Unit 4: Robot Mechanical Design Parameters

I. Orientation: Mechanical Description of Robot Motion

Robot mechanical design is governed by rigid-body motion, geometric constraints, and the relationship between forces, torques, and motion. Robot kinematics describes motion without considering forces; robot dynamics explains motion produced by forces and torques. A robot manipulator is modeled as a chain of rigid links connected by revolute or prismatic joints.

• Rigid-body assumption: Each link maintains constant shape and dimensions; deformation is neglected in basic analysis.
• Reference convention: A base frame is fixed to the robot, while link frames move with individual joints.
• Configuration: The vector (q) contains joint variables, such as revolute angles in radians or prismatic displacements in metres.
• Pose representation: A robot end-effector pose combines position and orientation, usually represented by a (4 \times 4) homogeneous transformation matrix.
• Degrees of freedom: The number of independent joint variables determines the robot's configuration freedom.
• Units and signs: Angles use radians in equations, distances use metres, forces use newtons, and torques use newton-metres.
• Model limitation: Ideal equations omit backlash, friction, flexibility, actuator saturation, and sensor errors unless these effects are added separately.

II. Robot Kinematics: Geometry of Motion

Robot kinematics establishes the mathematical relationship between joint variables and the position, orientation, velocity, and acceleration of links and end-effectors.

A. Introduction to robot kinematics

Introduction to robot kinematics provides the geometric foundation for describing a manipulator at a particular configuration.

• Core question: Kinematics determines where a link or tool is located for a given (q), without calculating the forces causing the motion.
• Joint variables: A revolute joint contributes an angle (\theta_i); a prismatic joint contributes a displacement (d_i).
• Configuration space: A robot with six independent joints has (q=[q_1,q_2,\ldots,q_6]^T), so its configuration space is six-dimensional.
• Main outputs: Position, orientation, velocity, and acceleration are obtained from geometric transformations and their time derivatives.
• Design relevance: Kinematic analysis reveals workspace, reachability, singularities, and possible collisions before mechanical construction.

B. Coordinate frames and transformations

Coordinate frames and transformations convert coordinates between the base, link, joint, and tool reference systems.

• Frame definition: A frame ({i}) consists of an origin and three mutually perpendicular unit axes, commonly (x_i,y_i,z_i).
• Rotation matrix: (R\in SO(3)) describes orientation and satisfies (R^TR=I) and (\det(R)=1).
• Homogeneous transformation: The pose of frame ({B}) relative to frame ({A}) is

T_A^B = [ R_A^B  p_A^B
             0      1  ]

where (p_A^B) is the position of frame B's origin expressed in frame A.

• Composition: Consecutive link transformations multiply as (T_0^n=T_0^1T1^2\cdots T{n-1}^n); multiplication order matters.
• Denavit-Hartenberg convention: Each link is described using (a_i) (link length), (\alpha_i) (link twist), (d_i) (link offset), and (\theta_i) (joint angle).
• Concrete implication: A point (p_B=[x,y,z,1]^T) becomes (p_A=T_A^Bp_B), allowing tool coordinates to be expressed in the base frame.

C. Degrees of Freedom (DOF)

Degrees of Freedom (DOF) measure the number of independent variables required to specify a robot's configuration.

• Joint counting: Each independent revolute or prismatic joint normally contributes one DOF; six serial joints can control a general spatial pose.
• Task-space requirement: A rigid body in three-dimensional space has six pose coordinates: (x,y,z) and three orientation coordinates.
• Underactuated case: With fewer than six independent DOF, a manipulator cannot generally achieve every arbitrary spatial pose.
• Redundant case: With more than six DOF, multiple joint configurations may produce the same end-effector pose.
• Constraint effect: Mechanical stops, coupled joints, or locked axes reduce effective DOF even when more physical joints exist.
• Design example: A planar two-link arm has two DOF and can independently control planar coordinates (x) and (y), but not arbitrary three-dimensional orientation.

D. Position and orientation of robots

Position and orientation of robots together define the end-effector pose required for a task such as drilling, welding, or gripping.

• Position vector: (p=[x,y,z]^T) specifies the tool origin relative to a chosen frame, in metres.
• Orientation alternatives: Orientation may use roll-pitch-yaw angles, Euler angles, an axis-angle pair, or a unit quaternion.
• Pose matrix: The complete pose is (T=[R\ p;0\ 1]), which avoids separating translation and rotation during frame composition.
• Orientation caution: Euler-angle descriptions can suffer gimbal lock; quaternions avoid this singularity but require normalization.
• Workspace: The reachable workspace contains positions attainable within joint limits; dexterous workspace additionally permits the required orientations.
• Mechanical parameter effect: Link lengths, joint offsets, and joint limits directly change reach, clearance, and tool accessibility.

III. Kinematic Solution Methods: From Joints to Tasks

Forward and inverse kinematics solve opposite mappings between joint space and task space.

A. Forward kinematics: Concept and applications

Forward kinematics: Concept and applications concerns calculating the end-effector pose from known joint variables.

• Formal mapping: The forward model is (x=f(q)), where (q) is the joint vector and (x) is the pose or task coordinate vector.
• Calculation process: Multiply the individual link transformations from the base to the tool: (T_0^n(q)=\prodiT{i-1}^i(q_i)).
• Known inputs: Joint encoder readings provide (q); link geometry supplies the constant dimensions and offsets.
• Application: Controller software uses forward kinematics to display tool position, check workspace limits, and detect predicted collisions.
• Worked example: For a planar two-link arm,

x = l1 cos(q1) + l2 cos(q1 + q2)
y = l1 sin(q1) + l2 sin(q1 + q2)

where (l_1,l_2) are link lengths and (q_1,q_2) are joint angles.

• Limitation: Forward kinematics is usually direct and may be evaluated for every control cycle, but it does not determine which joint values achieve a desired pose.

B. Inverse kinematics: Concept and applications

Inverse kinematics: Concept and applications calculates joint variables needed to achieve a specified end-effector pose.

• Formal mapping: The inverse problem solves (q=f^{-1}(x_d)), where (x_d) is the desired position and orientation.
• Multiple solutions: An elbow manipulator may have elbow-up and elbow-down configurations for the same point.
• No-solution condition: A target outside the workspace or violating joint limits is unreachable.
• Analytical solution: Geometric or algebraic equations can give fast exact solutions for specially structured robots.
• Numerical solution: Iterative methods update (q) using pose error and the Jacobian; they handle complex mechanisms but depend on initial guesses.
• Application: During palletizing, inverse kinematics converts each target pick-and-place pose into actuator commands.
• Practical constraints: Collision avoidance, joint limits, singularities, and preferred posture must be included when selecting among solutions.

C. Velocity kinematics and Jacobian (Introduction)

Velocity kinematics and Jacobian (Introduction) relate joint rates to linear and angular velocity of the end-effector.

• Velocity relation: The Jacobian (J(q)) maps joint velocity to spatial velocity:

v = J(q) q_dot

where (v=[v_x,v_y,v_z,\omega_x,\omega_y,\omega_z]^T), (q_dot) is the joint-rate vector, and (J) is the Jacobian matrix.

• Column meaning: For a revolute joint, the linear-velocity column is (z_{i-1}\times(pn-p{i-1})), and the angular column is (z{i-1}); for a prismatic joint, they are (z{i-1}) and (0).
• Singularity: If (J) loses rank, some Cartesian velocities cannot be produced; near-singular motion can require very large joint rates.
• Inverse velocity: When (J) is square and nonsingular, (q_dot=J^{-1}v); otherwise a pseudoinverse is commonly used.
• Application: Jacobians support resolved-rate control, manipulability analysis, and conversion between tool forces and joint torques.

IV. Motion Planning and Robot Dynamics

Motion planning specifies how a robot moves, while dynamics predicts the forces and torques required for that motion.

A. Robot trajectory and path planning (Basics)

Robot trajectory and path planning (Basics) determines a feasible route and its timing between start and goal configurations.

• Path versus trajectory: A path is geometric, such as a sequence of points; a trajectory adds time, giving (q(t)), (\dot q(t)), and (\ddot q(t)).
• Planning spaces: Planning may occur in joint space, where limits are easy to enforce, or Cartesian space, where tool motion is easier to specify.
• Interpolation: Cubic polynomials can connect boundary positions and velocities smoothly; trapezoidal velocity profiles limit acceleration.
• Collision checking: Obstacles, link geometry, and self-collision must be tested along intermediate samples, not only at endpoints.
• Feasibility: A trajectory must respect joint limits, maximum velocity, acceleration, jerk, and actuator torque.
• Application: Welding commonly uses a Cartesian path along a seam, while assembly may use joint-space motion to avoid obstacles efficiently.

B. Introduction to robot dynamics

Introduction to robot dynamics describes how mass, inertia, gravity, and external loads determine robot motion.

• Dynamic equation: A common manipulator model is

M(q) q_ddot + C(q, q_dot) q_dot + g(q) + F(q_dot) = tau + J(q)^T F_ext

where (M) is inertia, (C) represents Coriolis and centrifugal effects, (g) is gravity torque, (F) is friction, (\tau) is joint torque, and (F_{ext}) is an external tool wrench.

• Inertia matrix: (M(q)) is configuration-dependent, symmetric, and positive definite for a physically valid robot.
• Inverse dynamics: Given (q,\dot q,\ddot q), calculate required actuator torque (\tau).
• Forward dynamics: Given torque (\tau), calculate acceleration (\ddot q), then integrate to obtain velocity and position.
• Design use: Dynamics determines motor sizing, gearbox selection, structural strength, energy use, and controller feedforward compensation.

C. Newton–Euler formulation (Concept)

Newton–Euler formulation (Concept) derives manipulator dynamics by applying force and moment balance to each rigid link.

• Translational balance: For link mass (m), the resultant force is (F=ma), where (a) is the centre-of-mass acceleration.
• Rotational balance: The resultant moment is (N=I\alpha+\omega\times(I\omega)), where (I) is inertia, (\alpha) angular acceleration, and (\omega) angular velocity.
• Recursive structure: A forward recursion calculates link velocities and accelerations from the base outward; a backward recursion calculates forces and joint torques toward the base.
• Revolute joint torque: The actuator torque is obtained by projecting the net link moment onto the joint axis.
• Computational value: The recursive algorithm is efficient for serial chains and is widely used in real-time inverse-dynamics control.
• Included effects: Gravity can be introduced as base acceleration, while payload forces enter at the tool frame.

D. Lagrangian formulation (Concept)

Lagrangian formulation (Concept) derives equations from energy rather than writing force and moment balances link by link.

• Lagrangian definition: (L=T-V), where (T) is total kinetic energy and (V) is gravitational potential energy.
• Governing equation:

d/dt (∂L/∂q_dot_i) - ∂L/∂q_i = tau_i

where (q_i) is joint coordinate and (\tau_i) is the corresponding generalized actuator force or torque.

• Kinetic energy: For each link, translational energy is (\frac12mv^2) and rotational energy is (\frac12\omega^TI\omega).
• Potential energy: Near Earth, a link at height (h) has (V=mgh), with (g\approx9.81\ \text{m/s}^2).
• Strength: The method produces compact system equations and clearly exposes energy relationships.
• Limitation: Symbolic expressions become lengthy for many links, so recursive Newton-Euler methods are often preferred for numerical implementation.

V. Manipulator Loads, Balance, and Industrial Use

Forces, torques, and balance connect theoretical models to safe mechanical operation and industrial performance.

A. Forces and torques in robotic manipulators

Forces and torques in robotic manipulators are the loads transmitted through joints, links, bearings, tools, and payloads.

• Joint torque relation: A tool wrench maps to joint torque through (\tau=J^TF), where (F) contains three forces and three moments.
• Gravity load: A horizontal arm generally produces greater shoulder torque than a vertical arm because the perpendicular moment arm is larger.
• Moment equation: A force (F) applied at position (r) creates (\tau=r\times F); the magnitude is (rF\sin\phi).
• Payload effect: Increasing payload mass raises required actuator torque and may reduce acceleration or usable reach.
• Contact force: Assembly, polishing, and insertion require force control so that contact loads remain within tool and workpiece limits.
• Mechanical consequences: Excessive loads cause overheating, deflection, backlash, bearing wear, or structural failure.

B. Static and dynamic balance

Static and dynamic balance distinguish equilibrium under stationary loading from balance during accelerated motion.

• Static equilibrium: A stationary robot is balanced when (\sum F=0) and (\sum \tau=0); actuator torques primarily counter gravity and external loads.
• Centre of mass: Stability requires the projected centre of mass to remain within the support polygon for a fixed-base or mobile robot.
• Dynamic balance: During acceleration, inertial forces and moments act in addition to gravity; balance depends on (M(q)\ddot q) and velocity-dependent terms.
• Counterbalance mechanisms: Springs, gas struts, or counterweights reduce gravity torque, especially in vertical shoulder or elbow axes.
• Operational effect: A balanced arm needs less holding torque, consumes less energy, and experiences reduced brake and gearbox loading.
• Example: If a (10\ \text{kg}) payload acts (0.5\ \text{m}) from a joint, its static torque is approximately (10(9.81)(0.5)=49.05\ \text{N m}), before link mass and dynamic effects.

C. Applications of kinematics and dynamics in industrial robots

Applications of kinematics and dynamics in industrial robots include programming, control, sizing, safety, and process optimization.

• Kinematic programming: In welding, inverse kinematics converts seam poses into joint commands while maintaining torch orientation.
• Pick-and-place: Forward kinematics verifies the tool location; trajectory planning coordinates rapid movement without collisions.
• Machining: Jacobian and dynamics calculations help maintain tool orientation, stiffness, cutting force capacity, and vibration limits.
• Assembly: Force and torque models support compliant insertion and prevent excessive contact loads.
• Motor selection: Inverse dynamics estimates peak and continuous torque, allowing motors and gearboxes to be selected with a safety margin.
• Safety and monitoring: Model-predicted torque can be compared with measured current to detect collisions, overloads, or mechanical faults.
• Overall integration: Industrial robot performance depends on combining geometry, motion planning, load estimation, balance, sensing, and feedback control rather than using kinematics or dynamics alone.