Direct collocation methods are powerful tools to solve trajectory optimization problems in robotics. While their resulting trajectories tend to be dynamically accurate, they may also present large kinematic errors in the case of constrained mechanical systems, i.e., those whose state coordinates are subject to holonomic or nonholonomic constraints, such as loop-closure or rolling-contact constraints. These constraints confine the robot trajectories to an implicitly-defined manifold, which complicates the computation of accurate solutions. Discretization errors inherent to the transcription of the problem easily make the trajectories drift away from this manifold, which results in physically inconsistent motions that are difficult to track with a controller. This article reviews existing methods to deal with this problem and proposes new ones to overcome their limitations. Current approaches either disregard the kinematic constraints (which leads to drift accumulation) or modify the system dynamics to keep the trajectory close to the manifold (which adds artificial forces or energy dissipation to the system). The methods we propose, in contrast, achieve full drift elimination on the discrete trajectory, or even along the continuous one, without artificial modifications of the system dynamics. We illustrate and compare the methods using various examples of different complexity.

Kinodynamic rapidly-exploring random tree (RRT) planners are effective tools for finding feasible trajectories in many classes of robotic systems. However, they are hard to apply to systems with closed-kinematic chains, like parallel robots, collaborative arms manipulating an object, or legged robots keeping their feet in contact with the environment. The state space of such systems is an implicitly-defined manifold that complicates the design of the sampling and steering procedures, and leads to trajectories that drift from the manifold if standard integration methods are used. To address these issues, this article presents a kinodynamic RRT planner that constructs an atlas of the state space incrementally, and uses this atlas to generate random states, and to dynamically steer the system toward such states. The steering method exploits the atlas charts to compute locally optimal controls based on linear quadratic regulators. The atlas also allows the integration of the equations of motion using local coordinates, which eliminates any drift from the state space manifold and results in accurate trajectories. To the best of our knowledge, this is the first kinodynamic planner that explicitly takes closed kinematic chains into account. In this article, we illustrate the planner performance in significantly complex tasks involving planar and spatial robots that have to lift or throw a load using torque-limited actuators.

Kinodynamic RRT planners are considered to be general tools for effectively finding feasible trajectories for high-dimensional dynamical systems. However, they struggle when holonomic constraints are present in the system, such as those arising in parallel manipulators, in robots that cooperate to fulfill a given task, or in situations involving contacts with the environment. In such cases, the state space becomes an implicitly-defined manifold, which makes the diffusion heuristic inefficient and leads to inaccurate dynamical simulations. To address these issues, this paper presents an extension of the kinodynamic RRT planner that constructs an atlas of the state-space manifold incrementally, and uses this atlas both to generate random states and to dynamically steer the system towards such states. To the best of our knowledge, this is the first randomized kinodynamic planner that explicitly takes holonomic constraints into account. We validate the approach in significantly-complex systems.

Parallel robots exhibit the so-called forward singularities, which complicate substantially the planning and control of their motions. Often, such complications are circumvented by restricting the motions to singularity-free regions of the workspace. However, this comes at the expense of reducing the motion range of the robot substantially. It is for this reason that, recently, efforts are underway to control singularity-crossing trajectories. This paper proposes a reliable controller to stabilize such kind of trajectories. The controller is based on the classical theory of linear quadratic regulators, which we adapt appropriately to the case of parallel robots. As opposed to traditional computed-torque methods, the obtained controller does not rely on expensive inverse dynamics computations. Instead, it uses an optimal control law that is easy to evaluate, and does not generate instabilities at forward singularities. The performance of the controller is exemplified on a five-bar parallel robot accomplishing two tasks that require the traversal of singularities.