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actuator torques adjoint variables angular velocities bang-bang bang-bang control boundary conditions boundary value problem Cartesian space center of mass Chapter computed control for link control of robotic derivative determined displacement vector double pendulum dynamic equations dynamic programming Dynamic Systems energy minimization evaluate feedback given Hamiltonian IEEE Transactions implementation inequality constraints inertia iterations Jacobian joint displacements joint torques/forces joint velocities Journal of Dynamic kth time step Lagrangian linear manipulator dynamics maximum principle Measurement and Control minimization problem minimum motion non-linear off-line on-line controller optimal control optimal path planning Optimal torque optimal trajectory planning Optimal velocity profile outlined parameters penalty function performance index point k+1 possible velocity procedure profile of link qj(k robot manipulator segment shown in Figure Simulation Results solution solve specified stage terminal manifold torque for link trajectory planning algorithm trajectory planning problem vector velocity and acceleration velocity at point velocity constrained region velocity discretization velocity of joint