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A toy climbing robot
Matthew Bell
Dartmouth Computer Science Department
Hanover, NH 03755
Devin Balkcom
Dartmouth Computer Science Department
Hanover, NH 03755
Abstract— We built a simple toy climbing robot in order toexplore problems related to minimalist grasping, path planning,and robot control. The robot is capable of climbing a wall of pegs either under remote control, or on the basis of a set of pre-recorded keyframes. In addition, the robot can climb certainpeg configurations using a cyclic gait. All communications aresent through an infrared connection, and the tether to the robotconsists only of two power wires. Due to the minimalist, non-prehensile grasping method, the robot is capable of activelyremoving error while climbing, which is necessary to enable therobot to climb without sensing the environment.
I. INTRODUCTION
Our goal is to develop a simple, lightweight robot that
uses minimal computation and sensing in order to successfully
climb a wall of pegs. Our robot utilizes minimalist grasping
and a non-prehensile grip, allowing it to slide on the pegs to
actively remove error. Most robot grasping problems involve
grasping a small object with a large industrial manipulator; our
robot can be thought of as a small manipulator grasping the
entire climbing wall. Due to the lack of environmental sensing,
the robot cannot tell if it is off course; it is thus necessary to
plan motions in a way that provides stability and repeatability.
A major goal was to keep the robot as simple as possible to
make it feasible for the general public to buy an inexpensive
kit for building the robot. Our robot is made of hobby servos
and LEGO pieces (See Figure 1). There are three climbing
modes:
1) Manual remote control
2) Autonomous, with pre-recorded keyframes
3) Autonomous, using a simple cyclic gait
The robot is capable of climbing under manual control through
a Java interface and autonomously using a set of pre-recorded
keyframe positions. For appropriate wall configurations, a set
of cyclic keyframes exists that will make the robot climb
the wall with a cyclic gait. When executing pre-recorded
keyframes, the robot climbs open-loop, with no sensor feed-back.
Our mathematical model of the robot considers the arms and
legs to be rigid, and assumes that the servos can be locked into
position. We derived the forward and inverse kinematics for
use in the GUI, and used Reuleaux’s method [1] to analyze
stability geometrically to gain an intuitive understanding of
stable grasps. We represent the free motions of the robot with
a polyhedral convex cone, represented in matrix form. Using
Goldman [2] and Hirai [3], it is possible to determine if this
Fig. 1. The robot on the climbing wall
cone is empty, meaning that no free motions are possible and
the robot is stable.
Our toy robot is largely a preliminary exploration into
the challenges and limitations involved in building a simple
climbing robot. We explored several problems related to path
planning and minimalist grasping under uncertain conditions,especially in the area of error removal.
I I . RELATED WOR K
Our toy robot is not the first climbing robot; it is, however,
the simplest, one of the lightest, and the only one to reliably
remove error while climbing open-loop without sensing the
environment. Rus and Kotay developed the Inchworm [4],
a lightweight task-reconfigurable robot capable of climbing
on any ferrous surface that it can grasp with electromagnets.
Michigan State University has a pair of climbing robots
that use suction to climb on smooth walls and ceilings [9].
Nagakubo and Hirose [5] built a large, heavy quadruped robot
capable of navigation on horizontal and vertical surfaces usingsuction. Neubauer [6] developed a small robot for climbing
inside pipes. Neubauer’s robot is similar to ours in its use of
a non-prehensile grip. Linder and a group of undergraduate
students developed Tenzing [7], a large quadruped robot.
Tenzing is heavier than the toy robot and employs significant
sensing, including live video, to assist in motion planning.
Bretl at Stanford has developed a path-planning algorithm for
JPL’s LEMUR II robot [8]. This robot is capable of climbing
walls with arbitrarily shaped and angled handholds. The design
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(x,y,Θ)
δγ r
l 1
l 2
Θd1
Θd2
D
BA
C
pd2
Θc2
Θc1
α
Fig. 2. Variables used in mathematical analysis
of this robot is such that there is no definite “up” orientation
of the robot body, providing more flexible manueverability.
I I I . ROBOT DESIGN
The robot is built from eight small hobby servos and several
LEGO pieces, and is controlled by a Pontech SV203C board.The robot receives all of its control communications through
an infrared (IR) receiver, and as a result, the robot’s tether
consists only of two wires for power.
The IR communications rely on three main pieces of
software. The host system runs a Java application that sends
commands through a serial cable to the standalone transmitter
SV203C (Tx board). Embedded code runs on the Tx board,
and retransmits the commands along with a checksum to the
receiving SV203C (Rx board, located on the robot). Additional
embedded code on the Rx board interprets and verifies the
commands before moving the appropriate servo.
IV. MATHEMATICAL ANALYSIS
Using the variable definitions given in Figure 2, we analyzed
the forward and inverse kinematics of the robot. Each arm has
2 degrees of freedom, which are indicated by θx1 and θx2,
where x is the arm label. The position of arm D, represented
by pd2, is given by
pd2 = Rθ
x + rcδ + l1cd1 + l2cd1d2y + rsδ + l1sd1 + l2sd1d2
(1)
where Rθ is a rotation matrix, cij... = cos θ1 + θ2 + . . ., and
sij... = sin θ1 + θ2 + . . ..
The inverse kinematics for one arm are computed according
to standard methods. However, we only consider angles in theranges 0 ≤ θ1 ≤ π and 0 ≤ θ2 ≤ π to reduce the number of
solutions, as we have observed that these constraints hold for
climbing methods with the most stability.
θ1 = arccos
l22−l2
1−x2−y2
−2l1√
x2+y2
+ arctan2(y, x)
θ2 = arccosx2+y2−l2
1−l2
2
2l1l2
(2)
To compute the free motions of the robot, we first computed
the Jacobian matrix J from the forward kinematics. If the
(a) Initial position (b) Legs lifted
(c) Recentering position (d) Arms lifted
Fig. 3. Sequence of robot configurations during a single cycle of a gait
contact normals are given by a matrix N , then the free motions
of the robot are given by the set {q̇ : J N q̇ ≥ 0}. If this setis empty, then the robot is motionless and stable. This set
represents a polyhedral convex cone, and we can determine if
the cone is empty.
V. CLIMBING MODES
The robot is capable of climbing either under manual remote
control, or using interpolating motions between pairs of a
sequence of pre-recorded keyframes. In addition, a cyclic
gait has been developed for autonomously climbing a vertical
ladder configuration of the wall (See Figure 3).
A. Pre-recorded Keyframes
The robot’s primary climbing method uses a series of pre-recorded keyframes to guide the robot up the wall. These
keyframes are recorded by the human operator while the
robot is being navigated up the wall under manual remote
control. During playback, interpolation between the keyframes
is handled by moving all the arms at a constant angular rate
until they reach the position specified by the next keyframe,
resulting in a much smoother and quicker climb than under
manual control. The actual speed is dependent on the route
chosen by the human operator.
It was generally necessary to position the robot within about
0.5 cm horizontally of the initial position used when recording
the keyframes to ensure consistent success. Small variations
in position are insignificant due to the robot’s non-prehensilegrip, as the robot will not fall as long as some portion of
each of the four limbs is touching a peg. Introducing error-
correcting motions into the keyframes can increase the amount
of acceptable variation of the initial position.
B. Cyclic Gait
If the pegs are in a repeating configuration, it is possible to
climb using a cyclic gait. The configuration that we specifically
examined is a ladder formed from two sets of pegs in vertical
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Fig. 4. Position used to recenter the robot during a gait cycle
lines. The robot can climb this configuration at an upward
speed of about 4.1 cm/min. This type of configuration greatly
simplifies the problem of finding a path up the wall, as it
is only necessary to find one complete cycle. For a ladder
with perfectly vertical sides, the cycle only needs to result in
each arm of the robot being one handhold higher up the wall
(Figure 3). We developed a gait for the ladder configuration
by manually controlling the robot through one full cycle,
and recording the keyframes from this cycle. The motions
designated by the keyframes were then repeated multiple times
to drive the robot up through several cycles of the gait.
In order for a gait to be successful, the robot must not
shift from side to side as it climbs up the wall. The fact
that the robot is climbing open-loop in this mode requires
robust trajectories to ensure success. Thus, it is necessary to
perform some motion or sequence of motions that returns the
robot to some known configuration. The frequency of this
error-correcting motion depends on the complexity of the wall
configuration; however, it should occur at least once in every
cycle.
For the ladder configuration that we examined, we devel-
oped a maneuver that will successfully recenter the robot
into the position shown in Figure 4. In order to achieve this
position, the robot hooks its upper arms over the handholds
at angles that cause the robot to slide until the handholds
are at the elbow joints. For other configurations, it may be
necessary for the robot to fall slightly in order to become
recentered. These recentering methods require the robot to take
some action in order to remove error from the system. Due to
the recentering manuever executed during the gait, the robot
is much less sensitive to the intial position when climbing in
this mode.
V I . OPE N PROBLEMS
A major source of possible failure for the cyclic gait is
an improper recentering motion. Adding a minimal sensorpackage to the robot would permit detection of the correctness
of the recentering motion. However, another solution is to
make use of gravity. In Figure 4, if the angles of the arms
with respect to the horizontal are steep enough, the robot will
overcome static friction and slide into the recentered position.
An interesting area for further research is in climbing
using only local information. Full path planning requires prior
knowledge of the entire wall, which may not be available. With
either touch sensors or servo torque measurements, it should
be possible for the robot to wave its arms around to determine
where handholds are in its immediate vicinity, and to use this
information to climb locally. From this local climbing ability,
it should be possible to recursively climb the entire wall. The
robot can climb locally until it is unable to do so, and then it
can back down the wall for some distance, and try climbing in
a different direction. As it proceeds, it will slowly develop a
model of the entire wall from the local information it collects.
V I I . CONCLUSIONS
A toy climbing robot was successfully developed. Although
it is not capable of automatically pre-planned climbing, it can
climb a ladder using a cyclic gait in an open-loop mode. The
robot achieves this through the use of a recentering motion
that ensures that the robot is correctly positioned during each
cycle. The robot is sufficiently simple that it could be marketed
as a kit—it does not require a bulky tether to operate correctly,
and the host software’s GUI is fairly easy to use.
This project has also explored the mathematics involved in
robotic climbing, including the forward and inverse kinemat-
ics, and calculations of stability. The cyclic gait method has
been successfully implemented for one wall configuration, and
other gaits could be developed for other regular configurations.
It is possible that a general gait could be developed for a set
of similar configurations, provided that a generic recentering
strategy exists.
There are several potential areas of further exploration. A
full path planning algorithm could be developed to allow the
robot to climb any wall based only on knowledge of the
locations of the pegs. A general-purpose cyclic gait could exist
for a certain set of wall configurations. Sensors could be added
and incorporated into the climbing algorithm, and the robot
itself could be redesigned to allow it to climb blindly without
any knowledge of the wall.
REFERENCES
[1] F. Reuleaux, The Kinematics of Machinery. MacMillan, 1876, reprintedby Dover, 1963.
[2] A. J. Goldman and A. W. Tucker, “Polyhedral convex cones,” in Linear Inequalities and Related Systems, H. W. Kuhn and A. W. Tucker, Eds.York: Princeton Univ., 1956, pp. 19–40.
[3] S. Hirai, “Analysis and planning of manipulation using the theory of polyhedral convex cones,” Ph.D. dissertation, Kyoto University, Mar.1991.
[4] K. Kotay and D. Rus, “The inchworm robot: A multi-functional system.” Auton. Robots, vol. 8, no. 1, pp. 53–69, 2000.
[5] A. Nagakubo and S. Hirose, “Walking and running of the quadrupedwall-climbing robot,” in IEEE International Conference on Robotics and
Automation, vol. 2, 1994, pp. 1005–1012.[6] W. Neubauer, “A spider-like robot that climbs vertically in ducts or
pipes,” in IEEE/RSJ/GI International Conference on Intelligent Robots
and Systems, vol. 2, 1994, pp. 1178–1185.[7] S. P. Linder, E. Wei, and A. Clay, “Robotic rock climbing using computer
vision and force feedback,” in IEEE International Conference on Robotics
and Automation, 2005.[8] T. Bretl, S. Rock, J. C. Latombe, B. Kennedy, and H. Aghazarian, “Free-
climbing with a multi-use robot,” in International Symposium on Robotics
Research, 2004.[9] J. Xiao, J. Xiao, and N. Xi, “Minimal power control of a miniature
climbing robot,” in IEEE/ASME International Conference on Advanced
Intelligence Mechatronics, July 2003, pp. 616–621.