Forward Kinematics

Forward Kinematics#

A robot arm is a kinematic chain of rigid bodies (links) connected by actuated joints. One end of the kinematic chain is mounted to a base and the other end to an end-effector (gripper, tool). Joints have two tyipcal types (though we learned many): revolute and prismatic, as in Fig. 24. The pose of a robot arm is described by all joints (all DoFs). Each DOF is typically associated with a joint variable. Forward kinematics (FK) is to derive the relationship between joint variables and the pose of the end-effector.

../_images/end-effector_frame.jpg — Fig. 24 Description of the pose (position and orientation) of the end-effector frame#

Consider a robot arm of \(n\) joints. Define \(\boldsymbol{q}=[q_1, q_2, ..., q_n]\) as the \((n \times 1)\) vector of joint variables. With respect to a base frame \(O_{b}-x_{b} y_{b} z_{b}\), the FK is expressed by the homogeneous transformation

\[\begin{split}\boldsymbol{T}_{e}^{b}(\boldsymbol{q})=\left[\begin{array}{cccc} \boldsymbol{n}_{e}^{b}(\boldsymbol{q}) & \boldsymbol{s}_{e}^{b}(\boldsymbol{q}) & \boldsymbol{a}_{e}^{b}(\boldsymbol{q}) & \boldsymbol{p}_{e}^{b}(\boldsymbol{q}) \\ 0 & 0 & 0 & 1 \end{array}\right]\end{split}\]

where \(\{\boldsymbol{n}_{e}, \boldsymbol{s}_{e}, \boldsymbol{a}_{e}\}\) are the unit vectors of the end-effector frame, and \(\boldsymbol{p}_{e}\) is the position of the origin of the end-effector frame. If the end-effector is a gripper, the origin of the end-effector frame is located at the center of the gripper, \(\boldsymbol{a}_{e}\) is chosen in the approach direction to the object, \(\boldsymbol{s}_{e}\) is chosen in the sliding plane of the jaws, and \(\boldsymbol{n}_{e}\) is chosen to complete the right-handed rule.

Math Convention in a Robot Arm#

A robot arm composes of \(n+1\) links connected by \(n\) joints, where Link 0 is conventionally fixed to the ground/base. Each joint variable provides one DOF. Define a frame attached to each link, from Link 0 to Link \(n\). Then, the transformation of Frame \(n\) with respect to Frame 0 is (postmultiplication rule)

\[\boldsymbol{T}_{n}^{0}(\boldsymbol{q})=\boldsymbol{T}_{1}^{0}\left(q_{1}\right) \boldsymbol{T}_{2}^{1}\left(q_{2}\right) \ldots \boldsymbol{T}_{n}^{n-1}\left(q_{n}\right) .\]

Each homogeneous transformation \(\boldsymbol{T}_{i}^{i-1}\left(q_{i}\right)\) (for \(\left.i=1, \ldots, n\right)\) is a function of a corresponding joint variable \(q_i\).

Typically, the base frame can be different from the frame of Link 0, the end-effector frame is different from the frame of Link n.The FK describing the pose of the end-effector frame with respect to the base frame is

\[\boldsymbol{T}_{e}^{b}(\boldsymbol{q})=\boldsymbol{T}_{0}^{b} \boldsymbol{T}_{n}^{0}(\boldsymbol{q}) \boldsymbol{T}_{e}^{n}\]

where \(\boldsymbol{T}_{0}^{b}\) and \(\boldsymbol{T}_{e}^{n}\) are two constant homogeneous transformations describing the position and orientation of Frame 0 with respect to the base frame, and of the end-effector frame with respect to Frame \(n\), respectively.

New question: how to neatly choose frame for each link to faciliate the computation of \(\boldsymbol{T}_{i}^{i-1}\left(q_{i}\right)\)? This is why DH convention comes to help.

Denavit-Hartenberg (DH) Convention#

Denavit-Hartenberg (DH) Convention is to systematically and recursively define a frame to each link and obtain the transformation between two consecutive links.

../_images/DH_convention.jpg — Fig. 25 Denavit-Hartenberg convention and parameters for link Frame \(i\)#

Important

With reference to Fig. 25, joint \(i\) connects Link \(i-1\) to Link \(i\); DH convention sets the following rules to define link Frame \(O_{i}-x_{i} y_{i} z_{i}\), given previous link Frame \(O_{i-1}-x_{i-1} y_{i-1} z_{i-1}\):

Choose axis \(z_{i}\) along the joint \(i+1\) (righthand rule), and axis \(z_{i-1}\) along the joint \(i\).
Choose the origin \(O_{i}\) at the intersection of \(z_{i}\) with the common normal to \(z_{i}\) and \(z_{i-1}\) .
Choose \(x_{i}\) along the common normal to \(z_{i}\) and \(z_{i-1}\) with direction from Joint \(i\) to Joint \(i+1\).
Choose \(y_{i}\) to complete a right-handed frame.

Once the link frames are established by DH convention, the pose of link Frame \(O_{i}-x_{i} y_{i} z_{i}\) in the link Frame \(O_{i-1}-x_{i-1} y_{i-1} z_{i-1}\) can be determined by four basic parameters, with the help of an intermediate frame \(O_{i^{\prime}}-x_{i^{\prime}} y_{i^{\prime}} z_{i^{\prime}}\) shown in Fig. 25:

\(a_{i}\) distance between \(O_{i}\) and \(O_{i^{\prime}}\) — constant and depend only on the geometry of links
\(\alpha_{i}\) angle between axes \(z_{i-1}\) and \(z_{i}\) about axis \(x_{i}\) — constant and depend only on the geometry of links
\(d_{i}\) distance of between \(O_{i-1}\) and \(O_{i^{\prime}}\) along \(z_{i-1}\) — variable if Joint \(i\) is prismatic, otherwise constant
\(\vartheta_{i}\) angle between axes \(x_{i-1}\) and \(x_{i}\) about axis \(z_{i-1}\) — variable

To compute \(\boldsymbol{T}_{i}^{i-1}\left(q_{i}\right)\), we perform four basic transformations. First, translate the Frame \(O_{i-1}-x_{i-1} y_{i-1} z_{i-1}\) by \(d_{i}\) along axis \(z_{i-1}\). Second, Rotate the resulting frame by \(\vartheta_{i}\) about \(z_{i-1}\). These two steps lead to the pose of the intermediate Frame \(O_{i^{\prime}}-x_{i^{\prime}} y_{i^{\prime}} z_{i^{\prime}}\) in Frame \(O_{i-1}-x_{i-1} y_{i-1} z_{i-1}\):

\[\begin{split}\boldsymbol{A}_{i^{\prime}}^{i-1}=\left[\begin{array}{cccc} c_{\vartheta_{i}} & -s_{\vartheta_{i}} & 0 & 0 \\ s_{\vartheta_{i}} & c_{\vartheta_{i}} & 0 & 0 \\ 0 & 0 & 1 & d_{i} \\ 0 & 0 & 0 & 1 \end{array}\right]\end{split}\]

Third, translate the intermediate Frame \(i^{\prime}\) by \(a_{i}\) along axis \(x_{i^{\prime}}\), and fourth, rotate it by \(\alpha_{i}\) about axis \(x_{i^{\prime}}\). The last two steps lead to the pose of \(O_{i}-x_{i} y_{i} z_{i}\) in Frame \(O_{i^{\prime}}-x_{i^{\prime}} y_{i^{\prime}} z_{i^{\prime}}\):

\[\begin{split}\boldsymbol{A}_{i}^{i^{\prime}}=\left[\begin{array}{cccc} 1 & 0 & 0 & a_{i} \\ 0 & c_{\alpha_{i}} & -s_{\alpha_{i}} & 0 \\ 0 & s_{\alpha_{i}} & c_{\alpha_{i}} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\end{split}\]

In sum of the above four steps, the total transformation of link \(i\)’s Frame \(O_{i}-x_{i} y_{i} z_{i}\) in the link \(i-1\)’s Frame \(O_{i-1}-x_{i-1} y_{i-1} z_{i-1}\) is

\[\begin{split}\boldsymbol{T}_{i}^{i-1}\left(q_{i}\right)=\boldsymbol{A}_{i^{\prime}}^{i-1} \boldsymbol{A}_{i}^{i^{\prime}}=\left[\begin{array}{cccc} c_{\vartheta_{i}} & -s_{\vartheta_{i}} c_{\alpha_{i}} & s_{\vartheta_{i}} s_{\alpha_{i}} & a_{i} c_{\vartheta_{i}} \\ s_{\vartheta_{i}} & c_{\vartheta_{i}} c_{\alpha_{i}} & -c_{\vartheta_{i}} s_{\alpha_{i}} & a_{i} s_{\vartheta_{i}} \\ 0 & s_{\alpha_{i}} & c_{\alpha_{i}} & d_{i} \\ 0 & 0 & 0 & 1 \end{array}\right]\end{split}\]

The DH convention gives a nonunique definition of the link frame in the following cases.

For Frame 0, only the direction of axis \(z_{0}\) is specified; then \(O_{0}\) and \(x_{0}\) can be arbitrarily chosen.
For Frame \(n\), since there is no Joint \(n+1, z_{n}\) is not uniquely defined while \(x_{n}\) has to be normal to axis \(z_{n-1}\). Typically, Joint \(n\) is revolute, and thus \(z_{n}\) is to be aligned with the direction of \(z_{n-1}\).
When two consecutive joint axes intersect, the direction (+/-) of \(x_{i}\) is arbitrary.
When two consecutive joint axes parallel, \(x_{i}\) is chosen such that \(d_i=0\).
When Joint \(i\) is prismatic, the direction (+/-) of \(z_{i-1}\) is arbitrary.

In all such cases, the principle of choosing frames is to simplify the procedure; for instance, the axes of consecutive frames can be made parallel.

Important

To summarize, we write the following DH convention for a robot arm.

Identify each joint and set the directions of axes \(z_{0}, \ldots, z_{n-1}\)
Choose Frame \(O_{0}-x_{0} y_{0} z_{0}\). If possible, choose \(\{\boldsymbol{x}_{0}, \boldsymbol{y}_{0}, \boldsymbol{z}_{0}\}\) to coincide with the base frame.

Repeat step 3 from link \(i=1\) to link \(n-1\) :

Choose link i’s Frame \(O_{i}-x_{i} y_{i} z_{i}\) as follows.

First, choose \(O_{i}\) at the intersection of \(z_{i}\) with the common normal to \(z_{i-1}\) and \(z_{i}\). If \(z_{i-1}\) and \(z_{i}\) are parallel and Joint \(i\) is revolute, choose \(O_{i}\) so that \(d_{i}=0\); if Joint \(i\) is prismatic, choose \(O_{i}\) at as a mechanical limit. Second, choose \(x_{i}\) along the common normal to \(z_{i-1}\) and \(z_{i}\) with direction from Joint \(i\) to Joint \(i+1\), Third, choose axis \(y_{i}\) using right-handed rule.

Calculate transformation:

Choose final link’s Frame \(O_{n}-x_{n} y_{n} z_{n}\); if Joint \(n\) is revolute, then align \(z_{n}\) with \(z_{n-1}\), otherwise, if Joint \(n\) is prismatic, then choose \(z_{n}\) arbitrarily. choose axis \(x_{n}\) along the common normal to axes \(z_{n-1}\) and \(z_{n}\).
For \(i=1, \ldots, n\), establish the four DH parameter \(a_{i}, d_{i}, \alpha_{i}, \vartheta_{i}\), compute the consecutive transformation \(\boldsymbol{T}_{i}^{i-1}\left(q_{i}\right)\), and compute the transformation \(\boldsymbol{T}_{n}^{0}(\boldsymbol{q})=\boldsymbol{T}_{1}^{0} \ldots \boldsymbol{T}_{n}^{n-1}\).

Complete the assembly:

Given \(\boldsymbol{T}_{0}^{b}\) and \(\boldsymbol{T}_{e}^{n}\), compute the kinematics function as \(\boldsymbol{T}_{e}^{b}(\boldsymbol{q})=\) \(\boldsymbol{T}_{0}^{b} \boldsymbol{T}_{n}^{0} \boldsymbol{T}_{e}^{n}\).

Examples#

Three-link Planar Arm

../_images/3link_arm.jpg — Fig. 26 Three link robot arm#

Link

\(a_{i}\)

\(\alpha_{i}\)

\(d_{i}\)

\(\vartheta_{i}\)

1

\(a_{1}\)

0

0

\(\vartheta_{1}\)

2

\(a_{2}\)

0

0

\(\vartheta_{2}\)

3

\(a_{3}\)

0

0

\(\vartheta_{3}\)

\[\begin{split}\boldsymbol{T}_{3}^{0}(\boldsymbol{q})=\boldsymbol{T}_{1}^{0} \boldsymbol{T}_{2}^{1} \boldsymbol{T}_{3}^{2}=\left[\begin{array}{cccc} c_{123} & -s_{123} & 0 & a_{1} c_{1}+a_{2} c_{12}+a_{3} c_{123} \\ s_{123} & c_{123} & 0 & a_{1} s_{1}+a_{2} s_{12}+a_{3} s_{123} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\end{split}\]

Spherical Arm

Anthropomorphic Arm

Spherical Wrist

Stanford Manipulator

\[\begin{split}\boldsymbol{T}_{6}^{0}=\boldsymbol{T}_{3}^{0} \boldsymbol{T}_{6}^{3}=\left[\begin{array}{cccc} \boldsymbol{n}^{0} & \boldsymbol{s}^{0} & \boldsymbol{a}^{0} & \boldsymbol{p}^{0} \\ 0 & 0 & 0 & 1 \end{array}\right]\end{split}\]

with

\[\begin{split}\boldsymbol{p}_{6}^{0}=\left[\begin{array}{c} c_{1} s_{2} d_{3}-s_{1} d_{2}+\left(c_{1}\left(c_{2} c_{4} s_{5}+s_{2} c_{5}\right)-s_{1} s_{4} s_{5}\right) d_{6} \\ s_{1} s_{2} d_{3}+c_{1} d_{2}+\left(s_{1}\left(c_{2} c_{4} s_{5}+s_{2} c_{5}\right)+c_{1} s_{4} s_{5}\right) d_{6} \\ c_{2} d_{3}+\left(-s_{2} c_{4} s_{5}+c_{2} c_{5}\right) d_{6} \end{array}\right]\end{split}\]

\[\begin{split}\begin{aligned} & \boldsymbol{n}_{6}^{0}= {\left[\begin{array}{c} c_{1}\left(c_{2}\left(c_{4} c_{5} c_{6}-s_{4} s_{6}\right)-s_{2} s_{5} c_{6}\right)-s_{1}\left(s_{4} c_{5} c_{6}+c_{4} s_{6}\right) \\ s_{1}\left(c_{2}\left(c_{4} c_{5} c_{6}-s_{4} s_{6}\right)-s_{2} s_{5} c_{6}\right)+c_{1}\left(s_{4} c_{5} c_{6}+c_{4} s_{6}\right) \\ -s_{2}\left(c_{4} c_{5} c_{6}-s_{4} s_{6}\right)-c_{2} s_{5} c_{6} \end{array}\right] } \\ & \boldsymbol{s}_{6}^{0}=\left[\begin{array}{c} c_{1}\left(-c_{2}\left(c_{4} c_{5} s_{6}+s_{4} c_{6}\right)+s_{2} s_{5} s_{6}\right)-s_{1}\left(-s_{4} c_{5} s_{6}+c_{4} c_{6}\right) \\ s_{1}\left(-c_{2}\left(c_{4} c_{5} s_{6}+s_{4} c_{6}\right)+s_{2} s_{5} s_{6}\right)+c_{1}\left(-s_{4} c_{5} s_{6}+c_{4} c_{6}\right) \\ s_{2}\left(c_{4} c_{5} s_{6}+s_{4} c_{6}\right)+c_{2} s_{5} s_{6} \end{array}\right] \\ & \boldsymbol{a}_{6}^{0}=\left[\begin{array}{c} c_{1}\left(c_{2} c_{4} s_{5}+s_{2} c_{5}\right)-s_{1} s_{4} s_{5} \\ s_{1}\left(c_{2} c_{4} s_{5}+s_{2} c_{5}\right)+c_{1} s_{4} s_{5} \\ -s_{2} c_{4} s_{5}+c_{2} c_{5} \end{array}\right] \end{aligned}\end{split}\]

Anthropomorphic Arm with Spherical Wrist

../_images/anthropomorphic_manipulator.jpg — Fig. 31 Anthropomorphic manipulator#

Link

\(a_{i}\)

\(\alpha_{i}\)

\(d_{i}\)

\(\vartheta_{i}\)

1

0

\(\pi / 2\)

0

\(\vartheta_{1}\)

2

\(a_{2}\)

0

0

\(\vartheta_{2}\)

3

0

\(\pi / 2\)

0

\(\vartheta_{3}\)

4

0

\(-\pi / 2\)

\(d_{4}\)

\(\vartheta_{4}\)

5

0

\(\pi / 2\)

0

\(\vartheta_{5}\)

6

0

0

\(d_{6}\)

\(\vartheta_{6}\)

\[\begin{split}\boldsymbol{T}_{6}^{0}=\left[\begin{array}{cccc} \boldsymbol{n}^{0} & \boldsymbol{s}^{0} & \boldsymbol{a}^{0} & \boldsymbol{p}^{0} \\ 0 & 0 & 0 & 1 \end{array}\right]\end{split}\]

with

\[\begin{split}\begin{aligned} \boldsymbol{p}_{6}^{0}=&\left[\begin{array}{c} a_{2} c_{1} c_{2}+d_{4} c_{1} s_{23}+d_{6}\left(c_{1}\left(c_{23} c_{4} s_{5}+s_{23} c_{5}\right)+s_{1} s_{4} s_{5}\right) \\ a_{2} s_{1} c_{2}+d_{4} s_{1} s_{23}+d_{6}\left(s_{1}\left(c_{23} c_{4} s_{5}+s_{23} c_{5}\right)-c_{1} s_{4} s_{5}\right) \\ a_{2} s_{2}-d_{4} c_{23}+d_{6}\left(s_{23} c_{4} s_{5}-c_{23} c_{5}\right) \end{array}\right]\\ \boldsymbol{n}_{6}^{0}= & {\left[\begin{array}{c} c_{1}\left(c_{23}\left(c_{4} c_{5} c_{6}-s_{4} s_{6}\right)-s_{23} s_{5} c_{6}\right)+s_{1}\left(s_{4} c_{5} c_{6}+c_{4} s_{6}\right) \\ s_{1}\left(c_{23}\left(c_{4} c_{5} c_{6}-s_{4} s_{6}\right)-s_{23} s_{5} c_{6}\right)-c_{1}\left(s_{4} c_{5} c_{6}+c_{4} s_{6}\right) \\ s_{23}\left(c_{4} c_{5} c_{6}-s_{4} s_{6}\right)+c_{23} s_{5} c_{6} \end{array}\right] } \\ \boldsymbol{s}_{6}^{0}= & {\left[\begin{array}{c} c_{1}\left(-c_{23}\left(c_{4} c_{5} s_{6}+s_{4} c_{6}\right)+s_{23} s_{5} s_{6}\right)+s_{1}\left(-s_{4} c_{5} s_{6}+c_{4} c_{6}\right) \\ s_{1}\left(-c_{23}\left(c_{4} c_{5} s_{6}+s_{4} c_{6}\right)+s_{23} s_{5} s_{6}\right)-c_{1}\left(-s_{4} c_{5} s_{6}+c_{4} c_{6}\right) \\ -s_{23}\left(c_{4} c_{5} s_{6}+s_{4} c_{6}\right)-c_{23} s_{5} s_{6} \end{array}\right] } \\ \boldsymbol{a}_{6}^{0}= & {\left[\begin{array}{c} c_{1}\left(c_{23} c_{4} s_{5}+s_{23} c_{5}\right)+s_{1} s_{4} s_{5} \\ s_{1}\left(c_{23} c_{4} s_{5}+s_{23} c_{5}\right)-c_{1} s_{4} s_{5} \\ s_{23} c_{4} s_{5}-c_{23} c_{5} \end{array}\right] } \end{aligned}\end{split}\]

Workspace#

The robot arm workspace is the space reached by the origin of the end-effector frame when the robot arm’s joints take all allowable values. The workspace includes reachable workspace and dexterous workspace. The latter is the space that the origin of the end-effector frame can reach with different orientations, while the former is the space that the origin of the end-effector frame can reach with at least one orientation.

For an \(n\)-DOF robot arm, the reachable workspace is formally defined

\[\{ \boldsymbol{p}_{e}(\boldsymbol{q})\,|\, \quad q_{i m} \leq q_{i} \leq q_{i M} \quad i=1, \ldots, n, \}\]

where \(q_{i m}\left(q_{i M}\right)\) denotes the minimum (maximum) values at Joint \(i\). The workspace is typically reported in the data sheet of a robot manufacturer in terms of a top view and a side view.

In a robot arm, if actual mechanical parameters differ from the nominal value in data sheet, an error arises between actual position reached and theorical position computed via direct kinematics. Such error is called accuracy. Nowdays, accuracy level of a decent-sized robot arm is typically below 1mm.

Another parameter of a robot arm is repeatability which gives a measure of the robot’s ability to return to a previously reached position. Repeatability depends on external disturbance and also on internal controller; it is typically smaller than accuracy. For instance, for a robot arm with a maximum reach of \(1.5 \mathrm{~m}\), accuracy varies from 0.2 to \(1 \mathrm{~mm}\), while repeatability varies from 0.02 to \(0.2 \mathrm{~mm}\).

Kinematic Redundancy#

A robot arm is kinematically redundant when its DOF \(n\) is greater than dimension \(m\) of the task space (also named operational space), typically refering to the motion space of end-effector (frame). Redundancy is a concept relative to the task space; a robot arm can be redundant with respect to a task and nonredundant with respect to another.

Consider the three-DOF planar arm. If one only cares about the end effector’s position (i.e., the operation space is only the positional space of end effector), the robot arm is a functional redundancy \((n=3, n=2)\). When one care about both position and angle of the end-effector, then the robot is nonredundant, i.e., \(n=3, n=3\). On the other hand, a four-DOF planar arm is intrinsically redundant \((n=4, m=3)\).

At this point, a question may arise: Why to intentionally utilize a redundant robot arm? This is because redundancy can provide a robot arm with dexterity and versatility in its motion. The typical example is the human arm that has seven DOFs: three in the shoulder, one in the elbow and three in the wrist, without considering the DOFs in the fingers. Human arm is intrinsically redundant; in fact, if the base and the hand position and orientation are both fixed, the elbow can be moved, thanks to the additional available DOF. This can help us, for instance, avoid obstacles in the workspace. Further, if a joint of a redundant robot arm reaches its mechanical limit, there might be other joints that allow execution of the end-effector motion.

Modified DH parameters (Optional)#

../_images/DHParameter.png — Fig. 32 Modified DH parameters (or called Craig’s convention). Image from Wiki#

Some books use modified (proximal) DH parameters [John J. Craig, Introduction to Robotics: Mechanics and Control (3rd Edition)]. The difference between the classic DH parameters and the modified DH parameters are the locations of the coordinate to each links and the order of the performed transformations.

Compared with the classic DH parameters, the coordinates of frame, \(O_{i-1}\), is put on Joint \(i-1\), not the Joint \(i\) in classic DH convention;the coordinates of frame, \(O_{i}\), is put on Joint \(i\), not the Joint \(i+1\) in classic DH convention. Another difference is that according to the modified convention, the transform matrix is given by the following order of operations:

\[T^{i-1}_i=\text{Rot}_{x_{i-1}}(\alpha_{i-1})\text{Trans}_{x_{i-1}}(a_{i-1})\text{Rot}_{z_{i}}(\theta_{i})\text{Trans}_{z_{i}}(d_{i})\]

One example of using the above modified DH convention is Franka-Emika Panda robot arm, an increasingly popular robot for research and teaching. It has 7 joints which make it a redundant robot, that is, it has more joints than it needs to achieve an arbitrary position and orientation in the Cartesian workspace.

The DH parameter defined for Panda Robot arm is as follows: https://frankaemika.github.io/docs/control_parameters.html. Please find a great tutorial (using Python) about Panda Arm’s forward kinematics at

jhavl/dkt

../_images/dh-diagram.png — Fig. 33 Panda’s kinematic chain#

../_images/cover.png — Fig. 34 Panda’s kinematic chain (from Peter Corke’s Tutorial#

Link	\(a_{i}\)	\(\vartheta_{i}\)
1	\(a_{1}\)	\(\vartheta_{1}\)
2	\(a_{2}\)	\(\vartheta_{2}\)
3	\(a_{3}\)	\(\vartheta_{3}\)

Link	\(\alpha_{i}\)	\(d_{i}\)	\(\vartheta_{i}\)
1	\(-\pi / 2\)	0	\(\vartheta_{1}\)
2	\(\pi / 2\)	\(d_{2}\)	\(\vartheta_{2}\)
3	0	\(d_{3}\)	0

Link	\(\alpha_{i}\)	\(d_{i}\)	\(\vartheta_{i}\)
4	\(-\pi / 2\)	0	\(\vartheta\)
5	\(\pi / 2\)	0	\(\vartheta_{5}\)
6	0	\(d_{6}\)	\(\vartheta_{6}\)