Beyond Geometric Closure: Measuring and Diagnosing Mechanical Consistency in Visual Dynamics

Qibin Gu; Yunfeng Yan

cs.CVOct 17, 2026

Beyond Geometric Closure: Measuring and Diagnosing Mechanical Consistency in Visual Dynamics

Authors: Qibin Gu, Yunfeng Yan

Organizations: Nanjing Tech University · Zhejiang University

Abstract

Visual accuracy, geometric validity, and agreement with applied forces are distinct requirements for visual dynamics. We introduce a diagnostic framework that connects reaction-free SI measurements to controlled interventions on coordinate readout, training integration, and actuator response. Reference replay calibrates the measurement path, while continuous scoring exposes progress against fixed engineering targets. With three seeds per setting, we evaluate KeyCLD and CON across 343 planar assemblies at 64/256 pixels and V-JEPA + AC and LingBot-VA on 100 shared spatial test assemblies. CON reduces image MSE by 37.6%/37.2% relative to persistence, yet neither planar model attains valid native geometry. At 256 pixels, calibrated RGB replay places forecast errors far above the reference measurement floor. Exact-state KeyCLD has identical geometric pass rates under Euler and RK4 training, yet its score rises from 52.65 to 64.30 as force and energy errors decrease. In both spatial pathways, CAD/IK reconstruction restores geometry while mean actuator residuals exceed five times the engineering targets. These diagnoses distinguish coordinate correction, numerical improvement, and persistent force-response errors, guiding targeted improvements to visual dynamics.

Figures & tables

Figure 1: Four pathways from pixels to mechanics. (a) After reconstruction, blue/tan bars show measurable/unavailable fractions, with measurable/total counts and percentages. Effort/energy plots show cases, quartiles, ranges, and dashed tolerances. Below, six contributions sum to $S_{\rm comp}$ ; arrows show reconstruction gains and 50 marks the maximum kinematic contribution. Scores use task-specific targets with balanced seeds, families, and conditions; each spatial pathway contains 900 cases (LingBot FT: 50 updates). (b,c) Physical readouts feed kinematic and effort/energy tests using independent SI parameters ( $Bu_{A}=e_{1}\tau_{A}$ for planar chains).

Criterion	Torque	Servo	Spatial
Closure	$1$ mm	$1$ mm	$1$ mm
Velocity	$0.01$ m/s	$0.01$ m/s	$0.01$ m/s
Acceleration	$0.1$ m/s 2	$0.1$ m/s 2	$0.1$ m/s 2
Actuator RMS	$0.1$ N m	$0.25$ N m	$5\%$ rated
Energy defect	$0.02$ J	$0.05$ J	$1\%$ work
Point RMSE	$1$ cm	$1$ cm	$1$ cm

Table 1: Engineering targets fixed by task and shared by its models. Torque/servo columns are planar; spatial effort and work scales follow Section 7 .

Figure 2: Assembly corpus. (a) Three planar families and their 36 test identities, with Stewart/Delta examples. (b–d) Planar distributions: density contours cover 343 assemblies, test identities are ringed, and inertia is median over 41 poses. Torque and absolute work use 36 references per condition; boxes/whiskers show quartiles/ranges, and torque is normalized by actuator limits.

Figure 3: Matched resolution and measurement controls. (a) The first test case, seed 42, at both resolutions; dashed/solid skeletons show reference/native joints. (b) Labeled mean MSE ratios to same-resolution persistence; solid/hatched bars show RGB/motion mask, and lower is better. (c) Correct and generated RGB through one frozen reader and phase differentiator; dashed lines mark SI targets. (d) Reconstruction costs: diamonds show balanced means, boxes quartiles, whiskers full ranges.

	Native output: primary forecast assessment									Diagnostic geometric reconstruction
Training	Finite	RGB MSE ( $10^{-4}$ )	Phys.	G / V / A (%)	Torque (%)	Energy (%)	Point (mm)	Joint (%)	$S_{\rm comp}$ (0–100) $\uparrow$	Cov.	Correction (mm)	Torque (N m)	Energy (J)	Point (mm)
KeyCLD
Video 64 px	311	4.96	0	0/22.2/65.1	N/A	N/A	229.2	0	$15.73\pm 5.00$	311	120.49	6.383	0.921	194.80
Video 256 px	324	4.52	0	0/53.7/75	N/A	N/A	233.9	0	$22.30\pm 12.08$	324	128.13	6.229	1.201	198.19
State–Euler	324	–	324	100/100/100	8.33	0.00	139.1	0	$52.65\pm 0.02$	324	$<0.001$	4.564	22.453	139.06
State–RK4	324	–	324	100/100/100	7.41	0.00	16.4	0	$64.30\pm 0.07$	324	$<0.001$	0.715	2.925	16.37

Table 2: Planar audit: 36 assemblies

\times

three conditions

\times

three seeds per row. Finite / Phys. / Cov. count finite outputs / native SI coverage / reconstructed SI coverage. G/V/A are closure/velocity/acceleration pass rates; N/A marks unavailable mechanics.

S_{\rm comp}

is mean

\pm

seed SD. Correction is coordinate RMS. Kinematics supply 50.00 of KeyCLD’s 53.10 reconstructed points at 256 pixels. Point errors use maximum-channel RMSE on the native 240 Hz grid (Table S37 ), with available-case means balanced by seed/family/condition. State controls are resolution-independent. Persistence RGB MSE is 4.94/5.04 at 64/256 pixels.

Figure 4: Integration and resolution controls. (a–f) Resolution-independent Euler/RK4 pairs compare native point error and reconstructed force/energy; below-diagonal points favor RK4. (g–i) Six curves share one case, seed 42, time window, and geometric readout. Green bands mark tolerances; vertical dotted lines start forecasting.

Figure 5: Structural evidence. (a) Six checkpoints give zero odd torque on 648 states. The 100 Hz inset compares RMS odd acceleration (reference bars/dots, model crosses; symlog scale). (b) Each integrator pools 1,944 queries; the inset shows failed gain checks. (c) Mass fits use 44,280 poses, with separate/shared and family errors in the insets.

Figure 6: Spatial diagnostics. (a) LingBot video and CAD readouts. (b) Paired 50/0-update MSE ratios. (c) Finite point errors and mean bars (450 registered per setting/family). (d) Mechanics on 450 Stewart and 424 Delta pairs. Warm/green regions mark errors above reference/within tolerance; contours show density, diamonds means, and insets paired-ratio quartiles, ranges, and means.

		Native output			CAD intervention
Family	Step	RGB	Motion	$S_{\rm comp}$	$S_{\rm comp}$	Cov.
Stewart	0	41.67	2.59	$5.67\!\pm\!0.05$	$54.60\!\pm\!0.19$	450/450
Stewart	50	36.42	2.69	$6.01\!\pm\!0.15$	$54.31\!\pm\!0.51$	450/450
Delta	0	29.43	1.58	$2.80\!\pm\!0.08$	$43.37\!\pm\!1.17$	359/450
Delta	50	28.83	1.70	$3.39\!\pm\!0.02$	$51.92\!\pm\!1.17$	424/450
AC Stewart	386	–	–	$6.20\!\pm\!0.69$	$55.11\!\pm\!1.71$	450/450

Table 3: Spatial audit, 50 assemblies

\times

three conditions

\times

three seeds per row. Step counts updates. Scores are mean

\pm

seed SD; Cov. counts complete CAD measurements. RGB/motion MSE units:

10^{-4}/10^{-2}

		Effort RMS		Energy defect
Path / prediction	$n$	Ref.	Pred.	Ref.	Pred.
Planar / KeyCLD	324	0.116	6.565	0.057	10.835
Planar / CON	324	0.116	6.585	0.057	1.485
AC / Stewart	450	3.108	41.967	0.627	1.504
AC / Delta	432	2.083	12.425	0.985	2.462
LingBot / Stewart	450	3.009	40.513	0.605	1.640

Table 4: Matched reference replay and prediction. Each pair uses identical cases, a frozen reader, and the same reconstruction and derivatives. Planar rows use 256-pixel RGB with projected phase; spatial rows use CAD. Effort is N for Stewart and N m otherwise; energy is J. LingBot uses 50 updates. Reference energy passes on Stewart and Delta are 150/150 each for exact and 30 Hz sampled states, versus 48/150 and 63/150 through RGB (Table S32 ).

Appendix figures & tables46 assets

Supplementary material from the paper’s appendix.

Appendix

Condition	Change
Viscous	$b=(0.10,0.03,0.02,0.04)$ N m s/rad
Coulomb	$f_{c}=(0.008,0.005,0.005,0.007)$ N m
Mixed	Both preceding friction terms
Payload	Payload mass $0.25$ kg; cuboid size fixed
Density	Bar density $4050$ kg/m 3 ; payload fixed
Size	Link and ground lengths $\times 0.8$ ; along-coupler force/payload offsets $\times 0.8$ ; width and thickness fixed

Appendix

Table S1 : Changes from the nominal source condition. Unlisted physical parameters retain their source values.

Control	Rod-length RMSE ( $\mu$ m)	Tangent force RMS
State–Euler	0.121	0.977704 N m
State–RK4	0.449	0.459588 N m

Appendix

Table S2 : State-control force diagnostic on 50 original test trajectories. This uses a unit-norm null-space tangent

n

, alongside the crank-normalized

\gamma

used for actuator effort.

Figure S1 : Visual and integration diagnostics. (a) Reference, framewise reconstruction, and forecast at three fixed times. Reconstruction uses each true image; forecasts use three initial frames. (b) Fifty trajectories $\times$ three readouts; adjacent bars show mean coordinate RMSE $\pm$ SD (mm). (c) Paired state-supervised Euler/RK4 errors; diamond: mean.

Control	RGB MSE	Motion MSE	VPT
Framewise recon.	0.000118	0.000696	49.00
Video forecast	0.00894	0.0506	7.82
Last-frame persistence	0.00898	0.0533	7.98
Training-mean image	0.00483	0.0283	8.10

Appendix

Table S3 : Visual controls on the 50 torque-driven tests. MSE averages normalized RGB squared error. Motion MSE uses the temporal-variance mask. Framewise reconstruction receives each true image; the other controls use the observed prefix or a training-set statistic. VPT retains the author’s index convention.

Role	Conditions	Clips	Seeds
Source training	1	450	0–449
Source validation	1	50	450–499
Original held-out test	1	50	500–549
Extended transfer/probes	8	160	1000–1019
Torque calibration	8	96	2000–2011
Torque test	8	160	3000–3019

Appendix

Table S4 : Disjoint data roles used in the paper. Transfer probes and final scoring use different test seeds.

Condition	Torque RMS mean $\pm$ std (N m)	Energy error (J)	Position error (mm)	Torque passes
Nominal	$0.444\!\pm\!0.454$	3.397	61.6	10/20
Viscous	$4.318\!\pm\!4.148$	121.155	109.6	0/20
Coulomb	$0.442\!\pm\!0.448$	3.220	61.7	10/20
Mixed	$4.274\!\pm\!4.188$	120.860	111.5	0/20
Payload	$1.246\!\pm\!1.645$	1.955	59.9	9/20
Density	$0.537\!\pm\!0.536$	3.087	55.1	10/20

Appendix

Table S5 : Frozen State–RK4 component on the new test set. Each row contains 20 trajectories. Torque and position entries are means of trajectory-level RMS values; energy is the mean of each trajectory’s maximum cumulative residual. Position uses the maximum per-channel RMS. The torque standard deviation describes trajectories, not independent training seeds.

Condition	Positive	Median error	Mean error
Nominal	71/73	0.3516	3.4918
Payload	71/73	0.4156	2.6752
Density	71/73	0.5122	2.4735
Size	66/73	0.8542	5.6663
Slider	64/73	0.5185	3.6548

Appendix

Table S6 : Behavioral effective-inertia errors from calibrated

\pm 1

N m action queries at zero speed. Error is the dimensionless fraction

\lvert\widehat{\mathcal{I}}_{\rm resp}/\mathcal{I}-1\rvert

, evaluated on positive gains.

Condition	Rank	Normalized condition number	Maximum mass error	Test torque RMS (N m)
Nominal	12/12	483.3	$1.28\!\times\!10^{-6}$	$8.72\!\times\!10^{-8}$
Viscous	12/12	119.9	$0.67\!\times\!10^{-6}$	$7.03\!\times\!10^{-8}$
Coulomb	12/12	494.9	$0.56\!\times\!10^{-6}$	$7.64\!\times\!10^{-8}$
Mixed	12/12	123.4	$1.26\!\times\!10^{-6}$	$3.37\!\times\!10^{-8}$
Payload	12/12	300.1	$0.44\!\times\!10^{-6}$	$8.55\!\times\!10^{-8}$
Density	12/12	250.6	$0.90\!\times\!10^{-6}$	$8.54\!\times\!10^{-8}$

Appendix

Table S7 : Exact-state physical parameter control on the extended dataset, with known geometry, inertia-to-mass ratios, and SI torque. Mass error is the largest absolute relative error among four masses, expressed as a fraction.

Figure S2 : Independent physical errors of the frozen State–RK4 component on the torque-excitation scoring set. Each condition contains 20 paired trajectories. Points are individual trajectories, black marks show the median and interquartile range, and dashed lines are the fixed diagnostic tolerances. All trajectories pass the three geometric/differential constraint criteria, while physical residuals vary substantially with condition. Both axes are logarithmic.

Figure S3 : Torque-excitation pass profile, balanced across mechanism families and conditions. Hatched N/A cells have no valid physical readout and are distinct from measured zero-pass rates. All 160 state trajectories pass the three constraint criteria, while no trajectory passes work–energy balance.

Figure S4 : Constitutive probes of the frozen state component. (a) Matched velocity reversal gives nonzero reference odd acceleration and zero model response for all 292 pairs per condition. (b) Effective-inertia errors on positive input gains: both medians and means are shown on a log scale, with positive-gain counts below each condition. The large mean–median gaps retain the effect of near-zero gains.

Condition	True $\alpha$	Model $\alpha$	$r_{\tau}$
	(rad/s 2 )	(rad/s 2 )	(N m)
Nominal	590.32	613.06	0.08196
Mixed friction	48.46	141.58	0.53881
Slider	$-144.72$	95.43	0.96611

Appendix

Table S8 : Matched-state acceleration queries at the fixed first extended-test seed and

t=0.11

s. Each condition uses its corresponding true pose and speed. These are local constitutive probes, separate from trajectory scoring.

Family	Train	Val.	Cal.	Test
Crank-rocker	96	6	6	12
Crank-slider	92	6	6	12
Double-crank	83	6	6	12
Assemblies	271	18	18	36
Trajectories	813	54	54	108

Appendix

Table S9 : Multi-assembly partition. Each assembly contributes three operating conditions. Validation and numerical calibration occupy separate subsets of the original validation partition.

Arm	Seed	Finite	Timeouts	Physical readout	$S_{\rm comp}$
Video	42	108/108	0	0/108	14.78
Video	43	108/108	0	0/108	21.14
Video	44	95/108	13	0/108	11.28
State–Euler	42	108/108	0	108/108	52.66
State–Euler	43	108/108	0	108/108	52.62
State–Euler	44	108/108	0	108/108	52.66

Appendix

Table S10 : All multi-assembly test runs. Every checkpoint completes 40 epochs and 32,520 updates. Finite output and a complete physical readout are distinct requirements.

Arm	Family	Point (mm)	Torque (N m)	Energy (J)	Point pass (%)	Torque pass (%)
State–Euler	Crank-rocker	69.20	1.907	9.535	25.00	25.00
State–Euler	Crank-slider	81.31	3.172	38.040	0.00	0.00
State–Euler	Double-crank	266.68	8.612	19.785	0.00	0.00
State–RK4	Crank-rocker	12.91	0.320	1.544	42.59	17.59
State–RK4	Crank-slider	32.48	1.434	6.228	35.19	0.00
State–RK4	Double-crank	3.71	0.391	1.004	97.22	4.63

Appendix

Table S11 : Mechanism-level state-control results. Each row averages 108 predictions: 12 assemblies, three conditions, and three seeds. Errors are means of trajectory statistics. Point error is maximum-channel RMS; energy error is maximum absolute prefix imbalance. All rows have 100% passes on the three constraint criteria.

Training	G	V	A	F	E	P	$S_{\rm comp}$
State Euler	100.00	100.00	100.00	6.98	0.48	8.44	52.65
State RK4	100.00	100.00	100.00	29.93	0.58	55.29	64.30
$\Delta/6$ (points)	0.0002	0.0000	0.0000	3.82	0.02	7.81	11.65

Appendix

Table S12 : Exact-state KeyCLD continuous component scores. G/V/A are closure/velocity/acceleration, F/E are torque/energy, and P is tracking. Each component uses

100/(1+\rho^{2})

;

S_{\rm comp}

is their arithmetic mean. The final row gives each component’s contribution to the RK4-minus-Euler composite gain. Both arms have 324/324 coverage and zero joint passes.

Arm	Seed	Positive gains	Median inertia error	Mean inertia error	Friction RMS (N m)
State–Euler	42	445/648	74.24	220.28	0.039
State–Euler	43	458/648	73.68	325.58	0.039
State–Euler	44	443/648	73.90	243.99	0.039
State–RK4	42	570/648	115.39	210.19	0.039
State–RK4	43	570/648	113.45	206.87	0.039
State–RK4	44	575/648	120.12	224.87	0.039

Appendix

Table S13 : Counterfactual SI response of the corpus state controls. Inertia error is

\lvert\widehat{\mathcal{I}}_{\rm resp}/\mathcal{I}-1\rvert

, conditioned on positive gain. Friction error is the RMS equivalent torque of the missing velocity-odd response. Each checkpoint uses the same 648 physical states.

Model	Arms	Epochs	Batch	Updates	Sequence uses	Training integration
KeyCLD	Video, State–Euler, State–RK4	40	1	32,520	32,520	Euler / RK4 substeps
CON	Video, State–Dopri5	50	16	2,550	40,650	Dopri5, $0.01$ s steps

Appendix

Table S14 : 64-pixel video and state-control training configurations. Counts are per run. All arms use 813 training sequences and seeds 42, 43, and 44. A sequence use is one sequence in one epoch; temporal loss windows differ by model.

Interface	Physical	Geom.	Vel.	Acc.	Torque	Energy	Pred.	Joint	$S_{\rm comp}$ [95% CI]
KeyCLD
Video	0/324	0.00	22.22	65.12	N/A	N/A	0.00	0	15.73 [11.36, 20.97]
State–Euler	324/324	100.00	100.00	100.00	8.33	0.00	8.33	0	52.65 [52.30, 52.96]
State–RK4	324/324	100.00	100.00	100.00	7.41	0.00	58.33	0	64.30 [63.60, 65.03]
CON
Video + affine	0/324	0.00	0.93	33.02	N/A	N/A	0.00	0	7.23 [3.72, 9.73]

Appendix

Table S15 : Native criterion pass rates (%) and continuous

S_{\rm comp}

with seed/assembly bootstrap intervals for the five original assembly arms. Physical counts complete SI measurements. Table S12 separately gives continuous state-control components.

Arm	Seed	Finite	Physical	$S_{\rm comp}$	Point (mm)	RGB MSE	Reader val. (mm)
Video + affine	42	108/108	0/108	9.51	123.84	3.1102	90.39
Video + affine	43	108/108	0/108	3.53	124.75	3.1278	86.42
Video + affine	44	108/108	0/108	8.64	121.86	3.0027	93.54
State	42	108/108	0/108	1.18	167.48	N/A	N/A
State	43	108/108	0/108	2.75	128.01	N/A	N/A
State	44	108/108	0/108	1.80	131.68	N/A	N/A

Appendix

Table S16 : CON seed results. Finite and complete physical coverage refer to 108 test trajectories. Point error is mean maximum-channel future RMS. RGB MSE uses future frames 33–119 and is displayed in units of

10^{-4}

. Reader validation uses observed validation images and coordinate RMSE, not future forecast error. State has no RGB decoder or fitted reader.

Model	Reconstruction	Forecast	Finite forecast	Persistence	Reset Euler	Reset RK4	Reset hold
KeyCLD	2.7100	4.9595	311/324	4.9399	5.208	5.186	5.272
CON	2.7952	3.0802	324/324	4.9399	14.416	14.341	17.605

Appendix

Table S17 : Matched assembly-video diagnostics. RGB MSE is in units of

10^{-4}

on future frames 33–119. Four-step reset error measures self-consistency against later observed encodings, in millimeters through each model’s distinct fixed point readout. Means balance seed, family, and condition.

Figure S5 : Last forecast frame (119) of the first frozen test case, seed 42. Reconstruction uses the current true image to test visual representation, while forecasting uses frames 30–32 to test temporal prediction.

Model	Solver / steps	Finite	RGB MSE	Native difference
KEYCLD	euler / 1	311/324	5.17041	6.389e+15 (299)
KEYCLD	euler / 4	322/324	4.98690	1.25 (310)
KEYCLD	euler / 16	323/324	4.96358	0.3465 (311)
KEYCLD	rk4 / 1	321/324	4.97731	7.02e+11 (309)
KEYCLD	rk4 / 4	324/324	4.95350	0.02659 (311)
KEYCLD	rk4 / 16	324/324	4.95351	0.003339 (311)

Appendix

Table S18 : Frozen video inference sweeps. RGB MSE is in units of

10^{-4}

. Native difference is point-coordinate RMS in millimeters; parentheses count common finite cases. Finite trajectories have denominator 324. KeyCLD’s one-step Euler value,

6.389\times 10^{15}

mm, records numerical divergence; refinement to four/sixteen steps gives 1.25/0.347 mm.

Training solver	Point	Geometry	Raw physical
EULER	124.475	126.306	0/324
RK4	137.382	142.066	0/324

Appendix

Table S19 : Additional CON state-training controls, averaged over three seeds. Point error uses maximum-channel future RMS; geometry uses the trajectory maximum, both in millimeters. The original State–Dopri5 arm remains a separate reference.

Model / training	Raw physical	Readout physical	Correction (mm)	Torque (N m)	Energy (J)	Point (mm)
KeyCLD Video	0/324	311/324	120.4944	6.3830	0.9213	194.798
KeyCLD State–Euler	324/324	324/324	0.0009	4.5635	22.4534	139.062
KeyCLD State–RK4	324/324	324/324	0.0006	0.7148	2.9253	16.368
CON Video	0/324	308/324	73.1848	6.8593	17.9601	154.833
CON State–Dopri5	0/324	323/324	67.6754	6.5488	23.2825	132.278
CON State–Euler	0/324	317/324	54.7288	6.0279	11.8472	133.064

Appendix

Table S20 : Explicit geometric-readout intervention on seven assembly arms,

n=324

per arm. Coverage counts complete force/energy trajectories. Correction is per-trajectory coordinate RMS over the saved grid; point error is maximum-channel future RMS. Correction and residual means use available cases with equal seed/family/condition weighting.

Figure S6 : Independent force/energy evaluation after the common geometric readout, using the first frozen test case and seed 42; video arms use 64-pixel inputs. Panels show wrapped phase error in

[-\pi,\pi]

\lvert r_{\tau}\rvert

, and

\lvert r_{E}\rvert

. Dashed lines show the

0.25\,\mathrm{N\,m}

and

0.05

J criteria; the dotted line marks the first future frame. Work is anchored at the last observed frame.

Figure S7 : Spatial platform trajectories for the first registered Stewart and Delta test cases, seed 42. Black curves are reference positions; purple curves are V-JEPA 2.1 + AC with the frozen physical readout. The dotted line ends the three-frame observed prefix. All three coordinates and the full forecast horizon are shown.

Figure S8 : Five training camera views of Stewart (top) and Delta (bottom). Each row shows frame zero from the first registered training assembly in its low-speed condition. All views show the same physical state; only the camera changes. The full recorded frames are shown without cropping.

Family	Native $S_{\rm comp}$	CAD $S_{\rm comp}$	Mech.	Joint
Stewart	$6.20\!\pm\!0.69$	$55.11\!\pm\!1.71$	450/450	0/450
Delta	$10.40\!\pm\!5.34$	$50.05\!\pm\!2.71$	432/450	0/450

Appendix

Table S21 : V-JEPA + AC on 50 test assemblies per family, three conditions and three seeds. Scores give the mean

\pm

seed SD. CAD reconstruction is a separate intervention.

Family	Seed	Native $S_{\rm comp}$	CAD $S_{\rm comp}$	Mech.	Correction	Raw point	CAD point	Effort	Energy
Stewart	42	6.51	54.26	150/150	2.36	29.00	29.01	39.652	2.140
Stewart	43	6.68	57.07	150/150	1.95	30.47	30.15	42.015	0.971
Stewart	44	5.41	53.99	150/150	2.65	36.02	36.10	44.235	1.400
Delta	42	10.98	50.52	147/150	2.13	23.47	23.44	12.570	2.688
Delta	43	4.78	47.14	135/150	2.38	29.22	29.00	14.893	1.342
Delta	44	15.42	52.50	150/150	3.40	24.93	24.58	9.812	3.356

Appendix

Table S22 : V-JEPA + AC by seed on 150 trajectories per family. Point errors and correction costs are in mm; effort is in N for Stewart and N m for Delta, and energy is in J. CAD residuals average mechanically measurable cases.

Family	Readout	Input	Point RMSE		$S_{\rm comp}$		After CAD reconstruction
			Raw	CAD	Raw	CAD	Mech.	Effort	Energy
Stewart	Internal	Reference	3.59	4.52	15.63	71.32	450/450	8.882	2.186
Stewart	Calibrated	Reference	2.07	1.88	28.88	86.46	150/150	3.108	0.627
Stewart	Internal	Forecast	33.11	44.59	1.66	43.51	343/450	72.865	11.354
Stewart	Calibrated	Forecast	31.83	31.75	6.20	55.11	450/450	41.967	1.504
Delta	Internal	Reference	10.29	19.23	10.10	36.97	298/450	9.335	4.948

Appendix

Table S23 : Three independently trained AC predictors with internal and calibrated physical readouts. Reference and forecast features share each measurement head. Point errors are in mm; effort is in N for Stewart and N m for Delta, and energy is in J. Raw/CAD scores distinguish unconstrained decoded coordinates from their subsequent CAD/IK reconstruction. SI residuals average the mechanically measurable cases.

		Effort		Energy
Family	Pairs	Ref.	Pred.	Ref.	Pred.
Stewart	450	3.108	41.967	0.627	1.504
Delta	432	2.083	12.425	0.985	2.462

Appendix

Table S24 : Matched AC reference and forecast mechanics after calibrated readout and CAD/IK reconstruction. Each mean first averages the measurable cases within a seed, then averages seeds. Effort is in N for Stewart and N m for Delta; energy is in J.

Family	Updates	Seed	RGB	Motion	Native $S_{\rm comp}$	CAD $S_{\rm comp}$	Mech.	Effort	Energy
Stewart	0	42	38.84	2.55	5.64	54.38	150/150	39.548	1.153
Stewart	0	43	48.55	2.67	5.73	54.65	150/150	39.595	1.114
Stewart	0	44	37.62	2.54	5.65	54.75	150/150	39.401	1.074
Stewart	50	42	35.23	2.63	6.15	54.88	150/150	40.499	1.354
Stewart	50	43	37.19	2.72	5.85	53.88	150/150	40.278	1.931
Stewart	50	44	36.85	2.72	6.02	54.17	150/150	40.763	1.635

Appendix

Table S25 : LingBot results by seed on 150 test trajectories per family. RGB MSE uses

10^{-4}

units and motion-mask MSE

10^{-2}

. Force and energy means use the mechanically measurable CAD subset. Effort is in N for Stewart and N m for Delta; energy is in J.

Family	Updates	Readout	Reach.	Correction	Point	CAD point	$G$	$V$	$A$	$F$	$E$	$P$
Stewart	0	450/450	450/450	2.08	29.68	29.63	100.00	99.59	96.68	3.49	17.50	10.32
Stewart	50	450/450	450/450	2.11	30.01	29.92	100.00	99.76	98.32	3.37	14.14	10.26
Delta	0	374/450	372/450	7.64	22.80	22.01	82.67	80.34	71.43	0.30	11.15	14.34
Delta	50	442/450	428/450	9.06	23.23	23.24	95.11	94.23	90.38	0.37	15.88	15.58

Appendix

Table S26 : LingBot reconstruction and score components. Counts pool three seeds (450 registered cases per row). Correction and point errors are in mm and average available cases within seed, then across seeds.

G,V,A,F,E,P

are reconstructed closure, velocity, acceleration, effort, energy, and point scores on a 0–100 scale; their mean is

S_{\rm comp}

Reference image quality	Clean-trained	Augmented
Clean	1.81	1.90
Blur $\sigma=0.5$	2.92	2.04
Blur $\sigma=1$	13.22	2.64
Blur + $0.6\times$ contrast	64.80	2.73

Appendix

Table S27 : Planar reader ablation on 27 verification sequences. Entries are mean Euclidean point RMSE (mm). Augmentation targets image degradation without imposing closure.

RGB source	Finite	Point (mm)	Torque (N m)	Energy (J)
Reference RGB	108/108	2.13	0.73	0.57
KeyCLD, 42	108/108	185.68	7.00	29.87
KeyCLD, 43	108/108	183.16	6.69	8.06
KeyCLD, 44	95/108	191.94	6.86	23.96
CON, 42	108/108	167.70	6.49	7.46
CON, 43	108/108	165.23	6.91	19.38

Appendix

Table S28 : Matched public-reader ablation, 108 planar test clips per seed. Point: Euclidean RMSE from the common frozen RGB reader; torque and energy: the projected phase diagnostic below. Means use finite cases. Reference replay measures the instrument’s own residuals.

Figure S9 : Measurement controls. (a) Planar reader training with and without degradation augmentation. (b) Exact-coordinate differentiation errors relative to engineering targets. (c) Absolute-coordinate and CAD-coordinate heads at 256-pixel input resolution. (d) Coordinate-spline and cubic-phase differentiation of correct videos. Identical reference trajectories separate measurement accuracy from forecasting performance; configuration details and sensitivity controls are given in the text.

Configuration	Stewart	Delta	S cov.	D cov.
Absolute-coordinate head	11.02	12.37	6/6	3/6
CAD-coordinate head	2.65	4.82	6/6	6/6
+ known-pose alignment	2.96	3.44	6/6	6/6

Appendix

Table S29 : Coordinate-head verification, six clips per family. Point RMSE is in mm; coverage counts mechanically measurable reconstructions. The alignment row additionally uses a known initial pose.

Input / configuration	Point	Effort	Energy
Exact sampled states	0.00	0.01	0.02
Relative feature head	3.54	2.14	0.83
+ image-plane tracking	2.44	0.81	0.18
Held CAD reset	21.22	12.13	0.00

Appendix

Table S30 : Delta readout components on six independent verification clips. All rows use the same CAD reconstruction and derivative estimator. Point is full-marker Euclidean RMSE in mm; effort is in N m and energy in J. The held reset is a coordinate persistence control.

			Effort RMS		Energy defect
Family	Updates	$n$	Ref.	Gen.	Ref.	Gen.
Stewart	0	450	3.009	39.515	0.605	1.113
Stewart	50	450	3.009	40.513	0.605	1.640
Delta	0	359	0.733	12.295	0.209	0.613
Delta	50	424	0.733	12.319	0.203	1.206

Appendix

Table S31 : Matched LingBot mechanics after CAD reconstruction. Reference and generated videos use identical cases and processing. Values average per-seed available-case means;

n

pools the three seeds out of 450 registered cases. Effort is in N (Stewart) or N m (Delta); energy is in J.

Family	Input	Point	Effort	Energy	Effort pass	Energy pass
Stewart	Exact states	0.0000	0.0000	0.0064	150/150	150/150
Stewart	30 Hz differences	0.0000	0.0477	0.0253	150/150	150/150
Stewart	Reference RGB + CAD	1.8287	3.0090	0.6052	149/150	48/150
Delta	Exact states	0.0000	0.0000	0.0056	150/150	150/150
Delta	30 Hz differences	0.0000	0.0145	0.0253	150/150	150/150
Delta	Reference RGB + CAD	2.2475	0.7339	0.2020	91/150	63/150

Appendix

Table S32 : Spatial measurement controls on all 300 reference trajectories. Exact derivatives isolate the mechanics evaluator; 30 Hz configuration differences additionally test differentiation; reference RGB + CAD includes the frozen visual measurement path. Point RMSE is in mm, effort in N (Stewart) or N m (Delta), and energy in J. Pass counts use each family’s 150 distinct trajectories.

Model	Pixels	Parameters	Epochs	Batch	Updates	Sequence exposures	Forecast intervals
KeyCLD	$64^{2}$	596,508	40	1	32,520	32,520	4 per window
KeyCLD	$256^{2}$	2,378,268	40	1	32,520	32,520	4 per window
CON	$64^{2}$	67,271,267	50	16	2,550	40,650	48 per sequence
CON	$256^{2}$	1,075,870,307	50	16	2,550	40,650	48 per sequence

Appendix

Table S33 : Native video training at both resolutions. Counts are per seed; all rows use seeds 42–44 and the same 813 training sequences. Resolution changes image-layer capacity, while epochs, batches, sequence exposures, and the model-specific dynamical objectives stay fixed.

Model	Pixels	Seed	RGB MSE	Motion-mask MSE	Native $S_{\rm comp}$	Reconstructed $S_{\rm comp}$
KeyCLD	64	42	4.2396	5.5810	14.78	52.02
KeyCLD	64	43	5.5418	6.5761	21.14	54.32
KeyCLD	64	44	5.0970	6.0968	11.28	47.02
KeyCLD	256	42	4.8051	6.5578	33.37	54.32
KeyCLD	256	43	3.8003	4.9822	24.12	51.65
KeyCLD	256	44	4.9483	6.6122	9.41	53.33

Appendix

Table S34 : All twelve native video fits, with 108 test trajectories per seed. RGB MSE is in

10^{-4}

and motion-mask MSE in

10^{-3}

Table S35 : Continuous criterion scores (0–100), with equal seed/family/condition weighting. G/V/A are geometric, velocity, and acceleration consistency; F/E are effort and energy. The arithmetic mean of the six component columns is native

S_{\rm comp}

Measurement path	Future support	Coordinate statistic	Reference (mm)	KeyCLD (mm)	CON (mm)
Native-grid reconstruction	689 samples, 240 Hz	Maximum-channel RMS	–	198.19	167.35
Unprojected RGB reader	87 frames, 1.100–3.967 s	Euclidean RMS	0.334	194.994	208.454
Projected-phase RGB reader	80 frames, 1.167–3.800 s	Maximum-channel RMS	0.144	183.029	181.873

Appendix

Table S37 : Planar 256-pixel point-error definitions. All entries average trajectory RMSEs, with model results balanced across seeds, families, and conditions. The native-grid reconstruction uses each model’s declared physical readout; the other paths use the independent frozen RGB reader.

Model / reference	Track	Coverage	$0.5\times$	$1\times$	$2\times$	$0.5\times$	$1\times$	$2\times$
			$S_{\rm comp}$			Joint passes
KeyCLD Euler	State native	324/324	51.35	52.65	54.28	0	0	0
KeyCLD RK4	State native	324/324	57.67	64.30	72.19	0	0	0
KeyCLD 64 px	Native	0/324	10.68	15.73	21.56	0	0	0
KeyCLD 64 px	Reconstructed	311/324	49.79	51.12	52.96	0	0	0
KeyCLD 256 px	Native	0/324	18.85	22.30	26.21	0	0	0

Appendix

Table S38 : Target sensitivity on fixed measurements. Coverage is complete six-criterion availability and is constant across scales. Joint entries count cases passing every scaled target. CAD and reconstructed rows apply geometric interventions to saved forecasts.

Dynamical Systems

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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MechVerse: Evaluating Physical Motion Consistency in Video Generation Models

Rahul Jain, Mayank Patel, Asim Unmesh, Karthik Ramani

School of Electrical and Computer Engineering, Purdue University · School of Mechanical Engineering, Purdue University

Physical Consistency in Video Generation Image-to-Video Generation Video Generation Evaluation Geometric Consistency in Video Generation

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Text- and image-conditioned video generation models have achieved strong visual fidelity and temporal coherence, but they often fail to generate motion governed by kinematic and geometric constraints. In these settings, object parts must remain rigid, maintain contact or coupling with neighboring components, and transfer motion consistently across connected parts. These requirements are especially explicit in articulated mechanical assemblies, where motion is constrained by rigid-link geometry, contact/coupling relations, and transmission through kinematic chains. A generated video may therefore appear plausible while violating the intended mechanism, such as rotating a part that should translate, deforming a rigid component, breaking coupling between parts, or failing to move downstream components. To evaluate this gap, We introduce MechVerse, a benchmark for mechanically consistent image-to-video generation. MechVerse contains 21,156 synthetic clips from 1,357 mechanical assemblies across 141 categories, organized into three tiers of increasing kinematic complexity: independent articulation, pairwise coupling, and densely coupled multi-part mechanisms. Each clip is paired with a structured prompt describing part identities, stationary supports, moving components, motion primitives, direction, speed/extent, and inter-part dependencies. We evaluate proprietary, open-source, and fine-tuned image-to-video models using standard video metrics, instruction-following scores, and human judgments of motion correctness and kinematic coupling. Results show that current models can preserve appearance and smoothness while failing to generate mechanically admissible motion, with errors increasing as coupling complexity grows. MechVerse provides a benchmark for measuring and improving mechanism-aware video generation from image and language inputs.

Abstract

Dynamical Systems

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