Collaboration diagram for bell-shaped velocity trajectory:

Data Structures
struct	a_trajbell
	instance structure for bell-shaped velocity trajectory More...

Typedefs
typedef struct a_trajbell	a_trajbell

Functions
a_real	a_trajbell_gen (a_trajbell *ctx, a_real jm, a_real am, a_real vm, a_real p0, a_real p1, a_real v0, a_real v1)
	generate for bell-shaped velocity trajectory

a_real	a_trajbell_pos (a_trajbell const *ctx, a_real x)
	compute position for bell-shaped velocity trajectory

a_real	a_trajbell_vel (a_trajbell const *ctx, a_real x)
	compute velocity for bell-shaped velocity trajectory

a_real	a_trajbell_acc (a_trajbell const *ctx, a_real x)
	compute acceleration for bell-shaped velocity trajectory

a_real	a_trajbell_jer (a_trajbell const *ctx, a_real x)
	compute jerk for bell-shaped velocity trajectory

Detailed Description

Function Documentation

◆ a_trajbell_acc()

a_real a_trajbell_acc	(	a_trajbell const *	ctx,
		a_real	x )

compute acceleration for bell-shaped velocity trajectory

$\ddot{p} (t) = {\begin{cases} j_{m} t, & t \in [0, T_{a j}] \\ j_{m} T_{a j} = a_{m}, & t \in [T_{a j}, T_{a} - T_{a j}] \\ j_{m} (T_{a} - t), & t \in [T_{a} - T_{a j}, T_{a}] \\ 0, & t \in [T_{a}, T_{a} + T_{v}] \\ - j_{m} (t - T + T_{d}), & t \in [T - T_{d}, T - T_{d} + T_{d j}] \\ - j_{m} T_{d j} = d_{m}, & t \in [T - T_{d} + T_{d j}, T - T_{d j}] \\ - j_{m} (T - t), & t \in [T - T_{d j}, T] \end{cases}$

Parameters

[in]	ctx	points to an instance of bell-shaped velocity trajectory
[in]	x	difference between current time and initial time

Returns: acceleration output

◆ a_trajbell_gen()

a_real a_trajbell_gen	(	a_trajbell *	ctx,
		a_real	jm,
		a_real	am,
		a_real	vm,
		a_real	p0,
		a_real	p1,
		a_real	v0,
		a_real	v1 )

generate for bell-shaped velocity trajectory

If $p_{0} > p_{1}$ , then

$\begin{array}{r} p_{0} = - p_{0} \\ p_{1} = - p_{1} \\ v_{0} = - v_{0} \\ v_{1} = - v_{1} \end{array}$

If $(v_{m} - v_{0}) j_{m} < a_{m}^{2}$ , the maximum acceleration is not reached, then
$\begin{aligned} T_{a j} & = \sqrt{\frac{v_{m} - v_{0}}{j_{m}}} \\ T_{a} & = 2 T_{a j} \\ a_{m} & = + j_{m} T_{a j} \end{aligned}$
else
$\begin{aligned} T_{a j} & = \frac{a_{m}}{j_{m}} \\ T_{a} & = T_{a j} + \frac{v_{m} - v_{0}}{a_{m}} \\ a_{m} & = + a_{m} \end{aligned}$
If $(v_{m} - v_{1}) j_{m} < a_{m}^{2}$ , the maximum deceleration is not reached, then
$\begin{aligned} T_{d j} & = \sqrt{\frac{v_{m} - v_{1}}{j_{m}}} \\ T_{d} & = 2 T_{d j} \\ d_{m} & = - j_{m} T_{d j} \end{aligned}$
else
$\begin{aligned} T_{d j} & = \frac{a_{m}}{j_{m}} \\ T_{d} & = T_{d j} + \frac{v_{m} - v_{1}}{a_{m}} \\ d_{m} & = - a_{m} \end{aligned}$
Calculate the constant velocity phase
$T_{v} = \frac{p_{1} - p_{0}}{v_{m}} - \frac{T_{a}}{2} (1 + \frac{v_{0}}{v_{m}}) - \frac{T_{d}}{2} (1 + \frac{v_{1}}{v_{m}})$
If $T_{v} > 0$ , then there is a constant velocity phase and the phase calculation ends.
If $T_{v} <= 0$ , then $T_{v} = 0$ ,
$\begin{aligned} T_{a j} & = T_{d j} = T_{j} = \frac{a_{m}}{j_{m}} \\ T_{a} & = \frac{\frac{a_{m}^{2}}{j_{m}} + \sqrt{Δ} - 2 v_{0}}{2 a_{m}} \\ T_{d} & = \frac{\frac{a_{m}^{2}}{j_{m}} + \sqrt{Δ} - 2 v_{1}}{2 a_{m}} \\ Δ & = \frac{a_{m}^{4}}{j_{m}^{2}} + 2 (v_{0}^{2} + v_{1}^{2}) + a_{m} (4 (p_{1} - p_{0}) - 2 \frac{a_{m}}{j_{m}} (v_{0} + v_{1})) \end{aligned}$

a. If $T_{a} < 0$ , there is only deceleration phase, then $T_{a} = 0$ , $T_{a j} = 0$ ,
$\begin{aligned} T_{d} & = 2 \frac{p_{1} - p_{0}}{v_{1} + v_{0}} \\ T_{d j} & = \frac{j_{m} (p_{1} - p_{0}) - \sqrt{j_{m} (j_{m} (p_{1} - p_{0})^{2} + (v_{1} - v_{0}) (v_{1} + v_{0})^{2})}}{j_{m} (v_{1} + v_{0})} \end{aligned}$
b. If $T_{d} < 0$ , there is only acceleration phase, then $T_{d} = 0$ , $T_{d j} = 0$ ,
$\begin{aligned} T_{a} & = 2 \frac{p_{1} - p_{0}}{v_{1} + v_{0}} \\ T_{a j} & = \frac{j_{m} (p_{1} - p_{0}) - \sqrt{j_{m} (j_{m} (p_{1} - p_{0})^{2} + (v_{0} - v_{1}) (v_{1} + v_{0})^{2})}}{j_{m} (v_{1} + v_{0})} \end{aligned}$
c. If $T_{a} \geq 2 T_{j}$ , $T_{d} \geq 2 T_{j}$ , then
$\begin{aligned} a_{m} & = + j_{m} T_{a j} \\ d_{m} & = - j_{m} T_{d j} \\ v_{m} & = v_{0} + (T_{a} - T_{a j}) a_{m} = v_{1} - (T_{d} - T_{d j}) d_{m} \end{aligned}$
d. If none of the above conditions are met, let $a_{m} = α a_{m}, 0 < α < 1$ , and then repeat step 4.

Finally, use formulas to compute position, velocity and acceleration.

Parameters

[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

Returns: total duration

◆ a_trajbell_jer()

a_real a_trajbell_jer	(	a_trajbell const *	ctx,
		a_real	x )

compute jerk for bell-shaped velocity trajectory

$p^{(3)} (t) = {\begin{cases} j_{m}, & t \in [0, T_{a j}] \\ 0, & t \in [T_{a j}, T_{a} - T_{a j}] \\ - j_{m}, & t \in [T_{a} - T_{a j}, T_{a}] \\ 0, & t \in [T_{a}, T_{a} + T_{v}] \\ - j_{m}, & t \in [T - T_{d}, T - T_{d} + T_{d j}] \\ 0, & t \in [T - T_{d} + T_{d j}, T - T_{d j}] \\ j_{m}, & t \in [T - T_{d j}, T] \end{cases}$

Parameters

[in]	ctx	points to an instance of bell-shaped velocity trajectory
[in]	x	difference between current time and initial time

Returns: jerk output

◆ a_trajbell_pos()

a_real a_trajbell_pos	(	a_trajbell const *	ctx,
		a_real	x )

compute position for bell-shaped velocity trajectory

$p (t) = {\begin{cases} p_{0} + v_{0} t + j_{m} \frac{t^{3}}{6}, & t \in [0, T_{a j}] \\ p_{0} + v_{0} t + \frac{a_{m}}{6} (3 t^{3} - 3 t T_{a j} + T_{a j}^{2}), & t \in [T_{a j}, T_{a} - T_{a j}] \\ p_{0} + (v_{m} + v_{0}) \frac{T_{a}}{2} - v_{m} (T_{a} - t) + j_{m} \frac{(T_{a} - t)^{3}}{6}, & t \in [T_{a} - T_{a j}, T_{a}] \\ p_{0} + (v_{m} + v_{0}) \frac{T_{a}}{2} + v_{m} (t - T_{a}), & t \in [T_{a}, T_{a} + T_{v}] \\ p_{1} - (v_{m} + v_{1}) \frac{T_{d}}{2} + v_{m} (t - T + T_{d}) - j_{m} \frac{(t - T + T_{d})^{3}}{6}, & t \in [T - T_{d}, T - T_{d} + T_{d j}] \\ p_{1} - (v_{m} + v_{1}) \frac{T_{d}}{2} + v_{m} (t - T + T_{d}) \\ + \frac{d_{m}}{6} (3 (t - T + T_{d})^{2} - 3 (t - T + T_{d}) T_{d j} + T_{d j}^{2}), & t \in [T - T_{d} + T_{d j}, T - T_{d j}] \\ p_{1} - v_{1} (T - t) - j_{m} \frac{(T - t)^{3}}{6}, & t \in [T - T_{d j}, T] \end{cases}$

Parameters

[in]	ctx	points to an instance of bell-shaped velocity trajectory
[in]	x	difference between current time and initial time

Returns: position output

◆ a_trajbell_vel()

a_real a_trajbell_vel	(	a_trajbell const *	ctx,
		a_real	x )

compute velocity for bell-shaped velocity trajectory

$\dot{p} (t) = {\begin{cases} v_{0} + j_{m} \frac{t^{2}}{2}, & t \in [0, T_{a j}] \\ v_{0} + a_{m} (t - \frac{T_{a j}}{2}), & t \in [T_{a j}, T_{a} - T_{a j}] \\ v_{m} - j_{m} \frac{(T_{a} - t)^{2}}{2}, & t \in [T_{a} - T_{a j}, T_{a}] \\ v_{m}, & t \in [T_{a}, T_{a} + T_{v}] \\ v_{m} - j_{m} \frac{(t - T + T_{d})^{2}}{2}, & t \in [T - T_{d}, T - T_{d} + T_{d j}] \\ v_{m} + d_{m} (t - T + T_{d} - \frac{T_{d j}}{2}), & t \in [T - T_{d} + T_{d j}, T - T_{d j}] \\ v_{1} + j_{m} \frac{(T - t)^{2}}{2}, & t \in [T - T_{d j}, T] \end{cases}$

Parameters

[in]	ctx	points to an instance of bell-shaped velocity trajectory
[in]	x	difference between current time and initial time

Returns: velocity output

Data Structures

Typedefs

Functions

Detailed Description

Function Documentation

◆ a_trajbell_acc()

◆ a_trajbell_gen()

◆ a_trajbell_jer()

◆ a_trajbell_pos()

◆ a_trajbell_vel()