ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. CONTROL OF SATELLITES WITH ONBOARD ROBOTIC MANIPULATORS ROBERT BUCKELEW, ETHAN CATALANELLO, ANNALISA SCACCHIOLI (FACULTY ADVISOR) ✵ ABSTRACT Free-floating satellites with onboard robotic manipulators are subjected to widely varying loads resulting from the motion of the robotic manipula- tors. As there are no fixed supports in space, these loads will cause the satellite to move. By modelling the motion of the onboard robotic arms, determin- ing the necessary reaction loads (which must be sup- plied by the satellite to keep the arm fixed), and sim- ulating the resulting satellite dynamics, we designed a model of a satellite-arm system. We found that a Proportional-Integral-Derivative (PID) control scheme, with disturbance-estimating capabilities, was effective in maintaining satellite position and ori- entation during the operation of the onboard ro- botic manipulator. The MATLAB-based Simulink modeling environment was used to perform the sim- ulations of satellite dynamics and control. 1 INTRODUCTION The operational success of several fields, in- cluding communication technology and space ex- ploration, is dependent on a functional space satel- lite infrastructure. An advanced satellite network, such as a constellation of communications satellites, will require routine maintenance to maximize its functional lifetime[1]. The Defense Advanced Re- search Projects Agency (DARPA) has proposed tech- nology to perform robotic servicing of on-orbit sat- ellites[2]. The project is titled Robotic Servicing of Ge- osynchronous Satellites (RSGS) and involves satellite- mounted robotic arms which can perform opera- tions on other satellites. Potential missions for the “nurse” satellite include inspection, installation, and repair of on-orbit satellites as well as their relocation to new orbits. In general, the motion of robotic manipula- tors and the body to which they are mounted are coupled—that is, the motion of one tends to influ- ence the motion of the other. The magnitude of this effect depends on the inertial properties of the bod- ies and the interaction forces involved. In a micro- gravity environment such as Low Earth Orbit (LEO), no dissipative reaction forces exist unlike those pre- sent in operations performed on and near the Earth’s surface. Therefore, the motion of satellite-mounted robotic manipulators will have significant effects on satellite attitude (the direction the satellite is point- ing) [3]. It is desirable to reduce the effects of robotic manipulator disturbances on satellite position and attitude using an attitude control system (ACS). An ACS measures satellite position and orientation with respect to reference values. It uses mechanical actu- ators (e.g., reaction wheels, gas jet thrusters) to re- ject disturbances. In this project, we model the dy- namics of the FREND Mark II robotic arm and simu- late the motion of a base satellite subjected to dis- turbances from the robotic arm motion. Additionally, we designed a control scheme using a basic PID (Proportional-Integral-Derivative) controller. PID controllers feature three gains—proportional, inte- gral, and derivative—which determine the control in- put signal based on the magnitude, time integration, and time rate of change of the error signal respec- tively. The goal is to reduce the perturbations arising from operation of the robotic arm, resulting in a sat- ellite that maintains its position and attitude. While extensive work has been performed in this area uti- lizing complex, nonlinear controls, we wanted to evaluate the effectiveness of a much simpler, linear control scheme, as it was assumed the perturbations arising from the robotic arm would be very small. We believed that these small perturbations would be well approximated by linear control theory, and thus that linear controllers would be sufficient at stabiliz- ing the system. This belief/idea, combined with the ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II development of the modular simulation for control design, serves as the novel contribution of our pro- ject. We have created an environment that allows for control refinement without requiring rigorous math- ematical proofs. 2 METHODOLOGY The robotic arm proposed by DARPA for the Robotic Servicing of Geosynchronous Satellites (RSGS) program is the FREND Mark II (hereafter re- ferred to as the FREND), a 7-joint 3D manipulator which will be responsible for performing operations on the “patient” satellite. An illustration of the FREND arm is shown in FIGURE 1. The Robotics, Vision, and Control Toolbox, a third-party MATLAB-based toolbox for robotics ap- plications (hereafter referred to as “the Toolbox”) was used to develop a model of the FREND and to perform dynamics calculations. The results of these dynamics calculations, based on the FREND model, were used as inputs to the satellite dynamics simula- tion. These results effectively modeled the disturb- ances to the satellite arising from the motion of the robotic arm. FREND Arm Modelling The Toolbox is capable of modelling and performing calculations on any robotic arm so long as the arm’s relevant parameters are provided. The relevant parameters are inertial parameters, includ- ing the mass and inertia matrix and Denavit-Harten- berg parameters (which describe arm geometry). The Denavit-Hartenberg parameters used in the FREND model were defined according to the defini- tions shown in FIGURE 2 and measurements taken from FIGURE 3. The Denavit-Hartenberg parameters are shown in SUPPLEMENTAL TABLE S1. FIGURE 1: DARPA rendering of the FREND Mark II arm[4] IT IS DESIRABLE TO REDUCE THE EFFECTS OF ROBOTIC MANIPULATOR DISTURBANCES ON SATELLITE POSITION AND ATTITUDE USING AN ATTITUDE CONTROL SYSTEM (ACS). ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II None of the required parameters for the FREND were publicly accessible, so the measure- ments were approximated using pixel measure- ments from schematics and images that were pub- licly available. Knowing this information, an image editing software can be used to analyze the sche- matic in FIGURE 3 in order to define a pixel scale which relates real-world dimensions to pixel dimensions. The geometry of the arm was then calculated from the schematic in FIGURE 3 and used to compute the in- ertia tensor. Each link is assumed to be a cylinder of solid aluminum having a density of 2700 𝑘𝑘𝑘𝑘 𝑚𝑚3 and a center of mass located at the geometric center. This assumption of material composition represents a “worst case scenario” as the actual arm is not solid aluminum[4]. The computed dynamic parameters are shown in SUPPLEMENTAL TABLE S2 and then inertia matrices are shown in SUPPLEMENTAL TABLE S3. ABOVE: FIGURE 2: Graphic definition of Denavit-Hartenberg parameters[5] RIGHT: FIGURE 3: Schematic of the FREND Mark II showing the relative orientations of link reference frames[4] ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II [a] Gravity was neglected in our simulation because the satellite is in space. The Toolbox features a function that can numerically evaluate the robot dynamics equation (EQUATION (1)). This is a system of coupled differential equations that relates the joints’ angular kinematics (𝑞𝑞, �̇�𝑞, �̈�𝑞) of the robotic arm joints to the joint torques (𝑄𝑄) and other external loads (end effector wrench 𝑊𝑊 and gravity torque 𝐺𝐺(𝑞𝑞)[a]) acting on the arm. Solving these differential equations yields the joint torques and the reaction loads at the base of the arm. 𝑄𝑄(𝑞𝑞, �̇�𝑞, �̈�𝑞) = 𝑀𝑀(𝑞𝑞)�̈�𝑞 + 𝐶𝐶(𝑞𝑞, �̇�𝑞)�̇�𝑞 + 𝐹𝐹(�̇�𝑞) + 𝐺𝐺(𝑞𝑞) + 𝐽𝐽(𝑞𝑞)𝑇𝑇𝑊𝑊 The parameters of EQUATION (1) are defined in TABLE 1 below. The joint torques 𝑄𝑄(𝑞𝑞, �̇�𝑞, �̈�𝑞) obtained through the solution of the differential equation can be utilized to solve for the reaction forces as well as the moments the arm will induce on the satellite. Thus, these loads will be utilized in the simulation as the external disturbance the satellite is subjected to. Satellite Modelling To model the effects of robotic arm motion on satellite position and attitude, we utilized the Aerospace Blockset’s “6DOF” (six degree-of-freedom) block in the MATLAB-based Simulink modeling environment. This block implements the equations of motion described in EQUATIONS (2) and (3), which are the differential equations describing translational motion and angular motion, respectively. The parameters of these equations are defined explicitly in SUPPLEMENTAL TABLE S5. �̈�𝑥 = ∑ 𝐹𝐹𝑗𝑗(𝑡𝑡)𝑁𝑁𝑗𝑗=1 𝑚𝑚 �̈�𝜃(𝑡𝑡) = 𝐼𝐼−1 ���𝑀𝑀(𝑡𝑡) + 𝑟𝑟𝑗𝑗 × 𝐹𝐹𝑗𝑗(𝑡𝑡)� − �̇�𝜃(𝑡𝑡) × �𝐼𝐼�̇�𝜃(𝑡𝑡)� 𝑁𝑁 𝑗𝑗=1 � 𝑞𝑞, �̇�𝑞, �̈�𝑞 JOINT ANGULAR POSITION, VELOCITY, and ACCELERATION respectively 𝑄𝑄 VECTOR OF GENERALIZED JOINT TORQUES – torques required at each joint to achieve the motion described by (𝑞𝑞, �̇�𝑞, �̈�𝑞) 𝑀𝑀(𝑞𝑞) JOINT-SPACE INERTIA MATRIX – describes the moment of inertia of each link in matrix form 𝐶𝐶(𝑞𝑞, �̇�𝑞) CORIOLIS AND CENTRIPETAL COUPLING MATRIX – describes centripetal forces due to rotation of the links 𝐹𝐹(�̇�𝑞) FRICTION MATRIX – describes torques from friction present in the joints 𝐺𝐺(𝑞𝑞) GRAVITY LOAD – describes torques originating from gravity acting on the links 𝐽𝐽(𝑞𝑞) MANIPULATOR JACOBIAN MATRIX – describes the sensitivity of the end effector to the motion of each joint 𝑊𝑊 END EFFECTOR WRENCH – describes the load on the end effector TABLE 1: Definition of terms in EQUATION (1) (1) (2) (3) ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II Since there is no satellite design specified for the RSGS program, we assumed a generic small satellite configuration. We take the satellite to be a rectangular prism of 2𝑚𝑚 × 1𝑚𝑚 × 1𝑚𝑚 with two thin rectangular solar arrays fixed at opposing ends 3𝑚𝑚 × 1.5𝑚𝑚 × .0055𝑚𝑚 located . 5𝑚𝑚 from the satellite body. The density of the satellite was assumed to be 200 𝑘𝑘𝑘𝑘 𝑚𝑚3 , which is an approximate mass density for spacecraft of this size[9] and the solar panels were assumed to be identical to those manufactured by Spectrolab, a manufacturer of satellites for use in space, and have a planar density of 2.06 𝑘𝑘𝑘𝑘 𝑚𝑚2 [10]. This means that the entire satellite has a mass of 𝑚𝑚 = 411.9305362785 𝑘𝑘𝑘𝑘, and the inertia tensor is shown below: 𝐼𝐼 = (303.02687565826 0 0 0 170.14297228667 0 0 0 199.55068127826 )𝑘𝑘𝑘𝑘 ∙ 𝑚𝑚2 Controller Modelling Recall that our objective is to maintain the translational position as well as the angular position (or attitude) of the satellite throughout operation of the on-board robotic arm. To accomplish satellite position and attitude control, PID controllers, which represents a basic function for the control of linear systems, were implemented for each of the six de- grees of freedom, namely the three translational co- ordinates (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) of satellite position and the three Euler angles (𝜙𝜙, 𝜃𝜃, 𝜓𝜓) defining satellite roll, pitch, and yaw, respectively. The optimal values of the PID con- troller gains, 𝐾𝐾𝑃𝑃, 𝐾𝐾𝐼𝐼, and 𝐾𝐾𝐷𝐷 are determined automat- ically using the MATLAB PID Tuner tool. The optimal gains for the RSGS model satellite are shown in SUP- PLEMENTAL TABLE S4. 3 RESULTS As a result of this project, we have devel- oped a multiple component model in which a user can define a robotic arm configuration and trajectory as well as the inertial parameters of a satellite. The user can then simulate the motion response of the satellite to disturbances arising from motion of the arm and employ a predictive control scheme to maintain a reference satellite position and attitude. A simplified Simulink block diagram demonstrating the fundamental logic behind the model is shown in FIGURE 4. As shown in the figure, the controller (which is a PID controller in this case) receives the transla- tional position and angular position offset as well as an estimation of the incoming arm loads which it then uses to calculate the necessary control force/torque. 𝑥𝑥(𝑡𝑡) VECTOR OF SATELLITE POSITIONAL COORDINATES 𝜃𝜃(𝑡𝑡) VECTOR OF SATELLITE ANGULAR POSITION COORDINATES 𝐹𝐹𝑗𝑗(𝑡𝑡) VECTOR OF TRANSLATIONAL FORCE CAUSED BY ROBOTIC ARM 𝑗𝑗 𝑀𝑀𝑗𝑗(𝑡𝑡) VECTOR OF MOMENTS CAUSED BY ROBOTIC ARM 𝑗𝑗 𝑚𝑚 SATELLITE MASS 𝐼𝐼 SATELLITE INERTIA TENSOR 𝑟𝑟𝑗𝑗 VECTOR DEFINING THE BASE OF ARM 𝑗𝑗 RELATIVE TO THE SATELLITE COM TABLE 2: Definition of terms in EQUATIONS (2) and (3) ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II The objective of this work is to demonstrate the use of a PID controller with load estimative ca- pacity in the control of position and attitude of satel- lites with onboard robotic manipulators. The simula- tion was run for multiple scenarios: without control, with non-predictive control, and with the load esti- mating controller. For runs with the controller, it is assumed that the satellite has an ideal control actua- tor onboard that can instantly induce the required torques or forces determined by the PID controller. It is important to note that this is a preliminary as- sumption used to evaluate the effectiveness of the control scheme rather than to evaluate its perfor- mance in a realistic application. The position and at- LEFT: FIGURE 4: Simplified block dia- gram describing the logical rela- tions between the disturbance load, satellite, and controller BELOW: FIGURE 5: Disturbance forces and moments acting on the satellite throughout the arm’s trajectory The forces and moments of the satellite disturbance load are plotted against time in FIGURE 5. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II titude responses of the satellite both with and with- out control are compared in FIGURE 6. The left side of FIGURE 6 shows the uncontrolled case: notice that the angular and translational positions diverge from the reference (initial condition). The controlled case is shown on the right side of FIGURE 6. Notice how the satellite’s deviation from the reference was drasti- cally reduced compared to the uncontrolled case and that the satellite returned to the initial reference state (position of (0, 0, 0) and angular position of (0, 0, 0)) after the disturbance loads ceased. It is im- portant to note that this simulation represents a “worst case scenario” for the satellite, as several as- sumptions were made that negatively impact the control effectiveness. Despite these assumptions, the PID controller was very effective at stabilizing the system. Thus, we have succeeded in designing an ef- fective controller for the satellite. 4 DISCUSSION OF SIMULATION RESULTS The use of the Proportional-Integral-Deriva- tive control scheme with load prediction resulted in a very accurate satellite response. The maximum de- viations from reference position and attitude were . 07 𝑚𝑚𝑚𝑚 and 1.26° respectively. Without the use of the controller, the satellite motion would continue indefinitely following a disturbance. These results provide evidence that such a control scheme could effectively maintain satellite position and attitude during robotic servicing operations. The use of cal- culated robotic arm disturbances as load predictions is a reasonable assumption when considering the ac- tual implementation of robotic servicing missions. If a space organization were to implement satellite- mounted robotic arms for maintenance purposes, FIGURE 6: Comparison of system response between uncontrolled (LEFT) and controlled (RIGHT) satellite position (TOP) and angu- lar position (BOTTOM) ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II the simulation of the robot dynamics prior to the ex- ecution of the mission would result in a more accu- rately controlled system during operation, as the re- sult of the simulation would be used as the load es- timation. In addition to the load prediction, the PID controllers add robustness to the scheme, adjusting for additional perturbations arising from the opera- tional environment. 5 CONCLUSIONS AND FUTURE WORK The goal of this project was to create a dy- namics simulation of a robotic manipulator and sat- ellite in order to develop a model-based control al- gorithm for maintaining satellite position and orien- tation. Even with a very basic PID feedback control scheme, the results of the simulation were very promising. The arm load estimation technique al- lowed the controller to effectively eliminate transla- tional and angular displacement. We believe that this research offers a framework for future investiga- tion into more advanced control methods, including nonlinear approaches such at Lyapunov theory, which incorporates the nonlinear system model into the controls equation. We also hope to develop a more accurate satellite model that includes realistic actuators, such as reaction wheels and gas thrusters, to better evaluate the effectiveness of the control scheme. With this, we also hope to investigate vari- ous satellite/arm designs and to determine which configuration is most effective. This was inspired by DAPRA’s Robotic Ser- vicing of Geosynchronous Satellites (RSGS)[4] pro- posal, which aims to extend the operational lives of satellites located in geostationary orbit through is- sue diagnosis, repair, and upgrades. The implica- tions of this proposal include lowering the cost of operation for geosynchronous satellites, thereby en- couraging more private companies to contribute to space infrastructure. This, in turn, aligns closely with the NASA Space Technology Mission Directorate[18], as contributing to this Earth-based infrastructure will facilitate future missions carried out to the moon any beyond. We hope future work on this project will in- centivize more private companies to begin develop- ing autonomous, robotic satellite systems to further humanities reach beyond planet Earth∎ 6 ACKNOWLEDGEMENTS The authors would like to acknowledge the help of others who contributed to the success of this project. We thank our Aresty research advisor, Dr. Annalisa Scacchioli, for her continuous guidance and encouragement throughout the duration of the project. We thank Dr. Aaron Mazzeo for introducing us to the robotics toolbox. We thank Dr. Laurent Burlion for his discussion about control methodolo- gies and satellite attitude dynamics. We thank Mike Vivar for his recommendation of the robotics toolbox. Finally, we thank our undergraduate peers, especially Michael Higgins and Aaron Pfister, for their continued interest in our project. 7 REFERENCES [1] F. Garcia, L. Peret, G. Verfaillie, Deployment and Mainte- nance of a Constellation of Satellites: a Benchmark, 2003. [2] Defense Advanced Research Projects Agency. Robotic Ser- vicing of Geosynchronous Satellites (RSGS). HTTPS://WWW.DARPA.MIL/PROGRAM/ROBOTIC-SERVICING-OF-GEOSYNCHRO- NOUS-SATELLITES [3] A. Antonello, P. Tsiotras, A. Valverde, Free-flying Spacecraft- mounted Manipulators: A Tool for Simulating Dynamics and Control, 2019. [4] Defense Advanced Research Projects Agency. (2019, May 22). Robotic Servicing of Geosynchronous Satellites Propos- ers Day. [5] P.I. Corke, “Robotics, Vision & Control”, Springer 2017, ISBN 978-3-319-54413-7. [6] MathWorks. 6DOF (Euler Angles). HTTPS://WWW.MATHWORKS.COM/HELP/AEROBLKS/6DOFEULERANGLES.HTML [7] P.C. Hughes, “Spacecraft Attitude Dynamics”, Dover Publica- tions 2004, ISBN 0-486-43925-9 [8] Northrop Grumman. Space Logistics Services. HTTPS://WWW.NORTHROPGRUMMAN.COM/SPACE/SPACE-LOGISTICS-SER- VICES/ [9] Phipps, Claude & Albrecht, G. & Friedman, H. & Gavel, D. & George, E. & Murray, J. & Ho, Chunching & Priedhorsky, W. & Michaelis, M.M. & Reilly, J.P.. (1996). ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser. March 1996. [10] Spectrolab. Space Solar Panels. HTTPS://WWW.SPECTROLAB.COM/DATASHEETS/PANEL/PANELS.PDF [11] MathWorks. Open PID Tuner. 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HTTPS://WWW.MATHWORKS.COM/HELP/SIGNAL/REF/SGOLAYFILT.HTML https://www.darpa.mil/program/robotic-servicing-of-geosynchronous-satellites https://www.darpa.mil/program/robotic-servicing-of-geosynchronous-satellites https://www.mathworks.com/help/aeroblks/6dofeulerangles.html https://www.northropgrumman.com/space/space-logistics-services/ https://www.northropgrumman.com/space/space-logistics-services/ https://www.spectrolab.com/DataSheets/Panel/panels.pdf https://www.mathworks.com/help/slcontrol/ug/designing-controllers-with-the-pid-tuner.html https://www.mathworks.com/help/slcontrol/ug/designing-controllers-with-the-pid-tuner.html https://www.mathworks.com/help/signal/ref/sgolayfilt.html ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II [14] C.J. Damaren, A.H.J. de Ruiter, J.R. Forbes, “Spacecraft Dy- namics and Control: An Introduction”, Wiley 2013, ISBN 978- 1-118-34236-7 [15] S.A. Rawashdeh, Attitude Analysis of Small Satellites Using Model-Based Simulation, International Journal of Aerospace Engineering, 2019. [16] R. Hernandez-Alvarado, L.G. Garcia-Valdovinos, et. al., Neu- ral Network-Based Self-Tuning PID Control for Underwater Vehicles, Sensors, 2016. [17] H.E.Soken, C. Hajiyev, S. Sakai, Robust Kalman filtering for small satellite attitude estimation in the presence of measure- ment faults, European Journal of Control, 2013. [18] NASA. Space Technology Mission Directorate. HTTPS://WWW.NASA.GOV/DIRECTORATES/SPACETECH/ABOUT_US/INDEX.HTML https://www.nasa.gov/directorates/spacetech/about_us/index.html ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 8 SUPPLEMENTARY TABLES JOINT 𝑑𝑑 𝑎𝑎 𝛼𝛼 OFFSET 1 0.3469785 0 − 𝜋𝜋 2 0 2 0.2517295 0 − 𝜋𝜋 2 0 3 0.8776515 0 𝜋𝜋 2 0 4 0.2122692 0 − 𝜋𝜋 2 0 5 0.1592019 0 � 127 449 � 0 5A 0.6349236746 0 −� 127 449 � 0 5B 0.1074953 0 𝜋𝜋 2 0 6 0.1728089 0 − 𝜋𝜋 2 0 7 0.2000229 0 0 0 LINK 𝑚𝑚 (𝑘𝑘𝑘𝑘) 𝑟𝑟 (𝑚𝑚) 𝐿𝐿 (𝑚𝑚) 1 2.45581930238976 0.0530673 0.3469785 2 1.78167282722394 0.0530673 0.2517295 3 8.64176380422475 0.0625922 0.8776515 4 0.802308387477559 0.03877995 0.2122692 5 0.540058000435033 0.0367389 0.1592019 5A 2.15384119253858 0.0367389 0.6349236746 5B 0.364654547302287 0.0367389 0.1074953 6 0.425389148502536 0.0312961 0.1728089 7 0.536125954214708 0.0326568 0.2000229 SUPPLEMENTAL TABLE S1: Denavit-Hartenberg parameters for the FREND Mark II arm model. SUPPLEMENTAL TABLE S2: Mass, radius, and length of each cylindrical link object used in the definition of the FREND arm model ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II LINK 𝐼𝐼 (𝑘𝑘𝑘𝑘 ∙ 𝑚𝑚2) 1 (0.00778041772567878 0 0 0 0.0116706265885182 0 0 0 0.00778041772567878 ) 2 (0.0056446167813748 0 0 0 0.0084669251720622 0 0 0 0.0056446167813748 ) 3 (0.0380886296066415 0 0 0 0.0571329444099623 0 0 0 0.0380886296066415 ) 4 (0.00135740156152532 0 0 0 0.00203610234228798 0 0 0 0.00135740156152532 ) 5 (0.00082005923636261 0 0 0 0.00085043340488162 0.00010738583988220 0 0.00010738583988220 0.001199714686025 ) 5A (0.003270532724590 0 0 0 0.003391669964095 − 0.00042827260265995 0 − 0.00042827260265995 0.004784661847380 ) 5B (0.000553715210877318 0 0 0 0.000830572816315976 0 0 0 0.000553715210877318 ) 6 (0.000468726352717391 0 0 0 0.000703089529076087 0 0 0 0.000468726352717391 ) 7 (0.000643230468209275 0 0 0 0.000643230468209275 0 0 0 0.000964845702313913 ) 𝐾𝐾𝑃𝑃 𝐾𝐾𝐼𝐼 𝐾𝐾𝐷𝐷 FORCE 617.061827129926 35.6866537699081 2667.42085820047 MOMENT 102.843637854979 5.94777562634487 444.570143033412 SUPPLEMENTAL TABLE S3: Inertia tensors for each link of the FREND Mk. II arm model SUPPLEMENTAL TABLE S3: Proportional, integral, and derivative gains for the force and moment PID controllers