Nonholonomic Problem Research Articles

An Optimal Smooth-Path Motion Planning Method for a Car-like Mobile Robot

This paper proposes an optimal motion planning method consisting of a genetic algorithm (GA), potential field (PF), and Dubins curve for a Car-like mobile robot to solve the problem of finding the shortest and most feasible path in the global environment. Firstly, the GA finds the shortest path by evaluating, selecting, crossing over, and mutating from the initial population and finally provides the strongest individual evolution. Then the result from the GA is further applied with the PF algorithm to improve the ability of obstacle avoidance in the environment. Finally, the Dubins curve method is combined to smooth the path and helps the Car-like mobile robot solve the nonholonomic constraints problem. The major advantages of this method include finding the shortest path, improving avoidance obstacle ability, and smoothing the output path in an environment effectively. The simulation of the proposed method is executed on MATLAB to verify the ability to solve motion planning problems for a Car-like mobile robot.

Trajectory planning and base attitude restoration of dual-arm free-floating space robot by enhanced bidirectional approach

When free-floating space robots perform space tasks, the satellite base attitude is disturbed by the dynamic coupling. The disturbance of the base orientation may affect the communication between the space robot and the control center on earth. In this paper, the enhanced bidirectional approach is proposed to plan the manipulator trajectory and eliminate the final base attitude variation. A novel acceleration level state equation for the nonholonomic problem is proposed, and a new intermediate variable-based Lyapunov function is derived and solved for smooth joint trajectory and restorable base trajectories. In the method, the state equation is first proposed for dual-arm robots with and without end constraints, and the system stability is analyzed to obtain the system input. The input modification further increases the system stability and simplifies the calculation complexity. Simulations are carried out in the end, and the proposed method is validated in minimizing final base attitude change and trajectory smoothness. Moreover, the minute internal force during the coordinated operation and the considerable computing efficiency increases the feasibility of the method during space tasks.

Convergence of Lobatto-type Runge\u2013Kutta methods for partitioned differential-algebraic systems of index 2

In this paper a numerical scheme for partitioned systems of index 2 DAEs, such as those arising from nonholonomic mechanical problems, is proposed and its order for a certain class of Runge–Kutta methods we call of Lobatto-type is proven.

Application of variational calculus for optimization of optic gradient coatings

The design optimization for n-graded optic coatings based on variational calculus is discussed. In order to exemplify the approach, the Lagrange problem for minimizing refraction is stated and solved. The refraction law is generalized for alteration of the continuous refraction index. To solve problems of reflection minimization, an integro-differential equation of retarding potentials is proposed to use as a condition to the nonholonomic Lagrange problem.

What makes nonholonomic integrators work?

A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic integrator is a numerical method specifically designed for nonholonomic systems. It has been observed numerically that many nonholonomic integrators exhibit excellent long-time behaviour when applied to various test problems. The excellent performance is often attributed to some underlying discrete version of the Lagrange–d’Alembert principle. Instead, in this paper, we give evidence that reversibility is behind the observed behaviour. Indeed, we show that many standard nonholonomic test problems have the structure of being foliated over reversible integrable systems. As most nonholonomic integrators preserve the foliation and the reversible structure, near conservation of the first integrals is a consequence of reversible KAM theory. Therefore, to fully evaluate nonholonomic integrators one has to consider also non-reversible nonholonomic systems. To this end we construct perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map). Applying various nonholonomic integrators from the literature to these problems we observe that no method performs well on all problems. This further indicates that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators. A list of relevant open problems is given toward the end.

Trajectory planning and robust tracking control for a class of active articulated tractor-trailer vehicle with on-axle structure

Traditional articulated vehicles exhibit poor steering performance since there exist no direct rotating moment for the underactuated trailers, which cause such vehicles nearly cannot work well in the narrow space. To overcome this shortcoming associated with the traditional articulated vehicles, an active articulated structure is proposed, in which a steering motor is introduced at the articulated joint of tractor and trailer, in addition to utilizing mecanum wheels as trailer wheels. On the basis of this structure, a coordinated control method is designed to ensure the vehicle kinematics constraints and dynamic maneuverability. The coordination control consists of two level controllers. On the level of kinematics, model predictive control (MPC) is adopted as posture controller, which can solve the non-holonomic problem of the whole active articulated tractor-trailer vehicle system; On the dynamic level, a sliding mode control (SMC) is introduced to design a dynamic controller to track the desired velocities generated online, which can increase the robustness of the system. The simulation results show that the proposed active articulated tractor-trailer vehicle system has better maneuverability compared with the traditional one, and the proposed control strategy can ensure the required control effect.

Trajectory Planning in Time-Varying Adverse Weather for Fixed-Wing Aircraft Using Robust Model Predictive Control

The avoidance of adverse weather is an inevitable safety-relevant task in aviation. Automated avoidance can help to improve safety and reduce costs in manned and unmanned aviation. For this purpose, a straightforward trajectory planner for a single-source-single-target problem amidst moving obstacles is presented. The functional principle is explained and tested in several scenarios with time-varying polygonal obstacles based on thunderstorm nowcast. It is furthermore applicable to all kinds of nonholonomic planning problems amidst nonlinear moving obstacles, whose motion cannot be described analytically. The presented resolution-complete combinatorial planner uses deterministic state sampling to continuously provide globally near-time-optimal trajectories for the expected case. Inherent uncertainty in the prediction of dynamic environments is implicitly taken into account by a closed feedback loop of a model predictive controller and explicitly by bounded margins. Obstacles are anticipatory avoided while flying inside a mission area. The computed trajectories are time-monotone and meet the nonholonomic turning-flight constraint of fixed-wing aircraft and therefore do not require postprocessing. Furthermore, the planner is capable of considering a time-varying goal and automatically plan holding patterns.

Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and twisting of an n-dimensional balanced ball over a fixed sphere is considered. For a special inertia operator (depending on n parameters) we prove complete integrability when the radius of the ball is twice the radius of the sphere. In the case of symmetry, noncommutative integrability for any ratio of the radii is established.

Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin hamiltonization

In this note we consider the nonholonomic problem of rolling without slipping and twisting of an ??-dimensional balanced ball over a fixed sphere. This is a ????(??)?Chaplygin system with an invariant measure that reduces to the cotangent bundle ??*?????1. For the rigid body inertia operator r I? = I? + ?I, I = diag(I1,...,In) with a symmetry I1 = I2 = ... =Ir ? Ir+1 = Ir+2 = ... = In, we prove that the reduced system is integrable, general trajectories are quasi-periodic, while for ?? ? 1, ?? ? 1 the Chaplygin reducing multiplier method does not apply.

Dynamic Motion Planning and Adaptive Tracking Control for a Class of Two-Wheeled Autonomous Vehicle With an Underactuated Pendular Suspension

This paper investigates a dynamic motion planning approach and an adaptive tracking control scheme for a class of two-wheeled autonomous vehicle with an underactuated pendular suspension subject to nonholonomic constraint. Compared with the wheeled inverted pendulum system, this kind of two-wheeled pendular suspension (WPS) vehicle is more suitable for autonomous exploration in the complex unstructured environment. By Lagrange principle, a four-independent-coordinate dynamic model, which can describe the multivariate, nonlinear, and underactuated characteristics of the system, is first proposed. Besides, a reduced order dynamic is developed in the following so as to tackle the nonholonomic problem, and then the three-independent-coordinate reduced order dynamic is divided into an actuated part constituted by the rotational subsystem, and an underactuated part combined by the longitudinal and the swing angle subsystems. The sliding mode control (SMC) technique is utilized to construct the controller; especially, a composite sliding mode surface is proposed which can realize the velocity tracking and oscillation suppression for pendular suspension simultaneously. Furthermore, the adaptive mechanism is employed to update the true values of the inaccessible physical parameters which can enhance the adaptability of the WPS vehicle in unstructured environment. In addition, a dynamic motion planning method is presented, by aid of which the vehicle can track an arbitrary trajectory in Cartesian coordinate. The simulation results show the effectiveness and merits of the proposed control approaches.

Invariant measures of modified LR and L+R systems

We introduce a class of dynamical systems having an invariant measure, the modifications of well-known systems on Lie groups: LR and L+R systems. As an example, we study a modified Veselova nonholonomic rigid body problem, considered as a dynamical system on the product of the Lie algebra so(n) with the Stiefel variety V n,r , as well as the associated єL+R system on so(n) × V n,r . In the 3-dimensional case, these systems model the nonholonomic problems of motion of a ball and a rubber ball over a fixed sphere.

Randomized path planning on vector fields

Given a vector field defined on a robot’s configuration space, in which the vector field represents the system drift, e.g. a wind velocity field, water current flow, or gradient field for some potential function, we present a randomized path planning algorithm for reaching a desired goal configuration. Taking the premise that moving against the vector field requires greater control effort, and that minimizing the control effort is both physically meaningful and desirable, we propose an integral functional for control effort, called the upstream criterion, that measures the extent to which a path goes against the given vector field. The integrand of the upstream criterion is then used to construct a rapidly exploring random tree (RRT) in the configuration space, in a way such that random nodes are generated with an a priori specified bias that favors directions indicated by the vector field. The resulting planning algorithm produces better quality paths while preserving many of the desirable features of RRT-based planning, e.g. the Voronoi bias property, computational efficiency, algorithmic simplicity, and straightforward extension to constrained and nonholonomic problems. Extensive numerical experiments demonstrate the advantages of our algorithm vis-à-vis existing optimality criterion-based planning algorithms.

A Path Tracking Algorithm Using Future Prediction Control with Spike Detection for an Autonomous Vehicle Robot

Trajectory tracking is an important aspect of autonomous vehicles. The idea behind trajectory tracking is the ability of the vehicle to follow a predefined path with zero steady state error. The difficulty arises due to the nonlinearity of vehicle dynamics. Therefore, this paper proposes a stable tracking control for an autonomous vehicle. An approach that consists of steering wheel control and lateral control is introduced. This control algorithm is used for a non-holonomic navigation problem, namely tracking a reference trajectory in a closed loop form. A proposed future prediction point control algorithm is used to calculate the vehicle's lateral error in order to improve the performance of the trajectory tracking. A feedback sensor signal from the steering wheel angle and yaw rate sensor is used as feedback information for the controller. The controller consists of a relationship between the future point lateral error, the linear velocity, the heading error and the reference yaw rate. This paper also introduces a spike detection algorithm to track the spike error that occurs during GPS reading. The proposed idea is to take the advantage of the derivative of the steering rate. This paper aims to tackle the lateral error problem by applying the steering control law to the vehicle, and proposes a new path tracking control method by considering the future coordinate of the vehicle and the future estimated lateral error. The effectiveness of the proposed controller is demonstrated by a simulation and a GPS experiment with noisy data. The approach used in this paper is not limited to autonomous vehicles alone since the concept of autonomous vehicle tracking can be used in mobile robot platforms, as the kinematic model of these two platforms is similar.

Интегрируемость и стохастичность некоторых задач неголономной механики

Интегрируемость и стохастичность некоторых задач неголономной механики

On the Poisson structures for the nonholonomic Chaplygin and Veselova problems

We discuss a Poisson structure, linear in momenta, for the generalized nonholonomic Chaplygin sphere problem and the LR Veselova system. Reduction of these structures to the canonical form allows one to prove that the Veselova system is equivalent to the Chaplygin ball after transformations of coordinates and parameters.

Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support

We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.

Two non-holonomic integrable problems tracing back to Chaplygin

The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.

On some generalization of the area theorem with applications to the problem of rolling balls

This publication contributes to the series of RCD translations of Sergey Alexeevich Chaplygin’s scientific heritage. Earlier we published three of his papers on non-holonomic dynamics (vol. 7, no. 2; vol. 13, no. 4) and two papers on hydrodynamics (vol. 12, nos. 1, 2). The present paper deals with mechanical systems that consist of several spheres and discusses generalized conditions for the existence of integrals of motion (linear in velocities) in such systems. First published in 1897 and awarded by the Gold Medal of Russian Academy of Sciences, this work has not lost its scientific significance and relevance. (In particular, its principal ideas are further developed and extended in the recent article “Two Non-holonomic Integrable Problems Tracing Back to Chaplygin”, published in this issue, see p. 191). Note that non-holonomic models for rolling motion of spherical shells, including the case where the shells contain intricate mechanisms inside, are currently of particular interest in the context of their application in the design of ball-shaped mobile robots. We hope that this classical work will be estimated at its true worth by the English-speaking world.

Vehicle Path Tracking Using Future Prediction Steering Control

Trajectory tracking is one of the important aspect in autonomous vehicle. Therefore, this paper proposed a stable tracking control for autonomous vehicle. An approach that consists of steering wheel control and lateral control is introduced. This control algorithm is used for nonholonomic navigation problem which is tracking a reference trajectory in a close loop form. Future prediction point is used to calculate the vehicle lateral error is introduced. Controller based on steering wheel angle and yaw rate of the vehicle is introduced. The controller consists of relationship between future point lateral error, linear velocity, heading error and reference yaw rate. The effectiveness of the proposed controller is demonstrated by using simulation.

A 6-DOF adaptive parallel manipulator with large tilting capacity

In this paper, a novel 6 degrees of freedom (DOFs) adaptive parallel manipulator with large tilting capacity is presented. The manipulator consists of four identical peripheral limbs and one center limb connecting the base and the moving platform. Due to the special architecture, the doubly actuated center limb of the manipulator could have infinite inverse solutions. In every configuration of the end-effector, the manipulator can adapt its center limb to the position and orientation with best dexterity. An optimization equation for obtaining the optimized dexterity of the manipulator is introduced to solve this nonholonomic problem, which also makes the manipulator capable of large tilting capacity. Targeting for the application of five-face machining, the detailed kinematic analysis of the manipulator is developed, which includes the closed-form solutions of inverse position problems, the singularity, dexterity, workspace and tilting capability. The analysis developed in this paper shows that the proposed manipulator has large tilting capacity and thus a suitable candidate for five-face machining.