Feedback controller design for servo systems with dominant mechanical resonances

FEEDBACK CONTROLLER DESIGN FOR SERVO SYSTEMS WITH DOMINANT MECHANICAL RESONANCES Leo F. Casey

David M. Otten

Laboratory for Electromagnetic and Electronic Systems Massachusetts Institute of Technology Marthinus C. Van Schoor Facility for Experimental Structural Dynamics Massachusetts Institute of Technology

Even so, the closed loop controller is often designed with the second order dynamics initially ignored, and an attempt is then made to deal with them outside of the analysis. With an analog controller this is achieved by adding a notch filter at the resonant frequency. The same technique can also be employed with a discrete time controller if the resonance is sufficiently oversampled, but if this is not the case the closed loop bandwidth is invariably sacrificed to avoid exciting the resonance. Either way, while not exciting the resonance, the controller does not help stabilize or damp the resonance if it is excited by external disturbances. The approach taken in this paper is to address the resonance directly in the controller design.['] This requires accurate modeling of the drive dynamics, and the application of fairly sophisticated (but readily available) computer based controller design tools. The design methodology is quite simple to implement and the complexity of the resultant controller can be reduced to make it computationally acceptable. The effort expended was rewarded as a mechanical resonance a t only half the Nyquist frequency of our digital controller and was actively damped by the fourth-order state-estimator based regulator.

ABSTRACT An estimator-based linear discrete-time controller for a high-performance two-axis miniature wheelchair drive is presented. This drive has a dominant resonance at a p proximately half the Nyquist frequency of the controller. The controller actively damps this resonance while simultaneously achieving a higher closed loop bandwidth than a conventional design. The controller design methodology is based on Linear-Quadratic-Gaussian (LQG) optimal control. While apparently complex, this methodology is actually quite tractable. Further, transformations are presented that significantly simplify the implementation of the resulting fourth-order state estimator and linear control law. The drive is implemented with incremental position encoders, a microprocessor, switching converters and a pair of dc motors.

1.

Introduction

The conventional approach to servo system design is to use first-order models of the electromechanical system, and assume high-bandwidth sensing, control and drive electronics. While these simple analytic models are useful for preliminary design estimates, they invariably fail to account for critical dynamics, particularly the effects of undesirable mechanical (structural) and electomechanical resonances, which invariably limit the overall performance. Typical sources of limiting mechanical resonance are the interaction of rotational inertias through flexible couplings in rotary servos, and support arm vibrations in linear actuators['] and a common electromechanical resonance is the hunting transient12] encountered in stepper drives. While the effects of purely mechanical resonances may be by mechanical design changes, this compromises other design trade-offs and so is often not the preferred solution. Instead, the servo controller design must account for these resonances.

2.

Background - A MicroMouse Drive System

The controller described in this paper was designed for a MicroMouse (MITEE Mouse 11). A MicroMouse is a robot designed to explore and navigate a square maze 3 m on a side with corridors 16.8 c m wide separated by walls 5 c m high. The performance of the mouse is measured by the time required to traverse the maze from start to finish. Figures 1 and 2 show the basic chassis and drive components of MITEE Mouse 11. One drive wheel is located on either side of the mouse near the center of mass, much like a wheel chair. Each wheel is

139 CH2504-9/88/0000-0139$1.00 0 1988 IEEE

mounted on the shaft of a DC motor along with an incremental encoder to sense position. Casters at the front and back of the mouse provide additional support. NiCd batteries mounted adjacent t o the motors provide power for the mouse. A printed circuit board mounted above the chassis contains a microprocessor for control and two H-bridge switching converters to drive the DC motors.

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directions.['] Separate controllers are used for the forward and rotational movements, with the control signals combined to drive the motors. At any instant the motor torques result in a force acting on the mass of the mouse to change the forward motion and as a torque acting on the inertia of the mouse t o alter the rotational motion. Figure 3 shows how the wheel signals are separated into their component parts, these components then operated on by their respective controllers, and finally the outputs combined into the motor control signals. The mechanical resonance that limits the performance of the mouse is encountered in the rotational mode and so the rotational controller is the focus of this paper.

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Figure 2: Side View of MITEE Mouse I1 All control of the mouse is through the two drive wheels. The differential speed between the two wheels is used to steer the mouse, and changes in the commonmode component of the speed provides for forward acceleration and braking. Ultimately the position of each wheel determines the position and heading of the mouse in the maze. Fast, accurate, and stable control of each wheel is therefore a clear requisite of a high performance robot. The first step in modeling the mouse dynamics involves breaking the motion into two parts, forward motion where both wheels turn in the same direction, and rotational motion where the wheels turn in opposite

Figure 4 contains a block diagram of the rotational mode controller. The first order dynamic model of the mouse's rotational velocity has a single pole characteristic from the torque of the motor driving the inertia of the mouse. The conversion of the wheel speed to position creates a second pole at the origin. To create a stable feedback loop a lead-lag compensator ww deai,gned and implemented in the control microprocessor.[6] The design was performed in the continuous-time domain a y the goal was a closed-loop bandwidth of 120radls (E 20 Hz)and the sample rate was 1 k H z . The initial design was unstable, oscillating at a high frequency so the controller bandwidth was reduced to 60 rad/s ( N 10 Hz)and a new design was implemented. Figure 5 shows the step r e sponse of the resulting system. The implemented controller response matched the design except for the high frequency oscillations that can be seen on the leading edge of the step response. Further investigation revealed that the oscillations arose because the motor is not connected directly to the floor of the maze but is in fact connected through the torsional spring of the rubber tire. Figure 6 shows how the hub of the wheel can wind up under load without causing

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Under ideal circumstances the mouse wheels are never driven with a step, so the resonance is not excited. However during hard acceleration or braking, especially at high speed, the mouse has some vertical bounce which is excited by imperfections in the floor and wheels. This bounce causes the wheels t o slip with respect to the floor and results in a step change in the torque applied to the mouse. This step change excites the resonance, which the lead-lag controller does little to damp it, making it even more difficult for the tire to firmly grip the floor. To control the resonance we added wider tires to the mouse, as mentioned above. This created additonal problems, however. When the mouse is rounding a corner the inside edge of the tire moves with a different radius of curvature than the outside edge of the tire. Normally wheels a t different radii turn at different speeds so since they are part of the same wheel the edges of the tire must slip with respect to the floor. Most of the tirefloor

Figure 5: Step Response of Rotational Control with Lead-Lag Compensation any motion of the mouse, which effectively isolates the inertia of the motors from the inertia of the mouse at high frequencies. This torsional resonance is the source of the instability we encountered in our initial implementation and is of particular concern because it aggravates problems with tire traction as is discussed below. To stiffen this torsional spring wider tires were installed on the mouse. This improved the damping of the resonance aa shown in the step response of Figure 7. Because the resonant frequency was now higher we were able to increase the controller bandwidth without causing instability. Figure 8 shows the resulting step response with the controller bandwidth raised to 80 radls. While the bandwidth has been improved, the damping of

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Before a controller that accounts for the torsional resonance can be designed it is first necessary to develop an accurate model of the rotational motion that included the resonance. This model builds on the first-order model described in Section 2 and shown in Figure 4. The flexibility of the rubber wheel is added to the basic model as a simple single degree of freedom torsional spring, with torsional stiffness k,. The rubber of the wheels also adds damping to the system and this is included in the model as d,, where d, is adjusted to match experimentally observed damping characteristics. The equations of motion are now:

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Figure 8: Step Response of 80rad/s Rotational Control with Wide Tires Where 81 is the rotational position of the motor shaft, is the rotational position of the mouse body expressed in the frame of the shaft, 51 is the moment of inertia of the electrical motor, shaft and wheel, 52 is the moment of inertia of the mouse translated to the shaft frame, R is the motor resistance, Kt the torque constant, V the applied voltage, k, a stiffness added to account for cogging of the motor with d8 and k, characterizing the rubber as described above. Equations 4.1 can be written in matrix form as:

contact surface is therefore operating with the dynamic coefficient of friction instead of the static coefficient. The dynamic coefficient is lower than the static, so the wide tire has less traction in cornering than the narrower tire it replaced. Additionally, because the mouse determines its location in the maze from wheel position and because the point on the tire that is not scrubbing across the floor is unknown, this measurement is less accurate. A third difficulty with the tires is that they are thin. For a given type of rubber a thin tire makes a much stiffer torsional spring, but it also has less compliance radially. In fact the tires used compress only a small amount (.005”) under the load of the mouse. So if the mouse begins t o bounce as it travels through the maze the tires may become completely unloaded during part of each bounce, which again results in poor traction and ultimately allows the wheels t o break free. Further, the wider tires unload with smaller vertical bounce. The torsional resonance is therefore seen t o be both contributing to and aggravating a traction problem, and the opportunity to overcome i t in the mechanical design is limited. The result of these competing mechanical considerations is that it is desirable to control the torsional resonance through some means other than changes in the tire design. If the resonance can be controlled or tolerated by the controller it will allow greater freedom in the design and selection of the tire material and configuration, and ultimately result in a faster mouse. The current mouse tire can withstand 0.99 acceleration under static test but when the mouse is moving it loses traction at approximately 0.259. Additional latitude in the tire and/or suspension of the mouse is required to extend this performance and so this motivates the search for a better motor control algorithm.

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2 and 2 are the well known state-space plant coefficient and control matrices respectively, and 2 is the state vector. The shaft angle is measured with an angular resolver, so that the output y is given by:

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= &+5v where is the output matrix and 5 the feedforward matrix. Equations 5 and 6 were modeled with MATRIXx [‘I, an industry standard controller design and analysis package. The model was verified by measuring the frequency response of the mouse with an accelerometer while exciting the motors with a sinusoidal voltage signal. The coulombic damping of the rubber (db) and cogging stiffness (k,) were adjusted to yield the best comparison between the measured and predicted frequencies and damping ratios. Figures 9 and 10 show these measured and predicted responses for a position close to the front caster of the mouse.

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generally worse than the performance of the open loop system in terms of exciting and damping the resonance. To improve the performance in the high frequency region, active control of the high frequency resonance is required. Sufficient improvement in performance will permit more flexibility in the mechanical design (specifically softer tires) as discussed in Section 2, which will in turn allow us to improve the traction characteristics of the mouse. However, as the angular position of the motor shaft is the only available measurement, the possible improvement in performance is limited for simple controller structures. While the angular velocity of the shaft can be obtained by differentiating the encoder output, the quantized nature of the encoder results in noisy speed measurements which are unsuitable for direct rate feedback, so this motivates the use of an estimator based regulator. The states of the system can in fact be estimated using a mathematical model of the mouse that is derived from the model in the previous section. The pole locations of the estimator are selected to minimize the noise introduced into the system by the quantized resolver output and the PWMed motor amplifier input. This design technique, known as the ‘Optimal Estimator’ or ‘KalmanB ~ c y ’ [filtering ~] technique, yields a noise-robust estimator design. When implemented it yields an estimate of the four states of the rotational motion. These four states correspond to the four components of the state vector ( E ) of Equation 5 of Section 3. To the extent of our ability to model the system accurately the estimator provides us with details of the systems dynamics that includes the torsional resonance we are trying to control. The availability of these state estimates greatly enhances the control possibilities. Position and rate feedback, arbitrary pole placement and optimal regulator de-

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Controller Design and Implementation

Design Methodology The lead-lag compensator of Section 2 was found to destabilize the lightly damped resonance (wr N 210 H z ) . While a notch or low-pass filter placed in the feedback loop isolates the controller from components of the output due to the resonance this is not an acceptable solution. Although this filtering of the feedback signal does improve the controller response it does not result in a significant improvement over the open loop dynamics of the mouse. In fact as the filter passes some signal at the resonant frequency the closed loop performance of the lead-lag compensator utilizing a feedback loop filter is

143

signs are all possible. The LQG optimal regulator with its individual penalization of the states is particularly attractive. The designer implements this design by placing penalties on the state errors in such a manner as to retain their relative importance in the time response. Typically the designer decides on an allowed error for each state and uses the squared inverse of the error as a penalty weighting for that state. To perform a LQG design, control packages like MATRIX^ ,171 require; a mathematical model of the plant (in our case the model of Section 3)' estimates of the noise covariance matrices for the optimal observer, user selected state penalties to calculate the linear quadratic Gaussian regulator feedback gains. The design process is then applied itteratively to yield acceptable pole-locations. The precise location of the estimator poles are particularly important in discrete time controller implementations as discussed below.

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Quu = 8.2E - 71 The optimal regulator was iteratively designed. Iterations were required to ensure suitable pole locations in the digital domain. Too high penalties on the high frequency mode's states resulted in high feedback gains for these states. These high gains, the low sampling rate and possible model imperfections resulted in designs that, when implemented were unstable. The final penalties used were; l / ( O . l rad)' penalty on the motor shaft angular error, 1/(0.3rad)' penalty on the mouse body rotation angle, 1/(7 radls)' penalty on the mouse body angular position rate error and the penalty on the control effort in terms of the commanded voltage to be 1/(1.85 Volts)'. The resultant penalty matrices were:

Controller Parameters The digitized nature of the mouse's rotational controller, with a sampling rate roughly four times the maximum frequency of interest required the transformation of the problem into the digital domain. The transformation to the digital domain is a simple step in MATRIXx and the design of the optimal state estimator and regulator was done in the digital domain. The following design criteria was used to determine observer and regulator gains. The theory developed by W i d r o ~ [ to ~ ] estimate the noise associated with the quantized signals was used to estimate the noise covariance associated with the resolver and motor amplifier outputs. The state covariance due to quantization of a measured signal is:

These error covariance and penalty matrices were used in MATRIXx to obtain the full state optimal feedback design. Controller Implementation['ol The computational effort in implementing the observer design WBB reduced by transforming the observer into the modal form. The normal control sequence is:

(7) where yi is the error output from the shaft encoder at time instant i, 2i the observer estimation of the plant states and ui+l the new command to the motor amplifier at time instant i 1. Transforming the observer into modal form requires pre- and post-multiplication with the eigenvectors (?) of the observer matrix (Ae). That is, let:

and the covariance of the measured signal is:

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Where q is a measure of the quantization level and a is a measure of the time-constant. The accuracy of the resolver output is one pulse in two thousand or q = 1/2000. Using a = 0.04 as an estimate of the timeconstant, the state noise covariance matrix used is (in radians): 144

controller at both a 1kHz and a 2 k H z sampling rate. Because the new control output was calculated from the previous estimator state and the newly sampled output, as formulated in Equation 15, the delay between the sample and the new control command was only 4 0 p which is an acceptably small fraction of the sampling period.

The new command t o the motor can then be calculated as:

5.

Experimental Results

The mouse processor was programed to provide the excitation for the controller tests and to collect the data from the resulting transients as a background process, while running the control algorithm in the foreground. The shaft encoder information was sampled at 4 k H z (and stored in RAM), while the controller operated at 1 or 2 kHz. This ensured that the actual movement of the mouse was captured without aliasing. 1900 data points (4.8 seconds) were stored for each transient. Measuring the performance of the various controllers on the mouse proved to be a difficult task because only low level forces could be applied to the tires without causing slipping. This meant that maximum error signals were on the order of 10 shaft encoder pulses, making it difficult to get good data resolution. To improve the resolution, data from 50 tests was averaged for each response. The data taken in the 50 successive tests was added together by the mouse, prior to downloading to a personal computer for storage and display. Impulse and doublet excitations were used to further improve the repeatability of the successive tests, because with these excitations the mouse returns to the same starting position after each transient. In this case when 50 tests were performed in succession the mouse did not wander on the floor, and if the position did change it indicated that the tires were slipping and a lower excitation level was then used. Figure 11shows the results of a single impulse response measurement on the original lead-lag controller. The impulse was a 7.5V pulse applied for 1ms. Figure 12 shows the same transient with the results of 50 tests being averaged. As one can see, the averaging greatly improves the information content of the test. Note that the overall control response is very good but the torsional-resonance is aggravated by the controller. Figure 13 shows the impulse response of the LQG estimator-based controller with a 1kHz sampling rate. Not only is the control faster than the lead-lag response of Figure 12, but the resonance is much better damped. This measured response compares well with the response predicted by M A T R I X x , which is shown in Figure 14. To further investigate the effects of the different controllers on the damping of the torsional resonance, doublet tests were performed. The doublet was c h m n to

The number of non-zero entries in the matrix 2; is eight compared to sixteen non-zero entries in A,. This reduces the number of mutiplication and sum operations that the processor must perform at each time step from a maximum of 25 to a maximum of 17. The transformed state variables no longer have direct physical significance. The two modes are now decoupled from each other, and in addition to the reduction in computation the estimator is actually more robust in terms of quantization errors. Another simplification in the implementation was achieved by scaling the matrices so that conveniently scaled magnitudes of the transformed states were stored. This permitted one of the coefficients in each row or column of the matrix to be set t o unity, eliminating an additonal 4 multiplications. For many processors the shift instruction is faster than the multiply instruction. The scaling is therefore used to convert a number to an integral multiple of 2. Our final control design required 13 multiply-additions and 4 regular additions at each time step. All of the computations were performed with 1 6 b i t arithmetic. Four of the 16-bits were used as guard bits to protect against round off errors. The remaining 11bits-plus-sign provided a dynamic range of 1 out of 2048 which was adequate in this application. The output of the controller is a PWM amplifier with 256 output pulse widths (average amplitudes), so an additional 3 bits of dynamic range is actually provided by the controller. The control processor on MITEE Mouse I1 is an NEC 78310.["1 With a 12 M H z crystal, this chip has a minimum instruction time of 0 . 5 ~ 8 and , a &bit by 1 6 b i t multiply takes 6.5 ps. Unfortunately the multiply instruction is unsigned. Straight line coding (no subroutines) was used to implement all the control algorithms to maximize the speed. The program was 780 bytes long and required 360 - 4 1 0 ~ to s both calculate a new control output and update the estimator at each time step. It should be noted that the program was run from external memory (off chip RAM) for maximum flexibility. Data from external memory takes 4 cycles for the processor to fetch whereas data from internal memory (on chip ROM) takes only 1 cycle, so if the program were burned into the masked ROM on the chip it would run considerably faster. Nevertheless there was ample speed to run the

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Figure 12: Impulse response of lead-lag control (50 tests averaged) excite the torsional-resonance so that the high-frequency response of the mouse could be studied in isolation from the low-frequency dynamics. A voltage of 3.75V was a p plied for 2 ms followed by -3.75V for 2 ms. This drove the mouse with a fundamental excitation frequency near the 210 H z resonance of the mouse. Figure 15 shows the basic open-loop response of the mouse, to the doublet excitation. The first cycle is largely the driven response which is followed by the natural decay of the oscillation. The entire oscillation is seen to decay away in 70 msec. Figure 16 shows the response for the lead-lag controller and Figure 17 for the lkHt LQG controller. The leadlag controller severely aggravates the resonance, as compared to the LQG design. While the LQG controller is unable to respond to the disturbance during the first 3 half cycles, after that it reduces the amplitude by about 20% over the open-loop response, and damps out all oscillations within 50 m e .

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Figure 16: Doublet Response of Lead-Lag Controller The damping of the resonance is substantially improved and the bandwidth is higher. There is slightly more overshoot in the response, but as little attention was payed to this in the LQG design we feel that this will be improved in successive design iterations.

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able to damp the resonance in any substantial way but it does not aggravate it as is the case with the lead-lag design. A 2 k H z controller wa3 designed and implemented and the doublet response is shown in Figure 18. This controller achieves significant damping of both the 2nd and 3rd peaks of the transient response and then rapidly brings the error to within 1 pulse of zero. The quantbation of the error signal is a possible cause of the slower decay that follows. Unfortunately tests with higher amplitude excitations were not valid as the tires began to slip, but the test clearly demonstrated the ability to deal with the resonance within the controller providing the resonance is sufficiently oversampled. To confirm the ability of the LQG controller to improve the overall dynamics, and not merely the transient associated with the high-frequency resonance, we performed a step response measurement of the 1 k H e controller implementation. The result is shown in Figure 19, and the comparison should be made with Figures 5, 7, and 8.

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6. Conclusions The application of sophisticated modern-control design techniques was found to be a viable alternative to making sacrifices in the mechanical design of a highperformance servo system which has dominant mechanical resonances. The high-order estimator-based regulator was shown to implementable using a standard einglechip microcontroller, which lacks the advanced features and speed of a dedicated signal-processing integrated circuit. The digital LQG controller was able to actively damp a torsional-resonance at half of its Nyquist frequency, while simultaneously improving the bandwidth 147

9. B. Widrow, “A Study of Rough Amplitude Quantization by Means of Nyquist Sampling Theory,” IRE Trans. on Circuit Theory, CT-3,4, pp. 266276, 1956.

of the controller. The approach taken may be contrasted with digitally filtering the feedback signal, or ratefeedback using the differentiated position signal. In our application these approaches were found t o be undesirable. Ultimately, the additional effort expended in the controller design will hopefully pay off in higher performance for our servo system.

10. H. Hanselmann, “Implementation of Digital Controllers-A Survey,” Automatica, Vol. 23, No. 1, pp. 7-32, 1987. 11. pPD78310/312 Single-Chip Microcomputer, NEC Electronics Inc., Mountain View, CA., June 1986.

ACKNOWLEDGMENTS The authors would like t o acknowledge the contributions of the other members of the MITEE Mouse team, specifically Tony Caloggero, Andy Goldberg, Jeff Lang, Gerrardo Molina, Steve Umans, and George Verghese. The mouse team is indebted to Dean Gerald Wilson of MIT and Mr. Peter Lillios of International Totalizing Systems for their continued financial and moral support. We would like to thank the FESD Laboratory in MIT’s Aeronautical Engineering Department for the use of their facilities and equipment. We would also like to thank Yuki Kimura for her technical ’assistance.

REFERENCES 1. H. Hanselmann and W. Moritz, “High-Bandwidth Control of the Head-Pasitioning Mechanism in a Winchester Disk Drive,” Control Systems, Vol. 7, No. 8, Oct. 1987. 2. G.C. Verghese, J.H. Lang and L.F. Casey, “Analysis of Instability in Electrical Machines,” IEEE Trans. Industry Applications, Vol. IA-22, No. 5, pp. 853864, Sept./Oct. 1986. 3. H. Waagen, “Reduce Torsional Resonances in Incremental- Motion Servos,” Control Engineering, pp. 85-88, Apr. 1969. 4. K.J. k t r o m and B. Wittenmark, Computer Controlled Systems, Information and System Sciences Series, Prentice Hall, 1984.

5. J . H. Lang, The Pompous Definitive Tezt on Mouse Dynamics, internal memorandum, 1985. 6. J. Tal, Motion Control by Microprocessors, Galil Motion Control, ,Palo Alto, CA., 1984. 7. M A T R I X x User’s Guide, Integrated Systems Inc., Palo Alto, CA., 1986. 8. G.F. Franklin and J.D. Powell, Digital Control of Dynamic Systems, Addison- Wesley, 1980. 148