Lesson 1: Kinematic Joints and Degrees of Freedom in 3D Systems

• Motion limitations preventing required surgical maneuvers or assembly tasks
• Over-actuated designs wasting motors, cost, and control complexity
• Closure constraints causing unexpected loss of mobility in multi-robot cells
• Workspace restrictions from unaccounted external constraints (trocar ports, mounting)
• Inefficient grippers requiring multiple motors when one could suffice

Benefits of Systematic DoF Analysis:

• Optimal actuator count - knowing exactly how many motors needed
• Constraint accounting - predicting effective DoF under real operating conditions
• Design trade-offs - comparing alternatives quantitatively (size, force, complexity)
• Underactuation opportunities - exploiting passive compliance for adaptability
• Cost reduction - minimizing actuators while maintaining functionality

1. Count links and joints:

   • Links: n = 7 (6 moving links + ground)
   • Joints: j = 6 (all revolute)
   • Constraints per revolute joint: c = 5

2. Apply Grübler’s equation:

   ✅

3. Motion capability:

   • Translations: 3 (X, Y, Z)
   • Rotations: 3 (roll, pitch, yaw)
   • Total: Full spatial positioning ✅

4. Actuator count:

   Motors required: 6 (one per revolute joint) ✅

1. Count links and joints:

   • Links: n = 5 (4 moving links + ground)
   • Joints: j = 4 (3 revolute + 1 prismatic vertical)
   • Total constraints: 3(5) + 1(5) = 20

2. Apply Grübler’s equation:

   ✅

3. SCARA motion breakdown:

   • Horizontal plane: 2 translational DoF (X, Y)
   • Vertical: 1 translational DoF (Z)
   • Rotation: 1 about vertical axis (θ_z)

   Constraint: Tool axis remains vertical (no tilt) ✅

4. Actuator count:

   Motors required: 4 (3 rotary + 1 vertical linear) ✅

1. Count links and joints:

   • Links: n = 3 (2 moving segments + ground)
   • Joints: j = 2 (universal joints)
   • Constraints per universal joint: c_U = 4

   Reminder: Each universal joint allows 2 rotational DoF.

2. Apply Grübler’s equation:

   ✅

3. Rotation capability:

   • First universal joint: 2 rotational DoF
   • Second universal joint: 2 rotational DoF
   • Total: 4 rotational DoF for workpiece orientation ✅

4. Actuator count:

   Motors required: 4 (2 per universal joint assembly) ✅

1. Summary table:

   Subsystem	Config	Links (n)	Joints (j)	Constraints (Σc)	DoF	Actuators
   Robot 1 (6-Axis Welder)	6R	7	6	30	6	6
   Robot 2 (SCARA)	3R+1P	5	4	20	4	4
   Workpiece Positioner	2U	3	2	8	4	4
   TOTALS	-	15	12	58	14	14

   ✅

2. Verification using individual calculations:

   ✅

   Note: This assumes independent operation (robots not sharing workpiece).

3. Total actuator requirement:

   Independent operation: 14 motors ✅

   Control system needs:

   • 14 servo motor controllers
   • Real-time coordination computer
   • Synchronized motion planning software

4. Redundancy analysis:

   Robot 1 (6 DoF):

   • Task requires: 6 DoF (full welding orientation)
   • Available: 6 DoF
   • Redundancy: None (minimal configuration) ✅

   Robot 2 (4 DoF SCARA):

   • Task requires: 4 DoF (workpiece positioning)
   • Available: 4 DoF
   • Redundancy: None for full 4-DoF tasks ✅

5. Shared workpiece constraint analysis:

   When Robot 1 grips the workpiece held by the positioner:

   Closure constraints added: 6 (rigid connection removes 6 DoF)

   Modified DoF calculation:

   ✅

   Interpretation: The combined system (Robot 1 + Positioner + Workpiece) has only 4 controllable DoF when rigidly connected.

   Practical meaning:

   • Positioner must coordinate with Robot 1
   • Cannot move all 10 actuators independently
   • 6 actuators become dependent on the other 4

6. Critical design insights:

   Maximum motors needed: 14 (independent operation) ✅

   Effective DoF when coordinated:

   • Robot 1 + Positioner + Workpiece: 4 DoF
   • Robot 2 (SCARA): 4 DoF
   • Total coordinated: 8 DoF (not 14)

   Singularity concerns:

   • Robot 1: Wrist singularities (3 axes align), shoulder singularity
   • Robot 2 (SCARA): Elbow singularity (links collinear)
   • Positioner: Gimbal lock possible ✅

1. System configuration:

   • Proximal (outside patient): Insertion depth (1P) + shaft rotation (1R)
   • Distal (inside patient): Spherical wrist joint (1S at tool tip)
   • Links: n = 4 (insertion shaft + rotating shaft + wrist housing + end-effector + ground)
   • Joints: 1 prismatic + 1 revolute + 1 spherical

2. Apply Grübler’s equation:

   ✅

3. Motion capability breakdown:

   • Insertion (P): 1 translational DoF (depth control)
   • Shaft rotation (R): 1 rotational DoF (about insertion axis)
   • Spherical wrist (S): 3 rotational DoF (pitch, yaw, roll at tip)
   • Total: 2 translation + 3 rotation = 5 DoF ✅

   Missing: 1 translational DoF (cannot move laterally inside patient without pivoting about trocar)

4. Trocar constraint analysis:

   Remote Center of Motion (RCM) constraint:

   • Tool shaft MUST pivot through trocar point
   • This removes 2 translational DoF (X, Y perpendicular to shaft)
   • Effective internal DoF: 5 - 2 = 3 DoF usable inside patient ✅

   Note: Lateral motion achieved by pivoting entire instrument through trocar, not internal joints.

5. Actuator count:

   Motors required: 5 (1 linear for insertion + 4 rotary: 1 shaft + 3 in spherical joint) ✅

6. Advantages/Disadvantages:

   Advantages:

   • Compact spherical wrist design
   • All 3 rotational DoF at single point (simplified kinematics)
   • Minimum instrument diameter

   Disadvantages:

   • Spherical joint mechanically complex
   • Difficult to seal for sterilization
   • Higher friction in compact design

1. System configuration:

   • Proximal: Insertion (1P) + 2 revolute joints (2R for pitch/yaw angles)
   • Distal: Universal joint wrist (1U at tool tip)
   • Links: n = 5 (insertion + link 1 + link 2 + wrist + end-effector + ground)
   • Joints: 1P + 2R + 1U

2. Apply Grübler’s equation:

   ✅

   Note: Universal joint has 4 constraints (allows 2 rotational DoF).

3. Motion capability breakdown:

   • Insertion (P): 1 translational DoF
   • Proximal revolutes (2R): 2 rotational DoF (instrument pitch/yaw)
   • Universal wrist (U): 2 rotational DoF (end-effector orientation)
   • Total: 1 translation + 4 rotation = 5 DoF ✅

4. Trocar constraint effect:

   Same RCM constraint as Option A:

   • Effective internal DoF: 3 DoF (1 insertion + 2 wrist rotations) ✅
   • Proximal 2R joints used for positioning through trocar pivot

5. Actuator count:

   Motors required: 5 (1 linear + 4 rotary) ✅

6. Advantages/Disadvantages:

   Advantages:

   • Universal joint mechanically simpler than spherical
   • Easier to seal and sterilize
   • Better force transmission

   Disadvantages:

   • Slightly larger diameter than spherical design
   • 4 rotational DoF instead of 3 (one redundant)

1. System configuration:

   • Proximal: Insertion (1P) + shaft rotation (1R)
   • Distal: Three revolute wrist (3R - pitch, yaw, roll)
   • Links: n = 6 (insertion + shaft + wrist link 1 + link 2 + link 3 + end-effector + ground)
   • Joints: 1P + 4R total

2. Apply Grübler’s equation:

   ✅

3. Motion capability breakdown:

   • Insertion (P): 1 translational DoF
   • Shaft rotation (R): 1 rotational DoF
   • Wrist (3R): 3 rotational DoF (Euler angles: pitch-yaw-roll)
   • Total: 1 translation + 4 rotation = 5 DoF ✅

4. Trocar constraint effect:

   Effective internal DoF: 3 DoF (1 insertion + 2 wrist orientations, shaft rotation external) ✅

5. Actuator count:

   Motors required: 5 (1 linear + 4 rotary) ✅

6. Advantages/Disadvantages:

   Advantages:

   • All revolute joints (simplest mechanism)
   • Proven reliability in industrial robots
   • Easy maintenance and sterilization

   Disadvantages:

   • Largest diameter (three separate revolute joints)
   • Wrist singularity when axes align
   • Longer distal segment

1. Comparative summary table:

   Design	Config	DoF	Actuators	Diameter	Wrist Complexity	Force Trans.	Singularities
   Option A	1P+1R+1S	5	5	8 mm	High	Medium	Few
   Option B	1P+2R+1U	5	5	9 mm	Medium	High	Medium
   Option C	1P+4R	5	5	10 mm	Low	High	Many

   ✅

2. Workspace analysis:

   All three options provide equivalent effective DoF = 5 (before trocar constraint).

   After RCM constraint:

   • Insertion depth: 0-200 mm
   • Wrist rotation: ±90° pitch, ±90° yaw
   • Workspace volume: ~150 mm diameter sphere ✅

3. Force transmission comparison:

   Option A (Spherical):

   • Force through compact ball joint
   • Mechanical advantage: ~0.7 (friction losses)
   • Achievable grip force: 20N × 0.7 = 14N ⚠️ (below requirement)

   Option B (Universal):

   • Force through two perpendicular axes
   • Mechanical advantage: ~0.85
   • Achievable grip force: 20N × 0.85 = 17N ✅ (meets requirement)

   Option C (3R Wrist):

   • Force through series of revolute joints
   • Mechanical advantage: ~0.9
   • Achievable grip force: 20N × 0.9 = 18N ✅ (exceeds requirement)

4. Sterilization and reliability:

   • Option A: Difficult to seal spherical joint → infection risk ⚠️
   • Option B: Universal joint can be sealed → good ✅
   • Option C: All revolute joints easily sealed → excellent ✅

5. Final recommendation:

   Selected: Option B (1P + 2R + 1U)

   Justification:

   • Meets all DoF requirements (5 total, 3 effective inside patient) ✅
   • Acceptable diameter (9mm < 10mm trocar) ✅
   • Good force transmission (17N grip force) ✅
   • Medium complexity (balance of performance and reliability) ✅
   • Superior sealing for sterilization ✅

   Trade-off: Slightly larger than Option A, but significantly better force transmission and easier to sterilize than spherical joint. Simpler than Option C with fewer singularities.

6. Actuator implementation:

   5 motors total:

   • 1 linear actuator (insertion, external)
   • 2 rotary motors (pitch/yaw, external for trocar pivoting)
   • 2 rotary motors (universal wrist, cable-driven to distal end)

   Cable transmission ratio: 3:1 to achieve 20N output force ✅

1. Single finger configuration:

   • Links: n = 4 (proximal + medial + distal phalanges + palm/ground)
   • Joints: j = 3 (all revolute: θ₁, θ₂, θ₃)
   • Each revolute joint: c = 5 constraints

2. Apply Grübler’s equation (spatial):

   ✅

   Note: This assumes 3 independent actuators per finger (not our case).

3. Motion capability:

   • Joint 1 (proximal): Rotation θ₁
   • Joint 2 (medial): Rotation θ₂
   • Joint 3 (distal): Rotation θ₃
   • Total: 3 rotational DoF per finger ✅

4. Underactuation constraint:

   With tendon coupling:

   • All 3 joints driven by single tendon
   • Joints move sequentially (not independently)
   • Effective actuated DoF per finger: 1 ✅

   Passive joints: Joints 2 and 3 become passive after contact with object.

1. Total system without coupling:

   • Fingers: 4
   • Joints per finger: 3
   • Total joints: 4 × 3 = 12 revolute joints
   • Total links: n = 4 fingers × 3 phalanges + 1 palm = 13 links

   If all joints independent:

   ✅

2. With tendon coupling constraints:

   Coupling mechanism:

   • Single motor drives 1 main tendon
   • Differential pulley distributes force to 4 finger tendons
   • Each finger tendon drives 3 joints in sequence

   Additional constraints from coupling:

   • 4 fingers must close synchronously: 3 coupling constraints
   • 3 joints per finger coupled by single tendon: 2 constraints × 4 = 8 constraints
   • Total coupling constraints: 3 + 8 = 11 ✅

3. Modified Grübler’s equation with coupling:

   ✅

   Result: The entire 12-joint gripper has only 1 actuated DoF (the motor input).

4. Passive DoF during grasping:

   When fingers contact fruit:

   • Proximal joints stop when tendon tension balanced
   • Medial and distal joints continue closing (passive motion)
   • Self-adaptive shape conformance ✅

5. Actuator count:

   Motors required: 1 (single motor for entire gripper) ✅

   Mechanical advantage through pulleys:

   • Motor torque: 0.5 N·m
   • Pulley ratio: 4:1
   • Tendon force: (0.5 N·m) / (0.02 m radius) × 4 = 100 N
   • Force per finger: 100 N / 4 = 25 N
   • Fingertip force: 25 N / 5 (mechanical advantage) = 5 N ✅ (meets requirement)

1. Geometric parameters:

   • Proximal phalange length: L₁ = 40 mm
   • Medial phalange length: L₂ = 30 mm
   • Distal phalange length: L₃ = 25 mm
   • Maximum finger span (fully open): 120 mm diameter ✅
   • Minimum finger span (fully closed): 35 mm diameter

2. Joint angle ranges:

   • θ₁ (proximal): 0° to 90°
   • θ₂ (medial): 0° to 110°
   • θ₃ (distal): 0° to 120°

3. Grasping small fruit (40mm tomatoes):

   Contact sequence:

   • All 4 fingers close simultaneously
   • Proximal joints rotate to θ₁ ≈ 65°
   • First contact at proximal phalanges
   • Tendon tension stops proximal joints
   • Medial and distal joints continue passively
   • Final configuration: θ₁=65°, θ₂=25°, θ₃=10° ✅

   Grip diameter: 40 mm ✅ (matches fruit size)

4. Grasping large fruit (120mm tomatoes):

   Contact sequence:

   • Fingers open to maximum span
   • Proximal joints rotate to θ₁ ≈ 15°
   • First contact at distal fingertips
   • Tendon tension distributed across all joints
   • Final configuration: θ₁=15°, θ₂=5°, θ₃=5° ✅

   Grip diameter: 120 mm ✅ (matches fruit size)

5. Force distribution verification:

   For 40mm fruit (proximal contact):

   • Contact points: 4 (one per finger)
   • Force per contact: 5N / 4 = 1.25 N
   • Status: ✅ Within 2-5N requirement (no bruising)

   For 120mm fruit (distal contact):

   • Contact points: 4 (fingertips)
   • Moment arm disadvantage: ×1.5
   • Effective force: 5N / 1.5 = 3.3 N per contact
   • Status: ✅ Within 2-5N requirement

6. Workspace envelope:

   Achievable grip range: 35-120 mm ✅ Required range: 40-120 mm ✅ Margin: 5 mm (acceptable) ✅