Parameter Identification Methods for Free-Floating Space Robots with direct Torque Sensing

19th IFAC Symposium on Automatic Control in Aerospace September 2-6, 2013. Würzburg, Germany

Parameter Identification Methods for Free-Floating Space Robots with direct Torque Sensing Wolfgang Rackl ∗ Roberto Lampariello ∗ Alin Albu-Sch¨ affer ∗ ∗

Robotics and Mechatronics Center, German Aerospace Center (DLR), Oberpfaffenhofen, D-82234 Wessling, Germany (e-mail: [email protected], [email protected]) Abstract: In this paper we address the problem of parameter identification for free-floating space robots equipped with torque sensors. For the problem formulation we use two different methods. The first uses measurements of the acceleration of the base body, while the second doesn’t. These two torque dependent methods are compared with those based on the conservation of momentum or conservation of energy. The addressed identification problems relate to the base body, to the grasped target on the end effector or to both together. Furthermore the influence of orbital disturbances on the identification accuracy is addressed. The feasibility of the torque sensing methods is shown in simulation with addition of sensor noise and it results that the method which does not use the base body acceleration is the most accurate. Keywords: Parameter Identification, Space Robotics, Dynamic Modelling, Moments of Inertia, Nonlinear Models, Joint Trajectories 1. INTRODUCTION 1.1 Motivation Future on-orbit servicing (OOS) applications for freeflying robots will involve complex tasks which require high system performance like reliability, efficiency and safety, either in full or shared autonomy or in tele-operation. The knowledge of the dynamic model of the complete OOS system meaning satellite base, manipulator and eventually the grasped target, is a key issue for several reasons. To improve path planning and tracking capabilities as well as efficiency in energy consumption by reducing the control effort, the dynamic model must be known to a sufficient accuracy. Normally, the manipulator properties stay constant in space, but due to fuel consumption the parameters of the satellite base will change significantly. Furthermore, after grasping an unknown target object the complete dynamic behaviour of the new system will change dramatically. Last but not least, well identified target parameters are useful for a de-orbiting maneuver, since we need to direct the thruster force through the center of mass of the compound system. For this reasons, an identification method is indispensable to obtain the inertial parameters like mass, center of mass and moments of inertia of all system components, especially the satellite base and the unknown target. 1.2 Related Work The parameter identification for fixed base serial robots was studied very precise. In Atkeson et al. (1986) and Khalil et al. (2007) the recursive Newton-Euler approach with either direct torque sensing or torque calculation via 978-3-902823-46-5/2013 © IFAC

464

motor current was used. Other identification models like power model were tested e.g. in Gautier (1997). In the literature there are several studies listed for free-floating robots, too. Most of them are based on the conservation of momentum as you can find in Abiko and Yoshida (2004) and Ma et al. (2008). In Yoshida and Abiko (2002) the authors use the conservation of momentum in combination with reaction wheels or gravity gradient torque. In Lampariello and Hirzinger (2000) and Lampariello and Hirzinger (2005) the equation of motion with rheonomical joints is used. Murotsu et al. (1994) presented both, conservation of momentum and the equation of motion method. For the latter, the authors use only the measurements of linear and angular velocities and accelerations of joints and base. For the excitation trajectories the authors use simple single joint movements to excite only a few numbers of inertial parameters during one trajectory. The German robotic component verification experiment ROKVISS mounted at the outer side of the International Space Station proved that joint-torque equipped manipulators are feasible for use in space (see Albu-Schaeffer et al. (2006) and Landzettel et al. (2006)). In this paper, we present two methods of parameter identification for free-floating robots with the additional use of torque sensors. For reason of comparison the methods of conservation of momentum and conservation of energy were also implemented. Furthermore, the influence of orbital disturbances on the parameter identification is analyzed. The feasibility of the identification methods is shown by numerical simulations with simulated sensor noise.

10.3182/20130902-5-DE-2040.00121

2013 IFAC ACA September 2-6, 2013. Würzburg, Germany

2. FORMULATION OF THE DIFFERENT IDENTIFICATION METHODS A free-floating space manipulator is considered in this paper as a rigid multibody system with n + 1 bodies and n joints. The bodies are connected via rotational joints in a serial chain configuration and the first body is free to move in space. The unknown target object is considered to be grasped firmly and its relative position and orientation to the end effector does not vary. In Fig 1 a general example of a free-flying multibody system is shown. θn , τn θ3 , τ3 θ2 , τ2 rn

θ1 , τ1 P

Fe

sys

r1

γb

r sys

re

rb base F b

P

I

Fig. 1. Free-flying multibody system For the identification some measurements of the system states are assumed to be available, i.e. the base angular velocity measurements from the gyroscopes, accelerometers mounted on the satellite base to obtain the translational and rotational acceleration of the base, star sensors for the base orientation, joint position sensors and joint torque sensors. By differentiating or integrating those values with respect to time all necessary variables can be obtained. For the identification process, following assumptions are made in this paper • the manipulator is known and well identified on earth. This is neccessary due to the fact that the identification of the whole system in orbit is not possible, as shown in Lampariello and Hirzinger (2005). • rigid bodies without flexibility in joints or structures and no propellant sloshing • external forces on the system as orbital disturbances are not taken into account in the identification algorithms except the gravity gradient torque. However, the orbital disturbances are simulated for the measured sensor data we use for the identification • no initial momentum (P(t0 ) = L(t0 ) = 0) with respect to the orbital frame • the torque sensors of the robot are placed on the link side of the joint, like for the DLR LWR, such that joint friction does not play a role in the identification problem at hand As described above, the manipulator is considered as well identified and only the inertial parameters of the base and of the grasped target should be identified. These parameters are Φi = [mi , mi cx,i , mi cy,i , mi cz,i , · · · · · · Ixx,i , Ixy,i , Ixz,i , Iyy,i , Iyz,i , Izz,i ]T

(1)

with the index i either for the base, the target or base and target together within one identification process. Thus, for 465

a mission scenario, before grasping a target object the base should be identified first, after that the unknown target object should be grasped with the updated base-manipulator dynamic model, then the target can be identified either alone or with the base together. The general equation of motion for free-flying robots can be expressed as (2)        x ¨b Hb Hbm cb Fb + = (2) cm τ HTbm Hm θ¨m where Hb ∈ R6×6 , Hm ∈ Rn×n , Hbm ∈ R6×n are the inertia matrices of the base, manipulator and coupling inertia matrix between the base and the manipulator, respectively. The vectors cb ∈ R6×1 and cm ∈ R6×1 are the non-linear velocity dependent term on the base and on the manipulator, respectively. Fb ∈ R6×1 and Fe ∈ R6×1 are the force torque wrenches acting on the center of mass of the base body or the end effector, respectively. The term τ ∈ Rn×1 depicts the internal joint torques of the manipulator. The generalized coordinates for the base xb is T composed of the linear and angular position xb = [rb , γ b ] . As described above, for all identification methods presented here, there are no external forces included within the identification process (Fb = 0), except the gravity gradient torque. 2.1 Modified Recursive Newton-Euler Algorithm(MRNEA) This method is named the Modified Recursive NewtonEuler Algorithm (MRNEA) and is based on the linear formulation of the recursive Newton-Euler equation of motion for serial robots as described in Atkeson et al. (1986). Here, the inverse dynamics formulation is rearranged into a parameter linear form as described for reasons of simplicity for one rigid body as 

fii nii

 = Ai Φi

(3)

with   r¨i − g [ω ˙ i ×] + [ω i ×] [ω i ×] 0 Ai = 0 [(g − r¨i ) ×] [•ω ˙ i ] + [ω i ×] [•ω i ] (4) This inverse dynamics formulation can be easily modified for a free-floating base robot by connecting the floating base with a fictitious joint having six degrees of freedom (a 6-DoF joint). Such a joint does not have any motion constraints and therefore it has no influence on the freefloating base robot dynamics. Similar to serial fixed base robots as described in Atkeson et al. (1986), the modified linear form of the recursive Newton-Euler equation comes to   Φb τ = [ Πb Πm ] = ΠΛ (5) Ψm where τ ∈ R(6+n)×1 is the vector of joint torques, Πb ∈ R(6+n)×10 and Πm ∈ R6+n×10n are the so called observation matrices or regressor matrices of the free-floating base and manipulator respectively, Φb ∈ R10×1 and Ψ ∈ R10n×1 are the vectors of the base parameter and link parameters of the manipulator Φi .

2013 IFAC ACA September 2-6, 2013. Würzburg, Germany

Estimation Procedure With N measurements on a driven trajectory the over-determined system of linear equations can be set up as Y=ΠΛ+ρ (6) with the combined regressor matrix Π and the vector of all measured torques Y given by    τ (1) Π(1)  τ (2)   Π(2)     (7) Π=  ...  ; Y =  ...  τ (N ) Π(N ) and the vector of errors ρ between the measured torques and the model predicted torques. 

Such an over determined system of linear equations can be solved usually with a least-squares optimization φb = Arg.minkρk2 = ΠTb Πb


−1

Φ = Π+ b Υb

ΠTb Υb

(8) (9)

with Υb = (Y − Πm Ψm ) (10) Since the regressor matrix needs to be inverted for the system of equation, it is necessary to ensure, that it has no rank deficiency. The identifiability of the base and the target can be analyzed with this regressor matrix. Since we consider a known manipulator arm, the regressor matrix does not have any rank deficiency. The problem formulation for the identification of the target with a known base comes similar to (8) to φt = Π+ t Υt

(11)

with T

Πt = [Ubn , K1n , K2n , ..., Knn ]

(12)

Υt = Y − ΠΛ

(13)

Finally, for the case of identifying the base and the target in one step, the problem formulation can be written as φb,t = Π+ b,t Υb,t

(14)

with Πb,t = [Πb , Πb ]

(15)

Υb,t = Y − Πm Ψm

(16)

Excitation To find optimal exciting trajectories, we used the method described in Rackl et al. (2012) to solve the relative nonlinear optimization problem. The cost function of the latter was formulated as a sum of two scalars and is defined in detail as  −1 n X h X Γ = ξ1 λ + ξ2  τi,j  (17) i=0 j=0

where ξi are two weighting factors and λ is the condition number of the relevant regressor matrix. The second 466

summand depicts the reciprocal of the sum of the joint torques over a certain number of points h of the trajectory and was chosen to improve the signal-to-noise ratio for the measurements. Especially for a free-floating space robot with no gravitational forces and with the relatively slow motions which the orbital operational conditions impose, this term plays an important role for the identification. Furthermore, several constraints must be defined for the optimization problem. rmin ≤ re ≤ rmax

(18)

rscoll ≤ re

(19)

hi,min ≤ hi ≤ hi,max , i = 1, ..., n

(20)

where the end effector position re must not exceed on the one hand its defined workspace, on the other hand it must not violate the self collision space defined as sphere around the manipulator base with radius rscoll . Furthermore, the limits of the robot and the base must be fulfilled all the time, where h stands for x ⊂ {θi , θ˙i , θ¨i , τi , xb , x˙ b , x ¨b }. For the exciting trajectories we used a parameterization with B-splines for several reasons. B-spline functions can be adjusted locally without affecting the rest of the trajectory. By defining the order of the basis functions you are able to provide at least continuous jerk to avoid large impulses on the free-floating system. For more details, see Rackl et al. (2012). This trajectory was used for all identification methods. 2.2 Reduced Dynamics Algorithm (RDA) For the MRNE algorithm both base motion components vb and γ b are obtained from measurements. The second identification method with torque sensing is derived from the general inverse dynamics equation of motion for freefloating robots (2). By substituting x ¨b from its first line into the second line gives the general form of the equation as ˆ θ¨ + C ˆ=τ H

(21)

ˆ as with the generalized inertia matrix H ˆ = H m − H Tbm H −1 H bm H b

(22) ˆ and the generalized nonlinear velocity dependent term C as ˆ = cm − H Tbm H −1 cb C (23) b For this formulation, the base acceleration (¨ xb ) is not needed anymore. The obtained terms are only functions of ˆ = f (mm,i , cm,i , Im,i , γ b , ) H   ˙ θ ¨ ˆ = f mm,i , cm,i , Im,i , γ b , vb , ω b , θ, θ, C

(24) (25)

The included gravity gradient torque G here can be formulated after Yoshida and Abiko (2002) as 3µ r × Hr (26) R5 with the spatial vector to the centroid of the robot system from the center of the earth and the gravitational G=

2013 IFAC ACA September 2-6, 2013. Würzburg, Germany

constant µ. The advantage of this formulation is clearly the elimination of the dependence on the base body acceleration; nevertheless the equation is not linear in the parameters anymore and must be solved by a nonlinear optimization. To estimate the parameters, we used the inverse dynamics and solved the nonlinear identification problem with cost function Γ as the sum of all differences between the expected and the measured torques Γ=

n X N X

τi,j − τi,j,msr

(27)

i=1 j=1

2.3 Conservation of Momentum (CM)

The linear momentum P and the angular momentum L can be written as (28)

Γ=

x˙ 2i,k

−

x˙ 2i,k,msr

(31)

2.4 Conservation of Energy (CE) The principle of the conservation of energy of the complete system can also be used for the parameter identification. With the assumption of no external energy supply during the identification process, it stays constant. The total energy can be split up into a potential and a kinetic part. Since we assume no external forces and a rigid body system without dissipative elements, the potential energy can be set to zero. The total energy of the system depends therefore only on the kinetic energy and can be written as 1 T µ Hµ 2 with the velocity vector µ defined as

3. NUMERICAL SIMULATION To verify the proposed identification methods several simulation runs were made and are presented in this section. To simulate the orbital identification scenario, we set up first a rigid body model of the satellite base and robotic manipulator. To obtain a more realistic case sensor noise was added to the variables we defined as sensor. Additionally we simulated two different orbits with expected disturbances to the OOS.

For the rigid body satellite model we used a scaled DLR Light-Weight Robot LBR fixed on a simple cubic satellite base. The kinematic configuration is drafted in Fig. 2. The scaled LBR robot has a total weight of mm = 28.3 kg and a total kinematic length of 2.97 m. The inertial parameter θ6 d5 θ5

d3 θ3 θ2 w

d1 θ1

θ7

θ4

P

sys

P

h

b0

P

b

P

l

I

Fig. 2. Kinematic configuration of the free-floating system with mounted scaled DLR Light-Weight Robot LBR 3 to be identified either for the base or the target in low earth orbit (LEO) or in geostationary orbit (GEO) are listed in table 1.

i=1 k=1

T =

(34)

(30)

The nonlinear identificationn problem was solved with the cost function n X 6 X

Ti − Ti,msr

i=1

(29)

ˇ i are hereby the translational components The matrices H ˜ i stands for the of the complete inertia matrix H and H ˜ ˙ rotational part of it. The term Hbm θ describes the angular momentum caused by the robots movement. P and L are the initial linear and angular momentum, respectively. With the condition of zero initial momentum in the orbital frame the base movement can be formulated as ˙ x˙ b = −H−1 b Hbm θ

n X

3.1 Rigid Body Model

For reason of comparison we did also parameter identification via the principle of conservation of momentum and conservation of energy as it was done in the literature.

ˇ b vb + H ˇ bm θ˙ P =H ˜ b ωb + H ˜ bm θ˙ L=H

Γ=

(32)

˙T µ = [v b , ω b , θ] (33) With the assumption of no initial movement the total system energy equals zero, the cost function for the nonlinear optimization was defined as 467

Table 1. Simulation model parameter Parameter m[kg] cx [m] cy [m] cz [m] Ixx [kgm2 ] Ixy [kgm2 ] Ixz [kgm2 ] Iyy [kgm2 ] Iyz [kgm2 ] Izz [kgm2 ]

Base 1000 0.855 -0.408 0.441 937.5 50 70 937.5 -20 375

Target(LEO) 400 0.7 0.5 0.4 250 5 12 250 -8 250

Target(GEO) 1200 1.5 1.25 1.4 1058 20 30 920 -8 750

3.2 Simulated Sensor Noise To make the simulation more realistic, we add some noise to the sensor signals we need for the different identification methods. The noisy signals are θ, τ ,γ b , v˙ b and ω b . For

2013 IFAC ACA September 2-6, 2013. Würzburg, Germany

the generation, we used a band-limited white noise with a power of p = 0.1, a sampling time of tf = 0.1 ms, an initial seed of 23000 and a gain factor of fg = 0.01 to adjust the noise signal value.

Φi (1) Φi (1)

3.3 Orbital Disturbances In order to analyze the influence of orbital disturbances on the identification process, we simulated following disturbances during the measurement simulation • • • • • •

Table 3. Influence of orbital dist. to the identification process (RDA) for the target in LEO. Parameter values in kgm2

gravity gradient torque atmospheric force 3rd body force solar pressure force atmospheric torque solar pressure torque

For the disturbance model we used the JB2006 (ECSS) atmospheric indices with the accommodation coefficients αn = 1.0, αt = 0.3 and reflection coefficients with Ca = 0.5 and Cs = 0.5. The analyzed orbits are defined as the geostationary orbit and a low earth orbit with an attitude of h = 500 km over sea level. Both orbits are circular with inclination i = 0◦ 3.4 Optimized Excitation Trajectory For generating the trajectory, we defined following constraints re,x,y = ±3.5 m , γi,max = ±10◦ , ωi,max = ±10◦ /sec , ω˙ i,max = ±10◦ /sec2 rscoll = 1.5 m

re,z > 0 m θ˙i,max = ±10 ◦ /s θ¨i,max = ±5◦ /s2 τi,max = ±40 N m

Ixx 259.25 267.19

Ixy 9.54 12.00

Ixz 10.14 2.36

Iyy 249.21 254.58

Iyz 5.32 -15.47

Izz 253.42 261.95

4.2 Identification Methods Comparison The results of the different identification methods are described in this section. As mentioned before, all simulations contain sensor noise and orbital disturbances. Since the end effector position accuracy is an important criterion to solve on-orbit servicing tasks, the errors in position and orientation are listed in the tables 4-7. The mean error on a given verification trajectory and the maximum error are calculated for both position and orientation. In table 4 the base and the target seem to be well identified, whereas in the case of base & target the results are less accurate. For the reduced dynamics model in table 5 the calculated errors between the ideal and identified model look more accurate, especially for the target identification. The results for the method of momentum conservation in table 6 are less accurate than the previous methods with a maximum end effector position error of 0.2087 m. The results of the second method without torque sensing, the conservation of energy, are listed in table 7. This method shows the most inaccuracies in the end effector errors. One reason for this could be the fact, that the noisy velocity data go quadratically in the energy formulation and so the sensor noise has more influence on the total energy. Table 4. Error of end effector with MRNEA

whereas re and rscoll are refered to the manipulator base.

error [m, rad] mean pos E max pos E mean phi E max phi E

Base 0.0054 0.0186 0.0213 0.0671

Target 0.0269 0.1247 0.0271 0.0658

Base + Target 0.0399 0.0987 0.0567 0.3419

4. RESULTS Since the orbital disturbances in LEO are higher, especially the gravity gradient effect, we listed in this paper only the results of the simulation in LEO. 4.1 Influence of orbital Disturbances to the Identification Process The influence of the external forces and moments to the movement of the free-floating system are listed in table 2 and 3. Since the RDA method was supposed to be the most accurate one (see section 4.2), we tested the influence on this method. The presented results are the identified values of a simulation run with orbital disturbances (marked with 1) and without orbital disturbances (2). As it can be seen, the orbital disturbances have a measurable effect on the results of the identification. Table 2. Influence of orbital dist. to the identification process (RDA) for the target in LEO. Parameter values in kg, m Φi (1) Φi (2)

m 398.9 419.2

cx 0.781 0.821

cy 0.502 0.597

cz 0.419 0.448

Table 5. Error of end effector with RDA error [m, rad] mean pos E max pos E mean phi E max phi E

Base 0.0108 0.0044 0.0194 0.0620

Target 0.0150 0.0704 0.0163 0.0391

Base + Target 0.0601 0.1003 0.0657 0.1892

Table 6. Error of end effector with CM error [m, rad] mean pos E max pos E mean phi E max phi E

Base 0.0495 0.0923 0.0413 0.1421

Target 0.0342 0.1565 0.0328 0.0804

Base + Target 0.1021 0.2087 0.1954 0.1624

Table 7. Error of end effector with CE error [m, rad] mean pos E max pos E mean phi E max phi E

Base 0.0392 0.0787 0.0434 0.439

Target 0.0360 0.1661 0.0343 0.0843

Base + Target 0.1152 0.1961 0.0978 0.1899

In figure 3 the time history of the error of end effector position and orientation is shown for the case of target identification for all algorithms. 468

2013 IFAC ACA September 2-6, 2013. Würzburg, Germany

Error pos ee [m]

0.2

ACKNOWLEDGEMENTS

MRNEA RDA

0.15

The authors would like to thank SpaceTech GmbH, Immenstaad, who gave us the opportunity to use their orbital disturbance model.

CM 0.1

CE

0.05 0 0

10

20

30

40

50

REFERENCES

60

Error φ ee [rad]

0.1

0.05

0 0

10

20

30 time [s]

40

50

60

Fig. 3. End effector error 5. DISCUSSION AND CONCLUSION 5.1 Discussion The results presented in the previous section showed that torque measurements for the use of parameter identification either of the base, the load or both together is feasible with suitable accuracy. The MRNEA has the advantage that it is linear in the parameter to be identified and can be solved with stable and fast least squares techniques. On the other hand this method needs the measurement of the base accelerations (translational and rotational), which are generally very noisy. The RDA method do not need these base acceleration measurements, but the identification is formulated as a nonlinear optimization problem. Furthermore, the need of the translational velocity of the satellite base here is a disadvantage and should be eliminated in the future work. For the simulations, we implemented orbital disturbances, which effect the general motion of the free-floating robot in terms of external forces and moments. However, the identification algorithms with torque sensing do not take into account these external forces, except the gravity gradient torques. This causes errors in the identified parameters and should be addressed in the future. Furthermore, in the presented simulations, sloshing was not taken into account. However it is not yet clear if free-flyers will use cold gas thrusters or fuel tanks with membranes, which minimize the sloshing effect. We also will address the sloshing issue in more detail in the future. 5.2 Conclusion In this paper, two different identification algorithms with torque sensing are simulated and compared with the methods of momentum and energy conservation. Under the condition of a rigid body system without any flexibilities and sloshing a numerical simulation was used to set up a comparison of the methods. As a result the new approach of the identification formulation without the need of base acceleration measurements seems to be the most accurate. Furthermore, the influence of orbital disturbances on the identification process was addressed and showed a measurable effect to the identification results. 469

Abiko, S. and Yoshida, K. (2004). On-line parameter identification of a payload handled by flexible based manipulator. In Intelligent Robots and Systems, 2004. (IROS 2004). Proceedings. 2004 IEEE/RSJ International Conference on, volume 3, 2930 – 2935 vol.3. Albu-Schaeffer, A., Bertleff, W., Rebele, B., Schafer, B., Landzettel, K., and Hirzinger, G. (2006). Rokviss robotics component verification on iss current experimental results on parameter identification. In Robotics and Automation, 2006. ICRA 2006. Proceedings 2006 IEEE International Conference on, 3879 –3885. Atkeson, C., An, C.H., and Hollerbach, J.M. (1986). Estimation of inertial parameters of manipulator loads and links. The Inernational Journal of Robotics Research, 5(3), 101–119. Gautier, M. (1997). Dynamic identification of robots with power model. In Proc. IEEE Int. Conf. on Robotics and Automation, 1992–1927. Khalil, W., Gautier, M., and Lemoine, P. (2007). Identification of the payload inertial parameters of industrial manipulators. In IEEE International Conference on Robotics and Automation - ICRA’07, 4943–4948. Lampariello, R. and Hirzinger, G. (2000). Freeflying robots - inertial parameters identification and control strategies. In 6-th ESA Workshop on Advanced Space Technologies for Robotics and Automation (ASTRA 2000), ESTEC,Noordwijk, The Netherlands. Lampariello, R. and Hirzinger, G. (2005). Modeling and experimental design for the on-orbit inertial parameter identification of free-flying space robots. In ASME 2005 International Design Engineering Technical Conferences & Computers Information in Engineering Conference, Long Beach, California. Landzettel, K., Preusche, C., Albu-Schaffer, A., Reintsema, D., Rebele, D., and Hirzinger, G. (2006). Robotic on-orbit servicing - dlr’s experience and perspective. In Intelligent Robots and Systems, 2006 IEEE/RSJ International Conference on, 4587 –4594. Ma, O., Dang, H., and Pham, K. (2008). On-orbit identification of inertia properties of spacecraft using a robotic arm. Journal of Guidance, Control, and Dynamics, 31(6), 1761–1771. Murotsu, Y., Senda, K., and Ozaki, M. (1994). Parameter identification of unknown object handled by free-flying space robot. Journal of Guidance, Control, and Dynamics, 17(3), 488–494. Rackl, W., Lampariello, R., and Hirzinger, G. (2012). Robot excitation trajectories for dynamic parameter estimation using optimized b-splines. In Robotics and Automation (ICRA), 2012 IEEE International Conference on, 2042 –2047. Yoshida, K. and Abiko, S. (2002). Inertia parameter identification for a free-flying space robot. In Proc. of the AIAA Guidance, Navigation, and Control Conference, AIAA, 2002–4568.