基于微分几何的蛇形机器人动力学与控制统一模型

Dynamics-control unified model of a snakelike robot based on differential geometry

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.

Whereas the inputs for a snakelike robot are torques, the dynamics system is a nonlinear control system. With increasing modules in a snakelike robot, its nonlinear control system becomes complex and inconvenient for regulation and control. In this paper, the differential geometry method is used, and the Euler-Lagrange equations are extended to equations under any base. Thus, the dynamics equations are reduced to the standard affine control system, and the dynamics-control unified model is derived; this simplifies the regulation and control of the snakelike robot. Based on the unified model, a partial feedback linearization method is developed, and the head trajectory controller is designed. The configuration space of a snakelike robot corresponds to the manifold space, the velocity corresponds to the tangent space, the torque space corresponds to the cotangent space, and the kinematic energy provides a Riemann measure on the manifold. Thus, the dynamics of a snakelike robot can be described by Riemann geometry. Additionally, the passive wheels installed under the snakelike robot introduce the velocity constraint, which constrains the velocity space to a subspace of the tangent space. That is, the velocity space forms a distribution, and the dynamics system becomes a nonholonomic dynamics system.For a snakelike robot with passive wheels, the configuration is a Riemann manifold with a distribution. In the distribution, the appropriate base can be chosen to simplify the dynamics. In this paper, a base model is built based on the fiber bundle theory. Any set base is only a section in the fiber bundle. The orthogonal normalization technique is adopted to derive a set base that can simplify the dynamics calculation, and the dynamics-control unified model is derived. Finally, a nine-module snakelike robot is used as an example of the partial feedback linearization method.