Dynamics Of Morphing Robotic Arm With Space Debris Capture

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DYNAMICS OF MORPHING ROBOTIC ARM WITH SPACE DEBRIS CAPTUREArjun More(1) and Senthil Murugan(2)(1)Indian Institute of Technology, Madras, India. Email: ae19S006@smail.iitm.ac.in(2)Indian Institute of Technology, Madras, India. Email: drsen@iitm.ac.inABSTRACTThe space debris have been exponentially increasingwhich can damage active satellites by collision andtherefore, their removal becomes necessary. The usage ofa robotic arm to capture and remove active space debrisseems to be promising for medium-scale debris and is thefocus of this study. The existing robotic arms, attachedwith a chaser satellite, are designed as rigid structureswith fixed geometry. Therefore, while capturing, thesatellite has to come closer to debris which risks thelife of the satellite itself and need significant attitudecontrol. To avoid these issues, a “morphing roboticarm” is designed in this study. A robotic arm based ontelescopic type morphing-beam is designed such that thelength of the arm can be varied to make the capturingeasy without the spacecraft going close to debris. Inaddition, the robotic arm is designed such that thevibration due to the impact of debris can be controlled.For dynamical analysis, initially, the robotic arm isapproximated as a double pendulum with the variationsin length executed by an active control system whichresults in a parametric type system. Elastic stiffnessand mass distributions of robotic arm are modeled asequivalent bending spring and point mass of the doublependulum, respectively. Equations of motion derivedwith Euler-Lagrange formulation results in nonlinear,coupled, stiff-differential equations. A plastic-collisionis considered for contact-dynamics between the spacedebris and robotic arm during the capturing process.The dynamic response of the morphing robotic arm dueto debris capturing coupled with the variation of arm’slength is studied. The active control system designedwith linear model approximation shows the impact ofdebris capture can be minimized with least effort and thenumerical results of nonlinear system are discussed.Keywords: Robotic arm;Space debris removal;Morphingbeam.1.INTRODUCTIONSpace debris is human-generated objects in space mainlyin earth orbit which are not currently functional. Spacedebris is produced in many ways, such as nonfunctionalspacecraft, abandoned launch vehicle, hypervelocityimpacts with spacecraft wall, unburned particles fromsolid rocket motors or even paint flecks. Space debrisis very fast-moving (usually 10 km/s), and its volume inorbit is also high, possessing a risk to current and futurespace missions. They are usually noncooperative andthus different from the usual targets of orbit servicingmission and possess the greatest challenge of how tocapture and remove them without creating more reliably.J.C liou, through extensive simulation demonstratedthat Kessler syndrome [9] is already engaged, meaningthat debris would multiply in an unstoppable chainreaction without human intervention. To stabilize theenvironment, 5 to 10 space debris still needs to beremoved as shown by a predictive model of NASA [12].But currently, space debris is increasing fig [1]. Likelarge space debris, small space debris also has a high-riskfactor [1] as given in Table1. Altitude close to 800 km isthe most crowded orbit and altitude close to 600, 800 and1000 km are the massiest orbits as most space debris witha mass over 50 kg are located [11].According to French space agency (CNES),the actualdebris is divided as given in Table 1.Table 1. Debris Classification.SizeNoSmaller than 1 cm350,000,000Between 1cm and 10 cm300,000Bigger than 10 cm16,000RiskLow riskHigh riskModerate riskMany concepts have been proposed to capture andremove space debris, mainly divided into contact andcontactless. Contact type consists of single and multiarm;tentacle mechanism embraces the target and makes a stiffconnection between space debris and chase satellite [4].Net capturing is also a contact type capturing mechanismin which net is thrown at space debris for establishingcontact with them and then their removal [3]. Harpoonmechanism in which a tip is fixed from chaser satelliteto be thrown for penetration in space debris object soby pulling they can be removed or moved to graveyardorbit [15]. Some contactless mechanism involves dragaugmentation [2] and slingshot method [13].In this paperProc. 8th European Conference on Space Debris (virtual), Darmstadt, Germany, 20–23 April 2021, published by the ESA Space Debris OfficeEd. T. Flohrer, S. Lemmens & F. Schmitz, (http://conference.sdo.esoc.esa.int, May 2021)

Figure 2. Design of Space craft with robotic armFigure 1. Monthly no of objects in earth’s orbit [1]space in the launching vehicle compared to the traditionalrigid, fixed geometry robotic arm.design of a robotic arm is considered for space debrisremoval.The robotic arm technology has been used in manyon-orbit servicing missions such as canadarm2, orbitalexpress DARPA, and many others [7]. But the target hereis cooperative and non tumbling. The most importantresearch areas for space debris capture with a roboticarm are minimizing the impact influence, detumbling,and attitude synchronization. When using a roboticarm, contact will happen, and thus impact effect isof great concern. Tumbling of space debris due toresidual angular momentum also adds to difficulty [14].JAXA has shown tumbling rate below 3 degree/s can becaptured easily, and a tumbling rate between 3 degree/sand 30 degree/s can be detumbled by push contact.Attitude synchronization helps in directing robotic armstowards space debris during the critical capturing phase.Currently, many robotic technologies are in developmentfor capturing a non-cooperative and tumbling target;DLR has been developing it for a mission namedDEOS(Deutsche orbital servicing mission) [16], theFFERND arm is also designed, assembled, and tested[5], ATLAS [18] a two robotic arm controlled fromthe ground which can assemble space structure, roboticrefueling task, and space debris removal.This paper is organised as follows:-Section 2: Design ofrobotic arm, Section 3: Mathematical model of roboticarm , Section 4: Impact dynamics between robotic armand space debris, Section 5: Controller is derived for themodel, Section 6:Conclusion.2.Figure 3. Morphing robotic arm section3.DYNAMIC MODEL OF ROBOTIC ARMIn this section, dynamic model of the morphing armbased on telescopic type morphing beam is derived. Therobotic arm is assumed to be fixed to the chaser satelliteand the dynamics of base satellite base is not included inthe model. The robotic arm is approximated as doublemass pendulum with varying lengths and the springsattached to each link as shown in Fig [4]. The distributedmass of telescopic arm are approximated as two pointmasses M1 and M2 and the bending stiffness of eachbeam is represented by equivalent bending springs, K1and K2 . The length of the first arm, L1 is consideredas constant and the length of second arm L2 (t) isis considered to vary with time as shown in Fig [4].Now, Euler-lagrangian formulation is used to derive themathematical model of system as given below:DESIGN OF MORPHING ROBOTIC ARMKinetic and potential energies of systemThe robotic spacecraft consists of a morphing [6] armattached to the base satellite is shown in Fig. [2]. Withthe morphing capability of robotic arm, as shown inFig. [3], the distance between chaser satellite and debriscan be varied thus mimimizing the probability of debriscolliding with the chaser satellite. Further, this retractableand extendable nature of the robotic arm requires a lesserFor M1T1 0.5M1 (x 1 2 y 1 2 )

(M1 M2 )L21 θ̈ M2 L1 L2 cos(θ α)α̈ M2 L1 sin(θ α)L 2 2M2 L1 L 2 α̇ cos(θ α) M2 L1 L2 sin(θ α)(α̇)2 k1 (θ) k2 (θ α) f1(1)M2 L22 α̈ M2 L1 L2 cos(θ α)θ̈ M2 L1 L2 sin(θ α)(θ̇)2 2M2 L2 L 2 α̇ k2 (θ α) f2(2)Figure 4. Lumped model of robotic arm4.For M2T2 0.5M2 (x 2 2 y 2 2 )where the coordinates are given asx1y1x2y2 L1 sin(θ) L1 cos(θ) L1 sin(θ) L2 sin(α) L1 cos(θ) L2 cos(α)Now, differentiating the coordinates with respect to timex 1 L1 cos(θ)θ̇DEBRIS IMPACT MODELINGThe impact of the space debris on the robotic arm ismodeled in this section. The dynamics of capturingthe space debris is mainly analyzed through impact orcontact analysis. The impact is a complex phenomenonin which two bodies collide with each other. If theimpact is of brief duration, rapid dissipation of energyoccurs and the dynamic response of robotic arm decaysin short duration. However, if the contact occurs overa finite time, the dynamics will have secondary phasessuch as slipping, sticking, and reverse motion during thecapturing phase [8]. In this study, the collinear impactof bodies with e(coefficient of restitution) 0 for perfectlyinelastic collision is considered . Therefore, the energyconservation principle is applied to convert the impactof debris as the initial conditions applied to the Mass,M2 of the above system. Also, due to inelastic collisionassumption, the bodies are considered to stick togetherafter impact.y 1 L1 sin(θ)θ̇5. L 2 sin(α)x 2 L1 cos(θ)θ̇ L2 cos(α)(α) L 2 cos(α)y 2 L1 sin(θ)θ̇ L2 sin(α)(α)Strain energy in springsCONTROL SYSTEM FOR DEBRIS IMPACTIn this section, a control system is designed to mimimizethe dynamic response of the robotic arm induced bydebris impact. The linearized model is used to developthe LQR model and the control system is then verifiedfor the nonlinear model. The non-linear equations ofmotion are linearized near stationary points for analysis[17]. The system parameter values used for the analysisare given in Table 2.V 0.5K1 (θ)2 0.5K2 (θ α)2Now, the forces f1 and f2 are the actuator forces appliedat joint location for robotic arm control.Substituting the above expressions in Euler Lagrangianformula, the equation of motion is derived as non-linear,coupled differential equations given below:Now, using the Jacobian linearization, the stationaryvalues are found to beθ 0; θ̇ 0 ; α 0; α̇ 0; f1 0 ; f2 0 ;

[Table 2] System parameter values:ParameterM1 (kg)M2 (kg)L1 (m)L0 (m)kK2 (N-m)K1 (N-m)Value10.5110.550005000Figure 6. Controller applied to non-linear model with L 2 0An optimal regulator is considered for the linearizedsystem ẋ Ax Bu. Here, the matrix ’K’, for thecontrol law u(t) Kx(t), is found out such that itminimizeR the performance indexJ 0 (xT Qx uT Ru)dt.Here, the ’Q’ and ’R’ determines the relative importanceof the error and the expenditure of energy [10].Case 2: In the second case, the same controller derivedfor linear system is applied to non-linear system, whereit considered that L 2 0. That is, this case representthe nonlinear robotic arm, without the morphing process.The response is shown in Fig [6]. As shown in Fig. [6],the states of output are going to stable position with helpof designed LQR controller.The optimal matrix ’K’ is given by K R 1 B T Pwhere P is found by reduced riccati equation :AT P P A P BR 1 B T P Q 0Figure 7. Controller applied to non-linear model with L2 lo ksin(θ)Figure 5. Dynamics of linear system with initial value α̇ 0.1Now, the values of weight matrix are choosen for LQRcontroller based on the criteria that the optimal values ofenergy required for actuator and optimal time requiredfor reaching steady state. The values for the robotic armsystem are found asQ [500,0,0,0;0,500,0,0;0,0,500,0;0,0,0,500];R [5000,0;0;5000]Three cases of dynamics and control of robotic arm arestudied.Case 1: Initially, the dynamics of system with linearmodel with the derived control system is investigated.The dynamic response, for an initial disturbance due todebris, is shown in Fig 5. The system response decayswithin a short period. Also, the Eigen values of controlledsystem (A B K) are found to lie on the left half ofstability diagram making the system stable.Case 3: In the third case, the control of robotic armwith morphing dynamics is considered. The variation inthe arm L2 is taken as a function of θ, which implicitlydepends on time. Controller derived with linearizedsystem of case 1, is applied. The system dynamics isshown in Fig [7] which shows that the response becomesstable with the help of controller in short period. In allthree cases, the system response due to debris impact areminimized within a short duration.The above preliminary analysis and control show theeffectiveness and feasibility of the morphing robotic armfor space debris capture.6.CONCLUSIONA morphing robotic arm is designed to capture thespace debris with a flexibility of varying the lengthbetween the chaser satellite and debris. The roboticarm is approximately modeled as a double pendulumwith varying length consisting of lumped mass and

equivalent bending stiffness of the telescopic beam.The mathematical model of the system ends up as acoupled, nonlinear differential equations with varyingcoefficients. The impact of debris is modeld as inelasticcollision and the dynamic response due to debris impactis studied. An active control system is designed tominimize the response due to impact. The LQR (Linearquadratic regulator) optimum controller is derived withthe linearized model of the system. The controllerderived is then applied to the non-linear equation ofmotions with morphing robotic arm. The debris induceddynamic response is found to be minimized in shortduration. The initial morphing robotic arm designproposed in this study is found to be an effective wayto capture the debris. However, the improvements inmodeling such as continuous beam modeling of roboticarm, dynamic contact analysis rather than impact forceanalysis and coupled dynamics of robotic arm withsatellite base have to be included and are being studied.REFERENCES1. VV Adushkin, O Yu Aksenov, SS Veniaminov,SI Kozlov, and VV Tyurenkova. The small orbitaldebris population and its impact on space activitiesand ecological safety. Acta Astronautica, 176:591–597, 2020.2. MarianoAndrenucci,PierpaoloPergola,A Ruggiero, Joris Olympio, and Leopold Summerer.Active removal of space debris: Expanding foamapplication for active debris removal. Final Report,ESA, Contract No. 4000101449/10/NL/CBi, 2011.3. Bernd Bischof. Roger-robotic geostationary orbitrestorer.In 54th International AstronauticalCongress of the International AstronauticalFederation,the International Academy ofAstronautics, and the International Institute ofSpace Law, pages IAA–5, 2003.4. Alessandro Chiesa, Franco Fossati, GiovanniGambacciani, and Emanuele Pensavalle. Enablingtechnologies for active space debris removal: Thecadet project. In Space Safety is No Accident, pages29–38. Springer, 2015.5. Thomas Debus and Sean Dougherty. Overviewand performance of the front-end robotics enablingnear-term demonstration (frend) robotic arm. InAIAA Infotech@ Aerospace Conference and AIAAUnmanned. Unlimited Conference, page 1870,2009.6. José Lobo do Vale, Frederico Afonso, Fernando Lau,and Afzal Suleman. Chapter 4 - span morphingconcept: An overview. In Antonio Concilio, IgnazioDimino, Leonardo Lecce, and Rosario Pecora,editors, Morphing Wing Technologies, pages 125–144. Butterworth-Heinemann, 2018.7. Angel Flores-Abad, Ou Ma, Khanh Pham, and SteveUlrich. A review of space robotics technologies foron-orbit servicing. Progress in aerospace sciences,68:1–26, 2014.8. Gianni Gilardi and Inna Sharf. Literature surveyof contact dynamics modelling. Mechanism andmachine theory, 37(10):1213–1239, 2002.9. Donald J Kessler and Burton G Cour-Palais.Collision frequency of artificial satellites: Thecreation of a debris belt. Journal of GeophysicalResearch: Space Physics, 83(A6):2637–2646, 1978.10. E Vinodh Kumar and Jovitha Jerome. Robust lqrcontroller design for stabilizing and trajectorytracking of inverted pendulum.ProcediaEngineering, 64:169–178, 2013.11. J-C Liou. An active debris removal parametric studyfor leo environment remediation. Advances in spaceresearch, 47(11):1865–1876, 2011.12. J-C Liou, Nicholas L Johnson, and NM Hill.Controlling the growth of future leo debrispopulations with active debris removal.ActaAstronautica, 66(5-6):648–653, 2010.13. Jonathan Missel and Daniele Mortari. Removingspace debris through sequential captures andejections.Journal of Guidance, Control, andDynamics, 36(3):743–752, 2013.14. Shin-Ichiro Nishida and Satomi Kawamoto. Strategyfor capturing of a tumbling space debris. ActaAstronautica, 68(1-2):113–120, 2011.15. J Reed, J Busquets, and C White. Grappling systemfor capturing heavy space debris. In 2nd EuropeanWorkshop on Active Debris Removal, pages 18–19.Centre National d’Etudes Spatiales Paris, France,2012.16. D Reintsema, J Thaeter, A Rathke, W Naumann,P Rank, and J Sommer. Deos–the german roboticsapproach to secure and de-orbit malfunctionedsatellites from low earth orbits. In Proceedingsof the i-SAIRAS, pages 244–251. Japan AerospaceExploration Agency (JAXA) Japan, 2010.17. Jirka Roubal, Petr Husek, and Jan Stecha.Linearization: Students forget the operating point.IEEE Transactions on Education, 53(3):413–418,2009.18. Kazuya Yoshida. Achievements in space robotics.IEEE Robotics & Automation Magazine, 16(4):20–28, 2009.

Figure 2. Design of Space craft with robotic arm space in the launching vehicle compared to the traditional rigid, fixed geometry robotic arm. Figure 3. Morphing robotic arm section 3. DYNAMIC MODEL OF ROBOTIC ARM In this section, dynamic model of the morphing arm based on telescopic type morphing beam is derived. The robotic arm is assumed to .

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