On the Optimal Control of a Rolling Ball Robot Actuated by Internal Point Masses

Putkaradze, Vakhtang; Rogers, Stuart

doi:10.1115/1.4046104

Cited by 12 publications

(19 citation statements)

References 61 publications

(114 reference statements)

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“…This paper is a continuation of [1], providing numerical solutions of the controlled equations of motion for several special cases of the rolling disk and ball actuated by moving internal point masses. The paper [1] invokes Pontryagin's minimum principle to derive the theoretical background for the optimal control of the rolling disk and ball having general performance indexes. This paper implements the theory derived in [1] to solve several practical examples, such as trajectory tracking for the rolling disk and obstacle avoidance for the rolling ball.…”

Section: Overviewmentioning

confidence: 99%

“…The paper [1] invokes Pontryagin's minimum principle to derive the theoretical background for the optimal control of the rolling disk and ball having general performance indexes. This paper implements the theory derived in [1] to solve several practical examples, such as trajectory tracking for the rolling disk and obstacle avoidance for the rolling ball. The key contributions of this paper are listed below.…”

Section: Overviewmentioning

confidence: 99%

“…The reader is referred to [1] for further details about the properties and construction of (2.2). The optimal control problem for the rolling disk is…”

Section: Parametermentioning

confidence: 99%

“…The reader is referred to [1] for a more general description of the rolling disk's optimal control problem (2.7) and the associated controlled equations of motion (2.16).…”

Section: Parametermentioning

confidence: 99%

“…The reader is referred to [1] for a more general description of the ODE and DAE formulations, (3.4) and (3.18), of the rolling ball's optimal control problem and the controlled equations of motion (3.22) correpsonding to (3.4). The DAE TPBVP encapsulating the controlled equations of motion corresponding to the DAE formulation (3.18) of the rolling ball's optimal control problem were not investigated since a robust DAE TPBVP solver is not readily available in MATLAB.…”

Section: Parameter Valuementioning

confidence: 99%

See 4 more Smart Citations

Numerical simulations of a rolling ball robot actuated by internal point masses

Putkaradze¹,

Rogers²

2021

NACO

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Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

“…The reader is referred to [1] for further details about the properties and construction of (2.2). The optimal control problem for the rolling disk is…”

Section: Parametermentioning

confidence: 99%

“…The reader is referred to [1] for a more general description of the rolling disk's optimal control problem (2.7) and the associated controlled equations of motion (2.16).…”

Section: Parametermentioning

confidence: 99%

Section: Parameter Valuementioning

confidence: 99%

See 3 more Smart Citations

Numerical simulations of a rolling ball robot actuated by internal point masses

Putkaradze¹,

Rogers²

2021

NACO

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Geometric Kinematic Control of a Spherical Rolling Robot

Ohsawa

2019

J Nonlinear Sci

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We give a geometric account of kinematic control of a spherical rolling robot controlled by two internal wheels just like the toy robot Sphero. Particularly, we introduce the notion of shape space and fibers to the system by exploiting its symmetry and the principal bundle structure of its configuration space; the shape space encodes the rotational angles of the wheels, whereas each fiber encodes the translational and rotational configurations of the robot for a particular shape. We show that the system is fiber controllable-meaning any translational and rotational configuration modulo shapes is reachable-as well as find exact expressions of the geometric phase or holonomy under some particular controls. We also solve an optimal control problem of the spherical robot, show that it is completely integrable, and find an explicit solution of the problem. Main Results and Outline.We study the kinematics of the spherical rolling robot like the Sphericle or Sphero from the geometric point of view. Particularly, we exploit the symmetry of the kinematic model of the robot and the notion of shape space (see, e.g., Montgomery [12,13,14] and Kelly and Murray [9]), and analyze its controllability as well as its optimal control problem.We first formulate the kinematic equation describing the nonholonomic constraints of the system in Section 2. The resulting model is effectively the same as that of the Sphericle in Bicchi et al. [3].

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On the dynamics of a rolling ball actuated by internal point masses

Putkaradze

Rogers

2018

Meccanica

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The motion of a rolling ball actuated by internal point masses that move inside the ball's frame of reference is considered. The equations of motion are derived by applying Euler-Poincaré's symmetry reduction method in concert with Lagrange-d'Alembert's principle, which accounts for the presence of the nonholonomic rolling constraint. As a particular example, we consider the case when the masses move along internal rails, or trajectories, of arbitrary shape and fixed within the ball's frame of reference. Our system of equations can treat most possible methods of actuating the rolling ball with internal moving masses encountered in the literature, such as circular motion of the masses mimicking swinging pendula or straight line motion of the masses mimicking magnets sliding inside linear tubes embedded within a solenoid. Moreover, our method can model arbitrary rail shapes and an arbitrary number of rails such as several ellipses and/or figure eights, which may be important for future designs of rolling ball robots. For further analytical study, we also reduce the system to a single differential equation when the motion is planar, that is, considering the motion of the rolling disk actuated by internal point masses, in which case we show that the results obtained from the variational derivation coincide with those obtained from Newton's second law. Finally, the equations of motion are solved numerically, illustrating a wealth of complex behaviors exhibited by the system's dynamics. Our results are relevant to the dynamics of nonholonomic systems containing internal degrees of freedom and to further studies of control of such systems actuated by internal masses.

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On the Optimal Control of a Rolling Ball Robot Actuated by Internal Point Masses

Cited by 12 publications

References 61 publications

Numerical simulations of a rolling ball robot actuated by internal point masses

Numerical simulations of a rolling ball robot actuated by internal point masses

Geometric Kinematic Control of a Spherical Rolling Robot

On the dynamics of a rolling ball actuated by internal point masses

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