M009 · Moving bases

Polar-Coordinate Kinematics

Prescribe radial breathing and angular acceleration, inspect signed vector components on a rotating arm, then reconstruct motion from noisy Cartesian positions and expose angular aliasing.

Signed radial acceleration $\text{Ready}$

Signed transverse acceleration $\text{Ready}$

Radius second derivative $\text{Ready}$

Centripetal contribution $\text{Ready}$

Angular acceleration contribution $\text{Ready}$

Radial–angular cross term $\text{Ready}$

Polar fit acceleration RMS error $\text{Ready}$

Largest observed angle step $\text{Ready}$

Frame at selected time $\text{Ready}$

Selected time $\text{Ready}$

Selected speed $\text{Ready}$

Acceleration magnitude $\text{Ready}$

Signed tangential acceleration $\text{Ready}$

Normal acceleration $\text{Ready}$

Geometric curvature $\text{Ready}$

Osculating radius $\text{Ready}$

Sampled path length $\text{Ready}$

Acquired positions $\text{Ready}$

Cartesian fit speed-vector RMS error $\text{Ready}$

Cartesian fit acceleration RMS error $\text{Ready}$

PATH / CLOCK / RAW POSITIONS

Separate the rotating components

Initial radius

r_0

2.00\,m

Radial logarithmic growth over window

d

0.35\,

Radial breathing fraction

\epsilon_r

0.25\,

Initial signed angular velocity

\omega_0

1.00\,\frac{\mathrm{rad}}{\mathrm s}

Signed angular acceleration

\alpha

0.10\,\frac{\mathrm{rad}}{\mathrm{s^2}}

Initial angle

\theta_0

0.00\,{}^\circ

Traversal duration

T

6.00\,s

Selected time fraction

t/T

0.40\,

Position sampling rate

f_s

20.00\,Hz

Position noise bound

\eta

0.00\,mm

Quadratic fit window; odd stations

n_w

5.00\,

Drag the blue radius and breathing handles, or the copper phase and angular-acceleration handles in the two law plots. The radial law stays positive throughout the window. Signed angular velocity may reverse. Positive angle is counterclockwise in the physical horizontal plane; the scene applies one rigid rotation to display that plane.

The telescoping arm locates a prescribed point; it does not supply a simulated force law. Blue radial and copper transverse acceleration arrows share one fixed gain per record. Thin basis arrows are unit directions with a separate display length.

The cross term comes from differentiating rotating basis vectors. It contributes to inertial acceleration and is not an additional force. Total transverse acceleration can cancel even when its two terms are nonzero.

Polar reconstruction converts only acquired Cartesian positions into radii and nearest-branch unwrapped angles. It cannot recover turns missed between frames. A small observed angle step does not prove the absence of aliasing.

Acquisition records Cartesian position only, with exact time and bounded deterministic noise per coordinate in millimetres. A local least-squares quadratic estimates derivatives; endpoint windows are one-sided. There is no access to trajectory laws inside the estimator.

The selected odd window uses fewer stations if the record is short. Fewer than three positions cannot estimate acceleration. Larger windows suppress noise but bias curved or rapidly changing motion. RMS errors use the analytic trajectory only as a separate reference.

Geometry is in metres. Acceleration and velocity glyphs are lifted on a connector for visibility, preserving their directions and display gains. Markers and supports are schematic. There is no autoplay; scrub the gold clock. Editing the fit window preserves the acquired positions, while trajectory or sensor changes require a new acquisition.

Prescribed kinematics and simulated position observations; no force or safety certification.

Start with the essentials

Focus question: Where does the transverse cross term come from?
One-sentence intuition: Differentiate both the coefficient and its basis vector.

Core mathematical model

Positive radial law and signed rotation

\begin{aligned}r(t)&=r_0e^{dt/T}\left[1+\epsilon_r\sin\left(\frac{2\pi t}{T}\right)\right]\\\theta(t)&=\theta_0+\omega_0t+\tfrac12\alpha t^2\end{aligned}

Radius remains positive because the breathing fraction is strictly below one.

Differentiate the basis too

\dot{\mathbf e}_r=\dot\theta\mathbf e_\theta,\qquad\dot{\mathbf e}_\theta=-\dot\theta\mathbf e_r

A rotating direction changes even when its coefficient stays constant.

Velocity components

\mathbf v=\dot r\mathbf e_r+r\dot\theta\mathbf e_\theta

Radial change and rotation both contribute to the inertial velocity.

Acceleration components

\begin{aligned}\mathbf a&=(\ddot r-r\dot\theta^2)\mathbf e_r\\&\quad+(r\ddot\theta+2\dot r\dot\theta)\mathbf e_\theta\end{aligned}

The cross term is kinematic. These are signed components on instantaneous basis vectors.

Reconstruct from positions

\begin{aligned}\widehat r_i&=\sqrt{x_i^2+y_i^2}\\\widehat\theta_i&=\operatorname{unwrap}(\operatorname{atan2}(y_i,x_i))\end{aligned}

Fit local quadratics in time to the acquired radius and unwrapped angle. Nearest-branch unwrapping cannot identify missed full turns.

Common difficulties

Model versus observation

Typical misconceptionA smooth reconstruction or a dense plot proves the true acceleration.

Better mental modelSampling can miss turns, local fits introduce bias, and sampled maxima are not certified continuous bounds.

Run the experiment

01
Start with a circle
Select Uniform circle and scrub the clock.
What to observe: The radial acceleration points inward while the velocity is transverse.
02
Separate the cross term
Compare Radial only, Inward spiral and Breathing spiral.
What to observe: Radial motion and signed rotation determine the cross-term direction.
03
Reverse angular motion
Select Reverse the rotation; inspect the midpoint with zero angular velocity.
What to observe: Angular acceleration can remain nonzero at angular reversal.
04
Miss a turn
Acquire Angular aliasing, then increase frame rate and acquire again.
What to observe: The raw angle fit can be smooth and wrong; the analytic reference exposes the missed motion.