Complexity of the forward kinematic map

Pavešić, Petar

doi:10.1016/j.mechmachtheory.2017.07.015

Cited by 11 publications

(17 citation statements)

References 19 publications

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“…While the FK has a unique solution and its success depends on whether the joints are allowed to perform the desired transformations [11], [15], the IK may have zero, one or multiple solutions [14]. When the IK target point cannot be reached, the problem has no solution (meaning that a set of joint angles does not exist for that specific point), and it is said to be over-constrained.…”

Section: A the Inverse Kinematics Problemmentioning

confidence: 99%

Exploiting Multi-Level Parallelism for Run-Time Adaptive Inverse Kinematics on Heterogeneous MPSoCs

et al. 2020

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This paper presents a run-time solver for the inverse kinematics of a robotic arm implemented on a heterogeneous Multi-Processor System-on-Chip (MPSoC). The solver has been formulated as an optimization problem, in which two levels of algorithmic parallelism are proposed: i) the Nelder-Mead derivative-free method used as the optimization engine is modified to allow the evaluation of the cost function in multiple vertices simultaneously, ii) the trajectory is divided into non-overlapping segments, in which all the points are solved concurrently. Algorithmic parallelism is supported by a variable number of parallel instances of a custom hardware accelerator, which speeds up the computation of the forward kinematics equations of the robot required during the resolution of the inverse kinematics. This adaptable scheme provides run-time scalability in terms of trajectory accuracy, logic resources, dependability, and execution time. New design methodologies are used to unify the modeling of the software and hardware partitions of the controller while transparently providing adaptability. They are based on the dataflow Model of Computation (MoC), supported by the PREESM prototyping tool. This tool has been extended to support the use of dynamically reconfigurable hardware accelerators implemented using the ARTICo 3 framework. The proposal has been validated with a python-based robotic arm simulator. Experimental results show how the proposed parallelism, combined with hardware acceleration, enables the run-time resolution of the trajectory with adaptable performance using a Xilinx Zynq UltraScale+ MPSoC device.

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Section: A the Inverse Kinematics Problemmentioning

confidence: 99%

Exploiting Multi-Level Parallelism for Run-Time Adaptive Inverse Kinematics on Heterogeneous MPSoCs

et al. 2020

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“…, n admits a continuous partial section to the projection ξ u . This concept appeared in [7] where it was used in order to measure the manipulation complexity of a robotic device: X and Y were respectively the configuration space and the working space of a mechanical system (like a robot arm) and u : X → Y was interpreted as the forward kinematic map of the system. The theory was further developed in [8].…”

Section: Topological Complexity Of Mapsmentioning

confidence: 99%

Monotonicity of the Schwarz genus

Pavešić

2019

Proc. Amer. Math. Soc.

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Schwarz genus g(ξ) of a fibration ξ : E → B is defined as the minimal integer n, such that there exists a cover of B by n open sets that admit partial section to ξ. Many important concepts, including Lusternik-Schnirelmann category, Farber's topological complexity and Smale-Vassiliev's complexity of algorithms can be naturally expressed as Schwarz genera of suitably chosen fibrations. In this paper we study Schwarz genus in relation with certain type of morphisms between fibrations. Our main result is the following: if there exist a fibrewise map f : E → E ′ between fibrations ξ : E → B and ξ ′ : E ′ → B which induces an n-equivalence between respective fibres for a sufficiently big n, then g(ξ) = g(ξ ′). From this we derive several interesting results relating the topological complexity of a space with the topological complexities of its skeleta and subspaces (and similarly for the category). For example, we show that if a CW-complexes has high topological complexity (with respect to its dimension and connectivity), then the topological complexity of its skeleta is an increasing function of the dimension.

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“…Example 5.4. Our main motivating example is the complexity of the forward kinematic map of a robot as introduced in [23]. A mechanical device consists of rigid parts connected by joints.…”

Section: Complexity Of a Mapmentioning

confidence: 99%

“…In Section 3 we consider the properties of kinematic maps that are relevant to the study of topological complexity (cf. [23]). In particular, we determine the dimension of the configuration space, discuss when a kinematic map admits a continuous section (i.e.…”

Section: Introductionmentioning

confidence: 99%

A topologist’s view of kinematic maps and manipulation complexity

Pavešić¹

2018

Contemporary Mathematics

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In this paper we combine a survey of the most important topological properties of kinematic maps that appear in robotics, with the exposition of some basic results regarding the topological complexity of a map. In particular, we discuss mechanical devices that consist of rigid parts connected by joints and show how the geometry of the joints determines the forward kinematic map that relates the configuration of joints with the pose of the end-effector of the device. We explain how to compute the dimension of the joint space and describe topological obstructions for a kinematic map to be a fibration or to admit a continuous section. In the second part of the paper we define the complexity of a continuous map and show how the concept can be viewed as a measure of the difficulty to find a robust manipulation plan for a given mechanical device. We also derive some basic estimates for the complexity and relate it to the degree of instability of a manipulation plan.

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Complexity of the forward kinematic map

Cited by 11 publications

References 19 publications

Exploiting Multi-Level Parallelism for Run-Time Adaptive Inverse Kinematics on Heterogeneous MPSoCs

Exploiting Multi-Level Parallelism for Run-Time Adaptive Inverse Kinematics on Heterogeneous MPSoCs

Monotonicity of the Schwarz genus

A topologist’s view of kinematic maps and manipulation complexity

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