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外文翻譯 譯文題目 一種與移動機械臂的部分零件所受載荷相協(xié)調(diào)的運動結(jié)構(gòu)（2） 原稿題目 A kinematically compatible framework for cooperative payload transport by nonholonomic mobile manipulators（2） 原稿出處 Auton Robot (2006) 21:227–242 A kinematically compatible framework for cooperative payload transport by nonholonomic mobile manipulators （2） M.Abou-Samah1, C.P.Tang2, R.M.Bhatt2and V.Krovi2 (1)MSC Software Corporation, Ann Arbor, MI48105, USA (2)Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, NY14260, USA Received: 5August2005Revised: 25May2006Accepted: 30May2006Published online: 5September2006 AbstractIn this paper, we examine the development of a kinematically compatible control framework for a modular system of wheeled mobile manipulators that can team up to cooperatively transport a common payload. Each individually autonomous mobile manipulator consists of a differentially-driven Wheeled Mobile Robot (WMR) with a mounted two degree-of-freedom (d.o.f) revolute-jointed, planar and passive manipulator arm. The composite wheeled vehicle, formed by placing a payload at the end-effectors of two (or more) such mobile manipulators, has the capability to accommodate, detect and correct both instantaneous and finite relative configuration errors. The kinematically-compatible motion-planning/control framework developed here is intended to facilitate maintenance of all kinematic (holonomic and nonholonomic) constraints within such systems. Given an arbitrary end-effector trajectory, each individual mobile-manipulators bi-level hierarchical controller first generates a kinematically- feasible desired trajectory for the WMR base, which is then tracked by a suitable lower-level posture stabilizing controller. Two variants of system-level cooperative control schemes—leader-follower and decentralized control—are then created based on the individual mobile-manipulator control scheme. Both methods are evaluated within an implementation framework that emphasizes both virtual prototyping (VP) and hardware-in-the-loop (HIL) experimentation. Simulation and experimental results of an example of a two-module system are used to highlight the capabilities of a real-time local sensor-based controller for accommodation, detection and corection of relative formation errors. KeywordsComposite system-Hardware-in-the-loop-Mobile manipulator- Physical cooperation-Redundancy resolution-Virtual prototyping Kinematic collaboration of two mobile manipulators We now examine two variants of system-level cooperative control schemes—leader-follower and decentralized control—that can be created based on the individual mobile-manipulator control scheme. Leader-follower approach The first method of modeling such a system considers the midpoint of the mobile base (MP B) of the mobile-manipulator B to be rigidly attached to the end-effector of mobile manipulator A, as depicted in Fig. 4. Figure 4(b) depicts how the end-effector frame {E} of MP A is rigidly attached to the frame at MP B (separated by a constant rotation angle β). (15) Fig. 4 Schematic diagrams of the leader-follower scheme: (a) the 3-link mobile manipulator under analysis, and (b) the two-module composite system MP B now takes on the role of the leader and can be controlled to follow any trajectory that is feasible for a WMR. Hence, given a trajectory of the leader MP B , and the preferred manipulator configuration of , Eq. (5) can be rewritten as: (16) and correspondingly Eqs. (6)–(8) as: (17) Thus, the trajectory of the virtual (reference) robot for the follower MP A , and the derived velocities can now be determined. This forms the leader-follower scheme used for the control of the collaborative system carrying a common payload. Decentralized approach The second approach considers the frame attached to a point of interest on the common payload as the end-effector frame of both the flanking mobile manipulator systems, as depicted in Fig. 5. Thus, a desired trajectory specified for this payload frame can then provide the desired reference trajectories for the two mobile platforms using the similar framework developed in the previous section by taking and , where . This permits Eq. (5) to be rewritten as: (18) Fig. 5 Decentralized control scheme implementation permits the (a) composite system; to be treated as (b) two independent 2-link mobile manipulators and correspondingly Eq. (6)–(8) as: (19) Each two-link mobile manipulator now controls its configuration with reference to this common end-effector frame mounted on the payload. However, the locations of the attachments of the physical manipulators with respect to the payload reference frame must be known apriori. Implementation framework We examine the design and development of a two-stage implementation framework, shown in Fig. 6, that emphasizes both virtual prototyping (VP) based refinement and hardware-in-the-loop (HIL) experimentation. Fig. 6 Paradigm for rapid development and testing of the control scheme on virtual and physical prototypes Virtual prototyping based refinement In the first stage, we employ virtual prototyping (VP) tools to rapidly create, evaluate and refine parametric models of the overall system and test various algorithms in simulation within a virtual environment. 3D solid models of the mobile platforms and the manipulators of interest are created in a CAD package,4 and exported with their corresponding geometric and material properties into a dynamic simulation environment.5 Figure 7(a) shows an example of the application of such framework for simulating the motion of a mobile platform controlled by an algorithm implemented in Simulink.6 However, it is important to note that the utility of such virtual testing is limited by: (a) the ability to correctly model and simulate the various phenomena within the virtual environment; (b) the fidelity of the available simulation tools; and (c) ultimately, the ability of the designer to correctly model the desired system and suitably interpret the results. Fig. 7 A single WMR base undergoing testing within the (a) virtual prototyping framework; and (b) hardware-in-the-loop (HIL) testing framework Hardware-in-the-loop experimentation We employ a hardware-in-the-loop (HIL) methodology for rapid experimental verification of the real-time controllers on the electromechanical mobile manipulator prototypes. Each individual WMR is constructed using two powered wheels and two unactuated casters. Conventional disc-type rear wheels, powered by gear-motors, are chosen because of robust physical construction and ease of operation in the presence of terrain irregularities. Passive ball casters are preferred over wheel casters to simplify the constraints on maneuverability introduced by the casters. The mounted manipulator arm has two passive revolute joints with axes of rotation parallel to each other and perpendicular to the base of the mobile platform. The first joint is placed appropriately at the geometric center on top frame of the platform. The location of the second joint can be adjusted to any position along the slotted first link. The second link itself is reduced to a flat plate supported by the second joint. Each joint is instrumented with optical encoder that can measure the joint rotations. The completely assembled two-link mobile manipulator is shown in Fig. 1(c). The second mobile manipulator (see left module of Fig. 1(b) and (d)) employs the same overall design but possesses a motor at the base joint of the mounted two-link arm. The motor may be used to control the joint motion along a predetermined trajectory (which can include braking/holding the joint at a predetermined position). When the motor is switched off the joint now reverts to a passive joint (with much greater damping). The motor is included for permitting future force-redistribution studies. In this paper, however, the motor is used solely to lock the joint prevent self-motions of the articulated linkage for certain pathological cases (Bhatt etal., 2005; Tang and Krovi, 2004). PWM-output motor driver cards are used to drive the gearmotors; and encoder cards monitor the encoders instrumenting the various articulated arms. This embedded controller communicates with a designated host computer using TCP/IP for program download and data logging. The host computer with MATLAB/Simulink/Real Time Workshop8 provides a convenient graphical user interface environment for system-level software development using a block-diagrammatic language. The compiled executable is downloaded over the network and executes in real-time on the embedded controller while accessing locally installed hardware components. In particular, the ability to selectively test components/systems at various levels (e.g. individual motors, individual WMRs or entire systems) without wearing out components during design iterations was very useful. Figure 7(b) illustrates the implementation of such a system on one of the WMRs. Numerous calibration, simulation and experimental studies carried out with this framework, at the individual-level and system-level, are reported in Abou-Samah (2001). Experimental results For the subsequent experiments,9 we prescribe the initial configuration of the two-module composite system, as shown in Fig. 8. Specifically, we position the two WMRs such that MP A is located at a relative position of , and with a relative orientation difference of with respect to MP B. For fixed link-lengths this inherently specifies the values of the various configuration angles as shown in Table 1. Table 1 Parameters for the initial configuration of the two-module composite wheeled system (see Fig. 8 for details) Link lengths of the articulation L 1 0.28 m (11 in) L 2 0.28 m (11 in) Relative angles of the configuration of the articulation L 3 0.28 m (11 in) φ 1 333.98 φ 2 280.07 φ 3 337.36 Offset between the wheeled mobile bases φ 4 128.59 δ 0.00 0.00 m (0 in) 0.61 m (24 in) Fig. 8 Initial configuration of the two-module composite wheeled system Leader-follower approach A straight line trajectory at a velocity of 0.0254 m/s is prescribed for the leader, MP B. Given a desired configuration of the manipulator arm, the algorithm described in Section 4.1 is used to obtain a desired trajectory for MP A. A large disruption is intentionally introduced by causing one of the wheels of MP A to run over a bump, to evaluate the effectiveness of the disturbance accommodation, detection and compensation.The results are examined in two case scenarios – Case A: MP A employs odometric estimation for localization as seen in Fig. 9, and Case B: MP A employs sensed articulations for localization as seen in Fig. 10. In each of these figures, (a) presents the overall -trajectory of {M} of MPA with respect to the end-effector frame {E} (that is rigidly attached to the {M} of MPB) while (b), (c) and (d) present the relative orientation difference, X-difference and Y-difference as functions of time. Further in both sets of figures, the ‘Desired’ (–– line) is the desired trajectory typically computed offline; and ‘Actual’ (–o– line) is the actual trajectory followed by the system, as determined by post-processing the measurements of the instrumented articulations. However, in Fig. 9, the (–x– line) represents the odometric estimate while in Fig. 10 it stands for the articulation based estimate (which therefore coincides with the ‘Actual’ (–o– line) trajectory). Fig. 9 Case A: Odometric Estimation of Frame M, used in the control of MP A following MPB in a leader-follower approach, is unable to detect non-systematic errors such as wheel-slip. (a) XY trajectory of Frame M; (b) Orientation versus Time; (c) X position of Frame M versus Time; and (d) Y position of Frame M versus Time Fig. 10 Case B: Articulation based Estimation of Frame M, used for control of MPA with respect to MP B in a leader-follower approach is able to detect and correct non-systematic errors such as wheel-slip. (a) XY trajectory of Frame M; (b) Orientation versus Time; (c) X position of Frame M versus time; and (d) Y position of Frame M versus time In Case A, the introduction of the disruption causes a drift in the relative configuration of the system which remains undetected by the odometric estimation. Further, as seen in Fig. 9, this drift has a tendency to grow if left uncorrected. However, as seen in Fig. 10, the system can use the articulation-based estimation (Case B) to not only detect disturbances to the relative configuration but also to successfully restore the original system configuration. Decentralized control approach In this decentralized control scenario, a straight line trajectory with a velocity of 0.0254 m/s is presented for the payload frame. As in the leader-follower scenario, a large disruption is introduced by causing one of the wheels of MP A to run over a bump. The algorithm is tested using two further case scenarios—Case C: Both mobile platforms employ odometric estimation for localization as shown in Fig. 11, and Case D: Both mobile platforms employ sensed articulations for localization as shown in Fig. 12. Fig. 11 Case C: Odometric estimation of frames M of MP A and MP B, used in the control of MP A with respect to MP B in the decentralized approach, is again unable to detect non-systematic errors such as wheel-slip. (a) XY trajectory of frame M of MP A; (b) XY trajectory of frame M of MP B; (c) Relative orientation, between MP A and MP B, versus time; (d) X distance, between MP A and MP B, versus time; and (e) Y distance, between MP A and MP B, versus time. (f) Sequential photographs of the corresponding composite system motion (as time progresses from left to right along each row) Fig. 12 Case D: Articulation based estimation of frames M of MP A and MP B, used for the control of MP A and MP B with respect to a payload-fixed frame is able to detect and correct non-systematic errors such as wheel-slip. (a) XY trajectory of frame M of MP A; (b) XY trajectory of frame M of MP B; (c) Relative orientation, between MP A and MP B, versus time; (d) X distance, between MP A and MP B, versus time; and (e) Y distance, between MP A and MP B, versus time. (f) Sequential photographs of the corresponding composite system motion (as time progresses from left to right along each row) In each of these figures, subplots (a) and (b) presents the overall trajectories of frames {M} of MP A and MP B respectively with respect to their initial poses. Subplots (c), (d) and (e) present the relative orientation difference, X-difference and Y-difference of frames {M} of MP A and MP B respectively as functions of time. Further in both sets of figures, the ‘Desired’ (–– line) is the desired trajectory typically computed offline; and ‘Actual’ (–o– line) is the actual trajectory followed by the system, as determined by post-processing the measurements of the instrumented articulations. However, in Fig. 11, the (–x– line) represents the odometric estimate while in Fig. 12 it stands for the articulation based estimate. In Case C, both mobile platforms use the odometric estimation for localization—hence as expected, Fig. 11 reflects the fact that the system is unable to detect or correct for changes in the relative system configuration. However the data obtained from the articulations accurately captures the existence of errors between the frames of reference of MP B and MP A. Thus, using the articulation-based estimation of relative configuration for control as in Case D allows the detection of disturbances and successful restoration of the original system configuration as shown in Fig. 12. Note, however, while the relative system configuration is maintained, errors relative to a global reference frame cannot be detected if both WMRs undergo identical simultaneous disturbances. Detection of such absolute errors would require an external reference and is beyond the scope of the existing framework. Conclusion In this paper, we examined the design, development and validation of a kinematically compatible framework for cooperative transport of a common payload by a team of nonholonomic mobile manipulators. Each individual mobile manipulator module consists of a differentially driven wheeled WMR retrofitted with a passive two revolute jointed planar manipulator arm. A composite multi degree-of-freedom vehicle system could then be modularly created by attaching a common payload on the end-effector of two or more such modules. The composite system allowed payload trajectory tracking errors, arising from subsystem controller errors or environmental disturbances, to be readily accommodated within the compliance offered by the articulated linkage. The individual mobile manipulators compensated by modifying their WMR bases’ motion plans to ensure prioritized satisfaction of the nonholonomic constraints. The stabilizing controllers of the WMR bases could then track the recomputed “desired motion plans” to help restore the system-configuration. This scheme not only explicitly ensured maintenance of the kinematic compatibility constraints within the system but is also well suited for an online sensor-based motion planning implementation. This algorithm was then adapted to create two control schemes for the overall composite system— the leader follower approach and the decentralized approach. We evaluated both approaches within an implementation framework that emphasized both, virtual prototyping and hardware-in-the-loop using the case-study of a two module composite system. Experimental results verified the ability of the composite system with sensed articulations to not only accommodate instantaneous disturbances due to terrain irregularities but also to detect and correct drift in the relative system configuration. The overall framework readily facilitates generalization to treat larger systems with many more mobile manipulator modules. Appendix A The kinematic constraint equations shown in Eq. (3) may be written in the general form as: (20) Then the velocity constraint equation can be written as: (21) The general solution to this differential equation is: (22) By appropriate selection of the initial conditions within the experimental setup, one can create the condition , i.e., exactly satisfying the constraint at the initial time, which then permits the constraint to be satisfied for all time. However, one could also enhance this process by adding another term to the right-hand-side of Eq. (21) as: (23) where Φ is a positive-definite constant matrix. This results in a first order stable system: (24) whose solution for any arbitrary initial configuration is: (25) In such systems, there is no longer a requirement to enforce since the solution exponentially stabilizes to regardless of the initial conditions. This feature could potentially be easily added to transform Eq. (5) to further improve overall performance, but was not required. References Abou-Samah, M. 2001. A kinematically compatible framework for collaboration of multiple non-holonomic wheeled mobile robots. M. Eng. Thesis, McGill University, Montreal, Canada. Abou-Samah, M. and Krovi, V. 2002. Optimal configuration selection for a cooperating system of mobile m