Friction · LavenderSim

Friction is a limit, not a fixed force

Static friction chooses whatever tangential force is needed to prevent slip, up to a limit:

\lVert f_t\rVert\le\mu_s f_n

A resting block with no horizontal load has zero friction—not $\mu_s f_n$ . Add a 3 N push and friction supplies 3 N in the opposite direction. Once the required force exceeds the limit, sliding begins and a common model uses:

f_t=-\mu_d f_n\frac{v_t}{\lVert v_t\rVert},qquad \mu_d\le\mu_s

The discontinuity at zero slip is why simulators usually solve friction together with contact impulses instead of evaluating a simple smooth force law.

LavenderSim material mixing: each contact uses the geometric mean of the two surfaces' coefficients. If both surfaces use the same value, the effective coefficient is unchanged.

Experiment 1: cross the static threshold two ways

On an incline, gravity decomposes into $mg\sin\theta$ downhill and $mg\cos\theta$ normal to the surface. Static balance is possible while:

mg\sin\theta\le\mu_smg\cos\theta{}\quad\Longrightarrow\quad\theta\le\arctan\mu_s

For $\mu_s=0.6$ , the ideal critical angle is 30.96°. A horizontal push predicts the same threshold: $F\le\mu_smg=5.886$ N for a one-kilogram block.

Experiment 01verified

Tilt and push the same contact

Does measured motion begin near both analytical limits?

Block displacement across incline angle and horizontal push sweeps — The native block remains within a millimeter through 28°, then moves visibly by 30°. It sticks through 5.5 N and slides at 6 N. Manifold discretization and finite-step perturbations make the measured transition slightly earlier than the ideal 30.96°.

python

theta_critical = math.atan(mu_static)
static_limit = mu_static * mass * gravity

below = push_trial(5.5)  # sticks
above = push_trial(6.0)  # slides

After sliding starts, LavenderSim's measured tangential-force utilization drops toward $\mu_d/\mu_s=0.75$ . This is not weaker numerical convergence; it is the chosen dynamic coefficient.

Failure modes: always apply $\mu f_n$ , use $\mu_d$ for the onset threshold, ignore material mixing, or classify tiny contact transients as sustained slip.

Expected result: The command exits successfully and reproduces the numerical or visual relationship described in this card.
Failure modes: Non-finite values, a reversed trend, a failed assertion, or a materially different plot means the assumptions, seed, timestep, build, or backend should be inspected before continuing.

python examples/tutorials/simulation/ch10_friction/lavendersim_friction.py --quickView source ↗

Experiment 2: project a two-dimensional friction impulse

In 3D, a contact has two tangent directions. An isotropic Coulomb model limits their combined impulse:

\sqrt{\lambda_{t1}^2+\lambda_{t2}^2}\le\mu\lambda_n

The feasible cross-section is a disk. Projection preserves direction and limits magnitude. A cheaper approximation may constrain tangent components separately, producing a square cross-section and up to $\sqrt2$ too much diagonal grip.

Experiment 02verified

Compare circular and square projection

Should grip strength depend on how tangent axes happen to be oriented?

Trial tangential impulses projected onto a disk and square friction limit — The circular limit is isotropic. Independent component clamps form a square pyramid whose diagonal radius is √2 for the same axial limit.

python

trial = old_tangent_impulse + delta
if np.linalg.norm(trial) > static_limit:
    trial *= dynamic_limit / np.linalg.norm(trial)

LavenderSim uses this disk projection. Polyhedral pyramids can be useful in optimization, but their number and orientation of facets are modeling decisions—not the Coulomb law itself.

Expected result: The command exits successfully and reproduces the numerical or visual relationship described in this card.
Failure modes: Non-finite values, a reversed trend, a failed assertion, or a materially different plot means the assumptions, seed, timestep, build, or backend should be inspected before continuing.

python examples/tutorials/simulation/ch10_friction/friction.py --quickView source ↗

Experiment 3: give the contact patch an area

A mathematical point cannot resist rotation about its normal, and ideal pure rolling has no tangential slip. Real patches deform. Compact rigid-body engines often add bounded angular impulses:

|\lambda_{twist}|\le\mu_{twist},r_p\lambda_n,qquad \lVert\lambda_{roll}\rVert\le\mu_{roll},r_p\lambda_n

$r_p$ is a modeled contact-patch radius. These are useful phenomenological terms; they are not detailed tire or soft-material models.

Experiment 03verified

Roll a sphere and twist a grasped payload

Which motion survives when angular friction is zero?

Sphere speed for rolling friction sweeps and grasped payload spin for torsional friction sweeps — Higher rolling resistance shortens the sphere's coast. Two preloaded pads with nonzero torsional friction stop an axial payload spin that ordinary point friction barely sees.

0.82 m/sspeed after 2 s at μroll = 0

0.06 m/sspeed after 2 s at μroll = 0.12

8 → 0 rad/storsion μ = 0.1 grasp

Failure modes: confuse rolling resistance with sliding friction, omit patch radius from units, apply an unbounded resisting torque, or claim a patch approximation predicts real tire behavior.

Expected result: The command exits successfully and reproduces the numerical or visual relationship described in this card.
Failure modes: Non-finite values, a reversed trend, a failed assertion, or a materially different plot means the assumptions, seed, timestep, build, or backend should be inspected before continuing.

python docs-site/scripts/render_tutorial_ch10.pyView source ↗

Experiment 4: make slip observable

A controller needs more than a friction coefficient. LavenderSim's tactile summary reports normal force, shear force, post-solve tangential slip speed, static-limit utilization, a filtered slip score, a hysteretic slip flag, centroid, twist/rolling moments, and taxel forces.

u=\frac{\lVert\lambda_t\rVert}{\mu_s\lambda_n},qquad s_{raw}=\max\left(\frac{\lVert v_t\rVert}{v_{threshold}},\frac{u}{u_{threshold}}\right)

The score is low-pass filtered. The flag turns on above 1.0 and turns off below 0.58, preventing rapid chatter near a single threshold.

Experiment 04verified

Disturb a preloaded tactile grasp

Can the sensor separate high utilization from actual persistent slip?

Tactile slip speed utilization filtered score and hysteretic flag over time — A 4 m/s lateral disturbance exceeds the dynamic limit. Slip speed spikes, utilization sits near μd/μs = 0.75, the filtered score crosses one, and the hysteretic flag remains stable while the motion decays.

python

tactile = observation["tactile"]["pad_sensor"]["summary"]
slip_speed = tactile[3]
friction_utilization = tactile[4]
slip_score, slipping = tactile[5], bool(tactile[6])

Failure modes: infer slip from utilization alone, threshold raw velocity without force gating, forget filtering delay, or treat taxel pressure as an exact continuum stress field.

Expected result: The command exits successfully and reproduces the numerical or visual relationship described in this card.
Failure modes: Non-finite values, a reversed trend, a failed assertion, or a materially different plot means the assumptions, seed, timestep, build, or backend should be inspected before continuing.

pytest -q tests/tutorials/test_ch10_friction.pyView source ↗

Experiment 5: arrival is not settling

The ball-pushing task from the environment suite deliberately uses a low-friction board. That makes pushing easier but target control harder: the ball can cross the target with substantial remaining speed. A reward for position alone encourages fly-through behavior.

Experiment 05verified

Apply one identical push pulse

How does board friction change both destination and stopping time?

Ball speed and position after identical pushes on low and higher friction boards — After the 0.1 s push, the low-friction ball is still moving at 0.62 m/s after three seconds. The higher-friction ball has stopped around 1.06 m.

A useful task reward therefore combines target distance, low speed near the target, and dwell time. Friction is part of task design, not only engine configuration.

Expected result: The command exits successfully and reproduces the numerical or visual relationship described in this card.
Failure modes: Non-finite values, a reversed trend, a failed assertion, or a materially different plot means the assumptions, seed, timestep, build, or backend should be inspected before continuing.

python examples/tutorials/simulation/ch10_friction/lavendersim_friction.pyView source ↗

Paw-check: find the limit before tuning it

Repeat the incline derivation when the applied force is not parallel to the plane.
Rotate a four-sided friction pyramid and quantify direction-dependent error.
Set different block and ramp coefficients and verify geometric-mean mixing.
Plot dissipated kinetic energy for each rolling-resistance curve.
Halve contact-patch radius and predict torsional stopping time.
Change slip-filter gain and hysteresis thresholds without creating chatter.
Design a ball-target reward that values arrival, stopping, and stationary dwell separately.

What should I be able to say now?

Static friction is a bounded reaction that matches demand; dynamic friction opposes established slip. In 3D, a circular tangent-impulse limit is isotropic while pyramidal approximations introduce directional error. Rolling and torsional terms model finite contact-patch moments. Tactile telemetry turns friction state into a controllable signal, and low-friction task design must explicitly reward settling rather than mere target crossing.

By the end, you can…

Friction is a limit, not a fixed force

Experiment 1: cross the static threshold two ways

Experiment 2: project a two-dimensional friction impulse

Experiment 3: give the contact patch an area

Experiment 4: make slip observable

Experiment 5: arrival is not settling

Paw-check: find the limit before tuning it