Intelligent Control and Automation, 2011, 2, 8-23
doi:10.4236/ica.2011.21002 Published Online Febru ar y 2011 (http://www.SciRP .o rg/journal/ica)
Unified Modeling Approach of Kinematics, Dynamics and
Control of a Free-Flying Space Robot Interacting with a
Target Satellite
Murad Shibli
Mechanical Engineering Department, College of Engineering, United Arab Emirates Unive rsity, Al-Ain , UAE
Received November 28, 2010; revised December 12, 2010; accepted December 13, 201 0
Abstract
In this paper a unified control-oriented modeling approach is proposed to deal with the kinematics, linear and
angular momentum, contact constraints and dynamics of a free-flying space robot interacting with a target
satellite. This developed approach combines the dynamics of both systems in one structure along with holo-
nomic and nonholonomic constraints in a single framework. Furthermore, this modeling allows considering
the generalized contact forces between the space robot end-eff ecter and the target satellite as internal forces
rather than external forces. As a result of this approach, linear and angular momentum will form holonomic
and nonholonomic constraints, respectively. Meanwhile, restricting the motion of the space robot
end-effector on the surface of the target satellite will impose geometric constraints. The proposed momentum
of the combined system under consideration is a generalization of the momentum model of a free-flying
space robot. Based on this unified model, three reduced models are developed. The first reduced dynamics
can be considered as a generalization of a free-flying robot without contact with a target satellite. In this re-
duced model it is found that the Jacobian and inertia matrices can be considered as an extension of those of a
free-flying space robot. Since control of the base attitude rather than its translation is preferred in certain
cases, a second reduced model is obtained by eliminating the base linear motion dynamics. For the purpose
of the controller development, a third reduced-order dynamical model is then obtained by finding a common
solution of all constraints using the concept of orthogonal projection matrices. The objective of this approach
is to design a controller to track motion trajectory while regulating the force interaction between the space
robot and the target satellite. Many space missions can benefit from such a modeling system, for example,
autonomous docking of satellites, rescuing satellites, and satellite servicing, where it is vital to limit the con-
tact force during the robotic operation. Moreover, Inverse dynamics and adaptive inverse dynamics control-
lers are designed to achieve the control objectives. Both controllers are found to be effective to meet the spe-
cifications and to overcome the un-actuation of the target satellite. Finally, simulation is demonstrated by to
verify the analy tica l result s.
Keywords: Free-Flying Space Robot, Target Satellite, Servicing Flying Robot, Adaptive Control, Inverse
Dynamic Control, Hubble Telescope
1. Introduction
Free-flying space robots and free-floating space robots
have been under intensive consideration to perform many
space missions such as: inspection, maintenance, repair-
ing and servicing satellites in earth orbit. Particularly,
servicing satellite equipped with robot arms can be em-
ployed for recovering the attitude, charging the exhaust-
ing batteries, attaching new thrusters, and replacing the
failed parts l ike gyros, sola r panels o r antennas o f anoth-
er satellite.
There are two major classes of space robots can be
classified: 1) free-flyi ng sp ace rob ots a nd 2) free-floating
space robots. The manipulator system of the first type is
a system in which the reaction jets (thrusters) are kept
active so as to control the position and attitude of the
systems’ spacecraft. In opposition to the free flying robot,
a free-floating space robo tic system is a system in whic h

M. SHIBLI
9
the spacecrafts’ reaction thrusters are shut down to con-
serve attitude c ontrol fuel.
Comprehensive understanding of the kinematics and
momentum of space robots and their interaction with a
floating object is considered as a very essential part in
designing an efficient multi-body system with effective
control techniques of contact forces and motion trajecto-
ries. Many techniques in dynamic modeling of space
robots have been developed in [1-10]. Kinematics mo-
tion of a space robot system are developed based on the
concept of a Virtual Manipulator (VM) [10-14]. It as-
sumes imaginary mechanical links and it does not model
the angular momentum, then the attitude motion of the
base satellite has to be considered by other means. One
body of the space robotic system is used as the reference
fr a me wi t h a point on it to represent the tran sitional DOF
of t he s ys tem [2 ,8,9]. A tree topology of open chain mul-
ti-body system with the system center of mass as the
translational DOF is proposed in [6,7].
Many techniques in dynamic modeling of space robots
have been reviewed in [2,4,5]. Newton-Euler dynamic
approach of multi-body systems is proposed in [6,7].
This approach is characterized the use of a tree topology
of open chain multi-bod y system with the syste m Center
of Mass as the translational DOF. Barycenters are used
efficiently to formulate the kinematics and dynamics of
free-floating space robots. Another approach is called the
direct approach and it uses one body of the system to be
the reference frame with a point on it to represent the
transitional DOF of the system [2,8 ,9]. This approach is
simpler but results in coupled equations. A virtual mani-
pulator is proposed in [10-14] and used to simplify the
system dynamic s of sp ace robots. It decoupl es the s ystem
Center of Mass transactional DOF.
Free-flying space robots dedicated for maintenance or
rescue operations are involved in contact tasks. Many
studies on space-based robotic systems have assumed
zero external applied forces. Dynamics of space robots
by using what is so-called the virtual manipulator (VM)
is proposed in [10-14]. Multi-body systems approach
based on Newton-Euler dynamic is proposed in [6,7].
Achievements in Space Robotics are presented in [15]. In
this article three p arts are intro duced. I n the first part, the
achievements of orbital robotics technology in the last
decade are reviewed, highlighting the Engineering Test
Satellite (ETS-VII) and Orbital Express flight demon-
strations. In the second part, some of the selected topics
of planetary robotics from the field robotics research
point of view are described. Finally, technological chal-
lenges to asteroid robotics are discussed.
In work [16] three dynamical models of a two link
space robot are developed. One model treats the gravita-
tional field as constant over the volume of the robot and
another model uses 0th order Taylor series expansions of
a continuous gravitational field over the volume of the
robot. A third model neglects the effects of gravity. The
dynamics of a dual-arm space robot system was syste-
matically studied, and a dynamic model based on
Kane-Huston’s method and screw theory was presented
in [17]. The numerical example shows that acting mo-
ment of a composition unit of the robot can be solved for
given value of motion parameters with the exploitation
of the dynamic model, vice versa. A simulation system
of a three layer structure based on ADAMS, MATLAB
and VC++ is present in [18], which can simulate and
analyze the kinematics and dynamics of space robot in
the process of capturing and releasing space object. Veri-
fication results show that this system can well explain
space robot’s dynamic and kinetic characteristic in cap-
turing and releasing task under the space circumstances.
In research [19], the kinematics and dynamics of free-
floating coordinated space robotic system with closed
kinematic constraints are developed. An approach to
position and force control of free-floating coordinated
space robots with closed kinematic constraints is pro-
posed for the first time. Unlike previous coordinated
space robot control methods which are for open kine-
matic chains, the method presented here addresses the
main difficult problem of control of closed kinematic
chains. The controller consists of two parts, position
controller and internal force controller, which regulate,
respectively, the object position and internal forces be-
tween the object and end-effec tors. The inve rse kinemat-
ic control based on mutual mapping neural network of
free-floating dual-arm space robot system without the
basepsilas control is discussed in [20]. With the geome-
trical relation and the linear, angular momentum conser-
vation of the system, the generalized Jacobian matrix is
obtained. Based on the above result, a mutual mapping
neural network control scheme employing Lyapunov
functi ons is d esi gned to co ntr ol the end -effectors to trac k
the desired trajectory in workspace. The control scheme
does not require the inverse of the Jacobian matrix. A
planar dual-arm space robot system is simulated to verify
the proposed control scheme. In [21], the kinematics of
the FFSR is introduced firstly. Then the null space ap-
proach is used to reparameterize the path: the direction
and magnitude are decoupled and no direction error is
introduced. And the Newton iterative method is adopted
to find the optimal magnitude of the joint velocity. A
planar FFSR with a 2 DOFs manipulator is selected to
test the al gorithm and simula tion results illustrate that the
path following is realized precisely. The genetic algo-
rithm with wavelet approximation is applied to nonho-
lonomic motion planning in [22-25]. The problem of
nonholonomic motion planning is formulated as an op-

M. SHIBLI
10
timal control problem for a drift.
The problem of position control of robotic manipula-
tor s b ot h nonr e dund a nt a nd r e dund a nt i n t he ta s k sp ac e is
addressed in [26]. A computat ionall y simple class of task
space regulators consisting of a transpose adaptive Jaco-
bian controller plus an adaptive term estimating genera-
lized gravity forces is proposed. The Lyapunov stability
theor y is used to d er ive t he c ontr ol sc he me. I n [27 ] glo b-
al randomized joint-space path planning for articulated
robots that are subjected to task-space constraints is ex-
plored. This paper describes a representation of con-
strained motion for joint-space planners and develops
two simple and efficient methods for constrained sam-
pling o f j o int co n fi gur a tio ns : tange nt -space sa mpling (TS)
and first-order retraction (FR). In work [28], control-
moment gyroscopes (CMGs) are proposed as actuators
for a spacecraft-mounted robotic arm to reduce reaction
forces and torques on the spacecraft base. With the es-
tablished kinematics and dynamics for a CMG robotic
system, numerical simulations are performed for a gen-
eral CMG system with an added payload. In [29] the
problem of dynamic coupling and control of a space ro-
bot with a free -fl yin g ba se is discussed, which could be a
spacecraft, space station, or satellite. The dynamics of
the system systematically and demonstrate nonlinearity
of parameterization of the dynamics structure is formu-
lated. The dynamic coupling of the robot and base sys-
tem is studeid, and propose a concept, i.e., coupling fac-
tor, to illustrate the motion and force depe nd e ncies.
Dynamics and control of a flexible space robot cap-
turing a static tar get was pre sented in [30] . The dynamics
model of the robot system is derived with Lagrangian
formulation. The control method of flexible space during
capturing target was discussed. Work [31] proposes an
adaptive controller for a fully free-floating space robot
with kinemat ic and d ynamic mod el uncertai nty. In a dap-
tive control design for the space robot, because of high
dynamical coupling between an actively operated arm
and a passively moving end-point, two inherent difficul-
ties exist, such as non-linear parameterization of the dy-
namic equation and both kinematic and dynamic para-
meter uncertainties in the coordinate mapping from Car-
tesian space to joint space. Research [32] addresses
modeling, simulation and controls of a robotic servicing
system for the hubble space telescope servicing missions.
The simulation models of the robotic system include
flexible body dynamics, control systems and geometric
models of the contacting bodies. These models are in-
corporated into MDA’s simulation facilities, the multi-
body dynamics simulator “space station portable opera-
tions training s imulator (SPO TS)”.
Most previous studies describing the dynamics of a
space robo t neglect the coupled d ynamics with a floati ng
environment or consider only abstract external forces/
moments or impulse forces. The target has its own iner-
tial and nonlinear forces/moments that significantly in-
fluence the ones of the space robot and cannot be ignored.
Applying improper forces at the constraint surface may
cause a severe damage to the target and/or to the space
robot and its base satellite or cause the target to escape
away. To accomplish a capture in practice is not instan-
taneous, because the end-effecter needs to keep moving
and applying a force/moment on the surface of the target
until the target is totally captured. Moreover, from tra-
jectory planning point of view, not all trajectories and
displacements (velocities) are allowed due to the con-
servation of momentum and geometric constraints. In
this work, a unified control-oriented modeling approach
is proposed to deal with the kinematics, constraints and
dynamics of a free-flying space robot interacting with a
target satellite. This model combines the dynamics of
both systems together in one structure and handles all
holonomic and nonholnomic constraints in a single
framework. Moreover, this approach allows considering
the generalized constraint forces between the space robot
end-effecter and the target satellite as internal forces ra-
ther than external forces.
Most of the adaptive control algorithms assume the
absence of external forces acting on space robot. As it
can be seen most s tudies ignored considering constraints
imposed by linear momentum, angular momentum, and
contact constraints all together. The kinematics, dynam-
ics, the uncertainty of parameters of a free-flying space
robot and that of the target are considered separately. In
this paper the uncertainty of the combined system as a
whole is considered which gives more global results.
In this paper a unified control-oriented dynamics
model is developed by unifying dynamics of the space-
robot and the target satellite together along with all ho-
lonomic and nonholonomic constraints. Many space
missions can benefit from such a control system, for
example, autonomous docking of satellites, rescuing sa-
tellites, and satellite servicing, where it is vital to limit
the contact force during the robotic operation. It worthy
to monition that the advantage of this approach is consi-
dering the contact forces between the space robot
end-effector and the target as internal forces rather than
external forces. In this paper, inverse based-dynamics
and an adaptive inverse based-dynamics controllers are
proposed to handle the overall combined coupled dy-
namics of the based-satellite servicing robot and the tar-
get satellite all together with geometric and momentum
constraints imposed on the system. A reduced-order dy-
namical model is obtained by finding a common solution
of all constraints using the concept of orthogonal projec-
tion matrices. The proposed controller does not only

M. SHIBLI
11
show the capability to meet motion and contact forces
desired specifications, but also to cope with the under-
actuation problem [33,34].
The paper is organized as follows: In Section 2, mod-
eling of kinematics, linear and angular momentum, and
contact constraints are derived, and then a common solu-
tion for all co nstraints is proposed. In Section 3 , a n ove r-
all dyna mics model is develope d. In Section 4 an inverse
dynamics controller is proposed. An adaptive inverse
dynamics controller is presented In Section 5. Mean-
while, in Section 6, simulation results are demonstrated
to verify the analytical results, and finally in 7 summary
is concluded.
2. Kinematics and Momentum Modeling
2.1. Nomenclature
All generalized coordinates are measured in the inertial
frame unless ano ther frame is mentioned as follo ws
i
m
:
the mass of t he ith body
3
i
IR∈
: the inertia of the ith body
n
qR∈
: the robot joint variable vecto r
12
(,,,)
T
n
qq qq
3
b
RR∈
: the position vector of the centroid of the base
3
T
RR∈
: the position vector o f the target satellite cen-
troid
3
i
rR∈
: the position vector of the i-th joint
/
3
EE
T
RR ∈
: the relative position vector of the target
satellite centroid with respect to the end-effecter (EE)
3
b
VR∈
: the linear velocity of the base
3
b
RΩ∈
: the base angular velocity vector
3
U
: the 3 × 3 identit y matrix
n
R τ ∈
: the joint torque vector
( )
12
,,,
T
n
ττ τ 
6
b
FR∈
: forces and moments
( )
,
b
T
TT
b
f η
act on the
centroid o f ba se sate llite .
2.2. Kinematics
The purpose of this part is to model the kinematics of a
free-flying space robotic manipulator in contact with a
captured satellite as a whole. In this model the contact
between the space robot and the target satellite is as-
sumed established and not escaped.
Our combined system can be modeled as a multi-body
chain system composed of n + 2 rigid bodies. While the
manipulator links are numbered from 1 to n, the base
satellite (body 0) is denoted by b, in particular, and the
( )
1n th+
body (the target satellite) by T. Moreover,
This multi-body system is connected by n + 1 joints,
which are given numbers from 1 to n + 1. Where the
end-effecter is represented as the
( )
1n th+
joint as
sho wn in Figure 1.
We assume that all system bodies are rigid, the contact
surfaces are frictionless and known. Also the effect of
gravity gradient, solar radiation and aerodynamic forces
are weak and neglected. It is assumed also that the base
satellite is rea c tion-wheel actuated.
Referring to Figure 1, the position vector of the ith
body centroid with respect to the inertial frame can be
expressed as
ib ib
RRR= +
(1)
where the relative vector
/b
i
R
is the position of the ith
body centroid with respect to the base frame [35,36].
Upon differentiating both sides of (1) with respect to
time, the relationship between the ith body velo c ity
ibbib i
VVR v=+Ω ×+
(2)
where
i
v
is the linear velocities of the ith body in base
coordinates. Now in the case of any ith body of the ma-
nipulator, the velocity
i
v
can be expressed in terms of
the linear Jacobian matrix as
i
iL
v Jq=
(3)
where
() ()()
1 122
,, ,,0, ,0
i
Liii ii
Jz RrzRrzRr

=×− ×−×−


(4)
Link i
Link n
Link 2
Link 1
O
1
O
2
q
1
q
2
q
3
O
3
O
i
q
i
ib
R
O
i + 1
q
i + 1
O
n
q
n
O
n + 1
q
h
B
Σ
I
Σ
CG
Σ
T
Σ
T EE
R
Figure 1. F ree-fl o ating space robot in contact with a target
satellite.

M. SHIBLI
12
The end-effecter tip velocity is given by
EE
EEbbEE bL
V VRJq=+Ω ×+
(5)
Additionall y, the velocit y
T
V
of the target satellite i n
the reference frame can be obtained by der iving Equa tion
(1) as
T
TbbT bLTTEET
V VRJqRv ω =+Ω ×++×+ 
(6)
Since the target satellite is not stationary, (6) shows
the relative linear and angular velocities T
v
,
T
ω
be-
tween the end effecter and the target satellite and meas-
ured in the base frame.
Another relationship is needed between the ith body
angular ve locity
i
Ω
and joint angular velocity
i bi
ω Ω=Ω+
(7)
where i ω is the angular velocities of the ith body in
base coordinates and i ω
in case of the manipulator is
given by
i
iA
Jq ω =
(8)
where the angular Jacobian
[ ]
12
,, ,,0, ,0
i
Ai
J zz z=

(9)
While in the case of the target satellite, the absolute
angular velocity of can be expressed as
T
T bAT
Jq ω Ω =Ω++ 
(10)
Former analysis will be used in the next analysis to
derive the mome ntum of s free-flying space robot.
2.3. Linear and Angu lar Momentu m
The linear and angular momentum of a multi-body sys-
tem is a key par t in under stand ing the moti on of the s ys-
tem when it is not subjected to external forces. They may
impose kinematic-like constraints when the system is
free of any external force.
The linear momentum P and angular momentum L of
the whole s ystem is give n b y
1
0
n
ii
i
P mV
+
=
≡∑
(11)
( )
1
0
nBi iiii
i
LImR V
+
=
≡Ω+ ×
∑
(12)
By means of (2-10), linear and angular momentum in
(11-12) can then be represented in a compact form as
b bbb
Vb b
bb
b TbT
b TbT
VV Vq
b
Tbq
VVv T
T v
MM M
V
Pq
MM
LM
MM
v MM
ω
ω
ω
Ω
Ω
ΩΩ
ΩΩ




= + 

  Ω
   




+

 


(13)
where each block of the matrix is defined as follows
133
30
b
n
Vi
i
MU mR
+×
=
≡∈
∑
( 14)
/
1
33
0,
bb i
b
n
Vi
i ib
MmR R
+×
Ω= ≠

≡−×∈

∑ (15)
13
0,
bL
i
nn
Vq i
i ib
MmJ R
+×
= ≠
≡∈
∑
(16)
{ }
/
133
0, ()
bi
b
n
ii b
i ib
MImD RIR
+×
Ω=≠
≡ ++∈
∑ (17)
{ }
/
13
0,
bA bL
ii
n
Bn
qi ii
i ib
MIJmRJR
+×
Ω= ≠

≡+ ×∈

∑ (18)
1/
33
b TnEE
VT
M mRR
ω
+
×

≡−× ∈
 (19)
/
33
1
()
bT T
EE
b
in
MmD RIR
ω
×
Ω+
≡ +∈
(20)
33
31
bT
Vv n
MUm R×
+
≡∈
(21)
[ ]
1
33
1
bT n
vn
MmRR
+
×
Ω+
≡−× ∈
(22)
Note that the matrix function
[ ]
R×
for a vector
,,
T
xyz
R RRR  =
is defined as
[ ]
33
0
0
0
zy
zx
yx
RR
R RRR
RR
×

−

×≡− ∈


−

(23)
and
[ ][ ]
22
2 233
22
()
T
y zxyxz
xyx zyz
xzyzx y
DR RR
R RRRRR
RRR RRRR
RRRRRR
×
≡× ×

+− −

=−+− ∈


−− +

(24)
and the sub-matrices of the Jacobian of the ith body
representing the linear and angular parts are defined be-
for e .
Note that as in (13) the system is subjected to a non-
holonomic (non-integrable) constraint because of con-
servation of angular momentum in the absence of exter-
nal forces. Note that the momentum constraints are not
purely kinematical because of the inertial characteristics
it carries in. Thus, this constraint is called kinematics-
like. The physical meaning behind these constraints is
that they restrict the kinematically possible displace-
ments (possible velocities) of the individual parts of the
system. On the contrast, the linear momentum results i n a
holonomic (integrable) co nstraint.
Now assuming zero initial conditions then linear and
angular momentum is give by

M. SHIBLI
13
0
0
b bbb
Vb b
bb
b TbT
b TbT
VV Vq
b
Tbq
VVv T
T v
MM M
Vq
MM M
MM
v MM
ω
ω
ω
Ω
Ω
ΩΩ
ΩΩ




= + 

  Ω
   




+

 


(25)
Then it is possible that the relative linear and angular
velocities of the target satellite can calculated as
1
1
b bb
b TbT
Vb
b TbTbb
b TbTb
b TbTb
VV
V Vvb
TTb
Tv
V VvVq
vq
MM
MM V
MM vMM
MM M
q
MM M
ω
ω
ω
ω
ω
Ω
−
Ω
Ω
ΩΩ
−
ΩΩ Ω




= −



 Ω

  


 
− 
 
 

(26)
Equations (26 ) enables us to calculate the targe t veloc-
ities
[ ]
TT
v ω
witho ut me as ur e me nt s.
2.4. Contact Constraints
We assume that the end-effecter moves on a sub-surface
of the target satellite S and the profile of this surface is
known so that it can be d efined as
( )
:,,, , ,.SFxy zcconst αβγ = =
(27)
Let
χ
be the vector of generalized coordinates of the
robot end-effecter in the target frame. The end-effecter,
as a result of the contact with the target, is subjected to
holo no mic kinemati c c on st ra i nts de fi ned i n the c o nstr aint
frame as
( )
0 χ Φ=
(28)
where
():
nm
RRΦ⋅ →
is twice differe ntiable . T he robo t
joint and target coordinates are related through the for-
ward kienmatic function
( )
fq
χ = (29)
Now differe ntiating (28) with r e sp e c t to time gives
( )
0
χχ
χ
∂Φ =
∂
(30)
Also differentiating (29) with respect to time
( )
fqq
q
χ
∂
=∂

(31)
Substituting (31) into (30) (chain rule) yields to
( )()
0
fqq
q
χ
χ
∂Φ ∂=
∂∂

(32)
where the matrix
()( )
fq
Jq
θ
χ
χ
∂Φ ∂
=∂∂
is the Jacobian
matr ix
2.5. Common Solution of the Constraints
In this section we study the constraints on a space robot
in contact with a target satellite in one form. This entire
system is subjected to holonomic and nonholonomic
constraints at the same time. These Holonomic constraint
are usually given in algebraic form relating the genera-
lized variables (28). Now differentiating the holonomic
constraint at the velocity level a s in (30-32) leads to
( )
0J
θ
θθ =

(33)
where
( )
J
θ
θ
is the Jacobian of the holonomic con-
straint as defi ned in (32).
On the other hand, the conservation of momentum
holds two types of constraints: linear momentum which
is holnomic; and nonhol nomic co nstraints come into play
as a result of the conservation of the angular momentum.
These momentum constraints are not given in algebraic
form, but there are given in kinematical-like form as in
(13) and can be rewritten in a compact form as
( )
0
Bc θθ =

(34)
where 0 c represents the vector of the initial conditions
of the momentum. Equation (34) has k momentum con-
straint equations with
kN≤
, where N is number of
generalized coordinates. The purpose of representing
holonomic constraints in the form (33) is to treat both
holonomic and nonholonmic constraints at the same dif-
ferential level. But a difference exists in the matter of
initial conditio ns. Holono mic constraints are r estricted to
position initial conditions, but nonholonomic are only
restricted to their momentum conditions.
Now, all holonomic and nonholonomic constraints can
be combined together as
( )
( )
0
0 J
c B
θ
θθ
θ


=




 (35)
where the ne w combined matrix
( )( )
T
TT
JB
θ
θθ


is
of dimension
() ()
1km n+ ×+
. This implies a set of
( )
km+
linear equations with θ
 as vector of the gene-
ralized variables. Since matrices
( )
J
θ
θ
and
( )
B θ
have the same number of columns, we now seek for their
common solutions, if exist, expressing them in terms of
the solutio ns of (33) and (34). T he common solutions of
(33) and (34) are the solutions of the combined con-
straints (35). From the theory of linear algebra, the solu-
tions of Equations (33) and (34) constitute the intersec-
tion manifold
( )
{ }
( )
{ }
N JBcN B
θ
+
+
(36)
where
( )
NJ
θ
and
( )
NB
are the null space of
J
θ
and B, respectively. And the upper right script +

M. SHIBLI
14
represents the Pseudoinverse. Equation (36) is the set of
solutions of (33) and (34) is consistent if (36) is non-
empty. The n the common solution [ 36] is the manifolds
(a)
( )( ) ( )
( )
( ) ( )
0
NB NJ NJ
PPPBcNJN B
θθ
θ
++
+ ++
(37)
(b)
( ) ( )
( )
( ) ( )
0 ()0 NB NB
NJ
Bc PPPBc
NJ NB
θ
θ
+
++
=−+
++
(38)
(c)
( )
( ) ( )
0
JJBBBcN JN B
θθ θ
+
+ ++
=++
(39)
where
()N
P
⋅
is the projection matrix on the null space of
a given matrix ()⋅.
Since each of the manifolds given in (37)-(39) give a
solution of the combined system (35), these expressions
can also be used to get the generalized (pseudo-inverse)
of the combined matrices. Each of the following expres-
sions i s a
{ }
1,2,4 −
inverse of the combined matrix
( )()
T
TT
JB
θ
θθ


:
(d)
( )( ) ( )
( )
0NB
NJ NJ
XJP PPJB
θθ
θθ
+
+ ++
 
= ++−
 
(40)
(e)
( ) ( ) ( )
( )
0NB NB
NJ
YBP PPJB
θ
θ
+
+ ++
 
= −+−
 
(41)
(f)
( )
ZJJ BBJB
θθ θ
+
+ +++

= +
(42)
Moreover, if
()( )
{ }
0RJ RB
θ
∗∗
=
(43)
then each expressions of (40-42) is the Moore-Penrose
inverse of
( )( )
T
TT
JB
θ
θθ


.
3. Generalized Dynamics Modeling
To drive the dynamic equation of a space robot interact-
ing with a target, the total system kinetic energy as the
total summation of the transitional and rotational energy
of each body in the system can be expressed as
( )
1
0
1
2
i
nT Tb
iii ii
i
TmV VI
+
=
≡+Ω Ω
∑
(44)
where
i
V
and
i
Ω
is the transitional and rotational
velocities of i-th body , respectively, or it can be rear-
ranged
[ ]
1
2
b bTT
T V qv ω =Ω 
bbbbb TbT
Vbbb TbT
bb
VqqT T
bb
VqT TT
bT bTT
V vvqvvT
bT bTT TT
VVVq VVvb
Tqv
b
TT q qqv
TT TT
v
T TTTT
v
MM MMMV
MMMM M
M MMMMq
MM MMM
v
M MMMM
ω ωω
ω
ω
ω
ω
ωω
ω
Ω
Ω
Ω
Ω
Ω
ΩΩΩ Ω





Ω



×










(45)
where the block matrix in (45) is the inertia matrix and
the sub-matrices are defined previously in (14-22) and
also
{ }
( )()
111
0,
ii
nnn
T BT
qiLi LiAiA
i ib
MmJJIJ JR
++× +
= ≠
≡⋅+ ∈
∑
(46)
( )
/
33
11
TT
EE
nn
MIm DRR
ω
×
++
≡+ ∈ (47)
( )
31
11
TT EE
n
T TT
qn ATn LT
MIJmJ RR
ω
×+
++

≡ +×∈

(48)
( )
31
1
T
n
T
qvnLT
Mm JUR
×+
+
≡∈
(49)
33
1
TT
vnT EE
M mRR
ω
×
+

≡−×∈

(50)
33
31
T
vn
M UmR
×
+
≡∈
(51)
Note that the inertia matrix M defined in (45) is sym-
metric positive definite. Now define
TT
TTT TT
bb
V qv θω

= Ω


(52)
Then the total kinetic energy can be expresses in a
compact fro m
( )
1
2
T
TM θ θθ =
(53)
From the kinetic energy formulation, the dynamics
equations can be derived by using the Lagrangian ap-
proach. Since there is no potential energy accounted in
our system, the Lagrange function L is equal to the ki-
netic energy T then becomes
T
dT TJ
dt
τλ
θ
θ
∂∂
−=+
∂
∂
(54)
where
λ
is the vector of unknown Lagrangian multip-
liers.
The holonomic constraints are behind the generalized
constraint forces as a result of the contact between the
manipulator end-effecter and the surface of the target
satellite. The combined system dynamics model can be
represented as (assuming the target satellite is unac-
tauted)

M. SHIBLI
15
0
bbbbb TbT
Vbbb TbT
bb b
VqqT T
bb
T
VqT TT
bT bTT
T
V vvqvvT
bT bTT TT
VVVq VVvV
b
Tqv
b
TT q qqvq
TT Tw
T v
T TTTTv
v
MM MMMC
V
MMMM MC
M MMMMC
q
C
MM MMM
vC
M MMMM
ω ωω
ω
ω
ω
ω
ωω
ω
Ω
Ω
Ω
Ω
Ω
ΩΩΩ ΩΩ







Ω





+




















0
0
T
T
T
bL
bL T
bA
bA T
T
Tw
T
Tv
J
F
J
F
J
J
J
θ
λ τ








=+ 








(55)
The dynamic developed in (55) along with the com-
bined constraints in (35) completes the overall modeling
of a space ro bot interacting with a target satellite.
4. Inverse Dynamics Control
The basic idea of inverse dynamics control is to seek a
nonlinear dynamics control law that cancels exactly all
nonlinear terms in the system dynamics (55) so that the
closed loop dynamics is linear and decoupled [6].
Now as su ming zero initial conditions in (13) and (35),
the overall dynamics subjected to the constraints can be
expressed in a compact form as
c
MC F θ θτ +=+
 
(56)
where the inertia matrix M is defined in (55), the nonli-
near vector
( )
,C θθ

is the centrifugal/Coriolos forces
the generalized constraint forc es are
T
c
FJ λ =
,
m
R λ ∈
is the vector of unknown Lagrangian multipliers,
T
J
and
τ
are defined, respectively, as
t
t
T
bL
T
bA
T
Tq
T
Tw
T
Tv
J
J
J
J
J
J




=





,
0
0
L
A
b
b
F
F
ττ




=



as in (55), and finally the
constraint matrix
( )
( )
( )
J
AB
θ
θ
θθ

=


as given in ( 35).
In the constraint Equation (35) there are
( )
km+
li-
near equations and N of the generalized velocities θ

. It
clear that there are fewer equations than unknowns, this
implies the existence of infinite solutions. From the
theory of linear algebra, the solution of (35) can be given
by
( )
S θ θν =

(57)
where
( )
() ()
N Nkm
SR θ
× −−
∈
is an orthogonal projector of
full rank and belong to the null space of
( )
Aq
and the
vector
( ) Nkm
R ν
−−
∈

can be chosen arbitrary. It implies
that 0
T
SA=. [37]
Now differentiating (57) at the acceleration level with
respect to time yields
SS θνν = +
 
 
(58)
Upon substituting the velocities (57) and the accelera-
tion (58) into the dynamics (55) we obtain
( )
c
MSMS CSF ν ντ ++ =+

 
(59)
Let us define the controller as
τ
( )
( )
T
dD Pc
HSCSHSKeK eJ τ ννλ =++++ −
 
(60)
where the position tracking error is defined as
pd
e νν = −
and c λ is defined as
cdFFI F
KeKe dt λλ =−−
∫
(61)
where
Fd
e λλ
= − and the gain matrices P
K
,
D
K
,
F
K
and
I
K
are chosen as diagonal with positive ele-
ment s .
Note that the input to the proposed controller (60) are
the joint angles and velocities, angular velocity of the
base, relative velocities of both satellites, contact forces
and the output o f the controller is the joint torques. Note
also that TNkm sR τ
−−
∈ has the advantage of overcom-
ing the underactuation of the syste m as a res ult of target
satellite jet shutdown or failure, and the inputs provided
by the robot and the base is enough to control the whole
system. This is because the number of constraints
km+≥
the number of passive inputs of the target satel-
lite.
Now let us substitute the control law (60) into the dy-
namics (59), then, the closed loop dynamics is given by
( )
( )
0
TpDpPp
TT FFI F
SMSeKeKe
SJKeKedt
++
=−+
=
∫
 
(62)
Since J belongs to the null space of S, that is,
0
TT
SJ = and since by the virt ue of (5 7), the p rojectio n
matrix
( )
Sq
and its transpose are of full rank, and the
inertia matrix is symmetric positive definite, then
T
SMS is also a positive definite. Now we need to verify
the terms inside the brackets in (60) are zero. This condi-
tion can be guaranteed by choosing the proper positive
gains D K and P K such that
d
νν →
as
t→∞
. If
the gain matrices
D
K
and
P
K
are chosen as diagonal
with positive diagonal elements, then the resulted closed
loop dynamics is linear, decoupled and exponentially
stable. Global stability can then be guaranteed. The
closed loop dynamics natural frequency and damping
ration can be chosen to meet specific requirements. Also

M. SHIBLI
16
by inspecting the right hand side we can see that the
FFI F
KeKe dt+
∫
can be guaranteed to be zero by
choosing the suitable gain matrices
F
K
and
I
K
.
Now we can readily summarize the hybrid inverse-
dynamics controller in the following theorem:
Theor em 1 : Fo r the d ynamic s ystem given in (5 9 ) and
subjected to constraints (35), the inverse dynamics con-
trol law defined by (60)-(61) is globally stable and guar-
antees zero steady state and force tracking errors.
To further improve the dynamic response in case of
system parameters uncertainty, an adaptive controller
would serve that objective as in the nest section.
5. Adaptive Inverse Dynamics Control
Similar to the analysis followed in the previous section
and recalling (60)
( )
T
HSHS CSJ νντλ ++ =+

 
(63)
Now we assume that there are some uncertainties in the
syst e m p a ramet er s s uc h a s mas se s a nd i nertias a nd fo r this
reason an adaptive control approach will be investigated.
The dyna mics (63) can be represented b y benefiti ng fro m
the prope rty of l inearity in parameters as [38,39]
11
HC ννα +=Υ
 
(64)
where
1
H HS=
,
( )
1
CHS CS= +

, Y is an
( )
N Nm×−
matrix of known functions and known is the regressor,
and
α
is an
( )
Nm−
-dimensional vector of the sys-
tem parameters. After examining the structure of dy-
namics (55), three properties are obtained:
Property 1:
The modified inertia matrix
( )()( )
2
T
HSH S
θ θθ
=
is symmetric po sitive definite.
Property 2: If Proper ty 1 is verified, then
( )
22
2HC−
is skew-symmetric matrix where
21
T
C SC=
.
Property 3: The d ynamics (64) is linear in its parame-
ters.
The nonlinear co ntrol l aw is proposed to have th e fo rm
( )
11
ˆ ˆT
dD Pc
HKeKeCJ τ ννλ =+++−
   (65)
where c dFF
Ke
λλ = − and
F
K
is a positive definite
diagonal matrix for the force control feedback gain and
Fd
e λλ
= −, and where
1
ˆ
H
and
1
ˆ
C
are the estimates
of
1
H
and 1
C
, respectively. Note that the geometric of
the Jacobian in (56) is assumed to be determined.
Then the dynamics (55) can be modified to
11
ˆ ˆˆ
HC
ννα +=Υ
 
(66)
where
ˆ
α
is the estimated vector o f parameters
α
.
Upon substituting (65) into (63), and by adding and
subtracting at the same time the term
1
ˆ
Mv

on the left
hand side of (65) we get
()( )
1 1 11
11
ˆˆ
ˆ ˆ
TT
dD Pc
HHHC
HKeKeCSJ
ν ν νν
νν λλ
−++
=+++−−
 
 

(67)
Rearranging (58) and canceling out the similar terms,
yields
( )
1 111
11
ˆ ˆ
ˆˆ dDPPP
HHCC
HKe KeH
Y
ν ννν
νν ν
α
− +−
=−++−
=

  

(68)
or,
( )
11 1
ˆPDPPP
H CHeKeKeY νν α +=+ +=
  
 
(69)
where Pd
e
νν
= −
   , and
()( )() ˆ
⋅ =⋅−⋅

. The closed loop
dynamics error can be written as
( )
1PDPPP
eKeKeY α ++ =
  
(70)
where
1
11
ˆ
HY Y
−
=
.
It is possible now to express the error dynamics (70) in
a state space form as
xAx B
α =+ 

(71)
where
1
, ,
p
pPD
eOI O
xA B
eKK Y
  
= ==
  
−−
 


(72)
where A is a Hurwitz matrix, that is, the real parts of its
eigenvalues are negative, which guarantees globally ex-
ponentially stability. Based on the state space formula-
tion and Lyapunov techniques an adaptive control law
can be chosen as
11TT
YB Px α
−
= −Γ


(73)
where
0Γ>
and s ym met ri c , a nd P is a un ique po sitive
definite solution to the Lyapunov equation
T
A PPA+=
Q−
where Q is a positive definite symmetric.
Proof: Let the Lyapo nuv candidate func tion chosen as
TT
VxPx αα = +Γ

(74)
Now if take the time derivative of V along the trajec-
tories of (71) and by using the adaptation law (65), one
gets
( )( )
( )
( )
11
11
11
11
1
11
1
1 11
TT TT
TT
T
TTT TT
TTTTT TT
TT TTTT
T TTTT
TTTTT
VxPxxPx
Ax BYPx xPAx BY
YB PxYB Px
xAPxYB PxxPAxx PBY
x P BYYB Px
xA PPAxYBPx
x P BYx P BYY
αααα
αα
αα
αα
αα
α
α αα
−−
−−
=+ +Γ+Γ
=+ ++
− ΓΓ−ΓΓ
= +++
−ΓΓ−ΓΓ
= ++
+−−








 
TT
B Px
(75)

M. SHIBLI
17
By canceling out equivalent terms, this reduces to
0
T
VxQx
=−≤

(76)
Since
V

is negative semidefinite with regard to x
and the parameter error, and V is lower bounded by zero,
V remains bounded in the time interval
[ )
0,∞
. This fact
can be stated as
0Vdt
∞− <∞
∫ (77)
Now if we assume that
x

is bounded then from (74)
V

is bonded. If
V

is bounded then
V

is uniformly
continuous. If
V

is uniformly continuous and has a fi-
nite integral as given in (76) then by Barbalat’s lemma
0V→

as
t→∞
which i mplie s
0x→
as
t→∞
.
Substi tuting the control law (65) and (73) into the d y-
namics (55) yields
( )
Tc
JY λλ ασ − =−=

(78)
where
σ
is a bounded function. T hus
( )
1
TFF
JeK I σ
−
= +
(79)
and the force tracking error
( )
d
FF−
is bounded and
can be adjusted by changing the feedback gain
F
K
.
Thus, the previous adaptive algorithm can be concluded
in the following theor em:
Theorem 2: For t he dyna mic s yst e m gi ven in (6 3 ) a nd
subjected to constraints (35), the adaptive control law
defined by (69) and (73) is globally stable and guarantees
zero steady state and force tracking errors.
6. Simulation Results
This section will demonstrate the kinematics, dynamics
and controller presented in this paper as follows.
Part A (PD controller): A 6-DOF space robot arm
mounted on a base satellite is used to demonstrate the
analytical results. We assume that the end-effecter estab-
lished a co ntac t with a target satellite. This target satellite
is assumed to be totally floating and unactuated due to
the thrusters’ failure. The mass of the base servicing sa-
tellite is chosen as 30 0 kg, the masses o f the 6-robot arm
as [20 20 15 10 10 10] kg, and 1000 kg for the target
satellite. T he initial linear velocities of both the base and
the target are assumed to be 10 m/sec to keep a constant
linear relative velocity while conducting the task and to
avoid any damage. Two different PD controllers are used,
to control the base satellite reaction wheels and robot
arm as:
( )
,rwPbb desDb
kAA k τ =− +Ω
,
( )
12arm des
k qqkq τ = −−
,
respectively, where
[ ]
10 10 10
T
P
k=
,
[ ]
10 10 10T
D
k=
,
1
10k=
,
2
10k=
. The simulation results are shown in
Figures 2-6. Figures 2 and 3 shows a very slight varia-
tion in the base sate llite attitude and s mall increase i n its
linea r velocity over 3 minutes. O n the contrast, Figures 5
and 6 show that t he target dr ifts away, b ut a slig ht rise i n
its linear ve locity. The drift is due to the as sumption that
the target satellite is unactuated (passive) and there is no
regular control over its linear and angular motion. More
complicated control techniques other than the PD con-
troller should be investigated to cope with the unactua-
tion problem. Finally, Figure 4 shows that most robot
arm links approaches their desired values. Several simu-
lations are run and show that linear forces are preferable
on angular forces, and as long as the forces are relatively
small comparing to the target mass/inertia, its linear ve-
locities slig htly change.
Part B (Inv erse Dynamics Controller): For simulation
we assume that the end-effecter of the servicing space
robot manipulator has established a contact with a target
satellite. The robot arm is composed of 6-DOF and
mounted on base satellite is used to demonstrate the
ti me, sec
w
0
, rad/sec
× 10
-3
0.5
0
–0.5
–1
–1.5
–2
–2.5
–3
0 20 40 60 80 100 120
Figure 2. YRP-angles of the base satellite (10 e-3).
ti me, sec
v
0z
, m/sec
10.05
0 20 40 60 80 100 120
ti me, sec
0 20 40 60 80 100 120
ti me, sec
0 20 40 60 80 100 120
10
9.95
0.01
0
–0.01
0.02
0.01
0
v
0y
, m/sec
v
0x
, m/sec
Figure 3. Linear velocity of the base satellite.

M. SHIBLI
18
ti me, sec
q(t)
1.2
–0.2
0 20 40 60 80 100 120
1
0.8
0.6
0.4
0.2
0
Figure 4. Robot arm angles.
ti me, sec
w1, rad /sec
0.45
–0.05
0
20
40 60 80 100 120
0.05
0
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 5. YRP-angles of the target satellite.
v
tz'
, m/sec
10.05
ti me, sec
0 20 40 60 80 100 120
10
0.04
v
ty'
, m/sec
v
tx'
, m/sec
ti me, sec
0 20 40 60 80 100 120
ti me, sec
0 20 40 60 80 100 120
0.02
0
0.04
0.02
0
10.1
Figure 6. Linear velocity of the target satellite.
analytical results. This target satellite is assumed to be
totally floating and unactuated due to the thrusters’ fail-
ure. The mass of the base servicing satellite is chosen as
300 kg, the masse s of t he 6-robot arm as [10 10 10 10 10
10] kg, and 1500 kg for the target satellite. The initial
linear velocities of both the base and the target are as-
sumed to be 20 m/sec to keep a constant linear relative
velocity while conducting the task and to avoid any
damage. All other initial conditions are assumed to be
zero.
The desired values for the robot angular position are
chosen as
[ ]
0.3,0.2,0.4,0.45,0.5,0.0
des
q=
. The contact
forces are assumed to be linear and only in the
x-direction. T he motio n and force gain diagonal matrices
P
K
,
D
K
,
F
K
,
I
K
are chosen as
( )
30,30,30,30,30,30,30,30,30,30
P
K diag=
,
( )
25,25,25,25,25,25,25,25,25,25
D
K diag=
,
( )
50,50,50,50,50,50
I
K diag=
,
( )
10,10,10,10,10,10
F
K diag=
The simulation is used to verify the analytical results
and whether the proposed controller can track the desired
motion and the specified contact forces and, moreover,
overcome the under-actuation (passivity) in the target
satellite. The simulation results are shown in Figures
7-14. Figures 7 and 8 show, respectively, a fast response
for both linear and angular velocities of the base. Fig-
ures 9 and 10 represent, respectively, the error response
of robot arm angular position and velocities. The con-
troller is able to bring the links to the steady state posi-
tion at around 25 sec. Figure 11 shows the joints actua-
tors response which approaches zero after 25 sec. In
Figures 12 and 13, the linear and angular velocities error
response of the target satellite present a noticeable fast
response. Finally, Figure 14 shows the error in the La-
grangian force multiplier. The error gets close to zero at
time 20 sec.
ti me, sec
Base Satellite Angular velocity error, rad/s ec
× 10
-3
5
–5
0 5 10 15 20 25 30
4
3
2
1
0
–4
–3
–2
–1
Figure 7. Base satellite Ang ul ar velocity err or.

M. SHIBLI
19
ti me, sec
Base Satellite linear velocity error, m/sec
0
–0.002
0 5
10 15 20 25 30
–0.004
–0.006
–0.008
–0.01
–0.012
–0.014
–0.016
–0.018
Figure 8. Base satellite linear velocity error.
ti me, sec
Rob ot arm pos ition er r o r, rad/sec
0.2
0 5 10 15 20 25 30
–0.5
0.1
0
–0.4
–0.3
–0.2
–0.1
Figure 9. Spac e r obot arm angular position error.
ti me, sec
Robot arm velocity error, rad/sec
0.4
0 5 10
15 20 25 30
–0.05
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Figure 1 0. Space rob ot arm angular velocity error.
Part C (Adaptive Inverse Dynamics Controller): For
simulation we assume that the end-effecter of the servic-
ing space robot manipulator has established a contact
ti me, sec
Torque, N.m
16
0 5
10 15 20 25 30
–2
14
12
10
8
6
4
2
0
–4
Figure 1 1. Space robot arm actuation torque.
ti me, sec
Target Satellite Angular velocity error, rad/s ec
5
0 5 10 15 20 25 30
–1
× 10
-3
4
3
2
1
0
–2
–3
–4
–5
Figure 1 2. Target satellite angular velocity error.
ti me, sec
Target Satell it e li ne ar vel oc it y er r or, m/sec
14
0 5 10 15 20 25 30
× 10-3
12
10
8
6
4
2
0
-2
Figure 1 3. Target satellite l inear velocity error.
with a target satellite. The robot arm is composed of
6-DOF and mounted on base satellite is used to demon-
strate the analytical results. This Hubble Telescope is

M. SHIBLI
20
assumed to be totally floating and unactuated due to the
thrusters’ shutdown. The mass of the base servicing sa-
tellite is chosen as 3000 kg, the masses of the 6-robot
arm as [100 100 50 50 20 10] kg, and 11000 kg for the
Hubble Telescope as shown in Table 1. The initial rela-
tive linear velocities of both the base and the target are as-
sumed to be zero m/sec to keep a constant linear relative
veloci ty while conducting the task a nd to avoid any damage.
All other i nitial conditio ns are assumed to be zero. The d e-
sired values for the robot angular position are chos en as
[ ]
0.3 0.2 0.4 0.45 0.5 0.0
des
q=
.
The contact forces are assumed to be linear and only in
the y-direction and with desired value as 0
des
λ = .
The motion and force gain diagonal matrices
P
K
,
D
K
,
F
K
are in the simulation as:
( )
20,20,50,50,50,50,50,50,30,30
P
K diag=
( )
20,20,50,50,50,50,50,50,40,40
D
K diag=
( )
80,80,80,80,80,80
F
K diag=
.
time, s ec
Lagrangian error
2
0 5 10 15 20 25 30
1
0
–1
–2
–3
–4
–5
–6
–7
–8
Figure 1 4. Lagrang ian error.
Table 1. Simulated Combined system parameters.
Link i Mass (kg)
xx
I
(kg.m2)
yy
I
(kg.m2)
zz
I
(kg.m2)
Base Sat. 3000 1000 1000 1000
Hubble 11000 3000 3000 3000
Link 1 100 30 30 30
Link 2 100 30 30 30
Link 3 50 15 15 15
Link 4 50 15 15 15
Link 5 20 7 7 7
Link 6 10 3 3 3
We assumed that the space robot end-effector move on
the surface of a Hubble telescope in the z-direction as
sho wn in Figure 15.
The simulation is used to verify the analytical results
and whether the proposed controller can track the desired
motion and the specified contact forces and, moreover,
overcome the under-actuation (passivity) in the target
satellite. The simulation results are shown in Figures
16-23. Figures 16 and 17 show, respectively, a fast re-
sponse for both linear and angular velocities of the base.
Figures 18 and 19 represent, respectively, the error re-
sponse of robot arm angular position and velocities. The
controller is able to bring the links to the steady state
position at around 40 sec. Figure 20 shows the joints
actuators response which approaches zero after 40 sec. In
Figures 21 and 22, the linear and angular velocities error
response of the target satellite present a noticeable fast
response. Finally, Figure 23 shows the error in the La-
grangian force multiplier. The error gets close to zero at
time 40 sec.
Figure 15. A free-flying space robot conducting a main ten-
ance task on the surface of the Hubble Space Telescope.
ti me, sec
Base Satellite Angular velocity error, rad/sec
× 10
-3
0.5
–0.5
0 10 20 30 40 50 60
0
–1
–1.5
–2
–2.5
–3
–3.5
–4
Figure 1 6. Base satellite Angular velocity error.

M. SHIBLI
21
ti me, sec
Base Satellite linear velocity error, m/sec
× 10
-3
–0.5
0 10 20
30 40
50 60
0
–1
–1.5
–2
–2.5
–3
Figure 1 7. Base satellite linear veloci ty error.
ti me, sec
Rob ot arm pos ition er r o r, rad/sec
–0.5
0 10 20 30 40 50 60
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
Figure 1 8. Space robot ar m a ngular position error.
ti me, sec
Robot arm velocity error, rad/sec
–0.1
0 10 20 30 40 50 60
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
–0.05
Figure 1 9. Space robot arm angular velocity error.
7. Conclusions
An ove rall control-oriented modeling approach is devel
oped to deal with the kinematics, constraints and dy-
namics of a free-flying space robot interacting with a
target satellite. Treating kinematic constraints at the dif-
ferential level together with the constraints of linear and
angular momentum, a common solution is proposed.
Finally, based on the Lagrangian approach, a generalized
dynamical model suitable for control algorithms is de-
veloped. This framework allows considering the genera-
lized constraint forces between the end-effecter and the
target satellite as internal forces rather than external
forces. Future work will focus on designing a controller
using inverse dynamic and adap t ive/robust te chni ques.
The hybrid inverse-dynamics based controller pro-
posed in this paper is capable to track the desired motion
values and contact force specifications. The reduced-
order dynamics by using the orthogonal projector tech-
niques does not suffer of passivity or under-actuation.
ti me, sec
Robot ar m position err or, rad/sec
–5
0 10 20 30 40 50 60
25
20
15
10
5
0
–10
Figure 2 0. Space robot ar m a c tuation torq ue .
ti me, sec
Target Satellite Angular velocity error, rad/sec
× 10
-3
4.5
0 10 20 30 40 50 60
4
3.5
3
2.5
2
1.5
1
0.5
0
Figure 2 1. Hubble Telescop e angular velocity error.

M. SHIBLI
22
time, sec
Target Sate ll it e li ne ar vel oc it y err or, m/sec
× 10
-3
2
0 10 20 30 40 50 60
1.5
1
0.5
0
–0.5
–1
–1.5
Figure 2 2. Hubble Telescop e linear velocit y error.
ti me, sec
Lagrangian error
2
0 10 20 30 40 50 60
–12
0
–2
–4
–6
–8
–10
–14
Figure 2 3. Lagrangian multiplier error.
This controller deals with all geometric constraints and
momentum constraints. Future work will focus on de-
signing a controller using adaptive control approach.
An adaptive inverse-dynamics based controller proposed
in this paper is cap able to track the desired motion values
and contact force specifications and moreover, to over-
come the combined system parameters uncertainty.
Moreover, the reduced-order dynamics by using the or-
thogonal projector techniques does not suffer of passivity
or under-actuation.
The results of the approach proposed in this paper is
advantageous comparing to the most studies which de-
scribe the dynamics of a space robot and neglect the
coupled dynamics with a floating environment or they
consider only abstract external forces/moments or im-
pulse forces. Most of the adaptive control algorithms
assume the absence of external forces acting on space
robot. As it can be seen most studies ignored considering
constraints imposed by linear momentum, angular mo-
mentum, and contact constraints all together. This paper
work introduces a unified control-oriented modeling ap-
proach is proposed to deal with the kinematics, con-
straints and dynamics of a free-flying space robot inte-
racting with a target satellite along with parameters un-
certainty.
8. Acknowledgements
The authors would like to acknowledge the Canadian
Space Agency (CSA) for supporting this research.
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