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Autonomous Mobile Manipulation

Manipulation: Forward Kinematics, Manipulability

C. Papachristos

Robotic Workers (RoboWork) Lab

University of Nevada, Reno

CS-791

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Forward Kinematics

 

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Forward Kinematics

 

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Forward Kinematics

 

 

 

 

 

 

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Forward Kinematics

 

 

 

 

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Forward Kinematics

 

 

 

(zero-“pitch” i.e. ratio of linear/ angular velocity)

(unit)

 

 

 

 

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Forward Kinematics

 

 

 

 

(zero-“pitch” i.e. ratio of linear/ angular velocity)

(unit)

 

 

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Forward Kinematics

 

 

 

 

(zero-“pitch” i.e. ratio of linear/ angular velocity)

(unit)

 

 

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Forward Kinematics

 

 

 

 

(Point on rotation axis –�origin of revolute joint frame)

(Space Frame transformation �– Pre-multiply)

 

 

 

Revolute: Axis vector normal to rotation

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Forward Kinematics

 

 

(No rotation�– prismatic)

 

(Space Frame transformation �– Pre-multiply)

 

 

 

 

Prismatic: Axis is the vector by which joint extends

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Forward Kinematics

 

 

(Space Frame transformation �– Pre-multiply)

 

 

 

 

Revolute: Axis vector normal to rotation

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Frame:

 

(Body Frame transformation �– Post-multiply)

 

 

 

Revolute: Axis vector normal to rotation

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Forward Kinematics

 

 

Frame:

(Body Frame transformation �– Post-multiply)

 

 

 

 

Prismatic: Axis is the vector by which joint extends

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Forward Kinematics

 

Frame:

 

(Body Frame transformation �– Post-multiply)

 

 

 

 

Revolute: Axis vector normal to rotation

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Forward Kinematics

Velocity Kinematics – Jacobian (“Analytical”)

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Calculate End-Effector velocity as function of manipulator arm joint velocities

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  • Assuming we express Forward Kinematics in minimal coordinates form:

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    • Take time derivative (& apply chain rule):

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    • “Analytical” Jacobian of robot arm:

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A non-linear�equation of

some form

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Forward Kinematics

Velocity Kinematics – Jacobian (“Analytical”)

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Calculate End-Effector velocity as function of manipulator arm joint velocities

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  • Forward Kinematics in minimal coordinates form:

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    • Take time derivative (& apply chain rule):

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    • “Analytical” Jacobian of robot arm:

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Forward Kinematics

 

 

 

 

 

 

 

 

Have POE general formula

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Forward Kinematics

 

 

 

 

 

 

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Have POE general formula

 

 

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Forward Kinematics

 

 

 

 

 

:Screw axis calculated at Home

 

 

 

 

 

 

 

 

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Have POE general formula

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Forward Kinematics

 

 

 

 

 

 

 

 

 

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Have POE general formula

 

 

 

 

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Forward Kinematics

 

 

 

 

Note: At a Singularity, one or more of the�Manipulability Ellipsoid axes get “squashed”

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Forward Kinematics

 

 

Joint�Torques

End-Effector

Wrench

 

We may derive Force & Wrench Ellipsoids.

Note: “Reciprocal” relationship to Manipulability Ellipsoid.

 

 

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Forward Kinematics

 

 

 

Joint�Torques

End-Effector

Wrench

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Forward Kinematics

 

 

 

(i.e. some columns become�dependent on each other)

“Full Rank Jacobian”

“Singular Jacobian”

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Forward Kinematics

Kinematic Singularity

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Configuration where manipulator Jacobian (instantaneously) loses its maximal rank

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    • Jacobian provides a transformation from joint velocities to end-effector Twist and�end-effector Wrench to joint forces/torques

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  • Therefore its rank can ever be:

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  • Kinematic Characterization:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Forward Kinematics

Manipulability

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Jacobian provides a mapping from Joint velocity space topology to End-Effector velocity topology

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  • Simple 2R Robot example:
    • 2R Robot: Joint space velocity circle maps to End-Effector velocity ellipsoid

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Forward Kinematics

 

 

 

 

 

 

 

 

 

 

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Forward Kinematics

Manipulability

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Body(/End-effector) Jacobian provides a convenient parametrization to intuitively understand the Manipulability potential at a certain configuration

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  • Mapping relationship is w.r.t. Body(/End-effector) frame:

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  • Represent Body Jacobian as:

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    • Intuitive representation: “Separation” in End-effector frame �with same unit scaling (linear velocities / angular velocities)

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: Angular velocity Body Jacobian

: Linear velocity Body Jacobian

 

 

 

 

 

 

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Time for Questions !

CS-791

CS791 C. Papachristos