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.
Figures & tables
Fig. 1 : Platforms that benefit from guiding vector fields on SE(3) . Top left: a Voliro omnidirectional drone (image from https://voliro.com ). Top right: the octorotor design from [ 2 ] . Bottom left: an omnidirectional underwater vehicle [ 4 ] . Bottom right: a Kinova Gen3 manipulator (image from https://www.kinovarobotics.com ). All these systems accept six-dimensional velocity commands (twists).
Fig. 5 : The solid cyan line depicts the target curve, with target orientation frames shown by RGB axes. Intermediary configurations are shown as translucent representations, while the final configuration for each time period is opaque. The end effector orientation frame is shown by RGB axes.