Project completed.

## Inverse Kinematics for Redundant Robots

### Description of project

After the developement of methods to automatically generate the equations of the inverse kinematic problem (IKP) of serial link robots, we have focused on the applicability of the obtained results to redundant serial robots. This work also has been supported by the DFG.

The key idea of our approach is to decompose the joints of some redundant robot into redundant and non-redundant joints. The non-redundant joints have to be solved by the IKP. This has been done by using SKIP to generate the appropriate IKs. Furthermore, the integration of additional constraints for the end effector trajectories of serial link robots has been investigated, to get single-valued positions for the redundant joints. First, the algebraic approach of SKIP has been followed up to solve this problem. However, using this method, only algebraic constraints can be integrated. Thus, simple problems - like obstacle avoidance between a planar robot and a point shaped obstacle - lead to non practicable univariate polynoms of degree 8. Therefore, a hybrid local approach and a new method using a discretization of the configuration space have been developed.

In this section we will only explain the global discretization approach: Using the decomposition into redundant and non-redundant joints, the constraints, e. g. defined in the cartesian workspace, are transformed into a configuration space spanned by the redundant joints and the time axis. Due to high complexity, a description with parameterized surfaces isn't possible. Instead, a discretization of the work space is applied. Thus, the constraints can be transformed for each space element separately. This can be computed concurrently, because the computation is completely independent.

Subsequently, a path is searched inside the configuration space with a recursive filling algorithm, which connects the start and goal points. As planning is only allowed along the positive time axis, some simplifications are possible resulting in a time speed-up and in reduced memory requirements. An example of a planned collision free path for an articulated robot with 8 joints is shown in the following picture.

On the basis of closed form solutions of the IKP some efficient algorithms for redundant robots and other applications have been developed.

The key idea of our approach is to decompose the joints of some redundant robot into redundant and non-redundant joints. The non-redundant joints have to be solved by the IKP. This has been done by using SKIP to generate the appropriate IKs. Furthermore, the integration of additional constraints for the end effector trajectories of serial link robots has been investigated, to get single-valued positions for the redundant joints. First, the algebraic approach of SKIP has been followed up to solve this problem. However, using this method, only algebraic constraints can be integrated. Thus, simple problems - like obstacle avoidance between a planar robot and a point shaped obstacle - lead to non practicable univariate polynoms of degree 8. Therefore, a hybrid local approach and a new method using a discretization of the configuration space have been developed.

In this section we will only explain the global discretization approach: Using the decomposition into redundant and non-redundant joints, the constraints, e. g. defined in the cartesian workspace, are transformed into a configuration space spanned by the redundant joints and the time axis. Due to high complexity, a description with parameterized surfaces isn't possible. Instead, a discretization of the work space is applied. Thus, the constraints can be transformed for each space element separately. This can be computed concurrently, because the computation is completely independent.

Subsequently, a path is searched inside the configuration space with a recursive filling algorithm, which connects the start and goal points. As planning is only allowed along the positive time axis, some simplifications are possible resulting in a time speed-up and in reduced memory requirements. An example of a planned collision free path for an articulated robot with 8 joints is shown in the following picture.

On the basis of closed form solutions of the IKP some efficient algorithms for redundant robots and other applications have been developed.

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