We propose a learning framework, named Multi-Coordinate Cost Balancing (MCCB), to address the problem of acquiring point-to-point movement skills from demonstrations. MCCB encodes demonstrations simultaneously in multiple differential coordinates that specify local geometric properties. MCCB generates reproductions by solving a convex optimization problem with a multi-coordinate cost function and linear constraints on the reproductions, such as initial, target, and via points. Further, since the relative importance of each coordinate system in the cost function might be unknown for a given skill, MCCB learns optimal weighting factors that balance the cost function. We demonstrate the effectiveness of MCCB via detailed experiments conducted on one handwriting dataset and three complex skill datasets.
Trajectory optimization methods, such as @cite_2 and @cite_10 , focus on geometric features by minimizing costs specified using predefined norms. An optimization framework proposed in @cite_22 attempts to adapt multiple demonstrations to new initial and target locations by minimizing the distance between the demonstrations and the reproduction according to a learned Hilbert space norm. Indeed, learning an appropriate Hilbert space norm is related to finding an appropriate coordinate system based on the demonstrations. However, similar to the dynamical systems-based methods, the methods in @cite_2 @cite_10 @cite_22 are restricted to a single predefined or learned coordinate system and do not explicitly model and utilize the inherent time-dependent variations in the demonstrations.
@cite_10 @cite_22 @cite_2
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We formalize the problem of adapting a demonstrated trajectory to a new start and goal configuration as an optimization problem over a Hilbert space of trajectories: minimize the distance between the demonstration and the new trajectory subject to the new end point constraints. We show that the commonly used version of Dynamic Movement Primitives (DMPs) implement this minimization in the way they adapt demonstrations, for a particular choice of the Hilbert space norm. The generalization to arbitrary norms enables the robot to select a more appropriate norm for the task, as well as learn how to adapt the demonstration from the user. Our experiments show that this can significantly improve the robot's ability to accurately generalize the demonstration. Trajectory optimization is an essential tool for motion planning under multiple constraints of robotic manipulators. Optimization-based methods can explicitly optimize a trajectory by leveraging prior knowledge of the system and have been used in various applications such as collision avoidance. However, these methods often require a hand-coded cost function in order to achieve the desired behavior. Specifying such cost function for a complex desired behavior, e.g., disentangling a rope, is a nontrivial task that is often even infeasible. Learning from demonstration (LfD) methods offer an alternative way to program robot motion. LfD methods are less dependent on analytical models and instead learn the behavior of experts implicitly from the demonstrated trajectories. However, the problem of adapting the demonstrations to new situations, e.g., avoiding newly introduced obstacles, has not been fully investigated in the literature. In this letter, we present a motion planning framework that combines the advantages of optimization-based and demonstration-based methods. We learn a distribution of trajectories demonstrated by human experts and use it to guide the trajectory optimization process. The resulting trajectory maintains the demonstrated behaviors, which are essential to performing the task successfully, while adapting the trajectory to avoid obstacles. In simulated experiments and with a real robotic system, we verify that our approach optimizes the trajectory to avoid obstacles and encodes the demonstrated behavior in the resulting trajectory. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations. Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high-dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high-speed bipedal running on the FastRunner platform.