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A.G.L. Hoevenaars
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The dynamic performance of a parallel manipulator is often limited by its natural frequencies. These frequencies are a result of the controlled actuator stiffness and the mechanical design. Typically, only the lowest natural frequencies are of interest. However, existing methods to analyze natural frequencies of parallel manipulators with flexible links have been developed in a high-dimensional generalized coordinate space. This paper presents a novel natural frequency analysis method for parallel manipulators that focuses on the lowest natural frequencies and which expresses the corresponding eigenmodes in an end-effector Cartesian reference frame. The analysis method combines a flexible-body stiffness analysis method with a rigid-body inertia analysis method. As an example, the two lowest natural frequencies of the Heli4 robot are modeled and measured over a range of controlled actuator stiffness values for a single pose. The results show that structural compliance in the mechanical parts of the Heli4 robot is significant for actuator stiffness values above roughly 1000 Nm/rad. For an actuator stiffness value of 4000 Nm/rad, consideration of the structural compliance reduces the maximum prediction error for the two lowest natural frequencies from 33% to 6%.
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The dynamic performance of a parallel manipulator is often limited by its natural frequencies. These frequencies are a result of the controlled actuator stiffness and the mechanical design. Typically, only the lowest natural frequencies are of interest. However, existing methods to analyze natural frequencies of parallel manipulators with flexible links have been developed in a high-dimensional generalized coordinate space. This paper presents a novel natural frequency analysis method for parallel manipulators that focuses on the lowest natural frequencies and which expresses the corresponding eigenmodes in an end-effector Cartesian reference frame. The analysis method combines a flexible-body stiffness analysis method with a rigid-body inertia analysis method. As an example, the two lowest natural frequencies of the Heli4 robot are modeled and measured over a range of controlled actuator stiffness values for a single pose. The results show that structural compliance in the mechanical parts of the Heli4 robot is significant for actuator stiffness values above roughly 1000 Nm/rad. For an actuator stiffness value of 4000 Nm/rad, consideration of the structural compliance reduces the maximum prediction error for the two lowest natural frequencies from 33% to 6%.
Parallel manipulators with two end-effectors (PM2Es) enable the design of gripping robots with high dynamic performance. The gripping action is enabled by internal, relative degrees of freedom (DoFs) between the two end-effectors. Many standard methods for the analysis and control of parallel manipulators rely on a Jacobian, where a complete Jacobian analysis includes constraint relations. These constraint relations have not been consistently included in previous analyses of PM2Es, while they are specifically relevant for PM2Es because constraints play an important role in the static force analysis of a PM2E. This is because wrenches applied by the actuators can be transferred to the end-effectors through internal constraints, an effect which is not captured by kinematic relations alone. This paper presents a systematic approach to perform the Jacobian analysis of PM2Es, which is based on screw theory, and that takes all constraint relations into account. The approach is applied to a PM2E with three legs and one internal closed-loop chain. An example mechanism was built to experimentally validate the resulting Jacobian analysis using a static force analysis.
...
Parallel manipulators with two end-effectors (PM2Es) enable the design of gripping robots with high dynamic performance. The gripping action is enabled by internal, relative degrees of freedom (DoFs) between the two end-effectors. Many standard methods for the analysis and control of parallel manipulators rely on a Jacobian, where a complete Jacobian analysis includes constraint relations. These constraint relations have not been consistently included in previous analyses of PM2Es, while they are specifically relevant for PM2Es because constraints play an important role in the static force analysis of a PM2E. This is because wrenches applied by the actuators can be transferred to the end-effectors through internal constraints, an effect which is not captured by kinematic relations alone. This paper presents a systematic approach to perform the Jacobian analysis of PM2Es, which is based on screw theory, and that takes all constraint relations into account. The approach is applied to a PM2E with three legs and one internal closed-loop chain. An example mechanism was built to experimentally validate the resulting Jacobian analysis using a static force analysis.
A complete stiffness analysis of a parallel manipulator considers the structural compliance of all elements, both in designed degrees-of-freedom (DoFs) and constrained DoFs, and also includes the effect of preloading. This paper presents the experimental validation of a Jacobian-based stiffness analysis method for parallel manipulators with nonredundant legs, which considers all those aspects, and which can be applied to limited-DoF parallel manipulators. The experimental validation was performed by comparing differential wrench measurements with predictions based on stiffness analyses with increasing levels of detail. For this purpose, two passive parallel mechanisms were designed, namely, a planar 3DoF mechanism and a spatial 1DoF mechanism. For these mechanisms, it was shown that a stiffness analysis becomes more accurate if preloading and structural compliance are considered.
...
A complete stiffness analysis of a parallel manipulator considers the structural compliance of all elements, both in designed degrees-of-freedom (DoFs) and constrained DoFs, and also includes the effect of preloading. This paper presents the experimental validation of a Jacobian-based stiffness analysis method for parallel manipulators with nonredundant legs, which considers all those aspects, and which can be applied to limited-DoF parallel manipulators. The experimental validation was performed by comparing differential wrench measurements with predictions based on stiffness analyses with increasing levels of detail. For this purpose, two passive parallel mechanisms were designed, namely, a planar 3DoF mechanism and a spatial 1DoF mechanism. For these mechanisms, it was shown that a stiffness analysis becomes more accurate if preloading and structural compliance are considered.
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. ...
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. ...
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.
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.
The Cartesian stiffness analysis of mechanisms and robotic manipulators is often performed using Jacobian-based stiffness analysis methods. In general it is necessary to consider the effect of loading, which introduces a Jacobian-derivative term. However, early examples of stiffness analyses of parallel manipulators in which the effect of loading was considered resulted in asymmetric stiffness matrices, which go against the definition of stiffness. By replicating those first analyses using an alternative formulation of the stiffness analysis, this paper finds that the earlier observed asymmetry can be explained as a modeling inconsistency. If that inconsistency is corrected, symmetric matrices are obtained, which supports the notion that the Jacobian-derivative term is an integral part of the stiffness analysis of parallel manipulators and mechanisms.
...
The Cartesian stiffness analysis of mechanisms and robotic manipulators is often performed using Jacobian-based stiffness analysis methods. In general it is necessary to consider the effect of loading, which introduces a Jacobian-derivative term. However, early examples of stiffness analyses of parallel manipulators in which the effect of loading was considered resulted in asymmetric stiffness matrices, which go against the definition of stiffness. By replicating those first analyses using an alternative formulation of the stiffness analysis, this paper finds that the earlier observed asymmetry can be explained as a modeling inconsistency. If that inconsistency is corrected, symmetric matrices are obtained, which supports the notion that the Jacobian-derivative term is an integral part of the stiffness analysis of parallel manipulators and mechanisms.