Geometric modeling and local controllability of a spherical mobile robot actuated by an internal pendulum

Abstract

SUMMARYIn this paper, we present the modeling and local equilibrium controllability analysis of a spherical robot. The robot consists of a spherical shell that is internally actuated by a pendulum mechanism. The rolling motion of the sphere manifests itself as a nonholonomic constraint in the modeling. We derive the dynamic model of the system using Lagrangian reduction and the variational principle. We first compute the Lagrangian and identify the symmetry with respect to a group action. The system Lagrangian and the rolling constraint are invariant with respect to the group isotropy and hence permit a reduced dynamic formulation termed as the nonholonomic 'Euler-Poincaré' equation with advected dynamics. Using Lie brackets and symmetric products of the potential and control vector fields, local configuration accessibility and local (fiber) equilibrium controllability are presented.

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

“…More recently, [13] provides a detailed analysis of the trajectory of the Chaplygin sphere's contact point, and it has been shown that the dynamics of the Chaplygin top exhibit a strange attractor [14] and the phenomenon of reversal [15]. The dynamics of the rolling ball with dynamic internal structure is also an active topic in the nonholonomic mechanics literature [16,17,7,18,19,20,21,22,23].…”

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

“…where ξ ∈ R 2n+2 and ν ∈ R n+2+c , c ∈ {0, 1, 2}, are constant Lagrange multiplier vectors enforcing the initial and final conditions, (3.7) and (3.8), respectively. The Hamiltonian is 20) where λ ∈ R 2n+2 is a time-varying Lagrange multiplier vector enforcing the dynamics (3.6). Recall from (2.22) that the function describing the uncontrolled equation of motion for the rolling disk is κ (t, x, u) ≡ −rFe,1 + n i=0 miKi d2 + n i=0 mi (r sin φ + ζi,1) 2 + (r cos φ + ζi,3) 2 ,…”

“…More recently, [12] provides a detailed analysis of the trajectory of the Chaplygin sphere's contact point, and it has been shown that the dynamics of the Chaplygin top exhibit a strange attractor [13] and the phenomenon of reversal [14]. The dynamics of the rolling ball with dynamic internal structure is also an active topic in the nonholonomic mechanics literature [15,16,6,17,18,19,20,21,22].…”

“…A recent survey [40] provides an overview of the theory of mechanical systems having different types of nonintegrable velocity constraints, such as vakonomic, which is useful for control theory, and nonholonomic. Reference [6] investigates the controllability of a rolling ball actuated by internal rotors and sliders, while [20] studies the controllability of a rolling ball actuated by an internal pendulum and yoke. For a rolling ball actuated by internal rotors, reference [17] performs an analysis of controllability and trajectory tracking control, [18] analyzes the optimal control (in the sense of the variational Pontryagin's minimum principle) for some very specific performance indexes, and [19] investigates orientation and contact point trajectory tracking control by constructing feedback control laws.…”

“…In particular, [16] actuates the rolling ball by internal masses which reciprocate along spokes. Reference [6] actuates the rolling ball by a combination of internal rotors and sliders, [21] actuates the rolling ball by an internal gyroscopic pendulum as shown in Figure 1.3d, [28,29,30,31] actuate the rolling ball by an internal spherical pendulum as shown in Figure 1.3e, and [20] actuates the rolling ball by an internal pendulum and yoke. References [32,33] study the theory of and experiment with a rolling ball actuated by an internal omniwheel platform, which is a more sophisticated version of Sphero's driving mechanism depicted in Figure 1.3f.…”