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.

In general, it is impossible to cross a wrist singularity when controlling the robot in Cartesian space
In general, it is impossible to cross a wrist singularity when controlling the robot in Cartesian space

工业机器人可以控制在两个空间中:joint spaceandCartesian space。Hence, there are two sets of position-mode motion commands that make an industrial robot move. For joint-space motion commands (sometimes incorrectly called point-to-point commands), you simply specify — directly or indirectly — a desired set of joint positions, and the robot moves by translating or rotating each joint to the desired joint position, simultaneously and in a linear fashion. For Cartesian-space motion commands, you specify a desired pose for the end-effector AND a desired Cartesian path (linear or circular). To find the necessary joint positions along the desired Cartesian path, the robot controller must calculate the inverse position and velocity kinematics of the robot. Singularities arise when this calculation fails (for example, when you have division by zero) and must therefore be avoided.

尝试在联合空间中慢慢慢跑六轴机器人手臂,并且机器人将停止的唯一时间是当关节击中限制或机械干扰时。相比之下,尝试在笛卡尔空间中慢慢慢跑相同的机器人,并且机器人经常停止并拒绝在某些方向上进行,尽管它似乎远离您认为的工作空间边界。机器人奇异性是一种配置,其中机器人端部执行器在某些方向上被阻挡。

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

任何六轴机器人臂(也称为串行机器人或连续机械手)都有奇点。(实际上,正确的术语是six-degree-of-freedom,但让我们坚持流行,不科学的术语six-axis). Some robot arms have singularities that are extremely easy to identify. Other robotic arms have singularities that are impossible to describe without the use of lengthy and complex formulas. The complexity and types of singularity in a robot arm will depend on the number of joints, their types (linear or rotary), and their geometric arrangement.

因为奇点显着恶化了工业机器人手臂的性能,因此在使用笛卡尔空间运动命令时,您必须了解如何识别它们,并且永远不会靠近它们。

Robot singularities as degeneracies in velocity mapping

考虑以下最琐碎的示例,如下所示的六轴定位阶段,由三个正交线性引导件以及三个旋转级,在一个点相交的轴。让工具中心点(TCP)在该交叉点处,并让终端用件由图中所示的工具参考帧表示。这款笛卡尔机器人可以在黄色长方体内的任何地方带来TCP,并以任何方向定向其末端效应。它还可以连续沿着该工作空间内的任何6D路径置位其末端效应器。。。除非两个极端旋转关节的轴一致,如右侧的配置。后一种条件对应于该六轴机器人臂具有的唯一奇点(它还包括当接头5的位置偏移180°时出现的情况)。

在非奇异配置(左)和单数(右)中的六轴笛卡尔机器人
在非奇异配置(左)和单数(右)中的六轴笛卡尔机器人

在奇点,机器人不能沿着某些方向置换其末端执行器。在该特定示例中,右侧的配置中的机器人不能旋转其末端执行器围绕与三个旋转关节的轴线正常的轴线(这在奇点中的共面)。这种特定的奇点也被称为a万古锁

“在一个奇点,机器人手臂失去了一个或多个自由度。”

在奇点,机器人手臂失去了一个或多个自由度。机器人奇点是一种物理堵塞,而不是某种抽象的数学问题,尽管我们对它有一个简单的数学解释。六轴机器人臂的奇点可以通过以下逆速度运动学方程解释:

Q̇.=j-1V.

在哪里

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.

机器人奇点作为内部工作空间边界

机器人奇异不仅是逆速度运动学失败的配置:在奇异性中,机器人的逆位置运动方程也是如此。对于所需的末端效应器姿势,像MECA500这样的六轴机器人通常可以具有高达八种不同的用于接头位置的溶液,相应于相同的末端效应姿势,如下所示。(如果我们考虑到接头6往往是无限的,则可以进行更多的解决方案,但是让我们将其限制在一个完全旋转,以便讨论奇点)。这八种不同的解决方案对应于八种不同配置类型。Changing a configuration type requires passing through a singularity (see the top image in this tutorial). Thus, as we discuss in our tutorial on workspace, the Cartesian workspace of the typical six-axis robot arm is composed of eight 6D entities. The boundaries between these entities correspond to singularities. The remaining boundaries correspond to joint limits (and other mechanical interferences).

末端执行器姿势的一个例子对应于八个不同的关节位置
末端执行器姿势的一个例子对应于八个不同的关节位置

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°).

手腕奇点(中心)和无翻转(左)和翻转(右)条件
手腕奇点(中心)和无翻转(左)和翻转(右)条件

在手腕奇点,机器人不能移动the direction of the axis of joint 5. Consider the top-most figure in this tutorial where a robot is shown crossing a wrist singularity. In order for the TCP to follow a line through the singularity, joints 4 and 6 must simultaneously rotate 90° in opposite directions, at the singularity. Thus, crossing a wrist singularity while following a line is physically possible but, at the singularity, the end-effector remains motionless while joints 4 and 6 rotate. In other words, it is impossible for the end-effector to cross the singularity without stopping. That said, due to numerical problems, it is impossible to do this crossing while jogging in Cartesian space, even if you do not mind that the end-effector has to stop.

在腕部奇异性中,机器人的逆位置运动学有无限解决方案。如果{θ.1,θ.2,θ.3.,θ.4.,θ.5.,θ.6.是一个解决方案,那么{θ1,θ.2,θ.3.,θ.4.− β, θ5.,θ.6.+β}也是一种解决方案,其中β是任意角度。

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°。

肘部奇点(中心)和肘部(左)和肘部(右)条件
肘部奇点(中心)和肘部(左)和肘部(右)条件

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)).

在肘部奇点,两组逆位置运动液退化为一个。这个奇点是最不意想不到的,很容易避免。

肩部奇点

垂直铰接式机器人武器中的第三种和最后一个类型的奇点,其中包含内联手腕是shoulder singularity。当机器人手腕的中心位于穿过关节1和2的轴线(或穿过接头1的轴线并且平行于接头2的轴线)时发生。在MECA500中,机器人手腕的中心位于肩部奇异性的接头1的轴线上。这种奇点是最复杂的,因为它不依赖于单个联合位置,就像另外两个一样。

肩部奇点(center) and front (left) and back (right) conditions
肩部奇点(center) and front (left) and back (right) conditions

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。

穿过肩膀奇点
穿过肩膀奇点

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.

在肩部奇点,机器人的逆位置运动学存在无限解决方案。不幸的是,没有简单的公式来表达这些解决方案。

配置和重叠

请注意,您可以具有属于三种类型的奇点中的任何两个的奇点配置,甚至是所有三种。

另外,注意,通过上述三个配置条件的集合定义了配置类型,即{翻盖/无翻转,弯头/弯头,前/返回}。通过一种配置类型到另一个配置类型需要交叉奇点。因此,它必须势在必行,当使用笛卡尔空间命令时,指定所需的配置类型,而不仅仅是所需的末端执行器姿势。

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.

典型的六轴协同机器人(Cobot)中的奇点类型

如上所述,大多数所谓的合作机器人do not have a PUMA-type architecture. Indeed, the vast majority of six-axiscobots.在市场上与普遍机器人的UR3相同的六个旋转关节的安排,以此为例。即,关节2,3和4的轴是平行的,接头1的轴线相交并且是垂直于接头2的轴线,并且接头5的轴线与关节4和6的轴线相交并且是垂直的。

典型协作机器人手臂的奇点类型:腕(左),肘部(中心)和肩部(右)奇点
典型协作机器人手臂的奇点类型:腕(左),肘部(中心)和肩部(右)奇点

这些Cobots也具有简单的逆位置运动问题,允许多达八种不同的解决方案。然而,这些Cobots的奇点有点不同。当关节4和6的轴平行时,发生腕部奇点(上面的左图)。在UR3中,这对应于θ5.= 0°,θ5.=±180°或θ5.=±360°。此外,在腕部奇异性中,由关节2,3,4和5组成的机构可以在末端效应器保持静止的同时移动。当关节2,3和4的轴是共面的轴线时,发生弯头奇点,如上图所示。在UR3中,这对应于θ3.= 0°。最后,当关节5和6的轴的交叉点位于穿过关节1和2的轴线的平面中时,发生肩部奇点,如上图所示。在肩部奇点,θ的两个可能的解决方案1合并。与典型的六轴机器人臂不同,没有接合动作,导致静止的末端效应器。

示出了三种类型的机器人奇点this video由我的研究团队在Éts准备。

How to avoid robot singularities?

现在你知道哪些机器人配置ingular, the question is how to avoid them. Unfortunately, robot singularities can only be avoided by properly designing your robotic cell (and that includes the design of the adaptor plate for your end-effector). If you have poorly assigned your pick location so that it corresponds to a wrist singularity, for example, there’s very little you can do to solve your problem. Essentially, you can only hope that the desired pose can also be attained with another configuration that is non-singular, as illustrated in the figure below.

对应于奇异(左)和非单数(右)机器人配置的末端效应姿势
对应于奇异(左)和非单数(右)机器人配置的末端效应姿势

总之,在设计机器人细胞时,请记住机器人奇点。特别注意用于最终效应器的适配器板的设计。如果您不需要六个自由度来定位和定向您的末端效应器(例如,如果激光切割或插入圆形销),请利用您的冗余自由度摆脱奇点。最后,考虑使用脱机编程和仿真软件,如Robodk,这在检查机器人奇点时可能具有很大的帮助。

©Mexademade在全部或部分地重现本教程,严格禁止。