Spectral Decomposition of Inverse Dynamics for Fast Exploration in Model-Based Manipulation

Abstract

Planning long duration robotic manipulation sequences is challenging because of the complexity of exploring feasible trajectories through nonlinear contact dynamics and many contact modes. Moreover, this complexity grows with the problem's horizon length. We propose a search tree method that generates trajectories using the spectral decomposition of the inverse dynamics equation. This equation maps actuator displacement to object displacement, and its spectrum is efficient for exploration because its components are orthogonal and they approximate the reachable set of the object while remaining dynamically feasible. These trajectories can be combined with any search based method, such as Rapidly-Exploring Random Trees (RRT), for long-horizon planning. Our method performs similarly to recent work in model-based planning for short-horizon tasks, and differentiates itself with its ability to solve long-horizon tasks: whereas existing methods fail, ours can generate 45 second duration, 10+ contact mode plans using 15 seconds of computation, demonstrating real-time capability in highly complex domains.

Figures (8)

• Figure 2: Our method efficiently plans manipulation trajectories through several contact modes, discovering behavior like pushing, sliding, rolling, tumbling, and pivoting. In the physics-based simulator, an object (green prism) is moved with non-prehensile manipulation (red dot) through a contact-rich environment to a goal configuration.
• Figure 3: Top left: Variables for the problem setup. Bottom right: Simple example of the fingertip exhibiting extrinsic dexterity in order to move an object through a maze.
• Figure 4: An illustration of the fingertip controller. The dark green arrows are the linearized motion $\mathbf{v}_i$ projected forward $T_\text{proj}$ seconds, the gray arrows are the nonlinear motion, and the end of the red line is the fingertip reference point.
• Figure 5: Left: For a single configuration where the object is pinned between the fingertip and the environment, proposed motion directions $\mathbf{v}_i$ are shows with red arrows, while the true reachable set for the center of the box is shown in pink. Right: Same as the left image except for a set of 5 different initial configurations to capture sampling different contacts between the object and manipulator. The approximate reachable set for the object's COM is shown in gray, and efficiently captures a distinct subset of all possible motions in pink.
• Figure 6: Pictoral representation of the MDP tree for a simple maze problem. In grey are the possible motion paths for the center of mass, while in dark green is a possible path chosen by the RRT algorithm.