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Multifingered Hand Kinematics
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
In order to define the area of a surface, one needs to define the inner product between two tangent vectors on the surface. This defines the area of a parallelogram and the total area can then be calculated by integrating the infinitesimal areas generated by parallelogram-shaped patches on the surface. The first fundamental form for a surface describes how the inner product of two tangent vectors is related to the natural inner product on ℝ3. In a local coordinate chart, it is represented by a quadratic form Ip : ℝ2 × ℝ2 → ℝ which takes two tangent vectors attached at a point p = c(u, υ) and gives their inner product. If c is a local parameterization, then the matrix representation of the quadratic form is given by Ip=[cuTcucuTcυcυTcucυTcυ]. We will use the symbol Ip to represent both the quadratic form and its matrix representation. Note that each element of Ip is an inner product between vectors in ℝ3 and that Ip is symmetric and positive definite. If a parameterization is orthogonal, only the diagonal terms are nonzero.
Optimal control of a NASCAR – specification race car
Published in Vehicle System Dynamics, 2023
D. J. N. Limebeer, M. Bastin, E. Warren, H. G. Fensham
Our purpose here is to relate the parametric and body-fixed linear velocities. The velocity of P is given by which means that where u, v and w are the body-fixed components of the car's linear velocity. Dotting this equation with shows that since each of the remaining vectors is in the plane tangent to the road surface. We can now dot (18) with and , while recognising (19), to obtain In conformity with standard differential-geometric notation, is called the first fundamental form, which facilitates the calculation of metric properties such as ‘length’. It follows from (12) and (13) that and so constrains the way in which and can evolve given and . The assumed regularity of the road surface implies the invertibility of I [26].