Physical human-robot collaboration (pHRC) requires both compliance and safety guarantees since robots coordinate with human actions in a shared workspace. This paper presents a novel fixed-time adaptive neural control methodology for handling time-varying workspace constraints that occur in physical human-robot collaboration while also guaranteeing compliance during intended force interactions. The proposed methodology combines the benefits of compliance control, time-varying integral barrier Lyapunov function (TVIBLF) and fixed-time techniques, which not only achieve compliance during physical contact with human operators but also guarantee time-varying workspace constraints and fast tracking error convergence without any restriction on the initial conditions. Furthermore, a neural adaptive control law is designed to compensate for the unknown dynamics and disturbances of the robot manipulator such that the proposed control framework is overall fixed-time converged and capable of online learning without any prior knowledge of robot dynamics and disturbances. The proposed approach is finally validated on a simulated two-link robot manipulator. Simulation results show that the proposed controller is superior in the sense of both tracking error and convergence time compared with the existing barrier Lyapunov functions based controllers, while simultaneously guaranteeing compliance and safety.

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.