2. Robot Components
Before developing the mathematical theory of kinematics, it is useful to survey the physical building blocks of a manipulator. Every robot integrates four classes of components:
- Mechanical units — rigid links connected via joints; supporting structure (mobility), wrist (dexterity), end-effector (task execution).
- Actuation units — motors (electrical, hydraulic, pneumatic) and transmissions; motion control algorithms.
- Sensor units — proprioceptive (joint position and velocity) and exteroceptive (force, proximity, vision).
- Supervision units — task planning and control; artificial intelligence and reasoning.
1 Mechanical Structure
1.1 Links and joints
The individual bodies making up a mechanism are called links. Links are treated as rigid bodies to simplify modelling. At high speeds or heavy loads, material elasticity may become significant; in that case flexibility must be accounted for. Chains, cables, and belts may also be considered links.
The connection between two links is a joint, which constrains the relative motion between the connected members. A joint with \(f_i\) degrees of freedom imposes \(c_i = m - f_i\) constraints, where \(m\) is the DOF of a free rigid body (\(m = 3\) in 2D, \(m = 6\) in 3D).
Joint types and their DOF:
| Joint | Symbol | DOF \(f_i\) | Motion allowed |
|---|---|---|---|
| Revolute (hinge) | R | 1 | Rotation about a fixed axis |
| Prismatic (sliding) | P | 1 | Translation along a fixed axis |
| Helical (screw) | H | 1 | Coupled rotation + translation |
| Cylindrical | C | 2 | Independent rotation + translation on one axis |
| Universal | U | 2 | Two orthogonal revolute axes |
| Spherical (ball-and-socket) | S | 3 | Rotation about any axis through the centre |
1.2 Kinematic chains
A kinematic chain is a sequence of links connected by joints. Two topologies:
- Open chain (serial): a single sequence of links from base to end-effector. Each link connects to at most two neighbours.
- Closed chain (parallel): a loop of links; the end-effector is connected to the base by more than one path.
A kinematic chain becomes a mechanism when one link is fixed to the ground. A mechanism becomes a machine when an energy source (motor, hydraulic circuit) is added so that it can perform useful work.
1.3 Configuration space and degrees of freedom
The configuration of a robot is a complete specification of the position of every point of the robot. For a rigid-link robot, a finite number of coordinates suffices. The minimum such number is the degrees of freedom (DOF):
The DOF is the dimension of the configuration space (C-space) — the set of all possible configurations of the robot.
Examples: a door has \(\mathrm{DOF} = 1\) (the hinge angle \(\theta\)); a rigid body on a plane has \(\mathrm{DOF} = 3\) \((x, y, \theta)\).
1.3.1 Grübler’s formula
For a mechanism with \(N\) links (including the base), \(J\) joints, and joint freedoms \(f_i\):
\[ \boxed{\;\mathrm{dof} = m(N - 1 - J) + \sum_{i=1}^{J} f_i\;} \]
with \(m = 3\) (planar) or \(m = 6\) (spatial). The formula can yield zero (structure), negative (over-constrained), or positive (mechanism) values.
| Mechanism | \(m\) | \(N\) | \(J\) | \(\sum f_i\) | DOF |
|---|---|---|---|---|---|
| Four-bar linkage | 3 | 4 | 4 | 4 | 1 |
| Five-link chain | 3 | 5 | 5 | 5 | 2 |
| Gough–Stewart | 6 | 8 | 9 | 15 | 6 |
| Delta robot | 6 | 17 | 21 | 45 | 3 |
| Jansen linkage | 3 | 12 | 16 | 16 | 1 |
1.4 Task space and workspace
Task space is the space in which the robot’s task is naturally expressed — \(\mathbb{R}^2\) for pen-plotting, or the 6-DOF rigid-body pose space for general manipulation. The choice of task space is driven by the task, not by the robot.
Workspace is the subset of the environment the end-effector can reach. Its shape and volume depend on the manipulator geometry and on joint limits. A point in the workspace may be reachable by more than one robot configuration — the workspace does not fully specify the configuration.
2 Manipulator Geometries
Industrial manipulators are classified by the geometry of their first three joints, which determines the workspace shape.
2.1 Cartesian (Gantry) — PPP
Three mutually orthogonal prismatic joints. Each DOF corresponds directly to a Cartesian coordinate. High mechanical stiffness; uniform positioning accuracy throughout the workspace (a rectangular parallelepiped). Low dexterity since all joints are prismatic. Objects are approached from the side (or from above in a gantry arrangement). Applications: material handling, heavy-load assembly, machine tools.
2.2 Cylindrical — RPP
One revolute followed by two prismatic joints. High stiffness from the prismatic joints; accuracy decreases with increasing horizontal stroke. Workspace: portion of a hollow cylinder. The horizontal prismatic joint gives good access to cavities. Applications: large-object handling (hydraulic motors preferred).
2.3 Spherical — RRP
Two revolute joints followed by one prismatic. Lower stiffness than the above two. Workspace: portion of a hollow sphere (can include the floor). Applications: machining with electric motors.
2.4 SCARA — RRP (parallel axes)
Selective Compliant Arm for Robotic Assembly. Two revolute joints plus one vertical prismatic, all axes parallel (\(z\)-axes). High stiffness to vertical loads; compliant horizontally (hence selective compliance). 4 DOF total. Accuracy decreases with distance from the first joint axis. Applications: micro-assembly, pick-and-place of small objects.
2.5 Articulated (Anthropomorphic) — RRR
Three revolute joints with the first axis orthogonal to the other two (which are parallel). The second joint is the shoulder, the third the elbow. Most dexterous geometry; all joints revolute. Workspace: approximately a large spherical volume. Positioning accuracy varies across the workspace. The dominant industrial layout. Applications: welding, painting, assembly, machining.
2.6 Delta (Parallel)
A parallel robot with three kinematic chains connecting the base to a moving platform. High speed and acceleration (only the platform and end-effector move — motors stay on the base). High rigidity; errors from each chain do not accumulate. Smaller, less dexterous workspace than serial robots. 3 translational DOF (no platform rotation). Applications: high-speed pick-and-place (food, pharma, electronics).
2.7 Jansen mechanism
A single-DOF planar linkage (\(N=12\), \(J=16\), DOF\(=1\)) that converts crank rotation into a realistic walking gait. Used as a leg module in legged robots.
3 Actuation Units
3.1 Servomotor characteristics
A servomotor is a motor with a closed-loop position or velocity controller. Key characteristics for robotic use: low rotor inertia, high power-to-weight ratio, wide velocity range (up to ~2000 rpm), high position accuracy (~1/1000 turn), and power from 10 W to 10 kW.
3.2 Pneumatic actuators
Pneumatic actuators convert pressurised air into mechanical work via pistons. Simple and low-cost, but difficult to control precisely due to air compressibility. Main uses: binary gripper actuation, McKibben artificial muscles, soft-robotic applications.
3.3 Hydraulic actuators
Hydraulic actuators pump an incompressible fluid into a piston or rotary motor.
- Pros: self-lubricated, spark-free, very large torques, high power density, no overheating under static load.
- Cons: require reservoir and pump, large dimensions, high cost, oil-leak risk, lower efficiency than electric.
Force multiplication across pistons of areas \(A_1\), \(A_2\):
\[ F_2 = \frac{A_2}{A_1}\,F_1. \]
Piston dynamics (area \(A\), spring \(k_s\), damping \(b\), pressure \(P_m\), load \(F_L\)):
\[ P_m A = m_p\ddot{d} + b\dot{d} + F_L + k_s d. \]
3.4 Electrical actuators
Electric motors convert electrical energy into mechanical work. Pros: widely available power supply, low cost, high efficiency, no leaks, easy maintenance. Cons: static overheating; dangerous in flammable environments.
3.4.1 DC motor model
A brushed DC motor has a rotor (armature), stator (permanent magnets), and a commutator ring. The coupled electrical–mechanical equations are:
\[ V_a = L_a\frac{di}{dt} + R_a\,i + K_v\,\omega, \qquad K_T\,i = I_m\frac{d\omega}{dt} + b\,\omega + \tau_L, \]
where \(K_v\) is the back-EMF constant and \(K_T\) is the torque constant. Power balance gives:
\[ \boxed{\;K_v = K_T\;} \quad \text{(in SI units).} \]
3.4.2 Motor characteristic curves
The steady-state speed–torque curve is linear:
\[ \omega = \omega_0\!\left(1 - \frac{\tau}{\tau_{\text{stall}}}\right). \]
Peak mechanical power \(P_{\max} = \frac{1}{4}\omega_0\tau_{\text{stall}}\) occurs at the midpoint. Current is proportional to torque. Best practice: operate near peak efficiency, which occurs just below the power peak.
3.4.3 Brushless motors
Brushless motors move the magnets to the rotor and windings to the stator, eliminating the commutator. Advantages: reduced losses, less maintenance, better heat dissipation, more compact rotor, lower inertia — at higher cost due to electronic commutation.
4 Motion Transmissions
Transmissions serve three roles: (1) quantitative transformation (speed reduction / torque multiplication), (2) qualitative transformation (rotation to linear motion), and (3) structural benefit (locating motors near the base to reduce moving mass).
4.1 Gear ratio
\[ n = \frac{\dot\theta_1}{\dot\theta_2} = \frac{u_2}{u_1}, \qquad \tau_2 = n\,\eta_g\,\tau_1, \]
where \(u_i\) are pitch radii and \(\eta_g\) is gearbox efficiency.
4.2 Transmission types
| Type | Purpose | Key limitation |
|---|---|---|
| Spur / worm gears | Change direction or axis of motion | Deformation, backlash |
| Lead screw, rack and pinion | Rotation → translation | Friction, elasticity, backlash |
| Toothed belts and chains | Displace motor from joint axis | Belt compliance; chain vibration |
| Harmonic drive | In-line, compact, high ratio (up to 320:1) | Elasticity |
| Transmission shafts | Long in-link shafts | Alignment; flexible couplings needed |
Rack and pinion converts rotation to linear motion. Epicyclic (planetary) gears offer three gear ratios by fixing one of three components (ring, sun, or planet carrier). Sun-and-planet mechanisms convert reciprocating to rotary motion.
4.3 Backlash
Backlash is the angular gap between mating gear teeth, needed to accommodate manufacturing tolerances, thermal expansion, and lubrication. Excessive backlash causes positioning error and vibration under reversing loads; insufficient backlash causes overloading, heat, and accelerated wear. Backlash is eliminated in harmonic drives.
4.4 Harmonic drive
A harmonic drive achieves up to 320:1 reduction in the same volume a planetary gear yields ~10:1, through elastic deformation of a thin flexspline. Key features: no backlash, compact, lightweight, high torque, same-axis input/output. Widely used in robot joints, satellite mechanisms, and surgical robots.
4.5 Optimal gear ratio
Motor inertia \(J_M\), load inertia \(J_L\), gear ratio \(n\). The motor torque required for load acceleration \(a\) is:
\[ \tau_M = \left(J_M n + \frac{J_L}{n}\right) a. \]
Minimising over \(n\):
\[ \boxed{\;n_{\text{opt}} = \sqrt{J_L / J_M}\;} \]
This inertia-matching condition balances reflected load inertia against motor inertia. Increasing \(n\) beyond \(n_{\text{opt}}\) helps load inertia but penalises the motor; there is a clear trade-off.
5 Sensing Units
Proprioceptive sensors measure the robot’s internal state: joint position, velocity, torque, link acceleration. Exteroceptive sensors measure interaction with the environment: external force/torque, proximity, vision, and other physical quantities (gas, sound, humidity, temperature).
Key measurement-system properties: resolution (smallest detectable change), accuracy (closeness to true value), repeatability, linearity, bandwidth, and noise floor.
5.1 Position sensors
Linear displacement: potentiometers, LVDTs, inductosyns, Hall sensors. Angular displacement: potentiometers, resolvers, Hall sensors, digital encoders.
5.1.1 Digital encoder
A code disk rotates with the joint, alternating opaque and transparent sectors on concentric tracks. Infrared LEDs shine through the disk; photodetectors convert light pulses to electrical pulses. Resolution: \(360°/2^N\) per revolution for an \(N\)-track disk (e.g. \(N=12\) gives \({\approx}0.088°\)). Incremental encoders count from a reference; absolute encoders give a unique code at every position.
5.2 Accelerometers
MEMS accelerometers integrate a tiny proof mass on silicon. Two sensing principles:
- Capacitive: mass displacement changes the gap between mass and fixed plates, altering capacitance. Measured and converted to a voltage proportional to acceleration.
- Piezoresistive: mass displacement strains embedded piezoresistors, changing their resistance. Resistance change is converted to a voltage.
Compact, low-cost, low-power — enabling IMU integration with gyroscopes.
5.3 Force and torque sensors
5.3.1 Strain gauge and Wheatstone bridge
A strain gauge changes resistance with strain:
\[ \frac{\Delta R}{R} = \mathrm{GF}\cdot\varepsilon, \qquad \frac{V_o}{V_i} \approx \frac{\mathrm{GF}\cdot\varepsilon}{4}. \]
where \(\mathrm{GF}\) is the gauge factor and the voltage ratio is from a balanced Wheatstone bridge linearised for small strains.
5.3.2 Load cell
Strain gauges bonded to a metal elastic element (beam or column). Applied force causes a known elastic deformation; strain is measured and converted to force using the element’s known geometry and material modulus.
5.3.3 Torque cell
Gauges mounted at ±45° on a low-torsional-stiffness element (hollow shaft or flange). Torsional deformation generates shear strains detected by the gauges, connected to a Wheatstone bridge via a slip ring.
5.3.4 Six-axis force/torque sensor
Measures all six wrench components (\(F_x, F_y, F_z, M_x, M_y, M_z\)) simultaneously. The Maltese cross design connects a central hub to an outer ring via four thin spokes; gauges on each spoke respond to different load combinations. The full wrench is recovered via a calibration matrix:
\[ \boldsymbol{\gamma} = \mathbf{C}\,\mathbf{v}, \]
where \(\mathbf{v}\) is the vector of bridge voltages and \(\mathbf{C}\) is determined during factory calibration. Used at robot wrists for force-controlled assembly and safe human-robot interaction.
6 Control Hierarchy
Robot control spans multiple abstraction levels:
| Level | Function | Methods |
|---|---|---|
| High-level | Decision making, behaviour | Behaviour trees, AI |
| Localisation & mapping (mobile) | SLAM | |
| Task-space trajectory generation | Model Predictive Control | |
| Mid-level | Coordinate transformation | Inverse kinematics |
| Low-level | Joint servo control | PID, force/impedance control |
The low-level inner loop runs at ~1 kHz, controlling each servomotor from joint encoder feedback. The mid-level converts task-space references to joint-space commands via inverse kinematics. The high-level plans tasks, handles decisions, and — for mobile robots — builds and queries a map.