A Biarticulated Robotic Leg for Jumping Movements: Theory and

J. Babič
Department of Automatics,
Biocybernetics and Robotics,
Jožef Stefan Institute,
Ljubljana 1000, Slovenia
Bokman Lim
Mechanical and Aerospace Engineering,
Seoul National University,
Seoul 151-742, Korea
D. Omrčen
J. Lenarčič
Department of Automatics,
Biocybernetics and Robotics,
Jožef Stefan Institute,
Ljubljana 1000, Slovenia
F. C. Park1
A Biarticulated Robotic Leg for
Jumping Movements: Theory and
Experiments
This paper investigates the extent to which biarticular actuation mechanisms—springdriven redundant actuation schemes that extend over two joints, similar in function to
biarticular muscles found in legged animals—improve the performance of jumping and
other fast explosive robot movements. Robust numerical optimization algorithms that take
into account the complex dynamics of both the redundantly actuated system and frictional
contact forces are developed. We then quantitatively evaluate the gains in vertical jumping vis-à-vis monoarticular and biarticular joint actuation schemes and examine the
effects of spring stiffness and activation angle on overall jump performance. Both numerical simulations and experiments with a hardware prototype of a biarticular legged
robot are reported. 关DOI: 10.1115/1.2963028兴
Mechanical and Aerospace Engineering,
Seoul National University,
Seoul 151-742, Korea
e-mail: [email protected]
1
Introduction
Although recent biped robots have displayed impressive movement coordination skills that require balance and dexterity, as of
yet none can even remotely approach the ability of humans to
jump, run, kick, or perform other fast explosive movements. Several studies on jumping have been undertaken for one degree-offreedom mechanisms 关1,2兴, and for jumping robots with multiple
degrees of freedom 关3–5兴. Numerical optimization-based studies
of hopping and vertical jumping have also been performed 关6,7兴,
which take into account models for the ground-foot contact, actuators, and multibody dynamics 共of the human musculoskeletal
system in the latter case兲, and numerically determine optimal
jumps based on, e.g., user-specified center-of-mass trajectories,
contact time, air time, and take-off time.
An intriguing attempt at generating explosive robotic movements can be found in schemes that emulate the properties of
biarticular muscles such as those found in human legs. These
muscles extend over two joints—the human gastrocnemius
muscle, for example, extends from the knee to the ankle joints and
acts as both a knee flexor and ankle extensor 共Fig. 1兲—and have
been found to play a critical role in the generation of fast explosive human movements 关8兴. The gastrocnemius muscle also has
the interesting feature of being connected to the foot by an elastic
tendon; several biomechanical studies have confirmed the importance of elasticity in the tendons and muscle fibers in enhancing,
e.g., jumping and running motions 关9–12兴.
Inspired by these biomechanical findings, robotics researchers
have pursued both mechanical designs and motion control laws
that emulate the function of biarticular muscles. Saito et al.
关13,14兴 proposed a biarticular actuation mechanism based on a
1
Corresponding author.
Contributed by the Mechanisms and Robotics Committee for publication in the
JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 22, 2008; final
manuscript received July 2, 2008; published online August 5, 2008. Review conducted by Andrew P. Murray. Paper presented at the 2008 IEEE International Conference on Robotics and Automation, Pasadena, CA, May 19–23, 2008.
Journal of Mechanisms and Robotics
bilateral servosystem and conducted experiments involving a bipedal robot as well as an externally powered orthosis. Tahara et al.
关15兴 showed through simulation studies that biarticular actuation
schemes, when endowed with the nonlinear and time-varying
stiffness properties characteristic of muscles, can considerably
simplify feedback control—humanlike reaching movements for
serial arms, for example, can be achieved via a simple feedback
control law in task space, without the need for, e.g., inverse dynamics computations 共whether hardware realizations of such nonlinear muscle properties are possible is a separate and difficult
matter兲. While not directly making use of biarticular structures,
Scarfogliero et al. 关16兴 also suggest a bioinspired mechanism with
springs for a lightweight four-legged jumping robot.
More recently Babic et al. 关17,18兴 developed a planar jumping
robot, in which the thigh and heel are connected by a passive
biarticular link and spring-damper unit. Maximum height vertical
jumping motions for this robot are found under various simplifying assumptions, e.g., contact with the floor is modeled as a rotational joint, the hip joint velocity is linearly related to the knee
joint velocity, the ankle joint is active during contact but becomes
passive when airborne, etc.兲. These assumptions are made in order
to bypass the inherent complexities of modeling the dynamics of
the biarticular jumping robot—the foot-ground contact, in particular, requires careful attention, as does the dynamic model of the
leg, which involves a redundantly actuated closed chain with a
spring element. The ensuing motion optimization must also converge while satisfying these structure-varying discontinuous dynamic equations. This turns out to be quite challenging, given the
computational difficulties involved in, e.g., optimizing basic
reaching motions with no contact for even simple kinematic
chains 关19,20兴.
This paper attempts to more precisely determine the extent to
which passive biarticular actuation schemes can improve jumping
performance. Our approach takes into account the complex biarticular leg dynamics without any of the previous restrictive assumptions. We also determine maximum height vertical jumps
with respect to both the actuator input profiles and parameters
Copyright © 2008 by ASME
FEBRUARY 2009, Vol. 1 / 011013-1
•
•
•
Most of the simplifying assumptions in the previous work
are discarded.
The biarticular system’s optimized motions are compared
with the corresponding optimal motions obtained for
monoarticular and conventional legged robots.
The effects of the various parameters associated with the
biarticular structure, i.e., spring stiffness and activation
angle, are examined in detail.
The optimized motions themselves can be used as feedforward
reference trajectories in the implementation of control laws for
such systems. More broadly, these results offer guidelines on how
to design and implement biarticular actuation schemes for legged
robots.
Fig. 1 Biarticular gastrocnemius muscle and its kinematic
diagram
associated with the passive biarticular mechanism, i.e., spring
stiffness, and the joint angle at which the passive biarticular
mechanism is activated. The specific contributions of the paper
are as follows.
1.1 Dynamic Modeling and Optimization of Biarticular
Structures. When optimizing motions for biarticular actuated systems like our jumping robot, several additional complexities arise:
共i兲 the dynamics model involves not only a structure-varying
closed chain with redundant actuation and spring elements but
also frictional contact models between the feet and ground, 共ii兲 the
optimization involves discontinuities at several levels 共e.g., the
contact conditions, certain joints switching from passive to active,
etc.兲, and 共iii兲 the optimization variables involve diverse parameters 共e.g., spring stiffness, activation angle, etc.兲 beyond the usual
joint trajectory and input torque profiles. For problems of this
type, simply parametrizing the joint trajectories in terms of some
suitable spline function and applying a nonlinear optimization algorithm will generally fail. In this paper we provide a complete
and practical formulation, and also solution procedure, of the motion optimization problem for such systems. Diverse objective
functions such as maximum jump height or distance, minimum
effort, etc., can be admitted in our framework. Our approach further involves, e.g., using a hybrid dynamics algorithm with passive virtual joints to solve the dynamics and formulating posture
and torque limit constraints via penalty functions, so that all other
constraints are linear.
1.2 Evaluating the Effectiveness of Biarticular Structures.
Employing the optimization algorithms developed in this paper,
we answer the question of exactly how effective biarticular
mechanisms really are as a means of generating fast explosive
movements in robots. Using the work of Babic et al. 关17,18兴 as
our point of departure, we make the following specific contributions.
011013-2 / Vol. 1, FEBRUARY 2009
1.3 Experiments With a Hardware Prototype. To validate
our numerical studies, we design and construct a robotic leg prototype involving a passive realization of biarticular actuation.
Based on the earlier simulation results, we perform a series of
vertical jumping experiments on the prototype and compare the
jumping performance of robotic mechanisms with and without
biarticular links. We also compare the experimental results obtained for the hardware prototype with the simulation results.
The paper is organized as follows. In Sec. 2 we review biarticular actuation mechanisms and describe the planar jumping robot
model that will be the focus of our case studies. Section 3 describes the dynamic model and algorithm for generating maximal
height jumps. Section 4 presents numerical case studies on the
effects of design values 共spring stiffness and activation angle兲 on
jumping performance. Section 5 describes jumping experiments
performed with a hardware prototype. In Sec. 6 we summarize our
main findings and discuss their implications on the design of nextgeneration robots that employ biarticular mechanisms.
2
Biarticular Leg Structure
2.1 Biarticular Actuation Mechanism. Biarticular muscles
are muscles that extend over two joints and allow mechanical
energy to be directly transported from proximal to distal joints.
During human jumping this transportation effectively allows rotational motion of body segments to be transformed into translational motion of the body center of gravity 关21兴. For robotic jumping, biarticular actuation schemes divert some of the power of
proximal actuators to the distal joints, particularly when the proximal joints approach a singular configuration. The mechanism by
which this is achieved is the timely activation of the biarticular
muscles before the end of the push-off phase of the jump.
To examine this phenomenon in more detail, consider the pushoff phase of a vertical jumping motion. The influence of the
knee’s rotational motion on the jumper’s vertical velocity is
clearly the greatest in the fully squatted initial stance. As the jump
proceeds and the knee straightens 共approaching a kinematic singularity兲, rotational motion of the knee has progressively less effect on the vertical velocity.
At this stage extending the feet 共by rotating the ankle joint兲
contributes considerably more upward vertical velocity, and activating the biarticular gastrocnemius muscle 共Fig. 1兲 achieves precisely this effect: As the knee extends, the biarticular gastrocnemius muscle contracts isometrically 共i.e., contraction forces are
generated without any changes in the muscle length兲 and tries to
draw the thigh and heel closer together. Because the foot still
remains in contact with the ground, the muscle contraction now
causes the foot to extend via rotation of the ankle joint. In this
way part of the power generated by the knee extensors is now
delivered to the ankle joint, increasing the height of the jump.
However, as demonstrated by Bobbert et al. 关22兴, the precise timing of the gastrocnemius muscle activation is critical to jumping
performance; ill-timed activation of the gastrocnemius muscle can
Transactions of the ASME
Fig. 3 Robot jumping sequence
Fig. 2 Biarticular leg model
even degrade jumping performance in some cases. As we show
later, the magnitude of the gastrocnemius muscle force also significantly influences jumping performance.
2.2 Planar Jumping Robot Model. We model our planar
jumping robot as a three revolute-joint serial chain, as shown in
Fig. 2. The variables qa1, qa2, and qa3, respectively, denote the
positions of the actuated leg joints. The position and orientation of
the torso are parametrized by three coordinates, corresponding to
two prismatic joints and one revolute joint of a virtual serial
chain; these are, respectively, labeled q p1, q p2, and q p3, as indicated in the figure. Contact forces between the foot and ground are
assumed to be exerted at the toe and heel as shown.
The biarticular actuation mechanism is realized as a linear
spring connecting the thigh and heel links and passing through the
knee and ankle joints. The spring force is given by
f biarticular = − k共x − x0兲
共1兲
where k is the spring stiffness, x0 is the rest length of the spring at
the instant of activation, and x is the length of the spring. One can
also specify the instant at which the spring is activated 共that is, the
switch in Fig. 2 is closed兲 simply by the angle of the knee joint ␪0;
we call this angle the activation angle. The rest length of the
spring x0 is then determined solely as a function of the knee and
ankle joints: x0 = f共qa2 , qa3兲qa2=␪0, with ␪0 = qa2,activation. The activation angle ␪0 and stiffness value k completely characterize the
behavior of the biarticular actuation mechanism.
To better understand the role of the biarticular actuator, consider the vertical jumping motion sequence shown in Fig. 3. Beginning in the squat position with the biarticular mechanism inactivated 共the switch is open兲, the hip, knee, and ankle joints are
Journal of Mechanisms and Robotics
directly actuated to generate a vertically upward motion. When
the biarticular actuation mechanism is activated 共the switch is
closed兲 and the knee continues to extend, the spring begins to
exert a contraction force that tries to draw the thigh and heel
closer together. Because the foot is in contact with the ground at
this stage, the spring force has the effect of rotating the ankle joint
and extending the foot 共the robot is redundantly actuated during
this period of activation兲. Immediately after the robot pushes off
the ground, the biarticular actuator is deactivated.
Both the timing of the activation 共characterized by the activation angle兲 and the spring stiffness significantly influence jumping
performance. To illustrate, suppose we have a predetermined vertical jumping motion for a conventional leg structure 共without
biarticular actuation兲. If we now apply to this same motion a
biarticular force, then depending on when this force is activated,
the resulting jump motion may contain horizontal and rotational
components.
Also, choosing a large spring stiffness value will generate a
larger biarticular force, but the force itself will be applied for a
much shorter time. Increasingly larger spring stiffnesses will lead
to impulselike biarticular forces, whereas small stiffnesses will not
generate forces sufficient to improve vertical jumping. In the next
section we investigate in more detail the influence of the activation angle, and the tradeoffs involved in spring stiffness values, on
vertical jumping performance.
3
Dynamic Optimization
Let q = 共qa , q p兲 be the six-dimensional set of coordinates describing the kinematic configuration of the robot, where qa
苸 IR3 denotes the vector of actuated joints corresponding to the
hip, knee, and ankle joints, and q p 苸 IR3 denotes the three virtual
passive joints used to parametrize the position and orientation of
the torso. Denote by ␶a and ␶ p the torque 共or force兲 vectors, respectively, associated with qa and q p. Accounting for both contact
forces and biarticular muscle forces, the equations of motion then
assume the form
M共q兲
冉冊
q̈a
q̈ p
+ b共q,q̇兲 + JTc Fc + JTb Fb =
冉冊
␶a
␶p
共2兲
where M共q兲 苸 IR6⫻6 is the mass matrix, b 共q , q̇ 兲 苸 IR6 represents Coriolis, centrifugal, and gravity terms, Fc and Fb are the
wrenches corresponding to the contact and biarticular force, and
Jc and Jb are the respective constraint Jacobians associated with
the contact and biarticular wrenches.
In the inverse dynamics problem, given a prescribed motion of
the system in the form of inputs 兵q , q̇ , q̈a , ␶ p其, the objective is to
determine the output values 兵q̈ p , ␶a其. For this purpose we use the
FEBRUARY 2009, Vol. 1 / 011013-3
following hybrid dynamics algorithm 共for a general discussion of
hybrid dynamics algorithms for floating articulated body systems
and contact dynamics, see Refs. 关23–25兴兲:
Hybrid Dynamics Algorithm:
while t ⬍ t f do
forward iteration: i = 1 to n
Calculate displacement and velocity of link i
end
Calculate the biarticular force
backward iteration: i = n to 1
Calculate articulated body inertia of link i
Calculate bias force of link i
end
Check for contact between foot and ground
forward iteration: i = 1 to n
if i 苸 active then
Calculate the torque 共or force兲 of joint i
else if i 苸 passive then
Calculate acceleration of joint i
end
Integration: q̇ = q̇ + q̈ ⫻ timestep
if contact exists then
Formulate Linear Complementary Problem 共LCP兲
Solve LCP with Lemke’s solver 关26兴
forward iteration: i = 1 to n
Compensation by contact constraints:
if i 苸 active then
␶ = ␶ + ⌬␶ / timestep
else if i 苸 passive then
q̇ = q̇ + ⌬q̇
end
end
Integration: q = q + q̇ ⫻ timestep
t = t + timestep
end
Table 1 Robot model parameters
Link
Length
共mm兲
Center of mass
共y, z兲
Mass, inertia
共kg, kg mm2兲
Trunk
Thigh
Calf
Foot
a = 370
b = 254
c = 255, f = 30
d = 94, e = 200
共100, 788兲
共100, 476兲
共100, 221兲
共100, 47兲
共2.031, 20,366兲
共1.033, 7,440兲
共0.963, 7,487兲
共0.399, 1,057兲
cult. We therefore augment the objective function with appropriate
penalty terms that reflect these constraints. Joint angle limits are
easily bounded using B-spline properties, and joint velocity limits
can also be similarly constrained 关27兴. The resulting optimization
now only involves, besides the constrained dynamics equations,
purely linear equality and inequality constraints. For fixed spring
stiffness values, the objective function and constraints of Eq. 共3兲
can be expressed more concretely as
min J共P兲 = − Jperformance + Jtorque + Jposture
P
subject to
p1 = ¯ = p4 = qa0,
qគ a ⱕ pi ⱕ q̄a,
tf
Jtorque =
兺 共␶គ
a
− ␶a兲T共␶គ a − ␶a兲 + 共␶a − ¯␶a兲T共␶a − ¯␶a兲
共7兲
0
while Jposture is a postural stability criterion, e.g.,
subject to
Jposture = 储q p共t f 兲 − c1储2,
q̇គ a ⱕ q̇a ⱕ ¯q̇a
共3兲
␶p = 0
pzmp 苸 Pzmp
where pcom is the position of the robot center of mass, pzmp is the
zero moment point, and the overbar and underbar, respectively,
denote maximum and minimum values for the variables. To convert the above into a finite-dimensional optimization problem,
each active joint trajectory is parametrized using quintic
B-splines. The B-spline curve depends on the choice of basis functions Bi共t兲 and the control points P = 兵p1 , . . . , pm其, with pi 苸IRna,
where na is equal to the number of links in the leg. The active
joint trajectories then assume the form qa = qa共t , P兲, with
m
qa共t, P兲 =
共6兲
Jtorque is a penalty function bounding ␶a to within 关␶គ a¯␶a兴, e.g.,
k,qa
pcom 苸 Pcom,
共k − 1兲共pi − pi−1兲 ¯
ⱕ q̇a
ti+k−1 − ti
2
Jperformance = heightcom
max J共␶a, ␶ p,qa,q̇a,q̈a,q p,q̇ p,q̈ p,k兲
␶គ a ⱕ ␶a ⱕ ¯␶a,
共5兲
where Jperformance represents the performance index to be maximized; for maximal height vertical jumping it becomes
Given the above dynamic equations, our goal is to optimize
objective functions of the form
qគ a ⱕ qa ⱕ q̄a,
q̇គ a ⱕ
pm−3 = ¯ = pm = qaf
兺 B 共t兲p
i
i
共4兲
i=1
By parametrizing the trajectory in terms of B-splines, the original
optimal control reduces to a parameter optimization problem, in
which the optimization variables are the control points. Once the
optimal control point values are determined, trajectories for the
joint angle displacements, velocities, and accelerations can be derived straightforwardly from the B-spline basis functions.
The upper and lower bound constraints on the applied torques
␶a, when expressed in terms of the control points P, can become
highly nonlinear, making the numerical optimization more diffi011013-4 / Vol. 1, FEBRUARY 2009
or 储pcom共t f 兲 − c2储2
共8兲
The robot’s initial and final states 共as described by the displacement, velocity, and acceleration of the active joint angles兲 are
easily set by adjusting the B-spline control points as in Eq. 共5兲.
The posture can thus be checked by appropriately constraining the
joint information corresponding to the virtual passive joints. Postural stability can thus be ensured by checking the final kinematic
and dynamic state of the robot.
4
Numerical Studies of Vertical Jumping
This section considers numerical case studies for maximum
height vertical jumping. We determine the maximum height jumping motion subject to joint limits and constraints on the actuators
and also optimize jump height with respect to biarticular spring
stiffnesses and activation angles. All of the jumps are performed
once, starting from a stationary squatting configuration. The hybrid dynamics algorithm is implemented in MATLAB 7.0 and
MATLAB’s fmincon function with a stopping criterion of 兩Jk+1共P兲
− Jk共P兲兩 ⬍ ⑀, ⑀ = 10−4 is used in our optimization.
Table 1 lists the kinematic and inertial properties for our robot
model. For comparison purposes maximum height jumping motions are determined for the biarticular, monoarticular, and conventionally actuated robots, as shown in Fig. 4. The monoarticular
robot has a monoarticular spring actuation mechanism similar to
monoarticular muscles, connecting the calf and heel and extending over the ankle joint. The conventional legged robot does not
have any additional actuation mechanisms beyond direct actuation
of the joints.
Transactions of the ASME
Table 2 Boundary and other conditions used for vertical jump
optimization
Fig. 4 Robot model
The dominant factors affecting maximum height jumping performance include the input torque profiles, the spring stiffness,
and activation angles. The total time t f , provided it is chosen
sufficiently large, affects jump performance to a much lesser
extent—if t f is set large enough, and we select an initial motion
with no flight phase, then the flight phase of the optimized jump
will always have the same duration, with the jump simply being
initiated at a later time. On the other hand, if the chosen value of
Optimization conditions
共Hip, knee, ankle兲
Total jumping time
Integration sampling time
Friction coefficient
Number of variables
Performance index 共max兲
Joint angle limits 共rad兲
Joint velocity limits 共rad/s兲
Joint torque limits 共N m兲
Initial active joint values
Final active joint values
Boundary conditions
0.5 s
0.005 s
0.6
15
Jperformance = height2p 共t f 兲
com
q̄a = 共␲ , 0 , ␲兲, qគ a = −共␲ , ␲ , ␲兲
¯q̇a = 共20, 20, 20兲, q̇គ a = −q̇
¯a
¯␶a = 共20, 20, 20兲, ␶គ a = −¯␶a
qa0 = 共1.919, −2.094, 0.873兲
qaf = 共0.175, −0.262, −1.047兲
q̇a0 = q̈a0 = q̇af = q̈af = 0
t f is too small, then the optimized motion will necessarily be a fast
movement that does not achieve the maximum possible height.
Based on this observation, we repeatedly evaluate maximum
height jumps for the conventional robot with varying values of t f .
The objective function is chosen to be the maximum vertical
height of the robot’s center of mass. An initial motion in which the
foot pushes off the ground, but with no flight phase, is manually
constructed, as shown in Fig. 5共a兲. Each joint trajectory is parametrized using five B-spline control points. Our results indicate that
a value of t f = 0.5 s is sufficiently large enough for our purposes
and that larger values of t f have very little effect on the final form
of the optimal jump.
With this choice of t f , we determine optimal jumps for the three
cases 共biarticular, monoarticular, and conventional兲 with joint displacement and velocity limits and other various boundary conditions and actuator constraints, as shown in Table 2. The maximum
height jump for the biarticular case is achieved when the spring
stiffness is set to k = 7000 N / m and activation angle is set to ␪0
= −90 deg 共it should be noted that these values are determined by
obtaining maximum height jumps at discrete values of the spring
stiffness and activation angle and should only be considered to be
approximately optimal兲. For the monoarticular case, maximum
(a)
(b)
Fig. 5 Initial and optimized vertical jumps for the biarticular case
Journal of Mechanisms and Robotics
FEBRUARY 2009, Vol. 1 / 011013-5
Table 3 Maximum achievable vertical jump height for the three
cases
Jump height 共m兲
Conventional
Monoarticulated
Biarticulated
0.123
0.146
0.165
Table 4 Activation angle optimization for selected stiffnesses
„biarticular case…
Maximum jump height 共m兲
Activation angle 共deg兲
b
c
0.099
0.113
0.114
0.107
0.089
4.1 Activation Angle Optimization for Biarticular Leg.
Table 4 shows the results of the vertical jump optimization, for a
0
0.1
0.2
0.3
0.4
Knee joint torque (Nm)
i
vel
0.3
0.4
0.5
b
m
0.2
0
0.1
0.2
0.3
0.4
0.5
0.3
0.4
0.5
0.3
0.3
0.4
0.5
0.3
0.4
0.5
20
15
b
10
c
5
0
m
i
−5
0
0.1
0.2
10
5
0
−5
−15
c
i
b
m
−20
0.4
x: heel
o: toe
150
0.5
−25
Mono−/ Biarticular link force (N)
0.1
i
−10
i
c
0
−5
−10
Ankle joint torque (Nm)
0.2
c
m
0
25
c
m
0.1
b
5
−20
0.5
b
0
10
−15
175
Normal contact force (N)
0.075
0.076
0.151
0.130
0.112
−10
i
200
0
0.1
0.2
150
125
125
c
100
100
b
75
50
25
0
0.076
0.103
0.124
0.150
0.076
proximal hip and knee joint increase, while the torque in the ankle
joint decreases. This phenomenon can be traced to power transfer
from the proximal joints to the distal joints via the biarticular
mechanism. Biarticular actuation allows one to use smaller ankle
actuators, which can in turn reduce the mass of the calf and foot;
while not reported fully in this paper, preliminary studies seem to
suggest that transferring the mass of distal elements 共calf and foot兲
to the upper proximal elements 共torso兲 improves jumping performance even further.
Hip joint torque (Nm)
Hip joint velocity (rad/s)
m
Ankle joint velocity (rad/s)
5
2.5
0
−2.5
−5
−7.5
−10
−12.5
−15
k = 15,000
15
Knee joint velocity (rad/s)
15
12.5
10
7.5
5
2.5
0
−2.5
−5
k = 10,000
−70
−80
−90
−100
−110
height jumps are achieved for spring stiffness k = 5000 N / m and
activation angle ␪0 = −100 deg. The results of the optimization are
shown in Table 3 and Fig. 5. It can be seen that the biarticulated
robot reaches a maximum height that is nearly 13% higher than
that of the monoarticular robot.
Figure 6 shows the joint velocities and torques, normal contact
forces, and mono- and biarticular force trajectories for the optimized vertical jumps. It can be verified that the joint velocities
and torques stay within the prescribed limits. Observe that the
biarticulated case contains sudden and large increases in torque
values; such effects are much smaller in the conventional and
monoarticular cases. The torque profile discontinuities can be attributed to biarticular forces as well as changes in the contact
state.
Observe further that the maximum torque in the ankle joint is
lowest in the biarticular case. At the end of the stance phase, when
the biarticular element begins to exert a force, the torques in the
10
7.5
5
2.5
0
−2.5
−5
−7.5
−10
k = 5,000
0
0.1
0.2
0.3
0.4
75
50
c
25
0
b
0
0.5
Time (sec.)
0.1
0.2
Fig. 6 Joint velocities and torques, normal contact forces, and biarticular force trajectories for optimized vertical jumps „i: initial
motion, c: conventional robot, m: monoarticular legged robot, and b: biarticular legged robot…
011013-6 / Vol. 1, FEBRUARY 2009
Transactions of the ASME
Table 5 Stiffness optimization for selected activation angles
„biarticular case…
Maximum jump height 共m兲
Stiffness 共N/m兲
5,000
7,000
9,000
11,000
13,000
␪0 = −80 deg
␪0 = −90 deg
␪0 = −100 deg
0.103
0.124
0.149
0.076
0.095
0.124
0.165
0.124
0.125
0.079
0.150
0.111
0.107
0.098
0.113
selected range of stiffnesses, with respect to various choices of
activation angle. It can be observed that the optimal activation
angle values all lie in the range −90 deg to − 110 deg.
4.2 Stiffness Optimization for Biarticular Leg. Tables 4 and
5 show the results of the vertical jump optimization, for a selected
range of activation angles, with respect to various choices of
spring stiffness. The results indicate that for our particular biarticular model, a maximum jump height of 0.165 m is obtained for
a stiffness value of 7000 N / m and activation angle of −90 deg.
4.3 Activation Angle Optimization for Monoarticular Leg.
Table 6 shows the results of the vertical jump optimization, for a
selected range of spring stiffnesses, with respect to various
choices of activation angle for the monoarticular robot. Here too it
can be observed, like the biarticular case, that the optimal activation angles all lie in the range −90 deg to − 110 deg.
4.4 Stiffness Optimization for Monoarticular Leg. Tables 6
and 7 show the results of the jump optimization, for a selected
range of activation angles, with respect to various choices of
spring stiffnesses for the monoarticular robot. The results indicate
that a maximum jump height of 0.146 m is obtained for a stiffness
value of 5000 N / m and activation angle of −100 deg.
Comparing the results obtained for the biarticular and monoarticular cases, we can see that the maximum jumping heights
achieved for the biarticular case are far more diverse than for the
monoarticular case. While a greater height can be attained with
biarticular actuation 共0.165 m when k = 7000 and ␪0 = −90 deg兲,
the biarticular case is far more sensitive to variations in spring
stiffness and activation angle. Intuitively this makes sense, since
Table 6 Activation angle optimization for selected stiffnesses
„monoarticular case…
Maximum jump height 共m兲
Activation angle 共deg兲
k = 5,000
k = 10,000
k = 15,000
0.076
0.131
0.138
0.146
0.136
0.108
0.129
0.129
0.130
0.125
0.100
0.142
0.097
0.144
0.127
−70
−80
−90
−100
−110
Table 7 Stiffness optimization for selected activation angles
„monoarticular case…
Maximum jump height 共m兲
Stiffness 共N/m兲
5,000
7,000
9,000
11,000
13,000
␪0 = −80 deg
␪0 = −90 deg
␪0 = −100 deg
0.131
0.121
0.076
0.090
0.144
0.138
0.127
0.131
0.128
0.138
0.146
0.127
0.144
0.143
0.116
Journal of Mechanisms and Robotics
biarticular actuation simultaneously influences two joints, whereas
monoarticular actuation essentially has the effect of merely increasing the size of the ankle actuator.
5
Experiments With a Hardware Prototype
5.1 Design of the Hardware Prototype. To verify the results
of our numerical studies, we perform experiments with a hardware
prototype composed of four segments, representing the foot, calf,
thigh, and trunk. The link lengths and mass distributions are
scaled in accordance with the human lower extremity. The robot
has three active joints: the ankle joint, knee joint, and hip joint,
with an additional spring-actuated passive joint in the toes. The
main role of this spring-actuated passive toe joint is to absorb
impact forces during landing. This spring-attached passive joint
has a much smaller effect on the resulting robot motion compared
to the biarticular mechanism—the torsional force generated by the
small spring has very little effect on the overall motion compared
to the translational forces generated by the biarticular mechanism.
For this reason we do not include this additional passive toe joint
in our dynamic simulation model.
The biarticular actuation mechanism is attached between the
thigh and heel as in the simulation model. It is realized by a stiff
steel wire of diameter 1 mm that is connected to the heel via a
spring. The spring that is interchangeable-springs of varying stiffnesses 共determined by measurements兲 is available. At one end the
steel wire is wound on a 10 mm diameter grooved shaft that is
directly connected to a flat servomotor 共the spiral groove prevents
the wire from knotting itself兲. This motor functions solely as a
break: When the motor is not activated the biarticular link
smoothly follows the motion of the robot 共the force in the biarticular link is zero兲.
Each joint is activated by a servodrive mounted inside the
proximal segment with respect to the joint. The largest part of the
robot weight is in the trunk segment where, besides the motor that
activates the hip joint, a computer, a motion controller, all power
amplifiers, and an inclinometer are installed. To have a heavy
trunk is not desirable from a control point of view. However, it
can be shown that such a weight distribution can be beneficial for
improvement of fast and explosive movements such as vertical
jumping.
To achieve the high joint torques necessary for vertical jumps
with today’s lightweight electric motors, high gear ratios are usually necessary. However, high gear ratios are not desirable if one
wishes to maintain low friction in the joints. In addition, backdrivability of the joints cannot be achieved using gears with high
ratios. One means of simultaneously achieving high torques and
back-drivability is to use a motor with a low gear ratio and to
overload it for the short periods of time when high torques are
needed. Since explosive movements usually last for only a very
brief period of time, this overloading is usually not detrimental.
We use Maxon RE 40 dc servodrives with a stall torque of
2.5 N m and a weight of only 480 g. The gear ratio we choose is
1:8, which ensures both low friction and back-drivability of the
joint. Using this motor-gear combination, the maximal joint
torque is 20 N m, which can, based on the simulation results, be
sufficient to perform a 20 cm high vertical jump.
The prototype was constructed based on the computer aided
design 共CAD兲 model built using I-DEAS software. The real robotic
system and its comparison with the CAD model are shown in Fig.
7. All mechanical parts are made of aluminum, except for the
gears and axes, which are made of titanium. The total weight of
the robot is approximately 4.4 kg, and its height is 972 mm.
5.2 Experiments. Based on our earlier numerical results, we
perform vertical jumping experiments with the hardware prototype described above. Joint torques obtained in the optimization
study and shown in Fig. 6 are used as the desired joint torques for
the experiments. In the first case we perform squat vertical jumps
for a conventional robot that does not have a biarticular link. In
FEBRUARY 2009, Vol. 1 / 011013-7
Fig. 7 Hardware prototype and its CAD model
the second case we perform squat vertical jumps for a biarticular
legged robot. In this latter case the robot is equipped with a biarticular link that represents the gastrocnemius muscle in a human
leg.
The robot performs jumps on a force platform that is used to
measure ground reaction forces generated by the robot during the
push-off and landing phases of the vertical jump. By double integration of the measured forces that are proportional to the acceleration of the robot’s center of mass during the jump, the trajectory of the robot’s center of mass is determined during the vertical
jump experiments. By subtracting the height of the center of mass
just before the takeoff phase of the jump from the maximal height
of the center of mass during the flight phase of the jump, we
determine the relative height that the robot’s center of mass
reaches during the flight. In this way we obtain the height of the
jump of the robot without the biarticular link 共0.11 m兲 and the
height of the jump of the biarticular legged robot 共0.13 m兲. The
biarticular legged robot jumps approximately 18% higher than the
conventional robot.
If we compare these values with the simulation results of the
vertical jump optimization in Table 3 we can see that the height
obtained by the prototype is, in both cases, less than the height
achieved in the simulation study. For the conventional robot the
height obtained by the prototype is approximately 10% 共0.11 m兲
lower than the simulation result. For the biarticular legged robot
the difference is approximately 14% 共0.13 m兲. Considering the
mechanical issues that influence the motion of the real robotic
system and cannot be included into the modeling and simulation
of the vertical jump, these differences in jump height are not unreasonable. Among the most important mechanical issues are friction and backlash in the joints and gears, nonlinearity in the motors, the unknown transfer functions of the servocontrollers, and,
last but not least, the power-cables hanging from the robotic
system.
A typical sequence of the vertical jump performed by the biarticular legged robot is shown in Fig. 8. Images in the first row
show the robot between the beginning of the push-off phase of the
jump and the moment when the robot reached the maximal height
of the jump. In the second row of Fig. 8 the second part of the
flight phase and the landing phase is shown.
6
Conclusion
This paper has examined the extent to which biarticular mechanisms can improve the jumping performance of legged structures.
After developing detailed planar dynamic models of biarticular
legged structures, we formulate the ensuing optimization prob-
Fig. 8 A sequence of the biarticular legged robot performing a vertical jump experiment
011013-8 / Vol. 1, FEBRUARY 2009
Transactions of the ASME
lems for maximum height vertical jumping. Case studies involving a three degree-of-freedom biarticular leg structure are compared with those for an identical structure with conventional and
monoarticular actuation, and a quantitative comparison performed
for the three cases. Our results indicate that biarticular actuation
can noticeably improve jumping performance but that the choice
of activation angle and spring stiffness is critical.
Although the biarticular link we included in our study has the
most pronounced elastic properties among the various biarticular
muscles found in the human leg, it would be desirable to include
other biarticular muscles, such as the rectus femoris and the hamstrings, in the humanoid robot design. A special challenge would
be to design a humanoid lower extremity that includes all biarticular muscles of the human lower extremity, and to examine
their combined effect on vertical jump performance. Finally, future work on movement optimization with spatial dynamic models, involving more complex movements such as running high
jumps and hurdling, is underway.
Acknowledgment
J. Babic, D. Omrcen, and J. Lenarcic were supported by the
Slovenian Ministry of Higher Education, Science and Technology.
B. Lim and F. C. Park were supported in part by the KIST Center
for Intelligent Robotics and by IAMD-SNU.
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