M009 · Moving bases

Polar-Coordinate Kinematics

How do you find the velocity and acceleration of a point on an arm that stretches while it turns? Velocity is the stretching speed along the arm plus the radius times the turning rate across it. Differentiating again, the turning basis adds two terms to the radial and angular accelerations: an inward centripetal term, and a sideways cross term whenever the arm stretches while it turns.

Set how the radius breathes in and out and how the angle accelerates, then read the signed vector components on a rotating arm. Finally, rebuild the motion from noisy Cartesian positions and watch angular aliasing appear.

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 $\text{Ready}$

Set the path, its timing and the sampled 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\,

In the two law plots, drag the blue handles for radius and breathing, or the copper handles for phase and angular acceleration. The radius stays positive throughout the time window, but the angular velocity can change sign. Positive angles run counterclockwise in the physical horizontal plane, and the scene shows that plane after one rigid rotation.

The telescoping arm only marks where the prescribed point is; no force law is simulated. The blue radial and copper transverse acceleration arrows share one fixed scale within each record. The thin basis arrows show unit directions and have their own display length.

The cross term appears because the basis vectors rotate, so they must be differentiated too. It is part of the acceleration in the inertial frame, not an extra force. The total transverse acceleration can be zero even when neither of its two terms is.

The polar reconstruction uses only the acquired Cartesian positions. It converts them into radii and into angles unwrapped onto the nearest branch. It cannot recover whole turns missed between frames, so a small angle step between frames does not prove there is no aliasing.

An acquisition records only Cartesian position. Times are exact, and each coordinate carries a bounded, repeatable noise measured in millimetres. Derivatives come from a local least-squares quadratic fit, with one-sided windows at the ends. The estimator never sees the formula of the trajectory.

The fit window has an odd number of points and shrinks when the record is short. Acceleration needs at least three positions. A larger window suppresses noise but biases the estimate where the motion curves or changes quickly. The RMS errors use the analytic trajectory only as a separate reference.

Lengths are in metres. The velocity and acceleration arrows are raised on a connector so they are easy to see; their directions and display scales are unchanged. Markers and supports are schematic. Nothing plays automatically; drag the gold clock to move through time. Changing the fit window keeps the acquired positions, but changing the trajectory or sensor requires a new acquisition.

The motion is prescribed and the position observations are simulated. The Lab does not certify forces or safety.

Start with the essentials

Focus question: Where does the transverse cross term come from?
One-sentence intuition: In polar coordinates you differentiate both the coefficient and the basis vector it multiplies.

Core mathematical model

A radius that stays positive and a rotation of either sign

\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}

The radius stays positive because the breathing fraction is strictly less than 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 direction that rotates is changing, even when the coefficient in front of it stays constant.

Velocity components

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

Both the change in radius and the rotation contribute to the velocity in the inertial frame.

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 comes purely from the kinematics. These are signed components along the basis vectors at that instant.

Rebuilding the motion 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 measured radius and the unwrapped angle. Unwrapping always picks the nearest branch, so it cannot detect a full turn that was missed.

Common difficulties

A smooth curve is not the true acceleration

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

Better mental modelSampling can miss turns, local fits add bias, and the largest sampled value is not a guaranteed bound on the continuous curve.

Run the experiment

01
Start with a circle
Select Uniform circle and scrub the clock.
What to observe: The radial acceleration points inward, and the velocity points along the transverse direction.
02
Isolate the cross term
Compare Radial only, Inward spiral and Breathing spiral.
What to observe: The direction of the cross term depends on the radial motion and the sign of the rotation.
03
Reverse the rotation
Select Reverse the rotation; inspect the midpoint with zero angular velocity.
What to observe: At the moment the rotation reverses, the angular acceleration can still be nonzero.
04
Miss a turn
Acquire Angular aliasing, then increase frame rate and acquire again.
What to observe: The fit to the raw angle can look smooth and still be wrong. The analytic reference shows the motion that was missed.