M004 · Where to aim

Pursuit & Interception

Where should you aim to catch a moving target? Aim toward a future meeting point. A direct chase keeps turning as the target moves sideways; leading the motion can shorten the route. Steering limits, stale positions and a turning target can make either strategy miss.

Compare chasing, leading and proportional navigation. Move the target, limit the steering and delay its reported position; then solve a straight-target minimum-time challenge.

Current separation $3\,\mathrm{m}$

Closing speed $1\,\mathrm{m\,s^{-1}}$

Applied turning rate $0\,{}^{\circ}\,\mathrm{s^{-1}}$

Trial state $\text{Tracking}$

Straight-target minimum time $0\,\mathrm{s}$

Minimum separation in this trial $0\,\mathrm{m}$

Capture time $0\,\mathrm{s}$

Model time $0\,\mathrm{s}$

Predict a course, then watch it unfold

Chase the position or lead the motion?

Pursuer speed

v_p

1.8\,\mathrm{m\,s^{-1}}

Target speed

v_t

1.2\,\mathrm{m\,s^{-1}}

Maximum turning rate

\Omega_p

140\,\mathrm{{}^{\circ}\,s^{-1}}

Target report delay

\tau

0\,\mathrm{s}

Target turning rate

\Omega_t

0\,\mathrm{{}^{\circ}\,s^{-1}}

Guidance choice

j

0

Target initial horizontal position

x_t(0)

3.5\,\mathrm{m}

Target initial transverse position

y_t(0)

1\,\mathrm{m}

Target initial heading

\phi_t(0)

90\,\mathrm{{}^{\circ}}

Pursuer initial heading

\psi(0)

16\,\mathrm{{}^{\circ}}

Navigation gain

N

3

Trial time limit

T

12\,\mathrm{s}

Capture center-distance threshold

\rho

0.25\,\mathrm{m}

Replay progress

\eta

0\,\mathrm{\%}

A target follows a line or a circle at constant speed. All controllers receive the same delayed target position and velocity, without age compensation; own pose is current. Pursuit and lead use a 0.2-second heading response and the specified turn bound. The PN scalar uses closing speed and sight-line rate as a velocity-normal turning command. Midpoint integration and within-step proximity events are numerical. The exact minimum-time task assumes a straight target, zero report delay and freely chosen launch heading; it does not claim a global optimum for a turning target.

Model assumptions

Two constant-speed point agents move in a plane. Turning is bounded, target observations can be delayed, and the run ends at a specified center-distance threshold. Robot symbols are schematic. This is a kinematic navigation model with no collisions, forces or vehicle dynamics.

Start with the essentials

Focus question: Which part of a successful intercept comes from prediction, and which from turning ability?
One-sentence intuition: A steady sight-line direction with decreasing distance signals a meeting course; pointing at the target is a different condition.

Core mathematical model

Constant speed with bounded steering

\dot{\mathbf p}=v_p\begin{pmatrix}\cos\psi\\\sin\psi\end{pmatrix},\qquad |\dot\psi|\le\Omega_p

The robot changes direction at fixed speed. Its turning-rate limit is a kinematic constraint; no motor torque or tire force is modeled.

Geometry from delayed target reports

\begin{aligned}\mathbf r_o&=\mathbf q(t-\tau)-\mathbf p(t)\\\dot\lambda_o&=\frac{r_{o,x}\dot r_{o,y}-r_{o,y}\dot r_{o,x}}{|\mathbf r_o|^2}\end{aligned}

Target reports can be old; the robot knows its current pose. All three feedback rules use the same reports without correcting their age. The target has a defined pre-launch trajectory.

A constant-speed realization of proportional navigation

\begin{aligned}V_c&=-\frac{\mathbf r_o\cdot\dot{\mathbf r}_o}{|\mathbf r_o|}\\a_\perp&=N\max(V_c,0)\dot\lambda_o\\\dot\psi&=\mathrm{clip}_{[-\Omega_p,\Omega_p]}(a_\perp/v_p)\end{aligned}

The scalar sight-line-rate command becomes acceleration normal to the robot velocity. This reduced planar realization omits the sensor, attitude and autopilot dynamics of a real navigation system.

The earliest possible straight-target meeting

\begin{aligned}|\mathbf r_0+\mathbf v_tT_*|&=v_pT_*+\rho\\\psi_*&=\arg(\mathbf r_0+\mathbf v_tT_*)\end{aligned}

Choose the first nonnegative solution. Before this time, even a straight path at full speed cannot reach the target capture disk. A freely chosen initial heading attains the bound for a constant-velocity target with fresh initial information.

Common difficulties

Pointing at the target can waste distance

Typical misconceptionThe shortest route always points straight at the visible target.

Better mental modelA sideways-moving target leaves that point. A future meeting point accounts for the distance it will travel.

A steering limit can spoil a prediction

Typical misconceptionKnowing the meeting point guarantees capture.

Better mental modelThe robot may start in the wrong direction and need more time to turn. The exact minimum-time plan assumes it can choose its launch heading.

Delayed reports describe another instant

Typical misconceptionA perfectly accurate position report is always good enough.

Better mental modelAn old report can be accurate about the past. The robot still needs the target current motion to choose a useful course.

Run the experiment

01
Watch a direct chase
Choose Aim at now, then replay.
What to observe: The blue route bends as the gold target moves sideways.
02
Aim ahead
Choose Aim ahead with the same speed and turn limit.
What to observe: The initial correction points toward a meeting place ahead of the target. The path and capture time change.
03
Solve the shortest-course challenge
Choose Fastest straight course and open the prediction details.
What to observe: The robot starts with the analytically chosen heading. Its capture time reaches the straight-target bound within numerical error.
04
Turn the target
Choose Target turns, then change the target turning rate.
What to observe: Proportional navigation keeps adjusting from the sight-line rotation. A straight-target minimum-time guarantee no longer applies.
05
Slow the information
Choose Old position reports and increase the report delay.
What to observe: The controller steers from older target coordinates while the actual target continues moving.
06
Compare on equal terms
Open the simulated strategy comparison and keep the same physical settings.
What to observe: The three feedback rows use the same initial heading, speed, turn limit and delayed reports. An uncaptured trial reports its minimum separation.

Core mathematical model

Constant speed with bounded steering

Geometry from delayed target reports

A constant-speed realization of proportional navigation

The earliest possible straight-target meeting

Common difficulties

Pointing at the target can waste distance

A steering limit can spoil a prediction

Delayed reports describe another instant

Run the experiment

Watch a direct chase

Aim ahead

Solve the shortest-course challenge

Turn the target

Slow the information

Compare on equal terms