Unit 1: Robot Anatomy

I. Orientation

Robot anatomy describes how a robot is organized mechanically and how its joints determine the motion of its links. In laboratory analysis, the important relationship is between the robot’s physical structure, the allowable motion of each joint, and the resulting position and orientation of the end-effector. A robot is treated as a serial chain of rigid links connected by joints, with each independent joint motion contributing one degree of freedom (DOF).

• Link: A rigid member that maintains its shape while transmitting motion and force. Links are numbered from the base toward the end-effector.
• Joint: A mechanical connection that permits relative motion between two links. Revolute joints produce angular motion; prismatic joints produce linear motion.
• Joint variable: The quantity describing joint configuration: angle (\theta_i) for a revolute joint or displacement (d_i) for a prismatic joint.
• Degree of freedom: One independent coordinate needed to specify a robot’s configuration. A revolute joint normally contributes one rotational DOF, while a prismatic joint contributes one translational DOF.
• Configuration: The complete set of joint variables at a particular instant:
  
  TEXT

    q = [q1, q2, ..., qn]^T

  
  Here, (q) is the configuration vector and (n) is the number of independent joints.
• Workspace: The set of positions, or positions and orientations, that the end-effector can reach within the joint limits.
• Joint limits: The minimum and maximum permitted values of a joint variable:
  
  TEXT

    qi,min <= qi <= qi,max

• Forward kinematics: The calculation of end-effector position and orientation from known joint variables. It is the main mathematical basis for visualizing how link motion changes the robot’s pose.
• Home or reference position: A defined starting configuration used for consistent measurements. All laboratory readings should identify this reference clearly.
• Physical safety assumption: Motion is considered only within manufacturer or laboratory limits. A mathematical pose outside those limits is not an allowable robot position.

II. Sierena’s Perro — Link Limits and Configuration

Sierena’s Perro is examined as a physical robotic mechanism whose links and joints demonstrate how independent motions combine to produce end-effector movement. The laboratory task is to identify the permitted motion of each joint and relate that motion to the visible position of the links.

A. Determination of maximum and minimum position of links

This determination identifies the extreme positions reached by each link when its associated joint is moved through its permitted range. The result is a measured description of the robot’s mechanical envelope rather than an unrestricted theoretical motion.

• Identify the chain: Trace the mechanism from the fixed base through each link to the end-effector. Label the links (L_1, L_2, \ldots, L_n) and the joints (J_1, J_2, \ldots, J_n).
• Classify each joint: Record whether each joint is revolute or prismatic. For a revolute joint, measure an angle; for a prismatic joint, measure a linear displacement.
• Establish the zero position: Place the robot in its laboratory reference configuration and record:
  
  TEXT

    qi = 0

  
  This zero is a convention and need not represent the geometric center of the physical range.
• Measure joint limits: Move one joint at a time while holding the remaining joints fixed. Record the first mechanically or electronically permitted value as (q{i,\min}) and the last permitted value as (q{i,\max}).
  • Angular joint: Use degrees or radians, such as (-45^\circ \leq \theta_i \leq 45^\circ).
  • Linear joint: Use millimetres or metres, such as (0 \leq d_i \leq 120\text{ mm}).
• Observe link positions: At each limit, note the orientation and endpoint location of the corresponding link. A revolute joint sweeps an arc; a prismatic joint sweeps a line.
• Use a measurement table: A consistent record may contain the following fields:
  
  TEXT

    Joint | Type | Minimum | Maximum | Range | Link affected | Observation

  
  The joint range is calculated as:
  
  TEXT

    Ri = qi,max - qi,min

  
  where (R_i) is the available motion range.
• Separate joint limits from workspace limits: The limit of an individual link is determined by its joint, whereas the complete workspace depends on all link lengths, joint axes, joint limits, and possible collisions.
• Check mechanical interference: A nominal maximum may be unusable if a link strikes the base, another link, a fixture, or the robot’s own housing. The practical limit is the safe limit actually observed in the apparatus.
• Worked example: If a revolute joint is recorded from (-30^\circ) to (60^\circ), its available motion is:
  
  TEXT

    R = 60 - (-30) = 90 degrees

  
  The associated link can occupy positions throughout that measured angular sweep, subject to the other joints remaining fixed.

B. Visualization of DOF on Sierena’s Perro

Visualization of DOF makes each independent motion visible by changing one joint variable at a time and observing the resulting link movement. It prevents the common mistake of counting links, motors, or visible motions instead of independent coordinates.

• Use one-variable motion: Starting from the reference pose, vary (q_1) while keeping (q_2,\ldots,q_n) constant. Repeat for every joint. The experiment isolates the contribution of each DOF.
• Represent revolute motion: Mark the initial and final positions of a rotating link. The swept arc shows the joint’s rotational freedom, and the axis of rotation identifies the type of motion.
• Represent prismatic motion: Mark the two endpoint positions of a sliding link. The line segment between them shows the translational DOF.
• Count independent coordinates: If the robot configuration is described by (q=[q_1,q_2,q_3]^T), it has three joint DOF, provided none of the variables is mechanically dependent on another.
• Distinguish position from orientation: A joint may change the end-effector’s location, orientation, or both. A rotating wrist joint can alter orientation significantly while producing little translational displacement.
• Use a configuration sketch: Draw the base, joint axes, links, reference pose, and extreme poses. Label each joint variable directly on the sketch.
• Observe coupled endpoint motion: Although joints are tested individually, combined motion may produce a curved or complex end-effector path. For a planar two-link arm:
  
  TEXT

    x = L1 cos(theta1) + L2 cos(theta1 + theta2)
    y = L1 sin(theta1) + L2 sin(theta1 + theta2)

  
  Here, (L_1) and (L_2) are link lengths, (\theta_1) and (\theta_2) are joint angles, and ((x,y)) is the endpoint position.
• Record the visual result: For each DOF, state the moving joint, moving link, axis or direction, minimum pose, maximum pose, and effect on the end-effector.
• Interpret the result: The total DOF describes how many independent motions can be commanded. It does not automatically equal the number of Cartesian coordinates controlled at the tool.

III. Twist Robot — Joint Range and Motion Analysis

The Twist robot provides a second mechanism for comparing robot anatomy, joint limits, and motion visualization. Its analysis follows the same configuration-based method, but its link arrangement and joint-axis directions may produce a different workspace and a different relationship between joint motion and tool motion.

A. Determination of maximum and minimum position of links

For the Twist robot, the extreme link positions are obtained by applying the permitted limit to each joint and observing the resulting configuration. Accurate identification requires attention to joint axes, reference marks, and the physical stops of the model.

• Map the anatomy: Identify the base, successive links, joint axes, actuators, end-effector, and any transmission elements. The base is fixed; every other link moves relative to a preceding link.
• Record the joint variable: Use (\theta_i) for a twisting or rotating joint and (d_i) for a sliding joint. The notation should match the actual motion observed in the laboratory.
• Find the minimum position: From the reference configuration, move the selected joint toward its lower permitted boundary. Record the joint reading and the link’s final pose.
• Find the maximum position: Move the same joint toward its upper permitted boundary. Stop at the programmed limit, mechanical stop, or safe laboratory boundary, whichever is applicable.
• Do not force a stop: A mechanical stop is a limit of allowable motion, not a target for excessive force. Measurements should be taken without loading the linkage unnecessarily.
• Calculate the range: For every joint:
  
  TEXT

    Joint range = maximum value - minimum value

  
  For an angular joint, the result is in degrees or radians; for a linear joint, it is in millimetres or metres.
• Use repeatable readings: Return to the reference pose before measuring another joint. This reduces errors caused by accumulated positioning changes.
• Account for sign convention: Positive and negative directions depend on the selected coordinate frame and right-hand rule. A negative angle is not a smaller physical capability; it is a direction relative to the chosen zero.
• Include link geometry: The endpoint position depends on link lengths and joint variables. If a planar Twist configuration has two links, the same forward-kinematic form applies:
  
  TEXT

    p = f(q, L)

  
  Here, (p) is the end-effector position, (q) is the joint configuration, and (L) represents the link dimensions.
• State practical restrictions: Cable routing, actuator limits, joint backlash, collision zones, and payload can reduce the usable range below the geometric range.

B. Visualization of DOF on Twist robot

Visualization of DOF on the Twist robot connects the abstract configuration vector with observable motion. Each DOF is demonstrated by an independent change in one joint coordinate while the remaining coordinates are held constant.

• Prepare the reference pose: Photograph or sketch the robot before movement. Mark the base frame (O), link frames, joint axes, and end-effector reference point.
• Move joints sequentially: Apply the following laboratory sequence:
  
  TEXT

    set robot to reference pose
    for each joint Ji:
        hold all other joints fixed
        move Ji to minimum
        record pose
        return to reference
        move Ji to maximum
        record pose

  
  The variable (J_i) denotes the selected joint and “pose” includes both position and orientation.
• Visualize axis direction: A revolute DOF is shown by rotation about a fixed or moving axis. A prismatic DOF is shown by translation along an axis. The axis may be vertical, horizontal, or aligned with a link.
• Compare individual and combined effects: One joint may rotate a distal link together with all links attached after it. Thus, the visible motion can involve several links even though only one independent joint variable is being changed.
• Draw the reachable envelope: Connect or mark the endpoint positions obtained at the joint limits. For a rotating link of length (L) about a fixed point, the endpoint follows an arc of radius (L). Multiple joints generate a larger composite region.
• Identify orientation DOF: If the Twist robot includes a wrist or terminal rotation, its motion may primarily change tool orientation. Record this separately from translational movement.
• Use a DOF table: A suitable observation format is:
  
  TEXT

    DOF | Joint motion | Axis/direction | Links visibly affected | Tool effect

  
  This makes the distinction between joint motion and Cartesian motion explicit.
• Check independence: Two visible motions count as separate DOF only when they can be commanded independently. A mechanically synchronized pair driven by one actuator contributes one independent coordinate.
• Interpret singular or restricted poses: At some configurations, different joint motions can produce nearly the same endpoint motion. Such a pose reduces instantaneous mobility even though the robot’s structural DOF has not changed.

C. Comparative analytical dimension

Comparison of Sierena’s Perro and the Twist robot shows that DOF is a structural property, while workspace is a geometric and limit-dependent result.

• Structural comparison: Count independent joints and classify their motion types before judging reachability.
• Range comparison: Compare each joint’s measured (q{i,\min}), (q{i,\max}), and (R_i), using the same units and reference convention.
• Workspace comparison: A robot with fewer joints may reach a broad position region, while a robot with more joints may add orientation control or avoid obstacles more effectively.
• Accuracy limitation: Manual readings are affected by scale resolution, backlash, parallax, calibration error, and uncertainty in locating the exact joint boundary.
• Safety limitation: The observed envelope must exclude collision zones and unsafe configurations, even if those configurations are mathematically reachable.
• Final laboratory record: A complete anatomy analysis should contain the labelled mechanism, joint types, reference pose, measured limits, calculated ranges, DOF sketches, endpoint observations, and a clear distinction between structural DOF and usable workspace.