There and Back
Are Not the Same

Forward is not backward

The set of velocities a robot can actually hold at a configuration — its velocity envelope \(\mathcal{B}(q)\) — is a convex body that is not centered on the origin. Forward 1.2 m/s, backward 0.5, sideways 0.4. A Riemannian metric measures \(\|v\|_G=\|-v\|_G\), so it can express that sideways is expensive, but never that backward is expensive. It is structurally blind to exactly the asymmetry that legged robots have.

quadruped-class envelope · forward 1.2 / backward 0.5 / lateral 0.4 m/s · loops

The envelope is a metric

Take the gauge function of the envelope,

\[ F(q,v)=\inf\{\alpha>0:\ v/\alpha\in\mathcal{B}(q)\}, \]

which is positively one-homogeneous and convex but not even: \(F(q,v)\neq F(q,-v)\). That is a Finsler metric, and its forward distance is exactly the minimum travel time:

\[ T^\star(q_s,q_g)=\inf_{\gamma}\int_0^1 F(\gamma,\dot\gamma)\,dt \;=\; d_F(q_s,q_g). \]

Nothing here is a cost we designed. Measure the envelope from rollouts, and the time-optimal planning geometry follows. The minimum-time plan is a forward geodesic traversed on the envelope boundary.

The same goal, two geometries

Every video below compares planners that receive the same information and differ only in the geometry they plan with. Clocks run in real time. The comparison runs across four bodies: a humanoid, a drift field, a manipulator, and a mobile manipulator.

A goal behind the robot

SE(2) pose planning · learned flow integrated with RK4 · real time

On the robot

MuJoCo 3.13 · Unitree pretrained velocity-command policy at 50 Hz · commands at 20 Hz · real time

same identified envelope · same feedback rule · real time

Not only legs: a UR5, and effective mass

A manipulator looks like the symmetric case until you ask what limits it. Effective mass \(m_{\mathrm{eff}}(u)=(u^ op\Lambda^{-1}u)^{-1}\) is quadratic in \(u\), so by itself it is reversible. The asymmetry arrives with the constraint: capping the energy transferred on contact bounds how fast the arm may close on a workpiece, by \(\sqrt{2E_{\max}/m_{\mathrm{eff}}}\), and leaves retreat unbounded.

What that buys an arm is not the same as what it buys a walking robot, and the difference is worth being precise about. The symmetrised metric picks routes within 1.0 % of optimal here, so route selection barely changes. What it gets badly wrong is the clock.

MuJoCo UR5e (mujoco_menagerie) · joint angles solved against the model · real time

Gravity, by contrast, changes nothing about the routes either, and our own theory says why: gravitational power is the derivative of a potential, so the asymmetric part of the metric is a closed one-form and only travel times change. The asymmetries that do move routes are the ones with non-vanishing curl, which is what the legged and drift-field settings above have.

When the environment is the asymmetry

Asymmetry does not need legs. A head-wind channel makes eastward motion cost more than westward at the same place — a drift field, and Zermelo's classical navigation problem, whose solutions are Randers geodesics.

planar Randers field, β = 0.9 · goal-conditioned flow · real time

Geodesics in, one network pass out

The conditional probability paths of the flow model are the minimizing Finsler geodesics, traversed at constant Finsler speed:

\[ q_t=\mathrm{Exp}^F_{q_0}\!\big(t\,\mathrm{Log}^F_{q_0}(q_1)\big). \]

Integrating the trained field from a start pose is the plan; its Finsler length is its travel time, and its final speed predicts the time still to go. Two things break if you simply swap a Riemannian metric for a Finsler one, and both are easy to get wrong.

Both statements are specific to non-reversible geometry: in the Riemannian case \(g_v\equiv g(q)\), the gradient identity holds, and the metric-weighted loss is admissible.

Time-optimal, and fast to ask

Pose planning on SE(2)

A grid reference is an upper bound, so a ratio below one cannot by itself prove optimality. Re-scoring the same plans against exact shooting geodesics, on the poses where the grid certifies the geodesic minimal, removes that bias: Finsler 0.998, symmetric Riemannian 1.064, Euclidean 1.212. The remaining gap below one is the 0.2 m position tolerance, which lets a plan stop marginally short. Optimality here is verified, not inferred.

Learn the envelope from noisy rollouts, then plan with it

Throwing the asymmetry away costs 13 % on average and 74 % in the worst case. Even a crude one-term Randers fit, 9 % wrong on the envelope, plans within 3.6 % of the true optimum — time-optimal paths use the forward and turning faces, which every asymmetric fit captures.

What amortization buys

The sharper point is what the query ranges over. A grid reference is built for one metric: change the envelope — the robot picked up a payload, the terrain changed, the battery sagged — and the graph is rebuilt from scratch, which is the 0.14 s and 9.9 s above rather than the per-goal sweep. The envelope-conditioned field answers a start, a goal and an envelope it has never seen in the same 13–29 ms. Amortizing over capability is a difference in kind, not a constant factor, and it is the regime a robot whose envelope drifts during operation actually lives in.

Paper, supplement, code

The supplement carries the closed forms, the proofs, the loss-weight experiment, the full result tables and the grid convergence study. The code ZIP contains the Finsler library, the experiments and the MuJoCo pipeline; every number in the paper maps to a script in its README.

Cite

@inproceedings{finslerflowmatching,
  title     = {There and Back Are Not the Same:
               Finsler Flow Matching for Asymmetric Robot Planning},
  author    = {Anonymous},
  booktitle = {Under review},
  year      = {2026}
}

Research lineage