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Appendix: Kinematic Models

We solved the kinematic model for the standard six-axes anthropomorphic robot in detail in Chapter 3 and Chapter 4. In this appendix, we briefly show the solutions for the other kinds of industrial robots presented in Mechanical Configurations Section in Chapter 1.

Please note that, in order to understand the techniques and notations used in this appendix, you need to be familiar with the content introduced when solving the standard six-axes robot. I suggest you get a good grasp of that solution’s details before tackling these other robots.

We solve all the kinematic models using the base and TCP frames as reference. If you need to introduce an additional tool or zero frame, you can follow the same procedures described for the six-axes robot in the Sections on Zero Frame and Tool Frame in Chapter 3 (FK) and Chapter 4 (IK). When working with four-axes robots, the A and B angles do not exist, and we only consider the C angle for rotations around the vertical Z axis.

COBOT

Cobots (collaborative robots) are very popular these days because they are very simple to set up and deploy and especially because they are safe to deploy right beside human operators without being confined behind protection cages.

Their mechanical structure is a bit different than the standard six-axes manipulator that we analyzed in-depth in this book, and the kinematic model needs to be adjusted accordingly.

Figure A-1 shows a Cobot model in its home position and the names assigned to the mechanical parameters for each link. You can also choose a different home position and give different names to the parameters, but the concept does not change.

Figure A-1
Six-axes Cobot in its home position

The first joint J₁ rotates around the vertical Z axis (Figure A-2).

Figure A-2
The first joint rotates around the Z axis

Then we move along the Z and Y axes by a1z and a1y to reach the second joint J₂, which rotates around the Y axis (Figure A-3).

Figure A-3
The second joint rotates around the Y axis

We further move along the X and Y axes by a2x and a2y to find the third joint J₃, which also rotates around the Y axis (Figure A-4).

Figure A-4
The third joint rotates around the Y axis

Then again add a3x and a3y to reach the fourth joint J₄, also rotating around Y (Figure A-5).

Figure A-5
The fourth joint rotates around the Y axis

The fifth joint J₅ rotates around Z and is offset by a4z from J₄ (Figure A-6).

Figure A-6
The fifth joint rotates around the Z axis

Finally, we add a5y to reach the last joint J₆, which rotates around Y (Figure A-7).

Figure A-7
The sixth joint rotates around the Y axis

No fewer than four out of six joints rotate around the Y axis, while the other two rotate around the Z axis. There is no rotation around the X axis. Such a mechanical structure is quite different from the standard six-axes manipulator we are used to.

To solve the direct kinematics, we follow the usual procedure: we build the homogeneous matrix for each joint taking into account the rotation and translation caused by the corresponding mechanical link. See Chapter 2 for details.

For J₁, we simply need a rotation around Z.

(A-1)

For joints J₂, J₃, and J₄, we have rotations around Y and additional position offsets. Careful with the third joint: the value a2y is actually negative, because the link points backward, in opposite direction of the Y axis.

(A-2)

(A-3)

(A-4)

For J₅ we add a rotation around Z. Here also, make sure that a4z is negative because it points downward.

(A-5)

Finally, J₆ introduces one more rotation around Y.

(A-6)

To find out the TCP position as seen from the base frame, we need to chain-multiply the six homogeneous matrices together:

(A-7)

If you expand all the calculations, you can verify that the [X, Y, Z] position coordinates of the TCP are as follows:

(A-8)

(A-9)

(A-10)

Notice the compact notation s₂₃₄ = sin (J₂ + J₃ + J₄).

The total rotation of the TCP with respect to the base frame is the product of all the joint rotation matrices:

(A-11)

By expanding the product, you should find the following rotation matrix:

(A-12)

The TCP rotation matrix can be decomposed in either Euler angles or quaternions according to your needs, as explained in Chapter 2 and Chapter 5, respectively.

Notice how the mounting flange of the last joint is oriented along the Y axis, as opposed to facing the X axis for the standard manipulator. Keep that in mind when calculating offsets for the tool.

Solving the inverse kinematics can be done in different ways: some approaches are more geometric; others are more analytical. We present here one of the possible solutions, definitely a valid one, but not the only one.

This step is much harder than solving the direct transformations: it is not even guaranteed to have a solution, and when a solution exists, it is not necessarily unique. Actually, given a specific TCP pose, there are eight possible joints solutions (and many more if you consider all the ±2π offsets for each joint).

Figure A-8
Different joint configurations reach the same TCP pose

Figure A-8 shows a Cobot in different joint configuration reaching the same TCP pose. In the first two cases, the difference depends on the position of the first joint: we call them LEFT and RIGHT. In the third case, the configuration of the second and third joint is different: we call it DOWN, as opposed to UP for the first two examples.

Let’s start by finding the value of the first joint. The orientation frame of the TCP is given by the rotation matrix R_TCP = [n, o, a], which can be quickly composed from the given Euler angles [A, B, C]. Recall that [n, o, a] are the column vectors of the rotation matrix and represent the orientation of the frame axes at the TCP:

(A-13)

The column vector o is oriented along the local Y axis and is facing out of the mounting flange (see Figure A-9). If we move from the TCP backward along that direction by a distance of a5y, we find the wrist point:

(A-14)

Figure A-9
Finding the wrist point from the TCP

Observe that P₅ is mechanically restricted to lie on a plane that never intersects the origin of the robot. It is easier to see that from the top, as shown in Figure A-10.

Figure A-10
The wrist point moves on a plane that does not contain the base origin

We call ρ the distance between P₅ and the base origin of the robot on the two-dimensional XY plane. We can state that no inverse solution exists for values of ρ smaller than a6y, which is the sum of all the links along the Y axis (a1y + a2y + a3y), all taken with their correct signs (remember that a2y is usually negative).

(A-15)

In practice, that means that the robot cannot physically reach any TCP position which would force the wrist center point to be too close to the base origin. On the other hand, given a valid target position for P₅, we can quickly find the value of J₁.

There are actually two configurations that we can choose for the first joint, either with the arm LEFT or RIGHT, which are rotated at about 180 degrees from each other. Not exactly 180 degrees, because the plane of P₅ does not pass through the origin, as opposed to the case of the standard six-axes manipulator.

Let’s consider the standard LEFT configuration first (Figure A-11 left). The value of J₁ (in yellow) is the difference between α₁ (in purple) and α₂ (in green).

The angle α₁ is determined by the position of P₅:

(A-16)

The angle α₂ is constrained by a6y and ρ:

(A-17)

As usual, while one could solve via the arcsine function directly, I suggest implementing your code with the arctangent function for better computational stability.

The final solution for J₁ in the left configuration is as follows:

(A-18)

In case the selected solution must be the RIGHT configuration, J₁ has to turn all the way around (see Figure A-11 right).

(A-19)

If no forcing constraint is given by the operator, we can choose either one of the two solutions, typically the one closer to the current joint value at the start of the movement.

Given J₁ we can now find J₅. There are two different possible approaches that led to the same result: a geometric and an algebraic one. We choose the latter, because it is computationally more efficient. The idea is to extract the value of the joint from the robot’s rotation matrices.

Recall that the total TCP rotation matrix is the product of the six rotation matrices of each joint:

(A-20)

Multiply both sides by the inverse rotation matrix of the first joint:

(A-21)

We already derived all the individual rotation matrices while solving the direct transformations. Using the properties that the inverse of a rotation matrix is simply its transpose (R₁⁻¹ = R₁^T) and recalling Equation (A-13), we can quickly expand Equation (A-21) into the following expression:

(A-22)

(A-23)

We only focused on the second row ignoring the rest of the matrix, because we spot a lonely c₅, which will lead us to finding J₅. The elements on both sides of the identity must be equal, including the central element on the second column:

(A-24)

Since we already know J₁, we can immediately calculate J₅.

The sign of J₅ is not uniquely defined: there are two possible robot’s configurations, one with J₅ positive and another one with J₅ negative. Both solutions are technically valid, but we will choose only one later on, based on the values we find for the remaining joint angles.

We now move on to finding J₆. Let’s consider again the same matrix equivalence in Equation (A-23): since now both J₁ and J₅ are known, we can extract J₆ from the elements on the first and third columns:

(A-25)

(A-26)

Given the sine and cosine of J₆, we immediately can find its value with the arctangent function:

(A-27)

There is always one unique solution for J₆, as long as J₅ is not zero. Otherwise, the denominator is 0 and the robot is in a singularity. Mathematically, it means that there are infinite valid solutions for J₆. We can simply force it equal to any value we desire, typically the current value of the physical joint at the beginning of the movement.

The next step is to find J₂, J₃, and J₄, which requires quite a lengthy procedure. The trick is looking at the robot from the side and working on the projected plane along the first joint, as shown in Figure A-12. We observe that J₂, J₃, and J₄ form a kinematic chain with three rotational joints: we need to find the points P₂ and P₄ and from there solve the triangle.

Figure A-12
The projection plane to find J₂ and J₃

Just keep in mind that there are two possible options, either UP or DOWN. We will solve both configurations.

Let’s start from P₂. Observe in Figure A-13 that the first joint of the robot automatically fixes the position of point P₂, which lies on the rotation axis of J₂.

We use the forward kinematics to solve for P₂:

(A-28)

To find P₄, we move on the other end of the robot (Figure A-14). P₄ lies on the rotation axis of J₄, and its position and orientation are fixed by J₅ and J₆, both of which we already know.

Figure A-14
Calculating P₄ from the wrist point P₅

Starting from the orientation of the TCP and removing the rotations for J₅ and J₆, we find the orientation of the frame at point P₄:

(A-29)

Then we can also calculate the position of P₄ starting from P₅ and move back along the Z axis of the local frame of P₄:

(A-30)

At this point, we have both P₂ and P₄, and we can start solving the kinematic chain between the two of them. We call their geometrical distance P₄₂. The projection of P₄₂ on the side plane along the first joint is as follows:

(A-31)

a7y = a2y + a3y (A-32)

Figure A-15 shows a top view to clarify.

Figure A-15
Top view to find the projected distance between P₂ and P₄

The small difference along the Y axis (a7y) is caused by a mechanical offset, the sum of a2y and a3y. Remember that a2y is usually negative.

In order to have a physically valid solution, we need to make sure that λ can be reached using the two links a2x and a3x. Two conditions must be verified; otherwise, the inverse transformation function will fail:

(A-33)

(A-34)

Now that the three sides of the triangle [a2x, a3x, λ] are all known, solving for J₂ and J₃ is a matter of basic trigonometry (see Figure A-16).

Figure A-16
Geometrical solution for J₂ and J₃

We first calculate α and then derive β and γ using the law of cosines:

(A-35)

(A-36)

(A-37)

Following the right-hand rule for the angles, we obtain the following values for the joints in the default UP configuration:

(A-38)

(A-39)

For the DOWN configuration,

(A-40)

(A-41)

The only joint we are missing at this point is J₄. The solution is shown in Figure A-17 and requires finding the difference in orientation between the X axes of the frames in P₂ and P₄. We call that difference α₁₄, and also note that

(A-42)

In other words, once we know α₁₄, we can complete the solution of the inverse kinematics by applying the following:

(A-43)

Figure A-17
Finding the missing angle J₄

The vector x₁ is simply the direction of the local X axis at point P₂, which is parallel to ground and only depends on the first joint:

(A-44)

The vector x₄ is the direction of the local X axis at point P₄, which we calculated earlier on in Equation (A-29), when we derived the orientation of point P₄:

(A-45)

We also need the direction normal to the plane containing these two vectors. We call it n, and it is simply given by the local Y axis at point P₂:

(A-46)

The angle between two vectors in a three-dimensional space, lying on the same plane, is given by the following equivalence:

(A-47)

The determinant of the [x1, x4, n] matrix gives the sine of the angle between x₁ and x₄, while the dot product (x1 ∙ x4) gives the cosine. We use the arctangent function to extract the correct value of the angle.

It is critical to normalize all the three vectors before carrying on the calculations.

Using Equation (A-43), we can finally find J₄. Now we have the values for all the six joint angles, although remember that we end up with two different sets of solutions because of the two possible signs for J₅ after Equation (A-24). Select the one forced by the operator or the one closest to the current configuration of the robot before starting the movement.

SCARA

The SCARA robot can be built in different mechanical configurations according to its application requirements. The most typical option comes with four joints (three rotational and one translational) and offers four degrees of freedom: the TCP can be positioned along the [X, Y, Z] axes and rotated around the C coordinate.

Figure A-18 shows a SCARA model with four joint axes in its home position. It is conventional to place the base frame of the robot at the beginning of the first link avoiding any initial vertical offset from the base column.

Figure A-18
SCARA robot in its home position

The last parameter a4z is actually a mechanical coupling between the last two joints. Since most industrial SCARA robots are built that way, we include the parameter directly in the default transformations. If your robot does not have such a coupling, you can safely set the a4z coefficient equal to 0.

The first two joints J₁ and J₂ rotate around the vertical Z axis and are entirely responsible for positioning the [X, Y] coordinates of the TCP (see Figure A-19 and Figure A-20).

Figure A-19
The first joint rotates around the Z axis

Figure A-20
The second joint rotates around the Z axis

The third joint J₃ translates the TCP along the vertical Z axis (see Figure A-21).

Figure A-21
The third joint slides along the Z axis

Finally, the fourth joint J₄ introduces an additional rotation around the Z axis (see Figure A-22).

Figure A-22
The fourth joint rotates around the Z axis

Although the SCARA’s mechanical structure is simple enough to be quickly solved geometrically, we still adopt the common approach of composing all the homogeneous matrices for each joint to find the solution to the direct kinematics. See Chapter 2 for details.

The first joint J₁ introduces a rotation around Z.

(A-48)

The second joint J₂ also rotates around Z, after an offset along the X and Z axes.

(A-A9)

The third joint J₃ is of translational type and acts along the vertical Z axis.

(A-50)

Finally, the last joint J₄ introduces an additional rotation around Z. In most robots, this rotation is achieved by means of a screw-type mechanical shaft, which has the side effect of lifting and lowering the TCP while rotating. The additional vertical movement must be accounted for in the offset part of the homogeneous matrix. The coupling coefficient a4z specifies how much vertical displacement is caused by a complete rotation of J₄. Use 360 for degrees or 2π if your input values are in radians.

(A-51)

To find out the TCP position as seen from the base frame, we need to chain-multiply the four homogeneous matrices together:

(A-52)

Expanding the calculations leads to the following:

(A-53)

(A-54)

(A-55)

As usual, we adopted the compact notation s₁₂ = sin (J₁ + J₂).

To find the rotation of the TCP, we could multiply together the rotation matrices and extract the only relevant Euler angle C. However, it is much faster to observe the geometry of the problem on the horizontal [X, Y] plane (see Figure A-23), and notice that the TCP rotation is built up by the series chain of the three rotational joints:

(A-56)

The inverse transformations are best solved geometrically. Given the position and orientation [X, Y, Z, C] of the TCP, we need to find the four joint angles.

We start with the first two joints J₁ and J₂ and observe that they are responsible for determining the position of the TCP on the [X, Y] plane (see Figure A-24). We also notice that the solution is not unique: there are two possible options for the arm configuration, either LEFT or RIGHT, to reach the same TCP position.

We call ρ the two-dimensional distance between the TCP and the base frame on the [X, Y] plane.

(A-57)

It is possible to find a solution to the inverse transformations only if the two links a1x and a2x can reach the distance ρ.

(A-58)

(A-59)

The angle between the base X axis and the vector ρ is given by the following:

(A-60)

The two possible configurations of the arm are symmetric with respect to the vector ρ, forming two equal triangles with sides a1x, a2x, ρ. We call α the angle between a1x and ρ and β the angle between a1x and a2x (both angles are in green in Figure A-24).

Now we can express the values of the first two joints as functions of α and β. For the RIGHT pose, we have the following:

(A-61)

(A-62)

For the LEFT pose,

(A-63)

(A-64)

All we are missing at this point are the values for α and β. The angle β can be quickly found using the law of cosines in the triangle a1x, a2x, ρ.

(A-65)

As usual, I suggest using the arctangent for better numerical stability.

Once J₂ is known, we can derive the projections of the second link (l and h):

(A-66)

(A-67)

Finally, we find α:

(A-68)

Using either Equation (A-61) or (A-63), we can calculate J₁.

Solving the last two joints is much more straightforward. From Equation (A-56), we immediately find J₄:

(A-69)

Finally, J₃ is the only joint that affects the vertical displacement of the TCP from the base frame, with the addition of the fixed mechanical offsets (a1z, a2z) and the (optional) coupling effect from J₄.

(A-70)

PALLETIZER

The palletizer is a four-axes robotic arm typically deployed in loading/unloading applications, where large and heavy loads need to be transferred from conveyors to pallets. With only four joints, this robot can control four degrees of freedom of its TCP: three position coordinates [X, Y, Z] and one rotation angle C around the vertical Z axis. Figure A-25 shows a palletizer model in its home position.

The striking characteristic of its mechanical structure is that it is able to keep the load always parallel to ground regardless of the value of its joint axes. This feature is very helpful when handling packages, so that they can be picked or placed in any position in space without any need to electronically control their tilt. The TCP always stays horizontal to ground, thanks to a double parallelogram structure in the links of the robot. Figure A-26 and Figure A-27 show this feature when joints J₂ and J₃ are rotated.

Figure A-26
The second joint rotates around the Y axis

The first parallelogram also introduces a mechanical coupling between joints J₂ and J₃. Specifically, a rotation of J₂causes a rotation of the J₃ physical axis by the same angle in reverse direction, even when the motors in J₃ do not move. When solving the direct and inverse transformations, we will need to introduce a compensation for the effect of this mechanical coupling, as we learned in the related Sections in Chapters 3 and 4.

Figure A-27
The third joint rotates around the Y axis

The second effect of the mechanical parallelograms is to keep the TCP frame parallel to the base frame. The movement acts on the passive joint at the wrist point WP by introducing a rotation of −(J₂ + J₃), as shown in Figure A-28. There is no motor acting on this joint, but we still need to keep track of its passive rotation in the transformations.

Figure A-28
The mechanical links add a rotation opposite to (J₂ + J₃) at the WP

Finally, the first and last joints (J₁ and J₄) of the palletizer robot both rotate around the vertical Z axis (see Figure A-29).

Figure A-29
The first joint rotates around the Z axis

In particular, J₄ does not have any effect on the position of the TCP. It only affects its rotation angle C.

Despite being an open-chain kinematic, the extra mechanical links make this robot a hybrid between serial and parallel models. Nevertheless, the transformations are typically solved using a standard serial kinematic model and then adding coupling coefficients between the joints.

The direct kinematics could be solved geometrically because of the simple structure of the palletizer. However, we keep following the same notation as for the other robots, and we present here all the homogeneous transformations for each axis.

The first joint J₁ introduces a rotation around Z.

(A-71)

The second joint J₂ rotates around the Y axis, after an offset along the X and Z axes.

(A-72)

The third joint J₃ also rotates around the Y axis after a vertical offset along Z.

(A-73)

The mechanical links introduce a negative rotation around the Y axis centered at the wrist point. The magnitude of the passive rotation is the opposite of the sum of J₂ and J₃.

(A-74)

Finally, the last joint J₄ introduces an additional rotation around Z. The translational offset between WP and TCP can be both along the X and Z directions, depending on the mechanical built. Keep in mind that a4z is usually negative, translating toward ground.

(A-75)

To find out the TCP position as seen from the base frame, we need to chain-multiply the five homogeneous matrices together:

(A-76)

Expanding the calculations leads to the following:

(A-77)

(A-78)

(A-79)

When feeding the joint angle values to the direct transformation functions, remember to add the effect of the coupling between J₂ and J₃:

(A-80)

The rotation angle C can be extracted from the rotation matrix, or simply geometrically derived, by observing that only the first and last joint have any effect on it:

(A-81)

The inverse transformations can be derived geometrically observing the robot from the side, projected on a plane along the first joint (see Figure A-30). We start by deriving the position of the wrist point from the TCP coordinates [X, Y, Z]:

(A-82)

(A-83)

Figure A-30
Solving the inverse kinematics for the palletizer robot

Then, we find all the geometrical parameters we need in order to solve the triangle (a2z, a3z, ρ).

(A-84)

(A-85)

(A-86)

The condition we need to impose, for the transformations to return a valid solution, is that the arm links a2z and a3x can physically stretch enough to reach the wrist point.

(A-87)

(A-88)

We can now find all the unknown angles α, β, γ. The last two, using the law of cosines,

(A-89)

(A-90)

(A-91)

As usual, I suggest using the arctangent for better numerical stability.

At this point, we can quickly derive the values for J₂ and J₃:

(A-92)

(A-93)

Unlike most other robots, the palletizer does not allow different configurations of its joint to reach the same TCP pose. The values found in Equations (A-92) and (A-93) are unique.

The result of the inverse transformations for J₂ and J₃ must be adjusted to compensate for the mechanical coupling:

(A-94)

Solving for J₁ is straightforward, as it is the angle formed by the two coordinates [X, Y] of the TCP:

(A-95)

Finally, from Equation (A-81), we calculate the angle of the last joint:

(A-96)

DELTA

A Delta (or Tripod) robot is a parallel kinematic chain typically employed in applications with small workspace and high-speed requirements.

Figure A-31 shows a Delta robot with three joint axes in its home position. The three rotational joints J_i (i = 1..3) are located on the upper platform and directly actuate the upper arms h_i, which link points A_i to B_i. The lower arms k_i link points B_i to C_i and are built using parallelogram structures, which automatically keep the lower platform parallel to the upper one. Forcing the TCP to have a constant orientation makes the kinematic model much easier to solve.

Figure A-31
Delta robot in its home position

Most of the delta robots found in the industry feature an additional fourth axis to add a rotation capability to the TCP around the vertical Z axis. This extra joint acts directly on the TCP and does not affect the rest of the kinematic model, so we did not show it in the figures. The motor rotating the TCP is rarely mounted directly on the lower platform because that would increase its weight and reduce the dynamic performance of the robot. The motor is usually placed on the upper platform and connected to the TCP by means of a central flexible link.

The position of the TCP is often translated from the lower platform by an offset t. We call the center of the lower platform the wrist point (WP) and derive its position from the TCP as follows:

(A-97)

The center of the upper platform is the origin of the robot’s base frame. The first joint J₁is oriented along the X axis, while the two other joints J₂ and J₃ are oriented, respectively, at 120 and 240 degrees. The three axes thus form an equilateral triangle (see Figure A-32). The distance between the origin and the joints centers (A_i) is r. Similarly, on the lower platform, the distance between the WP and the (passive) joints centers (C_i) is s.

Figure A-32
Top view of the upper platform

A movement of the actuated joints causes a translational displacement of the lower platform along the positional [X, Y, Z] coordinates. See Figure A-33 for some examples.

Figure A-33
The lower platform always remains parallel to the upper one

Solving the direct kinematics consists in finding the position of the WP given the angles of the joint axes (see Figure A-34). The procedure requires the following steps:

Calculate the position of the B_i points.
Calculate the position of the P_i points (offset from B_i by s along the joint planes).
Find WP as the intersection of the three spheres centered in P_i of radius k.

Figure A-34
One intersection of the three spheres is the wrist point

Adopting the usual compact notation c₁ = cos J₁, the B₁ point along the X axis has coordinates:

(A-98)

Similarly, we find the B₂ and B₃ points, which lie on the joint planes rotated at ±120 degrees around the Z axis for the J₁ plane:

(A-99)

(A-100)

The passive universal joints in B_i allow rotations of the lower k_i links around an entire sphere. In other words, the position of the C_i points will be on spherical surfaces centered on B_i and of radius k. Since the WP is offset by s from C_i along the three joint axes planes, we first offset the points B_i by s to find the points P_i, and then we construct the three spheres centered in P_i. The intersection of the three spheres will be the WP.

(A-101)

(A-102)

(A-103)

There exist various possible ways of solving the problem of intersecting three spheres, both algebraic and geometric. One option involves solving a system of three quadratic equations in [X, Y, Z]. Another approach uses a technique called trilateration, which we show here in detail.

Given three spheres of radius k centered in P_i, let us find their intersection point [X, Y, Z]. It can be shown that there are actually two intersection points, both lying on a line perpendicular to the plane containing the three center points. Mathematically, there are also cases in which the spheres only intersect in one single point or do not intersect at all, in case the centers are spaced too far apart from each other. However, the arms of the delta robot are designed specifically to avoid these complications. The lower arms k_i are longer than the upper arms h_i, so the spheres always intersect. The fact that they intersect in two points is also irrelevant, because we always choose the location for WP lower than the upper base platform.

Figure A-35 shows the plane containing the three center points. We define the two vectors U = P₂ − P₁ and V = P₃ − P₁. From simple vector properties, we can derive the frame:

(A-104)

(A-105)

(A-106)

Figure A-35
Plane containing the three center points

The projections of V over the and axes are as follows:

(A-107)

(A-108)

It can be proven that the coordinates of the intersection point in the frame are as follows:

(A-109)

(A-110)

(A-111)

Notice that there are two solutions for the z coordinate. We already mentioned before that the only physically relevant value for the WP is the one below the upper platform of the robot. Also, the expression under the square root must be positive in order to return two distinct real values. That is always the case with standard industrial robots, because they are built to avoid mechanical violations.

Finally, we need to transform the WP coordinates back into the robot’s base frame:

(A-112)

Adding the last vertical offset will return the actual TCP position:

(A-113)

Solving the inverse transformations is somewhat simpler. The position of the wrist point is given, which fixes the lower platform in space. The upper platform is also fixed by default, so all we need to do is find the values of the joint angles J_i for which the upper arm and the lower arm connect with each other. Given the symmetry of the robot, the problem can be solved individually for each axis.

The upper arms can only rotate around their pivots A, forcing the points B to describe circles in space. The lower arms, on the other hand, can more freely swing around the universal joints in C, allowing the point B to describe spheres in space. The solution to the inverse kinematics for each joint is the intersection between the upper circle and the lower sphere, as shown in Figure A-36.

Figure A-36
Intersection between upper circle (yellow) and lower sphere (red)

Actually, if we project the geometry of the problem onto a plane containing the joint axis (e.g., the XZ plane for J₁), the lower sphere degenerates into a circle, and the solution becomes the intersection between two circles, which is straightforward to find. Notice that out of the two possible intersections, we pick the one with the robot’s arm pointing outward, not inward (see Figure A-37).

Figure A-37
Projecting the problem on the XZ plane

Let’s consider the XZ plane and solve for J₁:

The upper circle is centered in A = [r, 0] and has radius h.
The lower circle is centered in C^′ = [WP_X + s, WP_Z] and has radius .

The resulting equations of the two circles are as follows:

(A-114)

The system can be quickly solved by extracting z from the first equation and plugging it into the second to find x. If two distinct real solutions exist, we choose the one with the largest x coordinate to keep the robot’s arm on the outside of the platform. The unique result is point B = [B_X, B_Z], which forces the joint angle to be as follows:

(A-115)

If no real solution to the system is found, then the given TCP position is outside of the reachable robot’s workspace, and the inverse transformation function must return an error.

To find J₂ and J₃, we can simply follow the same procedure used for J₁, but each time the XZ plane needs to be rotated by 120 degrees around the vertical Z axis. The equation of the upper circle does not change. However, the coordinates of the wrist point used in the lower circle need to be adjusted to the new frames.

For J₂, use a rotation of 120 degrees from J₁ around Z:

(A-116)

For J₃ use a rotation of −120 degrees from J₁ around Z:

(A-117)

We can plug these values in Equation (A-114) and find the joint angles from Equation (A-115).

Index

AB quadrature signal

Absolute encoders

Acceleration

limited profile

profile

Actual mechanical configuration

Aggressive tuning

Algorithm

Aluminum

Angular speed

Anthropomorphic robot

Anti-windup

Asynchronous/passive decay

Autoencoders

Automatic movement programming

Back electromotive force (BEMF)

Back-propagation

Ball grid array (BGA)

Bang-bang approach

Bezier curves

Bezier profile

acceleration limit

computational complexity

constant time distance

cubic spline development

De Casteljau’s algorithm

dynamic values

equations

higher order spline

mechanical limitations

movement duration

path distance

peak acceleration

peak jerk

polynomial approach

seven-zones S-curve

smooth position profiles

total acceleration time

Bezier splines

Bilateral filter

BilateralFilter()

BitConverter

Bleeder resistor

Blurring

Bode plot

Body diode conductivity

Brakes

Braking resistors

Brushless motors

electronic commutation

industrial robots

internal structure

motor type comparison

PMSM

pole

servo loop

servomotors

slots

Buck converter

CAD models

Calibration

Canny edge detector

Capacitor

Capsules

Cartesian length

Cartesian movement length

Cartesian path length

Cartesian system

Cell calibration

calibrated frames, user program

coordinate system

frame’s origin position

frames poses

kinematic calculations

offline programming, robots

reference points, frame axes

solution

workpiece

Center of mass

Centrifugal and Coriolis effects

Centrifugal force

Centripetal force

Charge pump

Circle

arccosine operator

coefficient

collinear

definition

length

linear combination

movements

normalized vector

numerical values

parameters

position coordinates

vectors

Clarke-Park transformations

Classification problem

Closed form

equations

solution

CMOS sensor

Cobots

See, See also

SeeCollaborative robots (Cobots)

Codeless automation

Coils

Collaborative robots (Cobots)

different joint configuration, TCP pose

fifth joint rotates, Z axis

find missing angle

first joint rotates, Z axis

forward kinematics

fourth joint rotates, Y axis

geometrical solution

inverse kinematics

law of cosines

mechanical structure

projection plane

rotation matrix

second joint rotates, Y axis

six-axes Cobot, home position

sixth joint rotates, Y axis

solving for first joint

standard LEFT configuration

TCP orientation frame

TCP position

TCP rotation matrix

third joint rotates, Y axis

wrist point moves on plane

Collision detection

arbitrary value

bounding capsules

collision test

direction vectors

distance between capsules

distance between lines

distance between segments

geometrical intuition

global minimum point

intersection monitoring

minimum distance

multitude of bodies

paraboloids

parametric equations

planning time

quadratic function

Collisions

Column vectors

Commissioning

controllers’ parameters

goal

Commutation

Compact 300W servo drive

Compensation model

Composing rotations

Condition monitoring

Conservative tuning

Control algorithm

Control field (CF)

Controller Area Network (CAN)

bus architecture

CAN_HIGH

CAN_LOW

EtherCAT

microcontrollers

PROFIBUS

protocols

transceiver

Control software

Conveyors

Conveyor tracking

Convolution

Convolutional layers

Convolutional neural network (CNN)

automated script

classification objects

class imbalance

convolutional layers

advantages

feature extraction

inference process

kernel

kernel of filters

receptive field, neuron

weight sharing

deep CNN structure

discriminator

GANs

high-quality training dataset

industrial settings

neurons

object recognition

OCR

read digits and text

semantic segmentation

traditional contour finding techniques

training dataset

training process

transfer learning

uses, other fields

VGA image

Coriolis forces

Correction offsets

Cubic Bezier spline interpolation

Cubic spline

Cuboids

Current-sense amplifiers

Current-sense resistor

Current sensing

accurate motion control

CC6920 series, CrossChip

current-sense resistor

disadvantage

factors

filter

Hall effect sensors

high-side

individual measurements

in-line sensing

internal resistance

low-side/high-side, in-line measurements

measure in-line currents

measure motor phase currents

motor phases

phase current monitoring

PWM voltage profile

shunt resistor vs. Hall effect sensor, current measure

temperature drift

voltage

Data augmentation

Data transmission

DC brushed motors

DC bus voltage

braking resistors

bulk capacitance

capacitor bank

drawbacks, capacitance

electric cars

electrolytic capacitors

energy, motor

galvanic isolation

higher voltage level

implementations

braking resistors

in-rush current limiters

in-rush current

isolated gate driver

isolation solutions

large industrial robots

large power resistor

microcontroller monitors

MOSFETs

channel current vs. voltage characteristic

motor control bridges

N-channel

motor

output current

power stage, motor driver

primary purpose

robot

undervoltage behaviors

lack of power

pre-charge

short circuit

DC motors

DC-to-DC voltage regulators

DC voltage

De Casteljau’s algorithm

Decomposition

angles notation

application

arbitrary value fixing

arctangent function

control software

Euler angles notations

possible outcomes

rotation matrix

sines and cosines elements

singularity

Decoupling

Deep learning

autoencoders

back-propagation

capacity

classification head and regression head

classification problem

curve fitting problem

error function

feedback

gradient descent techniques

inference

input image vs. desired output

input layer

input-output relation

learning rate

modern optimization techniques

neural networks

object recognition

overfitting

regression problem

regularization

reinforcement

supervised

training process goal

error function behavior, training

VGA image

Degree of continuity

Degrees of freedom (DoFs)

Delta robot

center points, plane

industry feature

intersection between upper circle/lower sphere

inverse transformations

lower platform

parallel kinematic chain

TCP position

three joint axes, home position

top view, upper platform

trilateration

upper platform

WP position

WP transform

Differential kinematics

approximation

closed-form solution

control software

cycle time interval

equations

expression

individual columns

iterative procedure

Jacobian matrix

numerical approximate solution

partial derivatives

planned path

sophisticated approach

TCP path

translational components

Differential signaling

Digital feedback channel

Digital I2C interface

Digital input interface

Digital output circuit

Digital twin

individual robot simulation

kinematic model

numeric calculations

production parameters

requirements

software packages

three-dimensional model

Direct dynamics problem

Dispenser tool

Distance

DIY incremental encoder

Drain current

Dynamic braking

Dynamic constraints

Dynamic equations

Dynamic model

approximation

cross relations

derivation techniques

parameters

physical quantities

practical applications

Dynamic monitoring

Dynamic parameters

Dynamics

Edge devices

e-fuse

Electric actuator

Electrical angle

Electrical motors

Electric cars

Electric motors

actuators

BEMF

brakes

cobots

coil windings

components

equivalent circuit

flange

gearbox

inductance

inductor

industrial robots

input electrical power

input voltage

mobile vehicles

output mechanical power

power source

resistor

robotic applications

rotor

shaft

stationary section

Electric power

Electromagnetic compatibility (EMC)

Electromagnetic interference (EMI)

Electronic commutation

applications

brushed DC motors

brushless motors

centered-based PWM signals

Clarke-Park transformations

constant DC voltage

DC motor

direct/quadrature components, stator current vector

dynamic braking operation

fast decay

FOC

H-bridge

high-speed ON/OFF switching

internal structure, current controller

magnetic flux generate

operation states, H-bridge

output, PI controllers

permanent-magnet synchronous motor

phase-to-phase voltage

PI controllers

PWM

quadrature current component

rotor coordinate system

rotor orientation

stator current state vector

switches

three-leg bridge, drive three-phase brushless motor

UVW windings

voltage vector

windings currents

Electronic commutation algorithm

Electronic components

Electrostatic discharges (EDS)

Empirical calibration method

Encoders

absolute

characteristic

detect positions

feedback values

hall sensors

SeeHall sensors

incremental

low resolution

magnetic disc

magnetic vs. optical

mechanical axis, incremental encoders

optical sensors

robotic applications

robotic axes, absolute encoders

rotary vs. linear

sense position

Error function

ESP32

Ethernet

Euclidean space

Euler angles

definition

individual rotations

rotation in space

rotation matrix

strengths and weaknesses

interpolation

simplicity

uniqueness

three-tuple

Exclusive zones

advantage

definition

flag bit

multi-robot system

priority

robot movements

Expressing rotations

External force

External mechanical spring

External path corrections

Fabrication process

Fast decay

Feature extraction function

Feedback sensor

Fieldbus

CAN

ESP32

ESP-NOW protocol

feature

local area network

OCP UA

peer-to-peer messages

protocols

Field-oriented control (FOC)

Filter

Firmware

functions

robot

Fixed base coordinate system

Fixed base frame

Fixed geometrical parameters

Flange

Flexibility

Forbidden zones

axis

cases

cuboids

intersection points

requirement

2D equivalent problem

Force sensor

Forward kinematics (FKs)

SeeForward and inverse transformations

Forward transformations

base to TCP

description

homogeneous transformations

matrices

purpose

rotation matrix

steps

TCP frame

TCP position and orientation

Four-axes palletizers

Frame operations

Frame rotations

angles

coordinates calculation

right-hand rule

rotation matrix

Frames

column vectors

coordinate system

global coordinate system

global vs. local

kinematic models

local coordinate system

reference frames

tool coordinate system

transformation

translations

workpiece coordinate system

Friction

FRONT and BACK configurations

Full six-axes robot

Fully convolutional networks (FCNs)

GANFETs

Gate driver

capacitor

charge pump

EG2133 gate driver, EGmicro

fundamental requirements

gate resistors

high-side

integrated gate driver chips

low-side

microcontroller

motor control

requirement

safety features and configuration options

buck converter

cross conduction prevention

current-sense amplifiers

MCU interface

overcurrent protection

overtemperature shutdown

overvoltage and undervoltage lockout

slew rate

Gate resistors

Gauging

GaussianBlur()

Gaussian curve

Gaussian filter

Gear ratio

General purpose input/output (GPIO)

Generated trajectory

Generative adversarial networks (GANs)

Generative design

Generator

Generic Euler-Lagrange equation

Generic game engine

Generic rotation matrix properties

associative

commutative

orthogonal matrix

pre-multiplying

Generic speed profile

Geometrical intuitions

Geometrical paths

Geometrical tools

Gerber files

Glitch filter

Global coordinate system (GCS)

Glue-dispensing machine

Gradient descent techniques

Graphics processing units (GPUs)

Gravitational effects

Hall effect sensors

add software filters

angular velocity

complex commutation schemes

electrical angle

electrical cycle

ICs

motor’s permanent magnets

rotor position, brushless motor

rotor’s magnets

torque ripple

trapezoidal

H-bridge

Heat generation

Hierarchical learning

High-end control systems

High-frequency signals

High-inertia motors

High-side sensing

HMI display

architecture

low-cost Android tablet

Home position

Homogeneous matrix

combining process

expressing generic transformation

4x4 matrix

multiplication

pre-multiply

translations and rotations

Homogeneous transformation matrix

Horizontal movement

Householder transformation

Human-machine interface (HMI)

automatic movement programming

codeless automation

configuration

diagnostic

hardware

low-speed device

motor parameters configuration

movement programming

plotting data over time

robot vs. human operator

user interface layer

Xamarin

Hybrid control

Ideal diode

Identity matrix

Image processing

Imitation learning

Importing CAD models

Improper Euler angles

Incremental encoders

Individual joints

Industrial manipulator

Industrial robots

autonomously sorting packages

DoFs

kinematical configurations

mechanical structures

production line

repetitive tasks

robotic arms

sensors and cameras

shapes and sizes

TCP movements

Industrial servo drives

Industry 4.0

Inertia

Inference

Infinite negative acceleration

In-line sensing

Input/output (IO)

add input impedance voltage follower

characteristic

configurable IO expanders add flexibility to small MCUs

digital input interface

digital output circuit

GPIO

inductive loads

IO expanders chips

low-side N-channel switch

low-side switch circuit

microcontrollers

optocouplers

read analog signals

resistors

robots

robot’s controller

signal lines

software filter

switch on/off external devices

TVS

Integrated motion controllers

Integrated smart motion control

codeless programming

flexibility

IoT

modularity

software core

Integrated solution

condition monitoring functionalities

edge devices

hardware simplification

integrated motion controllers

linear motion transporters

simplify wirings and save cost

small SCARA robot

smart motion controller

software flexibility

Integrator

Intermediate control points

Interpolating positions

Interpolation

Interpreter

add M functions

application-related command

auxiliary commands

Cartesian length, movement

circular movement

circular movements

classical IF…ELSE

configuration

feed rate

GOTO command

infinite loop

interpolated movements

linear motion command ML

M commands

motion commands

movement

parameters, motion commands

path-interpolated movement

path/joint space

path movements

program execution

PTP movement

robotics programming language

series of movements

SET DO and RESET DO

splines

stepping mode

SUB command

TANG 1 command

tangential mode activation

trigger movements

TRK command

typical automatic execution

WAIT

Intricated regions

Inverse differential kinematics

Inverse dynamic

Inverse kinematic function

Inverse kinematics

closed form derivation

decoupling

function

joint angles position

joint axes values

joint space

mechanical coupling

mechanical parameters

nonlinear problem

nonunique solution

numerical test

path space

serial kinematic chain

singularities

solving arm

solving wrist

tool frame

zero frame

Inverse transformations, wrist center point

Inverted transformation

Inverting Equation

Jacobian

differentiations

function

Jerk

Jerk-limited trajectory

Jogging

Joints

index

motor torques

positions

space

speed limitations

speeds

positions function

values

Kernel

Kernel interface

Kinematic chains

Kinematic models

actual joint torques calculation

forward transformations

inverse transformations

mechanical structure

Kinematics

Lagrangian formulation

Lagrangian method

energy components

energy equilibrium constraints

energy expressions

Euler-Lagrange equation

generic variation

kinetic energy

link

mechanical system

motor’s inertia

principle of energy equilibrium

viscous friction

LEDs

Lens

Line

Linear encoders

Linear interpolation

Linearization

computational power

cuboids

detecting collisions

forbidden region

linear segments

practical simplifications

predicting collisions

Linear motion transporters

Linear motors

active cart

advantages

configurations

electric power

electronic commutation

linear transport systems

magnet pitch

movement accuracy

operation principle

passive cart

robotics

rotary motors

section

transport systems

zero reference electric position

Linear regulator vs. switching regulator

Local coordinate system

localEulerAngles property

Low-inertia motor

Low-side current sensing

Low-side switch circuit

Machine coordinate system (MCS)

Machine learning

artificial brain

artificial intelligence

complex task solved

condition monitoring

hierarchical learning

image recognition task

imitation

motion control

neural network

policy

reinforcement

robot control software

robotics

simulation environment

step-by-step approach

toolkit

training process

unity

vision

vs. traditional control

wind power field

Machine vision

CNN

SeeConvolutional neural network (CNN)

deep learning

SeeDeep learning

identification

industrial applications, advantage

measurement

positioning

quality control

smart camera

SeeSmart samera

tracking

vision functions

SeeVision functions

Magnetic encoders

Magnet pitch

Main controller

auxiliary axis movement

auxiliary commands

calculations

configuration

dry run

external auxiliary axes

frequency

high-end systems

IF branches

interpreter

interpreter operation, motion buffer

look-ahead

low-end systems

M-commands

microprocessor

motion buffer

path-planner

robot

robotics control system

six-axes manipulator mounted, linear actuator

six-axes robot transfers objects

software components

software control structure

time-critical calculations

trajectory generator

user program

Manipulability ellipsoid

MapleSim

Mass

Mathematical models

Matlab/Simulink

Maximum jerk

Maximum path speed

Maximum speed values

MCU interface

Mechanical coupling

axis rotation

coupling coefficients

direct transformations

external observer

industrial anthropomorphic robots

rotational joint axes

two-axes robot

Mechanical links

Mechanical offset

Mechanical parameters

Mechanical power

Mechanical structures

rotating axis

Mechanics

advanced materials

aluminum

3D metal printing

fabrication process

generative design

hobby robotic arm

materials

metal processing

motorized machine joint design

optimization algorithm

plastic processing

robot

robotic arms

steel

Mechanics CAD design

Metal process

Metal substrate

Microcontroller

controller firmware runs, RTOS

CPU/FPGA

ESP32

firmware upgrades

flash encryption protection

hobbyists

oscillator

PIC32

sufficient and stable power supply

32-bit

Microcontroller unit (MCU)

Micro-stepping

Minimalistic model

Mobile frame

Modern robots

Mono

Motion buffer

Motion commands

Motion trajectory

Motor controller

Motor driver

Motor drives

Motor inertia

Motor parameters configuration

Motor sizing

brake holding torque

cable cross section

configurations

gearbox

higher torque

high-inertia motor

high-speed laser cutting applications

inertia

low-inertia motor

mechanical weight

motor overheats

operation

palletizing applications

pole number

practical engineering

rated current

rated power

rated speed

rated torque

resistance/inductance

robot

robot movement

speed-torque curve, electric motor

stall current

stall torque

thermal margin

torque constant

typical motor datasheet

voltage constant

working point

Mounting point (MP)

Movement programming

Movements execution times

“MoveRobot.cs” script

N-channel power MOSFET

Neural network (NN)

Newton-Euler method

backward pass

closed-form solution

forward pass

inverse dynamics solving

linear components calculations

Nonlinearities

Nontangential transitions

Normalized linear interpolation (NLERP)

Notch filter

Numerical approach

Object visual analysis

Online path corrections

Online programming

OPC UA

Open-loop

Open Platform Communications Unified Architecture (OCP UA)

Open source computer vision (OpenCV)

Opposite vertexes

Optical character recognition (OCR)

Optical devices

Optical encoders

Optimal trajectory

Optimization algorithm

Optimization problem

Optimization task

Optocouplers

Orientation coordinates

Orientations

ORing controllers

Orthogonal matrix

Oscillator

Overfitting

Palletizer

direct kinematics

first joint rotates, Z axis

first parallelogram

four-axes robotic arm

fourth joint rotates, Z axis

geometrical parameters

home position

inverse kinematics

inverse transformations

inverse transformations result

joint angle values

mechanical parallelograms

mechanical structure

open-chain kinematic

second joint rotates, Y axis

TCP position

third joint rotates, Y axis

Palletizer robot

Parallel joints

Parallel kinematics

Parallel mechanics

Parameters identification

accuracy requirements

control quality

data-based problem

description

friction and elasticity

inverse dynamics

inverse problem

joint angle values

learning algorithm

linear system

non-square matrix

parametrized model

simplicity

torque values

Parametric equations

Path

acceleration

interpolation

length

planner

smoothing transition effect

space coordinates

Path-interpolated movements

coordinates

Euler orientation angles

fixed reference

linear interpolation

path space

positions

rotation matrix

singularity

starting and ending points

Path-planning

circle

coordinates

definition

linear interpolation

path movements

SeePath-interpolated movements

PTP

SeePTP movements

quaternions

SLERP

SeeSpherical Linear intERPolation (SLERP)

time-dependent trajectory

PCB design

board

components footprints

computer-generated preview

copper

copper layer thickness

decoupling capacitors

electronic components

electronic layout

EMC

EMI

emit radiation

fabrication files

factors

inventory

mounting type

operating temperature

price

size

tolerance

final board after components assembly

footprints

four-layer PCB layout

heat dissipation

copper areas

metal substrate

smart layout

thermal vias

layout design

manufactured board

process

board manufacturing

components assembly

component selection

layout design

size/complexity, board

vias

Peculiar condition

Permanent-magnet synchronous motors (PMSM)

PIC32

Pick-and-place applications

PI controllers

Plastic process

Point

Point-to-point (PTP)

Pole

Policy

Pose

Position control

Position coordinates

POSITIVE and NEGATIVE configurations

Power

Power management

DC bus voltage

SeeDC bus voltage

electric machine

input power section

power supply controller unit

primary function

secondary function

protection functions

SeeProtection functions, power management

stable and safe voltage source

voltage converter

Power section

Power source

Power switches

adjust switching

adjust switching frequency

breakdown voltage

diode current

drain current

expensive components

heat generation

higher switching losses

high-voltage applications

MOSFETs

H-bridge, drive DC brushed motor

rate drain current plot, case temperature

N-channel power MOSFET

on-state resistance

packaging

power transistors

reduce body diode losses

reduce switching transitions

servo drive

six N-channel power MOSFETs arranged, three legs

three independent half-bridges

total gate charge

transitions

use GANFETs

Power switches, high-side switches

Practical software implementation

Pre-multiplication

Printed circuit board (PCB)

Product coordinate system

Program execution

Programmed movement

Programming scripts

Protection functions, power management

bleeder resistor

bulk capacitance

circuit breaker

current limiter

discharge rate of bulk capacitor, bleeder resistor

e-fuse

high-power systems

high voltage

ideal diode

monitor current flow

MOSFET, reverse polarity protection

Over-and undervoltage

p-MOSFET

power requirements

rechargeable batteries

reverse current block

reverse polarity

connection

protection

safety feature

temperature monitor

use diode to block reverse voltage

voltage converter

PTP movements

Cartesian length

characteristic

geometrical path

inherently unsafe

jogging individual joint axes

joint angles

joint reconfiguration

joints

joint values control

linear interpolation

parametric equation

path space

position and orientation

robot positions

starting and ending points

Pulse width modulation (PWM)

Pure translation

PyTorch

QR factorization method

Quadrature

Quadrature AB pulses

Quartic spline

Quaternions

advantages

compact

dot product

elements

Euclidean size

expressing orientations

function

general expression

issues

multiplying

negative

normalized form

orientations

resulting product

scalar

vector

Random positions

Realistic solution

Real-time operative system (RTOS)

Reference frames

Reflow soldering

Region of interest (ROI)

Regression problem

Regularization

Reinforcement learning

Resolver

Reverse polarity connection

Reverse polarity protection

RMS torque

Robot

base frame

behavior

dynamic model

joint configuration

joints

main controller

mechanical structure

mechanical workspace

mechanics

movements distances

programming

Rotary encoders

visual interface

Robot calibration

compensation function

compensation model

kinematic model

kinematic parameters

mechanical parameters

mechanical sizes of links

optimal solution

parameters

Robot control system

artificial intelligence techniques

basic connection

controller

general structure

internal motion calculations

IOs

MCU

motor drives

smart motion controller

Robotics

applications

arms

joints

kinematics

links

kinematics

Robotics library

commands

commands structure

monitor

monitor structure

parameters

parameters structure

reference (homing) command

Rotary motors

Rotation matrix

angles’ notation

associative

column vector

decomposition

element

frames difference

inverse property

joint axes

orientation Euler angles

Rotor

coordinate system

position

Roto-translations

Round edges

Bezier curve

degrees of continuity

geometries

mathematical description

path smoothing transition effect

sharp 90 degrees corner

size

transition

Rotational torques

Safe cone

Safe orientation

Safety

acceleration limits

axes limits

commissioning procedure, primary goal

controller monitors, control system

controllers

emergency stops inputs

E-Stop

E-Stop signal

hardware modules

HW stop signal

I/O channels

issues

link inertia

main controller

motion safety functions

motor’s maximum torque

movement position limits

new software version

position limits

protective cell

protective gates

requirements

robots

set jerk limits

specialized HW safety module with configurable delay time

test E-Stop

torque limitations

unit scaling

Safe zones

SCARA robot

first joint rotates, Z axis

four joint axes, home position

fourth joint rotates, Z axis

inverse transformations

mechanical configurations

mechanical structure

parameter a4z

second joint rotates, Z axis

TCP position

third joint rotates, Z axis

top view

SCARA robots

Scripts

S-curve profile

acceleration limit

central zone length

constant-speed trajectory

deceleration section

dynamic limitations

four-zone profile

movement length

non-tangential transition points

parameters

positive and negative values

speed changing behavior

time-dependent equations

zones

zone 1

zone 2

zone 3

zone 4

zone 5-7

Selective Compliance Assembly Robot Arm (SCARA)

Self-collision

Semantic segmentation

Sensing resistor

Serial kinematic chain

Serial kinematics

Serial kinematic structures

Servo drives

anti-windup and forces

cascaded PID structure

communication interface

compact 300W

core hardware components

current controller

current sensing

SeeCurrent sensing

disadvantage, PID controller

feed-forward (open-loop) control signal

functions

gains

gate driver

SeeGate driver

individual controllers

industrial

industrial robots

integrator

linear proportional correction, disadvantage

logic stage

microcontroller

peak power values

physical interface

PI controllers

position controllers

power

power inverters

power stage

power supply

power switches

SeePower switches

proportional controller

receive set values, main controller

robot operates, zero-gravity mode

safety monitoring functions

switches

torque feed-forward

typical loop cycle times

Servomotors

Shaft

Sharp edges

Shoulder singularity

SHOULDER singularity

Simulation environments

Single matrix multiplication

Singular configurations

Singularities

arbitrary

complicated topic

critical points

determinant

ellipsoids

external torque

extreme configuration

joint space subsets

linear algebra

magnitude, spatial forces

planned path

relative magnitude

rotational axes

six-axes robot

speed mobility

static forces

TCP mobility

three-dimensional ellipsoid

values

wrench

wrist

Singular rotation matrix

Sinusoidal profile

advantage

constant-speed profile

derivatives

disadvantage

equations

maximum acceleration limit

parameters

speed profile integration

vs. S-curve

Six-axes cobot

Six-axes force sensor

Six-axes robot

anthropomorphic robot

closed-form solution

home position

kinematic model

mechanical couplings

mechanical parameters

TCP

Smart camera

calibration

camera frame vs. robot frame

CMOS sensor

configuration

industrial automation

LEDs

lens

solution

uniform vs. nonuniform scaling

vision sensor

Smart motion controller

Software implementation

Sophisticated model

Spatial velocity

Spherical Linear intERPolation (SLERP)

approximation

NLERP

quaternions

step-by-step procedure

TCP position

Spherical wrist

Splines

Bezier curves

connected path segment

control points

definition

distances

equation

middle points

movements

position coordinates

PTP movement

robots’ operators

rule of thumb

set points

tangential direction

transitions

Standard forbidden zones

Standard jerk-limited trajectory

Standard model

Start method

Statics

equilibrium

force control

TCP forces

Stator

Stator current state vector

Steel

Stepper motors

Stiffness

Supervised learning

Switches

Switching regulators

Synchronous (or active) decay

Synchronous Serial Interface (SSI)

absolute encoders

clock frequencies

communication protocol

controller

data transmission

hardware

incremental encoders

physical SSI cable

point-to-point communication

robot control system

wiring details

Tamagawa

communication protocol

control field (CF)

CRC field

data fields (DF0…7)

hardware component

interface

internal memory allocation, user data

microcontroller

multi-turn absolute encoder

read actual encoder position

received encoder position

SSI interface

status field (SF)

UART

Tangential

Target orientation

Target path accelerations

TcpClient

TCP/IP protocol

Teaching by hand method

TensorFlow

Testing

Thermal margin

Thermal vias

3D metal printing

Time-filters

advantages

control software

discrete variable

fixed filter

fixed time interval

Gaussian curve

Gaussian filter

Gaussian points

linear movement

optimal setting

TCP position

Time-optimal movement

dynamic limits

joints

maximum acceleration

trajectory

Time smoothing

Tool calibration

equations

geometrical properties

householder transformation

least-square problem

linear equations

MP position

procedure, identify tool size

pseudo-code, householder transformations

QR factorization method

record measurements, MP position

size

steps

TCP position

Tool center point (TCP)

accelerations

constant angular

coordinates

definition

Euler C angle

forces

frames

inverse kinematic function

kinematic domain

linear interpolation

linear movement

movements

orientation

path axes

path-interpolated movement

path movement

pose

position

power experienced

PTP movements

robotic arm

rotation matrix

target position

workpiece

wrench components

workspace

Tool coordinate system

Tool frame

Tool orientation

Torque feed-forward

complex kinematic structure

computational load

control

controller clock tick

individual controller

model-based knowledge

motor torques

robot control software

servo drive control

Torques

components

constraints

control

margin

optimal solution

ripple

Total joint speed

Touch panels

Tracking movements

Trajectory

definition

distances

geometrical curve

generating algorithm

generation task

limitations

optimization

sample points

transforms

Trajectory generator

Bezier profile

differential kinematics

external path corrections

path speed definitions

planned path

samples points

sampling frequency

S-curve

SeeS-curve profile

sinusoidal profile

time-filtering

SeeTime-filters

time-optimal movements

Transfer learning

Transformation matrix

Transform orientations

Transient voltage suppression (TVS)

Transitions

acceleration

between circle and line

between curves

curvature

degree of continuity

direction and curvature

linear segment

order of differentiation

path segments

points

Translation

effects

forces

offsets

Triangle inequality

Trilateration

Tuning process

aggressive

antiresonance frequency

Bode plot

compliance

compliant joint

compliant system

conservative

controllers

elastic system

employed model

fixed gain value

frequency response

gain margin

goal

increase gains

inertial mismatch

instabilities

large gain margin forces

limited bandwidth, controller

low-frequency range

low gains

notch filter

PID controller

resonance frequency

robotic axes

servo loop frequency response measurement

two-encoder controller loop

virtual encoder

2D geometry

Two-dimensional planar problem

Unit quaternion

Unity

add scripts

create UI

empty objects

import CAD models

machine learning

SeeMachine learning

programming scripts

robot control system

robotic projects

robot model import

scene

simulation

Unity 3D

CAD models

cross-platform applications

game engine

physics engine

script programming

simulate industrial applications

simulation environment

UnityEngine.UI

Universal Asynchronous Receiver/Transmitter (UART)

Unplanned path corrections

Unplanned paths

UP and DOWN configurations

Update method

User-friendliness

User interface (UI)

Valid parameterization

Variable link lengths

Vector control

Vertical interconnect access (vias)

Vertical offset

Vertical plane

Vertical position

Vertical translation

Virtual encoder

Vision functions

bilateral filter

blurring

calculates gradient, intensity profile

Canny edge detection result

Canny edge detector

contour detection result

edge detection algorithm

feature extraction function

findContour() function

glue-dispensing machine, automatic workpiece detection

minAreaRect() function

OpenCV

overview

quality inspection, metal pipes

robot

robotic arm, play ping-pong

2D spatial convolution

Visual servoing

Voltage vectors

W, X, Y

Wi-Fi

Wire-frame model

computational power

cuboid intersection function

desired accuracy

linear movements

point positions

robot’s TCP position monitoring

Wireless technology

Workpiece coordinate system

Workpiece location

Workspace monitoring

capsules

collision detection

SeeCollision detection

controllers

exclusive zones

external additive corrections

forbidden zones

linearization

logical elements

predicting collisions

programming errors

safe orientation

safety functions

safe zones

self-collision

wire-frame model

Wrench vector

Wrist

center point

configurations

joints

rotation

singularity

Wrist point (WP)

Zero frame

Zero-gravity mode

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