The kinematics, dynamics, and control of a unicycle moving without slipping on a plane has been extensively studied in the literature of nonholonomic mechanical systems. However, since planar motion can be seen as a limiting case of the motion on a sphere, we focus our analysis on the more general spherical case. This paper introduces a novel approach to path planning for a unicycle rolling on a sphere while satisfying the non-slipping constraint. Our method is based on a simple yet effective idea: first, we model the system as a linear time-varying dynamic system. Then, leveraging the fact that certain such systems can be integrated under specific algebraic conditions, we derive a closed-form expression for the control variables. This formulation includes three free parameters, which can be tuned to generate a path connecting any two configurations of the unicycle. Notably, our approach requires no prior knowledge of nonholonomic system analysis, making it accessible to a broader audience.

This paper demonstrates that for certain choices of mass distribution and addition of springs, an underactuated two-degree-of-freedom (2-DOF) \bmPRRRPsystem is static feedback linearizable, i.e., differentially flat as well. This paper is original and provides a ground breaking study in underactuated dynamical systems.

This paper investigates when a 2 degree-of-freedom PRRRP closed-chain system with a single actuator is both strongly accessible and feedback linearizable. It is demonstrated that for certain choices of mass distribution and addition of springs, an under-actuated 2 DOF PRRRP system is static feedback linearizable, i.e., also differentially flat.

The motion of a free-floating space robot is characterized by the principle of conservation of angular momentum. It is well known that these angular momentum equations are nonholonomic, i.e., are nonintegrable rate equations. If the base of the free-floating robot is partially actuated, it is difficult to attain trajectories of the joints that result in point-to-point motion of the entire robot system in its configuration space. However, if the drift-less system associated with the angular momentum conservation equations is shown to be differentially flat, point-to-point maneuvers of the free-floating robot in its configuration space can be constructed. However, an open research problem in the current literature is to show the property of differential flatness for a general space robot. The primary contributions of this paper are as follows: (i) study systematically the structure of the nonholonomic rate constraint equations of a free-floating open-chain space robot with arbitrarily oriented joints and two momentum wheels; (ii) establish the design conditions under which the system exhibits differential flatness; (iii) exploit these design conditions for point-to-point trajectory planning and control of the space robot

In this paper, the dynamic model of a wheeled inverted pendulum (e.g., Segway, Quasimoro, and Joe) is analyzed from a controllability and feedback linearizability point of view. First, a dynamic model of this underactuated system is derived with respect to the wheel motor torques as inputs while taking the nonholonomic no-slip constraints into considerations. This model is compared with the previous models derived for similar systems. The strong accessibility condition is checked and the maximum relative degree of the system is found. Based on this result, a partial feedback linearization of the system is obtained and the internal dynamics equations are isolated. The resulting equations are then used to design two novel controllers. The first one is a two-level velocity controller for tracking vehicle orientation and heading speed set-points, while controlling the vehicle pitch (pendulum angle from the vertical) within a specified range. The second controller is also a two-level controller which stabilizes the vehicle's position to the desired point, while again keeping the pitch bounded between specified limits. Simulation results are provided to show the efficacy of the controllers using realistic data.

The motion of free-floating space robots is characterized by nonholonomic, i.e., non-integrable rate constraint equations. These constraints originate from principles of conservation of linear and angular momentum. Trajectory planning of these systems is extremely challenging and computation intensive since the motion must satisfy differential constraints. However, under certain conditions, these drift-less control systems can be shown to be differentially flat. The property of flatness allows a computationally inexpensive way to plan trajectories for the dynamic system between two configurations as well as develop feedback controllers. In this paper, nonholonomic rate constraints for free-floating planar open-chain robots are studied together with auxiliary joint variable constraints to determine design conditions under which the system exhibits differential flatness. Sufficient conditions are derived for existence of flatness and are illustrated by examples.

The motion of free-floating space robots is characterized by nonholonomic, i.e., non-integrable rate constraint equations. These constraints originate from principles of conservation of linear and angular momentum. It is well known that these rate constraints can also be written as input-affine drift-less control systems. Trajectory planning of these systems is extremely challenging and computation intensive since the motion must satisfy differential constraints. However, under certain conditions, these drift-less control systems can be shown to be differentially flat. The property of flatness allows a computationally in-expensive way to plan trajectories for the dynamic system between two configurations as well as develop feedback controllers. Nonholonomic rate constraints for free-floating planar open-chain robots are systematically studied to determine the design conditions under which the system exhibits differential flatness. Under these design conditions, the property of flatness is used for trajectory planning and feedback control under perturbations in the initial state.