Degree Of Underactuation Research Articles

Motion planning in underactuated systems with impulsive phenomenon via dynamic shaping of virtual holonomic constraints

Rhythmic motions are traditionally achieved by developing predetermined paths for the states of the space system to follow. Since these paths are obtained offline, the dynamic behavior fails to adapt to changes in environmental conditions or user command desires. The solution we propose is a new strategy called dynamic shaping, in which the paths are formed online to allow the system to create an orbit with the characteristics we need. Hereupon, this paper focuses on applying this strategy to Dynamics with One Degree of Under-actuation and Impulsive Phenomenon (DSODUIP) to adapt the characteristics of outcomes to be in line with the demands.This research was conducted by leveraging the advantages of virtual holonomic constraints (VHCs) to establish these paths. Therefore, a novel two-level hierarchical control method is designed considering a stability criterion to avoid divergence. At the Low-Level, the controllers stabilize the output of system to follow the VHCs on the system. At the High-Level, the VHCs are modified to shape an orbit with our desired characteristic in the motion. As an illustrative example, the algorithm is implemented to adjust the average angular velocity of a devil stick and the hip velocity of a Three-Link biped robot. Their results vividly demonstrate smooth adjustments and efficient performance in achieving our desired outcomes.

Deep Reinforcement Learning-Based Control for Asynchronous Motor-Actuated Triple Pendulum Crane Systems With Distributed Mass Payloads

For transporting large cargoes in practice, the triple pendulum hoisting pattern is widely utilized, where the distributed mass payload (DMP) is suspended under the hanger, the hanger is suspended on the hook, and the hook is connected with the hoisting mechanism. However, existing studies mainly focus on single pendulum and double pendulum hoisting patterns, and few of them pay attention to triple pendulum hoisting patterns. Compared with single pendulum and double pendulum cranes, triple pendulum cranes, which control 4 degrees of freedom (DoFs) with merely 1 control input, have a higher underactuation degree, with higher control difficulties. In addition, the nonlinear characteristics of alternating current (AC) asynchronous motors, such as unknown nonlinear dead zones and time delays, will degrade tracking accuracy and control performance. To solve the above problems, a model-free deep reinforcement learning-based trajectory planning method is proposed for triple pendulum DMP cranes. Furthermore, an identification model for AC asynchronous motors is established. On this basis, a trajectory tracking method including an adaptive dead zone compensation (ADC) and a modified Smith predictor (MSP) is proposed. Finally, experiments are carried out on a self-built 2-ton crane experimental platform, which is actuated by AC asynchronous motors. Hardware experiments demonstrate that triple pendulum cranes can accurately track the planned trajectory, and the DMP residual swing is effectively suppressed.

Control system design for a class of non-cascade nonlinear under-actuated systems with an application to a rotary crane system

For cascade under-actuated systems, several effective control methods have been established. For non-cascade under-actuated systems with one under-actuation degree, several control methods, such as cascade transformation and energy-based control, have been studied. For non-cascade nonlinear under-actuated systems with two or more under-actuation degrees, approximate linearization is necessary to apply these methods, thus only stability at the equilibrium point has been guaranteed. In this paper, control system design and stability analysis are presented for a class of non-cascade nonlinear under-actuated systems with two or more under-actuated degrees. The stability condition obtained in the analysis is a conservative sufficient condition because norm estimation is used. Based on the sufficient condition, we propose a stabilization parameter search algorithm to numerically obtain the practical control parameters. The proposed algorithm is applied to a rotary crane system as one of the nonlinear systems with two under-actuated degrees. The effectiveness of the proposed control system design is verified by numerical simulations and experiments.

Switched Energy and Neural Network-Based Approach for Swing-Up and Tracking Control of Double Inverted Pendulum

The double inverted pendulum (DIP) system is a benchmark underactuated mechanical system. The control problem of the DIP system is challenging since it not only has high nonlinearity, but also has underactuation degree equal to two. In this article, a switched energy-based swing-up and neural network (NN) controller is proposed to solve the swing-up and tracking control problem of the DIP system when the desired position is time varying. The implementation of the proposed controller can be divided into three steps. In step one, the energy-based control method is adopted to drive the first pendulum from the downward position to the neighborhood of the upright position. In step two, the sliding mode control method is adopted to stabilize the first pendulum. Meanwhile, a method combining energy-based control and “equivalent cart” is adopted to swing up the second pendulum. In step three, on the basis of approximate nonlinear output regulation theory, the NN controller is used for achieving satisfactory position tracking performance. Finally, the effectiveness of our design is verified by experimental results.

Stabilization of a class of underactuated parallel robots via energy shaping: Application to cable driven manipulators

In this brief paper, potential energy shaping of underactuated parallel robots (UPRs) based on the method called interconnection and damping assignment passivity-based control (IDA-PBC) is presented. To this aim, to assign a new Hamiltonian to the closed-loop system, due to fewer actuators than degrees of freedom, a set of partial differential equations (PDEs) related to assigning a desired potential energy should be solved analytically, which are basically the stumbling block of IDA-PBC approach. Although in UPRs, the input mapping matrix is complex and configuration-dependent, a general solution for the potential energy PDE is derived such that it is independent of degrees of underactuation.

Robust output-feedback orbital stabilization for underactuated mechanical systems via high-order sliding modes

The manuscript deals with the robust orbital stabilization for a class of disturbed Euler–Lagrange systems with one degree of underactuation. The proposed strategy relies on the virtual holonomic constraints approach, using incomplete state measurements. First, a high-order sliding-mode extended observer estimates the state and the disturbances affecting the input channel. Then, proposing a new set of coordinates, the so-called virtual holonomic constraints, a robust output partial-feedback linearization approach takes the system into a double integrator with a particular zero dynamics. Thus, considering the general integral of motion of the zero dynamics, the orbital stabilization is reduced to stabilize a linear time-varying system. The resulting control law is a continuous signal. Therefore, the robustness to disturbances is addressed without the tarnishing effects of chattering. The closed-loop stability analysis is done using the Lyapunov theory. The feasibility of the method is illustrated experimentally in a cart-pendulum system.

Experimental Safety Analysis of R-Min, an Underactuated Parallel Robot

AbstractThe R-Min robot is an intrinsically safe parallel manipulator dedicated to pick-and-place operations. The proposed architecture is based on a five-bar mechanism, with additional passive joints in order to obtain a planar seven-bar mechanism with two degrees of underactuation, allowing the robot to reconfigure in case of a collision. A preload bar is added between the base and the end-effector to constrain the additional degrees-of-freedom. This article presents an analysis of the workspace and of the safety performances of the R-Min robot, and it compares them with those of the five-bar mechanism, in order to evaluate the benefits of introducing underactuation in a parallel architecture to obtain intrinsically safer robots. The geometrico-static model of the R-Min robot is formulated as an optimization problem. The direct and inverse kinemato-static models are derived from the geometrico-static model and they allow to express the singularity conditions of the R-Min robot. An analysis of the singularity loci is carried out among the robot’s workspace. A controller based on the dynamic model is proposed and experimentally validated on a prototype of the R-Min robot. Finally, the safety performances of the R-Min robot are evaluated experimentally and they are compared with that of an equivalent five-bar mechanism, using the maximum impact force as a safety criteria in accordance with recent international standards.

Virtual constraint generators for motion control of robots with degree of underactuation one

Virtual holonomic constraints (vhcs) are relations among the configuration variables of a mechanical control system that can be rendered invariant via feedback. A regular vhc is a vhc with the property that the accelerations imparted by the control forces are everywhere transverse to the constraint surface. This paper proposes a methodology for the design of regular vhcs for mechanical systems with degree of underactuation one based on the notion of a virtual constraint generator, a control system on the configuration manifold of the mechanical system whose orbits are all the regular vhcs for the mechanical system. Designing a regular vhc corresponds to designing controllers for the virtual constraint generator. We show how to design regular vhcs inducing two kinds of motions for the constrained dynamics: traversal of the vhc in one direction or oscillation back and forth along the vhc. We propose an optimal control problem for the virtual constraint generator whose solutions are regular vhcs inducing the kind of motions just described and, at the same time, meeting geometric requirements such as obstacle avoidance. We demonstrate the utility of our technique by designing a constraint for an overhead crane making the trolley move from left to right with bounded speed while ensuring that the cable angle remains within fixed limits, and a constraint for a cart–acrobot making the cart oscillate back and forth underneath an obstacle placed overhead while ensuring that the acrobot avoids the obstacle and remains as close as possible to being in the upright position.

Orbital stabilization of point-to-point maneuvers in underactuated mechanical systems

The task of inducing, via continuous static state-feedback control, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system. Namely, to a smooth curve in its state–control space which is consistent with the system dynamics and connects two (linearly) stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver as to combine two linearization techniques for this purpose: the Jacobian linearization at the equilibria on the boundaries and a transverse linearization along the orbit. This allows for the computation of stabilizing control gains offline by solving a semidefinite programming problem. The resulting nonlinear controller, which simultaneously asymptotically stabilizes both the orbit and the final equilibrium, is time-invariant, locally Lipschitz continuous, requires no switching, and has a familiar feedforward plus feedback–like structure. The method is also complemented by synchronization function–based arguments for planning such maneuvers for mechanical systems with one degree of underactuation. Numerical simulations of the non-prehensile manipulation task of a ball rolling between two points upon the “butterfly” robot demonstrates the efficacy of the synthesis.

A Constructive Procedure for Orbital Stabilization of a Class of Underactuated Mechanical Systems

In this note, we present a constructive procedure to solve the orbital stabilization problem of a class of nonlinear underactuated mechanical systems with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> degrees of freedom and underactuation degree one using the Immersion and Invariance technique. We define sufficient conditions to solve explicitly the partial differential equations arising in the Immersion and Invariance methodology, so that mechanical systems with gyroscopic terms can be considered. At the end, we use two practical systems as examples to illustrate the design procedure, as well as validating performance via simulations and experiments.

A new regulation control method for underactuated 3D overhead cranes

Interconnection and damping assignment passivity-based control (IDA-PBC) has been successfully applied to many mechanical, electrical, and electromechanical systems. For the control of such systems, existing IDA-PBC methods have concentrated on systems with underactuation degree one, such as the ball and beam system, the vertical takeoff and landing aircraft, and the pendulum on a cart. In this paper, we focus on the application of the IDA-PBC methodology to the regulation control of a three-dimensional (3D) overhead crane system with underactuation degree two. The goal of this study is to design and analyze a regulation controller that can drive the cart to the desired position precisely and eliminate the payload swing effectively. To achieve this challenging objective, we first find an ingenious way to solve partial differential equations for constructing a Lyapunov function candidate. Then, we devise a nonlinear controller on the basis of the constructed Lyapunov function and analyze the stability of the closed-loop system by using Lyapunov techniques and LaSalle’s invariance principle. Finally, both simulation and experimental results demonstrate the proposed strategy can achieve excellent positioning accuracy and significant swing elimination, and a comparison study between the proposed method and the existing PBC method is included as well.

Dynamic surface control with a nonlinear disturbance observer for multi‐degree of freedom underactuated mechanical systems

AbstractUnderactuated systems are extensively utilized in practice while attracting a huge deal of attention in theoretical studies. There are few robust control strategies for general underactuated systems because of the variety of their dynamic models. A dynamic surface control strategy with a nonlinear disturbance observer is proposed in this study, to stabilize multi‐degree of freedom underactuated systems. In such systems the number of underactuated degrees of freedom is not higher compared to the actuated ones. A disturbance observer is utilized to dispose of the uncertain disturbance and cross terms in dynamic model which may cause failure to the controller. Then, a dynamic surface control strategy is presented which is not sensitive to the diversity of dynamic models. The stability of whole system is proven by Lyapunov‐based method. The control law is successfully applied to nonlinear underactuated systems in benchmark cascade form such as two‐translational oscillator with rotational actuator, crane, wheeled inverted pendulum. The effectiveness of proposed controllers are illustrated by MATLAB simulation results. Finally, comparative studies are presented to verify the superiority of the proposed method.

A matrix algorithm based on controlled Lagrangians for stabilizing mechanical systems with underactuation degree one

A matrix algorithm based on controlled Lagrangians for stabilizing mechanical systems with underactuation degree one

Stabilization control of quadrotor helicopter through matching solution by controlled Lagrangian method

AbstractThis paper implements the controlled Lagrangian method on nonlinear quadrotor system with two degrees of underactuation. A Lagrangian system based on virtual angles is developed for the quadrotor. Based on the model, the matching conditions presented by nonlinear partial differential equations are explicitly solved for the nonlinear quadrotor system without decoupling or division. Besides, it is demonstrated that a constant controlled kinetic matrix could be utilized for energy shaping and simplify the matching conditions since gyroscopic force terms are omitted. Smooth state feedback control laws guaranteeing asymptotic stability of quadrotor system are obtained. Stability analysis and range of the control parameters are deduced based on the reshaped closed‐loop energy. Simulation for a quadrotor helicopter shows the feasibility and efficiency of the theoretical results.

Orbital synchronization of homogeneous mechanical systems with one degree of underactuation

AbstractA methodology for asymptotic orbital synchronization of a set of homogeneous mechanical systems with one degree of underactuation is proposed. A reference model generating a reference orbit is first constructed under the virtual holonomic constraints approach. Then, the synthesis is designed to, first, estimate the state vector of the homogeneous systems, provided that only position measurements are available, and second, to drive the state of the systems to that of the reference model. Due to the synthesis properties, the homogeneous systems converge asymptotically to the reference orbit when there are no disturbances affecting the systems nor the measurements. When disturbances are present, robustness is guaranteed provided that the gain of the closed‐loop systems remains lower than an attenuation level . Numerical results for a set of underactuated cart‐pendulums corroborate the proposed methodology.

Research on nonlinear coupled tracking controller for double pendulum gantry cranes with load hoisting/lowering

Gantry cranes have attracted extensive attention that are mostly simplified as nonlinear single pendulum systems without load hoisting/lowering. However, due to the existence of the hook in practice, gantry cranes produce double pendulum effect. With an extra underactuated degree of freedom, the anti-swing control of double pendulum gantry cranes becomes more difficult than that of single pendulum gantry cranes. Moreover, double pendulum gantry cranes with load hoisting/lowering may cause large swings, which lead to inaccurate positioning and low transportation efficiency. In this paper, a novel nonlinear coupled tracking anti-swing controller is proposed to solve these problems. The proposed controller can ensure the stable startup and operation of the trolley by introducing a smooth expected trajectory. In addition, a composite signal is constructed to suppress and eliminate the swing angles of the gantry crane system. The system stability is analyzed by utilizing Lyapunov techniques and Barbalat’s lemma. Theoretical derivation, simulation and experimental results indicate that the proposed controller suppresses and eliminates the hook/load swing angle effectively. Furthermore, it can achieve superior control effects and strong robustness against the changes of the load mass, trolley target displacement, initial rope lengths, initial system swing angles and external disturbances.

Mode-Reactive Template-Based Control in Planar Legged Robots

Translating the center of mass (CoM) while fixing the orientation of a rigid body supported by relatively massless, actuated limbs is a common problem setting in legged robotics. This paper proposes a hierarchical approach to such maneuvers that decouples CoM task planning from body orientation control in sagittal-plane models, thereby exposing a well-studied and computationally effective low dimensional dynamical system that can be used for CoM task planning. The resulting algorithms directly address the control authority (degree of underactuation) available at a given contact mode, enabling a-priori plans with these intuitive, robust pendular dynamics to be formally embedded at the (virtual) CoM of planar floating-torso models, while focusing high gain posture stabilizing feedback upon the body orientation. A series of numerical and empirical examples address single- and multi-legged leaping — transitional maneuvers where only a single brief stance mode is available to load energy into the CoM and guide its direction. We compare our hierarchical method to a model-based model-predictive controller in one of the tasks, demonstrating similar performance with a significantly smaller computational footprint.

On the matching equations of kinetic energy shaping in IDA-PBC

Interconnection and damping assignment passivity-based control scheme has been used to stabilize many physical systems such as underactuated mechanical systems through total energy shaping. In this method, some partial differential equations (PDEs) related to kinetic and potential energy shaping shall be solved analytically. Finding a suitable desired inertia matrix as the solution of nonlinear PDEs relevant to kinetic energy shaping is a challenging problem. In this paper, a systematic approach to solving this matching equation for systems with one degree of underactuation is proposed. A special structure for desired inertia matrix is proposed to simplify the solution of the corresponding PDE. It is shown that the proposed method is more general than that of some reported methods in the literature. In order to derive a suitable desired inertia matrix, a necessary condition is also derived. The proposed method is applied to three examples, including pendubot, VTOL aircraft, and 2D SpiderCrane.

Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

&lt;p style='text-indent:20px;'&gt;In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the &lt;i&gt;kinetic&lt;/i&gt; and &lt;i&gt;potential&lt;/i&gt; equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ C^{\infty} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.&lt;/p&gt;

One-Step Deadbeat Control of a 5-Link Biped Using Data-Driven Nonlinear Approximation of the Step-to-Step Dynamics

For bipedal robots to walk over complex and constrained environments (e.g., narrow walkways, stepping stones), they have to meet precise control objectives of speed and foot placement at every single step. This control that achieves the objectives precisely at every step is known as one-step deadbeat control. The high dimensionality of bipedal systems and the under-actuation (number of joint exceeds the actuators) presents a formidable computational challenge to achieve real-time control. In this paper, we present a computationally efficient method for one-step deadbeat control and demonstrate it on a 5-link planar bipedal model with 1 degree of under-actuation. Our method uses computed torque control using the 4 actuated degrees of freedom to decouple and reduce the dimensionality of the stance phase dynamics to a single degree of freedom. This simplification ensures that the step-to-step dynamics are a single equation. Then using Monte Carlo sampling, we generate data for approximating the step-to-step dynamics followed by curve fitting using a control affine model and a Gaussian process error model. We use the control affine model to compute control inputs using feedback linearization and fine tune these using iterative learning control using the Gaussian process error enabling one-step deadbeat control. We demonstrate the approach in simulation in scenarios involving stabilization against perturbations, following a changing velocity reference, and precise foot placement. We conclude that computed torque control-based model reduction and sampling-based approximation of the step-to-step dynamics provides a computationally efficient approach for real-time one-step deadbeat control of complex bipedal systems.