Vinicius M. Gonçalves

Felipe Bartelt, Luciano C. A. Pimenta, Weijia Yao et al.

Many robotic systems allow independent control of position and orientation (pose), including omnidirectional aerial vehicles, underwater robots, and manipulator end-effectors. In many applications, these systems must follow a continuous sequence of poses, leading to either trajectory-tracking or path following formulations. Compared to trajectory-tracking, path following offers important practical advantages. In particular, we focus on the problem of path following on Lie groups. Considering the robots as rigid bodies moving in the 3D space, this path-following problem can be posed as a problem of designing guiding vector fields on the matrix Lie group SE(3). In this paper, we develop a general vector-field framework for path following on connected matrix Lie groups, of which SE(3) is a prominent special case. The proposed vector field guarantees convergence to a desired parametric curve from almost all initial conditions while ensuring continuous motion along the path. Furthermore, another interesting feature is that, as opposed to previous works, the control input is "minimal" in terms of representation and closer to the engineering application (e.g., the body twist in the case SE(3)). After establishing the general case, the framework is then specialized to SE(3), of special interest in robotics, yielding an efficient algorithm suitable for real-time robotic control. Experiments with a robotic manipulator tracking complex pose paths demonstrate the effectiveness of the approach. An open-source implementation is also provided.

Vinicius M. Gonçalves, João Baião, Felipe Bartelt et al.

Vector-field-based methods are widely used for robot control and are often applied to the path-tracking problem. Some vector field approaches require repeatedly computing the distance between the robot configuration and the curve, as well as the corresponding closest point. Recently, vector fields have been extended to Lie Groups. In this case, this computation can be expensive, especially when performed at high control frequencies on embedded platforms. This paper proposes a method for efficiently computing the distance between a point and a curve represented as what is called a G-polynomial curve, which is a curve representation that generalizes polynomial curves to matrix Lie groups. The proposed approach exploits the structure of these curves to reduce the problem to a small number of polynomial root-finding computations. Simulation results show that the method significantly reduces computation time while maintaining accuracy compared to existing optimization-based approaches. Practical formulas are also provided for the case of the group SE(3), and the method is validated experimentally on a robotic manipulator. The methodology is implemented in a computational package, available online.

Brener G. Ferreira, Vinicius M. Gonçalves, Marcelo A. Santos et al.

This paper proposes a finite-horizon optimal control strategy for set-point tracking using a nonlinear model predictive control framework with integrated avoidance capabilities. The formulation employs a smooth point-to-cloud distance metric that ensures continuously differentiable and numerically well-conditioned gradients, even in the presence of regions with complex and nonconvex geometries. This smoothness allows safety constraints to be formulated consistently and differentiably through control barrier functions, resulting in a reliable avoidance behavior for the closed-loop system. Additionally, stationary artificial variables are introduced in the optimal control problem to preserve feasibility under changing set-points. The proposed approach is validated through numerical experiments of an aerial robot, demonstrating accurate tracking and smooth obstacle avoidance in complex environments.