Velocity similarity principle V(θ., ṡ, ub, Qb) = DabV (θ., ṡ, ua, Qa)D ab–1 is presented and used to derive several useful forms of the Jacobian matrix for the manipulator from its basic kinematic equations in 4 X 4 matrix form. The zero reference position representation is used and, therefore, the base system is the only coordinate system utilized in the derivations. For manipulators with a spherical wrist, a modified form of the Jacobian is presented in which the Jacobian columns corresponding to the regional structure are completely decoupled from those corresponding to the wrist structure.

Recent development of sensor knowledge representation by the use of a certainty grid has been extensive and shown the usefulness of the grid-based concept for robot navigation.

Yet the methodology was not perfect. This paper introduces the Bayesian formula into the certainty grid representation to overcome some difficulties of ad hoc formula that has been the only way of updating the grids. The complete derivation of the proposed updating formula is given and proved to be able to accurately identify the simulated world. Also, the paper suggests two updating models: context-sensitive and context-free. Both of them were shown to be usable through simulation in real world modeling.