Non-parametric regression for robot learning on manifolds

TL;DR

The paper addresses robot learning with manifold-valued data by developing a kernelised likelihood estimation (KLE) framework that performs regression directly on manifolds via local likelihood weighting. It unifies distributions tailored to each manifold, including ESAG on $S^{d-1}$, ACG on $S^{d-1}/Z_2$, and SPD lognormal on Sym^+(d), to estimate parameter functions $\hat{\theta}(t)$ and their dispersion. Empirical results on trajectories in $S^2$, $Sym^+(2)$, and $SE(3)$ show improved mean accuracy and meaningful uncertainty modeling over projection-based baselines like GMR and KMP, and demonstrate via-point adaptation and autonomous system applications. The approach is intrinsic and generic, eliminating reliance on tangent-space projections and enabling regression with manifold-valued predictors or responses, with practical impact for robust, flexible robot learning from demonstrations.

Abstract

Many of the tools available for robot learning were designed for Euclidean data. However, many applications in robotics involve manifold-valued data. A common example is orientation; this can be represented as a 3-by-3 rotation matrix or a quaternion, the spaces of which are non-Euclidean manifolds. In robot learning, manifold-valued data are often handled by relating the manifold to a suitable Euclidean space, either by embedding the manifold or by projecting the data onto one or several tangent spaces. These approaches can result in poor predictive accuracy, and convoluted algorithms. In this paper, we propose an "intrinsic" approach to regression that works directly within the manifold. It involves taking a suitable probability distribution on the manifold, letting its parameter be a function of a predictor variable, such as time, then estimating that function non-parametrically via a "local likelihood" method that incorporates a kernel. We name the method kernelised likelihood estimation. The approach is conceptually simple, and generally applicable to different manifolds. We implement it with three different types of manifold-valued data that commonly appear in robotics applications. The results of these experiments show better predictive accuracy than projection-based algorithms.

Figures (19)

• Figure 1: Examples of manifold-valued data in robotics. TOP-LEFT: A top-table grasp is an element of $\mathbb{R}^3\times\mathcal{S}^1$ which in this case parametrises the Schoenflies group of displacements, X(4). TOP-RIGHT: The orientation of the cutting tool of the 5-DOF Exechon XMini robot pc_exechon is a direction in $\mathcal{S}^2$. BOTTOM: The 7-DOF Franka Panda robot lifts a box in two different configurations but the same pose of end-effector. The direction of maximum force achievable is shown for each configuration. This direction is found by computation of the manipulability ellipsoid, an element of the manifold of SPD matrices, $\mathrm{Sym}^{+}(6)$
• Figure 2: Visual comparion between an intrinsic distribution fitted to axial data in $\mathcal{S}^2/\mathbb{Z}_2$, and Gaussians fitted to projections of the data in different tangent spaces
• Figure 3: Adaptation of scalar trajectories to $\mu^{*}=0.6$ at $t^{*}=0.7$. LEFT: $\mathscr{D}$ consists of 700 samples taken from 10 demonstrated curves. The resulting $\hat{\mu}(t)$ is shown in thick black curve. $\mathscr{E}$ is $S=10$ samples generated with $\mu^{*}=0.6$ and $\sigma^{*}=0.01$. RIGHT: The result of adaptation with the blue area showing $\pm 1.5\hat{\sigma}(t)$. BOTTOM RIGHT: The activation function correspond to $\alpha(t)=1-0.5\tanh(25(t-t^{*}+0.175))+0.5\tanh(25(t-t^{*}-0.175))$
• Figure 4: Regression for trajectories on $\mathcal{S}^2$. TOP: KLE method with ESAG model. MIDDLE: Riemannian GMR with 10 stages, the covariance for each stage is shown in coloured thicker curves. BOTTOM: KMP with 10 stages.
• Figure 5: Learning of trajectories on $\mathrm{Sym}^{+}(2)$. FIRST PLOT: Training data $\mathscr{D}$ (orange to yellow) and original mean $\mathbf{M}_j$ (blue). SECOND PLOT: Using KLE: estimated mean $\hat{\mathbf{M}}(t)$ (blue), original mean $\mathbf{M}_j$ (black outline), and samples from $LN(\hat{\mathbf{M}}(t), \hat{\bm{\Sigma}}(t))$ (orange to yellow). THIRD PLOT: Using Riemannian GMR: estimated mean $\hat{\mathbf{M}}(t)$ (blue), and original mean $\mathbf{M}_j$ (black outline). BOTTOM: eigenvalues of the original covariance matrix $\bm{\Sigma}_j$ (solid black), and KLE-computed $\hat{\bm{\Sigma}}(t)$ (dashed blue)