Energy-based Swing-up Controller Research Articles

Global analysis of energy-based swing-up control for soft robot

In this paper, we explore the energy-based swing-up control for a soft robot equipped with an actuated constant curvature soft pendulum and an unactuated rotational base joint. The aim is to swing the robot up towards its upright equilibrium point (UEP) with the pendulum at an upright position. We establish a necessary and sufficient condition to protect the control law from singularities and demonstrate the robot’s potential motion towards a homoclinic orbit or a closed-loop equilibrium point. After examining the robot’s closed-loop equilibrium points, we generate formulae to compute all such points and introduce two conditions concerning control parameters, eliminating all but the UEP and downward equilibrium point (DEP) with the robot in a downward position. We prove the instability of the equilibrium points with negative gravitational potential energy. This obviates the need for one of the two conditions concerning such equilibrium points. We also prove the robot’s linear controllability at the UEP. Our obtained results reveal that, regardless of its initial state, the soft robot can be swung-up towards its UEP using the energy-based controller, provided that it meets the proposed control parameter conditions, and can subsequently be balanced around the UEP using a locally stabilizing controller. Our simulation investigations validate the presented theoretical results.

A signal compensation-based balance control for the rotary inverted pendulum system

In this paper, we study the balance control for the rotary inverted pendulum system with two joints both subject to uncertainty. To facilitate the experiment, we apply a classical energy-based swing-up control, eschewing compensation for the uncertainty. In balance control, we employ a novel approach in the form of a modified signal compensation-based balance control, which models the uncertainty as its previously sampled term and its variation. The signal compensation-based balance control involves three steps. First, for the linearized discrete-time state-space model around the upright equilibrium point, we design the main control signal by using the pole-placement technique. Second, we estimate the discrete-time model of the system by using the least squares identification method, thereby unveiling key points in the identification procedure. The discrete-time model presents the uncertainty as a previously sampled term and its variation. Third, we design two discrete-time compensation signals for the previously sampled term and the variation. By combining the main control signal and the two compensation signals, we formulate the balance control law. We further optimize the signal compensation-based balance control by adding an extra parameter, successfully circumventing the critical stability problem, and analyze the performance of the closed-loop system. Numerical simulation and comparative experiments demonstrate its effectiveness and strong robustness against the uncertainty in frictions and modeling errors of the system.

Energy-based swing-up control for a two-link underactuated robot with flexible first joint

This paper concerns the swing-up control of a two-link robot moving in a vertical plane, which has a single actuator at the second joint and a linear torsional spring at the first joint. First, we present a necessary and sufficient condition such that the robot is linearly controllable at the upright equilibrium point (UEP, where two links are both upright). Second, we prove without any assumption that the robot is at an equilibrium point provided that its actuated joint angle is constant under a constant torque. Third, for the robot with its torsional stiffness of the spring being not greater than a value determined by the coefficients of its gravitational terms, we propose an energy-based swing-up controller without singular points. We conduct a global motion analysis for the robot under the proposed controller. For the case that the total mechanical energy of the robot converges to its desired value, we present the phase portrait of the closed-loop solution. For the case that the convergence is not achieved, we show that the closed-loop solution approaches an equilibrium point belonging to a set of equilibrium points, and give a sufficient condition to check its instability. From the motion analysis, we present a sufficient condition such that the robot can be swung-up close to the UEP under the proposed swing-up controller. Finally, we verify our theoretical results through a numerical simulation.

New analytical results of energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control solutions for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint. The control objective is to swing the Pendubot up to its unstable equilibrium point (at which two links are in the upright position). We improve the previous energy-based control solutions by analyzing the motion of the Pendubot further. Our main contributions are threefold. First, we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameter such that the up–down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. Second, we prove that the up–down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We show that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. Third, we clarify the relationship between the swing-up controller designed via the partial feedback linearization and that designed by the energy-based approach. We present the simulation results for validation of these results.

Energy-Based Swing-Up Control for a Remotely Driven Acrobot: Theoretical and Experimental Results

This brief concerns the energy-based swing-up control for a remotely driven acrobot (RDA) which is a 2-link planar robot with the first link being underactuated and the second link being remotely driven by an actuator mounted at a fixed base through a belt. An energy-based swing-up controller is designed via the Lyapunov stability theory. A global motion analysis of the RDA under the designed controller is provided focusing on the behavior of the closed-loop solution and the stability of the closed-loop equilibrium points. The conditions on control parameters for achieving a successful swing-up control are given. Furthermore, an experimental setup is described and experimental results are given to validate the presented theoretical results. The energy-based swing-up controller for the RDA is shown to be effective both theoretically and practically.

New Results of Energy-Based Swing-up Control for a Rotational Pendulum

In this paper, we revisit the energy-based swing-up control problem for a rotational pendulum. Different from the existing energy-based control solution, first, we present a necessary and sufficient condition for avoiding singular points in the control law and carry out a global motion analysis of the pendulum. Next, we remove the previous required constraint on the initial state of the pendulum and the control parameters for preventing the pendulum getting stuck at the downward equilibrium point by revealing that the point is saddle (hyperbolic and unstable). Specifically, we show that the Jacobian matrix evaluated at the point has two and two eigenvalues in the open leftand right-half planes, respectively. We prove that the pendulum will eventually be swung up into the basin of attraction of any locally stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero. Finally, we validate the presented theoretical results via numerical simulation of two rotational pendulums. Our simulation results show that the swing-up control can be achieved quickly under the improved conditions on the control parameters.

New Results of Energy-Based Swing-Up Control for Rotational Pendulum

AbstractIn this paper, we revisit the energy-based swing-up control problem for a rotational pendulum. Different from the existing energy-based control solution, first, we present a necessary and sufficient condition for avoiding singular points in the control law and carry out a global motion analysis of the pendulum. Next, we remove the previous required constraint on the initial state of the pendulum and the control parameters for preventing the pendulum getting stuck at the downward equilibrium point by revealing that the point is saddle (hyperbolic and unstable). Specifically, we show that the Jacobian matrix evaluated at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We prove that the pendulum will eventually be swung up into the basin of attraction of any locally stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero. Finally, we validate the presented theoretical results via numerical simulation of two rotational pendulums. Our simulation results show that the swing-up control can be achieved quickly under the improved conditions on the control parameters.

Swing-up control based on virtual composite links for [formula omitted]-link underactuated robot with passive first joint

This paper concerns the swing-up control of an n -link revolute robot moving in the vertical plane with the first joint being passive and the others being active. The goal of this study is to design and analyze a swing-up controller that can bring the robot into any arbitrarily small neighborhood of the upright equilibrium point with all links in the upright position. To achieve this challenging objective while preventing the robot from becoming stuck at an undesired closed-loop equilibrium point, we first address the problem of how to iteratively devise a series of virtual composite links to be used for designing a coordinate transformation on the angles of all the active joints. Second, we devise an energy-based swing-up controller that uses a new Lyapunov function based on that transformation. Third, we analyze the global motion of the robot under the controller and establish conditions on the control parameters that ensure attainment of the swing-up control objective; specifically, we determine the relationship between the closed-loop equilibrium points and a control parameter. Finally, we verify the theoretical results by means of simulations on a 4-link model of a gymnast on the high bar. This study not only unifies some previous results for acrobots and three-link robots with a passive first joint, but also provides insight into the energy- and passivity-based control of underactuated multiple-degree-of-freedom systems.

Swing-Up Control for a 3-DOF Gymnastic Robot With Passive First Joint: Design and Analysis

This paper concerns a swing-up control problem for a three-link gymnastic planar robot in a vertical plane with its first joint being passive (unactuated) and the rest being active (actuated). The objectives of this paper are to: (1) design a controller under which the robot can be brought into any arbitrarily small neighborhood of the upright equilibrium point, where all three links of the robot remain in their upright positions; and (2) attain a global analysis of the motion of the robot under the controller. To tailor the energy-based control approach to achieve the aforementioned objectives, first, this paper considers the links 2 and 3 as a virtually composite link, and proposes a coordinate transformation of the angles of active joints. Second, this paper constructs a novel Lyapunov function based on the transformation, and devises an energy-based swing-up controller. Third, this paper carries out a global analysis of the motion of the robot under the controller, and establishes some conditions on control parameters for achieving the swing-up control objective. To validate the theoretical results obtained, this paper provides simulation results for a three-link robot with its mechanical parameters being obtained from a human gymnast.