## Final Report Summary - COMMEC (Computational Geometry for efficient Motion Design by implementation through mechanisms)

Summary description of the project objectives

According to the research objectives described in part B / work plan, there are three major accomplishments to be sought. A geometric analysis of various sets of poses of a spatial system (or even a desired workspace of a joint articulated rigid body system) will build the groundwork for relating the interpolation tasks on these sets to the design of suitable mechanisms that can perform these tasks. Specifically, with emphasis on the number of poses, and on additional velocity and acceleration constraints (infinitesimal poses).

Afterwards, a systematic analysis of the workspaces of kinematic chains and mechanisms built up of kinematic chains with lower complexity will be achieved. For this purpose, computational methods and concepts coming from geometry, but also from algebra, will be adapted and developed. The inverse task of synthesising kinematic chains from their workspaces will be a key issue to be solved in order to connect the interpolation task with the design of suitable mechanisms.

Description of work performed since the beginning of the project

In order to achieve the research objectives described in part B / work plan, we started investigating a general motion task in spatial kinematics. Starting from spherical and planar kinematics and elabourating similarities and differences as spherical and planar kinematics are comprised by spatial kinematics. Deviating from the research objectives described in the work plan, we concluded that for reasons of applicability the general motion task shall be extended to motion varieties which are described by polynomial equations. They occur especially as constraint manifolds of kinematic chains as described in part B. Specifically, systems of merely linear equations as simple examples are subjects of our research, as described in work package B.

For one-parameter motions, we translated the k-th order neighborhood in an instant into a geometric language (of the kinematic point model), which allows the interpretation of k+1 infinitesimal poses of a spatial system. Furthermore, their connection with k-th order properties of the motion. In doing so, we could achieve a classification of the general motion task for k < 3.

The interpolation of a sequence of k+1 poses by rational one-parameter spline motions of low degree was also a matter during the past period. Additional velocity information are here interpretable as additional infinitesimal poses. The special construction allows the implementation of the motion interpolant by mechanisms / kinematic chains of low complexity (a minimal number of primer mover, that is here either one or two).

During the running time of the project, we worked also on two possible extensions of approaches that have been introduced for the one-parameter case. On the one hand, we started studying k-parameter spatial motion varieties which are described by merely linear equations. They might be used in the same manner: as rational approximate motion in a given neighborhood of a motion's instant preserving velocity and acceleration constraints. Furthermore, we believe that they are performable by mechanisms / kinematic chains of as low as possible complexity.

Our review activities for scientific journals such as Journal of Mechanism and Machine Theory made us aware of current developments in our research field. It was suggestive of one of the articles that we can combine interpolation tasks with low complexity implementation also for higher-degree one-parameter motions, which build a second direction in our further research.

Description of the main results achieved so far

We forecast high impact on the connection between motion interpolation and their performability by mechanisms of low complexity, as described in part B of the proposal. In this context, we could introduce one-parameter rational motions of low degree that approximate a general one-parameter motions at one instant preserving velocity and acceleration information. Their particular design, they occur as work space of certain mechanisms / kinematic chains of low complexity, ensures performability: only a minimal number of primer mover is necessary, that is here either one or two.

Another result achieved so far is the interpolation of data, consisting of sequences of k+1 poses (k can be chosen arbitrarily) of a spatial system, each equipped with velocity information, by one-parameter spline motions of low degree. Here, we applied a well-known principle, and complemented the generic case by classifying all possible motions with low degree. The approach is based on the classification of the spatial motion task. One feature of this interpolation method might be the piecewise performability of these motions by mechanisms / kinematic chains of low complexity: only a minimal number of primer mover are necessary, that is here either one or two.

Expected final results and their potential impact and use

Based on the project objectives described in part B (see summary above), universality theorems as overall results might be expected which state the connection between both the generation of suitable k-parameter rational motions associated with given data and their performability by certain mechanisms / kinematic chains of sufficiently low complexity.

Based on examples achieved so far, we suppose the question whether certain motion data allow a solution by mechanisms / kinematic chains low complexity or not might be of economic significance for mechanism design: up to now, numerous motion tasks are solved by fully mobile manipulators which are not necessary for reasons of economic expediency. The results we are focusing on would help to raise the well-know Burmester theory for planar kinematics onto the spatial case. That is, engineers might be enable to work on guidelines specifying mechanical solutions with low complexity for specific spatial motion tasks.

The results achieved so far confirm our expectations described in work plan.

According to the research objectives described in part B / work plan, there are three major accomplishments to be sought. A geometric analysis of various sets of poses of a spatial system (or even a desired workspace of a joint articulated rigid body system) will build the groundwork for relating the interpolation tasks on these sets to the design of suitable mechanisms that can perform these tasks. Specifically, with emphasis on the number of poses, and on additional velocity and acceleration constraints (infinitesimal poses).

Afterwards, a systematic analysis of the workspaces of kinematic chains and mechanisms built up of kinematic chains with lower complexity will be achieved. For this purpose, computational methods and concepts coming from geometry, but also from algebra, will be adapted and developed. The inverse task of synthesising kinematic chains from their workspaces will be a key issue to be solved in order to connect the interpolation task with the design of suitable mechanisms.

Description of work performed since the beginning of the project

In order to achieve the research objectives described in part B / work plan, we started investigating a general motion task in spatial kinematics. Starting from spherical and planar kinematics and elabourating similarities and differences as spherical and planar kinematics are comprised by spatial kinematics. Deviating from the research objectives described in the work plan, we concluded that for reasons of applicability the general motion task shall be extended to motion varieties which are described by polynomial equations. They occur especially as constraint manifolds of kinematic chains as described in part B. Specifically, systems of merely linear equations as simple examples are subjects of our research, as described in work package B.

For one-parameter motions, we translated the k-th order neighborhood in an instant into a geometric language (of the kinematic point model), which allows the interpretation of k+1 infinitesimal poses of a spatial system. Furthermore, their connection with k-th order properties of the motion. In doing so, we could achieve a classification of the general motion task for k < 3.

The interpolation of a sequence of k+1 poses by rational one-parameter spline motions of low degree was also a matter during the past period. Additional velocity information are here interpretable as additional infinitesimal poses. The special construction allows the implementation of the motion interpolant by mechanisms / kinematic chains of low complexity (a minimal number of primer mover, that is here either one or two).

During the running time of the project, we worked also on two possible extensions of approaches that have been introduced for the one-parameter case. On the one hand, we started studying k-parameter spatial motion varieties which are described by merely linear equations. They might be used in the same manner: as rational approximate motion in a given neighborhood of a motion's instant preserving velocity and acceleration constraints. Furthermore, we believe that they are performable by mechanisms / kinematic chains of as low as possible complexity.

Our review activities for scientific journals such as Journal of Mechanism and Machine Theory made us aware of current developments in our research field. It was suggestive of one of the articles that we can combine interpolation tasks with low complexity implementation also for higher-degree one-parameter motions, which build a second direction in our further research.

Description of the main results achieved so far

We forecast high impact on the connection between motion interpolation and their performability by mechanisms of low complexity, as described in part B of the proposal. In this context, we could introduce one-parameter rational motions of low degree that approximate a general one-parameter motions at one instant preserving velocity and acceleration information. Their particular design, they occur as work space of certain mechanisms / kinematic chains of low complexity, ensures performability: only a minimal number of primer mover is necessary, that is here either one or two.

Another result achieved so far is the interpolation of data, consisting of sequences of k+1 poses (k can be chosen arbitrarily) of a spatial system, each equipped with velocity information, by one-parameter spline motions of low degree. Here, we applied a well-known principle, and complemented the generic case by classifying all possible motions with low degree. The approach is based on the classification of the spatial motion task. One feature of this interpolation method might be the piecewise performability of these motions by mechanisms / kinematic chains of low complexity: only a minimal number of primer mover are necessary, that is here either one or two.

Expected final results and their potential impact and use

Based on the project objectives described in part B (see summary above), universality theorems as overall results might be expected which state the connection between both the generation of suitable k-parameter rational motions associated with given data and their performability by certain mechanisms / kinematic chains of sufficiently low complexity.

Based on examples achieved so far, we suppose the question whether certain motion data allow a solution by mechanisms / kinematic chains low complexity or not might be of economic significance for mechanism design: up to now, numerous motion tasks are solved by fully mobile manipulators which are not necessary for reasons of economic expediency. The results we are focusing on would help to raise the well-know Burmester theory for planar kinematics onto the spatial case. That is, engineers might be enable to work on guidelines specifying mechanical solutions with low complexity for specific spatial motion tasks.

The results achieved so far confirm our expectations described in work plan.