Unit 2: Mechanical Elements of Robotics

I. Orientation — Mechanical Design in Robotics

A robot’s mechanical system converts actuator output into controlled motion, supports operational loads, and enables interaction with the environment. Its design therefore links kinematics, force transmission, stiffness, accuracy, payload, speed, durability, and safety.

• Defining properties:
  • Degrees of freedom (DOF): The number of independent coordinates required to describe robot configuration; a typical articulated industrial robot has six DOF.
  • Workspace: The set of positions and orientations reachable by the end-effector, determined mainly by link dimensions and joint limits.
  • Payload: The maximum permitted mass or force at the end-effector under specified speed, reach, and acceleration conditions.
  • Stiffness: Resistance to elastic deformation, commonly expressed in N/m for translation or N·m/rad for rotation.
  • Accuracy and repeatability: Accuracy is closeness to a commanded pose; repeatability is the ability to return to the same pose.
  • Mechanical efficiency: The ratio of useful output power to input power:

η = P_out / P_in

Here, η is efficiency, P_out is output power, and P_in is input power.

• Design assumptions:
  • Rigid-body approximation: Links are often treated as rigid during kinematic analysis, although real links bend under load.
  • Ideal-joint approximation: Initial models may neglect backlash, friction, clearance, wear, and compliance.
  • Load consideration: Static weight, inertial force, shock, vibration, and actuator torque must all be considered.

II. Robot Structure — Motion-Producing Members

Robot structure is formed by links connected through joints; their arrangement determines mobility, workspace, and load paths.

A. Robot links and joints

Robot links are structural members, while joints permit constrained relative motion between adjacent links.

• Links:
  • Function: A link transmits forces and maintains a fixed geometric relationship between joints.
  • Types: The base is fixed; intermediate links form the arm; the terminal link carries the wrist or end-effector.
  • Design factors: A long, lightweight link improves reach but may reduce stiffness and increase vibration.
  • Load response: Axial deformation follows:

δ = FL / AE

Here, δ is deformation, F is axial force, L is link length, A is cross-sectional area, and E is Young’s modulus.

• Joints:
  • Revolute joint (R): Produces rotation about one axis; its coordinate is angle θ, measured in radians.
  • Prismatic joint (P): Produces linear translation along one axis; its coordinate is displacement d, measured in metres.
  • Other forms: Cylindrical, spherical, helical, and universal joints combine elementary rotational or translational motions.
  • Practical constraints: Bearings, hard stops, cables, and collision limits restrict the theoretical range of motion.

B. Kinematic chains

A kinematic chain is an ordered assembly of links and joints that transmits motion from the base to an output link.

• Serial chain:

  • Arrangement: Links form one path from base to end-effector, as in a six-axis articulated arm.
  • Advantages: Large workspace and simple mechanical construction.
  • Limitations: Position errors accumulate along the chain, and distal actuators increase moving mass.

• Parallel chain:

  • Arrangement: Two or more independent link paths connect the base to a moving platform, as in a Delta robot.
  • Advantages: High stiffness, speed, and load capacity because loads are shared.
  • Limitations: Smaller workspace and more complex kinematic solutions.

• Mobility: For a planar mechanism, the Kutzbach criterion is:

M = 3(n − 1) − 2j₁ − j₂

Here, M is mobility, n is the number of links including the frame, j₁ is the number of one-DOF lower pairs, and j₂ is the number of two-DOF higher pairs.

• Configuration: Joint variables form the vector q = [q₁, q₂, …, qₙ]ᵀ; forward kinematics maps q to end-effector pose.

III. Motion Transmission — Conversion of Actuator Output

Transmission mechanisms adapt motor speed and torque to the values required at robot joints or linear axes.

A. Mechanical transmission systems

A mechanical transmission carries power between an actuator and a load while modifying speed, torque, direction, or motion type.

• Power relationship:

P = Tω = Fv

Here, P is power in watts, T is torque in N·m, ω is angular velocity in rad/s, F is force in newtons, and v is linear velocity in m/s.

• Selection criteria:

  • Ratio: High reduction increases output torque but reduces output speed.
  • Backlash: Clearance between mating parts causes lost motion during reversal.
  • Compliance: Elastic deformation stores energy and may produce oscillation.
  • Efficiency: Friction converts input power into heat and raises actuator demand.
  • Positioning quality: Precision robots favor low backlash, high torsional stiffness, and low inertia.

• Common arrangements: Direct drives eliminate transmissions, whereas geared or screw-driven systems provide torque or force multiplication.

B. Gear trains

Gear trains transmit rotary motion through meshing teeth and provide an approximately constant velocity ratio.

• Simple gear pair:

ω_out / ω_in = N_in / N_out
T_out ≈ ηT_in(N_out / N_in)

Here, ω denotes angular speed, N denotes tooth count, T denotes torque, and η is transmission efficiency.

• Direction: Two external gears rotate oppositely; an idler changes direction without changing the overall ratio.
• Forms:
  • Spur gears: Efficient and economical for parallel shafts but noisy at high speed.
  • Helical gears: Smoother and quieter, although tooth inclination creates axial thrust.
  • Bevel gears: Transfer rotation between intersecting shafts, commonly at 90°.
  • Planetary gears: Provide high reduction and torque density in a compact, coaxial package.
  • Harmonic drives: Offer high ratios and very low backlash but have elastic compliance and limited shock tolerance.
• Example: A 20-tooth motor gear driving an 80-tooth gear gives a 4:1 reduction; ideally, output speed is one-quarter and torque is four times the input.

C. Belt and chain drives

Belt and chain drives transmit rotation between separated shafts and can reduce the need for multiple gear stages.

1. Belt drives:
   • Construction: A flexible belt runs over pulleys; toothed timing belts provide positive engagement.
   • Ratio:

ω₂ / ω₁ = D₁ / D₂

Here, ω₁ and ω₂ are pulley speeds, while D₁ and D₂ are pitch diameters.

• Advantages: Quiet operation, low mass, shock absorption, and inexpensive maintenance.
• Limitations: Flat and V-belts may slip; all belts stretch and require correct tension.

2. Chain drives:
   • Construction: A roller chain engages toothed sprockets, preventing steady-state slip.
   • Advantages: Higher load capacity and better resistance to heat or contamination than many belts.
   • Limitations: Lubrication, noise, wear, and polygonal speed variation reduce suitability for highly precise axes.

D. Lead screw and ball screw mechanisms

Lead screws and ball screws convert rotary motion into controlled linear displacement.

• Linear advance:

x = nL

Here, x is linear travel, n is the number of screw revolutions, and L is lead—the axial travel per revolution.

1. Lead screw:

   • Principle: Sliding contact occurs between screw and nut threads, often using trapezoidal or Acme profiles.
   • Characteristics: Low cost, good damping, and possible self-locking, but relatively high friction and wear.
   • Use: Suitable for moderate-speed positioning axes and gripper mechanisms.

2. Ball screw:

   • Principle: Recirculating balls roll between screw and nut, greatly reducing friction.
   • Characteristics: High efficiency, accuracy, speed, and life; preload can remove axial backlash.
   • Limitation: Greater cost and sensitivity to dirt; high efficiency usually prevents self-locking.

• Driving torque:

T = FL / 2πη

Here, T is input torque, F is axial load, L is screw lead, and η is efficiency.

IV. Support and Connection Elements — Controlled Motion Transfer

Bearings guide relative motion, while couplings connect shafts and accommodate practical alignment conditions.

A. Bearings

Bearings support radial or axial loads while reducing friction between moving components.

• Rolling-element bearings:

  • Ball bearings: Operate at high speed and support moderate radial and axial loads.
  • Roller bearings: Provide greater radial capacity because rollers have line contact.
  • Thrust bearings: Primarily support forces parallel to the shaft axis.
  • Cross-roller bearings: Resist radial, axial, and moment loads, making them useful in robot joints.

• Plain bearings: Use sliding contact and offer compactness, damping, and resistance to shock, but generally have greater friction.

• Life rating:

L₁₀ = (C / P)^p

Here, L₁₀ is rated life in millions of revolutions, C is dynamic load rating, P is equivalent load, and p is 3 for ball bearings or 10/3 for roller bearings.

• Installation factors: Lubrication, sealing, preload, fit, alignment, and operating temperature strongly affect life and precision.

B. Couplings

A coupling joins two shafts to transmit torque while managing alignment, vibration, or assembly requirements.

• Rigid couplings: Provide high torsional stiffness but demand accurate axial and angular alignment.
• Flexible couplings:
  • Elastomeric type: Absorbs shock and vibration but introduces torsional compliance.
  • Bellows type: Offers low backlash and good angular flexibility for servo systems.
  • Oldham type: Accommodates parallel offset through a sliding central disc.
  • Beam type: Uses helical cuts to permit small misalignments in light-duty drives.
• Selection factors: Rated torque must exceed peak accelerating or emergency-stop torque; excessive misalignment raises bearing load.
• Safety function: Torque limiters or slip couplings disconnect or slip when torque exceeds a preset value, protecting people and equipment.

V. Robot–Environment Interaction — Tooling and Grasping

The robot performs useful work through an end-effector selected for the geometry, material, load, and process.

A. Robot end-effectors

A robot end-effector is the device attached to the terminal wrist for manipulating objects or performing a process.

• Categories:
  • Gripping devices: Hold, move, orient, or release workpieces.
  • Process tools: Include welding guns, spray nozzles, drills, screwdrivers, and polishing spindles.
  • Sensing tools: Force–torque sensors, cameras, and probes support inspection or adaptive control.
• Interface requirements: Mechanical mounting, electrical power, pneumatic lines, data connections, and tool-center-point calibration must be provided.
• Performance factors: End-effector mass reduces available payload; its offset from the wrist increases moment:

M = Fr

Here, M is bending moment, F is applied force, and r is perpendicular distance from the wrist axis.

• Tool changers: Automatic couplers allow one robot to exchange tools while preserving repeatable mechanical and utility connections.

B. Grippers (mechanical, vacuum, magnetic)

Grippers create and maintain controllable contact between the robot and a workpiece.

1. Mechanical grippers:
   • Principle: Two or more fingers apply frictional or form-closure forces using pneumatic, electric, or hydraulic actuation.
   • Force condition: For a two-finger friction grip:

2μN ≥ Smg

Here, μ is friction coefficient, N is normal force per finger, S is safety factor, m is object mass, and g is gravitational acceleration.

• Strengths and limits: They handle varied shapes but require force control to avoid crushing delicate objects.

2. Vacuum grippers:
   • Principle: A pressure difference across a suction cup creates holding force:

F = ΔPA

Here, F is ideal holding force, ΔP is pressure difference, and A is effective sealed area.

• Use: Effective for glass, sheet metal, cartons, and smooth plastic.
• Limitations: Leakage, porous surfaces, acceleration, and seal wear reduce actual force.

3. Magnetic grippers:
   • Principle: Permanent magnets or electromagnets attract ferromagnetic materials such as steel.
   • Strengths: Rapid gripping, no external finger clearance, and operation on perforated surfaces.
   • Limitations: Restricted material range, residual magnetism, uncertain force on thin sheets, and possible power-loss hazards for electromagnets.

VI. Robot Construction — Material Selection

Material choice balances strength, stiffness, mass, manufacturability, cost, corrosion resistance, and vibration behavior.

A. Structural materials used in robots

Structural materials determine robot weight, deflection, natural frequency, durability, and attainable acceleration.

• Steel: Density is approximately 7,850 kg/m³ and Young’s modulus about 200 GPa; it offers high strength, stiffness, wear resistance, and low cost but adds substantial moving mass.
• Aluminium alloys: Density is approximately 2,700 kg/m³ and modulus about 69 GPa; they are lightweight, machinable, and corrosion-resistant, making them common for links and frames.
• Titanium alloys: Combine high specific strength and corrosion resistance with a density near 4,500 kg/m³, but material and machining costs are high.
• Carbon-fibre-reinforced polymer: Provides very high directional stiffness-to-mass ratio for long, fast links; drawbacks include anisotropy, difficult joining, and impact sensitivity.
• Engineering polymers: Nylon, acetal, and PEEK suit covers, guides, gears, and low-load components because of low mass and friction, although creep and temperature sensitivity limit structural use.
• Cast iron: Its mass and internal damping benefit stationary bases and precision machine frames by suppressing vibration.
• Selection principle: For moving links, high specific stiffness is desirable:

Specific stiffness = E / ρ

Here, E is Young’s modulus and ρ is density; a larger value supports lighter structures with improved resistance to elastic deflection.