含一弹性索的二索并联机构动力学系统的Brunovsky正则形式

Brunovsky Type Canconical Form for Direct Dynamics of a Two-cable-driven Parallel Mechanism with an Elastic Cable

提出一种含一弹性索的二索牵引并联机构,推导该机构的动力学系统。用微分代数及为经准静态线性化反馈来线性化的一般化控制器形式这两种不同方法推导获得了与原机构系统动力学状态方程等价的具有Brunovsky正则形式的线性系统。这两种方法本质上是一致的,侧重点都是在寻找原系统的一个微分k-同胚,然后在其基础上获得用新的状态坐标表示的与原系统对等的Brunovsky类型正则形式;差异在于研究工具和具体推导过程不同:第一种方法用微分域扩张表示机构动力学系统,而用代数闭合及微分闭合一致来获得与原系统对等的Brunovsky类型正则形式;第二种方法是用无限维微分几何将机构系统的状态方程限制成呈块三角结构的、带有一般化的控制系数非线性控制器形式,通过对该非线性控制器采用一种特殊的准静态反馈就,即可获得与原状态方程等价的具有Brunovsky正则形式的线性系统。可在具有Brunovsky正则形式的线性系统上开展该机构的末端轨迹跟踪控制问题,设计合适的反馈控制器。

A kind of 2T cable-driven parallel mechanism with one elastic cable and one massless and rigid strut was presented which can make the end-effector implement 2 directional translations.At first the inverse dynamics formulation was derived based on the motion equations of the end-effector,and the space-space representation of direct dynamics system was given.2 methods including differential algebra and generalized controller via quasi-state static feedback were used respectively to derive a linearized and decoupled system with a Brunovsky type canconical form which was equivalent to the previous direct dynamics system.The two methods were identical in essence,which focus on finding a differential k-isomorphism of the previous system,then a Brunovsky type canonical form with new states obtained from the differential k-isomorphism were obtained.The difference laid on the research tool and the derivation process:the first method used differential field extension to express the mechanism dynamics and obtain the Brunovsky type canonical form via the concepts of algebraic closure and differential closure,the second method used infinite dimensional geometry to restrict the state equations to a nonlinear controller form with generalized controllability indices as well as a block trigular structure,then a particular quasi-state feedback was used on the nonlinear controller to obtain the Brunovsky type canonical form.Results show that the linear system with Brunovsky type canonical form can be used to investigate the problem of trajectory tracking control of the ene-effector and suitable feedback controller can be designed.