Lecture 12velocity kinematics I

Katie DC

October 8, 2019

Modern Robotics Ch. 5.1-5.3

Admin

• HW6 is up and due on Friday 10/11

• Guest Lecture on Thursday 10/10

• Reflection due Sunday 10/20 at midnight (do it early!)

• Project Update 3 is due Sunday 10/13 at midnight

• Quiz 2 (required) Rigid Bodies and Kinematics, Oct 20 to Oct 22

• (optional) mid-semester self-assessment for participation due Friday 10/25

Who is Carl Gustav Jacob Jacobi?

• A German mathematician who contributed to elliptic functions, dynamics, differential equations, determinants, and number theory

• In Germany during the Revolution of 1848, Jacobi was politically involved with the Liberal club, which after the revolution ended led to his royal grant being temporarily cut off

• His PhD student, Otto Hesse, is known for the Hessian Matrix (second-order partial derivatives of a scalar-valued function, or scalar field)

• The crater Jacobi on the Moon is named after him

• Died of a smallpox infectionCredit: Wikipedia

Velocity Kinematics

Example

Jacobian Mapping

Jacobian and Manipulability

• Suppose joint limits are ሶ𝜃12 + ሶ𝜃2

2 ≤ 1

• The ellipsoid obtained by mapping through the Jacobian is called the manipulability ellipsoid

Jacobian and Statics

• Assume a force 𝑓𝑡𝑖𝑝 is applied on the end-effector. What torque 𝜏 must be applied at the joint

to keep the fixed position?

• By conservation of power:𝑓𝑡𝑖𝑝⊤ 𝑣𝑡𝑖𝑝 = 𝜏⊤ ሶ𝜃, ∀ ሶ𝜃

• Since 𝑣𝑡𝑖𝑝 = 𝐽 𝜃 ሶ𝜃, we have𝑓𝑡𝑖𝑝⊤ 𝐽 𝜃 ሶ𝜃 = 𝜏⊤ ሶ𝜃, ∀ ሶ𝜃

• Which gives:𝜏 = 𝐽⊤ 𝜃 𝑓𝑡𝑖𝑝

• If 𝐽⊤ 𝜃 is invertible, then given the limits of the torques at the joints, we can compute all the forces that can be counteracted at the end-effector

Computing the Jacobian (1)

Computing the Jacobian (2)

Computing the Jacobian: Example 1

Computing the Jacobian: Example 2

Computing the Jacobian: Example 2

𝜔𝑠4 = 𝑅𝑜𝑡 Ƹ𝑧, 𝜃1 𝑅𝑜𝑡 ො𝑥, −𝜃2

001

=

−𝑠1𝑠2𝑐1𝑠2𝑐2

𝜔𝑠5 = 𝑅𝑜𝑡 Ƹ𝑧, 𝜃1 𝑅𝑜𝑡 ො𝑥, −𝜃2 𝑅𝑜𝑡( Ƹ𝑧, 𝜃4)−100

=

−𝑐1𝑐4 + 𝑠1𝑐2𝑠4−𝑠1𝑐4 − 𝑐1𝑐2𝑠4

𝑠2𝑠4

𝜔𝑠6 = 𝑅𝑜𝑡 Ƹ𝑧, 𝜃1 𝑅𝑜𝑡 ො𝑥, −𝜃2 𝑅𝑜𝑡 Ƹ𝑧, 𝜃4 𝑅𝑜𝑡 ො𝑥, −𝜃5

010

=−𝑐5 𝑠1𝑐2𝑐4 + 𝑐1𝑠4 + 𝑠1𝑠2𝑠5𝑐5 𝑐1𝑐2𝑐4 − 𝑠1𝑠4 − 𝑐1𝑠2𝑠5

−𝑠2𝑐4𝑐5 − 𝑐2𝑠5

Body Jacobian

The relationship between the Space and Body Jacobian

Recall that:

𝒱𝑠 = ሶ𝑇𝑠𝑏𝑇𝑠𝑏−1 𝒱𝑏 = 𝑇𝑠𝑏

−1 ሶ𝑇𝑠𝑏𝒱𝑠 = 𝐴𝑑𝑇𝑠𝑏 𝒱𝑏 𝒱𝑏 = 𝐴𝑑𝑇𝑏𝑠 𝒱𝑠𝒱𝑠 = 𝐽𝑠 𝜃 ሶ𝜃 𝒱𝑏 = 𝐽𝑏 𝜃 ሶ𝜃

• Apply [𝐴𝑑𝑇𝑏𝑠] to both sides and recall that 𝐴𝑑𝑇𝑋 𝐴𝑑𝑌 = [𝐴𝑑𝑋𝑌]:𝐴𝑑𝑇𝑏𝑠(𝐴𝑑𝑇𝑠𝑏(𝒱𝑏)) = 𝒱𝑏 = 𝐴𝑑𝑇𝑏𝑠(𝐽𝑠 𝜃 ሶ𝜃)

𝐽𝑏 𝜃 ሶ𝜃 = 𝐴𝑑𝑇𝑏𝑠(𝐽𝑠 𝜃 ሶ𝜃)

• Since this holds for all ሶ𝜃:𝐽𝑏 𝜃 = 𝐴𝑑𝑇𝑏𝑠 𝐽𝑠 𝜃 = 𝐴𝑑𝑇𝑏𝑠 𝐽𝑠 𝜃

𝐽𝑠 𝜃 = 𝐴𝑑𝑇𝑠𝑏 𝐽𝑏 𝜃 = 𝐴𝑑𝑇𝑠𝑏 𝐽𝑏 𝜃

𝐴𝑑𝑇𝑠𝑏(𝒱𝑏) = 𝐽𝑠 𝜃 ሶ𝜃