’telescopic leg design’ is inferior to a leg with ’thigh’ and ’shank’.
The main issue is to find a constructive solution on how to apply a torque to
the knee while keeping the leg inertia small. A reevaluation of Toepfer’s models
(except the ’telescopic leg’) was done and extended by two additional designs
(Fig. 4).
Figure 4: Mechanical design variants
In future, we will refer to them as
• ballscrew design
• pantographic design
• motor knee design

2.2 First design evaluation and kinematic models 6
• cable design
It should be noticed that between the motor and leg an elastic element has to
be introduced to create series elastic actuation, which needs space, and might
add constraints on the construction.
Each design has a characteristic motor to end effector force relation Tm
. It is Fg
not only relevant how much torque Tm the motor can provide but also how much
force Fg can be delivered to the ground (Fig. 5). This ratio is characteristic
for each design depending on its kinematics, and is calculated using the power
balance.
Pin = Pout
(2.1)
Figure 5: Relevant for a high jump is actually not the motor torque but the
delivered force to the ground
2.2.1 Pantographic design
The pantographic design (Fig. 6) consists of four bars that are linked with each
other to a framework. The length of the thigh ( ¯ AB) and the shank ( ¯ BE) is l,
the extension of the shank measures d. If the bar ¯ AD is fixed and is twice as
long as the distance ¯ BC, the point E follows quite closely a straight trajectory
(Fig. 6), when the bar thigh ( ¯ AB) is rotated by φ around A.
With
The power balance derives the following torque to force ratio
the following equation is found
Pin = T · ω = T dφ
dt = Pout = F · v = F dy
dt
dy
dt
T
F
∂y ∂φ
= ·
∂φ ∂t
= v
ω
(2.2)
(2.3)
∂y
= . (2.4)
∂φ

2.2 First design evaluation and kinematic models 7
with
∂y
∂φ
(a) Pantographic
leg design
(b) Trajectory of pantographic leg
Figure 6: Pantographic leg design
The kinematic equation of the pantographic design E(φ) is (App. A.1)
y(φ) = yC −
l + d
d
· |−l cos φ − yC| (2.5)
yC = −2d cos φ2 + a
p (−l cos φ + 2d cos φ2) + h
p (l sin φ − 2d sin φ2)
�
(2.6)
p = (xB − xD) 2 + (yB − yD) 2
(2.7)
a = l2 − d 2 + p 2
2p
(2.8)
h = � l 2 − a 2 (2.9)
was derived numerically from the kinematic relationship.
2.2.2 Ballscrew design
Characteristic for the ballscrew design (Fig. 7) is a ballscrew motor along the
thigh. The motor pulls and pushes on the shank extension to rotate the shank
around the knee joint The actuator can rotate freely around the extension of
the shank and the hip.

2.2 First design evaluation and kinematic models 8
The power balance becomes
Figure 7: Ballscrew leg design
Fm ˙x1 = Fg ˙y (2.10)
∂x1 ∂φ
Fm
∂φ ∂t
Fm
Fg
∂y ∂φ
= Fg
∂φ ∂t
= ∂y
∂x1
and the kinematic equations for the ballscrew design are
Again, ∂y
∂x1
x1(φ) =
�
(l + d cos (π − 2φ)) 2 + (d sin (π − 2φ)) 2
(2.11)
(2.12)
(2.13)
y(φ) = −2l cos φ (2.14)
is calculated numerically.
2.2.3 Motor knee design
In the motor knee design (Fig. 8), the motor is placed in the thigh close to the
knee, so that the torque can be applied directly at the knee joint, for example
with bevel gears. For this design, the power balance and the kinematic equation
are
∂y
Tm ˙α = Fg ˙y = Fg ˙α (2.15)
∂α
y(φ) = −2l cos φ = −2l cos α
(2.16)
2
The torque to force relationship can be easily calculated analytically.
∂y α
= l sin = l sin φ (2.17)
∂α 2

2.2 First design evaluation and kinematic models 9
2.2.4 Cable design
Figure 8: Motor knee design
The motor is placed at the hip and with a cable a force or torque is transmitted
to the knee. This design has the same kinematic equation as the motor knee
design if the torque is transmitted using pulleys (Fig. 9(a)) or as the ballscrew
design if the cable is attached at two fixed points (Fig. 9(b)), so that for these
designs the torque to force ratio does not need to be evaluated separately.
(a) Cable design:
Pulley
(b) Cable design
2: Fixed
Figure 9: In the ’cable design’ the torque is provided at the hip and is applied
to the knee via a cable. In design (a) the cable is tightend to a pulley producing
a torque at the knee, whereas in design (b) the cable is fixed to the shank.
2.2.5 Design evaluation
Figure 10 compares the three kinematic results. Explicitely, they mean if the
motor provides a constant torque T the force in y-direction F at the end effector
is small where ∂y
∂φ is large. Or considering that a constant force is desired, T
must increase if ∂y
∂φ increases.
It is evident that the designs ’knee motor’ and ’cable motor’ with pulleys

2.3 Dynamic model 10
have the best torque to force transmission (dotted curve). The pantographic
design needs a doubled motor torque for a required torque at the knee due to
the framework ratio ¯
AD ¯
BC . The ballscrew design shows a fastly rising curve that
goes towards infinity if the angle φ gets close to 90 ◦ .
It can be seen also that the pantographic and the ballscrew design are both
limited in the angular range due to their construction.
Figure 10: Comparison of ∂y
∂φ
Table 1 compares other properties of the design variants. As there are no
clear advantage of the ballscrew or the pantographic design, the decision falls
either towards the cable or the motor knee design, as a result of the comparsion
of the torque analysis. They both have the same characteristic curve.
2.3 Dynamic model
Figure 11: Block diagram that shows the interdependence of the actuation
and the leg
The time-variant behavior of the system consisting of series elastic actuators
and the robotic leg is simulated with a dynamic model.
It consists of three parts. In the first part the motor is modeled. The second
block contains the equations for the elasticity, and thirdly, in SimMechanics, a

2.3 Dynamic model 11
Weight
distribution
Ballscrew
design
Knee motor in
thigh, spindle
of steel
Transmission Torque at the
knee doubles
through the
framework to
the hip
General remarks
Range
angle
(Fig 6, 7, 8)
Buckling of
spindle possible
Pantographic
design
Knee motor
at the hip,
aluminium
framework at
thigh
Direct torque
application
possible
Thigh and
shank motion is
coupled
Motor knee
design
Knee motor at
the knee
Linear
transmission
Torque to the
shank independent
of thigh
position
Cable design
Knee motor at
the hip, pulleys
necessary
Pulleys: See
motor knee
design
Fixed: See
Ballscrew
design
Cable might be
elastic
≈10 ◦ - 80 ◦ 23 ◦ - 78 ◦ 0 ◦ - 90 ◦ See: Ballscrew
and motor knee
Table 1: Design considerations
Simulink toolbox, the leg with the three bodies ’hip’, ’thigh’, and ’shank’ was
built (Fig. 11).
Input to the model is the motor voltage U. Depending on the angular
position of the gearbox φgb and the position of the leg φl the torque of the
elasticity Ts changes. The torque itself effects motor and leg. The velocity ˙y of
the body can be read from the model as output.
2.3.1 Motor
In this project, DC motors are used for actuating the three degrees of freedom of
the leg. Brush-type DC motors have a permanent-magnet stator so that a static
magnetic field excitation is generated. Through this setup a back electromotive
force (back emf) is induced in the rotor.
The electric circuit inside the motor is described by
Ua(t) = La · d
dt Ia(t) + Ra · Ia(t) + Ui(t), (2.18)
with the armature current Ia(t), the resistance Ra, the inductance La, and the
voltage Ua(t). Ui(t) is the induced back emf.
The induced voltage increases proportionally with the rotational speed ω,

2.3 Dynamic model 12
and can be expressed as
The torque of the motor is given by
Ui(t) = κi · ω (2.19)
Tm(t) = κa · Ia(t) (2.20)
From these to equations it can be followed that the current decreases with a
higher back emf, which has the effect of lowering the maximal torque.
The speed constant κi and the torque constant κa are in theory equal.
Rotational and other losses though are usually multiplied to the torque constant
resulting in a lower κa than κi.
According to Newton’s law, the motor torque Tm(t) acts on the motor shaft
(acceleration and deceleration) and the spring Ts(t)
Tm(t) = Jm ˙ω + Ts(t) (2.21)
with Jm being the moment of inertia of the motor load (i.e. gearbox and half
of the spring).
The gearbox multiplies the torque by the gear ratio γ, but introduces friction
losses ηgb.
2.3.2 Elasticity
Tgb = ηgbTmγ (2.22)
To achieve momentarily maximum torque, a spring is inserted between the
motor with a gearbox and the actuated body, as described in section 1.1. The
elasticity obeys the linear relationship
Ts = ks (φleg − φgb) . (2.23)
The spring constant is ks. φleg − φgb defines the compression of the elasticity.
2.3.3 Leg in SimMechanics
The origin of the world coordinate frame lies at the contact point of the shank
to the ground. The contact point was modeled as a revolute joint around
the z-axis. The other degrees of freedom were locked. Shank and thigh were
introduced as two beams with also a revolute joint in between. The hip was
modeled as a sphere of mass mh at the end of the thigh that can slide along the
vertical axis only (prismatic joint) and is connected to the thigh by a revolute

2.4 Optimization 13
joint. The effect of a motor mass at the knee was analyzed by adding another
sphere of mass mk there.
Figure 12: Simplified leg in SimMechanics
The inertia Js,t of the two beams (shank and thigh), and Jh,k of the spheres
were defined by
Js,t =
Jh,k =
⎛
⎜
⎝
⎛
⎜
⎝
r
ms,t
2 s,t
2 0 0
0 ms,t · (3r2 s,t +l2 )
12
0
0 0 ms,t · (3r2 s,t +l2 )
12
mh,kr 3 0 0
0 mh,kr 3 0
0 0 mh,kr 3
⎞
⎞
⎟
⎠
(2.24)
⎟
⎠ (2.25)
where m is the mass of the body, l is the length, and r is the radius.
2.4 Optimization
2.4.1 Parameter
The jump height was the criteria for choosing the optimal design. The jump
height depends only on the final velocity at the moment of take-off and can be
calculated by the equation for energy conservation. The change in the potential
energy at take-off to the maximum height must be equal to the change in the
kinetic energy.
∆Ekin = 1
2 mv2 (take − off) = mg∆h = ∆Epot
h(v) = v2
2g
(2.26)
(2.27)

2.4 Optimization 14
A few values in tab. 2 give an impression of the necessary velocity to achieve
a certain height.
v(take − off) [m/s] 1 1.4 2.0 2.4 2.8 3.15 3.45
h [cm] 5 10 20 30 40 50 60
Table 2: Some values comparing final velocity to jump height
It is therefore important that the final velocity before take-off is as large as
possible. Take-off occurs, when the vertical force on the ground becomes zero.
The last consideration that has to be made is that at take-off not every
part of the leg has the same velocity. Thus, slower parts will be accelerated,
while the hip mass is slowed down. This can be estimated with the equation
for conservation of momentum.
I = mtotvtot = mhipvhip + mlegvleg
(2.28)
Assuming a relatively homogeneous weight distribution in the thigh and the
shank, the average vertical velocity of the leg vleg can be approximated to be at
the middle of the leg at the knee and is then 0.5 · vhip. vtot is then approximated
with mtot = mhip + mleg as
vtot = vhip
�
1 − mleg
�
2mtot
(2.29)
indicating that a larger mass at the leg introduces velocity losses at take-off.
Still, both options without and with larger mass at the knee (motor knee design)
will be optimized for comparison.
At this point, the main challenge was the selection of suitable motors.
Smaller motors have higher dynamic but the maximum torque is limited. Larger
motors in turn provide more torque but become slow due to the high inertia
and the armature resistance.
Many parameters of the model and of the jump can be optimized: motor,
time-varying voltage input, gearbox parameter, spring, mass distribution in
the robot leg, and so on. To simplify the algorithm only three parameters were
chosen to be optimized: the transmission ratio of the gearbox γ, the linear
spring constant kspring k, and the initial leg position defined by the hip angle
φinit.
The main goal is to maximize the height of a jump, respectively the final
vertical velocity. The cost functional in the optimization
J = −1.5v 2 y hip (end) (2.30)

2.4 Optimization 15
gives good results.
The optimization was run for the six maxon motors given in tab. 3. DC
motors from Maxon were chosen because they offer very high quality, are easily
controllable, have a fast response, and also because the Autonomous Systems
Lab has already lots of experience with them. The motor parameter can be
found in the Maxon catalogue (App. A.3) and were inserted into to the motor
model derived in section 2.3.1. Only six motors were selected because larger
motors are too heavy whereas the smaller ones cannot be combined with gear-
boxes that can hold moments above 10 Nm, which were required too for stat-
ically holding the robots‘ mass. 7.5 Nm torque output of the gearbox means
that there is an additional transmission level of at least 1:1.5 necessary that
can be built implemented with bevel gears or pulleys.
The maximum current was set to 10 A, which is the maximum that can be
supplied by the power supply.
Label Gearbox Maximal Motor Mass [g] Nominal
torque [Nm] voltage [V]
1 GP 52 C 22.5-45 RE 40 1100-1250 24
2 GP 42 C 11.3-22.5 RE 36 710-810 32
3 RE 35 700-800 30
2 GP 32 C 7.5 RE 36 544 32
3 RE 35 534 30
4 RE 30 432 36
5 RE 26 344 36
6 RE 25 324 30
Table 3: Overview motors and gearboxes
At the initial leg position, the elasticity is assumed to be relaxed so that at
start the leg will fold until the motor and spring torque hold it and then extend
the leg again, accelerating the hip upwards. To reduce optimization parameters
the knee motor is assumed to be running with full torque at all time, keeping
in mind that the maximum torque changes depending on the motor velocity
(Sec. 2.3.1). The simulation is stopped when the vertical force on the ground
becomes positive, and thus the leg takes off from the ground. The velocity of
the hip at this instant, is inserted into eq. 2.29 to determine the jump height.
A simulation cycle is not only stopped, when either the force to the ground
becomes positive (take-off criterion), but also when the hip comes closer to the
ground than 5 cm (crash criterion), or when the leg is fully stretched (maximal
range criterion). Because the cost functional (Eq. 2.30) is certainly largest if
the simulation stops at the take-off criterion the optimization finds solutions

2.4 Optimization 16
for a maximal jump.
It was also investigated if there was an influence on the height of a jump if
the motor is placed at the knee (direct application of torque to the knee joint)
or at the hip (torque transmission with cable and pulleys).
2.4.2 Results of the optimization
Right at the beginning of the optimization, it turned out that the final velocity
of the hip is slightly higher (3.4 m/s compared to 3.1 m/s) when the mass at
the hip is larger, without having included the velocity losses shown in eq. 2.29
for a bigger mass at the knee. Considering also these losses, it can be followed
that a small mass at the knee is favorably. This led to the final decision on the
design that the cable design with pulleys is superior to the motor knee design,
and that this design would be constructed.
Evident became also that for a 5 kg leg, the motors 5 and 6 (RE 25 and RE
26) were too small for the admissible power supply. They pulled only 9.2 A,
and 7.6 A, respectively, when they are at zero speed.
Figure 13: Comparison of motor optimization. The circles indicate the results
with the voltage U = 36V
Figure 13 compares the optimization results of the other four motors, when
the mass of the hip is 4 kg and the mass at the knee is 1 kg. The plot to
the upper left shows the velocity of the hip, when the motors are charged with
the nominal voltage from the catalogue and when they are supplied with the

2.4 Optimization 17
Motor 1 2 3 4
h(Unom) [cm] 39 31 33 36
h(U = 36V ) [cm] 54 35 40 36
Table 4: Optimized jump height
maximal voltage of 36 V, which is short time possible for any motor. Motor 4
has its nominal voltage at 36 V. In tab. 4, the end velocity is calculated to the
maximal height of the jump.
Motor 1 (RE 40) reaches the highest velocity value. It speaks against it that
it has a very large mass. The high power consumption comes from the fact that
motor 1 has a large armature resistance, and always runs at the current limit
of 10A when supplied with 36 V, indicating that this motor clearly is oversized.
Thus, motor 1 is not considered.
It is remarkable that except for the spring constant k the parameter values
are relatively constant for motor 2, 3, and 4, and the decision which motor to
take is not evident.
Because motor 3, compared to motor 4, jumps slightly higher, the torque
is a bit smaller, and a larger gearbox that can hold 22.5 Nm can be combined
with it, which leaves open options for the construction, the decision falls to the
RE 35, outweighing the disadvantage of the larger mass.
Figure 14: Simulation of an optimal launch for a jump
Figure 14 shows the plots of the optimized jump for the DC Motor RE 35.

2.4 Optimization 18
The upper left plot demonstrates the initial compression of the spring leading
to a down movement of the hip, followed by the leg extension. The velocity
curve demonstrates well the acceleration until the moment of take-off.
In the lower left, the absolute value of the force to the ground shows a
maximum of 140 N.
The plot of the current is very characteristic. First it is running at the limit
until the spring is engaged followed by a sharp drop and then a smoothly rising
curve, as the torque rises.
Figure 15: Force vectors of the leg to the ground
The change of force direction before take-off is illustrated in fig. 15. It
is important to see that at take-off, there is a force left into the positive x-
direction, because the shank rotates around the knee joint, and the elasticity is
not yet fully relaxed.
In simulation, k is investigated. When leaving all the parameters the same
and only varying k, the velocity at take-off barely changes, if the variation lies
within ±1.5 Nm/rad.
2.4.3 Selection
The highest jump for 5 kg body mass, was achieved with the Maxon motor
RE 35 (30 V) (Section 2.4.2). The matching gearbox holding the maximum
torque is the planetary gearbox GP 42 C. The optimized transmission ratio of

2.4 Optimization 19
1:75 can be achieved with a gearbox of that ratio or with a gearbox of 1:26 and
additional a transmission ratio with pulleys of 1:3.
The maximal torque for the gearbox is 22.5 Nm. This torque defines the
maximal loading during a jump. All mechanical components will be designed
to withstand it.
The optimal spring constant lies in the range of 9-12 Nm/rad. Better con-
trollability of a harder spring tipped the scale towards an elasticity of 12 Nm/rad.

3 Mechanical Design 20
3 Mechanical Design
3.1 Introduction
In this Chapter a constructive solution for the robotic leg is introduced. Me-
chanical stability must be adequate to the values found in section 2.4, where the
maximum force on the ground was 140 N and the maximum motor torque was
22.5 Nm. The mechanical construction should be light-weight, and integrate
the motor RE 35 with a transmission of around 1:75, that can be realized with
the gearbox GP 42 C, and an elasticity of 12 Nm/rad.
3.2 Concepts
The key functionalities and partial functionalities are given in the morphological
box in appendix A.2.
The most important decisions in the selection process are explained in the
following sections. The major issues are how to position the motors and what
elastic element to include between shank and knee motor.
3.3 Spring selection
From simulation (Sec. 2.4), we obtained the rotational spring constant k =
12Nm/rad creating a maximum torque of Tmax = 22.5Nm. The maximum
rotation angle is thus φmax = Tmax
k
= 107◦ .
The simplest solution would be to have a torsional spring beside the knee
pulley. But torsional springs that can hold 22.5 Nm and twist 107 ◦ at a rea-
sonable weight could not be found.
Other solutions use a linear extension or compression spring transforming
the rotational into a linear elongation that can easily be done with a pulley.
The following equations lead to the depencency between the torsional spring
constant k, the spring constant c for a linear elasticity, and the pulley radius r.
M = k · φ (3.1)
F = c · x (3.2)
x = r · φ (3.3)
k = c · r 2
(3.4)
Equation 3.4 means that many linear springs with different spring con-
stants can be drawn into consideration by simply changing the pulley radius.
Of course, this will also change the maximum force in the cable and the neces-

3.4 Gear selection 21
sary spring deformation. A smaller radius always means higher cable tension.
Table 5 shows three possible elasticities from Gutekunst Federn [23].
Spring 1 2 3
Mmax [Nm] 22.5 22.5 22.5
k [Nm/rad] 12 12 12
r [mm] 26 22.9 15.2
c [N/mm] 17.6 22.9 51.5
Fmax [N] 865 983 1480
∆l [mm] 49 43 29
l0 [mm] 150 110 76
d [mm] 20 32 33
m [g] 130 165 148
Table 5: Comparison of three linear compression springs: radius of the pulley
r, spring constants k and c, pulley radius r, maximum force Fmax, elongation
∆l, relaxed spring length l0, spring diameter d, mass m
Spring 2 (App. A.3) was selected. It is shorter than spring 1, and the force
is not as large as in spring 3. The disadvantage is the mass, but compared to
other springs on the market it is still rather light weight.
Extension springs are drawn into consideration, too, but as they extend
under force, they are too long to fit into the shank.
3.4 Gear selection
Considered for motor gearboxes were Maxon planetary gears, harmonic drives,
and cable pulley constructions.
Harmonic drives have very high transmission ratios and no backlash but
were too heavy. So that planetary gears matching the Maxon motors are chosen.
Because pulleys were used anyway to apply torque to the knee provided from
a motor at the hip, a ratio of 1 to 3 of the pulley radi was added between
the hip pulley and the knee pulley which allowed the use of a smaller gearbox
(transmission ratio 1:26), therefore saving weight and efficiency.
The hip joint has two degrees of freedom: swinging forward/backward and
swinging to the side. A construction we call ’differential drive’ was designed. In
fig. 16 the functioning principle is shown. When the first hip motor T1 and the
second hip motor T2 turn in the same direction the leg moves sideways (torque
Thx). During a counter rotation, the torque Thz is built up that results in a
forward/backward swing of the leg.
The advantages over a construction that separates the two degrees of free-
dom lies mainly in the possibility to have two smaller motors working together

3.4 Gear selection 22
(a) Concept (b) Picture
Figure 16: Differential drive
for a movement. A leg swings a lot forward and backwards. If the two degrees
of freedom were decoupled, one motor would stay passive and must still be
carried around. Two motors working together must provide each only half of
the necessary torque reducing the size of motor and gearbox. Another advan-
tage is that both motors can be fixed to the body without losing one degree
of freedom. The added complexity to the construction is the major drawback
of the differential drive assembly. Other disadvantages are the weight, size and
the backlash of the bevel gear.
Figure 17: Load estimation hip: Offset torque F · lh because the leg is not
positioned on the axis of the differential drive
To check which option saves more weight a loading scenario is calculated.
Figure 17 shows the force F from the leg that accelerates the hip upwards.
Because this force is not acting on the axis of the differential drive the two hip
motors have to hold against the resulting torque Thx given by
Thx = lh · F = 10Nm (3.5)
when lh is 5 cm and the maximum force is estimated to be 200 N, which includes
a safety of 1.5 to the force that resulted from the jump analysis in section 2.4.2.
If two motors provide this torque, each has to provide 7.5 Nm. This has

3.5 Hip 23
Figure 18: Hip design
to be held also by the bevel gears of the differential drive. Table 3 shows that
from Maxon the gearbox GP 32 C has a maximum torque of 7.5 Nm with the
suitable motor RE 25. Nozag, a bevel gear manufacturer, offers spiral bevel
gears that can transmit 10 Nm, seen from the small bevel gear (App. A.3).
The teeth of the bevel gear are the critical point, so that two gears attaching
at different teeth do not influence the maximal torque. Together they can
hold Thx = 15Nm. The exceeding torque can therefore be used for dynamic
movements like actuating the thigh during a jump and not only to hold the hip
in place statically.
With the transmission of 1:2.5 inside the gearbox the maximum torque in
the forward and backward swing becomes 37.5 Nm, which is sufficient for any
loading scenario for this leg.
This option with two motors and three bevel gears weighs 1.11 kg (650 g+460 g).
If the two motors are decoupled, the gearboxes must provide alone 15 Nm
and 37.5 Nm. From the Maxon catalogue, the GP 42 C and the GP 52 C with
a suitable motor achieve the same strength. These two larger motors with gears
weigh together 2.05 kg (1.25 kg+700 kg).
Due to the much lower mass and the advantages described before, the deci-
sion falls towards the differential drive.
Having the answers to these three major decisions - what spring, what gears
and what motors to use -, the task was now to find a suitable construction
embedding these components.

3.5 Hip 24
3.5 Hip
Figure 19: Hip detail 1
The hip (Fig. 18) combines all three degrees of freedom -two of the hip and one
of the knee- in a very compact design.
The hole of the bevel gear was enlarged and the housing of the knee motor
is attached directly to the large bevel gear (Fig 19). Between motor shaft
and the shaft extension with an integrated pulley, a coupling was inserted that
compensates for small angle or displacement deviations.
Enclosing the shaft extension a hollow shaft (Fig. 20) was designed. It is
also attached directly to the large bevel gear and supports the shaft extension
of the motor. This support is needed because the radial force on the pulley
(shaft extension) is very large (983 N, Sec. 3.3). The hollow shaft is connected
rigidly to the side plates of the thigh. Therefore, the movement of the upper
leg is coupled directly to the large bevel gear. The cables enter through two
holes into the hollow shaft, where they are wound up around the pulley.
The small bevel gears are actuated by by two smaller motors (Maxon RE 25)
that are connected via torsion springs to the small bevel gears (Fig. 21).
An aluminium profile from Kanya[25] holds the two small motors and the
shafts that support the differential box. In the test setup (Section 5.2) the
profile is constrained to only up and down movement.
3.6 Thigh
The largest challenge of the thigh (Fig. 22(a)) design was the connection to
the hip and to the shank. The shank has a given width because of the spring.
When folding the leg the shank and the cable that is tightened from hip to knee
must neither touch the bevel gear at the hip, nor the tube of the shank at the
knee.

3.7 Knee 25
(a) Cutview hip
(b) Housing of the differential drive
Figure 20: Hip detail 2
The two side plates of the thigh are 44 mm apart, and the cable is guided
around a shaft so that the shank (width 42 mm) can fold close to the thigh.
The braces holding the shaft can be used at the same time to tighten the cable.
Figure 22(b) shows the guidance of the cable around the pulleys and the three
small shafts that guide the cable. The mid part of the thigh has a U-shape
giving the thigh torsional and bending stiffness.
3.7 Knee
The knee axle connects the thigh and the knee (Fig. 23(a)). It is blocked from
rotating inside the shank by a pin through the shaft, so that a potentiometer
can be placed on the shaft measuring the relative rotation against the thigh
(Fig. 23(b)).
The pulley is supported by two ball bearings on the knee shaft. Its diameter
is 56 mm as described in section 3.3 (spring 2). The cable is wound as shown
in fig. 24. The two endings from the shank are secured to the pulley after a
turning radius of 240 ◦ . The two cable endings leading from the hip shaft to the

3.8 Shank 26
Figure 21: Actuation of the differential drive
knee are wound 1.5 times around the pulley (if leg is stretched) and secured
there to the pulley.
At the fastening point, the cable leads through a hole into the the pulley
and a knot on the other side of the hole prevents the cable from gliding back.
3.8 Shank
Inside the shank, the compression spring is supported by two plates. The
bottom plate close to the foot can slide up and down and has two holes where
the cable is attached. To reduce the impacts on the bottom plate, when the
compression spring is rapidly relaxed and compressed, a small rubber damper
is located at the fastening point of the cable.
The foot is a simple half sphere that can smoothly roll on the floor.
3.9 Cable
Dyneema was selected for cable. The mechanical properties outperform steel
or other composites (Fig. 26)
3.10 Sensors
Encoders measure the angular position of the motor. The RE 25 use an optical
encoder and the RE 35 a magnetoresistive one, which is smaller, so that the
bush supporting the motor can be slid over the encoder.
Three potentiometer measure the rotation angle of the shank relative to the
thigh, and the rotation of the smaller bevel gears relative to the hip.
With the difference between the encoder and the potentiometer signals, the
spring deflection can be measured.

3.10 Sensors 27
(a) Thigh
(b) Cable guidance
Figure 22: Thigh

3.10 Sensors 28
(a)
(b)
Figure 23: Knee

3.10 Sensors 29
Figure 24: Cable winding around the knee pulley
(a) Shank (b) Cut view
Figure 25: Shank
Figure 26: Dyneema, source: www.swiss-composite.ch/pdf/t-dyneema.pdf

4 Control 30
4 Control
4.1 Introduction
In Chapter 2 a simple one-motor model has been introduced to estimate forces,
torques and to select motors, gearboxes and elastic elements. This model is
extended in this Chapter. The geometric and mass properties of the actual leg
developed in Chapter 3 is included, and the leg is actuated by all three motors.
An optimal feedforward control algorithm is proposed to realize a high jump,
and the influence of a combined knee and hip actuation is analyzed.
4.2 Improved model
To control the motors, Maxon EPOS controllers were used. These controllers
allow direct control of position, velocity, acceleration and deceleration. The
lower level control of current or voltage was done by the EPOS.
This means that the input variables of the model are the motor angles φ1, φ2
and φ3 on one end of the springs. This simplifies the complexity of the model.
The output variables remain the positions and the velocities of the bodies.
The leg is built of three major parts.
• Body 1 includes the aluminium profile with the two hip motors and the
smaller bevel gears. All these parts will be summarized as hip.
• Body 2 consists of the large bevel gear with the shaft and the knee motor
fixed to it, the parts of the thigh, and the pulley at the knee. This
assembly is referred to has thigh.
• The knee elasticity, the knee, the tube of the shank and the foot are
assembled to the shank forming body 3.
To define the initial state, the kinematic equations were derived to relate the
motor angles to the coordinates of the ground contact point and of the hip.
The geometrical properties of the parts are described in fig. 27. The inertial
reference frame is positioned vertically underneath the center of gravity of the
hip.

4.2 Improved model 31
with
Figure 27: Geometry and coordinate system of the leg
x0 = 0 (4.1)
y0(φ1, φ2, φ3) = − cos φhx · (−l sin φhz − l sin φk) + lh sin φhx
(4.2)
z0 = 0 (4.3)
xg(φ1, φ2, φ3) = l cos φhz − l cos φk
(4.4)
yg = 0 (4.5)
zg(φ1, φ2, φ3) = sin φhx · (−l sin φhz − l sin φk) + lh cos φhx
φhz = φ1 − φ2
2ndiff
φhx = φ1 + φ2
2
φk = φ3 + φhz
(4.6)
(4.7)
(4.8)
(4.9)
φhz and φhx describe the rotation around the x- and z-axis of the hip and φk
the rotation of knee in respect to the inertial reference frame. The transmission
of the bevel gears in the differential drive is ndiff .
The mass properties of the three bodies are summarized in tab. 6. The hip
itself has only one translational degree of freedom, thus the rotation is blocked.
The inertias Jh1 and Jh2 describe only the inertia of the motor shaft and the
bevel gear.
The analytical derivation of the dynamic model using for example the La-

4.2 Improved model 32
Mass [kg] Inertia [10 − 3 kg m 2 ]
mh = 2.2 Jh1 = Jh2 =
mt = 1.7 Jt =
ms = 0.4 Js =
⎛
1.8 0 0
⎞
⎝ 0 37.3 0 ⎠
⎛
0.2
0 0 37.3
⎞
−0.2 −3.6
⎝ −0.2 13.2 0.1 ⎠
−3.6
⎛
0.1
0.1
0
7.8
⎞
0
⎝ 0 1.2 0 ⎠
0 0 1.2
Table 6: Mass properties of the leg identified from the 3D-CAD model
grange method becomes too complex so that the numerical Matlab toolbox for
mechanical systems was used again. SimMechanics evaluates these geometric
and mass properties (Fig. 28).
Figure 28: SimMechanics model of the leg
The three series elastic actuators, each containing a motor, a transmission
and a spring, actuate the hip around the x- and the z-axis, and the knee around
the z-axis.
With the torques of the motors T1, T2, and T3, the actuation around the
three axes can be derived as
Thz = T1 − T2
ndiff
Thx = T1 + T2
Tk = T3
(4.10)
(4.11)
(4.12)

4.3 Optimized feedforward control 33
4.3 Optimized feedforward control
4.3.1 Standing
Standing appears to be an easy task. But in fact, this leg configuration is always
at a instable position. This can be seen in the following function
Tk(φ) = mglcos(φ) (4.13)
Tk is the required torque at the knee depending on the angle φ. m is the mass,
g the gravity constant and l the length of the thigh. Tk is defined in eq. 4.13,
but is simplified in this respect that the mass is distributed in all three bodies
and not only in the hip. The mass m is therefore substituted by the mass m ∗ ,
which is determined in simulation. m ∗ is slightly smaller than m but is constant
and independent of φ.
If the torque from the motor is too low, the leg moves downwards (φ de-
creases), where an even higher torque would be necessary, so the leg drops to
the floor. The inverse happens if the torque is slightly too high. The leg ex-
pands (φ grows) to an angle where the torque should be even lower. The result
is that the leg goes into full stretching. This means that the motor torque to
stand still has always to be controlled in a loop. The Maxon EPOS controller
does this if the angle and not the torque is controlled.
It is important to know what the initial motor position needs to be for a
required initial leg angle.
φm =
4.3.2 Optimization algorithm
�
φ + Tk
�
· γ (4.14)
ks
The goal to perform the highest possible jump is equivalent to maximizing the
final vertical velocity before take-off of the leg. For optimizing, constraints must
be considered. In this model there are three constraints that are explained next.
• Motor properties
• Foot is not allowed to slip
• Foot can only push towards the ground
Motors cannot turn at all speeds, and the maximum torque and the accel-
eration is limited. Angular velocity ω and maximum torque Tmax obey a linear
relationship. The back emf as described in section 2.3.1 increases linearly with

4.3 Optimized feedforward control 34
ω, lowering Tmax.
Tmax = κa · Ia,max − κi · ω (4.15)
The maximal torque is limited by the mechanical construction to 22.5 Nm.
Figure 29: Power constraint of the motor
The transmission ratio γ of the knee actuator of 1 : 78, thus the maximum
torque needed from the motor is 0.29 Nm, which can be supplied until the knee
pulley turns at n = 5634
78
= 72.2 rpm or 7.6 rad/s (Fig. 29). If the motor turns
faster the maximum torque is reduced. The motor constraint becomes
with
Tmax =
�
Tsim ≤ Tmax
22.5Nm, if n ≤ 7.6 rad
s
T0 − κi · ω if n > 7.6 rad
s
The motor velocity is ω, and the stalling torque is T0.
�
(4.16)
(4.17)
The dynamics of the inductance inside the motor are not accounted for in
this model. It must be checked later in experiments, if this is relevant for the
leg application (Section 5.3.2).
The leg dynamics and the motor torques do not only induce a vertical
force (y-axis) on the ground but also parallel forces. The second constraint is
therefore that the friction holding the foot in place must be always larger than
the horizontal forces. If µ is the friction coefficient the constraint can be written
as
with
FF ≤ µ |Fy| (4.18)
FF = � F 2 x + F 2 z
(4.19)

4.3 Optimized feedforward control 35
The last constraint is the stop criterion of the simulation. If the vertical
force on the ground becomes positive - foot wants to pull on the ground - which
is not possible. The simulation stops, and tend and vy(tend) are determined.
The following cost functional J for the optimization showed good results
J = −vy(tend) − v 3 � tend
y(tend) + cF
0
j�
cT jdt (4.20)
j being the number of motors used. cT and cF follow from the constraints and
are defined by
�
(Tsim − Tmax)
cT =
k �
, if Tsim > Tmax
0, if Tsim ≤ Tmax
�
(FF − µ |Fy|)
cF =
k �
, if µ |Fy| ≤ FF
0, if µ |Fy| ≥ FF
1
(4.21)
(4.22)
The power factor k is chosen to be 4 so that large exceeding of the maximal
torque increases cT by a power of four and not only linear. With this the
transition is continuous at the boundary of the constrained space and increases
fast when a constrained space is left by more than one unit.
The input variables to the model are the motor angles φ1(t), φ2(t) and φ3(t).
We know from section 2.4.2 that in the optimal trajectory the hip moves down
and accelerates then upwards. This trajectory means for the knee motor to
turn first in one direction and then to accelerate in the other. This behaviour
is approximated by a third order polynomial.
φ3(t) = a03 + a13t + a23t 2 + a33t 3 + φ03
(4.23)
φ03 is the initial motor angle (Eq. 4.14) so that without any other input the leg
would be standing still.
A negative a03 means that the torque in the spring is reduced right at
the beginning, when the motor is switched on, which provokes the initial and
characteristic downwards movement of the hip before accelerating upwards.
For the hip motors (j = 1, 2) the input trajectory is defined by a polynomial
as well.
φi(t) = a0i + a1it + a2it 2 + a3it 3
(4.24)
The twelve coefficients aij and the initial angle of the leg φinit are the
parameter to be optimized inside the algorithm.

4.3 Optimized feedforward control 36
All three motors should work in a way together that they give the most
thrust upwards and at the same time do not work against each other. In the
following subsection the influence of the hip motors is investigated.
4.3.3 Results
At first, the two hip torques were set to zero, and only the torque to the knee
was optimized, starting with three optimization parameters: φinit, a03 and a13.
Then the number of parameters were increased by a23 and a33. The foot position
was set exactly underneath the center of gravity of the hip. This tilts the hip
when the knee bends. Another option would be to set the leg into a vertical
position, which would set the foot 4.5 cm besides the other point. This distance
is introduced by the hollow shaft at the hip. The optimization results differed
by about 1 cm jump height or 3 N force, which can be neglected.
In the simulation, a33 has almost no effect on the maximal jump height. For
this reason, a33 was set to zero to keep the number of optimized parameters low.
Figure 30 shows the the angular position (the optimized curve), the velocity
trajectory, and the force trajectories of the optimized jump using only the knee
motor. The maximum velocity of the hip becomes 3.23 m/s, which corresponds
to a jump height of 42 cm. The calculation of the jump height takes into account
that the thigh and the shank have other velocities (Eq. 2.28). The maximal
force towards the ground is 164 N.
The next optimization step was to include the hip motors (i = 1,2). The op-
timized parameters from the knee motor were kept constant, while the number
of parameters (a0i ... a3i) for the hip motor were increased.
From the simulation it could be followed that a0i and a1i have no effect on
the outcome. The output of the optimization set them close to zero, and the
jump height decreased.
It is different for the parameters a2i and a3i, which do influence the jump
height positively. After optimizing the four knee motor parameter and the four
hip motor parameters all together (φinit, a03, a13, a23, a2i, a3i), the optimized
curve in fig. 31 resulted.
The velocity at take-off is 3.39 m/s and the calculated jump height is 46 cm.
It follows that in comparison to the case where only the knee was actuated, the
hip motors had only a slight influence on the jump(4 cm higher). One reason
might be that at some point they supported the jump action whereas at other
they reduced the force. This because the each hip motor trajectory was defined
by only two parameters.

4.4 Implementation 37
(a) Angular position of the knee motor (b) Position and velocity trajectories of
the hip
(c) Force trajectory on the ground (d) x-, y-, z- components of force
Figure 30: Optimized trajectories if only the knee motor applies a torque
4.4 Implementation
To find the optimal trajectories in simulation is only one part to the control
of the motors. It is also important that the communication with the hardware
works smoothly. The critical points here is the frequency of the data exchange.
We have three options
• Maxon software with RS 232-CAN cable connection
• Matlab with RS 232-CAN cable connection
• Matlab with CAN-CAN cable connection
The software from Maxon is very reliable and the user interface is very
easy and intuitive to use. The ASL developed a Matlab code using the same
functions as the Maxon software. The data exchange rate is not increased
compared to the Maxon software when using the RS 232 bus, but trajectories
can be implemented for forward control. The CAN-CAN connection has a
remarkably higher data exchange rate, and of course the trajectories can be

4.4 Implementation 38
(a) Angular position of the motors (b) Position and velocity trajectories of
the hip
(c) Force trajectory on the ground (d) x-, y-, z- components of force
Figure 31: Optimized trajectories if all motors apply torque
implemented just like in the second option. The disadvantage was, that the
software for the CAN-CAN connection was not very stable, and still under
developement.

5 Assembly and Test 39
5 Assembly and Test
5.1 Assembly
The leg could be assembled completely, though a few parts need to be modified
to enable a smooth assembly of the leg.
The circlip on one side of the knee cannot be mounted because the support
of the bearing to the thigh is protruding. The border securing the ball bear-
ings should be on the other side of the insert, which is supporting the bearing
(Fig. 32).
Figure 32: Circlip cannot be mounted to the knee shaft
The pin in the shaft of the knee to block the shaft from rotating does not
fit by 0.5 mm. The smaller pin that was used introduced several degrees of
backlash.
The torsion spring does not fit well into the groove of the coupling because
the radius of the spring is larger then expected. This coupling needs a whole
redesign (Fig. 33).
The support of the shaft extension of the hip motors block the leg from ro-
tating around the x-axis (rotation sideways), when the leg is completely folded.
The corners of this support block can be chamfered (Fig. 34) to prevent this.
The lower support of the compression spring slides up and down inside the
tube of the shank. The support and the tube are made of aluminum which is
not preferable, because this introduces remarkable friction losses. Possibly the
support can be made of a composite (Fig. 25).
The cable ripped initially at the edge where the cable is deflected to wind
up around the pulley. Thick double-sided adhesive tape on the pulley increased
the friction on the pulley and decreasing the strain over the edge (Fig. 35).

5.1 Assembly 40
Figure 33: Torsion spring does not fit into the groove
Figure 34: Corner should be chamfered

5.1 Assembly 41
Figure 35: Adhesive tape increases the friction of the pulley to avoid damage
to the cable

5.2 Test setup 42
5.2 Test setup
To test the leg to stand, to move up and down, and to jump, the aluminium
profile of the hip was attached to another profile of 1.5 m length leading to
the wall. It was secured there with hinge joints so that it could move on a
circular arc up and down leaving only one degree of freedom unconstrained.
The consequence of this defined path of the hip requires the hip to be able to
freely rotate around the x-axis. If this rotation were blocked the torsion springs
of the hip motors would be wound up creating a force towards the side of the
foot, which in the end brakes the acceleration.
Figure 36: Test setup: mounting
To prevent slipping of the foot, it was covered with a sticky coating, and
the ground was covered with a rubber mat (Fig. 36). This absorbeded also
potential side forces created by the hip motor. The controllers were set beside
the leg on the ground, and not mounted to the leg.
The whole leg including the profile leading to the wall but without the EPOS
controller weighs 4.3 kg. With the controller, the leg and the aluminium profile
have a mass of 5 kg.

5.3 Results 43
5.2.1 Motor control
The two hip motors were connected to the EPOS 24/5, and the knee motor to
the EPOS 70/10. The connection from the computer to the EPOS was either
using the RS 232 bus or the CAN bus (Section 4.4).
It resulted that the EPOS 70/10 cannot be combined in series with the
EPOS 24/5, so that for the tests either three EPOS 24/5 can be used when all
motors are mounted, or one EPOS 70/10 if only the knee motor is controlled
and both hip motors are dismounted.
To program the EPOS, like setting the controller gains, or defining the
maximum acceleration the software provided from Maxon proved to be very
useful. To follow a position trajectory the control using Matlab is necessary.
It was first tested, if the motor is able to follow the optimal trajectory
calculated in section 4.3.3. The connection used the RS 232 bus. First, the
trajectory was stretched, meaning that the same trajectory points had to be
passed, but in larger time intervals. At lower speed the desired trajectory was
perfectly matched by the motor. But at higher speed the motor was too slow
(Fig. 37).
Two possibilities exist to explain this. Either the motor is not able to
accelerate so quickly, or the data exchange rate is too low.
To increase the data exchange rate, a connection via the CAN bus could be
established.
5.3 Results
5.3.1 Preliminary tests with the leg
The leg was positioned with a 90 ◦ angle at the knee, and the foot lying under-
neath the hip (Fig. 36). Slowly, the hip was moved up and down.
It became evident that the cable lengthens by several centimeters because
it is braided. It did not seem to be elastic, though. This makes it necessary to
retension the cable several time after it is tightened the first time.
The cable is vulnerable at the points where it is guided over an edge, namely
at the position where it is guided through the holes of the pulley (Fig. 35).
Already at slow movement, the cable ruptures. This makes it necessary to
increase the friction on the pulley, so that the larger part of the force in the
cable is absorbed by friction.

5.3 Results 44
5.3.2 Maximum jump
As a first approach to realize a jump a start at the lowest position was chosen,
and then the leg was stretched with maximal power. The hip motors were not
used and therefore removed from the leg. The knee motor was controlled with
the EPOS 70/10.
For this setup the provided software from Maxon was sufficient. The end
position was an almost stretched leg (knee angle ≈170 ◦ ). And by increasing
the parameters of the motor control (current, voltage, velocity and acceleration
limitation) maximum power is approached. A final end position at less than
170 ◦ slowed down the extending leg too soon so that a jump was not possible.
At a voltage of 31 V and a current limit of 5.5 A, the leg lifted off the floor
(Fig. 38). After take-off, the thigh kept on rotating further around the hip.
The shank followed the thigh. It must be noticed that for this jump, there were
no hip motors mounted to stop this swinging movement.
In Fig. 38(i) two lines indicate the position of the aluminium profile at the
momement of take-off and at maximum altitude. From the picture the jumping
height can be estimated to 5.5 to 6 cm, which corresponds to the distance from
the top of the aluminium profile to the middle on the side of of the motor
support plate.
There are two main reasons why the 42 cm from simulation for one motor
(Sec. 4.3.3) was not reached. The leg was actuated with maximally 5.5 A
which is approximately one half of the power that was used in the simulation.
Secondly, the optimized control trajectory was not implemented. Inaccuracies
in the model and frictional losses might have reduced the maximal height, too.

5.3 Results 45
(a) Trajectory control twenty times slower
(b) Trajectory control two times slower
(c) Trajectory control normal speed
Figure 37: Trajectory control

5.3 Results 46
(a) (b) (c)
(d) (e) Lift off (f)
(g) (h) Top (i) Max. height 5.5-6 cm
Figure 38: Picture series of a jump

6 Summary and Future Work 47
6 Summary and Future Work
6.1 Summary
In this thesis, four mechanical designs for a robotic leg were evaluated. The
selected design option had the best motor torque to force ratio. To set most
mass at the hip (like the knee actuation motor) resulted in optimization to be
better then locating the motor at the knee. The torque transmission from the
motor at the hip to the knee has been realized with a cable pulley construction.
The leg was constructed with a series elastic actuator, to achieve maximum
peak power, to recover energy, and to increase the shock tolerance of the gear
and the motor. For the knee actuation, the spring and the motor with a gearbox
was found after optimization. A compression spring has been integrated into
the shank so that with the knee pulley a transformation from a linear to a
rotational elasticity could be realized.
A model with the geometric and mass properties of the leg has been used to
find and to optimize feedforward control trajectories for the knee and the hip
motors. In simulation, a maximum jump height of 46 cm was reached.
In first tests, the construction showed to be operating reliably. The leg
jumped 5.5 to 6 cm centimeters, starting from a folded leg position, and ac-
celerating with maximum available power. The leg did not jump 46 cm, as
predicted in the simulation. Reasons are the limited power supply of 31 V and
5.5 A. Also, the optimized trajectory has not yet been implemented. Another
reason is the inaccuracy of the model, due to losses of the backlash in the trans-
mission, friction of bearings, of the mounting to the wall, and of the spring and
the support plate inside the shank, which are not included in the model.
6.2 Future work
Mechanical improvements to reduce the weight and to alter or redesign the
parts described in section 5.1 must be made.
An Evaluation and comparison of the model and the actual leg will give
further insights to the dynamical behavior of the leg.
The feedforward control algorithm can be implemented, and enhanced by
more sophisticated control input or by feedback control. The developement of
a control strategy after take-off is necessary to have a controlled landing and
to make a series of jumps possible.
Ultimately, this thesis can be extended to other dynamic scenarios of or
more leg.

A Appendix 48
A Appendix
A.1 Derivation of kinematic equations for the pantographic design
The equations for the coordinate xC and yC originate from the calculation of
the intersection of two circles [26].
� � � �
A =
�
xA
yA
�
0
=
0
� �
B
xB(φ)
=
yB(φ)
� �
l sin φ
=
−l cos φ
�
�
D =
xD(φ)
yD(φ)
2d sin φ2
=
−2d cos φ2
�
p =
a = l2 − d 2 + p 2
(xB − xD) 2 + (yB − yD) 2
2p
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
h = � l2 − a2 (A.6)
� � �
xC(φ) xD +
C =
=
yC(φ)
a
p (xB − xD) + h
p (yD − yB)
yD + a
p (yB − yD) + h
p (xB
�
(A.7)
− xD)
� � �
xE(φ) xC −
E =
=
yE(φ)
|xB−xC|
�
d · (l + d)
(A.8)
· (l + d)
y(φ) = yC −
yC − |yB−yC|
d
l + d
d
(A.9)
· |−l cos φ − yC| (A.10)

A.2 Morphological box 49
A.2 Morphological box

A.2 Morphological box 50

A.2 Morphological box 51

A.2 Morphological box 52

A.3 Data sheets 53
A.3 Data sheets
• Maxon RE 35
• Maxon GP 42 C
• Maxon RE 25
• Maxon GP 32 C
• Gutekunst Federn D-357
• Federntechnik Knoerzer M45R21
• Nozag SS 131435
• Maedler Ausgleichskupplung LA 60293600

A.3 Data sheets 54
RE 35 �35 mm, Graphitbürsten, 90 Watt
M 1:2
Lagerprogramm
Standardprogramm
Sonderprogramm (auf Anfrage)
Bestellnummern
gemäss Massbild 273752 323890 273753 273754 273755 273756 273757 273758 273759 273760 273761 273762 273763
Wellenlänge 15.6 gekürzt auf 4 mm 285785 323891 285786 285787 285788 285789 285790 285791 285792 285793 285794 285795 285796
Motordaten (provisorisch)
Werte bei Nennspannung
1 Nennspannung V 15.0 24.0 30.0 42.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0
2 Leerlaufdrehzahl min-1 7070 7670 7220 7530 7270 6650 5960 4740 3810 3140 2570 2100 1620
3 Leerlaufstrom mA 245 168 123 92.7 77.3 68.7 59.7 44.7 34.2 27.1 21.6 17.2 12.9
4 Nenndrehzahl min-1 6270 6910 6420 6770 6490 5860 5150 3920 2970 2280 1710 1220 732
5 Nennmoment (max. Dauerdrehmoment) mNm 73.2 93.3 92.4 97.7 96.5 98.2 98.8 102 105 105 105 104 104
6 Nennstrom (max. Dauerbelastungsstrom) A 4.00 3.36 2.50 1.95 1.63 1.51 1.36 1.12 0.915 0.752 0.621 0.503 0.391
7 Anhaltemoment mNm 874 1160 949 1070 967 878 766 613 493 394 320 253 194
8 Anlaufstrom A 45.0 39.7 24.4 20.3 15.5 12.9 10.1 6.43 4.16 2.74 1.83 1.18 0.704
9 Max. Wirkungsgrad
Kenndaten
% 81 84 84 86 85 85 84 83 82 80 79 77 74
10 Anschlusswiderstand � 0.334 0.605 1.23 2.07 3.09 3.72 4.75 7.46 11.5 17.5 26.2 40.5 68.2
11 Anschlussinduktivität mH 0.085 0.191 0.340 0.620 0.870 1.04 1.29 2.04 3.16 4.65 6.89 10.3 17.1
12 Drehmomentkonstante mNm A-1 19.4 29.2 38.9 52.5 62.2 68 75.8 95.2 119 144 175 214 276
13 Drehzahlkonstante min-1 V-1 491 328 246 182 154 140 126 100 80.5 66.4 54.6 44.7 34.6
14 Kennliniensteigung min-1 mNm-1 8.43 6.79 7.76 7.16 7.62 7.67 7.89 7.85 7.84 8.08 8.19 8.46 8.55
15 Mechanische Anlaufzeitkonstante ms 5.97 5.60 5.50 5.40 5.38 5.38 5.39 5.38 5.37 5.38 5.39 5.39 5.41
16 Rotorträgheitsmoment gcm2 67.6 78.7 67.6 72.0 67.4 67.0 65.2 65.4 65.5 63.6 62.8 60.8 60.4
Spezifikationen
Thermische Daten
17 Therm. Widerstand Gehäuse-Luft 6.2 KW -1
18 Therm. Widerstand Wicklung-Gehäuse 2.0 KW -1
19 Therm. Zeitkonstante der Wicklung 30 s
20 Therm. Zeitkonstante des Motors 1050 s
21 Umgebungstemperatur -20 ... +100°C
22 Max. Wicklungstemperatur +155°C
Mechanische Daten (Kugellager)
23 Grenzdrehzahl 12000 min -1
24 Axialspiel 0.05 - 0.15 mm
25 Radialspiel 0.025 mm
26 Max. axiale Belastung (dynamisch) 5.6 N
27 Max. axiale Aufpresskraft (statisch) 110 N
(statisch, Welle abgestützt) 1200 N
28 Max. radiale Belastung, 5 mm ab Flansch 28 N
Weitere Spezifikationen
29 Polpaarzahl 1
30 Anzahl Kollektorsegmente 13
31 Motorgewicht 340 g
Motordaten gemäss Tabelle sind Nenndaten.
Erläuterungen zu den Ziffern Seite 47.
Option
Hohlwelle als Spezialausführung
Vorgespannte Kugellager
Betriebsbereiche Legende
n [min -1
]
Dauerbetriebsbereich
Unter Berücksichtigung der angegebenen thermischen
Widerstände (Ziffer 17 und 18) und einer Umgebungstemperatur
von 25°C wird bei dauernder
Belastung die maximal zulässige Rotortemperatur
erreicht = thermische Grenze.
Kurzzeitbetrieb
Der Motor darf kurzzeitig und wiederkehrend überlastet
werden.
Typenleistung
maxon-Baukastensystem Übersicht Seite 16 - 21
Planetengetriebe
�32 mm
0.75 - 4.5 Nm
Seite 230
Planetengetriebe
�32 mm
1.0 - 6.0 Nm
Seite 231
Planetengetriebe
�32 mm
8 Nm
Seite 233
Planetengetriebe
�42 mm
3 - 15 Nm
Seite 235
Empfohlene Elektronik:
ADS 50/5 Seite 268
ADS 50/10 269
ADS_E 50/5 269
ADS_E 50/10 269
EPOS 24/5 286
EPOS P 24/5 287
MIP 50 289
Hinweise 17
Encoder MR
256 - 1024 Imp.,
3 Kanal
Seite 251
Encoder HED_ 5540
500 Imp.,
3 Kanal
Seite 254 / 256
DC-Tacho DCT
�22 mm
0.52 V
Seite 263
Bremse AB 28
�28 mm
24 VDC, 0.4 Nm
Seite 300
Ausgabe April 2007 / Änderungen vorbehalten maxon DC motor 81
maxon DC motor

A.3 Data sheets 55
Planetengetriebe GP 42 C �42 mm, 3 - 15 Nm
Keramikversion
Lagerprogramm
Standardprogramm
Bestellnummern
Sonderprogramm (auf Anfrage)
Getriebedaten
203113 203115 203119 203120 203124 203129 203128 203133 203137 203141
1 Untersetzung 3.5 : 1 12 : 1 26 : 1 43 : 1 81 : 1 156 : 1 150 : 1 285 : 1 441 : 1 756 : 1
2 Untersetzung absolut 7
/2
49
/4 26 343
/8
2197
/27 156 2401
/16
15379
/54 441 756
3 Massenträgheitsmoment gcm2 14 15 9.1 15 9.4 9.1 15 15 14 14
4 Max. Motorwellendurchmesser mm 10 10 8 10 8 8 10 10 10 10
Bestellnummern 203114 203116 203121 203125 203130 203134 203138 203142
1 Untersetzung 4.3 : 1 15 : 1 53 : 1 91 : 1 186 : 1 319 : 1 488 : 1 936 : 1
2 Untersetzung absolut 13
/3
91
/6
637
/12 91 4459
/24
637
/2
4394
/9 936
3 Massenträgheitsmoment gcm2 9.1 15 15 15 15 15 9.4 9.1
4 Max. Motorwellendurchmesser mm 8 10 10 10 10 10 8 8
Bestellnummern 203117 203122 203126 203131 203135 203139
1 Untersetzung 19 : 1 66 : 1 113 : 1 230 : 1 353 :1 546 : 1
2 Untersetzung absolut 169
/9
1183
/18
338
/3
8281
/36
28561
/81 546
3 Massenträgheitsmoment gcm2 9.4 15 9.4 15 9.4 14
4 Max. Motorwellendurchmesser mm 8 10 8 10 8 10
Bestellnummern 203118 203123 203127 203132 203136 203140
1 Untersetzung 21 : 1 74 : 1 126 : 1 257 : 1 394 : 1 676 : 1
2 Untersetzung absolut 21 147
/2 126 1029
/4 1183 /3 676
3 Massenträgheitsmoment gcm2 14 15 14 15 15 9.1
4 Max. Motorwellendurchmesser mm 10 10 10 10 10 8
5 Stufenzahl 1 2 2 3 3 3 4 4 4 4
6 Max. Dauerdrehmoment Nm 3.0 7.5 7.5 15.0 15.0 15.0 15.0 15.0 15.0 15.0
7 kurzzeitig zulässiges Drehmoment Nm 4.5 11.3 11.3 22.5 22.5 22.5 22.5 22.5 22.5 22.5
8 Max. Wirkungsgrad % 90 81 81 72 72 72 64 64 64 64
9 Gewicht g 260 360 360 460 460 460 560 560 560 560
10 Mittleres Getriebespiel unbelastet ° 0.3 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5
11 Getriebelänge L1 mm 40.9 55.4 55.4 69.9 69.9 69.9 84.4 84.4 84.4 84.4
Gesamtlänge
Gesamtlänge
M 1:2
Technische Daten
Planetengetriebe geradeverzahnt
Abtriebswelle rostfreier Stahl
Abtriebswellenlagerung vorgespannte Kugellager
Radialspiel, 12 mm ab Flansch max. 0.06 mm
Axialspiel bei Axiallast < 5 N 0 mm
>5N max.0.3mm
Max. zulässige Axiallast 150 N
Max. zulässige Aufpresskraft 300 N
Drehsinn, Antrieb zu Abtrieb =
Empfohlene Motordrehzahl < 8000 min -1
Empfohlener Temperaturbereich -20 ... +100°C
erweiterter Bereich als Option -35 ... +100°C
Stufenzahl 1 2 3 4
Max. zul. Radiallast,
12 mm ab Flansch 120 N 150 N 150 N 150 N
Kombination
+ Motor Seite +Tacho Seite +BremseSeite = Motorlänge + Getriebelänge + (Tacho / Bremse) + Montageteile
RE 35, 90 W 81 111.9 126.4 126.4 140.9 140.9 140.9 155.4 155.4 155.4 155.4
RE 35, 90 W 81 MR 251 123.3 137.8 137.8 152.3 152.3 152.3 166.8 166.8 166.8 166.8
RE 35, 90 W 81 HED_ 5540 254/256 132.9 147.4 147.4 161.9 161.9 161.9 176.4 176.4 176.4 176.4
RE 35, 90 W 81 DCT 22 263 130.0 144.5 144.5 159.0 159.0 159.0 173.5 173.5 173.5 173.5
RE 35, 90 W 81 AB 28 300 148.0 162.5 162.5 177.0 177.0 177.0 191.5 191.5 191.5 191.5
RE 36, 70 W 82 112.2 126.7 126.7 141.2 141.2 141.2 155.7 155.7 155.7 155.7
RE 36, 70 W 82 MR 251 123.6 138.1 138.1 152.6 152.6 152.6 167.1 167.1 167.1 167.1
RE 36, 70 W 82 HED_ 5540 254/256 133.2 147.7 147.7 162.2 162.2 162.2 176.7 176.7 176.7 176.7
RE 36, 70 W 82 DCT 22 263 130.3 144.8 144.8 159.3 159.3 159.3 173.8 173.8 173.8 173.8
RE 40, 150 W 83 112.0 126.5 126.5 141.0 141.0 141.0 155.5 155.5 155.5 155.5
RE 40, 150 W 83 MR 251 123.4 137.9 137.9 152.4 152.4 152.4 166.9 166.9 166.9 166.9
RE 40, 150 W 83 HED_ 5540 254/256 132.7 147.2 147.2 161.7 161.7 161.7 176.2 176.2 176.2 176.2
RE 40, 150 W 83 HEDL 9140 259 166.1 180.6 180.6 195.1 195.1 195.1 209.6 209.6 209.6 209.6
RE 40, 150 W 83 AB 28 300 148.1 162.6 162.6 177.1 177.1 177.1 191.6 191.6 191.6 191.6
RE 40, 150 W 83 AB 28 301 156.1 170.6 170.6 185.1 185.1 185.1 199.6 199.6 199.6 199.6
RE 40, 150 W 83 HED_ 5540 254/256 AB 28 300 165.2 179.7 179.7 194.2 194.2 194.2 208.7 208.7 208.7 208.7
RE 40, 150 W 83 HEDL 9140 259 AB 28 301 176.6 191.1 191.1 205.6 205.6 205.6 220.1 220.1 220.1 220.1
EC 40, 120 W 163 111.0 125.5 125.5 140.0 140.0 140.0 154.5 154.5 154.5 154.5
EC 40, 120 W 163 HED_ 5540 255/257 129.4 143.9 143.9 158.4 158.4 158.4 172.9 172.9 172.9 172.9
EC 40, 120 W 163 Res 26 264 137.6 152.1 152.1 166.6 166.6 166.6 181.1 181.1 181.1 181.1
EC 40, 120 W 163 AB 28 300 141.8 156.3 156.3 170.8 170.8 170.8 185.3 185.3 185.3 185.3
EC 45, 150 W 164 152.2 166.7 166.7 181.2 181.2 181.2 195.7 195.7 195.7 195.7
EC 45, 150 W 164 HEDL 9140 259 167.8 182.3 182.3 196.8 196.8 196.8 211.3 211.3 211.3 211.3
EC 45, 150 W 164 Res 26 264 152.2 166.7 166.7 181.2 181.2 181.2 195.7 195.7 195.7 195.7
EC 45, 150 W 164 AB 28 301 159.6 174.1 174.1 188.6 188.6 188.6 203.1 203.1 203.1 203.1
Ausgabe April 2007 / Änderungen vorbehalten maxon gear 235
maxon gear

A.3 Data sheets 56
maxon DC motor
RE 25 �25 mm, Graphitbürsten, 20 Watt
Lagerprogramm
Standardprogramm
Sonderprogramm (auf Anfrage)
Bestellnummern
gemäss Massbild 118749 118750 118751 118752 118753 118754 118755 118756 118757
Wellenlänge 15.7 gekürzt auf 4 mm 302002 302003 302004 302005 302006 302007 302001 302008 302009
Motordaten
Werte bei Nennspannung
1 Nennspannung V 9.0 15.0 18.0 24.0 30.0 42.0 48.0 48.0 48.0
2 Leerlaufdrehzahl min-1 10000 9650 10200 9550 9860 11100 10300 8230 5050
3 Leerlaufstrom mA 110 60.7 53.9 36.9 30.5 25.2 20.1 15.2 8.51
4 Nenndrehzahl min-1 8980 8470 8890 8360 8680 9950 9190 7070 3870
5 Nennmoment (max. Dauerdrehmoment) mNm 11.1 20.6 23.1 26.7 27.2 27.6 28.4 29.4 30.8
6 Nennstrom (max. Dauerbelastungsstrom) A 1.50 1.50 1.47 1.17 0.983 0.799 0.667 0.548 0.352
7 Anhaltemoment mNm 244 237 233 257 263 299 280 222 136
8 Anlaufstrom A 30.7 16.6 14.3 11.0 9.21 8.39 6.38 4.03 1.52
9 Max. Wirkungsgrad
Kenndaten
% 77 83 84 86 86 88 88 87 85
10 Anschlusswiderstand � 0.293 0.902 1.26 2.19 3.26 5.00 7.53 11.9 31.6
11 Anschlussinduktivität mH 0.0275 0.0882 0.115 0.238 0.353 0.551 0.832 1.31 3.48
12 Drehmomentkonstante mNm A-1 7.97 14.3 16.3 23.4 28.5 35.7 43.8 55.0 89.7
13 Drehzahlkonstante min-1 V-1 1200 669 585 407 335 268 218 173 106
14 Kennliniensteigung min-1 mNm-1 44.1 42.3 45.3 38.1 38.2 37.5 37.4 37.6 37.5
15 Mechanische Anlaufzeitkonstante ms 5.36 4.58 4.49 4.28 4.20 4.13 4.11 4.10 4.09
16 Rotorträgheitsmoment gcm2 11.6 10.3 9.45 10.7 10.5 10.5 10.5 10.4 10.4
Spezifikationen
Thermische Daten
17 Therm. Widerstand Gehäuse-Luft 14 KW -1
18 Therm. Widerstand Wicklung-Gehäuse 3.1 KW -1
19 Therm. Zeitkonstante der Wicklung 12.4 s
20 Therm. Zeitkonstante des Motors 910 s
21 Umgebungstemperatur -20 ... +100°C
22 Max. Wicklungstemperatur +125°C
Mechanische Daten (Kugellager)
23 Grenzdrehzahl 14000 min -1
24 Axialspiel 0.05 - 0.15 mm
25 Radialspiel 0.025 mm
26 Max. axiale Belastung (dynamisch) 3.2 N
27 Max. axiale Aufpresskraft (statisch) 64 N
(statisch, Welle abgestützt) 270 N
28 Max. radiale Belastung, 5 mm ab Flansch 16 N
Weitere Spezifikationen
29 Polpaarzahl 1
30 Anzahl Kollektorsegmente 11
31 Motorgewicht 130 g
Motordaten gemäss Tabelle sind Nenndaten.
Erläuterungen zu den Ziffern Seite 47.
Option
Vorgespannte Kugellager
Betriebsbereiche Legende
n [min -1
]
Dauerbetriebsbereich
Unter Berücksichtigung der angegebenen thermischen
Widerstände (Ziffer 17 und 18) und einer Umgebungstemperatur
von 25°C wird bei dauernder
Belastung die maximal zulässige Rotortemperatur
erreicht = thermische Grenze.
Kurzzeitbetrieb
Der Motor darf kurzzeitig und wiederkehrend überlastet
werden.
Typenleistung
maxon-Baukastensystem Übersicht Seite 16 - 21
Planetengetriebe
�26 mm
0.5 - 2.0 Nm
Seite 226
Planetengetriebe
�32 mm
0.4 - 2.0 Nm
Seite 228
Planetengetriebe
�32 mm
0.75 - 6.0 Nm
Seite 229 / 231
Empfohlene Elektronik:
LSC 30/2 Seite 268
ADS 50/5 268
ADS_E 50/5 269
EPOS 24/5 286
EPOS P 24/5 287
MIP 10 289
Hinweise 17
M 1:2
Encoder MR
128 - 1000 Imp.,
3 Kanal
Seite 250
Encoder Enc
22 mm
100 Imp., 2 Kanal
Seite 252
Encoder HED_ 5540
500 Imp.,
3 Kanal
Seite 254 / 256
DC-Tacho DCT
�22 mm,
0.52 V
Seite 263
Bremse AB 28
�28 mm
24 VDC, 0.4 Nm
Seite 300
78 maxon DC motor Ausgabe April 2007 / Änderungen vorbehalten

A.3 Data sheets 57
Planetengetriebe GP 32 C �32 mm, 1.0 - 6.0 Nm
Keramikversion
Lagerprogramm
Standardprogramm
Bestellnummern
Sonderprogramm (auf Anfrage)
Getriebedaten
166930 166933 166938 166939 166944 166949 166954 166959 166962 166967 166972 166977
1 Untersetzung 3.7 : 1 14 : 1 33 : 1 51 : 1 111 : 1 246 : 1 492 : 1 762 : 1 1181 : 1 1972 : 1 2829 : 1 4380 : 1
2 Untersetzung absolut 26
/7
676
/49
529
/16
17576
/343
13824 421824
/125 /1715 86112
/175
19044 10123776
/25 /8575 8626176 /4375 495144
/175
109503
/25
3 Max. Motorwellendurchmesser mm 6 6 3 6 4 4 3 3 4 4 3 3
Bestellnummern 166931 166934 166940 166945 166950 166955 166960 166963 166968 166973 166978
1 Untersetzung 4.8 : 1 18 : 1 66 : 1 123 : 1 295 : 1 531 : 1 913 : 1 1414 : 1 2189 : 1 3052 : 1 5247 : 1
2 Untersetzung absolut 24
/5
624
/35
16224
/245
6877
/56
101062
/343
331776
/625
36501 2425488
/40 /1715 536406 1907712
/245 /625 839523
/160
3 Max. Motorwellendurchmesser mm 4 4 4 3 3 4 3 3 3 3 3
Bestellnummern 166932 166935 166941 166946 166951 166956 166961 166964 166969 166974 166979
1 Untersetzung 5.8 : 1 21 : 1 79 : 1 132 : 1 318 : 1 589 : 1 1093 : 1 1526 : 1 2362 : 1 3389 : 1 6285 : 1
2 Untersetzung absolut 23
/4 299 /14
3887 /49 3312 /25
389376 /1225 20631
/35 279841 9345024
/256 /6125 2066688 /875 474513 6436343
/140 /1024
3 Max. Motorwellendurchmesser mm 3 3 3 3 4 3 3 4 3 3 3
Bestellnummern 166936 166942 166947 166952 166957 166965 166970 166975
1 Untersetzung 23 : 1 86 : 1 159 : 1 411 : 1 636 : 1 1694 : 1 2548 : 1 3656 : 1
2 Untersetzung absolut 576
/25
14976
/175
1587
/10
359424
/875
79488
/125
1162213 /686 7962624 /3125 457056
/125
3 Max. Motorwellendurchmesser mm 4 4 3 4 3 3 4 3
Bestellnummern 166937 166943 166948 166953 166958 166966 166971 166976
1 Untersetzung 28 : 1 103 : 1 190 : 1 456 : 1 706 : 1 1828 : 1 2623 : 1 4060 : 1
2 Untersetzung absolut 138
/5
3588
/35
12167
/64
89401
/196
158171
/224
2238912 /1225 2056223 /784 3637933 /896
3 Max. Motorwellendurchmesser mm 3 3 3 3 3 3 3 3
4 Stufenzahl 1 2 2 3 3 4 4 4 5 5 5 5
5 Max. Dauerdrehmoment Nm 1 3 3 6 6 6 6 6 6 6 6 6
6 kurzzeitig zulässiges Drehmoment Nm 1.25 3.75 3.75 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5
7 Max. Wirkungsgrad % 80 75 75 70 70 60 60 60 50 50 50 50
8 Gewicht g 118 162 162 194 194 226 226 226 258 258 258 258
9 Mittleres Getriebespiel unbelastet ° 0.7 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
10 Massenträgheitsmoment gcm2 1.5 0.8 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
11 Getriebelänge L1 mm 26.4 36.3 36.3 43.0 43.0 49.7 49.7 49.7 56.4 56.4 56.4 56.4
Gesamtlänge
Gesamtlänge
M 1:2
Technische Daten
Planetengetriebe geradeverzahnt
Abtriebswelle rostfreier Stahl
Wellendurchmesser als Option 8 mm
Abtriebswellenlagerung Kugellager
Radialspiel, 5 mm ab Flansch max. 0.14 mm
Axialspiel max. 0.4 mm
Max. zul. Radiallast, 10 mm ab Flansch 140 N
Max. zulässige Axiallast 120 N
Max. zulässige Aufpresskraft 120 N
Drehsinn, Antrieb zu Abtrieb =
Empfohlene Motordrehzahl < 8000 min -1
Empfohlener Temperaturbereich -20 ... +100°C
erweiterter Bereich als Option -35 ... +100°C
Option: Geräuschreduzierte Ausführung
Kombination
+ Motor Seite +Tacho/BremseSeite Gesamtlänge [mm] = Motorlänge + Getriebelänge + (Tacho / Bremse) + Montageteile
RE 25, 10 W 76 81.0 90.9 90.9 97.6 97.6 104.3 104.3 104.3 111.0 111.0 111.0 111.0
RE 25, 10 W 76 MR 250 92.0 101.9 101.9 108.6 108.6 115.3 115.3 115.3 122.0 122.0 122.0 122.0
RE 25, 10 W 76 Enc 22 252 95.1 105.0 105.0 111.7 111.7 118.4 118.4 118.4 125.1 125.1 125.1 125.1
RE 25, 10 W 76 HED_ 5540 254/256 101.8 111.7 111.7 118.4 118.4 125.1 125.1 125.1 131.8 131.8 131.8 131.8
RE 25, 10 W 76 DCT 22 263 103.3 113.2 113.2 119.9 119.9 126.6 126.6 126.6 133.3 133.3 133.3 133.3
RE 25, 20 W 77 69.5 79.4 79.4 86.1 86.1 92.8 92.8 92.8 99.5 99.5 99.5 99.5
RE 25, 20 W 78 81.0 90.9 90.9 97.6 97.6 104.3 104.3 104.3 111.0 111.0 111.0 111.0
RE 25, 20 W 78 MR 250 92.0 101.9 101.9 108.6 108.6 115.3 115.3 115.3 122.0 122.0 122.0 122.0
RE 25, 20 W 78 Enc 22 252 95.1 105.0 105.0 111.7 111.7 118.4 118.4 118.4 125.1 125.1 125.1 125.1
RE 25, 20 W 78 HED_ 5540 254/256 101.8 111.7 111.7 118.4 118.4 125.1 125.1 125.1 131.8 131.8 131.8 131.8
RE 25, 20 W 78 DCT 22 263 103.3 113.2 113.2 119.9 119.9 126.6 126.6 126.6 133.3 133.3 133.3 133.3
RE 25, 20 W 78 HED_5540 / AB 28 300 132.2 142.1 142.1 148.8 148.8 155.5 155.5 155.5 162.2 162.2 162.2 162.2
RE 26, 18 W 79 85.3 95.2 95.2 101.9 101.9 108.6 108.6 108.6 115.3 115.3 115.3 115.3
RE 26, 18 W 79 MR 250 96.3 106.2 106.2 112.9 112.9 119.6 119.6 119.6 126.3 126.3 126.3 126.3
RE 26, 18 W 79 Enc 22 252 102.7 112.6 112.6 119.3 119.3 126.0 126.0 126.0 132.7 132.7 132.7 132.7
RE 26, 18 W 79 HED_ 5540 254/256 103.7 113.6 113.6 120.3 120.3 127.0 127.0 127.0 133.7 133.7 133.7 133.7
RE 26, 18 W 79 DCT 22 263 106.3 116.2 116.2 122.9 122.9 129.6 129.6 129.6 136.3 136.3 136.3 136.3
RE 30, 60 W 80 94.5 104.4 104.4 111.1 111.1 117.8 117.8 117.8 124.5 124.5 124.5 124.5
RE 30, 60 W 80 MR 251 105.9 115.8 115.8 122.5 122.5 129.2 129.2 129.2 135.9 135.9 135.9 135.9
RE 35, 90 W 81 97.4 107.3 107.3 114.0 114.0 120.7 120.7 120.7 127.4 127.4 127.4 127.4
RE 35, 90 W 81 MR 251 108.8 118.7 118.7 125.4 125.4 132.1 132.1 132.1 138.8 138.8 138.8 138.8
RE 35, 90 W 81 HED_ 5540 254/256 118.4 128.3 128.3 135.0 135.0 141.7 141.7 141.7 148.4 148.4 148.4 148.4
RE 35, 90 W 81 DCT 22 263 115.5 125.4 125.4 132.1 132.1 138.8 138.8 138.8 145.5 145.5 145.5 145.5
RE 35, 90 W 81 AB 28 300 133.5 143.4 143.4 150.1 150.1 156.8 156.8 156.8 163.5 163.5 163.5 163.5
RE 35, 90 W 81 HEDS 5540 / AB 28 254/300 150.6 160.5 160.5 167.2 167.2 173.9 173.9 173.9 180.6 180.6 180.6 180.6
RE 36, 70 W 82 97.7 107.6 107.6 114.3 114.3 121.0 121.0 121.0 127.7 127.7 127.7 127.7
RE 36, 70 W 82 MR 251 109.1 119.0 119.0 125.7 125.7 132.4 132.4 132.4 139.1 139.1 139.1 139.1
RE 36, 70 W 82 HED_ 5540 254/256 118.7 128.6 128.6 135.3 135.3 142.0 142.0 142.0 148.7 148.7 148.7 148.7
RE 36, 70 W 82 DCT 22 263 115.8 125.7 125.7 132.4 132.4 139.1 139.1 139.1 145.8 145.8 145.8 145.8
Ausgabe April 2007 / Änderungen vorbehalten 231
maxon gear

A.3 Data sheets 58
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A.3 Data sheets 59
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M45L20 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.00 29.41 75.45 10019.72 132.79 25.50 39.00 120.00 A
M45R20 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.00 29.41 75.45 10019.72 132.79 25.50 39.00 120.00 A
M45L21 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.25 30.54 79.23 10019.72 126.47 25.50 39.00 120.00 B
M45R21 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.25 30.54 79.23 10019.72 126.47 25.50 39.00 120.00 B
M45L22 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.50 31.67 83.00 10019.72 120.72 25.50 39.00 120.00 C
M45R22 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.50 31.67 83.00 10019.72 120.72 25.50 39.00 120.00 C
M45L23 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.75 32.81 86.77 10019.72 115.47 25.50 39.00 120.00 D
M45R23 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 5.75 32.81 86.77 10019.72 115.47 25.50 39.00 120.00 D
M45L32 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 7.50 40.72 113.18 10019.72 88.53 25.50 39.00 120.00 C
M45R32 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 7.50 40.72 113.18 10019.72 88.53 25.50 39.00 120.00 C
M45L40 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 8.00 42.99 120.73 10019.72 83.00 25.50 39.00 120.00 A
M45R40 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 8.00 42.99 120.73 10019.72 83.00 25.50 39.00 120.00 A
M45L50 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 11.00 56.56 166.00 10019.72 60.36 25.50 39.00 120.00 A
M45R50 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 11.00 56.56 166.00 10019.72 60.36 25.50 39.00 120.00 A
M45L60 1.4310 0 in Korb Daten CAD-3D 4.50 27.50 36.50 15.00 74.66 226.36 10019.72 44.26 25.50 39.00 120.00 A
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A.3 Data sheets 60
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A.3 Data sheets 61
Katalog
Antriebselemente und Normteile
Allgemeine Normteile
Auszugschienen
Buchsen
Federnde Druckstücke
Gewindestangen, Muttern, Schrauben
Gabelgelenke,Gabelköpfe und ES-Bo
Gelenkköpfe DIN 648 und Gelenklage
Griffe, Knöpfe und Bedienteile
Höhenverstellbare Elemente
Hub- und Verstellsysteme
Innenzahnkränze
Kantenschutzprofile
Kegelräder
Keilriemen und Keilriemenscheiben
Keilwellen und Keilnaben
Kettenräder
Kleinstoßdämpfer und Zubehör
Kugelrollen
Kupplungen
Ausgleichskupplungen KA
Ausgleichskupplungen LA
Zahn-Kupplungen BOS
Zahn-Kupplungen BOZ
Zahn-Kupplungen BW
Drehstarre Kupplungen HB
Drehstarre Kupplungen HF
Drehstarre Kupplungen HFD mit D
Drehstarre Kupplungen HU
Drehstarre Kupplungen HZ
Drehstarre Kupplungen HZD mit D
Elastische Kupplungen DX
Elastische Kupplungen ME
Elastische Kupplungen MU
Elastische Kupplungen RN
Elastische Kupplungen RN Edels
http://www.maedler.ch/katalog_ch/katalog/index.htm
Ausgleichskupplungen LA
Artikelnummer: 60293600
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Artikeldaten
Werkstoff 2)
Max. Betriebsdrehmomente Nm 10,00
B mm 8
L1 mm 38,10
L2 mm 11,00
D1 mm 25,40
Winkelversatz Grad 7
Parallelversatz mm 0,38
Gewicht g 44
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04.04.2008

A.4 Parts list 62
A.4 Parts list

References 63
References
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References 64
[12] D. Papadopoulos and M. Buehler. Stable running in a quadruped robot
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