Cutting Sequence Diffuser: Sim-to-Real Transferable Planning
for Object Shaping by Grinding

The grinding process is performed by local surface contact between the tool surface (belt) and the object shape. Ignoring removal resistance, shape transition can be viewed as a geometric splitting by the cutting surface (Figure2, top). Here, the robot’s action is equivalent to specifying the cutting surface. This step allows us to represent the shape transition model as a Geometric-Cutting Model (GCM), where cutting surface (robot action) $\mathbf{a}_{t}\in\mathcal{A}$ geometrically splits current shape $\mathbf{s}_{t}\in\mathcal{S}$ into next time-step shape $\mathbf{s}_{t+1}$ and removal shape $\mathbf{w}_{t+1}$. The shape transition by the cutting surface is defined as the following functions [2]:

These functions split the shape using geometric collision detection, simplifying their implementation. Figure2 (bottom) shows shape deformation by cutting surface $\mathbf{a}_{t}$.

In the actual grinding process, removal resistance is generated depending on the process conditions. The removal resistance generated by grinding is called grinding resistance $F_{t}$, which is basically proportional to removal volume $V_{t}$ and inversely proportional to belt rotation speed $S_{g}$, $F_{t}=\eta{{V}_{t}}/{S_{g}}$ [3]. Here $\eta$ is a coefficient, derived from the material type and the belt’s grinding performance. If the belt rotation speed is constant, the grinding resistance is proportional to the removal volume. Thus, a cutting action with a large removal volume increases the grinding resistance and the actual shape transition contradicts the GCM.

Given a shape transition model, we denote trajectory $\boldsymbol{\tau}\coloneqq[\mathbf{s}_{t},\mathbf{a}_{t},\ldots,\mathbf{s}_{t+H-1},\mathbf{a}_{t+H-1}]$ and cost function $c(\boldsymbol{\tau})$, optimal robot action sequences $\mathbf{a}_{t:t+H-1}^{*}$ can be formulated as follows:

Here $H$ is the planning horizon and $t$ is the task index. Deriving action sequences that satisfy Eq. 2 using conventional optimization-based action planning methods (e.g., random shooting method [20]), significantly increases the computational cost depending on the planning horizon. Hence, long-horizon action planning is time-consuming [7, 8].

Our framework generates trajectory $\boldsymbol{\tau}$ using diffusion models. They consist of a forward diffusion process that transforms the trajectory from data distribution into Gaussian noise and a denoising (inverse) process that transforms the Gaussian noise back to data distribution. Given that original data $\boldsymbol{\tau}^{0}\equiv\boldsymbol{\tau}$ and $\boldsymbol{\tau}^{i}$ denote the data after adding $i$ times noise, the forward diffusion process is defined: $q(\boldsymbol{\tau}^{1:N}|\boldsymbol{\tau}^{0})\coloneqq{\prod_{i=1}^{N}}q(\boldsymbol{\tau}^{i}|\boldsymbol{\tau}^{i-1}),q(\boldsymbol{\tau}^{i}|\boldsymbol{\tau}^{i-1})\coloneqq\mathcal{N}(\boldsymbol{\tau}^{i};\sqrt{{1-\beta_{i}}}\boldsymbol{\tau}^{i-1},\beta_{i}\mathbf{I}),$ where $\beta_{i}$ is a noise scheduler that controls the noise scale.

The denoising process is parametrized as a Gaussian form, and the mean and covariance are defined by a model with parameter $\theta$ which takes noise data $\boldsymbol{\tau}^{i}$ and diffusion steps $i$ as input:

Covariance $\boldsymbol{\Sigma}_{\theta}(\boldsymbol{\tau}^{i},i)$ denotes $\boldsymbol{\Sigma}_{\theta}(\boldsymbol{\tau}^{i},i)=\sigma_{i}\mathbf{I}=\tilde{\beta_{i}}\mathbf{I}$ and independent of parameter $\theta$. Where $\tilde{\beta_{i}}=\beta_{i}(1-\bar{\alpha}_{i-1})/(1-\bar{\alpha}_{i})$, $\bar{\alpha_{i}}=\prod_{s=1}^{i}\alpha_{s}$, and $\alpha_{t}=1-\beta_{t}$. In the learning process, instead of directly optimizing $\boldsymbol{\mu}_{\theta}$, noise $\epsilon$ added to data $\boldsymbol{\tau}^{i}$, can be learned by simplified loss function $\mathcal{L}_{\theta}=\mathbb{E}_{i,\epsilon,\boldsymbol{\tau}^{0}}\left[\|\epsilon-\epsilon_{\theta}(\boldsymbol{\tau}^{i},i)\|^{2}\right]$, since $\boldsymbol{\mu}_{\theta}$ can be represented by $\epsilon_{\theta}$ as $\boldsymbol{\mu}_{\theta}(\boldsymbol{\tau}^{i},i)=\frac{1}{\sqrt{\alpha_{i}}}(\boldsymbol{\tau}^{i}-\frac{1-\alpha_{i}}{\sqrt{1-\alpha_{i}}}\epsilon_{\theta}(\boldsymbol{\tau}^{i},i))$ [5].

By introducing, a binary variable to indicate the trajectory as optimal by $\mathcal{O}=1$ and non-optimal by $\mathcal{O}=0$, and using the Control as Inference, trajectory optimization can be formulated as a stochastic inference problem that samples the optimal trajectory from distribution by $\tilde{p}_{\theta}(\boldsymbol{\tau})=p(\boldsymbol{\tau}|\mathcal{O}=1)\propto p_{\theta}(\boldsymbol{\tau})p(\mathcal{O}=1|\boldsymbol{\tau}).$ Here $p_{\theta}(\boldsymbol{\tau})=\int p(\boldsymbol{\tau}^{N})\prod_{i=1}^{N}p_{\theta}(\boldsymbol{\tau}^{i-1}|\boldsymbol{\tau}^{i})d\boldsymbol{\tau}^{1:N}$ is a prior distribution of the trajectory, and $p(\mathcal{O}|\boldsymbol{\tau})$ is the likelihood for the optimality of the trajectory. As a common assumption, since $p$ belongs to an exponential family, we can arbitrarily write $p(\mathcal{O}=1|\boldsymbol{\tau})\propto\exp({-c(\boldsymbol{\tau}))}$.

The posterior distribution of the denoising process can be expressed in Gaussian form for each denoising process $p(\boldsymbol{\tau}|\mathcal{O}=1)=p_{\theta}(\boldsymbol{\tau}^{0:N})p(\mathcal{O}=1|\boldsymbol{\tau})$. Therefore, an approximate formulation is obtained as follows [4, 6]:

where $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ are given by Eq. 4: $\boldsymbol{\mu}=\boldsymbol{\mu}_{\theta}(\boldsymbol{\tau}^{i},.i),\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{\theta}(\boldsymbol{\tau}^{i},i)$. $g$ is derived from the gradient of the cost function:

Thus, by shifting mean $\boldsymbol{\mu}$ in the denoising process based on the gradient of a cost function designed for the task (e.g., state constraints, trajectory smoothness), we can generate trajectories guided by the cost function.

In addition, diffusion models are trained for the accuracy of the generated trajectories rather than single-step errors. Hence, they can quickly plan long-horizon action sequences without suffering from the significant increase in computational cost that is common in conventional optimization-based action planning methods [7, 8].

This section describes the types of cost functions used as guides for trajectory generation by a diffusion model. Total cost $c(\boldsymbol{\tau})$ for $t$ to $t+H-1$ is calculated by summing each cost: $c(\boldsymbol{\tau})=\lambda_{\mathrm{sm}}c_{\mathrm{sm}}(\boldsymbol{\tau})+\sum_{l=t}^{t+H-1}\sum_{k}\lambda_{k}c_{k}(\boldsymbol{\tau}[l])$, $k=\{\mathrm{state},\mathrm{col},\mathrm{vol}\}$. Where $\lambda$ are the weights of each cost.

State constrain cost: This cost generates a trajectory that satisfies the state constraint $\mathbf{c}_{l}$, such as current shape $\mathbf{s}_{\mathrm{current}}$ and target shape $\mathbf{s}_{\mathrm{target}}$. Observed state distributions in diffusion models are represented by a Dirac delta function. Hence, state constraint cost $c_{\mathrm{state}}$ is expressed as follows:

In the implementation, if $\mathbf{s}_{l}$ satisfies the state constraint, we directly replace $\mathbf{s}_{l}$ with a state constraint value $\mathbf{c}_{l}$ (e.g., $\mathbf{s}_{\mathrm{current}}$ or $\mathbf{s}_{\mathrm{target}}$) in all the denoising processes, as in [8].

Over-cutting cost: Since the grinding process is irreversible, over-cutting target shape $\mathbf{s}_{\mathrm{target}}$ is undesirable. Therefore, we introduce over-cutting cost $c_{\mathrm{col}}$:

Cut volume limit cost: This cost constrains the removal volume at one step. If removal volume exceeds $\delta_{\mathrm{vol}}$, the following cost $c_{\mathrm{vol}}$ is imposed:

Action smoothness cost: To suppress the vibrations of the robot’s end-effector and promote the planning of smooth cutting sequences, we constrain the travel length of the cutting surface as follows:

Where $a_{l}^{j}$ is the $j$-th action value at time step $l$ and ${a}^{j_{\text{max}}}$ is the upper bound of the travel length for each action dimension. The variable $j$ indicates the action index.

In the general grinding process, overcutting the target shape and exceeding the volume limit that can be removed in one step is fatal. Therefore, it would be preferable to set the weights to ensure that the $c_{\mathrm{col}}$ and $c_{\mathrm{vol}}$ costs are dominant.

The proposed method increases task horizon $T$ by constraining the removal volume to reduce the reality gap. Therefore, as shown in §IV-A, the task horizon is divided into practical planning horizon lengths $H$. However, the grinding process is a goal-conditioned trajectory planning problem where only current shape $\mathbf{s}_{t}$ and final target shape $\mathbf{s}_{\mathrm{target}}$ can be used as state constraints. Thus, if we naively set the final target shape as the end of planning horizon constraint $t+H-1$, CSD tries to achieve object shaping, which normally requires $T$ steps, within a planning horizon $H$. This typically occurs when the planning horizon is shorter than the task horizon (i.e., $t+H-1\ll T$). Hence, such an approach can generate cutting sequences that remove a large volume at once, potentially causing significant removal resistance.

To address this issue, we execute trajectory planning with two-step guidance. Initially, at $t=0$, we set planning horizon $H$ equal to task horizon $T$ and generate a trajectory for the entire task horizon to obtain the reference shape states for each time step. Next, we plan the $H$-horizon trajectory by referring to the obtained shape as the state constraint at the end of the planning horizon, $t+H-1$. Thus, we use the shape state obtained by planning the trajectory with the entire task horizon as the state constraint at the end of the planning horizon. This approach allows for the suppression of cutting actions with large removal volumes at once. As a result, it may reduce removal resistance and promote smooth trajectory generation.

Figure 5 (a) shows an example of the shape after processing Shape A with ASA. TABLE I compares the shape error and the action planning time for six trials of each method. In the following experiment, we used Chamfer discrepancy [25] for the shape error calculation:

where $\mathbf{x}$ and $\mathbf{y}$ are the positions of the individual points in the point clouds. This metric evaluates the error using the Euclidean distance between the nearest neighbor points of two point clouds. It indicates that the shape error is rarely zero unless the two point clouds are perfectly aligned. Const-RS and CSD were performed with $T=160$ task steps, a planning horizon of $H=32$, and $M=32$ control inputs. CSA-MBRL was performed with $T=40$, $H=5$, and $M=1$. CSA-MBRL defines a single-step action as follows: 1) Transitioning from the work origin to the target cutting surface, 2) Holding the target cutting surface for a specific duration, and 3) Returning to the work origin. This definition allows each step to cover a larger range of tasks, leading to shorter $T$, $H$, and $M$. Additionally, $T$, $H$, and $M$ were set based on [2] to maintain consistency with prior experiments and ensure comparability. These results show that our proposed method can quickly plan long-horizon action sequences and achieve comparable shape error as CSA-MBRL, which learns the shape transition model from real data.

TABLE IV compares the action planner within the proposed method. The former with CVAE had a shorter planning time than the diffusion model. However, it had larger shape error, indicating the suitability of employing a diffusion model for the proposed method, where grinding accuracy is important even if action planning takes some time.

Figures 5 (b)-(d) and TABLE I show the results of grinding Shape A with PC and Shapes B and C with ASA. These results indicate that the proposed method can be applied to environments with different grinding resistances and different initial and target shapes by combining actions with a small removal volume through flexible trajectory generation using a diffusion model.

TABLE II compares the task performances with/without a guide. Four evaluation metrics were used for the task performance. Over-cutting is the number of times the target shape was excessively cut. The cut volume limit is obtained by dividing the cumulative sum of the volumes that exceed the removal volume threshold $\delta_{\mathrm{vol}}$ by the number of task steps. Refer to Eqs. 12–13 concerning shape error and action smoothness. These results show that a guided trajectory generation with a cost function improves each evaluation metric.

TABLE III compares the cut volume limit with/without the two-step guide. Figure6 shows the transition of the cumulative sum of the volumes that exceeded the cut volume limit. In the case of a one-step guide, planning actions exceeded the cut volume limits around the initial step. However, the two-step guide suppressed the excesses of the cut volume limits.

Figure 7 (a) shows an example of the shape after processing Shape A with ASA. TABLE V compares the shape error and the action planning time for three trials of each method. Const-RS and CSD were performed with $T=160$ task steps, a planning horizon of $H=32$, and $M=32$ control inputs. CSA-MBRL was performed with $T=30$, $H=5$, and $M=1$. CSA-MBRL observed the shape every two steps to reduce the task execution time. The shape observation in the real robot experiments required the robot to be moved in front of the 3D vision sensor (Figure1) and to take multiple measurements to obtain an accurate shape, which was time-consuming. Therefore, the predicted shape (one step ahead) was used as the observed shape where no shape observation was performed, and the action was planned again. These results show that, similar to the simulation results, the proposed method can quickly plan long-horizon action sequences and achieve comparable shape error as CSA-MBRL, which learns the shape transition model from real data.

Figure 8 compares the observations, action planning, and action execution times required to process Shape A with ASA. Const-RS required longer time to optimize a long-horizon action. CSA-MBRL, required frequent shape observations to reduce the shape error. In contrast, the proposed method required less task execution time due to its ability to quickly plan long-horizon action sequences.

Figures 7 (b)-(d) and TABLE V show the results of grinding Shape A with PC and Shapes B and C with ASA. The proposed method can be applied to different materials and different initial and target shapes by combining actions with a small removal volume through flexible trajectory generation using a diffusion model.

TABLE VI compares the task performances with/without a guide. These results show that a guided trajectory generation with a cost function improves each evaluation metric. Additionally, TABLE VII compares the cut volume limit with/without the two-step guide. We confirmed that it suppressed the excesses of the cut volume limits.