[Modern Robotics] 강좌 3: 로봇 동역학 #5

 

 

 

Time-optimal time scaling

●Finding a time scaling that minimizes the time of motion $s(t)$ respecting the given path and the joint's torque limit.
●Of course it satisfies $s \in [0,1], s(0)=0, s(T)=1, \dot{s}(0)=\dot{s}(T)=0$ and $s(t)$ should be monotonically increasing.


Path-constrained inverse dynamics (My own explanation)

 

●The robot's inverse dynamics equation when the robot follows the given path $\theta(s)$
●Of course the joint torque should stay within the torque limit. $\tau_{min}\le\tau=m(s)\ddot{s}+c(s)\dot{s}^{2}+g(s)\le\tau_{max}$


Actuator constraint of time-scaling 
$\tau_{min}\le m(s)\ddot{s}+c(s)\dot{s}^{2}+g(s)\le\tau_{max} \longrightarrow \color{Red} L(s,\dot{s}) \le \ddot{s} \le U(s,\dot{s})$

●From the path-constrained inverse dynamics, we can derive the state-depending acceleration limit of the time-scaling from the torque limit and the path.
●This represents that the capable range of $\ddot{s}$ when the state($s,\dot{s}$) is given

●A state ($s,\dot{s}$) where $U(s,\dot{s})\lt L(s,\dot{s})$ is an unachievable state with the given torque limit and path.

 

 

 

 

 

$(s,\dot{s})$ phase plane

●A curve on the plane shows a time scaling.

●For the time-optimal time scaling problem, curves must start at (0,0) and end at (1,0).

●The larger the area below the curve, the shorter the time of the motion(T).

+)$T=\int_{0}^{T}1dt=\int_{0}^{1}\frac{dt}{ds}ds=\int_{0}^{1}\dot{s}^{-1}(s)ds$

●A tangent vector at the point of the curve represents $(\dot{s},\ddot{s})$, so it's direction contains the acceleration information.(Not good enough)

●Tangent vectors at (0,0) and (1,0) must be perpendicular to the s-axis since $\dot{s}(0)=\dot{s}(1)=0$.

 

 

●We can draw a cone at a point $(s,\dot{s})$ and it represents the possible $\ddot{s}$ range(actuator constraint) at that state $(s,dot{s})$

●So the cone contains the information about the robot's torque limit and the path. (My own explanation)

●There is an area where $U(s,\dot{s})\le L(s,\dot{s})$, so the time scaling curve cannot pass through it. (Velocity limit curve)

 

 

●The time-optimal time-scaling problem comes down to finding a possible time-scaling curve with the largest area.

●If there isn't any velocity limit, a curve with starting maximum acceleration $U$ followed by a curve of minimum acceleration $L$ is the time-optimal curve. (Bang-bang time scaling)

 

●If there is a velocity limit and the bang-bang curve intrudes into the inadmissible area, the time-scaling algorithm is needed to find the optimal curve.

 

※Chapter test와 Module test 일부 문제 완벽하게 이해 못함. 추후 추가 공부 필요. 특히 module test 2번 문제...

 

 

 

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