Model Tensor Planning
Authors: An Thai Le
Khai Nguyen
Minh Nhat VU
Joao Carvalho
Jan Peters
Venue: Accepted by TMLR ()

Abstract: Sampling-based model predictive control (MPC) offers strong performance in nonlinear and contact-rich robotic tasks, yet often suffers from poor exploration due to locally greedy sampling schemes. We propose \emph{Model Tensor Planning} (MTP), a novel sampling-based MPC framework that introduces high-entropy control trajectory generation through structured tensor sampling. By sampling over randomized multipartite graphs and interpolating control trajectories with B-splines and Akima splines, MTP ensures smooth and globally diverse control candidates. We further propose a simple $\beta$-mixing strategy that blends local exploitative and global exploratory samples within the modified Cross-Entropy Method (CEM) update, balancing control refinement and exploration. Theoretically, we show that MTP achieves asymptotic path coverage and maximum entropy in the control trajectory space in the limit of infinite tensor depth and width.

Our implementation is fully vectorized using JAX and compatible with MuJoCo XLA, supporting \emph{Just-in-time} (JIT) compilation and batched rollouts for real-time control with online domain randomization. Through experiments on various challenging robotic tasks, ranging from dexterous in-hand manipulation to humanoid locomotion, we demonstrate that MTP outperforms standard MPC and evolutionary strategy baselines in task success and control robustness. Design and sensitivity ablations confirm the effectiveness of MTP’s tensor sampling structure, spline interpolation choices, and mixing strategy. Altogether, MTP offers a scalable framework for robust exploration in model-based planning and control.

Published in Transactions on Machine Learning Research (08/2025)
Model Tensor Planning
An T. Lel.?, Khai Nguyen', Minh Nhat Vu João Carvalho', Jan Peters' 1,2,3,4 {an, joao, jon]@rrbot-leemmonn
Intelligent Autonomous Systems, Department of Computer Science, Technical University of Darmstadt, Germany Systems AI for Robot Learning, German Research Center for Al (DFKI), Germany Hessian Center for Artificial Intelligence (hessian.Al), Germany
Centre for Cognétive Science, Technical University of Darmstadt, Germany Automatioo & Control Institute, TU Wien, Austria
Austrian Institule of Technology (AIT), Vienna, Austria Robotiiss and VinUnéversity, Vietnam
Reviewed OpenReview: https://openreveee net/ formm id= fki2ZdXCE3
Abstract
Sampling-basee model prodictive control (MPC) offers strong performance in nonlinear and contact-rich robotic tasks, yet often suffers from poor exploration due to locally greedy sampling schemes. We propose Model Tensor Planning (MTP), novel sampling- based MPC framework that introduces high-entropy control trajectory generation through struc- tured tensor sampling. By sampling over randomized multipartite graphs and interpolating control trajectories with B-splines and Akima splines, MTP ensures smooth and globally diverse control candidates. We further propose a simple 3-mixing strategy that blends lo- cal exploitative and global exploratory samples within the modified Pross-Entropy Mothod (CEM) update, balancing control refinement and exploration. Theoretically, we show that MTP achieves asymptotic path coverage and maximum entropy in the control trajectory space in the limit of infinite tensor depth and width.
Our implementation is fully vectorized using JAX and compatible with MuJoCo XLA, supporting Jusl-in-lüme (JIT) compilation and batched rollouts for real-time control with online domain randomization. Through experiments on various challenging robotic tasks, ranging from dexterous in-hand manipulation to humanoid locomotion, we demonstrate that MTP outperforms standard MPC and evolutionary strategy baselines in task success and control robustness. Design and sensitivity ablations confirm the effectiveness of MTP's tensor sampling structure, spline interpolation choices, and mixing strategy. Altogether, MTP offers scalable framework for robust exploration in model-based planning and control.
1 Introduction
Sampling-bass based Model Predictive Control (MPC) (Mayne 2014 Lorenzen et al. 2017 Williams et al. 2017) has emerged as a powerful framework for controlling nonlinear and contact-rich systems. Unlike gradient- based or linearization approaches, sampling-based MPC model-agnostic and does not require differentiable dynamics, making it well-suited for high-dimensional, complex systems such as legged robots Slivarez-Padille et al. 2024) and dexterous manipulators (Li et al.) 2024). Moreover, its inherent parallelism enables elficient deployment modern hardware (e.g., GPUs), allowing for tigh-throughput simulation and online domain randomization in real-time control pipelines (Pezzato et al. 2025).
Despile these advantages, a fundamental limitation remains: sampling- search behavior. Most methods perturb a nominal trajectory or refine the current best samples, which makes them susceptible local minima and unable to consistently discover globally optimal solutions (Xuc) lt al. 2024). While the Cross- Cross-Entropy Method (CEM) (De Boer et al. 2005) has shown promise in high- dimensional control and sparse-reward settings (Pinneri et. al.] 2021
baasling-based based MPC is typically local in its
it. still suffers from mode collapse

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Published in Transactions on Machine Learning Research (08/2025)
when sampling locally (Zhang et al. 2022;. leading to suboptimal behaviors. The curse of dimensionality exacerbates this issue, as the number of samples required to explore control spaces grows exponentially with the planning horizon and control dimension, posing bottleneck if compute or memory is limited. These challenges motivate the need for a more effective, high-entropy sampling mechanism for control generation.
Evolutionary Strategies (ES) (Hansen 2016 Wierstra et al. 2014 Salimans et al., 2017 have also been applied in sampling-base based MPC settings to improve sampling exploration. While they improve over purely local strategies in some tasks, our experiments (cf. Section 4.1 reveal that ES still fails to systematically explore multimodal control landscapes, often yielding inconsistent performance on tasks requiring long-tern coordinated actions.
In this work, we introduce Model Tensor Plamming (MTP), a novel sampling-based MPC framework that enables globally exploratory control generation through structured tensor sampling. MTP reformulates control sampling as tensor operations over randomized multipartite graphs, enabling efficient generation of diverse control sequences with high entropy. To balance exploration and exploitation, we propose a simple yet effective 8-mixing mechanism that combines globally exploratory samples with locally exploitative refinements. We also provide theoretical analysis under coanndddvvriationn assumptions, showing that our sampling scheme achieves asymptotic path coverage, approximating maximum entropy in trajectory space.
MTP is designed with real-time applicability with matrix-based formulation, which is compatible with Just- in-time (JIT) compilation and vectorized mapping (e.g,, via JAX vmap (Bradbury et al. 2018)), allowing tigh-throughput sampling, batch rollout evaluation, and online domain randomization on modern simulators. Our main contributions are as follows:
We propose tensor sampling, a novel structured sampling strategy for control generation, and provide theoretical justification via asymptotic path coverage.
We introduce a simple 3-mixing mechanism that effectively balances exploration and exploitation by blending high-entropy and local samples within the modified CEM update rule.
We demonstrate that MTP is highly compatible with modern vectorized simulators, enabling efficient batch rollout evaluation and robust real-time control in high-dimensional, contact-rich environments.
2 Preliminary
Notations and Assumptions. We consider the problem of sampling-basee based MPC, Given a dynamics model t = f(x w), we consider the path sampling problem in the control space U C R" with the control u E U having n-dimensions at the current system state æ E X C Rdd Typically, a batch of control trajectories is sampled, rolled out through the dynamics model, and evaluated using a cost function. Let a control path be u: [0, + u, (() E U, we can define the path are length
sup TV(u) MEN+,0=h 00th....
EMT
(1)
We define as the set of all control paths that are uniformly continuous with bounded variation (V < 0o, e This assumption is common in many control settings, where the control trajectories are bounded in time and control space (i.e., both time and control spaces are compact). Throughout this pa-
per, we narrate the preliminary and the tensor sampling method in matrix definitions, discretizing continuous paths with equal time intervals.
2.1 Cross- Cross-Entropy Method for Campling-based MPC
Consider discretized dynamical system T++1 f(x,, uz), t 0,.., with horizon T, where æt e Rd, ue E R" denotes the state the control at time step t. Given the state cost function c(æ, u) and the terminal cost (z)), the objective is to minimize a cumulative cost function
U(T = Σ cl ux) + (T(T)) 8.t. Le-1 = f(æ,, ut)
(2)
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Published in Transactions on Machine Learning Research (08/2025)
Linear
B-soline
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Figure 1: Comparison of MTP interpolation methods versus CEM on PushT environment. The cost of PushT is the sum of the position and orientation error to the green target (without the guiding contact cost), and the initial T pose is randomized. The first row depicts the cost convergence over 10 seeds. In most seeds, CEM struggles to push the object due to the lack of exploration (e-g., mode collapsing), while MTP variants always find the correct contact point to achieve the task. Note that the control magnitude of MTP is high due to the global explorative samples (see second & third rows), compared to the white noise samples (blue). B-spline helps regulate the control magnitude due to its barycentric weightings, while retaining exploration behaviors. The last row illustrates the control trajectories between 64 tensor samples and 64 white noise trajectorios. Experiment videos are publicly available at httpp::/ssiten google. wtwp:/iww/w/eeeeeeeeeeeeee sampaoorrrmmmainngg//
where the state trajectory = [Fo, ( mrl (T++]xxd and control trajectory U = [uo.... uT- E RRxx are the dynamics rollout, c(w,, u,) is the immediate cost each time step, and cT(xT) is the terminal cost.
CEM optimizes the control sequence U iteratively by approximating the optimal control distribution using parametric probability distribution, typically Gaussian. Initially, the control inputs are sampled from an initial distribution parameterized by mean μ E R and standard deviation a E RTxx, At each iteration, CEM performs the following steps ileratively:
Sampling. Draw B candidate control sequences from the current Gaussian distribution:
U diaglo"))
k= L=.....
(3)
Evaluation. Rollout ((k) from the dynamics model and compute the cost ( for each sampled control sequence by sinulating the system dynamics.
Elite Selection. Choose the top-E B elite candidates of control sequencos that have the lowest cost, forming clite set E (U(4))
Distribution Update. Update the parameters (n, σ-) based on elite samples:
Nuew
Lu, o² new E-1
E
Mnew) )²
(4)
E
UEE
USE
An exponential smoothing update rule can be used for stabilily:
1 + au + (1 Muews
f- ao + (1 a)
(5)
where a E 10. is a smoothing factor.
Termination Criterion. The itcrative process continues until convergence criterion met or maximum
number of iterations is reached. The optimal control sequence approximated by the final mean μ of the Gaussian distribution.
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2.2 Spline-based based Controls
Splines provide powerful representation for trajectory generation in MPC due to their flexibility, continuity properties, and ease of parameterization (Jankowski et al.) 2023 Carvalho et al., 2025). In this work, we focus on splin-paramctriiation control trajectories. Spline-basck trajectories cnsure smooth and feasible control inputs that satisfy constraints and objectives inherent to MPC frameworks.
A spline is defined as a piecewise polynomial function u(t): [0 + U, which is polynomial within intervals divided by knots ,,, with continuity conditions enforced at these knots. In partieular,
Knots. A knot t E [0 is time point where polynomial pieces join. We have non-decreasing sequence of knots 0 = 4 S I tM = T, which partition the time interval [0, 7"] into pieces [4, 4i+ so that the path is polynomial in each piece. We may often have double or triple knots, meaning that several consecutive knots 12 = 4++1 are equal, especially at the beginning and end, as this can ensure boundary conditions for zero higher-order derivatives.
Waypoints. A waypoint ut) E U is point on the path, typically corresponding to utt).
Control Points. A set of control points 2 {zi|zz E [R parametrizes the spline via basis functions.
B-Spline Parameterization. In B-splines dd Boor 1973) of control points &; E Z
the path u is expressed as a linear combination
K
uut) = Σ (t)
s.t. Σ (t) =1
(6)
where Bipp: : R > R maps the time t to the weighting of the 8th control point, depicting the 5th control point weight for blending (i.e., as with a probability distribution over i). Hence, the control waypoint u(l) is always in the convex hull of control points. The B-spline functions (t) are fully specified by a non-decreasing series of time knots 0 = 11 .... S =t==T = and the integer polynomial degree E {0, 1.... by
(t, = 1t4 Ststt444 t+tl. for 1 KiSM--
(t) = t- 44
lipp - B (1. + 4+p-1 - 1 14p41 1,+1
Bit1p-1(t),
(7)
Big are binary indicators of E [ts, t44] with i S M The 1st- B-spline functions B have
support in 1 E [44, (44) with I i S M such that Σ i=1 (1) = holds. In general, degree p R-spline functions Bip have support in E [t,, with 1 I i < M p - 1. We need K = M p control points Σ1.K, which ensures the normalization property Σ i=1 (() = 1 for every degree.
K
Akima-Spline Parameterization The Akima spline (Akima 1974 is a piecewise cubic interpolation method that exhibits C' smoothness by using local points to construct the spline, avoiding oscillations or overshooting in other interpolation methods, such as B-splines. In other words, an Akima spline is piecewise cubic spline constructed to pass through control points with C' smoothness. Given the control point set Z with K- M, the Akima spline constructs a piecewise cubic polynomial u(I) for each interval [4,, 44
(t) d, (6--,) 4.)² +
+ b,(1 - (.) + a., - c1
(8)
where the coefficients a; bi, C2 d; E U are determined from the conditions of smoothness and interpolation.
Let m. (z++1 2,)/(t+41 t,), the spline slope is computed as
8; mi+1 m. mm |mi-1 m
(9)
m;+1 m. + m
m;-2l
The spline slopes for the first two points at both ends are 81 = m1, 82 = (m1 + m2) 8M-1 = (mm-1 m.M 22)2, SM= = mM-1- Then, the polynomial coefficients are uniquely defined
a: Ui b 8i. c₁ (3m; 2s; 8141)/(1+11 42), d: (8: 8:41 2m:)/ 4.)3 (10)
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Linear
B-spline p-2
B-spline p-3
Akima-spline
Figure/ 2: Illustration of different tensor interpolations evenly spaced graph with M 3, - 9. With higher B-spline degree, the control trajectories exhibit more smoothness and conservative behavior, while Akima-spline aggressively and smoothly tracks control-waypoints. Note that we do not consider boundary conditions for B-spline interpolation in tensor sampling.
Motivation. Spline representation provides several benefits. (i) It ensures smooth control trajectories (Watson & Peters 2023) with guaranteed continuity of positions, velocities, and accelerations under mild assumptions of the dynamics.
(ii) Spline simplifies complex trajectorics through a few control points and efficiently incorporales constraints and boundary conditions, enabling efficient learning and planning by di- mension reduction |Carvalho et al. 2025). (iii) It enables easy numerical optimization thanks differentiable and convex representations.
3 Method
We first propose lensor sampling a batch control path sampler having high explorative behavior, and investigate its path coverage property over the compact control space. Then, incorporate the tensor sampling with the modified CEM update rule, balancing exploration and exploitation in cost evaluation, forming an overall vectorized sampling-basee MPC algorithm.
3.1 Tensor Sampling
Inspired by (Le et al. 2025). we utilize the random multipartite graph as tensor discretization structure to approximate global path sampling.
Definition 1 (Random Multipartite Graph Control Discretization), Consider graph G(M, N) = WV on control space U, the node set V = { is a set of M layers. Each layer L; = {uj E U Uj Uniform(U)1 contains N control-waypoints sampled from a uniform distribution over control space (i.e., bounded by control limils). The edge set W defined by the union of (forward) poir-wise connections between layers
W ( u4)1 | Vuu E Li, us41 E Li41, 1sis
lending to complete M -partite directed graph.
Sampling from G(M, The graph nodes are represented as the control-waypoint tensor for all layers Z E BBMXN xxm within the control limits. To sample a batch of B E N+ control paths with horizon T' we subsample with replacement C Bx Mxn from the set of all combinatorial paths in & (cf. Algorithm I and further interpolate C into control trajectories U E BxTxn with different smooth structures, e.g., using Eq. (6) or Eq. (8). Sampling with replacement cheap O(MN), while sampling without replacement is O(N Sampling without replacement adds overheads due re-indexing tree-traversing to get batch of sequence indices (depending on the low-level implementation of sampling) but offers better diversity. In practico, we use sampling with replacement, which does not really affect diversity (see last row in Fig. [. is faster and scales well with JAX vectorized operations. To see this, we have NM combinatorial paths in G(M,N) N), each path in G(M, N) has uniform M/NM mass, and the probability of sampling the same paths is small.
Control Path Interpolation. Straight-line interpolation can be realized straightforwardly by simply probing an H [T/M] number of equidistant, points wetween layers, forming the linear coverage trajectories
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U E BxTxx However, there exist discontinuities at control waypoints using linear interpolation. Hence, we motivate spline tensor interpolations for sampling smooth control paths (ef. Fig.
Definition 2 (B-spline Control Trajectories). Given two time sequences ( and tÚ j (O...., T 1}, the B-Spline matrir Bp E RMxT ing Eq. 7 with index i,j corvesponding to the element Bipllz). Then, the comtrol brujeclories can be inler- poloted by performing einsum on the M dimension of Bp E I RXXT C BBXMx* resulting U RBxTx"
M+p)
be constructed recursive follow-
B-spline control trajectories exhibit conservative behavior they are strictly inside the control point convex hull. Alternalively, they can be forced to pass through all control points by adding multiplicity of per knot (de Boor 1973). However, this method wastes computation by increasing the B-spline matrix size to Mp N, and still cannot avoid the ovcrshooting problem. Thus, we further propose the Akima-spline control interpolation.
Definition 3 (Akima-spline Control Trajectories (Le et al. 2025 Given the time sequences t, = i/M, i E {O....,M .M - 1} representing the M layer time slices and the control points C E TR Bxxxxx the Akima poly- nomial parameters E Bx (M-1)xxx can be computed following Eq. 10 in batch. Then, given the time sequence I; = j/T, E {0. .0. 1}, the control trajectories U BxTxn are interpolated sollowing polynomial interpolation Eq.
the next section, we investigate whether the path distribution support of tensor sampling approximates the support of all possible paths in the control space, with linear interpolation. B-spline and Akima-spline variants are deferred for future work.
3.2 Path Coverage Guarantee
We analyze the path coverage property of ((M,N) for sampling control paths with lincar interpolation. In particular, we investigate that any feasible path in the control space can be approximated arbitrarily well by a path in the random multipartite graph G(M, N) as the number of layers M and the number of waypoints per layer N approach infinity.
Theorem 1 (Asymptotic Path Coverage). Let E F be any control path and G N) be a random multi- parlile gruph with M layers and N uniform samples per layer (ef- Defonition 7). Assuming a time sequence (t.e., knots) = t1 < t2 < .... tss = with equal intervals, associating with layers LM E g(M,N) respectively, then
lim M.N (E
min
1 - ||lbo = 0.
In intuition, Theorem D states the support of path distribution G(M, approximates F and converges to F as N N + c0. Thus, sampling paths from G(M,N) provides a tensorized mechanism to efficiently sample any possible path from F, which allows vectorized sampling.
Remark. As M,N + co, for any control path F, then g G(M,N). Hence, ((M,N) represents all homotopy classes in the limit M, N + oc, and sampling paths from ( N) approzimate sampling from all possible paths.
To quantify the exploration level of tensor sampling, one standard way is to investigate its path distribution entropy. In intuition, when M, N + co, tensor sampling entropy also approaches infinity due to uniform sampling per layer, which is further discussed in Appendix A.2 Note that this theory investigation serves as guiding principle, while practical settings trade off entropy with success rate and runtime feasibility.
Algorithm 1: Sampling Paths From G(M,N)
Control waypoints Z R MxNxn number of paths B Sampled control-waypoints C BxxR I + randint((B, M), N). // batch sample with replacement from I,..,,N with shape (B, (B,M) C < parse_index/ Z, I). // extract waypoints from sampled indices into C e BBxMx
Input:
Output:
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Practical Settings. We typically only set M < T. In principle, increasing N should enable finer-grained exploration over the trajectory space. However, we observe diminishing returns when N increases while keeping the total number of sampled trajectories B fixed (ef. Fig. S Intuitively, the underlying graph becomes denser, and the number of explored paths remains constant, resulting in only marginal performance improvements. Therefore, we recommend choosing N proportionally to B and within the bounds of available GPU memory, maintain computational efficiency without oversampling from small sample size.
3.3 Algorithm
Here, we present the overall algorithm combining tensor and local sampling with smooth structure options in Algorithm ?
We propose a simple mixing mechanism with 3 E [0, 1] by concatenating explorative and exploitative samples, forming a control trajectory tensor U (Line 4-8). We include the current nominal control for system stability at the fixed-point states (e.g., for tracking tasks) (Howell et al.) 2022). Using simulators that allow for parallel runs (Todorov et al. 2012 Makoviychuk et al. 2021], rollout and cost evaluation can be efficiently vectorized (Tine 9).
To lame the noise induced by tensor sampling, we modify the CEM update with softmax weighting on the elite set, for computing the new weighted control means and covariances similar to the MPPI update rule (Williams et al., 2017]. We observe that this greatly smoothens the update over timesteps (Line 11-13) (cf. Fig. 9). Finally, similar Howell al. (2022). we send the first control of the best candidate, since this control trajectory evaluated in the simulator rather than the updated mean μ.. Notice that we have fixed tensor shapes based on hyperparameters, for all subroutines of sampling, rolling out, and control distribution updates. Algorithm 2 can be JAX jit and vmap over a number of R model perturbations {f, u) for efficient online domain randomization, while maintaining real-time control (cf. Appendix A.5
Algorithm 2: Model Tensor Planning
Input: Model f(x, u), graph params N, mum samples B, mixing rate 3 E [0, 1], planning horizon T. CEM params a E [0, 1), À > 0, om > 0, E, which are smooth and temperature scalar, minimun variance, elite number, respectively.
Choose interpolation type Linear, or B-spline (Definition 2 or Akima (Definition
2 Init the nominal control μ E Rxxx and variance diag(o"), Txx.
while Task is not complete do
// tenser sampling
Uniformly sample Q E MxMxx on control space U. Sample control waypoints C E RxMMx with P [3B], using Algorithm !
Interpolate C' using with chosen interpolation method into control trajectories Ug E PxTxn. // local sampling
7
Sample B P local trajectories ULocal ~ (H, diag(o?)).
Stack ULocal Ug, into U E BBTxx
// Update routine using vectorized simulator
Batch rollout X E R Bxxxd from model f(x, u), and evaluate cost matrix S(X,U) E BxT
9
Sum cost Σ, S. E BB and sort top-E elite candidate indices ig.
10
exp( {
11
Select candidate U <- parse faaeee_ideexxx___ ii) and compute candidate weights w = Σ expl (
Compute mean μí wU and variance o' max(w (u- om)- Update μ + μt a(μ u²) and + o' + a(o o'). 11 Send the best evaluated control u² g u" <- Un
14
4 Experiments
In this section, investigate our proposed approach with the following research coostionsspooints.

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54
en
03
Sepe
Figure 3: Motivation comparison of MTP methods versus baselines with B = 256 on Navigation environ- ment. The environment is designed to be challenging to reach the green goal, requiring strong exploration to avoid large local minima the middle (sce task dctails in Appendix A.3 We plan with T = 20 with Al 0.05s. The figures show 5 random traces of white rollouts. (Teft) MTP-Akima rollouts reach the green goal very early due high-entropy tensor sampling, while (Right) OpenAI-ES struggles to generate a rollout exploring the way out of large local minima.
How does MTP's performance with Akima/B-spline control variants compare to standard sampling- based MPC baselines, and strong-cxploratory evolution strategies basclines?
How does MTP's performance vary with the number of clites and mixing rate 3 on interpolation methods (Linear, B-spline, Akima-spline)?
How does MTP's performance vary with (i) mixing rate B associating with levels of MTP exploration complex environments, and (ii) MTP-Bspline degree versus planning performance.
We study the cumulative cost J(T, U) Eq. over the control timestep for each task. For each experiment, we take the minimum cumulative cost in batch rollouts at each timestep, and plot the mean and standard deviation over seeds. Further ablations on sweeping graph paramcters M, N, softmax weighting, and JAX planning performance benchmark are presented in Appendix A.5
Practical Settings. All algorithms and environments are implemented in MuJoCo XLA Todorov et al, 2012 Kurtz 2024). to utilize the jit and vmap JAX operators for efficient vectorized sampling and rollout on multiple model instances. All experiment runs are sim-to-sim evaluated (MuJoCo XLA to MuJoCo). particular, we introduce some modcling crrors in MuJoCo XLA. Then, for online domain randomization, wo randomize a set of R models (2, /=1 then we perform batch sampling and rollout R trajectories. Finally, the cost evaluation is averaged on the domain randomization dimension. Note that, for this paper, deliberately design the task costs to be simple and set sufficiently short planning horizons to benchmark the exploratory capacity of algorithms. In practice, one may design dense guiding costs to achieve the tasks. Further task details are presented in Appendix A.3
Basclines. For explorative baselines, we choose OpenAT-ES (with antithetic sampling) (Salimans et al. 2017), which shows parallelization capability with a high number of workers in high-dimensionall model- based RT settings. Additionally, we choose the recently proposed Diffusion Evolution (DE) (Zhang et al., 2024), bridging the evolutionary mechanism with diffusion process (Ho al. 2020),, which demonstrates superior performances over classical baselines such CMA-ES (Hansen 2016). Both are implemented in evosax [Lange] 2023). For methods that take into account local information (exploitation methods), we compare MTP with standard MPPI (Williams et al. 2017 and Prodictive Sampling (PS) (Howell et al.
2022 to sanity check on task completion in sim-to-sim scenarios.
4.1
Motivating Example
Here, we provide experimental analysis of the baselines' exploration capacity on Navigation environment (cf. Fig. 3,, where the point-mass agent is controlled by an axis-aligned 2-dim velocity controller. We compare MTP-Bspline and MTP-Akima to evolutionary algorithms and standard MPC basclines, with maximum sampling noise settings (cf. Fig. 3). In particular, given the control limits [-1,1] on x-y axes, we set the standard deviation σ = for sampling noise of MPPI and PS, and population generation noise for OpenAl- ES and DE. Fig. 3 also shows the cost convergence and the cost entropy curves over timesteps, in which the entropy is computed as H - log P,. Pd exp(J,)/(E, exp(Ji)), where (JA1 is the batch of cunulative rollout costs. The entropy represents the diversity of rollout evaluation, implying the exploration
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Figure 4: Performance comparison of MTP variants against standard MPC methods (MPPI, PS) and evolu- tionary algorithms (OpenAI-ES, DE). The (horizontal) gray dashed line depicts the task success, while the (vertical) red line represents the timestep such that the first algorithm statistically succcods the task, or fails last in G1-Walk.
capability of algorithms. We observe that MTP-Akima has the highest entropy curve, inducing the lowest cost convergence over timesteps, while other baselines converge to the middle minima. In this case, MTP- Bspline also struggles due to conservative interpolation in tensor sampling, achieving moderate entropy, yet higher than evolutionary strategies.
4.2 Comparison Experiments
We analyze the performance comparison of MTP and the baselines over various robotics tasks representing different dynamics and planning cost settings (cf. Fig. In each task, we tune the baselines and set the same white noise standard deviation for MTP WTTP/MPPI/PS to study the performance gain by tensor sampling, while using the default pyperaarraeterss for evolutionary baselines.
Comparison Environments. PushT (Chi et al. 2023), Cube-In-Hand (Andrychowicz et al. 2020 require intricate manipulation environments where robust exploration critical due to complex contact dynam- ics and precise multi-slep manipulation requirements. G1-Standup, G1-Walk represent high-dimensional robotic tasks demanding substantial computational resources and sophisticated control strategies. Crane, Walker (Towess et al. 2024) present underactuated and nonlinear dynamic challenges. To ensure fair com- parison, for all baselines, tasks. All tasks are implemented in hydrax (Kurtz 2024 Further experiment details are in Appendix A..4 The PushT task, which involves pushing a T-shaped object precisely to a target location, particularly high- lights the advantage of MTP variants. While MPPI and PS frequently encounter mode collapse due to insufficient exploration, resulting in suboptimal or even failed attempts at solving the task, MTP-Bspline and MTP-Akima consistently achieve low-cost convergence. This underscores the significant benefit of strong exploration enabled by tensor sampling. Evolutionary algorithms like OpenAIEES and DE perform similarly to MTP in PushT, inherently show better exploration than MPPI and PS, but still fall short compared to MTP variants in Cube-In-Hand due to noisy rollouts. Cube-In-Be In-Hand requires strong exploration while main- taining intricate control to avoid the cube falling, thus emphasizing the effectiveness of the MTP 3-mixing strategy.
fix the same number of rollouts B = 16 Crane, and B 128 for all other
In the Crane environment, we apply heavy modeling errors of mass, inertia, and pullcy/joint damping. MTP variants excel by maintaining stable control trajectories with smooth transitions, effectively navigating the nonlinear dynamics and underactuation. The B-spline interpolation's conservative nature helps avoid

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Published in Transactions on Machine Learning Research (08/2025)
Linear
B-spline
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MPPI
TS- MPPII
MPPI
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0.050.25 0.6 0.9 1.0
0.0 0.050.25 0.4 0.75 Mixing Rate
0.050.25 0.6 0.9 1.0
Figure 5: Mixing rate 3 and number of elites E' sweep on PushT, G1-Walk tasks with B = 128 to investigate the algorithmic update rule Algorithm [ Line 9-12. The heatmap indicates accumulated cost over timesteps at termination, and the heat value range fixed for each lask/row.
overshooting and instability prevalent these tasks, thus outperforming both evolutionary algorithms and standard MPC methods that tend to produce erratic control imputs. The Walker task exhibits rather simple dynamics model and is less sensitive to the sampling distribution due to its relatively simple contact model. We apply no modeling error as a samity check. Indeed, MTP and classical MPPI/PS perform similarly in simple cases.
In G1-Standup, MTP-Akima demonstrates effective humanoid standup due aggressive yet smooth trajectory interpolation, enabling efficient exploration and rapid convergence. In contrast, OpenAI-ES and DE struggle with the dimensionality, often yielding higher cumulative costs and failing to adequately sample feasible trajectories, resulting in significant performance gaps. MTP variants have marginally higher perfor- mance than standard MPC basclines in G1-Standup, but show better control stability for longer G1-Wall before falling. These results underline the capability of MTP to balance exploration and exploitation in high-dimensional tasks systematically.
4.3 Design Ablation
We conduct an ablation study analyzing the effect of varying two crucial hyperparameters- the number of elites E and the mixing rate B on MTP algorithmic performance. Fig. 5 presents heatmaps indicating accumulated cost over time for each task and interpolation method (MTP-Linear, MTP-Bspline, MTP- Akima). Fach heatmap illustrates distinct algorithmic realizations at its corners. Specifically, the bottom-left corner represents PS, characterized by single clite and purely white noise sampling. Conversely, the bottom- right corner corresponds to Predictive Tensor Sampling (TS-PS), maintaining single elite but employing full tensor sampling. Due to the softmax update (Algorithm 2 Line 11-12), the top-left corner realizes MPPI (with adaptive sampling covariance). leveraging all candidate samples with local noise sampling, while the top-right corner reflects Tensor Sampling-MPPH (TS-MPPI), utilizing all candidates and full tensor sampling for maximum exploration.
We observe the consistent pattern that MTP performance degrades at the extremes of B. In PushT, moderate values of 3 lead significantly lower costs, while the absence tensor sampling (8 0) yields poor performance due to inadequate exploration. This observation reinforces the effectiveness of the B-mixing
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Linear 0.1
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Figure 6: Mixing scalar B sweep on PushT, Iobe-In-Hande G1 G1-Standup environments with B = 128 to investigate the sensitivity of MTP on explorative level. The dashed line represents the successful bar. In Cube-In-Hande some the cost curves increase due to the cube falling out of the LEAP hand.
strategy for balancing global and local sampling contributions. Interestingly, the number of elites E has limited impact in PushT, likely due to the task's s insensitivity to control stability. However, in G1-Walk, the choice of E' is crucial. Using single elite (E' = 1), corresponding to the PS control scheme, leads to unstable and jerky behavior, which aligns with the poor performance observed in Fig. ! At the other extreme, with 3 = (full tensor sampling), performance also degrades. This is attributed to the fixed rollout budget B; as M incrcases, global samples become too sparse to effectively capture the fino-grained control required for stable gait tracking. In this case, exploitation with local samples is essential to maintain intricate motion tracking control.
4.4 Sensitivity Ablation
Here, we perform sensitivity analyses for various critical algorithmic hyperparameters. Fig. evaluates sensitivity to the mixing rate for different tasks. For the PushT environment, results show minimal sensitivity across mixing rates, as the task inherently lacks significant failure modes, ensuring consistent success regardless of the inplliation-exploitatton balance. In contrast, the Cube-In-Hand task demonstrates high sensitivity, with larger mixing rates causing instability due to the cube falling out of grasp frequently. Optimal performance thus achievod with lower & values, suggesting careful management of exploration intensity. Furthermore, for the high-dimennional G1-Standup lask, smaller mixing rate helps stabilize the control, enabling the robot to achieve more consistent and stable stand-up performance.
5 Related Works
We review related efforts across two major dircctions: the voctorization of sampling-based MPC and sampling-based motion planning. While these approaches originate from different planning paradigms, dynamics-aware MPC versus collision-free geometric planning, they share common structure; interact- ing with an agent-environment model (e.g., dynamics model or collision checker) that can be vectorized for efficient, batched computation. Both domains benefit from high-throughput sampling, making them increasingly amenable to modern GPUs/TPUs.
Sampling-based MPC Vectorization.
Sampling-base based MPC Mayne 2014] has been successfully ap-
plied to high-dimensional, contact-rich control problems. Methods such Predictive Sampling (PS) (Howell al. 2022 Model Predictive Path Integral (MPPI) (Williams et al. 2017 Watson & Peters 2023 and
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CEM-based MPC (Pinneri et al. 2021 rely parallel sampling of control trajectories and subsequent rollouts using a system dynamics model. These methods naturally benefit from vectorized simulation back- ends, and recent works have extended them toward more structured and efficient exploration. For instance, inspired by the diffusion process, DIAL-MPC (Xue et al.) 2024 enhances exploration coverage and local refinement simultaneously, achieving high-precision quadruped locomotion and out perforrmming reinforcement learning policies (Schulman et al. 2017) climbing tasks. STORM (Bhardwaj et al.; 2022) demonstrates GPU-accelerated joint-space MPC for robotic manipulators, achieving real-time performance while handling task-space and joint-space constraints. Other recent efforts integrate GPU-parallelizabie simulators, such as TsaaeGym (Makoviychuk et al.] 2021), into the MPC loop (Pezzato et al., 2025) for online domain ran- domization, removing the need for explicit modeling and enabling real-time contact-rich control. In another line, CoVO-MPC (Yi el al., 2024)) enhances convergence speed by optimizing the covariance matrix during sampling, leading to performance gains in both simulated and real-world quadrotor tasks. These advances demonstrate that structurod, parallel control sampling can be effectively deployod in high-stakes robotics applications using vectorized dynamic models.
Motion Planning Vectorization. Recent advances in sampling-basee based motion planning have demonstrated that classical methods, such as RRT (Kuffner& & Valle 2000), can be significantly accelcrated through par- allel computation, while preserving theoretical guarantees like probabilistic completeness, which represents another form of maximum exploration. Farly work focused on accelerating specific subroutines like collision 2011 Pan & Manocha 2012), but more recent efforts have restructured planners
checking (Bialkowski et al. for full GPU-native execution.
Examples inchude GMT* (Lawson et al. 2020), VAMP (Thomason et al., 2025). and Kino-PAX (Perrault al., 2025 which achieve millisecond-scale
2024), pRRTC (Huang et al.
planning in high-dimensional configuration spaces by parallelizing sampling, forward kinematics, and tree 2025 pushes this even further by implementing the sampling, graph-building, and search pipeline as tensor operations over batch batch-planning instances, showcasing the feasibility of real-time planning across multiple environments.
expansions. GTMP (Le et al.
Complementing campling-based planning, trajectory optimization hods such as batch CHOMP (Zucker et al.
2013 StPP (Urain et al., 2022), cuRobo (Sundaralingam et al. 2023), and MPOT (Le
ct al.
2023
have embraced voctorization to solve hundreds of trajectory refinement problems in parallel.
Many of these systems are further enhanced by high-entropy initialization with learned priors (Carvalho et al.]
2023 Huang et al. 2024 Nguyen et al. 2025), allowing them to overcome challenging nonconvexities in cluttered environments, These developments collectively demonstrate that both motion planning and MPC can be reformulated as batchod, tensor-based pipelines suitable for modern accelerators.
Our work draws on these insights to propose a unified sampling-based control framework that operates entirely through tensorized computation, blending global exploration and local refinement in a single batched planning loop.
6 Discussions and Conclusions
In this work, we introduced Model Tensor Planming (MTP), robust sampling-based MPC approach designed to achieve global exploration via maximum entropy sampling. Theoretically, we demonstrated that in the limits of infinite layers M and samples per layer N, our tensor sampling method attains maximum entropy, thereby efficiently approximating the full trajectory space. Furthermore, MTP is intentionally designed to be practically feasible, enabling straightforward implementation for sampling high-entropy control trajectories (see Appendix A.5
While evolutionary strategies algorithms offer improved exploration capabilities (Salimans et al. 2017 Zhang
et al. 2024), compared to traditional MPC methods, our experiments highlight their limitations. The inherently noisy mutation processes often fail to achieve consistent high-entropy exploration, limiting their effectiveness in robotics tasks. In contrast, MTP's tensor sampling consistently explores smooth control possibilities and achieves robust performance.
Spline-based interpolations are central to the practical implementation of MTP, notably B-spline and Akima- spline. These interpolation methods effectively address discontinuities in simple linear interpolation, ensuring
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the generation of smooth, contimuous, and dynamically feasible control trajectories (Alvare-PPaiila et al, 2024). The experiments underscore the splines' critical role in enhancing trajectory quality, optimizing performance across diverse, complex tasks. We proposed a simple 3-mixing strategy for exploration while retaining intricate controls, effectively balancing exploration and exploitation within sampling- based MPC. This flexible strategy allows easy algorithm tuning to various tasks, significantly improving performance stability and robustness across environments with different exploration nceds.
From the vectorization standpoint, the matrix-based definition of MTP is specifically structured to leverage Just-in-time compilation jit and vectorized mapping vmap provided by JAX (Bradbury et al 2018) and MuJoCo XIA (Todorov et al. 2012). This design choice dramatically accelerates computations, enabling real-time implementation and seamless integration with online domain randomization with vmap, crucial for robust control. Overall, MTP offers an efficient, scalable solution for various robotic tasks that demand high exploration capacity and precise control optimization.
Limitations
While MTP demonstrates strong performance across diverse control tasks, it inherits several limitations typical of sampling-baser methods. First, its computational cost scales with the number of rollouts, mak- ing it challenging to deploy on hardware with limited parallel computing. Second, while our lensor-based sampler improves exploration coverage, it does not leverage task-specific priors or learning-basee based proposal distributions, which could further improve sample efficiency. Finally, MTP relies on a fixed dynamics model, limiting its robustness in partially observed stochastic environments.
Broader Impact Statement
This work contributes to the development of efficient sampling- based control by introducing scalable, high- entropy sampling mechanism for model predictive control (MPC). Model Tensor Planning (MTP) opens a promising direction in the design of exploratory algorithms that go beyond local refinements, allowing for global reasoning over control spaces. By enabling maximum entropy exploration via structured tensor operations, MTP provides a framework that may benefit. wide range of decision-making systems requir- ing robust performance in underexplored, high-dimensional environments, such dexterous manipulation, legged locomotion, autonomous vehicles operating under partial observability and uncertainty.
From a real-world deployment perspective, MTP maintains controls within physical limits, but its high-rate, tensorized control sequences may induce rapid variations that practical motor systems must robustly execute. While this fast-changing control is beneficial for agility and responsiveness, it necessitates attention to actu- ator dynamics and hardware safety. Therefore, safety-aware control filtering or actuator-awen aware smoothness constraints may be incorporated as extensions for deployment.
Furthermore, like other MPC approaches, MTP assumess access to reliable state estimation for initializing planning rollouts in the control loop. In practical deployment, this typically requires real-time state estimator and a simulation back-end that serves as a digital twin. For example, MuJoCo XLA can simulate hundrods of dynamics instances in parallel, making it suitable for real-time predictive control. However, realizing this in hardware introduces engineering challenges, such as ensuring low-latency communication between the physical robot and the simulator. We see this digital twin architecture as a promising frontier where algorithmic advances like MTP can be tightly integrated with system-level design for robust, real-time, and scalable autonomous control.
Acknowledgments
This work was funded by the German Federal Ministry of Education and Research Software Campus project ROBOSTRUCT (01S23067).
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A Appendix
A.1 Proof of Theorem 1
Let u E F be any path that is uniformly continuous and has bounded variation TV(u) < co. We begin by constructing a piecewise linear path approximating u, 9M [0, - U by dividing the interval [o, 1] into M subintervals, i.e., [2, [M with = l < L2 < ... < tu = On each subinterval
[ 4+ define the corresponding segment of g to approximate
g(t) =u(t)) u(t(+1) - utt,) t.), t [t,
(11)
441 - -4
Then, we define a control path in (MM
Definition 4 (Path In ((M,N)). A path u [0, 1] + U is in 9(M,N) N) (Le., G(M,N)) is and only if u. is piecewise linear having M - segments and v1 < i M, u(t;) E L.
Lemma (Piecewise Linear Path Approximation). Let J1 92 be a piecewise linear function having the same number of partition points {gi (t.)) (ta(t)) with t < l2 < .... < <tM = 1, 11g1 92llmo < E, if and only if (tg) ge(ta)l <, sclsss M.
Proof. Sufficiency. Given (11((4)-ge(4) <,, V1 Sis M, since 91 92 are piecewise linear functions, the linear interpolation ween partition points 4; 1441 ensures that the difference between 91, is maximized at the partition points. Consider 91, 92 segment [t,,441]
(() 92(1)| < max (1:) - (4:)- g2())) (191())41) - g2((++))113 <e
(12)
Henco, I191 - 92 max, (t) - gz(()]l <c
Necessity. Given || g2ll.0 <; then || (1:) g2(4:)l < selsis
Now, we investigate that any piecewise linear path g with M 1 equal subintervals, approximating u E F, uniformly converges to as M + co.
Lemma 2 (Convergence Linear Path Approximation). Let gMs be any piecewise linear path approzimaling u E having M equal subintervals of width = 1/M. Then,
lim M
||u l0 0.
Proof. Since u is uniformly continuous on [0, 1], for any O, there exists 6 > 0 such that for all t, E [0, 1]. if It - s| < 8, then
[e((t)
u(s) 2'
(13)
Also, the variation of u within each subinterval approaches zero as M >> xo due to the uniformly continuous property. Hence, for sufficiently large M, each subinterval length h = M < 6, and thus:
u(t) u((t) + u(tx+1) - u(ti) 4:)
sup [(((t) 9m (1)ll sup (E(t..
h
sup 1E
(u()) u(ta)) + sup
u((++1) - ui(ts) (t (t)) t.)
< e c =c. (14)
h
2
Taking the supremum over all / E [0, 1] (i.e., over M equal subintervals), we obtain:
| gasl aoo < c.
(15)
Since e > 0 is arbitrary and h + 0 as M + 0o, it follows that:
lim ||u gMll a0 0. M=+00
(16)
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We now prove that the random multipartite graph discretization is asymptotically dense in F. Specifically, as the number of layers M and the number of samples per layer N approach infinity, the graph will contain path that uniformly approximates any continuous path in F.
Theorem (Asymptotic Path Coverage). Let E F be any control path and N be a random multi- partite graph with M layers and N uniform samples per layer (ef. Defonition 7). Assuming a time sequence (t.e,, knots) = t < t2 < .... ts = 1 with equal intervals, associating with layers Li,..., LM E ((M,N) N) respectively, then
lim M.N 0o gE
min ||u-glla =0.
Proof. First, Lemma 2 implics that there exists sequence of linear piccewise 9M, having M equal intervals approximating u, converging to M + co.
Let 9M E N) be a control path in G (cf. Definition 4. Since the time sequence 0 = t < t2 <
(M = corresponding to layers Li,..., LM E G(M, has equal intervals, we can consider ÅM having M - segments approximating 9M without loss of generality.
Since U open, for cach = .., M, there exists a ball (e(((t:)) C U, <> 0. By definition 9ar, 9M has the same number of segments, the event ll gM a0 e is the event that, for each layer S i S M, there is at least one point gMl (t,) is sampled inside the ball (t.) The probability that none of the N samples in layer L; fall inside (e( is
(1- ((B.(u(())) (1
N S N
((U)
(17)
((U)
for some > From Lemma
1 the probability that every layer contains at least one such sample, such that llaw garllo oo <,, is at least Me-eN which converges to N oc.
From Lemma for sufficiently large M, have ||u l0 e. Now, due U is compact, apply the triangle incquality as N oc
I - M I ||u - Illloo + l1a - Iarloo <c+c= 2c.
(18)
Since was arbitrary, we conclude that
lim M.N dM EG(M,N)
min
|/u - Darlloo //--aalloo 0.
(19)
A.2 Exploration Versus Exploitation Discussion
We investigate the tensor sampling Definition (with linear interpolation) versus MPPI sampling with horizon T, corresponding to global exploration versus local exploitation behaviors from the current system state. In particular, we remark on the entropy of path distributions in both methods the discretized control setting with equal time intervals. Further investigation on the continuous control setting is left for future work.
Let discrete control path be T [u....., Un] E R E W. Let P(T) be the probability of sampling control path t under given planning method. The entropy is then defined as
H(P) = - Σ P(T) P(T),
(20)
TEBr
where Fr C is the set of all possible discrete control paths f of length T.
Consider 9(M,N) N), each node layer L2 is sampled independently from a uniform distribution over W.
and path candidates are equivalent to sequences of node indices T 5 (11, f2..,,.,.., E fl....,N) N)M (cf. Algorithm 1. Let 8 denote the set of all index sequences representing valid paths through the graph.
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The uniform distribution over S is given by PG(T) 1/|S|, where |S| N Hence, the entropy of tensor sampling is
- - E0/N""/0/(1///
log(AV)) M log
(21)
TES
Indeed, as M, N + co, the entropy ((pp) + oo, and the distribution over sampled paths in g becomes maximum entropy over Fr among all discrete path distributions. Theorem limplies that Fr - F as M,N + 0c, and thus tensor sampling distribution becomes maximum entropy over F.
Now, lypical MPPI implementation generates control paths by perturbing nominal Irajectory F with Gaussian noise (Vlahov et al. 2024) u, u, E,, e ~ (0, (O,E), Σ), and propagating the dynamics to generate state trajectory. The path distribution PMPPI concentrates around 7 and is generally non-uniform. The entropy is constant and computed in closed form
Tn (1+ log(( + log de)
2 (PMPP))) = 2
(22)
due to independent Gaussian noise over timestep (i.c., white noise kernel WWatson & Peters 2023
In general, tensor sampling serves as a configurable high-entropy sampling mechanism over control trajectory space, offering maximum exploration, while MPPI targets local improvement around a nominal trajectory, thereby performing exploitation. This distinction motivates the hybrid method, where we mix explorative (smooth) controls with local controls sampled from typical white noise kernel.
A.3 Task Details
Here, we provide task details on the task, cost definitions, and their domain randomization. There exist motion capture sensors in MuJoCo to implement the tasks. For this paper, deliberatcly design the task costs to be simple and set sufficiently short planning horizons to benchmark the exploratory capacity of algorithms. In practice, one may design dense guiding costs to make tasks easier
Navigation. A planar point mass moves in a bounded 2D space via velocity commands to reach a target while avoiding collisions. The stale cost is defined as
c(æ,, un) = 0i
+ a2 || x1, + 03 1/
where dreal(2:) is the distance to the closest wall, Fg is the goal position, and u, is the velocity control. Success is defined when the agent's distance to the target is sufficiently small. This task is extremely difficult for sampling-besee based MPC due to large local minima near the starting point.
Crane. The agent controls a luffing crane via torque inputs to move suspended payload to a target while minimizing oscillations. The cost function penalizes payload deviation and swing:
c(x,, u.) = a1 [ - x,,' 2 + 02 |
where x; is the payload tip position, xg is the target point, and o; is the tip velocity. Success is achieved when the payload tip is within a small radius of the target location. This task is difficult due to heavy modeling error and underactuation, which common in real crane applications.
Cube-In-Hand. Using velocity control of a dexterous LEAP hand, the agent must rotate a cube to match randomly sampled target orientation. The cost combines position and orientation crror:
c(x,, u.) a ( 2,g) + a2 1|
where xy and By are the current cube and target poses, dsæ(3) is the SE(3) distance metric between poses, and u, = aंg Success is defined when the combined position and orientation errors fall below a threshold. This task is difficult due to the high-dimensional, contact-rich, and failure mode of the falling cube.
Galw A Unitree G1 humanoid robot tracks motion-captured walking trajectory using position control. The cost is defined as the deviation from reference joint positions:
c( (22 u+) on ||ar, Brre( + 1)
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where z and k) are the current joint and reference joint positions, given the current control iteration k. Success is not binary but is measured by minimizing deviation from the reference joint configurations. Note that this cost is not designed for stable locomotion. The main challenge is maintaining motion tracking locomotion over long horizons with complex joint couplings.
The humanoid must rise from lying pose to an upright standing posture. The cost penalizes deviation from upright pose and instability:
GI-Standup.
c(a,; u.) an (h+ h² o2dso(3) RRooso R²y)² + 03 || Qnominal
where h h" is the torso height and the standing height threshold, R Rg are the orientation current and target torso. h2s R qi are elements of æt. Success is defined when the height of the torso exceeds target threshold. The task difficult due large initial instability and the need achieve balance high-dimensional dynamics.
PushT. A position--controlledd end effector to push a T-shaped block to goal pose. The cost measures block pose crror
cl2,, u.) = on dsE(3) (xt,
where and By are the current T-block and target poses. Success is achieved when the block's 's position and orientation errors are minimized. The task is challenging because contact dynamics is complex, requiring precise interaction strategies.
Walker. A planar biped must walk forward at a desired velocity while maintaining an upright torso. The cost function penalizes deviation from target velocity and orientation:
c(x,, u() = a1(h; - h²)²²²² - az(0, 8*) + az( - v²)² + a4 ||well?
where It, ht is the torso height and the standing height threshold, vt, u² is the forward and target velocity, 0,,0* 8* is the torso and target pitch angle. h₁ 8, 2 are elements of Vt. Success is measured by stable forward motion and velocity tracking. The difficulty lies generating stable gaits without explicit foot placement planning.
Table summarizes the environments used in our experiments, including their state and action space di- mensions, control modalities, and whether domain randomization or task randomization applied. We set the control horizon and sim step per plan such that they resemble realistic control settings.
Table 1: Summary of environment propertics.
Task
State Dim Action Dim
Control Type
Domain Randomization
Navigation
2
Velocity
Joint obs. noise, actuation gain, init position
Crane
24
3
Torque
Payload mass, inertia, joint damping, actuation gain
Cube-In-Hand
39
16
Velocity
Joint obs. noise, geom friction Joint obs. noise, geom friction Joint obs. noise, geom friction Geom friction, init pose None (fixed init)
G1-Walk
142
29
Position
Gl-Standup
71
29
Position
PushT
14
2
Position
Walker
6
Torque
A.4 Experiment Details & Discussions
Comparison Experiment. We summarize the simulation settings for comparison experiments (cf. Sec- tion 4.2 in Table 2 and the hyperparameters used for MTP variants in Table 3 Tables summarize the hyperparamcters used for MPPI and PS (temperature is not relevant for PS). The same noise is used for MTP local samples. For evolutionary strategies (DE and OpenAI-ES), we use default hyperparameters in evosax LLngg 2023
Design Ablation. We conducted the sweep on 8 10 and the number of clites to experimentally study the algorithmic design. The MTP hyperparameters in Section 4.3 are the same as in Table 3 Each setting evaluated with random seeds.
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Table Simulation Settings for Experiments
Task
Horizon At [s]
Horizon
Sim Step/Plan
Sim Hz
Num. Randomizations
Navigation
0.05
100
8
Crane
0.4
500
32
Jube-In-Hand
0.04
2
100
8
G1-Walk
0.1
100
4
G1-Standup
0.2
100
PushT
0.1
1000
Walker
0.15
15
200
1
Table 3: MTP Hyperparameters
Table PS/MPPI Hyperparameters
Task
M
O'iin
Elites
8
a
Task
Noise Std.
Temperature
Navigation
5
30
1.0
Navigation
1.0
0.1
Crane
2
30
0.05
0.5
0.0
Crane
0.05
0.1
Cube-In In-an
2
50
0.15
5
0.5
0.1
Cube-In-Hand
0.15
0.1
G1 G1-Standup
2
100
0.2
100
0.05
0.0
G1 01-Standup
0.2
0.1
G1-Walk
0.1
0.02
0.0
G1-Walk
0.1
0.01
PushT
3
50
0.1
20
0.5
0.0
PushT
0.3
0.1
Walker
2
50
0.3
20
0.5
0.5
Walker
0.3
0.1
Sensitivity Ablation. We conducted 3-mixing rate ablation study (cf. Section 4.4) by varying the B across different MTP variants. The MTP hyperparameters Section 4.4 are the same as in Table Each configuration was evaluated with 4 random seeds.
Experimental Discussions. In lasks with well-shaped dense reward structures and moderately non- linear dynamics, exploration becomes less critical and nominal sampling methods (e.g., MPPI, PS) often suffice-oxplaining MPPI's strong performance in G1-Standup, which benefits from fully actuated dynamics and informative rewards. However, in more challenging scenarios involving sparse rewards or highly non- linear dynamics, such as PushT and under domain shifts in Crane, these locally guided strategies tend to struggle with inadequate exploration, often converging to suboptimal solutions or showing high performance variance. A reprosentative case the Navigation task (Figure 3), where the agent must discover velocity sequences to bypass obstacles and reach the goal-a setting in which local Gaussian sampling clearly fails by getting trapped in local minima. To address these challenges, MTP introduces structured high-entropy sampling mechanism along with simple yet effective 8-mixing strategy that balances global exploration and local exploitation. With careful tuning of B, M, and N, MTP demonstrates robust and consistent performance across diverse tasks.
Mixing Rate Tuning. B determines ratio between exploratory (tensor sampling) and exploitative (nominal sampling) samples, which is delicate to tune. As shown in our ablations in Fig. 6 and Fig. 5 high values (e.g., 0.5) introduce strong exploration, which may benefit tasks with sparse rewards and not requiring delicate fixed-point stability (e-g., PushT, Cube-In-Hand, Navigation). Conversely, in tasks requiring high stability precise actuation, such C1-Standup G1-Walk, lower 3 values (e.g., 0.05-0.1) tend to yield better performance by favoring consistent behavior while still injecting enough exploration to escape suboptimal solutions.
A.5 Additional Ablation Performance Benchmarks
In this section, we conduct more MTP ablations understand how hyperparameters affect MTP perfor- mance, and also briefly benchmark the baseline JAX implementations to confirm the real-time performance.
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Crane
PustT
G1 G1-Walk
MTP-Bspline p-2
0.4
0.8
Bspline p-3
32
MTP Repine p=4
Cost
0.3
0.5
MTP PBppinn p-ó
0.1
0.2
0
Velocity Magnitude
7.5
0.8
5
2.5
20
40
60
80
100
0
20
40
80
80
100
200
400
600
800
1000
Figure 7: MTP- Bspline degree ablation. In G1 G1-Walk, the Unitree G1 controlled Bspline degrees all roughly fall at 500 time steps.
B-spline Degree Ablation. We investigate the sensitivity of MTP performance over B-spline interpolation degrees. The MTP hyperparameters are the same as Table Each setting was evaluated using random seeds.
In Fig. we investigate the sensitivity MTP performance over B-spline interpolation degrees. Results consistently show minimal performance differences across degrees (p 2 p 5) in terms of both cost and velocity magnitude curves across different tasks. Given this insensitivity, select the lower computational complexity B-spline 2 our default choice for the MTP-Bspline method.
Sweep M, N Ablation. We conducted an additional ablation to study the effect of varying M and N values of MTP variants on the Navigation task. We set S = to use full tensor sampling. Each configuration was evaluated with random seeds. According Fig. there exists a sweet point selecting the number of layers
Linear
B-spline
Akima-spline
10
10
0.8
8-
7-
8
4-
0.2
2-
90
20
80
0.0
N
N
Figure 8: Sweep M, N on Navigation enviromment with B = 256 to investigate the interplay between number of batch sample B, number of layer M, and number of control-waypoints per layer N. Each data point the success rate over seeds. The environment setting as in Appendix A.3
M in tensor sampling. Roughly, for all MTP variants, increasing M initially improves task performance, as more layers provide sufficient path complexity, allowing the planner to escape local minima and generate diverse, globally exploratory trajectories. However, beyond a certain point, further increasing M degrades performance. This is due to the exponential growth in the number of possible paths O(N while the rollout budget remains fixed. As a result, the sampled trajectory density becomes sparse relative to the vast number of paths, reducing effective coverage and leading to diminished exploration and performance. On the other hand, increasing N consistently improves performance by densifying the search each layer.
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However, this comes higher computational cost. Therefore, careful balance must be struck between M and N to maintain real-time control and effective control exploration.
Planning Performance. Table 5 benchmarks the performance of our JAX-based implementation by measuring the wall-clock time of the JIT-compiled planning function on G1-Standup task. This function includes a single sampling, single trajectory rollout, single cost evaluation, and single parameter update. Note that this benchmark only serves as an exemplary simulated planning function performance with JAX JIT, In practice, might have multiple search refinements, or multiple parameter updates in the control loops. The task setting is similar to Table 2 but we set the sim step per plan to with B 128. The results show that the initial JIT compilation incurs a significant one-time cost, as expected for MuJoCo XLA pipelines. However, after compilation, the per-step planning rates across MTP, MPPI, PS, and evolutionary baselines are roughly similar with the same batch sample B, as also reflected in Table [6] Table 7 and Table S The results confirm that the JIT [s) and Planning Time [ms] are algorithn-agnostic, which depends slightly on batch size B (i.c., Planning Time [ms] logarithmically increases with B) and on the environment dynamics. All algorithms remain real-time feasible GPU-accelerated hardware, when implemented with JAX and MuJoCo XLA.
Table 5: JAX implementation benchmark on G1-Standup, evaluated with 5 seeds on an Nvidia RTX 3090.
MTP-Rspline
MTP-Akima
PS
MPPT
OpenAI-ES
DE
JIT Time [s]
76.4 1.2
74.62 + 2.5
72.35 + 4.5
73.87 + 4.2
69.87 + 1.2
73.62 + 3.1
Planning Time [ms]
2.7 + 0.3
2.7 0.4
3.1 + 0.7
2.6 2.6 0.2
2.9 = 0.5
3.2 + 0.6
Table Planning performance of MTP-Akima. Averaged over seeds on an Nvidia RTX 4090.
JIT Time [s]
Planning Time [ms]
Batch Size B
PushT
Crane
Cube-In-Hand G1-Walk
PushT
Crane
Cube-In-Hand
G1-Walk
64
15.8
32.5
38.2
65.3
1.7 + 0.1
1.8 + 0.1
8.1 0.3
1.4 + 0.1
128
12.7
31.5
36.6
58.5
1.6 4 0.1
1.9 + 0.1
10.2 0.7
1.5 + 0.1
256
16.3
31.4
38.8
57.1
2.0 0.2
2.0 = 0.4
14.7 0.8
1.5 0.1
Table 7: Planning performance of MPPI. Averaged over 5 seeds Nvidia RTX 4090.
JIT Time [s]
Planning Time [ms]
Batch Size B
PushT
Crane
Cube-In- In-Hand
G1-Walk
PushT
Crane
Bube-In-Hame
G1-Walk
64
14.9
32.3
37.7
62.9
1.7 + 0.1
1.7 + 0.1
8.2 3.2:0.3. 0.3
1.4 + 0.1
128
18.4
29.9
36.7
58.2
1.7 0.1
188 0.1
10.4 0.5
1.4 0.1
256
14.9
30.3
37.5
55.7
1.9
1.8 0.1
15.2 + 0.9
1.4
Table 8: Planning performance of OpenAI-ES.
Averaged over 5 seeds on an Nvidia RTX 4090.
JIT Time [s]
Planning Time [ms]
Batch Size
PushT
Crane
Cube-In-Hand G1-Walk
PushT
Crane
Cube-In-Hee Hand G1-Walk
64
15.1
32.1
39.2
61.5
1.7 0.1
1.8 + 0.1
8.2 + 0.3
1.6 + 0.1
128
19.0
30.1
38.4
57.9
1.7
1.9 1.9 0.1
10.3 0.4
1.5 0.1
256
16.1
30.9
42.3
55.1
2.0 2.0:E0 0.1
1.8 + 0.1
14.9 O.7
1.5 0.1
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This demonstrates that MTP, despite its global exploration capabilities, remains suitable for real-time control applications. Our implementation benefits from efficient JIT and vmap vectorization in JAX and is compat with MuJoCo's XLA backend. These design choices ensure that sampling, rollout, and learning components of MTP are fully optimized and scalable, and they support advanced techniques such as online domain randomization. Overall, the benchmark confirms the practicality of deploying MTP high-performance robotic control loops.
Softmax Update Effect. Fig. illustrates the impact of applying sof tmax weighting clite candidates for updating the mean and standard deviation of control trajectories. The left plot demonstrates smoother and lower-variance control updates over time compared to updates without tmax weighting shown on the right. The smooth and stable updates afforded by softmax weighting are essential when effectively mixing global and local samples, highlighting its critical role in the MTP performance.
With softmax weighting
Without softmax weighting
0.4
80
.00
Time stop
0.4
0..9
-..6
-03
0.8
0.5
-0.9
-05
-33
0.3
0.5
Figure 9: Softmas weighting ablation on PushT environment. Both control update trajectories converge near-zero means with large variance at 100 timesteps, signifying task completion.
24


Reviews:
[{"review_id": "UT8iJTeYrd", "reviewer": "Reviewer_BBca", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strenghts)\n- Clearly point out the limitation of prior sampling-based MPC algorithms with motivating experiments.\n- Clearly demonstrate the robustness of the proposed method.\n\nWeakness)\n- Insufficient explanation on experiment results. \n- Lack of analysis on time-complexity.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose Model Tensor Planning (MTP), to effectively balance exploration and exploitation in sampling-based MPC. Authors first reformulate the sampling as tensor operations over randomized multipartite graphs to generate diverse candidates and introduce $\\beta$-mixing mechansim for exploration-exploitation trade-off. Authors conduct several experiments on high-dimensional real-time control tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I have several minor comments as follows:\n\n- Experiment results of Walker: It seems that there is no big difference between baselines and MTP variants in Walker task. While authors say it is due to absence of applying modeling error, more detailed discussion may required.\n\n- Experiments results of G1-Standup and G1-walk: Those benchmarks represent high-dimensional action spaces, which requires the capability of balancing exploration and exploitation systemically. However, it seems that there is no big difference in terms of performance between MPPI and MTP variants. It is quite strange as MPPI should suffer from balancing exploration-exploitation in high-dimensional spaces. It would be nice to interpret these observations clearly.\n\n- In my humble opinion, the implementation with JIT and vmap vectorization in JAX is one of the key contributions of the paper. However, i can find the time complexity analysis in the appendix with a single table. I recommend to compare wall clock time of algorithms across different benchmarks and also analyze the effect of B in terms of runtime, which might be interesting.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ekFOyduIVp", "reviewer": "Reviewer_2keG", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n1. MTP offers a simple and practical solution to the mode collapse problem in classical sampling-based planning methods, where the candidate distributions are often represented by Gaussian distributions. By converting the planning space in to an M-layer graph and maintaining a categorical distribution at each layer, MTP can plan over expressive, high-entropy distributions of trajectories without sacrificing optimality.\n2. MTP can be efficiently implemented using accelerated tensor libraries (e.g. JAX), which combined with accelerated simulations (e.g. BRAX) offer a significant speedup over classical methods.\n3. The paper performs thorough hyperparameter sweeps and ablation studies to help understand the contribution of each component.\n\n**Weaknesses**\n1. Despite theoretically converging to uniform coverage over the planning space in the limits, in practice the hyperparameters M and N are chosen to trade off fidelity and computation complexity.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces Model Tensor Planning (MTP), a tensorized sampling-based planning method designed to address the mode-collapse problem commonly encountered in classical methods, such as the Cross Entropy Method (CEM). The core idea is to discretize the planning space into a graph with M layers of N nodes each, where M corresponds to the planning horizon and N is the number of bins to discretize the control space. To generate a trajectory, one can sample a node from each layer and interpolate them using B-spline or Alkema-spline. The planning method maintains a catergorical distribution at each layer which is iteratively refined by refitting to the elite samples, similar to CEM. The paper proves that this tensorized planning method covers the full planning space as M and N approach infinity (achieving maximal entropy). Empirically, they show the effectiveness of MTP across a suite of simulated benchmarks, including Push-T, Crane, Hand-In-Cube, Walker, and Humanoid-Stand, where MTP demonstrates improved exploration behavior and faster convergence compared to continuous-space planning methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The paper is well-written and easy to follow. All the claims are supported by evidence. Therefore, I do not have requested changes.", "broader_impact_concerns": "The paper broadly applies to robotics problems. Despite evaluating purely in simulated environments, the potential of real-world applications warrants a Broader Impact Statement. I would kindly request the authors to add a Broader Impact Statement on the safety concerns of applying the method to real robots.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "9PT6zd1b6u", "reviewer": "Reviewer_fhfo", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\nThe approach is quite interesting and effectively encourages exploration by mixing in highly exploratory samples. Under certain conditions, these samples can achieve full coverage of the control space. From what I understand, this work offers a promising direction for future work to further refine the trade-off between exploration and exploitation.\n\n## Weaknesses\n\n1. The coverage result looks reasonable. However, its relevance here is questionable, particularly given how far the actual experimental setting is from the conditions $N,M\\rightarrow \\infty$ (see Table 3).\n\n2. (Question) The level of exploration in your method seems to be tunable with $N$ and $\\beta$. Here, $N$ affects how finely the exploration is, and $\\beta$ controls how many exploratory trajectories are. While you discuss the sensitivity of $\\beta$, how it should be chosen seems to be indecisive. I might have overlooked it, but there’s little mention of how to choose $N$. Is this important, given that you actually have quite different values in Table 3? Also, have you considered using a time-varying/decaying $\\beta$? Otherwise, your method always includes exploratory samples. I was just wondering how it affects the convergence or stability of $(\\mu, \\sigma^2)$.\n\n3. (Question) Can you explain a bit more about the complexity of sampling with/without replacement? It seems to be quite the opposite to me.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Sampling-based model predictive control (MPC) often suffers from limited exploration due to locally greedy sampling strategies. This paper aims to explicitly improve exploration in MPC. The proposed method generates a set of control sequences ${U^{k}}_{k=1}^B$ that combine Gaussian samples (from $\\mathcal{N}(\\mu, \\sigma^2)$) with highly exploratory paths generated from tensor sampling, where the portion of them is determined by the mixing parameter $\\beta\\in(0,1)$. The tensor sampling can be interpreted as uniformly sampling from an N-width, M-depth fully connected graph, where exactly one node is selected per layer. Such paths are structured and highly exploratory.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "### Main\n\n1. Respond to the questions in the weakness section.\n\n### Minor\n\n1. The notations $u$, $\\mathbf{u}$, $\\mathbf{U}$, and $u_t$ should be defined more clear - especially on page 2. In Definition 1, I assume $U(\\mathbb{U})$ denotes the uniform distribution over $\\mathbb{U}$, but this should be stated explicitly, especially given the heavy reuse of \"U\" throughout the paper.\n\n2. The cost function $c$ in Section 2.1 is used before definition.\n\n3. Should the average in (4) be $E$ or $B$?\n\n4. You reuse the notation $E$ for the number of elite candidates and the edge set. Would be better to make them different.\n\n5. The “path point” $\\mathbf{u}(t)$ in between Eq. (6) and (7) is better to call it “waypoint” here for the sake of consistency.", "broader_impact_concerns": "Not as far as I can see", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)