Robust Controller Design for Cancelling Biodynamic Feedthrough

Transcription

1 Robust Controller Design for Cancelling Biodynamic Feedthrough M. R. Sirouspour and S. E. Salcudean University of British Columbia Vancouver, BC, Canada Abstract: There are many manual control tasks in which the operator base moves in response to joystick movement. Base motion feedback can interfere with the joystick and operator dynamics, resulting in instability and reduced performance. This paper proposes a novel approach to the cancellation of such biodynamic feedthrough. A model-based approach is used to formulate a µ-synthesis robust performance design problem that is solved by D-K iterations. The resultant controllers are robustly stable with respect to variations in the arm/joystick and biodynamic feedthrough parameters, while optimizing relevant performance measures. Experimental studies demonstrate the effectiveness of the designed controllers. The approach developed in this paper can be applied to any system involving teleoperation from movable bases. 1. Introduction Human performance in manual control systems can degrade due to biodynamic interferences. Examples are controlling high speed vehicles over rough terrain or waves, maneuvering aircrafts under transonic buffeting, high gain fly-bywire systems, powered wheelchairs, and control of heavy hydraulic machines such as excavators, bulldozers, etc [1, 2]. In these applications, the human body s response to acceleration can introduce involuntary commands in addition to voluntary commands. This phenomena is referred to as biodynamic stick feedthrough. Biodynamic interferences can be classified as either openloop or closed-loop. In the open-loop case, the operator s base acceleration is uncorrelated with his/her voluntary command and is caused by external disturbances, e.g. in weapon aiming on-board moving platforms. In another category of manual control tasks, the base acceleration is a result of operator commands. This can not only degrade the performance but it may also cause instability. Piloted aircrafts and joystick-controlled heavy hydraulic machines belong to this category. Biodynamic feedthrough has been known to contribute to the pilot-coupled oscillations (PCO) and roll-ratchet phenomenon in aircrafts [3, 4] and also operator-induced oscillations in the operation of excavators [5]. Models for the body response in biodynamic environments have been developed in [4] and [6]. There is very little work in the literature regarding the cancellation of stick feedthrough dynamics. Prior work includes the adaptive filtering technique presented in [7] and [1], the acceleration feedforward used

2 Figure 1. Experimental setup. by [8], and increased joystick damping proposed by [5]. In this paper, a novel approach is proposed for the design of controllers that suppress closed-loop biodynamic feedthrough. The methodology is illustrated via an example. It is shown analytically that a heuristic control approach can result in instability. µ-synthesis-based controllers are then proposed that robustly stabilize the system with respect to uncertainties in the arm/joystick and feedthrough dynamics and at the same time optimize some performance measures. Experimental studies demonstrate the effectiveness of the approaches. This paper is organized as follows. In Section 2, the manual control task that is the subject of this paper and the experimental setup are introduced. The problem is formulated as a µ-synthesis robust performance design problem in Section 3. The stability of the system under a heuristic PD controller is analyzed in Section 4. Section 5 discuses identification of the arm/joystick parameters. The design of the controllers is addressed in Section 6. The experimental results are presented in Section 7. The paper is concluded in Section 8 where our plans for future work are also discussed. 2. Problem Statement Closed-loop biodynamic interferences occur in situations in which the operator is subject to accelerations due to his/her action, e.g. in joystick-controlled aircrafts and excavators. Details of these interactions differ from case to case, however, there are principal similarities between all cases. In this paper, a single-degree-of-freedom manual control task is considered (Figure 2). The operator uses a force-feedback joystick to position his/her base along the z- axis in order to follow a random target. This relatively simple task highlights the problems associated with biodynamic feedthrough and is used to illustrate the proposed approach.

3 Figure 2. Schematic of the manual control task. The experimental setup used in this study is shown in Figure 1. The components of this system are as follows: Motion : The hydraulic Stewart-type UBC motion simulator [9] was used to generate the base motion. Capable of generating 1g accelerations and speeds exceeding 1.5 m/s, the platform is controlled by link-space PD position controllers that use the link length measurements. The controller runs on a Themis Sparc 5 board under the VxWorks operating system with a sampling rate of 256Hz. Position commands in work-space coordinates are given and measured link lengths are returned. : The joystick used in our experimental setup has three degrees of freedom allowing for translation and rotation in a plane. The endpoints of two pantographs that move in offset parallel planes are coupled by means of a crankshaft connected to the interface handle. The device used is a slightly modified version of the one presented in [1]. In this application, gearheads were employed to increase its force/torque capability. Each of the four motor joint angles are measured by.9 degree resolution digital optical encoders placed on the motor side of the shafts. The joystick is controlled in work-space coordinates and the effect of gravity is compensated. The redundancy in actuation is exploited to minimize the norm of motor torques vector (motor currents). For the purpose of the experiments presented in this paper, the x and θ coordinates are locked using PD controllers. The joystick controller is implemented using the Matlab Realtime Workshop toolbox and Tornado development environment. The controller communicates with the platform controller, a graphical display, and its own Simulink block diagram. Graphical Display: A target position (frame) and the platform position (frame) with respect to the target are displayed by a projector on a screen in front of the operator. A schematic diagram of the manual control system is given in Figure 2. The following assumptions are made and will be used later in the paper: A1. The arm/joystick subsystem can be modeled as a mass-spring-damper sys-

4 tem with uncertain parameters m m, b m and k m : x m (s) = f h u m f bf m m s 2 + b m s + k m (1) where f h is the exogenous hand force, u m is the applied force by the motors and f bf, to be introduced in (2), models the feedthrough. m m = m + δm, b m = b + δb and k m = k + δk are the mass, damping and stiffness of the arm/joystick subsystem, respectively, with m, b and k denoting their nominal values. The parametric uncertainties δm, δb and δk account for errors in the linear model (1), and may also account for other unmodelled dynamics of the joystick and human arm. The arm/joystick subsystem is modeled by the block diagram given in Figure 3. A2. The effect of base acceleration on the arm/joystick motion with respect to the base can be modeled as an inertial force acting on the joystick, as done in [8, 11]. ωbf 2 f bf = m bf s 2 s 2 + 2ζ bf ω bf s + ωbf 2 x s (2) where m bf = m f + δm f is the biodynamic feedback mass that is usually equal to the total mass of the arm and joystick (m bf = m m ). δ mf (equal to δ m here) represents the uncertainty associated with the feedthrough mass. Inclusion of a second-order low-pass filter in the model reflects the fact that the compliance of the arm/joystick interface prevents high frequency platform accelerations from being conveyed to the operator s arm. It also makes the design problem well-posed. This model neglects the effects of body and shoulder motions on the arm. A3. The closed-loop dynamics of the platform controller can be approximated by first-order linear dynamics. 3. Robust Control Problem Formulation In order to assist the operator in accomplishing the task, controllers must be designed to coordinate the motion of the joystick with that of the platform. The two main objectives considered in the design are: 1. Kinematic correspondence between the motion of the joystick and that of the platform, enforced by making the following error small: e 1 = W e1 (x s c k x m ) (3) where the tracking error is weighted by the frequency dependent gain W e1. In general, c k can be any stable rational transfer function. For example, a constant c k corresponds to position mode control strategy and a rate mode controller is obtained by replacing c k (s) with c k s. 2. Perceived admittance of the arm/joystick, achieved by making the following error small: e 2 = W e2 (Y D f h x m ) (4)

5 Figure 3. System block diagram. where x m is the joystick position, f h is the hand force, W e2 is a frequency weight and Y D is a desired admittance that is found through human factors studies (e.g.,[11]), to facilitate manual control. Figure 3 shows a block diagram of the proposed control system. This system can be viewed as a two-channel bilateral teleoperation system with a movable base. It is assumed that the master (joystick) and slave (base) positions and also the base acceleration are measured. The master-side control action, u m, is the applied force/torque to the joystick by the motors. The slave-side control action, u s, is the position command to the base controller. In addition to performance measures e 1 and e 2, two more outputs ū m and ū s have been defined in Figure 3. These terms penalize excessive control inputs over given frequency ranges. ū m = W um u m ū s = W us u s (5) Measurement noise is also included in the model to prevent noise amplification and regularize the design problem (n m, n s, n a ). Frequency-dependent gains on the input signals, i.e. W f, W nm, W ns and W na are used to emphasize the frequency ranges at which their energy is concentrated. Communication delays are lumped into one block and are represented by e τ ds, a term that will be replaced by its P adé approximation in the controller design. The system shown in Figure 3 is redrawn in a standard µ-synthesis con-

6 Figure 4. System diagram in the µ-synthesis framework. figuration in Figure 4. in this diagram: d = [ f h n m n s n a e = [ e 1 e 2 ū m ū s w = [ w 1 w 2 w 3 w 4 z = [ z 1 z 2 z 3 z 4 u = [ u m u s y = [ x m x s x s where d is the exogenous input vector, e is the output vector, u is the control input vector and y is the measurement vector. Also, G op (s) is the open-loop transfer function, pert is the perturbation block and K(s) is the controller block. Note that an additional virtual perturbation block perf is added between the exogenous input and the output to pose the problem as a robust stability problem [12]. Then controller K achieves robust performance if and only if max µ (F L (G op, K)(jω)) < 1 (6) ω where = diag{ pert, perf } and F L (G op, K) is the transfer function from [ w d to [ z e when the loop is closed with controller K. In µ synthesis, the controller K is the solution to the following optimization problem [12]: min K stabilizing max ω µ (F L (G op, K)(jω)) (7) 4. Stability Analysis for a Heuristic Controller In this section, it will be shown that a heuristic controller design that achieves the control objectives could potentially lead to instability. The desired master admittance is assumed to be Y D = 1 m d s 2 + b d s + k d (8) and the desired kinematic correspondence is a position scaling by c k. For simplicity, it is assumed m d = m, so no acceleration measurements would be

7 required. The following control laws can achieve the control objectives for a frequency range below the platform controller bandwidth when no perturbation is present and the base does not move C m (s) = (b d b)s + k d k C 1 (s) = c k C s = C 4 = C bf = (9) Using the Nyquist criterion and assuming τ d =, it can be shown that the closed-loop system remains stable as long as ( ) md c k m f > b d b d + k d τ + τ (1) Similarly, if a position-velocity kinematic correspondence is desired, c k is replaced with c k s. In this case, assuming τ d =, the stability condition becomes: ( c k m f > b d 1 + k dτ 2 m d + b d τ ) (11) Therefore, in the absence of time delay, the system remains stable for all positive c k m f, whether position or rate control is used. In practice, there is always some delay in the loop. It is not difficult to show that the system can become unstable for both positive and negative c k m f in this case. This is more critical in case of velocity mode control. Note also how increasing the damping of the joystick improves the stability in both cases. 5. Identification of the Arm/ Parameters The nominal values of the arm/joystick parameters are required in the controller synthesis. To identify these, experiments were performed along the z-axis of the joystick (see Figure 2). One can write m z + bż + kz + f g = f z (12) where m, b, and k are the lumped mass, damping and stiffness of the arm/joystick, respectively, f g and f z are the graviational and applied forces along the z direction, respectively. Band-limited noise (bandwidth=4.6hz) was applied to the controller as the position command while the operator was holding the handle. The operator was instructed to relax his arm during the experiments each of which lasted 8 seconds. The applied motor torques (i.e. motor currents), and angular position of the motor shafts were measured. These measurements were used to compute the applied work-space force and the arm/joystick position using the device kinematics. The velocity and acceleration signals were computed by first applying a forward-backward low-pass filter and then the backward difference on the position signal. Finally, the Least Squares technique was employed to estimate the unknown parameters m, b, k, and f g. The estimated parameters are as follows m = 1.45kg b = 5.N.s/m k = 72.N/m f g = 11.N

8 Remark: In this study, the feedthrough mass m f is assumed be the same as m. A better way to identify m f might be to move the base and measure the arm/joystick position. This will be done in future. It is also worth noting that the dynamic mass m is slightly larger than the static mass f g /g. This might be due to the fact that the operator tends to hold to the joystick against gravity. 6. Controller Design The Matlab µ synthesis toolbox [13] was used to solve the robust control problem defined in (7). m d = 1.45kg b d = 1N.s/m k d = 85 N/m m f = 1.45kg δ mf = δ m, δ m kg c k = 1. δ b.4 5.N.s/m δ k.2 7N/m τ =.4s τ d =.8s w bf = 2π 3rad/sec ζ bf =.7 Note that c k = 1., which means that when the joystick is pulled up the platform moves down and vice-versa. The platform time constant has been chosen based on the specification of the platform position controller. The value of the time delay τ is based upon the estimated network communication delay between the joystick and the platform and also some artifical time delay added to resemble applications with larger delays. The weights used in the µ synthesis design are as follows: W e1 = 5 W e2 = 2 W um = s s + 1 W us = s s + 1 W nm = W ns = W na = (s +.1) s + 1 W 5 f = (.2s + 1) 2 A controller with 61 states was obtained after five D K iterations using function dkit from the µ-synthesis toolbox. Interestingly, all of the gains except C m and C 1 turned out to be effectively zero. The fact that the robust controller does not use platform position measurements is due to the absence of any uncertainties in the platform model. Introduction of a disturbance signal at the input or output of the platform model would make C s and C 4 nonzero. However, this is not a concern in the design problem presented in this paper. One may also expect that the controller would use the acceleration measurement to partially cancel biodynamic feedthrough. This turned out not to be the case here. Model order-reduction techniques were applied to reduce the order of the controllers to seven and five for C m and C 1, respectively. The robustness of the designed controller is compared with that of the heuristic PD controller in Figure 5 which shows the upper bounds on µ pert for the two controllers. Note that the PD controller is not robust with respect to uncertainties and can become unstable. In fact, for c k = 1, the nominal system under PD controller is very close to instability. If larger c k (in magnitude sense) had

9 5 4 PD Controller Robust Controller 5 Desired Frequency (rad/sec) Frequency (rad/sec) Desired Figure 5. Upper bounds on µ pert when δ m δ mbf. Figure 6. Performance of the robust controller Tracking Time (s) Time (s) Figure 7. Response of the robust controller (c k = 1.). Figure 8. Response of the PD controller (c k = 1.). been used (e.g., c k = 2), even the nominal system would be unstable. The nominal performance of the robust controller is presented in Figure 6 where Y d, x m /f h, and x s /f h are compared. Performance objectives are clearly met over the required frequency range. Note that, considering the simple biodynamic feedthrough model considered in this paper, it might seem reasonable to cancel the feedthrough by adding the force term ˆm bf ẍ s to the master control command. A robust control law then can be designed for the new system to account for uncertainties. For the design example of this paper, a PD controller with acceleration compensation can give more or less the same result as the robust controller with acceleration compensation. Nevertheless, numerical experiments revealed that for higher values of c k, the PD controller is not robust. Obviously, the controller presented in this paper has the advantage of not using acceleration measurements. The performance and robustness of the controllers designed in this section were validated through extensive numerical simulations using Matlab Simulink. The simulation results are not be presented here due to space constraints.

10 .5 Robust Controller Robust Controller with Acc. Compensation Tracking PD Controller Time (s) Figure 9. Response of the controllers with c k = Time (s) Figure 1. Response of the robust controller with acceleration compensation. 7. Experimental Results Experimental studies were carried out to evaluate the effectiveness of the proposed controllers. During the experiments, the operator was instructed to move the platform in step-wise and sinusoidal-wise motions. Both the robust controller and the PD controller were tested. The robust controller demonstrated an excellent response as it can be seen in Figure 7. Note that the sinusoidal motion in part of this experiment is intentional and is performed to show the tracking behavior of the system. The controller was found to be robust with respect to arm parameter variations (i.e., by taking off the hand from the joystick, using a different operator, etc.). The PD controller performed poorly and exhibited a nearly unstable response as shown in Figure 8. The operator had to loose the grasp on the joystick in order the stop the oscillations. Figure 9 shows the robustness of the proposed controller to changes in c k where the position command to the platform were multiplied by 1.1. Note how this worsens the response of the PD controller. The response of the robust controller with acceleration compensation was also examined. According to Figure 1 this controller also performs well and suppresses the biodynamic interferences. 8. Conclusions and Future Work This paper addressed the problem of closed-loop biodynamic feedthrough in joystick-controlled systems. The biodynamic stick feedthrough was modeled by a simple acceleration feedback term. It was shown via analysis that this feedback loop can cause instability if a PD controller is used. The µ-synthesisbased controllers proposed suppress biodynamic feedthrough, are robust with respect to variations in the system parameters, and at the same time optimize useful performance measures. The controllers were implemented and evaluated experimentally. The results demonstrate the effectiveness of the proposed approach. In the future, more complex models for biodynamic feedthrough will be developed, validated, and used in the controller design. Realistic manual control tasks involving multiple degrees of freedom will be considered. Performance measures will be chosen such that optimized controllers improve the

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