5.4.1 Single Integrator: Formation Acquisition

The control gain in (2.15) was set to while and 0 in ( 5.15 ) for all three vehicles. The position of each UGV as they formed the desired triangle is shown in Figure 5.7, where the markers denote the vehicle location at different instants of time during the interval s. The inter‐vehicle distance errors defined in (2.6) are depicted in Figure 5.8, showing that the desired formation was acquired in less than 5 seconds. Notice from Figures 5.7 and 5.8 that the vehicles exhibit a small oscillatory motion near the final formation configuration. This is due to the fact that the formation acquisition objective is a setpoint control problem, therefore the vehicles are operating near zero velocity where static friction effects are prominent. Furthermore, the vehicles have noticeable backlash in the mechanical transmission. Since the encoder was collocated with the DC motor's drive shaft, these small position perturbations are measured by the encoder, but are often unobserved at the vehicle level due to the flexibility of the gear train.

The control voltages applied to the each vehicle's drive‐wheel DC motor is shown in Figure 5.9, and are well below the maximum battery voltage of 16 V. The constant switching observed in the voltages after the UGVs reach the desired formation is due to the static friction compensation term, , in (5.10a). Due to imperfect, noisy measurements, “chatters” about zero in the steady state and causes this term to constantly switch. The steering angle commands to the servo motors that orient the UGV front wheels are given in Figure 5.10, showing that they quickly became saturated at their maximum value of approximately . This saturation can be attributed to the vehicles being regulated to a setpoint position, which results in the vehicle longitudinal speed approaching zero. As decreases over time, the steering angle becomes larger according to (5.10b).

Figure 5.11 displays the ‐ and ‐direction components of the velocity error ( 5.13 ) for each UGV. When tuning the control gain , a balance was struck between acceptable steady‐state performance and limiting the transient amplitude of the velocity errors.

5.4.2 Single Integrator: Formation Maneuvering

Two formation maneuvering experiments were conducted: one with formation translation only, and the other with translation and rotation. In both experiments, the UGVs started in the initial configuration of ( 5.17 ) while the control gains in (2.23) and ( 5.15 ) were set to , , and .

Formation Translation: In this experiment, the UGVs were commanded to move north‐east along a bearing of approximately with a constant translational speed of 0.5 m/s while maintaining the inter‐vehicle distance of 4 m in the triangular formation. As a result, the desired translational and angular velocities in (2.24) became m/s and .

Figure 5.12 shows the position of each UGV as they maneuver on the plane, including snapshots of the actual formation at s and s. Figure 5.13 shows the inter‐vehicle distance errors converging to a small steady‐state value after about 15 s.

The actual translational speed and heading (bearing) angle of each UGV are displayed in Figure 5.14. At first glance, we can see that UGVs 1 and 3 converge to the desired translational speed of m/s and the desired bearing angle of , while UGV 2 appears to be moving in the wrong direction. However, based on Figures 5.12 and 5.13, we know that the desired formation maneuvering is achieved. Upon closer inspection, we observe that UGV 2's heading is approximately , which is from the desired heading. Therefore, the vehicle is moving in reverse in the direction that promotes the formation maneuvering control objective. This occurs because the formation maneuvering controller is only concerned with the relative position of the vehicles and does not actively control the heading of each vehicle. As a result, the controller may determine that it is beneficial (based on the commanded velocity and the vehicle's current position/heading) to operate in reverse mode.

The DC motor voltage of each vehicle is shown in Figure 5.15. One can observe a steady‐state voltage of V on each of the vehicles. This bias voltage is required to keep the vehicles moving at the desired translational speed of 0.5 m/s. Note that a negative voltage indicates that the vehicle is moving in reverse. Indeed, from Figure 5.12 we can see the moment where UGV 2 transitions from an initial forward motion to the reverse motion, which it maintains until the end of the experiment. On the other hand, UGV 3 does the opposite maneuver: initial reverse motion followed by forward motion. Clearly, the formation translational speed can be increased significantly since each UGV only utilized approximately 2 V V of the battery voltage. The somewhat slow translational speed was specifically chosen to maintain the vehicles operating at low velocities in case of hardware failure, and to prevent the vehicles from overrunning the parking lot area.

The steering angles are displayed in Figure 5.16. Notice that, aside from a brief period during the transient, the steering angles do not saturate as in Figure 5.10 because in (5.10b) never approaches zero during formation maneuvering. The steering angles of each UGV attain a steady‐state value of zero since the front‐wheel direction needs to eventually align with the ‐axis for the vehicle to move in the desired direction.

Formation Translation and Rotation: Here, the formation had to move with the same translational velocity of the previous experiment while rotating about UGV 2 with an angular velocity of deg/s. Figure 5.17 depicts the propagation of the formation over time. Since UGV 2 is the axis of rotation, it merely translates in the north‐east direction while UGVs 1 and 3 trace a spiral‐like trajectory as they spin around UGV 2. The distance errors in Figure 5.18 indicate that the triangular formation was acquired by 10 s. In Figure 5.19, we can see the bearing rate of UGVs 1 and 3 converging to the desired angular velocity of 20 deg/s. This angle rate was calculated by numerically differentiating the vehicle's measured bearing with all discontinuities removed (i.e., when the vehicles' magnetic heading value transitioned from to ).

The velocity error of each UGV is shown in Figure 5.20. The smaller error for UGV 2 is expected since its desired velocity is constant (setpoint problem) whereas the desired velocities for UGVs 1 and 3 are time varying (tracking problem). The drive‐wheel motor voltages and steering angles are given in Figures 5.21 and 5.22, respectively. From Figure 5.21, we can see that UGV 2 was operating in reverse throughout the experimental trial since its voltage was always negative. Notice the nonzero steady‐state values of the steering angles for UGVs 1 and 3 required for producing the rotation about UGV 2. On the other hand, the steering angle of UGV 2 reaches zero at s when the vehicle becomes aligned with the direction of translational velocity vector .

5.4.3 Single Integrator: Target Interception

For this experiment a virtual vehicle was employed as the target since additional UGVs were not available. The target's translational speed was set to a constant value of m/s and its heading angle to an oscillatory motion of deg. The same triangular formation from Section 5.4.2 was used. The target velocity estimate in (2.45) was initialized to zero and was calculated using the trapezoidal rule of integration. The control gains were set to and in (2.53) and (2.54), and and in ( 5.15 ). Since only three vehicles were available, UGV 2 was selected as the leader of the formation, i.e., the agent responsible for chasing the target. As a result, the target interception error (2.44) was defined as , and the target was located at the boundary of the formation when intercepted.

Figure 5.23 shows the motions of the UGVs and target over time. The inter‐vehicle distance errors and the target interception error defined in (2.44) are given in Figures 5.24 and 5.25, respectively. From these figures, one can see that the target was successfully intercepted after approximately 10 s while the formation was acquired after 20 s.

Figure 5.26 displays the drive‐wheel motor voltage for each vehicle. As in the formation maneuvering experiment, the voltages reach a steady‐state value of V since the formation needs to track the target's translational velocity of  m/s. The steering angles, shown in Figure 5.27, have an oscillatory motion for s because of the sinusoidal heading movement of the target. That is, the oscillation period is 10 s, which corresponds to the frequency of the target's heading motion (0.1 Hz).

5.4.4 Single Integrator: Dynamic Formation

For the following experiment, the desired formation was set to a time‐varying version of the one used in the previous experiments. Specifically, we used an equilateral triangle with time‐varying inter‐vehicle distances specified as

(5.19)

In addition, we required the formation to maneuver according to the translation described in the first experiment of Section 5.4.2. The vehicles were again initially located according to ( 5.17 ), and the control gains were tuned to in (2.73), and and in ( 5.15 ).

The trajectory of each UGV along with snapshots of the formation at instants of minimum and maximum values for (5.19) are given in Figure 5.28. It is interesting to note that UGV 1 strictly followed the desired translation with virtually no oscillations, while UGVs 2 and 3 appear to carry the brunt of the effort to track the sinusoidal dynamic formation. From the distance errors in Figure 5.29, we see that the desired formation was acquired and tracked after about 15 s with a steady‐state error within cm.

The drive‐wheel motor voltages in Figure 5.30 again converge to V as needed to move the vehicles with m/s speed. In this experiment, the voltages were saturated to of the maximum battery voltage (8 V). Note that UGV 2 moved in reverse for the entire duration of the experimental run. The steering angles are displayed in Figure 5.31. It was mentioned previously that UGV 1 moved in a fairly straight positional track as compared to UGVs 2 and 3. This observation is substantiated by noting that UGV 1's steering angle has much smaller oscillations about as compared to UGVs 2 and 3.

5.4.5 Double Integrator: Formation Acquisition

In order to enable the proper comparison between the double integrator (DI)‐based formation acquisition controller and the single integrator (SI)‐based one, the control gain in (3.9) was set to the same value reported in Section 5.4.1, while we used in (3.8), which is the same value as of ( 5.15 ) in the experiment of Section 5.4.1.

The UGV trajectories from their initial configuration to the final configuration are displayed in Figure 5.32, while the inter‐vehicle distance errors are shown in Figure 5.34. In Figure 5.33, we replot the trajectories of both the DI‐ and SI‐based controllers to facilitate the comparison. First, one can see that the final formations are isomorphic according to Definition 1.4. Note that the DI‐based controller produced a more irregular response than the SI one. This is more clearly seen by the side‐by‐side comparison of the distance errors in Figure 5.35, where the response of the DI‐based controller is noticeably less damped. We believe that this is due to the control gains in (3.8) and (3.9) not being retuned for the DI system, whose closed‐loop dynamics have higher order than the SI dynamics. From the inset plots in Figures 5.8 and 5.34, we can see that the steady‐state errors of the DI‐based control were slightly smaller in their peak‐to‐peak amplitude than those of the SI‐based control. Again, better tuning of DI‐based control gains could result in even better steady‐state performance.

The motor voltages, which are shown in Figure 5.36, have larger transient magnitudes than those of the SI‐based controller (see Figure 5.9); however, the steady‐state magnitudes are only slightly larger. This suggests that the DI‐based formation acquisition controller required more control energy than its SI counterpart. The steering angles in Figure 5.37 are qualitatively similar to the ones for the SI‐based control (see Figure 5.10) in the sense that they operated in saturation for the duration of the experiment due to the UGV longitudinal speed quickly converging to zero.

5.4.6 Double Integrator: Formation Maneuvering

The same translation‐only maneuvering experiment described in Section 5.4.2 was run with the DI‐based controller. The same control gain values given in Section 5.4.5 were used in (3.17). In order to provide compensation for the uphill grade of the parking lot, the original DI‐based control (3.8) was augmented with the following integral feedback term:

(5.20)

where was set to 0.5 for this experiment. This term is akin to the one used in ( 5.15 ) for implementation of the SI‐based controllers.

Figure 5.38 displays the movement of the formation during the 40‐s experimental run, and Figure 5.39 compares it to the SI‐based control. The trajectories of UGVs 1 and 3 were very similar between the two schemes, while UGV 2 moved initially in opposite directions but then converged to a common trajectory. The distance errors are shown in Figure 5.40, and their comparison with the SI errors are given in Figure 5.41. We again observe the less damped response of the DI errors, which are attributed to the control gains not being fine tuned for the DI dynamics. Indeed, when we calculated the RMS value of the inter‐vehicle distance error vector, , defined by

(5.21)

where s is the final experiment time, we obtained for the DI controller as compared to 0.641 when the SI controller was employed.

The velocity error variable defined in (3.4) is shown in Figure 5.42 for each UGV and converges to zero as predicted by the theory. The motor voltages, displayed in Figure 5.43, have more oscillations and larger magnitude during the transient period than the SI voltages (see Figure 5.15), but naturally have the same steady‐state value of V corresponding to the 0.5 m/s translational speed of the maneuver. Note that the oscillations occurred about zero, indicating that the vehicles switched between forward and reverse motions during the first 8–10 s and then settled to one or the other (UGV 1 reverse; UGV 2 reverse; UGV 3 forward) until the end of the experiment. The steering angles in Figure 5.44 are much noisier than the SI ones in Figure 5.16. We believe the DI voltages and steering angles would look similar to their SI counterparts if the control gains were retuned for the DI dynamics.

5.4.7 Double Integrator: Target Interception

The following experiment utilized the same trial conditions as the experiment in Section 5.4.3 with the exception that the target velocity was assumed to be known by the vehicles as per the assumption in Section 3.4. The control law given by (3.19), (3.20), (3.21), and ( 5.12 ) was implemented with the addition of the integral feedback term (as was done in (5.20)) and with control gains set to , , , , and . Notice that the control gain that multiplies the discontinuous term sgn in (3.19) was set to zero in the experiment. During tuning trials, it was observed that larger values of caused the UGVs to jitter; thus, the variable structure‐type term was turned off. However, this term would be beneficial if the target motion was more aggressive since it would enable faster reactions from the chasing vehicles.

Figure 5.45 shows the trajectories of the UGVs and virtual target. Figures 5.46 and 5.47 display the inter‐vehicle distance errors and target interception error, respectively. These results show that while the desired formation was acquired relatively quickly (by 10 s), the target was only intercepted by UGV 2 after 25 s. The settling time of the target interception error may be explained by the behavior of the velocity error in Figure 5.48, which also has a settling time of 25 s. This settling time could be shorten by increasing the integral feedback gain from its original value of 0.5 (recall that consistency of control gain values between the various experiments was used to facilitate their comparison). In contrast to the SI target interception error, which exhibited some small steady‐state oscillations about zero (see Figure 5.25), the DI target interception error converged to zero with virtually no steady‐state offset. This is likely due to the target velocity being known to the DI‐based control whereas it was estimated in the SI‐based control. As for the inter‐vehicle distance errors, the SI and DI controllers yielded similar steady‐state values within cm.

The voltages and steering angles are given in Figures 5.49 and 5.50, respectively. In this experiment, the voltages were restricted to of the maximum battery voltage (approximately V) to limit the vehicles' velocity to within manageable limits ( m/s) so that the vehicles stayed within the testing area. As such, the voltages saturated during the first 3 s of the experiment since the transient period is usually the most demanding from the control energy perspective. The DI steering angles are noticeably noisier than the SI ones in Figure 5.27, although one can still distinguish the 10 Hz oscillation caused by the sinusoidal variation of the target heading angle.

5.4.8 Double Integrator: Dynamic Formation

This experiment was conducted using the same formation objectives as Section 5.4.4. The integral feedback term was also included (3.8), as in the previous two DI experiments with control gains set to , , and .

The results of the DI‐based dynamic formation controller are shown in Figures 5.51 to 5.56 along with comparisons with the SI‐based control. Unlike in the SI experiment, UGV 1 did not strictly follow the desired translation and exhibited small oscillations in its heading angle. This is also evidenced by the oscillations of its steering angle seen in Figure 5.56 in comparison to the other UGVs. The settling time and overshoot of the distance errors were smaller for the DI control, although the response was less damped. The DI control also produced smaller steady‐state errors, as can be seen by comparing the insets of Figures 5.53 and 5.29. A comparison of the RMS value of the errors given by (5.21) resulted in for the DI controller and 0.929 for the SI controller. From the drive‐wheel motor voltages in Figure 5.55, we can see that the vehicles had more irregular (forward‐reverse) motions than when the SI‐based control was used (see Figure 5.30 for comparison). Finally, the UGV steering angles of the DI‐based control had larger high‐frequency variations than those of the SI case.

5.4.9 Holonomic Dynamics: Formation Acquisition

Finally, we tested the adaptive controller presented in Section 4.3.2, which is based on the holonomic dynamics of the UGVs. The hand position of each UGV was set to m, , which is the same value as in (5.9). The torque‐level control law in (4.30a) was augmented with the integral feedback term . The control and adaptation gains in (4.30) and (2.15) were chosen as , , , and . In addition, the 18 components of the parameter estimate vector were initialized to zero.

The motion of the UGVs from the initial formation until the desired formation was acquired is shown in Figure 5.57. From Figure 5.58, we can see that the desired formation was reached within 5 s. The comparison with the SI‐ and DI‐based formation acquisition controllers in Figure 5.59 shows that the response of the adaptive controller was the most damped. In fact, the RMS value of the inter‐vehicle distance error given in ( 5.21 ) for the three controllers resulted in 0.835 (holonomic), 0.688 (SI), and 0.629 (DI). The higher value of the holonomic‐based controller is obviously due to its slower response. However, it is possible that the response can be sped up in a critically damped fashion by additional tuning of the control gains. That is, gain tuning is facilitated in this case by the fact that the adaptive controller—given its certainty equivalence‐like structure—is quasi‐feedback linearizing the hand dynamics (4.19).

For illustration purposes, the six parameter estimates of UGV 1 only are displayed in Figure 5.60. One can observe that the parameter estimates—most notably and —are drifting. The drift is caused by the adaptation law of (4.30b) being primarily driven by the velocity error signal which does not converge exactly to zero, as shown in Figure 5.61. Since in the pure formation acquisition problem the vehicles operate near zero velocity at steady state, they are subject to noticeable static friction affects. Although the low‐level controller attempts to compensate for static friction, the vehicles still tend to jitter about the desired setpoint, as can be seen from the steady‐state oscillations in the inset of Figure 5.58. As a result, these oscillations cause the parameter estimates to continue to update. If the experiment were conducted with formation maneuvering, one may not observe the parameter estimate drift. Furthermore, one can always modify the adaptation law with one of the numerous methods for bounding the parameter estimates (e.g., dead zone, ‐modification, ‐modification, and projection operators 123, 124).

The drive‐wheel voltages and steering angles commands are given in Figures 5.62 and 5.63, respectively. As expected, the voltages are constantly switching after the formation is acquired, albeit at a higher frequency than in the SI and DI formation acquisition experiments. This higher frequency can also be observed in the steering angles. This behavior means that the UGVs experienced a stronger jitter about the desired formation with the adaptive controller.

5.4.10 Summary

The above experimental results provide a baseline for understanding the performance of graph rigidity‐based formation controllers. Naturally, the results can be improved by employing better hardware (e.g., more powerful processor, higher resolution optical encoder, wireless router with higher baud rate), more exhaustive control gain tuning, and a more accurate vehicle model. Nevertheless, a few conclusions can be drawn from the experimental results.

First, the desired formation is easier to obtain when the vehicles are moving (formation maneuvering problem) because static friction is a prominent factor in the stationary case (pure formation acquisition problem). Therefore, adequate static friction compensation is imperative for high‐precision applications of UGV formations. Despite the control gains in this experimental study being tuned for the SI‐based controllers, the other controllers produced comparable performance. Therefore, there is greater room for improvement in the performance of the DI‐ and holonomic dynamics‐based controllers by better gain tuning. It should be noted that although the control gains have a PID‐like interpretation their tuning is not straightforward since the closed‐loop system is nonlinear.

Setpoint control problems, such as the formation acquisition problem, can lead to poor transient behavior due to the initial jump of step‐like command. One way of improving the transient performance of the formation acquisition controllers is to set the desired distance to the time‐varying function, where is a design parameter and is the final desired distance. This function creates a smooth step function with zero initial error.

Finally, independent of what control technique is employed, the importance of a reliable communication system for high‐performance formation control cannot be overstated.