On the false positives and false negatives of the Jacobian Matrix in kinematically redundant parallel mechanisms

The Jacobian matrix is a highly popular tool for the control and performance analysis of closed-loop robots. Its usefulness in parallel mechanisms is certainly apparent, and its application to solve motion planning problems, or other higher level questions, has been seldom queried, or limited to non-redundant systems. In this paper, we discuss the shortcomings of the use of the Jacobian matrix under redundancy, in particular when applied to kinematically redundant parallel architectures with non-serially connected actuators. These architectures have become fairly popular recently as they allow the end-effector to achieve full rotations, which is an impossible task with traditional topologies. The problems with the Jacobian matrix in these novel systems arise from the need to eliminate redundant variables when forming it, resulting in both situations where the Jacobian incorrectly identifies singularities (false positive), and where it fails to identify singularities (false negative). These issues have thus far remained unaddressed in the literature. We highlight these limitations herein by demonstrating several cases using numerical examples of both planar and spatial architectures

Self-Motions of General 3-RPR Planar Parallel Robots

This paper studies the kinematic geometry of general 3-RPR planar parallel robots with actuated base joints. These robots, while largely overlooked, have simple direct kinematics and large singularity-free workspace. Furthermore, their kinematic geometry is the same as that of a newly developed parallel robot with SCARA-type motions. Starting from the direct and inverse kinematic model, the expressions for the singularity loci of 3-RPR planar parallel robots are determined. Then, the global behaviour at all singularities is geometrically described by studying the degeneracy of the direct kinematic model. Special cases of self-motions are then examined and the degree of freedom gained in such special configurations is kinematically interpreted. Finally, a practical example is discussed and experimental validations performed on an actual robot prototype are presented

Dynamic Analysis of Parallel Manipulators under the Singularity-Consistent Parameterization

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Parallel robots with unconventional joints to achieve under-actuation and reconfigurability

The aim of the thesis is to define, analyze, and verify through simulations and practical implementations, parallel robots with unconventional joints that allow them to be under-actuated and/or reconfigurable. The new designs will be derived from the: * 6SPS robot (alternatively 6UPS or 6SPU, depending on the implementation) when considering the spatial case (i.e., robots with 3 degrees of freedom of rotation and 3 degrees of freedom of translation). * S-3SPS robot (alternatively S-3UPS or S-3SPU, depending on the implementation) when considering spherical robots (i.e., robots with 3 degrees of freedom of rotation). In both cases, we will see how, through certain geometric transformations, some of the standard joints can be replaced by lockable or non-holonomic joints. These substitutions permit reducing the number of legs (and hence the number of actuators needed to control the robot), without losing the robot's ability to bring its mobile platform to any position and orientation (in case of a spatial robot), or to any orientation (in case of a spherical robot), within its workspace. The expected benefit of these new designs is to obtain parallel robots with: * larger working spaces because the possibility of collisions between legs is reduced, and the number of joints (with their intrinsic range limitations) is also reduced; * lower weight because the number of actuators and joints is reduced; and * lower cost because the number of actuators and controllers is also reduced. The elimination of an actuator and the introduction of a motion constraint reduces in one the dimension of the space of allowed velocities attainable from a given configuration. As a result, it will be necessary, in general, to plan maneuvers to reach the desired configuration for the moving platform. Therefore, the obtained robots will only be suitable for applications where accuracy is required in the final position and a certain margin of error is acceptable in the generated trajectories.El objetivo de esta tesis es definir, analizar y verificar, mediante simulaciones e implementaciones prácticas, robots paralelos con articulaciones no-convencionales con el fin de incorporarles propiedades de sub-actuación y reconfigurabilidad. Los nuevos diseños se basaran en robots paralelos tipo: * 6SPS (alternativamente 6UPS o 6SPU, dependiendo de la implementación) para el caso de robot espacial (es decir, robots con 3 grados de libertad de rotación y de 3 grados de libertad de la traducción). * S-3SPS (alternativamente S-3UPS o S-3SPU, dependiendo de la implementación) para el caso de robot esférico (es decir, robots con 3 grados de libertad de rotación). En ambos casos, veremos cómo, a través de ciertas transformaciones geométricas, algunas de la articulaciones convencionales pueden ser sustituidas por articulaciones bloqueables o no holonómicos. Estas sustituciones permiten la reducción de la número de patas (y por tanto el número de actuadores necesarios para controlar el robot), sin perder la capacidad del robot para llevar su plataforma móvil a cualquier posición y orientación (en el caso de un robot espacial), o para cualquier orientación (en el caso de un robot esférico), dentro de su espacio de trabajo. El beneficio esperado de estos nuevos diseños es la obtención de robots paralelos con: * Espacios de trabajo mayores debido a que la posibilidad de colisiones entre las patas se reduce, y el número de articulaciones (con sus limitaciones intrínsecas de rango) también se reduce; * Menor peso debido a que el número de actuadores y de articulaciones se reduce; y * Un menor coste debido a que el número de actuadores y controladores también se reduce. La eliminación de un actuador y la introducción de una restricción de movimiento reduce, en uno, la dimensión del espacio de velocidades alcanzables para una configuración dada. Como resultado, será necesario, en general, planificar maniobras para llegar a la configuración deseada de la plataforma móvil. Por lo tanto, los robots obtenidos sólo serán adecuados para aplicaciones donde la precisión se requiera en la posición final y exista un cierto margen de error aceptable en las trayectorias generadasPostprint (published version

A general method for the numerical computation of manipulator singularity sets

The analysis of singularities is central to the development and control of a manipulator. However, existing methods for singularity set computation still concentrate on specific classes of manipulators. The absence of general methods able to perform such computation on a large class of manipulators is problematic because it hinders the analysis of unconventional manipulators and the development of new robot topologies. The purpose of this paper is to provide such a method for nonredundant mechanisms with algebraic lower pairs and designated input and output speeds. We formulate systems of equations that describe the whole singularity set and each one of the singularity types independently, and show how to compute the configurations in each type using a numerical technique based on linear relaxations. The method can be used to analyze manipulators with arbitrary geometry, and it isolates the singularities with the desired accuracy. We illustrate the formulation of the conditions and their numerical solution with examples, and use 3-D projections to visualize the complex partitions of the configuration space induced by the singularities.Preprin

Parallel robots with unconventional joints to achieve under-actuation and reconfigurability

The aim of the thesis is to define, analyze, and verify through simulations and practical implementations, parallel robots with unconventional joints that allow them to be under-actuated and/or reconfigurable. The new designs will be derived from the: * 6SPS robot (alternatively 6UPS or 6SPU, depending on the implementation) when considering the spatial case (i.e., robots with 3 degrees of freedom of rotation and 3 degrees of freedom of translation). * S-3SPS robot (alternatively S-3UPS or S-3SPU, depending on the implementation) when considering spherical robots (i.e., robots with 3 degrees of freedom of rotation). In both cases, we will see how, through certain geometric transformations, some of the standard joints can be replaced by lockable or non-holonomic joints. These substitutions permit reducing the number of legs (and hence the number of actuators needed to control the robot), without losing the robot's ability to bring its mobile platform to any position and orientation (in case of a spatial robot), or to any orientation (in case of a spherical robot), within its workspace. The expected benefit of these new designs is to obtain parallel robots with: * larger working spaces because the possibility of collisions between legs is reduced, and the number of joints (with their intrinsic range limitations) is also reduced; * lower weight because the number of actuators and joints is reduced; and * lower cost because the number of actuators and controllers is also reduced. The elimination of an actuator and the introduction of a motion constraint reduces in one the dimension of the space of allowed velocities attainable from a given configuration. As a result, it will be necessary, in general, to plan maneuvers to reach the desired configuration for the moving platform. Therefore, the obtained robots will only be suitable for applications where accuracy is required in the final position and a certain margin of error is acceptable in the generated trajectories.El objetivo de esta tesis es definir, analizar y verificar, mediante simulaciones e implementaciones prácticas, robots paralelos con articulaciones no-convencionales con el fin de incorporarles propiedades de sub-actuación y reconfigurabilidad. Los nuevos diseños se basaran en robots paralelos tipo: * 6SPS (alternativamente 6UPS o 6SPU, dependiendo de la implementación) para el caso de robot espacial (es decir, robots con 3 grados de libertad de rotación y de 3 grados de libertad de la traducción). * S-3SPS (alternativamente S-3UPS o S-3SPU, dependiendo de la implementación) para el caso de robot esférico (es decir, robots con 3 grados de libertad de rotación). En ambos casos, veremos cómo, a través de ciertas transformaciones geométricas, algunas de la articulaciones convencionales pueden ser sustituidas por articulaciones bloqueables o no holonómicos. Estas sustituciones permiten la reducción de la número de patas (y por tanto el número de actuadores necesarios para controlar el robot), sin perder la capacidad del robot para llevar su plataforma móvil a cualquier posición y orientación (en el caso de un robot espacial), o para cualquier orientación (en el caso de un robot esférico), dentro de su espacio de trabajo. El beneficio esperado de estos nuevos diseños es la obtención de robots paralelos con: * Espacios de trabajo mayores debido a que la posibilidad de colisiones entre las patas se reduce, y el número de articulaciones (con sus limitaciones intrínsecas de rango) también se reduce; * Menor peso debido a que el número de actuadores y de articulaciones se reduce; y * Un menor coste debido a que el número de actuadores y controladores también se reduce. La eliminación de un actuador y la introducción de una restricción de movimiento reduce, en uno, la dimensión del espacio de velocidades alcanzables para una configuración dada. Como resultado, será necesario, en general, planificar maniobras para llegar a la configuración deseada de la plataforma móvil. Por lo tanto, los robots obtenidos sólo serán adecuados para aplicaciones donde la precisión se requiera en la posición final y exista un cierto margen de error aceptable en las trayectorias generada

A general method for the numerical computation of manipulator singularity sets

The analysis of singularities is central to the development and control of a manipulator. However, existing methods for singularity set computation still concentrate on specific classes of manipulators. The absence of general methods able to perform such computation on a large class of manipulators is problematic because it hinders the analysis of unconventional manipulators and the development of new robot topologies. The purpose of this paper is to provide such a method for nonredundant mechanisms with algebraic lower pairs and designated input and output speeds. We formulate systems of equations that describe the whole singularity set and each one of the singularity types independently, and show how to compute the configurations in each type using a numerical technique based on linear relaxations. The method can be used to analyze manipulators with arbitrary geometry, and it isolates the singularities with the desired accuracy. We illustrate the formulation of the conditions and their numerical solution with examples, and use 3-D projections to visualize the complex partitions of the configuration space induced by the singularities.This work has been partially supported by the Spanish Ministry of Economy and Competitiveness under contract DPI2010-18449, and by a Juan de la Cierva contract supporting the fourth author.Peer Reviewe

Kinematics and Robot Design II (KaRD2019) and III (KaRD2020)

This volume collects papers published in two Special Issues “Kinematics and Robot Design II, KaRD2019” (https://www.mdpi.com/journal/robotics/special_issues/KRD2019) and “Kinematics and Robot Design III, KaRD2020” (https://www.mdpi.com/journal/robotics/special_issues/KaRD2020), which are the second and third issues of the KaRD Special Issue series hosted by the open access journal robotics.The KaRD series is an open environment where researchers present their works and discuss all topics focused on the many aspects that involve kinematics in the design of robotic/automatic systems. It aims at being an established reference for researchers in the field as other serial international conferences/publications are. Even though the KaRD series publishes one Special Issue per year, all the received papers are peer-reviewed as soon as they are submitted and, if accepted, they are immediately published in MDPI Robotics. Kinematics is so intimately related to the design of robotic/automatic systems that the admitted topics of the KaRD series practically cover all the subjects normally present in well-established international conferences on “mechanisms and robotics”.KaRD2019 together with KaRD2020 received 22 papers and, after the peer-review process, accepted only 17 papers. The accepted papers cover problems related to theoretical/computational kinematics, to biomedical engineering and to other design/applicative aspects