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Lecture 12velocity kinematics I
Katie DC
October 8, 2019
Modern Robotics Ch. 5.1-5.3
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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
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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
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Velocity Kinematics
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Example
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Jacobian Mapping
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Jacobian and Manipulability
• Suppose joint limits are ሶ𝜃12 + ሶ𝜃2
2 ≤ 1
• The ellipsoid obtained by mapping through the Jacobian is called the manipulability ellipsoid
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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
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Computing the Jacobian (1)
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Computing the Jacobian (2)
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Computing the Jacobian: Example 1
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Computing the Jacobian: Example 2
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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
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Body Jacobian
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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