Journal of Robotics

Research Article

A Riemannian-Geometry Approach for Modeling and Control of Dynamics of Object Manipulation under Constraints

Table 1

Hessian matrix of the potential

P .

\frac{\partial^{2} P}{\partial θ^{2}} = \frac{\partial S_{N}}{\partial θ} = \frac{f_{d}}{r_{1} + r_{2}} {\frac{\partial l}{\partial θ} (r_{1} \sin θ_{1} + r_{2} \sin θ_{2})

- \frac{\partial d}{\partial θ} (r_{1} \cos θ_{1} + r_{2} \cos θ_{2})}

= \frac{f_{d}}{r_{1} + r_{2}} {{(r_{1} \sin θ_{1} + r_{2} \sin θ_{2})}^{2} + {(r_{1} \cos θ_{1} + r_{2} \cos θ_{2})}^{2}}

= \frac{f_{d}}{r_{1} + r_{2}} {r_{1}^{2} + r_{2}^{2} + 2 r_{1} r_{2} \cos (θ_{1} - θ_{2})},

\frac{\partial^{2} P}{\partial p_{i}^{2}} = r_{i}^{2} {\frac{f_{d}}{r_{1} + r_{2}} + \frac{1}{γ_{i}}}, i = 1, 2,

\frac{\partial^{2} P}{\partial p_{i} \partial θ} = - \frac{f_{d}}{r_{1} + r_{2}} {r_{i}^{2} + r_{1} r_{2} \cos (θ_{1} - θ_{2})}, i = 1, 2,

\frac{\partial^{2} P}{\partial p_{2} \partial p_{1}} = - \frac{r_{1} r_{2} f_{d}}{r_{1} + r_{2}} \cos (θ_{1} - θ_{2}) .