A Statistical Learning Theory Approach for Uncertain Linear and Bilinear Matrix Inequalities

In this paper, we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities, which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results in statistical learning theory, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chervonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of these problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality. The effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters.Comment: 19 pages, 2 figures, Accepted for Publication in Automatic

Udwadia-Kalaba Approach for Three Link Manipulator Dynamics With Motion Constraints

Aiming to dynamic modeling of a three-link manipulator subjected to motion constraints, a novel explicit approach to the dynamical equations based on Udwadia-Kalaba (UK) theory is established. The motion constraints on the three-link manipulator can be regarded as external constraints of the system. However, it is not easy to obtain explicit equations for the dynamic modeling of constrained systems. For a multibody system subjecting to motion constraints, it is common to introduce Lagrange multipliers, but obtaining an explicit dynamical equation using traditional Lagrange multipliers is difficult. In order to obtain such equations more simply, motion constraints are handled using the UK equation. Compared with the Lagrange method, the UK approach can simplify the analysis and solution of a constrained system, without the need to introduce additional auxiliary variables to solve the Lagrange equation. Based on a more real-life nominal system (whose parameters are known) model considering the uncertain environment, this paper develops a nonlinear controller that satisfies the required trajectory. This controller allows the nonlinear nominal system to track the desired trajectory exactly without linearizations or approximations. These continuous controllers compensate extra force to eliminate the errors caused by uncertainties. The controllers are based on a generalization of sliding surfaces. Error bounds on tracking caused by uncertainties are analytically obtained. The numerical results show the simplicity and efficacy of the proposed methodology, and the reliability of the error bounds

Safe and Fast Tracking on a Robot Manipulator: Robust MPC and Neural Network Control

Fast feedback control and safety guarantees are essential in modern robotics. We present an approach that achieves both by combining novel robust model predictive control (MPC) with function approximation via (deep) neural networks (NNs). The result is a new approach for complex tasks with nonlinear, uncertain, and constrained dynamics as are common in robotics. Specifically, we leverage recent results in MPC research to propose a new robust setpoint tracking MPC algorithm, which achieves reliable and safe tracking of a dynamic setpoint while guaranteeing stability and constraint satisfaction. The presented robust MPC scheme constitutes a one-layer approach that unifies the often separated planning and control layers, by directly computing the control command based on a reference and possibly obstacle positions. As a separate contribution, we show how the computation time of the MPC can be drastically reduced by approximating the MPC law with a NN controller. The NN is trained and validated from offline samples of the MPC, yielding statistical guarantees, and used in lieu thereof at run time. Our experiments on a state-of-the-art robot manipulator are the first to show that both the proposed robust and approximate MPC schemes scale to real-world robotic systems.Comment: 8 pages, 4 figures

Force Sensorless Admittance Control for Teleoperation of Uncertain Robot Manipulator Using Neural Networks

In this paper, a force sensorless control scheme based on neural networks (NNs) is developed for interaction between robot manipulators and human arms in physical collision. In this scheme, the trajectory is generated by using geometry vector method with Kinect sensor. To comply with the external torque from the environment, this paper presents a sensorless admittance control approach in joint space based on an observer approach, which is used to estimate external torques applied by the operator. To deal with the tracking problem of the uncertain manipulator, an adaptive controller combined with the radial basis function NN (RBFNN) is designed. The RBFNN is used to compensate for uncertainties in the system. In order to achieve the prescribed tracking precision, an error transformation algorithm is integrated into the controller. The Lyapunov functions are used to analyze the stability of the control system. The experiments on the Baxter robot are carried out to demonstrate the effectiveness and correctness of the proposed control scheme