Continual Learning and Lifting of Koopman Dynamics for Linear Control of Legged Robots

Abulikemu Abuduweili; Changliu Liu; Feihan Li; Rui Chen; Weiye Zhao; Yifan Sun

REVIEW 4 major objections 6 minor 1 cited by

Continual Learning and Lifting of Koopman Dynamics for Linear Control of Legged Robots

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Learned Koopman models refined with failure data and higher latent dimensions let linear MPC control five high-dimensional legged robots.

desk verdict A genuinely useful incremental Koopman pipeline for legged robots, wrapped in a convergence theorem that assumes away the hard part of learning. read the letter →

arxiv 2411.14321 v3 pith:XRSENMDN submitted 2024-11-21 cs.RO

Feihan Li , Abulikemu Abuduweili , Yifan Sun , Rui Chen , Weiye Zhao , Changliu Liu This is my paper · ORCID

classification cs.RO

keywords Koopmanoperatormodelpredictivecontrolleggedrobotscontinuallearningextendeddynamicmodedecompositionlinearizationlocomotionlatentspacelifting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a learned linear Koopman model of a high-dimensional legged robot can be refined until it is accurate enough to serve as the dynamics model inside a linear MPC controller. The proposed incremental algorithm alternates between collecting failure-prone tracking data from its own current controller, expanding the training dataset, increasing the latent dimension, and retraining the embedding and linear dynamics end-to-end. The paper proves a convergence result: under stated assumptions on sampling and the embedding, the learned Koopman operator converges to the true one, and the paper claims the linear approximation error converges monotonically, with rate O(sqrt(ln(n)/m)) + O(1/sqrt(n)), so adding data and dimensions jointly shrinks the error. In simulation, linear MPC with these models tracks walking references across flat and rough terrain on five robots: Unitree G1, H1, A1, Go2, and ANYmal D. If the claim holds, it offers a scalable model-based alternative to nonlinear MPC and task-specific reinforcement learning for legged locomotion.

What carries the argument

The central object is the finite-dimensional Koopman model z_{t+1} = A z_t + B u_t in a latent state z = [x; g'(x)] that concatenates the original robot state with a neural-network embedding g'. The embedding and matrices are learned end-to-end with a discounted k-step prediction loss plus a small reconstruction term. The mechanism that carries the argument is the iterative loop: each round adds on-policy failed-tracking rollouts to the dataset, raises the latent dimension by $\Delta$ n, and retrains; theoretically, EDMD is read as an L2 projection of the true Koopman operator onto the span of the embedding functions, so the total error separates into a sampling error bounded by matrix Bernstein concentration and a projection error bounded by spectral decay of the Koopman eigenvalues. A QP-based linear MPC then solves the control problem in the lifted space, sending joint targets to a low-level PD controller.

What would settle it

Run the incremental algorithm on a system whose Koopman spectrum decays slowly, or on the robot datasets after checking whether the learned embedding functions are near-orthogonal and eigenfunction-like. If the measured linearization error plateaus as the latent dimension n increases despite scaling samples as m = $\Omega$(n log n), the O(1/$\sqrt$(n)) projection-error bound is violated and the central convergence claim fails even if the controller still works.

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Extended reading notes

Core claim

The central claim is that the incremental lifting scheme—expanding the dataset with trajectories the current controller fails to track and simultaneously growing the latent dimension—drives a finite-dimensional Koopman model of whole-body legged dynamics toward the true Koopman operator. Theorem 1 formalizes this: with i.i.d. samples, bounded latent states, and orthogonal embedding functions, the learned operator converges to the true operator as the sample size m scales as $\Omega$(n log n) and n grows; adding assumptions on spectral decay and on the embedding matching the dominant eigenfunctions gives the explicit rate O($\sqrt$(ln(n)/m)) + O(1/$\sqrt$(n)). The paper's experimental claim is that linear MPC using this learned model achieves high tracking performance on five high-dimensional legged robots over flat and rough terrain, with average survival close to the 200-step cap and substantially lower joint and root tracking errors than the compared baselines. The paper further claims this is the first application of linearized whole-body Koopman dynamics to locomotion control of high-dimensional legged robots.

Load-bearing premise

The rate and monotone-convergence conclusion assume the learned neural embedding functions are the first n eigenfunctions of the true Koopman operator and that the data are i.i.d. from a fixed distribution, while the method's own controller generates the incremental data; if the embedding misses those eigenfunctions, the claimed error bound does not follow.

Editorial extensions

If this is right

Linear MPC with an incrementally learned Koopman model can track whole-body walking references on five high-dimensional legged robots, on flat and rough terrain, with survival times near the 200-step cap.
Adding failure data and raising latent dimension are both necessary: ablations without them raise joint tracking error and cut survival steps sharply, with roughly seven times higher joint error when the data increment is removed.
The learned model keeps k-step prediction error low over horizons up to 15 steps, where the neural-network dynamics baselines show explosive error growth.
The theoretical rate gives a practical scaling rule: when the latent dimension is increased, sample count should grow at least as n log n to keep the approximation error decreasing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to treat the dimension increment Delta n as an adaptive quantity chosen from the measured approximation error, since the paper itself notes the latent dimension can explode if growth is unchecked.
The on-policy data collection creates a distribution shift that the i.i.d. assumption in the theorem ignores; re-weighting incremental data or annealing the dataset mixture would be a natural way to close that gap.
If the convergence proof transfers to other contact-rich, high-dimensional systems, the same failure-data-plus-lifting recipe could give a general route to linear whole-body models for manipulation or soft robots, though the paper only demonstrates locomotion.
The method's reliance on full-body reference trajectories means it would need tele-operation or retargeting data to move to real hardware, which the paper lists as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. First, the empirical recipe is real: alternating failed-tracking data collection with latent-dimension growth lets a learned linear Koopman model drive QP-based MPC for whole-body locomotion on five simulated legged robots, and the ablations show both increments matter. Second, the advertised convergence guarantee does not cover what the algorithm actually learns. The O(1/sqrt(n)) projection rate follows only if the learned embedding already equals the first n eigenfunctions of the true Koopman operator (Assumption 7); the end-to-end loss in Eq. 6 never enforces that, and the paper itself shows the rate is just spectral tail truncation once you grant the premise. So the theory is an idealized EDMD statement, not a justification for the trained model.

What's genuinely new here is the continual-lifting procedure and the whole-body Koopman/MPC demonstration. To my knowledge no one has used Koopman-linearized whole-body dynamics with linear MPC on high-dimensional legged robots at this scale. The code is public, the baselines (DKUC, DKAC, NNDM with NMPC, DKRL) are sensible, and the per-task tables in the appendix are consistent with the aggregate table. Credit where due: the ablations in Sections 4.4 and 4.5 support the claim that both dataset expansion and dimension expansion contribute, and the method beats the baselines by wide margins on most suites.

The soft spots, in order of seriousness. (1) Assumption 7 is a realizability assumption; without it the theorem reduces to a known result from Korda and Mezic. The paper calls the assumptions 'generally mild,' but this one is doing all the work. (2) Assumption 1 (i.i.d. samples) is inconsistent with the on-policy data collection in Algorithm 1, where the current MPC model generates the next dataset. This is fixable by framing as an approximation or by relaxing to ergodic trajectory data, but as stated it is a mismatch. (3) The main tracking tables report single-seed averages without variance; the survival metric caps at 200 steps, so the headline TSur numbers are close to ceiling. (4) The RL policy used to bootstrap D(0) is not included as a baseline, which weakens the comparison for practitioners.

If the theory section were rewritten to describe the ideal EDMD case and the monotonic-convergence claim were dropped or qualified, the paper would be stronger. As is, I would send it to review because the empirical direction is valuable and the flaw is in the interpretation, not in the core demonstration. The right referee would ask for variance estimates, a stronger baseline, and an honest statement of what the theorem does and does not prove.

Referee Report

4 major / 6 minor

Summary. The paper proposes an incremental (continual) Koopman-operator learning algorithm for legged-robot dynamics, in which the training dataset is repeatedly augmented with on-policy failure data and the latent state dimension is increased. The resulting linear model is used as the dynamics inside a linear MPC controller. The authors claim a theoretical guarantee that the linear approximation error converges monotonically as data size m and latent dimension n grow, with rate O(sqrt(ln(n)/m)) + O(1/sqrt(n)), and support the method with experiments on five simulated legged robots across flat and rough terrain, comparing against several Koopman and neural-dynamics baselines and ablating the data-increment and dimension-increment components.

Significance. If the empirical results hold, the paper makes a useful contribution: it demonstrates that a learned linear Koopman model, refined by an iterative data- and dimension-expansion procedure, can serve as the dynamics model for a whole-body MPC controller on high-dimensional legged robots, an application where Koopman methods have previously been limited to low-dimensional systems. The ablations in Section 4.4-4.5 provide evidence that both the dataset increment and the dimension increment contribute to the reported tracking performance. The paper also provides code at a public repository, which is a practical strength. However, the central theoretical claim — the advertised monotonic convergence guarantee and the O(1/sqrt(n)) rate — is established only under a realizability condition that the learned embedding exactly matches the dominant eigenfunctions of the true Koopman operator, a property the training loss does not enforce and the paper does not verify. The empirical claims are also weakened by the absence of reported variance and by the capped survival metric.

major comments (4)

[Appendix D.3, Assumption 7 and Eq. (44)-(49)] The O(1/sqrt(n)) projection-error rate in Theorem 1(b) is derived under Assumption 7, which states that the learned embedding functions Phi = [phi_1, ..., phi_n] are exactly the first n eigenfunctions of the true Koopman operator K. This is precisely the property that the end-to-end neural-network training with the loss in Eq. (6) is supposed to discover; nothing in the loss enforces orthogonality (Assumption 4) or that K Phi(s) = Phi(f(s)) on the dominant eigenspace, and the paper does not verify this condition after training. Consequently, the rate in Eq. (49) is a truncation bound for an exact eigenfunction basis, not a bound for the actually learned model. The statement in Section 3.4 that these assumptions are 'generally mild' is not defensible for Assumption 7. The theoretical convergence claim for the proposed algorithm is therefore not established as stated.
[Section 3.1 and Assumption 1] The main theorem (Theorem A.3/A.4) assumes that data samples s_1, ..., s_m are i.i.d. from a fixed distribution (Assumption 1). The algorithm described in Section 3.1, however, collects incremental data on-policy: the MPC controller built from the current dynamics T^(j) generates the next dataset D^(j+1)_incre. This changes the sampling distribution across iterations and creates dependence between the collected samples and the previously learned model. The theorem therefore does not directly apply to the iterative, on-policy data collection procedure that is central to the method. The authors should either prove the convergence result under an adaptive-sampling formulation or explicitly state the theorem as applying only to a simplified off-policy setting, with the on-policy procedure treated as an empirical heuristic.
[Theorem 1 and abstract] The abstract claims that 'the linear approximation error of our method converges monotonically,' but Theorem 1 does not establish monotonic convergence. It provides a limit statement (Eq. 7) and an asymptotic upper bound (Eq. 8) on the error for a given (n, m) pair; it does not show that the error sequence over the algorithm's iterations is monotone decreasing. The theory needs to be stated more precisely, and the wording should be aligned with what is actually proven.
[Section 4, Tables 1-2 and Appendix E.5] The main tracking results (Tables 1 and 2) and the per-task tables in Appendix E.5 report single-seed averages without standard deviations or confidence intervals. Since the comparisons include stochastic training and simulation, the reader cannot assess whether the reported improvements are statistically meaningful. In addition, the survival metric TSur is capped at 200 steps, and many entries — including the authors' own — are close to that cap, so the metric has limited sensitivity to differences in long-horizon robustness. Finally, the PPO policy used to generate the initial dataset D^(0) is not included as a control baseline; without it, the reader cannot separate the contribution of the Koopman-MPC controller from the quality of the initial data collection.

minor comments (6)

[Section 2.2, Eq. (5)] The MPC cost uses P z_{t+k} as the predicted state, but the state vector x_t is defined only in Section 4.1. It would be clearer to explicitly define the relation between x_t, z_t, and P before the cost is introduced.
[Section 3.3, Algorithm 1] The pseudocode initializes the optimizer and scheduler inside the TrainKoopman function, but the text in Section 3.3 does not describe how the latent-space dimension n^(j) is mapped to the network architecture, especially when the dimension is increased by Delta_n. A sentence clarifying this would help reproducibility.
[Section 4.1, metric definitions] The expression for E_pre(k) in the metrics paragraph uses x_t for the predicted state and x*_t for the reference, but the same notation is used later for the system state and reference trajectory. Please make the distinction explicit in the equation or in the text.
[Appendix D.1] In the definition of Theorem A.1, the statement 'for any g in H_N' uses the subscript N inconsistently with the earlier notation H_n; please use a consistent index.
[Section 5 and Appendix C.1] The statement in Section 5 that the method 'encounters an explosion in latent dimensionality' is not backed by any quantitative analysis of how n grows across iterations; the computational-cost comparison in Appendix C.1 only reports wall-clock times, not the dimensional growth. A brief analysis or a bound on the number of iterations would be useful.
[Throughout] There are a number of typographical errors and inconsistencies, e.g., 'catagorized' in Section 1, 'reconsrtuction' in Section 3.2, 'incre' for the incremental dataset in Section 3.1, and the inconsistent use of 'Tsur' and 'TSur'. A careful proofreading pass is recommended.

Circularity Check

1 steps flagged · score 7.0 of 10

Theorem 1(b)'s O(1/sqrt(n)) rate is derived under Assumption 7, which assumes the learned embedding is the true dominant Koopman eigenbasis; Eqs. (44)-(49) then reduce the error to a spectral truncation tail, making the central convergence claim conditional on the learning objective already being solved.

self definitional [Section 3.4, Theorem 1(b); Appendix D.3, Assumption 7 and Eqs. (44)-(49)]
"Assumption 7 (Accurate Learning of Dominant Modes). The embedding functions Φ(·) = [ϕ1, · · ·, ϕn]⊤ correspond to the first n eigenfunctions of K associated with the largest eigenvalues in magnitude. ... ϵproj(n) = ∥KnP µ n ψ − Kψ∥ = ∥P∞ i=n+1 ciλ⋆ i ϕ⋆ i ∥ ≤ B · (P∞ i=n+1 |λ⋆ i |2)^{1/2} ≤ BC/√n."

The advertised O(1/sqrt(n)) projection-error rate is proved under Assumption 7, which stipulates that the learned embedding functions are exactly the first n eigenfunctions of the true Koopman operator. Once that stipulation is granted, Eq. (45)-(48) show the error is just the L2 tail sum of the true spectral decomposition—a truncation bound that contains no quantity produced by the trained network. The actual training objective L_koopman in Eq. (6) is an end-to-end multi-step prediction loss that neither enforces orthogonality nor requires KΦ(s) = Φ(f(s)) on a dominant eigenspace. Thus the convergence rate holds only if the network has already solved the representation-learning problem the method is intended to discover.

full rationale

The empirical component is largely self-contained: baseline and proposed controllers are evaluated on the same IsaacLab test suites with reported metrics, and those comparisons do not reduce to a fitted parameter. No load-bearing self-citation was found: the EDMD L2-projection premise is imported from the external Korda and Mezić papers, and the paper's own citations do not carry the main proof. However, the central theoretical claim is circular in a definitional sense. Section 3.4 and Appendix D.3 derive the projection-error rate under Assumption 7, which asserts that the learned embedding functions are the dominant eigenfunctions of the true Koopman operator. Equations (44)-(49) make the reduction explicit: with Assumption 7, the error becomes the spectral tail ∥Σ_{i=n+1}^∞ c_i λ*_i φ*_i∥, bounded by BC/√n. This is an exact-basis truncation bound, not a statement about the end-to-end trained embedding g from Eq. (6). Nothing in the training loss verifies the eigenfunction property, so the theorem describes an ideal dictionary rather than the actual learned model. A separate correctness concern, not scored here as circularity, is that Assumption 1 (i.i.d. samples) is in tension with the on-policy incremental data collection of Section 3.1, where the current MPC model generates the next dataset. Overall, one load-bearing theoretical claim reduces by construction to its own success assumption, while the experimental results remain meaningful.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the EDMD/L2-projection framework taken from prior work (Theorem A.1), five stated assumptions (i.i.d. samples, bounded latent state, Gram invertibility, orthogonality, spectral decay), of which two (orthogonality, eigenfunction identity) are idealized and not enforced by training, and one (i.i.d.) is contradicted by the method's own on-policy data collection. The free parameters (n(0), Delta_n, alpha, epsilon_fail, horizon, MPC costs) are tuned per task and directly shape the reported metrics. No new physical entities are introduced; the lifted latent state is a methodological construct.

free parameters (6)

Latent dimension increment Delta_n = 100
Fixed step in Algorithm 1 chosen by computational trade-off; drives the claimed m = Omega(n log n) regime in Theorem 1.
Initial latent dimension n(0) = (512, 384, 384, 535, 214, 222, 453)
Per-suite initial latent dimension (Table 5). The capacity of the linear model, a central factor in the reported prediction and tracking metrics, is set by this hand choice.
Reconstruction loss weight alpha = 0.1
Set in Section 3.2 as a practical tuning choice ('shows better performance in practice'); shapes the learned dynamics through eq. (6).
Failure threshold epsilon_fail = 0.18 (Flat-Anymal-D), 0.10 (Flat-G1), 0.12 (Rough-Go2), etc.
Per-robot and per-terrain thresholds in Table 4, chosen when the robot is 'on the verge of falling'; directly defines the survival metric TSur.
MPC cost matrices Q, R, F = not given in text
The MPC objective eq. (5) depends on these matrices, but their numeric values are not provided, so the controller behavior is not fully specified.
Survival cap = 200 steps at 50 Hz
Caps TSur for all methods at 4 seconds, saturating the headline metric; the paper itself notes many cases 'remain robust enough for even longer tracking durations'.

assumptions (7)

domain assumption Samples are i.i.d. from a fixed measure mu (Assumption 1)
Invoked for the Matrix Bernstein bound in Appendix D.2, but contradicted by the algorithm's on-policy incremental data collection in Section 3.1, where the current MPC controller generates the next dataset.
domain assumption Bounded latent state: norm(Phi(s)) <= B (Assumption 2)
Used to bound the sampling error; not verified for the neural embeddings of a legged robot, which can diverge in failure states.
domain assumption Gram matrix invertible with lambda_min >= gamma (Assumption 3)
Standard EDMD regularity condition used to bound inverse Gram norms in eq. (19); unverifiable for learned embeddings at growing dimension n.
ad hoc to paper Embedding functions are orthogonal (Assumption 4)
Required for the eigenfunction-style expansion in Appendix D.3; the trained neural network enforces no orthogonality.
domain assumption Eigenvalue decay |lambda_i| <= C/i (Assumption 6)
Powers the O(1/sqrt(n)) projection rate; plausible for some Koopman systems but unverified for hybrid legged locomotion.
ad hoc to paper Learned embedding equals the first n eigenfunctions of K (Assumption 7)
The load-bearing premise for the projection error rate, eq. (44)-(48); it assumes the learning objective is already solved and is not enforced by the loss in eq. (6).
standard math EDMD estimator equals the L2 projection of K onto the dictionary span (Theorem A.1 from Korda-Mezic 2018a)
The foundation of the whole proof scheme, cited from prior work and acknowledged as such in Appendix D.

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Pith. "Pith review of Continual Learning and Lifting of Koopman Dynamics for Linear Control of Legged Robots." pith.science (2026). https://pith.science/paper/XRSENMDN

@misc{pith2026241114321,
  author       = {Pith},
  title        = {Pith review of: Continual Learning and Lifting of Koopman Dynamics for Linear Control of Legged Robots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XRSENMDN}},
  note         = {Machine review of arXiv:2411.14321}
}

read the original abstract

The control of legged robots, particularly humanoid and quadruped robots, presents significant challenges due to their high-dimensional and nonlinear dynamics. While linear systems can be effectively controlled using methods like Model Predictive Control (MPC), the control of nonlinear systems remains complex. One promising solution is the Koopman Operator, which approximates nonlinear dynamics with a linear model, enabling the use of proven linear control techniques. However, achieving accurate linearization through data-driven methods is difficult due to issues like approximation error, domain shifts, and the limitations of fixed linear state-space representations. These challenges restrict the scalability of Koopman-based approaches. This paper addresses these challenges by proposing a continual learning algorithm designed to iteratively refine Koopman dynamics for high-dimensional legged robots. The key idea is to progressively expand the dataset and latent space dimension, enabling the learned Koopman dynamics to converge towards accurate approximations of the true system dynamics. Theoretical analysis shows that the linear approximation error of our method converges monotonically. Experimental results demonstrate that our method achieves high control performance on robots like Unitree G1/H1/A1/Go2 and ANYmal D, across various terrains using simple linear MPC controllers. This work is the first to successfully apply linearized Koopman dynamics for locomotion control of high-dimensional legged robots, enabling a scalable model-based control solution.

Figures

Figures reproduced from arXiv: 2411.14321 by the authors.

**Figure 2.** Comparison results of k-step prediction error in our test suites. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Visualization of tracking performance on Flat-Unitree-G1. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Visualization of the data distribution by plotting the means of the joint-relative and root-relative [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: K-step prediction error comparison of ablation on di [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Computational cost(GPU occupancy, CPU occupancy and wall-clock time) comparison We compare the resource usage of all algorithms in terms of GPU and CPU consumption, as well as wall-clock time, using DKUC as a benchmark, as presented in [PITH_FULL_IMAGE:figures/full_f…

**Figure 7.** Figure 7: Legged robots and terrain in our test suites. [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

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ISSN 2405-8963. doi: https://doi. org/10.1016/j.ifacol.2022.11.219. URL https://www.sciencedirect.com/science/ article/pii/S2405896322028622. 2nd Modeling, Estimation and Control Conference MECC

2022 doi
[2023]

Korol: Learning visualizable object feature with koopman operator rollout for manipulation

Hongyi Chen, Abulikemu Abuduweili, Aviral Agrawal, Yunhai Han, Harish Ravichandar, Changliu Liu, and Jeffrey Ichnowski. Korol: Learning visualizable object feature with koopman operator rollout for manipulation. arXiv preprint arXiv:2407.00548,

arXiv
[2024]

Lars Grüne, Jürgen Pannek, Lars Grüne, and Jürgen Pannek

URL https://arxiv.org/abs/ 2406.10454. Lars Grüne, Jürgen Pannek, Lars Grüne, and Jürgen Pannek. Nonlinear model predictive control. Springer,

arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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These hyperparameters are selected when the robot is on the verge of falling and entering an unrecoverable state

It is worth noting that robots performing different tasks have distinct physical structures and movement patterns, necessitating different ϵf ailvalues. These hyperparameters are selected when the robot is on the verge of falling and entering an unrecoverable state. 24 CONTINUAL LIFTING OF KOOPMAN DYNAMICS Table 3: The settings of test suites environment ...

work page 2023

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Anusha Nagabandi, Gregory Kahn, Ronald S

doi: 10.1109/LRA.2023.3270034. Anusha Nagabandi, Gregory Kahn, Ronald S. Fearing, and Sergey Levine. Neural network dynamics for model-based deep reinforcement learning with model-free fine-tuning,

arXiv 2023

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Lixing Song, Junheng Wang, and Junhong Xu

URL https: //arxiv.org/abs/2408.04200. Lixing Song, Junheng Wang, and Junhong Xu. A data-efficient reinforcement learning method based on local koopman operators. In 2021 20th IEEE International Conference on Machine Learning and Applications (ICMLA), pages 515–520,

arXiv 2021

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URL https://doi.org/10.1080/01691864.2023

doi: 10.1080/01691864.2023.2168134. URL https://doi.org/10.1080/01691864.2023. 2168134. Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of computational mathematics, 12:389–434,

arXiv 2023

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URL http://dx.doi.org/ 10.1007/s00332-015-9258-5

doi: 10.1007/s00332-015-9258-5. URL http://dx.doi.org/ 10.1007/s00332-015-9258-5 . 13 LI ABUDUWEILI SUN CHEN ZHAO LIU Zeji Yi, Chaoyi Pan, Guanqi He, Guannan Qu, and Guanya Shi. Covo-mpc: Theoretical analysis of sampling-based mpc and optimal covariance design,

doi:10.1007/s00332-015-9258-5

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Peng Yin, Abulikemu Abuduweili, Shiqi Zhao, Lingyun Xu, Changliu Liu, and Sebastian Scherer

URL https://arxiv.org/abs/ 2401.07369. Peng Yin, Abulikemu Abuduweili, Shiqi Zhao, Lingyun Xu, Changliu Liu, and Sebastian Scherer. Bioslam: A bioinspired lifelong memory system for general place recognition. IEEE Transactions on Robotics,

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Robots with Attitude: Singularity-Free Quaternion-Based Model-Predictive Control for Agile Legged Robots

URL https: //arxiv.org/abs/2409.09940. Weiye Zhao, Feihan Li, Yifan Sun, Rui Chen, Tianhao Wei, and Changliu Liu. Absolute policy optimization: Enhancing lower probability bound of performance with high confidence. In Forty-first International Conference on Machine Learning. James Zhu, J. Joe Payne, and Aaron M. Johnson. Convergent ilqr for safe trajector...

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Convergent iLQR for Safe Trajectory Planning and Control of Legged Robots

URL https://arxiv.org/abs/2304.00346. 14 CONTINUAL LIFTING OF KOOPMAN DYNAMICS Appendix A. Related Works Controlling Nonlinear Dynamics. Nonlinear dynamic systems are generally controlled using two main approaches: model-free and model-based methods. Model-free methods, often associated with learning-based approaches, have become popular for addressing co...

work page Pith review arXiv 2018