Fite, Shao, and Goldfarb.pdf - Vanderbilt University

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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004 621 Fig. 2. Model of master/human dynamics. Fig. 3. Parsing the master/human dynamics. (a) Schematic of closed-loop force-controlled master manipulator interactingwith human. (b) Schematic of fully parsed voluntary and feedthrough effects. is essentially an exogenous input (i.e., not part of the feedback loop), is parsed from the teleoperator feedback loop. Consider, for example, a linear single-DOF master/human subsystem, which can be modeled as shown in Fig. 2 and described in [13]. In this model, the master manipulator is considered a simple mass-damper system (b m ;m m ) that is kinematically coupled to the human arm. The arm is modeled as a mass-spring-damper system (m h ;b 2;k 2) with an additional stiffness and damping (b 1 ;k 1 ) at the human/master interface (i.e., compliance in the human grip). Human voluntary input x hv is commanded directly into the base of the arm mass-spring-damper system, and is therefore filtered by these dynamics before resultingin motion of the master manipulator, a notion that is consistent with prior studies in human motor control [14]. The iconic model clearly indicates that slave motion command x h can result from either the voluntary human input x hv or the feedthrough force F h . In order to express the teleoperator system in terms of a classical loop with the human voluntary input as an exogenous command, the multiple-input single-output (MISO) human arm system depicted in Fig. 2 must be restructured as follows. Assumingthat the master closed-loop force controller is described by the transfer function C m , the master/human dynamics can be written in the block diagram form shown in Fig. 3(a), where G 1; G 2, and G 3 are transfer function matrices given by where A = G 1 = 0 G 2 = 0 0 0 b 2 s + k 2 m m 0 0 C m m m T T (1) (2) G 3 = C(sI 0 A) 01 (3) 0 1 0 0 0(k +k ) 0(b +b ) k b m m m m 0 0 0 1 k b 0k 0(b +b ) m m m m C =[0k 1 0b 1 k 1 b 1] (5) (4) Fig. 4. Slave/environment dynamics. (a) Schematic of closed-loop motion-controlled slave manipulator interactingwith environment impedance. (b) Restructuringof interaction, indicatingdependence of G on Z . (c) Use of local feedback of environment interaction force to decouple G from Z . (d) Schematic of resultingdecoupled dynamics. and where Y h is the admittance of the human operator, given by Y h = sX h F h [m h s 2 +(b 2 + b 1)s +(k 2 + k 1)]s = : (6) m h b 1s 3 +(m h k 1 + b 1b 2)s 2 +(k 1b 2 + k 2b 1)s + k 1k 2 All parameters in (1)–(6) are as defined in Fig. 2. The block diagram of Fig. 3(a) can be rearranged until the respective paths of the human voluntary input X hv and the feedthrough force F h contributing to the command motion X h are separated completely, as shown in Fig. 3(b). Specifically, the transfer function describing the force component actingon the human admittance resultingfrom human voluntary input is given by G hv , and the transfer function describingthe force component actingon the human admittance resultingfrom the feedthrough force is given by G m. Both transfer functions can be computed from (1)–(5) and from the expressions given in Fig. 3. The slave/environment dynamics depicted in Fig. 1 can be represented by the equivalent schematics of Fig. 4(a) and (b), both of which clearly indicate the dynamic couplingbetween the slave and environment. In the figures, Z e;Y s, and C s represent the environment impedance, slave manipulator admittance, and position controller, respectively, and G ~ s represents the closed-loop single-input, single-output (SISO) slave transfer function, which is clearly a function of the environment impedance. Reconfiguring the human-manipulators-environment system into a classical feedback loop, as described by Figs. 3(b) and 4(b), enables application of classical control techniques to address the transparency and stability of the system, as initially described in [12]. The proposed control approach incorporates two modifications to the two-channel bilateral
622 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004 Fig. 5. Two-channel telemanipulation architecture. Solid arrowheads represent signal interaction, whereas hollow ones represent physical interaction. structure shown in Fig. 1 to address the transparency and stability robustness of the system. First, the closed-loop slave dynamics includes local feedback of the interaction force between the slave manipulator and environment, as shown in Fig. 4(c). This component effectively compensates for the coupled slave/environment interaction, yielding closed-loop slave dynamics G s independent of the environment, as depicted in Fig. 4(d). Second, the control architecture includes a loop-shapingcompensator, G c , operatingon the command motion X h that enables frequency-domain manipulation of both the transparency and stability robustness properties of the teleoperation loop. The resultingteleoperator loop can be modeled as shown in Fig. 5, where G m;G hv , and G s are transfer functions as defined by Figs. 3 and 4. Assumingunity scalinggains and neglectingcommunication channel time delay, the transmitted impedance, which is the impedance of the teleoperator loop seen by the human operator, is given by Z t = F m sX h = G cG sG mZ e: (7) The transparency transfer function, defined as the ratio of the transmitted impedance to the environment impedance, is written as G t = G c G s G m : (8) The performance-control design objective is to choose a compensator G c that yields a transparency transfer function G t with a magnitude of unity and phase of zero within some desired bandwidth of operation. Note that without the previously mentioned local force feedback around the slave manipulator, the transfer function G s would be a function of the environment impedance, Z e , and as a result, the transparency transfer function G t would similarly change with varying environmental dynamics. Inclusion of the local slave force-feedback loop, however, renders G s , and thus G t , independent of changes in the environment dynamics (see Fig. 4). The stability robustness of the teleoperator loop is assessed by rearranging the loop into a Nyquist-like unity-feedback structure, enabled by parsingthe human voluntary force from the feedthrough force, as previously described. The resultingopen-loop transfer function governingthe stability robustness is given by G = 0G c G s G m Z e Y h : (9) The stability robustness of the loop is thus easily addressed by adding a desired amount of phase at the open-loop gain crossover frequency with the compensator G c. The design approach is thus to use the magnitude characteristics of G c to address the system transparency, and use the phase characteristics to address the system stability. Though Fig. 6. Top view of slave manipulator interacting with an environment stiffness. several possibilities for such compensator design exist, a reasonably general solution is of a lead-lag compensator of the form N i s +1 G c = kc5i=1 (10) i i s +1 where the parameters k c ;N; i , and i are used to shape the magnitude and phase characteristics of the compensator. IV. IMPLEMENTATION To verify the proposed control approach, the loop-shapingtechnique was experimentally implemented on a single-DOF telemanipulation system. Both the master and slave were kinematically single-DOF prismatic manipulators, with actuation provided by a DC brushed servomotor (PMI N12M4T) transformed to linear motion via a rack and pinion. A rotary potentiometer (Midori CPP-45B) was used for rotor position measurement. A rotational inertia was additionally mounted to each servomotor to represent the inertia of a typical manipulator. The master manipulator incorporated a handle mounted on the end of a cantilever beam coupled to the translatingrack to provide an interface with the human operator. Strain gauges mounted on the cantilever measured the interaction forces occurringbetween the human operator and the manipulator. The slave manipulator incorporated a cantilever beam that connected its endpoint to a pair of springs supported by a shaft mounted parallel to the linear motion. The springs imposed a bidirectional stiffness in series with the slave motion, providinga simple yet challenging environment with which to assess the teleoperative performance. Similar to the master, the slave incorporated strain gauges on the cantilever beam to measure the interaction forces between the slave and environment. Each manipulator was capable of exertinga maximum continuous force of approximately 45 N through a workspace of approximately 65 cm. The slave and master manipulators are pictured in Figs. 6 and 7, respectively. The control architecture was implemented with the real-time interface provided by MATLAB/Simulink (The MathWorks, Inc.) at a samplingrate of 1 kHz. A. Master and Slave Manipulator Control The nominal dynamics of the slave manipulator result from the rotational inertia previously described and the (assumed) viscous friction of the motor brushes and bearings, both transformed into translational equivalents via the rack and pinion. The nominal (translational)
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