The on-manifold constraint is included by using the first fundamental form of the surface for squared distance computations between the point cloud and the fitting curve.
In this subsection, we will derive a control law for (29) such that the constraint manifold (48) with the constraint functions defined in (47) is globally exponentially stable for the closed-loop system.
The proposed feedback control law is designed to exponentially stabilize the constraint manifold, i.e. to enforce the virtual holonomic constraints, which allows the convergence of the snake robot to the desired path.
The constraint manifold (48) is a six-dimensional submanifold of Q ̂, since we have three different constraint variables, i.e. η,ϕ o,p y.
To this end, we define a constraint manifold for the system, and we design the control input of (29) to exponentially stabilize the constraint manifold.
Previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space.
Hence, we formulate the computation of the unknown label matrix as a smoothing process on the manifold under the constraint that all observations represent an object of one single class.
Here, we model a protein as a kinematic linkage and present a new geometric method to characterize molecular rigidity from the constraint manifold Q and its tangent space TqQ at the current configuration q.
4) The complete framework of SSC-EKE organically incorporates multiple theories, such as joint manifold and pairwise constraint-based regularization, smoothness in the reproducing kernel Hilbert space, empirical risk minimization, and spectral methods, which facilitates the preferable classification accuracy as well as the generalizability of SSC-EKE.
In particular, we use input-output linearization to stabilize the constraint manifold (48).
In particular, we analytically design a smooth feedback control law to exponentially stabilize the constraint manifold.
