Second-order Nonholonomic Constraints Research Articles

Position-posture Control Strategy for Planar Underactuated Manipulators with Second-order Nonholonomic Constraint

Position-posture Control Strategy for Planar Underactuated Manipulators with Second-order Nonholonomic Constraint

Optimal Reference Trajectory for a Type Xn-1Rp Underactuated Manipulator

The aim of the present research is to find an optimal reference trajectory for an underactuated manipulator of type Xn-1Rp, where X is any type of joints and R is the last rotary joint, for n≥3. It is worth noting that in the case of absence of control of fully actuated manipulator, some second-order nonholonomic constraints may appear; these are known as acceleration constraints. The second-order nonholonomic constraint is a non-integrable differential equation. For this purpose, it was decided to combine two methods. The first one provides the open-loop control of the manipulator whatever the motion time is; in practice, the motion time should be minimal under the given geometric, technological, and dynamic constraints. To address this issue, a second method, based on the offline optimization approach, was used to achieve the time-optimal motion. It was revealed that the above combination gives an optimal control trajectory for an underactuated manipulator in which a reference trajectory can be utilized.

A General Position Control Method for Planar Underactuated Manipulators With Second-Order Nonholonomic Constraints.

The study on the stabilization of planar underactuated manipulators without gravity is well recognized as a major challenge since the system includes a second-order nonholonomic constraint when the passive link is not located at the first link. It is important to solve this difficulty for applications such as systems working in aerospace or underwater. This article presents a position control method based on bidirectional motion planning and intelligent optimization for this kind of system. The control objective is to move the end effector of the manipulator from its initial position to a given target position. The differential evolution algorithm is applied to solve the target angles of all links corresponding to the target position of the end effector. Then, bidirectional motion planning is performed, which consists of forward and backward motions. Each motion is planned by designing a trajectory for every active link based on their initial and target angles. During the forward motion, all active links except the first one are moved to their target angles, and the first active link and the passive link to the intermediate angles. For the backward motion, the first active link is moved to its target angle, the other active links remain at their target angles, and the passive link will be moved to its target angle at the same time. The planned trajectories are chosen based on the time-scaling method and differential evolution algorithm to make sure that the forward and backward planned motions can be connected smoothly. Finally, the trajectory tracking controllers for all active links are designed based on the sliding-mode control method. The proposed control method is verified on planar four-link underactuated manipulators with different passive joints. This strategy has the advantage that it works for planar underactuated manipulators with a second-order nonholonomic constraint whose passive link can be at different positions. Meanwhile, by combining intelligent optimization with bidirectional motion planning, the control process becomes simpler and more effective.

Simulation of motion of satellites after fixing the values of their accelerations

At the Department of Theoretical and Applied Mechanics of the Faculty of Mathematics and Mechanics of St. Petersburg State University [1] the theory of motion of nonholonomic systems with linear nonholonomic constraints of high order n<2 was created. The high-order constraints are considered as program and ideal ones, and their reaction force is considered as the required control force. A consistent system of differential equations with respect to unknown generalized coordinates and Lagrange multipliers is constructed to solve the problem. The report examines the motion of Soviet satellites of the systems “Cosmos”, “Molniya”, “Tundra” after fixing the values of their accelerations in apogees. This corresponds to imposing the nonlinear second-order nonholonomic constraints on the further motion of satellites [2, 3]. The equations of constraints are differentiated in time and presented as linear third-order constraints to make it possible to apply the above theory. The motions of satellites are studied in polar coordinates, the origin of this system coinciding with the center of the Earth. It turns out that after fixing the acceleration values in the apogees, the satellites begin to rotate between two concentric circles, alternately touching each of them.

Kinematic and Dynamic Characteristics of the Free-Floating Space Manipulator with Free-Swinging Joint Failure

For the free-floating space manipulator with free-swinging joint failure, motions among its active joints, passive joints, free-floating base, and end-effector are coupled. It is significant to make clear all motion coupling relationships, which are defined as “kinematic coupling relationships” and “dynamic coupling relationships,” inside the system. With the help of conservation of system momentum, the kinematic model is established, and velocity mapping relation between active joints and passive joints, velocity mapping relation between active joints and base, velocity mapping relation between active joints and end-effector. We establish the dynamic model based on the Lagrange equation, and the system inertia matrix is partitioned according to the distribution of active joints, passive joints, and the base. Then, kinematic and dynamic coupling relationships are explicitly derived, and coupling indexes are defined to depict coupling degree. Motions of a space manipulator with free-swinging joint failure simultaneously satisfy the first-order nonholonomic constraint (kinematic coupling relationships) and the second-order nonholonomic constraint (dynamic coupling relationships), and the manipulator can perform tasks through motion planning and control. Finally, simulation experiments are carried out to verify the existence and correctness of the first-order and second-order nonholonomic constraints and display task execution effects of the space manipulator. This research analyzes the kinematic and dynamic characteristics of the free-floating space manipulator with free-swinging joint failure for the first time. It is the theoretical basis of free-swinging joint failure treatment for a space manipulator.

Hierarchical sliding mode control for a two-dimensional ball segway that is a class of a second-order underactuated system

Ball segway, a ballbot-type personal carrier robot, is an actual under-actuated system with second-order nonholonomic velocity constraints and input coupling case. The two-dimensional ball segway system has two outputs consisting of ball position and body tilt angle, which needs to be controlled but has only one torque driving the ball. Actuators directly drive the ball and the body has no direct control. The main objective of this study is to design a robust controller for the ball segway so that the ball is transferred in a point-to-point motion while the body is maintained in the vertical position. The proposed controller is designed on the basis of hierarchical sliding mode control technique. Both simulations and experiments were conducted to verify the quality and stability of the control system.

Nonprehensile Manipulation of an Underactuated Mechanical System With Second-Order Nonholonomic Constraints: The Robotic Hula-Hoop

A mechanical system consisting of a hoop and a pole is considered, for which the corresponding dynamic model represents an underactuated system subject to second-order nonholonomic constraints. The pursued goal is to simultaneously track a trajectory in the unactuated coordinates and to stabilize the actuated ones. For the model under consideration, the well-known noncollocated partial feedback linearization algorithm fails since the corresponding zero dynamics is unstable. In this work, we show that the actuated coordinates, i.e., the pole can be stabilized by exploiting the null space of the coupling inertia matrix without affecting the performance in the underactuated coordinates tracking. We present a formal mathematical analysis, which guarantees ultimate boundedness of all coordinates. Performed simulations bolster the proposed approach.

Hybrid Collision Avoidance for Autonomous Surface Vehicles

Autonomous vehicles and collision avoidance (COLAV) systems are advancing rapidly. However, the majority of the COLAV methods developed are not designed for vehicles with second-order nonholonomic constraints, such as autonomous surface vehicles (ASVs). This paper proposes the hybrid dynamic window (HDW) algorithm, which in addition to acting as a reactive COLAV method, functions as a trajectory tracker in a hybrid COLAV architecture. The algorithm serves as an interface to any deliberate COLAV method which generates time parameterized trajectories, enabling vehicles to avoid local minima and track optimal trajectories towards the goal. Furthermore, a new distance function is developed for the dynamic window (DW) algorithm, improving the algorithm trajectory planning when operating close to obstacles. A case study is done for an ASV model using the HDW algorithm and the rapidly-exploring random tree (RRT) algorithm, which together form a hybrid COLAV system. The performance is evaluated through simulations using a defined set of performance metrics.

Master-slave teleoperation of underactuated mechanical systems with communication delays

This paper addresses the control problem of master-slave teleoperation of a class of underactuated mechanical systems with communication delays. The core of the proposed solution introduces a suitable coupling matrix, so the complete error dynamics is written in a linear part, and a nonlinear one, for which the matching condition is satisfied. Then, from this representation, discontinuous causal controllers are designed to achieve bilateral coordination in force and position. The proposal is valid for underactuated manipulators with second-order non-holonomic constraints, and represents, to our best knowledge, a contribution in teleoperation systems where these mechanisms have been barely considered. Numerical simulations and real-time experiments developed over the Internet are provided to show the effectiveness of the proposed control scheme.

Simultaneous balancing and trajectory tracking control for two-wheeled inverted pendulum vehicles: A composite control approach

This paper is concerned with a novel composite controller for an underactuated two-wheeled inverted pendulum vehicle with an unstable suspension subject to nonholonomic constraint. The presented composite controller is consisted with an adaptive sliding mode technique to construct an additional disturbance-like signal, and particularly a direct fuzzy controller to approximate the optimal velocity tracking control effort by the adaptive mechanism. Unlike the traditional control strategies, this composite control approach offers a coordinate control objective for the vehicle with an unstable equilibrium and second-order nonholonomic constraints. In addition, with the aid of a posture controller aiming to the tracking error dynamics, the nonholonomic vehicle can track an arbitrary trajectory given by the earth-fixed frame. Numerical simulation results confirm that the proposed controllers can maintain the balance of the unstable suspension, drive the vehicle globally to track any reference trajectory, and guarantee all the signals of the closed-loop system convergent within a compact set.

Drive-mode control for an underactuated MEMS vibratory rate gyroscope

It has been shown that the underactuated rate gyroscopes (URGs) hold appealing robust operational performance against the fluctuations of temperature, pressure and structure parameters. In order to further improve the performance of the URGs, the dynamics and the closed-loop control issues of the URGs are investigated in this paper. For a three-DOFs gyroscope with two drive-modes, single sense mode, and single actuator, it is shown that the dynamics of the drive subsystem of the URG can be described as a second-order nonholonomic constraint system, and the dynamics of the drive subsystem of the URG can be transformed into a fourth-order linear system with the Brunovsky's canonical forms by properly selecting the controlled output. Then a robust controller is presented to stabilize the underactuated system by solving a set of linear matrix inequalities. Additionally, the stability of the presented controller is analyzed and demonstrated by some simulations.

状態フィードバックによる2つのスラスターをもつ衛星の面内位置・姿勢制御

This paper proposes a feedback control technique for the in-plane motion of a satellite equipped with two thrusters, whose force directions are fixed to the satellite. Such systems have second-order nonholonomic constraints due to the fixed force directions, and moreover their thruster forces are restricted to be unilateral. To control the satellites' global position and orientation, this paper develops a tracking controller to a designed invariant manifold based on the Lyapunov's second method. The developed controller is effective to control the satellite's global motion, even when the model parameters include some range of errors. Simulation results are shown to demonstrate the effectiveness of the developed controller.

Analysis and Simulation of Optimal Vibration Attenuation for Underactuated Mechanical Systems

This paper deals with active optimal vibration attenuation of elastic structures modeled by finite elements. The system’s equations are linear with potentially large numbers of degrees of freedom, whereas the minimized performance index isquadratic. Theproblem isformulated inmodal space so thatthe dimension ofthe problem can belimitedtocontrollingsignificantmodesonly.Theirnumberisconsideredgreaterthanthenumberofindependent discrete actuators, making the system underactuated. The constraints resulting from underactuation are representedbythematrixofconstraints thatcouplesthe modalcontrols. Thismatrix, whichplaysanimportant role in predicting the systems controllability, is obtained by adding a set of dummy actuators. Themodal variables are in turn coupled via second-order nonholonomic constraints, which are satisfied with the help of time-dependent Lagrange multipliers. The optimality equations for the problem are derived in a compact form and solved by applying symbolic differential operators. The procedure, which applies standard finite element and mathematical software, renders the optimal actuation forces and the response of all controlled modes, or any selected degrees of freedom, for the entire control process. Two simulation examples are presented to illustrate the approach’s details and the use of controllability indicators derived from the matrix of constraints.

Nonlinear Feedback Control of a Gravity-Assisted Underactuated Manipulator With Application to Aircraft Assembly

A nonlinear feedback scheme for a gravity-assisted underactuated manipulator with second-order nonholonomic constraints is presented in this paper. The joints of the hyper articulated arm have no dedicated actuators but are activated by gravity. By tilting the base link appropriately, the gravitational torque drives the unactuated links to a desired angular position. With simple locking mechanisms, the hyperarticulated arm can change its configuration using only one actuator at the base. This underactuated arm design was motivated by the need for a compact snake-like robot that can go into aircraft wings and perform assembly operations using heavy end-effectors. The dynamics of the unactuated links are essentially second-order nonholonomic constraints for which there are no general methods to design closed-loop control. We propose a nonlinear closed-loop control law that is guaranteed to be stable in positioning one unactuated joint at a time. We synthesize a Lyapunov function to prove the convergence of this control scheme. The Lyapunov function also generates estimates of the domain of convergence of the control law for various control gains. The control algorithm is implemented on a prototype three-link system. Finally, we provide some experimental results to demonstrate the efficacy of the control scheme.

Simulating Active Vibration Attenuation in Underactuated Spatial Structures

The active optimal attenuation of vibrations in spatial structures, modeled by finite elements with a possibly large number of degrees of freedom, is considered. The problem is formulated as underactuated in modal space, such that the desired number of controlled modes may be greater than the number of independent discrete actuators. Consequently, modal variables are coupled via second-order nonholonomic constraints that impose limitations on the dynamically admissible trajectories that a given structure may undergo. The optimality equations for the problem with a quadratic performance index are derived in a compact form involving time derivatives of all modal variables and Lagrange multipliers, which are required to ensure that the constraints are satisfied. These equations are solved by applying symbolic differential operators. The procedure employs standard finite element and symbolic mathematical software to render the optimal actuation forces and trajectories of all controlled modes (or any selected degrees of freedom) to attenuate vibrations. For illustration the method, referred to as the constrained modal space optimal control, is used to synthesize the active controls and predict response of a mast structure.

Nonholonomic Motion Planning for Coupled Planar Rigid Bodies with Passive Revolute Joints

Motion planning for coupled rigid bodies in a horizontal plane is investigated. The rigid bodies are serially connected by passive revolute joints. The dynamic constraints on the system are second-order nonholonomic constraints. We attempted to control those n coupled rigid bodies by the translational acceleration inputs at the first joint. If each rigid body is hinged at the center of percussion, it is possible to compose a positioning trajectory by connecting rotational and translational trajectory segments. Each rigid body can be rotated about its center of percussion one after another. When all of the rigid bodies are aligned on a straight line, they can be translated. The algorithm for positioning is presented. Simulations show that the coupled planar rigid bodies can reach the desired configuration by the constructed inputs.

Nonholonomic Motion Planning for Coupled Planar Rigid Bodies with Passive Revolute Joints.

Motion planning for coupled rigid bodies in a horizontal plane is investigated. The rigid bodies are serially connected by passive revolute joints. The dynamic constraints on the system are second-order nonholonomic constraints. We attempted to control those n coupled rigid bodies by the translational acceleration inputs at the first joint. If each rigid body is hinged at the center of percussion, it is possible to compose a positioning trajectory by connecting rotational and translational trajectory segments. Each rigid body can be rotated about its center of percussion one after another. When all of the rigid bodies are aligned on a straight line, they can be translated. The algorithm for positioning is presented. Simulations show that the coupled planar rigid bodies can reach the desired configuration by the constructed inputs.

Tracking Control of Mechanical Dynamic System with Nonholonomic Constraints using Transformation of the Time Scale.

In this paper, we design tracking controller for mechanical system with second-order nonholonomic constraints using transformation of the time scale. First, we review the equivalence of trajectry in configuration space under consideration of time scaling and try to introduce a time scaling function for the second-order derivatives. We find that the time scaling function can be treated as a control input. Using linearization theory of nonlinear systems, we design a tracking controller that has one more additive control input and a linear form. Finally, the controller is applied to 3-link mechanism, simulations show that the proposed controller works effectively.

Sliding Mode Control of Nonholonomic Mechanical Systems: Underactuated Manipulators Case

The control of mechanical systems subject to a set of second-order nonholonomic constraints represents an important class of control problem. In this paper underactuated robots as an example of such nonholonomic systems are studied. Based on the results that smooth feedback stabilization to a single equilibrium point is not possible, a feedback scheme providing stability to a desired manifold is proposed.

Canonical equations for mechanical systems with second-order linear nonholonomic constraints

Canonical equations for mechanical systems with second-order linear nonholonomic constraints