trapezoidal velocity trajectory

Collaboration diagram for trapezoidal velocity trajectory:

Data Structures
struct	a_trajtrap
	instance structure for trapezoidal velocity trajectory More...

Typedefs
typedef struct a_trajtrap	a_trajtrap

Functions
a_real	a_trajtrap_gen (a_trajtrap *ctx, a_real vm, a_real ac, a_real de, a_real p0, a_real p1, a_real v0, a_real v1)
	generate for trapezoidal velocity trajectory
a_real	a_trajtrap_pos (a_trajtrap const *ctx, a_real x)
	compute position for trapezoidal velocity trajectory
a_real	a_trajtrap_vel (a_trajtrap const *ctx, a_real x)
	compute velocity for trapezoidal velocity trajectory
a_real	a_trajtrap_acc (a_trajtrap const *ctx, a_real x)
	compute acceleration for trapezoidal velocity trajectory

Detailed Description

Function Documentation

a_trajtrap_acc()

a_real a_trajtrap_acc	(	a_trajtrap const *	ctx,
		a_real	x )

compute acceleration for trapezoidal velocity trajectory

$$\ddot{p}(t)=\{\begin{array}{ll}a,& t\in [0,t_a)\\ 0,& t\in [t_a,t_d)\\ d,& t\in [t_d,T]\end{array}$$

Parameters

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

Returns

acceleration output

a_trajtrap_gen()

a_real a_trajtrap_gen	(	a_trajtrap *	ctx,
		a_real	vm,
		a_real	ac,
		a_real	de,
		a_real	p0,
		a_real	p1,
		a_real	v0,
		a_real	v1 )

generate for trapezoidal velocity trajectory

Assuming that there is no constant velocity phase, but only acceleration and deceleration phases, the maximum velocity in the motion is $v$, then we have

$$\frac{v^2-v_0^2}{2a}+\frac{v_1^2-v^2}{2d}=p_1-p_0$$

Solving for the maximum velocity is

$$|v|=\sqrt{\frac{a{v}_1^2-d{v}_0^2-2ad(p_1-p_0)}{a-d}}$$

1. If $|v|>|v_m|$, then there is a constant velocity phase.

   $$p_a=p_0+v_0{T}_a+\frac{1}{2}a{T}_a^2$$

   $$a_d=p_1-v_c{T}_d-\frac{1}{2}d{T}_d^2$$

   $$\{\begin{array}{l}{T}_a=\frac{v_m-v_0}{a}\\ T_c=\frac{p_d-p_a}{v_c}\\ T_d=\frac{v_1-v_m}{d}\end{array}$$

2. Otherwise there is no constant velocity phase.
   a. If $|v_0|<|v|\le |v_1|$, there is only an acceleration phase.

   $$|v_1|=\sqrt{v_0^2+2a(p_1-p_0)}$$

   $$\{\begin{array}{l}{T}_a=\frac{v_1-v_0}{a}\\ T_c=0\\ T_d=0\end{array}$$

   b. If $|v_0|\ge |v|>|v_1|$, there is only a deceleration phase.

   $$|v_1|=\sqrt{v_0^2+2d(p_1-p_0)}$$

   $$\{\begin{array}{l}{T}_a=0\\ T_c=0\\ T_d=\frac{v_1-v_0}{d}\end{array}$$

   c. If $|v|>|v_0|$, $|v|>|v_1|$, then there are acceleration and deceleration phases.

   $$\{\begin{array}{l}{T}_a=\frac{v-v_0}{a}\\ T_c=0\\ T_d=\frac{v_1-v}{d}\end{array}$$

3. Finally, the position and velocity are computed using the formula.

   Parameters

   [in,out]	ctx	points to an instance of trapezoidal velocity trajectory
   [in]	vm	defines the maximum velocity during system operation
   [in]	ac	defines the acceleration before constant velocity
   [in]	de	defines the acceleration after constant velocity
   [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_trajtrap_pos()

a_real a_trajtrap_pos	(	a_trajtrap const *	ctx,
		a_real	x )

compute position for trapezoidal velocity trajectory

$$p(t)=\{\begin{array}{ll}{v}_0+\frac{1}{2}a{t}^2,& t\in [0,t_a)\\ p_a+v_c(t-t_a),& t\in [t_a,t_d)\\ p_d+v_c(t-t_d)+\frac{1}{2}d(t-t_d{)}^2,& t\in [t_d,T]\end{array}$$

Parameters

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

Returns

position output

a_trajtrap_vel()

a_real a_trajtrap_vel	(	a_trajtrap const *	ctx,
		a_real	x )

compute velocity for trapezoidal velocity trajectory

$$\dot{p}(t)=\{\begin{array}{ll}{v}_0+at,& t\in [0,t_a)\\ v_c,& t\in [t_a,t_d)\\ v_c+d(t-t_d),& t\in [t_d,T]\end{array}$$

Parameters

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

Returns

velocity output