Taiga Yamasaki

Swing-up control based on virtual composite links for n-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.

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.

Dynamic stability and phase resetting during biped gait

Dynamic stability during periodic biped gait in humans and in a humanoid robot is considered. Here gait systems of human neuromusculoskeletal system and a humanoid are simply modeled while keeping their mechanical properties plausible. We prescribe periodic gait trajectories in terms of joint angles of the models as a function of time. The equations of motion of the models are then constrained by one of the prescribed gait trajectories to obtain types of periodically forced nonlinear dynamical systems. Simulated gait of the models may or may not fall down during gait, since the constraints are made only for joint angles of limbs but not for the motion of the body trunk. The equations of motion can exhibit a limit cycle solution (or an oscillatory solution that can be considered as a limit cycle practically) for each selected gait trajectory, if an initial condition is set appropriately. We analyze the stability of the limit cycle in terms of Poincaré maps and the basin of attraction of the limit cycle in order to examine how the stability depends on the prescribed trajectory. Moreover, the phase resetting of gait rhythm in response to external force perturbation is modeled. Since we always prescribe a gait trajectory in this study, reacting gait trajectories during the phase resetting are also prescribed. We show that an optimally prescribed reacting gait trajectory with an appropriate amount of the phase resetting can increase the gait stability. Neural mechanisms for generation and modulation of the gait trajectories are discussed.

Optimality of a kip performance on the high bar: An example of skilled goal-directed whole-body movement

Characteristics of simple goal-directed tasks, such as hand reaching movements, have been well-studied from the perspective of optimization principles. However, it is still unclear what characteristics or control mechanisms of such movements are shared with more general movements. This paper focuses on a gymnastic maneuver on the high bar, referred to as the kip movement, a difficult goal-directed whole-body movement under a nonholonomic constraint. The kip movement of an expert gymnast was simply represented by a three-link planar model, and attempted to be reproduced under three optimization criteria: the minimum angle jerk criterion; the minimum torque change criterion that has been reported to explain the double-joint reaching movements of the hand; and the minimum effort criterion. Numerical analysis shows that (i) none of the criteria that assume only an initial point and a final point could reproduce a measured movement of the gymnast; however, (ii) the minimum torque change criterion that assumes the initial and final points, and an appropriate via-point could almost reproduce the measured movement, which is the best prediction among the three criteria with a via-point. The results may suggest that the hand reaching movements and the specific sections of the kip movement share a common characteristic roughly explained by the minimum torque change criterion. Moreover, by comparison of the solutions under the minimum torque change criterion with different via-points, a possible criterion for explaining the via-point allocation in a hierarchical manner is speculated to reduce the effort cost of the whole movement.

Swing-up control for n-link planar robot with single passive joint using the notion of virtual composite links

In this paper, we concern a swing-up control problem for an n-link revolute planar robot with any one of the joint being a passive joint. 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, first, we address how to iteratively devise two series of virtual composite links separated by the passive joint to be used for designing a coordinate transformation on the angles of all 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 control parameters that ensure attainment of the swing-up control objective. The results obtained here unify some previous results for the Pendubot, the Acrobot, and three-link robots with a passive first joint. Finally, we validate the theoretical results via a numerical simulation investigation to a 4-link robot with a passive joint.

Estimation of muscle activity using higher-order derivatives, static optimization, and forward-inverse dynamics

We propose a new method to estimate muscle activity in a straightforward manner with high accuracy and relatively small computational costs by using the external input of the joint angle and its first to fourth derivatives with respect to time. The method solves the inverse dynamics problem of the skeletal system, the forward dynamics problem of the muscular system, and the load-sharing problem of muscles as a static optimization of neural excitation signals. The external input including the higher-order derivatives is required for a calculation of constraints imposed on the load-sharing problem. The feasibility of the method is demonstrated by the simulation of a simple musculoskeletal model with a single joint. Moreover, the influences of the muscular dynamics, and the higher-order derivatives on the estimation of the muscle activity are demonstrated, showing the results when the time constants of the activation dynamics are very small, and the third and fourth derivatives of the external input are ignored, respectively. It is concluded that the method can have the potential to improve estimation accuracy of muscle activity of highly dynamic motions.

Revisiting energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint (shoulder). Different from previous energy-based control solutions, we obtain the following results: 1) 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 parameters 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. 2) We prove that the up-down equilibrium point is saddle (hyperbolic and unstable) 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. This paper prove 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. The simulation results are provided to validate these results.

Analysis of a gymnastic maneuver on the horizontal bar as a goal-directed motion under a nonholonomic constraint

This paper addresses motion generation of a basic gymnastic maneuver on the horizontal bar, referred to as the kip motion, as a difficult goal-directed movement under a nonholonomic constraint. First, we show the results of the motion measurement of the kip movements performed by two expert gymnasts, where the movements of each subject are very similar among trials. Second, we attempt to elucidate whether the kip motion of the expert gymnasts can be understood by the criteria proposed for explaining the hand reaching movements or not. To this end, the body of the expert gymnasts is modeled as a rigid 3-link with an unactuated joint at the hand and actuated joints at the shoulder and the hip, where the angle profiles of the actuated joints are approximated by fifth spline functions. Moreover, it is confirmed that the models motion is approximately consistent with the measured motion. By using the above model, the kip movements are generated with the minimum torque change criterion explaining the hand reaching movements, and motion segmentation assuming the initial, via- and final postures. The results of the numerical analysis suggest that the measured movements of the expert gymnasts could be almost reproduced in this way. Thus, it is suggested that motion determination/generation of the hand reaching and kip movements could be understood by the common criterion

Energy based Swing-up Control for a 3-Link Robot with Passive Last Joint: Design and Analysis

This paper addresses a swing-up control problem for a 3-link underactuated robot in the vertical plane, whose first and second joints are active (actuated) and the third (last) joint is passive (unactuated). The objectives of this paper are: 1) to design a control law under which the robot can be brought to any arbitrarily small neighborhood of the upright (up-up-up) equilibrium point, where all three links remain in their upright positions; 2) to attain a global analysis of the motion of the robot under the control law. By designing a coordinate transformation on the actuated joint variables of joints 1 and 2 and constructing a new Lyapunov function based on the transformation, this paper proposes an energy based swing-up control law. For any initial state of the robot, this paper provides a necessary and sufficient condition for non-existence of any singular point in the control law for all future time, and shows how to choose the control parameters such that the state of the robot eventually approach either any arbitrarily small neighborhood of the upright equilibrium point, or the up-up-down equilibrium point, where the links 1, 2 and 3 are in upright, upright and downward positions, respectively. Moreover, this paper shows that the up-up-down equilibrium point is unstable

Trajectory Tracking Control of Pendulum with Variable Length by Partial Energy Shaping

Abstract This paper concerns a trajectory tracking control problem for a pendulum with variable length, which is an underactuated mechanical system of two degrees-of-freedom with a single input of adjusting the length of the pendulum. We aim to study whether it is possible to design a time-invariant control law to pump appropriate energy into the variable-length pendulum for achieving a desired swing motion (trajectory) with given desired energy and length of the pendulum. First, we show that it is difficult to avoid singular points in the controller designed by using the conventional energy-based control approach in which the total mechanical energy of the system is controlled. Second, we propose a tracking controller free of singular points by shaping only the kinetic energy of rotation and the potential energy of the pendulum and not utilizing the kinetic energy related to the motion along the rod. Third, we analyze globally the motion of the pendulum and clarify the stability issue of some closed-loop equilibrium points; and we also provide some conditions on control parameters for achieving the tracking objective. Finally, we present numerical simulation results to validate the presented theoretical results.