EXPERIMENTAL EVALUATION OF CONTACT/IMPACT DYNAMICS BETWEEN A
SPACE ROBOT WITH A COMPLIANT WRIST AND A FREE-FLYING OBJECT
N. Uyama, Y. Fujii, K. Nagaoka, and K. Yoshida
Department of Aerospace Engineering, Tohoku University
6-6-01 Aramaki Aza-Aoba, Aoba-ku, Sendai, Miyagi, 980-8579, Japanuyama@astro.mech.tohoku.ac.jp

ABSTRACTThis paper presents the experimental evaluation of contact/impact dynamics between a space robot with a compliant wrist and a free-flying object. In capturing a target
satellite by a free-flying space robot, a robot must prevent bounce-off of a target satellite from the operational
space of the robot due to hard collision. In order to avoid
hard collision, the authors propose the use of a stiff manipulator equipped with a compliant wrist because thiscombination is capable of elongating the contact duration
with less contact force peak, and hence, makes the contact control possible. In order to understand contact witha compliant wrist, this paper presents the contact dynamics of a uniaxial compliant wrist, which is modeled by amass-spring-damper system. Then, simplification is presented so that the dominant contact dynamics parameters
can be approximated by the stiffness and damping of thecompliant wrist. The collision experiment between a manipulator with a compliant wrist and a free-flying target
object is conducted using a two-dimensional microgravity emulator, called the air-floating test bed. The contact
duration and the coefficient of restitution were chosen as
evaluating parameters, and it was verified that both can beexpressed by the stiffness and damping of the compliant
wrist.
Key words: free-flying space robot, satellite capture, contact dynamics, compliant wrist, coefficient of restitution.

1.

major approaches to the introduction of virtual mechanical compliance. However, the active control approach has
been facing the difficulty in controlling contact/impact
due to the time delay effect because contact/impact is avery short duration phenomenon.
One of the solutions to overcome time delay effect isto increase controllable time by elongating contact duration. This is achieved by introducing passive mechanism,
namely mechanical compliance. Mechanical complianceplays an important role to reduce contact force peak andto increase contact duration. However, drawbacks of mechanical compliance are the end tip vibration of a roboticarm and the slow response to control input due to its flexibility.
In order to balance mechanical compliance with fast response, the use of a stiff manipulator with end tip compliance may ideally be suitable. The use of a stiff
manipulator makes it possible to obtain fast response,
which is suitable for force control. Recent works forsatellite capture using a stiff manipulator are, for example, the multiple impedance control by Moosavian andPapadopoulos[3], the impedance control by Yoshida andNakanishi et al.[4][5]. An example of a passive wrist wascreated by Salcudean and Hollis, called the IBM magic
wrist[6]. Examples of the use of both a stiff manipulator
and end tip compliance are the use of passive wrist forcontact control by Goswami et al.[7], and brash type ofend effector for space debris capture by Nishida et al.[8].
However, force control method in space by a single manipulator with a compliant wrist has not been fully established yet.

INTRODUCTION

In capturing a free-flying satellite in space, there is always possibility that the satellite may bounce off from aspace robot, and/or that the satellite may be damaged due
to hard collision. In order to avoid such a situation, soft
contact utilizing active compliance control and/or passive
mechanical compliance is of great interest, especially inautonomous satellite capture operation by a free-flying
space robot. For instance, the hybrid force/position controlled, originally proposed by Raibert and Craig[1], andthe impedance control, proposed by Hogan[2] are the two

In our previous paper, we showed the validity of the useof a stiff manipulator with a compliant wrist to obtain adesired coefficient of restitution by the impedance control
with a compliant wrist[9]. In this paper, we concluded
that the contact dynamics modeling is inevitable to improve the results of the proposed contact control method.
This paper presents the contact dynamics modeling ofa free-flying space robot with a compliant wrist. The
contact dynamics model is expressed by a mass-springdamper system. In order to predict post-contact behavior
by a simple algebraic form, the contact dynamics modelis simplified, and the contact dynamics parameters suchas stiffness and damping are represented by those of acompliant wrist. In this study, the coefficient of restitution is chosen as an algebraic expression of contact phenomena. The formulae of coefficient of restitution andcontact duration are derived from the simplified contact
dynamics model. The experimental verification of theproposed contact dynamics model is conducted using atwo-dimensional microgravity emulator, called the airfloating test bed. The results of experiment verify theproposed contact dynamics modeling.

2.

2.1.

mkM

1. The gravitational acceleration is zeroFc

8. The wrist mass is sufficiently smaller than the space
robot and the target object.
Based on the above assumptions, the schematic of contact
dynamics model for a one-dimensional case is shown inFig. 1.

2.2.

Exact Contact Dynamics Model

The exact contact dynamics model is derived as follows.
First, according to the assumption, the contact force canbe modeled as a linear spring-dashpot model. Thus, thec

= kc δ + cc δ̇= kc (xt − xw ) + cc (ẋt − ẋw )(1)

where xt and xw are the positions of the mass of the target mt and the compliant wrist mw , respectively, and kc
and cc are the stiffness and damping coefficients of actual contact surfaces, respectively. Then, the equationsof motion of each mass shown in Fig. 1 are derived asfollows:
M ẍhmw ẍw

4. Point contact is assumed.

7. The vibration frequency defined at contact surfaces
is sufficiently higher than the one defined at a wrist.

ccontact force is expressed in terms of penetration and rateof penetration as follows:

3. Links of manipulator, a wrist, and a free-flying target
object are expressed by a point mass.

6. A compliant wrist is modeled as a linear massspring-damper system.

wt

target body, and contact dynamics model.

2. Joints of manipulator are sufficiently stiff.

5. Contact force is modeled as a continuous springdashpot model.

mw

Figure 1. Schematic of a compliant wrist and a free-flying

Assumptions

In constructing the contact dynamics model of a space
robot with a compliant wrist, the following assumptions
are introduced.

cm
c

tkw

CONTACT DYNAMICS MODELING OF AUNIAXIAL COMPLIANT WRISTThis section presents the contact dynamics model of auniaxial compliant wrist. First, the assumptions usedin constructing the contact dynamics model are stated.
Then, the exact and simplified contact dynamics models
are presented.

Contact
model Target
Body

CompliantWrist

Stiff
Manipulator

mt ẍt= −kw (xh − xw ) − cw (ẋh − ẋw ) (2)= −kw (xw − xh ) − cw (ẋw − ẋh )−kc (xw − xt ) − cc (ẋw − ẋt ) (3)= −kc (xt − xw ) − cc (ẋt − ẋw )(4)where M is the equivalent mass projected at the end tipof manipulator; xh is the position of the end tip of manipulator; and kw and cw are the stiffness and damping
coefficients of the compliant wrist, respectively.

2.3.

Simplified Contact Dynamics ModelIn this subsection, a simplified contact dynamics modelis presented. A contact dynamics model expressing relative motion before and after contact is necessary to control contact in space. The goal of this contact dynamics modeling is to derive the relationship between precontact state and post-contact state of relative motion in
the form of the coefficient of restitution, which can beused for contact control. In order to accomplish this goal,
the contact dynamics model expressed in Eqs. (2)-(4) is
simplified to a single equation of motion.
Suppose that the system expressed in Fig. 1 has no damping, the natural frequency ω of the system is expressed inthe following form.
[}1{ 2ω =
ωw + ωc2 (1 + α)2]1/2√
122222±{ωw + ωc (1 + α)} − 4ωw ωc(5)2√kw
ωw =, ωc =m̃w
M mw
m̃w =,M + mw

√kc,m̃cm̃cm̃w
mw mtm̃c =mw + mt
α=(6)(7)

According to the assumption, the vibration frequency defined at contact surfaces is sufficiently larger than theone defined at a wrist. This condition is expressed asωw ≪ ωc . The wrist mass mw is also sufficiently smaller
than the chaser robot mass M and the target object mass
mt ; and hence, the mass ratio α becomes α ≈ 1.
In an actual system, there exists a damping effect. Whenthe assumption ωw ≪ ωc is satisfied, the vibration between the wrist mass and the target body mass damps out
much earlier than the vibration of the wrist itself. Therefore, the dominant natural frequency of the system expressed in Eq. (5) can be approximated as the lowest frequency of the system in the following form.
[1√ 2 21 2
ωw +(ωw )
ω ≈
22
√
kw
ωw =m̃
M (mw + mt )m̃ =M + mw + mt]1/2
≈ ωw

(8)

Eq. (13), the complete solution with initial conditions
y(0) = 0 (zero penetration), and ẏ(0) = v0 (non-zerorate of penetration), is given by the following:y(t) =Eq. (14) expresses the vibration of the compliant wrist.
From Eq. (14), the contact duration τ can be derived as
the half-period of oscillation (y(τ ) = 0) as follows:
τ=πT=√
21 − ζ 2 ωw

M ẍh = −kw (xh − xw ) − cw (ẋh − ẋw )(mw + mt )ẍw = kw (xh − xw ) + cw (ẋh − ẋw )

(11)
(12)

From Eq. (11) and Eq. (12), a single equation of relative
motion is derived as follows.
m̃ÿ = −kw y − cw ẏ

(16)τ → ∞ if ζ ≥ 1(17)Next, the coefficient of restitution ϵ can be expressed interms of the damping ratio ζ. Using the damping ratio
defined in Eq. (15), the time derivative of Eq. (14), andthe contact duration given by Eq. (16), the coefficient ofrestitution ϵ is also expressed as follows:
ϵ=−− √ ζπẏ(τ )= e 1−ζ2ẏ(0)if 0 ≤ ζ < 1(18)For the case ζ ≥ 1, Eq. (18) is no longer valid; instead,the coefficient of restitution becomes zero.
if ζ ≥ 1(19)3. EXPERIMENTAL VERIFICATION
In order to verify the proposed contact dynamics modeling method, the collision experiment between a compliant wrist attached at the tip of a stiff manipulator anda free-flying target object was conducted. The experimental setup is first explained in detail, followed by theexperimental conditions. The results and discussion ofcollision experiment are finally presented.

(13)

where y = xh − xw is the displacement of the compliantwrist.

2.4.

if 0 ≤ ζ < 1

where T is the period of oscillation. For the case ζ ≥ 1,
the contact duration is theoretically infinity.

ϵ=0When the above condition is satisfied, a free-flying target
object completely attaches to the end point of wrist andmoves together with the wrist mass during contact. Thisapproximation is expressed at the bottom of Fig. 2. Insuch a case, the equations of motion can be expressed in
the following forms.

(14)√where ωd =1 − ζ 2 ωw is the damped natural frequency, and the ζ is the damping ratio of the system asfollows.
cw
ζ= √(15)2 m̃kw(9)(10)

v0 −ζωw tesin ωd t
ωdContact Duration and Coefficient of Restitution

This simplification eases the theoretical formulation ofcontact duration and coefficient of restitution. From3.1.

Experimental Setup

The experimental setup and its schematic are shown inFigs. 3-4. A microgravity emulation was achieved by theair-floating test bed, which enables an object equippedwith air tanks to float on a smooth flat plane with negligibly small friction, utilizing the air bearing principle.
The floating target body was equipped with four air pads,

CompliantWrist

ContactModelkskc

Exact Contact Modelmw

mtcsccSimplified Contact Modelksmwmt(a) Air-floating test bed and robotic arm PA10cs+ ceq
Single body

Figure 2. Schematic of exact and simplified contact models.
and each pad was connected to the on-board air tanks bytubes. Each pad floats by static air pressure. The weightof the air-floating body can be changed by adding or subtracting partial weight. Hence, the position of center ofmass is adjustable. In this setup, an emulated environment was two-dimensional microgravity plane. The contact point on the air-floating body was designed as a hemisphere so as to realize point contact.
The compliant wrist used in this experiment was a uniaxial compliant wrist, where the motion of the wrist
mass was constrained by a linear guide, as shown inFig. 3(b). The compliant wrist was equipped with a 6axis force/torque sensor, made by Nitta Corporation, at
its end tip of movable part to measure the contact force.
As a space robot model, a ground-based robotic arm,called PA10, made by Mitsubishi Heavy Industry, wasused to represent a stiff manipulator. The stiffness ofjoints of PA10 robotic arm is 105 order while the stiffnessof the compliant wrist is 103 to 104 order. Therefore, thevibration of end tip of PA10 robotic arm was assumed tobe negligible compared to the one of the compliant wrist.
The motion measurement system used in this experiment
was a motion tracking cameras, called OptiTrack. Thissystem uses infrared cameras, each of which tracks thepositions of infrared rays reflected from reflective balls
attached on the air-floating target body and the compliantwrist.

3.2.

Experimental Conditions

This experiment was a one-dimensional collision experiment between the compliant wrist and the floating target body. The contact dynamics parameters are shown in
Tab. 1. Since the wrist used in this experiment had friction in its linear guides, the friction effect was estimated(b) Uniaxial compliant wrist at the tip of PA10
Figure 3. Experimental Setup of Collision Experimentand included as a form of equivalent viscous damping(ceq ), which was approximated as a constant value. For
each condition, 5 trials were conducted.
The end tip of PA10 robotic arm was set to an initial position with all servos locked. The air-floating target bodywas given various initial velocities by hand and collided
into the compliant wrist so that the post-contact motionof the air-floating target body did not induce rotation. Inthis experiment, the position of the center of mass of theair-floating target body was aligned to the direction ofcontact force so as to minimize the wrench as much aspossible.
In this experiment, the contact force and the motion ofboth the compliant wrist and the air-floating target body
were measured. The velocities of the wrist and the target body were computed by taking the time derivative of
the position histories. The pre- and post-contact velocities of the air-floating body were computed by taking theaverage of velocities 1-second before and after contact.

3.3.

Experimental Results and Discussion

The resulting coefficient of restitution and contact duration are shown in Fig. 5 and Fig. 6, respectively, where1

Infrared Motion Capture Camera

0.95

Wrist

Force/TorqueReflective Balls

SensorAir

Air-floatingTank

Test Bed

PA10
Robotic Arm

Coefficient of Restitution

Compliant

0.9
0.85
0.8
0.75
0.7
0.65
0.6

Experimental
Theoretical1020304050
Target + Movable Mass [kg]6070(a) Spring constant kw = 530 [N/m]Figure 4. Schematic of Experimental Setup of Collision
Experiment.

1

Table 1. Contact Dynamics Parameters
NameTarget Mass [kg]VariablemtChaser Mass [kg]Wrist Mass [kg]Wrist Stiff. [N/m]

M
mw
kw

Wrist Damp. [Ns/m]Contact Stiff. [N/m]Contact Damp. [Ns/m]
Equiv. Coeff. [Ns/m]

cw
kc
cc
ceq

Value
9.0, 23.2,35.7, 51.0 63.2
huge (109 )
0.7530, 1020,3600, 5100
1.4, 2.1, 2.8, 2.7
68,000unknown
12.5

Coefficient of Restitution

0.950.9
0.85
0.8
0.75
0.7
0.65
0.6

Experimental
Theoretical102030
4050
Target + Movable Mass [kg]6070(b) Spring constant kw = 1,020 [N/m]1The errors in the coefficient of restitution were mainly
due to the model error. In general, the coefficient ofrestitution is dependent on not only the characteristicsof mass, spring and damper, but also the initial colliding velocity, which is not formulated in the proposed linear modeling. The overall average initial colliding velocity was about 7 [cm/s]. In low spring constant casessuch as Fig. 5(a) and (b), the average velocity was about
10 [cm/s], which pushed up the resulting coefficient ofrestitution. Thus, the experiment showed that a velocitydependent form of coefficient of restitution is needed forfurther improvement.
Overall, it was concluded that the coefficient of restitution and the contact duration can be predicted with smalldeviation. One should pay attention to the prediction of
the coefficient of restitution when using the proposed linear contact dynamics modeling because of velocity dependency. The improvement in the prediction of the coefficient of restitution may be achieved by appropriately
including friction effect.

0.9
0.85
0.8
0.75
0.7
0.65
0.6

Experimental
Theoretical
1020304050
Target + Movable Mass [kg]6070(c) Spring constant kw = 3,600 [N/m]10.95
Coefficient of Restitution

dots with error bars are the experimental results, anddashed lines are the theoretical prediction by the simplified contact dynamics model given by Eq. (18). From thegraphs, both the resulting coefficient of restitution andcontact duration fairly matched the theoretical prediction.

Coefficient of Restitution

0.95

0.90.85
0.80.75
0.70.65
0.6

Experimental
Theoretical
1020304050
Target + Movable Mass [kg]6070(d) Spring constant kw = 5,100 [N/m]Figure 5. Resulting coefficient of restitution vs sum oftarget mass and movable mass of compliant wrist4. CONCLUSION

Contact Duration [s]

1.5

10.5

Experimental
Theoretical
01020304050
Target + Movable Mass [kg]6070(a) Spring constant kw = 530 [N/m]Contact Duration [s]1.5

15. ACKOWLEDGMENT
0.5Experimental
Theoretical
01020304050Target + Movable Mass [kg]60

Contact Duration [s]1.5

REFERENCES

Experimental
Theoretical

11020304050
Target + Movable Mass [kg]6070(c) Spring constant kw = 3,600 [N/m]1.5

Experimental
Theoretical

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0.50[1] M. Raibert, J. Craig. (1981). Hybrid Position/ForceControl of Manipulators, ASME Journal of DynamicSystems, Measurement, and Control. Vol. 102. No. 2.
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0.50This research was partially supported by Japan Society
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70(b) Spring constant kw = 1,020 [N/m]Contact Duration [s]This paper presented the contact dynamics modeling ofa space robot equipped with a compliant wrist at its end
tip. Utilizing a mass-spring-damper model, an exact contact dynamics model of a compliant wrist was formulated.
The simplification of contact dynamics model was performed with an assumption that the frequency definedat actual contact surfaces is much higher than the one
at a compliant wrist. Using the simplified contact dynamics model, the coefficient of restitution was formulated and became a function of wrist parameters and target mass. The collision experiment was conducted using
a two-dimensional microgravity emulator, called the airfloating test bed. The results and discussion of collisionexperiment verified the proposed contact dynamics modeling of a compliant wrist.

1020304050Target + Movable Mass [kg]6070(d) Spring constant kw = 5,100 [N/m]Figure 6. Resulting contact duration vs sum of target
mass and movable mass of compliant wrist[6] S. Salcudean, R. L. Hollis. (1988). A magneticallylevitated fine motion wrist: kinematics, dynamics andcontrol. Proc. of 1988 IEEE Int. Conf. on Robotics andAutomation. Philadelphia. PA. USA. pp.261-266. Vol.
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