Singularity-Free Dynamic Equations of Spacecraft-Manipulator Systems

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1 Singularity-Free Dynamic Equations of Spacecraft-Manipulator Systems Pål J. From Kristin Y. Pettersen Jan T. Gravdahl Engineering Cyernetics, Norwegian University of Science & Technology, Norway Astract: In this paper we derive the singularity-free dynamic equations of spacecraftmanipulator systems using a minimal representation. Spacecraft are normally modeled using Euler angles, which leads to singularities, or Euler parameters, which is not a minimal representation and thus not suited for Lagrange s equations. We circumvent these issues y introducing quasi-coordinates which allows us to derive the dynamics using minimal and gloally valid non-euclidean configuration coordinates. This is a great advantage as the configuration space of a spacecraft is non-euclidean. We thus otain a computationally efficient and singularity-free formulation of the dynamic equations with the same complexity as the conventional Lagrangian approach. The closed form formulation makes the proposed approach well suited for system analysis and model-ased control. This paper focuses on the dynamic properties of free-floating and free-flying spacecraft-manipulator systems and we show how to calculate the inertia and Coriolis matrices in such a way that this can e implemented for simulation and control purposes without extensive knowledge of the mathematical ackground. Keywords: Spacecraft-manipulator dynamics, root modeling, singularities, quasi-coordinates. 1. INTRODUCTION A good understanding of the dynamics of spacecraftmanipulator systems is important as these systems are emerging as an alternative to human operation in space Hughes, 22; Moosavian and Papadopoulos, 27). Operations include assemling, repair, refuelling, maintenance and operations of satellites and space stations. Due to the enormous risks and costs involved with launching humans into space, rootic solutions evolve as the most cost-efficient and reliale solution. Modeling spacecraft-manipulator systems is quite different from standard root modeling. Firstly, the manipulator is mounted on a free-floating unactuated) or free-flying actuated) spacecraft. There is thus no ovious way to choose the inertial frame. Secondly, the motion of the manipulator affects the motion of the ase, which results in a set of dynamic equations different from the fixedase case due to the dynamic coupling. Finally, we need to consider the effects of the free fall environment. Roustness of these systems is still a major concern for space operators. We therefore derive the singularity-free dynamic equations of spacecraft-manipulator systems using a Lagrangian framework. It is a well known fact that the kinematics of a rigid ody contains singularities if the Euler angles are used to represent the orientation and the joint topology is not taken into account. One solution to this prolem is to use a non-minimal representation such as the unit quaternion. These are, however, not generalized coordinates and can thus not e used in Lagrange s equations. This is a major drawack when it comes to modeling vehicle-manipulator systems as most methods used for root modeling are ased on the Lagrangian approach. It is thus a great advantage if also the vehicle dynamics can e derived from the Lagrange equations. The use of Lie groups and algeras as a mathematical asis for the derivation of the dynamics of multiody systems can e used to overcome this prolem Selig, 2). We then choose the coordinates generated y the Lie algera as local Euclidean coordinates which allows us to descrie the dynamics locally. For this approach to e valid gloally the total configuration space needs to e covered y an atlas of local exponential coordinate patches. The appropriate equations must then e chosen for the current configuration. The geometric approach presented in Bullo and Lewis 24) can then e used to otain a gloally valid set of dynamic equations on a single Lie group, such as a spacecraft with no root attached. Even though cominations of Lie groups can e used to represent multiody systems, the formulation is very complex and not suited for implementation in a simulation environment. In Kwatny and Blankenship 2) quasicoordinates was used to derive the dynamic equations of fixed-ase rootic manipulators using Poincaré s formulation of the Lagrange equations. In Kozlowski and Herman 28) several control laws using a quasi-coordinate approach were presented, ut only roots with conventional 1-DoF joints were considered. Common for all these methods is, however, that the configuration space of the system is descried as q R n. This is not a prolem when dealing with 1-DoF revolute or prismatic joints ut more complicated joints such as all-joints or free-floating joints then need to e modeled as compound kinematic joints Kwatny and Blankenship, 2), i.e., a comination of 1-DoF simple kinematic joints. For joints that use the

2 q 4 q 1 q 3 q 2 Ψ Ψ Fig. 1. Model setup for a root attached to a vehicle with coordinate frame Ψ and inertial reference frame Ψ. Euler angles to represent the orientation this leads to singularities in the representation. In this paper we follow the generalized Lagrangian approach presented in Duindam and Stramigioli 28) which allows us to comine the Euclidean joints and more general joints, i.e., joints that can e descried y the Lie group SE3) or one of its ten sugroups, and we extend these ideas to spacecraft-manipulator systems. There are several advantages in following this approach. The use of quasi-coordinates, i.e., velocity coordinates that are not simply the time derivative of the position coordinates, allows us to include joints or transformations) with a different topology than that of R n. For example, for a spacecraft we can represent the transformation from the inertial frame to the spacecraft ody frame as a freefloating joint with configuration space SE3) and we avoid the singularity-prone kinematic relations etween the inertial frame and the ody frame velocities that normally arise in deriving the spacecraft dynamics Hughes, 22). This approach differs from previous work in that it allows us to derive the dynamic equations of vehicle-manipulator systems for vehicles with a configuration space different from R n. The dynamics are expressed locally) in exponential coordinates φ, ut the final equations are evaluated at φ =. This has two main advantages. Firstly, the dynamics do not depend on the local coordinates as these are eliminated from the equations and the gloal position and velocity coordinates are the only state variales. This makes the equations valid gloally. Secondly, evaluating the equations at φ = greatly simplifies the dynamics and make the equations suited for implementation in simulationsoftware.wealsonotethattheapproachiswellsuited for model-ased control as the equations are explicit and without constraints. The fact that the configuration space of the spacecraft is a Lie group also simplifies the implementation. Even though the expressions in the derivation of the dynamics are somewhat complex, we have several tools from the Lie theory that allows us to write the final expressions in a very simple form. The paper is organized as follows. Section 2 gives the detailed mathematical ackground for the proposed approach. This section can e skipped and practitioners mainly interested in implementation can go straight to Section 4. Section 3 presents the state of the art in spacecraft-manipulator modeling and in Section 4 the dynamic equations for spacecraft-manipulator systems and the effects of the free-floating ase are treated in detail. To the authors est knowledge spacecraft-manipulator systems have not een studied in detail in literature using the framework presented here. The matrix representation of the dynamics and how to implement this is presented in great detail which allows the readers to implement this in a simulation or control environment without having to perform all the detailed computations themselves. 2. DYNAMIC EQUATIONS OF VEHICLE-MANIPULATOR SYSTEMS We extend the classical dynamic equations for a serial manipulator arm with 1-DoF joints to include the motion of the spacecraft. 2.1 Vehicle-Manipulator Kinematics Consider the setup of Figure 1 descriing a general n-link root manipulator arm attached to a vehicle. Choose an inertial coordinate frame Ψ, a frame Ψ rigidly attached to the vehicle, and n frames Ψ i not shown) attached to each link i at the center of mass with axes aligned with the principal directions of inertia. Finally, choose a vector q R n that descries the configuration of the n joints. Using standard notation Murray et al., 1994), we descrie the pose of each frame Ψ i relative to Ψ as a homogeneous transformation matrix g i SE3) of the form [ ] Ri p g i = i R ) with rotation matrix R i SO3) and translation vector p i R 3. This pose can also e descried using the vector of joint coordinates q as g i = g g i = g g i q). 2) The vehicle pose g and the joint positions q thus fully determine the configuration state of the root. We use g 6 DoF) to represent the vehicle configuration, ut the actual configuration space of the vehicle may e a suspace of SE3) of dimension m < 6. In our case, a spacecraft has configuration space SO3) with dimension m = 3. In the case of m < 6 we define a selection matrix H R 6 m such that the twist is given y V = HṼ, 3) where Ṽ Rm determines the m-dimensional velocity state of the vehicle. The spatial velocity of each link can e expressed using twists: V i = [ ] v i ω i = V +V i = Ad g HṼ +J i q) q where vi and ω i are the linear and angular velocities, respectively, of link i relative to the inertial frame, J i q) R 6 n is the geometric Jacoian of link i relative to Ψ, the adjoint is defined as Ad g := [ ] R ˆpR R R 6 6, and ˆp R 3 3 is the skew symmetric matrix such that ˆpx = p x for all p,x R 3. The velocity state is thus fully determined given the twist V of the vehicle and the joint velocities q. ) 4)

3 2.2 Vehicle-Manipulator Dynamics The previous section shows how the kinematics of the system can e descried in terms of the gloal) state variales g, q, V, and q. To derive the dynamics of the complete mechanism including the m-dof etween Ψ and Ψ ) in terms of these state variales, we follow the generalized Lagrangian method introduced y Duindam and Stramigioli 28). This method gives the dynamic equations for a general mechanism descried y a set Q = {Q i } of configuration states Q i not necessarily Euclidean), a vector v of velocity states v i R ni, and several mappings that descrie the local Euclidean structure of the configuration states and their relation to the velocity states. More precisely, the neighorhood of every state Q i is locally descried y a set of Euclidean coordinates φ i R ni as Q i = Φ i Q i,φ i ) with Φ i Q i,) = Q i. Φ i Q i,φ i ) defines a local diffeomorphism etween a neighorhood of R ni and a neighorhood of Q i. We start y deriving an expression for the kinetic coenergy of a mechanism, expressed in coordinates Q, v, ut locally parameterized y the coordinate mappings for each joint. For joints that can e descried y a matrix Lie group, this mapping can e given y the exponential map Murray et al., 1994). Let φ sen,r) e the Lie algera of SE3), then the exponential map expφ) is given y 2 ˆφ eˆφ = I + ˆφ+ 2 = ˆφ n 5) n! n= where I no suscript) is the identity matrix. The dynamics are thus expressed in local coordinates φ for configuration and v for velocity, and we consider Q a parameter. After taking partial derivatives of the Lagrangian function, we evaluate the results at φ = i.e. at configuration Q) to otain the dynamics expressed in gloal coordinates Q and v as desired. We note that even though local coordinates φ appear in the derivations of the various equations, the final equations are all evaluated at φ = and hence these final equations do not depend on local coordinates. The gloal coordinates Q and v are the only dynamic state variales and the equations are valid gloally, without the need for coordinate transitions etween various areas of the configuration space. Note also that taking the partial derivatives of the Lagrangian and evaluating at φ = greatly simplifies5) and the closed form expressions of the exponential map is not needed. This fact greatly simplifies the final equations. In general, the topology of a Lie group is not Euclidean. When deriving the dynamic equations for spacecraft from the Lagrangian, this is normally dealt with y introducing a transformation matrix that relates the local and gloal velocity variales. However, forcing the dynamics into a vector representation in this way, without taking the topology of the configuration space into account, leads to singularities in the representation or other deficiencies. To preserve the topology of the configuration space we will use quasi-coordinates, i.e. velocity coordinates that are not simply the time-derivative of position coordinates, ut given y a linear relation. Thus, there exist differentiale matrices S i such that we can write v i = S i Q i,φ i ) φ i for every Q i. For Euclidean joints this relation is given y the identity map while for joints with a Lie group topology we can use the exponential map to derive this relation. Given a mechanism with coordinates formulated in this generalized form, we can write its kinetic energy as T Q,v) = 1 2 vt MQ)v with MQ) the inertia matrix in coordinates Q and v the stacked velocities of the vehicle and the root joints. The dynamics then satisfy MQ) v +CQ,v)v = τ 6) with τ the vector of external and control wrenches collocated with v), and CQ, v) the matrix descriing Coriolis and centrifugal forces given y C ij Q,v) := k,l + k,l,m,s Mij S 1 kl 1 2 S 1 ki ) M jl v l 7) φ= S 1 Smj mi S ) kl) ms S 1 sk φ s φ M v l. j φ= We refer to Duindam and Stramigioli 28) for details. To applythisgeneralresulttosystemsoftheformoffigure1, we write Q = {g,q} as the set of configuration states where g is the Lie group SO3), and v = [ V) T q T] T as the vector of velocity states. The local Euclidean structure for the state g is given y exponential coordinates, while the state q is itself gloally Euclidean. Mathematically, we can express configurations g,q) around a fixed state ḡ, q) as g = ḡ exp j φ ) j, q i = q i +φ i i {1...n} j=1 with j the standard asis elements of the Lie algera so3). From expression 4) for the twist of each link in the mechanism, we can derive an expression for the total kinetic energy. Let I R 6 6 and I i R 6 6 denote the constant positive-definite diagonal inertia tensor of the ase and link i expressed in Ψ i ), respectively. The kinetic energy T i of link i then follows as T i = 1 ) V i TIi 2 i Vi i = 1 Ṽ 2 ) T H T + q T J i q) T) ) Ad T g i I i Ad gi HṼ +J i q) q = 1 [ Ṽ ) ] [Ṽ ] T 2 q T M i q) = 1 q 2 vt M i q)v 8) with M = [ I ] R m+n) m+n) for the vehicle and [ H T Ad T g M i q) := I i iad H gi HT Ad T g I i iad J ] gi i Ji T Ad T g i I i Ad gi H Ji T Ad T 9) g i I i Ad gi J i for the links. The total kinetic energy of the mechanism is given y the sum of the kinetic energies of the mechanism links and the vehicle, that is, T q,v) = 1 [I ] ) 2 vt + M i q) v 1) i=1 }{{} inertia matrix Mq) with Mq) the inertia matrix of the total system. Note that neither T q,v) nor Mq) depend on the pose g nor the choice of inertial reference frame Ψ. We can now write 6) in lock-form as follows [ ][ ] [ ][Ṽ ] [ ] MVV MqV T Ṽ CVV C + Vq τv = 11) M qv M qq q C qv C qq q τq Here the suscript V refers to the first m entries and q the remaining n m entries.

4 2.3 Vehicles with Configurations Space SE3) The configuration space of a free-floating vehicle, such as a spacecraft, can e descried y the matrix Lie group SE3). In this case we have the mapping Duindam, 26) V I 1 2 ad φ V + 1 ) 6 ad2 φ V... φ V 12) with ad p = = [ ] ˆp4...6 ˆp ˆp R 6 6 for p R 6 relating the local and gloal velocity variales. The corresponding matrices S i can e collected in one lock-diagonal matrix S R 6+n) 6+n) given y SQ,φ) = I 1 2 ad φ V + 1 ) 6 ad2 φ V ) I We note that when differentiating with respect to φ and sustituting φ = this simplifies the expression sustantially. To compute the matrix CQ,v) for our system, we can use the oservations that Mq) is independent of g, that SQ,φ) is independent of q, and that SQ,) I. Furthermore, the partial derivative of M with respect to φ V is zero since M is independent of g, and the second term of 7) is only non-zero for the C VV lock of CQ,v). Firstly,C VV dependsonoththefirstandthesecondterm in 7). We have i,j = Note that Mij = for k < 7 and Sij = for i,j,k > 6. This simplifies C VV to 6+n C ij Q,v) = M ij 1 M jk 2 φ i v k 14) k=7 + }{{} = Sij S ik φ= ) Mq)v) k. φ= Furthermore, if we write S = I 1 2 ad φ V ad2 φ V...) we note that after differentiating and evaluating at φ =, Sij is equal to 1 2 ad e k where e k is a 6-vector with 1 in the k th entry and zeros elsewhere. Similarly, S ik is equal to 1 2 ad e k. This is then multiplied y the k th element of Mq)v when differentiating with respect to φ k so that M VV C VV Q,v) = q k ad Mq)v)V 15) where Mq)v) V is the vector of the first 6 entries corresponding to V ) of the vector Mq)v. C Vq Q,v),i.e.,i = 1...6andj = n),isfoundina similar manner. First we note that M jk φ i = for i = and that Sij = and S ik = for j = n), so only the first part is non-zero and we get C Vq Q,v) = M Vq q k 16) Finally, the terms C qv and C qq depend only on the first part of 7) and can e written as From et al., 29) C qv = C qq = M qv q k 1 T 2 q M qq q k 1 T 2 q [MVV M T qv [MqV M T qq ] [ ]) V, 17) q ] [ ]) V. 18) q The C-matrix is thus given y M CQ,v) = q k 19) 1 2ad Mq)v)V T [MVV ] [ ]) 2 MqV T V T [MqV ] [ ]) M T V. qq 2.4 Vehicles with Configurations Space SO3) The dynamics of a vehicle-manipulator system for a vehicle with configuration space SO3) are derived in the same way. The velocity state is thus fully determined y only three variales and we choose H so that V = HṼ, H = [ 3 3 I 3 3 ] T. 2) The corresponding matrices S i R 3+n) 3+n) can e collected in one lock-diagonal matrix S given y SQ,φ) = I 1 2 ˆφ V + 1 ) 6 ˆφ 2 V ) I The precise computational details of the partial derivatives followthesamestepsasforthese3)caseexceptforc VV. Note that Mij = for k < 4 and Sij = for i,j,k > 3. When differentiating and evaluating at φ = the matrices Sij are equal to 1 2êk where e k is a 3-vector with 1 in the k th entry and zeros elsewhere. Similarly, S ik is equal to 2êk. 1 We then get C VV Q,v) = M VV q k Mq)v)Ṽ 22) where Mq)v)Ṽ is the vector of the first three entries of the vector Mq)v corresponding to Ṽ ). 3. STATE OF THE ART SPACECRAFT-MANIPULATOR DYNAMICS 3.1 State of the Art Spacecraft Dynamics The attitude of a spacecraft is normally descried y the Euler parameters, or unit quaternion. This is motivated y their properties as a nonsingular representation. We note that this is not the minimal representation, nor generalized coordinates, and thus not suited for the Lagrangian approach. Also, when transforming ack to Euler angles from the unit quaternion representation a singularity is present for θ = ± π 2. Any positive rotation ψ aout a fixed unit vector n can e represented y the four-tuple Q = [ η ɛ T] T, where η = cos ψ 2 ) R is the scalar part and ɛ = sin ψ 2 )n R3 the vector part. The kinematic differential equations can now e given y Hughes, 22) η = 1 2 ɛt ω, ɛ = 1 2 ηi +ˆɛ)ω 23)

5 where ω is the angular velocity of the ody frame with respect to the orit frame and I the spacecraft inertia matrix. The attitude dynamics are given y I ω + ˆω I ω = τ. 24) 3.2 State of the Art Spacecraft-Manipulator Dynamics The equations of motion of a spacecraft-manipulator system can e written as MQ) v +CQ,v)v = τ. 25) Here, v = [ ṙ T ω) T q T] T where r is the position of the center of mass of the spacecraft, ω the angular velocity of the spacecraft and q is the joint position of the manipulator. Alternatively we can use the center of mass of the whole system to represent the translational motion. Then v = [ṙt cm ω) T q T] T where ṙcm is the linear velocity of the center of mass of the spacecraft-manipulator system. This is decoupled from the angular velocity ω and the inertia matrix for a free-flying spacecraft-manipulator system can e written as Duowsky and Papadopoulos, 1993) mi M = M ωω Mqω T 26) M qω M qq where m is the total mass of the system. The Euler angle rates Θ relate to ω y Θ = T Θ Θ )ω. 27) Again T Θ Θ ) is singular at isolated points. The control torques are given y τ = [ ] τv T τω T τq T T where τv is the spacecraft forces, τ ω is the spacecraft moments, and τ q is the manipulator torques. Other models are also availale depending on the actuators availale to control the spacecraft. In the case where τ v,τ w free-flying space roots) the center of mass of the system is not constant, ut descried y the variale r cm of Equation 25) if we let v = [ ṙcm T ω) T q T] T. If no external forces act on the system and the spacecraft is not actuated, the center of mass does not accelerate, i.e., the system linear momentum is constant and ṙ cm =. This can e used to simplify the equations to an n-dimensional system with inertia matrix M r = M qq M qω MωωM 1 qω T and we get the reduced system y eliminating ω M r Q) q +C r Q,v) q = τ q. 28) The attitude of the spacecraft is then found from ω = MωωM 1 qω T q. 29) 4. THE PROPOSED APPROACH 4.1 Configuration space SE3) The configuration space of a spacecraft can e descried y the Lie group SE3) with respect to an orit-fixed frame. The dynamic equations can e written as [ ][ ] [ ][ ] [ ] MVV MqV T V CVV C + Vq V τv = 3) M qv M qq q C qv C qq q τq where M CQ,v) = q k 31) 1 2ad Mq)v)V T [MVV ] [ ]) 2 MqV T V T [MqV ] [ ]) M T V. qq The dynamics are singularity-free and with state variales Q = {g SE3),q R n } and v = [ V) T q T] T R 6+n, and valid for oth for actuated and unactuated spacecraft. 4.2 Configuration space SO3) For free-floating spacecraft we have τ V = and r cm is thus decoupled from the angular velocity of the spacecraft so we write the kinetic energy of link i of the system as T i = 1 [ ] [ ω 2 ) T q T] ω M i q) = 1 q 2 vt M i q)v 32) where [ H T Ad T g M i q) := I i iad H gi HT Ad T g I i iad J ] gi i Ji T Ad T g i I i Ad gi H Ji T Ad T. g i I i Ad gi J i and the inertia matrix is given y sustituting this into 1) and H given as in 2). The configuration space is then given y Q = {R,q}. The upper left part of the Coriolis matrix then ecomes following the mathematics of 2-22)) M VV C VV Q,v) = q k Mq)v)Ṽ q 33) k where Mq)v)Ṽ is the vector of the first three entries of the vector Mq)v corresponding to Ṽ = ω ). The dynamic equations can now e written y 3) with velocity state v = [ ω ) T q T] T and configuration space Q = {R,q}. Again we note that the singularity that normally arises when using the Euler angles is eliminated and the state space Q,v) is valid gloally. Alternatively we can re-write the mass matrix as M r = M qq M qv M VV MqV. T 34) The Coriolis matrix is then found y C r Q,v) = M r q k 1 T 2 q M rv) 35) withm r asin34)andthedynamicsaredescriedy28). Computing the Partial derivatives of Mq 1,...,q n ) The partial derivatives of the inertia matrix with respect to q 1,...,q n are computed y Mq 1,...,q n ) = 36) + [ ][ H T T Ad gi i=k i=k+1... J T i m m I i Ad gi +Ad T g i I i Ad gi T J i Ad T g q i I i Ad gi H... k ] [H J i ] H T Ad T J i g i I i Ad gi T J i Ad T g q i I i Ad gi J i +Ji T Ad T J. i g i I i Ad gi k )

6 Proposition 1. Express the velocity of joint k as V k 1) k 1)k = X k q k for constant X k. The partial derivatives of the adjoint matrix is given y Ad Ad gik 1) ad Xk Ad gk 1)j for i < k j gij = Ad gik 1) ad Xk Ad gk 1)j for j < k i otherwise Proof: To prove this, we write out the spatial velocity of frame Ψ k with respect to Ψ k 1) when i < k j: ˆX k q k = where ˆX := R k 1)k ˆV k 1) k 1)k = ġ k 1)kg 1 k 1)k = g k 1)k [ ˆXω X v g kk 1) q k ]. Comparing these terms we get = q ˆX p k 1)k ω R k 1)k, = k q ˆX ω p k 1)k +X v. k We can use this relation in the expression for the partial derivative of Ad gk 1)k : ] Ad gk 1)k q = [ Rk 1)k ˆp k 1)k R k 1)k q R k 1)k +ˆp k 1)k k R k 1)k [ ][ ] ˆXω ˆXv Rk 1)k ˆp = k 1)k R k 1)k ˆXω R k 1)k = ad Xk Ad gk 1)k 37) It is now straight forward to show that Ad gij Ad gk 1)k = Ad gik 1) Ad gkj = Ad gik 1) ad Xk Ad gk 1)j. 38) The proof is similar for j < k i. Implementation We first define the vector Mq)v) 1 ξ = Mq)v) V =. = [ ] [ ] M VV MqV T V. q Mq)v) m This gives the adjoint part of the second part of 19) as ξ 6 ξ 5 ξ 3 ξ 2 ξ 6 ξ 4 ξ 3 ξ 1 ξ ad ξ = 5 ξ 4 ξ 2 ξ 1 ξ 6 ξ. 39) 5 ξ 6 ξ 4 ξ 5 ξ 4 The lower part of the second term in 19) is found from Mv) 1 Mv) 2 Mv) 6 q 1 q 1 q 1 Mv) 1 Mv) 2 Mv) 6 T q Mq)v) V = q 2 q 2 q ) Mv) 1 Mv) 2 Mv) 6 q n q n q n where Mv)j is calculated as 6+n Mv) j M ji = v i. 41) q i=1 k The second part of 18) is computed in the same way. We thus only need to compute the partial derivative Mq) q i once and use the result in the oth in the first and second part of 19). 5. CONCLUSIONS In this paper the dynamic equations of spacecraftmanipulator systems are derived ased on Lagrange s equations. The main contriution is to close the gap etween manipulator dynamics which are normally derived ased on the Lagrangian approach and spacecraft dynamics which are normally derived using other approaches in order to avoid singularities. The proposed framework allows us to derive the dynamics of spacecraft using a minimal, singularity-free representation ased on Lagrange s equations which naturally extends to include also the manipulator dynamics. This is derived for the first time in this paper using the proposed framework. ACKNOWLEDGEMENTS P. J. From and J. T. Gravdahl wish to acknowledge the support of the Norwegian Research Council and the TAIL IO project for their continued funding and support for this research. The TAIL IO project is an international cooperative research project led y StatoilHydro and an R&D consortium consisting of ABB, IBM, Aker Solutions and SKF. During the work with this paper the first author was with the University of California at Berkeley. V. Duindam is sponsored through a Ruicon grant from the Netherlands Organization for Scientific Research NWO). REFERENCES Bullo, F. and Lewis, A.D. 24). Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems. Springer. Duowsky, S. and Papadopoulos, E. 1993). The kinematics, dynamics and control of free-flying and free-floating space rootic systems. IEEE Transactions on Rootics and Automation, 9, no. 5. Duindam, V. 26). Port-Based Modeling and Control for Efficient Bipedal Walking Roots. Ph.D. thesis, University of Twente. URL Duindam, V. and Stramigioli, S. 28). Singularityfree dynamic equations of open-chain mechanisms with general holonomic and nonholonomic joints. IEEE Transactions on Rootics, 243), From, P.J., Duindam, V., Gravdahl, J.T., and Sastry, S. 29). Modeling and motion planning for mechanisms on a non-inertial ase. International Conference of Rootics and Automation, Koe, Japan. Hughes, P.C. 22). Spacecraft Attitude Dynamics. Dover Pulications. Kozlowski, K. and Herman, P. 28). Control of root manipulators in terms of quasi-velocities. Journal of Intelligent and Rootic Systems, 53, no. 3. Kwatny, H.G. and Blankenship, G. 2). Nonlinear Control and Analytical Mechanics A Computational Approach. Birkhuser Boston. Moosavian, S.A.A. and Papadopoulos, E. 27). Freeflying roots in space: an overview of dynamics modeling, planning and control. Rootica, 25. Murray, R., Li, Z., and Sastry, S. 1994). A Mathematical Introduction to Rootic Manipulation. CRC Press. Selig, J.M. 2). Geometric fundamentals of rootics. Springer.