Summer 2013 - Andrea Rizzi

INDEX

THE GENERAL PROBLEM

The goal of the project is to set up an experiment that proves the effectiveness of an algorithm for collision avoidance at intersections. Three cars run on a test-bed following three different paths that all intersect in a single point. Cars are controlled by an on-board computer. Cars positions are measured by six cameras on the ceiling. Cars speed can be measured by an encoder mounted on the rear axis.

The final experiment will have an autonomous vehicle and 2 human-driven ones. The experiment will last few minutes and it will be considered successful if the vehicles will never bump into each other, given few constraints on the path following performance (e.g. the distance of the vehicle from its path cannot be greater than 30cm) that will have to be satisfied by both the autonomous and the human-driven vehicles.

The algorithm requires the prediction of the time at which a car enters and exits the intersection. The algorithm is robust to noise, meaning that it takes into account the existence of disturbances that cannot be controlled, given that those disturbances are limited and monotonic with respect to the position along the path y. It is assumed that the input signal (i.e. the torque applied by the motor) is also monotonic with respect to y. The main tasks of this work are thus to:

• determine and limit the disturbances which make the prediction uncertain to a degree that make the experiment unfeasible;
• identify a model of the system that is suitable for the long term prediction that the algorithm requires.

CURRENT LAB PROBLEMS

• Cameras should be screwed to the metal support. Now every time they move, the whole calibration procedure must be done again. An extrinsic calibration (at least) should probably be performed again after this job.
• The two chargers close to the whiteboard seem to have some wiring problem. Sometimes the led is green while cars are connected (i.e. they're done charging) but it turns red when you move the wires.
• Sometimes the connection between computers and cars is lost. There are 2 main cause.
  1. There are 2 critical functions in the CPS code in main.cpp that occasionally takes up to 300-400ms in computer 0. This does not happen (or happens extremely rarely) with computers 1 and 2. I think the fault may simply be the computer's hardware because computer 0 is the slowest among the three. The two critical functions are flycaptureGrabImage2() and cvWaitKey(). The CPS currently prints a warning when one of this two functions takes more than 0.1s (the car's clock). I changed the CPS so that computer 1 now sends the data to cars so that the connection is always active, still this kind of delay makes cars tracked by computer 0 to receive out-of-date information.
  2. Sometimes car 2 becomes unreachable to ping tests while moving. This causes the time between two clock cycles to reach even 1-2 seconds. I never found the same problem with car 1 and 3. Most of the times this happens, there is no timeout of the recvfrom() function which receives the data from computers. This suggests that the problem resides in the on-board computer more than in the connection. The reason could be originated by both hardware (the motherboard of car 2 is an older and slower model with respect to car 1 and 3) and software (out-of-date operating system or some background program that takes too much resources).
• Car 2 cannot handle high engine voltages for a long time. After about 2 minutes at constant PWM 200 the car's speed dropped exponentially. This does not happen with cars 1 and 3. Substituting the h-bridge and the battery did not help. Anyway, a motor controller that is not very stressful does not cause troubles.
• The encoder of car 1 often does not give signal at the beginning when the car starts running. It starts working only after about 10 seconds.
• There is a point on the part of path8CL tracked by camera 4 (the one closest to the computers and the door) in which there are "jumps" of the measured position even if the tracking of the car does not pass from a camera to another. I am not sure if this is the problem of the camera itself or of the calibration that must be redone.

TROUBLESHOOTING

• Router It happened to me once that the computers kept connecting and disconnecting to the local network making impossible to estabilsh a connection between them and with the cars. When this happened, removing the internet cable from the router (the one that is plugged to the wall) fixed the problem.
• Motherboard You can test if a car's motherboard is working by connecting it to the screen and turning the car on. If it doesn't, usually the cause is the power supply unit (PSU). Try substituting that piece.
• Brainstem ca2 console displays the message "Host-Stem Link: DOWN". Sometimes the baudrate of the brainstem resets itself for no apparent reason, so first try to set back the baudrate as explained in section "Hardware Reset Procedure" in the main page of the lab wiki. If you cannot make it work, proceed with an hardware reset (described in the same section). Note that if you do not have to configure a brand new card, there is usually no need to load the tea scripts again.
• Engine When a car is particularly slow, check that h-bridge connections to the power and the motor are solid.
• Encoder gives no signal, check that the serial cable is connected in the proper way (you can find the h-bridge datasheet http://www.acroname.com/robotics/parts/S11-3A-EMF-HBRIDGE_datasheet.pdf and encoder datasheet in the electronic components catalog on the shelf).
• Steer is biased towards a direction. The steer cannot be set straight only mechanically. You must set the offset parameter in RoboCar_Lib.tea.

SOFTWARE

All the files (source code and data) are saved in the folder "Dropbox/Shared Andrea/", from now on simply "Andrea/". If you move the files remember to update the documentation here. It contains 4 subfolders:

• "Andrea/src": contains the C code divided in 3 folders
  • "Andrea/src/car_src": contains the code specific for the car, the ca2 program (See #CA2).
  • "Andrea/src/sim_src": contains the code of the simulator (See #SIMULATOR).
  • "Andrea/src/util": contains the code shared between the car and the simulator (See #C UTILITIES).
• "Andrea/script": contains all the python scripts that can be used to compile the simulator, identify parameters and debug the experiment (See #SCRIPTS).
• "Andrea/paths": contains the new paths definition files used by the cars which specify also the steer compensator directives and the path-dependent disturbances functions (See #PATH SPECIFICATION).
• "Andrea/data": various data gathered during experiments on the test-bed.

SUMMARY OF CHANGES

Note about CPS usage. AndreaCPS is not compatible to previous versions of ca2. Viceversa my ca2 is not compatible with previous versions of CPS. This is because the communication protocol between CPS and ca2 is now a little bit different. In general I would suggest using AndreaCPS instead of KevinCPS or LeoCPS because of the bug that involved the initial target detection. I did not check CPS versions that are previous to KevinCPS but it is likely that those versions do not have this bug since computer 1 and 2 do not send their information to computer 0 before sending them to cars.

ca2

• Grouped utility functions in a separate folder that is shared with the simulator. This makes the code more modular.
• Implemented a new target detection algorithm that allow the car to work with the path with lots of points.
• Steer controller has been reworked, the monodimensional speed has been substituted by the one read from the encoder, the PD has been calibrated and a path-dependent compensator has been added. The compensator can be configured through the path specification file (See #COMPENSATOR DIRECTIVES).
• Implemented a new filter for the heading measure to cancel the effect of leaps (See #HEADING FILTERING).
• Set the limit of steer input to (-100,100) since the curvature radius saturates above it.
• Implemented the online estimation of the engine gain parameter (fitness) (See #FITNESS ESTIMATION).

CPS

• Fixed a bug that affected the initial target dectection for cars that were not tracked by computer 0 and buffer overflow bug.
• Implemented linear error correction for the computation of camera position. This visibly improves the path following of cars. See section #CAMERA MEASUREMENT ERROR CORRECTION.
• The computer that sends the data to cars is now computer 1 (not the slower computer 0 anymore). This is easily configurable through the constant SENDER_COMP_NUM in CPS.h.
• The sender computer now knows which camera is currently tracking which car and this information is send to cars.

Communication protocol

• Now the car sends its personal data (encoder speed, estimated fitness, desired pwm, etc.) to CPS that forwards this data to all the cars. If the encoder of a car does not work you can set to 0 the constant USE_ENCODER in "Andrea/src/car_src/const_car.h" and the CPS will estimate the car speed from differentials which is currently not that accurate (See #CPS MEASURED 2D SPEED).
• Each car listen on a different port (base_port_number + car_number). This was required to make the simulator handle multiple cars and I extended it to the CPS for coherence.

CONFIGURATION

There are 4 files which contains basically all the constants in the code. These should be the ones to check whenever you want to configure something.

• "Andrea/src/car_src/const_car.h": contains the compilation options that activate/deactivate the features of the code in the car and few other constants specific for the car code such as the car number, the initial pwm and when it has to stop.
• "Andrea/src/sim_src/const_sim.h": this contains very few simulator-specific constants which include the real fitness parameters of the cars, the paths followed by each car and whether to apply the measurement noise during the simulation.
• "Andrea/src/sim_src/model.c": contains the amplitudes of the noise signals that are applied to the physical quantities by the simulator.
• "Andrea/src/util/constants.h": this basically contains all the other constants that range from network configuration to control parameters to physical constants.

The specific effect of each constants is documented in the code.

SIMULATOR

Implemented a simulator in C for Linux. This is integrated with the code used on cars (ca2). It simulates the cars dynamics and the behavior of the Camera Positioning Systems (CPS). Besides being very useful to verify the system model, it makes easy to test changes to code and the controller before applying those changes to cars.

SCRIPTS

Implemented a script to visually debug cars behavior. It works both with the simulator and test-bed experiments data.

DISTURBANCES ANALYSIS

In this section a quantitative analysis of the disturbances in the system is attempted. In order to deal in an easier way with the assumption of monotonicity and with the coupling between the steer and the motor input, the model is linearized and the lateral and the longitudinal components are separated. The coupling effect is seen as a disturbance. In the following the coordinate system will be relative to the path as represented in Figure coord_sys.jpg. In other words, y represents the position along the path while x the lateral distance from the path.

LONGITUDINAL MODEL

A linear model of the car along the longitudinal component of the path is represented in figure long_block.jpg. Symbols represent

• PWM: the motor input signal.
• const: a constant term representing friction.
• W(s): transfer function between PWM signal and speed.
• v: 2D speed of the car.
• y': monodimensional speed obtained by projecting the 2D speed of the car on the path.
• y: monodimensional position of the car obtained by projecting its 2D position on the path.
• ym: monodimensional position of the car measured by cameras.
• Da: actuator disturbances on the control variable.
• Dslope: the test-bed is not perfectly flat but it is slightly inclined. This has been found to considerably affect the speed caused by slopes of the test-bed.
• Dslip: disturbances on the speed caused by the coupling with the lateral dynamics of the car (i.e. with the steer).
• Dproj: projecting the 2D speed on the path is modeled here as a disturbance.
• Djump: the path followed by cars is composed by a polygonal chain. This means that when the car goes from a segment to the next one, there is an interval of time during which the car position is projected on the same exact point of the path. For example, in Figure Djump.png the car goes through the trajectory A-B-C-D. While in the arc B-C, the vehicle position is projected on the path always in the same point P. Similar considerations can be done when a car cuts a corner. This disturbance is modeled with a negative (or positive in case the vehicle cut the corner) step that corrects the position each time the vehicle changes the segment of a path. Essentially, this is a measure of the difference between the length of the nominal path and the lenght of the actualy trajectory that the car follows.
• Dm: measurement error of the cameras.

Dslope and Dslip
Dslope and Dslip are path dependent. I plan to measure them experimentally and treat them as a path-dependent known disturbance.

Djump
The path fig8 is composed by 649 control points for a total length of 9465.53 millimeters. The angle difference between two consecutive segments is delta = 0.9 degrees. We set the maximum distance of the car from the path to be xmax = 30cm (which is realistic with the current steer control performances). Assuming the car x coordinate (distance from the path) to be constant at xmax, we can compute Djump = 649 * delta * xmax = 3058.34mm which is the 32.31% of the path length.

DISTRUBANCES ANALYSIS CONCLUSION

It is clear that the steer control performance affects considerably the behavior of the vehicle. Consequently lots of efforts should be done to improve it. Besides the designing of the controller, the ability of the car to stay close to the path is affected by the measurement error of cameras. Thus, limiting these disturbances should follow two directions:

• steer controller design;
• camera measurement error correction.

Work in Progress. The complete analysis still have to be performed. Still, the partial conclusions I found are driving the research. This led to the camera measurement connection and it is currently focusing my attention on the implementation of a new steer controller.

CAR MODEL

STEER INPUT - CURVATURE RADIUS RELATIONSHIP

Model
If one considers the car to have an ideal lateral dynamics then the vehicle can be represented as in Figure car.png, where w is the wheelbase and delta is the steer angle. Actually the dynamics is affected by some slip angles that I am going to assume negligible for the purpose of this model. I am interested in the curvature radius of the center of the car (i.e. R) because that is the position tracked by the CPS. I assume also that the steer angle delta is proportional to the steer input or, in other words, that delta = c * u, where c is the steer factor and u is the steer input which belongs to the interval (-100, 100).

With some simple trigonometry one can obtain
Rr = w / tan(c * u)
which means that
R = sqrt(Rr^2 + w^2/4) = sqrt(w^2 / tan(c * u) + w^2 / 4)

The wheelbase can be directly measured on car. The only unknown parameter of the model is the steer factor c.

Parameter identification
I run car 1 on circles with constant steer input and I manually measured the diameter of the circle it went through. Results are reported in the table below. All measures are reported in millimiters (with the exception of the input of course). Measured diameters for right and left turns is separated. The error in the table is the difference between the radius (not the diameter) predicted by the model and the real radius.

The steer factor was computed in order to minimize the average error which lead to the value c = 0.2116466582.

Model validation
The model has been validated with cars 2 and 3. Data is reported in the table below.

CAR 2

Average radius error: 70.57

CAR 3

Average radius error: -58.17

Notes

• It should be noted that the reported error refers to curvature radius not diameter. This means that when car 2 completes a semicircle, on average the model mispredicts its position by 7.057cm * 2 = 14.114cm. According to the partial data gathered, the misprediction can be up to 36cm. Still, it is not required for the model to be 100% correct. Indeed the main idea behind the steer controller is to associate a reference steer input given by the identified model with a PID or a PI that corrects its error over time. Considering that it takes about 4 seconds for the car with steer input 50 and motor input 150 to complete the semicircle, the PID will have 4 seconds to correct an error of 36cm.
• The trajectory followed by the car with constant steer is not a perfect circle but more similar to an ellipse with very low eccentricity.
• After some tests, it seems that the curvature radius is not affected by the motor input.
• The curvature radius of car 1 saturates for input above 100. Signals 110 and 120 gives exactly the same curvature radius. For cars 2 and 3 this happens even before at input 90.
• The current diameter of the curve part of path fig8 is about 148cm. This means that car 1 and 2 can follow it while car 3 cannot.

EFFECT OF PWM AND STEER INPUT ON THE SPEED DYNAMICS

I made car1 run on circles for 50 seconds with fixed PWM and steer input for a total of 16 runs. Every run was performed starting from the same battery voltage of 16.7V. The full data gathered and the detailed description of how the experiments were performed can be found in my folder on Dropbox in "../backup/data/circle_7-27-2013". All the images below are obtained by filtering the encoder signal with a moving average window to discard the large part of the noise. The oscillation of the speed is largely due to the fact that the test-bed is not perfectly flat but inclined in some areas.

Figure pwm.png shows the speed of the car obtained by keep the steer constant and varying the PWM. The relationship between PWM and velocity is quite linear. For some reason, when the steer input is high, the speed observed with PWM 140 is slightly lower than I would expect. On the other hand, the steer effect seems to be a little bit more complicated (see steer.png). Velocities observed for steer 92 and 120 are always very close. This may be due to the fact that curvature radius for the two steer input are very similar (the curvature radius is not linear with respect to the steer signal). Still the speed is not linear w.r.t. the curvature radius because the discrepancy between the speed for steer 36 and 64 (which I measured to have respectively a curvature radius of roughly 200cm and 100cm) tends to be reduced by decreasing the PWM.

As a side note, the low frequency oscillations should be noted. These oscillation are caused by the fact that the testbed is not completely flat, there are slight slopes that can be easily observed with a spirit level.

Work in Progress. These experiments were performed before the effect of the power filter capacitor on speed was clear and they will likely be done again for a longer time in order to get useful data.

EFFECT OF THE BATTERY AND THE POWER FILTER CAPACITOR ON THE SPEED DYNAMICS

I made an investigation on the effect that the battery has on the car speed. Data has been acquired by running car 1 on a circle with constant steer and engine input. The steer has always been set to -64. The starting position of the car has always been about (3000, 3600) in global coordinates (this if you want to make considerations connected to the test-bed slope). The voltage of the battery was measured both before turning on the car (while off) and just before run ca2 (while on). The full data gathered and the detailed description of how the experiments were performed can be found in my folder on Dropbox in "../backup/data/battery_7-30-2013".

• Experiment 1: the initial voltage was kept constant while varying the pwm signal. The car is run twice in a row for 250 seconds from a starting voltage of 16.8V (turned off). See figure experiment1.png. The speed always reaches a peak and then decrease to a steady state value. The second run starts more or less from the steady state value. The PWM does not seem to affect neither the time constant nor the steady state value that is more or less always 200 mm/s below the peak.
• Experiment 2: the pwm was kept constant while varying the initial voltage. The car is run only once. See figure experiment2.png. From starting voltage 16.9V to 16.5V the speed has basically the same trend. The steady state value and the constant time becomes lower below 16.4V.
• Experiment 3: the car is run 3 times in a row to check if the third time the speed dynamics changes again. See experiment 3.png. The third time the car has the same behavior of the previous run.
• Experiment 5: the car is left turned on for 25 minutes before running to check if the capacitor gets discharged even if the car is not running. See experiment 5.png. The car speed is not affected.
• Experiment 6: the car is run for 200 seconds, then it is left on for other 200 seconds without running and then run for other 200 seconds. See experiment 6.png. The car speed has the same peak as the previous run.

The behavior dynamic of the car speed is likely caused by the capacitor that filter the power source. When the car is not running the circuit is in steady state condition. When it runs the equilibrium moves and the capacitor slowly discharges. The same behavior was confirmed by experiments with both cars 2 and 3.

CPS

CAMERA MEASUREMENT ERROR CORRECTION

Problem
The position of the car on the test-bed computed by the CPS is affected by a considerable error. I made some manual measurement of this error by finding the real position with the measuring tape and checking the computed position on the CPS and I found it to be be up to 25cm. Moreover, when the tracking of a car pass from a camera to another, the global coordinates "leap" because the position error in the transition point is different for the two cameras. From the experiments on the path fig8, this leap can be up to 30cm. This negatively affects the ability of the car to accurately follow a path which is a crucial aspect in limiting disturbances.

I made some investigation on the trend of this error. The results are shown in error_trend.png. In the figure, "x_loc" indicate the x coordinate in pixels in the local coordinate system of the camera (i.e. the horizontal one), while "x_glob" is the x coordinate in cm in the global coordinate system. Remember that the axises of the local and the global coordinate systems are inverted. Data where gathered for two cameras.

Solution
I decided to apply a linear correction to the computed global coordinates error along both direction. The correction is thus applied in the form err = a*x_loc*y_loc + b*x_loc + c*y_loc + d. The parameters a, b, c, d are computed by the CPS on start by loading a file where the real global coordinates of four points must be saved. This is the procedure that must be followed to configure this file. This procedure must be repeated for each camera. It is calibration-independent, meaning that you do not have to repeat it if you have to perform a new instrinsic/extrinsic calibration, but if the camera is moved, the procedure must be done again.

1. Set the variable RECORD_OBJECT_DATA = 1 in the file CPS.h in the computer responsible for the camera that you want to configure and compile it. For information about compiling the CPS, check the original lab documentation.
2. Run CPS.exe. The RECORD_OBJECT_DATA mode was designed to take pictures of the cars symbols so it will ask you the car number and the section number, just put a negative number. The only thing you must insert correctly is the camera number that you want to configure.
3. Determine the four points to record. Considering the error trend, the four points should be chosen to form the broadest rectangle of interest, that is the rectangle with the biggest area contained in the camera view where the car can be tracked. To clarify, an example of how to choose the points is found in point_selection.png. Walls, obstacles and the end of the sections limit the rectangle. The console gives you information about the pixel coordinates of the top left corner of the small squared boxes, in the figure it is located close to (x2, y2). You can move that box with keys "a", "s", "d" and "w". Write down the pixel coordinates of those four points.
4. Now measure the global position of those four points with a measure tape and record them. You should now have written down 4 pixel coordinates (x1, y1, x2, y2) and 8 global coordinates, 2 for each point of the rectangle. Pay attention when writing down the global coordinates. Since the x and y axises of the local and the global coordinate system are inverted, it is easy to make confusion.
5. Open the folder "Desktop/camera_programs/Andrea CPS/calib_data". There 5 files called "error_camx.txt", where "x" is the number of the camera. Open the one that refers to the camera you are configuring. The format of the file is very easy to understand and consistent with this explanation. Write there the pixel coordinates of x1, y1, x2, y2 and the global coordinates of those points.
6. Now you can set RECORD_OBJECT_DATA to 0 and you are good to go.

Results
The error correction gave a visible improvement to the steer control performance. The car stays much closer to the given path. The error in camera 5 and 2 is constantly below 10cm and rarely above 5cm. Moreover the measured position "leaps" are reduced. I was not able to measure a leap above 15cm.

HEADING FILTERING

Leaps affect the measured heading too since it is computed from positions. I have thus implemented a model-based filter to eliminate heading jumps. The filter is applied in the car, not in the CPS because it needs the speed from the encoder. In short, the filter detects when a leap occurs and use the model heading instead of using the measurement when that happens. The algorithm computes the following quantities:

measured_heading = arctan((curr_x2 - prev_x2) / (curr_x1 - prev_x1))
This is the measured heading which is affected by leaps. The CPS actually computes and sends this but it makes some filtering first. When jumps are very small, this filtering causes the measured heading to become wrong but not enough to discard it. For consistency reasons then we keep the newly computed measured heading instead of the CPS one.

model_heading = prev_speed * dT / R = 2 * prev_speed * tan(c*prev_steer) * dT / (w * sqrt(4 * tan^2(c*prev_steer)))
Where R is the curvature radius obtained from the model described in section STEER INPUT - CURVATURE RADIUS RELATIONSHIP, c is the steer factor and w is the wheelbase of the car. This is the current heading according to the model.

measured_Dpos = sqrt((prev_x1 - curr_x1)^2 + (prev_x2 - curr_x2)^2)
The distance covered measured from the previous step.

model_Dpos = prev_speed * dT
The distance that the car should have covered. The algorithm is the following:

1. if |measure_heading - model_heading| > 30degrees
   1. keep model_heading
2. else if |measure_heading - model_heading| > 20degrees and |measure_Dpos - model_Dpos| > 0.35 * model_Dpos // this is the uncertain case
   1. keep model_heading
3. else
   1. keep measured_heading

Some leaps occurs more or less in the direction of the car but they can be still detected because the car covers a much longer (or shorter) distance than it should. The second if-statement detects this kind of leaps. Thresholds have been determined empirically. You can see this filter as a simple camera change detector that applies a Kalman filter with time-varying error covariance. The measurement error covariance is 0 when camera change is not detected (i.e. the Kalman filter keeps the measurement) and non-zero otherwise. On the contrary, the a priori estimate error covariance is 0 when camera change is detected (i.e. the Kalman filter keeps the model) and non-zero otherwise.

Figure Dheading.png shows Dheading = |filtered_heading - previous_heading| in time. As you can see basically all the leaps caused by the change of the tracking camera are removed. This improved the path following performances.

CPS MEASURED 2D SPEED

Work In Progress. The issues with CPS measured 2D speed detected in this testing are still unresolved. This is not currently a priority since all the encoders are working and can be used for the model identification. These issues will be deepen in case the 2D speed will actually be used for the prediction.

CPS now compute from a difference of position the 2D speed of cars. The reason I implemented this is because the encoder of car 2 (which is now fixed) was not working and I needed to measure its speed. I made a test to see if this new measurement was reliable. I run car2 with fixed steer and PWM in a circle whose diameter I manually measured to be about 200cm. The car completed 5 circles.

Results are in figure speed_cps_encoder.png. The left one is the raw speed, in the right one peaks are removed and the CPS speed is filtered with a average filter window (the window width is 5). The manually computed average speed is 835.66 mm/s, the encoder average speed is 857.69 mm/s and the cps average speed is 895.65 mm/s. In order for the encoder to be the real average speed, the actual diameter should be 5.4 cm greater. In case of the CPS one it should be 14.4cm greater which is less likely.

Few observations:

• peaks in CPS speed are likely to be caused by position leaps between cameras;
• CPS speed average is above the one measured with the encoder.

STEER CONTROLLER

LINEAR MODEL

The linear model used for the steer controller design is described by the block diagram in Figure lat_block.jpg. Symbols represent

• U(s): desired lateral distance from the path which is always 0.
• C(s) = c / s: constant compensation term of the controller computed by using the model steer input-curvature radius.
• M(s) = m / s: constant term that accounts for the error of the model used to determine the compensation.
• D(s) = d / s^2: disturbance term that models the curve in the path.
• w: the wheelbase.
• alpha: the steer factor.
• delta: steer input in (-100, 100).
• X(s): controlled variable, the lateral distance from the path.

The Laplace transform of the system is

X(s) = (-v0 * d * w + alpha * v0^2 * c + alpha * v0^2 * m) / (s * (w * s^2 + alpha * v0^2 * k2 * s + alpha * v0^2 * k1))

If we set c to cancel the effect of d, then the error is caused by m. By applying the final value theorem and cancel the effect of d with c, one obtains

lim x(t) = m / k2 for t -> inf

Thus the final error depends only on k2 as reasonable to expect. Recalling that m must be a steer input, it is easy to compute the maximum value of m. By looking at the maximum error in the validation data for the #STEER INPUT - CURVATURE RADIUS RELATIONSHIP I obtained m = 13.3045. For the derivation of the model check formulae.pdf. The model does not take into account the measurements disturbances (leaps included) and it considers negligible disturbances associated with the actuator, projection on the path and slip angles.

CONTROLLER CALIBRATION

The controller used is a PD in the form

delta = k1 * x' + k2 * x = k1 * (beta - gamma) + k2 * x

where beta and gamma are respectively the direction of the car and of the path in radians. The system has a pole in 0 and a pair of complex poles for

|k1| < +- sqrt(4 * w * k2 / (alpha * v0^2)

The pair of poles are

p = - alpha * v0^2 * k1 / 2w +- i * sqrt(4w * alpha * v0^2 * k_2 - alpha^2 * v_0^4 * k1^2) / 2w

We want the final error ef, the settling time ts and the frequency of oscillation wd to be low or, in other words, we want to keep low the quantities ts = -4.6 / Re(p) and wd = Im(p). With k1 = k2 = 0.3 and v0 = 800 we obtain ef = 44.34mm, ts = 3.37s and wd = 0.9314.

RESULTS

Figure distgamma.png shows the measured distance from the path and the difference between the car and the path direction (gamma). The controller keeps the error of the distance from the path below 10cm and gamma below 10 degrees. This does not happen only when leaps occur because of the change of the tracking camera. The (signed) average error during a curve is -3.57mm for the distance and -0.07 degrees for gamma which is very good for the predictor. The average squared error is 49.64mm for the distance and 5.83 degrees for gamma.

Figure distold.png shows the error of the distance with the old controller instead. In curve the average error was about 180mm. It should be noted that this is the distance measured with the cameras which is affected by an error.

PATH SPECIFICATION

COMPENSATOR DIRECTIVES

In order to apply the compensation, the controller must know the curvature radius of the path. You can specify it by adding to the directive to the path definition file. For example, in fig8CL.txt the first two lines are now

1. 1 648 750.07285
2. 651 1298 -750.07285

which means that the curvature radius of the path from target 1 to target 648 is 750.07285mm, while from target 651 to 1298 is -750.07285. The curvature radius specified must be positive if the car must turn right, negative to make the vehicle turn left.