Module 2/5 · Weeks 4–6 · 27 h

Robotics and kinematics

DRT 344 Automation, Robotics and Intelligent Control Systems

About 90 minDraft, awaiting reviewLast updated 26 September 2026

By the end of this module you will be able to

Define a robot as in ISO 8373 and classify robots by how they move
Explain the degrees of freedom (DOF) of robots and drones
Calculate forward and inverse kinematics of a two-link planar arm
Calculate the speed and turning radius of a differential-drive robot
Explain robot safety principles from ISO 10218 and ISO/TS 15066

Prerequisites: DRT 111 (trigonometry and vectors) · DRT 344 module 1

Why this matters

Many future jobs combine drones with ground robots: a warehouse mobile robot takes boxes from a drone, a robot arm swaps a drone’s battery at a dock, or a pipe-inspection robot works alongside a drone photographing from above. Designing these systems means answering basic questions such as “if the joints turn this much, where does the tip go?” and “if the wheels turn this much, how does the robot move?” This is kinematics.

Kinematics is like working out how to touch a glass with your finger. You do not think about muscle forces; you think about how far to bend your shoulder and elbow to put your hand exactly on the glass.

What is a robot?

ISO 8373:2021 defines a robot as a programmed actuated mechanism with a degree of autonomy to perform locomotion, manipulation or positioning, and a robot includes its control system. “Autonomy” is the ability to perform intended tasks based on current state and sensing, without human intervention. By this definition, a drone flying a mission by itself is a kind of robot.

Type	Examples	How it moves
Industrial arm	Pick-and-place arm, battery-swap arm	Fixed base; moves its tip within a workspace
Ground mobile robot	Warehouse carrier, patrol robot	Moves over the ground on wheels or legs
Aerial robot	Multirotor drone	Moves in three dimensions

Degrees of freedom (DOF) are the number of independent variables needed to specify a system’s configuration. A rigid body in space has 6 DOF (3 position, 3 rotation). A multirotor has 6 DOF but only 4 directly commanded inputs (total thrust, roll, pitch, yaw), so it must tilt before it can move sideways.

Kinematics of a two-link arm

A two-link planar arm: link one of length L1 at angle theta 1 from the horizontal, link two of length L2 at angle theta 2 relative to link one, with the end effector at x, y — Figure 1 Geometry of a two-link planar arm

Forward kinematics

Given the joint angles, find the tip position (Lynch & Park, 2017):

x = L_{1} cos θ_{1} + L_{2} cos (θ_{1} + θ_{2}), y = L_{1} sin θ_{1} + L_{2} sin (θ_{1} + θ_{2})

Inverse kinematics

Given a desired position, find the joint angles. This is harder because there may be several solutions or none. From the law of cosines:

cos θ_{2} = \frac{x ^{2} + y ^{2} - L _{1}^{2} - L _{2}^{2}}{2 L _{1} L _{2}}, θ_{1} = atan2 (y, x) - atan2 (L_{2} sin θ_{2}, L_{1} + L_{2} cos θ_{2})

If $cos θ_{2}$ falls outside $[- 1, 1]$ , the point is outside the workspace. Positive and negative $θ_{2}$ give two solutions (elbow up or elbow down).

Example 1 A battery-swap arm

The arm has $L_{1} = 0.40$ m and $L_{2} = 0.30$ m.

Forward, at $θ_{1} = 3 0^{\circ}$ , $θ_{2} = 4 5^{\circ}$ :

$x = 0.40 cos 3 0^{\circ} + 0.30 cos 7 5^{\circ} = 0.3464 + 0.0776 = 0.424$ m
$y = 0.40 sin 3 0^{\circ} + 0.30 sin 7 5^{\circ} = 0.2000 + 0.2898 = 0.490$ m

Inverse, to reach $(0.50, 0.30)$ m:

Distance from the base $0.5 0^{2} + 0.3 0^{2} = 0.583$ m lies between $∣ L_{1} - L_{2} ∣ = 0.10$ and $L_{1} + L_{2} = 0.70$ m, so it is reachable
$cos θ_{2} = \frac{0.25 + 0.09 - 0.16 - 0.09}{2 ( 0.40 ) ( 0.30 )} = \frac{0.09}{0.24} = 0.375$ , so $θ_{2} \approx 67.9 8^{\circ}$
$θ_{1} = atan2 (0.30, 0.50) - atan2 (0.30 sin 67.9 8^{\circ}, 0.40 + 0.30 cos 67.9 8^{\circ}) \approx 30.9 6^{\circ} - 28.4 9^{\circ} \approx 2.4 8^{\circ}$
Checking with forward kinematics gives $(0.500, 0.300)$ , as required

Differential-drive robots

Many mobile robots use separately driven left and right wheels (differential drive). With wheel radius $r$ , wheels $2 d$ apart, and wheel angular speeds $u_{L}$ and $u_{R}$ :

Figure 2 Differential-drive robot, top view

v = \frac{r ( u _{L} + u _{R} )}{2}, ω = \frac{r ( u _{R} - u _{L} )}{2 d}, R = \frac{v}{ω}

Equal wheel speeds drive straight ( $ω = 0$ ); equal and opposite speeds spin on the spot ( $v = 0$ ). This robot cannot move directly sideways, a constraint path planning must respect.

Example 2 A robot collecting boxes from a drone

Wheel radius $r = 0.05$ m, half the track width $d = 0.20$ m, right wheel 10 rad/s, left wheel 8 rad/s.

$v = 0.05 (8 + 10) /2 = 0.45$ m/s
$ω = 0.05 (10 - 8) / (2 \times 0.20) = 0.25$ rad/s (turning left, since the right wheel is faster)
Turning radius $R = 0.45/0.25 = 1.8$ m

Robot safety

ISO 10218-1:2025 sets safety requirements for industrial robots, and ISO/TS 15066:2016 is the technical specification for collaborative robots working near people (now under revision). Principles that also apply when drones and robots work together:

Separate spaces with fences or presence sensors; slow down or stop when a person enters
Limit force and power for robots that work close to people
Accessible emergency stops, and logic where stopping takes priority over running (module 1)
Assess the risk of the whole work cell, not just the robot (DRT 343 module 3)

Class activity

Activity: An automated battery-swap cell

Define the drone’s battery-bay position relative to the arm base and calculate the joint angles with inverse kinematics.
Check that every point is in the workspace and choose the elbow-up or elbow-down solution that avoids the structure.
Design the path of a differential-drive robot carrying the old battery to the charger, and calculate wheel speeds on the turn.
List safety measures for when a person walks into the cell.

Common mistakes

Watch out

Using degrees in functions that expect radians in code
Not checking the workspace before solving inverse kinematics
Using atan instead of atan2, giving the wrong quadrant
Assuming a differential-drive robot can move sideways
Assessing only the robot’s safety, not the whole cell

Summary

ISO 8373 defines a robot as a programmed actuated mechanism with a degree of autonomy
DOF are independent configuration variables; a drone has 6 DOF but 4 direct inputs
2R arm FK: $x = L_{1} cos θ_{1} + L_{2} cos (θ_{1} + θ_{2})$ ; IK uses the law of cosines and can have two solutions
Differential drive: $v = r (u_{L} + u_{R}) /2$ , $ω = r (u_{R} - u_{L}) /2 d$
Robot safety uses separation, force limits, emergency stops and whole-cell risk assessment

Check your understanding

Under the ISO 8373 definition, is a drone flying an automated mission a robot? Why?
With $L_{1} = 0.40$ m, $L_{2} = 0.30$ m, $θ_{1} = 0^{\circ}$ and $θ_{2} = 9 0^{\circ}$ , where is the tip?
Can the same arm reach a point 0.75 m from its base?
A differential-drive robot with $r = 0.05$ m has both wheels at 12 rad/s. What is its angular speed?
If the left wheel turns at −5 rad/s and the right at 5 rad/s, how does the robot move?

Answers

Yes. It is a programmed, actuated mechanism that moves by itself based on its current state and sensing
$x = 0.40 + 0.30 cos 9 0^{\circ} = 0.40$ m, $y = 0 + 0.30 sin 9 0^{\circ} = 0.30$ m, so $(0.40, 0.30)$
No. Its maximum reach is $0.40 + 0.30 = 0.70$ m
$ω = 0$ ; it drives straight
$v = 0$ ; it spins on the spot, counter-clockwise

Key formulas

Forward kinematics of a 2R arm	$x = L_{1} cos θ_{1} + L_{2} cos (θ_{1} + θ_{2}), y = L_{1} sin θ_{1} + L_{2} sin (θ_{1} + θ_{2})$
Inverse kinematics (elbow angle)	$cos θ_{2} = \frac{x ^{2} + y ^{2} - L _{1}^{2} - L _{2}^{2}}{2 L _{1} L _{2}}$
Differential drive	$v = \frac{r ( u _{L} + u _{R} )}{2}, ω = \frac{r ( u _{R} - u _{L} )}{2 d}, R = \frac{v}{ω}$

Key references

International Organization for Standardization. (2021). Robotics – Vocabulary (ISO 8373:2021). link
Lynch, K. M., & Park, F. C. (2017). Modern robotics: Mechanics, planning, and control. Cambridge University Press. link
Siegwart, R., Nourbakhsh, I. R., & Scaramuzza, D. (2011). Introduction to autonomous mobile robots (2nd ed.). MIT Press.
International Organization for Standardization. (2025). Robotics – Safety requirements – Part 1: Industrial robots (ISO 10218-1:2025). link
International Organization for Standardization. (2016). Robots and robotic devices – Collaborative robots (ISO/TS 15066:2016). link
Corke, P. (2023). Robotics, vision and control (3rd ed.). Springer.

In class / field

Lecture, case discussion and in-class problem solving

Learning evidence: Quiz results and submitted exercises

Module quiz

This is a formative self-check, not a graded exam

Knowledge domain: Automation, robotics and swarms

Robotics and kinematics

By the end of this module you will be able to

Why this matters

What is a robot?

Kinematics of a two-link arm

Forward kinematics

Inverse kinematics

Differential-drive robots

Robot safety

Class activity

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

Robot kinematics and motion planning

In class / field

Module quiz

Robotics and kinematics

Lesson

By the end of this module you will be able to

Why this matters

What is a robot?

Kinematics of a two-link arm

Forward kinematics

Inverse kinematics

Differential-drive robots

Robot safety

Class activity

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

Robot kinematics and motion planning

In class / field

Module quiz