M5-T3 — Curvature, Velocity and Acceleration Limits

Learning objectives

After this lesson, you should be able to:

• Compute the curvature of a sampled path and the speed limit it implies.
• Add a wheel-speed limit for a differential-drive robot and say which limit wins.
• Build a profile with a forward pass and a backward pass, and stop at the end of the path.
• Compute braking distance and the time to drive a profile.

The intuition

Driving a bend you do three things. You slow down for it before you reach it. You go through at a speed the car, and your passengers, can take. You speed up again when it opens out. A speed profile does exactly this, in two stages: first decide the highest speed allowed at every point on its own, then repair the profile so that the robot can actually get from one speed to the next.

The second step is where most mistakes are. If Scout needs to be at 0.8 m/s at a bend, it must start braking earlier. And if it starts from rest, it cannot be at 1.4 m/s one half-metre later. The profile has to be reachable going forwards and stoppable going backwards, and no single pass over the points does both.

The mathematics

Curvature of a sampled path

Curvature is 1/R, the reciprocal of the radius of the circle through the path. For three consecutive points a, b, c it can be computed without angles:

A turning angle at a point is not a curvature. Curvature is the turning angle per metre, so a heading change per point must be divided by the distance between points.

Speed limits from curvature

At speed v on a circle of radius R the sideways (lateral) acceleration is v²/R = v²·κ. Limit it to a_lat, which depends on the load (a tray of instruments, a patient on a stretcher) and the grip, and keep the wheel speeds in range:

a_lat

allowed sideways acceleration, in m/s².

b

track width, the distance between the drive wheels.

v_wheel_max

top speed of a drive wheel; on a bend the outer wheel runs at v·(1 + κ·b/2).

Acceleration and braking: two passes

Between two points a distance ds apart, constant acceleration a changes the speed by v₂² = v₁² + 2·a·ds. Work in v², so each step is exact:

1. Forward pass. Start at rest. For each next point: v² = min( limit², v_prev² + 2·a_max·ds ).
2. Backward pass. Force v = 0 at the last point. Going back: v² = min( v², v_next² + 2·d_max·ds ).

The result is the highest profile that respects every limit at once: no feasible profile is higher at any point. Time follows from the mean speed of each segment, T = Σ 2·ds / (v₁ + v₂).

Stopping and jerk

A jerk limit rounds the corners of the acceleration, which gives the smooth S-curve profile and costs a little time. It is worth adding when the load is fragile.

A worked example

One bend. Take a_lat = 0.5 m/s², track width b = 0.4 m and wheel limit 1.4 m/s.

Here the sideways acceleration always wins, so the wheel limit never binds. A robot with a lighter load or a wider track changes that.

Nine points. Points are 0.5 m apart. The limits are 1.4 m/s except 0.8 m/s at points 3 and 4 (a bend). The robot accelerates at 0.6 m/s², brakes at 0.8 m/s², and must stop at point 8 (the door).

The forward value at point 1 is sqrt(2 × 0.6 × 0.5) = 0.775 m/s. The backward value at point 7 is sqrt(2 × 0.8 × 0.5) = 0.894 m/s: any faster and Scout cannot stop in the last half-metre. Times: 2 × 0.5 / (0 + 0.775) = 1.29 s for the first segment, and the last, 2 × 0.5 / (0.894 + 0) = 1.12 s, so the whole run takes 5.50 s. Starting from 1.2 m/s with d_max = 0.8 m/s², braking needs 1.2² / 1.6 = 0.90 m; with d_max = 0.5 m/s² it needs 1.44 m.

The forward pass alone ends at 1.4 m/s at the door. That is the bug of lab M5-L2, stage 2.

Engineering implications and limitations

Look ahead along the path, not only at the point

A controller that sets its speed from the curvature under the robot reacts too late. The profile must be built from the whole path ahead, or the controller must look as far as the braking distance.

Pick limits that the hardware keeps

Use acceleration and braking limits below the real capability, and braking slightly stronger than acceleration only if the robot can really brake that hard with a load. A wet floor halves the usable limits.

Spacing of the samples

Curvature from three neighbouring points is noisy on a finely sampled, noisy path and blind on a coarse one. Resample at a spacing of a few tens of centimetres on a smoothed path (lesson M5-T2) before computing it.

Slow down for more than bends

The same two-pass logic handles other speed limits: a speed zone near a ward, a slower pace near the goal, a limit near people (lesson M6-T6). Each is another limit curve fed into the same forward and backward passes.

Common misconceptions

Engineering takeaways

1. Curvature is 1/R; compute it from three points with κ = 2|cross| / (|ab||bc||ca|).
2. Lateral acceleration v²κ gives v ≤ sqrt(a_lat/κ); the outer wheel adds v ≤ v_wheel/(1 + κb/2).
3. Build the profile in v² with a forward pass (acceleration) and a backward pass from the stop (braking).
4. Braking distance is v²/(2d); trip time is the sum of 2·ds/(v₁ + v₂).
5. The profile must be recomputed whenever the path changes.

References and further reading

Time-optimal path parameterisation and the velocity-planning methods that the forward and backward passes simplify.

• J. E. Bobrow, S. Dubowsky and J. S. Gibson, “Time-Optimal Control of Robotic Manipulators Along Specified Paths”, The International Journal of Robotics Research, 4(3), 1985.
• Q.-C. Pham, “A General, Fast, and Robust Implementation of the Time-Optimal Path Parameterization Algorithm”, IEEE Transactions on Robotics, 30(6), 2014.
• K. Kant and S. Zucker, “Toward Efficient Trajectory Planning: The Path-Velocity Decomposition”, The International Journal of Robotics Research, 5(3), 1986.

Knowledge check

Three conceptual questions. Write an answer, then reveal the explanation. Your answers stay in your browser.

What’s next

Next: the simplest controller that follows a path at these speeds.

Related content

Nothing below is required to finish this lesson.

• Before
  M5-T2: Smoothing: Shortcuts, Splines and Gradient Methods

  Remove zig-zags with shortcutting, splines and gradient smoothing, and keep the result collision-free.

• Lab
  M5-L2: Build a Velocity Profile

  Give a path a speed at every point so Scout slows for corners and stops exactly at the door.