Christoph Rösmann

Efficient trajectory optimization using a sparse model

The “timed elastic band” approach optimizes robot trajectories by subsequent modification of an initial trajectory generated by a global planner. The objectives considered in the trajectory optimization include but are not limited to the overall path length, trajectory execution time, separation from obstacles, passing through intermediate way points and compliance with the robots dynamic, kinematic and geometric constraints. “Timed elastic bands” explicitly consider spatial-temporal aspects of the motion in terms of dynamic constraints such as limited robot velocities and accelerations. The trajectory planning operates in real time such that “timed elastic bands” cope with dynamic obstacles and motion constraints. The “timed elastic band problem” is formulated as a scalarized multi-objective optimization problem. Most objectives are local and relate to only a small subset of parameters as they only depend on a few consecutive robot states. This local structure results in a sparse system matrix, which allows the utilization of fast and efficient optimization techniques such as the open-source framework “g2o” for solving “timed elastic band” problems. The “g2o” sparse system solvers have been successfully applied to VSLAM problems. This contribution describes the application and adaptation of the g2o-framework in the context of trajectory modification with the “timed elastic band”. Results from simulations and experiments with a real robot demonstrate that the implementation is robust and computationally efficient.

Timed-Elastic-Bands for time-optimal point-to-point nonlinear model predictive control

This contribution presents a novel approach for nonlinear time-optimal model predictive control (MPC) based on Timed-Elastic-Bands (TEB). The TEB merges the states, control inputs and time intervals into a joint trajectory representation which enables planning of time-optimal trajectories in the context of model predictive control. Model predictive control integrates the planning of the optimal trajectory with state feedback in the control loop. The TEB approach formulates the fixed horizon optimal control problem for point-to-point transitions as a nonlinear program. The comparative analysis of the TEB approach with state-of-the-art approaches demonstrates its computational efficiency. The TEB approach generates a trajectory that approximates the analytical time-optimal trajectory in few iterations. This efficiency enables the refinement of the planned state and control sequence within the underlying closed-loop control during runtime.

Integrated online trajectory planning and optimization in distinctive topologies

Abstract This paper presents a novel integrated approach for efficient optimization based online trajectory planning of topologically distinctive mobile robot trajectories. Online trajectory optimization deforms an initial coarse path generated by a global planner by minimizing objectives such as path length, transition time or control effort. Kinodynamic motion properties of mobile robots and clearance from obstacles impose additional equality and inequality constraints on the trajectory optimization. Local planners account for efficiency by restricting the search space to locally optimal solutions only. However, the objective function is usually non-convex as the presence of obstacles generates multiple distinctive local optima. The proposed method maintains and simultaneously optimizes a subset of admissible candidate trajectories of distinctive topologies and thus seeking the overall best candidate among the set of alternative local solutions. Time-optimal trajectories for differential-drive and carlike robots are obtained efficiently by adopting the Timed-Elastic-Band approach for the underlying trajectory optimization problem. The investigation of various example scenarios and a comparative analysis with conventional local planners confirm the advantages of integrated exploration, maintenance and optimization of topologically distinctive trajectories.

Planning of multiple robot trajectories in distinctive topologies

This contribution presents a novel approach for efficient online planning of topologically distinctive mobile robot trajectories. Trajectory optimization deforms an initial coarse path drafted by a global planner with respect to robot motion related objectives and constraints. The primary objective is to reach a goal state in minimal time on a collision free path that adheres to the kinematic and dynamic constraints of the mobile robot. Conventional local planners get often stuck in a local optimal trajectory as they are unable to transit across obstacles. Our approach seeks the globally optimal trajectory as it maintains and optimizes a subset of admissible candidate trajectories of distinctive topologies in parallel. In case of dynamic environments the planner switches to the current globally optimal trajectory among the candidate set. The online trajectory planning with timed elastic bands is tightly integrated with the robot motion feedback control. The comparative analysis with conventional local planners confirms the advantages of maintaining distinctive topologies to circumnavigate dynamic obstacles.

A real-time capable model predictive approach to lateral vehicle guidance

The paper at hand proposes a real-time capable approach to combined trajectory planning and control. One single prediction model is used to plan a feasible trajectory and to perform lateral guidance of the vehicle at the same time. Nonlinear model predictive control (NMPC) methods are applied to solve the optimal control problem, which incorporates environmental constraints leading to a model predictive planning and control approach (MPPC). Experiments are conducted utilizing a rapid prototyping system. The analysis shows the versatile application range of the developed algorithm, such as challenging emergency evasive maneuvers as well as automated steering as an approach to lateral guidance of the vehicle in general.

Online trajectory prediction and planning for social robot navigation

This paper addresses the safe and legible navigation of mobile robots in multi-agent encounters. A novel motion model provides the basis to predict, plan and coordinate agent trajectories in intersection scenarios. The approach establishes an implicit, non-overt cooperation between the robot and humans by linking the prediction and planning of agent trajectories within a unified representation in terms of timed elastic bands. The planning process maintains multiple topological alternatives to resolve the encounter in a manner compliant with the implicit rules and objectives of human proxemics. The trajectory is obtained by optimizing the timed elastic band considering multiple conflicting objectives such as fastest path and minimal spatial separation among agents but also global proxemic aspects such as motion coherence within a group. Cooperation is achieved by coupling predicted and planned agent trajectories to eventually reach an implicit agreement of the agents on how to circumnavigate each other. The parameters of the cost functions of the underlying motion model are identified by inverse optimal control from a dataset of 73 recorded encounters with up to five humans and a total of 283 individual trajectories. Playback simulations of recorded encounters and experiments with a robot traversing a group of oncoming humans demonstrate the feasibility of the approach to resolve general proxemic encounters.

Online Trajectory Optimization and Navigation in Dynamic Environments in ROS

This tutorial chapter provides a comprehensive step-by-step guide on the setup of the navigation stack and the teb_local_planner package for mobile robot navigation in dynamic environments. The teb_local_planner explicitly considers dynamic obstacles and their predicted motions to plan an optimal collision-free trajectory. The chapter introduces a novel plugin to the costmap_converter ROS package which supports the detection and motion estimation of moving objects from the local costmap. This tutorial covers the theoretical foundations of the obstacle detection and trajectory optimization in dynamic scenarios. The presentation is designated for ROS Kinetic and Lunar and both packages will be maintained in future ROS distributions.

Online Trajectory Planning in ROS Under Kinodynamic Constraints with Timed-Elastic-Bands

This tutorial chapter provides a comprehensive and extensive step-by-step guide on the ROS setup of a differential-drive as well as a car-like mobile robot with the navigation stack in conjunction with the teb_local_planner package. It covers the theoretical foundations of the TEB local planner, package details, customization and its integration with the navigation stack and the simulation environment. This tutorial is designated for ROS Kinetic running on Ubuntu Xenial (16.04) but the examples and code also work with Indigo, Jade and is maintained in future ROS distributions.

Time-Optimal nonlinear model predictive control with minimal control interventions

This paper presents a novel approach for timeoptimal model predictive control. In contrast to a global uniform time scaling, the underlying optimal control problem rests upon a dynamic, local temporal discretization of the shooting grid. The approach seeks for a grid partition with minimum overall transition time. Furthermore, a multi-stage optimization iteratively adapts the number of grid points during runtime to achieve a minimum number of control interventions. A comparative analysis with previous approaches for three nonlinear control problems demonstrates the superiority of the proposed scheme. The feasibility is experimentally demonstrated for position control of a servo drive operated at 200 Hz.

Model predictive trajectory set control for a proportional directional control valve

In practice, high quality control for mechatronic systems is often achieved by augmenting classical control architectures like PID controllers with numerous tailored nonlinear characteristic parameter curves and cascades. This complexity can be significantly reduced by utilizing advanced model predictive controllers (MPC). Furthermore, desired objectives like minimum control error and effort can be realized while explicitly adhering to state and control constraints. However, MPC is subject to iterative gradient-based online optimization algorithms which are computationally expensive. Hence, their application to mechatronic systems with fast dynamics is limited. It is worth mentioning that industrial systems often utilize low cost computational hardware. Accordingly, this contribution presents a model predictive trajectory set control (MPTSC) scheme that mimics a sub-optimal MPC by a rough discretization of the control input domain. A comparative analysis with a linear quadratic regulator demonstrates its ability to provide a sufficiently high control performance compared to the optimal reference. Furthermore, the approach is experimentally evaluated on a proportional directional control valve with a sample rate of 10 kHz. In addition to its efficiency the implementation of MPTSC is less complex and error-prone in comparison to MPC which is a reasonable advantage especially in industrial applications.