Every step, the flex block of the implicit effective metric M + K was
factorized by sparse Cholesky, because K depends on the configuration. On
model/flex/bag.xml, added here, that is roughly half the step, against a
comparable share for the constraint solve it exists to accelerate.
Keep only the metric's per-vertex 3x3 diagonal blocks, prefactored. Neither
consumer needs the exact inverse: the CG constraint solver only wants a
preconditioner, and qacc_smooth can come from an iterative solve using those
blocks. They are O(n) to build and to apply, but weaker, so CG runs about twice
the iterations and qacc_smooth becomes an iteration rather than a direct solve.
Net, the bag model steps roughly twice as fast.
The preconditioner, by metric state. Inactive, meaning no flex elasticity or an
explicit integrator: M^-1, unchanged. Bending only (nefmK == 0): M^-1 plus the
exact constant bending factor from mj_setConst on the dofs it covers,
unchanged; that factor is built at model compile time and costs nothing per
step. Per-step stiffness: M^-1 plus the 3x3 blocks, where before it was a
per-step sparse Cholesky, or, when M couples across the flex block, an inner
PCG of up to 50 iterations run once per outer CG iteration.
Only models carrying per-step stretch stiffness change in wall-clock. Both
ponchos hold their timing and take slightly fewer CG iterations than before,
because the preconditioner is now symmetric: it applies M^-1 and the covered
blocks to disjoint sets of dofs, where previously the two overlapped and the
operator was not symmetric, which PCG requires.
mjd_effSolve is the accurate solve of (M + K)x = b; what used to carry that
name only preconditions and is now mjd_effPrec. Its CG guarded the division by
pAp with mjMINVAL, an absolute floor on a quantity that scales with the square
of the right-hand side, so a small b aborted the solve while the curvature was
healthy: four flex models were quietly left short of tolerance. For an SPD
metric the guard is positivity, and with that the same solves converge. The qacc_smooth call site in
mj_fwdAcceleration is textually unchanged but now reaches the iterative solve,
which converges on opt.tolerance rather than a hardcoded threshold, floored in
mjUSESINGLE builds where the squared target is unreachable in float. Reaching
the iteration cap names the ill-conditioned flex stiffness and then reports it
through mjWARN_INERTIA, rather than returning an under-converged result.
Covered dofs are located by walking the covered rows of the stiffness matrix,
as they need not be 3-aligned from dof 0: any joint declared before a flexcomp
shifts them.
mjData.efm_L_rownnz, efm_L_rowadr and efm_L_colind described the sparsity of
the deleted factorization and are removed: left NULL with nonzero mjxmacro
extents they made the Python bindings hand back uninitialized arrays.
efm_active loses the value 2 for the same reason, nothing selects a solve path
on preconditioner exactness any more. Both are recorded under breaking changes.
model/flex/bag.xml is added because no shipped model carried per-step stretch
stiffness. The ponchos are bending-only and trampoline.xml uses an explicit
integrator, so the metric never activates there. It is excluded from
WriteReadCompareTest: stretch stiffness amplifies rest geometry that XML rounds
on save.
This CL replaces the post-hoc implicit flex correction (`flexInterp_cgsolve`) with a **linearly-implicit effective metric** `M̃ = M + (h² + h·damping)·K` carried by the CG constraint solver itself. Contact/friction forces and implicit flex elasticity are now computed against one consistent metric, instead of the solver seeing `M` and a post-solve correction changing `qacc` behind its back.
Gate (unchanged semantics): `solver="CG"` + implicit/implicitfast integrator + pyramidal cones + flex stiffness present. Newton and PGS are untouched. `solver="CG"` remains the user-facing contract — the factorization is an implementation detail of the preconditioner.
### What's in the metric
- **mjData `efm_*`** (arena, efc-like lifetime/skip semantics; built in `mj_fwdPosition`, value-refreshed in `mj_fwdVelocity`): the per-step stiffness CSR `efm_B_*`, its reverse-Cholesky factor `efm_dofid` + `efm_L_*` (nested-dissection ordered, separators-first for the reverse factorization), and the smooth-force shift `efm_c = h·K·qvel`.
- **`mjd_flexStiff_assemble`** now assembles stretch (Gauss–Newton), standard dim-2 bending, and — via the cached corotated stiffness `d->flexelem_krot` — interp stiffness (all node bodies on simple sliders: point Jacobian is I₃, `flex_centered` not required; fixed nodes drop like pins) into one dof-level CSR. `mjd_effMulAdd`/`mjd_effSolve` apply the metric, with matrix-free operator fallbacks where assembly does not apply.
- **mjModel `efm0_*`** (`nefm0dof`/`nefm0L`): the constant part of the metric factor — currently the dim-2 bending factor, computed once in `mj_setConst` — so bending-only models pay zero per-step factorization cost. Naming mirrors mjData's `efm_*` with the standard `0`-suffix (reference/constant) idiom, and is deliberately not bending-specific: future constant contributors extend it without renames.
- The solver consumes the metric through pre-shifted `qfrc_smooth` and the metric products `Ma`/`Mv`/`Mgrad`; `qacc_smooth` becomes the unconstrained minimizer of the implicit dynamics, which makes the no-constraint shortcut and the warmstart choice consistent by construction.
- **`mj_inverse` adds `B·qacc − c`**, making inverse dynamics discrete-consistent with the gated forward dynamics — exact, since the gated path has no qDeriv term (new test `ForwardTest.GatedFlexInverseConsistency`).
### Performance
All numbers: ms/step over the same 2000-step window, models as shipped on each side (old code with the old model settings vs this CL with the new ones).
The new solver path activates on exactly two shipped models — the ponchos, the only flex models that need an implicit integrator (poncho on Euler degenerates to >200 ms/step). For them, this CL trades speed for consistency: the implicit bending solve now runs inside every solver iteration, where the contact solve can see the stiffness, instead of once after the solve. Solver iterations drop because the curvature is visible, but each iteration pays for the implicit solve:
| model | before | after | solver iters/step |
|---|---|---|---|
| poncho | 2.47 | 3.30 (1.33×) | 16.8 → 11.8 |
| poncho_edgeequality | 1.96 | 2.72 (1.39×) | 13.2 → 10.0 |
What that price buys: contact forces consistent with the implicit elasticity (previously the post-hoc correction changed `qacc` after the constraint solve), discrete-consistent inverse dynamics, and the removal of the post-hoc special case from the integration path. Raising poncho's timestep from 2 to 5 ms leaves its per-step cost nearly flat, so the consistency price can be recovered by taking fewer steps where accuracy allows.
Every other flex model was measured stable on Euler at its shipped timestep and switches to it (these models predate the post-hoc integrator; implicit was never load-bearing for them). They end up equal or faster than before: bunny_multicell 0.47 → 0.40, trampoline 0.28 → 0.25, plate 1.02 → 0.99, pancake 0.34 → 0.33.
Finally, the per-step factorization makes configurations practical that the old code could only integrate explicitly: implicit stretch elasticity (`elastic2d="stretch"`/`"both"`, dim-3 solids) and factorized interp stiffness. No before/after exists for these — stock has no implicit treatment of stretch at all.
### Behavior changes
- With the post-hoc correction deleted, interp/bending models running `solver="Newton"` (or elliptic cones, or islands) now integrate flex elasticity **explicitly** (previously: post-hoc implicit). Affects e.g. `gripper_trilinear` (stable, and faster, but different semantics). Follow-up options: Newton-side metric support, or a documented fallback.
- With the gate on, `mj_forward` outputs are timestep-dependent for gated models (they answer the linearly-implicit discrete problem); `qacc_smooth` and `mj_inverse` change accordingly. Non-gated models are bit-identical (full suite green throughout).
### Validation
- 1737/1737 tests, including new: `FlexStretchDerivatives` (FD-validated GN operator), `FlexStiffAssemble`/`FlexStiffAssembleInterp` (CSR ≡ operators), `GatedFlexInverseConsistency` (fails pre-change), equivalence tests vs the old post-hoc treatment (bending matches to 2e-11).
- Fingerprint discipline throughout: bending-only models bit-exact across every refactor; permutation/kernel changes verified iteration-identical.
### Known follow-ups (not in this CL)
3×3-block sparse Cholesky kernel (the numeric factorization is index-bound; projected ~3× on the factor); mjModel persistence of the factor's symbolic pattern (rest-pose ND makes sizes compile-time); the general effective-metric mode (all solvers, all PSD-safe force classes, behind an enable flag).
PiperOrigin-RevId: 948561856
Change-Id: I8b8e32ebd0428042af71647d0470d10773bf6daf
The gyroscopic (bias) derivatives applied to standalone free bodies by the
implicitfast integrator provide comparable stability for spinning bodies,
with none of midpoint's restrictions: they apply under contacts, fluid
forces and constraints, and preserve the linear force-velocity relation
required by discrete-time inverse dynamics. The invdiscrete flag reverts to
its original single meaning and no longer affects forward dynamics.
Restore implicitfast coverage in the DiscreteInverseMatch test, removed
when midpoint made discrete inverse dynamics untestable.
Add implicit gyroscopic (bias) derivatives for free bodies in implicitfast.
The implicitfast integrator drops the RNE (bias) derivative to stay on the
symmetric Cholesky path, so fast-spinning free bodies integrate gyroscopic
forces explicitly and can gain energy. Symmetrizing the gyroscopic Jacobian
is not an option: its stabilizing content is the antisymmetric part, and
adding only the symmetric part is destabilizing.
Instead, exploit the fact that for a standalone free body the 6x6 block of
M - h*D is decoupled from the rest of the system (qDeriv sparsity is
tree-local): after the global solve, rebuild the block with the exact bias
derivative in closed form (mjd_freeBias_vel) and re-solve it with dense
unsymmetric LU, overwriting the block's rows of qacc. For lone spinning
bodies this makes implicitfast match implicit to rounding, at ~150ns per
eligible body: cheaper than the midpoint machinery it will replace.
Eligibility is structural only; contacts, fluid and constraints need no
gating. The same block is mirrored in discrete inverse dynamics
(mj_discreteAcc), making invdiscrete exact for spinning free bodies.
PiperOrigin-RevId: 948472495
Change-Id: I813ef3d98c7b399881bc8603b9f9208cfb02eb58
This gives a 3x speedup in implicitfast.
Also cleanup old code that was used in the dense factorization of the stiffness matrix before we switched to CG.
PiperOrigin-RevId: 915900315
Change-Id: Id6973c4bfd7d371a43ec6db982703969b23a3550
Standard flex (flex_interp=0) with thin-plate bending treated bending forces purely explicitly. This caused contact-induced vertex vibrations and non-physical energy injection for flat resting sheets, because the solver treated each vertex as an independent mass during contact and contact normals are orthogonal to stretch constraints.
Fix: extend the existing preconditioned CG solver to include the constant bending stiffness K_bend in the implicit operator via matrix-free mat-vec.
PiperOrigin-RevId: 914774020
Change-Id: I45e0d6749abb6f873566203bccae956514b2576b
with a preconditioned Conjugate Gradient (CG) solver that operates
on the full system matrix.
The previous approach extracted flex DOFs into a reduced banded system,
factored it separately, and overwrote the global solve. This required
precomputed bandwidth (makeFlexBandwidth), parent-joint detection,
coupling corrections, and a FlexInterpContext struct — and only worked
for standalone flex trees without parent joints.
The new CG solver uses the already-factored global system (M - h*qDeriv)
as a preconditioner and adds the flex stiffness contribution via
matrix-free products (mjd_flexInterp_mulKD/mulK). This handles any
kinematic configuration — including flexes attached to articulated
chains or with parent joints — without sparsity pattern restrictions.
Before (`bunny_multicell`):
```
Simulation time : 50.80 s
Steps per second : 197
Realtime factor : 0.20 x
Time per step : 5080.3 µs
CG iters / step : 3.16
Contacts / step : 31.04
Constraints / step : 124.15
Degrees of freedom : 178
Dynamic memory usage : 0.4% of 100M
```
After:
```
Simulation time : 9.52 s
Steps per second : 1051
Realtime factor : 1.05 x
Time per step : 951.7 µs
CG iters / step : 3.21
Contacts / step : 30.90
Constraints / step : 123.61
Degrees of freedom : 178
Dynamic memory usage : 0.3% of 100M
```
PiperOrigin-RevId: 913758038
Change-Id: If5aa617b2d535c86aec9bd71c9e0003a2b38bdd7
The flex interpolation stiffness matrix within the implicit/implicitfast solvers is now built and factorized in a banded format instead of a dense one. This involves:
- Calculating the bandwidth based on the sparsity of the mass/damping matrix and the connectivity within flex cells.
- Allocating and populating a banded matrix `H`.
- Using `mju_cholFactorBand` and `mju_cholSolveBand` for factorization and solving.
This change improves performance for flexes with many DOFs but local coupling.
PiperOrigin-RevId: 901297952
Change-Id: I3efe06353d1903ea65ab30dc49685cede228bb68
- Add `qH` and `qHDiagInv` to `mjData` to save factorized modified inertia.
- Add `mj_EulerSkip`, `mj_implicitSkip`, to `engine_forward.c`.
- Using the above functions, implement `mj_stepSkip` in `engine_derivative.c`.
- Add `mjd_stepFD` and `mjd_transitionFD` to `engine_derivative.c` to compute `mj_step` Jacobians.
- Exploit "Skip" functionality for speed.
- Correctly handle quaternion derivatives.
- Handle warmstarts and control limits.
PiperOrigin-RevId: 456584811
Change-Id: Iee8541f11e7b66feb8f431cb102d9bbe65461f79
Added analytic derivatives of smooth (unconstrained) dynamics forces, with respect to velocities:
- Centripetal and Coriolis forces computed by the Recursive Newton-Euler algorithm.
- Damping and fluid-drag passive forces.
- Actuation forces.
A new implicit-in-velocity integrator is implemented using the analytic derivatives. This integrator lies between the Euler and Runge Kutta integrators in terms of both stability and computational cost.
PiperOrigin-RevId: 450377010
Change-Id: Ie192b441876c22e732fb749333926f296e0a09cc
Alessio