This optional companion to Chapter 4 connects its faster-cornering experiment with broader ideas about trajectory planning. A path describes where to travel. A speed profile describes how fast to travel along it. Together they determine motion over time.
Chapter 3 uses \(v=\sqrt{\mu gR}\) as a basic steady-cornering estimate. The R in that formula is the radius of the car's path, not automatically the radius stored in a track segment.
A driver may use the width of a bend to follow a shallower path, especially through a small change of direction. But its path must connect to the entry and exit, fit the road with room for the car, and respect the next bend. An outside track edge radius alone does not establish any of those facts.
Our steering chooses centerline targets, but the resulting driven path depends on lookahead, speed, steering response, and tyre forces. Increasing a radius inside the speed model does not make the car follow that radius.
One way to estimate a more generous corner radius is to start with the outside radius and enlarge it for small-angle bends. Accumulate the arcs of consecutive segments turning in the same direction, so splitting a bend into several pieces does not automatically make each piece a short turn.
Writing that idea explicitly, with road width w and accumulated angle A:
\[ f=\frac{A}{\pi/2},\qquad R_{\mathrm{estimate}}=\frac{R+w/2}{\sqrt{f}}.\]
The angle is normalized by a quarter-turn, so f is dimensionless. Small f increases the estimated radius and therefore the proposed speed. As the car approaches the end of a bend, a smaller remaining angle can encourage earlier acceleration. This is an empirical idea, not a derivation of an optimal path.
USE_SHORT_TURN_HEURISTIC in speed4.cpp selects experimentalRadius:
\[ R_{\mathrm{experiment}}= \min\left(1.5R,\frac{R+w/2}{\sqrt{\operatorname{clamp}(f,0.05,1)}}\right).\]
Both bounds are teaching choices. Neither proves that a feasible trajectory exists.
For R = 40 m, w = 10 m, and A = π/4, the unbounded estimate is about 63.64 m. The experiment limits it to 60 m. Without aerodynamics this can raise the local model speed by at most \(\sqrt{1.5}\approx1.225\) before the existing margin and speed cap. The same ratio is not a universal bound with downforce: the aerodynamic denominator also changes with radius. The absolute speed cap still applies.
The smaller-angle limit prevents division by zero, and the loop bound prevents an endless search on a circular list. The estimate still has substantial limitations: it groups bends by turn direction, not by equal radius; it does not check the available swept corridor; and it does not model how one corner's exit position affects the next corner's entry. The normal braking lookahead continues to check the speed targets of subsequent segments.
Experiment: keep the 8 m / 0.33 s steering settings and Chapter 3's speed settings, then compare the switch off and on. Record lap times, road-edge excursions, and damage on both CG Speedway number 1 and E-Track 2. Watch whether the higher corner-speed target leaves enough room to exit each bend and approach the next one.
Consider a simple position-based throttle filter: cut throttle when the car is more than a quarter of the road width away from the centerline on a straight or on the outside of a bend. Permit acceleration on the inside and bypass the filter at low speed. Could this restore stability after raising corner speed?
There are useful ideas here: position should influence driving, inside and outside have different geometry, and a slow car needs enough power to rejoin. There are also important limits:
In a ground-fixed view, inward tyre force changes the velocity direction. If there is too little inward force, the car turns less sharply than desired and can run wide. An outward “centrifugal force” belongs to a rotating-frame description; it is not an extra external force we can rely on to place the car back on the centerline.
Tutorial 4 therefore uses the signed road clearance of four corners and lowers the speed target on either side. It interpolates from the planner's target to at most 5 m/s across the last metre of clearance. The existing pedal controller then decides between acceleration, coasting, and braking. This preserves motion at low speed while allowing braking when lifting alone would be insufficient. It remains reactive, and its current-corner samples are not a swept-path proof.
Our implementation requests a nominal steering angle equal to the target's heading error α, then divides by the steering lock to form a command. It is easy to follow, but it has no wheelbase in the calculation.
A geometric pure-pursuit controller instead fits a circle through a target relative to a chosen vehicle reference point, usually the rear axle center. Let D be the straight-line distance from that reference point to the target, α its direction relative to the vehicle, and b the wheelbase. For an ideal no-slip bicycle model:
\[ \kappa=\frac{2\sin\alpha}{D},\qquad \delta=\arctan(b\kappa).\]
κ is signed curvature in 1/m and δ is the front-wheel steering angle. Notice that D is not the centerline lookahead L used to select the target. A complete implementation must also use a consistent axle reference, map δ to the actual steering controls, and consider saturation and dynamic slip.
This is a useful next experiment, but changing just the formula while leaving the reference point unexplained would obscure its assumptions. The supplied example uses the heading-error method explained in Chapter 4.
Rules such as “aim farther ahead when faster” or “allow a larger radius for a short bend” are quick to implement and easy to experiment with. Improvements on one track can expose problems on another. Test tight bends, alternating directions, narrow roads, and changes of grip rather than selecting only the track where a rule works best.
Rémi Coulom's K1999 driver is an example of geometric curvature smoothing. Such approaches adjust a sequence of path points within road boundaries to reduce abrupt or large curvature demands. They can provide a useful candidate racing line efficiently.
Geometry alone does not include the car's power, braking ability, downforce, or grip variation. A path with gentle curvature is not necessarily the fastest: it may be longer, compromise a critical exit, or place acceleration where the tyres are already working hard to turn.
A spline provides a smooth curve through or near chosen control points. A simple construction starts with a few promising points, checks where the curve leaves the road, and inserts or adjusts points there. Smoothness is helpful, but a spline can overshoot between control points. Check the continuous curve or sufficiently bounded subdivisions, with room for the vehicle's footprint; checking only the control points misses the very problem being corrected.
A circular arc has constant curvature. Clothoids vary curvature linearly with distance, making them useful for transitions between straights and curves. Fitting arcs or clothoids within the road can produce understandable paths, but joining them requires attention to tangent direction, curvature continuity, available width, and the following turn. A collection of large circles that individually fit does not automatically form one drivable lap.
For a path parameterized by distance s and a positive speed profile v(s), lap time is:
\[ t_{\mathrm{lap}}=\int_0^{S}\frac{1}{v(s)}\,ds.\]
Minimizing this quantity requires constraints: road boundaries and vehicle footprint, lateral grip, longitudinal acceleration and braking, steering, and an appropriate vehicle model. Downforce, load transfer, or transient tyre behavior may matter depending on the intended accuracy. Modern numerical optimization can solve discretized versions, but the result depends on the model, resolution, initialization, and solver convergence. A local solution is not automatically a globally optimal racing line.
Computational difficulty depends on the exact planning problem. An inexpensive geometric candidate can be a useful starting point for a more demanding path-and-speed optimization with explicit vehicle constraints.
TORCS also provides a setting for learning steering, speed, or path decisions. Define what the agent observes, which controls it chooses, and how success is measured. A reward based only on progress can encourage cutting corners or accepting damage. Compare with this simple driver and evaluate on tracks and conditions not used for training. Report road departures, completed laps, damage, and consistency alongside lap time.
For a class project, comparing two modest, well-explained approaches on the same conditions is more informative than claiming an “optimal” driver without stating its assumptions.
Return to Chapter 4, or work through the geometry. You can also ask a question or share an experiment.
1.14.0 