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We derive an induced covariant derivative and metric on the tangent bundle using their counterparts on the base manifold. This result is a fundamental tool for our study of geometric attitude controllers on the SO(3) manifold.
The circle (left) is a manifold that can describe systems with circular motions such as a joint of a robotic arm or the steering angle of a car. In order to describe second-order dynamics on the circle, we need to consider velocities in addition to positions. Thus the tangent bundle (middle and right), also a manifold, constructed from the circle and all the tangent spaces is the configuration space of second-order dynamics. The middle image shows the tangent spaces (red lines) at the infinity many points on the circle. Equivalently, the tangent spaces can be re-arranged in a smooth manner with no overlaps to form an infinite cylinder (right). Thus, one can think of trajectories of the robot (positions and velocities) evolving on the surface of this infinite cylinder. (These images were taken from wikipedia.org)
Natural metrics provide a way to induce a metric on the tangent bundle from the metric on its base manifold. The most studied type is the Sasaki metric, which applies the base metric separately to the vertical and horizontal components. We study a more general class of metrics which introduces interactions between the vertical and horizontal components, with scalar weights. Additionally, we explicitly clarify how to apply our and other induced metrics on the tangent bundle to vector fields where the vertical component is not constant along the fibers. We give application to the Special Orthogonal Group SO(3) as an example.
A tangent vector at a point on the tangent bundle can be deconstructed into a horizontal and vertical subspace (shown by red and blue axes on right). The components can then be mapped to a tangent vector on the base manifold. Thus, tangent vectors on the tangent bundle can be represented by a pair of tangent vectors on the base manifold.
The main result of this work. The covariant derivative on the tangent bundle can be defined by the covariant derivative and curvature tensor on the base manifold.
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