liba 0.1.15
An algorithm library based on C/C++
|
Data Structures | |
struct | a_trajbell |
instance structure for bell-shaped velocity trajectory More... | |
Typedefs | |
typedef struct a_trajbell | a_trajbell |
Functions | |
double | a_trajbell_gen (a_trajbell *ctx, double jm, double am, double vm, double p0, double p1, double v0, double v1) |
generate for bell-shaped velocity trajectory | |
double | a_trajbell_pos (a_trajbell const *ctx, double x) |
calculate position for bell-shaped velocity trajectory | |
double | a_trajbell_vel (a_trajbell const *ctx, double x) |
calculate velocity for bell-shaped velocity trajectory | |
double | a_trajbell_acc (a_trajbell const *ctx, double x) |
calculate acceleration for bell-shaped velocity trajectory | |
double | a_trajbell_jer (a_trajbell const *ctx, double x) |
calculate jerk for bell-shaped velocity trajectory | |
double a_trajbell_acc | ( | a_trajbell const * | ctx, |
double | x ) |
calculate acceleration for bell-shaped velocity trajectory
\[ \ddot{p}(t)=\begin{cases} j_mt,&t\in[0,T_{aj}]\\ j_mT_{aj}=a_m,&t\in[T_{aj},T_a-T_{aj}]\\ j_m(T_a-t),&t\in[T_a-T_{aj},T_a]\\ 0,&t\in[T_a,T_a+T_v]\\ -j_m(t-T+T_d),&t\in[T-T_d,T-T_d+T_{dj}]\\ -j_mT_{dj}=d_m,&t\in[T-T_d+T_{dj},T-T_{dj}]\\ -j_m(T-t),&t\in[T-T_{dj},T] \end{cases} \]
[in] | ctx | points to an instance of bell-shaped velocity trajectory |
[in] | x | difference between current time and initial time |
double a_trajbell_gen | ( | a_trajbell * | ctx, |
double | jm, | ||
double | am, | ||
double | vm, | ||
double | p0, | ||
double | p1, | ||
double | v0, | ||
double | v1 ) |
generate for bell-shaped velocity trajectory
If \(p_0>p_1\), then
\begin{aligned}p_0=-p_0\\p_1=-p_1\\v_0=-v_0\\v_1=-v_1\end{aligned}
\begin{aligned}T_{aj}&=\sqrt{\frac{v_m-v_0}{j_m}}\\T_a&=2T_{aj}\\a_m&=+j_mT_{aj}\end{aligned}
else\begin{aligned}T_{aj}&=\frac{a_m}{j_m}\\T_a&=T_{aj}+\frac{v_m-v_0}{a_m}\\a_m&=+a_m\end{aligned}
\begin{aligned}T_{dj}&=\sqrt{\frac{v_m-v_1}{j_m}}\\T_d&=2T_{dj}\\d_m&=-j_mT_{dj}\end{aligned}
else\begin{aligned}T_{dj}&=\frac{a_m}{j_m}\\T_d&=T_{dj}+\frac{v_m-v_1}{a_m}\\d_m&=-a_m\end{aligned}
\[T_v=\frac{p_1-p_0}{v_m} -\frac{T_a}{2}\left(1+\frac{v_0}{v_m}\right)-\frac{T_d}{2}\left(1+\frac{v_1}{v_m}\right)\]
If \(T_v>0\), then there is a constant velocity phase and the phase calculation ends.\begin{aligned}T_{aj}&=T_{dj}=T_{j}=\frac{a_m}{j_m}\\ T_a&=\cfrac{\frac{a_m^2}{j_m}+\sqrt\Delta-2v_0}{2a_m}\\T_d&=\cfrac{\frac{a_m^2}{j_m}+\sqrt\Delta-2v_1}{2a_m}\\ \Delta&=\frac{a_m^4}{j_m^2}+2(v_0^2+v_1^2)+a_m\left(4(p_1-p_0)-2\frac{a_m}{j_m}(v_0+v_1)\right)\end{aligned}
\begin{aligned}T_d&=2\cfrac{p_1-p_0}{v_1+v_0}\\ T_{dj}&=\frac{j_m(p_1-p_0)-\sqrt{j_m(j_m(p_1-p_0)^2+(v_1-v_0)(v_1+v_0)^2)}}{j_m(v_1+v_0 )}\end{aligned}
b. If \(T_d<0\), there is only acceleration phase, then \(T_d=0\), \(T_{dj}=0\),\begin{aligned}T_a&=2\cfrac{p_1-p_0}{v_1+v_0}\\ T_{aj}&=\frac{j_m(p_1-p_0)-\sqrt{j_m(j_m(p_1-p_0)^2+(v_0-v_1)(v_1+v_0)^2)}}{j_m(v_1+v_0 )}\end{aligned}
c. If \(T_a\ge2T_j\), \(T_d\ge2T_j\), then\begin{aligned}a_m&=+j_mT_{aj}\\d_m&=-j_mT_{dj}\\v_m&=v_0+(T_a-T_{aj})a_m=v_1-(T_d-T_{dj})d_m\end{aligned}
d. If none of the above conditions are met, let \(a_m=\alpha a_m, 0<\alpha<1\), and then repeat step 4.[in,out] | ctx | points to an instance of bell-shaped velocity trajectory |
[in] | jm | defines the maximum jerk during system operation |
[in] | am | defines the maximum acceleration during system operation |
[in] | vm | defines the maximum velocity during system operation |
[in] | p0 | defines the initial position |
[in] | p1 | defines the final position |
[in] | v0 | defines the initial velocity |
[in] | v1 | defines the final velocity |
double a_trajbell_jer | ( | a_trajbell const * | ctx, |
double | x ) |
calculate jerk for bell-shaped velocity trajectory
\[ p^{(3)}(t)=\begin{cases} j_m,&t\in[0,T_{aj}]\\ 0,&t\in[T_{aj},T_a-T_{aj}]\\ -j_m,&t\in[T_a-T_{aj},T_a]\\ 0,&t\in[T_a,T_a+T_v]\\ -j_m,&t\in[T-T_d,T-T_d+T_{dj}]\\ 0,&t\in[T-T_d+T_{dj},T-T_{dj}]\\ j_m,&t\in[T-T_{dj},T] \end{cases} \]
[in] | ctx | points to an instance of bell-shaped velocity trajectory |
[in] | x | difference between current time and initial time |
double a_trajbell_pos | ( | a_trajbell const * | ctx, |
double | x ) |
calculate position for bell-shaped velocity trajectory
\[ p(t)=\begin{cases} p_0+v_0t+j_m\cfrac{t^3}{6},&t\in[0,T_{aj}]\\ p_0+v_0t+\cfrac{a_m}{6}(3t^3-3tT_{aj}+T_{aj}^2),&t\in[T_{aj},T_a-T_{aj}]\\ p_0+(v_m+v_0)\cfrac{T_a}{2}-v_m(T_a-t)+j_m\cfrac{(T_a-t)^3}{6},&t\in[T_a-T_{aj},T_a]\\ p_0+(v_m+v_0)\cfrac{T_a}{2}+v_m(t-T_a),&t\in[T_a,T_a+T_v]\\ p_1-(v_m+v_1)\cfrac{T_d}{2}+v_m(t-T+T_d)-j_m\cfrac{(t-T+T_d)^3}{6},&t\in[T-T_d,T-T_d+T_{dj}]\\ p_1-(v_m+v_1)\cfrac{T_d}{2}+v_m(t-T+T_d)\\ \quad+\cfrac{d_m}{6}\left(3(t-T+T_d)^2-3(t-T+T_d)T_{dj}+T_{dj}^2\right),&t\in[T-T_d+T_{dj},T-T_{dj}]\\ p_1-v_1(T-t)-j_m\cfrac{(T-t)^3}{6},&t\in[T-T_{dj},T] \end{cases} \]
[in] | ctx | points to an instance of bell-shaped velocity trajectory |
[in] | x | difference between current time and initial time |
double a_trajbell_vel | ( | a_trajbell const * | ctx, |
double | x ) |
calculate velocity for bell-shaped velocity trajectory
\[ \dot{p}(t)=\begin{cases} v_0+j_m\frac{t^2}{2},&t\in[0,T_{aj}]\\ v_0+a_m(t-\cfrac{T_{aj}}{2}),&t\in[T_{aj},T_a-T_{aj}]\\ v_m-j_m\cfrac{(T_a-t)^2}{2},&t\in[T_a-T_{aj},T_a]\\ v_m,&t\in[T_a,T_a+T_v]\\ v_m-j_m\cfrac{(t-T+T_d)^2}{2},&t\in[T-T_d,T-T_d+T_{dj}]\\ v_m+d_m(t-T+T_d-\cfrac{T_{dj}}{2}),&t\in[T-T_d+T_{dj},T-T_{dj}]\\ v_1+j_m\cfrac{(T-t)^2}{2},&t\in[T-T_{dj},T] \end{cases} \]
[in] | ctx | points to an instance of bell-shaped velocity trajectory |
[in] | x | difference between current time and initial time |