期望工具点按照带拋物线过渡的直线轨迹从 P1=( 0.0 0.0 )^T 到 P1=( 3.0 3.0 )^T 运动, P2= [2.0 1.0)^T 为中间点

期望工具点按照带拋物线过渡的直线轨迹从 P1=( 0.0 0.0 )^T 到 P1=( 3.0 ...

期望工具点按照带拋物线过渡的直线轨迹从 $P_1=\left(\begin{array}{lll}0.0 & 0.0\end{array}\right)^{\mathrm{T}}$ 到 $P_1=\left(\begin{array}{lll}3.0 & 3.0\end{array}\right)^{\mathrm{T}}$ 运动, $P_2=$ [2.0 1.0$)^{\mathrm{T}}$ 为中间点. 形期每一段的时间间隔为 $t_{d 12}=t_{S I}=1$, 加速度幅值为 $\dddot{x}=\ddot{y}=2$ 。画出该轨迹的 $x-y$ 坐标。

Key Concepts

Acceleration and Jerk Constraints

Acceleration and jerk constraints define the limits on the rates of change of velocity and acceleration, respectively. These constraints are critical in trajectory planning to ensure that the motion is physically plausible and to avoid causing dynamic stress or instability in the system.

Time Parameterization

Time parameterization is the assignment of time intervals to segments of the trajectory. This ensures that the motion not only follows the correct spatial path but also adheres to dynamic performance criteria, such as speed and acceleration profiles across different segments.

Cartesian Coordinate Trajectory Visualization

Displaying the trajectory in Cartesian coordinates provides a clear visual representation of the path. This visualization helps in verifying that the motion path is smooth, the transitions are as desired, and that the trajectory fulfills all spatial and temporal conditions.

Trajectory Planning

Trajectory planning involves creating a path that connects multiple points in space while meeting requirements such as smoothness and constraints on velocity, acceleration, and sometimes jerk. It is the process of designing the motion of a system so that transitions between different poses occur in a controlled, feasible way.

Parabolic Blends

Parabolic blends are used in trajectory planning to create smooth transitions between linear segments. They allow the system to transition smoothly by adjusting the curvature at the connection points, ensuring continuity in both velocity and acceleration.

Transcript

00:01 For this problem, we're asked to sketch the path r of t equals 1 minus t squared 1 minus t for t between negative 2 and positive 2, indicating the direction of motion.

00:08 We're then asked to draw the velocity and acceleration vectors at t equals 0 and t equals 1.

00:13 So we can take a few sample points.

00:15 For instance, r of negative 2.

00:17 You can see it would be equal to 1 minus negative 2 squared, so that would be 1 minus 4, or negative 3 in the x component.

00:24 And then 1 minus negative 2, so 1 plus 2, so that would be 3 in the y component.

00:30 Then at negative 1, we have 1 minus negative 1 squared.

00:35 So we'd have 0 in the x component.

00:37 1 minus negative 1 gives us 2 in the y component.

00:40 R of 0 clearly gives us the point 1 1.

00:45 R of 1 would then be equal to 1 minus 1 squared.

00:49 So again, 0 in the x component, and then 1 minus 1 or 0 in the y component.

00:54 And then at r of 2, we'd have 1 minus negative 2 squared.

00:59 So that gives us negative 3 in the x again but then we'd have 1 minus 2 in the y component so that would give us negative 1 so we can then create a rough little sketch here so we know that we start at the point negative 3 positive 3 so that would be over here then we go to x equals 0 y equals 2 then we go to x equals 1 y equals 1 then we go to the point 0 oops actually i missed it that's the point we want we go to 0 0 and then we go to negative 3 negative 1 so we'd have that our path is something like this then we're asked to well draw the velocity and acceleration vectors at t equals 0 and t equals 1 what we can do here is find the derivative of r we we have r prime of t will be equal to negative 2 t, negative 1...

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