在从 t=0 到 t=1 的时间区间使用一个单段三次样条曲线轨迹: θ(t)=10+5 t+70 t^2-45 t^3, 求其起始点和终止点的位置、速度和加速度。

Name: Problem 19, Chapter 7: 机器人学导论 introduction to robotics: mechanics and control
Uploaded: 2019-08-03T17:52:37-08:00
Duration: 4 min 15 s
Channel: Sahil Patel
Description: 在从 t=0 到 t=1 的时间区间使用一个单段三次样条曲线轨迹: θ(t)=10+5 t+70 t^2-45 t^3, 求其起始点和终止点的位置、速度和加速度。

step 1 So I'll keep this list of things that we have to do over here. So x prime is going to be 2t root

在从 t=0 到 t=1 的时间区间使用一个单段三次样条曲线轨迹: θ(t)=10+5 t+70 t^2-45 t^3,...

Key Concepts

Cubic Spline Trajectory

A cubic spline trajectory is a smooth, continuous function defined by a cubic polynomial over a given time interval. It is commonly used in motion planning and control to generate smooth paths. The cubic spline is advantageous because it ensures continuous position, velocity, and acceleration, making it suitable for applications requiring smooth transitions between motion states.

Kinematic Quantities

Kinematic quantities refer to the properties of motion such as position, velocity, and acceleration. Position is the location of a point at a given time, velocity is the rate of change of position with respect to time, and acceleration is the rate of change of velocity with respect to time. In trajectory planning, these quantities are critical in evaluating the performance and feasibility of the motion.

Differentiation of Trajectories

Differentiation of trajectories involves computing the first and second derivatives of the position function to obtain velocity and acceleration, respectively. This mathematical process is essential because it allows for an analysis of the motion's dynamics. For a polynomial function, differentiation is straightforward and provides accurate results for subsequent kinematic quantities.

Evaluation at Endpoints

Evaluation at endpoints involves substituting the starting and ending time values into the trajectory and its derivatives. This process yields the specific kinematic conditions—position, velocity, and acceleration—at the beginning and end of the motion. These boundary conditions are crucial for verifying that the motion meets the required specifications and constraints.

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