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Chapter 8: Path Planning

The cart vision system has an extremely simple minded approach to the world. It models everything it sees as clusters of points. If enough such points are found on each nearby object, this model is adequate for planning a non-colliding path to a destination.

The features in the cart's 3D world model can be thought of as fuzzy ellipsoids, whose dimensions reflect the program's uncertainty of their position. Repeated applications of the interest operator as the cart moves cause virtually all visible objects to be become modelled as clusters of overlapping ellipsoids.

To simplify the problem, the ellipsoids are approximated by spheres. Those spheres sufficiently above the floor and below the cart's maximum height are projected on the floor as circles. The cart itself is modelled as a 3 meter circle. The path finding problem then becomes one of maneuvering the cart's 3 meter circle between the (usually smaller) circles of the potential obstacles to a desired location.

It is convenient (and equivalent) to conceptually shrink the cart to a point, and add its radius to each and every obstacle. An optimum path in this environment will consist of either a straight run between start and finish, or a series of tangential segments between the circles and contacting arcs (imagine loosely laying a string from start to finish between the circles, then pulling it tight).

Superficially, the problem seems to be one of finding the shortest path in a graph of connected vertices. The tangential segments are the edges of the graph, the obstacles, along with the destination and source, are the vertices. There are algorithms (essentially breadth first searches, that repeatedly extend the shortest path to any destination encountered) which, given the graph, can find the desired path in $O(n^2)$ time, where $n$ is the number of vertices. On closer inspection, a few complications arise when we try to apply such an algorithm.

There are four possible paths between each pair of obstacles (Figure 8.1). because each tangent can approach clockwise or counterclockwise. Expanding each obstacle into two distinct vertices, one for clockwise circumnavigations, the other for counterclockwise paths, handles this.

Figure 8.1: The four tangential paths between circular obstacles A and B

Setting up the distance matrix of the graph involves detecting which of the tangential paths are not allowed, because they blocked by other obstacles (such blocked paths are represented by infinite distances). There are $O(n^2)$ tangent paths between obstacle pairs. Determining whether each particular path is blocked involves examining at least a fraction of the other obstacles, a process that takes $O(n)$ time. Thus generating the distance graph, whether explicitly before running the shortest path algorithm, or implicitly within the algorithm itself, takes $O(n^3)$ time. With this consideration, the algorithm is $O(n^3)$.

The obstacles are not dimensionless points. Arriving on one tangent and leaving on another also involves travel on the circular arc between the tangents. Furthermore, paths arriving at an obstacle tangentially from different places do not end up at the same place. Our circular obstacles occupy a finite amount of space. Both these considerations can be handled by noting that the there are only a finite number of tangent points around each obstacle we need consider, and these tangent points are dimensionless.

Each obstacle develops four tangent points because of the existence of every other obstacle. A path problem with $n$ circular obstacles can thus be translated exactly into a shortest path in graph problem with $4n(n-1)$ vertices, each edge in the graph corresponding to a tangent between two obstacles plus the arc leading from one end of the tangent path to the beginning of another one. The solution time thus appears to grow to $O(n^4)$. Fundamentally, this is correct, but significant shortcuts are possible.

Figure 8.2: The shortest path finder's solution to a randomly constructed problem. The route is from the lower left corner to the uper right. The numbered circles are the obstacles, the wiggly line is the solution.

Figure 8.3: Another path finder solution

Figure 8.4: A case where the approximate and exact methods differed. Top diagram is the exact solution, bottom one is the approximate algorithm's guess.

The distance matrix for the tangent points is extremely sparse. In our possible solution space, each tangent point leading from an obstacle connects to only about $2n$ others, out of the $4n(n-1)$ possible. This fact can be used to reduce the amount of work from $O(n^4)$ to about $O(n^3)$. Appendix 8 gives the details.

The algorithm just outlined finds the guaranteed shortest obstacle avoiding path from start to finish. It is rather expensive in time, and especially in space. It requires several two dimensional arrays of size $n$ by $n$. The number of obstacles sometimes grows to be about 100. Because both storage and running time needed conservation, the final version of the cart program used a simplified, and considerably cheaper, approximation to this approach.

The simplified program, also described in greater detail in Appendix 8, does not distinguish between different tangent points arriving at a single obstacle. Instead of a very sparse distance matrix of size $4n(n-1)$ squared, it deals with a dense matrix of dimension $2n$ by $2n$. Many of the arrays that were of size $n^2$ in the full algorithm are only of dimension $n$ in the cheap version. The arc lengths for travel between tangents are added into the computed distances, but sometimes too late to affect the search. If the obstacles were all of zero radius, this simple algorithm would still give an exact solution. As obstacle size grows, so does the probability of non-optimal solutions.

In randomly generated test cases containing about fifty typical obstacles, the approximation finds the best solution about 90% of the time. In the other cases it produces solutions only slightly longer.

A few other considerations are essential in the path planning. The charted routes consist of straight lines connected by tangent arcs, and are thus plausible paths for the cart, which steers like an automobile. This plausibility is not necessarily true of the start of the planned route, which, as presented thus far, does not take the initial heading of the cart into account. The plan could, for instance, include an initial segment going off 90° from the direction in which the cart points, and thus be impossible to execute.

The current code handles this problem by including a pair of “phantom” obstacles along with the real perceived ones. The phantom obstacles have a radius equal to the cart's minimum steering radius, and are placed, in the planning process, on either side of the cart at such a distance that after their radius is augmented by the cart's radius (as happens for all the obstacles), they just touch the cart's centroid, and each other, with their common tangents being parallel to the direction of the cart's heading. They effectively block the area made inaccessible to the cart by its maneuverability limitations.

In the current program the ground plane, necessary to decide which features are obstacles, and which are not, is defined a priori, from the known height of the cart camera above the floor, and the angle of the camera with respect to the horizontal (measured before a run by a protractor/level). Because the program runs so slowly that the longest feasible travel distance is about 20 meters, this is adequate for now. In later, future, versions the cart should dynamically update its ground plane orientation model by observing its own motion as it drives forward. The endpoints of each meter-long lurch define a straight line that is parallel to the local ground. The vector component of the ground plane model in the direction of the lurch can be tilted to match the observed cart motion, while the component perpendicular to that is be left unchanged. After moving in two non-colinear lurches, all ground-plane orientation parameters would be updated. This process would allow the cart to keep its sanity while traversing hilly terrain. Because the motion determination has short term inaccuracies, the tilt model should be updated only fractionally at each move, in the manner of exponential smoothing.

Path Execution

After the path to the destination has been chosen, a portion of it must be implemented as steering and motor commands and transmitted to the cart. The control system is primitive. The drive motor and steering motors may be turned on and off at any time, but there exists no means to accurately determine just how fast or how far they have gone. The current program makes the best of this bad situation by incorporating a model of the cart that mimics, as accurately as possible, the cart's actual behavior. Under good conditions, as accurately as possible means about 20%; the cart is not very repeatable, and is affected by ground slope and texture, battery voltage, and other less obvious externals.

Figure 8.5: An example of the simulator's behavior. The diagram is a plan view of the path executer's world model; the grid cells are one meter on a side. The cart's starting position and final destination and orientation are indicated by arrows. The two large circles, only portions of which are visible, represent the analytic two-arc path. It goes from Start through the tangent of the two circles to Finish. The heavier paths between the two points represent the iterations of the simulator as its parameters were adjusted to compensate for the cart's dynamic response.

The path executing routine begins by excising the first 0.75 meters of the planned path. This distance was chosen as a compromise between average cart velocity, and continuity between picture sets. If the cart moves too far between picture digitizing sessions, the picture will change too much for reliable correlations. This is especially true if the cart turns (steers) as it moves. The image seen by the camera then pans across the field of view. The cart has a wide angle lens that covers 60° horizontally. The 0.75 meters, combined with the turning radius limit (5 meters) of the cart results in a maximum shift in the field of view of 15°, one quarter of the entire image.

This 0.75 meter segment can't be followed precisely, in general, because of dynamic limits in the cart motion. The cart can steer reliably only when it is driving. It takes a finite time for the steering motor to operate. When the drive motors are energized the robot takes a while to accelerate to its terminal velocity, and it coasts for a half meter when the motors are turned off. These complications were too difficult to model in the obstacle path planning.

Instead the program examines the cart's position and orientation at the end of the desired 0.75 meter lurch, relative to the starting position and orientation. The displacement is characterized by three parameters; displacement forward, displacement to the right and change in heading. In closed form the program computes a path that will accomplish this movement in two arcs of equal radius, but different lengths. The resulting trajectory has a general “S” shape. This closed form has three parameters; the radius of the two arcs, the distance along the first arc and the distance along the second, just the right number for a constrained solution of the desired displacement.

Making the arcs of equal radius minimizes the curvature of the planned path, a desirable goal for a vehicle that steers slowly (as well as unreliably). Even with minimized curvature, the two-arc path can only be approximated, since the steering takes a finite amount of time, during which the robot must be rolling.

I was unable to find a closed form expressing the result of simultaneous steering and driving, so the program relys on a simulation. The on and off times for the drive motor necessary to cause the cart to cover the required distance are computed analytically, as are the steering motor on times necessary to set the cart turning with the correct radii. These timings are then fed to the simulator and the final position of the cart is examined. Because the steering was not instantaneous, the simulated path usually turns out to be less curvy than the requested one. The difference between the simulated final position and orientation and the desired one is used to generate a new input for the analytic solver (To clarify; if the simulation says the cart ends up one meter too far to the right, the next iteration will request a position one meter leftward. This process works well when the results of the simulation react nearly linearly to the initial requests). About five iterations of this step are usually sufficient to find an adequate command sequence. This sequence is then transmitted, and the cart moves, more or less as simulated.

Except for the endpoints, the path generated in this way differs, in general, from the one produced by the obstacle avoider algorithm. For 0.75 meter lurches, however, it stays within a few centimeters of it. The cart avoids each obstacle by a safety factor of about a half meter, so such inaccuracies can be tolerated. In any case, the mechanical precision of the cart's response is poor enough, and its seeing sparse enough, to require such a safety margin.

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