Parallel manipulators with two end-effectors: Getting a grip on Jacobian-based stiffness analysis

Robots that are developed for applications which require a high stiffness-over-inertia ratio, such as pick-and-place robots, machining robots, or haptic devices, are often based on parallel manipulators. Parallel manipulators connect an end-effector to an inertial base using multiple serial kinematic chains. This architecture enables the design of robots with all actuators located at the base, which greatly reduces the effective inertia with no detrimental effect on the stiffness, thus improving the stiffness-over-inertia ratio.

One of the limitations of current parallel manipulators is that gripping requires an additional, dedicated subsystem, which either degrades performance or can only handle objects with a flat surface. A promising alternative solution for future gripping robots are parallel manipulators with two end-effectors (PM2Es). These PM2Es facilitate mechanical gripping using an internal closed-loop chain, which enables gripping robots that have all their actuators located at the base. As such, PM2Es combine mechanical gripping with a favorable stiffness-over-inertia ratio. Because of the potential benefits in both cost and hygiene, the integration of compliant joints is identified as a second promising development for future gripping robots. The benefits of PM2Es and those of compliant joints can be enjoyed simultaneously if compliant PM2Es are designed.

However, it is currently not possible to effectively design compliant PM2Es, because existing stiffness analyses methods do not directly apply to PM2Es. To overcome this limitation, in this thesis a stiffness analysis method is developed that is also valid for compliant PM2Es.

In Chapter 2 insights from screw theory are used to develop a novel Jacobian-based stiffness analysis method for parallel manipulators, which is more general than existing methods. The stiffness analysis method takes the influence of actuators, compliant joints, as well as structural elements into account, and also considers the effect of loading. Stiffness in the constrained directions of individual serial chains is represented using virtual joints, which allows structural compliance to be considered for any parallel manipulator with non-redundant legs. This includes lower mobility parallel manipulators, which have less than six degrees of freedom. Loading is taken into account using a term that depends on the derivative of a Jacobian matrix.

The developed stiffness analysis method is experimentally validated in Chapter 3. It is shown that the accuracy of a stiffness analysis improves if the effect of loading is considered and also if structural compliance is included in the analysis. The example stiffness matrices in Chapter 3 are all symmetric, which is in line with the definition of a stiffness matrix.

In contrast, Griffis and Duffy (1993) found asymmetric stiffness matrices when they analyzed two mechanisms under loading. Loading cannot be part of a stiffness analysis if it can lead to asymmetric matrices, which explains the extensive discussions surrounding their work. Chapter 4 demonstrates that the asymmetry in the work by Griffis and Duffy can be explained as a modeling inconsistency. It is also shown that consistent modeling results in symmetric matrices, which supports the notion that loading is an integral part of a stiffness analysis of parallel manipulators.

Chapter 5 presents a systematic approach for the Jacobian analysis of PM2Es in order to extend the Jacobian-based stiffness analysis method of Chapter 2 to PM2Es. Previously, Jacobian analyses of PM2Es have only been performed for specific examples, but this thesis introduces structure to the Jacobian analysis of PM2Es by considering each end-effector as a rigid body with its own six-dimensional motion vector, and by representing a PM2E as an organization of serial chains. In Chapter 5 it is also shown that the structural compliance of internal serial chains must be considered in order to deal with relative degrees of freedom that are constrained between the two end-effectors.

In Chapter 6 the stiffness analysis method of Chapter 2 and the Jacobian analysis of Chapter 5 are integrated and the very first stiffness matrices of PM2Es are presented. An experimental verification of these stiffness matrices demonstrates that the developed stiffness analysis method also applies to compliant parallel manipulators with two end-effectors. Therefore, the method introduced in this thesis is ready to be used for the effective design of novel gripping robots.