关于6轴串联机器人，存在一个不可避免的特性，那就是奇点。在奇点时，无法通过逆向运算将笛卡尔坐标系转化为轴的角度，而且笛卡儿坐标系内一点微小的变化就会引起轴角度的剧烈变化。所以，做PTP运动时，不存在奇点问题。奇点只存在轨迹运动时。下面有文章和视频很形象演示了这种特性。LT翻译了一下（有些是意译，翻译水平还请见谅）：

原文链接： Singularities in six-axis vertically-articulated industrial robots http://coro.etsmtl.ca/blog/?p=107

Vertically-articulated six-axis serial robots exhibit three types of singularities: wrist singularities, shoulder singularities, and elbow singularities. A wrist singularity occurs when axes 4 and 6 are coincident. A shoulder singularity occurs when the wrist center lies on a cylinder centered about axis 1 and with a radius equal to the distance between axes 1 and 4. Finally, an elbow singularity occurs when the wrist center lies in the same plane as axes 2 and 3.

6轴串联关节机器人有三种奇点：腕部奇点，肩部奇点，肘部奇点。腕部奇点发生在4轴和6轴重合(平行)时。肩部奇点发生在腕部中心位于1轴旋转中心线时。肘部奇点发生在腕部中心和2轴3轴一条线 。

When a robot is controlled in cartesian mode and passes near a singularity, the speed of some joints becomes suddenly very large and the cartesian velocity of the end-effector is significantly reduced. For this reason, trajectories that make the robot pass near singular configurations should be avoided.

当机器人以笛卡尔坐标系运动时，经过奇点，某些轴的速度会突然变得很快，TCP点的路径速度会显著减慢。因此，应避免机器人的轨迹经过奇点附近。

The following video illulstrates the three types of singularities:
下面的视频演示这三种奇点：
http://v.youku.com/v_show/id_XNTU1OTA5Nzg4.html

An excellent reference on this topic is the following scientific paper:
关于此话题，有一个很好论文可参考：

M.J.D. Hayes, M.L. Husty, P.J. Zsombor-Murray, 2002, “Singular Configurations of Wrist-Partitioned 6R Serial Robots: a Geometric Perspective for Users”, Transactions of the Canadian Society for Mechanical Engineering, Vol. 26, No. 1, pp. 41–55. （链接）

扩展阅读：

KUKA 对应的三种奇点叫做：

• Wrist axis singularity    手轴奇点 α5
• Overhead singularity    顶置奇点 α1
• Extended position singularity    延展位置奇点 α2

关键字：奇点,  轨迹运动, 笛卡尔坐标系