University of Pennsylvania
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Department of Electrical & Systems Engineering
April 2000
A Brachiating Robot Controller
Jun Nakanishi
University of Michigan
Toshio Fukuda
Nagoya University
Daniel E. Koditschek
University of Pennsylvania, kod@seas.upenn.edu
Follow this and additional works at: htp://repository.upenn.edu/ese_papers
Recommended Citation
Jun Nakanishi, Toshio Fukuda, and Daniel E. Koditschek, "A Brachiating Robot Controller", . April 2000.
Copyright 2000 IEEE. Reprinted from IEEE Transactions on Robotics and Automation, Volume 16, Issue 2, April 2000, pages 109-123.
his material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the
University of Pennsylvania's products or services. Internal or personal use of this material is permited. However, permission to reprint/republish this
material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by
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NOTE: At the time of public, Daniel Koditschek was ailiated with the University of Michigan. Currently, he is a faculty member of the School of
Engineering at the University of Pennsylvania.
A Brachiating Robot Controller
Abstract
We report on our empirical studies of a new controller for a two-link brachiating robot. Motivated by the
pendulum-like motion of an ape's brachiation, we encode this task as the output of a "target dynamical
system." Numerical simulations indicate that the resulting controller solves a number of brachiation problems
that we term the "ladder," "swing-up," and "rope" problems. Preliminary analysis provides some explanation for
this success. he proposed controller is implemented on a physical system in our laboratory. he robot
achieves behaviors including "swing locomotion" and "swing up" and is capable of continuous locomotion
over several rungs of a ladder. We discuss a number of formal questions whose answers will be required to gain
a full understanding of the strengths and weaknesses of this approach.
Keywords
brachiation, dynamically dexterous robotics, limit cycles, locomotion, swing map, symmetry, target dynamics,
task encoding, underactuated system
Comments
Copyright 2000 IEEE. Reprinted from IEEE Transactions on Robotics and Automation, Volume 16, Issue 2,
April 2000, pages 109-123.
his material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way
imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal
use of this material is permited. However, permission to reprint/republish this material for advertising or
promotional purposes or for creating new collective works for resale or redistribution must be obtained from
the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all
provisions of the copyright laws protecting it.
NOTE: At the time of public, Daniel Koditschek was ailiated with the University of Michigan. Currently, he
is a faculty member of the School of Engineering at the University of Pennsylvania.
his journal article is available at ScholarlyCommons: htp://repository.upenn.edu/ese_papers/327
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
109
A Brachiating Robot Controller
Jun Nakanishi, Student Member, IEEE, Toshio Fukuda, Fellow, IEEE, and Daniel E. Koditschek, Member, IEEE
Abstract—We report on our empirical studies of a new
controller for a two-link brachiating robot. Motivated by the
pendulum-like motion of an ape’s brachiation, we encode this
task as the output of a “target dynamical system.” Numerical
simulations indicate that the resulting controller solves a number
of brachiation problems that we term the “ladder,” “swing-up,”
and “rope” problems. Preliminary analysis provides some explanation for this success. The proposed controller is implemented on
a physical system in our laboratory. The robot achieves behaviors
including “swing locomotion” and “swing up” and is capable of
continuous locomotion over several rungs of a ladder. We discuss
a number of formal questions whose answers will be required to
gain a full understanding of the strengths and weaknesses of this
approach.
Fig. 1. Brachiation of a gibbon: a picture taken from [2].
Index Terms—Brachiation, dynamically dexterous robotics,
limit cycles, locomotion, swing map, symmetry, target dynamics,
task encoding, underactuated system.
I. INTRODUCTION
A
BRACHIATING robot dynamically moves from handhold
to handhold like a long armed ape swinging its arms, as
depicted in Fig. 1. This paper concerns a simplified two-link
robot with one actuator at the elbow connecting two arms, each
of which has a gripper (see Fig. 2). Since the grippers cannot
impose torque on the handhold, this is an underactuated machine: it has fewer actuators than degrees of freedom. Designing
a brachiating controller for such a system is challenging since
the theory of underactuated mechanisms is not well established.
A growing number of robotics researchers have taken an interest in building machines that are required to interact dynamically with an otherwise unactuated environment in order to
achieve a designated task [1]. Brachiating robots take an interesting place within this larger category of robots that juggle,
bat, catch, hop, and walk. A brachiating and a legged locomotion system share the requirement of an oscillatory exchange
of kinetic energy and potential energy in the gravitational field.
Brachiation incurs the added problem of dexterous grasps: fumbles not only fail the task but incur a potentially disastrous fall
as well.
Manuscript received September 28, 1998; revised June 25, 1999. This paper
was recommended for publication by Associate Editor A. Bicchi and Editor
A. De Luca upon evaluation of the reviewers’ comments. The work of D. E.
Koditschek was supported in part by the National Science Foundation under
Grant IRI-9510673. This paper was presented in part at the 1997 IEEE International Conference on Robotics and Automation, Albuquerque, NM, April 1997,
and in part at the 1998 IEEE International Conference on Robotics and Automation, Leuven, Belgium, May 1998.
J. Nakanishi was with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2110 USA. He
is now with the Department of Micro System Engineering, Nagoya University,
Aichi 464-8603 Japan.
T. Fukuda is with the Center for Cooperative Research in Advanced Science
and Technology, Nagoya University, Aichi 464-8603, Japan.
D. E. Koditschek is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2110 USA.
Publisher Item Identifier S 1042-296X(00)03576-X.
Fig. 2. A two-link brachiating robot.
Problems of dexterous manipulation have given rise to a
growing literature concerned with explicit manipulation of an
environment’s kinetic as well as potential energy. Arguably,
the first great success in this domain must be attributed to
Andersson [3] whose ping pong playing robot developed
more than a decade ago was capable of beating many humans. The control architecture relied on meticulously crafted
aerodynamics and manipulator dynamics models recruited by
an elaborate black board style artificial intelligence decision
process. Saito et al. [4] first introduced to the robotics literature
the brachiation control problem. In fact, they built the physical
two-link brachiating robot that we use in this study. Their work
demonstrated empirically the validity of learning a feedforward
torque signal to effect successful underactuated control of
dynamically dexterous maneuvers [5]. Notwithstanding the
distinct advantages—no dynamical model is necessary—this
approach requires a long training period (about 200 experiments
with the physical robot) to generate a successful maneuver for
each configuration of the robot and given distance between the
branches. More recently, Mason and Lynch [6] have brought a
promising new view to the problem of dynamic underactuated
nonprehensile manipulation. Following a careful controllability
analysis, they have designed open-loop control laws for a
one-degree-of-freedom robot which performs dynamic tasks
such as snatching, rolling, throwing, and catching. Maneuvers
are accomplished by numerically generated solutions to appropriate optimal controls problems defined with respect to the
carefully modeled plant.
1042–296X/00$10.00 © 2000 IEEE
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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
Raibert’s landmark success in legged locomotion [7] represents the most important influence on the present work. Specifically, we adopt his appeal to biomechanically inspired design,
to the encoding of a dynamical task via a lower dimensional
target, and his systematic use of reverse time symmetry. The recourse to a parametrized family of feedback laws yields a far
more parsimonious action set than any of the approaches just
described, reducing the effort in commanding any specific task
to the selection of one or two key parameters. While this approach requires a calibrated dynamical model (a failing relative
to the pure learning approach of Saito et al.), the reliance on
feedback affords a far simpler model than required by Andersson or even Mason and Lynch. The Appendix outlines some
of the most important ways in which our physical apparatus departs significantly from the simple modeling assumptions built
into the controller. We describe in the body of the paper the relative ease with which control parameter adjustments can accommodate these discrepancies. Intriguing work by Atkeson and
Schaal on batting and juggling suggests that learning methods
may be used to develop similarly parsimonious stabilized [8]
dexterous maneuvers, but these considerations lie outside the
scope of the paper. Unlike the approaches of Andersson or Saito
et al.Raibert’s emphasis on solving tasks by recruiting natural
orbits offers a general foundation upon which formal insight
may be forthcoming across a broad class of robots and tasks
and scaling up to higher degrees of freedom. Unlike the optimal
control theoretic approach of Mason and Lynch, Raibert’s emphasis on regulation of energy toward those orbits yields controllers that are considerably less hostage to the exactitude of
the dynamical models they are built from.
In the present paper, we are most concerned with articulating a
version of the Raibert style approach and demonstrating its efficacy via physical experiments on a laboratory robot. Apart from
working out the details of the reverse time symmetry that govern
the choice of the lower dimensional target, we offer no stability
analysis beyond the kind of extensive numerical investigation
that characterizes much of the literature on Raibert-style hopping
robots [9], [10]. However, in a series of papers, the third author
and his colleagues have pursued a number of analytical studies of
simple hopping machines—revolute–prismatic (RP) kinematic
chainsdirectlymodeledonRaibert’smachines—addressingsuch
questions as regulation of hopping height [11], forward velocity
[12], and duty factor [13]. Moreover, we have shown how to
adapt these principles to the prismatic–prismatic kinematics of
batting [14], with generalization to far higher degree-of-freedom
settings [15], [16] and dynamical obstacle avoidance [17] problems as well. The brachiation problem represents in some sense
the most difficult setting in which to extend this body of ideas
since the underlying revolute–revolute (RR) kinematics induces
the most strongly coupled dynamics. There is a cyclic coordinate in prismatic–prismatic batting and in the “no stance gravity”
version of RP hopping as well [10], [12], [13]. The resulting integral invariants lay the basis for closed-form representations of
the Poincaré maps on which the existing stability analyses are
founded. In contrast, beyond energy conservation, there is no
hope of finding additional integrals in the RR problem. Generally speaking, the closed-loop systems that result from our controllers have the character of the restricted three-body problem
that led Poincaré to discover “chaos” [18]. It is not even clear to
us whether the recourse to closed-form approximants [19] that
have proven so useful [20] in the RP setting will be viable in
the brachiation task domain. Nevertheless, these ideas work and
may likely generalize across the brachiation domain similarly
to the manner in which they apply to various legged locomotion
and juggling tasks involving varying degrees of freedom.
Amidst the large and growing controls literature on underactuated mechanisms, this work is closest in method to Spong
and his colleagues’ studies of the “acrobot” [21]. They considered the swing-up problem of an underactuated system similar
to the two-link brachiating robot that we treat in this paper. Their
control algorithm pumps energy to the system in an instance of
Spong’s more general notion of partial feedback linearization
[22] directed toward achieving a kind of target dynamics whose
motions solve the swing-up problem. The controller that we introduce here bears many similarities to this, although the more
extended problems of slow brachiation require a rather differently conceived notion of target dynamics, and we are swinging
up to an (unstabilizable) handhold (refer to footnote 3)—not the
vertical equilibrium position as is common in much of the related literature. Thus, equilibrium motions (i.e., hybrid limit cycles) rather than equilibrium points are the regulated goal sets
in our problem.
II. PROBLEM SETUP
A. Physical Apparatus
Fig. 3 depicts the configuration of our experimental setup.
The length of each arm is 0.5 m and the total weight of the robot
is about 4.8 kg.
In Saito’s original version of this experimental setup [4], a
personal computer equipped with input/ouput (I/O) devices was
used to control the robot. We have replaced it with a VME
bus board computer, MVME 167 (Motorola, CPU MC68040,
33 MHz), with a real-time operating system, VxWorks 5.1, and
VME bus based I/O devices. The control law is evaluated exactly at a rate of 500 Hz.
The elbow joint is actuated by two DC motors with harmonic
gears (Harmonic Drive Systems, RH-14-6002). The stator of
each motor is fixed to a link, and their rotor shafts are directly
connected to each other. As a consequence, we can achieve a
total rotational speed at the elbow, which is two times faster
than the case where there is only one motor. This was necessary
since the rated rotational speed of these motors is 360 /s, while
we require that the rotational speed of the elbow be grater than
600 /s. An additional benefit of the symmetrical structure of this
design is better overall balance in the mechanism. Each gripper
is equipped with a DC motor that opens and closes it.
The angle of the first joint is measured by integrating its angular velocity, which is in turn obtained through a gyro (Murata,
ENV-05S) attached to the arm. The angle of the second joint and
the opening angle of the gripper are measured using optical encoders.
B. Model
The dynamical equations used to model the robot take the
form of a standard two-link planar manipulator as depicted in
NAKANISHI et al.: A BRACHIATING ROBOT CONTROLLER
111
Fig. 3. The experimental setup of the two-link brachiating robot.
Fig. 4. The mechanical model of the two-link brachiating robot used in this paper.
sent to a driver as
, where
is a positive constant.2
We use a lossless version of the model [
in (1)] for
the development of the controller and its analysis, but introduce
the losses in simulation.
the table in Fig. 4
C. Problem Statement
(1)
,
,1
is the
where
inertia matrix, is the Coriolis/centrifugal vector, and is the
gravity vector. and denote the viscous and coulomb friction
coefficient matrices, respectively. We assume that the elbow actuator produces torque proportional to a voltage command
X
X
X !Y)
1Throughout the paper, we shall use the tangent notation of Abraham and
is a manifold, then T
denotes the tangent vector space
Marsden [23]. If
at some x
and T =
T
denotes the tangent bundle. Moreover
T x will denote some point in T
and h :
Th : T
T
is
the derived “tangent map,” in coordinates, T h = (h(x); Dh(x)x_ ) where D
denotes the Jacobian.
2X
X
X
X
X! Y
Brachiation—arboreal locomotion via arms swinging hand
over hand through the trees—is a form of locomotion unique
to apes. Most commonly, the animals engage in “slow brachiation,” traveling at about the speed of the average human walking.
But when excited or frightened, apes can plunge through the
forest canopy at astonishing speeds, sometimes covering 30 ft
or more in a single jump without a break in “stride” (fast brachiation, ricocheting) [24]. In our reading of the biomechanics lit2To identify the dynamical parameters corresponding to the robot’s Lagrangian dynamics, we resorted to a rather simple identification procedure,
where the inertia parameters are obtained either by direct measurement or
from the manufacturer’s data, and the preliminary estimate of the friction
coefficients are obtained from the natural dissipation of the system.
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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
erature, we distinguish three variants of brachiation that we will
refer to in this paper as follows:
• ladder and swing-up problem;
• rope problem;
• leap problem.
The first arises when an ape transfers from one branch to another and controlling the arm position at next capture represents
the central task requirement. A robotics version of this problem
has been previously introduced to the literature by Saito [4],
[5]. They presented the robot that we consider here with a set
of discrete evenly spaced bars and the requirement to swing up
from rest, catch the next bar, and then swing from bar to bar
by pumping up energy in a suitable fashion. In our view, this
problem seems as much akin to that of throwing and catching
as to locomotion.
The second problem arises from brachiation along a continuum
of handholds—a branch or a rope—that seems most closely analogous to human walking. Since grasps are afforded at will, the
resulting freedom ofplacementcan be exploited to achieve a specified forward rate of progress. This is not possible for a two-degree-of-freedom machine on a ladder whose forward velocity is
essentially determined by the distance between the bars and its
own kinematics. To our knowledge, all previous work on robot
brachiationhasaddressedonly theladderandswing-upproblems.
We have found no studies concerned with the control of forward
velocity. We find the results of the rope problem in Section V
to be somewhat analogous to Schwind’s study on the forward
velocity control of simplified hopping robots [12].
The third problem arises in the context of fast brachiation
where the next branch is far out of reach, and the task cannot
be accomplished without a large initial velocity and a significant component of free flight. Solving this problem involves not
merely a swing phase, but a nonholonomic flight, as well where
the angular momentum of the system is conserved. Roughly
analogous to running quickly through a field of boulders, apes
can apparently achieve this movement with great regularity and
ease. We consider this a fascinating and challenging problem to
be addressed when the two previous simpler problems are better
understood.
We propose a control algorithm which is effective for the
first two “slow brachiation” problems—i.e., the ladder and
swing-up, and rope problems—inspired by our reading of the
biomechanics literature. The proposed controller is experimentally implemented on the physical two-link brachiating
robot described in Section II-A. Our experimental success
encompasses a number of brachiation tasks starting from a
variety of different initial hand positions. We have achieved
swing locomotion in the ladder problem, where both hands
are initially on the ladder, various swing-up behaviors from
a suspended posture, where only one hand is initially on the
ladder, and repeated locomotion over several rungs, where the
robot starts with either one or both hands on the ladder.
III. TASK ENCODING VIA TARGET DYNAMICS
forced to have the characteristic of a chosen target dynamics.
Thus, instead of tracking an exogenously designed reference trajectory, we force the system to generate and track its own reference motion.
Suppose a plant
(2)
(3)
is I/O linearizable. That is, given
(4)
if there can be found an implicit function such that for every
and
, then
(5)
implies
(6)
one calls (5) an I/O linearizing inverse controller in the sense
.
that
It is traditional in the underactuated robot control literature
to use the linearizing feedback (5) to force to track some ref. In the present article, we find it more
erence trajectory
useful to mimic a reference dynamical system
(7)
This behavior obtains by substituting
feedback law
for
in (5), yielding the
(8)
B. Target Dynamics and Its Associated Controller
According to the biomechanics literature [25], slow brachiation of apes resembles the motion of a pendulum. Although the
ape’s moment of inertia varies during the swing according to its
change of posture, the motion of a simplified pendulum gives
a fairly good approximation. Motivated by this pendulum-like
motion, we encode the robot brachiation task in terms of the harmonic oscillator
(9)
where , the natural frequency of the virtual pendulum, will
play the role of the task level control parameter in the sequel.
(9) serves as the target
Supporting this role, the function
dynamical system (7) for all the empirical work reported in this
paper.
Simulations suggest that any lossless mechanical oscillator of
the form
(10)
A. I/O Linearization and Target Dynamics
The notion of target dynamics represents a slight variant on
standard techniques of plant inversion. A system is inverted then
, an “artificial potential”
can encode brachiation when
function, is even and convex in the region of operation. For
NAKANISHI et al.: A BRACHIATING ROBOT CONTROLLER
113
where
is each element of
. Notice that
(15)
Fig. 5. Change of coordinates from RR to RP. We control to follow the
dynamics !
using a target dynamics controller.
+
=0
future reference, let
be the “pseudo” mechanical energy
defined by this oscillator,
(11)
The choice of output map (3) seems to be much more critical,
since it prescribes the combination of states that will be forced
to exhibit the selected target dynamics (10). In Fig. 5, we illustrate the local change of coordinates from joint space to polar
coordinates on
(12)
Intuitively, pursuing the analogy arising from biomechanical observation [25], the simplest pendulum to be found in the underlying RR kinematic chain obtains from its polar coordinate
“angle,” , motivating the choice
(13)
With these choices in place, the controller synthesis is formally complete. In summary, the virtual pendulum angle is
. Namely,
forced to follow the target dynamics
,
,
in (2), and
identify
and apply the control law formulated in (8) with respect to the target (10)
is satisfied in the pari.e., the invertibility condition of
ticular setting with the parameter values shown in the table in
Fig. 4.
In Section IV-A-1, we make the formal observation (Proposition 3) that any even potential together with an appropriately
“odd” choice of output map (3) will support the Raibert-style
reverse time symmetry [7] essential to the efficacy of our task
encoding. Indeed, in our numerical investigations, we have had
good experience with many choices for the artificial potential
function. However, in our empirical work, we have found it particularly convenient to adopt the specific Hooke’s Law potential
, (9), for two reasons.
First, the elliptic integral
(16)
is solvable in closed form using elementary functions when is
a Hooke’s law spring. This closed-form expression significantly
simplifies the computational effort incurred by the root finding
procedure of (31) required to tune the “natural frequency” in
the ladder and rope problems.
Second, numerical study addressing the swing-up problem
reveals that the “stiffness” (the second derivative of the potential ) plays an important role for reasons we do not yet understand well. Specifically, we require not only positive stiffness (i.e., convex potentials ), but find that some “stiffness
margin” profile is key to effective swing-up behavior. Generor
ally speaking, “hard” spring laws such as
work nicely. In contrast, consider
the effective torsional spring potential introduced by a gravity
, its stiffness beloaded simple pendulum,
comes zero at the boundary the domain of operation. Such “soft”
springs (i.e., potentials whose second derivative stiffness functions are not bounded away from zero over the domain of operation) characteristically result in “out of phase” swing ups that
fail the task. While hard springs work nicely in simulation, they
typically incur unavailably large torques. The Hooke’s Law potential enjoys benefits of positive stiffness and realistic torque
requirements.
IV. LADDER AND SWING-UP PROBLEM
A. Ladder Problem
(14)
In this section, the target dynamics method is applied to the
ladder problem. We show how a symmetry property of an appropriately chosen target system—(10) in the present case—can
solve this problem.
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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
1) Neutral Orbits : This section follows closely the ideas
originally developed in [12] and [13]. We discuss a reverse time
symmetry inherent in the brachiating robot’s dynamics. Here,
we are interested in orbits whose forward time motions from
the bottom states are a horizontal reflection of their backward
time motion from the same initial condition.
First, we show that the natural dynamics of the two-link
brachiating robot admit this reverse time symmetry . Then,
we give a condition under which feedback laws result in
closed loops that still admit . Last, following Raibert [7], we
introduce the notion of the neutral orbits of the symmetry and
show how they may be used to solve the ladder problem. In the
sequel, we will denote the integral curve of a vector field by
the notation .
admits a reverse time symmetry
Definition 1:
if and only if
.
Note that when is linear, after taking time derivatives, this
.
definition might be equivalently stated as
In this paper, we are concerned specifically with the symmetry
operator
Fig. 6. A ceiling configuration. The ceiling is parametrized by the distance
between the grippers d. A left branch c (d) and right branch c (d) are defined
in this manner.
2) Ceiling , and its Neutral Orbits: Define the “ceiling”
(21)
(17)
denotes the 2 2 identity matrix). When admits
(where
in (17), there exist orbits integrated forward in time from some
initial conditions, which are reflections of orbits backward in
time from the same initial conditions odd in angles and even in
and
.
velocities, i.e.,
, deNow, supposing we have chosen a feedback law
note the closed-loop dynamics of the robot as
(18)
admits . A straightforSay that “admits ” if and only if
ward calculation shows [26].
admits as in
Proposition 2: The closed-loop dynamics
if and only if
has
(17), i.e.,
.
the property
A similarly straightforward computation shows [26].
in (10)
Proposition 3: If the artificial potential function
, and if
, i.e., a smooth scalar
is even,
, then the
valued function, has the property
feedback law (14), , admits .
Define the fixed points of the symmetry to be
(19)
Fix
In the present case, i.e., for
in (17), note that
to be those configurations where the hand of the robot reaches
as depicted in Fig. 6.3
the height
In general, arms which drop from the ceiling, whether under
active torque control or not, will not pass through a bottom state
— they will not trace out a neutral orbit. In the special case that a
ceiling handhold (at zero velocity) lies on a neutral orbit, the arm
will drop, pass through a bottom state, and swing back up again
through an exactly mirrored trajectory that ends in the ceiling
on the opposite side of the handhold, the same distance away.
This fact is stated precisely in Proposition 4. Our problem now
is to find a virtual frequency matched to the desired distance
that renders this ceiling state neutral. This matter can, in turn, be
cast as a numerical root finding problem (31) as we now show.
The ceiling can be parameterized by two branches
(22)
of the maps,
(23)
In the sequel, we will be particularly interested in initial conditions of (18) originating in the zero velocity sections of the
. Now, note that
since
ceiling that we denote
Fix
(24)
Define the set of “neutral orbits” to be the integral curves which
go through the fixed point set
Fix
admits
and if
Proposition 4: If a feedback law
, then there can be found a time
such that for
, we have
(20)
Note that a neutral orbit has a symmetry property about its fixed
Fix , then
point—namely, if
(25)
3Notice that this handhold state, “ceiling,” cannot be made to be an equilibrium state under the influence of the gravity since we cannot find such that
(T q; )
0 in (1) when k (q ) = 0.
L
6
NAKANISHI et al.: A BRACHIATING ROBOT CONTROLLER
115
i.e., a time at which the left branch at zero velocity in the ceiling
reaches the right branch in the ceiling also at zero velocity.
Proof: By the definition of , there can be found a time
at which
Fix
(26)
Therefore
(27)
Applying the symmetry , we have
[from (24)]
(28)
But
[from (27)]
(29)
^ (d ). Target dynamics controller, ,
Fig. 7. Numerical approximation ! =
^ , that is designed to locate neutral orbits
is tuned according to this mapping,
originating in the ceiling.
hence
(30)
Thus, we conclude that any feedback law , which admits ,
solves the ladder problem, assuming we can find a such that
. Note that finding such a ceiling point
requires solving the equation
(31)
, for and
simultaneously. Of course,
where
in general, solving this equation is quite difficult. It requires a
two-dimensional “root finding” procedure for a function whose
evaluation entails integrating the dynamics (1).
3) Application of Target Dynamics: The feedback law
(14) arising from the target (9) and output map (13) admits
since Proposition 3 applies. The special target, (9), enjoys a very
nice property relative to the difficult root finding problem (31).
is given by
Namely, using this control algorithm,
(32)
In general, as discussed in Section I, we can expect no
or
, and we compute an
closed-form expression for
estimate
using a standard numerical scalar root finding
method (i.e., the “false position” or “secant” method) whose
convergence properties are well known [27].
We plot in Fig. 7 a particular instance of for the case where
the robot parameters are shown in the table in Fig. 4. is tuned
according to this mapping. We will use these parameter values
throughout the sequel for the sake of comparison between this
and subsequent figures.
for this
4) Simulation: Consider the case
parameter set above. The initial condition of the robot is
. From the numerical solution
. In this simulation,
depicted in Fig. 7,
the lossy model (1) is used and friction terms are added in the
inverse dynamics controller (14). Fig. 8 shows the resulting
movement of the robot. The joint trajectories and the voltage
command to the motor driver are shown in Fig. 9.
Note that the closed-loop dynamics of the system does not
strictly admit a reverse time symmetry discussed above, since
the uncanceled friction terms of the first joint enter the dynamics
of the unactuated degree of freedom. However, under these circumstances, numerical simulation shown in Figs. 8 and 9 suggests that the desired brachiation can be achieved. In practice,
we have found that model mismatch seems to affect behavior of
the physical robot rather considerably as discussed in the Appendix.
B. Swing-Up Problem
and we need merely solve (31) for . More formally,
we seek an implicit function
such that
. Of course, we are more likely in
practice to take an interest in tuning as a function of a desired
. Thus, we are most interested in determining
(33)
The swing-up problem entails pumping up from the suspended posture at rest and catching the next bar. In order to
achieve this task, it is necessary not only to add energy in
a suitable fashion but also to control the arm position at the
capture of the next target bar. This suggests the introduction of
a stable limit cycle to the system with suitable magnitude and
relative phase.
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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
Fig. 8. Movement of the robot (simulation). The symmetry properties of the
neutral orbit from the ceiling solves the ladder problem.
1) Swing-Up Controller: As we have mentioned, swing up
requires energy pumping in a suitable fashion. To achieve this,
we modify the target dynamics (9) as
(34)
as defined in (13),
is a posiwhere
is the desired
tive constant, is “pseudoenergy” (11), and
pseudoenergy level.
To achieve this target dynamics, the control law is formulated
as
(35)
Now, consider the time derivative of
along the motion
(36)
, then the pseudo-energy decreases, and if
, then increases. Therefore, will converge to the desired
level
eventually. This implies that the target dynamics has a
stable limit cycle whose trajectory is characterized by
on the phase plane of
.
Although the system’s motion projected onto the target subspace must exhibit the desired limit cycle, the swing-up task
still requires a coordination of the full four-dimensional robot
trajectory in order to guarantee the arm extension is correct at
the moment the “virtual pendulum” angle reaches the ceiling.
But for this task, in contrast to the ladder problem, we can make
no assumption regarding the robot’s initial conditions—the arm
If
might start out in any configuration (typically, at small velocity)
near the bottom state following a small “kick” of torque administered to break out of that passively stable equilibrium state.
In particular, there are no comparable means of appeal to a
tuned symmetry as before. Unfortunately, no general method is
presently known to stabilize a highly nonlinear underactuated
mechanical system around a specific (necessarily nonequilibrium) orbit. Hence, we are reduced to empirical tuning of the
pumping gain
in order to find task worthy values.
on the target system is quite straightforThe effect of
ward—(36) shows that it sets the time constant for convergence
to the specified lower dimensional target limit cycle, hence,
higher gains must result in quicker approach to the “virtual” steady state behavior. In contrast, the four-dimensional
closed-loop system can be expected to exhibit extremely
complex (RR kinematic chains are “chaotic”) orbits. Certainly, there is no reason to expect limit cycles from the true
four-dimensional system as its orbits accumulate toward the
three-dimensional limit set. Empirically, however, we find
, wherein the system’s
there are favorable regimes for small
motion tends toward “near-neutral” orbits resulting in very
slow swing up—that is, a relative phasing between the virtual
angle and extension that brings the gripper to the next handhold
at an acceptably small velocity. Fortunately, a numerical one
parameter search is quite simple to implement. We have found
it relatively straightforward to achieve effective swing-up
controllers both in simulation as well as in the lab by simply
incrementing the value of this pseudo-energy pumping gain
(starting from very small values), recording the favorable values
as they recur, and then running with a favorable value whose
associated pseudo-energy convergence rate is fast enough to
yield a viable handhold over three or four swings.
2) Simulation: What follows here is a presentation of different swing-up behaviors resulting from changes in the rate
. The next bar is loof energy pumping as characterized by
cated at the distance
, and we choose
according to the mapping depicted in Fig. 7. Since the
bottom condition with zero velocity is an equilibrium state of
the closed-loop dynamics, we give small initial velocity to the
second joint to initiate the swing motion in the desired direction.
In the following simulations, we assume that the robot can catch
the bar when it comes very close to the desired handhold.
Fig. 10 shows the typical motion obtained for a favorable
. In contrast, Fig. 11 exhibits the typical
small setting of
“chaotic” motion obtained for a large unfavorable setting of .
continues to increase, some particular choices may result
As
in robot trajectories which go through the next bar’s position
after a few swings as depicted in Fig. 12.
V. ROPE PROBLEM
In this section, we consider the rope problem: brachiation
along a continuum of handholds such as afforded by a branch or
a rope. First, the average horizontal velocity is characterized as a
result of the application of the target dynamics controller introduced above. Then, we consider the regulation of horizontal
velocity using this controller. An associated numerical “swing
NAKANISHI et al.: A BRACHIATING ROBOT CONTROLLER
(a)
Fig. 9.
117
(b)
Simulation results of the ladder problem. (a) Joint trajectories ( —solid, —dashed). (b) Voltage command to the motor driver.
(a)
(b)
Fig. 10. Slow swing-up behavior (simulation; K = 0:05). (a) Joint trajectories ( —solid, —dashed). (b) Voltage command to the motor driver. The robot
captures the bar at t = 19:2 s. Small choice of K achieves a near neutral orbit in the long time swing behavior.
(a)
(b)
Fig. 11. “Chaotic” swing behavior (simulation; K = 0:18). (a) Joint trajectories ( —solid, —dashed). (b) Voltage command to the motor driver. “Chaotic”
swing motion is observed when we let the robot keep swinging with large choice of K .
map” suggests that we indeed can achieve good local regulation
of the forward velocity through the target dynamics method.
A. Iterated Ladder Trajectory Induces a Horizontal Velocity
Supposing that the robot starts in the ceiling with zero velocity, then it must end in the ceiling under the target dynamics
. However,
controller, since follows the dynamics
, then the trajectory
if and are not “matched” as
,
ends in the tangent of the right branch of the ceiling,
but
and
. In [26], we present numerical
with
evidence suggesting that when
for small , then at
is also small. Assuming that any such small nonzero
velocity is “killed” in the ceiling, brachiation may be iterated by
opening and closing the grippers at left and right ends in the appropriately coordinated manner. Namely, imagine that the robot
concludes the swing by grasping firmly with its gripper the next
handhold in the ceiling and thereby damps out the remaining
kinetic energy. Imagine at the same instant that it releases the
gripper clutching the previous handhold and thereby begins the
next swing. We will call such a maneuver the iterated ladder trajectory (ILT).
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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
(a)
(b)
Fig. 12. Fast swing-up behavior (simulation; K = 0:328). (a) Joint trajectories ( —solid, —dashed). (b) Voltage command to the motor driver. The robot
captures the bar at t = 2:78 s.
It is natural to inquire as to how quickly horizontal progress
can be made along the ladder in doing so. When a gripper moves
in the course of the ladder trajectory, and if the
a distance
trajectory is immediately repeated, as described above, then the
each swing, hence, its
body will also move a distance of
average horizontal velocity will be
(37)
according to the discussion in Section IV-A. Numerical study
using in Section IV-A-3 and the robot parameters in the table
is
in Fig. 4 reveal that the ceiling-to-velocity map
momotone increasing.
B. Inverting the Ceiling-to-Velocity Map
Consider now the task of obtaining the desired forward veof brachiation. If is invertible, then
locity
and we can tune in the target dynamics as
(38)
to achieve a desired
where
is again the mapping (33).
Fig. 13. Swing map, , (solid) and identity (dashed) for the case
^ (d ), and the robot
h_ = 0:7; d = 0:6496; ! = 3:385 where ! =
parameters are in the table in Fig. 4. This swing map has an attracting fixed
point at d .
because of the symmetry properties of the
where
neutral orbits, demonstrated in Proposition 4.
Define a projection , from the ceiling’s tangents into the
zero velocity section
C. Horizontal Velocity Regulation
Consider the ceiling condition with zero velocity
(39)
(44)
Define the maps
and their inverses
as
(40)
(41)
A target dynamics controller (9) gives
where
where
since
then
follows the dynamics
In other words, is a map that “kills” any velocity in the ceiling.
We introduce this projection to model the ILT maneuver in cases
for
.4
when
We now have from (42)
. Now, if
(42)
,
(43)
hence, we may define a “swing map,”
of
into itself
(45)
, as a transformation
(46)
4The interested reader may consult [26] to see that magnitude of leftover velocity is not prohibitively large.
NAKANISHI et al.: A BRACHIATING ROBOT CONTROLLER
119
Fig. 14. Brachiation along the bar with the initial condition (49). Convergence of d
yields convergence to the desired average velocity, h_ .
Note that if
, then
(47)
that is, is a fixed point of the appropriately tuned swing map.
It is now clear that the dynamics of the ILT maneuver can
. Physically,
be modeled by the iterates of this swing map
suppose we iterate by setting the next initial condition in the
ceiling to be
(48)
This yields a discrete dynamical system governed by the iterates
of
Numerical evidence suggests that the iterated dynamics con, when is in the neighborhood
verges,
as depicted in Fig. 13 (local asymptotic stability of the
of
fixed point ). We plot the swing map calculated numerically
for the case where
, and the
robot parameters in the table in Fig. 4 are used (see Fig. 13).
D. Simulation
Suppose we would like to achieve the desired horizontal ve. The parameters shown in the table in Fig. 4 are
locity
.
used. For this case, is obtained as
Consider the case where we select
, but the initial
is wrong. We present simulation results with the initial condition
where
(49)
in Fig. 14. As the numerical swing map of (13) suggests, we
nevertheless achieve asymptotically the desired locomotion, i.e.,
.
With the assumption that any velocity in the ceiling is killed,
is fairly large
the size of the domain of attraction to under
according to the numerical evidence shown in Fig. 13.
VI. EXPERIMENTS
We present results of the experimental implementation of the
proposed controller in order to validate our control strategy.
The proposed algorithm is applied to the ladder and swing-up
!d
is illustrated as the numerical swing map (Fig. 13) indicates, and this
problem, however, the rope problem cannot be experimentally
carried out with the robot considered in this paper because of
the structure of the gripper.
A. Ladder Problem
In the experimental setting, the next bar is located at a distance
m. For this experimental setting, is tuned to
according to the mapping depicted in Fig. 7.
be
Early attempts to implement the controller (14) failed.
Swing motion close to the desired behavior was achieved,
but the gripper did not come close enough to the target
bar to catch it.5 A central component contributing to these
failures was the model mismatch. Therefore, we tuned the
parameters of the model manually. Some experience is helpful
in the refinement of these parameters: We choose to use
, and
instead of
the values in the table in Fig. 4 for the ladder problem. This
assignment yielded success.
A typical motion of the physical robot is plotted in Fig. 15,
while the joint trajectories and the voltage commands sent to the
driver are shown in Fig. 16. The mean locomotion time of ten
s error,6 which is very close to its
runs is 0.973 s with
analytically calculated value
s.
Notice that the symmetry of the neutral orbit is not perfectly
achieved in the motion of the robot. We discuss the discrepancy between the simulation and experimental results in the Appendix.
B. Swing-Up Problem
What follows is a presentation of the experimental results
of the different swing-up behaviors resulting from changes in
. The disthe rate of energy pumping, as characterized by
tance between the bars is 0.6 m. We consider three cases where
and
. These parameters are chosen manually based on our experience in numerical simulation and experiments. In order to successfully swing up, we have found
it necessary to slightly modify the desired pseudo-energy level
and some of the model parameters. The nominal pseudo-energy
so that the gripper
is chosen to be
5In practice, we need to consider the time lag in opening the gripper when the
robot initiates locomotion, something not taken into account in the analytical
work. It takes approximately 0.08–0.1 s to release the bar after the command to
open the gripper is sent. Empirically, we have observed that this time affects the
swing behavior of the robot. As a result, we choose to send the open command
of the gripper 0.08 s before the target dynamics controller is turned on.
6In the sequel, the error refers to the maximum deviation from the mean.
120
Fig. 15.
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
+
Movement of the robot (experiment). The target bar is located at a distance of 0.6 m marked by the “ .”
(a)
Fig. 16.
(b)
The experimental results of the ladder problem: left—joint trajectories and right—voltage command to the motor driver.
reaches the height of the bar, which corresponds to the condiwhen
. In the initial attempts using the nomtion
inal pseudo-energy level, we found that the gripper of the robot
came close to the bar, but did not reach the enough height up to
the ceiling to catch it. Thus, we introduce a slight modification
to increase the amplitude of the
to this value as
oscillation so that the gripper reaches the height of the bar, and
instead of the values
we choose to use
in the table in Fig. 4. The initial direction of the swing motion
depends solely upon the initial states of the system since the motion of the robot is governed by the closed-loop dynamics. Only
small deviation from the origin on the phase plane determines
this direction. Thus, we introduce an impulse-like initial torque
before the controller is turned on so that the robot starts its swing
motion in the desired direction at every run. The experimental
results of the swing-up problem do not exactly match those of
numerical simulations presented in Section IV-B. We investigate this matter in the Appendix.
Fig. 17 shows the joint trajectory and the voltage command
resulting in slow swing-up
to the motor driver when
behavior. The mean time of ten runs for this slow swing-up be-s error.
havior is 7.474 s with
Fig. 18 shows the joint trajectory and the voltage command
. This choice
yields
to the motor driver when
relatively fast swing up. The mean swing-up time of ten runs
s error.
for this swing up is 3.843 s with
Fig. 19 shows the joint trajectory and the voltage command
to the motor driver when
. This choice of
yields
a “faster” swing-up maneuver. The mean swing-up time of ten
-s error. In this
runs for this movement is 2.913 s with
case, the initial impulse-like torque is applied in the opposite
direction to the previous two cases in order to start swinging in
the counterclockwise direction.
C. Continuous Locomotion
Here, we exhibit the demonstration of continuous locomotion
over several rungs of the ladder. Fig. 20 depicts repeated locomotion of the robot initiated at the ceiling and moving from left
to right. This motion can be considered as the iteration of the
ladder trajectory. After each swing, the initial condition is reset,
and the function of each arm is switched. This switching is done
manually by looking at the motion of the robot to make sure that
the it does not fall off from the ladder by mistakenly releasing
the grasping bar before catching the next bar with some automated manner, which may result in serious damage to the robot.
Due to the symmetrical structure of the robot, the same model
is used in each swing where the configuration of the robot is
“flipped over.” In Fig. 21, we show a picture of continuous locomotion initiated from the suspended posture. This is a combination of the “faster” swing-up maneuver and the iterated ladder
trajectory. First, the robot swings three times—going forth (1)
NAKANISHI et al.: A BRACHIATING ROBOT CONTROLLER
(a)
121
(b)
Fig. 17. Experimental results of slow swing-up behavior (K = 0:03). (a) Joint trajectories and (b) voltage command to the motor driver. The robot captures
7:5 s.
the bar when t
(a)
(b)
Fig. 18. Experimental results of fast swing-up behavior (K = 0:47). (a) Joint trajectories and (b) voltage command to the motor driver. The robot captures the
bar when t
3:8 s.
(a)
(b)
Fig. 19. Experimental results of faster swing-up behavior (K = 0:9). (a) Joint trajectories and (b) voltage command to the motor driver. The robot captures the
bar when t
2:9 s.
and back (2) to gain momentum, and again swinging forward
)
(3) to catch the bar—with the swing-up controller (
described above. Then the control law is switched into the locomotion controller.
In these experiments, we have observed that disturbances
caused by the cable, which hangs down from above, can
occasionally have a detrimental effect on the robot’s motion.
In particular, sometimes the robot has difficulty reaching the
bar because of the dragging effect of the cable. Thus, some
care has to be taken so that the influence of the cable can be
reduced. Nonetheless, we feel that these experimental results
demonstrate the relevance of our strategy despite the many
practical issues which have not been formally treated, such as
model mismatch, inaccuracy of sensors and actuators, and the
presence of various disturbances.
VII. CONCLUSION
We have presented empirical studies of a new brachiating
controller for a simplified two-link robot. The algorithm uses a
target dynamics method to solve the ladder, swing-up, and rope
problems. These tasks are encoded as the output of a target
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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 2, APRIL 2000
B. Future Work
Fig. 20. A picture of continuous locomotion started in the ceiling. The robot
iterates brachiation three times moving from left to right.
Fig. 21. A picture of continuous locomotion initiated from the suspended
posture. First, the robot swings three times—going 1) forth and 2) back to gain
momentum, and again swinging 3) forward to catch the bar—with the swing-up
controller (K = 0:9) described above. Then the control law is switched into
the locomotion controller.
dynamical system inspired by the pendulum-like motion of an
ape’s (slow) brachiation. We provide numerical simulations
suggesting the effectiveness of the proposed algorithm. We
also present our empirical success in the implementation of
the target dynamics method to a physical two-link brachiating
robot. The proposed algorithm is applied to achieve the ladder
and swing-up behaviors. We achieve swing locomotion in the
ladder problem and various swing-up behaviors with different
. We demonrates of energy pumping, as characterized by
strate repeated locomotion over several rungs of the ladder as
well. The experimental success bears out the validity of our
control strategy in spite of the presence of model mismatches
and physical effects previously unconsidered, although some
manual tuning is required to implement these ideas.
In Section VII-A, we review some of the open questions this
raises, and in Section VII-B, we address future work.
A. Open Problems
These numerical simulations and experimental results suggest that the proposed algorithm is effective for solving robot
brachiation problems. However, a formal mathematical analysis will be required in order to truly understand how these
ideas work. Most importantly, we need to consider the internal
boundedness of the states of the closed-loop system. The unactuated dynamics of our closed loop take the form of a one-degree-of-freedom mechanical system forced by a periodic input.
Such problems of parametric resonance are known to be complex. A second open problem concerns the swing map. Numerical studies suggest the local stability of the fixed point , but
this must be verified analytically, and the extent of the domain
of attraction must be characterized.
The controller developed in this paper requires exact
model knowledge of the robot. “Passive” and, hence, less
model-dependent strategies will be addressed in our future
work pursuing the analogy between the brachiation problem
and the control of hopping robots. This analogy becomes
particularly useful as we begin to contemplate studies of
robot brachiation using more complicated models with higher
degrees of freedom, where modeling of such systems is much
more difficult. Specifically, in Schwind’s study on the control
of simplified spring-loaded inverted pendulum (SLIP) hopping
robots [12], [13], a particular choice of a spring law allows
us to integrate the system’s dynamical equation of the stance
phase analytically, and gives us the stance map in closed form.
We suspect a similar approach may make the slow brachiation
problem more analytically tractable.
Finally, the study of the fast brachiation—the leap
problem—seems compelling. For reasons discussed in the
introduction, this problem lies in the more distant future.
We believe there are generalizable principles of brachiation
which may be established through the study of this simplified
two-degree-of-freedom model.
In the longer run, we believe that the ideas presented in this
paper may have wider application to such areas of robotics as
dexterous manipulation, legged locomotion, and underactuated
mechanisms.
APPENDIX
DISCREPANCY BETWEEN SIMULATIONS AND EXPERIMENTS
We believe that the discrepancy between the physical robot
motions and simulated trajectories can be attributed to unmodeled nonlinear characteristics of the harmonic drive DC motors.
The I/O linearizing controller aggressively cancels nonlinearities in the plant dynamics to achieve the target dynamics, which
requires exact model knowledge of the system. Harmonic drives
bear complicated nonlinear dynamics [28]. Consider slightly
modified friction model which includes coulomb friction, linear
and cubic viscous friction and stiction
(50)
is the coulomb friction coefficient,
and
are the
where
represents stiction torque and
viscous friction coefficients,
denotes the lubrication coefficient [29]. The introduction of
the cubic term in (50) seems to be reasonable as described in
the study on the modeling of harmonic drive gear transmission
mechanisms [28]. In fact, as discussed in [28], harmonic drives
have other complicated characteristics such as variation of friction depending on the position of harmonic-drive output and
dramatic increase of dissipation at some operating ranges that
excite system resonance, which are difficult to model. Furthermore, [28] points out that friction in some drives can actually
decrease over some velocity ranges as reported in [30]. We assume that torque produced by the actuator saturates when torque
commands exceeds the regular operating range of the motor.
Preliminary numerical simulations using the revised actuator
NAKANISHI et al.: A BRACHIATING ROBOT CONTROLLER
model (50) do, indeed, match closely observed experimental results [26].
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Jun Nakanishi (S’95) received the B.E. and M.E. degrees both in mechanical engineering from Nagoya
University, Nagoya, Japan, in 1995 and 1997, respectively. He also studied in the Department of Electrical
Engineering and Computer Science at the University
of Michigan, Ann Arbor, from 1995 to 1996. He received the Ph.D. degree in engineering from Nagoya
University in 2000.
He is currently a Research Associate with the Department of Micro System Engineering, Nagoya University. His research interests include nonlinear control of underactuated Lagrangian systems.
Toshio Fukuda (M’83–SM’93–F’95) graduated
from Waseda University in 1971 and received the
M.E. and Dr.Eng. degrees from the University of
Tokyo, Tokyo, Japan, in 1973 and 1977, respectively.
Meanwhile, he studied at Yale University, New
Haven, CT, from 1973 to 1975.
In 1977, he joined the National Mechanical Engineering Laboratory in 1977, and was a Visiting Research Fellow at the University of Stuttgart from 1979
to 1980. He joined the Science University of Tokyo
in 1981, and then joined Nagoya University in 1989.
Currently, he is a Professor at Center for Cooperative Research in Advanced Science and Technology, Nagoya University, Japan, mainly engaging in the fields
of intelligent robotic systems, mechatronics, and microrobotics.
Dr. Fukuda received the IEEE Eugene Mittelmann Award in 1997. He has
been Vice President of the IEEE Industrial Electronics Society since 1990, IFSA
Vice President since 1997, as well as past President of the IEEE Robotics and
Automation Society.
Daniel E. Koditschek (S’80–M’83) received the
Ph.D. degree in electrical engineering from Yale
University, New Haven, CT, in 1983.
He was appointed to the faculty at Yale University,
in 1984, and moved to the University of Michigan,
Ann Arbor, in 1993, where he is presently a Professor in the Department of Electrical Engineering
and Computer Science with a joint appointment in
the Artificial Intelligence Laboratory and Control
Systems Lababoratory. His research interests include
robotics, the application of dynamical systems
theory to intelligent mechanisms, nonlinear control, and applications of
real-time distributed control technology.
Dr. Koditschek is a member of the AAAS, ACM, AMS, MAA, SIAM, and
Sigma Xi.