-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathindex.html
More file actions
892 lines (787 loc) · 40.4 KB
/
Copy pathindex.html
File metadata and controls
892 lines (787 loc) · 40.4 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta name="description" content="ContractionPPO: Certified Reinforcement Learning via Differentiable Contraction Layers">
<meta name="keywords" content="ContractionPPO, reinforcement learning, contraction theory, legged locomotion, quadruped robots, certified control, stability guarantees, PPO, robotics, AI, autonomous systems, control theory, machine learning">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="author" content="Vrushabh Zinage, Narek Harutyunyan, Eric Verheyden, Fred Hadaegh, Soon-Jo Chung">
<meta name="robots" content="index, follow">
<meta name="language" content="English">
<meta name="revisit-after" content="7 days">
<meta name="distribution" content="global">
<meta name="rating" content="general">
<meta name="subject" content="Reinforcement Learning, Contraction Theory, Legged Locomotion, Certified Control">
<meta name="coverage" content="Worldwide">
<meta name="target" content="all">
<meta name="HandheldFriendly" content="True">
<meta name="format-detection" content="telephone=no">
<meta name="theme-color" content="#3498db">
<meta name="msapplication-TileColor" content="#3498db">
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="default">
<meta name="apple-mobile-web-app-title" content="ContractionPPO">
<meta name="application-name" content="ContractionPPO">
<meta name="msapplication-TileImage" content="static/images/favicon.ico">
<meta name="msapplication-config" content="none">
<link rel="canonical" href="https://contractionppo.github.io">
<link rel="alternate" hreflang="en" href="https://contractionppo.github.io">
<link rel="alternate" hreflang="x-default" href="https://contractionppo.github.io">
<meta property="og:title" content="ContractionPPO: Certified Reinforcement Learning via Differentiable Contraction Layers"/>
<meta property="og:description" content="ContractionPPO is a certified reinforcement learning approach that uses differentiable contraction layers for provably stable and robust policy optimization."/>
<meta property="og:url" content="https://contractionppo.github.io"/>
<title>ContractionPPO: Certified Reinforcement Learning via Differentiable Contraction Layers</title>
<link rel="icon" type="image/x-icon" href="static/images/favicon.ico">
<meta property="og:image" content="https://contractionppo.github.io/assets/first_photo.jpg" />
<meta name="twitter:title" content="ContractionPPO: Certified Reinforcement Learning via Differentiable Contraction Layers">
<meta name="twitter:description" content="ContractionPPO is a certified reinforcement learning approach that uses differentiable contraction layers for provably stable and robust policy optimization.">
<!-- Path to banner image, should be in the path listed below. Optimal dimenssions are 1200X600-->
<meta name="twitter:image" content="https://contractionppo.github.io/assets/first_photo.jpg">
<meta name="twitter:card" content="summary_large_image">
<!-- Keywords for your paper to be indexed by-->
<!-- <meta name="keywords" content="KEYWORDS SHOULD BE PLACED HERE"> -->
<!-- Google tag (gtag.js) -->
<script async src="https://www.googletagmanager.com/gtag/js?id=G-6H457V6YSW"></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag(){dataLayer.push(arguments);}
gtag('js', new Date());
gtag('config', 'G-6H457V6YSW');
</script>
<link href="https://fonts.googleapis.com/css?family=Google+Sans|Noto+Sans|Castoro" rel="stylesheet">
<link rel="stylesheet" href="./static/css/bulma.min.css">
<link rel="stylesheet" href="./static/css/bulma-carousel.min.css">
<link rel="stylesheet" href="./static/css/bulma-slider.min.css">
<link rel="stylesheet" href="./static/css/fontawesome.all.min.css">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1/css/academicons.min.css">
<link rel="stylesheet" href="./static/css/index.css">
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.5.1/jquery.min.js"></script>
<script defer src="./static/js/fontawesome.all.min.js"></script>
<script src="./static/js/bulma-carousel.min.js"></script>
<script src="./static/js/bulma-slider.min.js"></script>
<script src="./static/js/index.js"></script>
<script type="module" src="https://gradio.s3-us-west-2.amazonaws.com/4.36.1/gradio.js"></script>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\\[', '\\]']]
}
};
</script>
<script type="text/javascript" id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script type="application/ld+json">
{
"@context": "https://schema.org",
"@type": "ScholarlyArticle",
"headline": "ContractionPPO: Certified Reinforcement Learning via Differentiable Contraction Layers",
"description": "ContractionPPO is a certified reinforcement learning approach that uses differentiable contraction layers for provably stable and robust policy optimization.",
"keywords": "ContractionPPO, reinforcement learning, contraction theory, legged locomotion, quadruped robots, certified control, stability guarantees, PPO, robotics, AI, autonomous systems, control theory, machine learning",
"author": [
{
"@type": "Person",
"name": "Vrushabh Zinage",
"url": "https://vrushabh27.github.io/vrushabh_zinage/"
},
{
"@type": "Person",
"name": "Narek Harutyunyan",
"url": "https://linkedin.com/in/narek-harutyunyan-428069233"
},
{
"@type": "Person",
"name": "Eric Verheyden",
"url": "https://linkedin.com/in/eric-verheyden"
},
{
"@type": "Person",
"name": "Fred Hadaegh",
"url": "https://cast.caltech.edu/people/fred-hadaegh"
},
{
"@type": "Person",
"name": "Soon-Jo Chung",
"url": "https://www.eas.caltech.edu/people/sjchung"
}
],
"publisher": {
"@type": "Organization",
"name": "California Institute of Technology"
},
"datePublished": "2025",
"dateModified": "2025",
"mainEntityOfPage": {
"@type": "WebPage",
"@id": "https://contractionppo.github.io"
},
"url": "https://contractionppo.github.io",
"image": "https://contractionppo.github.io/assets/first_photo.jpg"
}
</script>
<style>
.line-height-adjust {
line-height: 1.1;
}
.button-shadow {
box-shadow: 4px 4px 8px rgba(0, 0, 0, 0.2);
}
.blockMobile {
display: none; /* Hide the component by default */
}
.image-overlay {
position: absolute;
right: 0;
top: 50%;
transform: translateX(30%) translateY(-50%);
}
@media only screen and (min-width: 768px) {
.blockMobile {
display: block; /* Show the component for screen width 768px and above (desktop) */
}
}
.showMobile {
display: none; /* Hide the component by default */
}
.bar-image {
max-width: 100px;
height: auto;
}
@media only screen and (max-width: 768px) {
.showMobile {
display: block; /* Show the component for screen width 768px and above (desktop) */
}
.image-overlay {
position: relative; /* Move the image below the text */
top: 0;
transform: none; /* Remove vertical centering */
margin-top: 20px; /* Space between text and image */
margin-right: 0; /* Remove right alignment */
}
.bar-image {
display: none;
}
}
#dynamic-content {
position: relative; /* Ensure the content sections are positioned relative to this container */
min-height: 200px; /* Optional: Set a minimum height to prevent collapsing before content is displayed */
}
#dynamic-content .content-section {
position: absolute;
top: 0;
left: 0;
width: 100%;
opacity: 0;
visibility: hidden;
transition: opacity 0.5s ease, visibility 0.5s ease;
}
#dynamic-content .content-section.active {
opacity: 1;
visibility: visible;
position: relative; /* Allow active content to influence the height of #dynamic-content */
}
/* Dropdown content styling */
.dropdown-content {
text-align: left !important;
}
.dropdown-item {
text-align: left !important;
}
/* Button alignment fix */
.publication-links .link-block {
display: inline-block;
vertical-align: middle;
margin: 0 2px;
}
.publication-links .link-block .button,
.publication-links .link-block .dropdown-trigger .button {
height: 2.5em !important;
min-height: 2.5em !important;
display: inline-flex !important;
align-items: center !important;
vertical-align: middle !important;
}
.dropdown {
display: inline-block;
vertical-align: middle;
}
/* Fix dropdown button alignment */
.dropdown-trigger .button {
height: 2.5em !important; /* Match height of other buttons */
align-items: center;
display: inline-flex !important;
}
.container.is-max-desktop {
max-width: 100% !important;
padding-left: 1rem;
padding-right: 1rem;
}
</style>
</head>
<body>
<nav class="navbar" role="navigation" aria-label="main navigation">
<div class="navbar-brand">
<a role="button" class="navbar-burger" aria-label="menu" aria-expanded="false">
<span aria-hidden="true"></span>
<span aria-hidden="true"></span>
<span aria-hidden="true"></span>
</a>
</div>
<div class="navbar-menu">
<!-- <div class="navbar-start" style="flex-grow: 1; justify-content: center;">
<a class="navbar-item" href="https://contractionppo.github.io/">
<span class="icon">
<i class="fas fa-home"></i>
</span>
</a>
</div> -->
</div>
</nav>
<section class="hero">
<div class="hero-body">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column has-text-centered">
<h1 class="title is-1 publication-title"><span style="color: #3b82f6;">ContractionPPO</span>: Certified Reinforcement Learning via Differentiable Contraction Layers</h1>
<!-- <h3 class="is-size-5 has-text-weight-bold" style="color: orange;">
RAL
</h3> -->
<!-- <br> -->
<div class="is-size-6 publication-authors" style="color: #3b82f6;">
<span class="author-block">
<a href="https://vrushabh27.github.io/vrushabh_zinage/" target="_blank" style="color: #3b82f6;">Vrushabh Zinage</a>*<sup>1</sup>,</span>
<span class="author-block">
<a href="https://linkedin.com/in/narek-harutyunyan-428069233" target="_blank" style="color: #3b82f6;">Narek Harutyunyan</a>*<sup>1</sup>,</span>
<span class="author-block">
<a href="https://linkedin.com/in/eric-verheyden" target="_blank" style="color: #3b82f6;">Eric Verheyden</a>*<sup>1</sup>,</span>
<span class="author-block">
<a href="https://cast.caltech.edu/people/fred-hadaegh" target="_blank" style="color: #3b82f6;">Fred Hadaegh</a><sup>1</sup>,</span>
<span class="author-block">
<a href="https://www.eas.caltech.edu/people/sjchung" target="_blank" style="color: #3b82f6;">Soon-Jo Chung</a><sup>1</sup>
</span>
</div>
<div class="is-size-6 publication-authors">
<span class="author-block"><sup>1</sup>California Institute of Technology</span>
</div>
<div class="is-size-6 publication-authors">
* Equal Contributions
</div>
<div class="column has-text-centered">
<div class="publication-links">
<!-- PDF Link. -->
<a href="https://google.com" target="_blank">
<span class="link-block">
<span class="button is-normal is-rounded is-dark">
<span class="icon">
<i class="fas fa-file-pdf"></i>
</span>
<span>Paper</span>
</span>
</span>
</a>
<!-- ArXiv Link -->
<a href="https://arxiv.org/pdf/2603.19632" target="_blank">
<span class="link-block">
<span class="button is-normal is-rounded is-dark">
<span class="icon">
<i class="ai ai-arxiv"></i>
</span>
<span>arXiv</span>
</span>
</span>
</a>
<!-- Video Link. -->
<a href="https://www.youtube.com/watch?si=ZiBLBlgqiZ1OZgOp&v=2BsRQfMgIFo&feature=youtu.be">
<span class="link-block">
<span class="button is-normal is-rounded is-dark">
<span class="icon">
<i class="far fa-images"></i>
</span>
<span>Video</span>
</span>
</span>
</a>
<!-- Github -->
<a href="https://github.com/contractionppo/ContracionPPO" target="_blank">
<span class="link-block">
<span class="button is-normal is-rounded is-dark">
<span class="icon">
<i class="fa-brands fa-github"></i>
</span>
<span>Github</span>
</span>
</span>
</a>
<!-- Related Works -->
<!-- <a href="https://google.com" target="_blank">
<span class="link-block">
<span class="button is-normal is-rounded is-dark">
<span class="icon">
<i class="fas fa-link"></i>
</span>
<span>Related Research</span>
</span>
</span>
</a> -->
</div>
</div>
</div>
</div>
</div>
</div>
</section>
<section class="hero abstract_fig" style="padding: 0; margin-top: -2rem;">
<div class="container is-max-desktop">
<div class="hero-body" style="padding: 0;">
<video autoplay loop muted style="width: 100%; height: 500px; border-radius: 8px; object-fit: cover;">
<source src="assets/video_cppo.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<!-- <p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666; text-align: center;">We present ContractionPPO</p> -->
</div>
</div>
</section>
<section class="section" style="background-color: #f8f9fa;">
<div class="container is-max-desktop">
<!-- Abstract. -->
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h2 class="title is-3">Abstract</h2>
<div class="content has-text-justified">
<p>
Legged locomotion in unstructured environments demands not only high-performance control policies but also formal guarantees to ensure robustness under perturbations. Control methods often require carefully designed reference trajectories, which are difficult to construct in high-dimensional, contact-rich systems like quadruped robots. In contrast, Reinforcement Learning (RL) directly learns policies that implicitly generate motion, and uniquely benefits from access to privileged information, such as full state and dynamics during training, that is not available at deployment. We present <strong>ContractionPPO</strong>, a framework for certified robust planning and control of legged robots by augmenting Proximal Policy Optimization (PPO) RL with a state-dependent contraction metric layer. This approach enables the policy to maximize performance while simultaneously producing a contraction metric that certifies incremental exponential stability of the simulated closed-loop system. The metric is parameterized as a Lipschitz neural network and trained jointly with the policy, either in parallel or as an auxiliary head of the PPO backbone. While the contraction metric is not deployed during real-world execution, we derive upper bounds on the worst-case contraction residual and show that these bounds ensure the learned contraction metric generalizes from simulation to real-world deployment. Our hardware experiments on quadruped locomotion demonstrate that contraction augmented PPO enables robust, certifiably stable control even under strong external perturbations.
</p>
</div>
</div>
</div>
<!--/ Abstract. -->
</div>
</section>
<section class="hero is-small">
<div class="hero-body">
<div class="container is-max-desktop">
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h2 class="title is-3">ContractionPPO Architecture</h2>
<div class="content has-text-justified">
<img src="assets/contractionppo_pipeline.jpg" alt="ContractionPPO Architecture" style="width: 100%; height: auto;" />
<p style="font-size: 0.875rem;">
Architecture of ContractionPPO. The PPO policy processes raw observations $\mathbf{o}$ and outputs desired joint poses $\mathbf{x}_d$, which are executed by a low-level PD controller. In parallel, the contraction metric MLP $M_\phi$ receives privileged and raw observations $\hat{\mathbf{x}}$ and outputs a positive definite metric $M_\phi = \Theta^\top \Theta$. The contraction loss is evaluated using the Lyapunov condition $\dot{V} + \alpha V \leq -\epsilon_\alpha$, where $\alpha$ specifies the desired contraction rate and $\epsilon_\alpha$ quantifies the approximation margin between the learned value function $V_\phi$ and the true contraction Lyapunov function $V$. A larger $\alpha$ leads to faster convergence guarantees but also requires a larger $\epsilon_\alpha$, making optimization more challenging. This joint training setup ensures that the policy not only maximizes task reward but also satisfies certifiable incremental stability guarantees during locomotion.
</p>
</div>
</div>
</div>
</div>
</section>
<section class="hero is-small is-light">
<div class="hero-body">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column is-four-fifths">
<div style="text-align: center; margin: 3rem 0;">
<video autoplay loop muted
style="width: 85%; max-width: 900px; height: 420px; border-radius: 12px; object-fit: cover; box-shadow: 0 10px 30px rgba(0,0,0,0.15);">
<source src="assets/first_video.mp4" type="video/mp4">
</video>
<p style="font-size: 0.95em; margin-top: 0.6rem; color: #666;">
</p>
</div>
<section class="section" style="background-color: #ffffff;">
<div class="container is-max-desktop">
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<!-- <h2 class="title is-3">Why Contraction Matters</h2> -->
<div class="content has-text-justified">
<p>
Classical reinforcement learning can discover highly capable locomotion policies, but it typically does not provide formal guarantees on how the robot will behave under perturbations, modeling error, or observation noise. Contraction theory addresses this by certifying that nearby trajectories of the closed-loop system converge exponentially toward one another.
</p>
<p>
In ContractionPPO, this idea is embedded directly into training through a learned state-dependent contraction metric. The result is a controller that is not only performant in nominal conditions, but is also driven toward <strong>provably stable and robust behavior</strong> throughout learning.
</p>
</div>
</div>
</div>
</div>
</section>
<h2 class="title is-2 has-text-centered" style="margin-bottom: 2rem;">ContractionPPO Deployment</h2>
<h3 style="font-size: 1.3em; font-weight: 600; margin: 2em 0 1em 0; color: #34495e; text-align: center;">Comparisons with Baseline Methods</h3>
<div style="display: grid; grid-template-columns: 1fr 1fr; grid-template-rows: 1fr 1fr; gap: 20px; margin: 2rem 0;">
<!-- Video 1 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/contraction_0.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;"><strong>ContractionPPO</strong></p>
</div>
<!-- Video 2 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/nature.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;">TumblerNet</p>
</div>
<!-- Video 3 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/rma.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;">Rapid Motor Adaptation</p>
</div>
<!-- Video 4 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/ppo.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;">PPO</p>
</div>
</div>
<!-- <div style="text-align: center; margin: 1rem 0;">
<p style="font-size: 0.875em; margin: 0.5rem 0 0 0; color: #666; text-align: center;">Quadruped trajectories during handstand under wind disturbances of varying magnitudes. As the disturbance intensity increases, transient deviations from the initial position become more evident. Note that ContractionPPO was never trained on external disturbances. Despite this, ContractionPPO policy ensures bounded trajectories and lies inside the circle marked by the black curve. This highlights the robustness and incremental stability guarantee of ContractionPPO, even under strong external perturbations.</p>
</div> -->
<!-- Caption for baseline comparison videos -->
<!-- <div style="text-align: center; margin-top: 1rem;">
<p style="font-size: 0.9em; color: #666; font-style: italic;">Comparisons with baseline methods</p>
</div> -->
<!-- Comparisons baselines image -->
<div style="text-align: center; margin: 1rem 0;">
<img src="assets/comparisions_baslines.jpg" alt="Comparisons with baseline methods" style="width: 100%; max-width: 800px; height: auto; border-radius: 8px;">
<p style="font-size: 0.875em; margin: 0.5rem 0 0 0; color: #666; text-align: center;">Comparison of trajectories for quadruped handstand using ContractionPPO (ours), TumblerNet, RMA and PPO where robots (for all baseline algorithms) were trained with identical reward functions that encourage remaining close to their initial position. While PPO, TumblerNet and RMA leaves the region and ultimately fails to remain on platform, ContractionPPO guarantees that the robot lie inside the circle (marked in black). This illustrates the core advantage of our approach i.e., provably stable and robust behavior.</p>
</div>
<h3 style="font-size: 1.3em; font-weight: 600; margin: 2em 0 1em 0; color: #34495e; text-align: center;">Wind Experiments</h3>
<div style="display: grid; grid-template-columns: 1fr 1fr; grid-template-rows: 1fr 1fr; gap: 20px; margin: 2rem 0;">
<!-- Video 1 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/contraction_4.8.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;">Wind Speed: 4.8m/s</p>
</div>
<!-- Video 2 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/contraction_6.4.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;">Wind Speed: 6.4m/s</p>
</div>
<!-- Video 3 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/contraction_8.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;">Wind Speed: 8.0m/s</p>
</div>
<!-- Video 4 -->
<div style="text-align: center;">
<video autoplay loop muted style="width: 100%; height: 200px; border-radius: 8px; object-fit: cover;">
<source src="assets/contraction_9.6.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666;">Wind Speed: 9.6m/s</p>
</div>
</div>
<!-- Caption for wind experiments -->
<div style="text-align: left; margin-top: -1.5rem; margin-bottom: 1rem;">
<p style="font-size: 0.9em; color: #666; font-style: italic;">*Wind disturbances were never seen during training</p>
</div>
<div style="text-align: center; margin: 3rem 0;">
<video autoplay loop muted
style="width: 75%; max-width: 800px; height: 380px; border-radius: 12px; object-fit: cover; box-shadow: 0 10px 30px rgba(0,0,0,0.15);">
<source src="assets/biped.mp4" type="video/mp4">
</video>
<p style="font-size: 0.95em; margin-top: 0.6rem; color: #666;">
Humanoid stabilization </p>
</div>
<!-- Wind speed image -->
<div style="text-align: center; margin: 1rem 0;">
<img src="assets/wind_speed_image.jpg" alt="Wind speed experiments" style="width: 100%; max-width: 800px; height: auto; border-radius: 8px;">
<p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666; text-align: center;">Quadruped trajectories during handstand under wind disturbances of varying magnitudes. As the disturbance intensity increases, transient deviations from the initial position become more evident. Note that ContractionPPO was never trained on external disturbances. Despite this, ContractionPPO policy ensures bounded trajectories and lies inside the circle marked by the black curve. This highlights the robustness and incremental stability guarantee of ContractionPPO, even under strong external perturbations.</p>
</div>
</div>
</div>
</div>
</div>
</section>
<section class="hero is-small is-light">
<div class="hero-body">
<div class="container is-max-desktop">
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h3 style="font-size: 1.3em; font-weight: 600; margin: 0.5em 0 1em 0; color: #34495e; text-align: center;">Results</h3>
<div class="content has-text-justified">
<div style="overflow-x: auto; margin: 0.5rem 0;">
$$\begin{array}{|l|c|c|c|c|c|c|}
\hline
\textbf{Method} & \textbf{10} & \textbf{20} & \textbf{30} & \textbf{40} & \textbf{50} & \textbf{Combined} \\
\hline
\textbf{ContractionPPO} & \textbf{0.00} & \textbf{0.00} & \textbf{0.00} & \textbf{0.00} & \textbf{0.00} & \textbf{0.00} \\
\hline
\text{TumblerNet} & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
\hline
\text{RMA} & 0.23 & 0.23 & 0.19 & 0.19 & 0.16 & 0.20 \\
\hline
\text{PPO} & 0.98 & 0.98 & 0.98 & 0.99 & 0.99 & 0.99 \\
\hline
\end{array}$$
<p style="font-size: 0.875em;">
<em>Comparison of methods across control points and violations. Numbers represent failure ratio over all episodes. Combined is 2500 episodes, and each control point is 500. Lower is better.</em>
</p>
</div>
</div>
</div>
</div>
</div>
</div>
</section>
<!-- <section class="section" id="videoJump">
<div class="container is-max-desktop">
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h2 class="title is-3">Video</h2>
<!-- <video autoplay loop muted style="width: 100%; height: 400px; border-radius: 8px; object-fit: cover;">
<source src="assets/first_video.mp4" type="video/mp4">
Your browser does not support the video tag.
</video> -->
<!-- <iframe style="width: 100%; height: 450px; border-radius: 8px; object-fit: cover;" src="https://www.youtube-nocookie.com/embed/o2ctFPs7OD4?si=ywKpSTpv4qw3zV7j" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
<!-- <p style="font-size: 0.9em; margin: 0.5rem 0 0 0; color: #666; text-align: center;">We present ContractionPPO</p> -->
<!-- </div> -->
<!-- </div> -->
<!-- </div> -->
<!-- </section> -->
<!-- <section class="hero is-small is-light">
<div class="container is-max-desktop">
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h2 class="title is-3">Acknowledgments</h2>
<div class="content has-text-centered">
<p>
This work was supported by the National Institute of Advanced Industrial Science and Technology (AIST),
DSTA contract No. DST000EC124000205, NSF GRFP under Grant No. DGE1745016, a hardware grant from Nvidia,
and by Institute of Information & communications Technology Planning & Evaluation (IITP) grants by the Korean government (MSIT) (No.RS-2022-00143911,
AI Excellence Global Innovative Leader Education Program; No.RS-2020-II201336, Artificial Intelligence graduate school support (UNIST); and No.RS-2024-00341055).
</p>
</div>
</div>
</div>
</section> -->
<!-- <section class="section" id="BibTeX">
<div class="container is-max-desktop content">
<h2 class="title">BibTeX</h2>
<pre><code>
@article{zinage2025contractionppo,
title={ContractionPPO: Certified Reinforcement Learning via Differentiable Contraction Layers},
author={Zinage, Vrushabh and Harutyunyan, Narek and Verheyden, Eric and Hadaegh, Fred and Chung, Soon-Jo},
journal={IEEE Robotics and Automation Letters (RAL)},
year={2025}
}
</code></pre>
</div>
</section> -->
<style>
#blocks {
width:100%;
height:60px;
margin:0 auto;
}
#block1 {
height:33.33%;
width:30%;
float: left;
}
#block2 {
height:33.33%;
width:40%;
float: left;
}
#block3 {
height:33.33%;
width:30%;
float: right;
}
</style>
<footer class="footer">
<div class="container">
<div id="blocks">
<div id="block1">
<a href="https://aerospacerobotics.caltech.edu/" target="_blank">
<img src="static/images/file.svg" alt="AirLab Logo" style="width:60%;">
</a>
</div>
<div id="block2">
<center>
<a class="button is-text" href="https://aerospacerobotics.caltech.edu/" target="_blank">
<i class="fas fa-globe fa-lg" style="height:100%;"></i>
</a>
<a class="button is-text" href="https://www.youtube.com/user/AeroboticsUIUC" target="_blank">
<i class="fab fa-youtube fa-lg"></i>
</a>
<br>
<br>
<p class="" style="font-size: .65rem;">This webpage was adapted from the
<a href="https://github.com/eliahuhorwitz/Academic-project-page-template" target="_blank">Academic Project Page Template</a>
template, which is licensed under a
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a>.
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">
<img alt="Creative Commons License" style="border-width:0; height: 0.65rem;" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png" />
</a>
</p>
<p class="" style="font-size: .65rem;">If you use the <a href="https://github.com/ia-tigris/ia-tigris.github.io">source code</a> of this webpage,
please link back to this page in the footer.
</p>
<!-- <p>
This page was built using the <a href="https://github.com/eliahuhorwitz/Academic-project-page-template" target="_blank">Academic Project Page Template</a> which was adopted from the <a href="https://nerfies.github.io" target="_blank">Nerfies</a> project page.
You are free to borrow the source code of this website, we just ask that you link back to this page in the footer.
</p> -->
</center>
</div>
<div id="block3">
<a href="https://www.caltech.edu/" target="_blank">
<img src="static/images/Caltech_Logo.svg" alt="Caltech Logo" style="float: right;">
</a>
</div>
</div>
</div>
</footer>
<script>
function showContent(option,section) {
// Remove the active class from all sections
// const parentContainer = document.getElementById(`dynamic-content-${section}`);
const button = event.target;
const parentContainer = button.closest(".hero");
// Remove the active class from all content sections within this container
const sections = parentContainer.querySelectorAll(".content-section");
sections.forEach(contentSection => {
contentSection.classList.remove("active");
});
// Add the active class to the selected section
const selectedSection = document.getElementById(`content-${section}-${option}`);
if (selectedSection) {
selectedSection.classList.add("active");
}
}
// Video Carousel Functions
let currentSlideIndex = 0;
const totalSlides = 4;
function pauseAllVideos(carouselId) {
const carousel = document.getElementById(carouselId);
const videos = carousel.querySelectorAll('video');
videos.forEach(video => {
video.pause();
});
}
function playCurrentVideo(carouselId, slideIndex) {
const carousel = document.getElementById(carouselId);
const videos = carousel.querySelectorAll('video');
if (videos[slideIndex]) {
// Reset video to beginning and play
videos[slideIndex].currentTime = 0;
// Ensure volume is set to 10%
videos[slideIndex].volume = 0.1;
videos[slideIndex].play();
}
}
function changeSlide(carouselId, direction) {
const carousel = document.getElementById(carouselId);
const dots = carousel.parentElement.querySelectorAll('.dot');
// Pause all videos first
pauseAllVideos(carouselId);
currentSlideIndex += direction;
// Handle wrapping
if (currentSlideIndex >= totalSlides) {
currentSlideIndex = 0;
} else if (currentSlideIndex < 0) {
currentSlideIndex = totalSlides - 1;
}
// Update carousel position
carousel.style.transform = `translateX(-${currentSlideIndex * 100}%)`;
// Update dots
dots.forEach((dot, index) => {
dot.classList.toggle('active', index === currentSlideIndex);
dot.style.background = index === currentSlideIndex ? 'rgba(255,255,255,0.8)' : 'rgba(255,255,255,0.5)';
});
// Play the current video
playCurrentVideo(carouselId, currentSlideIndex);
}
function currentSlide(carouselId, slideNumber) {
const carousel = document.getElementById(carouselId);
const dots = carousel.parentElement.querySelectorAll('.dot');
// Pause all videos first
pauseAllVideos(carouselId);
currentSlideIndex = slideNumber - 1;
// Update carousel position
carousel.style.transform = `translateX(-${currentSlideIndex * 100}%)`;
// Update dots
dots.forEach((dot, index) => {
dot.classList.toggle('active', index === currentSlideIndex);
dot.style.background = index === currentSlideIndex ? 'rgba(255,255,255,0.8)' : 'rgba(255,255,255,0.5)';
});
// Play the current video
playCurrentVideo(carouselId, currentSlideIndex);
}
// Auto-play functionality (optional)
function startAutoPlay(carouselId) {
setInterval(() => {
changeSlide(carouselId, 1);
}, 5000); // Change slide every 5 seconds
}
// Initialize carousel when page loads
document.addEventListener('DOMContentLoaded', function() {
// Set all videos to 10% volume
const carousel = document.getElementById('windExperimentsCarousel');
const videos = carousel.querySelectorAll('video');
videos.forEach((video, index) => {
// Set volume to 10%
video.volume = 0.1;
video.addEventListener('ended', function() {
// Only auto-scroll if this is the currently active video
if (index === currentSlideIndex) {
console.log('Video ' + index + ' ended, moving to next slide');
// Move to next slide (wraps to first slide after last)
setTimeout(() => {
changeSlide('windExperimentsCarousel', 1);
}, 300);
}
});
// Also ensure video is ready to play
video.addEventListener('loadedmetadata', function() {
video.volume = 0.1;
});
});
// Play the first video when page loads
playCurrentVideo('windExperimentsCarousel', 0);
// Add intersection observer to pause videos when not visible
const observerOptions = {
root: null,
rootMargin: '0px',
threshold: 0.5 // Video needs to be at least 50% visible
};
const videoObserver = new IntersectionObserver((entries) => {
entries.forEach(entry => {
const video = entry.target;
if (entry.isIntersecting) {
// Video is visible - resume if it's the current slide
const videoIndex = Array.from(videos).indexOf(video);
if (videoIndex === currentSlideIndex) {
video.play();
}
} else {
// Video is not visible - pause it
video.pause();
}
});
}, observerOptions);
// Observe all videos in the carousel
videos.forEach(video => {
videoObserver.observe(video);
});
// Uncomment the line below if you want auto-play
// startAutoPlay('windExperimentsCarousel');
});
</script>
</body>
</html>