v3/paper_source/appendix.tex

\appendix
\onecolumn
\allowdisplaybreaks
\section{Proof for Proposition \ref{prop:performance}} \label{app:proof1}
% Proposition 1 (Performance Monotonicity). Let $\pi_{\text{sched}}$ be the policy induced by the scheduler which selects action $k$ at state $o_t$, and $\pi_{\text{base}}$ be the base policy which selects $A_t$. If the scheduler satisfies the zero-violation constraint such that $c_q(s_t, k_t) \le 0$ for all visited states, then the performance of the accelerated policy is guaranteed to be no worse than the base policy, i.e., $J(\pi_{\text{sched}}) \ge J(\pi_{\text{base}})$.
We aim to prove that: $\forall s,Q^{\pi}(s, A^k) \ge Q^{\pi}(s, A)$ (zero-violation), implies $V^{\pi'}(s_0) \ge V^{\pi}(s_0)$ (success rate guarantee). In the following derivation, we assume $\gamma=1$.

First, we must rigorously define what the ``Action Chunk'' $A$ and its Q-value represent in terms of atomic, low-level control actions. Let an action chunk $A$ consist of a sequence of atomic actions $u$ over a physical duration $L$.\begin{itemize}\item Original Chunk $A$: Sequence $\{u_1, u_2, \dots, u_L\}$.\item Accelerated Chunk $A^k$: Sequence $\{u'_1, u'_2, \dots, u'_{L'}\}$ where $L' = L/k$.\end{itemize}The execution of a chunk is not a single jump, but a trajectory of atomic state transitions. Let $s_{\tau, i}$ denote the state at the $i$-th atomic step within the execution of chunk $A_\tau$ (where $\tau$ is the chunk index).The Q-value of a chunk $A$ under policy $\pi$ is defined as the sum of atomic rewards within the chunk plus the value of the state after the chunk finishes:\begin{equation} \label{eq:atomic_expansion}Q^{\pi}(s_\tau, A) = \mathbb{E}\left[ \underbrace{\sum_{i=1}^{L} r(s_{\tau, i}, u_i)}_{\text{Intra-chunk Reward}} + V^{\pi}(s_{\tau+1}) \right],\end{equation}where $s_{\tau+1}$ is the state reached after executing the last atomic action $u_L$. Similarly, for the accelerated chunk $A^k$:\begin{equation}Q^{\pi}(s_\tau, A^k) = \mathbb{E}\left[ \sum_{j=1}^{L'} r(s'_{\tau, j}, u'_j) + V^{\pi}(s'_{\tau+1}) \right].\end{equation}

We now prove that $V^{\pi'}(s) \ge V^{\pi}(s)$ for all states, which directly implies the final required inequality. We use mathematical induction (or recursive expansion) over the sequence of chunks. Let $V^{\pi'}(s)$ be the value of following the scheduler policy $\pi'$ (which always selects $A^k$). By definition:\begin{equation}V^{\pi'}(s_0) = Q^{\pi'}(s_0, A_0^k) = \mathbb{E} \left[ \sum_{j=1}^{L'_0} r(s'_{0, j}, u'_j) + V^{\pi'}(s_{1}) \right].\end{equation}We now show that $V^\pi(s_0)\leq V^{\pi'}(s_0)$ with recursive expansion of the value function: 
\begin{align}
    V^{\pi}(s_0) 
    &= Q^{\pi}(s_0, A_0) \le Q^{\pi}(s_0, A_0^k) \\
    &= \mathbb{E}_{s_1\sim \pi'} \left[ \sum_{j=1}^{L'_0} r(s'_{0,j}, u'_{0,j}) + V^{\pi}(s_1) \right] \\ 
    &= \mathbb{E}_{s_1\sim \pi'} \left[ \sum_{j=1}^{L'_0} r(s'_{0,j}, u'_{0,j}) + Q^{\pi}(s_1, A_1) \right]   \\
    &\le \mathbb{E}_{s_1\sim \pi'} \left[ \sum_{j=1}^{L'_0} r(s'_{0,j}, u'_{0,j}) + Q^{\pi}(s_1, A_1^k) \right] \\
    &= \mathbb{E}_{s_1, s_2\sim \pi'} \left[ \sum_{j=1}^{L'_0} r(s'_{0,j}, u'_{0,j}) + \sum_{m=1}^{L'_1} r(s'_{1,m}, u'_{1,m}) + V^{\pi}(s_2) \right]   \\
    &\quad \vdots \quad   \\
    \label{eq:expansion_logic}
    &\le \mathbb{E}_{\tau \sim \pi'} \left[ \sum_{t=0}^{\infty} \sum_{j=1}^{L'_t} r(s'_{t,j}, u'_{t,j}) \right] = V^{\pi'}(s_0).  
\end{align} 
\textbf{Remarks on $\gamma=1$}: We emphasize that the undiscounted setting ($\gamma=1$) is both physically motivated and mathematically essential for our derivation. First, since our primary metric is the task success rate under sparse rewards, setting $\gamma=1$ ensures that the value function $V^\pi(s)$ directly represents the probability of success, i.e., $V^\pi(s) = \mathbb{P}(\text{Success} | \pi, s)$. Second, $\gamma=1$ is a necessary condition for the validity of the telescoping sum in Eq. \ref{eq:expansion_logic}. In a variable-duration setting where the scheduler accelerates execution, the physical arrival time at any future state $s_{t+1}$ differs from that of the base policy. If $\gamma < 1$, the discount factors associated with $s_{t+1}$ would not align between the two policies, preventing the intermediate terms in the performance difference expansion from canceling out. By assuming $\gamma=1$, the value of a state becomes invariant to $L$ and $L'$, allowing for a rigorous proof of global performance preservation despite temporal downsampling.

\section{Proof for Proposition \ref{prop:penalty}}
\label{app:proof2}
Let $Q^*(s, k)$ denote the optimal action-value function. The maximum possible value of a safe trajectory is bounded by $V_{\max} = \sum_{t=0}^\infty \gamma^t K_{\max} = \frac{K_{\max}}{1-\gamma}$.
Consider an arbitrary state $s$.
If an action $k_{v}$ violates the constraint (i.e., $h(s, k)=1$), its Q-value is bounded by:
\begin{equation}
Q^*(s, k_v) = -\Omega + \gamma \mathbb{E}[V^*(s')] \le -\Omega + \gamma V_{\max}.
\end{equation}
Conversely, since valid actions yield at least a reward of 1 (assuming $k \ge 1$), the Q-value of the optimal safe action $k_{safe}$ satisfies $Q^*(s, k_{safe}) \ge 1+\gamma$.
To ensure the optimal policy never selects a violation, we require $Q^*(s, k_{safe}) > Q^*(s, k_v)$. It suffices to show:
\begin{equation}
1+\gamma > -\Omega + \frac{\gamma K_{\max}}{1-\gamma} \implies \Omega > \frac{\gamma K_{\max}}{1-\gamma} - 1-\gamma.
\end{equation}
Thus, setting $\Omega > \frac{\gamma K_{\max}}{1-\gamma}$ (a strictly stronger condition) guarantees that any violating action has a lower value than any valid action, compelling the optimal policy to strictly satisfy the safety constraint.
\newcommand{\tb}[1]{\textbf{#1}}
\definecolor{HighLightColor}{rgb}{1.0, 0.92, 0.8}

\begin{table*}[t!]
    \centering
    \caption{Performance comparison of different methods in all Bigym tasks. Best success rates and shortest lengths are \textbf{bolded}. Method with best success rate per task is \colorbox{HighLightColor}{highlighted}.}
    \label{tab:bigym_all_result_3}
    \setlength{\tabcolsep}{4pt} % 稍微调整列间距以适应
    \small
    \resizebox{\linewidth}{!}{
    \begin{tabular}{clccccc}
        \toprule
        \multicolumn{2}{c}{\textbf{Method}} & {\footnotesize Sandwich Remove} & {\footnotesize Take Cups} & {\footnotesize Put Cups} & {\footnotesize Dishwasher Open Trays} & {\footnotesize Move Plate}\\
        \midrule
        \multirow{4}{*}{ACT} & -base & (0.45, 340.5) & (0.15, 288.3) & (0.28, 320.3)  & ( 1.0, 275.0) & \cg{( \tb{0.58}, 194.8)}\\
                             & -ds2  & (0.48, 186.3) & (0.13, 178.0) & (0.36, 175.9)  & ( 1.0, 169.0) & (0.50, 155.0)\\
                             & +\textit{DemoSpeedup}  & (0.56, 171.5) & (0.10, 183.9) & (0.33, 169.3)  & (1.0, 156.0) & (0.28, \tb{71.7})\\
                             & +\textit{SuP(Ours)}  & \cg{( \tb{0.64}, \tb{155.9})} & \cg{( \tb{0.20}, \tb{176.6})}  & \cg{( \tb{0.38}, \tb{156.0})}  & \cg{(1.0, \tb{131.0})} & (0.49, 167.2)\\
        \midrule
        \multirow{4}{*}{DP}  & -base & (0.40, 376.8) & (0.07, 284.6) & \cg{( \tb{0.28}, 307.7)} & (0.57, 351.2) & (0.33, 261.1)\\
                             & -ds2  & (0.40, 209.6) & (0.12, 218.0) & (0.22, 181.5) & (0.48, 310.2) & (0.36, 203.3)\\
                             & +\textit{DemoSpeedup}  & (0.35, 199.3) & \cg{( \tb{0.21}, 239.5)} & (0.25, \tb{143.9}) & (0.94, \textbf{108.8}) & (0.38, 178.8) \\
                             & +\textit{SuP(Ours)} & \cg{(\tb{0.42}, \tb{179.9})} & (0.19, \tb{203.0}) & (0.27, 200.8) & \cg{( \tb{0.97}, 267.4)} & \cg{( \tb{0.39}, \tb{164.8})}\\
        \bottomrule
        \multicolumn{2}{c}{\textbf{Method}} & {\footnotesize Saucepan to Hob} & {\footnotesize Flip Cutlery} & {\footnotesize Cupboards Close All} & {\footnotesize Sandwich Flip} & {\footnotesize Dishwasher Close Trays}\\
        \midrule
        \multirow{4}{*}{ACT} & -base & (0.78, 334.1) & (0.35, 243.6) & ( 1.0, 449.8)  & (0.20, 406.7) & ( 1.0, 200.3)\\
                             & -ds2  & (0.47, 248.7) & (0.22, 257.6) & ( 1.0, 234.0)  & (0.19, 239.4) & ( 1.0, 117.0)\\
                             & +\textit{DemoSpeedup}  & (0.78, \tb{154.5}) & \cg{( \tb{0.39}, \tb{90.8})} & \cg{( \tb{1.0}, \tb{202.1})}  & (0.18, \tb{156.8}) & ( 1.0, 95.0)\\
                             & +\textit{SuP(Ours)}  & \cg{( \tb{0.88}, 174.3)} & (0.27, 138.7)  & ( 1.0, 212.1)  & \cg{( \tb{0.22}, 194.5)} & \cg{( \tb{1.0}, \tb{86.0})}\\
        \midrule
        \multirow{4}{*}{DP}  & -base & \cg{( \tb{0.69}, 426.1)} & (0.06, 368.0) & (0.90, 544.0) & (0.06, 436.0) & (0.94, 210.4)\\
                             & -ds2  & (0.58, 285.0) & (0.12, 356.7) & (0.75, 270.3) & (0.03, \tb{157.3}) & (0.53, 146.4)\\
                             & +\textit{DemoSpeedup}  & (0.60, \tb{157.8}) & (0.10, \tb{145.9}) & (0.60, 231.0) & \cg{(0.11, 294.5)} & \cg{(\tb{0.96}, \tb{108.3})} \\
                             & +\textit{SuP(Ours)} & (0.66, 332.7) & \cg{( \tb{0.16}, 389.9)} & \cg{( \tb{0.91}, \tb{146.2})} & ( \tb{0.11}, 311.5) & (0.72, 175.4)\\
        \bottomrule
        \multicolumn{2}{c}{\textbf{Method}} & {\footnotesize Pick Box} & {\footnotesize Drawers Close All} & {\footnotesize Drawers Open All} & {\footnotesize Dishwasher Close} & {\footnotesize Wall Cupboard Open}\\
        \midrule
        \multirow{4}{*}{ACT} & -base & ( 0.25, 372.0) & ( 1.0, 100.0) & ( 1.0, 325.5)  & ( 1.0, 175.0) & ( 1.0, 148.8)\\
                             & -ds2  & (0.04, 182.0) & ( 1.0, 52.0) & (0.99, 190.0)  & ( 1.0, 90.9) & ( 1.0, 74.5)\\
                             & +\textit{DemoSpeedup}  & (0.02, 173.0) & ( 1.0, 54.0) & (0.99, 163.0)  & \cg{( \tb{1.0}, \tb{83.9})} & (0.92, 64.0)\\
                             & +\textit{SuP(Ours)}  & \cg{(0.25, \tb{317.9})} & \cg{( \tb{1.0}, \tb{40.0})}  & \cg{( \tb{1.0}, \tb{152.2})}  & (1.0, 84.0) & \cg{( \tb{1.0}, \tb{65.0})}\\
        \midrule
        \multirow{4}{*}{DP}  & -base & (0.0, -) & ( 0.66, 118.9) & \cg{( \tb{0.89}, 478.1)} & \cg{( \tb{0.99}, 178.9)} & (0.89, 260.0)\\
                             & -ds2  & (0.0, -) & (0.54, 64.9) & (0.12, 412.7) & (0.54, \tb{64.9}) & (0.88, \tb{131.0})\\
                             & +\textit{DemoSpeedup}  & (0.0, -) & (0.37, \tb{49.2}) & (0.44, \tb{193.2}) & (0.9, 158.4) & ( 0.91, 147.2)\\
                             & +\textit{SuP(Ours)} & (0.0, -) & \cg{(\tb{0.66}, 65.2)} & (0.30, 442.9) & (0.95, 147.3) & \cg{( \tb{0.91}, 146.2)}\\
        \bottomrule
        \multicolumn{2}{c}{\textbf{Method}} & {\footnotesize Store Box} & {\footnotesize Wall Cupboard Close} & {\footnotesize Dishwasher Open} & {\footnotesize Sandwich Toast} & {\footnotesize Flip Cup}\\
        \midrule
        \multirow{4}{*}{ACT} & -base & (0.51, 454.4) & ( 1.0, 100.0) & ( 1.0, 389.0)  & (0.08, 596.9) & \cg{( \tb{0.53}, 312.7)}\\
                             & -ds2  & (0.41, 229.2) & ( 1.0, 52.0) & ( 1.0, 377.0)  & (0.07, 261.9) & (0.32, 182.4)\\
                             & +\textit{DemoSpeedup}  & (0.33, \tb{222.2}) & \cg{( \tb{1.0}, \tb{46.0})} & ( 1.0, 163.3)  & (0.09, \tb{157.1}) & (0.30, \tb{149.8})\\
                             & +\textit{SuP(Ours)}  & \cg{( \tb{0.53}, 231.2)} & ( 1.0, 54.0)  & \cg{(1.0, \tb{157.4})} & \cg{( \tb{0.13}, 171.2)} & (0.46, 181.0)\\
        \midrule
        \multirow{4}{*}{DP}  & -base & (0.25, 456.0) & ( 1.0, 96.0) & (0.57, 354.1) & (0.04, 426.0) & (0.01, 312.0)\\
                             & -ds2  & (0.39, 328.6) & ( 1.0, 57.5) & (0.56, 281.3) & (0.03, \tb{178.7}) & (0.03, 552.0)\\
                             & +\textit{DemoSpeedup}  & (0.14, \tb{296.0}) & \cg{( \tb{1.0}, \tb{47.5})} & (0.38, \tb{113.5}) & (0.01, 180.0) & (0.04, \tb{150.3})\\
                             & +\textit{SuP(Ours)} & \cg{( \tb{0.39}, 336.6)} & ( 1.0, 50.6) & \cg{( \tb{0.57}, 267.4)} & \cg{( \tb{0.07}, 240.9)} & \cg{( \tb{0.05}, 164.5)}\\
        \bottomrule
    \end{tabular}
}
\end{table*}

\section{Whole Results of Bigym}
\label{sec:whole_bigym}
We report the (Success Rate, Episode Length) pairs for all 20 tasks across both ACT and DP architectures. As shown in Tab. \ref{tab:bigym_all_result_3}, SuP achieves the best balance between efficiency and success rate in most tasks, outperforming both static downsampling and DemoSpeedup baselines across a wide range of manipulation skills.




% 定义高亮颜色 (仿照图片中的浅橙色)





\section{Simulation Experiment Detail}
\label{app:sim_exp_detail}
\subsection{Bigym}
Here, we provide details of the BiGym tasks: we utilize a total of 20 tasks, all set in a kitchen scenario. The task descriptions (which can serve as language prompts if required) are listed below:
(1) Sandwich Remove: Take the sandwich out of the frying pan.
(2) Take Cups: Take two cups out from the closed wall cabinet and put them on the table.
(3) Put Cups: Pick up cups from the table and put them into the closed wall cabinet.
(4) Dishwasher Open Trays: Pull out the dishwasher’s trays with the door initially open.
(5) Move Plate: Move the plate between two draining racks.
(6) Saucepan to Hob: Take the saucepan from the closed cabinet and place it on the hob.
(7) Flip Cutlery: Take the cutlery from the static holder, flip it, and place it back into the holder.
(8) Cupboards Close All: Close all drawers and doors of the kitchen set.
(9) Sandwich Flip: Flip the sandwich in the frying pan using the spatula.
(10) Dishwasher Close Trays: Push the dishwasher’s trays back with the door initially open.
(11) Pick Box: Pick up a large box from the floor and place it on the counter.
(12) Drawers Close All: Close all sliding drawers of the kitchen cabinet.
(13) Drawers Open All: Open all sliding drawers of the kitchen cabinet.
(14) Dishwasher Close: Push back all trays and close the door of the dishwasher.
(15) Wall Cupboard Open: Open doors of the wall cabinet.
(16) Store Box: Move a large box from the counter to the shelf in the cabinet below.
(17) Wall Cupboard Close: Close doors of the wall cabinet.
(18) Dishwasher Open: Open the dishwasher door and pull out all trays.
(19) Sandwich Toast: Use the spatula to put the sandwich on the frying pan and toast it.
(20) Flip Cup: Flip the cup initially positioned upside down on the table to an upright position.



\textbf{1. Observation Space (State Space)} \\
BiGym's observation space is hybrid, combining visual inputs, proprioceptive data, and (for bi-manual mode) base state, which is defined as:
\[
O = \{I_{\text{head}}, I_{\text{left}}, I_{\text{right}}, s_{\text{proprio}}\}
\]
\begin{itemize}
    \item \textbf{Visual Observations}: RGB images ($I_{\text{head}}, I_{\text{left}}, I_{\text{right}}$) from three cameras (forehead, left wrist, right wrist), with a default resolution of $84 \times 84$.
    \item \textbf{Proprioceptive State ($s_{\text{proprio}}$)}: The state space adopts the Bi-manual mode, with the low-dimensional state ranging from 60 to 70 dimensions, including joint angles, joint velocities, and base states (where the leg control is configured in floating base mode), etc.
\end{itemize}

\vspace{5pt}
\textbf{2. Action Space} \\
The action space $A \in \mathbb{R}^{16}$ in Bigym can be formularized as three parts:
\begin{equation*}
    A = \{A_{\text{arms}} (\mathbb{R}^{10}), A_{\text{base}} (\mathbb{R}^{4}), A_{\text{grip}} (\mathbb{R}^{2})\},
\end{equation*}
where $\{A_{\text{arms}}$ controls the qpos of the robot arm, $A_{\text{base}}$ controls the floating base (i.e. legs) of the robot and $A_{\text{grip}}$ controls the left and right gripper of the robot arm.
\vspace{5pt}

\textbf{3. Training of $\pi_{\text{base}}$} \\
% Biygm中，我们follow Demospeedup开源仓库中对ACT和DP的实现方式[https://github.com/lingxiao-guo/DemoSpeedup/tree/main/robobase]，由于没有现成checkpoint release，所以我们按照源代码在20个环境中重跑了算法，并选取在评估中胜率最高的模型作为我们的base policy，ACT和DP的训练超参数如这两张表所示，我们发现Bigym中DP的性能表现不如ACT方法。我们也尝试在Bigym上基于$\pi_{0.5}$训练模型，但发现完成任务的成功率普遍弱于ACT和DP，这可能说明该模型不适合用于全身控制中。
In the Bigym environment, we followed the implementation of ACT and DP from the DemoSpeedup's open-source repository (specifically the robobase folder)\footnote{\url{https://github.com/lingxiao-guo/DemoSpeedup/tree/main/robobase}}. 
Due to the absence of pre-released checkpoints, we retrained the algorithms across 20 environments according to the original source code. 
We selected the models that achieved the highest win rates during evaluation as our base policies. 
The training hyperparameters for ACT and DP are detailed in Tab.~\ref{tab:act_dp_hyperparams}, respectively. 
Our experiments revealed that the performance of DP in Bigym was generally inferior to that of ACT. 
We also attempted to train a model based on $\pi_{0.5}$, but we found that the success rates were lower than those of both ACT and DP in a lot of tasks.
This suggests that the model may not be suitable for whole-body control tasks. Consequently, we did not attempt to accelerate the VLA base policy in Bigym.

\begin{table*}[t]
    \centering
    \caption{Hyperparameters for ACT and DP in Bigym.}
    \label{tab:act_dp_hyperparams}
    
    % --- 左侧: ACT Hyperparameters ---
    \begin{minipage}{0.48\textwidth}
        \centering
        \caption*{ACT Hyperparameters}
        \begin{tabular}{lc}
            \toprule
            Hyperparameter & ACT \\
            \midrule
            Learning Rate & 1e-5 \\
            Weight Decay & 1e-4 \\
            Batch Size & 64 \\
            Chunk Size ($k$) & 24 \\
            Feedforward Dim & 3200 \\
            Hidden Dim & 512 \\
            Encoder Layers & 4 \\
            Decoder Layers & 7 \\
            Attention Heads & 8 \\
            Dropout & 0.1 \\
            \bottomrule
        \end{tabular}
    \end{minipage}
    \hfill % 填充中间间距，使两个表格撑满一行
    % --- 右侧: DP Hyperparameters (No Aloha) ---
    \begin{minipage}{0.48\textwidth}
        \centering
        \caption*{DP Hyperparameters}
        \begin{tabular}{lc}
            \toprule
            Hyperparameter & DP \\
            \midrule
            Learning Rate & 1e-4 \\
            Weight Decay & 1e-6 \\
            Batch Size & 64 \\
            Observation Horizon & 2 \\
            Action Horizon & 24 \\
            Diffusion Steps & 100 \\
            Noise Scheduler & DDPM \\
            Kernel Size & 5 \\
            Vision Model & MVT \cite{seo2023multi} \\
            Down Dims & [256,512,1024] \\
            \bottomrule
        \end{tabular}
    \end{minipage}
\end{table*}

% \textbf{3. Initial State Distribution} \\
% Initial states are task-specific but follow a consistent structure across all 40 household manipulation tasks (e.g., target reaching, dishwasher interaction, kitchen counter tasks):
% \begin{itemize}
%     \item \textbf{Robot Initialization}: The humanoid is spawned at a fixed base position/orientation relative to task objects (e.g., $0.5$–$1.0$m from tables/dishwashers). Joints are set to neutral positions, and grippers are fully open ($s_{\text{grip}} = [1.0, 1.0]$).
    
%     \item \textbf{Object Initialization}: Task objects (e.g., plates, cups, cutlery, drawers) are placed in predefined, task-relevant configurations:
%     \begin{itemize}
%         \item Rigid objects (e.g., plates, blocks) are positioned on tables/draining racks with random minor perturbations ($\pm 5$cm in $x/y$, $\pm 5^\circ$ in orientation) to ensure diversity.
%         \item Articulated objects (e.g., dishwashers, drawers) are initialized in task-appropriate states (e.g., closed for "open" tasks, partially open for "load/unload" tasks) with joint angles sampled near $0$ (closed) or $1$ (open) with small noise ($\pm 0.05$).
%     \end{itemize}
    
%     \item \textbf{Observability}: Cameras are calibrated to face task-relevant regions (e.g., wrists toward objects, head forward) at initialization, ensuring partial observability (consistent with POMDP formulation).
% \end{itemize}

% All initial states are sampled from deterministic distributions with controlled noise to balance reproducibility and real-world realism, aligned with BiGym's focus on demo-driven mobile bi-manual manipulation.



\subsection{Libero}

The Libero suite comprises four specialized sub-suites, each designed to isolate or integrate specific types of knowledge transfer for robot manipulation tasks, with distinct focuses and standardized language instruction patterns. Libero-spatial focuses on the transfer of declarative knowledge about spatial relationships, using instructions that specify spatial descriptors and target objects; Libero-object targets declarative knowledge about object concepts, with instructions centered on object names and containers; Libero-goal concentrates on procedural knowledge about task goals, featuring instructions that outline action-oriented tasks; Libero-long consists of long-horizon tasks involving entangled declarative and procedural knowledge transfer, with multi-step instructions that combine spatial, object, and goal concepts. Below is a detailed breakdown of the observation and action spaces common to or specific to each sub-suite, along with their core characteristics.

\textbf{1. Observation Space (State Space)} \\
Libero's observation space is hybrid, combining visual inputs, proprioceptive data, which is defined as:
\[
O = \{I_{\text{top}}, I_{\text{wrist}}, s_{\text{proprio}}\}
\]
\begin{itemize}
    \item \textbf{Visual Observations}: RGB images ($I_{\text{top}}, I_{\text{wrist}}$) from two cameras (top, wrist), with a default resolution of $224 \times 224$.
    \item \textbf{Proprioceptive State ($s_{\text{proprio}}$)}: The state space is an 8-dimensional low-dimensional joint state space.
\end{itemize}

\vspace{5pt}
\textbf{2. Action Space} \\
The action space is 7-dimensional, with 6-dimensional delta-EEF control, and 1-dimensional Gripper control.

\textbf{3. Detail of $\pi_{\text{base}}$} \\
% 在Libero中，我们使用已经开源好的Libero模型参数，对于$\pi_{0.5}$，遵循其开源仓库的instruction [https://github.com/Physical-Intelligence/openpi/tree/main/examples/libero] 就可以获取对应的模型参数。对于VLA-Adapter，遵循其开源仓库的instruction [https://github.com/OpenHelix-Team/VLA-Adapter] 就可以。$\pi_{0.5}$为4个task suite使用相同的模型权重，而VLA-Adapter则使用独立的权重。
In the Libero environment, we utilized the officially released pre-trained model checkpoints. 
For $\pi_{0.5}$, we obtained the corresponding model parameters by adhering to the instructions provided in its open-source repository\footnote{\url{https://github.com/Physical-Intelligence/openpi/tree/main/examples/libero}}. 
Similarly, for the VLA-Adapter, we followed the instructions outlined in its respective repository\footnote{\url{https://github.com/OpenHelix-Team/VLA-Adapter}}. 
Specifically, $\pi_{0.5}$ employs a shared set of model weights across all four task suites, whereas the VLA-Adapter utilizes independent weights for each suite. The training demonstration data is downloaded directly via HuggingFace\footnote{\url{https://huggingface.co/datasets/openvla/modified_libero_rlds}}.



\begin{figure}
    \centering
    \includegraphics[width=0.8\linewidth]{figures/real_device.png}
    \caption{ARX5 illustration. (a) The master-follow design for data collection (b) The actual scene of our real-world experiment.}
    \label{fig:real_device}
\end{figure}

\section{Real-world Experiment Detail}
\label{app:real_exp}
\subsection{Hardware Setup}
The hardware configuration is detailed in Fig. \ref{fig:real_device}. We utilize the ARX5 robotic platform, a dual-arm system analogous to Aloha, consisting of two master arms and two puppet arms. Both arms were actively employed for dual-arm teleoperation and data collection. To provide visual feedback, a top-mounted RealSense D435i camera captures the RGB image observations required for the experiments.

\subsection{Details of Real-world Tasks}
\textbf{Fold Towel.}~~The scene consists of two towels of different colors or patterns placed on the tabletop. The robot is required to identify the target towel specified by a linguistic instruction and execute a folding sequence. This task tests the policy's ability to handle deformable objects and its grounding of language instructions in a multi-object scene.

\textbf{Arrange Table.}~~This task involves three plates and five objects initially distributed across them (arranged in a $2, 2, 1$ pattern). The robot must follow a three or four-step instruction to pick and place specific objects into designated plates. This task represents a long-horizon challenge requiring precise spatial reasoning and high-level planning.

\textbf{Stack Plates.}~~Three plates are placed separately on the table. The robot must stack them into a single pile following a specific order provided in the instruction (e.g., bottom-to-top sequence). This task emphasizes contact-rich manipulation and the strict maintenance of operational order.

\textbf{Training of $\pi_{\text{base}}$.}~~We collected a total of 200 high-quality demonstrations using teleoperation, with 50 trajectories in Fold Towel, 100 trajectories in Arrange Table and 50 trajectories in Stack Plates. We then utilized the $\pi_{0.5}$ model as the foundation. The model was fine-tuned on task-specific trajectoris to serve as $\pi_{\text{base}}$, ensuring reliable execution of the fundamental manipulation primitives.
\section{Details of SuP}
\label{app: detail of SuP}
In this section, we provide the detailed implementation of SuP, including how to downsample gripper action, how to calculate state deviation and the architecture of Recurrent World Model and scheduler. 
\subsection{Gripper Action Compensation for Downsampling}

Our methods rely on action chunk downsampling strategy that remain semantically aligned with the original one. While the downsampling strategy described in Sec. \ref{bg:ds} ensures that the robot's arm waypoints remain spatially consistent in the sense of they desired, gripper actions require separate consideration due to their binary nature and specific physical constraints. In most simulation environments, gripper actions are represented as binary signals (e.g., $<0$ for closed, $>0$ for open). Regardless of whether absolute or relative position control is used, standard downsampling causes a mismatch in the cumulative physical displacement of the gripper. For example, if a full grasp requires several consecutive closure commands, reducing the action frequency results in the gripper failing to reach the intended state in time, leading to failed grasps.To resolve this inconsistency and maintain the success rate after downsampling, we applied the following task-specific compensations:
\begin{itemize}
    \item BiGym: We followed the method described in DemoSpeedup \cite{guo2025demospeedup} by increasing the control gain of the gripper, ensuring it responds more aggressively to the reduced number of commands.
    \item Libero: We doubled the magnitude (velocity) of each gripper action command. 
\end{itemize}
For instance, if a single original action resulted in a 0.1 cm closure, the adjusted action for a downsampling factor of 2 ($N=2$) produces a 0.2 cm closure. Although this adjustment is specifically tailored for a downsampling rate of 2, we found it to be a highly effective heuristic for maintaining physical state consistency. The necessity of gripper action compensation is quantitatively validated in Tab. \ref{tab:gripper}. Without the fix, naive downsampling ($N=2$) leads to a significant performance degradation, with the average success rate dropping from 96.9\% to 84.2\%, particularly in the libero-spatial task where the gripper often fails to secure objects due to insufficient closure displacement. By applying our proposed compensation—adjusting the gripper's response magnitude—the ``-ds2'' variant recovers the average success rate to 92.6\% while maintaining a high inference speedup (1.72$\times$). This results in a much more robust balance between efficiency and task reliability.

\begin{table}[h]
    \centering
    \caption{Ablation study of gripper action compensation on Libero benchmarks. We compare the original policy ($pi_{0.5}$) with downsampled versions ($N=2$) before and after applying the gripper fix.}
    \label{tab:gripper}
    \begin{tabular}{lccccc}
        \toprule
        \textbf{Method} & \textbf{spatial} & \textbf{long} & \textbf{goal} & \textbf{object} & \textbf{Average} \\
        \midrule
        $\pi_{0.5}$ (Original) & 0.988, 105.3 & 0.924, 267.9 & 0.980, 113.1 & 0.982, 138.1 & 0.969, 1.00$\times$ \\
        \midrule
        No Grip Fix (-ds2) & 0.708, 77.0 & 0.818, 167.4 & 0.888, 66.2 & 0.954, 84.4 & 0.842, 1.58$\times$ \\
        With Grip Fix (-ds2) & 0.914, 67.9 & 0.874, 153.4 & 0.952, 67.6 & 0.970, 75.0 & 0.928, 1.72$\times$ \\
        \bottomrule
    \end{tabular}
\end{table}
\subsection{Calculation of State Deviation}
\label{app:state_deviation_calc}
In this section, we detail calculation of the state deviation metric $\mathcal{E}$, which serves as the core criterion for the switching logic within our Speedup Patch (SuP) framework. The calculation of state deviation relies on the formal representation of the robot's spatial configuration via the End-Effector (EEF) pose. The EEF pose is defined as a combination of its 3D Cartesian coordinates $(x, y, z)$ and its orientation, represented internally as a unit quaternion to avoid singularities. To evaluate the fidelity of the robot's motion during downsampled execution with a rate $k$, we determine the "expected" state at any intermediate sub-step $i \in \{1, \dots, k-1\}$ through pose interpolation between two consecutive reference waypoints $e_t$ and $e_{t+k}$ produced by the base policy. Specifically, the reference position is obtained via linear interpolation, while the reference orientation is computed using Normalized Linear Interpolation (NLERP) \cite{shoemake1985animating}. This approach ensures that the interpolated orientation remains on the unit hypersphere by normalizing the result of a linear interpolation between the two reference quaternions, providing a computationally efficient approximation of the shortest rotation path.

% 使用插值对齐降采样前后的EEF轨迹后，我们现在叙述如何计算两个EEF之间的距离$d(e_{curr},e_{ref})$. EEF距离由欧几里得位置和旋转距离组成。其中旋转距离定义为由有当前姿态到目标姿态的最小旋转距离，可以通过四元数计算得到。具体来说，两个EEF之间的距离定义为：
% $$d_{t+i} = \sqrt{(x_{curr} - x_{ref})^2 + (y_{curr} - y_{ref})^2 + (z_{curr} - z_{ref})^2} + \lambda \cdot \arccos(|\langle q_{curr}, q_{ref} \rangle|)$$
% 其中。。。
To evaluate the fidelity of the generated trajectories, we define a composite distance metric $d(e_{curr}, e_{ref})$ that measures the discrepancy between the current and reference end-effector (EEF) states. This distance comprises two components: the Euclidean distance for translational position and the geodesic distance for rotational orientation. The total distance at step $t+i$ is formulated as:$$d_{t+i} = \frac12\sqrt{(x_{curr} - x_{ref})^2 + (y_{curr} - y_{ref})^2 + (z_{curr} - z_{ref})^2} + \cdot \arccos(|\langle q_{curr}, q_{ref} \rangle|)$$where $\mathbf{p} = [x, y, z]^\top$ represents the Cartesian coordinates and $q$ denotes the orientation expressed as a unit quaternion. The rotational term calculates the minimum angular displacement between the two orientations, using the absolute value of the inner product $\langle q_{curr}, q_{ref} \rangle$ to account for the antipodal property of quaternions.

From an implementation perspective, these geometric operations—including NLERP and geodesic distance calculations—are natively and efficiently supported by the \textit{scipy.spatial.transform.Rotation module} in the SciPy library.

\subsection{Network architecture}
\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{figures/network.pdf}
    \caption{Network architecture of Recurrent World Model and Scheduler Policy.}
    \label{fig:architecture}
\end{figure}
\vspace{-1mm}

\textbf{Recurrent World Model.} The Any-Step Dynamics Model (ADM) is designed to predict future trajectories while bypassing the recursive error accumulation typical of auto-regressive transitions. The process begins by mapping the initial observation $o_t$ to a latent representation $h_t = \text{enc}_{\theta}(o_t)$ using a state encoder. This latent vector serves as the initial hidden state for a Gated Recurrent Unit (GRU), denoted as $g_{\theta}$. For each step $i \in \{0, \dots, L-1\}$, the GRU updates the hidden state via $h_{t+i+1} = g_{\theta}(h_{t+i}, a_{t+i}^k)$, conditioned on the previous latent state and the external action $a_{t+i}^k$. Crucially, a transition decoder $\text{dec}_{\theta}$ maps each latent state directly to a predicted observation $\hat{o}_{t+i+1}$. By decoupling the latent dynamics from the observation space—specifically by ensuring predicted observations are never fed back as inputs—the model maintains high trajectory fidelity and provides a stable foundation for counterfactual evaluation.

\textbf{Scheduler Policy.} The scheduler policy is implemented within the Implicit Q-Learning (IQL) framework, comprising separate Q and V networks. To handle the variable-length nature of the action sequences $A^k$, the Q-network employs a dual-stream architecture: a GRU processes the temporal dependencies of the action sequence, while a standard Multi-Layer Perceptron (MLP) encodes the current environment state $o$. The resulting features are concatenated and passed through a secondary MLP to produce the final Q-value. In contrast, the V-network utilizes a simplified architecture, consisting of a single MLP that maps the environment state $o$ directly to a state-value estimate. This design ensures the policy can effectively evaluate complex, multi-step action plans against the current environmental context.

\section{Hyperparameter of SuP}
% SuP
\begin{table}[h]
\centering
\label{tab:hyper}
\caption{Hyperparameter configurations of SuP in different experiment settings.}
\begin{tabular}{lccc}
\toprule
\textbf{Hyperparameter} & \textbf{BiGym} & \textbf{Libero} & \textbf{Real-world} \\ \midrule
Learning rate           & $3 \times 10^{-4}$    & $1 \times 10^{-4}$     & $1 \times 10^{-4}$         \\
Batch size              & 512            & 512             & 512                 \\
GRU hidden dimension    & 256            & 256             & 256                 \\
GRU layers              & 3              & 3               & 3                   \\
Chunk length            & 24             & 10              & 20                  \\
$k_{min}$               & 2              & 1               & 2                  \\
$k_{max}$               & 4              & 2               & 4                 \\
Epsilon ($\epsilon$)    & 0.01-0.02      & 0.01-0.02       & 0.02-0.04         \\
Expectile ($\tau$)      & 0.95           & 0.95            & 0.95                 \\
Penalty ($\Omega$)      & -5             & -2              & -1                  \\
Gamma ($\gamma$)        & 0.9            & 0.1             & 0.9                \\ \bottomrule
\end{tabular}
\end{table}


\section{Additional Visualization Results}
\label{app: Additional Visualization Results}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/arrange_table_combined.pdf}
    \caption{Arrange Table}  % 按需修改标题
    % \label{APP:}
\end{figure}


\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/stack_plate_combined.pdf}
    \caption{Stack Plate}  % 按需修改标题
    % \label{APP:}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/sandwich_toast_combined.pdf}
    \caption{Sandwich Toast}  % 按需修改标题
    % \label{APP:}
\end{figure}
% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/libero_long_cropped.pdf}
%     \caption{Libero Long}  % 按需修改标题
%     % \label{APP:}
% \end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/libero_long_frame_k_combined.pdf}
    \caption{Libero Long}  % 按需修改标题
    % \label{APP:}
\end{figure}
% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/cupboards_close_all_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/dishwasher_close_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/dishwasher_close_trays_combined.pdf}
%     \caption{Dishwasher Close Trays}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/dishwasher_open_combined.pdf}
%     \caption{Dishwasher Open}  % 按需修改标题
%     % \label{APP:}
% \end{figure}


% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/dishwasher_open_trays_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}


% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/drawers_close_all_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/drawers_open_all_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/flip_cup_combined.pdf}
%     \caption{Flip Cup}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/flip_cutlery_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}


% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/move_plate_combined.pdf}
%     \caption{Move Plate}  % 按需修改标题
%     % \label{APP:}
% \end{figure}


% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/pick_box_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}



% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/put_cups_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/sandwich_flip_combined.pdf}
%     \caption{Sandwich Flip}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/sandwich_remove_combined.pdf}
%     \caption{Sandwich Remove}  % 按需修改标题
%     % \label{APP:}
% \end{figure}


% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/sandwich_toast_combined.pdf}
%     \caption{Sandwich Toast}  % 按需修改标题
%     % \label{APP:}
% \end{figure}


% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/saucepan_to_hob_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}



% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/store_box_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/take_cups_combined.pdf}
%     \caption{Take Cups}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/wall_cupboard_close_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}

% \begin{figure}[H]
%     \centering
%     \includegraphics[width=0.75\textwidth]{figures/many_sup_samples/wall_cupboard_open_combined.pdf}
%     \caption{Arrange Table}  % 按需修改标题
%     % \label{APP:}
% \end{figure}