Robotics Fundamentals

Robotics sits at the intersection of geometry, mechanics, electronics, and software. A practical understanding starts with how a robot is shaped and moves, then progresses to how it is controlled, how it senses the world, and how modern systems are organized using frameworks such as ROS. This article covers the core ideas that show up across industrial arms, mobile robots, and research platforms: robot geometry, kinematics and dynamics, trajectory planning, PID control, and sensor integration.

Robot geometry and coordinate frames

A robot is fundamentally a collection of rigid bodies connected by joints. Geometry in robotics is less about drawing and more about being precise with coordinate frames.

Links, joints, and degrees of freedom

• Links are rigid segments.
• Joints allow relative motion between links. The most common are revolute joints (rotation) and prismatic joints (translation).
• Degrees of freedom (DoF) count independent motions. A typical industrial arm might have 6 DoF to position and orient its tool in 3D space.

The arrangement of joints determines what motions are possible, where singularities occur, and how easy it is to plan paths.

Frames and transformations

Robots use coordinate frames to describe positions and orientations: a base frame, frames attached to each link, and an end-effector frame at the tool. A pose is commonly represented by a rotation and translation, packaged in a homogeneous transformation matrix:

$$T=\left[\begin{array}{cc}R& p\\ 0& 1\end{array}\right]$$

Here $R$ is a $3\times 3$ rotation matrix and $p$ is a 3D position vector. Chaining transformations is how robots compute where the tool is relative to the base.

Forward kinematics and inverse kinematics

Kinematics describes motion without considering forces. It is the backbone of motion planning, control, simulation, and perception.

Forward kinematics (FK)

Forward kinematics answers: given joint variables $q$ (angles or displacements), where is the end effector?

For a serial robot arm, FK is computed by multiplying link transformations from base to tool:

$$T_{0n}(q)=T_{01}(q_1)T_{12}(q_2)\cdots T_{(n-1)n}(q_n)$$

FK is deterministic and typically straightforward once the robot’s geometry is defined. In practice, FK is used constantly: to display the robot in a simulator, to predict tool position, and to compute Jacobians for control.

Inverse kinematics (IK)

Inverse kinematics asks the harder question: given a desired tool pose $T^{\ast }$, what joint variables $q$ achieve it?

IK often has:

• Multiple solutions (elbow-up vs elbow-down configurations).
• No solution if the target is outside the workspace or violates joint limits.
• Sensitivity near singularities, where small pose changes require large joint changes.

Closed-form IK exists for some common arm geometries, but many robots rely on numerical IK, which iteratively adjusts $q$ to reduce pose error. In applied robotics, IK is rarely “set and forget”; it must respect joint limits, collision constraints, and task preferences such as keeping the wrist away from awkward angles.

Dynamics: why forces and inertia matter

Dynamics extends kinematics by accounting for forces, masses, and inertia. Even if a controller is designed in joint space, the robot still lives in the physical world: heavy links resist acceleration, friction steals torque, and gravity pulls on extended arms.

A common model form is:

$$\tau =M(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)+{\tau }_f$$

• $M(q)$: inertia matrix
• $C(q,\dot{q})$: Coriolis and centrifugal effects
• $g(q)$: gravity torques
• ${\tau }_f$: friction and other disturbances

Understanding dynamics informs actuator selection, achievable speeds, and the difference between a robot that tracks smoothly and one that oscillates or overheats. Even when you do not implement full model-based control, gravity compensation and basic friction handling can dramatically improve performance.

Trajectory planning: from goals to motion

Trajectory planning turns a desired motion into a time-parameterized command. It is not only about getting from A to B, but doing so safely, smoothly, and within limits.

Path vs trajectory

• A path is geometric: it describes where the robot should go.
• A trajectory adds time: it specifies position, velocity, and sometimes acceleration as functions of time.

A robot arm might need to move the tool tip along a straight line in Cartesian space, while joints follow a coordinated motion that avoids joint limits and keeps velocities within bounds.

Smooth profiles and constraints

Real actuators cannot jump instantly to a new velocity. Common motion profiles include trapezoidal velocity profiles and S-curve profiles, both designed to limit velocity and acceleration (and, for S-curves, jerk). Planning typically respects:

• Joint position limits
• Max joint velocities and accelerations
• Workspace constraints
• Collision avoidance (with the environment and self-collisions)

In mobile robotics, trajectory planning often adds nonholonomic constraints (such as a car-like robot that cannot move sideways) and is coupled with localization.

Control fundamentals: PID and beyond

Planning generates reference trajectories. Control is the layer that makes the robot follow them despite modeling errors and disturbances.

PID control in robotics

The workhorse controller is PID:

$$u(t)=K_{p}e(t)+K_i\int e(t)dt+K_d\frac{de(t)}{dt}$$

• $e(t)$ is the tracking error (position error in joint or task space).
• $K_p$ drives responsiveness.
• $K_i$ removes steady-state error but can cause windup.
• $K_d$ damps motion and reduces overshoot.

PID is widely used because it is simple and effective when properly tuned. On a joint-controlled robot arm, each joint might have its own PID loop, often running at high frequency. In practice, successful PID control depends on sensor quality, sampling rate, actuator bandwidth, and careful tuning to avoid oscillation.

Practical tuning considerations

• Start with proportional gain to get basic responsiveness.
• Add derivative action to reduce overshoot and vibration.
• Introduce integral action cautiously and add anti-windup if commands saturate.
• Validate performance across the robot’s workspace, not only in one pose.

More advanced strategies include feedforward terms (like gravity compensation), computed torque control, and impedance control for safe interaction. Even then, the fundamentals of error, feedback, and stability remain the same.

Sensors and actuators: closing the loop

Robots are only as capable as their ability to sense and act reliably.

Common sensors

• Encoders measure joint position and sometimes velocity.
• IMUs provide angular velocity and acceleration for mobile robots and balancing systems.
• Cameras enable perception tasks such as object detection and visual servoing.
• LiDAR supports mapping and obstacle detection in navigation.
• Force/torque sensors enable compliant manipulation and contact-rich tasks.
• Proximity and tactile sensors improve safety and grasp feedback.

Sensor integration is not just wiring. It includes calibration (intrinsic and extrinsic), time synchronization, filtering, and dealing with noise and drift. A simple example is using an IMU to stabilize orientation while fusing wheel encoders to reduce long-term drift in a mobile robot.

Actuators and drive systems

Robotic actuators range from DC motors with gearboxes to torque-controlled brushless motors, hydraulics for high power, and pneumatics for fast, simple motion. The choice affects controllability and safety:

• High gear ratios can increase torque but reduce backdrivability and sensitivity.
• Torque control enables compliant interaction but requires better sensing and more careful control design.

ROS introduction: organizing real robot software

The Robot Operating System (ROS) is a standard ecosystem for building robotic applications. It is not an operating system in the traditional sense; it is a middleware and tooling environment that helps structure complex systems.

Core concepts

• Nodes: separate processes that perform tasks (control, perception, planning).
• Topics: publish/subscribe message streams for data like sensor readings or commands.
• Services and actions: request/response or long-running tasks like “plan and execute.”
• TF (transforms): managing coordinate frames over time, essential for relating sensor data to the robot and world.

ROS encourages modular design. A camera driver can publish images, a perception node can publish detected objects, a planner can publish trajectories, and a controller can execute them, all while tools visualize the state and log data for debugging.

Putting it together: a typical robotics pipeline

A real system ties these fundamentals into a loop:

1. Define robot geometry and frames.
2. Use forward kinematics to compute current tool pose.
3. Plan a trajectory that respects limits and avoids collisions.
4. Use inverse kinematics to convert a tool goal into joint targets when needed.
5. Control joints with PID (often with feedforward compensation).
6. Fuse sensors to estimate state and correct drift or noise.
7. Coordinate modules through ROS so each part can be tested and improved independently.

Robotics fundamentals are not abstract prerequisites; they are the daily tools used to make robots move predictably, safely, and repeatably. Mastering geometry, kinematics, planning, control, and sensing creates a foundation that scales from a two-wheel robot to