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On the dynamics of a novel manipulator with slewing and deployable links
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Acta Astronautica Vol. 51, No. 12, pp. 821–830, 2002 ? 2002 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0094-5765(02)00034-6 0094-5765/02/$ - see front matter
ON THE DYNAMICS OF A NOVEL MANIPULATOR WITH SLEWING AND DEPLOYABLE LINKS V. J. MODI†‡, J. ZHANG§ and C. W. DE SILVA Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4
and A. K. MISRA¶ Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 (Received 18 June 1999)
Abstract—Using a novel space platform-based manipulator with slewing and deployable links, the paper addresses two issues of considerable importance: (a) How important is it to model Dexibility of the system? (b) How many modes are needed to adequately represent the elastic character? Results suggest that the fundamental mode is able to capture physics of the response quite accurately. Due to its massive character, the platform dynamics is virtually unaHected, even by severe maneuvers of the manipulator. Hence, treating the platform as rigid would save the computational cost without aHecting the accuracy. Although the link Dexibility does aHect the manipulator’s tip vibration, the joint and platform vibrations remain negligible. The revolute joint Dexibility appears to be an important parameter aHecting both the joint as well as tip responses. The information should prove useful in the design of this new class of manipulators. ? 2002 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION
The Space shuttle-based Canada arm has vividly demonstrated its application in launching of satellites as well as retrieval of disabled spacecraft for repair. The Mobile Servicing System (MSS) on the International Space Station will assist in its construction, operation, and maintenance. There have been proposals for free Dying robotic systems with appropriate instrumentation to monitor health of spacecraft, identify problems and even perform corrective measures. Most of these applications involve multilink manipulators with revolute joints for which there is a vast body of literature. This has been reviewed quite eHectively by Morita and Modi [1], Modi and Chan [2], Modi et al. [3,4], Nagata et al. [5] and several other investigators [6,7].
†Corresponding author. Tel.: +1-604-822-2914; fax: +1-604-822-2403. E-mail address: [email protected] (V.J. Modi). ‡Professor Emeritus; Academy Member IAA; Fellow, AIAA. §Visiting Scholar. Natural Sciences and Engineering Research Council Chair Professor of Industrial Automation; Fellow IEEE. ¶Professor; Associate Fellow, AIAA. 821
On the other hand, manipulators with revolute and prismatic joints, permitting slewing as well as deployment=retrieval of links, have received relatively little attention [7]. Such variable geometry, snake-like manipulators have distinct advantages of reduced coupling eHects leading to simpler equations of motion and inverse kinematics, less number of singularity conditions, and ease of obstacle avoidance [8]. Using the order-N formulation of the problem, Caron et al. [9] studied dynamics of such orbiting platform-based systems when subjected to a variety of initial disturbances and maneuvers. Analysis of large scale elastic systems has raised, in the past, two major issues of importance: (a) For a continuum system, Dexibility is often represented by a synthesis of component or system modes. The amount of computational eHort involved relates to the number of modes used to represent Dexible character of the system. So the question arises: How many modes are required to capture the Dexibility eHects? (b) Inclusion of elastic eHects enormously complicates the formulation and increases the computational eHort. How important are the
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V. J. Modi et al. Payload
Payload
Prismatic Joint Slewing/Deploying Manipulator Unit Revolute Joint Slewing-Deployable Link Unit
Orbit Center of Mass
Trajectory
Fig. 1. Schematic diagram of a Dexible variable geometry manipulator, based on a space platform, considered for study.
Mobile Base
Slewing Link Platform
m
Orbit
Fig. 3. Connecting modules, each with slewing and deployable links, would result into a snake-like manipulator.
Kj Cj Rotor
m
Stator Revolute Joint
Initial Position of Rotor
Fig. 2. Geometry of the revolute joint showing rigid body and Dexible motions coordinates m and m , respectively.
elastic eHects? To what extent they aHect the response? The present paper attempts to address these issues. 2. SYSTEM CONFIGURATION AND APPROACH
The system under consideration is schematically shown in Fig. 1. It represents a platform, supporting a manipulator near its tip, in an arbitrary orbit around earth. The manipulator consists of one unit with two links: one is free to slew while the other is permitted to deploy or retrieve. The platform as well as links are treated as Dexible and modeled as Euler–Bernoulli beams. The revolute joint is also treated as Dexible, i.e. with Pnite torsional stiHness (Fig. 2). The prismatic joint, being located on the slewing link, deforms with it. Connecting such modules in series would lead to a snake-like manipulator as shown in Fig. 3.
The governing equations of motion for planar dynamics of this class of systems were obtained, using the Lagrangian principle, by Caron who also developed a numerically eQcient order-N algorithm for their integration [7,9]. This is a signiPcant improvement recognizing that, normally, computational eHort varies as N 3 , where N is the number of degrees of freedom. In the present study, this approach is used to advantage. The elastic deformations of the platform and manipulator are discretized using the conventional assumed modes for each component of the system [10], i.e. by a product of spatially varying admissible shape functions and time-dependent generalized coordinates. The platform is treated as a free–free beam while the links are taken to be cantilever beams with tip masses. It is often useful to specify some of the generalized coordinates. For example, the mobile base may be placed at a given location on the platform; or the manipulator may be commanded to perform prescribed maneuvers. This is achieved through constraint relations, which are introduced in the equations of motion through Lagrange multipliers. In the case of the prescribed maneuvers, acceleration of the speciPed coordinate, qSs , is dePned as the desired proPle. In the present study, a sine-on-ramp proPle is selected as it assures zero velocity and acceleration at the beginning and end of the maneuver, thereby reducing the structural response. Thus
Manipulator dynamics
the maneuver time–history can be written as U 2 Uqs − sin ; qSs = U 2 U
(1)
where qs is the speciPed or constrained coordinate, Uqs is its desired variation, is the time, and U is the time allowed for the maneuver. 3. SIMULATION CONSIDERATIONS
The system performance is governed by a large number parameters. Some of the important variables are listed in Table 1. Obviously, a systematic change of above variables would lead to a large volume of information. However, it would also demand considerable amount of time, eHort and computational cost. Hence, one is forced to focus on cases that are likely to provide useful trends. As the study aims at assessing the eHect of system Dexibility, the attention was particularly directed towards parameters that would contribute to that end. These include the manipulator position; platform, link and joint Dexibility; number of modes; proPle and speed of maneuver; and mass of the payload. Even with these selected parameters, the task is formidable. Hence only a few typical results are presented for conciseness. Numerical values used in the analysis, unless speciPed otherwise, are indicated below: Orbit: Circular orbit at an altitude of 400 km; period = 92:5 min. Platform: Geometry: circular cylindrical with diameter = 3 m; axial to transverse inertia ratio of 0.005; mass (mp ) = 120; 000 kg; length (lp ) = 120 m; Dexural rigidity (EIp ) = 5:5 × 108 N m2 (changed in some cases). Manipulator position (d): d = 0 or 60 m.
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Manipulator module (lm ): Initial length of the manipulator module (i.e. ls + deployed length, 7:5 + ld ) is taken as 10 m, i.e. the deployable link is initially extended by 2:5 m. Here ls , ld represent lengths of slewing and deployable links, respectively. Manipulator links (slewing and deployable): Geometry: circular cylindrical with axial to transverse inertia ratio of 0.005; mass (ms ; md )=200 kg; length (ls ; ld; max ) = 7:5 m; Dexural rigidity (EIs ; EId ) = 5:5 × 105 N m2 (changed in some cases). Revolute joint: Mass = 20 kg; moment of inertia (Izz )=10 kg m2 ; stiHness (Kj )=104 N m=rad (changed in some cases); note, the prismatic joint is treated as a part of the slewing link. Payload: The manipulator carries a 2000 kg payload, i.e. the payload ratio ≈ 5. Modes: Fundamental mode for a cantilever beam with tip mass (number of modes used changed in some cases). Note, subscripts d, j, m, p, and s correspond to deployable link, revolute joint, manipulator, platform and slewing link, respectively. Initially, the platform is in equilibrium along the local vertical. The manipulator links are aligned with the platform, i.e. they are also along the local vertical before the maneuver. The manipulator undergoes a combined maneuver of ◦ 90 in slew and 5 m deployment in 0.01 orbit, i.e. 0:925 min. The damping is purposely assumed to be zero in all components to obtain conservative estimate of the response, i.e. Ψp Local Vertical
ep
αm x
βm
Table 1. Important factors aHecting the system performance. Parameters
Orbit eccentricity Mass of: platform; links; joints; payload StiHness of: platform; links; joints Damping of: platform; links; joints Link length
Initial conditions
Platform attitude Manipulator location and orientation Deformation of platform, manipulator links, joints
Maneuvers
Type: slewing; deployment; retrieval; base translation Amplitude Speed
Discretization
Shape of admissible functions Number of modes
em
d Centerline of the Platform
Deformed Manipulator
C. M. Orbit y
Underformed Platform
To Earth Center
Fig. 4. Schematic diagram showing important parameters appearing in the response study.
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Fig. 5. EHect of the manipulator location, during a slewing and deployment maneuver, on system response and tip trajectory dynamics.
Cj : damping coeQcient for joints, considered zero; m : structural damping coeQcient for manipulator links, considered zero; p : structural damping coeQcient for the platform, considered zero. More important speciPed and response parameters are summarized below and indicated in Fig. 4: d:
position of the base from the center of the platform; em : manipulator tip vibration amplitude; ep : platform tip vibration amplitude; x; y: body Pxed coordinate system with x aligned with the undeformed axis of the platform and y normal to x in the orbital plane; m : rigid body rotation, with respect to the platform, during a slew maneuver;
m : contribution due to Dexibility of the revolute joint; p : pitch libration. 4. RESULTS AND DISCUSSION
4.1. Manipulator location
To begin with the eHect of manipulator location on the platform is assessed during its simultaneous ◦ 90 slew and 5 m deployment maneuver in 0.01 orbit. Three cases are considered: Dexible manipulator with the base located at the center of the platform (d = 0); Dexible manipulator with the base at the tip of the platform (d = 60 m); and the rigid manipulator located at the tip. Figure 5 shows the resulting system response. As can be expected, location of the manipulator at the platform tip would induce a relatively larger moment about the system center of mass (approximately at the platform center), and hence
Manipulator dynamics
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Fig. 6. InDuence of higher modes on the system response.
constitute a more severe disturbance. This is clearly reDected in the platform responses p and ep . Note, ◦ the platform libration reaches nearly 0:8 when the manipulator is at the platform tip (Fig. 5). This is rather large and may be unacceptable as, depending on the mission, the pointing accuracy re◦ quired may be as small as 0:1 . In practice, this will be regulated through a suitably designed control system using momentum gyros. On the other hand, the librational response remains essentially unaffected by the system Dexibility. One may anticipate that the manipulator response due to the maneuver should remain essentially unaHected by its location on the platform, except for minor coupling eHects between the massive platform and the manipulator (Fig. 5; em ; m ). However, Dexibility eHects are quite signiPcant suggesting a need for damping and active control. Figure 5 also shows the payload (i.e. the manipulator tip) trajectories for the two Dexible cases, d = 0 and 60 m. The coordinate value in parentheses correspond to the d = 0 case, i.e. when the manipulator is located at the center of the platform. As before, the manipulator location has virtually no inDuence on its tip dynamics. However, Dexibility of the manipulator, no matter where it is
located, is important and leads to a large amplitude tip oscillations (≈ 4 m) following the maneuver. To summarize: (i) tip location of the manipulator accentuates the platform’s librational response due to the maneuver; (ii) location of the manipulator has virtually no eHect on its own response; (iii) manipulator Dexibility eHects on the platform dynamics are rather small, however, elastic character of the manipulator aHects its own dynamics quite signiPcantly. In the subsequent studies, the manipulator is located at the platform tip (d = 60 m) to impose the most severe disturbance through its prescribed maneuver. 4.2. E(ect of higher order modes
As mentioned earlier, the system Dexibility, i.e. elastic character of the members such as platform, links, joints, etc., was discretized using assumed modes. One of the important problems faced by dynamicists is the number of modes needed to predict the system response accurately. This is important as the number of modes directly aHect the computational eHort. To get insight into this aspect, the Dexibility of the platform and each of the two links was represented by up to Prst Pve modes. The
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Fig. 7. System response as aHected by variation of the revolute joint stiHness.
system response was evaluated for the same maneuver as before for a variety of cases involving change in system parameters, maneuver speed, etc. In all the cases, the eHect of second and higher modes was found to be negligible. A typical set of results are presented in Fig. 6.c It is apparent that the fundamental mode is able to capture the system dynamics quite well. In the subsequent studies, the member Dexibility is represented by the fundamental mode. 4.3. In,uence of the revolute joint sti(ness
The prismatic joint, being a part of the slewing link, has its Dexibility accounted through the link elasticity. Here, the focus is on the revolute joint. Four diHerent values of the joint stiHness were considered, including the joint treated as rigid. It should be emphasized that the value of Km = 1 × 104 N m=rad, considered in Fig. 5 during the manipulator base location study, is quite low compared to the conventional value. This low value was purposely chosen to accentuate the eHect
of maneuver-induced disturbances. Thus, Kj = 5 × 103 N m=rad represents an extremely Dexible joint. At the other end of the spectrum is the ideal rigid joint. As can be expected, the joint Dexibility aHects the manipulator dynamics signiPcantly, however its inDuence on the platform response is relatively small. The platform pitch libration (p ) as well as tip vibration (ep ) are now modulated because of the joint Dexibility, however ep continues to remain negligibly small (≈ 4 mm maximum amplitude) while the p response remains essentially unchanged. On the other hand, amplitude of the joint vibration can reach as high as 30◦ . Although em is measured with reference to the link-Pxed coordinate system, it increases signiPcantly with decrease in the joint Dexibility suggesting coupling between the two. Figures 7(e) and (f) better depict the joint Dexibility eHects on the manipulator tip dynamics with reference to the platform-based x; y-coordinate system. Note, the steady-state transverse oscillation
Manipulator dynamics
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Fig. 8. EHect of platform Dexibility on the system response to a manipulator maneuver.
amplitude can exceed 5 m for Kj = 5 × 103 case, primarily because of the very low joint Dexibility, which also leads to the axial component of the tip vibration of around 2:5 m. This suggests careful planning of the joint stiHness in the manipulator design, and need for active control to meet mission speciPcations. 4.4. E(ect of platform sti(ness
The platform stiHness was varied over two orders of magnitude, EIp = 5:5 × 107 –5:5 × 109 N m2 . Even the idealized case of rigid platform (EIp =∞) was also considered. It is of interest to note that the Dexural rigidity of the platform had virtually no eHect on the platform libration, joint vibration and the manipulator tip response (Fig. 8). Even the payload trajectory remained essentially unaffected. Of course, the platform tip deDection, being the parameter directly related to the platform Dexibility, showed some eHect. However, it was virtually negligible until the platform Dexibility reached a value of 5:5 × 107 N m2 , the lowest considered in the study. This is an important piece of information and can be used to advantage in the system design. It
suggests that the platform Dexibility has insigniPcant inDuence on the manipulator dynamics. Thus treating the platform as rigid can give information of considerable importance with a large saving in the computational eHort. 4.5. Flexibility of links
The link stiHness was changed by an order of magnitude above and below the nominal values EIs = EId = 5:5 × 105 N m2 . The results are presented in Fig. 9. The cases with rigid links as well as the whole system considered rigid are also included (Figs. 9(e) and (f)). As anticipated, the link stiHness has virtually no eHect on the system response in p , ep and m degrees of freedom except for the phase shift, which is of little consequence in design. However, as can be expected, the manipulator tip deDection increases signiPcantly, in a nonlinear fashion, as the links’ Dexural rigidity diminishes. The corresponding increase in the period of tip oscillations is also apparent. Note, for a decrease in the link Dexibility by an order of magnitude from its nominal value, the tip oscillation amplitude increases from ≈ 0:4 to ≈ 3:6 m. Although in the real-life situation there will be
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Fig. 9. InDuence of the link Dexibility on the system dynamics.
some structural damping present, the system design is usually a compromise between several competing factors. For example, increased Dexural rigidity to reduce tip deDection would demand an increase in weight. Thus, the results suggest that there are situations where active control of the system may be necessary. Figures 9(e) and (f) show x and y components of the payload trajectory with reference to the platform-based coordinate system. For the entire system considered rigid, there are no tip vibrations induced by the maneuver and the manipulator tip (i.e. payload) reaches the equilibrium value of x = 60 m and y = 16:5 m (lm = 15 m; 1:5 m radius of the platform’s cylindrical structure). The rigid link case essentially depicts the eHect of joint Dexibility. The link Dexibility does aHect the payload trajectory and hence its x and y components, quite signiPcantly, particularly for the case of EIs = EId = 5:5 × 104 N m2 . Note, now the contributions from joint and link Dexibilities are, approximately, similar in magnitude. This is
in contrast to the earlier cases where the joint Dexibility dominated the payload trajectory. 4.6. E(ect of maneuver speed
Figure 10 shows the eHect of maneuver speed on the resulting system dynamics. To recapitulate, the study so far considered manipulator located at the tip of the platform undergoing simultaneous ◦ slewing (from m = 0 to 90 ) and deployment (from lm = 10 m to 15 m) in 0.01 orbit (55:5 s). The objective now is to assess the inDuence of faster and slower maneuvers compared to the nominal one. Intuitively, one would expect the faster maneuver to accentuated the response. This is borne out by the simulation results. Note, the faster maneuver aHects both the platform as well as the manipulator dynamics, although the eHect on the former is relatively small. The joint vibration ◦ amplitude can reach nearly ≈ 50 , while the manipulator tip oscillation amplitude exceeds 1 m. The results suggest inherent danger associated
Manipulator dynamics
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Fig. 10. System response as aHected by the maneuver speed.
with relatively faster maneuvers and thus provides useful information for the system operation.
5. CONCLUDING REMARKS
Using the Lagrangian procedure and order-N formulation for a novel manipulator with slewing as well as deployable links, the eHect of Dexibility on system dynamics is studied in detail. The response of the system to a speciPed maneuver has been evaluated through a planned variation of parameters likely to aHect both librational and vibrational degrees of freedom. Based on the results, following general conclusions can be made: (i) Position of the manipulator at the tip of the platform represents the critical location in terms of severity of the disturbance. Location of the manipulator on the platform has little eHect on its own dynamics. (ii) Flexibility of the joints, links and platform has very little eHect on the platform’s librational response. (iii) Flexibility of the manipulator does not aHect the platform’s vibrational dynamics signiP-
cantly, however, its own dynamical response changes substantially. (iv) The fundamental mode is able to capture system response rather well. (v) The speed of maneuver and joint Dexibility appear to be two major parameters aHecting the system dynamics. (vi) The results provide information useful in design and operation of this novel manipulator system. REFERENCES
1. Morita, Y. and Modi, V. J., Dynamics of Dexible orbiting platform with Mobile Remote Manipulator System (MRMS). Institute of Space and Astronautical Science, Tokyo, Japan, Report No. 626, February 1988. 2. Modi, V. J. and Chan, J. K., Acta Astronautica, 1991, 25, 149 –156. 3. Modi, V. J., Ng, A. C. and Karray, F., Nonlinear Dynamics, 1994, 5, 71–91. 4. Modi, V. J., Mah, H. W. and Misra, A. K., Acta Astronautica, 1994, 32, 419 – 439. 5. Nagata, T., Modi, V. J. and Matsuo, H. In Collection of Technical papers, AIAA=AAS Astrodynamics Conference, Scottsdale, Arizona, USA, 1994.
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AIAA Publisher, Washington DC, USA, Paper No. AIAA-94-3756-CP, 1994, pp. 366 –375. 6. Nagata, T., Dynamics of Dexible multibody systems: a formulation with applications. Ph. D. Thesis, University of Tokyo, 1995. 7. Caron, M., Planar dynamics and control of space-based Dexible manipulators with slewing and deployable links. M. A. Sc. Thesis, University of British Columbia, 1996.
8. Chu, M. S. T., Design, construction and operation of a variable geometry manipulator. M. A. Sc. Thesis, University of British Columbia, 1997. 9. Caron, M., Modi, V. J., Pradhan, S., de Silva, C. W. and Misra, A. K., Journal of Guidance, Control, and Dynamics, 1998, 24, 572–580. 10. Meirovitch, L., Elements of Vibration Analysis, 2nd edn. McGraw-Hill Book Company, Inc., New York, 1986, pp. 255 –256, 282–290.
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Acta Astronautica Vol. 51, No. 12, pp. 821–830, 2002 ? 2002 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0094-5765(02)00034-6 0094-5765/02/$ - see front matter
ON THE DYNAMICS OF A NOVEL MANIPULATOR WITH SLEWING AND DEPLOYABLE LINKS V. J. MODI†‡, J. ZHANG§ and C. W. DE SILVA Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4
and A. K. MISRA¶ Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 (Received 18 June 1999)
Abstract—Using a novel space platform-based manipulator with slewing and deployable links, the paper addresses two issues of considerable importance: (a) How important is it to model Dexibility of the system? (b) How many modes are needed to adequately represent the elastic character? Results suggest that the fundamental mode is able to capture physics of the response quite accurately. Due to its massive character, the platform dynamics is virtually unaHected, even by severe maneuvers of the manipulator. Hence, treating the platform as rigid would save the computational cost without aHecting the accuracy. Although the link Dexibility does aHect the manipulator’s tip vibration, the joint and platform vibrations remain negligible. The revolute joint Dexibility appears to be an important parameter aHecting both the joint as well as tip responses. The information should prove useful in the design of this new class of manipulators. ? 2002 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION
The Space shuttle-based Canada arm has vividly demonstrated its application in launching of satellites as well as retrieval of disabled spacecraft for repair. The Mobile Servicing System (MSS) on the International Space Station will assist in its construction, operation, and maintenance. There have been proposals for free Dying robotic systems with appropriate instrumentation to monitor health of spacecraft, identify problems and even perform corrective measures. Most of these applications involve multilink manipulators with revolute joints for which there is a vast body of literature. This has been reviewed quite eHectively by Morita and Modi [1], Modi and Chan [2], Modi et al. [3,4], Nagata et al. [5] and several other investigators [6,7].
†Corresponding author. Tel.: +1-604-822-2914; fax: +1-604-822-2403. E-mail address: [email protected] (V.J. Modi). ‡Professor Emeritus; Academy Member IAA; Fellow, AIAA. §Visiting Scholar. Natural Sciences and Engineering Research Council Chair Professor of Industrial Automation; Fellow IEEE. ¶Professor; Associate Fellow, AIAA. 821
On the other hand, manipulators with revolute and prismatic joints, permitting slewing as well as deployment=retrieval of links, have received relatively little attention [7]. Such variable geometry, snake-like manipulators have distinct advantages of reduced coupling eHects leading to simpler equations of motion and inverse kinematics, less number of singularity conditions, and ease of obstacle avoidance [8]. Using the order-N formulation of the problem, Caron et al. [9] studied dynamics of such orbiting platform-based systems when subjected to a variety of initial disturbances and maneuvers. Analysis of large scale elastic systems has raised, in the past, two major issues of importance: (a) For a continuum system, Dexibility is often represented by a synthesis of component or system modes. The amount of computational eHort involved relates to the number of modes used to represent Dexible character of the system. So the question arises: How many modes are required to capture the Dexibility eHects? (b) Inclusion of elastic eHects enormously complicates the formulation and increases the computational eHort. How important are the
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V. J. Modi et al. Payload
Payload
Prismatic Joint Slewing/Deploying Manipulator Unit Revolute Joint Slewing-Deployable Link Unit
Orbit Center of Mass
Trajectory
Fig. 1. Schematic diagram of a Dexible variable geometry manipulator, based on a space platform, considered for study.
Mobile Base
Slewing Link Platform
m
Orbit
Fig. 3. Connecting modules, each with slewing and deployable links, would result into a snake-like manipulator.
Kj Cj Rotor
m
Stator Revolute Joint
Initial Position of Rotor
Fig. 2. Geometry of the revolute joint showing rigid body and Dexible motions coordinates m and m , respectively.
elastic eHects? To what extent they aHect the response? The present paper attempts to address these issues. 2. SYSTEM CONFIGURATION AND APPROACH
The system under consideration is schematically shown in Fig. 1. It represents a platform, supporting a manipulator near its tip, in an arbitrary orbit around earth. The manipulator consists of one unit with two links: one is free to slew while the other is permitted to deploy or retrieve. The platform as well as links are treated as Dexible and modeled as Euler–Bernoulli beams. The revolute joint is also treated as Dexible, i.e. with Pnite torsional stiHness (Fig. 2). The prismatic joint, being located on the slewing link, deforms with it. Connecting such modules in series would lead to a snake-like manipulator as shown in Fig. 3.
The governing equations of motion for planar dynamics of this class of systems were obtained, using the Lagrangian principle, by Caron who also developed a numerically eQcient order-N algorithm for their integration [7,9]. This is a signiPcant improvement recognizing that, normally, computational eHort varies as N 3 , where N is the number of degrees of freedom. In the present study, this approach is used to advantage. The elastic deformations of the platform and manipulator are discretized using the conventional assumed modes for each component of the system [10], i.e. by a product of spatially varying admissible shape functions and time-dependent generalized coordinates. The platform is treated as a free–free beam while the links are taken to be cantilever beams with tip masses. It is often useful to specify some of the generalized coordinates. For example, the mobile base may be placed at a given location on the platform; or the manipulator may be commanded to perform prescribed maneuvers. This is achieved through constraint relations, which are introduced in the equations of motion through Lagrange multipliers. In the case of the prescribed maneuvers, acceleration of the speciPed coordinate, qSs , is dePned as the desired proPle. In the present study, a sine-on-ramp proPle is selected as it assures zero velocity and acceleration at the beginning and end of the maneuver, thereby reducing the structural response. Thus
Manipulator dynamics
the maneuver time–history can be written as U 2 Uqs − sin ; qSs = U 2 U
(1)
where qs is the speciPed or constrained coordinate, Uqs is its desired variation, is the time, and U is the time allowed for the maneuver. 3. SIMULATION CONSIDERATIONS
The system performance is governed by a large number parameters. Some of the important variables are listed in Table 1. Obviously, a systematic change of above variables would lead to a large volume of information. However, it would also demand considerable amount of time, eHort and computational cost. Hence, one is forced to focus on cases that are likely to provide useful trends. As the study aims at assessing the eHect of system Dexibility, the attention was particularly directed towards parameters that would contribute to that end. These include the manipulator position; platform, link and joint Dexibility; number of modes; proPle and speed of maneuver; and mass of the payload. Even with these selected parameters, the task is formidable. Hence only a few typical results are presented for conciseness. Numerical values used in the analysis, unless speciPed otherwise, are indicated below: Orbit: Circular orbit at an altitude of 400 km; period = 92:5 min. Platform: Geometry: circular cylindrical with diameter = 3 m; axial to transverse inertia ratio of 0.005; mass (mp ) = 120; 000 kg; length (lp ) = 120 m; Dexural rigidity (EIp ) = 5:5 × 108 N m2 (changed in some cases). Manipulator position (d): d = 0 or 60 m.
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Manipulator module (lm ): Initial length of the manipulator module (i.e. ls + deployed length, 7:5 + ld ) is taken as 10 m, i.e. the deployable link is initially extended by 2:5 m. Here ls , ld represent lengths of slewing and deployable links, respectively. Manipulator links (slewing and deployable): Geometry: circular cylindrical with axial to transverse inertia ratio of 0.005; mass (ms ; md )=200 kg; length (ls ; ld; max ) = 7:5 m; Dexural rigidity (EIs ; EId ) = 5:5 × 105 N m2 (changed in some cases). Revolute joint: Mass = 20 kg; moment of inertia (Izz )=10 kg m2 ; stiHness (Kj )=104 N m=rad (changed in some cases); note, the prismatic joint is treated as a part of the slewing link. Payload: The manipulator carries a 2000 kg payload, i.e. the payload ratio ≈ 5. Modes: Fundamental mode for a cantilever beam with tip mass (number of modes used changed in some cases). Note, subscripts d, j, m, p, and s correspond to deployable link, revolute joint, manipulator, platform and slewing link, respectively. Initially, the platform is in equilibrium along the local vertical. The manipulator links are aligned with the platform, i.e. they are also along the local vertical before the maneuver. The manipulator undergoes a combined maneuver of ◦ 90 in slew and 5 m deployment in 0.01 orbit, i.e. 0:925 min. The damping is purposely assumed to be zero in all components to obtain conservative estimate of the response, i.e. Ψp Local Vertical
ep
αm x
βm
Table 1. Important factors aHecting the system performance. Parameters
Orbit eccentricity Mass of: platform; links; joints; payload StiHness of: platform; links; joints Damping of: platform; links; joints Link length
Initial conditions
Platform attitude Manipulator location and orientation Deformation of platform, manipulator links, joints
Maneuvers
Type: slewing; deployment; retrieval; base translation Amplitude Speed
Discretization
Shape of admissible functions Number of modes
em
d Centerline of the Platform
Deformed Manipulator
C. M. Orbit y
Underformed Platform
To Earth Center
Fig. 4. Schematic diagram showing important parameters appearing in the response study.
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Fig. 5. EHect of the manipulator location, during a slewing and deployment maneuver, on system response and tip trajectory dynamics.
Cj : damping coeQcient for joints, considered zero; m : structural damping coeQcient for manipulator links, considered zero; p : structural damping coeQcient for the platform, considered zero. More important speciPed and response parameters are summarized below and indicated in Fig. 4: d:
position of the base from the center of the platform; em : manipulator tip vibration amplitude; ep : platform tip vibration amplitude; x; y: body Pxed coordinate system with x aligned with the undeformed axis of the platform and y normal to x in the orbital plane; m : rigid body rotation, with respect to the platform, during a slew maneuver;
m : contribution due to Dexibility of the revolute joint; p : pitch libration. 4. RESULTS AND DISCUSSION
4.1. Manipulator location
To begin with the eHect of manipulator location on the platform is assessed during its simultaneous ◦ 90 slew and 5 m deployment maneuver in 0.01 orbit. Three cases are considered: Dexible manipulator with the base located at the center of the platform (d = 0); Dexible manipulator with the base at the tip of the platform (d = 60 m); and the rigid manipulator located at the tip. Figure 5 shows the resulting system response. As can be expected, location of the manipulator at the platform tip would induce a relatively larger moment about the system center of mass (approximately at the platform center), and hence
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Fig. 6. InDuence of higher modes on the system response.
constitute a more severe disturbance. This is clearly reDected in the platform responses p and ep . Note, ◦ the platform libration reaches nearly 0:8 when the manipulator is at the platform tip (Fig. 5). This is rather large and may be unacceptable as, depending on the mission, the pointing accuracy re◦ quired may be as small as 0:1 . In practice, this will be regulated through a suitably designed control system using momentum gyros. On the other hand, the librational response remains essentially unaffected by the system Dexibility. One may anticipate that the manipulator response due to the maneuver should remain essentially unaHected by its location on the platform, except for minor coupling eHects between the massive platform and the manipulator (Fig. 5; em ; m ). However, Dexibility eHects are quite signiPcant suggesting a need for damping and active control. Figure 5 also shows the payload (i.e. the manipulator tip) trajectories for the two Dexible cases, d = 0 and 60 m. The coordinate value in parentheses correspond to the d = 0 case, i.e. when the manipulator is located at the center of the platform. As before, the manipulator location has virtually no inDuence on its tip dynamics. However, Dexibility of the manipulator, no matter where it is
located, is important and leads to a large amplitude tip oscillations (≈ 4 m) following the maneuver. To summarize: (i) tip location of the manipulator accentuates the platform’s librational response due to the maneuver; (ii) location of the manipulator has virtually no eHect on its own response; (iii) manipulator Dexibility eHects on the platform dynamics are rather small, however, elastic character of the manipulator aHects its own dynamics quite signiPcantly. In the subsequent studies, the manipulator is located at the platform tip (d = 60 m) to impose the most severe disturbance through its prescribed maneuver. 4.2. E(ect of higher order modes
As mentioned earlier, the system Dexibility, i.e. elastic character of the members such as platform, links, joints, etc., was discretized using assumed modes. One of the important problems faced by dynamicists is the number of modes needed to predict the system response accurately. This is important as the number of modes directly aHect the computational eHort. To get insight into this aspect, the Dexibility of the platform and each of the two links was represented by up to Prst Pve modes. The
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Fig. 7. System response as aHected by variation of the revolute joint stiHness.
system response was evaluated for the same maneuver as before for a variety of cases involving change in system parameters, maneuver speed, etc. In all the cases, the eHect of second and higher modes was found to be negligible. A typical set of results are presented in Fig. 6.c It is apparent that the fundamental mode is able to capture the system dynamics quite well. In the subsequent studies, the member Dexibility is represented by the fundamental mode. 4.3. In,uence of the revolute joint sti(ness
The prismatic joint, being a part of the slewing link, has its Dexibility accounted through the link elasticity. Here, the focus is on the revolute joint. Four diHerent values of the joint stiHness were considered, including the joint treated as rigid. It should be emphasized that the value of Km = 1 × 104 N m=rad, considered in Fig. 5 during the manipulator base location study, is quite low compared to the conventional value. This low value was purposely chosen to accentuate the eHect
of maneuver-induced disturbances. Thus, Kj = 5 × 103 N m=rad represents an extremely Dexible joint. At the other end of the spectrum is the ideal rigid joint. As can be expected, the joint Dexibility aHects the manipulator dynamics signiPcantly, however its inDuence on the platform response is relatively small. The platform pitch libration (p ) as well as tip vibration (ep ) are now modulated because of the joint Dexibility, however ep continues to remain negligibly small (≈ 4 mm maximum amplitude) while the p response remains essentially unchanged. On the other hand, amplitude of the joint vibration can reach as high as 30◦ . Although em is measured with reference to the link-Pxed coordinate system, it increases signiPcantly with decrease in the joint Dexibility suggesting coupling between the two. Figures 7(e) and (f) better depict the joint Dexibility eHects on the manipulator tip dynamics with reference to the platform-based x; y-coordinate system. Note, the steady-state transverse oscillation
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Fig. 8. EHect of platform Dexibility on the system response to a manipulator maneuver.
amplitude can exceed 5 m for Kj = 5 × 103 case, primarily because of the very low joint Dexibility, which also leads to the axial component of the tip vibration of around 2:5 m. This suggests careful planning of the joint stiHness in the manipulator design, and need for active control to meet mission speciPcations. 4.4. E(ect of platform sti(ness
The platform stiHness was varied over two orders of magnitude, EIp = 5:5 × 107 –5:5 × 109 N m2 . Even the idealized case of rigid platform (EIp =∞) was also considered. It is of interest to note that the Dexural rigidity of the platform had virtually no eHect on the platform libration, joint vibration and the manipulator tip response (Fig. 8). Even the payload trajectory remained essentially unaffected. Of course, the platform tip deDection, being the parameter directly related to the platform Dexibility, showed some eHect. However, it was virtually negligible until the platform Dexibility reached a value of 5:5 × 107 N m2 , the lowest considered in the study. This is an important piece of information and can be used to advantage in the system design. It
suggests that the platform Dexibility has insigniPcant inDuence on the manipulator dynamics. Thus treating the platform as rigid can give information of considerable importance with a large saving in the computational eHort. 4.5. Flexibility of links
The link stiHness was changed by an order of magnitude above and below the nominal values EIs = EId = 5:5 × 105 N m2 . The results are presented in Fig. 9. The cases with rigid links as well as the whole system considered rigid are also included (Figs. 9(e) and (f)). As anticipated, the link stiHness has virtually no eHect on the system response in p , ep and m degrees of freedom except for the phase shift, which is of little consequence in design. However, as can be expected, the manipulator tip deDection increases signiPcantly, in a nonlinear fashion, as the links’ Dexural rigidity diminishes. The corresponding increase in the period of tip oscillations is also apparent. Note, for a decrease in the link Dexibility by an order of magnitude from its nominal value, the tip oscillation amplitude increases from ≈ 0:4 to ≈ 3:6 m. Although in the real-life situation there will be
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Fig. 9. InDuence of the link Dexibility on the system dynamics.
some structural damping present, the system design is usually a compromise between several competing factors. For example, increased Dexural rigidity to reduce tip deDection would demand an increase in weight. Thus, the results suggest that there are situations where active control of the system may be necessary. Figures 9(e) and (f) show x and y components of the payload trajectory with reference to the platform-based coordinate system. For the entire system considered rigid, there are no tip vibrations induced by the maneuver and the manipulator tip (i.e. payload) reaches the equilibrium value of x = 60 m and y = 16:5 m (lm = 15 m; 1:5 m radius of the platform’s cylindrical structure). The rigid link case essentially depicts the eHect of joint Dexibility. The link Dexibility does aHect the payload trajectory and hence its x and y components, quite signiPcantly, particularly for the case of EIs = EId = 5:5 × 104 N m2 . Note, now the contributions from joint and link Dexibilities are, approximately, similar in magnitude. This is
in contrast to the earlier cases where the joint Dexibility dominated the payload trajectory. 4.6. E(ect of maneuver speed
Figure 10 shows the eHect of maneuver speed on the resulting system dynamics. To recapitulate, the study so far considered manipulator located at the tip of the platform undergoing simultaneous ◦ slewing (from m = 0 to 90 ) and deployment (from lm = 10 m to 15 m) in 0.01 orbit (55:5 s). The objective now is to assess the inDuence of faster and slower maneuvers compared to the nominal one. Intuitively, one would expect the faster maneuver to accentuated the response. This is borne out by the simulation results. Note, the faster maneuver aHects both the platform as well as the manipulator dynamics, although the eHect on the former is relatively small. The joint vibration ◦ amplitude can reach nearly ≈ 50 , while the manipulator tip oscillation amplitude exceeds 1 m. The results suggest inherent danger associated
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Fig. 10. System response as aHected by the maneuver speed.
with relatively faster maneuvers and thus provides useful information for the system operation.
5. CONCLUDING REMARKS
Using the Lagrangian procedure and order-N formulation for a novel manipulator with slewing as well as deployable links, the eHect of Dexibility on system dynamics is studied in detail. The response of the system to a speciPed maneuver has been evaluated through a planned variation of parameters likely to aHect both librational and vibrational degrees of freedom. Based on the results, following general conclusions can be made: (i) Position of the manipulator at the tip of the platform represents the critical location in terms of severity of the disturbance. Location of the manipulator on the platform has little eHect on its own dynamics. (ii) Flexibility of the joints, links and platform has very little eHect on the platform’s librational response. (iii) Flexibility of the manipulator does not aHect the platform’s vibrational dynamics signiP-
cantly, however, its own dynamical response changes substantially. (iv) The fundamental mode is able to capture system response rather well. (v) The speed of maneuver and joint Dexibility appear to be two major parameters aHecting the system dynamics. (vi) The results provide information useful in design and operation of this novel manipulator system. REFERENCES
1. Morita, Y. and Modi, V. J., Dynamics of Dexible orbiting platform with Mobile Remote Manipulator System (MRMS). Institute of Space and Astronautical Science, Tokyo, Japan, Report No. 626, February 1988. 2. Modi, V. J. and Chan, J. K., Acta Astronautica, 1991, 25, 149 –156. 3. Modi, V. J., Ng, A. C. and Karray, F., Nonlinear Dynamics, 1994, 5, 71–91. 4. Modi, V. J., Mah, H. W. and Misra, A. K., Acta Astronautica, 1994, 32, 419 – 439. 5. Nagata, T., Modi, V. J. and Matsuo, H. In Collection of Technical papers, AIAA=AAS Astrodynamics Conference, Scottsdale, Arizona, USA, 1994.
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AIAA Publisher, Washington DC, USA, Paper No. AIAA-94-3756-CP, 1994, pp. 366 –375. 6. Nagata, T., Dynamics of Dexible multibody systems: a formulation with applications. Ph. D. Thesis, University of Tokyo, 1995. 7. Caron, M., Planar dynamics and control of space-based Dexible manipulators with slewing and deployable links. M. A. Sc. Thesis, University of British Columbia, 1996.
8. Chu, M. S. T., Design, construction and operation of a variable geometry manipulator. M. A. Sc. Thesis, University of British Columbia, 1997. 9. Caron, M., Modi, V. J., Pradhan, S., de Silva, C. W. and Misra, A. K., Journal of Guidance, Control, and Dynamics, 1998, 24, 572–580. 10. Meirovitch, L., Elements of Vibration Analysis, 2nd edn. McGraw-Hill Book Company, Inc., New York, 1986, pp. 255 –256, 282–290.