At the end of this video, you should be able to:• Describe the mobility of spatial mechanism joint types• Determine the number of degrees‐of‐freedom of a spatial 

mechanism• Describe the motion of a spherical mechanism

Introduction to Spatial Mechanisms

Source: Qu, H., Guo, S., and Zhang, Y., 2016, “A novel relative degree‐of‐freedom criterion for a class of parallel manipulators with kinematic redundancy and its applications,” Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science.

Examples of Spatial Mechanisms

Hooke (Cardan) Joint Swashplate

Bone Cutter

Lower Pair Joint Types

Spatial Mechanism Naming Convention

Harrisberger’s “Most Useful” 1 DOF List

Mobility Defined by Kutzbach Criterion

Kutzback Criterion Example

Source: Qu, H., Guo, S., and Zhang, Y., 2016, “A novel relative degree‐of‐freedom criterion for a class of parallel manipulators with kinematic redundancy and its applications,” Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science.

Planar and Spatial CorrelationsPlanar Theory Spherical Theory Spatial Theory

Gruebler’s Equation Kutzbach Criterion Kutzbach Criterion

Pole Rotation Axis Displacement Screw

Instant Center Instantaneous Rotation Screw

Instantaneous Screw

Image Pole Image Rotation Axis Image Screw

Pole Triangle Spherical Screw Triangle

Screw Triangle

Center Point Curve Fixed R-Axis Cone Fixed C-Axis Congruent

Circle Point Curve Moving R-Axis Cone Moving C-Axis Congruent

Burmester Point Pairs 5 Position R-R Axes 5 Position C-C Axes

Euler-Savary Equation Spherical Euler-SavaryEqn

Spatial Euler-SavaryEqn

Spherical 4‐bar

Spherical 4‐bar

Spherical Mechanisms

α – central angle of crankβ – central angle of outputγ – central angle of couplerζ – central angle of ground

Spherical Coupler Curves

Source: Chu, J. and Sun, J., 2010, “Numerical Atlas of Path Generation of Spherical Four‐Bar Mechanism,” Mechanism and Machine Theory, v. 45, p. 867‐879.

Spherical Roth‐Burmester Curves

At the end of this video, you should be able to:• Describe

Representing Spatial Mechanisms: Denavit‐Hartenberg Parameters

Source: irobotkits.blogspot.com

DH Parameters

4 D-H Parameters:ai,i+1: dist along xi+1 from zi to zi+1

αi,i+1: angle from zi to zi+1 (as seen from xi+1)θi,i+1: angle from xi to xi+1 (as seen from zi)si,i+1: dist along zi from xi to xi+1

Conventions:1. Number joints consecutively

– start at the input2. Choose joint axis such that: xi

is perpendicular to zi-1 and zi

3. Origin xi, yi, zi is fixed in link w/ joints i-1 and i

4. In closed chain, joint 1 references last joint, joint N

DH Parameter Example

DH Parameter TableJoint a s

0 a , , , ,

1 a , , , ,

2 a , , , ,

ai,i+1: dist along xi+i from zi to zi+1

αi,i+1: angle from zi to zi+1 (as seen from xi+1)θi,i+1: angle from xi to xi+1 (as seen from zi)si,i+1: dist along zi from xi to xi+1

Source: irobotkits.blogspot.com

DH Parameters

ai,i+1: dist along xi+i from zi to zi+1

αi,i+1: angle from zi to zi+1 (as seen from xi+1)θi,i+1: angle from xi to xi+1 (as seen from zi)si,i+1: dist along zi from xi to xi+1

DH Parameters: Closed Chain

•Label the appropriate DH parameters•Write the symbolic matrix transformation eqn

R1

R2

R3

R4

z1

z2

z3

z4

x1

3 2

1

4

Spherical Four-Bar1. Label the x-axes2. Draw the appropriate DH parameters3. Write the symbolic matrix

transformation equation

R1

R2

R3 

R4 

z1z3 

z4 

x1 

32

1

4

α34

α41 

α12

α23

θ12θ41

x3

x2x4

z2

Hooke (Cardan) Joint Analysis

Hooke / Cardan Joint

Driveshaft Phasing