Motion Control of Mobile Robots From Static Targets to Fast Drives in Moving Crowds

Transcription

1 Autonomous Robots 12, , 2002 c 2002 Kluwer Academic Publishers. Manufactured in The Netherlands. Motion Control of Mobile Robots From Static Targets to Fast Drives in Moving Crowds CHRISTFRIED WEBERS Fraunhofer Gesellschaft, Schloß Birlinghoven, Sankt Augustin, Germany christfried.webers@ieee.org UWE R. ZIMMER Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia uwe.zimmer@ieee.org Abstract. Motion control of vehicles under uncertain, noisy, and discontinuous positioning is essential in autonomous navigation in unknown environments. This article suggests two methods for motion control, where the initial parameters of the on-line control are physically explainable, the resulting trajectory as well as the control parameters are asymptotically converging and glitches in the localization are handled robustly. The differences to a known method based on Lyapunov functions are discussed theoretically as well as in terms of actual motion measurements. Physical experiments with landbound vehicles show the reliability and limitations of these different methods in setups employing static attractors, systematically moving targets and fast, unpredictable moving targets in highly dynamical environments. Mainly due to the physical meaning of the control parameters the adaptation to actual kinematics and dynamics is significantly simplified. Keywords: mobile robots, tracking, dynamical environments, navigation, motion control 1. Motivation A common task in mobile robotics is to drive the robot to a certain position and orientation as fast as possible given the limits of the static and dynamic properties of the robot setup. Autonomous robots may choose targets in a discontinuous and unpredictable manner (e.g. while exploring unknown or dynamical environments). Recent scenarios out of this class of applications are control of autonomous deep sea vehicles or planetary explorations. Therefore, a robust behaviour in reaching noisy, drifting, or even stochastically moving targets is a necessary condition for successful applications of autonomous robots. The motivation for this article is the stochastic manner in which most autonomous mobile robots with local world model abilities are choosing their targets and the influence of these reactive components on motion control. Although the terms autonomous and unpredictable are mostly synonym in the context of autonomous, self-contained, and mobile robots, it is widely omitted in experiments in order to keep these experiments simple and reproducible. This article tries to approach this problem by a series of experiments with increasing stochastical components. A common reactive modelling and navigation structure of this class might be found in Brooks and Stein (1994). Since autonomous systems have only approximate information about the environment through their sensory system, continuous corruptions and corrections of the position measurement appear on the motion control level indiscriminately from moving targets. Here, targets are generated based on global landmarks or based on dynamical internal world models. In that sense, robust posing control for an autonomous robot

2 174 Webers and Zimmer is, in fact, a target following problem, which includes here the additional case of a static goal. Lyapunov based approaches (Astolfi, 1996; Krstić et al., 1995; Laumond, 1998) and one approach superimposing differential control equations ( superimposed dynamics ) are employed to study the dynamic behaviour of a physical land robot in tracking a moving target. Both approaches use closed loop controllers without global states (represented by autonomous differential equations). The proposed controllers have to take into account the physical constraints of the vehicle, first of all the limited lateral acceleration and the bounded curvature. The employed on-line sensor readings in the experiments stemming from (laser and sonar) range finders, gyroscopes, accelerometers, and encoders on a landbound mobile robot have been additionally (artificially) disturbed in order to demonstrate the robustness of the discussed methods. In Section 2 the kinematic model and physical constraints of the robot are introduced. The Lyapunov approach used in Aicardi et al. (1995) and Indiveri (1999) for the convergence to a point is discussed in Section 3. A refinement to that Lyapunov approach with an explicit bound on the linear velocity is presented in Section 4. An alternative control law for the convergence to a ball is developed in Section 5. Results of experiments with a real robot employing the discussed three methods are described in Section 6 where the robot is approaching a static goal, following a moving target and driving through a fast changing environment. The final Section 7 gives some conclusions drawn from these experiments. 2. The Kinematic Model and Physical Constraints The model, describing the motion of the cartesian unicycle vehicle is given by ẋ = u cos φ ẏ = u sin φ (1) φ = ω Figure 1. Unicycle kinematic model. generality the goal can be chosen to be (x G, y G,φ G ) = (0, 0, 0). Since on-line, reactive, and realtime systems are considered here, u and ω cannot explicitly depend on time but only on the state variables thereby reducing the control system to a set of autonomous differential equations of the state variables. Brockett s Theorem (Brockett, 1983) proves that the system (1) cannot be stabilized using smooth timeinvariant feedback. On the other hand, if the state itself is not defined in the goal, Brockett s Theorem does not prevent this stabilization any longer. This can be achieved in fact by a nonlinear coordinate transformation. A suitable choice for this transformation are the following coordinates which were introduced in Aicardi et al. (1995) e = x 2 + y 2 θ = atan( y, x) (2) α = θ φ where atan(y, x) ( π,π)is the four quadrant inverse tangent function describing the signed angle between the positive x-axis and a line through the origin (0, 0) and (x, y). With the new coordinates e,α,θ the kinematic model (1) is transformed to with u being the linear velocity in the direction of φ and ω the angular velocity (Fig. 1). In a point-to-point navigation task, the vehicle starts at point (x S, y S ) with heading φ S and should be driven with appropriate u and ω to the goal. Without loss of ė = ucos α θ = u sin α e α = ω + u sin α e (3)

3 Motion Control of Mobile Robots 175 Any real vehicle has limitations which depend on the vehicle itself or on its interaction with the environment. The following ones are considered here: (a) Bounded linear and angular velocities u u max ; ω ω max. Normally, the low level motor controllers prevent any dangerous settings of the controls which would break the gears of the vehicle. A more serious problem is the use of control setting ranges which can not be physically realized in the vehicle and would result in an invalid experimental setup. (b) Bounded lateral acceleration a = u ω a max. Motion control experiments depend on the precision of the odometry. If in a curve the lateral acceleration of the vehicle is too strong, the wheels loose close contact to the ground and the odometry data will no longer be meaningful. (c) Bounded curvature c = ω u c max. Additionally to the above restrictions which apply to every vehicle, a large class of vehicles can not turn on the spot, i.e. the curvature is bounded. (d) Only forward moving vehicles u 0. Moving is assumed to be possible in one direction only in order to avoid additional bifurcations. 3. Linear Velocity Lyapunov Approach (LV) Under the assumption for the linear velocity and u = γe γ>0 (4) shown that even under this condition, the convergence of the system can be preserved. For practical use the approach has two drawbacks: The resulting path scales proportionally in both directions, thus demanding a very wide side space. (For instance, with a distance of 10 m and the vehicle pointing away from the goal, more than 6 m space to the side is necessary.) In order to keep the lateral acceleration in curves bounded to the maximum, the velocity has to be set to a much smaller value then the vehicle would allow on less curved path segments. The overall speed of this approach is therefore very low. For this reasons, improvements to the Lyapunov approach and an alternative approach using superimposed dynamics have been developed. 4. Bounded Velocity Lyapunov Approach (BV) While (4) introduced the velocity as depending linearly on the distance, the following assumption for the velocity u = u max tanh(κe) (7) establishes the existence of an upper bound for the velocity in the Lyapunov approach. Here, κ is a measure for the deceleration of the vehicle when it approaches the goal. With a similar Lyapunov function and first derivative as in (5) V = 1 2 (α2 + hθ 2 ) h > 0 V = γβα 2 β>0 (5) V = 1 2 (α2 + hθ 2 ) h > 0 V = u max βα 2 β>0 (8) for the Lyapunov function and its derivative a nonlinear, time-invariant, globally and asymptotically converging control law for ω of the form ( ) hθ sin α ω = γ sin α + + βα α 1 < h; 2 <β<1 + h (6) the angular velocity now becomes ω = u max ( tanh(κe) sin α e + hθ tanh(κe) sin α αe ) + βα (9) This results in the state equations for the BV approach can be found (Aicardi et al., 1995; Indiveri, 1999). Assumption (4) cannot be realized on a real vehicle because it leads to either huge velocities or, with reduced γ, to very slow motion. Therefore, in experimental setups u will be bounded to u max. It can be ė = u max tanh(κe) cos α ( ) hθ tanh(κe) sin α α = u max βα + αe tanh(κe) sin α θ = u max e (10)

4 176 Webers and Zimmer Similar to the LV approach global convergence to the goal can be proven via the Local Invariant Set Theorem (Slotine and Li, 1991; Indiveri, 1999). The goal in the origin must be approached on the negative x-axis. Furthermore for (e, θ, α) (0, 0, 0) the curvature c = ω u = sin α e + hθ sin α αe + βα tanh(κe) (11) must approach 0 without oscillations. The approximation of the state Eq. (10) at some point (e,θ,α)close to the origin provide the linearized state equations and ė = u max κe (12) [ ] βumax hκu max α = α θ κu max 0 θ (13) The eigenvalues of the matrix in (13) are λ 1,2 = u max ( β ± β2 4hκ 2 2) (14) The curvature will approach 0 if the dominant eigenvalue of (14) governing the decrease of (θ, α) is strictly larger than the factor governing the decrease of e in (12), meaning that the angles (θ, α) are approaching (0, 0) faster than the distance e from the origin is going to 0 u max κ< u max ( β β2 4hκ 2 2) (15) A second condition results from the fact that no oscillations are allowed β 2 4hκ 2 > 0 (16) Finally, the conditions (15) and (16) for the asymptotic convergence of the bounded velocity approach can be expressed as 1 < h; 2 hκ<β<(1 + h)κ (17) In order to keep the lateral acceleration bounded to a max, both u and ω are multiplied by a reduction factor r = a max /u ω, (18) if u ω > a max. The curvature and thereby the resulting path is left unchanged. For large distances, the angular velocity ω in (9) is dominated by the factor u max βα. In the case of large α, which means the vehicle is pointing away from the goal, ω can not be bounded to an arbitrarily small ω max with a given u max because β has a lower bound given by (17). These contradicting constraints reflect the fact that one global Lyapunov function and chosen Lyapunov derivative govern very different dynamic situations like turning to the goal, moving toward the goal and converging into the goal. In order to keep ω bounded, and to preserve the curvature, proportional reduction of both u and ω as in the adjustment of the lateral acceleration is used. The following parameters have been used in all experiments with the BV approach: u max = 1.6 [m/s] maximal u a max = 0.4 [m/s 2 ] max lateral acceleration ω max = 80 [ /s] maximal ω h = 2.0 κ = 1.0 β = 2.9 (19) In order to gain the full benefits from the Lyapunov approach for the whole control system, all physical constraints should be included in (7) or (8) directly, instead of pruning the resulting system with (18). Unfortunately, by doing so, the complexity of the equations is exploding and numerical problems conflict with stability and realtime requirements quickly. 5. Superimposed Dynamics (SD) Since it turned out that very different control strategies apply depending mainly on the distance to the goal they are first identified individually and formulated as individual and complete control equations in the first place. The sets of differential equations are then superimposed in order to gain a closed formulation for the control system in the end. By blending the individual aspects instead of switching between them, the problem of artificially introduced phase transitions is avoided. This approach could thus be sketched as a superposition of dynamics (SD), where the following aspects are considered (e,α,φ as introduced above): Approaching the goal directly: Far from the goal, only the difference in the heading towards the goal

5 Motion Control of Mobile Robots 177 α is considered to control the vehicle with: cw α ω = ω max tanh ω max (20) u = u max (u min + (1 u min ) max {0, cos α} c l ) (21) Approaching the goal with specific orientation: Closer to the goal, the trajectory needs to consider the difference to the requested final orientation φ as well as the distance to the goal e with: π α + φ δ close = c m 2 tanh 1 c m (22) approximating the direction to a point on the negative x-axis in distance c m e from the goal, which would be precisely for α <π/2: cm sin α atan 1 c m cos α The derived control equations are: cω (δ close + α) ω = ω max tanh ω max (23) (24) u = u max tanh(ec b ) (25) At the goal: When so close to the goal that the uncertainties in the positioning are of the same dimension than the actual remaining distance, the direction to the goal is no longer influencing the dynamics. This is especially important, if large changes in α (which are unavoidably increasing as the goal gets closer) should not lead to arbitrary large changes and thus instabilities in ω. Thus the control laws at the goal consider φ and e only: cω φ ω = ω max tanh ω max (26) u = u max tanh(ec b ) (27) Note that these final approach strategy is not reaching the goal exactly, but offers a stable way to get close to the goal only. By superimposing these aspects of the control task, a closed representation can be formulated, where the robustness of the simple individual parts is preserved. First, the currently required deviation δ from the direct heading to the goal is expressed with: π α + φ δ = c m 4 tanh (1 tanh(c s (e c d ))) 1 c m (28) where c m gives the strength which attracts the vehicle to a straight line into the goal (i.e. the smoothness or precision of the trajectory can be controlled here), c d sets the distance at which the intended goal orientation is started to be considered, and c s gives the speed of the transition from straight towards the goal to the final approach behaviour. Second, the linear velocity reduction and the transition to the relaxed being there dynamic of the control can be formulated as: where u fo = 1 (α d (1 tanh(ec b ))) (29) a fo = 1 (α d (1 (tanh(ec a )) 6 )) (30) α d = max {0, cos((2α/3) 4 )} (31) Finally the closed control laws for ω and u can be defined as: ωamp δ o ω = ω max tanh (32) ω max a1 u = u max u fo tanh (33) ω u max where δ o = a fo (δ + α + φ) φ (34) The parameter a l determines the tolerated lateral acceleration. Equations (29) and (30) reflect the fact that deceleration and relaxation are reasonable only, if the goal is in front of the vehicle, where c a and c b control the angular relaxation and the linear deceleration respectively. The parameter ω amp, u max and ω max are the overall velocity amplifications and limits. This approach has no singularities (beside the obvious bifurcation, when the goal is exactly behind the vehicle) and can be adjusted according to the physical constraints of the setup directly. The vehicle will be lead only close to the goal considering the uncertainties of the available position information. Therefore,

6 178 Webers and Zimmer instabilities due to overestimations of the position reliability or precision are avoided. Parameters chosen for the physical experiment are: u max = 1.6 [m/s] maximal u ω max = 60 [ /s] maximal ω ω amp = 3 [1/s] amplification in ω c a = 10 [1/m] angular relaxation c b = 1 [1/m] deceleration c m = 0.6 [0, 1] smoothness of final turn c d = 2 [m] starting final approach c s = 0.5 [1/m] smoothness of bending away a l = 0.4[m/s 2 ] max lateral acceleration One related method, superimposing dynamics separated in activation and target dynamics can be found in Jaeger (1996). Another super positioning method based on connectionist techniques is introduced in Brooks and Viola (1990). Interesting experimental results of an action fusion approach, where collision avoidance is being taken into consideration additionally, might also be found in Egerstedt et al. (1999). Another kind of superposition is demonstrated in Egerstedt et al. (1998) ( virtual vehicle control), where a geometrical path generation and following approach is combined with classical track control methods. 6. Experimental Setup The physical system employed for all experiments (Fig. 2) has a width of 0.5 m, a length of 0.8 m, a height of 0.6 m and a weight of 60 kg. It is equipped with the following sensor systems and actuators: 3-axis gyroscope: stability: 1 /s; sampling frequency: 176 Hz. 3 linear accelerometers: resolution: 5 mg; sampling frequency: 176 Hz. 2 encoders: resolution: ticks per wheel revolution; sampling frequency: 58 Hz. 4 wheel drive with differential steering, a maximal linear speed of 1.6 m/s and a maximal angular speed of 150 /s; control frequency: 193 Hz. Gyroscopes, accelerometers and encoders are combined to stabilize for glitches in the encoders (wheel slip) and drifts in the gyroscopes. Since the robustness of the approaches against uncertainties is to be shown the resulting position measurement is deteriorated by adding ±10 mm/sample uniform noise on the linear forward movement and ±3 /sample uniform noise on the orientation information, as measurement by the encoders Approaching a Static Goal In order to evaluate the different approaches, the vehicle is requested to approach a goal 5.4 m behind the starting position in the same orientation as it started. In Fig. 3, the driven paths as recorded by odometry (and projected as a bird s eye view) are plotted, where the starting point is on the right side with the vehicle facing to the right. The constraints are to reach the goal as fast as possible but keep the accelerations and velocities in reasonable limits (tolerated lateral acceleration: 0.4 m/s 2 ; top speed: 1.6 m/s and angular velocity: 110 /s at most). The durations, lateral accelerations, as well as the employed space in y, and the maximal curvature Figure 2. Employed autonomous mobile robot, see text for specifications. Figure 3. Comparison of generated trajectories (static goal).

7 Motion Control of Mobile Robots 179 Table 1. Durations, accelerations, space and curvature. Max. lateral Employed Orientation Maximal Total acceleration space (y) at goal curvature duration (s) (m/s 2 ) (mm) ( ) ( /m) LV BV SD needed for reaching the goal within a range of 3 cm and the orientations close to the goal position are shown in Table 1. The maximal curvature along the paths occurs with the LV and BV method at the goal position, where the strongest angular corrections are forced. The curvature of the LV method is not bounded by any means thus the vehicle tries to turn almost on the spot close enough to the goal. Since the curvature of the BV method is bounded explicitly it is still very low, allowing for corrections in the borders of the vehicles capabilities only. The maximal curvature in the SD method occurs in the last phase of the path also, but significantly before the goal position, because angular corrections are suppressed closer to the goal. It is not explicitly bounded here, nevertheless the method allows naturally for limited curvature paths only (due to the maximal lateral acceleration at high velocities (33) and the fact that the angular velocity is reduced faster than the linear velocity, while approaching the goal (29)(30)). A closer look at the final approach phase can be found in Fig. 4, where strong corrections can be detected in the LV and BV methods trying to reach the goal exactly and turning (in case of LV) the vehicle by 65 off the intended orientation in 29 mm off the goal. In the superimposed dynamics method the vehi- cles s orientation is controlled to zero before reaching the goal and variations in α are less considered closer to the goal. Therefore the orientation and the overall behaviour can be kept stable until the very end, but convergence to the goal position is not forced. Due to the high lateral starting acceleration of the LV method the maximal speed needed to be set to 0.5 m/s here (Fig. 5), which is then constant for most of the path and linearly reduced near the goal. Nevertheless a top lateral acceleration of 0.9 m/s 2 needed to be tolerated (Fig. 6) in order not to disqualify this method with regard to the overall travel duration. Both other methods keep in the given limit of 0.4 m/s 2 (Figs. 7 and 8) and using the tolerated top speed of 1.6 m/s (Figs. 9 and 10). Differences throughout the path are larger variations in omega and lateral acceleration for the BV method, but therefore reaching the goal approximately 2 seconds earlier than the superimposed dynamics method. Figure 5. Linear velocity profile with the LV method. Figure 4. Comparison of end approaches (static goal). Figure 6. Lateral acceleration with the LV method.

8 180 Webers and Zimmer Figure 7. Lateral acceleration with the BV method. Figure 10. Linear velocity profile with the SD method. The O(1) complexity enables these control methods for all hard realtime control tasks Following a Moving Goal Figure 8. Figure 9. Lateral acceleration with the SD method. Linear velocity profile with the BV method. All shown methods were executed under realtime constraints with a permanent control frequency of 193 Hz. All control laws are one step direct evaluations without any recursions or loops (the computational complexity of all approaches is O(1)) and the actual computation time constant depends on the evaluation of the individually required trigonometric functions only. The setup of these experiments has been guided by the long term research goal of applying the developed closed loop controllers to autonomous systems. In such systems discontinuities can be expected due to localization glitches or inconsistencies during local spatio-temporal model updates. The control mechanism should react as smooth as possible to this discontinuous changes in the goals of the overall system. The only assumption in the approaches described above is, that a certain corridor from the start to the goal and a turning space around the goal is not occupied by obstacles. Experiments which are easily reproducible and comparable to other systems and methods can be realized by a cyclic change of goals where the timing of these goal changes has a great impact on the dynamic properties of the robot. One single goal is defined by the position in space and the orientation of the robot when driving into this position. For each experiment an ordered set of four goals is defined which are approached in a cyclic manner. The relative change of position and orientation between all adjacent goals are chosen the same in order to avoid limit circles in the dynamic behaviour which are artificially introduced by periodically varying goals. A new goal is chosen either after a fixed time interval (used in all experiments shown here) or by the distance to the present goal as measured by the inertial system and the dead-reckoning of the robot. The goals can be chained forward or backward as shown in Fig. 11. In the following experiments backward chained goals

9 Motion Control of Mobile Robots 181 Figure 11. Schematic of target settings for: (a) forward chained goals, (b) backward chained goals. are analysed only. These experiments disclose a richer dynamics compared to the forward shifted goal. If the system shows stable behaviour for the case of the backward chained goal, it will be also stable for the case of the forward chained goal but not vice verca. For all following experiments the next goal relative to the current goal as viewed from the center of the robot is defined as in Fig. 11(b) SD Timed Moving Goal. Predictability under real world influences is considered the central criterion of these experiments. Specifically, it is evaluated whether the motion control system behaves similar in similar situations. Figure 12(a) shows the simple case in which the goal is switched slowly (every 6 s). Therefore the system follows synchronously. As soon as the switch time is less than a critical limit at which all four goals may not be approached individually any longer, the system turns to several cyclic behaviours. A significant variance in the peak speeds and the actual trajectories can be observed in the x, y-traces (Fig. 12(b) and (c)) or the u,ωplots (Fig. 15(b) and (c)). Nevertheless cyclic attractors are established, demonstrating that small changes in physical situations are not influencing the global behaviour even if the goals may no longer be actually approached. The wide variety of the individual (but similar) trajectories due to unpredictable disturbances and the robust overall motion pattern becomes especially obvious in Fig. 12(b). Reducing the switch time even further, the state of cyclic attractors is left finally, leading to the state of stochastical reorientation towards quickly changing goals (Figs. 12(d) and 15(d)). The covered space as well as the occurring speeds and accelerations are nevertheless strictly bounded, ensuring that even in this case the control system keeps well behaved. Figure 12. Recorded trajectories, SD-method, circles indicate target switches, target switch intervals: (a) 6 s, (b) 5 s, (c) 4 s, (d) 3 s BV Timed Moving Goal. The BV method manages a faster turn to the goal compared to the SD approach which results in a larger curvature around the goal points as shown in Fig. 13(a). Figure 13. Recorded trajectories, BV-method, circles indicate target switches, target switch intervals: (a) 5 s, (b) 4 s, (c) 3 s, (d) 2 s.

10 182 Webers and Zimmer Figure 14. Linear and angular velocity patterns in stable trajectories, BV-method, target switch times: (a) 3 s, (b) 2 s. With a smaller switch time (Fig. 13(b)) two different phases emerge depending on the current attitude of the robot. In case the robot is already close to the goal when the next goal is presented, the phase is the same as in Fig. 13(a). But if the robot is not yet close enough to the current goal the new presented goal is approached with a turn to the right. This takes a certain time and before the robot can really move close to that goal the next goal is presented which the robot is now able to closely approach in time. Which of the two phases are realized depends on the exact conditions where the bifurcation appears and is stochastical in nature. With yet smaller switch time only the second of the just described phases can be observed (Fig. 13(c)). Further reduction of the switch time leads yet to another phase (Fig. 13(d)). Very similar to the results from the SD method, in all cases explored the covered space as well as the velocities and accelerations are strictly bounded and the tracking system is well behaved. In highly dynamic situations, where fast direction changes are necessary, the BV method constrains the angular velocity to ω max, keeping the curvature and lateral acceleration bounded. This results in regulating mostly the velocity u as shown in Fig. 14(a) and (b). The SD methods employs both u and ω (see Fig. 15(a) (d)) Wandering in the Crowd The final tests leave the paths of reproducible experiments and exposes the system to a typical real world situation: When moving quickly in a highly dynamical Figure 15. Linear and angular velocity patterns in stable trajectories, SD-method, Target switch times: (a) 6 s, (b) 5 s, (c) 4 s, (d) 3 s.

11 Motion Control of Mobile Robots 183 Figure 16. Sample of the local, robot-centric range model. Figure 17. Recorded trajectory (line), and targets (clusters) are changing with 4.7 Hz, see text for detailed explanation. environment, the estimation of the current target is unstable and changes frequently. In order to test this situation in a real world setup, a maximal escape target generation algorithm was developed. The generated local range model representing local obstacles (as for instance in Fig. 16) is searched for the deepest open corridor and the target is set inside this corridor. Since the local range model is updated every 4.7 Hz, so are the calculated targets. Allowing for a linear speed of 1.6 m/s, an angular speed of 60 /s, and a maximal lateral acceleration of 0.4 m/s 2 the system was set in an open space, with people walking at the front-left and at the front-right side in parallel to the robot s path guiding the robot in an approximate S-curve over 12 m distance (what is by the way not an easy task for this dynamical human environment operating with such a fast reacting system actually the robot almost escaped from the guiding persons once during these test runs). In this setup we could not identify any further differences between the SD, and the BV method, therefore only the data from the SD method is shown here. The tests with the LV method could be performed with reduced speeds only. Besides this drawback, the LV method delivered similarly satisfactory results. Figure 17 gives the path of one of these dynamical motion experiments together with the targets (appearing as bow shaped clusters). The robot starts at the lower left corner (0 m, 0 m) and is being guided at the highest possible speed (but not along a straight line) to approximately (10 m, 6 m). The shape of the target clusters stems from the fact that the angular ambiguity is stronger than the range jitter. At position (5.57 m, 2.90 m) and time stamp ms the target changes drastically into the cluster at the left side. 209 ms later the targets are changed again to the right side and towards the goal. During this episode the linear velocity is reduced (Fig. 18(a)) and the angular velocity changes its sign (Fig. 18(b)). The unintended excursion was triggered by the fact that the vehicle, with a reaction frequency of 4.7 Hz and a top speed of 1.6 m/s, is hard to control by interactive changes of the environment only (human interaction with the robot). The important observation from all practical wandering in the crowd experiments with its unpredictable changing targets is nevertheless, that the motion control never tended to oscillate or overreact, while the motion is being kept fast and the control is completely stiff all the time Effect of Noise All experiments presented up to here are subject to strong disturbances partly from physics, partly from added noise on several channels. In this section the effect of the additional noise is discussed briefly. Enlarging the middle section of the stable tracking in Fig. 12(a), it can be seen in Fig. 19(a) that there is a noticeable variance in the actual tracks, but each individual track is smooth and (as discussed before) the overall behaviour is still identical. By removing the artificial noise and performing the experiments due to

12 184 Webers and Zimmer Figure 18. Linear and angular velocities along the trajectory of Fig. 17. sensor data) certificated sufficient robustness and reactivity for both methods. In order to compare essential abilities of different autonomous mobile robots it is very desirable to define general and adequate measurement criteria for navigation/control and to establish common experimental setups in non-trivial environments. The authors are highly interested in collaborated work leading to such criteria. 1 Figure 19. Trajectory changes due to more precise position measurement (by skipping artificial noise): (a) with artificial noise, (b) without artificial noise. physical constraints only, Fig. 19(b) demonstrates that the variance in the tracks is reduced and the tracks themselves are even smoother. Nevertheless, an effect on the overall behaviour (bifurcations or differences amplified over time) could never be observed. 7. Conclusion The dynamic behaviour and the robustness of two closed loop control methods for track control under real world constraints have been tested. Both methods are stable in a broad range of dynamical situations. The computational complexity of the methods is O(1) which makes them very suitable for fast closed loop control. Different control characteristics have been revealed in controlled reductionist experiments, while practical experiments with the fast 60 kg vehicle in highly dynamical environments (with even artificially disturbed Acknowledgments The authors would like to express their high appreciation to Bärbel Herrnberger for pinpointing weaknesses in the argumentation and to Giovanni Indiveri for inspiring discussions. Finally, the comments from the anonymous reviewers have been very productive. Note 1. Implementations of all algorithms and overall control architectures are available from the authors. References Aicardi, M., Casalino, G., Bicchi, A., and Balestrino, A Closed loop steering of unicycle-like vehicles via Lyapunov techniques. IEEE Robotics and Automation Magazine, 2: Astolfi, A Discontinuous control of nonholonomic systems. Systems & Control Letters, 27: Brockett, R.W Asymptotic stability and feedback stabilization. In Differential Geometric Control Theory, R.W. Brockett, R.S. Millmann, and H.J. Sussmann (Eds.), Birkhauser: Boston, USA, pp

13 Motion Control of Mobile Robots 185 Brooks, R.A. and Stein, L.A Building brains for bodies. Autonomous Robots, 1(1):7 25. Brooks, R.A. and Viola, P.A Network based autonomous robot motor control: From hormones to learning. In Advanced Neural Computers, R. Eckmiller (Ed.), Elsevier Science/North- Holland: Amsterdam, pp Egerstedt, M., Hu, X., and Stotsky, A Control of a car-like robot using a virtual vehicle approach. In Proc. 37th IEEE Conf. on Decision and Control, Tampa, FL, USA, pp Egerstedt, M., Hu, X., and Stotsky, A A hybrid control approach to action coordination for mobile robots. In Proc. of IFAC 99: 14th World Congress, Beijing, China. Indiveri, G Kinematic time-invariant control of a 2D nonholonomic vehicle. In Proc. 38th Conference on Decision and Control, CDC 99, Phoenix, USA. Jaeger, H The dual dynamics design scheme for behaviourbased robots: A tutorial. GMD Technical report # 966, St. Augustin, Germany. Krstić, M., Kanellakopoulos, I., and Kokotović, P Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, USA. Laumond, J.-P. (Ed.) Robot Motion Planning and Control, Lecture Notes in Control and Information Sciences, Vol. 229, Springer: Berlin. Slotine, J.-J.E. and Li, W Applied Nonlinear Control, Prentice- Hall: Englewood Cliff, NJ, USA. statistical kinetic equations in the Academy of Sciences in Leipzig. He moved to Japan in 1984 and worked as director in the Tokyo Office of the German National Research Center for Information Technology. He planned and established an international research laboratory in Kitakyushu, Japan, together with Uwe. R. Zimmer. His current research interest is in motion control for autonomous systems and in the behaviour of groups of robots. Uwe R. Zimmer is currently a fellow at the Australian National University at the Faculty of Engineering and Information Technology and at the Research School of Information Sciences and Engineering. He received his Dr. rer. nat. degree in 1995 at the University of Kaiserslautern, Germany. Building up an underwater research group in Germany at the former German National Research Center for Information Technology, and setting up a new laboratory for autonomous robust systems in Kyushu, Japan together with Christfried Webers have been the other major activities since then. His major research interests are currently in autonomous spatiotermporal modelling and understanding, with application to real-time environments. Christfried Webers received a diploma in physics from Leipzig University, Germany, in After graduation he investigated quantum