Topics in Trajectory Generation for Robots

A fundamental problem in robotics is generating the motion for a task. How to translate a task to motion or a series of movements is a non-trivial problem. The complexity of the task, the structure of the robot, and the desired performance determine the sequence of movements, the path, and the course of motion as a function of time, namely the trajectory. As we discuss in this thesis, a trajectory can be acquired from a human demonstration or generated by carefully designing an objective function. In the first approach, we examine a number of robotic setups which are suitable for human demonstration. More notably, admittance control as a new dimension to the robot-assisted teleoperation is investigated. We also describe a free-floating behavior which makes robust lead-through programming possible. As a way to utilize these setups, we present some ideas for developing a high-level language for an event-based programming common to assembly tasks.

Since immediate reaction to variations in the target state and/or robot state is desirable, we reformulate the trajectory generation problem as a controller design problem. Using the Hamilton-Jacobi-Bellman equation, we derive a closed-loop solution to the fixed-time trajectory-generation problem with a minimum-jerk cost functional. We show that the resulting trajectory coincides with a fifth-order polynomial function of time that instantaneously updates due to changes in the reference signal and/or the robot states.

A short comparison is made between kinematic and dynamic models for generating optimal trajectories. The conclusion is that given conservative kinematic constraints, both models behave in a similar way. Having this in mind, we derive an analytic solution to the problem of fixed-time trajectory generation with a quadratic cost function under velocity and acceleration constraints. The advantage of the analytic solution compared to an on-line optimization approach lies in the efficiency of the computation.

To extend the idea of closed-loop trajectory generation, we adapt the Model Predictive Control (MPC) framework. MPC is traditionally applied to tracking problems, i.e., when there is an explicit reference signal. Thus, it is a common practice to have a separate layer that generates the reference signal. We propose an integrated approach by introducing a final state constraint in the formulation. Additionally, we give the interpretation that the difference between tracking and point-to-point trajectory-planning problems is in the density of the specified desired reference signal. We utilize a strategy to reduce the discretization time successively. This way, we respect the real-time constraints for computation time while the accuracy of the solution is gradually improved as the deadline approaches. We have verified our proposed MPC approach to trajectory generation in a ball-catching experiment.