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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.
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Examples of Spatial Mechanisms
Hooke (Cardan) Joint Swashplate
Bone Cutter
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Lower Pair Joint Types
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Spatial Mechanism Naming Convention
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Harrisberger’s “Most Useful” 1 DOF List
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Mobility Defined by Kutzbach Criterion
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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.
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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
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Spherical 4‐bar
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Spherical 4‐bar
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Spherical Mechanisms
α – central angle of crankβ – central angle of outputγ – central angle of couplerζ – central angle of ground
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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.
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Spherical Roth‐Burmester Curves
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At the end of this video, you should be able to:• Describe
Representing Spatial Mechanisms: Denavit‐Hartenberg Parameters
Source: irobotkits.blogspot.com
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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
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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
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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
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DH Parameters: Closed Chain
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•Label the appropriate DH parameters•Write the symbolic matrix transformation eqn
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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
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R1
R2
R3
R4
z1z3
z4
x1
32
1
4
α34
α41
α12
α23
θ12θ41
x3
x2x4
z2
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Hooke (Cardan) Joint Analysis
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Hooke / Cardan Joint
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Driveshaft Phasing