Trigonometric traces--harmonic traces

Trigonometric traces – harmonic traces

Harmonic trajectory

gihub project source code: OTPL

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Harmonic motion is characterized by an acceleration curve that is proportional to the position curve, with opposite signs.

set point$q$for points$p$Projection on the diameter of the circle, if the point$p$moving uniformly on the circle, then the point$q$The motion is called harmonic motion. Can be described by the following formula

$$s(\theta)=R(1-\text{cos} \theta) \tag{1}$$
here $R$ is the radius of the circle. In a more general form, harmonic motion can be defined as
$$q(t)=\frac{h}{2}\left ( 1-\text {cos} \frac{\pi(t-t_0)}{T}\right)+q_0 \tag{2}$$
Therefore
$$\begin{cases} \dot q(t)=\frac{\pi h}{2T}\text{sin}\left ( \frac{\pi(t-t_0)}{T}\right )\\ \ddot q(t)=\frac{\pi^2 h}{2T^2}\text{cos}\left ( \frac{\pi(t-t_0)}{T}\right )\\ q^{(3)}(t)=-\frac{\pi^3 h}{2T^3}\text{sin}\left ( \frac{\pi(t-t_0)}{T}\right )\\ \end{cases}$$

example: the given condition is$t_0=0,t_1=8,q_0=0,q_1=10$, call the harmonic motion planning module (the code and test examples are saved under the OTPL project), and get the planning result as shown in the figure below.

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references:

[1]Biagiotti L, Melchiorri C. Trajectory Planning for Automatic Machines and Robots[M]. Springer Berlin Heidelberg, 2009.