Fundamental Kinematic Equations For Industrial Robots

In robotics, a kinematic model is the mathematical relationship that maps a robot’s joint variables (angles or displacements) to the pose (position + orientation) of its body or end-effector — and vice-versa — without using forces, torques, masses, or inertia.

 

It belongs to Robot Kinematics rather than Robot Dynamics. 

What the model actually does

It answers two fundamental problems:

1) Forward kinematics (FK)

joint values q→pose (x,y,z,R)

Given motor encoder readings → where is the tool tip?

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2) Inverse kinematics (IK)

$$\text{desired pose} \;\rightarrow\; \text{joint values}$$

Given a target position → what should each motor angle be?

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Example — mobile robot (differential drive)

State:

$$(x, y, \theta)$$

Wheel speeds: $v_L, v_R$

$$v = \frac{r}{2}(v_R + v_L)$$ $$\omega = \frac{r}{L}(v_R - v_L)$$ $$\dot{x} = v\cos\theta,\quad \dot{y} = v\sin\theta,\quad \dot{\theta} = \omega$$

This lets the robot:

• estimate its trajectory (odometry)

• do localization

• plan paths

No motor torque model required.

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Example — robot arm

A 6-axis industrial manipulator:

Joint vector:

$$q = [\theta_1,\theta_2,\dots,\theta_6]$$

Pose:

$$T_{0}^{6} = \begin{bmatrix} R & p \\ 0 & 1 \end{bmatri$$

Computed using Denavit–Hartenberg parameters.

This is the core of:

• pick-and-place

• CNC robots

• surgical robots

• manipulators

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What is NOT included

A kinematic model ignores:

• torque

• inertia

• friction

• motor current

• gravity compensation

Those belong to dynamics.

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Why robotics always starts with kinematics

Because most robot software needs only geometry:

Task	Needs kinematics?	Needs dynamics?
Localization	✔	✖
Path planning	✔	✖
SLAM	✔	✖
Visual servoing	✔	✖
Motion control (low speed)	✔	✖
Torque control	✖	✔

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A robotic kinematic model is the geometric mapping between actuator motions and the robot’s pose in space, used to predict and control motion without modeling physical forces.

  

The Denavit–Hartenberg (DH) convention is a standardized method used in Robot Kinematics to systematically assign coordinate frames to the links of a serial robot and express the robot’s forward kinematics using a small set of parameters.

It converts a complex 3D mechanism into a sequence of simple homogeneous transformations.

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Purpose

Instead of writing geometry manually for each robot, the DH convention provides a repeatable procedure:

$$\text{Robot geometry} \;\longrightarrow\; \text{table of parameters} \;\longrightarrow\; \text{transformation matrices}$$

From that table you can compute:

$$T_0^n = A_1 A_2 A_3 \cdots A_n$$

which gives the pose of the end-effector.

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The four DH parameters

For each joint $i$, only four numbers describe the relative pose between two consecutive links.

Parameter	Symbol	Meaning	Type
Link length	$a_i$	distance between joint axes along $x_i$	geometry
Link twist	$\alpha_i$	angle between joint axes around $x_i$	geometry
Link offset	$d_i$	distance along $z_{i-1}$	prismatic variable
Joint angle	$\theta_i$	rotation around $z_{i-1}$	revolute variable

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Transformation matrix

Each joint produces one homogeneous transformation:

$$A_i = \begin{bmatrix} \cos\theta_i & -\sin\theta_i\cos\alpha_i & \sin\theta_i\sin\alpha_i & a_i\cos\theta_i \\ \sin\theta_i & \cos\theta_i\cos\alpha_i & -\cos\theta_i\sin\alpha_i & a_i\sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

The end-effector pose:

$$T_0^n = \prod_{i=1}^{n} A_i$$

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Key idea

The convention forces every joint motion to occur along a z-axis and every link length along an x-axis.
This drastically simplifies robotic modeling.

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Why it is important

The DH convention allows:

• automatic forward kinematics

• inverse kinematics derivation

• Jacobian computation

• simulation and control implementation

• universal description of any serial manipulator

It is the most widely used geometric modeling method in industrial robotics.

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The Denavit–Hartenberg convention is a standardized four-parameter representation that describes the relative pose between consecutive robot links and enables systematic computation of a manipulator’s kinematics.