【作者】 黄晓华；

【导师】 王德伦；

【作者基本信息】 大连理工大学 ， 机械设计及理论， 2006， 博士

【摘要】 机器人操作性能涉及到机器人机构的几何学、运动学与动力学性能以及运动规划和最优控制方式,是机器人设计与控制的理论基础,越来越受到机器人学界的重视。在国家自然科学基金(No.59975013)的资助下,本文应用微分流形、黎曼几何以及李群李代数的思想与方法,对机器人操作臂的运动学、动力学和轨迹规划等问题进行了研究。 首先应用微分几何中自然标架的思想,在机器人各个连杆上建立独立自然标架系,在新标架系中,建立各个连杆之间的变换关系,从而建立机器人新的运动学方程,得到机器人运动学正解和逆解的递推公式,并以XHK5140刀架机械手和PUMA560机器人为例,证明了新运动学方程的正确性。该方法比较简单的解决了操作臂末端执行器相对于固定参考坐标系的空间描述。依据上面建立的多维空间自然标架系,应用机器人运动学运动变换的不变特性以及协变导数,建立了自然标架、李群和关节映射关系,并对机器人运动学的速度、加速度进行分析,得到速度、加速度的递推公式以及在不同操作空间中的雅克比矩阵,然后依据雅克比矩阵来分析机器人的奇异特性。 其次,利用李群和李代数方法,在机器人动力学牛顿—欧拉方程和拉格朗日方程的基础上,定义了广义力和广义速度,推导出机器人动力学逆问题以及拉格朗日动力学方程在关节空间和操作空间中的李群表达式,该方法简化了计算过程,降低了计算的复杂性和计算量,同时由于动力学方程中相关的参数矩阵是常数矩阵,可以采用离线计算,大大减少了控制系统实时运算时间,满足实际控制的实时性要求。通过得到的惯性矩阵,应用黎曼曲率有效地分析了机器人动力学操作性能。 最后,应用近似化方法和黎曼度量方法,研究了机器人最优轨迹规划的问题。近似化方法是利用泰勒公式以及多项式插值等相互结合,对控制系统进行近似化处理,将非线性控制系统转换为近似的线性控制系统,实现了机器人能耗最小的最优轨迹规划。以两自由度的操作臂为例,通过系统仿真,得到关于能耗最小的最优轨迹以及各个关节之间的耦合关系。黎曼度量方法是将机器人的运动学和动力学性质与抽象的几何曲面相对应,通过研究这些黎曼曲面上不变量的几何性质,来研究机器人的最优操作性能。譬如将机器人操作臂末端执行器的系统动能进行黎曼度量,这样就可以将机器人运动学和动力学特性等同映射为相应的黎曼曲面,该黎曼曲面上的短程线就代表能耗最小。在此基础上,文中提出了基于短程线的机器人动力学最优轨迹规划方法,通过求解不同黎曼曲面的短程线微分方程,并将其映射到机器人的各个关节变量,充分利用各个关节之间的耦合关系,得到机器人运动学和动力学最优规划轨迹。文中给出了几种二自由度和三自由度机器人最优轨迹规划的算例,对所得到的最优轨迹规划结果进行了仿真,验证了该方法的正确性。 总之,本文采用了微分流形、黎曼几何、李群李代数等几何方法同机器人的运动学、动力学、最优轨迹规划等方面相互融合,比较有效地解决了机器人操作性能的一些相关问题,并对机器人操作性能的研究产生了一定的推动作用。更多还原

【Abstract】 Robot manipulability includes the geometry, kinematics and dynamics, trajectory planning and the mode of optima] control of framework, which is the base of the design and control of robot. Many specialists attach importance to it. In this dissertation, using techniques and notation which come from the theory of modern differential manifold, Riemannian geometry, Lie groups and Lie algebras, we have researched the some problems about robot manipulability and trajectory planning under the support by National Natural Science Foundation of China (No.59975013).In the first place, natural frames are established on the end of linkages; According to the transformation between the new natural frames, a new robot forward kinematics and back kinematics formulas are got, which is proved to be correct through the kinematics formulas of the XKH5140 and PUMA560. It describes the movement and configuration of system and avoids the singularity of robot, In the next place, on the base of the natural frame vector field, a mapping is established among the natural frame, Lie groups and joints of robot, which has the characteristic of robot and explicit signification of geometry and kinematics. The velocity and the acceleration have been studied by using the left invariant and right invariant of motion and covariant derivative. The recursive formulations and the Jacobian matrix of the joint space and the operational space have been gained and the singularity of robot has been analyzed through the Jacobian matrix in the meaning time.In succession, based on the recursive Newton-Euler equations and Lagrange equations, the general force and the general velocity are defined, and we utilize the theory of Lie groups and Lie algebras to derive the back-dynamics equations and Lagrange equations of robot in the joint space and operational space. The method makes it easy to compute and reduces the computational complexity. In the meaning time, the real-time control of robot comes true. Through using the inertial matrix, we analyse the manipulation of robot by the Riemannian curvature.In the end, the problem of robot trajectory planning is investigated by the linearization method and Riemannian metric. On the one hand, the optimal control system is linear programming by the Talyor equation and the polynomial. The nonlinear control system is transformed into the linear one. Several trajectory planning examples of 2-DOF and 3-DOF robot are given and the simulative results verified the trajectory planning and the couple among differential joints. On the other hand, the kinetics and dynamics of the system are regarded as Riemannian metrics, so the robot kinematics and dynamics are mapped into different Riemannian surfaces. The geodesic on the surfaces is invariant and it is necessary condition of the shortest distance between two points on the surface. The geodesic on the dynamic surface corresponds to the optimal kinetics trajectory. Through solving differential equations of geodesic on these surfaces, the optimal kinematics and dynamics trajectories are obtained. Through the couple relations between joints of the geodesic, the optimal trajectory is got directly. At last, several trajectory planning examples are given and the simulation results verified the trajectory planning method successfully based on geodesic.In a word, robot kinematics, robot dynamics and robot trajectory planning are studied respectively by differential geometry, Riemannian geometry and Lie groups and Lie algebras. Some problems about robot manipulability are solved effectively.更多还原