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蛇形机器人进行力矩控制时,其动力学系统为非线性控制系统.当蛇形机器人模块数增大时,非线性系统变得冗繁而不易于计算和控制.本文利用微分几何的方法,将欧拉—拉格朗日(EulerLagrange)动力学方程进行了扩展,得到了任意基底下的动力学方程,将蛇形机器人的动力学方程化简成标准的仿射控制系统方程,得到动力学与控制统一的模型,使得蛇形机器人的动力学方程变得简洁易于计算和控制.基于此模型,利用部分反馈线性化的方法将控制方程进行线性化,并设计了头部轨迹跟踪控制器.蛇形机器人的构型空间对应着流形空间,速度空间对应着切空间,力矩空间对应着余切空间,动能提供了流形空间上的一个黎曼(Riemann)度量,因此蛇形机器人的动力学可以用黎曼几何来描述.而带有被动轮的蛇形机器人引入了速度约束,使得速度空间不是整个切空间而是切空间的一个子集,即对应着一个分布,与分布相对应的动力学为非完整动力学.所以带有被动轮的蛇形机器人对应着带有分布的黎曼流形.而在分布上选取恰当的基底可以简化动力学计算,本文根据纤维丛的理论将这些基底空间建模为纤维丛,任意一个基底只是纤维丛上的一个截面.我们通过正交归一化和线性变换规则得到具有物理意义且可以简化动力学计算的截面,从而实现了微分几何框架下的动力学与控制统一模型.最后我们以9模块蛇形机器人为例,验证了部分反馈算法的有效性.
When the serpentine robot carries out torque control, its dynamic system is a nonlinear control system.When the number of serpentine robot increases, the nonlinear system becomes redundant and difficult to calculate and control.In this paper, using the method of differential geometry, Euler Lagrange dynamic equations were extended to obtain the kinetic equation under any substrate, the dynamic equation of the serpentine robot is simplified into a standard affine control system equation, and the dynamics and control The uniform model makes the dynamics of the serpentine robot simple and easy to calculate and control.Based on the model, the feedback equation is linearized by the partial feedback linearization method, and the head trajectory tracking controller is designed. The robot’s configuration space corresponds to the manifold space, the velocity space corresponds to the cutting space, the torque space corresponds to the residual space, and the kinetic energy provides a Riemann metric on the manifold space. Therefore, the dynamics of the serpentine robot can Use Riemannian geometry to describe, while a serpentine robot with a passive wheel introduces a speed constraint such that the velocity space is not the entire tangent space but a space cut Set, that corresponds to a distribution, and distribution corresponding to the dynamics of nonholonomic dynamics. So with a passive round of the serpentine robot with a corresponding distribution of the Riemannian manifold. And in the distribution of the appropriate base can be simplified to simplify the motivation Based on the theory of fiber bundles, we model these basement spaces as fiber bundles, and any one basement is just a cross section of the fiber bundles.We obtain the physical meaning through the normalization and linear transformation rules and can simplify the kinetics And the unified model of dynamics and control under the differential geometry is realized.Finally, we take the 9-module serpentine robot as an example and verify the validity of the partial feedback algorithm.