IEEE TRANSACTIONS ON ROBOTICS 1 Dynamically Consistent Online Adaptation of Fast Motions for Robotic Manipulators Alexander Pekarovskiy, Student Member, IEEE, Thomas Nierhoff Sandr Hirche, Senior Member, IEEE, and Martin Buss, Fellow, IEEE Abstract—The planning and execution of real-world robotic tasks largely depend on the ability to generate feasible motions online in response to changing environment conditions or goals. A spline deformation method is able to modify a given trajec- tory so that it matches the new boundary conditions, e.g. on positions, velocities, accelerations, etc. At the same time, the deformed motion preserves velocity, acceleration, jerk or higher derivatives of motion profile of precalculated trajectory. The deformed motion possessing such properties can be expressed by translation of original trajectory and spline interpolation. This spline decomposition considerably reduces the computational complexity and allows the real-time execution. Formal feasibility guarantees are provided for the deformed trajectory and for the resulting torques. These guarantees are based on the special properties of Bernstein polynomials used for the deformation and on the structure of the chosen computed torque control scheme. The approach is experimentally evaluated in a number of planar volleyball experiments using 3-DoF robots and human participants. Index Terms—Manipulation planning, motion adaptation, mo- tion control, path planning for manipulators. I. I NTRODUCTION F OR many years robotic tasks used preplanned repetitive motions for industrial applications. However, nowadays there is a deep need for robots with the capability of doing a variety of tasks with desired encoded behavior but with- out offline recomputation of the whole motion. It becomes increasingly important to generate motions in accordance with changing goals and adapt to the changing environments. Thus, motion planning for reactive real-world scenarios needs to fulfill several requirements. On the one hand, it needs to comply with the imposed constraints such as position, velocity or acceleration limits, be able to avoid obstacles and at the same time prevent torque saturation. On the other hand, Manuscript received December 30, 2016; accepted September 23, 2017. This paper was recommended for publication by Associate Editor P. Fraisse and Editor A. Billard upon evaluation of the reviewers’ comments. This work was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. [267877], partly from the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement n. [643433], project “Robotic Assistant for MCI Patients at home (RAMCIP)“ and in part by the Technical University of Munich - Institute for Advanced Study (www.tum-ias.de), funded by the German Excellence Initiative.(Corrsponding author: Alexander Pekarovskiy.) A. Pekarovskiy and M. Buss are with the Chair of Automatic Control Engineering and the TUM Institute for Advanced Study, Technical University of Munich, Lichtenbergstr. 2a, 85748 Garching, Germany {a.pekarovskiy, mb} at tum.de T. Nierhoff and S. Hirche are with the Chair of Information-oriented Control, Technical University of Munich, D-80333 Munich, Germany {tn, hirche} at tum.de resulting solutions should be generated online, if not real-time, as a response to the updated sensory data. That is particularly important for dynamic manipulation, motion imitation and autonomous driving. There exist different conceptual approaches to motion gen- eration, however, none of them is capable of satisfying all the requirements. Learning methods for motion generation can be fast, but they generally do not include robot model and guaranteed online constraint satisfaction. On the contrary, planning and replanning based on classical optimal control may address all type of constraints including the nonlinear robot dynamics, i.e. they are dynamically consistent. However, it is computationally expensive and in realistic cases too slow for online application. Instead of complete replanning we propose to resort to efficient approximate methods. The key scientific challenge is to incorporate the different types of constraints for a modified motion under specified time limits. The subject of this article is online motion deformation which preserves the derivative profile of an initial (planned or learned) motion, incorporates desired boundary conditions and checks violation of trajectory and torque constraints. The resulting approach provides a reactive motion generation suit- able for robotic manipulators with non-negligible dynamics. A. Related Work Existing motion generation methods only partially possess the desired features for online motion adaptation. 1) Optimal Control and Optimization Methods: Optimal control allows setting task goals, imposing a large variety of constraints and achieving the desired performance. Op- timal Control Problems (OCPs) are oftentimes solved with numerical approximate methods such as direct collocation or multiple shooting [1]. Due to the complexity of OCPs, such a recalculation cannot be done online, especially for non- linear dynamic systems and nonlinear constraints. Therefore, some relaxation of OCP conditions is required to improve the computation time. For instance, the method for creating optimal motions with the satisfaction of continuous inequality constraints offline and replanning them for humanoid robots is implemented in [2]. For these safe motions, solutions are found using semi-infinite programming which is computationally costly, whereas replanning them might be faster but it works only in the vicinity of the safe motions in its joint space. Another option is to relax the original OCP to a trajectory op- timization problem, essentially assuming that the control will do the job of perfectly tracking the trajectory. The nonlinear