An algebraic method to check the singularity-free paths for parallel robots

Trajectory planning is a critical step while programming the parallel manipulators in a robotic cell. The main problem arises when there exists a singular configuration between the two poses of the end-effectors while discretizing the path with a classical approach. This paper presents an algebraic method to check the feasibility of any given trajectories in the workspace. The solutions of the polynomial equations associated with the tra-jectories are projected in the joint space using Gr{\"o}bner based elimination methods and the remaining equations are expressed in a parametric form where the articular variables are functions of time t unlike any numerical or discretization method. These formal computations allow to write the Jacobian of the manip-ulator as a function of time and to check if its determinant can vanish between two poses. Another benefit of this approach is to use a largest workspace with a more complex shape than a cube, cylinder or sphere. For the Orthoglide, a three degrees of freedom parallel robot, three different trajectories are used to illustrate this method.Comment: Appears in International Design Engineering Technical Conferences & Computers and Information in Engineering Conference , Aug 2015, Boston, United States. 201

On the determination of cusp points of 3-R\underline{P}R parallel manipulators

This paper investigates the cuspidal configurations of 3-RPR parallel manipulators that may appear on their singular surfaces in the joint space. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a non-singular change of assembly mode. In previous works, the cusp points were calculated in sections of the joint space by solving a 24th-degree polynomial without any proof that this polynomial was the only one that gives all solutions. The purpose of this study is to propose a rigorous methodology to determine the cusp points of 3-R\underline{P}R manipulators and to certify that all cusp points are found. This methodology uses the notion of discriminant varieties and resorts to Gr\"obner bases for the solutions of systems of equations

A new 3-DOF 2T1R parallel mechanism: Topology design and kinematics

This article presents a new three-degree-of-freedom (3-DOF) parallel mechanism (PM) with two translations and one rotation (2T1R), designed based on the topological design theory of the parallel mechanism using position and orientation characteristics (POC). The PM is primarily intended for use in package sorting and delivery. The mobile platform of the PM moves along a translation axis, picks up objects from a conveyor belt, and tilts them to either side of the axis. We first calculate the PM's topological characteristics, such as the degree of freedom (DOF) and the degree of coupling, and provide its topological analytical formula to represent the topological information of the PM. Next, we solve the direct and inverse kinematic models based on the kinematic modelling principle using the topological features. The models are purely analytic and are broken down into a series of quadratic equations, making them suitable for use in an industrial robot. We also study the singular configurations to identify the serial and parallel singularities. Using the decoupling properties, we size the mechanism to address the package sorting and depositing problem using an algebraic approach. To determine the smallest segment lengths, we use a cylindrical algebraic decomposition to solve a system with inequalities.Comment: IDETC-CIE 2023 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, ASME, Aug 2023, Boston, Franc

An efficient finite element formulation of dynamics for a flexible robot with different type of joints

If two adjacent links of a flexible robot are connected via a revolute joint or a fixed prismatic joint, the relative motion of the next link will depend on both the joint motion and the elastic displacement of the distal end of the previous link. However, if the two adjacent links are connected via a sliding prismatic joint, the relative motion of the next link will depend additionally on the elastic deformation distributed along the previous link. Therefore, formulation of the motion equations for a multi-link flexible robot consisting of the revolute joints, the fixed prismatic joints and the sliding prismatic joints is challenging. In this study, the finite element kinematic and dynamic formulation was successfully developed and validated for the flexible robot, in which a transformation matrix is proposed to describe the kinematics of both the joint motion and the link deformation. Additionally, a new recursive formulation of the dynamic equations is introduced. As compared with the previous methods, the time complexity of the formulation is reduced by O(2η), where η is the number of finite elements on all links. The numerical examples and experiments were implemented to validate the proposed kinematic and dynamic modelling method

Time-Optimal Control by Iterating Forward and Backward in Time

When costumers look for a robot for their factory two things are important. The robot should be as efficient as possible, but still be cheap. In order to make the robot efficient the robot-controller has to know the dynamics of the robot and its limits. Based on these it can then generate a time-optimal plan (trajectory) for each movement. The standard way of generating a time-optimal trajectory with the exact dynamics and limits is very computationally heavy. Today most approaches do the planning based on the hardest limits on each axis of the robot. These values are then used even though the current limits might allow the robot to move faster. This means that the full capacity of the robot will not be utilized and because of this the efficiency is lowered. In this thesis a different method for generating time-optimal trajectories is tested. The approach is based on iterating forward and backward in time and finding the point where the two paths meet. This approach has the advantage of that it is based on simulating the system. Therefore more complex dynamics can be included in the planning by just calculating a value instead of complicating the optimization problem. Another benefit is that robot manufacturers usually create simulation models of new robots already. This means that very little extra effort would be needed to create the trajectory generator using this approach and this reduces development costs. Four different approaches for patching together the forward and backward paths are discussed in the thesis. The different techniques are tested on a simplified model of one servo axis of a robot and compared against a known time optimal solution for the simplified model. One of the techniques shows very good results and generates trajectories that are time-optimal

Large-scale automated investigation of free-falling paper shapes via iterative physical experimentation

Free-falling paper shapes exhibit rich, complex and varied behaviours which are extremely challenging to model analytically. Physical experimentation aids in system understanding but is time-consuming, sensitive to initial conditions and reliant on subjective visual behavioural classification. In this study robotics, computer vision and machine learning are used to autonomously fabricate, drop, analyse and classify the behaviours of hundreds of shapes. The system is validated by reproducing results for falling disks, which exhibit four falling styles: tumbling, chaotic, steady and periodic. A previously-determined mapping from a non-dimensional parameter space to behaviour groups is shown to be consistent with these new experiments for tumbling and chaotic behaviours. Steady or periodic behaviours, however, are observed in previously unseen areas of the parameter space. More complex hexagon, square and cross shapes are investigated, showing that the non-dimensional parameter space generalises to these shapes. The system highlights the potential of robotics for the investigation of complex physical systems, of which falling paper is one example, and provides a template for future investigation of such systems

Path Planning for Shepherding a Swarm in a Cluttered Environment using Differential Evolution

Shepherding involves herding a swarm of agents (\emph{sheep}) by another a control agent (\emph{sheepdog}) towards a goal. Multiple approaches have been documented in the literature to model this behaviour. In this paper, we present a modification to a well-known shepherding approach, and show, via simulation, that this modification improves shepherding efficacy. We then argue that given complexity arising from obstacles laden environments, path planning approaches could further enhance this model. To validate this hypothesis, we present a 2-stage evolutionary-based path planning algorithm for shepherding a swarm of agents in 2D environments. In the first stage, the algorithm attempts to find the best path for the sheepdog to move from its initial location to a strategic driving location behind the sheep. In the second stage, it calculates and optimises a path for the sheep. It does so by using \emph{way points} on that path as the sequential sub-goals for the sheepdog to aim towards. The proposed algorithm is evaluated in obstacle laden environments via simulation with further improvements achieved

ROBOTRAN: a powerful symbolic gnerator of multibody models

The computational efficiency of symbolic generation was at the root of the emergence of symbolic multibody programs in the eighties. At present, it remains an attractive feature of it since the exponential increase in modern computer performances naturally provides the opportunity to investigate larger systems and more sophisticated models for which real-time computation is a real asset. <br><br> Nowadays, in the context of mechatronic multibody systems, another interesting feature of the symbolic approach appears when dealing with enlarged multibody models, i.e. including electrical actuators, hydraulic devices, pneumatic suspensions, etc. and requiring specific analyses like control and optimization. Indeed, since symbolic multibody programs clearly distinguish the modeling phase from the analysis process, extracting the symbolic model, as well as some precious ingredients like analytical sensitivities, in order to export it towards any suitable environment (for control or optimization purposes) is quite straightforward. Symbolic multibody model portability is thus very attractive for the analysis of mechatronic applications. <br><br> In this context, the main features and recent developments of the ROBOTRAN software developed at the Université catholique de Louvain (Belgium) are reviewed in this paper and illustrated via three multibody applications which highlight its capabilities for dealing with very large systems and coping with multiphysics issues