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Controlling the network of underactuated Euler–Lagrange (EL) systems is challenging because of their coupled inertia matrices and time-variant control input matrices. We present generalized multi-coordinates transformation that renders the network of underactuated Euler-Lagrange dynamics in particular forms, whose mechanical properties should be preserved. The network of N nonidentical Euler–Lagrange EL-systems is modeled as a weighted interconnection graph where each EL-system is a node, and the control action at each node is a function of its state and the states of its neighbors. Second, we propose an online optimally centralized control mechanism with the prime objective of energy efficiency. The result is applied to the network of underactuated vertical takeoff and landing aircraft with strong input coupling, including the effect of the weight of the rotors into the dynamical system models. In this regard, we obtain very simple and powerful state-feedback solutions.
Steering large-scale particle or robot systems is challenging because of their high dimensionality. We use a centralized stochastic approach that allows for optimal control at the cost of a central element instead of a decentralized approach. Previous works are often restricted to the assumption of fully actuated robots. Here we propose an approach for underactuated robots that allows for energy-efficient control of the robot system. We consider a simple task of gathering the robots (minimizing positional variance) and steering them towards a goal point within a bounded area without obstacles. We make two main contributions. First, we present a generalized coordinate transformation for underactuated robots, whose physical properties should be considered. We choose Euler-Lagrange systems that describe a large class of robot systems. Second, we propose an optimal control mechanism with the prime objective of energy efficiency. We show the feasibility of our approach in robot simulations.
In this paper, we propose a procedure to solve the controlled design for a class of underactuated mechanical systems. Our proposed method can be viewed as a sub-method of the IDA-PBC or Controlled Lagrangian approaches, with a particular focus on shaping the potential energy. By emphasizing potential energy shaping, we can effectively tackle the bottleneck presented by the matching equation in these approaches. Moreover, our method leverages a suitable coordinate transformation that is inspired by the physics of the system, further enhancing its efficacy. Therefore, our design procedure is based on a coordinate transformation plus potential energy shaping in the new coordinates, and its existence and possibility of potential energy shaping can be verified via some algebraic calculations, making it constructive. To illustrate the results, we consider the cart-pole system and a recently introduced underactuated mechanical system named swash mass pendulum (SMP) [1]. The SMP consists of a pendulum made of a rigid shaft connected to a pair of cross-shafts where two swash masses can move under the action of servo-mechanisms.