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A toy climbing robot

Matthew Bell

Dartmouth Computer Science Department

Hanover, NH 03755

[email protected]

Devin Balkcom

Dartmouth Computer Science Department

Hanover, NH 03755

[email protected]

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



(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



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.

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