[Modern Robotics] 강좌 3: 로봇 동역학 #3

 

 

 

 

 

 

Open chain dynamics in the Task space: Given ${\mathcal{V}}_{tip}$and $\theta$, the relation between ${\dot{\mathcal{V}}}_{tip}$ and ${\mathcal{F}}_{tip}$ (※${\mathcal{F}}_{tip}$ is the wrench generated by the robot movement, not an external force)

 

$\tau =M(\theta )\ddot{\theta }+h(\theta ,\dot{\theta })⟶{\mathcal{F}}_{tip}={\mathrm{\Lambda }}_{tip}(\theta ){\dot{\mathcal{V}}}_{\mathcal{tip}}+{\eta }_{tip}(\theta ,\mathcal{V})$

 

End effector mass matrix ${\mathrm{\Lambda }}_{tip}(\theta )$ : Refer to [강좌3, 로봇 동역학 #1]

 

${\eta }_{tip}(\theta ,\mathcal{V})=J^{-T}h(\theta ,\dot{\theta })-\mathrm{\Lambda }(\theta )\dot{J}{J}^{-1}\mathcal{V}=J^{-T}h(\theta ,J^{-1}\mathcal{V})-\mathrm{\Lambda }(\theta )\dot{J}{J}^{-1}\mathcal{V}$

 

●Derived by using ${\mathcal{V}}_{tip}=J_{tip}(\theta )\dot{\theta }$ and ${\dot{\mathcal{V}}}_{tip}={\dot{J}}_{tip}(\theta )\dot{\theta }+J_{tip}(\theta )\ddot{\theta }$

●Note that Jacobian $J$, twist $\mathcal{V}$, wrench $\mathcal{F}$ all are expressed in end effector frame {tip}

●Cannot replace $\theta$ with end effector configuration $X$ since the open chain inverse kinematics can have multiple solutions.

 

 

 

Friction term ${\tau }_{fric}$: The torque generated by the motor's friction(gears, bearing).

●The previous open chain dynamics didn't account for the friction. Therefore, to model precisely, torques by friction should be included.

●Generally there are two terms: Static friction which is constant and Viscous friction which is proportional to $\dot{\theta }$

 

 

 

 

I'll skip section 8.7 Constrained Dynamics since I haven't fully understood the chapter....

 

 

 

 

 

DC (servo) motor basics

●Torque is proportional to current with the torque constant: $\tau =k_{t}I$

●Simply put, torque and angular velocity have a linear relation: $\omega =-\frac{R}{k_t^2}\tau +\frac{V}{k_t}$ for constant $V$ > Speed-torque curve of the DC motor.

●No-load speed ${\omega }_0$: Maximum motor speed without making any torque.

●Stall torque ${\tau }_{stall}$: Maximum motor torque without rotating.

●The lower the voltage, the lower the stall torque and no-load speed.

●Below the curve represents the continuous operating region, $({\tau }_{cont},{\omega }_{cont})$

 

●The gear system is used to achieve higher torque.

●For the geared DC motor with gear ratio $G$ and efficiency $\eta$, speed becomes $\frac{1}{\eta G}$ times and torque becomes $\eta G$ times.

●To decouple the dynamics of each joint of the open chain robot, using a motor with a higher gear ratio can be a solution, as the mass matrix $M(\theta )$, which includes the motor inertia, becomes diagonalized. (My own explanation, uncertain)

 

 

 

========================================================================
정확한 정보 전달보단 공부 겸 기록에 초점을 둔 글입니다.
틀린 내용이 있을 수 있습니다.
틀린 내용이나 다른 문제가 있으면 댓글에 남겨주시면 감사하겠습니다. : )
========================================================================