text: Inverse dynamics control and feedforward linearization techniques are typically used to convert the complex nonlinear dynamics of robots to a set of decoupled double integrators, and then a standard, outer-loop controller can be used to calculate the commanded acceleration for the linearized system. However, these methods typically depend on having a very accurate system model, which is not available in practice. While this challenge has been addressed in the literature using different learning approaches, most of these approaches do not provide safety guarantees in terms of stability of the learning-based control system. In this paper, we provide a novel, learning-based control approach based on Gaussian processes (GPs) that ensures both stability of the closed-loop system and high-accuracy tracking. In our proposed approach, we use GPs to approximate the error between the commanded acceleration and the actual acceleration of the system, and then use the predicted mean and variance of the GP to calculate an upper bound on the uncertainty of the linearization. This uncertainty bound is then used in a robust, outer-loop controller to ensure stability of the overall system. Moreover, we show that using our proposed strategy, the tracking error converges to a ball with a radius that can be made arbitrarily small through appropriate control design. Furthermore, we verify the effectiveness of the proposed approach via simulations on (i) a 2 degree-of-freedom (DOF) planar manipulator, and (ii) a 6 DOF industrial manipulator using the Gazebo simulator.
  @cite_1: Learning of inverse dynamics modeling errors is key for compliant or force control when analytical models are only rough approximations. Thus, designing real time capable function approximation algorithms has been a necessary focus towards the goal of online model learning. However, because these approaches learn a mapping from actual state and acceleration to torque, good tracking is required to observe data points on the desired path. Recently it has been shown how online gradient descent on a simple modeling error offset term to minimize tracking at acceleration level can address this issue. However, to adapt to larger errors a high learning rate of the online learner is required, resulting in reduced compliancy. Thus, here we propose to combine both approaches: The online adapted offset term ensures good tracking such that a nonlinear function approximator is able to learn an error model on the desired trajectory. This, in turn, reduces the load on the adaptive feedback, enabling it to use a lower learning rate. Combined this creates a controller with variable feedback and low gains, and a feedforward model that can account for larger modeling errors. We demonstrate the effectiveness of this framework, in simulation and on a real system.
  @cite_14: High performance tracking control can only be achieved if a good model of the dynamics is available. However, such a model is often difficult to obtain from first order physics only. In this paper, we develop a data-driven control law that ensures closed loop stability of Lagrangian systems. For this purpose, we use Gaussian Process regression for the feed-forward compensation of the unknown dynamics of the system. The gains of the feedback part are adapted based on the uncertainty of the learned model. Thus, the feedback gains are kept low as long as the learned model describes the true system sufficiently precisely. We show how to select a suitable gain adaption law that incorporates the uncertainty of the model to guarantee a globally bounded tracking error. A simulation with a robot manipulator demonstrates the efficacy of the proposed control law.
  @cite_4: 
summary: Focusing our attention on safe, learning-based inverse dynamics control, we refer to @cite_4 @cite_1 @cite_14 . In @cite_4 , a model reference adaptive control (MRAC) architecture based on Gaussian processes (GPs) is proposed, and stability of the overall control system is proved. While the approach in @cite_4 is based on adaptive control theory, our approach is based on robust control theory. In particular, in @cite_4 , the mean of the GP is used to exactly cancel the uncertainty vector, while in our approach, we use both the mean and variance of the GP to learn an upper bound on the uncertainty vector to be used in a robust, outer-loop controller. Hence, unlike @cite_4 , in our approach, the uncertainty of the learning module is not only incorporated in the stability analysis but also in the outer-loop controller design. Intuitively, the less certain our GPs are, the more robust the outer-loop controller should be for ensuring safety. When more data is collected and the GPs are more certain, the outer-loop controller can be less conservative for improved performance. While the results of @cite_4 are tested in simulations on a two-dimensional system, we test our results experimentally on a 6 DOF manipulator.