Evaluation of a robotic (serial or parallel) manipulator’s static and kinematics performance requires the solution of closed and complicated non-closed form systems of equations for both the forward and inverse problem types. In this work, a robotic system or manipulator is represented as a network whose motion generating kinematic pairs are represented as the network’s inter-connected nodes. Within this network theoretic context, we develop a formulation employing higher dimensional multivectors defined in Clifford Algebra that approximates the computational outcomes of such complicated systems of equations, for both inverse and forward problem types. The statics and kinematics performance of these mechanical networks (serial and parallel robots) are rapidly evaluated without the need to solve their respective traditional systems of equations using what we call ‘screw hypervolumes’, which capture the algebro-geometric structures generated by a robot’s physical design parameters and end-effector position in terms of kinematics and statics. These dimensionless hypervolume functions enable the specialist to produce a very computationally efficient function/index that approximates the kinematic and static output spaces of robots. We find that this formulation inadvertently presents a curious mathematical analogue to these systems of equations. A 6R serial robot, the Delta robot and the 3-RRR manipulator are analysed using the hypervolume methods for the case studies. This approach can be adapted and applied to other multi-agent physical systems displaying non-commutative interactions as well.

中文翻译：

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基于几何代数的机器人操纵机构静力学和运动学的高维近似方法

要评估机器人（串行或并行）机械手的静态和运动学性能，需要解决正向和逆向问题类型的方程组的封闭式和复杂式非封闭式方程组。在这项工作中，机器人系统或操纵器表示为网络，其运动生成运动学对表示为网络的互连节点。在此网络理论背景下，我们开发了一种使用在Clifford代数中定义的高维多重矢量的公式，该公式对于逆向和正向问题类型都近似于此类复杂方程组的计算结果。可快速评估这些机械网络（串行和并行机器人）的静力学和运动学性能，而无需使用我们所谓的“螺杆超体积”来求解它们各自的传统方程组，该“超体积”捕获了机器人物理产生的代数几何结构运动学和静力学方面的设计参数和末端执行器位置。这些无量纲的超量函数使专家能够产生非常高效的计算功能/指数，可以近似于机器人的运动学和静态输出空间。我们发现，这种表述无意间对这些方程组提出了一种好奇的数学模拟。使用超大容量方法对6R串行机器人，Delta机器人和3-RRR机械手进行了分析。