Use of observers for the inversion of nonlinear maps: an application to the inverse kinematic problem

The authors define in a general framework several problems related to inverse kinematics already present in the literature. Six problems are introduced: given an extended reference trajectory, the first three problems consist in finding, both in an exact and in an approximate way, an extended inverse reference trajectory. The other three problems consist in finding, both in an exact and in an approximate way, an extended inverse reference trajectory without the knowledge of the time derivatives of the reference trajectory. It is then shown how a solution to the above problems can be obtained by constructing an observer for a certain time-varying nonlinear system, which can be associated with the direct kinematics and the given reference trajectory, and by proposing some observer structures for such a nonlinear system. All the definitions and the theory reported are applied to a simple example: a two-link robot arm