12-6台体型Stewart冗余并联机构正向运动学研究

针对六自由度并联机构的正向运动学无全解析解或全解析解推导困难、不便于程式化、存在多解选择的现状,提出一种含混合单开链支路的12-6台体型Stewart冗余并联机构,推导了适用于实时反馈控制的正向运动学全解析算法。通过单开链的方位特征集运算及路线分解,对机构的拓扑构型进行了剖析,并解算了耦合度,为运动学方程的构造及同构处理指明了方向。基于动平台上4个共面特征点的拓扑关系,对15个二次相容方程进行同构运算,推导了12个一次多项式,得到正向位姿方程的全解析解,具有形式简洁、对称,且根能够唯一确定的特点。根据速度基点法,建立特征点速度之间的矢量关系,解决了机构的速度正解问题。结合杆长协调方程及速度映射方程,并运用Jacobian代数法解析出机构的3组奇异曲面方程。实验结果表明,位姿正解的计算值与测量值完全一致,速度正解的最大相对误差为0.08%,它们的效率指标值分别为0.21和0.32,均满足实时性要求。

Forward Kinematics of General 12-6 Stewart Redundant Parallel Mechanism

Aiming at the present situation that forward kinematics of most 6-DOF (degrees of freedom) parallel mechanisms cannot be described with whole analytical solutions, and a very few of them can be described with whole analytical solutions but also with these difficulties, including calculation, programming, and sorting the suitable solution, a general 12-6 Stewart redundant parallel mechanism with hybrid single opened chains was presented, and a whole analytical decoupling algorithm of forward kinematics, which was needed by the closed-loop real-time feedback control system, was constructed. By calculating the position and orientation characteristics set and decomposing the routes of single opened chains, the topological configuration of new mechanism was analyzed, and then the coupling degrees was solved, which were useful to show clearly direction to construct and process the kinematics equations. Based on the topological relations between four coplanar feature points on the moving platform, the combination algorithms of 15 quadratic isomorphic compatible equations were designed, and then the whole analytical solution, which was succinct, symmetric and unique, of forward displacement equations was derived. Based on the method of base points, velocity relations between feature points were established, and then forward velocity equations were solved. Based on compatibility equations and velocity equations, the Jacobian matrix of mechanism was derived, and then three types of singular equations were also derived. Experimental results indicated that the calculated values of forward displacement equations were well consistent with the measured values, and calculation error of forward velocity equations was 0.08%, and the calculation/sampling time ratios of these two equations were 0.21 and 0.32, respectively, which met the real-time demand of control algorithms.