This paper presents a new kinematic model based on the Darboux-frame for motion control and planning. In this work, we show that an underactuated system as a spin-rolling sphere on a plane with three inputs and five states can be transformed into a fully-actuated model by the given Darboux-frame transformation. This nonlinear state transformation is a geometric model that is different from conventional state-space models . First, a kinematic model of the Darboux frame at the contact point of a rotating object i.e., the sphere, is established. Next, we propose a virtual surface that is trapped between sphere and plane surfaces . This virtual surface generates arc-length-based inputs for controlling the  trajectories on the sphere and  plane. Finally, we discuss the controllability of this new system after our introduced transformation . In the future, we will design a proper geometric path planning method for the obtained kinematic model.
This paper presents a new kinematic model based on the Darboux frame for motion control and planning. In this work, we show that an underactuated model of a spin-rolling sphere on a plane with five states and three inputs can be transformed into a fully-actuated one by a given Darboux frame transformation. This nonlinear state transformation establishes a geometric model that is different from conventional state-space ones . First, a kinematic model of the Darboux frame at the contact point of the rolling sphere is established. Next, we propose a virtual surface that is trapped between the sphere and the contact plane . This virtual surface is used for generating arc-length-based inputs for controlling the contact trajectories on the sphere and the plane. Finally, we discuss the controllability of this new model . In the future, we will design a geometric path planning method for the proposed kinematic model.
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Darboux-frame
Darboux frame
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system as
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three inputs and five states
five states and three inputs
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model by the given Darboux-frame
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models
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a rotating object i.e., the sphere,
the rolling sphere
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sphere and plane surfaces
the sphere and the contact plane
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generates
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contact
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the
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system after our introduced transformation
model
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