If you intend to use a six-axis robot arm, such as Mecademic's Meca500, the example featured in this tutorial, you will probably need to do more than just position and orient the robot结束效应在各种姿势。您可能还需要最终效应器遵循胶合中的规定路径，或者插入引脚时。如果是这种情况，那么你必须了解机器人奇点那because these special configurations will often impede the Cartesian movements of your robot end-effector. You must therefore know how to keep away from robot singularities by properly designing your robot cell.

"A robot singularity is a configuration in which the robot end-effector becomes blocked in certain directions."

## Robot singularities as degeneracies in velocity mapping

“在一个奇点，机器人手臂失去了一个或多个自由度。”

Q̇.=j-1V.

V.=[Z.̇ωXωyωZ.]T.

is the笛卡尔速度矢量of the end-effector, q̇ is the vector ofjoint velocitiesandj是一个6×6矩阵，称为雅各比亚矩阵。T.he Jacobian matrix is function of the joint positions (q) and the robot geometry. When this matrix becomes singular (at certain joint positions), the above equation is not defined and finding joint velocities for certain Cartesian velocity vectors becomes impossible. In other words, the robot becomes blocked in certain directions, and we say that it is in a singularity.

"The problem with singularities is not only the impossibility of crossing them, but also the high joint velocities resulting from passing close to them."

T.he problem with singularities is not only the impossibility of crossing them, but also the high joint velocities resulting from passing close to them. A robot is said to be close to singularity when the determinant of its Jacobian matrix is close to zero, which yields the effect of division by a very small number. Such high joint velocities may be unexpected and can pose safety risks in the case of big, fast industrial robots. Furthermore, when following a specific Cartesian path and passing close to a singularity, the feasible end-effector velocities are significantly reduced. Finally, due to control problems, the path accuracy of a robot controlled in Cartesian space deteriorates significantly in the vicinity of singularities.

## T.he three types of singularity in a wrist-partitioned, vertically-articulated robot arm

T.he vast majority of six-axis industrial robots consist of six revolute joints arranged as in the Meca500. Namely, the axes of joints 2 and 3 are parallel, the axes of joints 1 and 4 are normal to the axes of joints 2 and 3, the axis of joint 5 is normal to the axes of joints 4 and 6, and these last three axes intersect at one point. This architecture is often referred to as aV.ertically-articulated robot arm。1978年通过宣传开发的Puma机器人首次采用。此外，机器人臂，其中最后三个关节的轴在一点中被称为手腕划分or as havinginline wrists。这种流行的架构的主要优点之一是描述其运动学的数学方程是相当简单的。只有油漆机器人没有内联手腕，因为轴之间的偏移允许机器人具有更大的方向能力。许多所谓的协作机器人也没有内联手腕，但我们稍后会考虑这些特殊机器人。

### 手腕奇点

T.he most frequently-encountered singularity in vertically-articulated robot arms with inline wrists is the手腕奇点。它发生的时候axes of joints 4 and 6 become coincident. In most robots, this condition corresponds to θ5.= 0°。In the figure below, the middle configuration corresponds to a wrist singularity whereas the other two correspond to two different sets of configuration types. In the left configuration, we have the so-calledno-flipcondition (θ5.> 0°) whereas, in the right configuration, we have the翻动condition (θ5.< 0°).

### 肘部奇点

T.he second type of singularity in vertically articulated robot arms with inline wrists is theelbow singularity。它发生的时候手腕中心(the point where the axes of joints 4, 5 and 6 intersect) lies on the plane passing through the axes of joints 2 and 3. We can say that, in an elbow singularity, the arm is fully stretched. (Due to mechanical interferences, most robot arms cannot be fully folded, which would be the other elbow singularity.) An elbow singularity is determined only by the position of joint 3. For example, in the Meca500, the elbow singularity occurs when θ3.= - arctan（60/19）≈~~72.43°。

In the above figure, the middle configuration corresponds to an elbow singularity whereas the other two correspond to two different sets of configuration types. In the left configuration, we have the so-called肘部condition (θ3.> −arctan(60/19)) whereas, in the right configuration, we have theelbow-downcondition (θ3.< −arctan(60/19)).

### 肩部奇点

In the above figure, the middle configuration corresponds to a shoulder singularity whereas the two others correspond to two different sets of configuration types. In the left configuration, we have theFR.ontcondition whereas, in the right configuration, we have the背部健康）状况。当然，数学公式决定了这两个条件，但它是一个比特复杂的，并且这里不会呈现。

In a shoulder singularity, the robot cannot move in the direction of the axis of joint 2. Consider the above figure where a robot is shown crossing a shoulder singularity. In order for the TCP to follow a line through the singularity, joints 1 and 4 must simultaneously rotate 90° in opposite directions (other joints need to rotate too), while the end-effector remains stationary. Thus, it is physically possible to cross a shoulder singularity while following a line but, at the singularity, the end-effector remains motionless while some of the joints rotate. In other words, it is impossible to cross a shoulder singularity without having the end-effector stop. That said, due to numerical problems, it is impossible to do this crossing while jogging in Cartesian space.

### 配置和重叠

Finally, the three types of singularity present in vertically-articulated robot arms with inline wrists are illustrated in the video below. Note that in the sequence where the robot passes near a wrist singularity, the minimum value for θ5.是0.2° - 非常接近腕部奇点。此外，在说明弯头奇异性的序列中，θ的值3.is quickly oscillating ±12° around the singular value of θ3.≈ −72.43°, but the end-effector is almost stationary.