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Forward Model for Sonar Maps Produced With the Polaroid Ranging Module Roman Kuc
Abstract—Conventional Polaroid time-of-flight (TOF) sonars provide object range measurements, but their wide beam widths limit the object bearing accuracy. We describe a first-order model that predicts the TOF readings from an object at given range, bearing, and reflecting strength produced by the Polaroid 6500 series sonar ranging module connected to a 600 series electrostatic transducer. The model provides a simple method for solving the inverse problem of estimating these object parameters from conventional TOF data. Experimental results demonstrate model performance. Our model provides a reasonable fit for main-lobe data observed for objects with reflecting strengths that vary by 25 dB. An example of an inverse problem solution demonstrates the model can improve object range and bearing measurements, and can also estimate object reflecting strength, a landmark feature useful for robot navigation tasks. Index Terms—Forward model, inverse problem, landmark identification, reflecting strength, robot navigation, sonar, sonar map, time-of-flight.
(a)
(b) Fig. 1. Simulation of Polaroid ranging module operation. (a) Processed echo waveform. (b) Expanded time and amplitude scale around threshold crossing point.
I. INTRODUCTION Sonar is an attractive sensing modality for mobile robotics because it is simple to implement and process, and has low cost and energy consumption. The most common sonar in mobile robots employs the Polaroid 6500 series ranging module [1], [2]. This sonar determines range to the nearest echo-producing object by extracting the echo time-of-flight (TOF) using threshold detection. Object location is typically interpreted to lie either along the transducer axis or within a predetermined beam angle [3]–[5]. The wide sonar beam is problematic because, while the object range is accurate, its bearing is resolvable to only within the beam width, with errors as large as 624 , as shown below. Probabilistic methods have been proposed to improve resolution by moving the sonar [4]. These methods are appropriate for random errors introduced by dynamic inhomogeneities in the transmission medium, which cause echoes to exhibit arrival time jitter [6], [7]. However, since the errors produced by beam effects are not random, but systematic, deterministic approaches prove to be more accurate [8]–[11]. This paper shows how object range and bearing estimates can be improved by modeling how the Polaroid ranging module detects echoes. This and three previous papers [12]–[14] indicate how to extract additional information from conventional Polaroid sonars. We describe an echo detection model that is specific to the TOF value produced by the Polaroid sonar, which extends our previous model [3] by including the emission waveform and the detection process, which uses rectification, filtering, and threshold detection, a form of nonlinear detection common to active sensing systems. Hence, our model applies to the sonar class using threshold detection. Our model does not apply to correlation methods [15], which require more elaborate processing, to zero-crossing methods [16], which employ nonstandard Polaroid emissions and processing, or to methods that more accurately model echoes [19], which require analog-to-digital conversion.
Manuscript received February 11, 2002; revised July 24, 2002. This paper was recommended for publication by Associate Editor M. Buehler and Editor S. Hutchinson upon evaluation of the reviewers’ comments. This work was supported by the National Science Foundation under Grant IRI-9504079. The author is with the Intelligent Sensors Laboratory, Department of Electrical Engineering, Yale University, New Haven, CT 06520-8284 USA (E-mail:
[email protected]). Digital Object Identifier 10.1109/TRA.2003.809586
The Polaroid module operation is modeled in the next section. Model predictions and experimental results are compared in Section III. An example of solving the inverse problem is given in Section IV. II. POLAROID RANGING MODULE PROCESSING Most conventional sonars employ Polaroid 6500 ranging modules [1] connected to the Polaroid 600-series electrostatic ultrasound transducer. The module is controlled with digital signals on two input lines (INIT and BLNK) and the TOF reading occurs on its output line (ECHO). A logic transition on INIT causes the transducer to emit a pulse lasting for 16 cycles at 49.4 kHz. The same transducer detects echoes after a short delay to allow transmission transients to decay. Another interrogation pulse is typically emitted only after all the echoes produced by the previous pulse have decayed below a detection threshold. For most environments this occurs within 100 ms. The module processes echoes by performing rectification and lossy integration. Fig. 1 illustrates a simulation of the processed waveform applied to the threshold detector. While the echo arrives at time To after the emission, ECHO exhibits a transition at measured TOF time Tm , the first time the processed echo signal exceeds a detection threshold . By convention, the range r of the reflecting object is calculated by r=
c
2 Tm 2
(1)
where c is the speed of sound in air, usually taken as 343 m/s. Fig. 1(b) shows details around the threshold detection point including the residual high-frequency ripple. Note that a slight reduction in echo amplitude causes a half-period delay in Tm . We model the Polaroid transducer as a vibrating piston to compute the transmitted pressure beam [18]. It is only an approximation because no real transducer behaves exactly like a piston vibrating within an infinite baffle [18], but the prediction is surprisingly good when the transducer is excited with 16 cycles, a good approximation to narrow-band excitation. The beam pattern for a piston having the transducer diameter d = 3:6 cm and frequency Fo = 49:4 kHz is modeled by a well-known diffraction pattern whose first off-axis nulls occur at sin01 (1:22 c=Fo d) = 14 . Since the same transducer receives echoes, the reciprocity principle indicates that the transmitter/receiver beam pattern is the square of the transmitter beam [18], which yields the curve shown in Fig. 2. To simplify the analysis, we
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Fig. 3. TOF values T and T for idealized processed echo waveforms having two amplitudes. Solid line indicates larger amplitude echo.
III. EXPERIMENTAL RESULTS
Fig. 2. Piston model transmitter/receiver pattern for Polaroid 600-series transducer. Gaussian approximation with SD = 5:25 is shown in dashed line. Equivalent threshold levels for plane, pole, and rod objects are shown as dot-dashed lines.
approximate the peak of the transmitter/receiver beam profile with a Gaussian function in Fig. 2 to determine the echo amplitude as a function of bearing , or A = Ao exp
2
0 22
(2)
where Ao is the on-axis amplitude and is a measure of beam width. The value = 5:25 provides a good fit around the peak of the diffraction pattern. We previously used a Gaussian model for describing the beam pattern for wide-bandwidth pulses emitted by impulsively exciting this transducer [19]. In this latter case, the empirical beam pattern did not contain side lobes and exhibited a larger value for . Here the Gaussian model is reasonable only over the central section of the main lobe that produces detectable echoes. We assume the echo arrival time To does not change significantly with transducer orientation (object bearing), although we have investigated this effect previously [3]. In contrast, the measured TOF, de0 in Fig. 3, is amplitude dependent and is a function noted Tm and Tm of object bearing, which effects echo amplitude by Fig. 2. The module processes the detected echo waveform by rectification and lossy integration, as shown in Fig. 1. For a useful analytic model we assume the integration is lossless and the rectified echo to be a unit-step function with amplitude A. This approximates the processed waveform shown in Fig. 1(b) by a linear function around time Tm , shown in Fig. 3. The model ignores the residual ripple and the decreasing slope of the waveform shown in Fig. 1. The impact of this simplification is discussed below. The linear function with slope proportional to the echo amplitude is given by A (t 0 To ), for t To . This function exceeds the threshold at Tm = To +
= To + exp A Ao
2 2 2
:
(3)
For a fixed , the incremental delay in Tm is a function of bearing and inversely proportional to the echo amplitude. When echo amplitude A volts is applied to the integrator, the slope of the linear output is A volts per second, with typical slope values on order of A = 105 V=s. If = 0:10 V, =A = 1006 s = 1 s.
Experiments were conducted with a Polaroid 600-series transducer connected to a model 6500 ranging module. The sonar mounts 8 cm above the center of a rotating table driven by a stepper motor to have 0.3 increments. The table was located 1 m above the floor to avoid multiple reflection artifacts [3]. Interrogation pulses occur every 100 ms. The ranging module connects to a 133 MHz Pentium computer through a general digital input/output board. A program written in C executes under the real-time MS-DOS operating system. After producing an emission, our system repeatedly samples the ranging module output line to determine the occurrence of an ECHO indication. Tm is computed from the number of cycles at which the ECHO transition signal is observed. This query cycle period is 2 s, corresponding to a range increment of 0.3 mm. Assuming the actual Tm occurred at a time that was uniformly distributed over the previous interrogation cycle, the range measurement error has a bias of 0.15 mm with p standard deviation (SD) equal to the discretization error SD, 0:3 mm= 12 0:1 mm. To verify our model, the Polaroid module was operated conventionally to generate Tm values as a rotational scan is performed. Objects include a 1-m wide plane, a 8.9-cm diameter pole, and an 8-mm diameter rod, all located at 1.5-m range. A corner can be treated as a plane and a sharp edge produces echoes only when the range is less than 0.5 m [11]. All objects extended above and below the transducer beam. A rotational scan was performed from 040 to +40 in 0.3 steps. At each angle, 100Tm values were recorded. The mean deviations from the Tm when the object is on the sonar axis ( = 0) were determined and the SD values were computed. There were no other objects in proximity to the object being scanned. Echoes from objects beyond 2 m were eliminated by a range gate. Fig. 4(a) shows data for the plane, Fig. 4(b) for the pole, and Fig. 4(c) for the rod. The values are shown relative to the Tm value observed at 0 bearing. The angular extent of the arc over which echoes were detected equals 45 for the plane, 22.8 for the pole, and 18.6 for the rod. Tm increases as the echo amplitude decreases, which happens as the object bearing increasingly deviates from zero. The side lobes produced by the plane are visible and have small echo amplitudes, which cause their Tm values to be retarded in time. Dynamic changes in the transmission medium [6], [7] cause random jitter in Tm , with S:D: = 5 s (0.9 mm) at zero bearing. The SD increases with deviation from zero bearing because smaller echoes exceed the threshold later in the processed waveform. The smaller slope of the latter part of processed echo waveform (rf: Fig. 1) causes greater Tm differences for a given variation in echo amplitude, thus increasing the SD. Two features in the data not described by the model are due to residual ripple in the processed waveform and the decreasing slope in the waveform low-frequency trend, shown in Fig. 1. The ripple causes
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(a)
(b)
(c) Fig. 4. TOF data from object at 1.5 m range. Means of 100 measurements with bars indicating one standard deviation. Dashed lines indicate model predictions. (a) 1-m wide plane ( =A = 0:15 s). (b) 8.9-cm diameter pole ( =A = 0:67 s). (c) 8-mm diameter rod ( =A = 2:68 s).
jumps in TOF readings equal to half period (10 s) added to the value predicted by (3). These jumps are clearly evident in the mean values of Fig. 4. The decreasing slope causes the actual TOF values to be greater than predicted by (3). However, echoes that first exceed the threshold in the second half of the total echo waveform are weak, typically occur at the edges of the detected arc, and exhibit large SD values (a consequence of their smaller slope). Echo amplitudes that would have produced the respective arcs according to the piston model are indicated in Fig. 2. For the plane, the threshold level relative to the maximum echo amplitude equals 038 dB, for the pole 025 dB, and for the rod 013 dB. Since the ranging module threshold at 1.5 m range is the same for each object, the difference in levels indicates the relative echo strength from each object, i.e., the plane echo is 13 dB (a factor of 4.5) greater than the pole echo, and the pole echo is 12 dB (factor of 4) greater than the rod echo. To illustrate the predictive quality of our model, we fit (3) to the pole TOF data by finding a =A value that produces a reasonable fit ( =A = 0:67 s). The =A value for the plane is decreased by a factor of 4.5, determined from the beam model, to yield =A = 0:15 s for the plane. The =A value for the rod is increased by a factor of 4, to give =A = 2:68 s for the rod. Fig. 4 shows Tm values predicted for the plane and rod by (3) in dashed line. The predictions agree with the data around the main lobe, except for the ripple-caused jumps as the processed waveform amplitude decreases [3] and the delay caused by the decreased slope. The side lobes that appear for the strong plane
reflector are also not included. However, the model does provide a reasonable description of the observed TOF values around the main lobe for objects that vary by 25 dB in reflecting strength. IV. AN INVERSE PROBLEM SOLUTION EXAMPLE The solution to the inverse problem provides values for object range, bearing and reflecting strength given the data and the model. The mean and SD data in Fig. 4 provide a means of performing weighted leastsquares fit to the data around their main lobes to determine Tm , bearing, and object =A parameter value. We can then estimate the range by computing To from =A and Tm in (3) and using To in (1). For example, consider the TOF data observed during a single scan over a pole, plane and rod shown in Fig. 5. The pole is located at (r = 1:46 m, = 017 ), the plane at (r = 1:5 m, = 4 ), and the rod at (r = 1:47 m, = 22 ). Fig. 5(a) shows the conventional TOF map, which does not appear very informative. Fig. 5(b) shows an expanded time scale referenced to the first Tm value. The single-scan data display typical TOF jitter caused by dynamic fluctuations in the transmission medium, which is typically treated as noise [7]. While the sonar map shows essentially a single arc, the TOF data clearly shows three distinct regions. The rod data exhibits the greatest curvature, the pole data smaller curvature and the plane data are essentially constant. Data are segmented by first finding contiguous segments that are consistent with the expected “U” shapes. The figure contains two such
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forward. Optimal segmentation and fit methods are currently being examined. While the inverse calculations provide accurate values for the pole and rod, the bearing estimate for the plane would not be as accurate because the plane TOF data lack significant features. The observed plane data were caused by large echoes from the normally-incident section of the plane and exhibit almost constant Tm values, ignoring jitter. Such limited-arc data would produce a broad minimum plagued by jitter-driven local minima in the objective function. Application of the model to mobile robots operating in general environments involves several considerations.
(a)
(b) Fig. 5. TOF data from a single sector scan of a pole, plane, and rod. (a) Sonar map with sonar located at (0,0). (b) TOF values referenced to smallest observed T value. Dashed lines indicate model predictions.
segments, the first centered around 017 and the second around 22 . Remaining segments typically lie at farther ranges and are partially occluded by nearer objects, the case for the plane. Finding best fit model with parameter vector (Tm , , =A) improves the object location and estimates the reflecting strength of the object. The fits shown by dashed line produce vector values (8,523 s, 017 , 0.67 s) for the pole, (8,746 s, 4 , 0.15 s) for the plane, and (8,583 s, 22 , 2.68 s) for the rod. The echo arrival times are then computed as = 8;522 s To; pole =Tm; pole 0 A pole To; plane =Tm; plane 0 = 8;746 s A plane To; rod =Tm; rod 0 = 8;580 s: (4) A rod Using these values in (1) yields range estimates rpole = 1:462 m, rplane = 1:5 m, rrod = 1:471 m. These range values are more accurate than those obtained using Tm values. V. DISCUSSION The pole and rod TOF data conveniently masked the troublesome side lobes in the plane, making the inverse problem solutions straight-
• General reflecting objects. The model applies to general cluttered environments. The important model feature is the smoothness of the TOF data as the sonar scans over an object. Observed Tm data are typically smooth (ignoring jitter) when the nearest-range reflector is stronger than other objects in its vicinity. Strong reflectors exceed within the first few cycles of the echo, so objects lying more than a few centimeters farther in range do not interfere with the strong echo producing Tm [12]. Absence of strong reflectors produces nonsmooth Tm readings over bearing, which are easily recognized [13]. • Need for scanning. Scanning is a time-consuming operation. One alternative is to employ multiple sensors that sense echoes at a variety of azimuth angles [19]–[22]. The last reference uses the number of transducers detecting an echo to extract echo amplitude and object bearing, while this paper uses the form of the TOF curve. • System tolerances. Most robot sonars employ some form of calibration to relate the Tm measure (e.g., cycle counts) to physical range. TOF errors have been documented extensively [6], [7]. The effect of these errors is reduced by smoothing Tm data with the parametric fit of expected TOF values predicted by our model. It is interesting to note that the Tm produced by the Polaroid sonar is insensitive to echo amplitude for large amplitudes (rf: (3)). Hence, it operates robustly over a typical 10% variation of ranging board and transducer parameter values. Repeated experiments with different transducers and modules indicate that this variation effects the range and reflecting strength estimates, but not the bearing estimate, the component most needing improvement. • Time-variable gain. The objects above were placed around a 1.5-m range. The 6500 module applies a time-variable gain to compensate for dispersion and absorption losses, thereby maintaining a constant beam width. This implies a constant Ao and applicability of the model over range. However, instantaneous gain changes occur at specific range values [1], which would cause artifacts in Tm smoothness. Measurements at these ranges should be treated as special cases. • Tilted objects. Planar object tilt can occur about the vertical and horizontal axes. Tilts about a vertical axis affect the bearing and are measurable as long as the normally incident section of the plane, recognized by a minimum in the range readings, occurs within the scan. Otherwise, a partial scan or occlusion occurs, requiring a match to only a part of the “U” template. Objects tilted about a horizontal axis form less efficient echo producers, similar to off-axis bearing, and would produce smaller reflecting strengths. However, their parametric Tm curves would still accurately localize the echo-producing section of the object. Further, Tm curves are repeatable, making the model useful for landmark recognition and robot localization. Recent results using the Polaroid sonar consider more general environments [12], [13]. • Moving objects. The results above considered stationary sonar and objects. If the scanning were done while the sonar moved, Tm measurements would exhibit linear trends added to the
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patterns shown above. A known velocity relative to the object could predict and remove this trend, allowing our model to be applied to moving sonars. Stationary sonars sensing moving objects would also exhibit Tm values that do not have the expected parametric form. A conservative system choice may be to revert to the conventional TOF interpretation, that is, an object somewhere within the beam at the measured range. Scans produced by moving sonars are currently being examined. VI. SUMMARY This paper described a model of the Polaroid sonar that predicts the observed TOF values while performing a sector scan of an object having a given reflection strength. Our model provided a reasonable fit for the main lobe data observed for objects that had reflection strengths that varied by 25 dB. The model did not predict the observed jumps in TOF as the processed echo waveform exceeds the threshold at the next half-cycle as the amplitude decreases or the side lobes produced by strong reflectors. We demonstrated that our forward model can improve range and bearing estimates of objects and estimate their reflecting strength, offering a means of identifying landmarks for robot navigation. REFERENCES [1] C. Biber, S. Ellin, E. Sheck, and J. Stempeck, “The Polaroid ultrasonic ranging system,” presented at the 67th Audio Engineering Society Conv., New York, NY, Oct. 31-Nov. 3, 1980. [2] J. Borenstein, H. R. Everett, and L. Feng, Navigating Mobile Robots. Wellesley, MA: A. K. Peters, 1996. [3] R. Kuc and M. W. Siegel, “Physically-based simulation model for acoustic sensor robot navigation,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 766–778, June 1987. [4] A. Elfes, “Sonar-based real world mapping and navigation,” IEEE Trans. Robot. Automat., pp. 249–265, 1987. [5] J. Borenstein and Y. Koren, “Obstacle avoidance with ultrasonic sensors,” IEEE Trans. Robot. Automat., pp. 213–218, 1988. [6] M. K. Brown, “Feature extraction techniques for recognizing solid objects with an ultrasonic range sensor,” IEEE Trans. Robot. Automat., vol. RA-1, pp. 191–205, Aug. 1985. [7] A. M. Sabatini, “A stochastic model of the time-of-flight noise in airborne sonar ranging systems,” IEEE Trans. Ultrason., Ferroelect. Freq. Contr., vol. 44, pp. 606–614, May 1997. [8] M. Beckerman and E. M. Oblow, “Treatment of systematic errors in processing wide-angle sonar sensor data for robotic navigation,” IEEE Trans. Robot. Automat., vol. 6, pp. 137–145, Apr. 1990. [9] O. Bozma and R. Kuc, “Building a sonar map in a specular environment using a single mobile robot,” IEEE Trans. Pattern Anal. Machine Intell., vol. 13, pp. 1260–1269, 1991. [10] J. J. Leonard and H. F. Durrant-Whyte, “Dynamic map building for an autonomous mobile robot,” Int. J. Robot. Res., vol. 11, pp. 286–298, 1992. [11] R. Kuc, “A spacial sampling criterion for sonar obstacle detection,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 686–690, July 1990. [12] , “Pseudo-amplitude sonar maps,” IEEE Trans. Robot. Automat., vol. 17, pp. 767–770, Oct. 2001. [13] , “Transforming echoes into pseudo-action potentials and classifying plants,” J. Acoust. Soc. Amer., vol. 110, no. 4, pp. 2198–2206, 2001. [14] , “Binaural sonar electronic travel aid provides vibrotactile cues for landmark, reflector motion, and surface texture classification,” IEEE Trans. Biomed. Eng., vol. 49, pp. 1173–1180, Oct. 2002. [15] L. Kleeman and R. Kuc, “Mobile robot sonar for target localization and classification,” Int. J. Robot. Res., vol. 14, no. 4, pp. 295–318, 1995. [16] T. Yata, L. Kleeman, and S. Yuta, “Fast-bearing measurement with a single ultrasonic transducer,” Int. J. Robot. Res., vol. 17, no. 11, pp. 1202–1213, 1998. [17] B. Barshan and R. Kuc, “A bat-like sonar system for obstacle localization,” IEEE Trans. Syst., Man, Cybern., vol. 22, pp. 636–646, Apr. 1992.
[18] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics. New York: Wiley, 1982. [19] B. Barshan and R. Kuc, “Differentiating sonar reflections from corners and planes by employing an intelligent sensor,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 560–569, June 1990. [20] R. Kuc and Y. D. Di, “Intelligent sensor approach to differentiating sonar reflections from corners and planes,” in Intelligent Autonomous Systems, L. O. Hertzberger, Ed. Amsterdam, The Netherlands: Elsevier, 1986, pp. 329–333. [21] R. Kuc, “Biomimetic sonar system recognizes objects using binaural information,” J. Acoust. Soc. Amer., vol. 102, no. 2, pp. 689–696, 1997. [22] T. Yata, A. Ohya, and S. Yuta, “Use of amplitude of echo for environmental recognition by mobile robots,” in Proc. IEEE/RSJ Int. Conf. Intelligent Robots and Systems, vol. 2, 2000, pp. 1298–1303.
Time-Scaling Control for an Underactuated Biped Robot C. Chevallereau
Abstract—This paper presents a control law for the tracking of a cyclic reference trajectory by an underactuated biped robot. The robot studied is a five-link planar robot. The degree of underactuation is one during the single support phase. The control law is defined in such a way that only the geometric evolution of the robot is controlled, not the temporal evolution. To achieve this objective, we consider a time-scaling control. But, for a given reference path, the temporal evolution during the geometric tracking is completely defined and can be analyzed through the study of the dynamic model. A simple analytical condition is deduced that guarantees convergence toward the cyclic reference trajectory. This condition implies temporal convergence after the geometric convergence. This condition is defined on the cyclic reference path. The control law is stable if the angular momentum around the contact point is greater at the end of the single support phase than at the beginning of the single support phase. Index Terms—Biped robot, control, limit cycle, stability, underactuated system, walking.
I. INTRODUCTION The reduction of the number of actuators of a walking robot is a step toward simpler and cheaper robots. The suppression of the feet and of the actuated ankle seems to be a reasonable way to simplify the mechanical design of the robot. But a consequence is that only purely dynamic walking can be achieved, i.e., motion without mechanical equilibrium (neither static nor dynamic). We choose to study a planar biped with only four actuators, two on the haunch, and two on the knees. During the single support phase, the configuration of the robot is defined by five independent variables, but there are only four actuators. Hence, the robot is underactuated. This simplification in terms of mechanics makes the design of the control law difficult. One classical way to control a robotic system consists of two steps. During the first step, an open-loop joint reference trajectory is designed. In the second step, a control law is defined to track this reference trajectory. In such a context, a reference trajectory was obtained
Manuscript received April 15, 2002; revised September 30, 2002. This paper was recommended for publication by Associate Editor H. Arai and Editor A. De Luca upon evaluation of the reviewers’ comments. This work was supported by the Centre National de la Recherche Scientifique (CNRS) Robea Program. The author is with the Institut de Recherche en Communications et Cybernétique de Nantes, 44321 Nantes Cedex 3, France (e-mail:
[email protected]). Digital Object Identifier 10.1109/TRA.2003.808863
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