In this paper a motion planner for nonholonomic stratified systems was proposed. Those systems may arise easily when reliable systems are designed to be robust against failures in difficult servicing environments. For a special class of the systems, a strong controllability condition was introduced, and a criterion to satisfy the condition was formulated and used to plan the motion of free-floating space manipulators. Modules of the planner were enumerated and their roles were emphasized. Some features of the planner were examined and discussed based on simulation results performed on two models of space manipulators.

Spherical robot is a special kind of nonholonomic system that cannot be converted to chained form, which means most of the well-known control methodologies are not suitable for this system. For the trajectory tracking of BHQ-1, a spherical robot designed by our lab, a two-state trajectory tracking controller is proposed in this paper. First, the kinematic model of the robot is built using screw theory and exponential method and the controllability is proved based on the differential geometric control theory. Then to solve the two-state trajectory tracking problem of BHQ-1, a shunting model of neurodynamics and Lyapunov's direct method are combined to design a two-state trajectory tracking controller, of which the Lyapunov stability is validated. Finally, typical simulation examples, such as tracking linear, circular, and sinusoidal trajectories, are introduced to verify the effectiveness of the proposed controller. In this paper the proposed method can also be applied to the control of other spherical robots.

A path planning algorithm for a spherical mobile robot rolling on a plane is presented in this paper. The robot is actuated by two internal rotors that are fixed to the shafts of two motors. These are in turn mounted on the spherical shell in mutually orthogonal directions. The system is nonholonomic due to the nonintegrable nature of the rolling constraints. Further, the system cannot be converted into a chained form, and neither is it nilpotent nor differentially flat. So existing techniques of nonholonomic path planning cannot be applied directly to the system. The approach presented here uses simple geometrical notions and provides numerically efficient and intuitive solutions. We also present the dynamic model and derive motor torques for execution of the algorithm. Along the proposed paths, we achieve dynamic decoupling of the variables making the algorithm more suitable for practical applications.

In this paper, a velocity space method of motion planning for nonholonomic systems is presented. This method, based on Lie algebraic principles and locally around consecutive current states, plans a motion towards a goal. It is effective as most of the computations can be carried out analytically. This method is illustrated on the unicycle robot and the inverted pendulum.

We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset A of a metric space is the minimum number of balls of a given radius ε needed to cover A. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-Riemannian manifold as the infimum of the lengths of all trajectories contained in an ε-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold. We construct first a norm $\| \cdot \|_{\varepsilon}$ on the tangent space that depends on a parameter ε > 0. Our main result states then that the entropy of a path is equivalent to the integral of this ε-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen.

This paper presents a collision-free path planner for mobile robot navigation in an unknown environment subject to nonholonomic constraints. This planner is well adapted for use with embarked sensors, because it uses only the distance information between the robot and the obstacles. The collision-free path-planning is based on a new representation of the obstacles in the velocity space. The obstacles in the influence zone are mapped as linear constraints into the velocity space of the robot, forming a convex subset that represents the velocities that the robot can use without collision with the objects. The planner is composed by two modules, termed “reaching the goal” and “boundary following”. The major advantages of this method are the very short calculation time and a continuous stable behavior of the velocities. The results presented demonstrate the capabilities of the proposed method for solving the collision-free path-planning problem.

This paper presents a new adaptive controller as a solution to the problem of stabilizing nonholonomic mechanical systems in the presence of incomplete information concerning the system dynamic model. The proposed control system consists of two su bsystems: a slightly modified version of the kinematic stabilization strategy of M'Closkey and Murray which generates a desired velocity trajectory for the nonholonomic system, and an adaptive control scheme which ensures that this velocity trajector y is accurately tracked. This approach is shown to provide arbitrarily accurate stabilization to any desired configuration and can be implemented with no knowledge of the system dynamic model. The efficacy of the proposed stabilization strategy is illustr ated through extensive computer simulations with nonholonomic mechanical systems arising from explicit constraints on the system kinematics and from symmetries of the system dynamics.

This paper considers the problem of controlling the motion of nonholonomic mechanical systems in the presence of uncertainty regarding the system model and state. It is proposed that a simple and effective solution to this problem can be obtained by first using a reduction procedure to obtain a lower dimensional system which retains the mechanical system structure of the original system, and then adaptively controlling the reduced system in such a way that the complete system is driven to the goal configuration. This approach is shown to be easy to implement and to ensure accurate motion control despite measurement and model uncertainty. The efficacy of the proposed control strategy is illustrated through computer simulations and preliminary hardware experiments with nonholonomic mechanical systems arising from both explicit kinematic constraints and symmetries of the system dynamics.

This paper introduces the notion of kinematic redundancy in nonholonomic mechanical systems, identifies some of the interesting properties which result because of the presence of the redundancy, and initiates a study of the control and application of these systems. It is shown that kinematic redundancy in nonholonomic system can be exploited both to simplify the problem of controlling these systems and to enhance their performance capabilities. Moreover, it is demonstrated that these results can be obtained even in the presence of considerable uncertainty regarding the system model. The proposed ideas are illustrated through the study of three example systems: a space robot, a mobile manipulator, and a tractor-trailer system with two steering inputs (fire truck).